Life System Modeling and Intelligent Computing: International Conference on Life System Modeling and Simulation, LSMS 2010, and International ... Computer Science and General Issues)

Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max Planck Institute for Informatics, Saarbruecken, Germany

6328

Kang Li Minrui Fei Li Jia George W. Irwin (Eds.)

Life System Modeling and Intelligent Computing International Conference on Life System Modeling and Simulation, LSMS 2010 and International Conference on Intelligent Computing for Sustainable Energy and Environment, ICSEE 2010 Wuxi, China, September 17-20, 2010 Proceedings, Part I

13

Volume Editors Kang Li The Queen’s University of Belfast, Intelligent Systems and Control School of Electronics, Electrical Engineering and Computer Science Ashby Building, Stranmillis Road, Belfast BT9 5AH, UK E-mail: [email protected] Minrui Fei Shanghai University, School of Mechatronical Engineering and Automation P.O.Box 183, Shanghai 200072, China E-mail: [email protected] Li Jia Shanghai University, School of Mechatronical Engineering and Automation P.O.Box 183, Shanghai 200072, China E-mail: [email protected] George W. Irwin The Queen’s University of Belfast, Intelligent Systems and Control School of Electronics, Electrical Engineering and Computer Science Ashby Building, Stranmillis Road, Belfast BT9 5AH, UK E-mail: [email protected]

Library of Congress Control Number: 2010933354 CR Subject Classification (1998): J.3, I.6, I.2, C.2.4, F.1, I.2.11 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-642-15620-7 Springer Berlin Heidelberg New York 978-3-642-15620-5 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2010 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper 06/3180

Preface

The 2010 International Conference on Life System Modeling and Simulation (LSMS 2010) and the 2010 International Conference on Intelligent Computing for Sustainable Energy and Environment (ICSEE 2010) were formed to bring together researchers and practitioners in the fields of life system modeling/simulation and intelligent computing applied to worldwide sustainable energy and environmental applications. A life system is a broad concept, covering both micro and macro components ranging from cells, tissues and organs across to organisms and ecological niches. To comprehend and predict the complex behavior of even a simple life system can be extremely difficult using conventional approaches. To meet this challenge, a variety of new theories and methodologies have emerged in recent years on life system modeling and simulation. Along with improved understanding of the behavior of biological systems, novel intelligent computing paradigms and techniques have emerged to handle complicated real-world problems and applications. In particular, intelligent computing approaches have been valuable in the design and development of systems and facilities for achieving sustainable energy and a sustainable environment, the two most challenging issues currently facing humanity. The two LSMS 2010 and ICSEE 2010 conferences served as an important platform for synergizing these two research streams. The LSMS 2010 and ICSEE 2010 conferences, held in Wuxi, China, during September 17–20, 2010, built upon the success of two previous LSMS conferences held in Shanghai in 2004 and 2007 and were based on the Research Councils UK (RCUK)-funded Sustainable Energy and Built Environment Science Bridge project. The conferences were jointly organized by Shanghai University, Queen's University Belfast, Jiangnan University and the System Modeling and Simulation Technical Committee of CASS, together with the Embedded Instrument and System Technical Committee of China Instrument and Control Society. The conference program covered keynote addresses, special sessions, themed workshops and poster presentations, in addition to a series of social functions to enable networking and foster future research collaboration. LSMS 2010 and ICSEE 2010 received over 880 paper submissions from 22 countries. These papers went through a rigorous peer-review procedure, including both pre-review and formal refereeing. Based on the review reports, the Program Committee finally selected 260 papers for presentation at the conference, from amongst which 194 were subsequently selected and recommended for publication by Springer in two volumes of Lecture Notes in Computer Science (LNCS) and one volume of Lecture Notes in Bioinformatics (LNBI). This particular volume of Lecture Notes in Computer Science (LNCS) includes 55 papers covering 7 relevant topics.

VI

Preface

The organizers of LSMS 2010 and ICSEE 2010 would like to acknowledge the enormous contributions from the following: the Advisory and Steering Committees for their guidance and advice, the Program Committee and the numerous referees worldwide for their significant efforts in both reviewing and soliciting the papers, and the Publication Committee for their editorial work. We would also like to thank Alfred Hofmann, of Springer, for his continual support and guidance to ensure the highquality publication of the conference proceedings. Particular thanks are of course due to all the authors, as without their excellent submissions and presentations, the two conferences would not have occurred. Finally, we would like to express our gratitude to the following organizations: Chinese Association for System Simulation (CASS), IEEE SMCS Systems Biology Technical Committee, National Natural Science Foundation of China, Research Councils UK, IEEE CC Ireland chapter, IEEE SMC Ireland chapter, Shanghai Association for System Simulation, Shanghai Instrument and Control Society and Shanghai Association of Automation. The support of the Intelligent Systems and Control research cluster at Queen’s University Belfast, Tsinghua University, Peking University, Zhejiang University, Shanghai Jiaotong University, Fudan University, Delft University of Technology, University of Electronic Science Technology of China, Donghua University is also acknowledged.

July 2010

Bohu Li Mitsuo Umezu George W. Irwin Minrui Fei Kang Li Luonan Chen Li Jia

LSMS-ICSEE 2010 Organization

Advisory Committee Kazuyuki Aihara, Japan Zongji Chen, China Guo-sen He, China Frank L. Lewis, USA Marios M. Polycarpou, Cyprus Olaf Wolkenhauer, Germany Minlian Zhang, China

Shun-ichi Amari, Japan Peter Fleming, UK Huosheng Hu,UK Stephen K.L. Lo, UK Zhaohan Sheng, China

Erwei Bai, USA Sam Shuzhi Ge, Singapore Tong Heng Lee, Singapore Okyay Kaynak, Turkey Peter Wieringa, The Netherlands

Cheng Wu, China Guoping Zhao, China

Yugeng Xi, China

Kwang-Hyun Cho, Korea

Xiaoguang Gao, China

Shaoyuan Li, China Sean McLoone, Ireland Xiaoyi Jiang, Germany Kok Kiong Tan, Singapore Tianyuan Xiao, China Donghua Zhou, China

Liang Liang, China Robert Harrison, UK Da Ruan Belgium Stephen Thompson, UK Jianxin Xu, Singapore Quanmin Zhu, UK

Steering Committee Sheng Chen, UK Tom Heskes, The Netherlands Zengrong Liu, China MuDer Jeng, Taiwan, China Kay Chen Tan, Singapore Haifeng Wang, UK Guangzhou Zhao, China

Honorary Chairs Bohu Li, China Mitsuo Umezu, Japan

General Chairs George W. Irwin, UK Minrui Fei, China

International Program Committee IPC Chairs Kang Li, UK Luonan Chen, Japan

VIII

Organization

IPC Regional Chairs Haibo He, USA Wen Yu, Mexico Shiji Song, China Xingsheng Gu, China Ming Chen, China

Amir Hussain, UK John Morrow, UK Taicheng Yang, UK Yongsheng Ding, China Feng Ding, China

Guangbin Huang, Singapore Qiguo Rong, China Jun Zhang, USA Zhijian Song, China Weidong Chen, China

Maysam F. Abbod, UK Vitoantonio Bevilacqua, Italy Yuehui Chen, China

Peter Andras, UK Uday K. Chakraborty, USA Xinglin Chen, China

Costin Badica, Romania

Minsen Chiu, Singapore Kevin Curran, UK Jianbo Fan, China

Michal Choras, Poland Mingcong Deng, Japan Haiping Fang, China Wai-Keung Fung, Canada Xiao-Zhi Gao, Finland Aili Han, China Pheng-Ann Heng, China Xia Hong, UK Jiankun Hu, Australia

Tianlu Chen, China Weidong Cheng, China Tommy Chow, Hong Kong, China Frank Emmert-Streib, UK Jiali Feng, China Houlei Gao, China Lingzhong Guo, UK Minghu Ha, China Laurent Heutte, France Wei-Chiang Hong, China Xiangpei Hu, China

Peter Hung, Ireland

Amir Hussain, UK

Xiaoyi Jiang, Germany Tetsuya J. Kobayashi, Japan Xiaoou Li, Mexico Paolo Lino, Italy Hua Liu, China Sean McLoone, Ireland Kezhi Mao, Singapore Wasif Naeem, UK Feng Qiao, China Jiafu Tang, China Hongwei Wang, China Ruisheng Wang, USA Yong Wang, Japan Lisheng Wei, China Rongguo Yan, China Zhang Yuwen, USA Guofu Zhai, China Qing Zhao, Canada Liangpei Zhang, China Shangming Zhou, UK

Pingping Jiang, China

IPC Members

Huijun Gao, China Xudong Guo, China Haibo He, USA Fan Hong, Singapore Yuexian Hou, China Guangbin Huang, Singapore MuDer Jeng, Taiwan, China Yasuki Kansha, Japan Gang Li, UK Yingjie Li, China Hongbo Liu, China Zhi Liu, China Fenglou Mao, USA John Morrow, UK Donglian Qi, China Chenxi Shao, China Haiying Wang, UK Kundong Wang, China Wenxing Wang, China Zhengxin Weng, China WeiQi Yan, UK Wen Yu, Mexico Peng Zan, China Degan Zhang, China Huiru Zheng, UK Huiyu Zhou, UK

Aim`e Lay-Ekuakillel, Italy Xuelong Li, UK Tim Littler, UK Wanquan Liu, Australia Marion McAfee, UK Guido Maione, Italy Mark Price, UK Alexander Rotshtein, Ukraine David Wang, Singapore Hui Wang, UK Shujuan Wang, China Zhuping Wang, China Ting Wu, China Lianzhi Yu, China Hong Yue, UK An Zhang, China Lindu Zhao, China Qingchang Zhong, UK

Organization

IX

Secretary-General Xin Sun, China Ping Zhang, China Huizhong Yang, China

Publication Chairs Xin Li, China Wasif Naeem, UK

Special Session Chairs Xia Hong, UK Li Jia, China

Organizing Committee OC Chairs Shiwei Ma, China Yunjie Wu, China Fei Liu, China OC Members Min Zheng, China Yijuan Di, China Qun Niu, UK

Banghua Yang, China Weihua Deng, China Xianxia Zhang, China

Yang Song, China Tim Littler, UK

Reviewers Renbo Xia, Vittorio Cristini, Aim'e Lay-Ekuakille, AlRashidi M.R., Aolei Yang, B. Yang, Bailing Zhang, Bao Nguyen, Ben Niu, Branko Samarzija, C. Elliott, Chamil Abeykoon, Changjun Xie, Chaohui Wang, Chuisheng Zeng, Chunhe Song, Da Lu, Dan Lv, Daniel Lai, David Greiner, David Wang, Deng Li, Dengyun Chen, Devedzic Goran, Dong Chen, Dongqing Feng, Du K.-L., Erno Lindfors, Fan Hong, Fang Peng, Fenglou Mao, Frank Emmert-Streib, Fuqiang Lu, Gang Li, Gopalacharyulu Peddinti, Gopura R. C., Guidi Yang, Guidong Liu, Haibo He, Haiping Fang, Hesheng Wang, Hideyuki Koshigoe, Hongbo Liu, Hongbo Ren, Hongde Liu, Hongtao Wang, Hongwei Wang, Hongxin Cao, Hua Han, Huan Shen, Hueder Paulo de Oliveira, Hui Wang, Huiyu Zhou, H.Y. Wang, Issarachai Ngamroo, Jason Kennedy, Jiafu Tang, Jianghua Zheng, Jianhon Dou, Jianwu Dang, Jichun Liu, Jie Xing, Jike Ge, Jing Deng, Jingchuan Wang, Jingtao Lei, Jiuying Deng, Jizhong Liu, Jones K.O., Jun Cao, Junfeng Chen, K. Revett, Kaliviotis Efstathios, C.H. Ko, Kundong Wang, Lei Kang,

X

Organization

Leilei Zhang, Liang Chen, Lianzhi Yu, Lijie Zhao, Lin Gao, Lisheng Wei, Liu Liu, Lizhong Xu, Louguang Liu, Lun Cheng, Marion McAfee, Martin Fredriksson, Meng Jun, Mingcong Deng, Mingzhi Huang, Minsen Chiu, Mohammad Tahir, Mousumi Basu, Mutao Huang, Nian Liu, O. Ciftcioglu, Omidvar Hedayat, Peng Li, Peng Zan, Peng Zhu, Pengfei Liu, Qi Bu, Qiguo Rong, Qingzheng Xu, Qun Niu, R. Chau, R. Kala, Ramazan Coban, Rongguo Yan, Ruisheng Wang, Ruixi Yuan, Ruiyou Zhang, Ruochen Liu, Shaohui Yang, Shian Zhao, Shihu Shu, Yang Song, Tianlu Chen, Ting Wu, Tong Liang, V. Zanotto, Vincent Lee, Wang Suyu, Wanquan Liu, Wasif Naeem, Wei Gu, Wei Jiao, Wei Xu, Wei Zhou, Wei-Chiang Hong, Weidong Chen, WeiQi Yan, Wenjian Luo, Wenjuan Yang, Wenlu Yang, X.H. Zeng, Xia Ling, Xiangpei Hu, Xiao-Lei Xia, Xiaoyang Tong, Xiao-Zhi Gao, Xin Miao, Xingsheng Gu, Xisong Chen, Xudong Guo, Xueqin Liu, Yanfei Zhong, Yang Sun, Yasuki Kansha, Yi Yuan, Yin Tang, Yiping Dai, Yi-Wei Chen, Yongzhong Li, Yudong Zhang, Yuhong Wang, Yuni Jia, Zaitang Huang, Zhang Li, Zhenmin Liu, Zhi Liu, Zhigang Liu, Zhiqiang Ge, Zhongkai Li, Zilong Zhao, Ziwu Ren.

Table of Contents – Part I

The First Section: Intelligent Modeling, Monitoring, and Control of Complex Nonlinear Systems Stabilization of a Class of Networked Control Systems with Random packet Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minrui Fei, Weihua Deng, Kang Li, and Yang Song

1

Improved Nonlinear PCA Based on RBF Networks and Principal Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xueqin Liu, Kang Li, Marion McAfee, and Jing Deng

7

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guangrui Wen, Xining Zhang, and Ming Zhao

16

Analyzing Deformation of Supply Chain Resilient System Based on Cell Resilience Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yonghong Li and Lindu Zhao

26

Multi-objective Particle Swarm Optimization Control Technology and Its Application in Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Jia, Dashuai Cheng, Luming Cao, Zongjun Cai, and Min-Sen Chiu

36

Online Monitoring of Catalyst Activity for Synthesis of Bisphenol A . . . . Liangcheng Cheng, Yaqin Li, Huizhong Yang, and Nam Sun Wang

45

An Improved Pyramid Matching Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jun Zhang, Guangzhou Zhao, and Hong Gu

52

Stability Analysis of Multi-channel MIMO Networked Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dajun Du, Minrui Fei, and Kang Li

62

Synthesis of PI–type Congestion Controller for AQM Router in TCP/AQM Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junsong Wang and Ruixi Yuan

69

A State Identification Method of Networked Control Systems . . . . . . . . . . Xiao-ming Yu and Jing-ping Jiang Stabilization Criterion Based on New Lyapunov Functional Candidate for Networked Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qigong Chen, Lisheng Wei, Ming Jiang, and Minrui Fei

77

87

XII

Table of Contents – Part I

Development of Constant Current Source for SMA Wires Driver Based on OPA549 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yong Shao, Enyu Jiang, Quanzhen Huang, and Xiangqiang Zeng High Impedance Fault Location in Transmission Line Using Nonlinear Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Min-you Chen, Jin-qian Zhai, Zi-qiang Lang, Ju-cheng Liao, and Zhao-yong Fan Batch-to-Batch Iterative Optimal Control of Batch Processes Based on Dynamic Quadratic Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Jia, Jiping Shi, Dashuai Cheng, Luming Cao, and Min-Sen Chiu

95

104

112

Management Information System (MIS) for Planning and Implementation Assessment (PIA) in Lake Dianchi . . . . . . . . . . . . . . . . . . . Longhao Ye, Yajuan Yu, Huaicheng Guo, and Shuxia Yu

120

Integration Infrastructure in Wireless/Wired Heterogeneous Industrial Network System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Haikuan Wang, Weiyan Hou, Zhaohui Qin, and Yang Song

129

Multi-innovation Generalized Extended Stochastic Gradient Algorithm for Multi-Input Multi-Output Nonlinear Box-Jenkins Systems Based on the Auxiliary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jing Chen and Xiuping Wang

136

Research of Parallel-Type Double Inverted Pendulum Model Based on Lagrange Equation and LQR Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jian Fan, Xihong Wang, and Minrui Fei

147

A Consensus Protocol for Multi-agent Systems with Double Integrator Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fang Wang, Lixin Gao, and Yanping Luo

157

A Production-Collaboration Model for Manufacturing Grid . . . . . . . . . . . . Li-lan Liu, Xue-hua Sun, Zhi-song Shu, Shuai Tian, and Tao Yu

166

The Second Section: Autonomy-Oriented Computing and Intelligent Agents Parallel Computation for Stereovision Obstacle Detection of Autonomous Vehicles Using GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhi-yu Xu and Jie Zhang

176

Framework Designing of BOA for the Development of Enterprise Management Information System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiwei Ma, Zhaowen Kong, Xuelin Jiang, and Chaozu Liang

185

Table of Contents – Part I

XIII

Training Support Vector Data Descriptors Using Converging Linear Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hongbo Wang, Guangzhou Zhao, and Nan Li

196

Research on Modeling and Simulation of an Adaptive Combat Agent Infrastructure for Network Centric Warfare . . . . . . . . . . . . . . . . . . . . . . . . . . Yaozhong Zhang, An Zhang, Qingjun Xia, and Fengjuan Guo

205

Genetic Algorithm-Based Support Vector Classification Method for Multi-spectral Remote Sensing Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yi-nan Guo, Da-wei Xiao, and Mei Yang

213

Grids-Based Data Parallel Computing for Learning Optimization in a Networked Learning Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lijun Xu, Minrui Fei, T.C. Yang, and Wei Yu

221

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yongzhong Li, Chunwei Jing, and Jing Xu

233

Single-Machine Scheduling Problems with Two Agents Competing for Makespan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Guosheng Ding and Shijie Sun

244

Multi-Agent Asynchronous Negotiation Based on Time-Delay . . . . . . . . . . LiangGui Tang and Bo An

256

The Third Section: Advanced Theory and Methodology in Fuzzy Systems and Soft Computing Fuzzy Chance Constrained Support Vector Machine . . . . . . . . . . . . . . . . . . Hao Zhang, Kang Li, and Cheng Wu

270

An Automatic Thresholding for Crack Segmentation Based on Convex Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chunhua Guo and Tongqing Wang

282

A Combined Iteration Method for Probabilistic Load Flow Calculation Applied to Grid-Connected Induction Wind Power System . . . . . . . . . . . . Xue Li, Jianxia Pei, and Dajun Du

290

Associated-Conflict Analysis Using Covering Based on Granular Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuang Liu, Jiyi Wang, and Huang Lin

297

Inspection of Surface Defects in Copper Strip Based on Machine Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xue-Wu Zhang, Li-Zhong Xu, Yan-Qiong Ding, Xin-Nan Fan, Li-Ping Gu, and Hao Sun

304

XIV

Table of Contents – Part I

BIBO Stability of Spatial-temporal Fuzzy Control System . . . . . . . . . . . . . Xianxia Zhang, Meng Sun, and Guitao Cao

313

An Incremental Manifold Learning Algorithm Based on the Small World Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lukui Shi, Qingxin Yang, Enhai Liu, Jianwei Li, and Yongfeng Dong

324

Crack Image Enhancement of Track Beam Surface Based on Nonsubsampled Contourlet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chunhua Guo and Tongqing Wang

333

The Class-2 Linguistic Dynamic Trajectories of the Interval Type-2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liang Zhao

342

The Fourth Section: Computational Intelligence in Utilization of Clean and Renewable Energy Resources Strategic Evaluation of Research and Development into Embedded Energy Storage in Wind Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . T.C. Yang and Lixiong Li

350

A Mixed-Integer Linear Optimization Model for Local Energy System Planning Based on Simplex and Branch-and-Bound Algorithms . . . . . . . . Hongbo Ren, Weisheng Zhou, Weijun Gao, and Qiong Wu

361

IEC 61400-25 Protocol Based Monitoring and Control Protocol for Tidal Current Power Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jung Woo Kim and Hong Hee Lee

372

Adaptive Maximum Power Point Tracking Algorithm for Variable Speed Wind Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moo-Kyoung Hong and Hong-Hee Lee

380

Modeling and Simulation of Two-Leaf Semi-rotary VAWT . . . . . . . . . . . . . Qian Zhang, Haifeng Chen, and Binbin Wang

389

The Fifth Section: Intelligent Modeling, Control and Supervision for Energy Saving and Pollution Reduction Identification of Chiller Model in HVAC System Using Fuzzy Inference Rules with Zadeh’s Implication Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yukui Zhang, Shiji Song, Cheng Wu, and Kang Li An Improved Control Strategy for Ball Mill Grinding Circuits . . . . . . . . . Xisong Chen, Jun Yang, Shihua Li, and Qi Li

399 409

Table of Contents – Part I

Sliding Mode Controller for Switching Mode Power Supply . . . . . . . . . . . . Yue Niu, Yanxia Gao, Shuibao Guo, Xuefang Lin-Shi, and Bruno Allard

XV

416

The Sixth Section: Intelligent Methods in Developing Vehicles, Engines and Equipments Expression of Design Problem by Design Space Model to Support Collaborative Design in Basic Plan of Architectural Design . . . . . . . . . . . . Yoshiaki Tegoshi, Zhihua Zhang, and Zhou Su Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles . . . . Behnam Ganji, Abbas Z. Kouzani, and H.M. Trinh

425 434

The Seventh Section: Computational Methods and Intelligence in Modeling Genetic and Biochemical Networks and Regulation Supply Chain Network Equilibrium with Profit Sharing Contract Responding to Emergencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ating Yang and Lindu Zhao

445

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yijuan Di, Minrui Fei, Xin Sun, and T.C. Yang

456

Robust Semi-supervised Learning for Biometrics . . . . . . . . . . . . . . . . . . . . . Nanhai Yang, Mingming Huang, Ran He, and Xiukun Wang

466

Research on Virtual Assembly of Supercritical Boiler . . . . . . . . . . . . . . . . . Pi-guang Wei, Wen-hua Zhu, and Hao Zhou

477

Validation of Veracity on Simulating the Indoor Temperature in PCM Light Weight Building by EnergyPlus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chun-long Zhuang, An-zhong Deng, Yong Chen, Sheng-bo Li, Hong-yu Zhang, and Guo-zhi Fan Positive Periodic Solutions of Nonautonomous Lotka-Volterra Dispersal System with Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ting Zhang, Minghui Jiang, and Bin Huang An Algorithm of Sphere-Structure Support Vector Machine Multi-classification Recognition on the Basis of Weighted Relative Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shiwei Yun, Yunxing Shu, and Bo Ge Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

486

497

506

515

Table of Contents – Part II

The First Section: Advanced Evolutionary Computing Theory and Algorithms A Novel Ant Colony Optimization Algorithm in Application of Pheromone Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peng Zhu, Ming-sheng Zhao, and Tian-chi He Modelling the Effects of Operating Conditions on Motor Power Consumption in Single Screw Extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chamil Abeykoon, Marion McAfee, Kang Li, Peter J. Martin, Jing Deng, and Adrian L. Kelly Quantum Genetic Algorithm for Hybrid Flow Shop Scheduling Problems to Minimize Total Completion Time . . . . . . . . . . . . . . . . . . . . . . . Qun Niu, Fang Zhou, and Taijin Zhou Re-diversified Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . Jie Qi and Shunan Pang Fast Forward RBF Network Construction Based on Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jing Deng, Kang Li, George W. Irwin, and Minrui Fei A Modified Binary Differential Evolution Algorithm . . . . . . . . . . . . . . . . . . Ling Wang, Xiping Fu, Muhammad Ilyas Menhas, and Minrui Fei

1

9

21

30

40

49

Research on Situation Assessment of UCAV Based on Dynamic Bayesian Networks in Complex Environment . . . . . . . . . . . . . . . . . . . . . . . . Lu Cao, An Zhang, and Qiang Wang

58

Optimal Tracking Performance for Unstable Processes with NMP Zeroes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jianguo Wang, Shiwei Ma, Xiaowei Gou, Ling Wang, and Li Jia

69

Typhoon Cloud Image Enhancement by Differential Evolution Algorithm and Arc-Tangent Transformation . . . . . . . . . . . . . . . . . . . . . . . . . Bo Yang and Changjiang Zhang

75

Data Fusion-Based Extraction Method of Energy Consumption Index for the Ethylene Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhiqiang Geng, Yongming Han, Yuanyuan Zhang, and Xiaoyun Shi

84

XVIII

Table of Contents – Part II

Research on Improved QPSO Algorithm Based on Cooperative Evolution with Two Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Longhan Cao, Shentao Wang, Xiaoli Liu, Rui Dai, and Mingliang Wu

93

Optimum Distribution of Resources Based on Particle Swarm Optimization and Complex Network Theory . . . . . . . . . . . . . . . . . . . . . . . . . Li-lan Liu, Zhi-song Shu, Xue-hua Sun, and Tao Yu

101

The Model of Rainfall Forecasting by Support Vector Regression Based on Particle Swarm Optimization Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . Shian Zhao and Lingzhi Wang

110

Constraint Multi-objective Automated Synthesis for CMOS Operational Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jili Tao, Xiaoming Chen, and Yong Zhu

120

Research on APIT and Monte Carlo Method of Localization Algorithm for Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jia Wang and Fu Jingqi

128

Quantum Immune Algorithm and Its Application in Collision Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jue Wu, LingXi Peng, LiXue Chen, and Lei Yang

138

An Artificial Bee Colony with Random Key for Resource-Constrained Project Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yan-jun Shi, Fu-Zhen Qu, Wang Chen, and Bo Li

148

The Second Section: Advanced Neural Network and Fuzzy System Theory and Algorithms Combined Electromagnetism-Like Mechanism Optimization Algorithm and ROLS with D-Optimality Learning for RBF Networks . . . . . . . . . . . . Fang Jia and Jun Wu

158

Stochastic Stability and Bifurcation Analysis on Hopfield Neural Networks with Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuewen Qin, Zaitang Huang, and Weiming Tan

166

EMD-TEO Based Speech Emotion Recognition . . . . . . . . . . . . . . . . . . . . . . Xiang Li, Xin Li, Xiaoming Zheng, and Dexing Zhang A Novel Fast Algorithm Technique for Evaluating Reliability Indices of Radial Distribution Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mohammoud M. Hadow, Ahmed N. Abd Alla, and Sazali P. Abdul Karim

180

190

Table of Contents – Part II

An Improved Adaptive Sliding Mode Observer for Sensorless Control of PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ran Li and Guangzhou Zhao Clustering-Based Geometric Support Vector Machines . . . . . . . . . . . . . . . . Jindong Chen and Feng Pan A Fuzzy-PID Depth Control Method with Overshoot Suppression for Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhijie Tang, Luojun, and Qingbo He Local Class Boundaries for Support Vector Machine . . . . . . . . . . . . . . . . . . Guihua Wen, Caihui Zhou, Jia Wei, and Lijun Jiang

XIX

199

207

218

225

Research on Detection and Material Identification of Particles in the Aerospace Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shujuan Wang, Rui Chen, Long Zhang, and Shicheng Wang

234

The Key Theorem of Learning Theory Based on Sugeno Measure and Fuzzy Random Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minghu Ha, Chao Wang, and Witold Pedrycz

241

Recognition of Fire Detection Based on Neural Network . . . . . . . . . . . . . . Yang Banghua, Dong Zheng, Zhang Yonghuai, and Zheng Xiaoming

250

The Design of Predictive Fuzzy-PID Controller in Temperature Control System of Electrical Heating Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ying-hong Duan

259

Stability Analysis of an Impulsive Cohen-Grossberg-Type BAM Neural Networks with Time-Varying Delays and Diffusion Terms . . . . . . . . . . . . . Qiming Liu, Rui Xu, and Yanke Du

266

The Third Section: Modeling and Simulation of Societies and Collective Behaviour Characterizing Multiplex Social Dynamics with Autonomy Oriented Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lailei Huang and Jiming Liu

277

A Computational Method for Groundwater Flow through Industrial Waste by Use of Digital Color Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Takako Yoshii and Hideyuki Koshigoe

288

A Genetic Algorithm for Solving Patient- Priority- Based Elective Surgery Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yu Wang, Jiafu Tang, and Gang Qu

297

XX

Table of Contents – Part II

A Neighborhood Correlated Empirical Weighted Algorithm for Fictitious Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hongshu Wang, Chunyan Yu, and Liqiao Wu

305

Application of BP Neural Network in Exhaust Emission Estimatation of CAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linqing Wang and Jiafu Tang

312

Dynamic Behavior in a Delayed Bioeconomic Model with Stochastic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yue Zhang, Qingling Zhang, and Tiezhi Zhang

321

The Fourth Section: Biomedical Signal Processing, Imaging, and Visualization A Feature Points Matching Method Based on Window Unique Property of Pseudo-Random Coded Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Chen, Shiwei Ma, Hao Zhang, Zhonghua Hao, and Junfeng Qian

333

A Reconstruction Method for Electrical Impedance Tomography Using Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Min-you Chen, Gang Hu, Wei He, Yan-li Yang, and Jin-qian Zhai

342

VLSI Implementation of Sub-pixel Interpolator for AVS Encoder . . . . . . . Chen Guanghua, Wang Anqi, Hu Dengji, Ma Shiwei, and Zeng Weimin

351

The Fifth Section: Intelligent Computing and Control in Distributed Power Generation Systems Optimization of Refinery Hydrogen Network . . . . . . . . . . . . . . . . . . . . . . . . . Yunqiang Jiao and Hongye Su Overview: A Simulation Based Metaheuristic Optimization Approach to Optimal Power Dispatch Related to a Smart Electric Grid . . . . . . . . . . Stephan Hutterer, Franz Auinger, Michael Affenzeller, and Gerald Steinmaurer

360

368

Speed Control for a Permanent Magnet Synchronous Motor with an Adaptive Self-Tuning Uncertainties Observer . . . . . . . . . . . . . . . . . . . . . . . . Da Lu, Kang Li, and Guangzhou Zhao

379

Research on Short-Term Gas Load Forecasting Based on Support Vector Machine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chao Zhang, Yi Liu, Hui Zhang, Hong Huang, and Wei Zhu

390

Network Reconfiguration at the Distribution System with Distributed Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gao Xiaozhi, Li Linchuan, and Xue Hailong

400

Table of Contents – Part II

XXI

The Sixth Section: Intelligent Methods in Power and Energy Infrastructure Development An Autonomy-Oriented Computing Mechanism for Modeling the Formation of Energy Distribution Networks: Crude Oil Distribution in U.S. and Canada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benyun Shi and Jiming Liu

410

A Wavelet-Prony Method for Modeling of Fixed-Speed Wind Farm Low-Frequency Power Pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel McSwiggan and Tim Littler

421

Direct Torque Control for Permanent Magnet Synchronous Motors Based on Novel Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sizhou Sun, Xingzhong Guo, Huacai Lu, and Ying Meng

433

The Seventh Section: Intelligent Modeling, Monitoring, and Control of Complex Nonlinear Systems A Monitoring Method Based on Modified Dynamic Factor Analysis and Its Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xueyan Yin and Fei Liu

442

A Novel Approach to System Stabilization over Constrained Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weihua Deng, Minrui Fei, and Huosheng Hu

450

An Efficient Algorithm for Grid-Based Robotic Path Planning Based on Priority Sorting of Direction Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aolei Yang, Qun Niu, Wanqing Zhao, Kang Li, and George W. Irwin

456

A Novel Method for Modeling and Analysis of Meander-Line-Coil Surface Wave EMATs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shujuan Wang, Lei Kang, Zhichao Li, Guofu Zhai, and Long Zhang

467

The Design of Neuron-PID Controller for a Class of Networked Control System under Data Rate Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lixiong Li, Rui Ming, and Minrui Fei

475

Stochastic Optimization of Two-Stage Multi-item Inventory System with Hybrid Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuli Zhang, Shiji Song, Cheng Wu, and Wenjun Yin

484

Iterative Learning Control Based on Integrated Dynamic Quadratic Criterion for Batch Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Jia, Jiping Shi, Dashuai Cheng, Luming Cao, and Min-Sen Chiu

493

XXII

Table of Contents – Part II

Impedance Measurement Method Based on DFT . . . . . . . . . . . . . . . . . . . . . Xin Wang

499

A 3D-Shape Reconstruction Method Based on Coded Structured Light and Projected Ray Intersecting Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Chen, Shiwei Ma, Bo Sun, Zhonghua Hao, and Liusun Fu

505

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

515

Stabilization of a Class of Networked Control Systems with Random Packet Loss Minrui Fei1, , Weihua Deng1 , Kang Li2 , and Yang Song1 1

School of Mechatronical Engineering and Automation, Shanghai University, Shanghai, 200072, China [email protected], [email protected], y [email protected] 2 School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, United Kingdom [email protected]

Abstract. This paper addresses the problem of stabilizing of a class of Networked Control Systems (NCSs) based on both the communication and network properties. Random packet loss together with Signal-toNoise Ratio (SNR) constrained channels are studied and stabilization conditions are given in discrete time domain. A numerical example is presented and shown that unstable poles and packet dropouts significantly affect the SNR requirement for stabilizing the NCSs.

1

Introduction

Network Controlled Systems (NCSs)[1-2] have drawn a lot of interest in recent years and a number of progress has been made on its theoretic. Currently, the majority of researches ignore the limitations on the communication and mainly consider the characteristics of the network such as the network-induced delays[3], data loss[4] and so on. From the stability point of view of NCSs, the impacts of communication, for instance, the quantization error[5], constraint of SNR[6], etc. are equally important. So it is important to consider the two factors, i.e. the network and communication in the design of NCSs. Recently the factors of communication have been taken into account as well. SNR[7-9] is a major issue that were discussed in these research papers. This problem was firstly studied in [7] and discussed in detail in [8]. In [8] the conditions of stabilization over SNR constrained channel are derived in both continuous and discrete time domain. But the characteristics of networks such as network induced delays and dropouts of packet were not analyzed together with SNR attribute in their above works. Further, it is not easy to compute the numerical solutions due to the absence of proper linear matrix inequality (LMI) formulation. In this paper, we extend the aforementioned works. LMI approach is used to simplify the numerical implementation of the result derived in [8]. And ran

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 1–6, 2010. c Springer-Verlag Berlin Heidelberg 2010 

2

M. Fei et al.

Fig. 1. Control over an AWGN channel in network environment

dom packet dropouts is considered together with the SNR constraint and the stabilization criterion of NCS is established. Notations: The closed unit disk and its complement are denoted as D and ¯ C respectively. The expectation operator is denoted as E. The size of signal is D measured by L2 and L∞ norm, n(k)2L∞ = sup E{nT (t)n(t)} and n(k)2L2 = ∞  2 n(k) . k=0

2

Control over an AWGN Channel with Random Packet Dropouts

Here we will consider the packet dropouts together with the channel SNR constraint. The discrete-time NCS in Fig 1 is given by x(k + 1) = Ax(k) + B νˆ(k)

(1)

where νˆ(k) = θ(k)ν(k) is control input. Hence the system can be further reformulated as x(k + 1) = Ax(k) + Bθ(k)ν(k)

(2)

where θ(k) is a 0-1 random variable with a probability distribution given by  α i=0 Pr{θ(k) = i} = 1−α i=1 where 0 ≤ α ≤ α ¯ ≤ 1 is the probability of packet losses. The controller and channel are chosen as u(k) = −Kx(k).

(3)

v(k) = n(k) + u(k)

(4)

Stabilization of a Class of NCSs with Random Packet Loss

3

where u(k) is the channel input, v(k) is the channel output, and n(k) is zeromean AWGN with power spectral density Sn (ω) = Φ, namely, its variance. The power in the channel input is defined as u(k)pow = E{uT (k)u(k)} and satisfies u(k)pow < P

(5)

where P > 0 is predetermined input power level. Thus the closed-loop system is given as x(k + 1) = (A + BKθ(k))x(k) + θ(k)Bn(k).

(6)

If the system is stable, then power spectral density of u(t) as [8] can be rewritten as 2 Su (ω) = |T (s)| Sn (ω) where T (s), s = jw is the continuous-time closed loop transfer function from n(t) to u(t). u(t)pow has the following definition u(t)pow =

1 2π



∞

−∞

Su (ω)dω .

Thus (5) becomes P > upow = T 2H2 Φ

∞ 1 where T H2 = ( 2π |T (jw)|2 dω)1/2 is H2 norm of T (s). It is obviously that −∞ (5) is equivalent to the form of SNR constraint 2

T H2 <

P . Φ

(7)

The left side of (7) is just the SNR of the channel. We have the follow definition and problem. Definition 1: (a) The discrete-time NCS (1) with n(k) = 0 is mean-square stable; (b) if given γ, under zero initial conditions, for all nonzero n(k) ∈ L2 [0, ∞] and the probability of packet losses 0 ≤ α ≤ α ¯ , u(k)2L∞ < γ 2 n(k)2L2 is satisfied, then the closed loop system (53) is mean-square stable with H2 performance γ. From Definition 1 and (7) we have ρ<

P Φ

(8)

where ρ = γ 2 . Problem 1: Use LMI approach to find a stabilizing state feedback controller gain of the discrete-time NCS (1) such that H2 performance ρ satisfies the constraint (8) and to compute the optimal numerical solution of ρ.

4

M. Fei et al.

Theorem 1. Consider the discrete-time NCS (1), if A has different eigenvalues ¯ C ; i = 1, · · · , m}, then for the given α {pi ∈ D ¯ ρ∗ =

min ρ

(9)

s.t.(11)−(13)

and Problem 1 is solvable if and only if the admissible SNR satisfies P > ρ∗ Φ

(10)

Proof. It is apparent that the (9) is equivalent to the following problem: Given scalar γ > 0, 0 ≤ α ≤ α ¯ ≤ 1, the discrete-time NCS (51) exists a state ¯ satisfies feedback H2 control law , if there exist Q > 0, Z > 0 and K ⎡ ⎤ ¯ T BT −Q QAT 0 QAT + K ⎢ ⎥ A −α−1 Q 0 0 ⎢ ⎥<0 (11) T ⎣ ⎦ 0 0 −1 B −1 ¯ AQ + B K 0 B −(1 − α) Q

¯ −Z K T ¯ K −P¯

<0

(12)

Trace(Z) < ρ

(13)

V (k) = xT (k)P x(k)

(14)

ΔV (k) = E[xT (k + 1)P x(k + 1)|x(k) ] − xT (k)P x(k) = E[(Ax(k) + B νˆ(k))T P (Ax(k) + B νˆ(k))|x(k) ] − xT (k)P x(k) = (1 − α)(Ax(k) + Bν(k))T P (Ax(k) + Bν(k)) +α(Ax(k))T P Ax(k) − xT (k)P x(k)

(15)

We first consider function

then

Thus it is easy to prove discrete-time NCS (1) with n(k) = 0 is mean-square stable by (11). So (a) in Definition 1 is satisfied. Next we will prove condition (b). If Let k−1  nT (q)n(q) Jd = V (k) − Mα =

q=0

 T   αAT P A − P 0 + (1 − α)( A + BK B P A + BK B 0 −1

then the condition (b) is satisfied by (12) and (13). Thus according to Definition 1 the discrete-time NCS (1) is mean-square stable with H2 performance γ and through the optimal problem (9) the optimal H2 performance ρ∗ is obtained.

Stabilization of a Class of NCSs with Random Packet Loss

5

Table 1. The relation of unstable poles and the minimal SNR under random packet dropouts p 1.1329 1.1429 1.1529 1.1629 1.1729 ρ∗ 0.28353 0.52012 1.0826 1.872 2.8661 0.6

0.6 x1(k)

x1(k)

x2(k)

0.4

x1(k)

4

x2(k)

0.4

x2(k) 3

0.2

0.2

0

0 x(k)

x(k)

x(k)

2

−0.2

−0.2

−0.4

−0.4

1

0

−1 −0.6

−0.6 −2

−0.8

0

50

100

150 k

200

250

(a)

300

−0.8

0

50

100

150 k

(b)

200

250

300

0

50

100

150 k

200

250

300

(c)

Fig. 2. The differences in effectiveness under three cases

To further illustrate the proposed theory the following example [10] is given. Example 1. The parameters of discrete-time NCS (1) are given by



a 0.0301 −0.0001 A= ,B = 0.5202 a −0.0053 where a = 1.0078. To show the relation of unstable poles and the minimal SNR clearly we first define a ∈ {1.0078, 1.0178, 1.0278, 1.0378, 1.0478} and obtain the unstable poles of the system correspondingly p ∈ { 1.1329, 1.1429, 1.1529, 1.1629, 1.1729 }. If the probability of packet losses α ¯ is 0.25, then by solving (9) gives the results in Table 1. Remark 1. Table 1 reveals that the random packet dropouts has a significant impact on the minimum of SNR for stabilizability. Compare with the other previously published results of research on SNR, the contribution in this paper is unique and complimentary. It not only enables convenient numerical convenience of numerical implementation simulation but also takes into consideration of network characteristic. T In the example, we we further choose p = 1.1329, P Φ = 40dB and x0 = [0.5 0.5] , we can clearly show the difference in effectiveness under three cases as in Fig. 2: (a) without considering AWGN channel; (b) only considering AWGN channel; (c) considering both AWGN channel and data loss. From Fig. 2, it is obvious that the random data loss and the AWGN have significant influences and the system in the example is stable because the minimal SNR for stabilization is satisfied with the constraint of (7), that is, 0.28353 < P Φ = 40.

3

Conclusions

In this paper we investigate the stabilization of NCS under the influence of both the network and communication characteristics. The stabilization conditions

6

M. Fei et al.

are formulated and given using the LMI approach. Compared with the existing results, The first advantage is that the numerical computation can be realized easily, and another important merit is that the channel SNR constraint are taken into account together with the network-induced factors when analyzing the NCSs, which is practically more applicable.

Acknowledgment The work of Minrui Fei, Weihua Deng and Yang Song was supported by the National Science Foundation of China under Grant 60834002,60774059,60904016, The Excellent Discipline Head Plan Project of Shanghai under Grant 08XD14018, The graduate student innovation fund of Shanghai University under Grand SHUCX101016, Shanghai Key Laboratory of Power Station Automation Technology and The Mechatronics Engineering Innovation Group project from Shanghai Education Commission; and the work of Kang Li was supported by the Research Councils UK under Grant No. EP/G042594/1.

References 1. Yang, T.C., Yu, H., Fei, M.R.: Networked control systems: a historical review and current research topics. Measurement and Control 38(1), 12–17 (2005) 2. Hespanha, J., Naghshtabrizi, P., Xu, Y.: A survey of recent results in networked control systems. Proceedings of the IEEE 95(1), 138–162 (2007) 3. Lin, C., Wang, Z.D., Yang, F.W.: Observer-based networked control for continuoustime systems with random sensor delays. Automatica 45(2), 578–584 (2009) 4. Schenato, L.: To Zero or to Hold Control Inputs With Lossy Links. IEEE Transactions On Automatic Control 54(5), 1093–1099 (2009) 5. Tsumuraa, K., Ishii, H., Hoshin, H.: Tradeoffs between quantization and packet loss in networked control of linear systems. Automatica 45(12), 2963–2970 (2009) 6. Rojas, A.J., Braslavsky, J.H., Middleton, R.H.: Control over a Bandwidth Limited Signal to Noise Ratio constrained Communication Channel. In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain, pp. 197–202 (2005) 7. Braslavsky, J.H., Middleton, R.H., Freudenberg, J.S.: Feedback stabilization over signal-to-noise ratio constrained channels. Technical Report (2003) 8. Braslavsky, J.H., Middleton, R.H., Freudenberg, J.S.: Feedback stabilization over signal-to-noise ratio constrained channels. IEEE Transactions On Automatic Control 52(8), 1391–1403 (2007) 9. Braslavsky, J.H., Middleton, R.H., Freudenberg, J.S.: Effects of time delay on feedback stabilization over signal-to-noise ratio constrained channels. In: IFAC (2005) 10. Gao, H.J., Meng, X.Y., Chen, T.W.: Stabilization of networked control systems with a new delay characterization. IEEE Transactions On Automatic Control 53(9), 163–167 (2008)

Improved Nonlinear PCA Based on RBF Networks and Principal Curves Xueqin Liu1 , Kang Li1 , Marion McAfee2 , and Jing Deng1 1

2

School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, BT9 5AH, U.K. [email protected] Department of Mechanical and Electronic Engineering, Institute of Technology Sligo, Sligo, Ireland

Abstract. Nonlinear PCA based on neural networks (NN) have been widely used in different applications in the past decade. There is a difficulty with the determination of the optimal topology for the networks that are used. Principal curves were introduced to nonlinear PCA to separate the original complex five-layer NN into two three-layer RBF networks and eased the above problem. Using the advantage of Fast Recursive Algorithm, where the number of neurons, the location of centers, and the weights between the hidden layer and the output layer can be identified simultaneously for the RBF networks, the topology problem for the nonlinear PCA based on NN can thus be solved. The simulation result shows that the method is excellent for solving nonlinear principal component problems.

1 Introduction Multivariate statistical methods have been widely used for fault detection and diagnosis in industrial processes. Among them, principal component analysis (PCA) [6] is the most popular one due to its simplicity. However, the traditional PCA is a linear projection method, it may not be suitable to model the nonlinear dynamics efficiently [12]. This has naturally motivated the development of nonlinear PCA for the nonlinear problem. Nonlinear extensions of PCA were first developed by Kramer [8], which rely on multi-layer auto-associative neural networks (ANNs). Given the complex network topology, ANNs are difficult to train and the size of the network topology is difficult to determine [5]. Reference [2] incorporated principal curves into the neural network structure to overcome some of these deficiencies. Other nonlinear PCA methods include an input-training neural network [7, 11] and the use of genetic programming [4]. Despite recently reported success of nonlinear PCA monitoring applications, the following problems still remain: (1) the original nonlinear PCA based on ANNs are difficult to train and the size of the network topology is difficult to determine; (2) how to construct the network effectively and efficiently. This paper tackles these problems by incorporating principal curves and RBF networks with the recently reported Fast Recursive Algorithm (FRA) [9], and proposes an improved nonlinear PCA monitoring scheme, which is termed as NPCA. In the proposed NPCA, principal curves are incorporated into RBF networks, which simplifies the 5-layer ANN topology to two separate 3-layer networks, aiming to solve the first issue. The paper addresses the second issue K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 7–15, 2010. c Springer-Verlag Berlin Heidelberg 2010 

8

X. Liu et al.

by utilizing the recently proposed reported FRA, where the number of neurons, the location of centers, and the weights between the hidden layer and the output layer can be identified simultaneously and efficiently for the RBF networks. The paper is organized as follows. Preliminaries, including PCA-based monitoring, principal curves, and FRA are given in Sect. 2. This is followed by the proposed NPCA monitoring scheme based on principal curves and RBF networks in Sect. 3. Section 4 then presents a simulation example. A concluding summary is given in Sect. 5.

2 Preliminaries This section gives a summary of PCA, the concept of principal curves and the multiinput and multi-output (MIMO) RBF networks based on FRA. 2.1 Principal Component Analysis and Principal Curves Principal Component Analysis. Assuming that a data matrix X ∈ RN ×m is composed with N samples and m variables, the PCA decomposition allows the construction of two univariate statistics for a sample vector x ∈ Rm , a Hotelling’s T2 statistic and a Q statistic:   (1) T 2 = xT PΛ−1 PT x = tT Λ−1 t, Q = xT I − PPT x for which confidence limits can be obtained as discussed in [6], Λ is a diagonal matrix consisting of r eigenvalue of covariance matrix of scaled X. t = xT P is a score vector, and P ∈ Rm×r (r ≤ m is the number of retained PCs) is a loading matrix. Principal Curves. A principal curve, which is proposed by Hastie and Stuetzle [3] (HSPC), is a smooth 1-D curve passing through the middle of an m-D data set. It satisfies the self-consistent property, that is a point on the curve is the average of all data points that project on to it. As illustrated in Fig. 1, the HSPC gives a generalization of the first linear principal component, which assumes that the intrinsic middle structure of data is a curve instead of a straight line. As presented in [3], 1-D principal curve embedded in Rm is a smooth and continuous function f : τ → Rm , and the variable τ ∈ Γ which is the arc-length along the curve parameterizes the curve f (τ ) = (f1 (τ ), f2 (τ ), · · · , fm (τ )), where fi (τ ) (i = 1, 2, · · · , m) is called the coordinate function. For any point x, the projection of that point onto the curve τ is defined by a corresponding projection index τf and τf (x) = sup{τ : x − f (τ ) = inf x − f (μ)} τ

μ

(2)

where μ is a project index. The projection index τf (x) of x is the value of τ for which f (τ ) is closest to x. The distance from x to the curve f (τ ) is f (τf (x)): d2 (x, f ) = x − f (τf (x))22 . Based on the above concepts, an iterative HSPC algorithm was presented in [3] to find the principal curve for a given data distribution.

Improved Nonlinear PCA Based on RBF Networks and Principal Curves

9

1.4 1.2 1

x2

0.8 0.6 0.4 0.2 0 -0.2

0

0.2

0.4

0.6

x1

0.8

1

1.2

1.4

Fig. 1. Principal curve (curved line) and linear PC (straight line) for a given data set

2.2 MIMO RBF Networks Based on Fast Recursive Algorithm MIMO RBF Network Model. Consider a general MIMO RBF network with m inputs, q outputs, n hidden nodes and an output bias as: x ˆ=

n 

θk φk (x; wk ) + e

(3)

k=1

where x and x ˆ ∈ Rq is the RBF network input and output vector respectively, φk (x; wk ) denotes the kth radial basis function, and wk = [σk , ck ] ∈ Rm+1 is the hidden layer parameter vector which includes the width σk ∈ R1 and centers ck ∈ Rm . Moreover, θk ∈ Rq represents the output layer weights for each RBF node, and e ∈ Rq is the modelling error. Suppose a set of N data samples {x, x ˆ}N are used for model training, then (3) can be written in the matrix form  = ΦΘ + Ξ X (4)  = [ˆ where Φ = [φ1 , . . . , φn ] ∈ RN ×n is the regression matrix, X xT (1), . . . , x ˆT (N )]T T T T N ×q n×q T T T N ×q ∈R , Θ = [θ 1 , . . . , θn ] ∈ R and Ξ = [e1 , . . . , eN ] ∈ R . The above problem in (4) can be solved based by mini on the least-square method  2 T  − ΦΘ) (X  − ΦΘ) , where  · F mizing the loss function J(Θ) = ΞF = tr (X is the Frobenius norm. Fast Recursive Algorithm. Suppose k RBF basis vectors φ1 , · · · , φk have been selected from all M candidates. FRA is given by defining a recursive matrix Mk  T ΦTk Φk and a residual matrix Rk  I−Φk M−1 k Φk , where R0  I, Φk = [φ1 , · · · , φk ]. According to [9] and [10], Rk , k = 1, · · · , M can be updated recursively: Rk+1 = Rk −

Rk φk+1 φTk+1 RTk , k = 0, 1, · · · , M − 1 φTk+1 Rk φk+1

(5)

10

X. Liu et al.

 T Rk X).  In the forward stepThe cost function can be rewritten as: J(Θ, Φk ) = tr(X wise stage, the RBF centers are selected one-by-one. Thus, suppose at the k-th step, one more center φj , k + 1  j  M from the candidate term pool is to be selected, the net contribution of φj to the cost function can be calculated as  

2 (k) T (k) (k) T (k)  ΔJk+1 (Θ, Φk , φj ) = tr (X ) φj /(φj ) φj (6) (k)  (k)  Rk X.  By defining an auxiliary matrix A ∈ Rk×M and where φj  Rk φj , X M×q ax ∈ R with their elements 0, 1≤j
(i−1) T  (k) (φi ) X ,1≤i≤k (8) ai,x  (k)  (k) , k < i ≤ M (φi )T X  the cost function can be updated recursively: ΔJk+1 (Θ, Φk , φj ) = tr (aj,x )T (aj,x )/ aj,j ). The model term that gives the largest contribution is then selected. After a satisfactory network with n RBF centers has been constructed, the model coefficients can

 be computed recursively θˆj = aj,x − ni=j+1 θˆi aj,i /aj,j , j = k, k − 1, · · · , 1.

3 Nonlinear Process Monitoring Based on NPCA Nonlinear extensions of PCA were first proposed by Kramer [8], which rely on a 5layer ANN to reconstruct the network input data matrix X as accurately as possible using a reduced bottleneck nodes, as illustrated in Fig. 2. However, Kramer’s method is not a proper generalization of linear PCA due to (1) the ANN topology is complex and difficult to train; (2) it is difficult to determine the number of nodes in the mapping layer, the de-mapping layer and the bottleneck layer; (3) the bottleneck layer doesn’t necessarily imply the nonlinear principal components [2, 11]. An alternative approach of nonlinear PCA was developed by [2] to simplify the original complex 5-layer ANN by separating it into two 3-layer networks to represent the mapping and the de-mapping functions. As illustrated in Fig. 3, the input of the mapping network is the original data matrix X and the output are the nonlinear scores T, while the inputs and outputs of the de-mapping network are the scores T and the re respectively. As presented in [2], principal curves are used constructed data matrix X,  2 , and eased the first to obtain the nonlinear scores instead of optimizing E = X − X and the third problems stated above. This paper takes the advantage of the nonlinear PCA structure developed in [2] and applies FRA [9] to solve the second problem. Using the HSPC algorithm, every data point in data matrix X can be projected orthogonally onto the principal curve and results an ordered arc-length vector, which is defined as the nonlinear score. For the given data matrix X with N data samples, a general nonlinear form of nonlinear PCA is obtained as follows: X = F(T) + E

(9)

Improved Nonlinear PCA Based on RBF Networks and Principal Curves

X

11

X

T

Fig. 2. Topology of nonlinear PCA based on a 5-layer ANN

X

X

T

(a) Mapping

T

(b) De-mapping

Fig. 3. The architecture of NPCA based on RBF networks (a) mapping (b) de-mapping

where E is the residual; F() is the nonlinear principal loading function, which represents the interrelationships between the latent variables T = [t1 , t2 , · · · , tr ] and the original data X. Here, we use the percentage of explained variance to choose the number of nonlinear principal components r.  and the associated scores are obtained based After the approximated data matrix X on the HSPC algorithm, a nonlinear principal component model in the sense of nonlinear principal loading can be generated using RBF networks based  on FRA, as illustrated  in Fig. 3. In this paper, the Gaussian kernel function φk = exp −1/2σk x − ck 2 is used in the basis functions of the RBF networks. The corresponding outputs of the mapping network, i. e. the nonlinear scores, are given by n  θ k φk (x; ck ; σk ) (10) ti = k=1

12

X. Liu et al.

where ti ∈ Rr is the sample score vector, i = 1, 2, · · · , N . The corresponding outputs of the de-mapping network, i. e. the corrected data vector x ˆi , are given by x ˆ = n  ˆ θ k φk (ti ; ˆ ck ; σ ˆk ). k=1

Thus, the Hotelling’s TN2 P CA and QN P CA statistics for a new centered sample z can be formulated: TN2 P CA = zT Sxx −1 z n n   QN P CA = z − θk φk (z; ck ; σk ); ˆ ck ; σ ˆk )22 θˆk φk ( k=1

(11) (12)

k=1

It should be noted that the nonlinear scores may not follow Gaussian distribution any more. Therefore, it is appropriate to use kernel density estimation for the confidence limit determination [1]. For both statistics, the 95% confidence limits are used in this paper.

4 Simulation Example This section shows the application procedure of the above proposed improved nonlinear PCA algorithm (NPCA) on a simulated process, along with conventional PCA. 4.1 Process Description The process under investigation comprises of 3 process variables, x1 , x2 and x3 that depend on a parameter a: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ x1 a e1 x = ⎝ x2 ⎠ = ⎝ a2 − 3a ⎠ + ⎝ e2 ⎠ (13) −a3 + 3a2 x3 e3 where e1 , e2 and e3 were independent and identically-distributed Gaussian sequences of zero mean and variance 0.01, which represented measurement uncertainty. a was a uniformly distributed stochastic sequence within the range of [ 0.01 2 ]. From the above process, a first data set of 1000 samples was generated representing normal process operations. A second data set including 100 samples was generated under the same conditions with some disturbances and was used as the testing data set. The disturbances were introduced as follows: (1) a step bias of x2 by -0.5 from the 21st to 50th sample; and (2) a ramp change of x2 with increments of 0.03 from the 31st to 50th sample. 4.2 Process Monitoring with PCA/NPCA for Nonlinear System Section 3 has shown that establishing a NPCA model requires the selection of the RBF basis functions for two RBF networks, including the width parameter σ, the centers c, the number of nodes and the number of PCs. As stated in the previous section, the Gaussian kernel function is used in this paper. Based on ‘the trial and error’ method, in this

Improved Nonlinear PCA Based on RBF Networks and Principal Curves

13

7 6

150

TNPCA

4 3

100

2

T2

PCA

5

2

50

1 0

2

QNPCA

3 2.5

1.5

Q

PCA

0 2.5

1 0.5 0

2 1.5 1 0.5

20

40

60

Sample Number

80

0

100

20

40

60

80

Sample Number

100

Fig. 4. Univariate monitoring statistics based on PCA and NPCA for step-type fault 7 150

6

NPCA

4 3

T2

T2

PCA

5

2

100

50

1 0

0

34

2.5 0.7 0.6

QNPCA

Q

PCA

2 1.5 1 0.5 0

0.5 0.4 0.3 0.2 0.1

20

40

60

Sample Number

80

100

10

20

30

40 46

60

Sample Number

70

80

90

100

Fig. 5. Univariate monitoring statistics based on PCA and NPCA for ramp-type fault

example, σ = 4 for both RBF networks gave good performance. In addition, 4 nodes including the 898th, 158th, 450th, 35th sample vector of the original data, and 4 nodes including the 696th, 450th, 688th, 869th sample vector of the scores were selected as the centers of candidate hidden nodes for the mapping network and de-mapping network , respectively. In the meantime, FRA determined the weights between the network layers, ΘT = [ −200.18 4.55 28.82 192.34 ], with a bias 5.36 for the mapping network, for example. One nonlinear principal component (PC) which can capture 99.9% of the variance of the original 3 system variables was chosen for the NPCA model. In contrast, one linear PC only captured 95.7% of the total variance. The monitoring results for the testing samples with the step and ramp disturbances are shown in Fig. 4 and Fig. 5, respectively. Both TP2 CA and QP CA statistics calculated

14

X. Liu et al.

by the PCA model (the left plots) for the testing samples are not sensitive to the disturbances. However, the statistics calculated by the NPCA model showed rapid response to the above disturbances, as shown in the right plots in Fig. 4 and Fig. 5. Based on the above simulation results, it seems that PCA is not sensitive to non-significant step- or ramp-type faults. By contrast, the proposed NPCA method has better performance and is more sensitive than PCA under the same conditions.

5 Conclusions This paper has studied the monitoring of nonlinear processes based on neural networks. Existing methods in this topic rely on the use of either a complex 5-layer ANN topology, or principal curves to simplify the former into two 3-layer networks. However, existing work has not addressed the following issues: (1) how to determine the size of the network topology; (2) how to determine the number of nodes, the locations of the centers in the network effectively and efficiently. This paper has addressed the above issues by (1) incorporating principal curves and RBF networks to simplify the original 5-layer ANN topology to two 3-layer networks; (2) utilizing the advantage of FRA to determine the number of neurons, the locations of the centers, and the weights between the network layers simultaneously and efficiently for the RBF networks. Finally, this paper introduced a monitoring scheme by incorporating principal curves, RBF networks and FRA into the NPCA-based monitoring framework. This scheme has been demonstrated by an simulation study that it is more reliable than linear PCA method.

Acknowledgement This work was partially supported by the EPSRC, UK, under grant EP/F021070/1, and RCUK under grant EP/G042594/1.

References [1] Chen, Q., Wyne, R., Goulding, P.R., Sandoz, D.J.: The application of principal component analysis and kernel density estimation to enhance process monitoring. Control Engineering Practice 8(5), 531–543 (2000) [2] Dong, D., McAvoy, T.J.: Nonlinear principal component analysis-based on principal curves and neural networks. Computers and Chemical Engineering 20(1), 65–78 (1996) [3] Hastie, T.J., Stuetzle, W.: Principal curves. Journal of American Statistical Association 84, 502–516 (1989) [4] Hiden, H., Willis, M., Tham, M., Montague, G.: Nonlinear principal components analysis using genetic programming. Computers and Chemical Engineering 23(3), 413–425 (1999) [5] Hinton, G.E., Salakhutdinov, R.R.: Reducing the dimensionality of data with neural networks. Science 313(5786), 504–507 (2006) [6] Jackson, J.E.: A Users Guide to Principal Components. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York (1991) [7] Jia, F., Martin, E.B., Morris, A.J.: Nonlinear principal components analysis with application to process fault detection. International Journal of Systems Science 31(11), 1473–1487 (2000)

Improved Nonlinear PCA Based on RBF Networks and Principal Curves

15

[8] Kramer, M.A.: Nonlinear principal component analysis using autoassociative neural networks. AIChE Journal 37(3), 233–243 (1991) [9] Li, K., Peng, J., Irwin, G.W.: A fast nonlinear model identification method. IEEE Transactions on Automatic Control 50(8), 1211–1216 (2005) [10] Li, K., Peng, J., Bai, E.: A two-stage algorithm for identification of nonlinear dynamic systems. Automatica 42(7), 1189–1197 (2006) [11] Tan, S., Mavrovouniotis, M.L.: Reducing data dimensionality through optimizing neural network inputs. AIChE Journal 41(6), 14711480 (1995) [12] Wilson, D.J.H., Irwin, G.W., Lightbody, G.: Rbf principal manifolds for process monitoring. IEEE Transactions on Neural Networks 10(6), 1424–1434 (1999)

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing Guangrui Wen1,2, Xining Zhang1, and Ming Zhao1 1

Research Institute of Diagnostics & Cybernetics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China 2 Xi’an Shaangu Power Co., Ltd, Xi’an, Shaanxi 710611, China {grwen,zhangxining,mzhao}@mail.xjtu.edu.cn

Abstract. Based on the Holo-balancing theory of shaft system, a new multiobjective optimization balancing method including load mass, uniformity and maximum of the residual vibration is proposed by building a multi-objective fuzzy evaluation function and application of particle swarm optimization algorithm. The advantage of the proposed method is studied by comparing with the traditional genetic algorithms optimization. And the shortcoming of influence coefficient methods failing to restrict the load mass and guaranteeing the uniformity of the residual vibration is conquered. Finally, the validity and effectiveness of the proposed method is verified through a field power generator set balancing case. Keywords: Particle swarm optimization, Holo-balancing, power generator set, Rotating machine.

1 Introduction A large amount of statistical data indicates that the mass unbalance of the rotors is usually the major cause of excessive vibrations of large capacity turbine. In these huge power plants, a rotor is always consisted of several flexible shafts, which are coupled together by rigid or flexible couplings. Although the shafts would all have been balanced at a high speed in manufactory, after assembled together on-site, their balance conditions change due to coupling-affect between each of them. Unbalance is a common and frequent fault of power generator set. It will cause deflection and internal stress of generator set, speed up the wearing of bearing and shaft seal, and decrease the work efficiency of the machine. Moreover, it is root causes of various accidents [1]. Because of the difference between rotor set balancing and single rotor balancing in power generator set, the influence of vibration measurement section and supporting condition of bearing, it is very difficult to get the optimal result by solving the balancing weight directly. Multipurpose optimizing of power generator set balancing is studied, and the particle swarm optimization method is employed to solve the problem here. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 16–25, 2010. © Springer-Verlag Berlin Heidelberg 2010

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing

17

Influence coefficient [2] and modal balance method [3] are widely used methods for balancing in industry field. Holospectrum [4][5][6] based balancing technique is a relatively new method for balancing [7]. It takes the differences of stiffness of power generator set system in each direction into consideration by employing the holospectrum in balancing. Therefore, it possesses unique advantageous in balancing. Holospectrum based balancing method is adopted in weights optimization in the paper. The interrelated concepts of rotating frequency ellipse, initial phase point of rotating frequency ellipse and transfer matrix are involved. The rotating frequency ellipse is the synthesis ellipse of rotating frequency on holospectrum [8]. The initial phase point is the position of rotor centre in rotating frequency ellipse when the keyphase slot aligns the key-phase sensor. The vector from the center of rotating frequency ellipse to the initial phase point is defined as the initial phase vector. Transfer matrix is the corresponding vibration response matrix of the holospectrum parameters when trial weight of 1000g is added at zero degree on rotor. Once the transfer matrix of generator set is obtained, the original vibrations at each measuring section of rotors can be decreased or eliminated by adjusting the weights and the corresponding phase angles.

2 Holo-Balancing Technique Traditional filed balancing method, which is based on the hypothesis that the isotropic stiffness of the rotor system is equal, uses a single sensor to capture the vibration signals and calculate the dynamic balancing on a measuring plane. While the isotropic stiffness of the rotor system has distinct differences, this method will bring errors and reduce the balancing precision[9]. The holospectrum technique, which is based on the principle of information theory, could organically integrate and fuse the information captured by multi-sensors, and make full use of the multiway vibration signals to fully and factually reflect the vibration state of the rotor or machine unit. 2.1 Rotation Frequency Ellipse Rotation frequency ellipse is one time frequency ellipse which is compounded of x , y two directions information on each measuring section of the rotor. Rotation frequency ellipse uses right hand coordinate system, namely, when the observer dead against the arrowhead of z-axis, the counterclockwise direction of the movement of the rotor center on the rotation frequency ellipse is from x to y. The motion equation is: ⎧ x = A sin(ωt + α ) ⎨ ⎩ y = B sin(ωt + β )

(1)

⎧ x = sx ⋅ sin ωt + cx ⋅ cos ωt ⎨ ⎩ y = sy ⋅ sin ωt + cy ⋅ cos ωt

(2)

18

G. Wen, X. Zhang, and M. Zhao

The rotation frequency ellipse of whole rotor system could expressed by coefficient matrix. Namely the matrix form of 3D- holospectrum is:

⎡ sx1 ⎢ sx ⎢ 2 ⎢ M ⎢ ⎢⎣ sxn

cx1 cx2 M cxn

sy1 sy2 M syn

cy1 ⎤ cy2 ⎥⎥ M ⎥ ⎥ cyn ⎥⎦

(3)

The coordinate of initial phase point could be expressed as follows: (cxi , cyi )

; i = 1L n.

(4)

where n is the total number of measuring sections. The initial phase point, which has close connection with the key of rotor, is the feature point of the rotation frequency ellipse when the rotor on its revolution and is the location of the rotor center on the rotation frequency ellipse when the key phase slot aligns the key-phase sensor. 2.2 Transfer Matrix Transfer matrix is the vibration responding matrix caused by adding 1000 g < 0。 to some counterweight plane. It could be expressed as follows:

⎡ sx1 ⎢ sx ⎢ 2 ⎢ M ⎢ ⎣⎢ sxn

cx1 cx2 M

sy1 sy2 M

cxn

syn

cy1 ⎤ cy2 ⎥⎥ M ⎥ ⎥ cyn ⎦⎥

(5)

Each counter weight plane corresponds to a transfer matrix, which expresses the 3Dholospectrum of the vibration response of standard trial weight. Each row of the 3Dholospectrum is a 2D-holosprectrum and the i th row means that the vibration response caused by the trial weight acting on the i th bearing surface. Then, after obtaining the transfer matrix, the initial vibration could be reduced by adjusting the size and the angle of the trial weight on the weighting plane.

3 The Optimization of Balancing Weights Suppose one generator set with M balancing planes, N vibration measuring sections and one balancing speed is considered. Now we expect to seek a group of proper balancing weights Pij to make the vibrations of rotor set meet the following equations: M

Ai 0 + ∑ cij p j = 0 j =1

(i = 1, 2,..., N ;

j = 1, 2,..., M )

(6)

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing

19

Where, Ai 0 is the original vibration in i th measuring section, cij is the influence coefficient of weight on j th balancing plane to i th measuring section in transfer matrix. p j is the weight on j th balancing plane. The traditional influence coefficient balancing method get the balancing weights by minimizing the square sum of residual vibrations according to Goodman’s least square method[3]. Holospectrum based balancing method for generator set determines the weights preliminarily according to the duality principle of unbalancing response of generator set. Usually, the number of balancing plane M is smaller than that of the measuring section N . So it is impossible to completely eliminate the vibration at each measuring section in field balancing. Different balancing results will be obtained by different balancing weight plans. Therefore, it is necessary for us to do further optimization to get the optimal balancing weights 1. 3.1 The Selection of Optimization Object The least square of residual vibrations is the main and common used method to solve the balancing weights. However, the minimum of square sum of residual vibrations is by no means the minimum of all the residual vibrations in each measuring section. Therefore, the larger vibration may still exist in certain measuring section. Considering the above situation, the minimum square sum of residual vibrations min F ( X ) = min ∑ x 2 , the i 1 maximum of residual vibration min F2 ( X ) = min(max xi ) and diversity of residual

vibrations between each measuring section min F3( X ) = min(max xi − min xi ) are selected to measure and evaluate the effects of balancing. Then the solution of the optimal weight scheme turns to a multipurpose optimization. To solve the multipurpose optimization problem, generally different weights is set to each item in optimization object function according to their importance first. Then the multipurpose optimization problem turns to a single object optimization problem by the method of linear weighting summation. Because of the differences of magnitude and dimension among each item in object function, it is liable to happen that some item in object function is always in dominant position while other item remains unchanged and will not be optimized [9]. To settle this problem the membership function is introduced for each item in object function based on the fuzzy mathematics. Half trapeziform function is selected as the membership function for each function item. ⎧ 1 ⎪ − M F ⎪ i(X ) μ ( Fi ( X )) = ⎨ i − M ⎪ i mi ⎪ 0 ⎩

Fi ( X ) = mi mi < Fi ( X ) < M i Fi ( X ) = M i

(7)

20

G. Wen, X. Zhang, and M. Zhao

Where, M i and mi are the maximum and minimum value of the function item Fi (X) respectively. It is obvious that the higher the value of membership function, the closer the solution to the optimum value. The overall optimization object function is defined as weighting summation of each membership function. H ( X ) = λ μ ( F ( X )) + λ μ ( F ( X )) + λ μ ( F ( X )) 1 1 2 2 3 3

(8)

，

Where λi i = 1, 2,3; is the weight coefficient of each membership function, and it is determined by the importance of each function item. Considering of the leading role of Fi (X) , its weight coefficient are set as λ = 0.7 1

,

λ 2 = 0.2

,

λ3 = 0.1 in the paper. The

optimal-balancing scheme will be the solution that will achieve maximum of H (X) . 3.2 The Principle of Particle Swarm Balancing Weight Optimization

The particle swarm optimization method [10-12] was proposed by Kennedy and Eberhart in 1995. The original intention was to simulate the movement of organisms in a bird flock or fish school. In fact, it is a population-based optimization algorithm. Each particle is an individual, and the swarm is composed of a certain number of particles. During the process of optimization, evolutionary thought is introduced. In particle swarm optimization, the problem solution space is formulated as a search space. Each position in the search space is a correlated solution of the problem, which is corresponding to one individual bird in flock. The individual bird is called “particle” and has its own position, speed and fitness value. Each particle remembers and follows the current optimal particle and researches in the solution space. Suppose a continuous optimization problem with D variables is considered, then the solution space is formulated as a D-dimensional search space, and the value of jth variable is formulated as the position on jth dimension. Each particle moves according to its velocity. The velocity is randomly generated toward pbest and gbest positions. pbest is the particle best position found by each particle itself. gbest is the global best position found by the whole swarm so far. Each particle updates its speed and position according to following two equations:

Where,

vidk +1 = wvidk + c1rand1k ( pbestidk − xidk ) + c2 rand2k ( gbestdk − xidk )

(9)

xidk +1 = xidk + vidk +1

(10)

k τ X ik = ( xik1 , xik2 ,L, xidk ,L , xiD )

position and speed of particle i .

xidk

k τ and Vi k = (vik1 , vik2 , L, vidk , L, viD )

are the

is the current position of particle i at iteration k. vidk

is the speed of particle i at iteration k. c1 and c2 are the coefficients of acceleration, which control the influence of pbest and gbest on the search process. According interrelated references, here c1 = c2 = 2.05 . rand1 and rand2 are the random numbers in [0, 1].

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing

21

The inertia weight w is used to control memory. The particles maintain high velocities with a larger w, and low velocities with a smaller w. A larger w can prevent particles from being trapped in local optima, and a smaller w encourages particles searching the local area. The dynamic inertial factor is adopted in order to get the good searching ability in earlier stage and high convergence speed in the later stage. During iteration procedure inertial factor w decreases from 1 to 0.2 in the paper. The speed of particle in each dimension is confined in [−vd max ,+ vd max ] to prevent the particle flying away from the best solution or falling into the local optima. Proper value should be set to vd max . In the paper, vd max is set to be 0.1xd max . Suppose the number of balancing plane is K , the process of particle swarm optimization can be described below: 1) Design particle as X = [m1 , α1 , m2 , α 2 , L , mk , α k ] . Where

mi and α i are the

weight and phase on the ith plane. 2) Calculate the balancing weight according to the transfer matrix and original vibration. Determine the searching space. 3) Initialize the population, and determine the maximum M i and minimum

mi of

each dimension i. 4) Determine the residual vibrations according to the original vibration, transfer matrix and the weight of each particle. 5) Evaluate the fitness value of each H ( X ) = λ1 μ ( F1 ( X )) + λ2 μ ( F2 ( X )) + λ3 μ ( F3 ( X )) .

particle

in

population,

6) Compare the particle’s fitness value to determine the pbest of each particle and gbest of all particles. 7) Adjust the speed and position of individual particle, and generate the new generation population.

8) Go to step 4 to do the iteration, until the optimal solution is found. 4 Application of Particle Swarm Optimization in Power Generator Set Field Balancing The following is a field balancing application of a generator in power plant. The power generator set of turbine was balanced due to the high vibrations in highpressure rotor and intermediate pressure rotor. The block diagram of the generator is shown in Figure 1. The target of the balancing was to reduce the vibrations in highpressure and intermediate pressure rotors. The flange edge at bearing 1#, the flanged edge at bearings 2# and 3#, and the flanged edge at bearing 4# and 5# were selected as balancing plane, which are remarked with A, B and C in figure 1. Vibration

22

G. Wen, X. Zhang, and M. Zhao

measuring sections were selected at bearings of 1#, 2#, 3# and 4#. The transfer matrix of balancing plane was calculated by the previous balancing experiments.

Fig. 1. Block diagram of the generate set

The optimized balancing weight is shown in Table 1. The vibrations before and after the balancing are given in Table 2 and Figure 2. After balancing, vibrations in the high-pressure cylinder and intermediate pressure cylinder are decreased greatly. In the mean time, the vibrations in each bearing are well distributed especially after the optimization. In order to further verify the performance of particle swarm optimization, genetic algorithm optimization was compared. According to the reference 13, the crossing rate and mutation rate of genetic algorithm were set to 0.4 and 0.01 respectively. The population size of the two methods was 150. Average processing with 30 optimization experiments was introduced to eliminate the randomness in result. Figure 3 shows the convergence curves of two optimizing processes. It is clear that on the condition of the same population size the convergence rate of particle swarm optimization is higher than that of genetic algorithm. Table 1. Balancing weight after optimization Table Type Styles Balancing Plane Balancing weight (g °)

∠

A

B

C

∠

984.34 151.76

∠

966.27 -61.3

1057.49 93.13

∠

Table 2. Vibrations before and after balancing Vibration Test Plane 1# 2# 3# 4#

∠°) ∠°) ∠°) ∠°) ∠°) ∠°) ∠°) ∠°)

X(um Y(um X(um Y(um X(um Y(um X(um Y(um

Vibration Before Balancing

∠136 ∠28.0 ∠114.0 ∠65.0 ∠110.0 ∠18.0 ∠-84.0 18.5∠176.0

116 64.0 54.0 79.0 117.0 110.0 49.0

Vibration After Balancing

∠68 ∠-56 ∠139 ∠34 ∠82 ∠-9 ∠104 37.2∠25

64.2 57.4 19.2 36.3 49.6 78.8 54.9

Vibration after Multipurpose Optimization

∠-69.4 ∠10.3 ∠21.6 ∠29.5 ∠39.2 ∠83.3 ∠22.9 34.4∠153.3 45.9 35.5 22.2 39.8 45.7 45.6 45.9

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing

a) Vibrations before balancing

23

b) Vibrations after balancing

c) Vibrations after multiobjective optimization Fig. 2. 3D holospectrum of vibrations before and after balancing (rotating frequency ellipses of vibrations at bearing 1#, 2#, 3# and 4# are arranged from lower left to top right in turn)

Fig. 3. Comparison of convergence progress between particle swarm method and genetic algorithm

24

G. Wen, X. Zhang, and M. Zhao

5 Conclusion The requirement of field dynamic balancing is analyzed, and the maximum, uniformity and square summation of residual vibration are selected to be the optimization sub-objects for generator set balancing. Fuzzy mathematics theory is employed to build the multipurpose optimization function. The principle of optimization of balancing weight with particle swarm algorithm is studied. Furthermore it is applied to implement the weight optimization of power generator set. The effectiveness and advantageous of the proposed method is verified by a practical field balancing application of generator set.

Acknowledgement The research was financially supported by the project 2007AA04Z432 of High Technology Program of China, the project 50775175 of Natural Science Foundation of China, and the project (2009CB724405) of State Basic Research Development Program of China.

References 1. Wu, S.T.: Study on field balancing method of flexible rotor sets. Master thesis of Xi’an Jiaotong University (2003) 2. Grobel, L.P.: Balancing turbine generator rotors. General electric review 56(4), 22 (1953) 3. Goodman, T.P.: A least-squares method for computing balance corrections. ASME Transactions. Journal of engineering for industry 86(3), 273–279 (1964) 4. Qu, L.S., Liu, X., Chen, Y.D.: Discovering the holospectrum. Noise & Vibration Control Worldwide 20(2), 58–62 (1989) 5. Qu, L.S., Liu, X., Peyronne, G., Chen, Y.D.: The holospectrum: a new method for rotor surveillance and diagbosis. Mechanical Systems and Signal Processing 3(3), 255–267 (1989) 6. Qu, L.S., Chen, Y.D., Liu, J.Y.: The holospectrum: a new FFT based rotor diagnostic method. In: Proceedings of the 1st International Machinery Monitoring & Diagnostics Conference, Las Vegas, Nevada, vol. 9, pp. 196–201 (1989) 7. Qu, L.S., Qiu, H., Xu, G.H.: Rotor balancing based on Holospectrum analysis: principle and practice. China mechanical engineering 9(1), 60–63 (1998) 8. Qu, L.S., Wang, X.F.: An introduction to Holo-balancing technique. Shaanxi Electric Power 1, 1–5 (2007) 9. Jia, Z.H.H., Chen, H.P., Sun, Y.H.: Multi-objective Particle Swarm Optimization Algorithm for Flexible Job Shop Scheduling. Journal of Chinese Computer Systems 29(5), 885–889 (2008) 10. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the 1995 IEEE International Conference on Neural Networks, pp. 1942–1948. IEEE Press, New Jersey (1995)

Application of Partical Swarm Optimization Algorithm in Field Holo-Balancing

25

11. Eberhart, R.C., Shi, Y.: Particle swarm optimization: developments, applications and resources. In: Kim, J.H., Zhang, B.-T., Fogel, G., Kuscu, I. (eds.) Proceedings of the 2001 IEEE Congress on Evolutionary Computation, pp. 81–86. IEEE Press, New Jersey (2001) 12. Yang, W., Li, Q.Q.: Survey on Particle Swarm Optimization Algorithm. Engineering Science 16(15), 87–94 (2004) 13. Li, J.Q., Shi, G.Z.: A Study of the Relationship of Crossover Rate and Mutation Rate in Genetic Algorithm. Journal of Wuhan University of Technology 27(1), 97–99 (2003)

Analyzing Deformation of Supply Chain Resilient System Based on Cell Resilience Model Yonghong Li and Lindu Zhao* Institute of Systems Engineering, Southeast University, Nanjing, Jiangsu 210096, P.R. China [email protected], [email protected]

Abstract. Resilience is a basic attribute of a supply chain system, and affects the core competence of the system. In the paper, based on the supply chain resilience analysis method, it built a mathematical model of supply chain resilient system with two members, and researched the changing rules of supply chain system deformation with time under the impact of sustained accumulation risks. It provided a novelty way for supply chain research. At last, a numerical simulation on the theoretic analysis was carried out by Matlab. The results showed the system deformation driven by supply chain resilience would increase gradually with some certain relationship. Keywords: supply chain resilient system, system deformation, risks stress.

1 Introduction With the increase of market complexity and uncertainties, the demand structure becomes diversified. Facing such complicated environment, some supply chains could maintain their functions to keep normal operation while some are in the brink to break down. The main reason is the deformation ability of supply chain system driven by resilience. The “9.11” event in America and the earthquake disaster in China are two examples showing the importance of deformation ability of a supply chain resilient system. The concept of resilience appeared first in material mechanics field which represented the material ability to return to the original state or location after a deformation. Later, the word “resilience” was used to characterize the response ability of organizations to unexpected disruptions. Mallak (1998) defined resilience as the individuals or organization ability to minimize the negative influence [1]. Carpenter et al. (2001) defined three properties of resilience: the change amount that could be bear by the system, the extent for system self-organization ability and the system study and self-adaptive extent [2]. Meanwhile, Anderson (2003) pointed out that resilience represented not only the recovered ability, but also meant the organizations adaptive ability in response to positive and negative influences [3]. And Vogus and Sutcliffe *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 26–35, 2010. © Springer-Verlag Berlin Heidelberg 2010

Analyzing Deformation of SC Resilient System Based on Cell Resilience Model

27

(2007) defined resilience as the maintenance of positive adjustment under challenging conditions such that the organization emerged from those conditions strengthened and more resourceful [4]. Based on the complex systems theory, Goldspink and Kay (2003) regarded organizations as a network structure and analyzed the coexistence and consistency among linkage unit, for explaining the dynamic evolution rules of organizations, especially the dynamic evolution rules of resilience [5]. Therefore, the adaptive ability was an element of resilience, which reflected the system study rules in response to disturbance [6]. When the concepts of agility, responsiveness and flexibility were introduced into the literatures on supply chains [7], researchers began to emphasize that as a network, a supply chain should make rapid response to change just like an organic system. And then, the concept of supply chain resilience emerged. Similar to the resilience definition in ecological system [8], Christopher (2004) defined resilience as the ability that a supply chain system returned to its original state or transferred to a new and more satisfying state after disruptions, flexibility, adaptability and agility included [9]. He held that the best mode to obtain resilience for a supply chain was to create a network responding rapidly to states change. Lee (2004) summarized the standard of high efficient supply chain resilience [10]. Sheffi (2005) pointed out that resilience referred to the ability of a company to bounce back from a large disruption , including the speed with which it returned to normal performance levels (production, services, fill rate) [11]. Meanwhile, he thought that resilience could be realized by redundancy or constructing flexibility, which was the same with the opinion of Rice and Caniato (2003) [12]. Moreover, Snyder and Shen (2006) compared different effects of suppliers and pointed out that redundancy supplier strategy made a critical effect for preventing supply disruption and how to recover normal supply [13]. Different from the work of Rice and Caniato, Muckstadt and Murray (2001) proposed that the supply chain resilience could be created by enhancing the cooperative relationship and cooperation intensity of supply chain members and reducing uncertainty of operation environment [14] and Haywood and Peck (2003) found changing supply chains would suffer attack more easily while researching supply chain reliability. Therefore, they suggested that it was the key of enhancing supply chain resilience to enhance the management of supply chain change period [15]. Under the premise of analyzing supply chain risk sources and corresponding managerial decisions, Kong (2008) pointed out the key points to obtain resilience were knowledge management resources, enterprise response management, firm visibility and planning [16]. In the cell biology field, cell deformation is important for many human diseases, such as the tumor. Cell resilience is an important property to determine cell deformation. As the basic structure for organisms to live, a cell could exchange materials and information with the environment. When suffering from external stress, a cell would show good active deformation ability and passive deformation ability [17-21]. Zhao (2009) pointed out a supply chain system with excellent resilience could be regarded as a cell, and the internal organizational structures influenced the system resilience. The system possessed the ability of good active deformation and

28

Y.H. Yang and L.D. Zhao

passive deformation. However, different from the cell resilience, a supply chain could enhance its resilience by self-learning and self-organization ability. He defined supply chain resilience as self-adaption and self-recovery abilities and pointed out a supply chain resilience framework based on cell resilience [22]. Based on the supply chain resilience definition in Ref. [22], Zhao and Li (2009) considered the whole supply chain system as a cell, constructed a supply chain resilience model and researched deformation of supply chain resilience system quantitatively [23]. Different from Zhao and Li’s work, in the paper, we considered each supply chain member as a cell, constructed a supply chain resilience model and analyzed the system deformation based on the cell resilience model.

2 Supply Chain Resilience Model Based on the opinion that a supply chain could exchange materials and information with the environment and deform actively and passively as a cell [22], we consider a two-echelon supply chain and there is no information sharing between them. Each member is regarded as a cell. Based on the similarity between supply chain and cell organism, a supply chain resilience model is shown as Fig. 1. 2.1 Model Parameters In Fig. 1, the supply chain consists of two members M1 and M2 which are series relationship. Each member Mi (i=1,2) consists of two Maxwell elements Mi1 and Mi2. Elements M11 and M21 describe the viscoelasticity of members M1 and M2 responding to internal uncertainties, while elements M12 and M22 describe the viscoelasticity of members M1 and M2 responding to external uncertainties. All elements together reflect the deformation of supply chain resilient system when suffering risks.





 























Fig. 1. Supply chain resilience model

By Ref. [22], we define the supply chain resilience as the self-adaption and selfrecovery abilities, where the self-adaption ability is regarded as the adaptable and learning abilities. In Fig. 1, elements Ei1 and Ei2 represent recovery abilities of

Analyzing Deformation of SC Resilient System Based on Cell Resilience Model

29

member Mi (i=1,2) when facing internal and external risks. Elements μi1 and μi2 represent internal and external learning abilities of member Mi (i=1,2), respectively. Suppose that the total stress of risks shocks suffered by a supply chain resilient system is σ, and the corresponding strain is ε. The stress suffered by member Mi (i=1,2) is σi and the corresponding strain is εi. 2.2 Model Analysis Based on the relationships of supply chain members, parameters satisfy relationships:

σ = σ1 = σ 2

(1)

ε = ε1 + ε 2

(2)

Similar to Ref. [23], from Equations (1) and (2) we can obtain the relationship between the stress and strain of supply chain members M1 and M2, respectively:

μ11μ12 (E11 + E12 )

d2ε1 dε d2σ dσ + E11E12 (μ11 + μ12 ) 1 −[μ11μ12 21 + (μ11E12 + μ12E11) 1 + E11E12σ1] = 0 2 dt dt dt dt

(3)

μ21μ22(E21 + E22)

d2ε2 dε2 d2σ2 dσ + E E ( μ + μ ) − [ μ μ +(μ21E22 + μ22E21) 2 + E21E22σ2] = 0 21 22 21 22 21 22 2 2 dt dt dt dt

(4)

For the risks suffered by a supply chain often increase gradually in the real world, we assume that the risks shock is a linear relationship: ⎧ kt ⎩ kt0

σ = ⎨

t ≤ t0

(5)

t > t0

From Equations (3) and (4), we can obtain the following expressions when t≤t0: d 2 ε E 21 E 22 ( μ 21 + μ 22 ) d ε E E ( μ + μ 12 ) E 21 E 22 ( μ 21 + μ 22 ) d ε 1 + + [ 11 12 11 − ] 2 dt μ 21 μ 22 ( E 21 + E 22 ) dt μ 11 μ12 ( E11 + E12 ) μ 21 μ 22 ( E 21 + E 22 ) dt −

k ( μ 11 E12 + μ 12 E11 ) + E11 E12σ 1 k ( μ 21 E 22 + μ 22 E 21 ) + E 21 E 22σ 2 − =0 μ 11 μ12 ( E11 + E12 ) μ 21 μ 22 ( E 21 + E 22 )

（

（

(6)

（

μ 21 μ 22 E 21 + E 22 ) d 2 ε μ11 μ12 E11 + E12 ) μ 21 μ 22 E 21 + E 22 ) d 2 ε 1 d ε + [ − ] + E 21 E 22 ( μ 21 + μ 22 ) dt 2 E11 E12 ( μ11 + μ12 ) E 21 E 22 ( μ 21 + μ 22 ) dt 2 dt k ( μ11 E12 + μ12 E11 ) + E11 E12σ 1 k ( μ 21 E 22 + μ 22 E 21 ) + E 21 E 22σ 2 − − =0 E11 E12 ( μ11 + μ12 ) E 21 E 22 ( μ 21 + μ 22 )

(7)

Using notations A = E 1 1 E 1 2 ( μ 1 1 + μ 1 2 ) and B = E 21 E 22 ( μ 21 + μ 22 ) , we can μ 21 μ 22 ( E 21 + E 22 ) μ 1 1 μ 1 2 ( E1 1 + E1 2 ) rewrite Equations (6) and (7) as follows:

30

Y.H. Yang and L.D. Zhao

kE11E12 kE21E22 d 3ε d 2ε dε + ( A + B ) + AB − − dt 3 dt 2 dt μ11μ12 ( E11 + E12 ) μ21μ22 ( E21 + E22 ) k (μ E + μ E ) + E11E12σ1 k (μ21E22 + μ22 E21 ) + E21E22σ 2 − AB[ 11 12 12 11 + ]=0 E11E12 (μ11 + μ12 ) E21E22 (μ21 + μ22 )

(8)

The homogeneous linear equation of Equation (8) can be expressed as:

d 3ε d 2ε dε + ( A + B ) 2 + AB =0 3 dt dt dt

(9)

So the corresponding characteristic equation of Equation (9) is:

（ ）

λ 3 + ( A + B ) λ 2 + AB λ = 0

(10)

The latent roots of Equation (10) are λ1

= 0 , λ2 = −A and λ3 = − B . And one λ t λ t fundamental solution is ε 1 (t ) = 1 , ε 2 ( t ) = e 2 and ε 3 ( t ) = e 3 . On is ε

*

the

other

hand,

assuming

that

one

solution

of

Equation

(8)

(t ) = t (C t + D ) , we can obtain the parameters as follows: *

*

C* =

D* =

k ( μ 11 + μ 12 + μ 21 + μ 22 ) 2 ( μ 1 1 + μ 1 2 )( μ 2 1 + μ 2 2 )

k k k (μ E + μ E ) + + 11 12 12 11 B(μ11 + μ12 ) A(μ21 + μ22 ) E11E12 (μ11 + μ12 )

k (μ21E22 + μ22 E21 ) k ( A + B)(μ11 + μ12 + μ21 + μ22 ) + − E21E22 (μ21 + μ22 ) AB(μ11 + μ12 )(μ21 + μ22 )

(11)

(12)

Therefore, the general solution of Equation (8) is:

ε (t ) = C1 + C 2 e λ t + C 3 e λ t + t ( C *t + D * ) 2

3

(13)

From Ref. [23], we can know that

ε 1 (0 ) = ε 2 (0 ) =

( E1 1 + E1 2 )

σ 1 (0) E1 1 + E1 2

σ 2 (0 ) E 21 + E 22

d ε1 dσ 1 E E E |t = 0 = |t = 0 + 1 2 σ 1 (0 ) + ( 11 − 1 2 ) E 1 1ε 1 (0 ) dt dt μ12 μ 11 μ 1 2

(14)

(15)

(16)

Analyzing Deformation of SC Resilient System Based on Cell Resilience Model

( E 21 + E 22 )

dε2 dσ 2 E E E |t = 0 = |t = 0 + 2 2 σ 2 (0 ) + ( 2 1 − 2 2 ) E 2 1ε 2 (0 ) dt dt μ 22 μ 21 μ 22

31

(17)

Therefore, from Equation (6) and Equations (14)-(17), we can obtain initial conditions of Equation (8) as follows:

ε (0 ) =

σ 1 (0 ) E 11 + E 1 2

+

σ 2 (0 ) E 21 + E 22

=0

dε dε dε k ( E11 + E12 + E 21 + E 22 ) |t = 0 = 1 |t = 0 + 2 |t = 0 = dt dt dt ( E11 + E12 )( E 21 + E 22 ) 2 2 k ( μ11 E122 + μ12 E112 ) k ( μ 21 E 22 + μ 22 E 21 ) d 2ε | = + t =0 dt 2 μ11 μ12 ( E11 + E12 ) 2 μ 21 μ 22 ( E 21 + E 22 ) 2

(18)

(19)

(20)

On the other hand, from Equation (13), it is well known that:

ε (0) = C1 + C 2 + C 3 dε |t = 0 = λ 2 C 2 + λ 3C 3 + D * = − AC 2 − BC 3 + D * dt d 2ε |t = 0 = λ 22 C 2 + λ32 C 3 + 2C * = A 2 C 2 + B 2 C 3 + 2 C * dt 2

(21) (22) (23)

From Equations (18)-(23), we can obtain parameters C1, C2and C3 as follows:

C1 = −

D* k(E11 + E12 + E21 + E22 ) 1 k(μ11E122 + μ12E112 ) + + [ A A(E11 + E12 )(E21 + E22 ) AB μ11μ12 (E11 + E12 )2

k(μ E2 + μ E2 ) Ak(E11 + E12 + E21 + E22 ) + 21 22 22 212 + − 2C* − AD* ] μ21μ22 (E21 + E22 ) (E11 + E12 )(E21 + E22 )

C2 =

D* k(E11 + E12 + E21 + E22 ) k(μ11E122 + μ12E112 ) 1 − − [ A A(E11 + E12 )(E21 + E22 ) AB − A2 μ11μ12 (E11 + E12 )2

k(μ E2 + μ E2 ) Ak(E11 + E12 + E21 + E22 ) + 21 22 22 212 + − 2C* − AD*] μ21μ22 (E21 + E22 ) (E11 + E12 )(E21 + E22 ) C3 =

2 1 + μ22 E212 ) k ( μ11 E122 + μ12 E112 ) k ( μ21 E22 [ + B 2 − AB μ11μ12 ( E11 + E12 ) 2 μ21μ22 ( E21 + E22 ) 2

Ak ( E11 + E12 + E21 + E22 ) + − 2C * − AD* ] ( E11 + E12 )( E21 + E22 )

(24)

(25)

(26)

2 2 2 2 Using notations E = E11 + E12 + E21 + E22 and F = (μ11E12 + μ12 E11 ) + (μ21E22 + μ22 E21 ) , we 2 μ11μ12 ( E11 + E12 ) μ21μ22 ( E21 + E22 )2 (E11 + E12 )(E21 + E22 )

can know when the supply chain suffers a linear risks stress σ=kt (t≤t0), the

32

Y.H. Yang and L.D. Zhao

relationship between the deformation value of the supply chain resilient system and time can be calculated:

D* Ek 1 D* Ek 1 + + (AEk + F −2C* − AD*) +[ − − (AEk + F −2C* − AD*)]eλ2t A A AB A A AB− A2 (27) 1 * * λ3t * * + 2 [AEk + F −2C − AD ]e +t(C t + D ) B − AB

ε(t) =−

Because When t>t0, the risks stress is constant (σ=σ1=σ2≡kt0). We can obtain the system differential equation and the corresponding initial conditions:

d 3ε d 2ε dε + ( A + B ) + AB − 2 ABC * t 0 = 0 dt 3 dt 2 dt

(28)

ε ( t 0 ) = C1 + C 2 e λ t + C 3 e λ t + t 0 ( C * t 0 + D * )

(29)

dε |t =t = λ2C2 eλ2t0 + λ3C3eλ3t0 + 2C *t0 + D* dt 0

(30)

d 2ε | = λ22C 2 e λ2 t0 + λ32 C3e λ3t0 + 2C * 2 t = t0 dt

(31)

2 0

3 0

Simply, from Equations (28)-(31), the relationship between the deformation value of the supply chain resilient system and time can be calculated when t>t0:

Ek 1 e−λ2t0 ( AEk + F ) + t0 (D* − C*t0 ) + 2C*t0t + [C2 − (2C* + BD* )]eλ2t + A AB AB − A2 e−λ3t0 (2C* + AD* )]eλ3t +[C3 + 2 B − AB

ε (t ) =

(32)

To summarize, we can obtain the relationship between the deformation value of the supply chain resilient system and time when suffering risks stress as Equation (5): ⎧ D* Ek 1 D* Ek 1 λ2t * * * * ⎪− A + A + AB ( AEk + F − 2C − AD ) + [ A − A − AB − A2 ( AEk + F − 2C − AD )]e ⎪ ⎪+ 1 [ AEk + F − 2C* − AD* ]eλ3t + t (C*t + D* ) t ≤ t0 ⎪⎪ B2 − AB ε (t ) = ⎨ −λ2t0 ⎪ Ek + 1 ( AEk + F ) + t (D* − C*t ) + 2C*t t + [C − e (2C* + BD* )]eλ2t 0 0 0 2 ⎪ A AB AB − A2 ⎪ −λ3t0 ⎪+[C + e t > t0 (2C* + AD* )]eλ3t 3 2 ⎪⎩ B − AB

(33)

3 A Numerical Simulation Assume the internal study abilities of the two supply chain members are μ11=10 and μ21=15, the external study abilities are μ12=5 and μ22=15, the recovery abilities when

Analyzing Deformation of SC Resilient System Based on Cell Resilience Model

33

facing internal risks are E11=45 and E21=40, the recovery abilities when facing external risks are E12=30 and E22=30, and the stress satisfies k=0.1 and t0=40. Fig 2 is a numerical simulation, which describes the changing relationship between the deformation value of a supply chain resilient system and time when suffering a risks stress.

 Fig. 2. Curve of deformation value with time increasing

Fig. 2 shows when a supply chain resilient system suffers a risks stress, a system deformation would take place immediately. From the assumption, it is well known that when t ≤ 40 , the risks stress is increasing gradually from zero. And the deformation value of the system caused by the risks stress increases gradually from zero as well. The relationship between the deformation value and time is close to quadratic polynomial. Therefore, it is easy to know that the relationship between the deformation value and the risks stress is close to quadratic polynomial as well. This section of the curve illustrates that when supply chain risks occur and increase gradually, how a supply chain resilient system reacts under the influence of the study ability and recovery ability of the supply chain members. When t > 40 , the risks stress is constant. That is, supply chain risks reach the maximum and are not changing with time. Meanwhile, the deformation value of the system arisen from the risks increases rapidly and the speeds of deformation value increment are faster than those when t ≤ 40 . The relationship between the deformation value and time is close to linearity. Therefore, it is easy to know that the relationship between the deformation value and the risks stress is close to linearity as well. And this section of the curve illustrates the reaction of a supply chain resilient system when the supply chain risks develop to a stable state. The numerical simulation results accord with the theoretical analysis. From the analysis and numerical simulation results, it is well known that when a supply chain resilient system suffers risks, change on the system states would occur. When the risks increase gradually, the states change increases as well. Moreover, the states change is related with the sustained time of risks, the risks amount and the system resilience (study ability and recovery ability).

34

Y.H. Yang and L.D. Zhao

4 Conclusions When facing complex external environments and internal structure, a supply chain resilient system would come through a process from growth to crisis, then conversion, and at last renovation, which is called spiral rise process [24]. In the paper, based on the similarity between supply chain system and cell organism, we build a supply chain resilient model with two members, and research quantitatively the relationship between the deformation value of the supply chain resilient system and time when suffering a gradually increasing risks stress, coming to an accurate mathematical expression of the two factors. The research results also present the relationship between the risks influence and the supply chain system resilience (learning ability and recovery ability). From the results, the managers could see the change process of the supply chain system and how the supply chain system respond when suffering risks and find out strategies to reduce the negative influences. It provides a new method for supply chain risks management. Research on supply chain resilience is a complex work, and only some situations are analyzed and discussed in this paper. For further research, we can see three aspects: to analyze the deformation of supply chain system with more members; to build a supply chain resilient model when there is information sharing among members; to research the influence of study ability and recovery ability on the system deformation.

Acknowledgment This work was supported by the National Key Technology R&D Program of China (No. 2006BAH 02A06).

References 1. Mallak, L.: Putting organizational resilience to work. Ind. Manage. 40, 8–13 (1998) 2. Carpenter, S., Walker, B.: From metaphor to measurement: Resilience of what to what? Ecosyst. 4, 765–781 (2001) 3. Anderson, S.: Business + IT = Resilience. Contingency Plann. Manage. 10, 216–232 (2003) 4. Vogus, T.J., Sutcliffe, K.M.: Organizational resilience: towards a theory and research agenda. In: IEEE International Conference on Systems, Man and Cybernetics, Montreal, ISIC 2007, pp. 3418–3422 (2007) 5. Goldspink, C., Kay, R.: Organizations as self-organizing and sustaining systems: A complex and autopoietic systems perspective. Int. J. Gen. Syst. 32, 459–474 (2003) 6. Gunderson, L.: Ecological resilience-In theory and application. Annu. Rev. Ecol. Syst. 31, 425–439 (2000) 7. Reichhart, A., Holweg, M.: Creating the customer-responsive supply chain: a reconciliation of concepts. Int. J. Oper. Prod. Manage. 27, 1144–1172 (2007)

Analyzing Deformation of SC Resilient System Based on Cell Resilience Model

35

8. Holling, C.S.: Simplifying the complex: The paradigms of ecological function and structure. Eur. J. Oper. Res. 30, 139–146 (1987) 9. Christopher, M., Peck, H.: Building the resilient supply chain. Int. J. Logist. Manage. 15, 1–29 (2004) 10. Lee, H.L.: The Triple-A Supply Chain. Harvard Bus. Rev. 82, 102–112 (2004) 11. Sheffi, Y.: Preparing for a big one. Manuf. Eng. 84, 12–15 (2005) 12. Rice, J.B., Caniato, F.: Building a secure and resilient supply network. Supply Chain Manage. Rev. 7, 22–30 (2003) 13. Snyder, L.V., Shen, Z.-J.M.: Managing disruption to supply chains. Forthcoming in The Bridge (National Acad. Eng.) 36 (2006) 14. Muckstadt, J.A., Murray, D.H.: Guidelines for collaborative supply chain system design and operation. Inf. Syst. Frontiers: Special Issue on Supply Chain Manage. 3, 427–453 (2001) 15. Haywood, M., Peck, H.: Improving the management of supply chain vulnerability in UK aerospace manufacturing. In: Proceedings of the 1st EUROMA/POMs Conference, Lake Como, pp. 121–130 (2003) 16. Kong, X.Y., Li, X.Y.: Creating the resilient supply chain: The role of knowledge management resources, Wireless Communications, Networking and Mobile Computing. In: 4th International Conference WICOM 2008, Dalian, pp. 1–4 (2008) 17. Chien, S., Sung, K.L., Skalak, R., Usami, S., Tözeren, A.: Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane. Biophys. J. 24, 463–487 (1978) 18. Schmid-Schönbeina, G.W., Sung, K.-L.P., Tozeren, H., et al.: Passive mechanical properties of human leukocytes. Biophys. J. 36, 243–256 (1981) 19. Chien, S., Sung, K.-L.P., Schmid-Schönbeina, G.W., et al.: Rheology of leukocytes. Ann. N. Y. Acad. Sci. 516, 333–347 (1987) 20. Schmid-Schönbeina, G.W.: Leukocyte kinetics in the microcirculation. Biorheology 24, 139–151 (1987) 21. Sunga, K.L., Donga, C., et al.: Leukocyte relaxtion properties. Biophys. J. 54, 331–336 (1988) 22. Zhao, L.D.: Analysis on supply chain resilience based on cell resilience model. Logist. Technol. 28, 101–104 (2009) (in Chinese) 23. Li, Y.H., Zhao, L.D.: Analyzing supply chain risk response based on resilience model. Syst. Eng. Theory Methodol. Appli. (in press) (2010) (in Chinese) 24. Zhao, L.D: Supply chain risks management. China Materials Press, Beijing (2008) (in Chinese)

Multi-objective Particle Swarm Optimization Control Technology and Its Application in Batch Processes Li Jia1, Dashuai Cheng1, Luming Cao1, Zongjun Cai1, and Min-Sen Chiu2 1 Shanghai Key Laboratory of Power Station Automation Technology, Department of Automation, College of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China 2 Faculty of Engineering, National University of Singapore, Singapore [email protected]

Abstract. In this paper, considering the multi-objective problems in batch processes, an improved multi-objective particle swarm optimization based on pareto-optimal solutions is proposed. In this method, a novel diversity preservation strategy that combines the information on distance and angle into similarity judgment is employed to select global best and thus guarantees the convergence and the diversity characteristics of the pareto front. As a result, enough pareto solutions are distributed evenly in the pareto front. Lastly, the algorithm is applied to a classical batch process. The results show that the quality at the end of each batch can approximate the desire value sufficiently and the input trajectory converges; thus verify the efficiency and practicability of the algorithm. Keywords: Batch process; Muti-objective; Pareto-optimal solutions; Particle swarm optimization.

1 Introduction Recently, batch processes have been used increasingly in the production of low volume and high value added products, such as special polymers, special chemicals, pharmaceuticals, and heat treatment processes for metallic or ceramic products [1]. For the purpose of deriving the maximum benefit from batch processes, it is important to optimize the operation policy of batch processes. Therefore, Optimal control of batch processes is of great strategically importance. In batch processes, production objective changes dynamically with custom demands characterized by presence of multiple optimal objectives which often conflict with each other. This raises the question how to effectively search the feasible design region for optimal solutions and simultaneously satisfy multiple constraints. In the field of multiple optimizations, there rarely exists a unique global optimal solution but a Pareto solution set. Thus the key is how to find out the satisfied solutions from Pareto solution set. To solve it, the traditional and simple method is to transfer the muti-objective functions into single objective function, and then the mature methods used in single objective optimization can be employed. But the drawback of this K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 36–44, 2010. © Springer-Verlag Berlin Heidelberg 2010

Multi-objective Particle Swarm Optimization Control Technology

37

method is that only one solution can be available at each time, which makes it difficult for finding out enough satisfied solutions distributing in the Pareto solution set. Recently the particle swarm optimization (PSO) algorithm has showed its powerful capacity for multi-objective optimization, which is a stochastic global optimization approach derived from the feeding simulation of fish school or bird flock [1]. But at first PSO is designed for numerical problems, considering nothing about muti-objective problems. If the particle swarm optimization (PSO) algorithm is directly adopted into muti-objective problems, there will be some shortcomings, such as the choosing of Pareto set and the global best update strategy. To circumvent those drawbacks, a novel iterative muti-objective particle swarm optimization technique for batch processes is proposed in this paper. In the proposed method, the selection of the global optimal solution by using similarity judgment is presented to direct next decision such that it can acquire the Pareto front which is closer to the true Pareto optimum. The advantage is that there are enough Pareto solutions being distributed evenly in the Pareto front. This paper is organized as follows. The preliminaries knowledge is described in Section 2. The proposed novel iterative muti-objective particle swarm optimization is discussed in Section 3. Batch simulation results are presented in Section 4. Finally, concluding results are given in Section 5.

2 Preliminaries 2.1 Multi-objective Optimization Problem in Batch Processes The objective in operating batch process is to maximize the amount of the final product and minimize the amount of the final undesired species. This can be formulated as a multi-objective optimization problem

min ⎡⎣ f1 ( x ) , f 2 ( x ) , L , f m ( x ) ⎤⎦ s.t. gi ( x ) ≤ 0 i = 1, 2,L , k hi ( x ) = 0

(1)

i = 1, 2, L , p

Where x = [ x1 , x2 , L , xn ] ∈ R n , fi ( x ) i = 1, 2,L , m , gi ( x ) ≤ 0 i = 1, 2,L , k and T

hi ( x ) = 0 i = 1, 2, L , p are respectively decision vector, M objective functions k inequality constraints and p equality constraints. In this study, the batch time is divided into T segments of equal length. Without loss of generality, the multi-objective optimal control problem can now be stated to determine the optimal control policy in the time interval 0 ≤ t ≤ t f .

Before proceeding, the following definitions are given. Definition 1-Pareto solution: X ∗ ∈ Ω is a Pareto solution, if ∀X ∈ Ω , fi ( X ) > f i X ∗ holds i = 1,L, n , where n and Ω are decision vector dimension

( )

and feasible domain, respectively.

38

L. Jia et al.

Definition 2-Pareto dominance: A decision vector u is said strictly dominate vector v (denoted u p v ) if and only if ∀i ∈ {1,L, n} : fi ( u ) ≤ f i ( v ) and

∃j ∈ {1,L , n} : f j ( u ) < f j ( v ) . u weakly dominate v (denoted u p v ) if and only if

∀i ∈ {1,L , n} : fi ( u ) ≤ fi ( v ) . Definition 3- Pareto optimal set: The set of all Pareto solutions in the decision variable space is called the Pareto optimal set. Definition 4- Pareto front: The corresponding set of objective vectors is called the Pareto front. 2.2 Foundation of Particle Swarm Optimization

The PSO heuristic algorithm was first proposed by Kennedy and Eberhart, which was inspired by simulation of a simplified social model[5]. It is one of the modern heuristic algorithms and can generate a high-quality solution with shorter calculation time and stable convergence characteristic than other stochastic methods, such as genetic algorithm (GA) and evolutionary strategy (ES). Another advantage is that PSO algorithm does not use crossover and mutation operation which makes the computation simpler. Moreover, in PSO algorithm, a fixed population of solutions is used and all the solutions randomly distributed in the feasible domain. In PSO, the member of the population is called particle, which modifies its memorized values as the optimization routine advance. The recorded values are: velocity, position, best previous performance and best group performance. The ith particle is commonly represented as pi = ( pi1 ,pi2 ,L ,piD ) ; its velocity is represented as vi = ( vi1 ,vi2 ,L , vid ,L ,viD ) ; its best position in history is represented as pi = ( pi1 , pi 2 ,L , pid ,L , piD ) named particle best ( pbest ) .the whole position’s best position is represented as p g = pg1 ,pg2 ,L , pgd ,L ,pgD named global best ( gbest ). The general algorithms for the adjustment of these velocities and positions are:

(

)

(

)

(

vidk +1 = ω vidk + c1r1 pid − zidk + c2 r2 p gd − zidk

)

pidk +1 = pidk + vidk +1

(2)

(3)

where ω is the inertia of a particle, c1 and c2 are constraints on the velocity toward

global and local best, r1 , r2 ∈ U ( 0,1) , Usually c1 and c2 are set 2.

ω = ωmax −

ωmax − ωmin kmax

×k

(4)

ωmax is the initial weight, ωmin is final weight, kmax is maximum number of generations, k is current number of iterations.

Multi-objective Particle Swarm Optimization Control Technology

39

3 Pareto-optimal Solutions Based Multi-objective Particle Swarm Optimization (PSO) Control Multi-objective PSO algorithms should guarantee the convergence and the diversity characteristics of the Pareto front, thus the mission of it is to set an appropriate diversity strategy for the Pareto front and global best update mechanism. In this section, we design a novel method to choose global best location ( gbest ) by using the notations of distance and angle. 3.1 The Similarity of Two Points

In the preceding section, we propose a new definition of the similarity of two points. This definition includes information on distance and angle. To this end, the following similarity number, si , is defined: −

1

si , q = γ

2πσ

d ( xi − xq )

e

( )

( )

+ (1 − γ ) cos θi ,q , if cos θi ,q ≥ 0

2σ 2

(5)

where γ is a weight parameter and is constrained between 0 and 1, σ is mean of maximum value range of xi .and the Euclidian between xi and xq can be defined as:

(

)

And the angle θi , q for point xi and where xi =

( xi − xq ) ( xi − xq ) xq defined as: cos (θi ,q ) = ( xi ' xq ) ( xi '

d xi , xq =

(6) xq

)

xi ' xi .

xq ( )

cos θ i, q < 0

xi

( )

cos θi, p > 0

xp

θi,q θi,p

xq Fig. 1. Illustration of angle measure

It is important to note that Eq.(5) will not be used to compute the similarity particle si between xq and xi if cos θ i ,q is negative. For simplicity, this point is illustrated in the two dimensional space as shown in Fig.1, where xq denotes the vector

( )

40

L. Jia et al.

perpendicular to xq . it is clear that x p which lies in the right of xq is more similar to xq then xq which lies in the left of xq .the use of cosine function to discriminate the directionality between xi and xq indicates that these two vectors are dissimilar if cosine x is negative, and hence the q in the reference group will be discarded. 3.2 The Selection of the Global Optimal Solution ( gbest )

By using the similarity among particles in pareto optimal set, the proposed selection of global best gbest is different from those that select the particle with maximum or minimum objective, and it needs to acquire a Pareto front with evenly distribution. Firstly, the similarities between the ith particle and all particles in Pareto optimal set will be calculate. We denote si , j , j = 1, 2,..., N , N is the number of particles in the Pareto optimal set. Next the particle k in the Pareto optimal set which has the minimum similarity with the ith particle as the gbest of the ith particle. f2 ( x )

f1 ( x )

Pareto particles in Pareto Front Particles in feasible domain Fig. 2. The Selection of the global optimal solution

For a given problem, the tuning of γ results more possible particles to choose a gbest , which will ensure the convergence and the diversity of the Pareto front. In Fig.4, in order to facilitate the analysis, suppose that there are only two points A and B on the Pareto front. Those particles that have more choice lie in the shadow area which is enclosed by line L1 , line L2 and Pareto front. It is noted that all points in L1 has an equal angle to A and B , while all points in L2 has an equal distance to A and B . Evidently, the proposed algorithm guarantees the convergence and the diversity characteristics of the Pareto front by using the notation of similarity. 3.3 Algorithm Steps

In summary, the following describes the steps to find out the optimal control solution by using above-mentioned approach and the workflow of the proposed algorithm is shown in Fig.5:

Multi-objective Particle Swarm Optimization Control Technology

41

f2 ( x )

L2

q1

q2

L1

q3

0

B

f1 ( x )

Pareto particles in Pareto Front Particles in feasible domain Fig. 3. The Selection of the global optimal solution

Step 1: Initialization. Set N, pbesti , gbesti , xi and vi . Step 2: Calculate the maximin fitness for all particles; sort all particles as the non-dominated solutions whose maximin value is less than zero. Update the non-dominated set based on the domination relation. Step 3: Renew pbesti based on the particles’ history information. Step 4: Iterate through the whole swarm population. For the ith particle, calculate its similarity with all the particles in the non-dominated set. And choose particle whose similarity is max as it’s gbesti . Step 5: Update the population based on the particle history information and the whole population history information. Step 6: Go back to Step 2, until a termination criterion is not met. If terminated, then calculate performance measures in the final iteration.

4 Application to Batch Processes The process reactor used in this section was taken from papers [10-13], the mechanism k1 k2 can then be expressed through the short notation A ⎯⎯ → B ⎯⎯ → C , and the mechanistic model of the fed-batch ethanol fermentation process is described as follow:

42

L. Jia et al.

dx1 = −4000 exp ( −2500 / T ) x12 dt dx2 = 4000 exp ( −2500 / T ) x12 − 6.5 × 105 exp ( 5000 / T ) x2 dt where x1 and x2 represent the dimensionless concentrations of A and B , T is temperature of the reactor and is scaled to dimensionless value Td = (T − Tmin ) (Tmax − Tmin ) , t f is fixed 0.1 h , and the values of Tmin and Tmax

(7)

the as are

298 k and 398 k ,respectively. The initial conditions are x1 ( 0 ) = 1 and x2 ( 0 ) = 1 , and the constraint on the control u = Td is 0 ≤ Td ≤ 1 . As far as we know that in the related articles the performance index is to maximise the concentration of B at the end of batch. They all ignore the concentration of A , and hypothesize that concentration of A meet the requirements under all conditions. In reality, for some reason we don’t want too much C , so we should limit the amount of A , so as to reduce the concentration of C . Or we have not enough A . Pareto curve 0.7 0.65 0.6

fun2

0.55 0.5 0.45 0.4 0.35

( A)

0.1

0.15

0.2

0.25

0.3 fun1

0.35

0.4

0.45

0.5

0.75 concentration of A 0.7 0.65

valueof object

0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25

( B)

0

0.1

0.2

0.3

0.4

0.5 time

0.6

0.7

0.8

0.9

1

0.65 concentration of B 0.6

valueof object

0.55 0.5 0.45 0.4 0.35 0.3 0.25

(C )

0

0.1

0.2

0.3

0.4

0.5 time

0.6

0.7

0.8

0.9

1

Fig. 4. (A)Productivity-yield Pareto-optimal front for case batch process,(B)concentration of A with optimal time of operation,(C) concentration of B with optimal time of operation

Multi-objective Particle Swarm Optimization Control Technology

43

We consider a multi-objective optimal control problem where maximum the concentration of B and minimum the concentration of B simultaneously. The multi-objective optimal control problem can now be formally defined as:

⎧⎪ max imum Ctf ( B) ⎨ ⎪⎩ min imun Ctf ( A )

(8)

For the convenience of calculation, a little trick which put a maximum problem to a minimum problem is used

⎧⎪ max imum C ( B) desired − Ctf ( B) ⎨ min imun Ctf ( A ) ⎪⎩

(9)

C ( B )desired is the expected the concentration of C . When C ( B)desired − Ctf ( B ) = 0 , it means that the concentration of C has reached its maximum. The application of the multi-objective optimization algorithm for maximum generations of 30 gives the results are shown in Fig.6. The Pareto front between the C ( B ) desired − Ctf ( B ) and the Ctf ( A ) is shown in Fig.6 A and Fig.6 B and C present the variation of the C ( B)desired − Ctf ( B) and the Ctf ( A ) with the time of operation, respectively.

5 Conclusions In batch processes there exist lots of multi-objective problems. Traditionally, simple method based on gradient theory is applied to obtain the solution of such problems using weighted approaches. But it can not obtain all possible solutions. Recently particle swarm optimization (PSO) algorithm has showed its powerful capacity for multi-objective optimization because of their capability to evolve a set of Pareto solutions distributed along the Pareto front. According to the situation of batch process, an improved multi-objective algorithm is proposed in this paper. Firstly, we utilized the concept of similarity of two particles to improve multi-objective particle swarm algorithm. The similarity concept that combines distance and angle element can guarantee the convergence and the diversity characteristics of the Pareto front. Then some benchmark functions were used to test the algorithm. The results show that the improve algorithm is able to produce a convergence and spread of solutions on the Pareto-optimal front. Lastly, a batch process example illustrates the performance and applicability of the proposed method. Acknowledgement. Supported by Research Fund for the Doctoral Program of Higher Education of China (20093108120013), Shanghai Science Technology commission (08160512100, 08160705900, 09JC1406300 and 09DZ2273400), Grant from Shanghai Municipal Education commission (09YZ08), Shanghai University, "11th Five-Year Plan" 211 Construction Project and Innovation Project for Postgraduate Student Granted by Shanghai University (SHUCX102221).

44

L. Jia et al.

Reference 1. Xin, L.J.: PSO-based Multi-objective Optimization Algorithm Research and Its Applications (2006) 2. Hu, X., Eberhart, R.: Multiobjective optimization using dynamic neighborhood particle swarm optimization. In: Proceedings of the Evolutionary Computation, pp. 1677–1681 (2002) 3. Parsopoulos, K., Vrahatis, M.: Particle swarm optimization method in multiobjective problems. ACM, 603–607 (2002) 4. Mostaghim, S., Teich, J.: Strategies for finding good local guides in multi-objective particle swarm optimization (MOPSO), pp. 26–33 (2003) 5. Kenned, Y.J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, Piscataway, NJ, USA, pp. 1942–1948. IEEE, Los Alamitos (1995) 6. Fieldsend, J., Everson, R., Singh, S.: Using unconstrained elite archives for multiobjective optimization. IEEE Transactions on Evolutionary Computation 7, 305–323 (2003) 7. Senior Member in IEEE, Leung, Y. W., Wang, Y.P.: U-measure a quality measure for multiobjective programming. IEEE Transactions on Systems, Man, and Cybernetics. Part A, Systems and humans (2003) 8. Zhang, L.B.: Research on Optimization Algoruth Base on Particle Swarm Optimization and Differential Evolution (2007) 9. Wei, J.X.: Evolutionary Algorithms for Single-Objective and Multi-Objective Optimization Problems (2009) 10. Liu, H., Jia, L., Liu, Q.: Batch-to-batch control of batch processes based on multilayer recurrent fuzzy neural network. In: Proceedings of the International Conference on Intelligent Systems and Knowledge Engineering. Atlantis Press, France (2007) 11. Lu, N., Gao, F.: Stage-based process analysis and quality prediction for batch processes. Industrial and Engineering Chemistry Research 44, 3547–3555 (2005) 12. Li, J.: Run-to-run product quality control of batch processes. Journal of Shanghai University (English Edition) 13, 267–269 (2009) 13. Ray, W.: Advanced process control. McGraw-Hill Companies, New York (1981)

Online Monitoring of Catalyst Activity for Synthesis of Bisphenol A Liangcheng Cheng1, Yaqin Li1, Huizhong Yang1, and Nam Sun Wang2 1

Institute of Measurement and Process Control, Jiangnan University, Wuxi, Jiangsu, 214122, P.R. China 2 Department of Chemical & Biomolecular, University of Maryland, College Park , MD, 20742, US {Cheng Liangcheng,Li Yaqin,Yang Huizhong, Nam Sun Wang,yhz}@jiangnan.edu.cn

Abstract. In the synthesis of bisphenol A, it is necessary to monitor the catalytic activity of ion exchanger catalyst online. A new online method to monitor the catalyst activity is proposed. Factors affecting catalyst activity are taken into consideration to compute its value using support vector machine and mathematical regression. Simulation and real operation results confirm the effectiveness of this method. Keywords: bisphenol A, catalyst activity, online monitoring, soft sensing.

1

Introduction

Bisphenol A (BPA) is an important starting material for producing epoxy resins and polycarbonates, which is manufactured by ion exchanger resin catalyzed acetone and phenol[1]. In the real plant, it is needed to know the produced BPA content to guide the field operation of BPA production[2-3]. Traditionally laboratory analysis is applied to get the BPA content. Because of this large time delay, laboratory analysis can provide little useful information for real-time plant operation. To solve this problem, soft sensing method[4] is used to achieve online measurement of BPA content. By combining kinetic model and support vector machine (SVM) method, the model of BPA production process can be constructed and BPA content monitored. To achieve higher reaction rate, ion exchanger resin is used as catalyst. Presence of the ion exchanger resin increases the reaction rate significantly and improves the selectivity also[5-7]. The catalyst activity should be included in the soft sensing model of BPA production process and its value needs to be online monitored. Despite the significance of catalyst activity, in literature it is possible to find surprisingly small number of papers devoted to technological aspects of BPA catalyst[8-12]. This paper proposes a novel method of online monitoring the catalyst activity for synthesis of BPA by constructing soft sensing model of catalyst activity using mathematical regression and SVM method. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 45–51, 2010. © Springer-Verlag Berlin Heidelberg 2010

46

L. Cheng et al.

2 Experiment For the BPA production is used solid acid ion exchanger resin as the catalyst. The synthesis is carried out in the fixed-bed reactor. Phenol and acetone flow into the reactor. The mixture of those raw materials passes through the fixed-bed reactor of acid ion exchanger resin catalyst. The synthesis yields bisphenol A and water as well as some reaction by-products. By using all kinds of sensors and DCS systems, most important process data such as the flow of phenol and acetone, reactor temperature and pressure can be measured online. Those real time data are displayed on monitors and meanwhile recorded into database. It is needed to analyze sample mixture in the laboratory to get these data necessary for the study on BPA reaction.

3 Results and Discussion 3.1 Reaction Kinetics Many studies have devoted to the reaction kinetics of BPA synthesis by considering the above mentioned factors and quite a lot of reaction mechanisms have been proposed. Some mechanisms have proved effective by laboratory experiments. However, few of them can be applied to online computation of produced BPA content and some other data. It is because that most variables in the kinetic equations cannot be measured online through existent sensors. For the online monitoring of BPA content, after considering the reaction mechanism[8-12] and making mathematical transformation and simplification, a new kinetic equation of BPA reaction can be proposed. The BPA content can be expressed as follows,

X =

at 1 + bt

(1)

where X is BPA content, a and b are two variables, and t is the reaction time. Then the reaction velocity r can be given by r=

dX a = dt (1 + bt )2

(2)

The remaining time E is defined as

E (t ) =

1

τ

e−t τ .

(3)

So the BPA content can be written as ∞

∞

0

∞

a 1 −t τ a e −t τ e dt + C = ∫ dt + C 2 τ 0 (1 + bt ) 2 0 (1 + bt ) τ

X = ∫ rE (t ) dt + C = ∫

where C is the concentration of BPA at the inlet.

(4)

Online Monitoring of Catalyst Activity for Synthesis of Bisphenol A

Let v = e −t τ , t = −τ ln v, dt =

47

−τ dv v

Then

X =

−τ 1 dv + C = a ∫ dv + C τ ∫1 (1 − bτ ln v) 2 v − b (1 τ ln v) 2 0 a

0

1

v

(5)

Taken the reaction temperature T into consideration, the variable a is expressed as

a = k0 exp(

−E −K ) = k0 exp( ) RT T

(6)

where k0 , K , E , R are all constants. Thus Eq.5 changes to 1

1 dv + C τ (1 ln v) 2 − b 0

X = k0 e − K T ∫

(7)

With the catalyst activity α considered, Eq.7 finally changes to 1

1 dv + C − b (1 ln v) 2 τ 0

X = α k0 e − K / T ∫

(8)

In the above kinetic equation, all the variables expect α and b can be measured online. To make this equation used in the real plant, soft sensing models are constructed to compute α and b . This paper discusses the modeling of catalyst activity α . 3.2 Catalyst Characters

For the BPA production is used solid acid ion exchanger resin as the catalyst. According to theories it has been pointed out that about five factors can cause the deactivation of the acid ion exchanger resin catalyst, including metal ions, methanol, hydrogen donor, high temperature and water[13-18]. 3.3 Numerical Simulation of Catalyst Activity 3.3.1 Restrictions for Modeling and the Solutions It is well known that the data selected as assistant variables of soft sensing models should be measurable online. All the above mentioned factors except reaction temperature cannot be measured by sensors. In this paper the support vector machine method (SVM)[19] is used to build a model describing their relationship. SVM is a novel learning method based on Statistical Learning Theory which can solve small sample learning problem better. SVM is now widely used in industrial processes where nonlinearity and limited number of data exist. Compared to the above mentioned factors, the service time of the catalyst has clearer relationship with its activity. Mathematical regression is used to derive an equation to describe this relationship.

48

L. Cheng et al.

3.3.2 Model Description For the modeling of the ion exchanger resin catalyst activity were available the following operational data: whole reaction mixture input (kg/h), phenol input (kg/h), acetone input (kg/h), temperature at the reactor inlet (T), production of BPA (%wt). Using the above operational data besides the service time of the catalyst, the soft sensing model of catalyst activity is expressed as follows,

α = f (t ) + g ( flow, rate, T )

(9)

In Eq. α is the catalyst activity which is substituted by the conversion rate of acetone. t is the service time of the catalyst. f (t ) is a quadratic polynomial derived by mathematical regression describing the relationship of service time and catalyst activity. flow , rate and T stand for the flow of mixture input, the flow ratio of phenol to acetone and reaction temperature respectively. g ( flow, rate, T ) represents the effect of load, ratio of phenol to acetone and reaction temperature on the catalyst activity, which is in the form of a SVM model. In the real plant after inputting these process parameters into the model the catalyst activity can be computed online. With the increasing of its service time the catalyst activity decreases. This trend is irreversible, which can be expressed through a decreasing function. Considering its simple form and practical usage, a quadratic polynomial is applied in this paper,

α1 = f (t ) = a1t 2 + a2 t + a3

(10)

where α1 is catalyst activity and an (n = 1, 2,3) are coefficients which need to be computed. To calculate the unknown coefficients in the quadratic polynomial, nonlinear regression is used. Nonlinear regression is a useful technique for creating models to explain relationship between variables[20]. The value of three coefficients were obtained, which provides basis for the soft sensing model. Then in the output of the SVM model is the difference between computed and real catalyst activity and the inputs are the load of the catalyst, reaction temperature and the flow ratio of phenol to acetone. 3.3.3 General Results A batch of process data records of the assistant variables and the corresponding laboratory analysis of the acetone conversion rate at the outlet of the reactor with the service time of the catalyst are selected as data samples for modeling. Using the SVM algorithm, soft sensing model based on SVM is constructed through training data. The mean relative error, maximum relative error and mean squared error (MSE) are shown in Table 1. Table 1.Training error of soft sensing model

catalyst activity BPA content

mean relative error 0.835% 1.21%

max relative error 1.15% 3.29%

MSE 0.0086 0.29

Online Monitoring of Catalyst Activity for Synthesis of Bisphenol A

49

Table 2. Testing error of soft sensing model

catalyst activity BPA content

mean relative error 0.847% 1.28%

max relative error 2.38% 3.27%

MSE 0.0095 0.30

To evaluate the generalization performance of the SVM model, some process data records are employed to test the model. The testing errors are shown in Table.2. For soft sensing models, the ability of generalization is a more important criterion for judging the performance of models. To confirm the model performance in real production, twenty process data records collected in October are employed. The results are shown in Fig.1, Fig.2 and Table.3. 1

22

21

catalyst activity

BPA content(wt% )

0.95

0.9

0

2

4

6

8 10 12 sample number

14

16

18

19

18

17

real estimated 0.85

20

20

Fig. 1. Estimated and real value of catalyst activity

16

real estimated 0

2

4

6

8 10 12 sample number

14

16

18

20

Fig. 2. Estimated and real value of BPA

Table 3. Error of soft sensing model

catalyst activity BPA content

mean relative error 0.92% 1.76%

max relative error 2.56% 3.82%

MSE 0.011 0.32

From Fig. 1, Fig.2 and Table.3, it can be found that the performance of the soft sensing model is satisfactory for the commercial production in the real plant, which can reflect the change of the catalyst activity and BPA content at the reactor outlet. As a result the BPA production becomes more efficient and the business more profitable.

4 Conclusions The catalytic activity of ion exchanger resin is studied and the method of its online monitoring proposed. The kinetic mechanisms of BPA reaction and catalyst deactivation are the basis of soft sensing model. Nonlinear regression is applied to describe the decreasing trend of catalyst activity. Its difference with the real value is

50

L. Cheng et al.

assumed to be caused by the changes of load, reaction temperature and flow ratio of phenol to acetone. Then a soft sensing model based on SVM is constructed taken those three factors as inputs to estimate that difference. Combined with the quadratic polynomial obtained through nonlinear regression the soft sensing model can estimate the catalyst activity online. The simulation results and real production prove the satisfactory performance of the model for the online monitoring of catalyst activity.

References 1. Peng, Q.: Technical progress and application of bisphenol a at civil and abroad. Thermosetting Resin 19(3), 25–28 (2004) 2. Prokop, Z., Hankova, L., Jerabek, K.: Bisphenol A synthesis- modeling of industrial reactor and catalyst deactivation. Reactive & Functional Polymers 60, 77–83 (2004) 3. Kawase, M., Inoue, Y., Araki, T., Hashimoto, K.: The simulated moving-bed reactor for production of bisphenol A. Catalysis Today 48, 199–209 (1999) 4. Lin, B., Reckeb, B., Knudsen, J.K.H., Jørgensen, S.B.: A systematic approach for soft sensor development. Computers and Chemical Engineering 31, 419–425 (2007) 5. Jerabek, K., Wanghun, L.: Comparison for the kinetics of bisphenol A synthesis on promoted and unpromoted ion exchanger catalysts. Collect Czech Chem. Commun. 54(2), 321–325 (1989) 6. Mingyang, H., Qun, C., Xin, W.: Study on Stability of Catalyst for Bisphenol A Synthesis. Chemical Industry and Engineering Progress 24(3), 274–277 (2005) 7. Hsu, J.-P., Wong, J.-J.: Kinetic modeling of melt transesterification of diphenyl carbonate and bisphenol-A. Polymer 44, 5851–5857 (2003) 8. Kruegera, A., Balcerowiaka, W., Grzywa, E.: Reasons for deactivation of unpromoted polymeric catalysts in bisphenol A synthesis. Reactive & Functional Polymers 45, 11–18 (2000) 9. Wang, L., Jiang, X., Liu, Y.: Degradation of bisphenol A and formation of hydrogen peroxide induced by glow discharge plasma in aqueous solutions. Journal of Hazardous Materials 154, 1106–1114 (2008) 10. Chen, C.-C., Cheng, S., Jang, L.-Y.: Dual-functionalized large pore mesoporous silica as an efficient catalyst for bisphenol-A synthesis. Microporous and Mesoporous Materials 109, 258–270 (2008) 11. Tai, C., Jiang, G., Liu, J., Zhou, Q., Liu, J.: Rapid degradation of bisphenol A using air as the oxidant catalyzed by polynuclear phthalocyanine complexes under visible light irradiation. Journal of Photochemistry and Photobiology A: Chemistry 172, 275–282 (2005) 12. Baohe, W., Nianyong, Y., Jing, Z., Xin, J., Zongli, Z.: Reaction kinetics of bisphenol A synthesis catalyzed by thiol-promoted resin. Journal of Tianjin University 39(4), 428–431 (2006) 13. Hongming, F., Gang, J.: Studies on optimization of bisphenol A production process with ion exchanger as catalyst. Technology & Testing 10, 38–40 (2001) 14. Gates, B.C.: Catalytic Chemistry [M]. Wiley, New York (1992) 15. Xiwang, Q., Hongfang, C.: Reaction kinetics of bisphenol A synthesis catalyzed by sulfonic acid resin. Petrochemical Technology 25(9), 620–624 (1996) 16. Jinlai, Q., Wenwen, Z., Mingyang, H., Qun, C.: Catalysts for Synthesis of Bisphenol A. Chemical Industry and Engineering Progress 23(7), 710–717 (2004)

Online Monitoring of Catalyst Activity for Synthesis of Bisphenol A

51

17. Li, F.B., Li, X.Z., Liu, C.S., Liu, T.X.: Effect of alumina on photocatalytic activity of iron oxides for bisphenol A degradation. Journal of Hazardous Materials 149, 199–207 (2007) 18. Jerabek, K., Hankova, L., Prokop, Z., Lundquist, E.G.: Relations between morphology and catalytic activity of ion exchanger catalysts for synthesis of bisphenol A. Applied Catalysis A: General 232, 181–188 (2002) 19. Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer, NY (1995) 20. Morada, S., Shachamb, M., Brenner, A.: Utilization of collinearity in regression modeling of activated sludge processes. Chemical Engineering and Processing 46, 222–229 (2007) 21. Yan, W., Shao, H., Wang, X.: Soft sensing modeling based on support vector machine and Bayesian model selection. Computers and Chemical Engineering 28, 1489–1498 (2004)

An Improved Pyramid Matching Kernel Jun Zhang, Guangzhou Zhao, and Hong Gu College of Electric Engineering, Zhejiang University, Hangzhou, China, 310027 {kenny007,zhaogz,ghong}@zju.edu.cn

Abstract. The pyramid matching kernel (PMK) draws lots of researchers’ attentions for its linear computational complexity while still having state-of-theart performance. However, as the feature dimension increases, the original PMK suffers from distortion factors that increase linearly with the feature dimension. This paper proposes a new method called dimension partition PMK (DP-PMK) which only increases little couples of the original PMK’s computation time. But DP-PMK still catches up with other proposed strategies. The main idea of the method is to consistently divide the feature space into two subspaces while generating several levels. In each subspace of the level, the original pyramid matching is used. Then a weighted sum of every subspace at each level is made as the final measurement of similarity. Experiments on dataset Caltech-101 show its impressive performance: compared with other related algorithms which need hundreds of times of original computational time, DP-PMK needs only about 4-6 times of original computational time to obtain the same accuracy. Keywords: dimension partition, bags of features, SVM, pyramid matching, kernel function, object recognition.

1 Introduction In the object recognition field, much recent work has shown the great strength of bags (sets) of local features [1-4]. Global image features, such as color or grayscale histograms or even raw pixels of image, are sensitive to real world imaging conditions. Whereas local descriptors (i.e., SIFT, PCA-SIFT), since its invariant to common image transformation, it can be more reliably detected and matched across images of same object under different view points ,poses, or lighting conditions. However, one set of features, i.e., an image of people’s face may be represented by a set of local descriptors with different facial parts, may be expected that the number of features will be vary across examples due to different imaging conditions, or inconsistent detections by the interest operator, and the size of each set may be too large to endure even square time complexity in number of size. Kernel based learning methods, which include SVM, kernel PCA, and Gaussian Processes, are widely used for their generalization ability and efficiency. However, traditional kernels are designed for fixed-length vector input, not for bags of varying number features. Existing kernel-based approaches particularly designed for bags of features [5-7], each suffers from some of the following drawbacks: two bags need the K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 52–61, 2010. © Springer-Verlag Berlin Heidelberg 2010

An Improved Pyramid Matching Kernel

53

same number features; needs square or even cubic time that makes large feature set sizes impractical; kernels that are not positive-defined. Hence, how to approximate correspondences between bags of features precisely and rapidly remains a challenging work.

2 Related Work Recently, Grauman and Darrell [1, 8, 9] proposed pyramid matching kernel (PMK) algorithm to find an approximate correspondence between two sets. The pyramid matching simply maps bags of unordered features to multi-resolution histograms and computes a weighted histogram intersection aimed to find implicit correspondences based on the finest resolution histogram cell where a matched pair features first appears. Since the pyramid matching’s linear time computation and performance state-of-the-art. It quickly draws a lot of researchers’ attentions [2-4, 10-15]. The original PMK lost all images’ spatial information. So spatial pyramid matching (SPM) algorithm is proposed to take images’ geometric information into count[3]. Experiments demonstrate SPMK’s good performance in natural scene categories. Since SPM may need to cluster 200 or more types, this largely increases the computational time, and the structure of the algorithm is much more complicated too. As the feature dimension increases, original PMK suffers from distortion factors that increase linearly with the feature dimension. Grauman and Darrell[2] proposed a new pyramid matching kernel—Vocabulary Guide-PMK. Because of hierarchy clustering, the algorithm becomes more complicated and the computation time is much longer. Recent work in object recognition and image classification has shown that significant performance gains can be achieved by carefully combining multi-level positive-defined kernel function. Siyao Fu et al. [14]proposed a multiple kernel learning method which alternatively solving SVM and optimizing multi-kernel weights to get a good kernel weights. Yi Liu et al.[4]proposed Dimension-Amnesic PMK which utilizes Gaussian distribution to generate multiple matrixes, and then uses these matrixes to map features to low dimension feature space, then do traditional PMK operates in it. Junfeng He et al. [15]presented a method which also cares about the weights optimization utilizing QP method.

3 Improved Pyramid Matching Kernel 3.1 Pyramid Matching Kernel Pyramid matching kernel approximates the optimal partial matching by computing a weighted intersection over multi-resolution histograms. Specifically, G G G G G G let X = {x1 , x2 ," , xm } , Y = { y1 , y2 ," , yn } be two bags of d-dimensional features in In this example, two 1-D feature sets are used to form two histogram pyramids. Dotted lines are bin boundaries, dashed line indicates a match already formed at a finer resolution level, and solid line represents a new pair matched at this level. If we

54

J. Zhang, G. Zhao, and H. Gu

Fig. 1. The 1-D pyramid matching example

use weights suggested by Grauman & Darrell, we get the similarity between the two L −1 1 1 1 sets: K = ∑ wi N i = 1× 3 + × 1 + × 1 + × 1 = 3.875 . 2 4 8 i =0 G G feature space S , where, xi , yi are local features. We define feature extraction function Ψ as:

Ψ ( X ) = [ H 0 ( X )," , H L −1 ( X )]

(1)

Where, X ∈ S , L = ⎢⎡log 2 D ⎥⎤ + 1 , D is the range of feature space. H i ( X ) is a

histogram vector formed over points in X using d-dimensional bins of side length 2i ,

H i ( X ) has dimension ri = ( D 2i ) , so Ψ ( X ) is pyramid histogram: the bottom bins d

in the finest-level H 0 are small enough that each unique point falls into its own bin, and at the top coarsest level H L −1 , all points fall into a single bin. Hence the pyramid matching kernel function is defined as: L −1

K ( X , Y ) = ∑ wi Ni

(2)

N i = I ( H i ( X ), H i (Y )) − I ( H i −1 ( X ), H i −1 (Y )),

(3)

i =0

j = ri

I ( H i ( X ), H i (Y )) = ∑ min( H i ( X )( j ) , H i (Y )( j ) )

(4)

j =1

Here, wi is a weight at level i , H i ( X )( j ) denotes the count of the j th bin of H i ( X ) . Since the pyramid match has been proved [8] satisfying Mercer’s condition, so it is valid for use as a kernel in any existing learning methods that require Mercer kernels. 3.2 Improved Pyramid Matching Kernel

According to Grauman & Darrell[8], the expected value of the pyramid match cost is bounded:

An Improved Pyramid Matching Kernel

E[ K ( X , Y )] ≤ (Cd log 2 D + d )Θ( M ( X , Y ; π * ))

55

(5)

Where C represents a constant, and Θ( M ( X , Y ; π * ) indicates the optimal partial match cost. That means the pyramid match cost is linear in the feature dimension. When dealing with high dimensional feature spaces, the expected cost is large, accuracy may decrease. Several algorithms have been proposed to solve it[2-4]. For instance, G G specifically, suppose two 128-dimensional vectors: x = (1,1," ,1, 63) , y = (1,1," ,1,1) , according to the original PMK algorithm, let D=64. So at level 0, two vectors will fall G G into their own bins; at level 1, x will fall into bin x = (2, 2," , 2, 64) , while y will fall into bin y = (2, 2," , 2, 2) ; and continue, until at level 6—the coarsest level, they won’t fall into the same bin and match (as if the two vectors haven’t matched with other vectors). Such vectors like these, between each other have a large similarity, but, actually they will just match at coarsest level, which makes little contribution to similarity between bags of features.

Fig. 2. A model of hierarchical random partition (4 levels)

As a matter of fact, to any two feature vectors, which level will match is depend on the dimension which has largest distance. To previous example, other dimensions except last one all have zero distance. Nevertheless, the last dimension has distance 62, which determines their matched level (coarsest). But in high dimension feature space, some large distances dimensions occur easily. Base on this, some dimension reduction strategies—such as PCA, LLE—are proposed. Yi Liu et al[4] presented a dimension amnesic method aimed to solving high dimension problem. However, its increasing computation time ruined PMK’s strength: speed; Grauman & Darrell’s VG-PMK still has to deal with clustering time. The join of hierarchy cluster largely increases the computational time; SPMK always do PMK operator on two-dimension coordinates, but its former’s traditional cluster to all feature space takes much time. All above strategies may solve high dimension feature problem, but their largely increased computational time making them imperfect. Therefore, this paper proposes another strategy called Dimension Partition PMK— DP-PMK, which need only little couples of original PMK computation time, but still comes up with all other strategies in accuracy. Its main idea is to consistently divide the high dimension feature space into several low dimension feature space, and do the PMK operate in each dividing feature space, then make a weighted sum as the

56

J. Zhang, G. Zhao, and H. Gu

correspondence between the original bags of features. As the DP-PMK doesn’t add new complex algorithms, its only extra computation time is spend on partition of the feature, which is small compared with other proposed algorithms. Specifically, DPPMK makes hierarchical random partition to generate T levels feature spaces. At level 0, feature space consists of original bags of features with dimension d , we do PMK operates; and at level 1, we randomly divide original bags of features into two subspaces; and so on, at level t, we choose the largest dimension subspace from sets of subspaces generated from level t-1, and divide it into two subspaces: Original feature space makes the top level, and at each level, divide the largest dimension of previous level leading one more in number of subspace than previous level. L −1 ⎧ ′ = K wit ( dt1 ) Nit ( dt1 ) ∑ t ⎪ ⎪ i =0 , (dt1 + dt 2 ) = max{d( t −1)π 0 , d( t −1)π1 ," , d (t −1)π t−1 } ⎨ L −1 ⎪ K ′′ = wt ( dt 2 ) N t ( dt 2 ) i i ⎪⎩ t ∑ i =0

(6)

Where, d ( t −1)π j covers all previous subspaces’ dimensions. Since the pyramid match is determined by the largest distance between two vectors, for instance, suppose G G G x = (1, 2," , 62,1), y = (1, 2," , 62, 63), and z = (63, 62," , 2,1) , these three vectors will G all match at coarsest level between, so we can’t obtain that z is alien. But if we divide feature space into two subspaces, we can easily get the large similarity between x and y, leaving z still abnormal. Hence we propose our matching kernel function: T

K = ∑ α t ( K t′ + Kt′′− ωt Kt∑−1 ) + Kbase

(7)

t =1

Where, α t is weight for each level, and Kt∑−1 is the kernel function from level t-1 which will be divided by level t because of its largest dimension, ωt is coefficient of that kernel with ωt ∈ [0, 2] . Kbase is a base kernel to assure its non-zero diagonal elements of K. 3.3 Analysis of DP-PMK 3.3.1 Efficiency A key aspect of our improved method is that we maintain the original PMK’s linear time computation, and doesn’t increase too much–will show in experiment, while still have high accuracy. According to Grauman & Darrell’s paper, the computational T

i =t

complexity of original PMK is O(dmL) , so DP-PMK has O(∑∑ dti mL ) , and t =1 i = 0

T

i =t

T

i =t

since ∑∑ dti mL =(∑∑ dti )mL = TdmL , 0 ≤ T ≤ 4 , when it comes to the partition t =1 i = 0

t =1 i = 0

time, if we don’t create new memory, pyramid matching always be done in d-

An Improved Pyramid Matching Kernel

57

dimension feature space, different dimensions use different indices. Hence DP-PMK only increases T times of original computational time. 3.3.2 Kernel Enhancements Only positive semi-define kernels guarantee an optimal solution to kernel-based algorithms based on convex optimization, which includes SVM. According to Mercer’s theorem, a kernel is p.s.d if and only if it corresponds to an inner product in some feature space [16]. First let us rewrite the kernel function at each level as: K t′ + Kt′′− ω K t∑−1 = Kt′ + Kt′′ − 2 K t∑−1 + (2 − ωt ) K t∑−1 , here we set ωt ∈ [0, 2] . Suppose there are two bags of features having the same size, so the pyramid matching is one-one matching: Every feature has its unique matching feature. If we divide features into two, since matching level is depend on the largest distance of dimension between, so after partition, these two sub-features will one stay at original level and the other fall into a finer level—if they don’t match with others—making the kernel function much larger, and if they do match with others this means the matches can only be finer. When these two bags of features have different number of features, the matching kernel also always makes better. Hence, for any two being matched bags of features K t′ + K t′′ ≥ 2 Kt∑−1 . So if we set ωt = 2 , for example, an input set compared against itself—no changes after dividing—will get zero value, and some couples of vectors like previous x and y, will get large value, while x and z still small, so, any kernel mostly measured by changes of number of the similar dimensions after partition which is more reliable compared with kernel measured by largest distance of dimension, and such kernel can somehow overcomes the memorize effect[8]; and if we set ωt = 0 , we get a purely Mercer kernel, since kernels are closed under summation and scaling by a positive constant [16], and as is shown in [8], pyramid kernel is a Mercer kernel. Hence ωt is a coefficient trades off between a purely Mercer kernel and a kernel whose diagonal elements equal to zero (without base kernel). If we set ωt = 2 , we will get all diagonal elements equal to zero, that means selfsimilarity is zero which is unacceptable; hence we add a base kernel for the use of SVM. In experiments we set

K base = β K orignal

(8)

Where, K orignal is the original kernel of PMK, since K orignal has diagonal that is significantly larger than off-diagonal entries[8]. β is the coefficient, β ∈ (0,1] . According to the matrix theory, increasing diagonal elements make the leading principal minors of the matrix greater or equal to zero, which makes a positive semi-define kernel. In applications, we set a lowest dimension (dim=8) to avoid partitioning an over too low dimension.

4 Experiments The Caltech-101 dataset contains 9,197 images comprising 101 different classes, plus a background class, collected via Google image search by Fei-Fei et al.[17]. Caltech-101

58

J. Zhang, G. Zhao, and H. Gu

is one of the most diverse object databases available today; some sample images are shown in Fig. 3. For this dataset, a Harris detector[18] is used to find interest points in each image, and SIFT-128[19] descriptor is used to compose the feature sets. In the following, we provide results to empirically demonstrate our matching’s accuracy and efficiency on real data, and we compare it with the original pyramid matching kernel and VG-PMK. For each method, a one-versus-all SVM classifier is trained for each kernel type, and the performance is measured via 10 runs of kernel computation. In experiments, we set α t = 1 and wit = 1 d 2i , the same as[8].

Fig. 3. Example images from Caltech-101 dataset. Three images are shown for each of 15 of the 101 classes. 0.5 PMK DP-PMK 0.45

Mean Accuracy (10 runs)

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0 0

2

4

6

8

10

12

14

16

18

20

Each Sample of Ramdomly Selected 15 Classes

Fig. 4. A preliminary study of DP-PMK

Here, we randomly select 15 classes from Caltech-101, and use at most 60 images of each class to compose SIFT-128 feature sets, the number of training images is 30. β = ωt = 1 , T=2. From the figure, DP-PMK can easily get at least 10% higher accuracy than the original PMK. Next, we randomly select 15 classes from Caltech-101 (called Caltech-15)—shown in table 1—to optimize related parameters, here we choose an order and optimized each parameter separately before moving to the next. Our goal is to find parameter values that could be used for any dataset, so our choosing of parameter values is careful.

An Improved Pyramid Matching Kernel

59

Table 1. Selected Caltech-15

revolver garfield cannon

joshua_tree octopus louts

headphone strawberry emu

okapi ceiling-fan schooner

Motorbikes stegosaurus cougar-face

450

Total Computational Time (s)

Mean Accuracy (10 runs)

0.5

0.4

0.3

0.2

0.1

0 0

1

2

3

4

5

6

400

350

300

250

200

150

100

50

0

7

0

1

2

Level T

3

4

5

6

7

Level T

Fig. 5. Different levels to Caltech-15

Fig.5. shows different levels to Caltech-15, left plot shows the mean accuracy according to different levels. As level increases, the accuracy increases. But consider the related time spend (right plot), we choose T=4. Base Kernel to Caltech-15 0.8

0.56

0.7

Mean Accuracy (10 runs)

Mean Accuracy (10 runs)

Wt to Caltech-15 0.58

0.54

0.52

0.5

0.48

0.46

0.44

0.42 0

0.5

1

Wt

1.5

2

0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.2

0.4

0.6

0.8

1

Beta

Fig. 6. The result of parameter tuning for enhancements to DP-PMK. From the figure, we set ωt = 2 and β = 0.6 .

Finally we make a comparison with PMK and VG-PMK using SIFT-128 descriptors. For VG-PMK , we set k=10 and L=5, suggested by[2]. From table 2, to obtain the same accuracy, DP-PMK spends only about 6 times of original computational time while VG-PMK needs about 250 times of original computational time, which shows DP-PMK’s impressive performance.

60

J. Zhang, G. Zhao, and H. Gu Table 2. A comparision with PMK and VG-PMK

Pyramid matching method PMK DP-PMK VG-PMK

Mean accuracy 0.229293 0.623569 0.621212

Total computation time (s) 41.58 248.76 1082.2

5 Conclusion We have introduced an improved pyramid matching kernel with much less computation time, while still maintains high accuracy, when dealing with high dimension feature (such as SIFT-128) space. Our DP-PMK consistently divides the feature space into several low dimension feature subspaces to avoid distortion factors, in each subspace, the original PMK operates there, and the similarity is measured by a weight sum to each subspace. Compared with other state-of-the-art approaches which need hundreds of times of original computational time, DP-PMK needs only about 46 times of original computational time to obtain the same accuracy. However, DP-PMK’s performance doesn’t improve much using PCA-SIFT descriptor, one reasonable explanation may be that PCA-SIFT is a low dimension feature (<32), and through principal components analysis, PCA-SIFT may not suitable for dimension partition. Future work will be a more carefully choosing parameter values for seeking more perfect DP-PMK suitable for different high dimension feature descriptors. Acknowledgments. This work is supported by National Natural Science Foundation of China (60872070); Zhejiang Province Key Scientific and Technological Project (Grant No. 2007C11094, No. 2008C21141); Zhejiang Provincial Natural Science Foundation of China (Grant No. Y1080766).

References 1. Grauman, K., Darrell, T.: The pyramid match kernel: Discriminative classification with sets of image features. In: Proceedings of the IEEE International Conference on Computer Vision, Beijing, China, vol. 2 (2005) 2. Grauman, K., Darrell, T.: Approximate correspondences in high dimensions. In: Scholkopf, B., Platt, J.C., Hofmann, T. (eds.) Advances in Neural Information Processing Systems, Cambridge, MA (2007) 3. Lazebnik, S., Schmid, C., Ponce, J.: Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, New York City, NY, vol. 2 (2006) 4. Liu, Y., Wang, X., Zha, H.: Dimension amnesic pyramid match kernel. In: Proceedings of the 23rd National Conference on Artificial Intelligence, Chicago, vol. 2, pp. 652–658 (2008) 5. Grauman, K., Darrell, T.: Fast contour matching using approximate earth mover’s distance. In: Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Washington D.C., vol. 1 (2004)

An Improved Pyramid Matching Kernel

61

6. Berg, A., Berg, T., Malik, J.: Shape matching and object recognition using low distortion correspondences. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, San Diego, CA, vol. 1 (2005) 7. Lyu, S.: Mercer kernels for object recognition with local features. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, San Diego, CA, vol. 2 (2005) 8. Grauman, K., Darrell, T.: The pyramid match kernel: Efficient learning with sets of features. Journal of Machine Learning Research 8, 725–760 (2007) 9. Grauman, K.: Matching sets of features for efficient retrieval and recognition. PhD thesis. MIT (2006) 10. Kapoor, A., Shenoy, P., Tan, D.: Combining brain computer interfaces with vision for object categorization. In: IEEE Conference on Computer Vision and Pattern Recognition, Alaska, pp. 1–8 (2008) 11. Saenko, K., Darrell, T.: Filtering Abstract Senses From Image Search Results. MIT CSAIL, Cambridge (2009) 12. Bosch, A., Zisserman, A., Munoz, X.: Representing shape with a spatial pyramid kernel. In: Proceedings of the 6th ACM International Conference on Image and Video Retrieval, Amsterdam, p. 408 (2007) 13. Yeh, T., Lee, J., Darrell, T., MIT, C.: Adaptive vocabulary forests for dynamic indexing and category learning. In: Proceedings of the IEEE International Conference on Computer Vision, Rio de Janeiro, Brazil, pp. 1–8 (2007) 14. Fu, S., ShengYang, G., Hou, Z., Liang, Z., Tan, M.: Multiple kernel learning from sets of partially matching image features. In: Proceedings of the IEEE International Conference on Pattern Recognition, Florida, pp. 1–4 (2008) 15. He, J., Chang, S., Xie, L.: Fast kernel learning for spatial pyramid matching. In: IEEE Conference on Computer Vision and Pattern Recognition, Alaska, pp. 1–7 (2008) 16. Shawe-Taylor, J., Cristianini, N.: Kernel methods for pattern analysis. Cambridge University Press, Cambridge (2004) 17. Fei-Fei, L., Fergus, R., Perona, P.: Learning generative visual models from few training examples: an incremental Bayesian approach tested on 101 object categories. In: Workshop on Generative Model Based Vision, Washington, D.C. (2004) 18. Harris, C., Stephens, M.: A combined corner and edge detector. In: Proceedings of the 4th Alvey Vision Conference, Manchester, UK, vol. 15, pp. 147–151 (1988) 19. Lowe, D.: Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision 60, 91–110 (2004)

Stability Analysis of Multi-channel MIMO Networked Control Systems Dajun Du1 , Minrui Fei1 , and Kang Li1,2 1

2

School of Mechatronical Engineering and Automation, Shanghai University, Shanghai, 200072, China [email protected], [email protected] School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, United Kingdom [email protected]

Abstract. This paper studies the stability of multi-input multi-output networked control systems (MIMO NCSs) with multiple channels. A general model for multi-channel MIMO NCSs with many independent sensors and actuators is first proposed. Based on Lyapunov stability theory combined with linear matrix inequalities (LMIs) techniques, a sufficient condition is then derived for multi-channel MIMO NCSs to be asymptotical stable. Finally, simulation results confirm the effectiveness of the proposed method.

1

Introduction

Networked control systems (NCSs) have been widely used in DC motor systems and large-scale transportation vehicles, etc. [1],[2]. NCSs are spatially distributed systems in which the communication among various sensors, actuators and controllers occurs over communication networks. However, communication networks are not always reliable, and network-induced delay, data packet out-of-order and data packet dropout are inevitably introduced [3],[4]. Therefore, the negative effects caused by communication networks must be taken into account in analyzing and designing multi-channel MIMO NCSs. In most of NCSs, the communication for both the sensor-to-controller (S/C) and the controller-to-actuator (C/A) occur through one common channel. In this architecture, all sensors output signals are encapsulated into single packet and transmitted to the controller, and the control signals are encapsulated into single packet and transmitted to all the actuators [3],[5]. However, various sensors and actuators are usually distributed at different spatial locations, the communication for both the S/C and the C/A occurs through multi-channel. Therefore, the traditional assumption that the multiple signals are encapsulated into single packet and transmitted through one common channel may not hold. This case is more complex, e.g. there exist data packet dropout, data packet out-of-order and network-induced delay in every channel. These problems have stimulated several research works. Yan et al [6] proposed a continuous model of MIMO NCSs with distributed time-delays and uncertainties and gave some delay-dependent K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 62–68, 2010. c Springer-Verlag Berlin Heidelberg 2010 

Stability Analysis of Multi-channel MIMO Networked Control Systems

63

asymptotic stability criteria. Zhang et al [7] investigated a MIMO LTI plant subject to medium access constraints that can be modeled as a linear time-varying (LTV) system, and proved that the closed-loop NCS was uniformly exponentially stable under the proper periodic communication sequences. However, the aforementioned results have been achieved by only considering a limited number of nondeterministic issues. How to model and prove the stability of MIMO NCSs considering the whole multi-channel transmission mechanism is still an open problem. In this paper, a general model is proposed to describe the MIMO NCSs, and a sufficient condition is derived for MIMO NCSs to be asymptotical stable. Finally, a numerical example is given to demonstrate the effectiveness of the proposed control scheme.

2

Problem Formulation and Modeling

Consider a MIMO discrete-time linear time-invariant (LTI) system in the following state-space form: x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)

(1)

where x = [x1 , · · · , xn ]T ∈ Rn , u = [u1 , · · · , un ]T ∈ Rm , y = [y1 , · · · , ym ]T ∈ Rp are the plant’s state, input, and output, respectively, A, C and C are know matrices with proper dimensions. The plant contains p sensors and m actuators which are spatially distributed. The sensors are connected to the controller through the network with p output channels, while the controller is connected to the actuators through the network with m input channels. For each input and output channel, there exist the network-induced delay, data packet out-of-order and data packet dropout, and i j (1 ≤ i ≤ p) and τca (1 ≤ j ≤ m) as the network-induced delay of denote τsc the output channel i and the input channel j, respectively. Unlike previously proposed methods [5] in dealing with network-induced delay, data packet outof-order and data packet dropout separately, we take into account these issues within one framework by a discard-packet strategy [4]. To better describe the modeling procedure for MIMO NCSs, two definitions are given as follows. Definition 1: Generalized sensor state: the state at which the network transmission of sensor data is valid or invalid at every sampling instant. Definition 2: Generalized actuator state: the state at which the network transmission of actuator data is valid or invalid at every sampling instant. Without loss of generality, the following assumptions are also made. Assumption 1: The actions of sensors, controller and the actuators are timedriven and have the same sampling frequency. Assumption 2: The pair (A, B) is completely controllable, and the pair (A, C) is completely observable. Here, buffering strategy is introduced to handle the data packet dropouts, and buffers are introduced to the controller and actuator nodes. Based on the

64

D. Du, M. Fei, and K. Li

discard-packet strategy, a sensor output, says yi (k), is available to the controller only when the sensor can successfully transfer data to the controller within one sampling period through network. If the sensor fails to transmit data or the data i > 1) or the arrives at the controller using more than one sampling period (τsc data arrives at the controller is out-of-order based on the time stample, then the controller will read out the most recent data yi (k − 1) from the buffer and utilize it as y¯i (k). A diagonal matrix Mp (k) is defined to describe every generalized sensor status at sampling instant k by  1 valid (2) [Mp (k)]ii = 0 invalid then, for the buffers in the controller node, we have y¯(k) = Mp (k)y(k) + (I − Mp (k))y(k − 1).

(3)

where y(k), y¯(k) denote the plant’s output, and the output received by the controller, respectively. Similarly, we can define a matrix Mσ (k) to describe all generalized actuators status at the sampling instant k by  1 valid [Mσ (k)]jj = (4) 0 invalid then, for the buffers in the actuator node, we have u(k) = Mσ (k)¯ u(k) + (I − Mσ (k))¯ u(k − 1).

(5)

where u ¯(k), u(k) denote the control signal from the controller, and the control signal received by the plant at the sampling instant k, respectively. From (1)-(5), the MIMO NCS model with multi-channel buffering strategy can be obtained ⎧ x(k + 1) = Ax(k) + Bu(k) ⎪ ⎪ ⎨ y(k) = Cx(k) (6) u(k) + (I − Mσ (k))¯ u(k − 1) ⎪ u(k) = Mσ (k)¯ ⎪ ⎩ y¯(k) = Mp (k)y(k) + (I − Mp (k))y(k − 1) From (6), the data that the controller can make use of are some sampling values from previous sampling periods. The state observer can be used to estimate the state vector, and the estimated values are then used to compute the control law. Consider the following state observer x ˆ(k + 1) = Aˆ x(k) + B u ¯(k) + L(¯ y(k) − yˆ(k)).

(7)

where L is the observer gain. We use a state feedback law u ¯(k) = K x ˆ(k).

(8)

where K is feedback gain. Considering the state observer (7) and using (8) and (6) gives x ˆ(k + 1) = (A + BK − LC)ˆ x(k) + LMp (k)Cx(k) + L(I − Mp (k))Cx(k − 1).

(9)

Stability Analysis of Multi-channel MIMO Networked Control Systems

65

Define e(k + 1) = x(k + 1) − x ˆ(k + 1), from (6) and (9) we obtain ¯ σ (k)K − LM ¯ p (k)C)x(k) + (A − B M ¯ σ (k)K − LC)e(k) e(k + 1) = (B M ¯ ¯ ¯ σ (k)Ke(k − 1). (10) + (−B Mσ (k)K + LMp (k)C)x(k − 1) + B M ¯ σ (k) = Mσ (k) − I, M ¯ p (k) = Mp (k) − I. where M Using (8) and e(k), x(k + 1) becomes x(k + 1) = (A + BMσ (k)K)x(k) − BMσ (k)Ke(k) ¯ σ (k)Ke(k − 1). ¯ σ (k)Kx(k − 1) + B M − BM

(11)

Combining (10) and (11), we have 

     x(k + 1) A00 A01 A¯ x(k) = + ¯00 A10 e(k + 1) A10 A11 e(k)

A¯01 A¯11



x(k − 1) e(k − 1)

 (12)

¯ σ (k)K − LM ¯ p (k)C, where A00 = A + BMσ (k)K, A01 = −BMσ (k)K, A10 = B M ¯ ¯ ¯ ¯ ¯ ¯ p (k)C, A11 = A−B Mσ (k)K −LC, A00 = −B Mσ (k)K, A01 = −B Mσ (k)K +LM ¯ ¯ ¯ ¯ A10 = B Mσ (k)K, A11 = B Mσ (k)K. From (12), the closed-loop system can be viewed as a switched system with time varying parameters. If we can find a Lyapunov function V that makes the ΔV always less than 0, then the system is asymptotically stable.

3

Stability Analysis of Multi-channel MIMO NCSs

Theorem 1: If there exist symmetric positive-definite matrices P , Q, M , N and real matrices K , L such that ⎡ M −P 0 0 0 ((A + BMσ (k)K)T )P ⎢ ∗ N −Q 0 0 ((−BMσ (k)K)T )P ⎢ ¯ σ (k)K)T )P ⎢ ∗ ∗ −M 0 ((−B M ⎢ ¯ ⎢ ∗ ∗ ∗ −N ((B Mσ (k)K)T )P ⎢ ⎣ ∗ ∗ ∗ ∗ −P ∗ ∗ ∗ ∗ ∗ ⎤ (13) ¯ p (k)C)T )Q ¯ σ (k)K − LM ((B M ¯ σ (k)K − LC)T )Q ⎥ ((A − B M ⎥ ¯ σ (k)K)T )Q ⎥ ((B M ⎥<0 ⎥ 0 ⎥ ⎦ −Q hold for all Mp (k), Mσ (k) , then (12) is asymptotically stable. Proof: Choosing the following Lyapunov function  T    x(k) P 0 x(k) V (k) = e(k) 0 Q e(k)  T    x(k − 1) M 0 x(k − 1) + e(k − 1) 0 N e(k − 1)

66

D. Du, M. Fei, and K. Li

where P , Q, M , N are symmetric positive-definite matrices. Then from (12), we have ⎡

⎤T ⎛⎡ T T ⎤ x(k) A00 A10   ⎢ e(k) ⎥ ⎜⎢ AT01 AT11 ⎥ P 0 A00 A01 ⎥ ⎜⎢ T T ⎥ ΔV (k) = ⎢ ⎣ x(k − 1) ⎦ ⎝⎣ A¯00 A¯10 ⎦ 0 Q A10 A11 e(k A¯T01 A¯T11 ⎤⎞ ⎡ ⎡ − 1) ⎤ M −P 0 0 0 x(k) ⎢ 0 ⎟⎢ ⎥ N −Q 0 0 ⎥ ⎥⎟ ⎢ e(k) ⎥ +⎢ ⎣ 0 ⎦ ⎠ ⎣ 0 −M 0 x(k − 1) ⎦ 0 0 0 −N e(k − 1)

A¯00 A¯10

A¯01 A¯11



(14)

According to Lyapunov stability theory, if ΔV (k) < 0, the closed-loop system (12) is asymptotically stable. So we have ⎡ ⎤ ⎡ T T ⎤ M −P 0 0 0 A00 A10    ⎢ AT01 AT11 ⎥ P 0 A00 A01 A¯00 A¯01 ⎢ 0 N −Q 0 0 ⎥ ⎢ ⎥< 0 (15) ⎢ T T ⎥ ⎣ A¯00 A¯10 ⎦ 0 Q A10 A11 A¯10 A¯11 +⎣ 0 0 −M 0 ⎦ 0 0 0 −N A¯T01 A¯T11 According to Schur complement, (15) becomes ⎡ M −P 0 0 0 AT00 ⎢ ∗ N −Q 0 0 AT01 ⎢ ⎢ ∗ ∗ −M 0 A¯T00 ⎢ ⎢ ∗ ∗ ∗ −N A¯T01 ⎢ ⎣ ∗ ∗ ∗ ∗ −P −1 ∗ ∗ ∗ ∗ ∗

⎤ AT10 AT11 ⎥ ⎥ A¯T10 ⎥ ⎥<0 A¯T11 ⎥ ⎥ 0 ⎦ −Q−1

(16)

pre- and post-multiplying both sides of (16) by diag(I,I,I,I,P,Q) and using the definition A00 , A01 , A10 , A11 , A¯00 , A¯01 , A¯10 , A¯11 , we can obtain (13). This completes the proof. Inequality (13) is a set of BMIs for matrix variables P , Q, M , N , L and K, which is difficult to solve. However, for fixed matrix P , Q, M and N , BMIs (13) are linear matrix inequalities (LMIs) for L and K. Further, for fixed L and K, BMIs (13) become LMIs for P , Q, M and N . Therefore the feasible solution of BMIs (13) can be derived by using the following ‘alternate’ algorithm [8].

4

Numerical Example

Consider the discrete two-input, two-output system with the following system matrices: 

     −0.2800 −0.3100 0.0079 0.2350 0 0.1100 A= ,B = ,C = . (17) −0.3000 0.3600 3.0452 0.1260 0.2500 0.3900

Stability Analysis of Multi-channel MIMO Networked Control Systems

2 Actuator 1 status

Sensor 1 status

2 1.5 1 0.5 0

0

20

40

60

80

1 0.5 0

20

40

60

80

100

0

20

40 60 Sample k

80

100

2 Actuator 2 status

Sensor 2 status

1.5

0

100

2 1.5 1 0.5 0

67

0

20

40

60

80

100

1.5 1 0.5 0

Sample k

(a)

(b)

Fig. 1. Generalized senor and actuator state 1.5 x1 x2

1

State

0.5 0 −0.5 −1 −1.5

0

20

40 60 Sample k

80

100

Fig. 2. State x1, x2, and x3

Since the system with two inputs and two outputs, there are four cases for the two generalized sensor and actuator states, respectively, i.e., Mp = diag {(0, 0), (1, 0), (0, 1), (1, 1)} , Mσ = diag {(0, 0), (1, 0), (0, 1), (1, 1)} . By setting initial values P = Q = diag(10−5 , 10−5 ), M = N = diag(10−6 , 10−6 ), the feedback and observer gains are derived according to Theorem 1 of the proposed method and the ‘alternate’ algorithm as follows:     0.0111 −0.0188 −0.0587 −0.3090 K= ,L = . 0.1310 0.0702 −0.1500 0.2705 For every channel in both the S/C and the C/A, the discard-packet strategy is first used to view long-delayed data packets and out-of-order data packets as data packet dropout. The two generalized sensor and actuators states are shown in Fig. 1(a) and (b), where the vertical lines ‘0.5’, ‘1’ indicate the invalid instants of the generalized sensor state (data packet dropout) and the valid instants of the generalized sensor state (the success of data transmission), respectively. The initial state of the system is assumed as x0 = [1, 0.4]T . The state response of

68

D. Du, M. Fei, and K. Li

the multi-channel MIMO NCSs under the proposed control law is illustrated in Fig. 2. It is obvious that the system is asymptotically stable under the proposed control strategy.

5

Conclusions

This paper has studied the stability of MIMO NCSs with multiple channels. Based on the defined generalized sensor and actuator state, a more general and practical modeling method has been proposed to describe the networked nondeterministic characteristics which can be different among various channels in both the S/C and the C/A. A sufficient condition has then been derived for multi-channel MIMO NCSs to be asymptotical stable. Simulation results show the feasibility and effectiveness of the proposed method.

Acknowledgment This work was supported in part by the National Science Foundation of China under Grant No.60774059 and No.60834002, the Innovation Fund Project for Shanghai University, Mechatronics Engineering Innovation Group Project from Shanghai Education Commission and the Research Councils UK under Grant No. EP/G042594/1 .

References 1. Li, H., Chow, M., Sun, Z.: EDA-based speed control of a networked DC motor system with time delays and packet losses. IEEE Trans. Industrial Electronics 56(5), 1727–1735 (2009) 2. Li, Y.H., Yang, L.M., Yang, G.L.: Network-based coordinated motion control of large-scale transportation vehicles. IEEE/ASME Trans. Mechatronics 12(2), 208–215 (2007) 3. Cloosterman, M.B.G., Wouw, N.V.D., Heemels, W.P.M.H., Nijmeijer, H.: Stability of networked control systems with uncertain time-varying delays. IEEE Trans. Automatic Control 54(7), 1575–1580 (2009) 4. Du, D., Fei, M.: A two-layer networked learning control system using actor-critic neural network. Applied mathematics and computation 205(1), 26–35 (2008) 5. Shi, Y., Yu, B.: Output feedback stabilization of networked control systems with random delays modeled by markov chains. IEEE Trans. Automatic Control 54(7), 1668–1674 (2009) 6. Yan, H., Huang, X., Wang, M., Zhang, H.: Delay-dependent stability criteria for a class of networked control systems with multi-input and multi-ouput. Chaos Solitons & Fractals 34, 997–1005 (2007) 7. Zhang, L., Hristu-Varsakelis, D.: Communication and control co-design for networked control systems. Automatica 42(6), 953–958 (2006) 8. Goh, K.C., Turan, L., Safonov, M.G., Papavassilopoulos, G.P., Ly, J.H.: Baffine matrix inequality properties and computational methods. In: Proceedings of the American Conference, Baltimore, Maryland, pp. 850–855 (1994)

Synthesis of PI–type Congestion Controller for AQM Router in TCP/AQM Network Junsong Wang and Ruixi Yuan Tsinghua University, Hai-dian District, 100084 Beinjing, China [email protected]

Abstract. This paper considers the problem of stabilizing network using a proportional-integral (PI) based congestion controller in active queue management (AQM) router, a necessary and sufficient condition on the network is determined for the existence of stabilizing PI controllers, which will present some general guidelines for the design of a stable PI AQM controller. The complete stabilizing set in the plain of proportional-integral gains can then be drawn and identified immediately based on the presented method, rather than computed mathematically. Finally, the results are illustrated by using MATLAB simulations, which demonstrates that it is able to choose an appropriate PI control parameter based on the stability conditions derived in this paper, to achieve satisfactory network performance. Keywords: active queue management (AQM), PI, congestion control, stabilizing regions.

1

Introduction

Congestion control plays a crucial role on the success of the modern Internet, Active queue management (AQM), as one class of packet dropping/marking mechanism in the router queue, has been a very active research area in the Internet community. A fluid-based a delay differential equations model of wired networks is derived in [1], up to now, which is extensively employed to analyze or to design various AQM algorithms from the viewpoint of control theory and control engineering [2-9], and a great progress has been achieved for that problem. Among the existing congestion control and avoidance approaches, random early detection (RED) [2, 3, 4, 10, 11] is perhaps the most well-known queue-based active queue management (AQM) scheme, which introduces a control mechanism for randomized packet dropping with a queue length averaging technique. The stability issue of the TCP/RED system is discussed in [4] by applying the classical control systems theory, and a local stability conditions for the RED system is derived. However, there are several tightly coupled parameters in RED, which have to be turned very carefully; otherwise, the performance of RED will be degraded. In order to improve the performance of the naive RED, the authors of [5] use classical control system techniques to develop proportional–integral (PI) controllers, K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 69–76, 2010. © Springer-Verlag Berlin Heidelberg 2010

70

J. Wang and R. Yuan

which are shown to have better theoretical properties than the well known RED controller, and the simulation results demonstrate that the PI-type controller outperforms RED significantly. Other PI–type AQM scheme includes [12–15], and the similar conclusions as that in [5] are drawn. In [16], a PD-type AQM scheme is proposed, and the simulations demonstrate the improvement in performance of PD AQM over RED AQM. Other control theoretic based design of AQM includes [7, 8, 9, 17]. A kind of AQM controller is designed using robust control theory respectively in [7, 8, 9], with regard to the complexity of implementation, it is still a challenging issue in practical application of robust control. PID controller is a popular and powerful tool used in power and speed regulation because of their robust performance and simplicity [5-6]. Moreover, PI is a particular form of PID, which is widely used since derivative action is not used very often [17].Given this background, in this paper, we aim at developing a PI control approach based AQM algorithm for the TCP/AQM networks, which will provide an effective and systematic way, and some general guidelines for the quantitative design of a stable PI congestion controller for the TCP/AQM network.

2

Improved Fluid-Model of TCP/AQM Network

In [1], a dynamic model of TCP behavior was developed using fluid-flow and stochastic differential equation analysis. Simulation results demonstrated that the model accurately captured the dynamics of TCP. To design congestion controller, a linear model is simplified as Fig.1, and the transfer function as follows:

G p' ( s ) = Where k =

ke − R0 s (T1 s + 1)(T2 s + 1)

( R0 C ) 3 R 2C , T1 = R0 , T2 = 0 , and 2 2N 4N

round-trip time (secs),

(1)

C is link capacity (packets/sec), R0

N load factor (number of TCP sessions).

Fig. 1. Block-diagram of the fluid model of TCP/AQM network

Synthesis of PI–type Congestion Controller for AQM Router in TCP/AQM Network

71

It is noticed that time delays in inner feedback loop have been ignored during the linearization process [18]. The linearized dynamics of the TCP/AQM network are illustrated in a block diagram form in Fig.2, and the transfer function of the improved model of TCP/AQM network is G p (s) = Where k =

ke − R0 s (T1 s + 1)(T2 s + 1 + e − R0 s )

(2)

( R0 C ) 3 R0 2 C , , . T = R T = 1 0 2 2N 4N 2

Fig. 2. Block-diagram of the improved fluid model of TCP/AQM network

3

Stability Synthesis of PI–type AQM Congestion Controller

The block-diagram of a congestion control system of TCP/AQM network is shown as Fig.3, where C ( s ) and Gp ( s) is the transfer function of congestion controller and the improved network model respectively. In our study, C ( s ) represents PI controller.

_

C(s)

Gp(s)

Fig. 3. Block-diagram of a congestion control system of TCP/AQM network

C ( s ) is the PI type controller of the form

C ( s) = k p +

ki ki + k p s = s s

(3)

72

J. Wang and R. Yuan

Where k p and ki are the proportional-gain and the integral-gain of PI controller, respectively. The aim of the paper is to determine the parameter set ( k p , ki ) of the PI controller (3) for which the closed-loop system of the PI controller based congestion control system is stable. 3.1

Stability Analysis

According to Eqs. (2) and (3), the open loop transfer function of the congestion control system of TCP/AQM is given by C ( s )G p ( s ) =

ki + k p s s

ke − R0 s (T1 s + 1)(T2 s + 1 + e − R0 s )

(4)

From the open loop transfer function of the plant (Eq.4) the closed loop characteristic polynomial of the congestion control system can be generated as Δ* ( s ) = (T1 s + 1)(T2 s + 1 + e− R0 s ) s + k (ki + k p s )e − R0 s

Eq.5 multiplied by e

R0 s

(5)

:

Δ ( s ) = (T1 s + 1)(T2 s + 1) se R0 s + k (ki + k p s ) + s (T1 s + 1)

(6)

Assuming s = jy ， and y > 0 ， as the coefficients of the characteristic quasipolynomial Eq.(6) and τ are real, if jy is the root of Eq.(6), then, so is the complex conjugate of it. Hence, in the following one only considers y > 0 . Substitution s = jy into Eq.(6) yields

δ r ( y ) = − y (1 − T1T2 y 2 ) sin yR0 − (T1 + T2 ) y 2 cos yR0 + kki − T1 y 2 = 0 (7)

δ i ( y) = (1 − T1T2 y 2 ) y cos yR0 − (T1 + T2 ) y 2 sin yR0 + y(kk p + 1) = 0 where δ r ( y ) and

δi ( y)

are the real and imaginary part of the characteristics Δ ( s )

respectively. According to the Implicit Function Theorem, if the Jacobin matrix

⎡ ∂δ r ⎢ ∂k i J =⎢ ⎢ ∂δ i ⎢ ⎣⎢ ∂ki

∂δ r ⎤ ∂k p ⎥ ⎥ ∂δ i ⎥ ⎥ ∂k p ⎦⎥

(8)

Synthesis of PI–type Congestion Controller for AQM Router in TCP/AQM Network

73

is nonsingular, Eq. (8) has a unique local solution curve (ki ( y ), k p ( y )) . Furthermore, the following proposition holds [19]. Proposition 1. Following this curve in the direction of increasing y , the critical roots

are in the right-half plane if the point in the parameter space, relative to the selected values of ki and k p , lies at the left side of the curve (ki ( y ), k p ( y )) when det J < 0 ; and to the right when det J > 0 . Here J is the Jacobin matrix defined in Eq. (8). Remark 1. Since the critical roots lie in the right-half plane which means that the

control system is unstable, the right side of the curve (ki ( y ), k p ( y )) is the stabilizing parameter region following this curve in the direction of increasing y , when

det J < 0 ; and to the left when det J > 0 . According to Eq. (7), it follows that

∂δ r ∂δ ∂δ ∂δ r =0， = k ， i = ky ， i = 0 ∂k p ∂k p ∂ki ∂ki So

det J =

∂δ r ∂ki

∂δ r ∂k p

∂δ i ∂ki

∂δ i ∂k p

= k2 y > 0

(9)

Proposition 2. For the PI controller based congestion control system studied in this

paper, following this curve (ki ( y ), k p ( y )) in the direction of increasing y ( y > 0 ), the parameter space to the left of the curve is stabilizing parameter region. 3.2

Stability Region and Simulations

In this Section, the stabilizing parameter region of the PI controller for TCP/AQM network is discussed below. According to Eq. (7), it follows that

1 1 k p = [(T1 + T2 ) y sin yR0 − (1 − T1T2 y 2 ) cos yR0 ] − k k 1 ki = [ y (1 − T1T2 y 2 )sin yR0 + (T1 + T2 ) y 2 cos yR0 + T1 y 2 ] k

(10)

74

J. Wang and R. Yuan



   







































1  + )  K

  Fig. 4. Stabilizing region of parameters ( ki , k d ) of PI congestion controller

!""#$%& 

Synthesis of PI–type Congestion Controller for AQM Router in TCP/AQM Network

75













  





Fig. 5. Simulation result of PI congestion controller of TCP/AQM network

According to Proposition 2, the stability boundary in k p − ki plane is defined by Eq. (10), so we can plot the stability boundary and actual stability region (shaded) in

k p − ki plane, corresponding to the following network parameter: link capacity C=3750packet/s, round-trip time R0 = 0.15 s, number of TCP sessions N=60, shown as Fig. 4., where Fig. 4.b is enlarged plot of Fig. 4.a, and the shaded region in Fig. 4.b is the stability region of PI congestion controller. We selected the value of PI controller parameter lying in the center of the stabilizing region, i.e., k p = 0.00002 , ki = 0.000144 , the step response of the congestion control system are conducted. Simulation result is shown as Figs.5, which demonstrates that the PI controller designed here enhances the TCP/AQM network to high performance despite the limitations imposed by the time delay in the network.

4

Conclusions

In this paper, a graphical approach has been employed to determine of the boundaries of the values of PI controller, the proposed method has further been used to find the stabilizing region of the PI congestion controller of TCP/AQM network with a time delay. Some entire stability conditions have been derived, under which the PI based congestion control system is stable, which will provide some general guidelines for the design of a stable PI controller. The method presented does not need linear programming to solve a set of inequalities [12, 13], which results that the computation

76

J. Wang and R. Yuan

time and complexity increases in an exponential manner as the order of the system increases; therefore, the method has advantages over existing results. The examples given clearly show that the presented method is valid and effective.

References 1. Misra, V., Gong, W.B., Towsley, D.: Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED. In: Proc. ACM SIGCOMM, pp. 151– 160 (2000) 2. Hollot, C.V., Misra, V., Towsley, D., Gong, W.B.: A Control Theoretic Analysis of RED. In: Proc. IEEE INFOCOM (2001) 3. Tan, L.S.H., Yang, Y., Zhang, W., Zukerman, M.: On Control Gain Selection in DynamicRED. IEEE Commun. Lett. 9, 81–83 (2005) 4. Tan, L.S.H., Peng, G., Chen, G.R.: Stability of TCP/RED Systems in AQM Routers. IEEE Trans. Autom. Contro. 51, 1393–1398 (2006) 5. Hollot, C.V., Misra, V., Towsley, D., Gong, W.B.: Analysis and Design of Controllers for AQM Routers Supporting TCP Flows. IEEE Trans. Autom. Control 47, 945–959 (2002) 6. Li, Y., Ko, K.T., Chen, G.R.: Stable parameters for PI-control AQM scheme. Electronics Lett. 42 (2006) 7. Quet, P.F., Ozbay, H.: On the design of AQM supporting TCP flows using robust control theory. IEEE Trans. Autom. Control 49, 1031–1036 (2004) 8. Chen, Q., Yang, O.W.W.: Robust Controller Design for AQM Router. IEEE Trans. Autom. Control 52, 938–943 (2007) 9. Zheng, F., Nelson, J.: An H∞ approach to the controller design of AQM routers supporting TCP flows. Automatica 45, 757–763 (2009) 10. Floyd, S., Jacobson, V.: Random Early Detection Gateways for Congestion Avoidance. IEEE/ACM Trans. Netw. 1, 397–413 (1993) 11. Low, S.H., Paganini, F., Wang, J.T., Doyle, J.C.: Linear stability of TCP/RED and a scalable control. Computer Networks 43, 633–647 (2003) 12. Chang, X.L., Muppala, J.K.: A stable queue-based adaptive controller for improving AQM performance. Computer Networks 50, 2204–2224 (2006) 13. Hong, Y., Yang, O.W.W.: Design of an adaptive PI rate controller for streaming media traffic based on gain and phase margins. IEE Proc.-Commu. 153, 5–14 (2006) 14. Hong, Y., Yang, O.W.W.: Design of Adaptive PI Rate Controller for Best-Effort Traffic in the Internet Based on Phase Margin. IEEE Trans. Parallel and Distributed Systems 18, 550–561 (2007) 15. Chen, Q., Yang, O.W.W.: On Designing Self-Tuning Controllers for AQM Routers Supporting TCP Flows Based on Pole Placement. IEEE Journal on Selected Areas in Communications 22, 1965–1974 (2004) 16. Sun, J.S.H., Ko, K.T., Chen, G.R., Chan, S., Zukerman, M.: PD-RED: To Improve the Performance of RED. IEEE Commun. Let. 7, 406–408 (2003) 17. Aström, K.J., Hägglund, T.: The future of PID control. Control Eng. Pract. 9, 1163–1175 (2001) 18. Wang, J.S., Gao, Z.H.W.: Improved Fluid-Model of TCP/AQM Network for Congestion Control. The Open Automation and Control Systems Journal 2 (2009) 19. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay equations: Functional-, complex- and nonlinear analysis, Applied Mathematical Science. Springer, New York (1995)

A State Identification Method of Networked Control Systems Xiao-ming Yu and Jing-ping Jiang College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China [email protected], [email protected]

Abstract. Various control strategies achieve results in different aspects of both research and practice on Network Control Systems (NCSs). Aiming at NCSs with short delay which is less than one sampling period, from the system state identification point of view, the concept of network delay noises (NDNs) is presented, Kalman filter based on NCSs is deduced, the major factors impacting on the error variance of Kalman filter based on NCSs are explained, convergence formula of error variance of a priori estimate and convergence value of error variance of a posteriori estimate are given. At last, the simulation proves that the Kalman filter based on NCSs is feasible. Keywords: NCSs, Kalman filter, NDNs, a priori estimate, a posteriori estimate.

1

Introduction

Various control strategies are introduced to solve all kinds of problems of NCSs where information transmission is disturbed by random network delays. Augmented deterministic discrete-time model method proposed by Halevi and Ray is an available control method on a cyclic service network; Queuing method is proposed by Luck and Chan aiming at compensation and prediction of random network delays; Optimal stochastic control method by Nilsson treats the effects of random network delays in NCSs as a LQG problem; Perturbation method can be applied to the NCSs which network is restricted to be priority-based networks[1]; Scheduling-protocol-oriented NCSs is studied widely too[2]. Different from the control strategies above, research on NCSs from the perspective of system state identification is made in this paper and the Kalman filter based on NCSs is deduced which makes use of the available theory and technology of classical Kalman filter theory.

2 2.1

NDNs Introduction of NDNs

The key of Kalman filter based on NCSs is to transform the effects of random network delays to system dynamic process into NDNs, that is, NDNs is proposed to K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 77–86, 2010. © Springer-Verlag Berlin Heidelberg 2010

78

X.-m. Yu and J.-p. Jiang

map random network delays. By this kind of mapping, the state identification problem of NCSs with random network delays is transformed into a state identification problem of unNCSs disturbed by random NDNs at which Kalman filter theory is just very good. The method of transforming random network delays into NCNs is proposed, the mathematical expression of NCNs is deduced, and the property of NCNs is analyzed in this chapter. The linear continuous system x

y

Ax

Cx

Bu

can be discretized into a discrete system below with sampling period

under the

circumstances that the random network delay is d in the kth sampling period. The situation d

is considered only in this paper. x

e x

e dλ Bu y

e dλ Bu

(1)

Cx

where random network delays sequence d

is Gauss white noises process below:

d ~N υ, σ E d

Ed

d

Ed

0, i σ ,i

j j

Time-sequence of the system discretization process is shown in Fig.1. When the kth sampling period of control process starts out, sensors of plant will send sensor detection signals which include all information about plant used to calculate control signals in controller. Controller calculates control signals according to sensor detection signals received from sensors and sends them to actuators. The elapsed time is d from sensors sending sensor detection signals to actuators receiving control signals in the kth sampling period, and d is often referred to as random network delay in the kth sampling period, because the time spent by calculating in controller is far shorter than the time spent by signals transmission in the network. Actuators still adopt old control signal u

of previous sampling period to execute control action

before new control signal u of the kth sampling period arrives.

A State Identification Method of Networked Control Systems

79

Sensor Controller Actuator

k-1

k

k+1

Fig. 1. Sampling time sequence of the kth sampling period

The discretization of NCSs above is based on certain assumptions below: ─ Random network delays of NCSs are always less than the sampling period . ─ Nodes of controller and actuators are event-driven, whereas nodes of sensors are time-driven mode. ─ Information transmitted by sensors and controller can be encapsulated in one packet. Formula (1) is transformed as follows by mathematical equivalence transformation, the purpose of mathematical transformation is to make transformed formula (2) conform to classical form of Kalman filter.

x

e

x

e dλ Bu

e dλ Bu

e dλ Bu

e dλ Bu

e dλ Bu e

x

e dλ Bu

e dλ Bu

e

x

e dλ Bu

e dλ B u

u

Introduce variables as follows G H

ω

e dλ B u

e e dλ B

u

A

e

e

B u

u

80

X.-m. Yu and J.-p. Jiang

reformulate NCSs

x

Gx y

Hu Cx

ω (2)

The coefficient matrix and system matrix are both time-invariant matrices without any random elements and they are both determined by dynamic characteristics of system and sampling period T together. Then, all nondeterminacy of system comes down to NDNs sequence ω . When sampling period

is small (sampling period

minimal time constant), we have e

AT

is close to one-tenth of

I [3]. Consequently, the expression of

NDNs becomes

ω

2.2

d Bu

u

(3)

Characteristics of NDNs

As a stochastic process NDNs sequence ω random network delays sequence d

is obviously a linear function of

which is a Gauss white noises sequence.

Consequently, NDNs is also a Gauss white noises sequence ω ~N

E ω

υB u

Eω

ω

u , σ u

Eω

σ u

u

B B

0, i u

j B B ,i

j

Different from white noises sequence of unNCSs, NDNs sequence ω random network delays is in direct proportion to the variation u

generated by u

of system

input. Consequently, numerical characteristics of NDNs are also inevitably related to the variation u

u

of system input, that is, the mean and variance of NDNs

increase with increase in the variation u the variation u

u

u

of system input. The influence of

of system input on NDNs is peculiar to NCSs and is also

the important factor which we have to consider when estimating system state of NCSs.

A State Identification Method of Networked Control Systems

3 3.1

81

Kalman Filter Based on NCSs Introduce of Kalman Filter Based on NCSs

In this chapter, Kalman filter based on NCSs is deduced aiming at NDNs. State equation (2) describes system dynamics and measurement method. NDNs generated by random network delays is considered in system dynamics, in addition, there is no more other noises considered in the state transition and measurement process. The purpose of Kalman filter based on NCSs is to estimate system state and give the error covariance matrix of the estimate according to equation (2).

Time

k

k+1

Fig. 2. Relationship of a priori and a posteriori estimate

We follow the custom of classical Kalman filter of unNCSs[4-5] and deduce the Kalman filter based on NCSs. As in Fig.2 given by literature [3], if we have all of the measurements before but not including time k of x

1 available for use in our estimate

, we can form a priori estimate denoted as x

a priori estimate is denoted as P including time k

, the error covariance matrix of

; if we have all of the measurements up to and

1 available for use in our estimate of x

posteriori estimate denoted as x estimate is denoted as P mean value of state x

, we can form a

, the error covariance matrix of a posteriori

. The way to obtain the state estimate is to compute the conditioned on all the measurements available. x

P

E x E x

x P

x E x

E x

|y , y , x

|y , y , x

,y

x

x ,y x

The difference between a priori and a posteriori estimate is whether the current (time k

1 ) system measurement y

is adopted in the state estimate. The relationship

is depicted in Fig.2. There is not any additional measurements available to participate

82

X.-m. Yu and J.-p. Jiang

in a priori estimate between time k estimate x

at time k

and time k

1 , so we just achieve a priori

1 based on a posteriori estimate x

at time

and the

knowledge of system dynamics, a priori estimate is also named time update. x

Gx P

Hu

E ω

GP G

(4)

P ω

(5)

The measurement y of time k 1 is applied to correct the a priori estimate x of time k 1 . The result of correction is a posteriori estimate x of time k 1 and the process we achieve a posteriori estimate x is also called data update. x

x

P

P

where coefficient matrix K

K

y

K

cx

cP

I

(6)

cK

(7)

is called gain matrix, the principle of determining gain

matrix is to minimize the error covariance matrix P

of a posteriori estimate. To

achieve this aim, derivative of matrix is applied to make derivative of error covariance matrix P

to gain matrix K

equal 0, that is

dP dK

0

by solving equation above, we know the condition to minimize error covariance matrix P

of a posteriori estimate is K

P

c cP

c

P

I

K

c P

(8)

thereby, we have (9)

Finally, the Kalman filter based on NCSs is composed of formulae (4) (5) (6) (8) and (9). Before any available measurements are obtained, a reasonable and commonly adopted method is to regard initial state value x of system as a posteriori estimate x of time 0, that is x

x

P

P

A State Identification Method of Networked Control Systems

3.2

83

Characteristics of Kalman Filter Based on NCSs

Because NDNs is different from white noises, Kalman filter based on NCSs for characteristics of NDNs is certainly different from classical Kalman filter for unNCSs. y

cx

is used as correction item to correct a priori estimate x

in formula (6). As a result, more accurate a posteriori estimate x estimate x

than a priori

is obtained. Measurement y can accurately reflect the actual state of

the system without measurement error in measurement equation. Consequently, a posteriori estimate x

can converge to actual system state x

correcting, that is, elements of error covariance matrix P

by correction term

of a posteriori estimate

will all converge to 0. The relationship of error covariance matrix of a posteriori estimate and a priori estimate in formula (5) shows that error covariance matrix of a priori estimate will converge to covariance matrix P ω

of NDNs when error

covariance matrix of a posteriori estimate converges to 0. Formulae (5) and (9) show that error variance of Kalman filter based on NCSs is also influenced by variance of noises like unNCSs. Whereas, the variance of NDNs described by formula (3) is obviously related to three factors: system input matrix

,

system input u and the variance σ of random network delays sequence d .

4

Simulation

The linear continuous system below is adopted to simulate to verify Kalman filter theory of NCSs and its characteristics. x

0 2 1

1 3 1 y

0 0 x 3

0 1 u 2

001 x

Continuous system above is discretized with sampling period T random network delays sequence d

is 0.05, that is, δ

1s, the variance

0.05.

Fig.3 is the simulation result of error variance of Kalman filter based on NCSs. The dash line and solid line indicate error variance of a priori and a posteriori estimate respectively. Three small pictures from top to bottom in Fig.3 are error variances of three state estimate respectively. Because a posteriori estimate uses more information than a priori estimate, it can be seen from Fig.3 that error variance of a posteriori estimate is always less than error variance of a priori estimate. And as stated

84

X.-m. Yu and J.-p. Jiang

Fig. 3. Estimate error variance of Kalman filter based on NCSs

previously, it can also be seen from Fig.3 that error variance of a posteriori estimate always approaches 0 and error variance of a priori estimate always approaches variance of NDNs which are diagonal elements of covariance matrix P ω . P ω

σ u

u

B B

0 0 0

0 0.05 0.1

0 0.1 0.2

Fig. 4a) shows the error variance curves of estimate for various rate of change of input u , rates of change of input u are 0, 0.5, 1 and 1.5 from bottom to top respectively. Obviously, as the rate of change of system input u increases, the error variance of estimate also increases and the convergence rate is slower. Especially, NDNs identically equal to 0 when rate of change of input u equaling to 0. It just takes a posteriori estimate one sampling period to converge to 0 in this case, that is, a posteriori estimate can get the exact state value after one sampling period. In the same situation, it takes a priori estimate two sampling period to converge to 0. The reason why velocity of convergence of a priori estimate is slower than a posteriori estimate is that a priori estimate has to deal with error variance P of initial state estimate and it has no self-correction ability.

A State Identification Method of Networked Control Systems

85

Fig. 4b) shows that the error variance curves of estimate for various variances σ of random network delays sequences d

which are 0.01, 0.05 and 0.1 from bottom

to top respectively. As expected, as the variances σ of random network delays sequences d

increases, the error variance of estimate also increases.

a) Influence of rate of change of input u on error variance of estimate

b) Influence of variance δ of random network delays on error variance of estimate

Fig. 4. Factors influencing error variance of Kalman filter based on NCSs

5

Conclusion

This paper applies classical Kalman filter theory to research on NCSs from state identification point of view. Firstly, NDNs as the key factor of state identification of NCSs is proposed. Secondly, Kalman filter based on NCSs is deduced on the basis of characteristics of NDNs. Thirdly, the key factors affecting accuracy of Kalman filter based on NCSs are studied. Finally, simulation results prove that Kalman filter based on NCSs is feasible. Acknowledgments. The research is supported by Special Scientific Research Fund of National Doctoral Programme Disciplines (20030335002) and Science and Technology Department of Zhejiang Province (2004C31084). I am thankful to my tutor Professor Jing-ping Jiang, who encourages me to face hard work and help me to produce and realize new idea. Finally I acknowledge the love and support of my parent.

86

X.-m. Yu and J.-p. Jiang

References 1. Mo-Yuan, C., Yodyium, T.: Network-Based Control Systems: A Tutorial, IECON’01. In: The 27th Annual Conference of the IEEE Industrial Electronics Society (2001) 2. Li, H.B., Sun, Z.Q., Sun, F.C.: Networked control systems: an overview of state-of-the-art and the prospect in future research. Control Theory & Applications 27, 238–243 (2010) 3. Liu, B., Tang, W.S.: Modern control Theory. China machine press, Beijing (2006) 4. Dan, S.: Optimal State Estimation. A John Wiley & Sons, Inc., Publication, Chichester (2006) 5. Greg W., Gary B.: An Introduction to the Kalman Filter, TR 95–041, Department of Computer Science University of North Carolina at Chapel Hill, NC 27599-3175 (2006)

Stabilization Criterion Based on New Lyapunov Functional Candidate for Networked Control Systems Qigong Chen1, Lisheng Wei1, 2, Ming Jiang1, and Minrui Fei2 1

School of Electrical Engineering, Anhui Polytechnic University, Wuhu City, Anhui Province, 241000 P.R. China [email protected] 2 Shanghai Key Laboratory of Power Station Automation Technology School of Mechatronics and Automation, Shanghai University Shanghai City, 200072 P.R. China [email protected]

Abstract. The issue of stability criterion based on new Lyapunov functional candidate for networked control systems (NCSs) is researched, where its network transmission is connected with network-induced delay both sensor-tocontroller and controller-to-actuator. The complete mathematical model of NCSs is given. The sufficient condition for asymptotical stability is derived, and the criteria of delay-dependent asymptotical stability for systems are analyzed by using new Lyapunov functional candidate and free-weighting matrices techniques. The merit of the proposed design methods lies in their less conservativeness. Keywords: Networked Control Systems (NCSs), Stabilization Criterion, Lyapunov Functional Candidate, Free-weighting Matrices.

1 Introduction Since 1980s, the accelerated integration and convergence of communications, computing, and control has inspired researchers and practitioners from a variety of disciplines to become interested in the emerging field of networked control systems (NCSs) [1]. NCSs consist of sensors, actuators, and controllers whose operations are distributed at different geographical locations and coordinated through information exchanged over transmission network. Compared with conventional control systems, the main advantages of NCSs are low cost, reduced weight, simple installation and maintenance, and high reliability [1-3]. It is well known that NCSs have been widely applied to many complicated control systems, such as, space explorations, terrestrial exploration, factory automation, remote diagnostics, tele-robotics, automobiles, aircraft, manufacturing plant monitoring [2-3]. A challenging problem in NCSs is the network-induced delay effects. The delay does not only degrade the performance of the control system, but they also can destabilize the system. Recently, many researchers have studied stability analysis of NCSs in the presence of network-induced delay both sensor-to-controller and K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 87–94, 2010. © Springer-Verlag Berlin Heidelberg 2010

88

Q. Chen et al.

controller-to-actuator [4-9]. There are some preliminary results studying the stabilization problems of NCSs in the existing literature. Luo et al. establish the continuous time model of NCSs, and consider the delay-independent stability of linear differential delay NCSs Based on the 2nd Lyapunov method [8]. Liu et al. present an interval model of networked control systems with time-varying sampling periods and time-varying network-induced delays and discuss the problem of stability of NCSs using Lyapunov stability theory. And the sufficient stability condition is obtained by solving a set of linear matrix inequalities [9]. As we can see, the stability analysis methods presented above are usually based on linear matrix inequalities (LMIs). However, the relationship issues of states with maximum delays are not considered in Lyapunov functional candidate. In order to obtain much less conservative results, a new Lyapunov functional candidate and free weighting matrix approach are used to derive the asymptotical stabilization criterion for NCSs in this paper. The remainder of this paper is organized as follows. In Section 2, the complete mathematical model of NCSs is introduced. In Section 3, the sufficient condition for convergence of the proposed method is analyzed. And the criterion is derived by using Lyapunov stability theory combined with free-weighting matrices techniques. Finally conclusion remarks are presented in Section 4.

2 Networked Control Systems Modeling Consider a class of LTI system described by the following equations

x& (t ) = Ax(t ) + Bu(t ) y(t ) = Cx(t ) x(t ) = φ (t ) = 0,

t ∈ [ −τ ,0]

(1)

where x(t ) ∈ R n is the state vector, u (t ) ∈ R m is the control input vector, y(t ) ∈ R r is the measurable output vector, A , B , C are known constant real matrices with proper dimensions. φ (t ) is a given continuous vector valued initial function and subsection differential on [−τ , 0] . For the convenience of investigation, without loss of generality we make the following assumptions [11-15]: 1) The sensors are time-driven, the sample periods are T , and the time gas among the sensors' nodes can be neglected, the controllers and the actuators are event-driven. 2) The data is transmitted with one data packet through the communication network. That is, a single packet is enough to transmit the plant output at every sampling period. Then the real memory-less state feedback control input u (t ) realized through zerothorder hold in Equation (1) is a piecewise constant function.

u (t + ) = Kx (t − τ k ),

t ∈ {ik T + τ k , k = 1, 2, L}

(2)

Stabilization Criterion Based on New Lyapunov Functional Candidate for NCSs

89

If we consider the effect of the network-induced delay and network packet dropout on the NCSs, then the real control system (1) with Equation (2) can be rewritten as follows[10]. t ∈ [ i k T + τ k , i k + 1T + τ k + 1 )

x& ( t ) = Ax ( t ) + BK x ( ik T ) y (t ) = C x (t ) x ( t ) = φ ( t ) = 0,

(3)

t ∈ [ −τ , 0 ]

where T is the sampling period, ik ( k = 1, 2, L) are some integers and {i1 , i2 , L} ⊆ {1, 2,L} , it is not required that ik +1 > ik , τ k is the time delay, which denotes the time from the instant ik T when the sensor nodes sample sensor data from a plant to the instant when actuators transfer data to the plant. When {i1 , i2 , L} = {1, 2, L} , it means that no packet dropout occurs in the transmission. If ik +1 = ik + 1 , it implies that T + τ k +1 > τ k . Obviously, U∞k =1[ik T + τ k , ik +1T + τ k +1 ) = [ 0, ∞ ) . In this paper, we assume that u (t ) = 0 before the first control signal reaches the plant and a constant τ > 0 exists such that (ik +1 − ik )T + τ k +1 ≤ τ ( k = 1, 2, L) . Equation (3) can be viewed as a general form of the NCSs model, where the effect of the network-induced delay and data packet dropout are simultaneously considered. With these models, the stabilization criterions are derived in the next section.

3 Main Results Before the development of the main result, the following lemma will be used. Lemma1 [12](Schur complement theorem): Given any symmetric matrix S = S T . ⎡S S = ⎢ 11 ⎣ S 21

S12 ⎤ S 22 ⎥⎦

(4)

r×r

where S11 ∈ R , S12 , S 21 , S 22 are known real matrices with proper dimensions. The following three conditions are equivalent (1) S < 0 (2) S11 − S12 S 22−1S12T < 0 and S22 < 0

(5)

(3) S 22 − S12T S11−1 S12 < 0 and S11 < 0

、

Lemma2 [12]: Given any symmetric matrix Q = QT , and constant matrix H E with appropriate dimension, if F (t ) is an uncertain matrix function with Lebesgue measurable elements and satisfies F T (t ) F (t ) ≤ I , in which I denotes the identity matrix of appropriate dimension. Then

Q + HF (t ) E + E T F T (t ) H T < 0

(6)

If and only if there exists positive constant ε > 0 such that

Q + ε −1 HH T + ε E T E < 0

(7)

90

Q. Chen et al.

In the following, the sufficient condition for convergence is given. Theorem 1: considering the closed-loop system (3), given a positive constant τ , if there exist constant matrix P = PT > 0 , Q = QT > 0 , R = RT > 0 and Z = Z T > 0 with proper dimensions such that the LMI (10) holds.

⎡Ξ τ AKT ⎤ <0 ⎢ −1 ⎥ ⎣ * −τ Z ⎦

(8)

Then system (3) is asymptotically stable. Where * denotes the symmetric terms in a symmetric matrix, and

⎡ PA + AT P + Q + R ⎢ Ξ=⎢ ∗ ⎢ ∗ ⎣

0 ⎤ ⎥ 0 ⎥ , AK = [ A BK − R ⎥⎦

PBK −Q ∗

0]

Proof: Use time-delay system theory and Construct a Lyapunov-Krasovskii functional candidate is presented as follows [13-14]:

V (t) = V1 (t) +V2 (t) +V3 (t) +V4 (t)

where P

(9)

、 Q 、 R and Z are symmetric positive matrices, and t

V1 (t ) = xT (t ) Px(t ) , V2 (t ) =

∫

t

xT ( s )Qx( s )ds , V3 (t ) =

0

V4 (t ) =

t

∫∫

∫x

T

( s ) Rx( s )ds

t −τ

ik T

(10) x& T ( s ) Zx& ( s )dsdθ

−τ t +θ

Calculating the derivative of V (t ) along the solutions of system (3), we have

V& (t ) = 2 xT (t ) Px& (t ) + xT (t )Qx(t ) − xT (ik T )Qx(ik T ) + xT (t ) Rx(t ) − xT (t − τ ) Rx(t − τ ) + τ x& T (t ) Zx& (t ) −

t

∫τ x&

T

( s ) Zx& ( s )ds

t−

= 2 xT (t ) P [ Ax (t ) + BKx (ik T ) ] + xT (t ) ( Q + R ) x(t ) − xT (ik T )Qx(ik T ) − x (t − τ ) Rx(t − τ ) − T

t

∫ x&

T

( s ) Zx& ( s )ds

t −τ

+ τ [ Ax(t ) + BKx(ik T ) ] Z [ Ax(t ) + BKx(ik T ) ] T

= ς T (t ) ⎡⎣ Ξ + τ AKT ZAK ⎤⎦ ς (t ) −

t

∫τ x&

t−

T

( s ) Zx& ( s )ds

(11)

Stabilization Criterion Based on New Lyapunov Functional Candidate for NCSs

91

where

ς (t ) = ⎡⎣ xT (t ) xT (ik T ) xT (t − τ ) ⎤⎦ , AK = [ A BK 0] , T

⎡ PA + AT P + Q + R ⎢ Ξ=⎢ ∗ ⎢ ∗ ⎣

PBK −Q ∗

0 ⎤ ⎥ 0 ⎥ − R ⎥⎦

(12)

Because Z is the symmetric positive matrice, so if the following inequation holds, system (3) is asymptotically stable by using the Lyapunov-Krasovskii functional theorem. Ξ + τ AKT ZAK < 0

(13)

Now, according to Equation (11), and using Lemma 1(Schur complement theorem) to simplify and rearrange (13), we have ⎡ Ξ τ AKT ⎤ Ξ + τ AKT ZAK < 0 ⇔ ⎢ <0 −1 ⎥ ⎣ * −τ Z ⎦

( Schur complement theorem)

(14) ■

The proof of Theorem 1 is completed.

In order to obtain much less conservative results, free weighting matrix approach is used to derive the asymptotical stabilization criterion for NCSs in next part. Theorem 2: considering the closed-loop system (3), given a positive constant τ , if there exist constant Tmatrix P = PT > 0 , Q = QT T> 0 , R = RT > 0 , Z = Z T > 0 , N = ⎡⎣ N1T N 2T N 3T ⎤⎦ and M = ⎡⎣ M 1T M 2T M 3T ⎤⎦ with proper dimensions such that the LMI (15) holds.

⎡Π τ AKT ⎢ −1 ⎢ * −τ Z ⎢* * ⎢ * ⎢⎣ *

τN 0 −τ Z *

τM ⎤

⎥ 0 ⎥ <0 0 ⎥ ⎥ −τ Z ⎥⎦

(15)

Then system (3) is asymptotically stable. Where * denotes the symmetric terms in a symmetric matrix, and Π = Π1 + Π 2 + ΠT2 , AK = [ A BK ⎡ PA + AT P + Q + R PBK ⎢ Π1 = ⎢ ∗ −Q ⎢ ∗ ∗ ⎣

0] 0 ⎤ ⎥ 0 ⎥ , Π2 = [ N − R ⎥⎦

−N + M

−M ]

92

Q. Chen et al.

Proof: The proof of this theorem is similarly to theorem 1. Use time-delay system theory and Construct a Lyapunov-Krasovskii functional candidate is presented as follows [13-14]:

V (t) = V1 (t) +V2 (t) +V3 (t) +V4 (t)

where P

、 Q 、 R and Z are symmetric positive matrices, and t

V1 (t ) = xT (t ) Px(t ) , V2 (t ) =

∫x

t

T

( s )Qx( s )ds , V3 (t ) =

0

T

( s ) Rx( s )ds

t−

ik T

V4 (t ) =

∫τ x

t

∫τ ∫θ x&

T

( s ) Zx& ( s )dsdθ

− t+

On the one hand, from the Leibniz-Newton formula, we have t

∫

x(t ) = x(ik T ) +

x& (s)ds ; x(ik T ) = x(t − τ ) +

ik T

ik T

∫ x&(s)ds

(16)

t −τ

Then the following equation is true for any free-weighting matrices N and M with appropriate dimensions. t ⎡ ⎤ 2 ⎡⎣ xT (t ) N1 + xT (ikT ) N 2 + xT (t − τ ) N 3 ⎤⎦ × ⎢ x (t ) − x (ik T ) − ∫ x& ( s )ds ⎥ = 0 ik T ⎣⎢ ⎦⎥ ik T ⎡ ⎤ 2 ⎡⎣ x (t ) M 1 + x (ik T ) M 2 + x (t − τ ) M 3 ⎤⎦ × ⎢ x (ik T ) − x(t − τ ) − ∫ x& ( s )ds ⎥ = 0 ⎢⎣ ⎥⎦ t −τ T

T

(17)

T

Calculating the derivative of V (t ) along the solutions of system (3), we have V& (t ) = 2 xT (t ) Px& (t ) + xT (t )Qx(t ) − xT (ik T )Qx(ik T ) + xT (t ) Rx(t ) − xT (t − τ ) Rx(t − τ ) + τ x& T (t ) Zx& (t ) −

t

∫τ x&

T

( s ) Zx& ( s )ds

t−

= 2 xT (t ) P [ Ax(t ) + BKx(ik T )] + xT (t ) ( Q + R ) x(t ) − xT (ik T )Qx(ik T ) − xT (t − τ ) Rx(t − τ ) −

ik T

∫τ

t−

t

x& T ( s ) Zx& ( s )ds −

∫ x&

T

( s ) Zx& ( s )ds

ik T

+ τ [ Ax(t ) + BKx(ik T )] Z [ Ax(t ) + BKx(ik T ) ] T

t ⎡ ⎤ + 2 ⎡⎣ xT (t ) N1 + xT (ik T ) N 2 + xT (t − τ ) N3 ⎤⎦ × ⎢ x(t ) − x(ik T ) − ∫ x& ( s )ds ⎥ ik T ⎣⎢ ⎦⎥ ik T ⎡ ⎤ + 2 ⎡⎣ xT (t ) M 1 + xT (ik T ) M 2 + xT (t − τ ) M 3 ⎤⎦ × ⎢ x(ik T ) − x(t − τ ) − ∫ x& ( s )ds ⎥ t −τ ⎣⎢ ⎦⎥

(18)

Stabilization Criterion Based on New Lyapunov Functional Candidate for NCSs

93

≤ ς T (t ) ⎡⎣Π + τ AKT ZAK + τ NZ −1 N T + τ MZ −1 M T ⎤⎦ ς (t ) −

ik T

∫τ ⎡⎣ς

(t ) M + x& T ( s ) Z ⎤⎦ Z −1 ⎡⎣ς T (t ) M + x&T ( s ) Z ⎤⎦ ds

T

(t ) N + x& T ( s ) Z ⎤⎦ Z −1 ⎡⎣ς T (t ) N + x& T ( s ) Z ⎤⎦ ds

t− t

−

∫ ⎡⎣ς

T

T

T

ik T

where

ς (t ) = ⎡⎣ xT (t ) xT (ik T ) xT (t − τ ) ⎤⎦ , AK = [ A BK 0] , Π = Π1 + Π 2 + ΠT2 T

⎡ PA + AT P + Q + R PBK ⎢ Π1 = ⎢ ∗ −Q ⎢ ∗ ∗ ⎣

0 ⎤ ⎥ 0 ⎥ , Π2 = [ N − R ⎥⎦

−M ]

−N + M

Because Z is the symmetric positive matrice, so if the inequation (18) holds, system (3) is asymptotically stable by using the Lyapunov-Krasovskii functional theorem. Π + τ AKT ZAK + τ NZ −1 N T + τ MZ −1 M T < 0

(19)

By the Schur complement theorem, and according to Equation (19), we have

Π + τ AKT ZAK + τ NZ −1 N T + τ MZ −1 M T < 0 ⇔ Π − ⎡⎣τ AKT

⎡ −τ Z −1 ⎢ τ N τ M ⎤⎦ ⎢ * ⎢ * ⎣

⎡Π τ AKT ⎢ * −τ Z −1 ⇔⎢ ⎢* * ⎢ * ⎢⎣ *

τN 0 −τ Z *

τM ⎤

0 −τ Z *

−1

0 ⎤ ⎡ τ AK ⎤ ⎥ 0 ⎥ ⎢⎢τ N T ⎥⎥ < 0 −τ Z ⎥⎦ ⎢⎣τ M T ⎥⎦

(20)

⎥ 0 ⎥ < 0 ( Schur complement theorem) 0 ⎥ ⎥ −τ Z ⎥⎦

The proof of Theorem 2 is completed.

■

As we can see, when N = M = 0 , LMI (15) is equation to LMI (8). By using freeweighting matrices techniques to choose the proper dimension matrices N and M , we can obtain much less conservative results.

4 Summary In this paper, the stabilization criterion for networked control systems with random communication network-induced delays has been proposed. Based on Lyapunov stability theory, the stabilization and robust stabilization criterion is derived by introducing some free-weighting matrices which can be selected properly to lead to

94

Q. Chen et al.

less conservativeness results. In terms of the derived criteria, which are expressed as a set of linear matrix inequalities, stability of the considered NCSs can be obtained. Acknowledgments. This work was supported by the Science Foundation of Science and Technology Commission of Shanghai Municipality under grant 09DZ2273400, National Natural Science Foundation of China under grant 60774059 and 60843003, the Priming Scientific Research Foundation for the Introduction Talent in Anhui University of Technology and Science 2009YQQ003.

References 1. Hespanha, J.P., Naghshtabrizi, P., Xu, Y.G.: A Survey of Recent Results in Networked Control Systems. Proceedings of IEEE 95, 138–162 (2007) 2. Gao, H., Meng, X., Chen, T.: Stabilization of Networked Control Systems with a New Delay Characterization. IEEE Transactions on Automatic Control 53, 2142–2148 (2008) 3. Yang, T.C.: Networked control system: a brief survey. IEEE Proceedings of Control Theory Applications 153, 403–412 (2006) 4. Wei, L.S., Fei, M.R., Hu, H.S.: Modeling and stability analysis of grey–fuzzy predictive control. Neurocomputing 72, 197–202 (2008) 5. Li, H.B., Chow, M.Y., Sun, Z.Q.: Optimal Stabilizing Gain Selection for Networked Control Systems With Time Delays and Packet Losses. IEEE Transactions on Control Systems Technology 17, 1154–1162 (2009) 6. Dritsas, L., Tzes, A.: Robust stability bounds for networked controlled systems with unknown, bounded and varying delays. IET Control Theory Application 3, 270–280 (2009) 7. Huang, J., Guan, Z.H., Wang, Z.D.: Robust stabilization of uncertain networked control systems with data packet dropouts. Acta Automatica Sinica 31, 440–445 (2005) 8. Luo, D.S., Wang, Z.W., Guo, G.: Modeling and Stability Analysis of MIMO Networked Control Systems. In: Proceedings of the 27th Chinese Control Conference, Kunming, Yunnan, China, July16-18, pp. 240–242 (2008) 9. Liu, F.C., Yao, Y., He, F.H., Chen, S.L.: Stability analysis of networked control systems with time-varying sampling periods. Journal of Control Theory and Applications 6, 22–25 (2008) 10. Yue, D., Han, Q.L., Peng, C.: State feedback controller design of networked control systems. J. IEEE Trans. on Circuits and Systems-II 51, 640–644 (2004) 11. Peng, C., Yue, D., Tian, E.G., Gu, Z.: A delay distribution based stability analysis and synthesis approach for networked control systems. Journal of the Franklin Institute 346, 349–365 (2009) 12. Yu, L.: Robust Control- Linear Matrix Inequalities. Press of Qinghua University, Beijing (2002) 13. He, Y., Wang, Q.G., Xie, L.H., Lin, C.: Further Improvement of Free-Weighting Matrices Technique for Systems With Time-Varying Delay. IEEE Trancactions on Automatic Control 52, 293–299 (2007) 14. Wu, M., He, Y., She, J.H.: New delay-dependent robust stability criteria for uncertain neutral systems. IEEE Transactions on Automatic Control 49, 2266–2271 (2004)

Development of Constant Current Source for SMA Wires Driver Based on OPA549 Yong Shao, Enyu Jiang, Quanzhen Huang, and Xiangqiang Zeng Department of Automation, School of Mechatronics Engineering & Automation, Shanghai University; Shanghai Key Laboratory of Power Station Automation Technology, Shanghai, 200072, P.R. China [email protected]

Abstract. A wide output range multi-channel constant current source is designed for the research of active structural vibration control using shape memory alloy wires. The corresponding design circuit can discharge large constant current for low voltage load and realize the precise discharging control. With digital control, the multi-channel constant current could switch each channel according to the control method with intelligent management employed. By testing, the range of current is 0 8A, within 1% control precision, while the range of operating temperature can reach -40 ~+125 . Experiment results show that, with over-current and over-temperature protection, the designed constant current source could drive the SMA wires for smart structures, and the effects is good. The designed current source is also fit for other research that multi-channel large current supply is needed.

～

℃

℃

Keywords: smart structures, shape memory alloy, constant current driver, design proposal.

1 Introduction Intelligent devices such as dampers or actuators made up by smart material has vast application prospects in structural vibration control, with the characteristic of simple mechanical structure, easy adjustment, low power consumption, quick response. Shape memory alloy (SMA), piezoelectric materials, magnetostrictive materials and other smart materials are widely used in recent intelligent structure research. With shape memory effect, superelasticity, high energy consumption, good anticorrosion property, good fatigue resistance, SMA has been considered as one of most promising materials for sensing, actuating in structure control. Active structural vibration control can be achieved based on the thermal characteristic of SMA. In current research principles and methods to realize active structural vibration control, SMA is embedded into the structure to change the mechanics feature and stiffness of composite material by current drive. Thus the feasibility and efficiency of control system highly depends on the quality of constant current driver for the SMA[1-2]. This paper emphasizes on giving a design proposal for wide range multi-output constant current source to drive SMA, which can supply multiplexed output K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 95–103, 2010. © Springer-Verlag Berlin Heidelberg 2010

96

Y. Shao et al.

adjustable constant flow source, with the characteristic of power supply output high current stability. Digital control with good intelligent management capabilities is adopted to improve practicality. TI's OPA549 Precision Amplifier is adopted as the amplifier chip while Microchip’s PIC18F4620 is chosen as the core MCU. Combining software development and function design, the practical and reliable multi-channel constant current sources are achieved. This paper provides a basic experiment method and essential component unit for the active vibration control of the intelligent structure based on the SMA smart materials.

2 The Design Principle of Constant Current Source Which Was Used for SMA To design a multiple output constant current source driving SMA, designers have to consider stability of the large constant current, as well as the amplifier’s biggest conversion rate and the output power. Thus the choosing of the amplifier’s parameters is of vital importance for the constant current source design. On the basis of analysis and comparison, we choose TI’s high-voltage large-current amplifier OPA549. It has the following characteristics: 1. 2. 3. 4. 5.

Maximum stable current output: 8A with 10A peak; Wide power supply range: single supply: 60 V; dual supply: ±30V; Fully protected: thermal shutdown; adjustable current limit; High slew rate: 9V/us; The biggest power consumption: 90W;

The system is consisted of power amplifier, man-machine interface, CRT, MCU PIC18F4620, RS-485 communication interface, protective circuit and other devices. Protective circuit and devices contains thermal shutdown and over current protection. The block diagram of system is shown in Fig.1.

Fig. 1. Block diagram of the system

Development of Constant Current Source for SMA Wires Driver Based on OPA549

97

3 Hardware Design 3.1 The Design Principle of Main Circuit of Current Source Main circuit of current source is divided into three parts[3]: the sampling circuit, the control circuit and the output circuit. The sampling circuit and the output circuit is shown in Fig.2.

Fig. 2. Schematic diagram of the main circuit of current source

As shown in Fig 2, the positive inputted voltage of OPA549 U + , is provided by MCU through D/A converter TLC5615 and operational amplifier OP07. U + can be expressed as following equation:

U+ =

R2 ×U I R1

(1)

The sampling resistance Rs connected with the inverting input of OPA549 consists of a negative current feedback circuit[4-6]:

U − = I × Rs

(2)

U+ = U−

(3)

R2 × U I = I × Rs R1

(4)

Considering

We can get

98

Y. Shao et al.

While the current flowing through SMA is equal to the current that flows through the sample resistance Rs . Thus if the MCU gives U I , the current flowing through SMA and sampling resistance is:

I=

R2 × U I / Rs R1

(5)

Considering that the slow heat dissipation capability of SMA, multi-output constant current source is designed to drive different SMA wires. When one channel of the SMA wires is cooling, another channel can be driven by PIC18F4620. PIC18F4620 controls TLC5615 through SPI bus, while one channel of PIC18F4620 can control four constant current sources channels. 3.2 The Character of Main Device The main control circuit of constant current source amplifier OPA549 is a low-cost high-voltage/high-current operational amplifier ideal for driving a wide variety of loads. This laser-trimmed monolithic integrated circuit provides excellent low-level signal accuracy and high output voltage and current. The OPA549 operates from either single or dual supplies for design flexibility. The input common-mode range extends below the negative supply. TI’s TLC5615 is chosen as the D/A converter chip, with 10-Bit CMOS voltage output, 3-wire serial Interface, update rate of 1.21MHz. With 16 bites RISC instruction, the Harvard structure, two data bus, 64KB flash cache, 4KB RAM, internal watchdog, internal EEPROM and abundant internal resources, Microchip’s PIC18F4620 is chosen as the arithmetic unit and control unit for the constant current source, as it has the advantage of high anti-interference ability low power consumption and high speed. Sample resistance and the load are connected in series. By measuring the current flowing through the sample resistance, the change of current signal is transformed into voltage signal to feed back into the negative input of OPA549. Thus the stability of the sample resistance directly affects the performance of constant current source, if the power of the sample resistance is not big enough, the resistance may be burned out. Considering the above factors, we choose high precise resistance made of manganin materials, with the temperature coefficient -3~20 ×10-6/ . The value of sample resistance and the wire measured through electrical bridges is 36.7 mΩ [7].

（

）

℃

4 Software Design Software design mainly includes three parts: the control of the current source and D/A converter, man-machine interface, 485 communication interfaces for communicating with DSP or PC.

Development of Constant Current Source for SMA Wires Driver Based on OPA549

99

4.1 Software Realization of the Constant-Current Source Element The software for constant-current source element mainly includes the initialization procedure, TLC5615 sending and receiving procedure, timely switching procedure. Timer interrupt is adopted to meet the real-time requirements of switching the current. At the beginning of the program, PIC18f4620 initialize the SPI control port and determine the SPI transmission rate.PIC18F4620 choose the slave address of the constant-current element through I/O and impose control to the D/A converter through SPI and Timer. In this way, constant current will flow in the load in accordance with certain time intervals. The software realization of the constant current unit is shown in Fig.3.

Fig. 3. Software realization of constant current unit

4.2 Man-Machine Interface for Touch Screen With resistive touch screen covering on its surface, the man-machine interface is made up of an OPWAY’s 320*240 LM2088EFW-C touch screen, embedded with EPSON’s S1D13700 chip and AD7843. AD7843 is a widely used 4-wire touch screen controller from AD Company. In this program, the parameter of current is shown on the touch screen, while the touch screen is also used as the input of man-machine interface measuring the coordinate value (X and Y) to judge the user’s input. The flow chart of reading data from the touchpad is shown in Fig.4.

100

Y. Shao et al.

Fig. 4. Flow chart of reading data from touchpad

4.3 485 Communication Interface with DSP/PC In order to realize DSP/PC control multiple current source, we adopt 485 communication interfaces for controlling multi-channel current source. The schematic diagram of multiple control use 485 communication interfaces are shown in Fig.5.

Fig. 5. Schematic diagram of multiple control use 485

Development of Constant Current Source for SMA Wires Driver Based on OPA549

101

The baud rate of a data communication system is 19200 bits/s in this program, the data frame is shown inTable1. Table 1. The data frame of 485

prefix characters

Slave Address

Date length

current value

Start time

sustain time

check

After calculating the constant current value, the beginning time and the time of duration, DSP or PC sends commands according with the data frame shown in Table 1 current, start time, sustain time through certain algorithm to right address current source. The right address current source receives data frame and resolve it, and outputs the corresponding current. Other current sources keep waiting while they don’t receive right address. In order to improve the response speed of current source, receiving and sending order is put into the interrupt. The main program is used to send the control order, while interrupt service routine is used to receive data and send data to the corresponding data area, and to judge whether to deal with the data immediately. The flow chart of interrupt service routine is shown in Fig.6

Fig. 6. Flow chart of interrupt service routine

102

Y. Shao et al.

5 Experiment and Result Table 2 shows the relation of the given voltage, theory current and actual current that flow passes sampling resistance. Table 2. A group of sampling data

Voltage (mV)

0 50 80 120 160 200 220 250

Sample resistance (mΩ) 36.7 36.7 36.7 36.7 36.7 36.7 36.7 36.7

Theory current (A)

Actual current

0.00 1.36 2.18 3.27 4.36 5.45 5.99 6.81

0.00 1.35 2.20 3.30 4.35 5.41 6.05 6.75

(A)

Fig. 7. Voltage and Current of sampling resistance

Fig. 8. Error curve of current

Inaccuracy (%) 0.00 -0.91 0.93 0.92 -0.22 -0.73 0.93 -0.91

Development of Constant Current Source for SMA Wires Driver Based on OPA549

103

Fig.7 shows comparison of the given voltage, theory current and actual current. We can find out that the theory current matches well with the actual current in this program. After long time testing, it is proved that this system has high stability to be used in the situation that high current stability and high current accuracy. The error curve of actual current and theory current is shown in Fig.8.

6 Conclusion To drive the smart material SMA for active structural vibration control, a multi-output constant current source with good constant-current effect and high control precision is designed. It shows fine stability and high reliability under long time reliability testing and running. Using the constant-current source of multi-channel distribution for SMA driver, the efficiency of the active vibration control experiment using SMA as composite structure of smart material, is improved. This paper also provides useful solutions for other applications that need multi-channel output and large current constant current source. Acknowledgments. This work is supported by National Nature Science Foundation of China (No.90716027), Science and Technology Commission of Shanghai Municipality Yangtze River Delta cooperation Project (09595811000),Shanghai University “11th Five-Year Plan” 211 Construction Project and Shanghai Key Laboratory of Power Station Automation Technology.

References 1. Kaori, Y.Y.K.: Development and experimental consideration of SMA/CFRP actuator for vibration control. Sensors and Actuators A 122, 99–107 (2005) 2. Xu, M.B., Song, G.: Adaptive control of vibration wave propagation in cylindrical shells using SMA wall joint. Joural of Sound and Vibration 278, 307–326 (2004) 3. Zhang, H.H., Teng, Z.S., Lin, H.J., Wen, H.: Design of precision voltage-controlled constant current source with low power dissipation single-supply. Chinese Journal of Scientific Instrument 12, 2678–2682 (2008) 4. Nikolic, A.B.: Direct torque control strategy based on constant switching frequency applied in current source inverter fed induction motor drive. International Review of Electrical Engineering-IREE 2, 827–834 (2007) 5. Jiang, B.S., Jia, H.Z., Fan, X.T.: A double-feedback constant current source suitable for LDs and LEDs. Electronics World 113, 40–43 (2007) 6. Walker, T., Dillman, N., Weiss, M.L.: A constant current source for extracellular microiontophoresis. Journal of Neuroscience Methods 63, 127–136 (1995) 7. Xi, Y.B., Gu, T.X.: High-Accuracy Strain Measuring Technology Based on the Method of Dual Invariable-Current. Journal of UEST of china 34, 407–409 (2005)

High Impedance Fault Location in Transmission Line Using Nonlinear Frequency Analysis Min-you Chen1, Jin-qian Zhai1, Zi-qiang Lang2, Ju-cheng Liao3, and Zhao-yong Fan3 1

State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400044, China 2 Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK 3 Chongqing Electric Power Company

Abstract. The detection and location of high impedance faults on power system has been one of the most difficult problems in power transmission and distribution systems. According to a very highly nonlinear behavior of high impedance faults, a methodology is presented to locate high impedance faults in power system. The proposed technique is based on the fact that high impedance faults in power line can make the whole system behave nonlinearly. Consequently, the location of high impedance fault can be determined by detecting the position of nonlinear component in power line systems. The effectiveness of the method has been verified through simulation studies. Keywords: high impedance fault, nonlinear frequency analysis, transmission line, power line carrier.

1 Introduction A high-impedance ground fault occurs when a primary conductor makes electrical contact with a road surface, sidewalk, sod, tree limb, or with some other surfaces or object, which restricts the flow of fault current to a level below that to be reliably detectable by conventional over-current devices [1]. Many high impedance faults can be difficult to locate simply because they occur in a subtle and erratic manner [2]. When these faults persist, it is hazardous for both human beings and electrical equipment, especially when they are associated with arcs [3]. Therefore, from both public safety and operational reliability viewpoints, detection and location of HIFs are critically important. Various methods have been proposed by researchers and protection engineers. These techniques can be divided into two groups, time domain and frequency domain algorithms. In time domain, ratio ground relay, proportional relay algorithm [4], and a smart relay[5], fault current flicker and half-cycle asymmetry[6] has been proposed. In frequency domain, several articles have been published based on Kalman filtering approach [7], the fractal theory [8] and neural networks [9,10]. Recently wavelet transform has been proposed to achieve a better solution [11]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 104–111, 2010. © Springer-Verlag Berlin Heidelberg 2010

High Impedance Fault Location in Transmission Line

105

According to the nonlinear characteristics of high impedance arc faults, the method proposed in this article uses nonlinear frequency analysis [12] to detect the position of high impedance faults in transmission line without turning off the electric power. In outlet side of substation, high frequency signal is injected onto power line through coupling equipment, this high frequency signal flow through power line, which was received by receiving device installed in each tower. By wireless network, the equipment sends this signal to control center. In monitoring center, by analyzing all receiving signals, the position of HIFs in transmission line is found. The purpose of this work is to evaluate the applicability of nonlinear frequency analysis to high impedance fault location.

2 Modeling of High Impedance Arc Faults To obtain dynamic features of high impedance arc fault, the nonlinear highimpedance model is shown in Fig.1 [13].

Fig. 1. Simplified two-diodes fault model of HIFs

This HIF model is based on arcing in sandy soil. The model includes two dc sources: 1) V p and 2) Vn which present the arcing voltage of air oil and/or between the trees and line[14]. During the positive half cycle, the current flows through

Vp

and

during the negative, through Vn . When the phase voltage is greater than the positive DC voltage V p , the fault current flows toward the ground. The fault current reverses when the line voltage is less than the negative DC voltage Vn . For values of the phase voltage between Vn and V p no fault current flows. Two resistances— R p and Rn present the resistance of trees and/or earth resistance.

3 Nonlinear Frequency Analysis Technique for Detecting High Impedance Faults 3.1 Derivation of the Technique For the power transmission system without HIFs, the equivalent circuit of power transmission line is shown in Fig.2[15].

106

M.-y. Chen et al. i1 R1

iJ −1 L1

us

iJ LJ −1

RJ −1

u1

C1

iJ +1

RJ

LJ

u J −1

CJ −1

RJ +1 C0

in LJ +1

in +1

uJ

CJ +1

Rn +1

Ln

Rn

u J +1

Cn

un

Ln +1

Z load

Fig. 2. The equivalent circuit of power line system without HIFs

Assuming all initial conditions are zero, according to the current and voltage laws, the mathematical model of transmission line system can be written in time domain as: AU (t ) + BU (t ) + CU (t ) + DU (t ) = U s (t )

(1)

where ⎡C1 L1L2 ⎢ 0 A=⎢ ⎢ M ⎢ ⎣ 0

⎤ ⎥ ⎥ ⎥ O M ⎥ 0 Cn Ln Ln+1 ⎦

L

0

C2 L2 L3 L

0

0 M 0

⎡ L1C1 R2 + R1C1 L2 + G1 L1 L2 ⎢ 0 B=⎢ ⎢ M ⎢ 0 ⎣

0

⎤ ⎥ ⎥ ⎥ O 0 ⎥ L Ln Cn Rn+1 + RnCn Ln+1 + Gn Ln Ln+1 + Ln Cn Z load ⎦

0 −L2

L O

M

−L1 ⎡RRC 1 2 1 +LGR 1 1 2 +RGL 1 1 2 +L2 +L1 ⎢ RRC −L3 2 3 2 +LGR 2 2 3 +RGL 2 2 3 +L3 +L2 ⎢ C=⎢ 0 O ⎢ M O ⎢ ⎢ 0 0 ⎣

⎡ R 1 + (1 + G 1 R 1 ) R 2 ⎢ − R3 ⎢ D = ⎢ 0 ⎢ M ⎢ ⎢ 0 ⎣

L

0

0

L2C2 R3 + R2C2 L3 + G2 L2 L3 L

0

O O −Ln Rn−1RC n n−1 +Ln−1Gn−1Rn +Rn−1Gn−1Ln +Ln +Ln−1 0 −Ln+1

− R1

0

R 2 + (1 + G 2 R 2 ) R 3 O O

− R2 O − Rn

0

L

⎤ ⎥ ⎥ ⎥ 0 ⎥ −Ln−1 ⎥ ⎥ RR C LGR RGL L L ( RC LG ) Z + + + + + + n n+1 n n n n+1 n n n+1 n+1 n n n n n load ⎦ 0 M

L

R n −1

0

O O + (1 + G n − 1 R n − 1 ) R n − ( R n +1 + Z l o a d )

M 0 − R n −1 R n + (1 + G n R n ) ( R n + 1 + Z lo a d

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ) ⎥⎦

are the system parameter matrixes, respectively. U = (u1 ,L , un )′ is the voltage vector, and U s = ( L2u&s + R2us , 0,L , 0)′ is the external input vector acting on the system. Obviously, this system is a linear system. When HIF occurs in this transmission line system, the power line system can be described as AU (t ) + BU (t ) + CU (t ) + DU (t ) = NU + U s (t )

(2)

The system described by (Eq.9) is a typical locally nonlinear Multi-Degree Of Freedom (MDOF) system. Obviously, the specific forms of matrices A, B, C, D in models (7) and (9) are essentially the same as the models (1) and (4) MDOF vibrating system in [16] in the cases of with and without a nonlinear component, respectively. So, the nonlinearity location approach in [16] can be applied to detect the location of nonlinearity in system (9), that is, the position of HIF in transmission line systems. By applying the nonlinearity detection approach in [16] to model (9), a nonlinear frequency analysis technique for locating HIF of transmission line is proposed. The procedure of the technique can be described as follows [16]:

High Impedance Fault Location in Transmission Line

107

(1) Excite transmission line system separately using two different input voltages and measure the corresponding voltage response at each receiving site of

us( q ) (t ), q = 1, 2 ,

transmission line to obtain ui( q ) (t ), q = 1, 2, i = 1,L , n . (2) Calculate the FFT spectrum of us( q ) (t ), q = 1, 2 and ui( q ) (t ), q = 1, 2, i = 1,L , n to produce U s ( q ) ( jω ), q = 1, 2

and U i( q ) ( jω ), q = 1, 2, i = 1,L , n .

(3) Evaluate the functions of frequency E i ,i+1 ( jω ) , i = 1,L , n − 1 from the results obtained in Step (2) as follows −1

⎡U s (1) ( jω ) , U i +1(1) ( jω ) ⎤ ⎡U i (1) ( jω ) ⎤ E i ,i +1 ( jω ) = [1 0] ⎢ (2) ⎥ ⎢ (2) ⎥ (2) ⎣⎢U s ( jω ) ,U i +1 ( jω ) ⎦⎥ ⎣⎢U i ( jω ) ⎦⎥

(3)

(4) Evaluate E i ,i+1 for i = 1,L , n − 1 as ω2

E

i ,i +1

=

∫ω

1

E i ,i +1 ( jω ) d ω

(4)

ω2 max i∈(1,L,n−1) ⎡ ∫ E i ,i +1 ( jω ) dω ⎤ ⎢⎣ ω1 ⎥⎦

where [ω1 , ω2 ] is a frequency band within the frequency range of the input spectrum U s ( q ) ( jω ) , q=1,2. (5) Examine E i ,i+1 for i = 1,L , n − 1 . If E i ,i+1 is found to change sharply in the iˆ th section, it can be concluded that HIF is located at the iˆ th section of transmission line system. 3.2

Implementation of the Technique Using Power Line Carrier Method

The idea of the implementation of the proposed technique using power line carrier is illustrated in Fig.3 where a transmission line system is divided into five sections. The detection procedure requires exciting the distribution systems twice using two Power Line HV

MV

Section 1

Section 2

Section 3

Section 4

Section 5

...

load

Transformer Receiver1

Coupling device

Receiver2

Data acquisition module

Receiver3

…

Receiver4

…

…

Receiver5

Data acquisition module

GPRS network INTERNET Power Line Carrier

Wireless receive device

Nonlinear Frequency Analysis

Monitoring centre

Fig. 3. Schematic of power line based on nonlinear frequency analysis

108

M.-y. Chen et al.

sinusoidal inputs separately by power line carrier. The continuous-wave (CW) monitoring signals, which are high frequency signals, are injected onto the Power Line by using the existing power line carrier infrastructure. Then, these high frequency signals are flowing through the power line to every decoupling device, which are installed in several towers of transmission line. By through decoupling device, the 50Hz signals from the power line are filtered out and only high frequency signals are received by data acquisition boards. For each inspecting test, all the measurements and the corresponding carrier signal are collected. The results from two separate inspecting tests where the carrier signals are different as required are used as the information collected in Step (1) of the technique. Then, these high frequency signals are transmitted into wireless receive device by GPRS network. Then, received signals are stored in the computer installed at the substation of the power utility. These signals were analyzed by power utilities technician, fault features can be extracted and recognized based on nonlinear frequency analysis. Thereby, online monitoring and location of high impedance arc fault of power line can be achieved.

4 Simulation Studies In order to verify the effectiveness of the proposed approach, simulation studies using the Matlab/Simulink facilities for transmission line systems were conducted. 35 kV power line were used for the simulation studies. The transmission line system was considered to consist of five sections; four are 1km long and fault free and one is also 1km long but with HIF. For power line system parameters, line length is 5km, line type is ACSR LGJ-70. Simulation study is used in the configuration system shown in Fig.3. The results obtained from case studies with HIF, where a nonlinear 5-dof system was considered, are presented as follows. In this case, the second section of transmission line has HIF as described by model with Rp = 1250Ω , Rn = 500Ω , V p = 5000V , Vn = 7000V , and other system parameters hold the line. The proposed technique was implemented as follows: (1) Two sinusoidal voltage inputs us (1) (t ) = 12 sin(2 * π *50000 * t ) , us (2) (t ) = 12 sin(2 * π * 50000 * t + 90)

were applied to excite this system, respectively, to generate two sets of output responses on the 5 transmission line sections. (2) The spectra of the two sets of system responses at the driving frequency ωu = 2π × 50000 were evaluated to yield the results U i( q ) ( jωu ) , i = 1,K ,5, q = 1, 2. (3) E i ,i+1 ( jωu ) , i = 1,K , 4 , were evaluated from equation (14) using the results obtained in Step (2). i ,i +1 (4) Evaluate E , i = 1,K , 4 using the results obtained in Step (3). (5) The results obtained were given in Table.1, and illustrated in Fig.4. From Table.1 or Fig.4, it can be found that iˆ = 2 . Therefore, it can be concluded that HIF takes place at the second section of transmission line, which is obviously correct.

High Impedance Fault Location in Transmission Line

Table 1. E

i ,i +1

109

i = 1,K , 4 evaluated for a case where HIF is located at Section 2 in Case Study I E

1, 2

E

1.000 0

2,3

E

0.040 1

3, 4

E

0.039 5

4,5

0.038 8

1 0.9 0.8 0.7 0.6 0.5

E i ,i +1

0.4 0.3 0.2 0.1 0

1

2

3

4

i

Fig. 4. The illustration of E

i ,i +1

i = 1,K , 4 in Table 1

Table 2/Fig 5 shows the analysis results by the proposed technique for the cases where HIF was located in the fourth section of transmission line system. Obviously the proposed technique can also correctly locate the position of HIF for this case. Table 2. E

i ,i +1

i = 1,K , 4 evaluated for a case where HIF is located at Section 4 in Case Study I

E

1, 2

E

1.0 000

2,3

E

0.7 116

3, 4

E

0.5 493

4,5

0.0 222

1 0.9 0.8 0.7 0.6

E i ,i +1

0.5 0.4 0.3 0.2 0.1 0

1

2

3

4

i

Fig. 5. The illustration of E

i ,i +1

i = 1,K , 4 in Table 2

110

M.-y. Chen et al.

5 Conclusions In this paper, the application of a nonlinear frequency analysis based approach is proposed to locate HIF in transmission line system. The power carrier signal technology has been suggested for the practical implementation of the proposed technique. Numerical simulation studies have been conducted. The results verified the effectiveness of the new technique and demonstrated the potential to apply the technique in practice to resolve the important HIF location problem. Further research will be focused on laboratory tests to make necessary preparations for future experimental studies on real transmission line systems. The restriction of the proposed approach is the position of HIF can be determined by the proposed approach when more than HIF occur. More details of these will be reported in a future publication. Acknowledgements. The authors gratefully acknowledge the support from the Project (2007DA10512709402) from State Key Laboratory of Power Transmission Equipment & System Security and New Technology, and National “111” Project of China (B08036). ZQL would also like to thank the UK EPSRC for supporting this research study.

References 1. Etemadi, A.H., Sanaye-Pasand, M.: High-impedance Fault Detection using Multiresolution Signal Decomposition and Adaptive Neural Fuzzy Inference System. IET Generation, Transmission & Distribution 2, 110–118 (2008) 2. Aucoin, B.M., Jones, R.H.: High Impedance Fault Detection Implementation Issues. IEEE Transactions on Power Delivery 11, 139–148 (1996) 3. Elkalashy, N.I., Lehtonen, M., Darwish, H.A., Izzularab, M.A., Taalab, A.-M.I.: Modeling and Experimental Verification of High Impedance Arcing Fault in Medium Voltage Networks. IEEE Transactions on Dielectrics and Electrical Insulation 14, 375–383 (2007) 4. Calhoun, H., Bishop, M.T., Eiceler, C.H., Lee, R.E.: Development and Testing of An Electro-mechanical Relay to Detect Fallen Distribution Conductors. IEEE Trans. Power App. Syst. 100, 2008–2016 (1998) 5. Sharaf, A.M., Abu-Azab, S.I.: A Smart Relaying Scheme for High Impedance Faults in Distribution and Utilization Networks. In: Canadian Conf. Electrical Computer Engineering, pp. 740–744. IEEE Press, Los Alamitos (2000) 6. Sultan, A.F., Swift, G.W., Fedirchuk, D.J.: Detecting Arcing Downed-wires Using Fault Current Flicker and Half Asymmetry. IEEE Trans. Power Del. 9, 461–470 (1994) 7. Girgis, A.A., Chang, W., Makram, E.B.: Analysis of High-impedance Fault Generated Signals Using a Kalman Filtering Approach. IEEE Trans. Power Del. 5, 1714–1724 (1990) 8. Mamishev, A.V., Russel, B.D., Benner, C.L.: Analysis of High Impedance Faults Using Fractal Techniques. IEEE Trans. Power Syst. 11, 435–440 (1996) 9. Ebron, S., Lubkeman, D.L., White, M.: A Neural Network Approach to the Detection of Incipient Faults on Power Distribution Feeders. IEEE Trans. Power Del. 5, 905–914 (1990) 10. Snider, L.A., Yuen, Y.S.: The Artificial Neural Networks Based Relay for the Detection of Stochastic High Impedance Faults. Neurocomput. 23, 243–254 (1998)

High Impedance Fault Location in Transmission Line

111

11. Shyh-Jier, H., Cheng-Tao, H.: High-impedance Fault Detection Utilizing a Morlet Wavelet Transform Approach. IEEE Trans. Power Del. 14, 1401–1410 (1999) 12. Lang, Z.Q., Billings, S.A.: Output Frequency Characteristics of Non-linear System. Int. J. Control. 64, 1049–1067 (1996) 13. Ibrahim, D.K., Eldin, E.S.T., El-Din Abou EI-Zahab, E.M., Saleh, S.M.: Unsynchronized Fault-Location Scheme for Nonlinear HIF in Transmission Lines. IEEE Transactions on Power Delivery 25, 631–637 (2010) 14. Lai, T.M., Snider, L.A., Lo, E., Sutanto, D.: High-impedance Fault Detection using Discrete Wavelet Transform and Frequency Range and RMS Conversion. IEEE Transactions on Power Delivery 20, 397–407 (2005) 15. Lonngren, K.E., Bai, E.W.: Simulink Simulation of Transmission Lines. IEEE Transaction on Circuits and Devices Magazine 12, 10–16 (1996) 16. Lang, Z.Q., Peng, Z.K.: A Novel Approach for Nonlinearity Detection in Vibrating Systems. Journal of Sound and Vibration 314, 603–615 (2008)

Batch-to-Batch Iterative Optimal Control of Batch Processes Based on Dynamic Quadratic Criterion Li Jia1, Jiping Shi1, Dashuai Cheng1, Luming Cao1, and Min-Sen Chiu2 1 Shanghai Key Laboratory of Power Station Automation Technology, Department of Automation, College of Mechatronics Engineering and Automation, Shanghai University, Shanghai 200072, China 2 Faculty of Engineering, National University of Singapore, Singapore

Abstract. A novel dynamic parameters-based quadratic criterion-iterative learning control is proposed in this paper. Firstly, quadratic criterion-iterative learning control with dynamic parameters is used to improve the performance of iterative learning control. As a result, the proposed method can avoid the problem of initialization of the optimization controller parameters, which are usually resorted to trial and error procedure in the existing iterative algorithms used for the optimization of batch process. Next, we make the first attempt to give rigorous description and proof to verify that a perfect tracking performance can be obtained. Lastly, examples are used to illustrate the performance and applicability of the proposed method. Keywords: batch processes, iterative learning control, quadratic criterion.

1

Introduction

Recently, batch processes have been used increasingly in the production of low volume and high value added products, such as special polymers, special chemicals, pharmaceuticals, and heat treatment processes for metallic or ceramic products [1-10]. For the purpose of deriving the maximum benefit from batch processes, it is important to optimize the operation policy of batch processes. Therefore, optimal control of batch processes is very significant. The key of optimal control depends on obtaining an accurate model of batch processes, which can provide accurate predictions. Usually, developing first principle models of batch processes is time consuming and effort demanding. For example, Terwiesch et al. pointed out that the effort for building a useful kinetic model that is still just valid for a limited range of operating conditions often exceeds one man year [2]. This may not be feasible for batch processes where frequent changes in product specifications occur and a type of product is usually manufactured for a limited time period in response to the dynamic market demand [3]. Motivated by the previous works, a novel dynamic parameters-based quadratic criterion-iterative learning control is proposed in this paper. The contributions of this paper are shown as follows. 1) Quadratic criterion-iterative learning control with K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 112–119, 2010. © Springer-Verlag Berlin Heidelberg 2010

Batch-to-Batch Iterative Optimal Control of Batch Processes

113

dynamic parameters is used to improve the performance of iterative learning control; 2) We work on data based iterative learning control and make the first attempt to give rigorous description and proof to verify that a perfect tracking performance can be obtained under the proposed constraint conditions in NF model based iterative learning control, namely the changes of the control policy converge to zero as the batch index number tends to infinite, which are normally obtained only on the basis of the simulation results in the previous works. The paper is structured as follows. Section 2 presents the neuron-fuzzy model with nonlinear for batch processes. Section 3 presents the optimization control strategy based on the neuro-fuzzy model proposed above. Simulation examples are given in Section 4, followed by the concluding remarks given in Section 5.

2 Batch Process Descriptions Suppose that t f is the batch end time, which is divided into T equal intervals. And define that U k = [uk (1),L , uk (T )] is a vector of control policy variables during T

k-th batch and Yk = [ yk (1),L , yk (T )]

T

is a vector of product quality variables

during k-th batch, where k represents the batch.

y ∈ Rn

and

u ∈ R m are,

respectively, the product quality and control action variables. In this paper, data based model (DM) is employed to build the nonlinear relationship between

uk

and

yk ,

namely

yˆk (t +1) = DM[ yk (t), yk (t −1),L, yk (t − ny +1), uk (t), uk (t −1)L, uk (t − nu +1)] where n y and

nu

(1)

are integers related to the model order.

In this work, only the product quality at the end of a batch, y d (t f ) , is concerned. So the proposed DM is used to predict the product quality at the end of the k-th batch,

yˆ k ( t f

),

for the giving initializations and input sequence of k -th batch

U k = [uk (0),L , uk (T − 1)]T , noted by

114

L. Jia et al.

yˆ k ( t f ) = f DM (U k ) where

(2)

f DM is the nonlinear function related to DM. The Fig. 1 describes how the data

is generated in batch processes for DM. There are three axes in this figure, namely t ,

k and x . In the direction of axes k , there are K batches of data vectors, in which one vector is with the dimensions of n y + nu , and in the direction of axes t , it is divided into T equal intervals. Then the data are unfolded up from the axes of t and thus the number of the data is K × T (noted by KT ). Therefore, Eq 1 can be rewritten to

yˆ( kt + 1) = DM [ x ( kt )]

(3)

where

x ( kt ) = [ y ( kt ), y ( kt − 1),L , ykt (t − n y + 1), u ( kt ), u ( kt − 1),L , ukt (t − nu + 1)]T and

3

kt represents the data point， kt = 1, 2,L, KT . Dynamic Parameters Based Quadratic Criterion-Iterative Learning Control for Batch Processes

The formulation of the proposed dynamic parameters based quadratic criterioniterative learning control scheme is depicted in Fig.1, where k is the batch index and t

Fig. 1. The proposed NFS model-based iterative optimal control scheme

Batch-to-Batch Iterative Optimal Control of Batch Processes

115

is the discrete batch time, uk and yk are, respectively, the input variable of the product qualities and product quality variable of batch processes, U k is the input sequence of a batch,

yˆ k (t ) is the predicted values of product quality variable at time t of the k-th

batch, y% k +1 (t f ) is the modified predictions of the data based model at the end of the k+1-th batch and DM is employed to predict the values of product quality variable at the end of the k+1-th batch yˆ k +1 (t f ) . In this study, based on the information from previous batch, the optimization controller can find an updating mechanism for the input sequence U k +1 of the new batch using improved iterative optimal control law derived by the rigorous mathematic proof method discussed shortly. At the next batch, this procedure is repeated to let the product qualities at the batch end convergence asymptotically towards the desired product qualities y d (t f ) . Owing to the model-plant mismatches, the process output may not be the same as the one predicted by the model. The offset between the measured output and the model prediction is termed as model prediction error defined by

eˆk (t ) = yk (t ) − yˆ k (t )

(4)

Since batch processes are repetitive in nature, the model prediction at the end of the kth batch, yˆ k +1 (t f ) , can be corrected by

y% k +1 (t f ) = yˆ k +1 (t f ) + α k +1eˆk (t f )

(5)

where α k ( 0 ≤ α k ≤ 1) is a bias correction parameter and eˆk (t f ) is the average model prediction errors of all the previous batches as follows

1 k ∑ eˆi (t f ) k i =1 1 k = ∑ ( yi (t f ) − yˆi (t f ) ) k i =1

eˆk (t f ) =

(6)

116

L. Jia et al.

Considering that the objective is to design a learning algorithm to implement the control trajectory U k so that the product qualities at the batch end convergence asymptotically towards the desired product qualities. Thus, instead of using the whole reference sequences for product qualities, which is less important for batch processes

( )

product quality control, only the desired product quality yd t f at the end of a batch is used. Here the following quadratic criterion with dynamic parameters is employed:

min J (U , k ) = e%kf+1 U k +1

2 Q

+ ΔU k +1

2

(7)

R (k )

f where e%k +1 = yd (t f ) − y% k +1 (t f ) is the difference between desired product qualities

y d (t f ) and modified predictions of the neuro-fuzzy model at the end of the k+1th batch y% k +1 (t f ) . The updating function of optimization controller is defined as

ΔU k +1 = U k +1 − U k

(8)

u low ≤ uk +1 (t ) ≤ u up y low ≤ yk +1 (t f ) ≤ y up

(9)

where ulow and u up are the lower and upper bounds of the input trajectory, y low and

y up are the lower and upper bounds of the final product qualities, Q and R(k ) are both weighting matrices and defined as Q = λq × I n and R ( k ) = λr ( k ) × I n where

λr (k ) and λq are

the factors. In this paper we use SQP algorithm to solve this

optimization problem. The convergence property of the proposed dynamic parameters based quadratic criterion-iterative learning control is described as follows. Theorem 1. If the limitation of α k

( lim α ) k →∞

k

exists and

λr (k ) is

bounded by

positive constants r1 and r2 , namely r2 ≥ λr ( k ) ≥ r1 > 0 , the proposed dynamic parameters based quadratic criterion-iterative learning control policy with the following constraint condition (10) converges, namely ΔU k → 0 as k → ∞ .

Batch-to-Batch Iterative Optimal Control of Batch Processes

0≤

ydi (t f ) − y% ki +1 (t f ) ≤ bk ydi (t f ) − y% ki (t f )

117

(10)

where bk ≤ 1 , yd and y% k represent the i-th element of the vector yd and y% k . Proof: The proof is omitted. i

4

i

Example： An Ethanol Fermentation Process

The fed-batch fermentation process was taken from paper [11] and the mechanistic model of the fed-batch ethanol fermentation process is described as follow:

dx1 dt dx2 dt dx3 dt dx4 dt With ζ =

= ζ x1 −

x1 u x4

= −10ζ x1 + x = ξ x1 − 3 u x4

(150 − x2 ) u x4

=u

0.408 x2

(1 + x3 /16 )( 0.22 + x2 )

and ξ =

x2

(1 + x3 / 71.5)( 0.44 + x2 )

where x1 , x2 , x3 , x4 respectively represent cell mass concentration, substrate concentration, product concentration and liquid volume of the reactor. The initial

[

condition is specified as x(0) = 1 150

0 10] , with the feed rate to the T

0 L / h ≤ u ≤ 12 L / h , the liquid volume of the reactor is limited by the 200 L vessel size: x4 ≤ 200 L , and the batch time t f is fixed as

reactor u constrained by

63.0 h. The control objective of this process is to maximize the yield of the reactor by

choosing an appropriate feeding policy u in the time interval 0 ≤ t ≤ t f as follow:

max J (u ) = x3 (t f ) x4 (t f ) u (t )

In this simulation, the reactor u , product concentration are first normalized using u =

x3 and liquid volume x4

x x u , x3 = 3 , x4 = 4 which are all bounded in 12 110 200

118

L. Jia et al.

[ 0,1] . The reactor

the region

u is divided into 10 equal intervals, and in each of

which the control trajectory is a constant. The following initialization parameters in batch process are assumed:

v

α = 0 , λq = 10 ， λr = 3 , U 0 = 0 ， eˆ0 (t f ) = 0 .

The run-to-run control strategy is implemented with the following parameters:

yd (t f ) = 0.94 , τ 1 = 1× 105 , τ 2 = 2 , τ 3 = 1 , τ 4 = 5 ×106 .

Fig.2 and Fig.3 show the concentration and 20th batches.

x3 and volume x4 in the 1st, 3rd, 10th

120

Product Concentration(g/l)

100

80

60

40 1th batch 3th batch 10th batch 20th batch

20

0

0

10

20

30 Time (h)

40

50

60

Fig. 2. The product concentration x3 in the 1st, 3rd, 10th and 20th batches of ethanol fermentation process 250

Volume(L)

200

150

100

1th batch 3th batch 10th batch 20th batch

50

0

0

10

20

30 Time (h)

40

50

60

Fig. 3. The volume x4 in the 1st, 3rd, 10th and 20th batches of ethanol fermentation process

5

Conclusion

In this paper, a novel dynamic parameters-based quadratic criterion-iterative learning control is proposed. Firstly, quadratic criterion-iterative learning control with dynamic

Batch-to-Batch Iterative Optimal Control of Batch Processes

119

parameters is used to improve the performance of iterative learning control. Next, we make the first attempt to give rigorous description and proof to verify that a perfect tracking performance can be obtained under the proposed constraint conditions in the iterative learning control, which are normally obtained only on the basis of the simulation results in the previous works. Lastly, examples illustrate the performance and applicability of the proposed method.

Acknowledgement Supported by Research Fund for the Doctoral Program of Higher Education of China (20093108120013), Shanghai Science Technology commission (08160512100, 08160705900, 09JC1406300 and 09DZ2273400), Grant from Shanghai Municipal Education commission (09YZ08) and Shanghai University, "11th Five-Year Plan" 211 Construction Project.

Reference 1. Bonvin, D.: Optimal operation of batch reactors: a personal view. Journal of Process Control 8, 355–368 (1998) 2. Terwiesch, P., Argarwal, M., Rippin, D.W.T.: Batch unit optimization with imperfect modelling: a survey. J. Process Control 4, 238–258 (1994) 3. Zhang, J.: Batch-to-batch optimal control of a batch polymerisation process based on stacked neural network models. Chemical Engineering Science 63, 1273–1281 (2008) 4. Cybenko, G.: Approximation by superposition of a sigmoidal function. Math. Control Signal Systems 2, 303–314 (1989) 5. Girosi, F., Poggio, T.: Networks and the best approximation property. Biological Cybernetics 63, 169–179 (1990) 6. Park, J., Sandberg, I.W.: Universal approximation using radial-basis-function networks. Neural Comput. 3(2), 246–257 (1991) 7. Su, H.T., McAvoy, T.J., Werbos, P.: Long-term prediction of chemical process using recurrent neural networks: a parallel training approach. Ind. Eng. Chem. Res. 31, 1338– 1352 (1992) 8. Tian, Y., Zhang, J., Morris, A.J.: Modeling and optimal control of a batch polymerization reactor using a hybrid stacked recurrent neural network model. Ind. Eng. Chem. Res. 40, 4525–4535 (2001) 9. Tian, Y., Zhang, J., Morris, A.J.: Optimal control of a batch emulsion copolymerisation reactor based on recurrent neural network models. Chemical Engineering and Processing 41, 531–538 (2002) 10. Xiong, Z., Zhang, J.: Modelling and optimal control of fed-batch processes using a novel control affine feedforward neural network. Neurocomputing 61, 317–337 (2004) 11. Xiong, Z.H., Zhang, J.: Neural network model-based on-line re-optimisation control of fed-batch processes using a modified iterative dynamic programming algorithm

Management Information System (MIS) for Planning and Implementation Assessment (PIA) in Lake Dianchi Longhao Ye1, Yajuan Yu1,*, Huaicheng Guo2, and Shuxia Yu3 1 Beijing Key Laboratory of Environmental Science and Engineering, School of Chemical Engineering & Environment, Beijing Institute of Technology, Beijing, China, 100081 [email protected] 2 College of Environmental Sciences and Engineering, Peking University, Beijing, China, 100871 3 College of Resource and Environment, Huazhong Agricultural University, Wuhan, China, 430070

Abstract. A kind of Management Information System for Planning and Implementation Assessment (PIA-MIS) is studied. The system is applied to the Lake Dianchi Watershed. MapInfo is selected as the system platform, Access is chosen as its database. MapBasic and Visual Basic are used as secondary development tools. PIA-MIS contains five parts: user management module; research module; statistical analysis module; database management module; display and output control module. The system can not only standardize the management of data monitoring and implementation status of the water pollution prevention program of Lake Dianchi, but also simulate and forecast the pollution condition of the whole lake. The system is meaningful to the assessment, anticipation, control and supervision of water quality. It provides a dependable basis for the future decision. Keywords: Management Information System; Lake Dianchi Watershed; Planning Implementation Assessment.

1 Introduction Lake Dianchi watershed in Yunnan Province, China, is the largest freshwater lake in Southwest China. During the past 20 years, with the rapid development of economy and society around the watershed, the area faces more and more pressure from water pollution. The whole watershed is under the pressures of over exploitation of land, the risen contradiction between supply and demand, long-lasting agricultural non-point source pollution. As a result, the water quality has gradually deteriorated. The water quality indicators in the existing 12 monitoring points of the lake are all worse than Grade V of National Water Quality Standard. The native residents and the native *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 120–128, 2010. © Springer-Verlag Berlin Heidelberg 2010

Management Information System (MIS) for PIA in Lake Dianchi

121

municipal government have taken a lot of countermeasures to control the pollution. Some of the measures are water pollution treatment projects, while others are administrant ones. No matter what kind of projects they are, their benefits or effects haven’t been directly validated [1]. In order to better take advantage of the water quality motoring data and control the water pollution as well as eutrophication, a kind of Management Information System for Planning and Implementation Assessment (PIA-MIS) is studied in this paper. The system has both database and management functions, which could show the status of the water pollution prevention program of Lake Dianchi. It’s meaningful to the assessment [2], anticipation, control and supervision of water quality, and provides a dependable basis for the future decision making [3].

2 Objective and Research Process of PIA 2.1 Objective of PIA PIA of Lake Dianchi watershed is mainly concerned about the water pollution prevention program. The objectives of planning and implementation assessment are: 1) Analyzing the achievement and open questions of the water pollution prevention program of the lake; 2) Analyzing the effectiveness of implementation for the water pollution prevention program, integrating the increasing tendency of social economy. 2.2 Research Process of PIA Main research steps of Lake Dianchi water pollution control planning and implementation assessment (Fig.1): 1) Finding out the information of the water pollution prevention planning of Lake Dianchi, which includes the planning goal, strategies, and planning projects with their budgets. 2) Collecting relevant planning and implementation data, which include the execution condition and financial state of the planning projects, as well as the relative policies for guarantee of the implementation. 3) Analyzing the implementation of the water pollution prevention planning of Lake Dianchi, including water quality control objective, total amount control objective, implementation status of pollution-control projects, implementation status of motoring and supervision. 4) Integrating water quality and development of socioeconomic conditions into a comprehensive system. Analyze the variation tendency of contamination index based on the increase of social economy. 5) Summarizing the achievement of water pollution control and treatment program of Lake Dianchi Watershed. It is to show the effect of the Planning, in both water quality amelioration and economical development.

122

L. Ye et al. Data of the water pollution prevention programme of Lake Dianchi

Besic information of Lake Dianchi o o o o o o o o

Geography Climate Water system Hydrology Geology and landform Ecological environment Society and economy Soil

o

Data collection

The Ninth Five-Year Plan for the water pollution prevention programme of Lake

o

Dianchi The Ninth Five-Year Plan for the water pollution prevention programme of Lake

o

Dianchi The eleventh Five-Year Plan for the water pollution prevention programme of Lake Dianchi

Implementation status of the water pollution prevention programme of Lake Dianchi

Total amount control objective

Water quality control objective

Implementation status of pollution-control projects

Implementation status of motoring and supervision

Trend analysis of fluctuation of water quality based on the increase of society and economy Water quality assessment of lake inlet

Water quality assessment of Lake Dianchi

Summary of result of the water pollution prevention programme of Lake Dianchi Problem analysis Improvement and suggestion of the water pollution prevention programme of Lake Dianchi

Fig. 1. Technical Approach for Lake Dianchi water pollution control planning and implementation assessment

3 Constructing Principals and Scheme of PIA-MIS 3.1 Constructing Flow Chat of PIA-MIS The system can make better use of the geographic information system to collect, save, analyze and reproduce spatial information (Fig.2). By constructing a database and keeping updating using relative technologies, spatial data, water quality data, text data, image data and view data can be reappeared on the monitoring points on Dianchi digital map which contains all monitoring points of lake inlets and lake Dianchi. 3.2 Software Needed MapInfo is selected as the system platform, Access software is chosen as database as its ease of use and widely-use. MapBasic and Visual Basic are used as secondary development tools, which is achievable from technical aspects.

Management Information System (MIS) for PIA in Lake Dianchi

The environmental protection department

Surveying and mapping department

123

other departments

Map information of Lake Dianchi watershed, water quality monitoring data of Lake Dianchi

Spatial data of Lake Dianchi

Spatial data of Lake inlet

attribute data of Lake Dianchi

Through scanner,digitizer and software

attribute data of Lake inlet

Through keyboard and mouse

Spatial database

attribute database Connection MapInfo Secondary development by VB and MapBasic Water quality MIS of Lake Dianchi

Fig. 2. Constructing flow chat of PIA-MIS

3.3 Objective of MIS Achieve the input, search, management and output of monitoring information of Lake Dianchi and implementation status of water pollution prevention program, including accomplishment of water quality control objective and total amount control objective, implementation status of pollution-control projects and implementation status of motoring and supervision.

4 General Structure of MIS MIS (Management Information System) is constructed basing on modularization, which contains five parts [4]: user management module, research module, statistical analysis module, database management module, display and output control module (Fig.3). The different modules’ functions are as following. (a) User management module: achieve the management of user. Only administrator and relative people can enter the system. (b) GIS function module: achieve basic operation of map and layer management [5-6], with multiple layers. (c) Searching module: achieve the research for monitoring point, obtaining from map to attribute and from attribute to map [7-9].

124

L. Ye et al.

(d) Statistical analysis module: achieve statistical analysis according to users’ need, such as histogram comparison analysis of pollutant concentration [10-13]. (e) Database management module: achieve the updating and sorting of data. (f) Display and output control module: achieve the display and output of research result and analysis result. Tools like zoom in and zoom out are provided for map display. Basic operation of map Basic GIS functions User

Layer control Layer management

Database management

Data updating Data sorting

Authentication system

Information searching Multimedia interface

Spatial information searching Implementation status of programs Water quality monitoring data searching Water quality anticipation

Water quality modeling

Water quality analysis Result interpretation Map display

Data output and display

Map print Data print

Fig. 3. General Structure of MIS for Planning and Implementation Assessment

5 PIA for Lake Dianchi Watershed The Lake Dianchi, is comprised of two parts: Waihai (in the south) and Caohai(in the north) as showed in Fig.4. As the Caohai is near the urban areas of Kunming City, it receives more pollution load than Waihai does. Besides, the volume of Waihai is larger than Caohai. As a result, the water quality of Waihai is better than that of Caohai. In order to demonstrate the implementation effect of Water Pollution Control Planning of Lake Dianchi, a kind of Effect Index (EI) for PIA is introduced here, signed

Management Information System (MIS) for PIA in Lake Dianchi

125

as EI-PIA. The EI-PIA is defined as the water quality protection effect versus the growth of economy. When water quality is expressed through indicators, such as COD or NH3-N, while the economy growth is expressed as Gross Domestic Product (GDP) of the relative watershed, the EI-PIA could be gained from water quality indicators of unit GDP.

Fig. 4. Monitoring points of Lake Dianchi Watershed

126

L. Ye et al.

Table 1 shows the EI-PIA of Lake Dianchi. Fig.5 and Fig.6 show the tendency of contamination indicators per unit GDP in Lake Caohai and Lake Waihai from year 1993 to 2008. Table 1. EI-PIA (Contamination index per unit GDP) in Lake Caohai and Lake Waihai year 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

COD Per unit GDP(106mg/L·GDP) Lake Caohai Lake Waihai 37.91 27.14 26.68 24.23 19.56 16.21 17.33 13.50 16.54 10.96 12.43 10.13 12.47 12.80 8.49 8.80 8.13 9.02 10.90 8.99 7.82 8.19 6.98 7.08 5.65 5.75 5.05 5.80 3.77 4.77 3.24 3.74

NH3-N Per unit GDP(108mg/L·GDP) Lake Caohai Lake Waihai 234.29 9.95 115.02 12.11 89.81 2.38 83.06 3.67 89.68 7.15 78.36 6.04 71.05 3.10 103.61 3.65 118.32 2.31 99.29 4.95 84.00 3.84 91.14 3.10 89.84 2.38 92.84 2.72 82.37 1.94 79.48 1.94

The tendency of EI-PIA shows the water quality is going to be better comparing with the economical development.

Fig. 5. EI-PIA (COD per unit GDP) of Lake Dianchi (unit: 106mg/L·GDP)

Management Information System (MIS) for PIA in Lake Dianchi

127

Fig. 6. EI-PIA(NH3-N per unit GDP) of Lake Dianchi (unit: 108mg/L·GDP)

6 Prospects and Concluding Remarks Lake Dianchi water quality MIS provides functionality for data capture, data editing, pre-processing, statistical analysis and result interpretation. The use of MIS can improve and enhance the rapid identification of pollutant, improving response time and mitigation strategies. However, the constructing of Dianchi PIA-MIS is a rather complicated project, which needs great expense and long time. The system constructing is a project that cannot complete at once but progress step by step and be improved gradually. Problems are found and solved while its application, which make the system more feasible for practical application. Acknowledgement. The paper is subsided by: (1) “National major Science and Technology Program – Water Body Pollution Control and Remediation” (2008ZX07102-001)”; (2) National Natural Science Foundation of China (40701066).

References 1. Zhang, Y., Barten, P.K.: Watershed Forest Management Information System (WFMIS). J. Environmental Modeling & Software 24, 569–575 (2009) 2. Martin, P.H., LeBoeuf, E.J., Daniel, E.B., Dobbins, J.P., Abkowitz, M.D.: Development of a GIS-based Spill Management Information System. J. Journal of Hazardous Materials B112, 239–252 (2004) 3. Chen, C.-F., Ma, H.-w., Reckhow, K.H.: Assessment of water quality management with a systematic qualitative uncertainty analysis. J. Science of the Total Environment 374, 13–25 (2007) 4. Fedra, K.: Urban environmental management: monitoring, GIS, and modeling. J. Computers, environment and urban systems 23, 443–457 (1999) 5. Santé-Riveira, I., Crecente-Maseda, R., Miranda-Barrós, D.: GIS-based planning support system for rural land-use allocation. J. Computers and Electronics in Agriculture 63, 257–273 (2008)

128

L. Ye et al.

6. Wang, X., Yu, S., Huang, G.H.: Land allocation based on integrated GIS-optimization modeling at a watershed level. J. Landscape and Urban Planning 66, 61–74 (2004) 7. Leon, L.F., Lam, D.C., McCrimmon, C., Swayne, D.A.: Watershed management modeling in Malawi: application and technology transfer. J. Environmental Modeling & Software 18, 531–539 (2003) 8. Villarreal, E.L., Semadeni-Davies, A., Bengtsson, L.: Inner city stormwater control using a combination of best management practices. J. Ecological Engineering 22(4-5), 279–298 (2004) 9. Gal, G., Hipsey, M.R., Parparov, A., Wagner, U., Makler, V., Zohary, T.: Implementation of ecological modeling as an effective management and investigation tool: Lake Kinneret as a case study. J. Ecological Modeling 220, 1697–1718 (2009) 10. Sarkar, S.K., Saha, M., Takada, H., Bhattacharya, A., Mishra, P., Bhattacharya, B.: Water quality management in the lower stretch of the river Ganges, east coast of India: an approach through environmental education. J. Journal of Cleaner Production 15, 1559–1567 (2007) 11. Hellström, D., Jeppsson, U., Kärrman, E.: A framework for systems analysis of sustainable urban water management. J. Environmental Impact Assessment Review 20, 311–321 (2000) 12. Assaf, H., Saadeh, M.: Assessing water quality management options in the Upper Litani Basin, Lebanon, using an integrated GIS-based decision support system. J. Environmental Modelling & Software 23, 1327–1337 (2008) 13. Chi-Feng, C., Hwong-wen, M., Reckhow Kenneth, H.: Assessment of water quality management with a systematic qualitative uncertainty analysis. J. Science of the Total Environment 374, 13–25 (2007)

Integration Infrastructure in Wireless/Wired Heterogeneous Industrial Network System* Haikuan Wang1, Weiyan Hou 2,1, Zhaohui Qin2, and Yang Song1 1

School of Mechatronics and Automation, University Shanghai, Shanghai, 200072, China 2 School of Information Engineering, University Zhengzhou, Zhengzhou, 450052, China [email protected]

Abstract. As various wired Field-bus systems are still pervasive in present industrial and factory environments, it is very unlikely that wireless communications will be able to replace current wired Field-bus systems in many industrial environments completely. The integration of wireless and wired segment (networks) was studied in this paper. Three integration patterns on application layer, physical layer, data-link layer were discussed respectively. Then, an industrial network-oriented universal conversion infrastructure of heterogeneous network is proposed. In light of features of industrial Field-bus data communication and special requirements for real-time properties for the automation in factory environment, a mid-layer model (DCM) is presented which uses united data reflection zone instead of conventional frame forwarding. Through the building and flexible configuration of DCM modules for different wired or wireless segments, seamless access of terminal nodes of a variety of wireless and cable networks cab be realized, and data cooperation could be attained. Keywords: Industrial network; Wireless/Wired Heterogeneous system; Integration pattern; Real-time.

1 Introduction Wireless communication technologies have been becoming more and more pervasive in the past years. It is going to be a new tendency that the most advanced short-range wireless communication techniques are introduced to industrial automation field in factory environment. At present researching on the application of wireless networks in industrial control field has been paid more and more attentions [1][2][3]. Compared to wired networks, the reliability and timing requirement of wireless networks for industrial control are significantly more difficult to meet. The growth of wireless technologies in industrial applications is expected to be much slower than in other areas for several reasons, such as efficiency, safety, cost and security, etc. *

This work is supported by Chinese 863-High-Tech Plan under No:2006AA04030405, 2007AA04Z174, National Natural Science Foundation of China under No:60974097, 60904016.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 129–135, 2010. © Springer-Verlag Berlin Heidelberg 2010

130

H. Wang et al.

Wireless networks would undoubtedly play a crucial role in many next-generation industrial and producing systems, but it’s far from being considered well enough for current industrial control applications comparing with wired system. To be fair, various wired Field-bus systems are still pervasive in present industrial and factory environments. It is very unlikely that wireless communications will be able to replace current wired Field-bus systems adopted in many industrial environments completely, for many reasons of efficiency, safety, cost and security, etc. Thus, it will be most advisable and feasible that the integration of wireless with the existing wired network systems is adopted to achieve flexibility, efficiency and performance for the overall network system [4][5]. In this paper the research of integration architecture between wired and wireless networks are presented in detail. In section 2, some challenges of integration are discussed. In section 3, several integration models are analyzed. In section 4, the new protocol integration architecture was proposed in detail. On analysis of integration patterns based on OSI protocol, an industrial network-oriented universal conversion structure of heterogeneous network protocol is proposed, and it can integrate a variety of wireless and cable networks. In light of features of industrial Field-bus data communication and special requirements for real-time properties of protocol conversion, a method is presented which uses united data reflection zone instead of conventional frame forwarding. Conclusions are summarized in section 5.

2 Solutions for Integration of Wireless and Wired Networks Integrating wireless networks with wired Field-bus systems is to design and realize an interconnection device so that terminal nodes in the different network segments could communicate with each other effectively, but not affect the existing wired Field-bus control systems [6]. However, the interconnection will bring about conversion time delay, which could affect the real-time performance of industrial control systems. So far as a wireless/wired heterogeneous network is concerned, the real-time performance depends not only on media access delay (efficiency of MAC protocol), but also on the pattern and efficiency of protocol conversion between wireless and wired networks. The gateway of Internet is a protocol converter based on the TCP/IP protocol [12]. However, the protocol converter in industrial field aims at Field-bus, Industrial Ethernet and wireless networks, which are very different from Physical Layer to Application Layer. Information transmitting in the industrial control networks can be divided into periodic process data and non-periodic management configuration data. Periodic process data through the protocol converter require small time delay value, small Jitter, no packet losing, and no important information loss when the transmission speed of respective port has big difference. So it is a great challenge to assure the realtime performance of heterogeneous networks and realize the information cooperation. Generally there are several integration models in heterogeneous network systems, for instance, OPC, modem, gateway, etc. These integration models have their own advantages and disadvantages. In recent years, some researches for integration of wireless and wired networks have been done.

Integration Infrastructure in Wireless/Wired Heterogeneous Industrial Network System

131

2.1 Integration on Application Layer As showed in Fig.1, OPC Sever is used for information exchanging between wireless networks and wired Field-bus networks. The advantages include the easy implication, no need to change the system architecture, and easy for configuration and control with software tools in the running stage [9]. However, this method did not assure the realtime performance in system control because of the isolation between networks protocols.

Fig. 1. Model of Integration on Application Layer

2.2 Integration on Physical Layer Generally, a modem is used to receive the information from wireless stations, convert the format of frame and relay the information, which is linked with the Physical Layer of a wired station [7]. Thus, wireless stations and wired stations could work in the same protocol. The model is showed in Fig.2. This method is able to assure the real-time performance and be realized easily. However it only realizes the communication of point to point.

Fig. 2. Model of Integration on Physical Layer

2.3 Integration on Data Link Layer Gateways are the device to convert one protocol stack into another among different communication networks, and can be used to realize the integration of diverse networks on Data Link Layer [8][11]. As showed in Fig.3 , an access point for wireless network segment is added to the existing wired network station, and used to convert one protocol into another when the information need to be exchanged between the different network segments. However, the gateway solution may not offer satisfied real-time performance, because the time delay of information exchanging cross the gateway is undetermined.

132

H. Wang et al.

Due to the difference of wired and wireless networks in MAC protocol, link-layer, data rate, etc, such an approach may require large timeout value. And the design of hardware and software of a gateway is also full of challenges.

Fig. 3. Model of Integration on Data Link Layer

3 Universal Conversion Structure of Heterogeneous Network In order to solve the problems in the course of the integration of wireless and wired heterogeneous networks, a universal conversion structure of heterogeneous network protocol is presented which uses a united data reflection zone instead of conventional frame forwarding. Through flexible configuration of DCM modules of different networks, seamless access of terminal nodes of a variety of wireless and cable networks is realized, and information cooperation could be attained. 3.1 Design of Universal Conversion Structure A universal conversion structure is presented to realize the integration between wireless and wired networks and assure real-time performance and reliability of the whole heterogeneous network system. In the new conversion structure, the information of different network segments could be processed by respective DCM, and then the information cooperation could be attained by a common Data Image Cache under the control of the Interconnection Manegement Layer. Thus, communication among different network segments is realized by a gateway. The universal conversion structure is showed in Fig.4. 3.2 Design of Interconnection Manegement Layer Protocol and DCM Module In the model of protocol conversion, a new layer, also called Mid Layer, is added between Data Link Layer and Application Layer, as showed in Fig.5. The Mid Layer can be divided into two parts. The part on the top is bottom are DCMs of different subsystems, which could convert the data frame of one protocol into another, and then put the final data into the common Data Image Cache. Then an example is given to describe the structure of the protocol conversion module. In respective network segment the protocol converter communicate with other Terminals just as a normal node. In the wireless network segment, DCM is responsible for assembling and disassembling the data packets. In the design of DCM in wireless segment, a Token-based control mechanism is presented to manage and control the CSMA protocol [9][11]. Actually, the protocol converter, acting as a coordinator in the

Integration Infrastructure in Wireless/Wired Heterogeneous Industrial Network System

133

wireless segment, could control the Token-based mechanism and manage the interconnection list of terminal nodes. As to the DCM of wired network segment, it aims at Industrial Ethernet which is based on the peer-to-peer mechanism of CSMA and PROFIBUS which is based on the token mechanism.

Fig. 4. Conversion sturcture of heterogeneous network protocol

3.3 Data Image Cache In order to realize information cooperation of different network segments, the Data Image Cache is designed on the basis of the structure of slot and block, which could reflect the data change of the terminal nodes in the different segments. The information of Image Cache is updated by respective DCM under the control of the Interconnection Management Layer protocol. An example of data updating in the Image Cache when converting protocol among different segments is showed in Fig.6.

Fig. 5. Conversion module of heterogeneous network protocol

So far as the base station of a wired segment is concerned, the Data Image Cache actually is a data zone of terminal nodes. And data of wireless terminals can be read or written by the wired base station through accessing to the Image Cache.

Fig. 6. Cycle of data updating

134

H. Wang et al.

Fig.7 describes the situation of data exchange among several network segments. Considering features of short data frame and prominent periodicity of industrial data communication, and coordination of the data updating period in different segments, Interconnection Management Layer Protocol can be utilized to manage and control the data updating of different segments so as to provide a dynamic QoS and reduce the affection of Jitter. So the chief research topic is how to assure a satisfied response time delay when the amount of terminals keeping changeless.

Fig. 7. Information coorperation of Dara Image Cache

4 Conclusions In this paper, we proposed an industrial network-oriented universal conversion infrastructure of heterogeneous network protocol to realize the information exchange between wireless and wired networks, so that wireless networks could be integrated with the existing wired systems. In the universal protocol conversion structure proposed above, using united data reflection zone instead of conventional frame forwarding, seamless access of terminal nodes of a variety of wireless and wired networks is realized, and information cooperation could be attained through flexible configuration of DCM modules of different networks. Several application scenarios have been designed to evaluate its time performance in next time. The first is to design the protocol converter satisfied with the special time requirements of industrial application independently by us, including wireless nodes design and configuration software design. Secondly, a wireless and wired heterogeneous network system will be constructed on the basis of the protocol converter in laboratory and in the practical industrial environments to test real-time performance and reliability in a power factory.

References 1. Willig, A., Matheus, K., Wolisz, A.: Wireless technology in industrial networks. Proceedings of the IEEE 93, 1130–1151 (2005) 2. Neumann, P.: Communication in industrial automation–What is going on? Control Engineering Practice 15, 1332–1347 (2007)

Integration Infrastructure in Wireless/Wired Heterogeneous Industrial Network System

135

3. Willig, A.: Recent and Emerging Topics in Wireless Industrial Communications: A Selection. IEEE Transactions on Industrial Informatics 4, 102–124 (2008) 4. Sleman, A., Alafandi, M., Moeller, R.: Integration of wireless Fieldbus and wired Fieldbus for health monitoring. In: Digest of Technical Papers International Conference on ICCE 2009, in Consumer Electronics, pp. 1–2 (2009) 5. Touati, Y., Ali-Cherif, A., Achili, B.: Smart Wheelchair design and monitoring via wired and wireless networks. In: IEEE Symposium on ISIEA 2009 in Industrial Electronics & Applications, pp. 920–925 (2009) 6. Cena, G., Valenzano, A., Vitturi, S.: Wireless Extensions of Wired Industrial Communications Networks. In: 2007 5th IEEE International Conference on Industrial Informatics, pp. 273–278 (2007) 7. Sousa, P., Ferreira, L.L., Alves, M.: Repeater vs. Bridge-Based Hybrid Wired/Wireless PROFIBUS Networks: a Comparative Performance Analysis. In: IEEE Conference on ETFA 2006 in Emerging Technologies and Factory Automation, pp. 1065–1072 (2006) 8. Zhong, T., Haibin, Y., Hong, W.: Implementation of hybrid wired and Wireless Foundation Fieldbus. In: 7th World Congress on WCICA 2008 in Intelligent Control and Automation, pp. 4325–4330 (2008) 9. Kobayashi, M., Nakayama, H., Ansari, N., Kato, N.: Reliable Application Layer Multicast Over Combined Wired and Wireless Networks. IEEE Transactions on Multimedia 11, 1466–1477 (2009) 10. Hou, W.-y., Liu, W.-c., Fei, M.-r.: A Token-based MAC oriented Wireless Industrial Control Networks. In: IEEE International Conference on Information Acquisition, pp. 22– 25 (2006) 11. So, A., Liang, B.: Efficient Wireless Extension Point Placement Algorithm in Urban Rectilineal WLANs. IEEE Transactions on Vehicular Technology 57, 532–547 (2008) 12. Zarimpas, V., Honary, B., Darnell, M.: Channel adaptive multiple I/O protocol for mixed wireless and wired channels. Communications, IET 1, 982–989 (2007)

Multi-innovation Generalized Extended Stochastic Gradient Algorithm for Multi-Input Multi-Output Nonlinear Box-Jenkins Systems Based on the Auxiliary Model Jing Chen1 and Xiuping Wang2 1

2

Control Science and Engineering Research Center, Jiangnan University, Wuxi, PR China 214122 [email protected] Wuxi Professional College of Science and Technology, Wuxi, PR China 214028 [email protected]

Abstract. An auxiliary model based multi-innovation generalized extended stochastic gradient algorithm is developed for multivariable nonlinear Box-Jenkins systems. The basic idea is to construct an auxiliary model using the measured data and to replace the unknown terms in the information vector with their estimates, i.e., the outputs of the auxiliary model. The proposed algorithm can give high accurate parameter estimation compared with existing stochastic gradient algorithms. A simulation example is given. Keyword: Stochastic gradient, Multi-input multi-output systems, Multiinnovation identification, Box-Jenkins systems, Recursive identification.

1

Introduction

Many conventional algorithms can estimate system parameters, see, e.g., [1– 3], most of them assume that the system with single-input single-output, but in process industries, multiple-input multiple-output (MIMO) systems exist widely. Earlier work on MIMO system identification exists: some methods estimated the parameters of the difference equation models obtained from state-space models, e.g., [4–7]; others estimated the parameters of state-space models by using the two-stage bootstrap identification technique [8–10]. There are two important parameter estimation approaches: least squares (LS) [11–19] and stochastic gradient (SG) methods [20–25]. The SG algorithm requires less computational effort than the recursive least squares (RLS) algorithm, but has slower convergence rate than the RLS algorithm [22, 26]. In order to improve the convergence rate of the SG algorithm, Ding et al presented the multi-innovation identification theory [26–36], multi-innovation SG algorithms for linear or pseudo-linear regression models [26–28, 32, 37] and auxiliary model 

This work was supported by the National Natural Science Foundation of China.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 136–146, 2010. c Springer-Verlag Berlin Heidelberg 2010 

Multi-innovation Generalized Extended Stochastic Gradient Algorithm

137

based multi-innovation least squares algorithms for single-output output error systems [35, 36]. The auxiliary model based least squares algorithm can identify the parameters of the systems with unknown variables (namely, unavailable noise-free outputs, noises, etc.), its basic idea is to replace the unmeasurable variables of the system with the outputs of an auxiliary model [27, 38–44]. Ding and Chen presented the auxiliary model based least squares algorithm [41], the auxiliary model based stochastic gradient algorithm [38], and the auxiliary finite impulse response model based identification algorithm [39] for dual-rate systems. Recently, Wang et al proposed an auxiliary model-based RELS and MI-ELS algorithms for Hammerstein OEMA systems and an auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems [45, 46]. Recursive or iterative identification algorithms are very related to the iterative solutions for linaer matrix equations [47–56]. In this area, a series of recursive or iteraive identification methods were presented for linear or nonlinear systems [57–65]. For example, Chen, Zhang and Ding presents an auxiliary model based multi-innovation extended stochastic gradient algorithm for multivariable nonlinear systems with moving average (MA) noise [66]. On the basis of the work in [66], this paper studies the auxiliary model based multi-innovation generalized extended stochastic gradient identification problems for multivariable nonlinear Box-Jenkins systems by expanding the innovation vector to an innovation matrix, treating the autoregressive moving average (ARMA) noise. The rest of the paper is organized as follows. Section 2 develops a novel SG algorithm for the nonlinear MIMO systems and Sections 3 studies the auxiliary model based multi-innovation extended stochastic gradient algorithm for the nonlinear MIMO systems. Section 4 provides an illustrative example. Finally, concluding remarks are given in Section 5.

2

The Auxiliary Model Based Generalized Extended Stochastic Gradient Algorithm

Let us introduce some notations first. The symbol I stands for an identity matrix of appropriate sizes; the norm of a matrix X is defined as X := tr[XX T ] = tr[X T X]; the superscript T denotes the matrix transpose. Consider an MIMO nonlinear Box-Jenkins system: y(t) = A−1 (z)B(z)f (u(t)) + C −1 (z)D(z)v(t),

(1)

where u(t) = [u1 (t), u2 (t), · · · , ur (t)]T ∈ Rr is the system input vector, y(t) ∈ Rm is the system output vector, v(t) ∈ Rm a stochastic white noise vector with zero mean, A(z), B(z), C(z) and D(z) are polynomial matrices in the unit backward shift operator [z −1 y(t) = y(t − 1)] and A(z) = I + A1 z −1 + A2 z −2 + · · · + Ana z −na , Ai ∈ Rm×m , B(z) = B 1 z −1 + B 2 z −2 + · · · + B nb z −nb , B i ∈ Rm×r ,

138

J. Chen and X. Wang

C(z) = I + C 1 z −1 + C 2 z −2 + · · · + C nc z −nc , C i ∈ Rm×m , D(z) = I + D 1 z −1 + D 2 z −2 + · · · + D nd z −nd , D i ∈ Rm×m . The nonlinear function f (u(t)) ∈ Rr is a nonlinear vector function [66]: f (u(t)) = [f1 (u1 (t)), f2 (u2 (t)), · · · , fr (ur (t))]T ∈ Rr , fi (ui (t)) is a nonlinear function of a known basis (γ1 , γ2 , · · · , γl ): fi (ui (t)) = c1 γ1 (ui (t)) + c2 γ2 (ui (t)) + · · · + cl γl (ui (t)), ci are unknown parameters. Any pair (αf (u(t)), α−1 B(z)) for some nonzero constant α would produce identical input and output measurements. In other words, none of the identification schemes can distinguish between (f (u(t)), B(z)) and (αf (u(t)), α−1 B(z)). Therefore, one of the gains of f (u(t)) and B(z) has to be fixed. There are several ways to normalize the gains. Here, we adopt the assumption used in [1, 20, 44, 66– 69]: The first coefficient of the function f (·) equals 1, i.e., c1 = 1. Define the intermediate variables: x(t) : = A−1 (z)B(z)f (u(t)),

(2)

w(t) : = C −1 (z)D(z)v(t).

(3)

Then from (1)–(3) we have y(t) = x(t) + w(t).

(4)

Define the parameter matrix θ and the information vector ϕ(t) as: θT := [θTs , θ Tn ] ∈ Rm×n , n := mna + lrnb + mnd + mnc , θTs := [A1 , A2 , · · · , Ana , B 1 c1 , B 2 c1 , · · · , B nb c1 , B 1 c2 , B 2 c2 , · · · , B nb c2 , · · · , B 1 cl , B 2 cl , · · · , B nb cl ] ∈ Rm×(mna +lrnb ) , θTn := [C 1 , C 2 , · · · , C nc , D1 , D2 , · · · , Dnd ] ∈ Rm×(mnd +mnc ) ,   ϕs (t) ∈ Rn , ϕ(t) := ϕn (t) ϕs (t) := [−xT (t − 1), −xT (t − 2), · · · , −xT (t − na ), γ T1 (u(t − 1)), γ T1 (u(t − 2)), · · · , γ T1 (u(t − nb )), γ T2 (u(t − 1)), γ T2 (u(t − 2)), · · · , γ T2 (u(t − nb )), · · · , γ Tl (u(t − 1)), γ Tl (u(t − 2)), · · · , γ Tl (u(t − nb ))]T ∈ Rmna +lrnb , ϕn (t) := [−wT (t − 1), −wT (t − 2), · · · , −wT (t − nc ), v T (t − 1), vT (t − 2), · · · , v T (t − nd )]T ∈ Rmnd +mnc , γ i (u(t)) := [γi (u1 (t)), γi (u2 (t)), · · · , γi (ur (t))]T ∈ Rr , where the subscripts (Roman) s and n represent the first letters of the words “system” and “noise”, respectively. Then we can get x(t) = θ Ts (t)ϕs (t),

(5)

Multi-innovation Generalized Extended Stochastic Gradient Algorithm

w(t) = θ Tn (t)ϕn (t) + v(t),

139

(6)

y(t) = θ Ts (t)ϕs (t) + θTn (t)ϕn (t) + v(t) = θ T (t)ϕ(t) + v(t).

(7)

Minimizing the cost function J(θ) := E[y(t) − θT (t)ϕ(t)2 ], we can obtain the following SG identification algorithm for estimating the parameter matrix θ: ˆ = θ(t ˆ − 1) + ϕ(t) eT (t), θ(t) r(t) T ˆ (t − 1)ϕ(t), e(t) = y(t) − θ r(t) = r(t − 1) + ϕ(t)2 , r(0) = 1.

(8) (9) (10)

Because the information vector ϕ(t) on the right-hand sides of (8)–(10) contains unknown inner variables x(t − i) and unmeasurable noise terms v(t − j), w(t − k) which makes this SG algorithm in (8)–(10) impossible to implement for practical systems. The solution is based on the auxiliary mode identification idea [38, 39, 41, 44]: These unknowns x(t − i) are replaced with the outputs xa (t − i) of an auxiliary (or reference) model x(t) = θ Tas (t)ϕas (t), ˆ s (t) of θs (t) and ϕ (t) to be the regressive then we take θas (t) to be the estimate θ s vector of xa (t) and γ i (u(t − j)), and use ϕas (t) as ϕs (t). Of course, there are other ways to choose auxiliary models, e.g., using the finite impulse response model [36, 38, 39, 44]. According to the auxiliary model identification principle: the unknown variables x(t− i) in ϕs (t) are replaced by the output xa (t− i) of the auxiliary model, ˆ (t − i), w(t ˆ − i), then v(t − i), w(t − i) are replaced by the estimated residuals v it is easy to obtain an auxiliary model based generalized extended stochastic gradient (AM-GESG) algorithm: ˆ ˆ = θ(t ˆ − 1) + ϕ(t) eT (t), θ(t) r(t) ˆ T (t − 1)ϕ(t), ˆ e(t) = y(t) − θ 2 ˆ r(t) = r(t − 1) + ϕ(t) , r(0) = 1,   ˆ s (t) ϕ ˆ ϕ(t) = , ˆ n (t) ϕ

(11) (12) (13) (14)

ˆ s (t) = [−xTa (t − 1), −xTa (t − 2), · · · , −xTa (t − na ), ϕ γ T1 (u(t − 1)), γ T1 (u(t − 2)), · · · , γ T1 (u(t − nb )), γ T2 (u(t − 1)), γ T2 (u(t − 2)), · · · , γ T2 (u(t − nb )), · · · , γ Tl (u(t − 1)), γ Tl (u(t − 2)), · · · , γ Tl (u(t − nb ))]T ,

(15)

140

J. Chen and X. Wang

ˆ n (t) = [−w ˆ T (t − 1), −w ˆ T (t − 2), · · · , −w ˆ T (t − nc ), ϕ T T T ˆ (t − 1), v ˆ (t − 2), · · · , v ˆ (t − nd )]T , v ˆ T (t)ϕ ˆ (t), xa (t) = θ s

s

ˆ w(t) = y(t) − xa (t), ˆ T (t)ϕ ˆ (t) = w(t) ˆ ˆ n (t). v −θ n

3

(16) (17) (18) (19)

The Auxiliary Model Based Multi-innovation GESG Algorithm

In order to enhance the convergence rate of the AM-GESG algorithm, the objective of this paper is to extend the AM-GESG algorithm such that the parameter estimation accuracy can be improved. Such an algorithm is just the multi-innovation stochastic gradient (MISG) algorithm. The MISG algorithm is a tradeoff between the SG and RLS algorithms. At time t, the AM-GESG algorithm only uses the current data (y(t), ϕ(t)) thus has slow convergence rate. Next, we derive a new algorithm by expanding the single innovation vector e(t) ∈ Rm to an innovation matrix T

T

ˆ (t − 1)ϕ(t), ˆ (t − 1)ϕ(t ˆ ˆ − 1), · · · , E(p, t) = [y(t) − θ y(t − 1) − θ T ˆ (t − 1)ϕ(t ˆ − p + 1)] ∈ Rm×p y(t − p + 1) − θ ˆ which uses the past data {y(t−i), ϕ(t−i), i = 1, 2, · · · , p−1}, where p represents the innovation length. ˆ t) and stacking output matrix Y (p, t) By defining the information matrix Φ(p, as ˆ t) : = [ϕ(t), ˆ ˆ − 1), · · · , ϕ(t ˆ − p + 1)] ∈ Rn×p , Φ(p, ϕ(t Y (p, t) := [y(t), y(t − 1), · · · , y(t − p + 1)] ∈ Rm×p , the innovation matrix E(p, t) can be expressed as: T

ˆ (t − 1)Φ(p, ˆ t). E(p, t) = Y (p, t) − θ Then we can obtain an auxiliary model based multi-innovation generalized extended stochastic gradient algorithm with the innovation length p for MIMO Box-Jenkins systems (the AM-MI-GESG algorithm for short) as follows: ˆ ˆ = θ(t ˆ − 1) + Φ(p, t) E T (p, t), θ(t) r(t) T ˆ (t − 1)Φ(p, ˆ t), E(p, t) = Y (p, t) − θ ˆ s (t)2 + ϕ ˆ n (t)2 , r(0) = 1, r(t) = r(t − 1) + ϕ Y (p, t) = [y(t), y(t − 1), · · · , y(t − p + 1)],

(20) (21) (22) (23)

Multi-innovation Generalized Extended Stochastic Gradient Algorithm

 ˆ t) = Φ(p,

 ˆ s (p, t) Φ ˆ n (p, t) , Φ

141

(24)

ˆ s (p, t) = [ϕ ˆ s (t), ϕ ˆ s (t − 1), · · · , ϕ ˆ s (t − p + 1)], Φ ˆ ˆ n (t), ϕ ˆ n (t − 1), · · · , ϕ ˆ n (t − p + 1)], Φn (p, t) = [ϕ

(25) (26)

ˆ s (t) = [−xa (t − 1), −xa (t − 2), · · · , −xa (t − na ), ϕ T

T

T

γ T1 (u(t − 1)), γ T1 (u(t − 2)), · · · , γ T1 (u(t − nb )), γ T2 (u(t − 1)), γ T2 (u(t − 2)), · · · , γ T2 (u(t − nb )), · · · , γ Tl (u(t − 1)), γ Tl (u(t − 2)), · · · , γ Tl (u(t − nb ))]T , ˆ n (t) = [−w ˆ T (t − 1), −w ˆ T (t − 2), · · · , −w ˆ T (t − nc ), ϕ T T T ˆ (t − 1), v ˆ (t − 2), · · · , v ˆ (t − nd )]T , v ˆ T (t)ϕ ˆ (t), xa (t) = θ s

s

ˆ w(t) = y(t) − xa (t), ˆ T (t)ϕ ˆ (t) = w(t) ˆ ˆ n (t). v −θ n

(27) (28) (29) (30) (31)

Because E(p, t) ∈ Rp×m is an innovation matrix, namely, multi-innovation, the algorithm in (20)–(31) is known as the multi-innovation identification one. As p = 1, the AM-MI-GESG algorithm reduces to the AM-GESG algorithm in (11)– (19). ˆ by the AM-MIThe steps of computing the parameter estimation matrix θ(t) GESG algorithm are listed in the following: ˆ ˆ (i) = 1m /p0 , w(i) ˆ = 1. To initialize, let t = 1, θ(0) = I/p0 , xa (i) = 1m /p0 , v 1m /p0 for i ≤ 0, p0 = 106 , and set the innovation length p. 2. Collect the input-output data u(t) and y(t), and compute γ i (u(t)). ˆ s (p, t) by (25) and Φ ˆ n (p, t) by (26). ˆ s (t) by (27), ϕ ˆ n (t) by (28), Φ 3. Form ϕ ˆ t) by (24) and Y (p, t) by (23). 4. Form Φ(p, 5. Compute r(t) by (22) and E(p, t) by (21). ˆ 6. Update θ(t) by (20). ˆ ˆ (t) by (31). by (30) and v 7. Compute xa (t) by (29), w(t) 8. Increase t by 1 and go to step 2.

4

Example

Consider the following 2-input 2-output nonlinear system, −1     1.540 −0.050 0.000 1.000 + 0.128z −1 y1 (t) = 0.220 0.200 y2 (t) 0.000 − 0.116z −1 1.000 − 0.030z −1  −1   2 −1 u1 (t − 1) + 0.500u1(t − 1) 0.000 + 0.033z −1 1.000 + 0.050z + u2 (t − 1) + 0.500u22(t − 1) 0.000 + 0.013z −1 1.000 + 0.016z −1    1.000 − 0.015z −1 0.000 − 0.050z −1 v1 (t) . 0.000 + 0.110z −1 1.000 + 0.043z −1 v2 (t) 

142

J. Chen and X. Wang

The inputs u1 (t) and u2 (t) are taken as two uncorrelated persistent excitation signal sequences with a zero mean and unit variances, v1 (t) and v2 (t) as two white noise sequences with a zero mean and variances σ12 = 0.32 for v1 (t) and σ22 = 0.32 for v2 (t). Applying the AM-GESG algorithm and AM-MI-GESG algorithm to estimate the parameters of this system, the parameter estimation errors δ := ˆ θ(t) − θ/θ versus t are shown in Figure 1 with p = 1, 5 and 10. The auxiliary model is as follows: 

x1a (t) x2a (t)



ˆ T (t)ϕ ˆ s (t), = xa (t) = θ s

ˆ s (t) = [−x1a (t − 1), −x2a (t − 1), u1 (t − 1), u2 (t − 1), u21 (t − 1)), u22 (t − 1)]T , ϕ ˆ T (t) = [A ˆ 1 (t), B ˆ 1 (t), B  θ 1 c2 (t)]. s

0.6

0.5

0.4

δ

AM−GESG (AM−MI−GESG, p = 1) 0.3

0.2

AM−MI−GESG, p = 5

0.1

AM−MI−GESG, p = 10 0

0

500

1000

1500 t

2000

2500

3000

Fig. 1. The parameter estimation errors δ versus t

From Figure 1, we can draw the conclusions: 1) The AM-MI-GESG algorithm with p = 5 and p = 10 have a higher estimation accuracy than the AMGESG algorithm; 2) The parameter estimation errors by the AM-MI-GESG algorithm become smaller and smaller and go to zero with the data length t increasing.

5

Conclusions

The AM-MI-GESG algorithm for MIMO Box-Jenkins systems with nonlinear input is presented by using the multi-innovation identification theory in this paper, the developed algorithm can improve the parameter estimation accuracy and reduce the computational burden. The hierarchical identification methods [65, 70–73] can be extended to this class of nonlinear systems.

Multi-innovation Generalized Extended Stochastic Gradient Algorithm

143

References 1. Ding, F., Chen, T.: Identification of Hammerstein nonlinear ARMAX systems. Automatica 41(9), 1479–1489 (2005) 2. El-Sherief, H., Sinha, N.: Bootstrap estimation of parameters and states of linear multivariable systems. IEEE Transactions on Automatic Control 24(2), 340–343 (2002) 3. Prasad, R.M., Sinha, A.K.: On bootstrap identification using stochastic approximation. IEEE Transactions on Automatic Control 22(4), 671–672 (1977) 4. Gauthier, A., Landau, I.D.: On the recursive identification of multi-input muiltoutput systems. Automatica 14(6), 609–614 (1978) 5. El-Sherief, H., Sinha, N.K.: Choices of models for the identification of linear multivariable discrete-time systems. IEEE Proceedings, Part D, Control Theory and Applications 126(12), 1326–1330 (1979) 6. Sinha, N.K., Kwong, Y.H.: Recursive estimation of the parameters of linears multivariable systems. Automatica 15(4), 471–475 (1979) 7. El-Sherief, H.: Parameters identification of a state-space model of multivariable systems using the extended leastsquares method. IEEE Transactions on Systems, Man and Cybernetics 11(3), 223–227 (1981) 8. Goodwin, G.C., Sin, K.S.: Adaptive Filtering Prediction and Control. PrenticeHall, Englewood Cliffs (1984) 9. Overshee, P.V., N4SID, De Moor, B.: Subspace algorithm for the identification of combined determinestochastic systems. Automatica 30(1), 75–93 (1994) 10. Verhaegen, M.: Identification of the deterministic part of MIMO state space models given in innovations form inputoutput data. Automatica 30(1), 61–71 (1994) 11. Ding, F., Chen, T.: Performance bounds of the forgetting factor least squares algorithm for time-varying systems with finite measurement data. IEEE Transactions on Circuits and Systems–I: Regular Papers 52(3), 555–566 (2005) 12. Ding, F., Shi, Y., Chen, T.: Performance analysis of estimation algorithms of non-stationary ARMA processes. IEEE Transactions on Signal Processing 54(3), 1041–1053 (2006) 13. Ding, F., Shi, Y., Chen, T.: Amendments to Performance analysis of estimation algorithms of non-stationary ARMA processes. IEEE Transactions on Signal Processing Part I 56(10), 4983–4984 (2008) 14. Ding, F., Liu, X.P., Shi, Y.: Convergence analysis of estimation algorithms of dualrate stochastic systems. Applied Mathematics and Computation 176(1), 245–261 (2006) 15. Ding, J., Ding, F.: The residual based extended least squares identification method for dual-rate systems. Computers & Mathematics with Applications 56(6), 1479–1487 (2008) 16. Xiao, Y.S., Ding, F., Zhou, Y., Li, M., Dai, J.Y.: On consistency of recursive least squares identification algorithms for controlled auto-regression models. Applied Mathematical Modelling 32(11), 2207–2215 (2008) 17. Ding, J., Han, L.L., Chen, X.M.: Time series AR modeling with missing observations based on the polynomial transformation. Mathematical and Computer Modelling 51(5-6), 527–536 (2010) 18. Ding, F., Chen, T.: Least squares based self-tuning control of dual-rate systems. International Journal of Adaptive Control and Signal Processing 18(8), 697–714 (2004)

144

J. Chen and X. Wang

19. Ding, F., Chen, T., Iwai, Z.: Adaptive digital control of Hammerstein nonlinear systems with limited output sampling. SIAM Journal on Control and Optimization 45(6), 2257–2276 (2006) 20. Mahmoudi, A., Karimi, M.: Estimation of the parameters of multi- channel autoregressive signals from noisy observations. Signal Processing 88(11), 2777–2783 (2008) 21. Zhao, S.K., Man, Z.H., Khoo, S.Y., Wu, H.R.: Variable step-size LMS algorithm with a quotient form. Signal Processing 89(1), 67–76 (2009) 22. Ding, F., Yang, H.Z., Liu, F.: Performance analysis of stochastic gradient algorithms under weak conditions. Science in China Series F–Information Sciences 51(9), 1269–1280 (2008) 23. Ding, F., Liu, P.X., Yang, H.Z.: Parameter identification and intersample output estimation for dual-rate systems. IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans 38(4), 966–975 (2008) 24. Ding, F., Chen, T.: A gradient based adaptive control algorithm for dual-rate systems. Asian Journal of Control 8(4), 314–323 (2006) 25. Ding, J., Shi, Y., Wang, H.G., Ding, F.: A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems. Digital Signal Processing 20(4), 1238–1249 (2010) 26. Ding, F., Chen, T.: Performance analysis of multi-innovation gradient type identification methods. Automatica 43(1), 1–14 (2007) 27. Ding, F., Liu, P.X., Liu, G.: Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises. Signal Processing 89(10), 1883–1890 (2009) 28. Zhang, J.B., Ding, F., Shi, Y.: Self-tuning control based on multi-innovation stochastic gradient parameter estimation. Systems & Control Letters 58(1), 69– 75 (2009) 29. Han, L.L., Ding, F.: Multi-innovation stochastic gradient algorithms for multiinput multi-output systems. Digital Signal Processing 19(4), 545–554 (2009) 30. Han, L.L., Ding, F.: Identification for multirate multi-input systems using the multi-innovation identification theory. Computers & Mathematics with Applications 57(9), 1438–1449 (2009) 31. Wang, D.Q., Ding, F.: Performance analysis of the auxiliary models based multiinnovation stochastic gradient estimation algorithm for output error systems. Digital Signal Processing 20(3), 750–762 (2010) 32. Ding, F.: Several multi-innovation identification methods. Digital Signal Processing 20(4), 1027–1039 (2010) 33. Xie, L., Yang, H.Z., Ding, F.: Modeling and identification for non-uniformly periodically sampled-data systems. IET Control Theory & Applications 4(5), 784–794 (2010) 34. Liu, Y.J., Yu, L., Ding, F.: Multi-innovation extended stochastic gradient algorithm and its performance analysis, Circuits. Systems and Signal Processing 29(4), 649–667 (2010) 35. Liu, P.X., Liu, G.: Multi-innovation least squares identification for linear and pseudo-linear regression models. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 40(3), 767–778 (2010) 36. Ding, F., Chen, H.B., Li, M.: Multi-innovation least squares identification methods based on the auxiliary model for MISO systems. Applied Mathematics and Computation 187(2), 658–668 (2007)

Multi-innovation Generalized Extended Stochastic Gradient Algorithm

145

37. Liu, Y.J., Xiao, Y.S., Zhao, X.L.: Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model. Applied Mathematics and Computation 215(4), 1477–1483 (2009) 38. Ding, F., Chen, T.: Parameter estimation of dual-rate stochastic systems by using an output error method. IEEE Transactions on Automatic Control 50(9), 1436–1441 (2005) 39. Ding, F., Chen, T.: Identification of dual-rate systems based on finite impulse response models. International Journal of Adaptive Control and Signal Processing 18(7), 589–598 (2004) 40. Han, L.L., Sheng, J., Ding, F., Shi, Y.: Auxiliary model identification method for multirate multi-input systems based on least squares. Mathematical and Computer Modelling 50(7-8), 1100–1106 (2009) 41. Ding, F., Chen, T.: Combined parameter and output estimation of dual-rate systems using an auxiliary model. Automatica 40(10), 1739–1748 (2004) 42. Liu, Y.J., Xie, L., Ding, F.: An auxiliary model based recursive least squares parameter estimation algorithm for non-uniformly sampled multirate systems. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 223(4), 445–454 (2009) 43. Ding, F., Ding, J.: Least squares parameter estimation with irregularly missing data. International Journal of Adaptive Control and Signal Processing 24(7), 540–553 (2010) 44. Ding, F., Shi, Y., Chen, T.: Auxiliary model based least-squares identification methods for Hammerstein output-error systems. Systems Control Letters 56(5), 373–380 (2007) 45. Wang, D.Q., Chu, Y.Y., Ding, F.: Auxiliary model-based RELS and MI-ELS algorithms for Hammerstein OEMA systems. Computers & Mathematics with Applications 59(9), 3092–3098 (2010) 46. Wang, D.Q., Chu, Y.Y., Yang, G.W., Ding, F.: Auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems. Mathematical and Computer Modelling 52(1-2), 309–317 (2010) 47. Ding, F., Chen, T.: Gradient based iterative algorithms for solving a class of matrix equations. IEEE Transactions on Automatic Control 50(8), 1216–1221 (2005) 48. Ding, F., Chen, T.: Iterative least squares solutions of coupled Sylvester matrix equations. Systems & Control Letters 54(2), 95–107 (2005) 49. Ding, F., Chen, T.: On iterative solutions of general coupled matrix equations. SIAM Journal on Control and Optimization 44(6), 2269–2284 (2006) 50. Ding, F., Liu, P.X., Ding, J.: Iterative solutions of the generalized Sylvester matrix equations by using the hierachical identification principle. Applied Mathematics and Computation 197(1), 41–50 (2008) 51. Xie, L., Ding, J., Ding, F.: Gradient based iterative solutions for general linear matrix equations. Computers & Mathematics with Applications 58(7), 1441–1448 (2009) 52. Ding, J., Liu, Y.J., Ding, F.: Iterative solutions to matrix equations of form AiXBi=Fi. Computers & Mathematics with Applications 59(11), 3500–3507 (2010) 53. Ding, F.: Transformations between some special matrices. Computers & Mathematics with Applications 59(8), 2676–2695 (2010) 54. Ding, F., Liu, P.X., Liu, G.: Gradient based and least-squares based iterative identification methods for OE and OEMA systems. Digital Signal Processing 20(3), 664–677 (2010)

146

J. Chen and X. Wang

55. Han, H.Q., Xie, L., Ding, F., Liu, X.G.: Hierarchical least squares based iterative identification for multivariable systems with moving average noises. Mathematical and Computer Modelling 51(9-10), 1213–1220 (2010) 56. Liu, Y.J., Wang, D.Q., Ding, F.: Least-squares based iterative algorithms for identifying Box-Jenkins models with finite measurement data. Digital Signal Processing 20(5), 1458–1467 (2010) 57. Xiao, Y.S., Zhang, Y., Ding, J., Dai, J.Y.: The residual based interactive least squares algorithms and simulation studies. Computers & Mathematics with Applications 58(6), 1190–1197 (2009) 58. Wang, L.Y., Xie, L., Wang, X.F.: The residual based interactive stochastic gradient algorithms for controlled moving average models. Applied Mathematics and Computation 211(2), 442–449 (2009) 59. Shi, Y., Ding, F., Chen, T.: Multirate crosstalk identification in xDSL systems. IEEE Transactions on Communications 54(10), 1878–1886 (2006) 60. Shi, Y., Ding, F., Chen, T.: 2-Norm based recursive design of transmultiplexers with designable filter length. Circuits, Systems and Signal Processing 25(4), 447–462 (2006) 61. Wang, D.Q., Ding, F.: Input-output data filtering based recursive least squares identification for CARARMA systems. Digital Signal Processing 20(4), 991–999 (2010) 62. Liu, Y.J., Sheng, J., Ding, R.F.: Convergence of stochastic gradient algorithm for multivariable ARX-like systems. Computers & Mathematics with Applications 59(8), 2615–2627 (2010) 63. Ding, F., Chen, T., Qiu, L.: Bias compensation based recursive least squares identification algorithm for MISO systems. IEEE Transactions on Circuits and Systems– II: Express Briefs 53(5), 349–353 (2006) 64. Ding, F., Xiao, Y.S.: A finite-data-window least squares algorithm with a forgetting factor for dynamical modeling. Applied Mathematics and Computation 186(1), 184–192 (2007) 65. Wang, L.Y., Ding, F., Liu, P.X.: Consistency of HLS estimation algorithms for MIMO ARX-like systems. Applied Mathematics and Computation 190(2), 1081–1093 (2007) 66. Chen, J., Zhang, Y., Ding, R.F.: Auxiliary model based multi-innovation algorithms for multivariable nonlinear systems. Mathematical and Computer Modelling (2010) (in press) 67. Ding, F., Chen, T.: Author’s reply to Comments on ‘Identification of Hammerstein nonlinear ARMAX systems. Automatica 43(8), 14–97 (2007) 68. Ding, F., Shi, Y., Chen, T.: Gradient based identification algorithms for nonlinear Hammerstein ARMAX models. Nonlinear Dynamics 45(1-2), 31–43 (2006) 69. Wang, D.Q., Ding, F.: Extended stochastic gradient identification algorithms for Hammerstein-Wiener ARMAX Systems. Computers & Mathematics with Applications 56(12), 3157–3164 (2008) 70. Ding, F., Qiu, L., Chen, T.: Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems. Automatica 45(2), 324–332 (2009) 71. Ding, F., Chen, T.: Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica 41(2), 315–325 (2005) 72. Ding, F., Chen, T.: Hierarchical least squares identification methods for multivariable systems. IEEE Transactions on Automatic Control 50(3), 397–402 (2005) 73. Ding, F., Chen, T.: Hierarchical identification of lifted state-space models for general dual-rate systems. IEEE Transactions on Circuits and Systems 52(6), 1179–1187 (2005)

Research of Parallel-Type Double Inverted Pendulum Model Based on Lagrange Equation and LQR Controller Jian Fan1,2, Xihong Wang1, and Minrui Fei1 1

Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, Shanghai, 200072, P.R. China 2 School of Information Science and Technology, JiuJiang University, Jiangxi Province, 332005, P.R. China [email protected], [email protected], [email protected]

Abstract: Car inverted pendulum system is often used as a benchmark for verifying the performance and effectiveness of a new control method. However, it’s very unusual to use parallel-type double inverted pendulum. Based on Lagrange equation, this paper presents a novel state space model for the parallel-type double inverted pendulum system. In contrast to the modeling of traditional car inverted pendulum, the technical challenge in this system is that the analysis of the spring increases research complexity. Furthermore, this paper designs a LQR controller with the Lagrange modeling method. Simulation results are presented to demonstrate the feasibility and performance of the proposed model and controller. Keywords: Lagrange equation; Parallel-type double inverted pendulum; Linear quadratic regulator (LQR).

1 Introduction The control problem of car inverted pendulum is considered as the well known control puzzle in the control theory [1]. As a typical nonlinear, multi-variable, essentially unstable system, car inverted pendulum system is often used as a benchmark for verifying the performance and effectiveness of a new control method. There are mainly two methods in establishing mathematical model, namely Lagrange equation and Newtonian mechanics. Lagrange equation is the motion equation established from the view of energy. It does not need to analyze the force, force moment and accelerate speed which act on every mass point and rigid body. With the general Newton’s laws of motion, we need to resolve a large number of differential equations. Above all, the modeling process will be largely simplified with the Lagrange equation [2]. So far, quantities of research have been made about the modeling of car inverted pendulum system with Lagrange equation at home and abroad. The mathematical model of single car inverted pendulum had been established by using the Lagrange equation [3]. Paper [4] gave out the process of modeling double inverted pendulum with Lagrange equation in detail. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 147–156, 2010. © Springer-Verlag Berlin Heidelberg 2010

148

J. Fan, X. Wang, and M. Fei

Paper [5] presented the modeling of triple inverted pendulum employing Lagrange equation. In paper [6], through experiments, Chen proved that Lagrange method simplified the model establishment of four inverted pendulum greatly. Article [2] analyzed the movement of single, double and triple car inverted pendulum model with Lagrange method, and valuably gave out the general model of car N level inverted pendulum. Parallel-type double inverted pendulum was first put forward by Zhao [7], and he build the system model with Newtonian mechanics method [8]. Unlike the double inverted pendulum proposed by Yi [9], the lengths of the two pendulums are the same in our system. Also, unlike the parallel dual inverted pendulums presented by Zhu [10], in which there is no spring connecting the two pendulums. It is worth noting that the tension of the spring will have resulting torques on the pendulums.

2 Modeling of Parallel-Type Double Inverted Pendulum Without considering air resistance and friction, the physical parameters of paralleltype double inverted pendulum system are shown in Fig.1.

Fig. 1. Structure and physical parameters of parallel-type double inverted pendulum

Where θ is positive in the clockwise direction. Cart position x is positive towards the right direction. Point C and D in the figure are the projections of point A and B ─the two end points of the spring. Oi is the chain joint point of pendulum i. The length of AO1 is equal to that of BO2. The meaning of the parameters is described in table 1. Table 1. Parameters of the parallel-type double inverted pendulum Symbol x1 , x2

Physical meaning displacement of car 1 and car 2

Value m

θ1 , θ2

angle included between pendulum and vertical direction

rad

M1 , M 2

mass of car 1 and car 2

1.096Kg

Research of Parallel-Type Double Inverted Pendulum Model

149

Table 1. (continued)

m1 , m2

mass of pendulum 1 and pendulum 2

0.109Kg

l1 , l2

0.25m

J1 , J 2

length of pendulum 1 and pendulum 2 from center of mass to pivot moment of inertia for pendulum 1 and pendulum 2

0.009083Kg·m2

L0 , L

original length of spring; length of draught spring

L0=0.181m

Δw

d h ks ,g

distance between the two projection points C and D distance between two slide rails length from spring end point A and B to pivot O1 and O2 elastic coefficient of spring; acceleration of gravity

m 0.187m 0.4m 24.5N/m; 9.8m/s2

Sh

component force of spring along the direction of slide rail

N

N1 , N1

component force of car acting on pendulum along the horizontal direction component force of car acting on pendulum along the vertical direction

P1 , P2

N N

Now we will create the model of parallel-type double inverted pendulum with inputs being the acceleration of car 1 and car 2. The basic form of Lagrange Equation is as follows: ⎛ ⎞ d ⎜ ∂L ⎟ ∂L − = fi dt ⎜ ∂ q. ⎟ ∂qi ⎝ i⎠ L( qi , qٛi ) = T ( qi , qٛi ) − V ( qi , qٛi )

(1)

i = 1,2,3,...n

(2)

Where L is the Lagrangian operator, qi is the generalized coordinate of the system under research; f i represents the generalized force. T is kinetic energy of the system, and V is potential energy. In the parallel-type double inverted pendulum system, the generalized coordinates are the displacement of car 1 and car 2, rotor mechanical angle of pendulum 1 and pendulum 2, marked as x1 , x2 ,θ1 , θ 2 respectively. System energy is as follows [2, 11]:

(1) Energy of No.1 car 1 2 =0.

Kinetic energy: TM = M 1 xٛ12 . 1

Potential energy: VM

1

(2) Energy of No.2 car 1 2 = 0.

Kinetic energy: TM = M 2 xٛ22 . 2

Potential energy: VM

2

(3) Energy of No.1 pendulum

150

J. Fan, X. Wang, and M. Fei 1 ⎧⎪ d

2

2

⎫⎪

Kinetic energy: Tm = J 1θ12 + ⎨ ⎡⎢ ( x1 − l1 sin θ1 ) ⎤⎥ + ⎡⎢ ( l1 cosθ1 )⎤⎥ ⎬ . 2 2 ⎩⎪ ⎣ dt ⎦ ⎣ dt ⎦ ⎭⎪ 1

ٛ

d

1

4 3

Where J1 = m1l12 is the moment of inertia for pendulum 1. Potential energy: Vm = m1 gl1 cosθ1 . 1

(4) Energy of No.2 pendulum 1 ⎪⎧ d

2

2

⎪⎫

Kinetic energy: Tm = J 2θ 22 + ⎨ ⎡⎢ ( x2 − l2 sin θ 2 )⎤⎥ + ⎡⎢ ( l2 cos θ 2 )⎤⎥ ⎬ . 2 2 ⎣ dt ⎦ ⎣ dt ⎦ 1

ٛ

d

⎩⎪

2

⎭⎪

4 3

Where J 2 = m2l2 2 is the moment of inertia for pendulum 2. Potential energy: Vm = m2 gl2 cosθ 2 . 2

(5) Elastic potential energy of spring Vspring =

1 2 k s ( L − L0 ) . 2

According to the equations above, the kinetic energy and potential energy of the parallel-type double inverted pendulum system can be obtained as follows: T = TM1 + TM 2 + Tm1 + Tm2

(3)

V = VM1 + VM 2 + Vm1 + Vm2 + Vspring

(4)

There are no external forces along the generalized ordinates θ1 and θ 2 , so the following equations can be obtained: when qi = θ1 ,

d ⎛ ∂L ⎞ ∂L =0 ⎜ ٛ⎟ − dt ⎝ ∂θ1 ⎠ ∂θ1

(5)

when qi = θ 2 ,

d ⎛ ∂L ⎞ ∂L =0 ⎜ ٛ⎟− dt ⎝ ∂θ 2 ⎠ ∂θ 2

(6)

After expansion and deduction of Eqs.(5), (6), Eqs.(7), (8) are followed: ٛ 7 1 ∂ ⎡ 2 m1l12θ1 − m1l1 xٛ1 cosθ1 − m1 gl1 sin θ1 + k s ( L − L0 ) ⎤⎦ = 0 3 2 ∂θ1 ⎣

(7)

ٛ 7 1 ∂ ⎡ 2 m2 l2 2θ 2 − m2 l2 xٛ2 cos θ 2 − m2 gl2 sin θ 2 + k s ( L − L0 ) ⎤⎦ = 0 ⎣ 3 2 ∂θ 2

(8)

Where, L = d 2 + Δw2 , Δw = ( x2 − h sin θ 2 ) − ( x1 − h sin θ1 ) is the distance between the projective points C and D along the direction of the rail shown in Fig.1, then L = d 2 + [( x2 − h sin θ 2 ) − ( x1 − h sin θ1 )]2 .

Research of Parallel-Type Double Inverted Pendulum Model

∂ ⎡ ∂L 2 ( L − L0 ) ⎤⎦ = 2( L − L0 ) ∂θ1 ⎣ ∂θ1

(

)

= 2( L − L0 )

∂ ∂θ1

= 2( L − L0 )

1 ∂ ( d 2 + [( x2 − h sin θ2 ) − ( x1 − h sin θ1 )]2 ) 2 L ∂θ1

d 2 + [( x2 − h sin θ 2 ) − ( x1 − h sin θ1 )]2

151

(9)

⎛ ⎞ L0 = 2 ⎜1 − ⎟ ( x2 − x1 − h sin θ 2 + h sin θ1 )h cosθ1 2 2 d + Δw ⎠ ⎝ ⎛ L ⎞ ≈ 2 ⎜1 − 0 ⎟ ( x2 − x1 − h sin θ 2 + h sin θ1 )h cos θ1 d ⎠ ⎝

Similarly, ∂ ⎡ L 2 ( L − L0 ) ⎤⎦ ≈ 2 ⎛⎜1 − 0 ⎞⎟ ( x2 − x1 − h sin θ 2 + h sin θ1 )(−h cosθ2 ) ∂θ 2 ⎣ d ⎠ ⎝

(10)

What we concern is the dynamic changes of the two pendulums in the vicinity of the vertical direction, therefore, linearization can be implemented in the neighborhood of the balance point where the two pendulums are in the vertical upward direction without falling down. Assume θ1 ≈ 0 , θ 2 ≈ 0 , after approximation, we have cosθ1 = 1 , cos θ 2 = 1 , sin θ1 = θ1 , sin θ 2 = θ 2 , (

dθ1 2 dθ ) = 0 , ( 2 ) 2 = 0 . Incorporate Eqs.(9), dt dt

(10) into Eqs.(7), (8) respectively, the following equations can be satisfied: ٛ 7 m1l12θ1 − m1l1 xٛ1 − ( m1 gl1 − β h )θ1 − β x1 + β x2 − β hθ 2 = 0 3

(11)

ٛ 7 m2 l2 2θ 2 − m2 l2 xٛ2 − (m2 gl2 − β h )θ 2 + β x1 − β x2 − β hθ1 = 0 3

(12)

Where β = k s h ⎛⎜1 − ⎝

L0 ⎞ ⎟. d ⎠

After further transformation, Eqs. (11), (12) can be summarized as follows: ٛ

J 1θ1 = λ [ β x1 + (m1 gl1 − β h )θ1 − β x2 + β hθ 2 + m1l1 xٛ1 ] ٛ

J 2θ 2 = λ [ − β x1 + β hθ1 + β x2 + (m2 gl2 − β h )θ 2 + m2 l2 xٛ2 ]

Where λ =

(13) (14)

4 . 7

The state-space model of the parallel-type double inverted pendulum system is given as: ٛ X = AX + BU . Y = CX + DU .

152

J. Fan, X. Wang, and M. Fei ٛ

ٛ

Where X = ⎡⎣ x1 θ1 xٛ1 θ1 x2 θ 2 xٛ2 θ 2 ⎤⎦ , U = [ xٛ1 xٛ2 ] is the acceleration vector of No.1 car T

T

and No.2 car. More specifically, according to Eq.(13) and Eq.(14), the mathematical model in state-space form is derived as follows: ⎡ 0 ⎡ xٛ1 ⎤ ⎢ 0 ⎢ θٛ ⎥ ⎢ ⎢ 1⎥ ⎢ 0 ⎢ xٛ1 ⎥ ⎢ λβ ⎢ ٛ⎥ ⎢ ⎢θ1 ⎥ ⎢ J 1 ⎢ xٛ2 ⎥ = ⎢ 0 ⎢ ٛ⎥ ⎢ ⎢θ 2 ⎥ ⎢ 0 ⎢ xٛ ⎥ ⎢ 0 ⎢ ٛ2 ⎥ ⎢ ⎢⎣θ 2 ⎥⎦ ⎢ −λβ ⎢ ⎣⎢ J 2

0

1 0

0

0

0

0 1

0

0

0

0 0

0

0

0 0

−λβ J1

λβ h

0

0 0

0

0

0

0 0

0

0

0

0 0

λ (m1 gl1 − β h ) J1

λβ h

0 0

J2

J1

0

0

λβ

λ ( m2 gl2 − β h )

J2

J2

0 0⎤ ⎡ 0 0 0⎥ ⎡ x1 ⎤ ⎢ 0 ⎥⎢ ⎥ ⎢ 0 0⎥ ⎢θ1 ⎥ ⎢ 1 ⎥ ⎢ xٛ ⎥ ⎢ λl m 1 0 0⎥ ⎢ ٛ ⎥ ⎢ 1 1 ⎥ ⎢θ1 ⎥ ⎢ J 1 +⎢ ⎥ 1 0 ⎥ ⎢ x2 ⎥ ⎢ 0 ⎢ ⎥ 0 1⎥ ⎢θ 2 ⎥ ⎢ 0 ⎥ ⎢ ٛ⎥ ⎢ 0 0 ⎥ x2 ⎢ 0 ⎢ ⎥ ⎥ ⎢θٛ ⎥ ⎢ 0 0⎥ ⎣ 2 ⎦ ⎢ 0 ⎦⎥ ⎣⎢

⎤ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ٛ ⎥ ⎡ x1 ⎤ ⎥ 0 ⎥ ⎢⎣ xٛ2 ⎥⎦ 0 ⎥ ⎥ 1 ⎥ λ m2l2 ⎥ ⎥ J 2 ⎦⎥ 0 0

Substitute the value of the parameters mentioned in Table 1 into matrices A and B, the matrices as follows can be obtained: 0 ⎤ ⎡ 0 0 1 0 0 0 0 0⎤ ⎡ 0 ⎢ 0 ⎢ 0 ⎥ 0 ⎥⎥ 0 0 1 0 0 0 0⎥ ⎢ ⎢ ⎢ 1 0 ⎥ ⎢ 0 0 0 0 0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ 1.7143 0 19.782 8.8879 0 0 −19.782 7.9128 0 0 ⎥ , ⎥ B=⎢ A=⎢ ⎢ 0 ⎢ 0 0 ⎥ 0 0 0 0 0 1 0⎥ ⎢ ⎥ ⎢ ⎥ 0 ⎥ 0 0 0 0 0 0 1⎥ ⎢ 0 ⎢ 0 ⎢ 0 ⎢ 0 1 ⎥ 0 0 0 0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0 ⎢⎣ −19.783 7.9128 0 0 19.782 8.8879 0 0 ⎥⎦ 1.7143⎥⎦

3 LQR Controller for the Inverted Pendulum System For the above system, we will apply a linear quadratic regulator to stabilize it. The aim of LQR is to find an optimal control law which minimizes the following cost function: J=

1 ∞ T [ x (t )Qx(t ) +u T (t ) Ru (t )]dt 2 ∫t0

(15)

Where Q ≥ 0 and R > 0 are weight matrices for state variable x(t ) and controlled variable u (t ) respectively. Q and R are arbitrary; the main task of designing LQR controller is to adjust these two matrices, especially Q [12]. According to the principle of minimum value, the optimal control law can be obtained as follows: u ∗ (t ) = − R −1 BT Px(t ) = − kx(t )

(16)

Where k = R −1BT P is the optimal feedback gain matrix; P > 0 is a constant matrix and satisfies Riccati equation:

Research of Parallel-Type Double Inverted Pendulum Model

PA + AT P − PBR −1BT P + Q = 0

153

(17)

According to Eq.(16) and Eq.(17), it follows that the design of u ∗ (t ) comes to the problem of solving Riccati algebraic equation. Without loss of generality, R is set to I . The set of Q is crucial, since it is of great influence on the stability of the system. Assume Q = diag (Q11 , Q22 , Q33 , Q44 , Q55 , Q66 , Q77 , Q88 ) , then as for the parallel-type double inverted pendulum system, the cost function Eq.(15) becomes: J=

1 ∞ (Q11 x12 + Q22θ12 + Q33 x12 + Q44θ12 + Q55 x22 + Q66θ 22 + Q77 x22 + Q88θ22 + u12 + u22 )dt 2 ∫t0

(18)

It follows that Qii is the weight for the square of state variable. To simplify, we only study the weight coefficients of deviating arguments, namely Q11 , Q 22 , Q55 , Q66 , and that of velocity arguments are set to 0. After a series of system simulation, the value of Qii below is desirable: Q = diag (900,230,0,0,900, 230,0,0). ⎡ 15.1984 55.6099 −10.7308 14.8908 −45.1984 38.9376 −12.0474 7.8772 ⎤ K=⎢ ⎥. ⎣−45.1984 38.9376 −12.0474 7.8772 15.1984 55.6099 −10.7308 14.8908⎦

4 Simulation Results and Analysis The simulation parameters are set as follows: z z z z z

Initial state value X10=[0.06 0.1 0 0 -0.04 -0.15 0 0], X20=[0.04 -0.1 0 0 0.04 0.15 0 0]. The sampling period T is 5s. The sampling interval t is 0.005s. Weight matrix of state variable is Q = diag (900, 230,0,0,900,230,0,0) Outputs are placement of car1 and car 2, angle of pendulum1 and pendulum 2.

What displayed in Fig.2 is the step response of the system without LQR controller. Obviously, the output responses are divergent, thus the system is unstable. In Fig.3, the outputs are convergent to steady-state values, namely the system is successfully stabilized by using LQR controller. The comparison of Fig.2 and Fig.3 verifies the effectiveness and feasibility of LQR controller in parallel-type double inverted pendulum model based on Lagrange equation. The simulation results with different initial state values X10 and X20 are shown in Fig.4 and Fig.5. The system can reach stable state quickly, in the third second approximately. A fact that the system can reach stable state under different initial values is proved. Then, experiment is executed in the case of disturbance. In the 5th second, add a disturbance to the system, assuming the initial state X0=[0.06 0.1 0 0 -0.04 -0.15 0 0]. The experimental result is shown in Fig.6. The parallel-type double inverted pendulum system controlled by LQR has the capacity of resisting disturbance.

154

J. Fan, X. Wang, and M. Fei

The actual parallel-type double inverted pendulum in experiment is shown in Fig.7. The two pendulums keep upright, and the two cars come back to the center of the slide rail, our aim of controlling the system is achieved.

Fig. 2. Step response of the pendulums and cars without LQR controller

Fig. 3. Step response of the pendulums and cars with LQR controller

Fig. 4. Simulation curve with initial state X10

Research of Parallel-Type Double Inverted Pendulum Model

155

Fig. 5. Simulation curve with initial state X20

Fig. 6. Simulation curve with a disturbance

Fig. 7. Actual parallel-type double inverted pendulum system

5 Conclusions As a new experimental object, it’s very unusual to use parallel-type double inverted pendulum. In this paper, the model of parallel-type double inverted pendulum is established on the basis of Lagrange equation. An important feature of Lagrange

156

J. Fan, X. Wang, and M. Fei

equation is that it is very simple for complex car inverted pendulum model. Furthermore, we apply LQR controller to the system to demonstrate the feasibility of the proposed modeling method. The Simulation results are presented to demonstrate the feasibility and performance of the proposed model and controller. It’s believed that employing the parallel-type double inverted pendulum to verify the performance and effectiveness of many other algorithms seems to be an obvious direction for future research. Acknowledgements. This work is supported by Shanghai University, “11th FiveYear Plan” 211 Construction Project, Key Project of Science and Technology Commission of Shanghai Municipality under Grant 08160512100 and 08160705900, Innovation Fund of Shanghai University.

References 1. Li, Z.S., Tu, Y.Q.: Human Simulate Intelligent Control. National Defence Industrial Publishing House, Beijing (January 2003) 2. Zhang, L.J., Tu, Y.Q.: Research of Car Inverted Pendulum Model Based on Lagrange Equation. In: Intelligent Control and Automation. The Sixth World Congress, pp. 820– 824. IEEE Press, Dalian (2006) 3. Xiao, L.L., Peng, H.: Up Swing and Stabilization Control of a Single Inverted Pendulum System Based on the Lagrangian Modeling. J. Technique of Automation & Applications 26(4), 4–7 (2007) 4. Guo, Y.: Study on the Double Inverted Pendulum Controlled by Intelligent Predictive Control, pp. 31–34. D. Hebei University of Technology (2007) 5. Gao, J.W., Cai, G.Q., Ji, Z.J.: Adaptive Neural-Fuzzy Control of Triple Inverted Pendulum. J. Control Theory & Applications 27(2), 278–282 (2010) 6. Chen, H.L., Liang, H.B., Huo, Y.H.: Modeling Four-Stage Inverted Pendulum System Based on Lagrange Equation. J. Journal of Guangdong University of Technology 21(2), 59–63 (2004) 7. Fei, M.R., Wei, L.S., Zhao, W.Q.: Networking Parallel-Type Double Inverted Pendulum. P. CN101477764, 2009-07-08 8. Zhao, W.Q., Wei, L.S., Fei, M.R.: Modeling and Simulation of Parallel-type Double Inverted Pendulum System Based on LQR. J. Process Automation Instrumentation 30(4), 1–4 (2009) 9. Yi, J.Q., YUBAZAKI, N.Y.S., HIROTA, K.R.: Stabilization Fuzzy Control of ParallelType Double Inverted Pendulum System. In: The Ninth IEEE International Conference, pp. 817–822. IEEE Press, San Antonio (2000) 10. Zhu, D.F., Zhou, D.: Synchronization Control of Parallel Dual Inverted Pendulums. In: Automation and Logistics, pp. 1486–1490. IEEE Press, Los Alamitos (2008) 11. Lin, Y.S., Qian, J.X., Xue, A.K., Wang, J.H.: Simple Multi-pd Control Algorithm of Double Inverted Pendulum. In: The 10th IEEE Region Conference on Computers, Communications, Control and Power Engineering, pp. 1428–1431. IEEE Press, Los Alamitos (2002) 12. Starin, S.R., Yedavalli, R.K., Sparks, A.G.: Design of a LQR Controller of Reduced Inputs for Multiple Spacecraft Formation Flying. In: American Control Conference, pp. 1327– 1332. IEEE Press, Los Alamitos (2001)

A Consensus Protocol for Multi-agent Systems with Double Integrator Model Fang Wang, Lixin Gao, and Yanping Luo Institute of Operations Research and Control Sciences, Wenzhou University, Zhejiang 325035, China [email protected]

Abstract. In this paper, we present a consensus protocol for continuoustime multi-agent systems with fixed and switching topologies. The agent dynamics is expressed in the form of a double integrator model. The consensus protocol is provided based on the information of each agent’s neighbors. By using the graph theory and the Lyapunov function, a sufficient condition is obtained to guarantee that each agent can follow the leader if the leader moves at an unknown constant acceleration. Finally, a numerical simulation is given to show the effectiveness of our theoretical results.

1

Introduction

In recent years, the coordination problem of multiple autonomous agents has attracted many researchers because of a broad range of disciplines including biology, physics, computer science and control engineering including cooperative control of unmanned air vehicles, flocking of birds, schooling for underwater vehicles, distributed sensor networks, attitude alignment for cluster of satellites, collective motion of particles, and distributed computations [1–5]. A critical problem for coordinated control is to design appropriate protocols and algorithms such that the group of agents can reach consensus on the shared information in the presence of limited and unreliable information exchange and dynamically changing interaction topologies [6]. The neighbor-based rules feedback scheme is inspired originally by the aggregates of individuals in nature. Flocks of birds and schools of fish achieve coordinated motions of large numbers of individuals without a central controlling mechanism [1]. [2] proposed typical neighbor-based consensus control strategies which is a simplified Vicsek model ([4]), and it was shown that the multi-agent system could reach consensus under the jointly-connected switching interaction topologies. [6] extended the work of Jadbabaie et al. [2] to the case of directed graphs. Similar or generalized consensus problems have also been studied in [7–9]. In many applications, especially in sensor networks and robot networks, some unmeasured variables which may lead the system to achieve a prescribed group behavior can not be obtained by agent directly. In [10], it assumed that the velocity of an active leader cannot be measured and each follower agent with first K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 157–165, 2010. c Springer-Verlag Berlin Heidelberg 2010 

158

F. Wang, L. Gao, and Y. Luo

order dynamics used an neighbor-based estimation rule to estimate the active leader’s unmeasurable velocity. [11] assumed the velocity information of all following agents also can be obtained directly but the velocity leader cannot be measured by each following agent. In a recent paper, [12] proposed distributed observers for the second-order follower-agents together with the neighbor-based control rules under the assumption that the velocity of the active leader cannot be measured. Unfortunately, the main result given in [12] is not right, because it misses a coefficient in its model transformation. The similar model transformation in this paper given in (7). [12] misses the coefficient k1 of term v¯ˆ, which makes the proof of the main result not right. In this paper, we use the similar neighbor-based estimation rules and the neighbor-based control rules proposed by [12] directly, and analyze goal consensus stability of multi-agent system by constructing a new parameter-dependent Lyapunov function under fixed and switching interconnection topological cases. The rest paper is organized as follows. In section 2, we provide some results in graph theory and introduce the consensus protocol. Then in section 3, the results on the consensus stability are obtained for multi-agent system with varying interconnection topologies. In section 4, simulation examples are presented. Notation: R is the real number set. I is an identity matrix with compatible dimension. AT is denoted as transpose of a matrix A. For symmetric matrices A and B, A > (≥ B) means A − B is positive (semi-) definite. λ(A) represents an eigenvalue of A. For symmetric matrices A, λmin (A) and λmax (A) represent the minimum and maximum eigenvalue of A respectively. diag{. . .} represents a block-diagonal matrix. ⊗ is the denotes the Kronecker product, which satisfies (1) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD); (2) If A ≥ 0 and B ≥ 0, then A ⊗ B ≥ 0.

2 2.1

Problem Formulation Graph Theory

Stability analysis of the group of agents is based on several results of algebraic graph theory. In this section, we first introduce some basic concepts and notations in graph theory that will be used throughout this paper (see [10]). Denote the graph by G = {V, ε}, where V = {vi , i ∈ I} is the set of vertices (with I an index set) and ε ⊂ V × V is the set of edges of the graph. Throughout this paper, we assume all the graphs have no edges {vi , vi } from a node to itself, and each edge is denoted by ek = (vi , vj ) ∈ ε, i, j ∈ I. A graph is said to be undirected if ∀(vi , vj ) ∈ ε =⇒ (vj , vi ) ∈ ε. Otherwise, the graph is called a directed graph. If (vi , vj ) ∈ ε, then vj said to be a neighbor of vi and we denote the set of all neighbor vertices of vertex vi denoting Ni = {j|(vi , vj ) ∈ ε}. A path from vertex vi to vertex vj is a sequence of distinct vertices starting vi and ending with vj such that consecutive pair of vertices makes an edge of graph. A graph is simple if it has no self-loops or repeated edges. If there is a path between any two vertices of a graph G, then G is connected, otherwise disconnected. A = [aij ] ∈ Rn×n is its weighted adjacency matrix, where aij = 0

A Consensus Protocol for Multi-agent Systems

159

and aij = aji ≥ 0, with aij > 0 meaning that there is an edge between agent i n×n and agent j . Its degree matrix D is a diagonal matrix, = {d1 , d2 , ..., dn } ∈ R where diagonal elements di = aij for i = 1, ..., n. Then the Laplacian of G j∈Ni

is defined as L = D − A, which is symmetric. Here, we consider a leader-following problem for a multi-agent system in which there are n follower-agents (labeled by vi , i = 1, 2, · · · , n) and a leader (labeled by v0 ). A simple and undirected graph G describes the topology relation of these n follower-agents, and G¯ contains G and v0 with directed edges from some agents to the leader v0 . The graph G is allowed to have several components, within every such component all the agents are connected via undirected edges. The graph G¯ is said to be connected if at least one agent in each component of G is connected to the leader by a directed edge. Moreover, the information topology G¯ is time-varying. Denote S¯ = {G¯1 , G¯2 , · · · , G¯N } as a set of the graphs of all possible topologies and denote P = {1, 2, · · · , N } as its index set. To describe the variable interconnection topology, we define a switching signal σ : [0, ∞) → P, which is piecewise constant. Let t1 = 0, t2 , t3 , · · · be an infinite time sequence at which the interconnection graph of the considered multi-agent system switches. Therefore, Ni and the connection weight aij (i, j = 1, · · · , n) are time-varying, and moreover, Laplacian Lσ(t) (σ(t) ∈ P) associated with the switching interconnection graph is also time-varying, though it is a time-invariant matrix in any interval [ti , ti+1 ). 2.2

Consensus Protocol

In this section, we consider the graph G¯ associated with the system consisting of n agents and one leader, where the leader indexed by 0 and the other agents indexed by 1, ..., n. We intend to coordinate n mobile agents with each agent expressed in the form of a double integrator:  x˙ i = vi , xi , vi , ui ∈ Rm , i = 1, ..., n (1) v˙ i = ui where xi is the position of agent i, vi its velocity and ui its control input. We assume that the leader is active. Its velocity is hard to measure in real time. The state of the leader agent keeps changing without influence by other agents. Its dynamical behavior can be described as follows:  x˙ 0 = v0 (2) , x0 , v0 , u0 ∈ Rm v˙ 0 = u0 (t) where u0 (t) is control input, which may be regarded as some given policy known to all the agents. Particularly, u0 (t) = 0 expresses that the leader move at constant velocity. We say that local control law can solve the consensus problem if and only if lim (xi (t) − x0 (t)) = 0 and lim (vi (t) − v0 (t)) = 0 for any given initial values t→∞

t→∞

xi (0) and vi (0) (i = 1, · · · , n).

160

F. Wang, L. Gao, and Y. Luo

Since v0 (t) of the leader can not be measured directly, we should use the information obtained from its neighbors to estimate v0 (t). We denote vˆi (t) as an estimate of v0 (t) by ith (i = 1, ..., n) agent. Thus, a neighbor-based rule which consists of two parts can be expressed as follows: A neighbor-based control rule :  aij (t)(xi − xj ) + bi (t)(xi − x0 )]. (3) ui = u0 − k1 (vi − vˆi ) − k2 [ j∈Ni (t)

A neighbor-based dynamics system to estimate v0 :  vˆ˙ i = u0 − k3 [ aij (t)(xi − xj ) + bi (t)(xi − x0 )].

(4)

j∈Ni (t)

where k1 , k2 and k3 are positive constants and regarded as “control” parameters. At time t , the aij (t) and bi (t) are chosen by  αij if agent i is connected to agent j (5) aij (t) = 0 otherwise  bi (t) =

βi if agent i is connected to the leader 0 otherwise

(6)

where αij > 0(i, j = 1, ..., n) is connection weight constant between agent i and agent j, and βi > 0(i = 1, ..., n) is connection weight constant between agent i and leader. We also assume that αij = αji . For the convenience, define x¯ = (x1 , x2 , ..., xn )T − x0 1, v¯ = (v1 , v2 , ..., vn )T − ˆ = (ˆ v1 , vˆ2 , ..., vˆn )T − v0 1, where 1 = (1, 1, ..., 1)T . We can obtained v0 1 and v¯ the error dynamic of the closed-loop system (1-4) in this form of the matrix as follows ⎧ ⎨ x¯˙ = v¯ x − k1 v¯ + k1 v¯ˆ v¯˙ = −[k2 (Lσ + Bσ ) ⊗ Im ]¯ (7) ⎩ ¯˙ vˆ = −[k3 (Lσ + Bσ ) ⊗ Im ]¯ x where the switching signal σ : [0, ∞) → P = {1, ..., N } is a piecewise constant. Lσ is the Laplacian for the n agents, and Bσ is a n × n diagonal matrix whose ith diagonal elements is bi (t) at time t. For simplicity, let Hσ = Lσ + Bσ . The error dynamics of (7) can be expressed as follows: (8) ε(t) ˙ = (Fσ ⊗ Im )ε(t) ⎡ ⎤ ⎡ ⎤ x ¯ 0 I 0 where ε(t) = ⎣ v¯ ⎦ , Fσ = ⎣ −k2 Hσ −k1 I k1 I ⎦ v¯ ˆ −k3 Hσ 0 0 Here are some well known results, which will be used later.

A Consensus Protocol for Multi-agent Systems

161

Lemma 1. [14] ( Routh-Hurwitz theorem ) Assume that a polynomial of degree n has the form a0 xn + a1 xn−1 + ... + an−1 x + an = 0. Here a0 > 0. The determinants a1 a0 0 . . . 0 a3 a2 a1 . . . 0 a1 a0 , ..., n =

1 = a1 , 2 = . = an Δn−1 . . .. . . .. .. a3 a2 . . .. a2n−1 a2n−2 a2n−3 . . . an Here if i > n, ai = 0. All the roots of the polynomial have negative real parts if and only if the following inequalities hold true: a1 > 0, 2 > 0, ..., n−1 > 0, an > 0.

 A11 A12 Lemma 2. [13] Consider a symmetric matrix A = , where A11 and AT12 A22 A22 are square. The matrix A is positive definite if and only if A22 and A11 > T A12 A−1 22 A12 is positive definite. Lemma 3. [10] If graph G¯ is connected, then the symmetric matrix Hσ is positive definite.

3

Main Results

In this section, we concentrate on the analysis of the multi-agent systems with a leader. Obviously, the consensus problem of the multi-agent system (7) is converted into the stability problem of the 3n dimension linear system (8). In the sequel, we will demonstrate the convergence of the dynamics system (8). We first discuss the special fixed interconnection topology case. 3.1

Networks with Fixed Topology

In this subsection, we will focus on the convergence analysis of a group of dynamic agents with fixed interconnection topology. Theorem 1. Take positive constants k1 , k2 , k3 satisfying k2 > k3 . If the interconnection topology graph is fixed and connected for any time t, then the system (8) is stable under any given initial condition x(0) and v(0) , which implies that lim (xi (t) − x0 (t)) = 0, lim (vi (t) − v0 (t)) = 0. t→∞

t→∞

Proof : If the interconnection topology is fixed, the subscript σ(t) can be dropped. The system (8) is expressed as ⎡

ε(t) ˙ = (F ⊗ Im )ε(t) ⎤

0 I 0 where F = ⎣ −k2 H −k1 I k1 I ⎦ and H = L + B. 0 0 −k3 H

(9)

162

F. Wang, L. Gao, and Y. Luo

By Lemma 3, the matrix H is positive definite since the fixed graph G¯ remains being connected. Let U be a schur transformation matrix of H, meaning the unitary matrix U such that U T HU = T := diag{λ⎡1 (H), λ2 (H), ..., λ⎤ n (H)}. Take 0 I 0 P = diag{U, U, U }. Therefore, we have P T F P = ⎣ −k2 T −k1 I k1 I ⎦. −k3 T 0 0 By using simple calculations, it is not difficult to get the eigenvalue polynomial  of matrix F that is ni (λ3 +k1 λ2 +k2 λi (H)λ+k1 k3 λi (H)), where λi (H) denotes the ith eigenvalue of matrix H. For all i, we can obtain that all the roots of the equality λ3 + k1 λ2 + k2 λi (H)λ + k1 k3 λi (H) = 0 have negative real parts when k2 > k3 by using the result of Lemma 1. Thus, F is stable matrix when k2 > k3 , which means F ⊗ Im is also a stable matrix. Therefore, for any k2 > k3 , if the interconnection topology graph is fixed and connected, then the closed system (9) is stable under any given initial condition x(0) and v(0). The proof is now completed. 3.2

Networks with Switching Topology

As [10] pointed out that Fσ can not be expressed as a stochastic matrix, and therefore, the method reported in [2] cannot be applied here directly. In what follows, we will propose an approach based on the common Lyapunov- function for sys¯ := max{λ(Hp )|G ¯ p is connected} and λ ¯p tem (8). Denote λ := min{λ(Hp )|G p∈P

p∈P

is connected}. Based on Lemma 3, we know that λ(Hp ) > 0. Noticing the fact ¯ are well-defined, which are positive that the set P is finite, we know that λ and λ and depend directly on the constants αij and βi for i, j = 1, 2, ..., n given in (5) and (6). Now we give our main result for switched interconnection topology as follows. Theorem 2. Suppose the switched interconnection graph is connected for any interval [tj , tj+1 ). If the positive constants k1 , k2 and k3 satisfy ⎧ k2 > 11.6k3 ⎪ ⎪ ⎪ √ ⎪ ⎪ k12 ((2k2 − 19k3 ) − 0 ) ⎨ λ> 4(k2 − 5k3 )2 ⎪ √ ⎪ 2 ⎪ ⎪ k1 ((2k2 − 19k3 ) + 0 ) ⎪ ¯ ⎩ λ< 4(k2 − 5k3 )2

(10)

where 0 = 3.6k22 − 72k2 k3 + 351k32, then the system (8) is stable under any given initial condition x(0) and v(0), that is, lim (xi (t)− x0 (t)) = 0, lim (vi (t)− t→∞

t→∞

v0 (t)) = 0 Proof: To prove the theorem, we consider the dynamics in each interval at first. In any interval [tj , tj+1 ), the matrices Lσ , Bσ and Hσ associated with G are time-invariant matrices. Thus, we can assume that σ(t) = p, p ∈ P, t ∈ [tj , tj+1 ).

A Consensus Protocol for Multi-agent Systems

Construct a Lyapunov-function V (ε) = εT (P ⊗ Im )ε, with ⎡ 1 2 1 ⎤ 10 k1 I 10 k1 I −k1 I 1 1 k1 I −I ⎦ P = ⎣ 10 5I −k1 I −I kk23 I

163

(11)

which is positive definite if k2 > 11.6k3 . Then, the derivative of V (ε) is given by V˙ (ε) = εT [(FpT P + P Fp ) ⊗ Im ]ε = −εT (Qp ⊗ Im )ε, where ⎤ 1 2 k1 I ( k52 − 2k3 )k1 Hp ( k52 − k3 )Hp − 10 1 Qp = ⎣ k I − 15 k1 I ⎦ ( k52 − k3 )Hp 5 1 1 2 1 − 10 k1 I − 5 k1 I 2k1 I ⎡

(12)

 − 15 k1 I Noticing that > 0, we have Qp is positive definite if and only 2k1 I if the following matrix Kp is positive definite 1 5 k1 I − 15 k1 I

  k2 1 2 − 2k3 )k1 Hp − ( k52 − k3 )Hp − 10 k1 I 5

1 −1 k  k1 I − 51 k1 I ( 52 − k3 )Hp 5 × 1 2 − 15 k1 I 2k1 I − 10 k1 I

Kp : = (

=−

50 k2 1 k3 ( − k3 )2 Hp2 + (2k2 − 19k3 )k1 Hp − 1 I 9k1 5 9 180

Due to k2 > 11.6k3 , we have Δ = ( 19 (2k2 − 19k3 )k1 )2 − Furthermore, we can obtain −

4k13 50 k2 180 ( 9k1 ( 5

− k3 )2 ) > 0.

50 k2 1 k3 ( − k3 )2 (λi (Hp ))2 + (2k2 − 19k3 )k1 λi (Hp ) − 1 > 0. 9k1 5 9 180

¯ satisfy (10). Then, one can see Kp is positive definite. Thus Qp is also If λ and λ a positive definite matrix by Lemma 2. It’s well-known that λmin (Qp ⊗ Im ) λmin (Qp ) εT Qp ε ≥ = . εT P ε λmax (P ⊗ Im ) λmax (P ) Let β :=

min λmin (Qp )

p∈P

. We also know that β is fixed and β > 0 because of the finite of P. Therefore, we have V˙ (ε) ≤ −2βV (ε) which implies V (ε(t)) ≤ V (ε(ti ))e−2β(t−ti ) , ∀ t ∈ [tj , tj+1 ). Consequently, we can get λmax (P )

V (ε(t)) ≤ V (ε(0))e−2β(t)

(13)

Thus we have lim ε(t) = 0 which means that the multi-agent system achieves t→∞ consensus. The proof is now completed. Remark 1. By using similar analysis method of [10], it is not too difficult to get the following result. In each time [ti , ti+1 ) , if the total period over which

164

F. Wang, L. Gao, and Y. Luo

the entire graph is connected is sufficiently by large, then there exists a constant cδ > 0 such that lim (xi (t) − x0 (t)) ≤ cδ , lim (vi (t) − v0 (t)) ≤ cδ for the multit→∞

t→∞

agent system (7). In other words, the tracking error of each agent is bounded. Furthermore, we also can discuss the case that the control input is bounded disturbances and can get similar result.

4

Simulation

In this section, to illustrate our theoretical results derived in the above section, we will provide a numerical simulation example. Without loss of generality, we take m = 1 in numerical simulation. The multi-agent system includes one leader and five followers. The interconnection topology is time-varying of switching period 0.5 among three graphs G¯i (i = 1, 2, 3) described as follows. The laplacian matrices Li (i = 1, 2, 3) for the three subgraphs Gi (i = 1, 2, 3) are ⎡

⎤ ⎡ ⎤ ⎡ ⎤ 2 −1 0 0 −1 3 −1 0 −1 −1 2 0 −1 0 −1 ⎢ −1 2 −1 0 0 ⎥ ⎢ −1 2 0 −1 0 ⎥ ⎢ 0 2 0 −1 −1 ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ L  1 = ⎢ 0 −1 2 −1 0 ⎥ L2 = ⎢ 0 0 1 0 0 ⎥ L3 = ⎢ ⎢ −1 0 2 −1 0 ⎥ ⎣ 0 0 −1 2 −1 ⎦ ⎣ −1 −1 0 3 −1 ⎦ ⎣ 0 −1 −1 2 0 ⎦ −1 0 0 −1 2 −1 0 −1 −1 2 −1 −1 0 0 2 and the diagonal matrices for the interconnection relationship between the leader and the followers are B1 = diag{0, 1, 0, 0, 1}, B2 = diag{0, 1, 1, 0, 0}, B3 = diag{0, 1, 1, 1, 0}. Take k1 = 11, k2 = 20, k3 = 1. All the conditions of theorem 2 are satisfied. The initial positions and velocities of the all agents are randomly produced. The position and velocity errors in Fig. 1 and Fig. 2 are defined as xi − x0 and vi − v0 respectively. The position tracking errors are given in Fig. 1 and velocity tracking error are given in Fig. 2. It show that the follower-agents can track the leader from Fig. 1 and Fig. 2. 40

120 100

30 80 60 Velocity error

position error

20

10

0

40 20 0 −20

−10

−40 −20 −60 −30

0

5

10

15 Time

20

25

30

Fig. 1. Position tracking errors of five followers

−80

0

5

10

15 Time

20

25

30

Fig. 2. Velocity tracking errors of five followers

A Consensus Protocol for Multi-agent Systems

5

165

Conclusion

In this paper, we present a consensus protocol for continuous-time multi-agent systems with fixed and switching topologies. In this study, each agent updated its value based on the information of its neighbors and the conditions were given to achieve the coordination of these agents with the proposed local control strategies. Finally, a numerical simulation example was provided to demonstrate the effectiveness of our theoretical results. The protocol studied in this paper assumes that there is no delay in receiving the information of other agents. Using the analysis method as in [8], it is not too difficult to analyze the multi-agent system with communication delays.

Acknowledgment This work is supported by National Nature Science Foundation of China under Grant 60674071.

References 1. Okubo, A.: Dynamical aspects of animal grouping, swarms, schools, flocks and herds. Advances in Biophysics 22, 1–94 (1986) 2. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile agents using nearest neighbor rules. IEEE Transactions on Automatic Control 48, 988–1001 (2003) 3. Olfati-Saber, R., Murray, R.M.: Consensus Problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control 49, 1520–1533 (2004) 4. Vicsek, T., Czirok, A., Jacob, E.B., Cohen, I., Schochet, O.: Novel type of phase transitions in a system of self-driven particles. Physical Review Letters 75, 1226– 1229 (1995) 5. Murray, R.M.: Recent Research in Cooperative Control of Multivehicle Systems. ASME Journal of Dynamic Systems, Measurement, and Control 129, 571–583 (2007) 6. Ren, W., Beard, R.W.: Consensus seeking in multi-agent systems using dynamically changing interaction topologies. IEEE Transactions on Automatic Control 50, 655–661 (2005) 7. Gao, L., Cheng, D., Hong, Y.: Control of group of mobile autonomous agents via local strategies. Journal of Control Theory and It’s Applications 6, 357–364 (2008) 8. Hu, J., Hong, Y.: Leader-following coordination of multi-agent systems with coupling time delays. Physical A 374, 853–863 (2007) 9. Hong, Y., Gao, L., Cheng, D., Hu, J.: Lyapunov-based Approach to LeaderFollowing Convergence with Jointly-Connected Interconnection Topology. IEEE Transactions on Automatic Control 52, 943–948 (2007) 10. Hong, Y., Hu, J., Gao, L.: Tracking control for multi-agent consensus with an active leader and variable topology. Automatica 42, 1177–1182 (2006) 11. Peng, K., Yang, Y.: Leader-following consensus problem with a varying-velocity leader and time-varying delays. Physical A 388, 193–208 (2009) 12. Hong, Y., Chen, G., Bushnell, L.: Distributed observers design for leader-following control of multi-agent networks. Automatica 44, 846–850 (2008) 13. Horn, R., Johnson, C.: Matrix Analysis. Cambbridge Univ. Press, New York (1985) 14. Hirich, M., Smale, S.: Differetial Equations, Dynamical Systems and Linear Algebra. Academic Press, New York (1974)

A Production-Collaboration Model for Manufacturing Grid Li-lan Liu, Xue-hua Sun, Zhi-song Shu, Shuai Tian, and Tao Yu Shanghai Enhanced Laboratory of Manufacturing Automation And Robotics, Shanghai University, Shanghai, 200072, China [email protected]

Abstract. Based on the analysis of current research on manufacturing grid (MG), a model of complex network is firstly proposed to simulate dynamic movements of resource nodes in MG. Then based on Scale-free Network and collaboration networks, with the characteristics of production collaboration and the complexity of self-organized campaign in MG taken into account, a production-collaboration model of MG, MPC Model, is described in this article. Then the formula of the distribution is deduced with mean field theory. Finally, through computer simulation, the distribution of nodes in the model shows the characteristics of power-law tail, of which the attenuation index is consistent with the results of calculated studies under certain conditions. Keywords: Complex Network, Manufacturing Grid, Production Collaboration Network, MPC Model.

1 Introduction To achieve the dynamic sharing and collaborative work of distributed and heterogeneous manufacturing resources, many national research institutions and enterprises immediately paid close attentions to the Manufacturing Grid (MG) once its concept was introduced. At present, the studies of MG is mainly focused on the structure of system platform, breakthrough of cell technology, and demonstration of applications, including setting up a open and standard system architecture, packaging and managing resources based on the Web Services Resource Framework (WSRF), exploiting collaborative environment based on the Computer Supported Cooperative Work (CSCW), developing and implementing tasks and workflow management and so on [1, 2]. This article is aiming to solve how to provide support and management for different stages and various activities of product life cycle, with the ultimate goal of providing an integrated platform and operating environment of collaborative product development for members of cross-boundary and cross-enterprise. However, it is widely known that it is extremely difficult for the theory of MG to make a practical application to industry. As shown in figure 1, MG is such a complex system caused by many factors, one of the key factors is the resource node, which is mainly confined by: (1) Large-scale. The category of manufacturing resources K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 166–175, 2010. © Springer-Verlag Berlin Heidelberg 2010

A Production-Collaboration Model for Manufacturing Grid

167

includes technology, production equipments, plants, human and information resources etc. The variety of resource may directly lead to the large-scale of resource nodes. (2) Strongly dynamic random changes. MG nodes comply with self-organization model, which always join or quit dynamically. (3) The dynamic relationship changes between cooperation and competition. Various resources in the organization management belong to different enterprises, the relationships among which are competitive to maximize the benefits; however, numbers of resources often compose a dynamic virtual organization to achieve win-win by taking mutual cooperation during the tasks, while the virtual organization would be immediately dismissed after the assignment completed.

Fig. 1. Practical application of MG

Therefore, based on broad network of cooperation and biological networks, applying complex network theory to MG has been tried in our research, such as putting forward a dynamic collaboration model by considering product as the main line so as to analyze the dynamic self-organization cooperative relations among MG resource nodes, finally to build a architecture of manufacturing resources’ optimizing and controlling. The rest of this paper is organized as follows: The Scale-free Network and its applications in Collaboration networks are introduced in section 2. And the Production-Collaboration model for MG (MPC model) is introduced concisely in section 3. Then the process of mathematical deduction and the simulation results of the MPC model are described in section 4. Also the experiments and results will be discussed in section 5. Finally the conclusion is drawn in the end.

2 Complex Network Theory and Collaboration Network Complex networks describe a wide range of systems in nature and society, such as the World Wide Web (WWW), Internet, genetic networks, ecological networks, citation

168

L-l. Liu et al.

networks, social networks and so on, which play an important role in people’s daily life. This is why researchers are getting more and more interested in unraveling the structure and dynamics of complex networks [3, 4]. Scale-free networks was firstly introduced by Barabási and Albert in their pioneering works, who had explored several large databases describing the topology of the large networks, including the WWW, the actor collaboration network, the citation network, and so on. The empirical results show that the degree distribution

p (k ) in these networks decays as a power law, that is following p (k ) ∼ k − r with the degree exponent r scattering between 2.1 and 3, which firstly offered the evidence that some large networks can self-organize into a stationary scale-free state (which called BA model)[5, 6]. Collaboration network is a kind of complex network, which can be represented as bipartite graphs with two types of vertices. On one side, act nodes that may be represented as a special kind of vertices. On the other side, the actor nodes (normal vertices) that are linked to the collaboration acts in which they participate. Examples of this kind of network can be found in movie actors related by costarring the same movie, scientists related by coauthoring a scientific paper, etc [7]0. However, these bipartite graphs are empirically studied more in the one-mode way than both of the two modes in existed projections. The vertices representing the acts of collaboration are excluded while actors of the same act are connected with each other. Newman [8]0 studied a variety of nonlocal statistics for scientific collaboration networks, such as typical distances between scientists through the network, and measures of centrality such as closeness and betweenness. Furthermore, weighted collaboration networks were proposed to measure the variation in the strength of collaborative ties. Ramasco, et al0.[9] studied the collaboration networks in terms of evolving, selforganizing bipartite graph models, then proposed the RDP model, which combines preferential edge attachment with the bipartite structure, generic for collaboration networks. RDP model describes many of the main topological characteristics (degree distribution, clustering coefficient, etc.) of one-mode projections of several real collaboration networks, without parameter fitting. The simulation results from this model are in rather good agreement with the empirical results, which RDP obtained for a scientific co-authorship network and Hollywood movie actor collaboration network. He Da-ren, et al. [10, 11] suggested to extend the ‘‘collaboration network’’ concept to non-social systems, which called “generalized collaboration networks”, and a model was presented to describe the evolution of this kind of networks. The analytic and numerical analyses of the model show that the degree distribution can take different forms: a power law or an exponential decay in the extreme cases, but an SED (Stretched Exponential Distribution) in between. And they studied a number of systems, including: Bus Route Networks of Beijing and Yangzhou, the Travel Route Network of China, Huai-Yang recipes of Chinese cooked food, and the Collaboration Network of Hollywood Actors, the results related to the degree distribution, act-

A Production-Collaboration Model for Manufacturing Grid

169

degree distribution and act-size distribution are in good agreement with the analytic and numerical results obtained from the model. Roger Guimerá, et al0. [12] proposed that a model for the self-assembly of creative teams has its basis in three parameters: team size, the fraction of newcomers in new productions, and the tendency of incumbents to repeat previous collaborations. The model suggests that the emergence of a large connected community of practitioners can be described as a phase transition. They concluded that team assembly mechanisms determine both the structure of the collaboration network and team performance for teams derived from both artistic and scientific fields. At the same time, they studied data from both artistic and scientific fields (i) the Broadway musical industry (BMI) and (ii) the scientific disciplines of social psychology, economics, ecology, and astronomy, and identify the conclusions. Bruce Kogut, et al0. [13] using methods of complex graphs, analyzed 159,561 transactions over nearly 45 years to demonstrate the rapid emergence of a national network of syndications follows power law trails of scale free phenomenon. Of which a node denotes a VC firm, an edge is formed when a VC firm has built with another VC firm through syndicated deals. Also, they emphasize the importance of dynamics and complex weighted graphs for the analysis of social and economic behaviors.

3 Production-Collaboration Model for Manufacturing Grid The characteristics of product manufacturing can be concluded as follows: (1) Production in the enterprise constantly develops new products, which the gradual increasing number of products directly leads to the increasing number of devices for the purpose of completing new products. Meanwhile, in order to obtain higher output than input, less equipment for new products will be added as possible. (2)Significant repeated connections: repeated connections exist among borer and internal external grinding machine, double housing planer and internal external grinding machine, double housing planer and borer. This shows an unbalanced utilization of capacity, with some key equipment overloaded while some others are obviously insufficient. The ABC method is often used to divide equipment to distinguish their importance in enterprises. (3) The main driving force of network evolution is the growth of the equipment duplication. New facilities purchased for the increasing number of new products are mostly the copy of existing facilities, which lead to the characteristics of partial reproduction. Based on BA model and generalized collaboration network model, a simple evolving model of production collaboration (MPC Model) listed as below. Step 1: There are m0 initial devices (t=0) existing in the network, formed a fully connected graph (with each two devices connected). Step 2: A new product will be added to the network each step of the evolution, with (T+1) devices needed to complete the production.

170

L-l. Liu et al.

Step 3: In order to expatiate clearly, the (T+1) devices are separated into 2 groups, only an added vertex named v forms one and other existed T vertices form the other. The vertex v may be generated with probability q by copying a vertex u randomly chose from the existing (m0+t) devices or else a totally new one. Step 4: The probability that vertex v is a copied one

of a random vertex u is q, and in such an instance δ of T vertices are randomly chosen from those connected with u and the rest of T vertices in proportional to its own degree from all existed vertices, that is to say, connect with a preferential attachment. Step 5: If vertex v is not a copied one in step 4 but a newly added one with probability (1-q), then T vertices are chosen from all existed vertices randomly with probability p, or else the T vertices should be chosen from all existing vertices with a preferential attachment. Step 6: Connect v with these T vertices, which have been chosen in step 3 or 4. Step 7: Repeat step 2 to step 6. According to mean-field theory, we could deduce the formula of degree distribution from step 3 as below:

∂ki / ∂t = q × Α + (1 − q ) × B

(1)

A and B in the first formula each represents a copied evolving in step 4 and a newlyadded evolving in step 5. In the followed part we will derivate the expression of A. Considering vertex i with ki vertices connected, while the networks keeps evolving, it may play its role as a randomly chosen vertex u with probability 1/t or a neighbor of u in the premise that it had not been chose as u.

Fig. 2. The process of copying vertex one

A Production-Collaboration Model for Manufacturing Grid

171

As we can see in figure 2, vertex i may never connect with more vertices as being chosen as u (vertex 1 in the figure) but only when becomes vertex u’s neighbor such as 2, 3 or 4. So we compose the expression of A as below:

A = ( (1/ t ) ⋅ 0 + (1 − (1/ t )) ⋅ X )

(2)

According to step 4, apparently we could express X as below:

X = δ × X 1 + (T − δ ) × X 2 Namely,

A = ( (1/ t ) ⋅ 0 + (1 − (1/ t )) ⋅ (δ × X 1 + (T − δ ) × X 2 ) )

(3)

Now let’s anatomize this model, as the process illustrated in step 1 and 2, there are

CT2+1 lines initially with T lines added each step in the network, so after having evolved for t times, there would be ( t + T + 1 ) vertices and ( Tt + CT +1 ) lines. 2

However, we may take no account of the initial network to simplify the calculation. That is to say, there would be t vertices and Tt lines after having evolved for t times. Therefore the average number a vertex connects is 2T. Next let’s analyze the vertex i with ki vertices connected. In randomly chosen instances, any vertex which has connected with vertex i may be chosen with probability 1 2T , moreover as the average number a vertex connects with is 2T, this vertex as the chosen one’s neighbor would be chosen with probability ki

/ t ; while in

preferential instances, as there are 2Tt degrees totally in the network, this vertex i would be chosen with a probability ki / (2Tt ) . So we may conclude as below:

A = (1 / t ) ⋅ 0 + (1 − (1 / t ) )[δ ⋅ (k i / t ) /(2T ) + (T − δ ) ⋅ (k i /(2Tt ))]

(4)

Referred as above, we should get the express of B easily as below:

B = p ⋅ T / t + (1 − p) ⋅ T ⋅ k i /(2Tt )

(5)

Substitute all parameters back into the equation (1), we may get the final equation as below:

∂ki / ∂t = q ⋅ (1/ t ) ⋅ 0 + q (1 − (1/ t ) ) [δ ⋅ (1/ 2T ) ⋅ ( ki / t ) + (T − δ ) ⋅ ( ki / 2Tt ) ] + (1 − q ) [ p ⋅ (T / t ) + (1 − p ) ⋅ T ⋅ ( ki / 2T ⋅ t ) ]

That is,

∂ki / ∂t = [q + (1 − q)(1 − p)] × ki / 2t + p(1 − q)T / t + o(1/ t )

172

L-l. Liu et al.

Solve this differential equation, we may get:

p(k ) ~ ⎡⎣( k + (1 − q) pT / β ) / 2 β ⎤⎦

β

and

γ

−r

(6)

in the formula represents:

β = q + (1 − q )(1 − p ) / 2 ，

γ = 1 + (1 / β ) = ( q + (1 − q )(1 − p ) + 2) /( q + (1 − q )(1 − p )) From the outcome we could see γ is ranging from 3 to infinity when p and q changes between 0 and 1. In the following part, we’ll take some simulation to see if our theoretic results tally with the simulated results. However, here we allow some deviations which are unable to avoid engendering while using the mean-field theory to suppose the average number one vertex connect is 2T.

4 MPC Analyzing by Simulation The growing process of the model above is simulated with Java and Matlab. Some parameters are initialized as follows: m0=6, T=5, t=100000 and b=3, with p, q varies between 0 and 1, the simulated values γ are listed as follows. (1) q=1, b=T, the model turns into a stochastic duplication model, and the result

γ = 2.782;

(2) When q=1, b=0, the model turns into a preferential duplication model, and

γ =2.813;

(3) When p=0, q=0, apparently it becomes a B-A model, the value of quite closed to the theoretical valve 3.0.

γ

is 2.972,

Table1. The other results are followed as below

0
δ

=3,(theoretical values by formula 6 are listed in brackets)

，p=0.2

p, q

q=0.2

γ

3.189(3.38)

p, q

q=0.5

γ

2.929(3.222)

p, q

q=0.8

γ

2.872(3.043)

，p=0.2 ，p=0.2

q=0.2

，p=0.5

4.101(4.333) q=0.5

，p=0.5

3.391(3.667) q=0.8

，p=0.5

2.86(3.222)

q=0.2

，p=0.8

6.426(6.556) q=0.5

，p=0.8

4.133(4.333) q=0.8

，p=0.8

3.101(3.381)

A Production-Collaboration Model for Manufacturing Grid

173

Fig. 3. Simulation diagram(The degree distribution obtained by simulation is shown as solid blue line; by the theoretical formula method is show as solid green line. The settings for the inserts are q = 0. 2, p = 0. 2 with r =3.189 for the bottom left and q = 0. 2, p = 0.8 with r = 6.426 for the top right insert and q = 0. 2, p = 0.5 with r= 4.101 in the middle).

Fig. 4. Simulation diagram(The degree distribution obtained by simulation is shown as solid blue line; by the theoretical formula method is show as solid green line. The settings for the inserts are q = 0. 5, p = 0. 2 with r =2.929 for the bottom left and q= 0. 5, p = 0. 8 with r =4.133 for the top right insert and q = 0. 5, p = 0. 5 with r= 3.391 in the middle).

174

L-l. Liu et al.

Fig. 5. Simulation diagram(The degree distribution obtained by simulation is shown as solid blue line; by the theoretical formula method is show as solid green line. The settings for the inserts are q = 0.8, p = 0.2 with r =2.872 for the bottom left and q = 0.8, p = 0. 8 with r =3.101 for the top right insert and q = 0.8, p= 0. 5 with r= 2.86 in the middle).

Apparently, our simulation is quite closed to our numerical outcome.

5 Conclusions By analyzing the complexity of manufacturing grid resource nodes’ self-organizing campaigns and production collaborative relationships between resources, production collaboration model of manufacturing grid (MPC Model) are introduced in this paper. The model has both the characteristics of merit-based growth and partial duplication. Adding a node by the BA model would mainly reflected as a partial duplication while copying the existed one would mainly reflected the preferred choice. By simulating the growth process of the model with Java and Matlab programming tools in this article, several results have been obtained showing that the range of power-law index is 3 to ∞ with time step t=100000, while δ =3, p and q takes different values. At the same time, in order to verify the reasonableness of the model, also four related production data of the Shanghai Machine Tool Plants are analyzed, obtained the correctness of our results.

A Production-Collaboration Model for Manufacturing Grid

175

References 1. Fan, Y.S., Zhao, D.Z.: Manufacturing grids: needs, concept, and architecture. In: Li, M., Sun, X.-H., Deng, Q.-n., Ni, J. (eds.) GCC 2003. LNCS, vol. 3032, pp. 653–656. Springer, Heidelberg (2004) 2. Liu, L.L., Yu, T., Shi, Z.B., Fang, M.H.: Self - organization Manufacturing Grid and Its Task Scheduling Algorithm. Computer Integrated Manufacturing Systems 9, 449–455 (2003) 3. Strogatz, S.H.: Exploring complex networks. Nature 410, 268–276 (2001) 4. Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002) 5. Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509– 512 (1999) 6. Barabási, A.L., Albert, R., Jeong, H.: Mean-field theory for scale-free random networks. Physica A 272, 173–187 (1999) 7. Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, Cambridge (1994) 8. Newman, M.E.J.: Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Physical Review E 64, 1–7 016132 9. José, J.R., Dorogovtsev, S.N., Romualdo, P.S.: Self-organization of collaboration networks. Physical Review E 70, 1–10 036106 (2004) 10. Zhang, P.P., Chen, K.: Model and empirical study on some collaboration networks. Physica A 360, 599–616 (2006) 11. Zhang, P.P.: A model describing the degree distribution of collaboration networks. Hacta Physica Sinicah 55, 60–67 (2006) 12. Roger, G., Brian, U., Jarrett, S., Luís, A.N.A.: Team Assembly Mechanisms Team Assembly Mechanisms Structure and Team Performance. Science 308, 698–701 (2005) 13. Bruce, K., Pietro, U., Walker, G.: Emergent Properties of a New Financia Market. American Venture Capital Syndication 53, 1181–1198 (1960-2005)

Parallel Computation for Stereovision Obstacle Detection of Autonomous Vehicles Using GPU Zhi-yu Xu and Jie Zhang School of Electronics and Information Engineering, Tongji University, Shanghai, 201804, China [email protected]

Abstract. Due to the parallelism on general-purpose computation, the graphics processing unit (GPU) is applied in autonomous vehicles and a stereovision obstacle detection system is developed. We first perform census transform on the rectified stereo image pair; then employ the epipolar constraint to facilitate the visual correspondence matching. Based on the dense disparity map, the 3D coordinate of each pixel is calculated, according to which obstacles are identified. Furthermore, several techniques are listed for exploring the specific functionalities of GPU to boost the overall performance. A prototype system is finally implemented and integrated in the onboard PC of autonomous vehicles. Experimental results validate the real-time accuracy under various illuminations and road conditions. Since the low level image-processings are run by GPU in parallel, the proposed design not only merits high speed and efficiency, but also frees up CPU so as to focus on the decision and control. Keywords: graphics processing unit; parallelism; disparity; obstacle detection; autonomous vehicle.

1 Introduction Statistics discloses that most of the traffic trams and accidents are caused by the mistakes of human drivers. With the aim of reducing injury and ensuring safety, computers are employed to make the right decision at the right moment and autonomous vehicles are developed. As the mainstream technique of the robot perception, the binocular stereovision extracts the information of environment from a pair of images captured simultaneously but from different viewpoints. Although possessing the advantages of large field of view, low costs and interferenceless [1], this passive optical method requires heavy computations for online image processing. And parallel solutions (ASIC, FPGA, etc.) are proposed to free up the CPU from low level calculations [2]. The graphics processing unit (GPU) is a programmable processor in which vertex shaders and fragment shaders perform computations in highly parallel by utilizing the rendering pipeline. This paper exploits the general-purpose computation capability of GPU [3] and develops a stereovision obstacle detection system for autonomous vehicles. The remainder of this paper is organized as follows: Section 2 outlines the fundamentals of stereovision and discriminates the obstacles by its coordinate. Section K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 176–184, 2010. © Springer-Verlag Berlin Heidelberg 2010

Parallel Computation for Stereovision Obstacle Detection of Autonomous Vehicles

177

3 applies the census transform to handle the visual correspondence and presents the technical details of the GPU implementation. In Section 4, several specific functionalities/structures are proposed for hardware acceleration. A prototype system is established, whose efficiency and accuracy in real-time are validated by experimental results listed in Section 5. Finally we draw the conclusion.

2 Preliminary 2.1 Binocular Disparity and 3D Reconstruction The stereo camera (left and right), shown in Fig.1, has the focus length f , baseline b and the two optical axes are in parallel. A point P ( X , Y , Z ) in 3D space is projected as pl ( ul , vl ) and pr ( ur , vr ) on the two image planes of the stereo pair. Since

OL X L overlaps OR X R , we have ul ≠ ur , vl = vr = v , the disparity d is defined as d = ul − ur YL

(1) P ( X ,Y , Z )

YR ZL

ul vl p′r

vr pr

or pl

JL d

OL

ZR

ur ol

f

XL

JR XR

OR

b

Fig. 1. Disparity indicates the spatial depth information

Given

a

projection

(u, v )

and

its

disparity d

,

the

coordinate

of

point P ( X , Y , Z ) could be calculated by

⎛ X ⎞ ⎛1 ⎜ ⎟ ⎜ Y 0 Q⎜ ⎟ = ⎜ ⎜ Z ⎟ ⎜0 ⎜ ⎟ ⎜ ⎝ 1 ⎠ ⎝0

0 0 − W 2 ⎞⎛ u ⎞ ⎟⎜ ⎟ 1 0 − H 2 ⎟⎜ v ⎟ 0 0 f ⎟⎜ d ⎟ ⎟⎜ ⎟ 0 1b 0 ⎠⎝ 1 ⎠

(2)

where W , H are the width and height of the image, respectively. 2.2 Disparity Map and Visual Correspondence The key to disparity map is the stereo matching, namely, for each point pl on one image J L , to find its correspondence pr on the other image J R . Relevant algorithms are

178

Z.-y. Xu and J. Zhang

classified as feature-based, area-based, etc. Although the latter is superior on obtaining dense disparity map[4], mismatches occur at the discontinuous areas of disparity since the area-based method only surveys the intensity of a certain region. Thus [5] develops non-parametric solutions which evaluate the correlation among pixels and describe the local image structures.

3 System Design 3.1 Functions of the System The paper aims to develop a system which realizes the following functions: camera calibration, image acquiring and rectification, stereo matching, disparity map extraction, 3D reconstruction and obstacle identification. The stage of stereo matching dominates the overall performance and computation. Since the SIMD (Single Instruction Multiple Data) of GPU, a matching algorithm characterized the SIMD nature should be selected to maximize the advantages of GPU parallelism. 3.2 Stereo Matching Algorithms The census transform is a non-parametric local transform and quite suitable for hardware implementations[6]. As for census transform, the matching is based on the relative intensity of the pixel with respect to its surroundings, rather than its absolute intensity. In Fig.2, the intensity of pixel p is I ( p ) , a correlation window is of size w × h and

centered with p , N ( p ) represents the neighbors of p in the window. For

∀p′ ∈ N ( p) , we define

，

⎧⎪1 if I ( p ) > I ( p′ ) ⎪⎩0, else

ξ ( p, p ′ ) = ⎨

(3)

and the census transform

ε ( p ) = ⊗ ξ ( p, p ′ )

(4)

P ′∈N ( P )

Census

h

p

Transform w

0 1 1 1 0 1 1 0 0

w × h bits

p : 011101100 Fig. 2. Census transform expresses the local image structure with a bitstring

Parallel Computation for Stereovision Obstacle Detection of Autonomous Vehicles

179

Namely, compare the intensity of the center pixel p with all the other pixels p′ of the window, and obtain a bitstring. Consider a pixel pl ∈ J L , for each candidate of its correspondence

p1r , pr2 ,L , prm ∈ J R , compute the matching cost M ( plδ ) =

∑ ε ( p ) ⊕ ε ( pδ ), l

r

δ = 1, 2,K , m

(5)

CTW

where ⊕ is the logic operator “exclusive or”, ∑ is the Hamming distance between two

( ).

bitstrings ε ( pl ) and ε prδ

( )

The best matching of pl is the one prδ which

minimizes M plδ and its disparity is ul − urδ . Since the algorithm stated above deals with each pixel separately, it is suitable and feasible for the GPU implementation in parallel. Remark 1. The information of the center pixel is lost during the census transform and a modification is proposed[7]. Although its robustness to illumination differences, the modified census transform is unnecessary in this application since the stereo camera of autonomous vehicles always work under the same illumination. Remark 2. The census transform which uses ’1’ and ’0’ to express the relative intensity, belongs to the qualitative description, while the extended census transform tries a quantitative analysis and uses a byte string to store the intensity difference[8]. But the accuracy results in tremendously heavy computation. Balance the algorithm efficiency, real time constraint and resource limitations, this paper adopts the census transform for stereo matching. 3.3 Memory Allocation [9] provides the GPU implementation of SSD and SAD. As for the census transform, the arrangement of memory should be taken in account. As shown in Fig. 3, consider an image of size W × H , it requires w × h bit memory to store the local image structure of each pixel if a window of size w × h is selected. We figure out the following two approaches. A1. Extension in color channels. The bitstring is allocated in the RGBA channels in a pre-defined order. The programming would be sophisticated since enough high-order bits of each channel should be reserved for the possibility of overflow caused by accumulation [10]. A2. Expansion of image size. The bitstring is allocated in a region of w × h , thus the output image size is Ww × Hh rather than the original W × H . The programming is simple and high efficient but the enlarged size is limited by the max texture resolution of GPU. A2 is considered as a superior option as far as the feasibility and flexibility are concerned.

180

Z.-y. Xu and J. Zhang

Ww

W

W

Hh

H

H

p w × h bits p : 011101100

0 1 1 1 0 1 1 0 0 w

h

Fig. 3. Two approaches for allocating the bitstring generated by census transform

3.4 Process Flow Fig.4 illustrates the proposed system process. Programming with the OpenGL Shading Language. i) Camera calibration obtains the camera internal parameters: focus f and baseline b , etc. ii) The stereo image pair J Lraw , J Rraw of size W × H are captured simultaneously, transferred to the RAM of GPU and bind as two textures. iii) The color textures are rectified by using the look-up table determined by calibrated parameters and converted to gray scaled textures J Lrct , J Rrct of size

W ×H .

iv) Let the correlation window is of size w × h , the census transform generates two

textures J LCT , J RCT of size Ww × Hh . v) Set the hypothesis disparity d as the fragment program input and depth test is lowpass. vi) Employ the epiploar constraints, compute I L ( u, v ) − I R ( u + d , v ) 1 vii) Compute the matching cost2

M=

+h 2

⎛

+w 2

∑⎜∑

v =− h 2

⎝ u =− w 2

⎞ I L ( u, v ) − I R ( u + d , v ) ⎟ ⎠

(6)

viii) Set d and M as the intensity and depth of out fragment respectively.

1

2

Since the intensity of J LCT , J RCT is either 0 or 1, and 1 −1 = 0 − 0 = 0, 1 − 0 = 0 − 1 = 1 , the absolute difference has the same effect as the logic ‘exclusive-or’. Step(vii) accumulates in the region of size w × h , which shrinks the image size from Ww × Hh back to W × H .

Parallel Computation for Stereovision Obstacle Detection of Autonomous Vehicles

181

ix) Repeat Step(vi)~(viii) in the searching region d = 1, 2,L , D could generate the disparity map texture J Ldsp of size W × H since the lowpass filter of depth test minimizes the matching cost for each pixel. x) Invoke the fragment program whose inputs are f , b,W , H to compute eq(2) to obtain the 3D coordinate ( X , Y , Z ) of each point of J Ldsp . xi) Overlay RED color to indicate obstacles in J Lrct . xii) Copy J Lrct to PC main memory, wait for new instructions or start processing the next image pair. camera calibration

camera calibration image acquire

left camera parameters

J Lraw

J Rraw

right camera parameters

image rectification

J Lrct

window size

J Rrct

w× h

census transform

J LCT

J RCT area-based matching

J

dsp L

3D reconstruction

external parameters

coordinate obstacle identification

J

image flow numeric flow

obs L

Fig. 4. Process flow of disparity-based obstacle detection

Notice that all the procedures are executed by GPU in parallel except step (ii) and (xii), which involve the main memory read/write, therefore the system efficiency is dramatically improved while leaving the CPU idle.

4 Performance Enhancements The system performance could be further promoted if the dedicated structures/functionalities of GPU are utilized and the computation potential is fully exploited [9].

182

Z.-y. Xu and J. Zhang

■ The P-buffer supported by new generation GPUs enables the fragment shader access GPU RAM directly. The time consumption is greatly reduced in multi-step rendering since the previous output texture could be directly processed in the next rendering. ■ The RGBA channels of GPU could be employed in Step (ix) to store the matching cost of d , d + 1, d + 2, d + 3 , respectively. The circulation length of correspondence searching is reduced to 1/4. ■ In the area-based matching, large windows neglects the details of images, while small windows results in mismatches. The MIPMAP of GPU could implement the multi-resolution analysis. ■ The box-filter

J uk,+v1 =

2 v +1 2 u +1

∑∑J

q =2v p =2u

k p,q

(7)

could be employed in Step(vii) for parallel summation, where J 0 is the original image

and ( u , v ) , ( p, q ) are pixel coordinates. But notice that this technique is most suitable for large dimensions and of size 2n × 2n .

5 Experimental Evaluations 5.1 Prototype Design

Fig. 5. Hardware setup of the prototype system

The prototype system setup is as follows: NVIDIA Quadro FX540 graphic card, 300 MHz clock, 128 MB RAM, 8.8 GB/s bandwidth, max texture resolution is 2048 × 1536 ; STH-DCSG-VAR stereo camera, baseline (5~20) cm adjustable, output texture resolution is 320 × 240 / 640 × 480 (pixel). The stereo camera is rigidly fixed on the front platform of the autonomous vehicle, with a height of 580 mm to the ground. The XOZ plane is parallel with the ground plane and the optical axis OZ is straightforward. Software calibration shows that the camera focus length is 562.22 pixel and the baseline is 186 mm. The graphic card and camera interface the on-board PC with PCI-Express and IEEE1394 standard, respectively. (Shown in Fig. 5)

Parallel Computation for Stereovision Obstacle Detection of Autonomous Vehicles

183

5.2 Field Test The experiment setup is as follows: camera resolution is 320 × 240 , correlation window size is 5 × 5 , correspondence searching range is (0~32) pixels. Field tests were carried out in the courtyard of INRIA-Roquencourt and a number of real-time processing results were recorded, among which six groups are shown in Fig.6. The experimental results indicate that the prototype system obtain dense disparity map and correctly identify obstacles under various luminous and road conditions.

Fig. 6. Six groups of experimental results in field test (In each group: top-left/right original images, bottom left-disparity map, bottom right-obstacle marked result)

6 Conclusions The stereovision perception of autonomous vehicle inevitably causes the heavy computation of real-time image processing. Aiming to perform a hardware acceleration, this paper proposes a novel solution which maps the majority of graphic transformation into the rendering pipeline of GPU, including image conversion, image rectification, visual correspondence, 3D coordinate reconstruction and obstacle identification. Thus the algorithms are running completely in parallel. Census transform is adopted for stereo matching. Furthermore, specific structures and functionalities of GPU, such as P-buffer, RGBA channels, etc. are also discussed for the system promotion. The field test results confirm the reliability and efficiency of the prototype system under various conditions. Compared to its embedded counterparts such as FPGAs, GPU is the core of the commodity graphic board so easier to be integrated to the onboard PC of autonomous vehicles. Acknowledgements. The first author gratefully acknowledges the research opportunity provided by Prof. C. Laurgeau of ENSMP and Prof. M. Parent of INRIA. Sincere thanks to IMARA team colleagues: R. Benenson, N. Simon, A. Yvet, P. Bodu, Y. Dumortier, et al. for the valuable collaborations and discussions.

184

Z.-y. Xu and J. Zhang

References 1. Seki, A., Okutomi, M.: Robust Obstacle Detection in General Road Environment Based on Road Extraction and Pose Estimation. In: IEEE Intelligent Vehicles Symposium, pp. 437–444. IEEE Press, New York (2006) 2. Khaleghi, B., Ahuja, S., Wu, Q.: An Improved Real-Time Miniaturized Embedded Stereo Vision System (MESVS-II). In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pp. 1–8. IEEE Press, New York (2008) 3. Owens, J.D., Houston, M., Luebke, D., Green, S., Stone, J.E., Phillips, J.C.: GPU Computing. Proceedings of the IEEE 96, 879–899 (2008) 4. Gong, M., Yang, R., Wang, L., Gong, M.: A Performance Study on Different Cost Aggregation Approaches Used in Real-Time Stereo Matching. Int. J. Computer Vision 75, 283–296 (2007) 5. Zabih, R., Woodfill, J.: Non-Parametric Local Transforms for Computing Visual Correspondence. In: 3rd European Conference on Computer Vision, pp. 151–158. Springer, Heidelberg (1994) 6. Murphy, C., Lindquist, D., Rynning, A.M., Cecil, T., Leavitt, S., Chang, M.L.: Low-Cost Stereo Vision on an FPGA. In: 15th Annual IEEE Symposium on Field-Programmable Custom Computing Machines, pp. 333–334. IEEE Press, New York (2007) 7. Froba, B., Ernst, A.: Face Detection with the Modified Census Transform. In: 6th IEEE International Conference on Automatic Face and Gesture Recognition, pp. 91–96. IEEE Press, New York (2004) 8. Ambrosch, K., Humenberger, M., Kubinger, W., Steininger, A.: Extending Two Non-Parametric Transforms for FPGA Based Stereo Matching Using Bayer Filtered Cameras. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pp. 1–8. IEEE Press, New York (2008) 9. Yang, R., Pollefeys, M.: A Versatile Stereo Implementation on Commodity Graphics Hardware. Real-Time Imaging 11, 7–18 (2005) 10. Zach, C., Klaus, A., Hadwiger, M., Karner, K.: Accurate Dense Stereo Reconstruction using Graphics Hardware. In: EUROGRAPHICS, pp. 227–234 (2003)

Framework Designing of BOA for the Development of Enterprise Management Information System Shiwei Ma*, Zhaowen Kong, Xuelin Jiang, and Chaozu Liang School of Mechatronic Engineering & Automation, Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, No.149, Yanchang Rd., 200072 Shanghai, China [email protected]

Abstract. This paper put forward the definition of business oriented architecture (BOA) for the development of enterprise management information system and its technical components, as well as the designing method of reusable component based on business oriented model (BOM). This method uses JDOM to generate and analyze the properties of configuration files with XML format, and realizes the interaction between basic middleware and configuration software by using AJAX. It can give different page styles using JavaScript and has properties of short period of designing and good flexibility. An example in the development of a management information system based on BOA for a machine manufacturing factory is introduced. In which, most of the components was designed with the proposed method. It manifests that the BOA framework can be a good choice to meet the needs of different customers and to reduce system development cycle. Keywords: Management Information System; BOA; BOM; Reusable Component.

1 Introduction Along with the development of computer and network technology, particularly the rapid development of Internet application, large scale distributed heterogeneous systems came out in all stages of enterprise management information system or ERP system. Enterprise information resources show features of heterogeneous, distribution and loose coupling. Traditional integrated information processing system can not yet adapt to the actual needs. In addition, in order to meet the variable business needs of customers, traditional waterfall or spiral software design patterns often require designers to develop software system based on its original model again. So these patterns can not afford the increasingly difficult tasks of large scale enterprise management software. In traditional service oriented architecture (SOA) technique, which has been commonly used in the framework designing of enterprise management information *

Paper supported by Shanghai Science and Technology Foundation No.085958209, No. 09dz2201200, No.09dz2273400.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 185–195, 2010. © Springer-Verlag Berlin Heidelberg 2010

186

S. Ma et al.

system or ERP system, information systems in different position and those with different functions are integrated together in the form of client-server. But it still has some disadvantages. First, each application just works on its own and forms a lot of information islands. The lack of information exchange between these islands makes the seamless integration of applications and businesses impossible to be realized. Second, the structure and functions of SOA are fixed and difficult to be customized. Third, there are lacking techniques or design methods to help operators making creativities in operation and management. Fourth, the lack of unified system UI for operators makes working inefficient. These disadvantages can increase the risk of lose of control for system security management, with the increase of application and complexity in management. Since most of these shortages derives from restriction of user’s demand, development idea, limitation of methodology and techniques. These could be solved by business oriented architectures (BOA). In recent years, the designing approaches, which use BOA and reconfigurable business oriented modeling (BOM) to greatly improve designing efficiency, are adopted more and more in the development of ERP software system [1~5]. This paper gives the conception and advantages of BOA and workflow technology, propose a designing method of reusable components based on BOM which can be used to construct enterprise management information system software simply and flexibly. Its application example in the development of an actual management information system software for a manufacture factory is introduced and discussed.

2 BOA and Its Technical Realization Different from most of technical oriented software frameworks, BOA is a business oriented technology with coarse granularity, loose software framework which contains three levels, as in Fig.1, basic middleware, integrated middleware and vertical solution package (VSP) [1].

Fig. 1. Three layer of BOA framework

In this framework style, basic middleware which is also the BOM divides users’ business into a series of procedures and services with coarse granularity,

Framework Designing of BOA for the Development of Enterprise Management

187

independence and reusability. Hence, BOM is the technical foundation and basic unit of integrated middleware. Meanwhile BOM supports the customized application and VSP level. Integrated middleware is generated after service providers conducting certain functional configurations of BOM model. So, it is a functional product of basic middleware and an important technical solution to application system. However, VSP layer is a kind of business abstraction facing a profession area, which combines integrated middleware according to business sequence and can be used to solve common problems in the area. Nowadays, enterprise business tends to be more and more flexible and much attention should be paid to business modeling during the development of ERP system. Since most of existing software frameworks are neither adapt to the increasingly complex system integration nor meet the needs of business changes. However, the BOA framework, which possesses characteristics of loosely coupling and reusable component, will be conducive to solve these problems. The advantages of BOA architecture include: 1)

2)

3) 4)

Directly reflect and express the application requirements of users. Every user can easily understand and operate the system without the help of technicians, thus erase the gape between demand and software realization, and make a quicker and better development. The tasks in BOM are much more "soft" and "light" than the components, which are coded by software developers. The tasks can be defined by normal users. And its easy to configure, thus costs less and provides more adaptations to different needs. BOA gives the application layer much more reusability. Well defined tasks can work well with other tasks,just like how the building blocks do. BOM do not need the support of application server and base structures like middleware. BOM has less requirements in environments and user both in development and operation, thus it's easy to use.

To construct a complete framework of BOA, one must first construct a BOM. It is generally consists of several abstract concepts: front applications, supporting services and data. The front application is used for the operations of all the management system. Equivalent to the View layer in Web design pattern, it is used as an interface for user to see and interact. The supporting services are used to provide users with standards of functions, usage and constraints. The data is provided for supporting the configuration of services. Its general format is transmitted by using XML. Various business characteristics can be generated by binding different configuration files and BOM middleware. Thus, different business can be realized by this kind of BOM configuration and the configured BOM is just the layer of integrated middleware mentioned above. Complicated changes of enterprise’s business also need BOM module to be rearranged and combined. Therefore, VSP layer is always needed to manage all the modules to make them coarse granularity, loose coupling, reusable, inter operability and combined together as a whole. The realization of VSP layers should introduce the concept of workflow. In view of computer, the workflow is a computerized model which defines all kinds of parameters used to complete the whole process. These

188

S. Ma et al.

parameters cover the definition of each step, the execution sequence and condition between steps, the establishment of data flow, the application program needed for each activity, etc. Workflow model can be divided into six kind of model: sequential AND branching, AND connecting, OR branching, OR connecting and cycling. Generally, complex business process can be constructed by these models

3 Design of BOM The first goal of BOM is to align with the business so as to support the rapid change of services, then is to realize the flexibility of framework and the reusability of module. Therefore, BOA framework can meet user’s needs greatly by transforming business components into service models taking advantage of coarse granularity business services modeling. Fig.2 gives the realization structure of BOM. In our work, based on B/S structure, the tool of Sun Java Studio Creator was employed to construct visualization program so as to quickly develop required enterprise management information system software and to allow system developers focus on analyzing and designing its business process. Furthermore, JSF was used to construct basic middleware, JDOM to generate XML configuration files from configuration data stored in database, and AJAX to transmit XML configuration files to realize the binding and reading of JSF model toward them and to render reusable pages.

Fig. 2. Realization structure of BOM

The design idea of this paper was as following. First taking client and market analysis, then determine the developed object and making its virtualizition model. After that, the development and configuration of management module would be done. The designing principle of management module configuration was shown in Fig.3. There are two parts, one is BOM basic module register UI, the other is BOM configuration UI. The basic module is registered and managed by register UI, in which basic module must be registered before configuration, so the whole system could identify the module and make an configuration file for the module. The BOM basic module could be configured with the help of BOM configuration UI, after it was registered. Then a configure file of XML format would be made from the configuration data of Database by JDom. And the file would be transferred with

Framework Designing of BOA for the Development of Enterprise Management

189

AJAX techniques. Thus the binding and accessing of configuration file with JSF module, display of pages for reusing and services organization structure, middleware making of BOM, all of these have been done.

Fig. 3. Realization structure of BOM middleware layer

3.1 Configuring the Configuration Files XML (Extensible creating Language) is an open, self described data structure. It can describe not only the data content but also the data structure so as to reflect relationship between one data and another. Due to these characteristics, it can be used for producing BOM. In our work, the root node of XML configuration file was specified as the label of <xml_config>, the name of each child node as the name of each user configuration attribution, the ID of each child node as the ID of corresponding object, and the attributions of each child node as the user defined ones. This can realize the analysis and binding of front pages to each attribute. To analysis and bind the configuration files, we first put JDOM packet into the project, then programmed a JavaBeans file by calling JDOM library functions. This JavaBeans file can not only selectively read the configuration data form SQL2005 database and form a correspondingly XML file, but also analyse the generated configuration files and get all of its attributes to react them on front pages by using JavaScript, as shown in Fig.4. Due to this kind of bilateral effect, the BOM can be generated and configured with good flexibility. 3.2 Transmitting the Configuration Files Usually, the transmission between front web pages and data is triggered by server response after user clicking the corresponding web control By using such traditional way to transmit XML configuration files developer cannot achieve the goal of configuring pages flexibly, since the web page was also refreshed according to corresponding response once the web control was triggered and hence the state of page could not meet users' demand. To solve this problem, one can employ AJAX to transmit these files. Normally, in order to transmit XML files by using AJAX, one need firstly using JavaScript to create a XmlHttpRequest object whose Open function can be used to send data or XML documents to server synchronously or asynchronously with POST or GET method.

190

S. Ma et al.

Fig. 4. XML configuration files created by JDOM

Fig. 5. XML JSF way of AJAX

Different from the traditional transmission mode of AJAX, this paper use AJAX mode combining with JSF (JavaServer Faces) method. Its principle is shown in Fig.5, where the XmlHttpRequest object was used to send strings or XML files to a Servlet which is written in the JSTL code of JSF page. The Servlet was then used to call operational functions in JavaBeans to achieve the required operation purpose. The following gives the key code segment used to write such Servlet:

Framework Designing of BOA for the Development of Enterprise Management

191

<jsp:scriptlet> response.addHeader("Cache-Control", "no-cache"); response.addHeader("Expires", "Thu, 01 Jan 1970 00:00:01 GMT"); request.setCharacterEncoding("UTF-8"); String date1=null; date1=(String)request.getParameter("method"); if(date1!=null){ String mystr=null; ajaxbean.setRequest(request); mystr=ajaxbean.getResponseXML(); response.setContentType("text/html;charset=UTF -8"); response.getWriter().print(mystr); response.getWriter().close(); } It can be seen that the Servlet code open up a channel for data transfer between the front and JavaBean. The using of request and response objects can facilitate the data transfer. Then, the background operations will be performed in the JavaBean, and the results will be returned to front client. The methodology designed here is used to fetch data between BOM Configurator and the background database. 3.3 Construction and Configuration of Middleware Integrated middleware was constructed in the design idea of large scale components customize techniques, after the whole business was divided according to application and organized by its particular function. The divided granularity should be proper. Smaller granularity would break an integral business function or process, which would do harm to its reassemble and extend in business level. The larger ones would cover too many business functions in it, which would reduce flexibility and suitability. After a well business separation, the basic middleware could be configured to meet business needs. The normal time sequences for the configuration is shown in Fig.6. First the UI send a configure command to JSP by using AJAX. The Servlet give a redirection heading for JavaBean controller to the command. In the controller, there would be a configuration process. First identify and determine which basic module would be configured from the command info, here the basic module registration make the identification and the fetch of XML file easy to do. After analysis the XML file, the property of basic module could be configured according to the file by using Javascript. Different configuration corresponds different business function, which should meet the demand of the reconfiguration.

192

S. Ma et al.

Integrated middleware can work with the interaction between BOM register and BOM configurator. The function of BOM register is for the registration of components library. The basic middleware and its data and configuration file are tied together with the help of BOM register.

Fig. 6. Operational time sequence of BOM configurator

4 Application Example The above mentioned designing method was successfully used in the development of a management information system for a large scale coal machine manufacturing factory. The business of this factory has following characteristics: 1)

2)

3) 4)

Carrying out the designing and manufacturing of customers’ order oriented goods, which express multiple variety with short-run production and long production cycle. Along with the shortening period of the order, the temporary supplementary orders will increase in number. Therefore, in order to shorten the period of whole production of the factory, the general or semi-general purpose components and parts with long production cycle must be produced in advance. It will result in the increase in stock of components and parts. Since there are lot of situations of components and parts borrowing during the production, the statistics of lacked components and parts is difficulty. Due to complex and variety of working procedure for units production, the accounting of working hours is extremely difficulty.

Hence, it is necessary to create a corresponding business information model, that is the layer of VSP, for the development of management information system software for this factory. The proposed BOM method just provides the foundation for this work.

Framework Designing of BOA for the Development of Enterprise Management

193

In this system, a BOA based production management information software system was developed, which combined with model driven architecture to generate application software framework from the management and administration model of this factory. It employed visualized interface development tools and implemented ten functions and processes, shown in Fig.7.

Fig. 7. Schematic diagram of business in the application example

As a factory level application, the system was three layer framework using J2EE platform based B/S structure as a whole, shown in Fig.8. The AJAX technology was used for communication between clients and server. The database was developed by using SQLSERVER 2005 with XML function to be competent at non-structural data in the application, such as the structure of product’s fabrication, the routine of part’s machining and user’s work direction tree, etc.

Fig. 8. J2EE application used in the example

194

S. Ma et al.

As a production management system, we use the List component which is a list box with different optional components. With the design method of BOA, we built different modules according to different businesses. The internal process of the list is shown in Fig.9. It triggered by events of customs to choose different options, then to call backstage function method and to read data from database.

Fig. 9. Flowcharts of List

Except the list component, the table component and some other components are also redeveloped for a better fitting in this system. The table component of JSF have good performance and appearance. But it can not work well with BOA designings. Improved table component, which could cooperate well with AJAX, as it can be refreshed to update cells in both client and server, can satisfy the design method of BOA. And, a well designed report print component was developed in the BOA framework. It can solve the problem of the weakness in customize and adaptiveness of requirements in web report generation. In addition, some special components for this system were also designed. Which include: the order form generation component that would work with client agreement; the client information organization component that would add update or delete items according to the information of client company; the basic sales price component that would organize the products prices; the store components for input and output of the products which can generate bills and current account changes; the web IM component which can make the communication between workers in production management easier. All these special components were integrated together and made the system suited for this factory application. Since most of manufacturing factories now have multiple products and large number of complicated standard parts, even distinguishing feature on their business processes, it is difficult using general-purpose standardized ERP software to adapt their specific characteristics. However, due to its heterogeneous, loosely coupled distributed characteristics, the BOA framework can be a good choice to meet the needs of different customers and to reduce development cycle.

5 Conclusion The presented three-layer structure model of BOA and its designing method employed SunJavaStudioCreator as development platform for its BOM and configuration software. It combined JSF visualized components into AJAX heterogeneous transmission technology and used JDOM to generate and analyse the properties configuration files of XML format, which can produce different page styles by using JavaScript tool and realize reusable modules with same business processes

Framework Designing of BOA for the Development of Enterprise Management

195

or functions and reduce software development cycle. It was successfully applied in the development of an enterprise management information system for a large scale coal machine manufacturing factory.

References 1. Dong, C.L., Guo, S.S.: The research of ERP reconfigurable mode based on the theory of evolution. Manufacturing Automation Journal 6, 15–18 (2008) 2. The Business Architecture of BOA, http://www.tongtech.com/products/boadetail.jsp 3. Johnson, D., White, A., Charland, A.: Enterprise Ajax: building robust Ajax applications. Prentice Hall PTR, New Jersey (2008) 4. Zhou, H.B.: Convert database to XML programming by JDOM. J. Computer Knowledge and Technology 2, 182–183 (2006) 5. Dudey, B., Lehr, J., Willis, B., Mattingly, L.R.: Mastering JavaServer Faces. Publishing House of Electronics Industry, Indiana (2005) 6. Jacobi, J., Fallows, J.R.: Pro JSF and AJAX: Building Rich Internet Components, pp. 3–20. USA Press (2006) 7. Gallardo, M.d. , Martínez, J., Merino, P., Rosales, E.: Using XML to implement abstraction for Model Checking. In Proceedings of the 2002 ACM Symposium on Applied Computing, SAC 2002, pp.1021–1025. ACM, New York (2002)

Training Support Vector Data Descriptors Using Converging Linear Particle Swarm Optimization Hongbo Wang, Guangzhou Zhao, and Nan Li College of Electrical Engineering, University of Zhejiang, Hangzhou, Zhejiang Province, China {whongbo,zhaogz,happyapple}@zju.edu.cn

Abstract. It is known that Support Vector Domain Description (SVDD) has been introduced to detect novel data or outliers. The key problem of training a SVDD is how to solve constrained quadratic programming (QP) problem. The Linear Particle Swarm Optimization (LPSO) is developed to optimize linear constrained functions, which is intuitive and simple to implement. However, premature convergence is possible with the LPSO. The LPSO is extended to the Converging Liner PSO (CLPSO), which is guaranteed to always find at least a local optimum. A new method using CLPSO to train SVDD is proposed. Experimental results demonstrate that the proposed method is feasible and effective for SVDD training, and its performance is better than traditional method. Keywords: Support Vector Domain Description; Constrained Quadratic Programming; Linear Particle Swarm Optimization; Premature Convergence; Converging Linear Particle Swarm Optimization.

1

Introduction

Support Vector Machines (SVM)[1,2]are a recently developed class of classifier architectures, derived from the structural risk minimization principle, which have a remarkable growth in both the theory and practice in the last decade. Support Vector Data Description (SVDD)[3] can be viewed as a smooth extension of SVM. It represents one class of known data samples in such a way that for a given test sample it can be recognized as known, or rejected as novel. Training SVM requires solving a quadratic problem (QP) with box and linear equation constraints. This problem often involves a matrix with an extremely large number of entries, which requires huge memory and computational time for large data applications. Practical techniques decompose the problem into manageable subproblems, which may be solved by numeric packages. Chunking methods suggested in [4] reduce the matrix’s dimension and is suitable to large problems. Decomposition methods [5, 6] break down the large QP problem into a series of smaller subproblems, and need a numeric QP optimizer to solve each of these problems. The most extreme case of decomposition is Sequential Minimal Optimization (SMO)[7], K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 196–204, 2010. © Springer-Verlag Berlin Heidelberg 2010

Training Support Vector Data Descriptors Using Converging LPSO

197

which solves the smallest possible optimization problem at each step. Most of the above approaches have been employed successfully to solve the learning of SVM. Particle Swarm Optimization (PSO)[8,9] is a swarm intelligent optimization method. It can find solution quickly in high dimensional space by its stochastic and multipoint searching ability. Using PSO to train SVM has been put forward recently, and this is really a novel tentative. For the non-constraint optimization problem, the linear particle swarm optimization (LPSO)[10] for linearly constrained problem was presented, and applied for training SVM[11]. In this paper, LPSO is introduced for SVDD training. To address the issue of premature convergence, a new parameter was introduced to LPSO, which guarantees the LPSO always finding a local minimum[12,13]. The new method was called Convergence Linear Particle Swarm Optimizer (CLPSO).

2

Support Vector Data Description

Given a set S = {x1 , , xn } , SVDD attempts to find a hyper-sphere (i.e. F ( R, a) ), with centre a and radius R , that contains all the patterns in S . The volume of this hypersphere R should be minimized. To allow for the presence of outliers in the training set, slack variables ξi ≥ 0 are introduced. The primal optimization problem is defined as

min

F ( R, a ) = R 2 + C ∑ ξ i

∀i

(1)

i

where the variable C gives the trade-off between simplicity (volume of the sphere) and the number of errors (number of target objects rejected). This has to be minimized under the constrains xi − a

2

≤ R 2 + ξi

∀i

(2)

The optimization problem can be translated into Lagrange problem L( R, a, α i , ξi ) = R 2 + C ∑ ξi − ∑ α i {R 2 + ξ i − ( xi2 − 2axi + a 2 )} − ∑ γ i ξi i

i

i

(3)

with Lagrange multipliers α i ≥ 0 and γ i ≥ 0 . By solving the partial derivatives of L, we obtain

∑α i

i

= 1 , C − α i − γ i = 0 , a = ∑ α i xi i

(4)

Equation (3) can be rewritten as Lagrange form of ai with above constraints max

L = ∑ α i ( xi ⋅ xi ) − ∑ α iα j ( xi ⋅ x j ) i

i

(5)

In order to have a flexible data description, as opposed to the simple hyper-sphere discussed above, a kernel function K ( xi , x j ) = Φ ( xi ) ⋅ Φ ( x j ) is introduced. After

198

H. Wang, G. Zhao, and N. Li

applying the kernel and using Lagrange optimization, the SVDD function using kernels becomes max

L = ∑ α i K ( xi , xi ) − ∑ α iα j K ( xi , x j ) i

i

0 ≤ αi ≤ C, ∑αi = 1 i

(6)

(7)

It can be observed that only data points with non-zero ai are needed in the description of the data set, therefore they are called support vectors of the description. Given the support vectors xi , a test sample z is accepted when K ( z , z ) − 2∑ α i K ( z , xi ) + ∑ α iα j K ( xi , x j ) ≤ R 2 i

i, j

(8)

Optimizing the functions in (5) and (6) is a Quadratic Programming (QP) problem with linear constraints (7). Generally the SVDD is used to describe large data sets. Solving the above problem via standard QP techniques becomes intractable, because the quadratic form of (6) needs to store a matrix whose size is equal to the square of the number of training samples.

3 3.1

Particle Swarm Optimization and Improvement Basic Particle Swarm Optimization

PSO differs from traditional optimization methods, in which a population of potential solutions is used in the search. The direct fitness information instead of function derivatives or other related knowledge is used to guide the search. This search is based on probabilistic, rather than deterministic, transition rules. The basic PSO is as follows. Each particle pi is a potential solution to the collective goal, usually to minimize a fitness function f . Each particle has a corresponding position and velocity, denoted as pi = ( pi1 , , pin ) and vi = (vi1 , , vin ) , i = 1, 2 , m respectively, where m is the number of particles and n is the dimension of the solution. PSO uses the following equations to update the particle’s velocity and position during evolution.

vit,+j1 = wt vit, j + c1r1 j ( pbestit, j − pit, j ) + c2 r2 j ( pg tj − pit, j )

(9)

pit,+j1 = pit, j + vit,+j1

(10)

Where t is the current generation, w is the inertia weight, c1 and c2 are coefficients, and r1 j , r2 j are evenly distributed random number in [0,1] , which embodies the stochastic nature of the algorithm. pbesti = ( pbesti1 , , pbestin ) is the best solution found by particle i , whereas pg = ( pg1 , , pg n ) is the best found by the whole population up to now.

Training Support Vector Data Descriptors Using Converging LPSO

3.2

199

Linear Particle Swarm Optimization

A new PSO algorithm, the Linear PSO, is developed specifically with linear constraints. Traditionally, the velocity and position update equations, shown in equations (7) (8), are specified separately for each dimension of a particle. In LPSO, particles update their velocity using the under-mentioned formula. vit +1 = wt vit + c1r1t ( pbestit − pit ) + c2 r2t ( pg tj − pit )

(11)

pit,+j1 = pit, j + vit,+j1

(12)

where the same random number r1 , r2 are adopted to update each component of the velocity vector. This approach has the advantage that the flight of particles is defined by standard linear operations on vectors, and hence it is referred to as a Linear Particle Swarm Optimizer (LPSO). 3.3

Improvement of Particle Swarm Optimization

Suitable inertia weight w provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution. In general, w is set according to the following equation w = 0.5

itermax − iter + 0.4 itermax

(13)

Where iter and itermax are the current and the maximum number of iterations respectively. A constriction factor K has been introduced to PSO, which improves the ability of PSO to constrain and control velocities[14]. K=

2 2 − r − r 2 − 4r

(14)

where r = r1 + r2 , r > 4 , and the LPSO is then vit +1 = K ( wt vit + c1r1t ( pbestit − pit ) + c2 r2t ( pg t − pit ))

(15)

If vit,+j1 is too high, particles might fly past good solutions. If vit,+j1 is too small, particles may not explore sufficiently. The parameters Vmax and Vmin = −Vmax should be determined. In many experiences, Vmax was often set at 10% ∼ 20% of the max position. The original PSO algorithm has a potentially dangerous property: if a particle’s current position coincides with the global best position, the particle will only move away from this point. This may lead to premature convergence of the algorithm. To address this issue, the Convergence Particle Swarm Optimizer (CPSO) was developed in [12] [13]. In this algorithm, the velocity update for the global best particle is

200

H. Wang, G. Zhao, and N. Li

changed to force it to search for a better solution in an area around the position of that particle. Let τ be the index of the global best particle, then pbestτ = pg

(16)

Change the velocity update equation (11) for the global best particle τ using vτt +1 = − xτt + pg t + ρ tϑτt

(17)

Where ρ t is a scaling factor and ϑ t ∼ UNIF ( −1,1) n with the property that Aϑ t = 0 . Since xτt +1 = vτt +1 + xτt = pg t + ρ tϑτt

(18)

The new position of the global best particle will be its personal best pg t , with a random vector ρ tϑτt . It is only the global best particle that is moved with equations (17) and (18), and all other particles in the swarm are still moved with the original equations (11) and (12). This new algorithm, referred to as CLPSO is guaranteed, in the limit, to always find at least a local minimum.

4

SVDD Training Based on CLPSO

From the mathematical model of SVDD (6) (7), we can learn that the variables need to be optimized are the coefficients of support vectors, so the particle can be coded as p = (α1 , , α n ) and the fitness function is selected as f ( p ) = ∑ α i K ( xi , xi ) − ∑ α iα j K ( xi , x j ) i

i

(19)

For the training of SVDD is a quadratic programming problem with box and linearly equation constraints according to the requirements of CLPSO, the position of the particle must satisfy constraints (7). We initialize the particle’s position with pij = C ⋅ rand () , where rand () is evenly distributed random number in [0,1] , and the following is the method to tune up the position to fulfill the box restriction. n

d = ∑ pij ; j =1

delta = d − 1 ; while delta > ε

{

if

delta > ε

{

s = delta / n;

if

{

pij + s < C

pij = pij + s;

}

and

pij + s > 0

Training Support Vector Data Descriptors Using Converging LPSO

201

} else

{

s = − delta / n ; if

}

{

pij − s < C

pij = pij − s ;

and

pij − s > 0

}

n

d = ∑ pij Y j ; j =1

delta = d − 1 ;

}

The velocity of particle is set to be zero. Thus, the Lagrange multipliers p satisfy constraints (7). The procedure of CLPSO to train SVDD is as shown below: 1. Initialize 1) Set all parameters of CLPSO. 2) Set current generation t = 0 . 3) Initialize the velocity vector to zero; Initialize the particle’s position randomly and then adjust each particle to satisfy the constraints (7) by above method. 4) Evaluate values of fitness function of f ( p ) for each particle; Set pbesti0 = pi0 and

pg 0 = max( pbesti0 , i = 1,

, m)

.

2. Optimize 1) Update inertia weight w using (11). 2) Generate random number r1 and r2 ; Calculate constriction factor K using (14). 3) Update the velocity of the global best particle using (17), and update the velocity of the other v > Vmax particles using (15); For any components, if i , j , vi , j = Vmax vi , j < Vmin vi , j = Vmin then ; If , then .

202

H. Wang, G. Zhao, and N. Li

4) Update particle position using (10); For any component, if pi , j > C , then pi , j = C ; If pi , j < 0 , then pi , j = 0 ; and then using the same method as in initialization to tune up the other position to satisfy the linear equation constraint. 5) Evaluate values of fitness function of f ( p ) for each particle; Update pbestit and pg t . 6) The current iteration number is set to t + 1 ; If t > itermax , then pg t is the solution of the problems. Else, return to Step 1) and repeat the above procedure again.

5

Experimental Results

In order to show the performance of the proposed method and its efficiency, we present two groups of experiments both on artificial ellipse dataset and real dataset from UCI Repository of Machine Learning Databases. The kernel functions of SVDD in two x − xj experiments are selected as Gaussian kernel function K ( xi , x j ) = exp(− i 2) .

σ

The limiting value of C was chosen to be 1, and the other parameters are selected as r1 = 2.8 , r2 = 1.3 , r = r1 + r2 = 4.1 , Vmax = 0.3 , Vmin = −0.3 . As we know that the population size has influence on the convergence and performance of the PSO. Artificial Ellipse Dataset (Fig. 1) was used for analyzing the effect of the population size. Ellipse Data 4

y

3

2

1

0

0

5

10 x

15

20

Fig. 1. Artificial Ellipse Dataset

Fig.2 shows the comparison of the running time and iteration steps for population sizes of 5 to 100 particles. The Max Iteration Steps was selected as 500. As it can be

Training Support Vector Data Descriptors Using Converging LPSO

3

203

Training Iteration Training Time

2 1 0 -1 -2 -3

0

20

40

60

80

100

Fig. 2. The dashed line denotes the training iteration steps of CLPSO. The solid line denotes the training time of CLPSO.

seen that the more particles in the population, the smaller the iteration steps. However, increasing the population sizes resulted in more training time. The experiments showed that a population of 20 to 50 is best in terms of both sides. We selected 409 (16-dimensional) training points (known class = ’A’) from Letter Image Recognition Data and 171(19-dimensional) training points of ‘BUS’ features from Vehicle Silhouettes Data Set. Correctly classified rate was defined as CR = correctly classified / Total Testing . As shown in Table 1, in all of the two datasets, the LPSO can obtain better result of CR than SMO. Table 1. Comparison of SMO and CLPSO on UCI data set

UCI dataset

Dim

Size of training set

Size of testing set

Letter Recognition

16

409

2013

Vehicle

6

18

171

Name of algorithm

Correct Rate (FR)%

SMO

86.38

CLPSO

92.00.

SMO

85.11

CLPSO

89.89

188

Conclusion

SVDD training is a quadratic programming (QP) problem. This paper proposed a method to solve the optimization by PSO, which finds solution by its stochastic and multipoint searching ability. The results of experiment show the new way is successful and accurate. Acknowledgments. This work is supported by National Natural Science Foundation of China (60872070) and Key Scientific and Technological Project of Zhejiang Province (Grant No. 2007C11094, No. 2008C21141).

204

H. Wang, G. Zhao, and N. Li

Reference 1. Vapnik, V.N.: The nature of statistical learning theory. Springer, Heidelberg (2000) 2. Cortes, C., Vapnik, V.: Support-vector networks. Machine Learning 20, 273–297 (1995) 3. Tax, D.M.J., Duin, R.P.W.: Support vector data description. Machine Learning 54, 45–66 (2004) 4. Vapnik, V.N., Kotz, S.: Estimation of dependences based on empirical data. Springer, New York (2006) 5. Osuna, E., Freund, R., Girosi, F.: Support vector machines: Training and applications (1997) 6. Osuna, E., Freund, R., Girosi, F.: An improved training algorithm for support vector machines. Citeseer (1997) 7. Platt, J.C.: Fast training of support vector machines using sequential minimal optimization (1999) 8. Kennedy, J., Eberhart, R.: Particle swarm optimization (1995) 9. Yuan, H., et al.: A modified particle swarm optimization algorithm for support vector machine training (2006) 10. Paquet, U., Engelbrecht, A.P.: New particle swarm optimiser for linearly constrained optimization 11. Paquet, U., Engelbrecht, A.P.: Raining support vector machines with particle swarms (2003) 12. Paquet, U., Engelbrecht, A.P.: Particle swarms for equality-constrained optimization. Submitted to IEEE Transactions on Evolutionary Computation 13. Bergh, F., Engelbrecht, A.P.: A new locally convergent particle swarm optimizer (2002) 14. Eberhart, R.C., Shi, Y.: Comparing inertia weights and constriction factors in particle swarm optimization. IEEE, Piscataway (2000)

Research on Modeling and Simulation of an Adaptive Combat Agent Infrastructure for Network Centric Warfare Yaozhong Zhang, An Zhang, Qingjun Xia, and Fengjuan Guo Department of Electronic and Information, Northwestern Ploytechnical University, Xi’an 710129, China {zhang_y_z,zhangan}@nwpu.edu.cn, [email protected], [email protected]

Abstract. In order to shift from platform centric warfare to Network Centric Warfare (NCW) for the military simulation, a new adaptive combat agent model is designed that with some special sub-agent advisor. The sub-agent advisor can perform like a specialist to carry out the management of sensors, situational assessment, tactical decision-making, combat mission, and communication management. So we can demonstrate realistic and valid behaviors of military entities in a Computer Generated Forces (CGF) simulation. It provides an effective training environment to exhibit advanced coordination and cooperation capability of military units in NCW with high resolution simulation. Keywords: Network Centric Warfare; tactical decision-making; computer generated force; simulation; Agent.

1 Introduction Agent-based models borrow heavily on a biological analogy. Now it has widely used in all kinds of complex system, such as ecosystem, economic system, society system and the military system [1]. An agent is a physical or virtual entity that has the ability: a) which is capable of acting in an environment; b) which can communicate directly with other agents; c) which is driven by a set of tendencies; d) which possesses resources of its own; e) which is capable of perceiving its environment; f) whose behavior tends towards satisfying its objectives [2]. So agent-based models are very appropriate for simulate the military entities, such as artificial soldiers. We can recognizing warfare as a complex adaptive system, it requires a set of agents that are individually autonomous; it means that each agent can determines its actions based on its own state and the state of the environment, without explicit external command. It has opened the doors for a number of agent-based distillation combat systems to emerge. This includes the Irreducible Semi-Autonomous Adaptive Combat (ISAAC) and the Enhanced ISAAC Neural Simulation Toolkit (EINSTein) from the US Marine Corps Combat Development Command, the Conceptual Research Oriented Combat Agent Distillation Implemented in the Littoral Environment (CROCADILE) [3]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 205–212, 2010. © Springer-Verlag Berlin Heidelberg 2010

206

Y. Zhang et al.

However existing agent-based distillation combat systems were developed mainly on platform centric warfare and current agent architectures, which limit their ability to test the character of NCW. So we need to develop a new framework for the adaptive combat agent which can reflect the very character of information war and can leading to the fundamental shift from platform centric warfare to NCW.

2 Adaptive Combat Agent Infrastructure In NCW the soldiers rich with variegated information. The sensor-to-shooter time is greatly reduced. The soldiers in the battle field may take actions based on their own raw intelligence from sensor's displays or coordination each other instead of waiting for commands. They must operate in task forces, coalition forces, and joint operations where unit and equipment capabilities vary greatly[4]. The technology shift from platform centric warfare to NCW brings much embarrassment to the combat simulation. The primary architecture of agent is difficult to demonstrate realistic and valid behaviors of military entities. So we need a new systematic Infrastructure that can support the challenge like sensors manage, situation awareness, decision making and missions manage. In our research a new Adaptive Combat Agent (ACA) is adopted which with a special sub-agents layer. 2.1 Top Structure of ACA with Special Advisor In NCW the growing trend in the military simulation community is for greater degrees of system interoperability, resource distribution, entity and aggregate behavioral complexity. So our ACA must be designed and implemented for these varying degrees of integration. The new infrastructure of ACA in our research is shown in Fig.1.

Fig. 1. Top structure of combat agent with advisor

The ACA has a sub-agents layer that is building up with four specialist advisor. These advisors roughly correspond to the reality that a soldier's work flows in NCW. This design allows the implementation of different artificial intelligence techniques to be applied to their appropriate categories of problems. It also provides facilities to abstract the interfaces for specifically function and simulations. The key to this approach is the implementation of advisors which are sub-agents with special knowledge and skills.

Research on Modeling and Simulation of an Adaptive Combat Agent Infrastructure

207

2.2 Microstructure of the Sub-agent Advisor The four advisors in Fig.1 have the same common microstructure as shown in Fig.2.

Fig. 2. Microstructure of Sub-Agent Advisor

The advisor controls the interfaces to receives and stores external information or to interact with other ACA. The data received through interfaces can be translated into internal representations which represent its perceptions of the battlefield environment. The choice of internal structures can depend on the level of abstraction required or the advisor’s special ability in order to deal the information efficiently and effectively. The advisor can receive external command from its direct leadership which has the higher privilege. The inference engine is the key to the advisor which derived from expert system shells and processing control rules to utilize the battlefield information. This open architecture is very convenient for other features such as neural networks, genetic algorithm, case-based reasoning, and machine learning which can be easily added to the inference engine. 2.3 Sensor Manage Advisor Sub-agent The functional frame of Sensor Manage Advisor (SMA) is shown in Fig.3.

Fig. 3. Sensor manager advisor functional frame

208

Y. Zhang et al.

The SMA sub-agent can collect the battlefield data (including terrain databases, hostile unit etc) according to the detect task. Raw data from SMA can be translated into valid battlefield data and send to the Situation Advisor (SA) where the Common Operation Picture (COP) is constructed. The command of sensor strategy control or work mode control is from the Mission Mange Advisor (MMA). In NCW the sensor’s ability is expand rapidity, so we can test the new features of the sensor without to develop the simulation from scratch. We can only do this by adding new services to the SMA. 2.4 Situation Advisor Sub-agent The functional frame of SA is shown in Fig.4.

Fig. 4. Situation advisor functional frame

Here we can see that the battlefield data from SMA first be translated into the internal representation which be called battlefield information. From this information and the history information the inference engine of SA can acquire the situation awareness and the prediction of the situation. Finally the SA can construct the Common Operation Picture (COP) and submit to the Tactical Decision Advisor (TDA). In NCW the agent rich with variegated information from the information network. How to verify the information each other and construct the COP are becoming complex and difficult. With SA we can test the new fusion arithmetic and other theory for the situation awareness. 2.5 Tactical Decision Advisor Sub-agent The functional frame of TDA is shown in Fig.5.

Fig. 5. Tactical decision advisor functional frame

As we know TDA works under conditions best described as "the fog and friction of war". TDA must first interpret his understanding as a set of alternative courses of actions (this is called decision-making). Then he must be able to select the best course of action from those alternatives (this is called mission-making). In NCW TDA must also submit the mission’s collaboration and cooperation request to the MMA.

Research on Modeling and Simulation of an Adaptive Combat Agent Infrastructure

209

2.6 Mission Manage Advisor Sub-Agent The functional frame of MMA is shown in Fig.6. Sensor Manage Advisor

Mission Effectiveness Tactical Decision Advisor

Mission Collaboration

Mission Cooperation

Action selection

Action

Mission Manage Advisor Other Agents

Fig. 6. Mission manage advisor functional frame

In NCW how to exhibit advanced coordination and cooperation of military units (which are found on a real battlefield) is paramount to the MMA. The MMA deals with diverse types of missions in different contexts. MMA can submit the best action using the optimization techniques and it can also monitor the mission effectiveness.

3 Software Design of the Adaptive Agent Combat System Since the beginning of computer simulations, military analysts and programmers have developed different combat simulations for every scale of combat operations[5][6]. Adaptive Agent Combat System (AACS) in our research is belong to high-resolution combat models, which are detailed models of warfare, represent individual combatants as separate entities with numerous attributes. Each agent in AACS has four advisors as described above which can act as a real solder in network centric warfare environment.

Fig. 7. Software structure of the AACS

210

Y. Zhang et al.

The AACS try to explore combat as a self-organized emergent phenomenon, and take a bottom-up, synthesis approach, vice the more traditional top-down approach. It is developed with object oriented C++ code. The software structure is shown in Fig.7. Its running snapshots are shown in Fig.8.

Fig. 8. Snapshots of the AACS

4 Scenario Analysis In our simulation the blue agent force was selected as two squads; each squad has 25 entities and their common target. The red agent was also selected as two squads; each squad has 25 entities and their common target to be protected. The detailed parameters are shown in table1. Table 1. Simulation parameters

Agent Blue Group1 Blue Group2 Blue Group1 Blue Group2

Velocity Detect Detect Weapon Hit Fire Range Cycle Equip prob range (s) (unit) (unit)

Target position

Initially amount

Initially position

25

(300,100)

1

100

5

3

0.1

100

(300,480)

25

(150,100)

1

100

5

3

0.1

100

(450,480)

25

(200,300)

1

100

5

3

0.1

100

25

(350,300)

1

100

5

3

0.1

100

With this scenario, we analyzed the combat outputs with four different settings of blue agent's sensor ability in the same conditions while holding all other parameters the same. The different settings of blue agent’s sensor ability including the detect range and detect cycle. As we know the ability of reconnaissance is the key to NCW, so with this scenario we can easily contrast the advantage that the changes of sensor’ ability to the combat. The simulation was run 500 times for each setting and the result was shown in Fig.9.

Research on Modeling and Simulation of an Adaptive Combat Agent Infrastructure

211

Fig. 9. Agent loss graph according to vary sensor capability

From the analyses results we can see that as the blue agent's sensor ability increases, its loss decreases while the red agent's loss increases. Because with the dominance of sensor information the blue agent can predict possible courses of actions, they can provide an assessment of possible future actions. The goal of our research is to provide a useful tool to the military simulation for the NCW. From the result we can see that the AACS had the potential to be a useful aid to situation awareness and decision-making. With AACS we can easily analysis the advanced characteristics such as the superiority of information, the coordination and cooperation capability of military units in NCW with high resolution simulation.

5 Conclusion The ACA in our research can demonstrate realistic and valid behaviors of military entities. It can provide an effective training environment or predictive capability for the military simulation in NCW. The ACA simulation software is very convenient to exhibit advanced coordination and cooperation ability of military units that required in NCW. From the simulation result we can see that the Infrastructure of ACA not only coincides with real-world models but it also provides many software engineering advantages as well.

References 1. Parunak, H., Sven, B.: Entropy and Self-Organization in Multi-Agent Systems. In: Proceedings of the International Conference on Autonomous Agents, pp. 124–130 (2001) 2. Jilson, E.W., Mert, E.: Modeling Conventional Land Combat in a Multi-Agent System Using Generalizations of the Different Combat Entities and Combat Operations. Master’s Thesis, Naval Postgraduate School (2001) 3. Yang, A., Abbass, H.A., Sarker, R.: Landscape dynamics in multi-agent simulation combat systems. In: Webb, G.I., Yu, X. (eds.) AI 2004. LNCS (LNAI), vol. 3339, pp. 39–50. Springer, Heidelberg (2004)

212

Y. Zhang et al.

4. Functional Description of a Command Agent. Lockheed Martin Corporation, ADST-IICDRL-023R- 96OO237A (1997) 5. Hencke, R.B.: An Agetn-based Approach to Analyzing Information and Coordination in Combat. Master’s Thesis, Naval Postgraduate School (1998) 6. David, T., Lee, L.: Testing a New C2 Model of Battlefield Information Sharing and Coordination. In: The 14th ICCRTS, Washington, DC (2009)

Genetic Algorithm-Based Support Vector Classification Method for Multi-spectral Remote Sensing Image Yi-nan Guo, Da-wei Xiao, and Mei Yang School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou, Jiangsu221008 P.R. China [email protected]

Abstract. The traditional classification methods based on asymptotic theory for multi-spectral remote sensing image need the infinite training samples, which is impossible to be satisfied. Support vector classification (SVC) method based on small samples overcome above difficulty. However, the values of hyperparameters in SVC directly determine the method’s performance, which are randomly selected. In order to obtain the optimal parameters, genetic algorithms (GAs) are introduced. Experimental results indicate that this method can not only save time for classification, but also improve the generalization of the SVC model. Keywords: remote sensing image, classification, genetic algorithms, parameters selection.

1 Introduction In remote sensing images, different cultures are reflected by different brightness and spatial position. Each objects has its own spectral and spatial information, hence feature vectors, which characterize various types of objects, distribute in different feature space [1]. Therefore, classification for remote sensing image is to partition different cultures by computers. Useful features are extracted through analyzing the spectrum and spatial information. Then these features are classified into nonoverlapping feature space. At last, each image pixel are mapped into corresponding feature space[2]. Multi-spectral images have high resolution, multi-band and massive data. So they more require high quality classification [3]. In traditional classification methods for remote sensing image: minimum distance method [4-5] is easy to fall into local optimum because only the local characteristics are considered. But it has better performance for block objects. Maximum likelihood classification method [4] needs the probability distribution function of each class to be normal distribution. However, the selected samples may deviate from normal distribution. Although the radial basis neural network [5] is good for classification of block object, it is not ideal in classification of confusion region. Above conventional methods based on asymptotic theory for multi-spectral remote sensing image need the infinite training samples, which is impossible to be satisfied. Support vector machine (SVM) based on structural risk minimization [6] K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 213–220, 2010. © Springer-Verlag Berlin Heidelberg 2010

214

Y.-n. Guo, D.-w. Xiao, and M. Yang

has stronger generalization. It is fit for solving the nonlinear or high dimensions problems with small samples. However, the learning ability of SVM depends on the values of parameter in SVM models. By now parameters selection of SVM-based classification method for remote sensing image is always determined by experience. This lead to unstable and worse classification results. So genetic algorithm is introduced to obtain optimal parameters of SVC model for multi-spectral remote sensing image in the paper. GA-based classifier can not only save time, but also improve the generalization of models.

2 Least Square Support Vector Machine (LS-SVM) 2.1 The Principle of LS-SVM Traditional SVM with the inequality constraints is based on structural risk minimization. It is used to solve nonlinear, high dimension problems with small samples. However, the computational complexity is increasing as the training samples is more. In order to decrease the computational complexity, LS-SVM is proposed. It uses ε i2 as loss function to convert the traditional SVM with inequality constraints into equality constraints. Suppose a given set has n samples. xi ∈ X = Rn and

yi ∈Y = {1,− 1} respectively denote the input and output of i-th sample.

， y )，L ，( x ， y

T = {( x1

1

n

n

)} ∈ ( X × Y ) n

(1)

Based on above set, LS-SVM can be described by the following optimal problem: ω

1 1 n 2 ω + C ∑ ε i2 2 2 i =1 y i (( ω • x i ) + b ) = 1 − ε i

min

， b， ε s.t

； i = 1，L ， n

(2)

Its dual form is: min α

1 2 n

s .t

∑

i=1

n

n

∑ ∑ i=1

y iα

y i y jα iα

j

(Φ ( x i ) • Φ ( x

j

)) −

j=1

i

n

∑

α

j

j =1

= 0

，

C ≥ α

i

≥ 0

，

， ，

i = 1 L

n

(3)

Here, C is the hyper-parameter. Its value directly influences the performance of LSSVM. So how to obtain the optimal value of hyper-parameter is a key problem. 2.2 Multi-classification Method Classification for remote sensing images using LS-SVM is a multi-classification problem in fact. However, LS-SVM is usually used in binary classification problems. So it is necessary to convert a multi-classification problem into many binary classification problems. At present, some specific methods have been given[7-9], such as One-Against-All(OAA), One-Against-One(OAO), Error-Correcting Output Code(ECOC), Minimum Output Code( MOC) and so on.

Genetic Algorithm-Based SVC Method for Multi-spectral Remote Sensing Image

215

In these methods: 1)There exists impartible area using OAA because it is asymmetric between the number of training samples belong to positive class and negative class. But the binary classifiers needed to be trained are less. 2)The impartible area is less whereas the trained binary classifiers are more when OAO is adopted. 3)When ECOC is used, the training speed is quicker. But the accuracy is inconsistent as it is repeatedly used under the same conditions. 4)MOC method is fastest, best precision and recognition rate.

3 GA-based Parameter Optimization for LS-SVM Model 3.1 Parameters Optimization for LS-SVM Model According to LS-SVM dual form shown in formula(3), we know that two parameters are needed to be optimized. They are C and kernel parameter. However, there are different kernel parameters in different kernel functions. Here, Gaussian radial basis function (RBF) is used as kernel function. Its expression is shown as follows:

K (x

，x′) = exp( − x − x′

2

σ 2)

(4)

where σ is variance. It controls radial coverage of RBF. In this paper, C and σ are parameters needed to be optimized.In order to use GA for parameter selection in LS-SVM model, the methods mentioned in Section2.2 are adopted to pre-process the training samples and test samples so as to convert multiclassification problem into binary classification problems. 3.2 Design GA Operations Code. Two parameters in LS-SVM classification method are formed into individual in genetic algorithm. Considering the optimized parameters are real-number, each individual is coded by two real-numbers shown as x = (C , σ ) . Selection Operator. Random rank selection and elitist preserved strategy are used as selection operator. Detailed selection operation is shown as follows: Step1: The individuals with the highest or lowest fitness values are finded from current population. If the best individual in current population is better than the individuals in preserved set, this individual will be saved in preserved set. Then the worst individual in current population is substituted by the best individual in preserved set. Step2: Aiming at any individual, N individuals are randomly selected from the population and compared with it. The individuals with less dominated individuals are preserved into the population in next generation. Step3: Repeated above procedure for m times, the population in next generation is obtained. Here, m is the population size.

216

Y.-n. Guo, D.-w. Xiao, and M. Yang

Crossover Operator. Arithmetic crossover operator is adopted. Two new individuals are generatd by linear combination with two individuals. Suppose two individuals are

x A (t ) , x B (t ) . The new individuals after crossover operation are: ⎧⎪ x A ( t + 1) = ax B ( t ) + (1 − a ) x A ( t ) ⎨ ⎪⎩ x B ( t + 1) = ax A ( t ) + (1 − a ) x B ( t )

(5)

where a ∈ (0,1) is constant. Mutation Operator. Non-uniform mutation operator is used. Taken C as example, '

mutation operator is shown as follow. Here, C and C denote the parameter before and after mutation. Suppose [C min , C max ] is the bound of this variable. ⎧⎪ C + Δ ( t C' = ⎨ ⎪⎩ C + Δ ( t

，C − C) ，C − C )

， ( 0， 1) = 1

if random ( 0 1 ) = 0

max

if random

min

(6)

Let y expresses C max − C or C − C min . Δ (t , y ) ∈ [0, y ] denotes stochastic number satisfying non-uniform distribution. Fitness Function. The fitness function is used to evaluate individuals. Here, error rate of LS-SVM model is used as the fitness function, expressed as follows: N

min st

f (C i , σ i ) =

∑ YP j =1

j

≠ Yj i = 1, 2 , L , m

N

C min ≤ C ≤ C max ; σ

min

≤σ ≤σ

(7)

max

where N is the number of training samples. YPj and Yj are the predictive and true classification result. [Cmin , Cmax] and [σmin,σmax] are searching bound of C and σ. Termination Condition. The genetic algorithm is stopped when the deviation satisfies a given value expressed by ε or the maximum generation is reached. Then we get the optimal solution.

4 Simulation and Analysis In order to verify the rationality of the proposed method, we apply it to the classification for remote sensing image. Aiming at the optimization model in Setion 3.1, GA-based SVC model is realized adopting MATLAB. In all experiments, we select the same training and testing samples. In multi-spectral remote sensing images, massive data information are contained. So the cultures are easy to be recognised. However, the redundant data increases the difficulty for image processing. Therefore, they need to be pre-processed. The processed original remote sensing images used in experiments are shown in Fig.1. Here, the size of each picture is 197×119.

Genetic Algorithm-Based SVC Method for Multi-spectral Remote Sensing Image

(a) band 1

(b)band 2

(e) band 5

(c )band 3

(f) band 6

217

(d)band 4

(g) band 7

Fig. 1. 7 bands of original remote sensing image

(a)Cross-validation Optimization Method We use cross-validation method to select hyper-parameters in LS-SVM classification model. Four coding methods, including OAO,OAA,ECOC,MOC are adopted. Based on ten traning samples, simulation results show in Tab 1. Table 1. Simulation result adopting cross-validation method Code

MOC

ECOC

OAA

OAO

Time(s)

62.156

91.281

137.08

334.11

In general, multi-classification problems are converted to multi-binary classification problems. According to the coding theory of multi-classifier, there are three binary classifiers needed to be trained in LS-SVM model using MOC coding method. In OAA-based classification method, seven binary classifiers are trained. Twenty one binary classifiers are trained in OAO-based classification method. However, in ECOC-based classification method, the number of decision functions expressed by L determines the number of binary classifiers. In this paper, L=4. Considering the number of training binary classifiers and the running time shown in Table 1, we know that the domination rank of above four coding methods are MOC f ECOC f OAA f OAO. It is obvious that the coding method influences the hyper-parameter selection in the model. Aiming at original multi-spectral remote sensing images in Section 4.1, the classification result adopting above four coding methods are shown in Fig 2. In all figures, the black parts are impartible. From Fig 2, we know that MOC-based classification method is the best one. There are too many impartible area using OAA and OAO coding methods. In addition, OAO coding method needs much running time. Therefore, we use MOC and ECOC to compare the classification performances.

218

Y.-n. Guo, D.-w. Xiao, and M. Yang

(a)MOC

(b)ECOC

(c)OAA

(d)OAO

Fig. 2. Classification results adopting MOC,ECOC, OAA, OAO

Taking different number of training samples, the accuracy rate of different coding methods are shown in Tab 2. Table 2. The Accuracy rate of different coding methods Number of training Data Coding method

Accuracy

10

20

OAA

MOC

OAA

MOC

0.9

0.8857

0.8786

0.9143

The classification results adopting cross-validation method with MOC and ECOC coding methods are shown in Fig 3 and Fig 4.

(a )MOC

(b ) ECOC

Fig. 3. Classification results adopting cross-validation method with MOC and ECOC(the number of training samples=10)

(a )MOC

(b ) ECOC

Fig. 4. Classification results adopting cross-validation method with MOC and ECOC(the number of training samples=20)

Genetic Algorithm-Based SVC Method for Multi-spectral Remote Sensing Image

219

From Tab 2 and Fig 3-4, we know that no matter how the accuracy rate of MOC method is, it’s classification result is much better than ECOC because there are less impartible area. Therefore, this coding method is suitable to be used in LS-SVM classification model for remote sensing images. It is obvious that the coding method used in SVC model significantly influences the classification performances for remote sensing images. (b)Comparison between cross-validation method and GA-based method. Parameters in GA-based LS-SVM classification method are shown in Tab 3. From Tab 1, we know running time of MOA-based cross-validation method is least. So we use it to compare the performance with GA-based classification method. Simulation results are shown in Tab 4. Table 3. GA’s parameters Crossover Probability

Mutation Probability

Population Size

Accuracy

Termination Generation

0.8

0.03

10

0.001

50

[Cmin, Cmax] [σmin,σmax] [1,1000]

[0,1]

Table 4. Comparison of performance between cross-validation and GA method number of training Data

10

methods

Crossvalidation

running time(s)

62.156

20

60

GA

Crossvalidation

GA

GA

9.625

290.78

19.906

173.42

It is obvious that running tims of GA-based method is less than cross-validation method aiming at the same training data. When the training data is three times to cross-validation method, GA-based method also has the better performance. Therefore, it is better to use GA to select the LS-SVM hyper-parameters. By comparing simulation results shown in Fig 5, the classification effect of methods adopting optimal hyper-parameters are better than experience-based method without hyper-parameters optimization. Through compared with Fig.1, GA-based method has the most precise classification. The impartible area is obviously less. Therefore, it is necessary to optimize the hyper-parameters in LS-SVM classification model for remote sensing images.

(a)Experience method

(b)Cross-validation method

(c) GA-based method

Fig. 5. Classification result with three methods

220

Y.-n. Guo, D.-w. Xiao, and M. Yang

In conclusion, we know that GA-based LS-SVM classification method for remote sensing images not only has less running time, but also has best classification results. It also improves the generalization of the model. So it is important for multiclassification problem to optimize hyper-parameters of SVC model and adopt suitable code method.

5 Conclusions The traditional classification methods based on asymptotic theory for multi-spectral remote sensing image need the infinite training samples, which is impossible to be satisfied. Support vector classification method based on small samples can overcome above difficulty. However, the values of hyper-parameters in SVC directly determine the method’s performance, which are randomly selected. In order to obtain the optimal parameters, genetic algorithms are introduced. Experimental results indicate that this method can not only save time for classification, but also improve the generalization of the SVC model. The new optimization method used in SVC for remote sensing image will be the next work. Acknowledgments. This work was supported by National Natural Science Foundation of China under Grant 60805025 and Qinglan Project of Jiangsu.

References 1. Du, F.L., Tian, Q.J., Xia, X.Q.: Expectation and Evaluation of the Methods on Remote Sensing Image Classifications. Remote Sensing Technology and Application 19, 521–524 (2004) 2. Zhou, F., Pan, H.P., Du, Z.S.: The classification of RBF neural network based on principal component analysis to multispectrum satellite remote-sensing images. Computing Techniques for Geophysical and Geochemical Exploration 30, 158–162 (2008) 3. Yang, Z.S., Guo, L., Luo, X., Hu, X.T.: Research on segmented PCA based on ban selection algorithm of hyperspectral image. Engineering of Surveying and Mapping 15, 15–18 (2006) 4. Liu, Q.J., Lin, Q.Z.: Classification of remote sensing image based on immune network. Computer Engineering and Application 44, 24–27 (2008) 5. Tan, K., Du, P.J.: Hyperspectral Remote Sensing Image Classification Based on Radical Basis Function Neural Network. Spectroscopy and Spectral Analysis 28, 2009–2013 (2008) 6. Vapnik, V.: The Nature of Statistical Learning Theory. Springer, New York (1995) 7. LI, X.C.: A Remote-sense Image Classification Technology Based on SVM. Communications Technology 42, 115–117 (2009) 8. Mathur, A., Foody, G.M.: Multiclass and binary SVM classification: Implications for training and classification users. Geoscience and Remote Sensing Letters 5, 241–245 (2008) 9. Gou, B., Huang, X.W.: SVM Multi-Class Classification. Journal of Data Acquisition & Processing 21, 334–339 (2006)

Grids-Based Data Parallel Computing for Learning Optimization in a Networked Learning Control Systems* Lijun Xu, Minrui Fei1,**, T.C. Yang2, and Wei Yu3 1

Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, Shanghai 200072, China 2 University of Sussex, UK 3 CSK Systems(Shanghai) Co., LTD, Shanghai 200002, China [email protected], [email protected], [email protected], [email protected]

Abstract. This paper investigates a fast parallel computing scheme for the leaning control of a class of two-layered Networked Learning Control Systems (NLCSs). This class of systems is subject to imperfect Quality of Service (QoS) in signal transmission, and requires a real-time fast learning. A parallel computational model for this task is established in the paper. Based on some of grid computing technologies and optimal scheduling, an effective scheme is developed to make full use of distributed computing resources, and thus to achieve a fast multi-objective optimization for the learning task under study. Experiments of the scheme show that it indeed provides a required fast on-line learning for NLCSs. Keywords: Grids, Parallel computing, Optimization, Real-time.

1 Introduction A networked control system (NCS)[1] is the control system whose sensors, actuators and controllers are closed over communication networks, which has been widely used in control field due to various advantages including modular system design with greater flexibility, low cost of installation, powerful system diagnosis and ease of maintenance. We have already proposed two-layer networked learning control system (NLCS[2-3]) architecture that is suitable for complex plants. Due to complexity and varieties of data sources, learning agents consume a large amount of computing resources and cannot satisfy real-time [4] requirements. Grid as a shared service-oriented computing is becoming a focus, which has more powerful computing capabilities, having broad application prospects [5]. Here, grid computing *

This work is supported by National Natural Science Foundation of China under Grant 60774059, the Excellent Discipline Head Plan Project of Shanghai under Grant 08XD14018, Shanghai Science and Technology International Cooperation Project 08160705900 and Mechatronics Engineering Innovation Group project from Shanghai Education Commission. ** Corresponding author. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 221–232, 2010. © Springer-Verlag Berlin Heidelberg 2010

222

L. Xu et al.

is established into two-layer NLCS to meet real-time and QoS requirements of the system through learning scheduling tasks parallelization and fast grid computing. Genetic algorithm (GA) [6] is a class of stochastic optimization inspired by the behavior of Darwin organic evolution. To improve the performance of original GA and avoid trapping to local excellent situations, we construct a new quantum solutions expression and adaptive crossover and mutation probability for multi-objective optimization to seek the optimal decision of network resources allocation. The rest of the paper is organized as follows: Section 2 introduces a two-layer architecture networked control system. Section 3 presents a learning scheduling optimization scheme and efficient solving method of quantum-inspired adaptive genetic algorithm. Section 4 establishes learning task parallel computing. Section 5 provides the experimental results. Finally, a brief conclusion is given in Section 6.

2 System Description In this paper, two-layer networked learning control system architecture [2-3] is introduced as shown in Fig.1,with the objectives to achieve better control performance, better interference rejection and to increase adaptability to varying environment. Local controller connects with sensors and actuators, which are attached to the complex plant via the bottom layer communication network, typically some kind of field bus dedicated to real-time control. Local controller also communicates with computer system which typically functions as a learning and scheduling agent through the upper layer communication network. The general architecture can be used in many industrial applications for distributed plants especially in power systems.

Fig. 1. Two-layer networked learning control system (NLCS)

The learning and scheduling agent update knowledge continuously online, finding the underlying laws. Online optimization improved control performance. Because of the complexity of learning control algorithm and consumption of a large number of

Grids-Based Data Parallel Computing for Learning Optimization in a NLCSs

223

computing resources, and network is typically some kind of existing local area network (LAN), wide area network (WAN), or possibly the Internet, network service quality (QoS) and delay are uncertainties, real-time requirements are not satisfied. Grid and grid scheduling, to some extent, can integrate these network nodes over the existing IT architecture to achieve dynamically sharing and collaboration of CPU, memory and database resources. Through the parallel task scheduling and grid services, the learning and scheduling unit in upper layer network reached high-speed computing to enhance service performance and real-time requirements.

3 Optimization of Networked Learning Control System The scheduling of a NLCS mainly allocates network resources and schedules the information transmission of control loops. Using the scheduling scheme, the control system may satisfy the requirements of real time and some restricted control tasks. In addition, it can ensure that the information of control tasks is transmitted during a certain communication time in order to improve the network utilization. 3.1 Multi-objective Optimization Problem in NLCS Multi-objective optimization(MOO) method dynamically adjusts the sampling of each loop, and using possible smallest bandwidth requirements to achieve optimal control performance. The MOO problem according to [7] with described rules becomes:

min

n

n

i =1

i =1

J = ∑ (α i + β i ti ) + γ i ∑ bi

(1)

subject to C T b ≤ U d (1 − ω )

(2)

bimin ≤ bi ≤ bimax

(3)

bi = bimin , if ei ≤ eith

(4)

We use Rights Law compounding system control performances called integral absolute error (IAE) and bandwidth consumption, aiming at minimize demand for bandwidth under the best system performance to the single-objective function. Where γ i is the weight coefficient. (2) is global availability of bandwidth in NLCS, where

b = [ b1 , b2 " bn ] , C = [1,1, " ,1] and percentage ω is introduced which is the T

T

proportion of network bandwidth reserved for other functions. (3) describes bandwidth consumption of subsystem

bi is bound by [bimin , bimax ] . Some expert

knowledge[8] is applied to the rules as part of constraints. Control loops require only

224

L. Xu et al.

less bandwidth to maintain the transmission of information near the equilibrium point.

eith expresses a sufficiently small error thresholds of the i th control loop. Obviously, the conventional mathematical programming is very difficult to solve the optimization problems and achieving real-time except some heuristic modern optimization algorithms. In this paper, an adaptive genetic algorithm will be given to solve the multi-objective scheduling optimization problem and data parallelization and grid computing technology will be utilized to achieve real-time control in NLCS. 3.2 Solving Method of A Quantum-Inspired Adaptive Genetic Algorithm In the analysis of network load distribution, the key is to find a optimal allocation scheme for subsystems. Genetic Algorithm (GA)[9] imitates biological evolution based on natural selection, which is a stochastic optimization method of biological evolution process imitation. Due to the adaptation of nonlinear multi-objective optimization and characteristics of parallelizable, we present a quantum-inspired adaptive genetic algorithm to solve the above optimization problems in NLCS. The crossover probability Pc and mutation probability Pm of our adaptive genetic algorithm are respectively:

⎧ b3 ( f max − f ' ) ' ⎧ b1 ( f max − f ) , f ≥ f , f ≥ f avg avg ⎪ ⎪ Pc = ⎨ f max − f avg , Pm = ⎨ f max − f avg ⎪b , ⎪b , f < f avg f ' < f avg ⎩ 2 ⎩ 4 Where

(5)

f max is the largest fitness value of population, f avg is the average , f the

larger fitness value in two individuals for crossover, for mutation and

f ' is fitness value of individuals

b1 , b2 , b3 , b4 are empirical constants. The crossover and mutation

probability of adaptive GA can automatically change with fitness. From the perspective of quantum mechanics, a new genetic algorithm model is proposed. This model is based on the DELTA potential, which has quantum behavior. The new genetic algorithm is simple and easy to achieve and have advantages of less adjustable parameters, with good stability and convergence. The basic unit of information in quantum computation [10] is the qubit. A qubit is a two-level quantum system and can be represented by a unit vector of a two dimensional Hilbert space

(α , β ∉ ^) :

ψ =α 0 + β 1 ,

α + β =1 2

2

(6)

The state of a qubit can be changed by the operation with a quantum gate. Inspired by the concept of quantum computing, PSO is designed with a novel Q-bit

Grids-Based Data Parallel Computing for Learning Optimization in a NLCSs

225

representation, a Q-gate as a variation operator, and an observation process. A Q-bit individual as a string of n Q-bits is defined as (7).

⎡α tj1 α tj 2 α tjm ⎤ q =⎢ t t " t ⎥ ⎢⎣ β j1 β j 2 β jm ⎥⎦ t j

(7)

In this paper, PSO use the appropriate Q-gate, by which operation the Q-bit should be updated as the step of mutation. The following rotation gate is used as a Q-gate in PSO, such

⎡cos(Δθ ) − sin(Δθ ) ⎤ U (Δθ ) = ⎢ cos(Δθ ) ⎥⎦ ⎣sin(Δθ )

(8)

θi = s(α i , βi ) × Δθi

(9)

1 (α i∗ , β i∗ )

Δθ

(α i , βi )

0

Fig. 2. Polar plot of the rotation gate for qubit individuals

⎡α i∗ ⎤ ⎡α i ⎤ ⎡ cos(Δθi ) − sin(Δθi ) ⎤ ⎡α i ⎤ ⎢ ∗ ⎥ = U (Δθi ) ⎢ ⎥ = ⎢ cos(Δθi ) ⎥⎦ ⎢⎣ β i ⎥⎦ ⎢⎣ β i ⎥⎦ ⎣ βi ⎦ ⎣sin(Δθi )

(10)

Where Δθ is a rotation angle of each Q-bit toward state depending on its sign. 3.3 Parallel Task Model Data parallel calculation mode[11] is single program multiple data (SPMD). The parallel task load of parallelizable quantum-inspired adaptive genetic algorithm will be divided into finite tasks, which were the smallest division of original mission, being mapped to independent resources and no causal relationship implementation.

226

L. Xu et al.

Fig. 3. Dynamic Data Parallel Task Model

Figure 3 shows the dynamic data parallel mode. Subtask segmentation unit is responsible for a clear delineation of rules, who divides total task into sub-task groups. Subtask scheduling unit provides allocation mapping for each subtask execution unit. Subtask execution unit processes sub-task groups and send results to integration unit. Subtasks integration unit receive all the partial results from subtask execution units and collecting using the analysis results to assembly achieve specific application logic. Reasonable population Individual evaluation parallel computing achieved distributed desktop systems and cluster resource sharing.

4 Grids 4.1 Grid Model Grid system is defined as finite set

R in available grid computing resource M :

R = {R1 , R1 ," , RM }

Fig. 4. Grid Services Architecture

(11)

Grids-Based Data Parallel Computing for Learning Optimization in a NLCSs

227

Under the condition of performance and time constraints, parameters of resources depend on the executive function of specific subtasks and computational complexity. Grid computing resources are multi-site and distributed workstations, parallel machines or clusters connected through the wide area network to achieve local and remote applications. Figure 4 shows the services architecture of the grid resources. If the number of subtasks is ni ∈ N 0 and resource is Ri , Tpi is processing time :

Tpi = pi + t pi ni Where

(12)

pi is the execution initial offset and t pi is the proportionality constant of

processing time and the scale of sub-mission among allocate resources. We get the communication time:

Tci = di + tci ni Where

(13)

d i is the communication initial offset and tci is the proportionality constant

of execution time and the scale of sub-mission among allocate resources. Therefore, we will obtain the total execution time Ti as follows:

Ti = Tpi + Tci = pi + d i + t pi ni + tci ni = ( pi + di ) + (t pi + tci )ni

(14)

⎧e + t n with ni > 0 ⇒ Ti = ⎨ i i i , ei = pi + d i , ti = t pi + tci with ni = 0 ⎩0

(15)

4.2 Grid Matching Strategy Scheduling server served as a provider to achieve job service, which needed to make rational use of the grid resources needed to complete the job request that the users submitted. Scheduling is composed by multiple independent processes and each process periodically checks the database state to check the task partitioning, scheduling, results validation, results of integration and delete files and so on. MultiQoS constraints based grid selection task scheduling approach packaged the individual fitness evaluation process into moderate-sized independent task unit to assign to computing nodes, achieving the control of the multi-objective genetic algorithm implementation process. (1) Tasks Division Scheduling server divides the population of each generation and allocates computing nodes to each individual through the real-time requirements. (2) Processing the Requests of Computing Nodes The daemons of computing nodes send timing requests to the scheduling server. The server returns a response message after receiving the request, which contains a set of tasks unit descriptions and URL address of related data files. According to this

228

L. Xu et al.

address, computing nodes download the specified file of objective function value at the local calculation via the standard HTTP protocol. When task is completed, results are automatically uploaded to the server, and at the same time sending a request to the server to apply for a new task unit. (3) Multi-QoS Constraints Based Grid Choice Task Scheduling Tasks with QoS requirements are of different groups, processing according to their QoS levels. Relatively high QoS of tasks always have the higher priority to access to resources, avoiding low QoS tasks occupy resources given to high QoS task which may lead to be forced to wait in a queue for the higher one. According to the Prediction of system performance, we obtain the expectations of the completion time and its standard deviation for the tasks in different groups which meet the requirements of QoS, then calculated relative standard deviation on the overall mean value by the standard deviation value. Relative standard deviation can effectively express the dispersion of the task. Min-Min algorithm is utilized scheduling system when the degree of dispersion is relatively small; while large, the scheduler option is of using Max-Min algorithm. Finally, different QoS groups via cycle and round robin selection get a QoS guided more excellent performance Min-Min results. (4) Implement of Optimal Control Algorithm Scheduling server traces the task units and the corresponding results of the state track records in the database. Once checked the state change, the scheduling server send them to the appropriate processes to deal with and sort out the results after finishing calculating all the objective functions. Scheduling server decide whether to produce the next generation or sort, starting a new round of iteration according to the above results. Optimized shape parameters are satisfied within a reasonable range.

5 Experimental Analysis In this section, we present the experimental analysis and results in order to operate and evaluate scheduling optimization strategy and show the validity of quantuminspired adaptive genetic algorithm in NLCS. Also we compared the effectiveness of parallel computing grid with no introducing condition.

Fig. 5. Experimental Network Topology

Grids-Based Data Parallel Computing for Learning Optimization in a NLCSs

229

Table 1. System Configuration of Grid Computing Resource Nodes Grid Nodes

Learning Unit 2 Intel E3200 2.4G

Scheduling Server Intel Xeon 2.0

OS

Learning Unit 1 Intel Core(TM)2 Quad Q8300 @ 2.5GHz 4GB SATA 600G 10M/100M Adaptive RHEL 5

2GB SATA 320G 10M/100M Adaptive RHEL 5

4GB 200G Raid 1 Dual NIC 100M RHEL 5

Software Virtualization

JDK1.5 GT4 No

JDK1.5 GT4 No

JDK1.5 GT4 No

IP

218.242.4.25

58.198.109.24 7

222.66.167.22 9

CPU

RAM HD NIC

Grid resources Intel Xeon 2.3

8GB 300G Raid 1 Dual NIC 100M Windows Server 2008 JDK1.5 GT4 Hyper-V (2 nodescluster) 222.66.167.230~32

Network topology of the experimental operating environment is shown in Fig. 5, using four computers (Table 1) as the data nodes and computing nodes which are configured in three different network segments of Internet. Two hosts are distributed learning units who act as grid tasks clients. Meanwhile the hosts deploy parallel computing-related services in the grid service container. Therefore, the learning units are both clients and computing resources. The scheduling server manages grid service with the installation for the Web server of Apache Tomcat v5.0. Another computer runs three virtual machines via Hyper-V management as grid resources nodes. Globus Toolkit4 GT4 grid platform is utilized in the experimental environment. For quantum-inspired adaptive genetic algorithm, the number of individuals takes 50 and the maximum evolutionary generation is 500. The crossover probability coefficients are b1 = 0.5, b2 = 0.9 and mutation ones are b3 = 0.02, b4 = 0.05 ,

（ ）

Discrete accuracy is set 0.01 in the test environment. Table 2. Dynamic Process Simulation of Grid Computing Resource Nodes

Time 5s 10s Virtual Machine(Grid Nodes) 1 √ √ 2 √ √ 3 √

15s

20s

25s

30s

√

√ √

√ √ √

√

The first experiment sticks to 50 population size to the requirements of quantuminspired adaptive GA. We test the resources of grid nodes on one host of virtual machines(VM). Table 2 established whether the VM is available online, servering as a grid node in the experimental time, where “√” means the VM is working.

230

L. Xu et al.

Figure 6 shows the resources utilization of all grid nodes for the corresponding experiments time. Dimension of longitudinal coordinates is percentage, which expresses the resource occupancy. We test 6 different switching states, simulating the dynamic process of grid nodes, that is, new entrants to grid nodes and unexpected circumstances for the offline grid node machine and cannot be used as grid resources.

Fig. 6. Resources utilization of grid nodes for six states

Figure 7 is the timing diagram of total resource consumption for the three virtual hosts, simulating the above experimental conditions of the rate for total grid resources nodes, which reflects the service capability of dynamic grid, that is, resources providing capacity for the client requests in the circumstances of the dynamic changing of grid nodes. In the case of one to three working fictitious host computer respectively, we can see the output overall resource utilization held stationary.

Fig. 7. Total resources utilization of all grid nodes

Grids-Based Data Parallel Computing for Learning Optimization in a NLCSs

231

Table 3. Parallel computing grid optimization results comparison

Parallel grid number (population size 50) 1 2 3 Population size (the number of parallel grid is 3) 10 30 MPI (Population size is 50 and the number of parallel grid is 3)

Optimizati on time (s) 12.906s 8.456s 6.834s

Network utilization 23.8% 30.4% 34.7%

4.872s 5.768s

33.1% 33.5%

8.936s

26.8%

Statistical results (Table 3) show that the average efficiency achieved 30.5% under the grid environment and meanwhile the number of parallel efficiency increases varying with the increase of population. Through the experiments, we found advantages of GT4-based grid data parallel computing for optimization in NLCS are: (1) Fast calculation and can realize real time; (2) High-performance computing resources in wide area network is utilized for distributed coordination; (3) A reasonable division of tasks and dynamic computing nodes designation for those different computational tasks can reasonably be allocated to appropriate computing nodes; (4) No extra trenchant demanding in the configuration. Relative to the parallel computing tools MPI, grid-based platform for parallel processing have been improved of its limitations. The top advantage of the designed grid computing platform based parallel optimization service are achieving learning optimization parallel online computing on wide-area specifically for the characteristics of wide-area. Meanwhile, grid monitoring components can real-time monitor and register the performance of all computing nodes in grid system, providing suitable computational capabilities for parallel computing and a rational allocation of computing tasks to computing nodes. Grid service is proved to improve QoS and real time of the system.

7 Conclusion An integrated optimization design of grid based data parallel computing in a twolayer architecture networked learning control system is proposed. After the establishment of learning task parallel computing model, distributed computing resources are integrated into high-performance computing environment via grid technology and optimization scheduling. Approached to grid services, we realized multi-objective optimization and grid fast computing, solving the problems of data parallel computing, learing real-time and network QoS in two-layer NLCS. Experimental results show the scheme improves the network utilization and control

232

L. Xu et al.

performance of the system .Consequently, our next steps are proposing more efficient algorithm and considering some uncertainties and more restrictive conditions in grid model, such as such as limited time and cost grid, heterogeneous network of Windows and Linux, virtualization, multi-task concurrency and internet-based grid security requirements, intending to solve more complex and time-consuming problems according to reality in NLCS.

References 1. Yang, T.C.: Networked control system: a brief survey. IEEE Proc.-Control Theory Appl. 153, 403–412 (2006) 2. Du, D.J., Fei, M.R., Li, K.: A two-layer networked learning control system using actorcritic neural network. Applied Mathematics and Computation 205, 26–36 (2008) 3. Du, D.J., Fei, M.R., Hu, H.S.: Two-layer networked learning control using self-learning fuzzy control algorithms. Chinese Journal of Scientific Instrument 28, 2124–3131 (2007) 4. Battisha, M., Elmaghraby, A.: Adaptive tracking of network behavioral signals for real time forensic analysis of service quality degradation. IEEE Transactions on Network and Service Management 5, 105–117 (2008) 5. Grimshaw, A., Morgan, M.: An Open Grid Services Architecture Primer. IEEE Computer 42, 27–34 (2009) 6. Dieter, D.: A Decomposition-Based Genetic Algorithm for the Resource-Constrained Project-Scheduling Problem. Operations Research 55, 457–469 (2007) 7. Xu, L.J., Fei, M.R.: A Hybrid Quantum Clone Evolutionary Algorithm-based Scheduling Optimization in a Networked Learning Control System. In: Proceedings of the 22th Chinese Control and Decision Conference (CCDC 2010), Xuzhou, China (2010) 8. Li, Z.X.: Optimal control and scheduling of system with resource constraints. Control Theory & Applications 26, 97–102 (2009) 9. Wang, C.S., Chang, C.T.: Integrated Genetic Algorithm and Goal Programming for Network Topology Design Problem With Multiple Objectives and Multiple Criteria. IEEE Transactions on Networking 16, 680–690 (2008) 10. Andrea, M., Enrico, B., Tommaso, C.: Quantum Genetic Optimization. IEEE Trans. on Evolutionary Computation 12, 231–241 (2008) 11. Nadia, R.: Time and Cost-Driven Scheduling of Data Parallel Tasks in Grid Workflows. IEEE Systems Journal 3 (2009)

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent* Yongzhong Li1, Chunwei Jing1, and Jing Xu2 1

School of Computer Science and Engineering, Jiangsu University of Science and Technology 212003 Zhenjiang, China 2 College of Information Engineering,Yancheng Institute of technology,Yancheng, China [email protected]

Abstract. Intrusion detection system based on mobile agent has overcome the speed-bottleneck problem and reduced network load. Because of the low detection speed and high false positive rate of traditional intrusion detection systems, we have proposed an immune agent by combining immune system with mobile agent. In the distributed intrusion detection systems, the data is collected mostly using distributed component to collect data sent for processing center. Data is often analyzed in the processing center. However, this model has the following problems: bad real time capability, bottleneck, and single point of failure. In order to overcome these shortcomings, a new distributed intrusion detection method based on mobile agent is proposed in this paper by using the intelligent and mobile characteristics of the agent. Analysis shows that the network load can be reduced and the real time capability of the system can be improved with the new method. The system is also robust and fault-tolerant. Since mobile agent only can improve the structure of system, dynamic colonial selection algorithm is adopted for reducing false positive rate. The simulation results on KDD99 data set have shown that the new method can achieve low false positive rate and high detection rate. Keywords: Immune agent; Mobile agent; Network security; Distributed intrusion detectiont.

1 Introduction An intrusion detection system (IDS) is used to detect unwanted attempts at accessing, manipulating, and disabling computer system by either authorized users or external perpetrators, mainly through a network, such as the Internet. The IDS responds to the suspicious activity by resetting the connection or by reprogramming the firewall to block network traffic from the suspected malicious source [1]. It can remedy the deficiency of firewall effectively. Intrusion detection has experienced four stages: based on host, based on multihosts, based on network, and based on distributed intrusion detection. Traditional *

This paper is supported by Research fund of University of Jiangsu Province and Jiangsu University of Science and Technology’s Basic Research Development Program (No. 2005DX006J).

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 233–243, 2010. © Springer-Verlag Berlin Heidelberg 2010

234

Y. Li, C. Jing, and J. Xu

intrusion detection systems are centralized and based on a monolithic architecture. Data are collected on a single machine by looking at log files or network flow and are analyzed on a single computer, which has some defects both in the structure and in the detection technology. So distributed intrusion detection system (DIDS) appears. It becomes a research focus in the field of intrusion detection. Reference [2] presented a distributed information-gathering step, but centralized on analyzing process. The Graph-based Intrusion Detection System (GrIDS) [3] and Event Monitoring Enabling Responses to Anomalous Live Disturbances (EMERALD) [4] are IDS that use a hierarchical approach in a more sophisticated way. The hierarchical approach seems to show better scalability by allowing local analyses at distributed monitoring areas. However, a monitor operating at the highest level may induce single point of failure. When the topology of current network is changed, it causes a change of network hierarchy, and the whole mechanism for aggregation of local analysis reports must be changed. Autonomous Agent for Intrusion Detection (AAFID) is the first attempt to use autonomous agents for network intrusion detection by Spafford and Crosbie in [5]. In AAFID, nodes of the IDS are arranged in a hierarchical structure in a tree. Agents in AAFID were not mobile. Current DIDS mostly use distributed component to collect data, and then send collected data to processing center. These models solve the problem of distributed data acquisition effectively in wide bandwidth network. However, they have bad real time capability, bottleneck problem, and single point of failure because of the central processing node. The above-mentioned problems can be solved by utilizing the intelligent, mobile, and self-adaptive characteristics of agent and its distributed collaborative calculation capability [6] − [7]. False positive rate and false negative rate are other import aspects that IDS must consider. In [8], the authors stated similarities between the defenses of natural immune systems and computer security: both must discriminate self and non-self to protect a complex system from inimical agent. Be inspired of immune system, Kim and Bentley in [9] have proposed dynamic clonal selection algorithm and shown that this algorithm could reduce false positive rate. In this paper, dynamic clonal selection algorithm is adopted. Detectors are embedded in agents. With their communication mechanism, detection agents can detect cooperatively. Using the mobile characteristic of agent, detection agent can move to local host, and thus it can reduce network load and improve real-time capability. The model is full distributed. The rest of the paper is structured as follows: In section 2, we study about Immune system and mobile agent. The overall design model based on immune mobile agent is described in Section 3. Simulation results are shown in Section 4. Section 5 is the conclusions.

2 Relative Knowledge 2.1 Immune System The immune system [10] − [13] is a complex network of organs and cells responsible for the organisms defense against alien particles. Among a large number of different

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent

235

innate and acquired cells, lymphocytes play a central role. Lymphocytes are classified into two main types: B-cells and T-cells. B-cells are antibody-secreting cells and T-cells kill antigens or help or suppress the development of B-cells. Both originate from bone marrow, and they are developed by the bone marrow and the thymus, respectively. Before leaving the bone marrow and the thymus, maturing B- and T-cells have to pass the last test-negative selection. Mature B- and T-cells that pass the negative selection are released from the bone marrow and thymus, respectively. The development of B-cells and T-cells are shown in Fig. 1.

Fig. 1. Development of B-cells and T-cells

Both B-cells and T-cells continuously circulate around the body in the blood and encounter antigens for activation and evolution. The antibodies of B-cells, which recognize harmful antigens by binding to them, are activated directly or indirectly. When B-cell antibody receptors bind to antigen epitopes with strong affinity above a threshold, they are directly activated. When B-cell antibody receptors fasten to antigen epitopes with weak affinity, MHC molecules try to find some hidden antigens inside cells. When MHC molecules find them, they transport them on the surface of B-cells. The receptors of T-cells are genetically structured to recognize the MHC molecule on the B-cell surface. When the T-cell binds to an MHC molecule with strong affinity, it sends a chemical signal to the B-cell, which allows it to activate, grow, and differentiate. Clonal selection is immediately followed with B-cells activate. The more specific antigens B-cells bind to, the more chance being selected for cloning they have. When antigens activated B-cells, they produce memory cells for the reoccurrence of the same antigens in the future. Therefore, the secondary response is quickly. Fig 2 shows the generation of memory cells via clonal selection of B-cell, which are direct activated and indirect activated, respectively.

236

Y. Li, C. Jing, and J. Xu

Fig. 2. Generation of memory cell

2.2 Mobile Agent Mobile agent is a type of software agent, with the feature of autonomy, social ability, learning, and most import, mobility [14]. Mobile agent can transport its state from one environment to another with its data intact and still be able to perform appropriately in the new environment. When a mobile agent decides to move, it saves its own state, transports this saved state to the next host and resumes execution from the saved state. Fig. 3 illustrates the work processing of mobile agent.

Fig. 3. Work processing of mobile agent.

Mobile agent has many advantages. Mobile agent makes computation move to data, it can reduce network load. Because the actions are dependent on the state of the host environment, it is dynamic adaptation. It can operate without an active connection between client and server, so it has the capability of faults tolerance to network. Because of the mobility and cooperative of agent, mobile agent has extensive application range, such as network management, information retrieval. However, communication security is a problem that must face. It can be solved by encrypting when transfer. This is not belonging to the range of discussion of this paper.

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent

237

Mobile agent neither brings new method to detect for IDS nor increases detection speed for some kind of attracting. Nevertheless, it improves the design, construct, and execute of IDS obviously.

3 The Intrusion Detection Model Based on Immune Mobile Agent 3.1 System Architecture Be inspired of immune system, this paper combines immune mechanism with mobile agent, and constructs some immune agents to monitor and detect attraction on the network. Each immune agent can be regarded as immune cell. Like B-cells and Tcells circulating around the body in the blood and preventing the body by suppressing or killing the foreign invaders, immune agents roam on the network, and monitor and detect attacking. Fig.4 presents the architecture of intrusion detection based on immune mobile agent. It composes of central control agent (C-agent), detection agent (B-agent), memory agent (M-agent), and response agent (K-agent). C-agent runs in the server and plays a role of manger. B-agent and M-agent travel though the network in order to detect attacking. If any attacking is detected by B-agent or M-agent, K-agent is activated and responds to it immediately.

Fig. 4. Architecture of intrusion detection system based on immune agent

The function of each component in this model is described as follows: C-agent. C-agent is a kind of agents, which mainly manage, coordinate, and control roaming agent on the network. Its function is similar to that of bone marrow and thymus. It can create, dispatch, and recall agent. Once B-agent is created, it can work continually without the connection between server and client. Although we adopt server and client model, it does not induce single point of failure.

238

Y. Li, C. Jing, and J. Xu

B-agent. Each B-agent contains a set of mature detectors. The function of B-agent is similar to that of B-cells. B-agent strays on the network to monitor intrusion. If antigen comes, B-agent is activated, and it will move to the local host to detect whether intrusion occurs. M-agent. Each M-agent contains a set of memory detectors. It imitates the mechanism of secondary response in immune system. If antigen comes and M-agent exists, M-agent is activated, and they will be detected by M-agent firstly. If it does not match these antigens, B-agent will detect continually. It can improve the speed of detecting known intrusion. K-agent. The function of K-agent is analogous to that of T-cells. If any intrusion is detected by B-agent or M-agent, K-agent will be activated immediately. It will act in response to it by disconnecting suspicious connection, locking account, or restricting login. Collect-agent. The function of collect-agent is to collect data, which are foundation of intrusion detection system. It can collect data based on host and based on network. Collect-agent in this paper mainly captures network packet. In order to improve efficiency of detection, collect-agent needs to extract useful property of data packet besides of capturing data packet. 3.2 Generation of Detectors Detectors play an important role in intrusion detection. The more attacking features these detectors have, the higher detection rate the system has. The less normal network features these detectors contain, the less false positive rate the system has. Kim presents dynamic clonal selection algorithm and experiment shows it can reduce false positive rate with high detection rate [9]. When detectors are generated, they are embedded in mobile agent. Suppose that there are N mature detectors in total and each B-agent can carry n detectors, the system will generate ⎡⎢ N / n ⎤⎥ B-agents. These agents with detectors roam on the network and realize the distributed computing. The dynamic selection algorithm is described in detail as follows: Step1: create an immature detector population with random detectors, and perform negative selection by comparing immature detectors to self-antigens. Because of negative selection, the immature detectors binding any self-antigen are deleted from the immature detector population and then new immature detectors are generated until the number of immature detectors becomes the maximum size of a non-memory detector population. The same processes continue for the tolerance period (T) number of generations. When the total number of generations reaches T, some immature detectors whose age reaches T, which were born at generation 1, become mature detectors. Step2: at generation T+1, mature detectors start to detect new antigen set. If any mature detector matches an antigen, the match count of a mature detector increases by one. After all the given antigen are compared with all the existing mature detectors, the system will check whether the match counts of mature detectors are larger than a

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent

239

pre-defined activation threshold (A) and whether the ages of mature detectors meet a pre-defined life span (L). If the match count of a mature detector is larger than A, it becomes a memory detector. If the age of a mature detector meets L, the mature detectors are deleted from the mature detector population. Any antigens detected by activated mature detectors are deleted, and the remaining antigens are presented to the immature detectors for negative selection. If the number of existing immature detectors and mature detectors is less than the maximum size of a non-memory detector population, generate immature detectors with random detectors until they are equal. Step 3: at generation T+2, when memory detectors match any antigen and the detected antigen bind any self-antigen, the memory detector are added to immature detectors. In addition, if the detected antigen does not bind self-detectors, it is removed directly. The remaining antigens are matched by mature detectors and the process is the same as the period of T+1.

4 Simulations 4.1 Design Guidelines In this section, we present the details of results on KDD 99 data set, and the achieved false positive rate and detection rate. Data sets. In order to survey and evaluate research in intrusion detection, the 1998 DARPA Intrusion Detection Evaluation Program was managed by MIT Lincoln Labs. Lincoln Labs set up an environment to acquire nine weeks of raw TCP dump data for a local-area network (LAN) simulating a typical U.S. Air Force LAN. They operate the LAN as if it was a true Air Force environment, but peppered it with multiple attacks. KDD99 data set is the data set, which was obtained from the 1998 DARPA. The data set is composed of a big number of connection records. Each connection is labeled as either normal or as an attack with exactly one specific attack type. Attacks in the data set can be classified into four main categories namely Denial of service (DOS), Remote to User (R2L), User to Root (U2R) and Probe. In our experiment, we only used 10 percents of the raw training data (kddcup.data_10_percent) for training and the test data set (corrected.gz) for testing. It is important to note that the test data is not from the same probability distribution as the training data, and it includes specific attack types not in the training data. The test data contains of 20 types of training attack and 17 types unknown attacks. The 37 types attacks are classified into four categories as follows: DOS: {back, land, Neptune, pod, smurf, teardrop, processtable, udpstorm, mailbomb, apache2} R2L: {ftp_write, guess_passwd, imap, multihop, phf, warezmaster, sendmail, xlock, snmpguess, named, xsnoop, snmpgetattack, worm} U2R: {buffer_overflow, loadmodule, perl, rootkit, xterm, ps, httptunnel, sqlattack} Probing: {ipsweep, nmap, portsweep, satan, mscan, saint} For each connection, there are 41 features. Among them, 32 features are continuous variables and 9 features are discrete variables. Table 1 demonstrates the network data

240

Y. Li, C. Jing, and J. Xu

feature labels. Among these features, some are redundant and some contribute little to the intrusion detection process [15]–[16]. Considering efficiency, we select features 1, 2, 3, 5, 6, 7, 8, 9, 15, 16, 18, 19 and 20 to compose of detector and choose statistical features 20 to 41 except for 30 to be collaborative signal. Among the aforementioned features, 1, 5, 6, 8, 9, 16, 18, 19, 20 and all-statistical features are continuous. We should make them discrete. Take the feature of src-bytes for example: we can use much less, less, normal, more and much more to express bytes from source to destination. Moreover, the value of them is 000, 001, 010, 011, and 100, respectively. Fig. 5 shows its membership functions. Table 1. Network data feature labels Label 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Network data Network data Label Label Network data feature feature feature duration 15 su_attempted 29 same_srv_rate protocol-type 16 num_root 30 diff_srv_rate service 17 num_file_creations 31 srv_diff_host_rate flag 18 num_shells 32 dst_host_count src_bytes 19 num_access_files 33 dst_host_srv_count dst_bytes 20 num_outbound_cmds 34 dst_host_same_srv_rate land 21 is_host_login 35 dst_host_diff_srv_rate wrong_fragment 22 is_guest_login 36 dst_host_same_src_port_rate urgent 23 count 37 dst_host_srv_diff_host_rate hot 24 srv_count 38 dst_host_serror_rate num_failed_login 25 serror_rate 39 dst_host_srv_serror_rate logged_in 26 srv_serror_rate 40 dst_host_rerror_rate num_compromised 27 rerror_rate 41 dst_host_srv_rerror_rate root_shell 28 srv_rerror_rate

p( x − μ ≥ ε ) ≤

σ2 ε2

(1)

According to Chebyshev's inequality (1) and the proportion of normal and attack data in KDD99, the values of variables in Fig. 5 are shown as follows: a = μ − 2σ

， b = μ − σ ， c = μ ， e = μ + σ ， d = μ + 2σ

(2)

For all collaborative signals, we use normal, suspicious and abnormal to express them, and the process is the same as the above. The value is 00,01 and 10 respectively. When the inherence character of the detector matches one antigen, this antigen is not always pathogen. Only in the abnormal circumstance, which is likely induced by it, it will be considered an attack. That is why the collaborative signals are adopted in this paper. When a detector matches an antigen, and there is an abnormal or suspicious feature in the collaborative signals, this antigen will be taken for an attack. Otherwise, it will be taken for a normal behavior. Considering possibility of incomplete studying, for the sake of reducing undetected rate, when the number of abnormal and suspicious feature exceed threshold, the antigen will be taken for attack behavior.

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent

241

This paper implemented on IBM’s aglet toolkit, which is composed of a set of java-based mobile software agents that carry out specific tasks on the network and collaborate on the problem of network intrusion detection. In the simulation, aglet is installed in three computers. Among them one is as sever, and others are as clients. When detectors are generated, they are embedded in agents. Utilizing aglet, agents can be dispatched, cloned, and recalled.

Fig. 5. Features of detector

4.2 Simulation Results Test of Robust and Fault-tolerant of the system. When the system is start-up, Bagent in one host is broken off in order to observe its effect to the system. Experiment shows that system can discover the invalidated agent and then create and dispatch new agent to this host. One node invalidate does not induce disability of the system. This indicates that the system is robust and fault-tolerant. Detection result. The size of non-memory detectors is defined as 100000, the training data is divided into self-antigen set and non-self antigen set. In addition, in our experiment, self-antigen set and non-self antigen set are classified into four antigen clusters. Moreover, the iterative generation is set 200. Analysis shows that the value of T is inversely proportional to detection rate and false negative rate. The larger value of T equals the less activated detectors the system has. However, if the value of T is small, immature detectors do not have enough time to experience the process of self-tolerance. Experiment shows that when T increases, the drop in FP is much shaper than the drop in TP. The value of L and N are inversely proportional to false negative rate and proportional to detection rate. In addition, the larger value of A can reduce false negative rate. Simulation results show the best results, as shown in table 2, when T, L, N and A equal to 5, 20, 1 and 5, respectively. In table 2, comparing with winning entry of KDD’99 Classifier Learning Contest, the proposed approach has a good performance in detecting DOS, Probe, U2R attack and Normal behavior. Nevertheless, the performance of detecting R2L is poor. This is because the packet of R2L is slightly different from the packet of normal. How to improve the ability of the detecting R2L and U2R is the future work.

242

Y. Li, C. Jing, and J. Xu Table 2. Comparison with winning entry of KDD’99 Classifier Learning Contest Detection Result Data TP of the winning entry TP of the proposed approach Normal(60593)

99.5%

98.127%

DOS(229853)

97.1%

97.565%

Probe(4166)

83.3%

90.494%

U2R(228)

13.2%

71.491%

R2L(16189)

8.4%

0.371%

5 Conclusions In this paper, a new distributed intrusion detection model based on immune mobile agent is proposed. Dynamic clonal selection algorithm and mobile agent are described in detail. The simulation results showed that our model is efficiently to classify the anomaly profile from the normal profile. Our model has following advantages. First, the model realized that computing move to data by utilizing mobile agent. Therefore, real time capability is improved and bottleneck problem is overcome. Second, compared with other hierarchical model, it surmounts single point of failure. Dependability of the system is enhanced. In addition, the system is robust and faulttolerant. Third, false positive rate is low and true positive rate is high by adopting dynamic clonal selection algorithm. Acknowledgments. This paper is supported by Research fund of University of Jiangsu Province and Jiangsu University of Science and Technology’s Basic Research Development Program (No. 2005DX006J ).

References 1. Axelsson, V.: Intrusion detection systems: A survey and taxonomy. Technical Report No 99–15, Chalmers University of Technology, Sweden 2. Hunteman, V.: Automated information system–(AIS) alarm system. In: 20th NIST-NCSC National Information Systems Security Conference, pp. 394–405. IEEE Press, New York (1997) 3. Staniford Chen, S., Cheung, S., Crawford, R., et al.: Grids: a graph based intrusion detection system for large networks. In: 19th National Information System Security Conference. National Institute of Standards and Technology, vol. 1, pp. 361–370. IEEE Press, Los Alamitos (1996) 4. Porras, P.A., Neumann, P.G.: MERALD: event monitoring enabling responses to anomalous live disturbances. In: 20th National Information Systems Security Conference, National Institute of Standards and Technology, pp. 11–13. IEEE Press, Los Alamitos (1997)

A New Distributed Intrusion Detection Method Based on Immune Mobile Agent

243

5. Spafford, E.H.: Intrusion detection using autonomous agent. Computer Networks 3(4), 547–570 (2000) 6. Dasgupta, D., Brian, H.: Mobile security agent for network traffic analysis. In: DARPA Information Survivability Conference and Exposition II (DISCEX-II), Anaheium, CA, pp. 332–340. IEEE Press, Los Alamitos (2001) 7. Jansen, W., Mell, P., Karygiannis, T., Marks, D.: Mobile agents in intrusion detection and response. In: 12th Annual Canadian Information Technology Security Symposium, pp. 12– 18. IEEE Press, Ottawa (2000) 8. Hofmeyr, S.A., Forrest, S., Somayaji, A.: Intrusion detection using sequences of system calls. Journal of Computer Security 6, 151–180 (1998) 9. Kim, J., Bentley, P.: Towards an artificial immune system for network intrusion detection: an investigation of dynamic clonal selection. In: Congress on Evolutionary Computation, Honolulu, pp. 1015–1020 (2002) 10. Kim, J., Bentley, P., Aickelin, U., et al.: Immune system approaches to intrusion detectiona review. Natural Computting 6, 413–466 (2007) 11. Aickelin, U., Greensmith, J., Twycross, J.: Immune system approaches to intrusion detection–a review. In: Nicosia, G., Cutello, V., Bentley, P.J., Timmis, J. (eds.) ICARIS 2004. LNCS, vol. 3239, pp. 316–329. Springer, Heidelberg (2004) 12. Glickman, M., Balthrop, J., Forrest, S.: A machine learning evaluation of an artificial immune system. Evolutionary Computation 13(2), 179–212 (2005) 13. Gomez, J., Gonzalez, F., Dasgupta, D.: An immune-fuzzy approach to anomaly detection. In: 12th IEEE International Conference on Fuzzy Systems (FUZZIEEE), vol. 2, pp. 1219– 1224. IEEE Press, Los Alamitos (May 2003) 14. Carver, C., Hill, J., Surdu, J., Pooch, U.: A methodology for using intelligent agents to provide automated intrusion response. In: IEEE Systems, Man, and Cybemetics Information Assurance and Security Workshop, West Point, New York, pp. 110–116 (2000) 15. Zainal, A., Maarof, M.A., Shamdudd, S.M.: In.: Feature selection using rough set in intrusion detection. In: Proc. IEEE TENCON, p. 4 (2006) 16. Kim, B.j., Kim, I.k.: Kernel based intrusion detection system. In: Proc. IEEE ICIS, pp. 6 (2005)

Single-Machine Scheduling Problems with Two Agents Competing for Makespan Guosheng Ding1,2 and Shijie Sun1 1

Department of Mathematics, Shanghai University, Shanghai, China 2 School of Science, Nantong University, Nantong, China

Abstract. This paper considers the single-machine scheduling problems in which two distinct agents are concerned. Each agent has a set of jobs with different release times, and both of them expect to complete their respective jobs as early as possible. We take the makespan of each agent as its own criterion and take the linear combination of the two makespans as our objective function. In this paper, both off-line and online models are considered. When preemption is allowed, we present an exact algorithm for the off-line model and an optimal algorithm for the on-line model. When preemption is not allowed we point out that the problem is NP-hard for the off-line model and give a (2+1/θ)-competitive algorithm for the on-line model. We also prove that a lower bound of the competitive ratio for the later model is 1 + θ/(1 + θ), where θ is a given factor not less than 1.

1

Introduction

In recent years, management problems in which multiple agents compete on the usage of a common processing resource are receiving increasing attention in different application environments, such as the scheduling of multiple flights of different airlines on a common set of airport runways, of berths and material/people movers (cranes, walkways, etc.) at a port for multiple ships, of clerical works among different “managers” in an office and of a mechanical/electrical workshop for different uses [1]; and in different methodological fields, such as artificial intelligence, decision theory, and operations research. A number of papers investigate the multi-agent scheduling problems, and mainly in the two-agent setting. Baker and Smith [5] analyze the computational complexity of combining linearly the agents’ criteria into a single objective function. Agnetis et al. [1] and Agnetis et al. [2] study both the problems of finding a single Pareto-optimal solution and enumerating the whole set of Pareto-optimal solutions for two agents. Agentis et al. [3] extend their research to multi-agent problems. Ng et al. [14] studied minimum weighted completion time for one agent subject to a given permitted number of tardy jobs for the second agent. Cheng et al. [6] studied in this context the objective of minimum number of tardy jobs. Cheng et al. [7] obtained several complexity results for single machine problems 

This work is partially supported by Grant 08KJD110014.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 244–255, 2010. c Springer-Verlag Berlin Heidelberg 2010 

Single-Machine Scheduling Problems

245

with max-form criteria. Liu and Tang [9] and Liu et al. [10] provided polynomialtime solutions for single machine problems with deteriorating jobs. And Liu et al. [11] introduce an aging effect and a learning effect into the two-agent single machine scheduling. Wan et al. [16] also considered several two-agent scheduling problems with deteriorating jobs, in which process times of jobs of one agent are deteriorating. Lee et al. [8] focused on minimizing total weighted completion time, and introduced fully polynomial-time approximation schemes and an efficient approximation algorithm with a reasonable worst case bound. Agnetis et al. [4] introduced an efficient branch and bound algorithm for minimizing total weighted completion of one agent subject to various constraints on the second agent’s performance. Mor and Mosheiov [13] introduced a polynomial time solution for the problem of minimizing maximum earliness cost of one agent, subject to a bound on the maximum earliness cost of the second agent. On the other hand, there have been a number of research on on-line scheduling because it establishes a bridge between deterministic and stochastic scheduling. And in practice, it is often the case that a very limited amount of information is available when a decision must be made. But until now the on-line model of the multi-agent scheduling has not been discussed. In this paper We take the makespan of each agent as its own criterion and take the linear combination of the two makespans as our objective function, and both off-line and on-line models are considered. This paper is organized as follows. In section 2, we introduce the model and the notation. Section 3 discusses off-line models where preemption is allowed or not. In section 4, we consider the on-line models and provide an optimal on-line algorithm when the jobs are preemptive, and a (2 + 1/θ)-competitive on-line algorithm when preemption is not allowed, where θ is a given weighted factor which is not less than 1. Finally, in section 5, we make a conclusion and some suggestions for future research.

2

Model and Notation

There are two competing agents, called agent A and agent B. Each of them has a set of jobs to be processed on a common machine. The agent A has to execute the job set J A = {J1A , J2A , . . . , JnAA }, whereas the agent B has to execute the job set J B = {J1B , J2B , . . . , JnBB }. For each job JhA (JkB ) there is a processing B A B time pA h (pk ) and a release time rh (rk ). Let σ indicate a feasible schedule of the n = nA + nB job, i.e., a feasible assignment of starting times to the jobs of both agents. The start times of job JhA and JkB in σ are denoted as ShA (σ) and SkB (σ), and the completion times are denoted as ChA (σ) and CkB (σ), or simply ShA , SkB , ChA , and CkB whenever these do not generate confusion. A B We use Cmax (σ) and Cmax (σ) to denote the completion times of agent A and A B A agent B in σ, or simply Cmax and Cmax . So, Cmax = max{C1A , C2A , . . . , CnAA }, B B B B Cmax = max{C1 , C2 , . . . , CnB }. A B We take Cmax +θCmax as our objective function, where θ > 0 is a given factor that helps us to distinguish the two agents’ relative importance, and use Cmax

246

G. Ding and S. Sun

to denote the makespan of all jobs which belong to both agent A and agent B. In the remainder of our paper, if all jobs of a certain agent are completed, then we say that the agent is completed. And jobs are numbered by nondecreasing release date.

3 3.1

Off-line Models A B 1 | rj , pmtn | Cmax + θCmax

A B Consider an arbitrary instance of the problem 1 | rj , pmtn | Cmax + θCmax . We assume, of an optimal schedule, the completion times of agent A and agent B A∗ B∗ ∗ ∗ are Cmax and Cmax , and the makespan of all the jobs is Cmax . Thus, Cmax = ∗ ∗ A B max{Cmax , Cmax }

Algorithm 1. At first, regardless agent A, make agent B completed as early as possible. Namely, every time there is a B-jobs available, we process the schedulable jobs of agent B according to the nondecreasing order of their release times, while putting agent A aside. Secondly, according to the nondecreasing order of release times of the jobs of agent A, process them on the machine as early as possible. Then obtain a A B (σ1 ) + θCmax (σ1 ). schedule, called σ1 . Denote objective1 = Cmax Analogously, schedule agent A firstly and then schedule agent B, both in the nondecreasing order of their jobs release times. Then obtain σ2 and denote A B (σ2 ) + θCmax (σ2 ). objective2 = Cmax A B Finally, Cmax (σi∗ ) + θCmax (σi∗ ) = min{objective1 , objective2}, is the optimal value. A + Theorem 1. Algorithm 1 yields an optimal schedule for 1 | rj , pmtn | Cmax B θCmax .

Proof. Assume that σ ∗ is an optimal schedule. Then the optimal objective value is A B A B (σ ∗ ) + θCmax (σ ∗ ). Since preemption allowed, max{Cmax (σ ∗ ), Cmax (σ ∗ )} = Cmax A B A B max{Cmax (σ1 ), Cmax (σ1 )} = max{Cmax (σ2 ), Cmax (σ2 )}. As algorithm 1 says, agent B is completed in σ1 as early as possible. So is agent A in σ2 . Thus B B A A Cmax (σ ∗ ) ≤ Cmax (σ1 ) and Cmax (σ ∗ ) ≤ Cmax (σ2 ). Therefore, whichever agent ∗ A ∗ B A B is completed firstly in σ , Cmax (σ ) + θCmax (σ ∗ ) ≤ Cmax (σi∗ ) + θCmax (σi∗ ) = min{objective1 , objective2} 3.2

A B + θCmax 1 | rj | Cmax

1 2 Theorem 2. The problem 1 | rj | Cmax + θCmax is NP-hard.

Proof. The proof is based on the fact that the Partition problem [12] can be 1 2 reduced to 1 | rj | Cmax + θCmax . a2 , . . . , at , does there exist a subset S ⊂ Partition. Given positive integers a1 ,   T = {1, 2, . . . , t} such that i∈S ai = i∈T −S ai ?

Single-Machine Scheduling Problems

247

J12 F 2

F 2

F +1

+1

1 2 + θCmax is NP-hard Fig. 1. 1 | rj | Cmax

 Define F = i∈T ai . Assume that agent A has t jobs with p1i = ai and ri = 0, (i ∈ T ), agent B has a job with p21 = 1 and r12 = F/2, and y = 1+F +θ(F/2+1). If Partition has a solution, then the job of agent B may start its processing immediately after it is released, while the jobs of agent A are just divided into two 1 2 +θCmax = 1+F +θ(1+F/2) = y. equal parts, as illustrated in Figure 1. So Cmax 1 2 If 1 | rj | Cmax + θCmax has a solution, then the jobs of agent B can not start 1 2 its processing strictly after time F/2, otherwise Cmax + θCmax > 1 + F + θ(1 + F/2) = y, that is, the job of agent B has to start its processing at time F/2, and all the jobs of agent A have to completed not later than F + 1. Then, Partition has a solution.

4

On-line Models

4.1

A B 1 | online, rj , pmtn | Cmax + θCmax

A B In this section, for the objective function Cmax + θCmax , we use OP T, ALG and ALG2 to denote the values corresponding to an optimal schedule, a schedule constructed by an arbitrary on-line algorithm and a schedule constructed by algorithm 2 respectively.

Theorem 3. The competitive ratio of any on-line algorithm for the problem A B 1 | online, rj , pmtn | Cmax + θCmax is at least 1 + θ/(1 + θ + θ2 ). Proof. Consider the following instance. At time 0, agent A has a job J1A with B B B r1A = 0 and pA 1 = a; agent B has a job J1 with r1 = 0 and p1 = b = θa. A If some on-line algorithm schedules these jobs in order of J1 , J1B , then job J1B can not be completed before time t = a + b. In this case, at time t = a + b, a new job of agent A, J2A , is released , and its processing time is ε (ε is an arbitrary small number). Corresponding to this case, ALG ≥ a+ b + ε + θ(a+ b) ≈ a+ b + θ(a+ b) ALG ≥ a+b+θ(a+b) = 1 + 1+ bθ+θ b = and OP T = a + b + ε + θb ≈ a + b + θb. So, OP T a+b+θb a

a

θ 1 + 1+θ+θ 2. Analogously, if the on-line algorithm schedules these two jobs released at time 0 in order of J1B , J1A , then job J1A can not be completed before time t = a + b. In this case, at time t = a + b, a new job of J2B is released, and its processing time is ε. Now, ALG ≥ a + b + θ(a + b + ε) ≈ a + b + θ(a + b) and OP T = a+b+θ(a+b) ALG 1 θ a+θ(a+b+ε) ≈ a+θ(a+b). So, OP T ≥ a+θ(a+b) = 1+ a +θ a +θ = 1+ 1+θ+θ 2 .

Thus,

ALG OP T

≥1+

θ 1+θ+θ 2 .

b

b

A B Assume jobs J1A , J2A , . . . , JA of agent A and J1B , J2B , . . . , JB of agent B have t t been known at a certain time t. Schedule agent A jobs, schedule agent B jobs,

248

G. Ding and S. Sun

and schedule both agent A and agent B jobs, respectively, on one machine for minimizing makespan. Three sequences are obtained. And their makespans are denoted by at , bt , and Dt . Algorithm 2. At any time t, if the jobs of only one agent are known and to be processed, then process this agent’s jobs. If there remain both agent A and agent B jobs to be processed, then compare (Dt − at )/θ and Dt − bt . When (Dt − at )/θ ≥ Dt − bt , process agent A jobs; otherwise process agent B jobs, where at , bt > 0. Theorem 4. Algorithm 2 is (1 + A B Cmax + θCmax .

θ 1+θ+θ 2 )-competitive

for 1 | online, rj , pmtn |

Proof. First, for an arbitrary instance known at time t, we prove that the agent whose jobs are completed last in the optimal schedule is the same as the one whose jobs are completed last in the resulting schedule of algorithm 2. If the jobs of only one agent are known and to be processed, then this agent’s jobs are last completed in both the optimal schedule and the resulting schedule of algorithm 2. If there remain both agent A and agent B jobs to be processed at time t, then at , bt > 0. In the optimal schedule, if agent B jobs are last completed, then its objective value must be no more than that of the other schedule in which agent A jobs are last completed. So, at + θDt ≤ Dt + θbt , i.e., (Dt − at )/θ ≥ Dt − bt . In this case, algorithm 2 make agent B jobs completed last. Similarly, If agent A jobs are last completed in the optimal schedule, then they are also last completed in the resulting schedule of algorithm 2. In the following of the proof, assume agent B jobs are last completed in the optimal schedule. So they are also last completed in the resulting schedule of algorithm 2. If agent A jobs are last completed in the optimal schedule, the proof process is similar. In the worst case, algorithm 2 postpone agent A jobs, such that the ratio 

CA

+θC B





CA

+θC B

∗

max max max ρ = ALG2 = Cmax is very large. Thus how large can ρ be A∗ +θC B ∗ A∗ B∗ OP T = Cmax max max +θCmax in the worst instance? In order to make ρ larger, the worst instance has the follow properties:

1. there must be all agent A jobs and some small agent B jobs released at time 0, such that (Dt − at )/θ < Dt − bt , (t = 0); moreover some small agent B jobs are released continuously in this later time, such that this inequality holds for the longest time; 2. there must be only a block, which consists of both agent A and B jobs in the optimal schedule, where a block means a batch of jobs processed contiguously; also there is only a block in the resulting schedule of algorithm 2. ∗

A (= But for the worst instance, the last agent A job is delayed at most θCmax A∗ θat=0 ). In fact, the inequality (Dt − at )/θ < Dt − bt holds from time 0 to θCmax . A∗ And from time 0 to θCmax , those small agent B jobs are processed continuously.

Single-Machine Scheduling Problems

249

Because agent B jobs are completed last in both the optimal schedule and the B∗ A∗ A∗ resulting schedule of algorithm 2, Cmax ≥ Cmax + θCmax . So, 

ρ=

∗

∗

∗

∗

∗

A B A A B Cmax + θCmax θCmax + Cmax + θCmax θC A ≤ = 1 + A∗ max B ∗ ∗ ∗ ∗ ∗ A B A B Cmax + θCmax Cmax + θCmax Cmax + θCmax ∗

A θ θCmax = (1 + ) . ∗ ∗ A A A∗ ) Cmax + θ(Cmax + θCmax 1 + θ + θ2 The proof is completed and we also know that algorithm 2 is optimal for A B 1 | online, rj , pmtn | Cmax + θCmax .

≤ 1+

4.2

A B + θCmax 1 | online, rj | Cmax

Theorem 5. The competitive ratio of any on-line algorithm for problem 1 | A B online, rj | Cmax + θCmax is at least 1 + θ/(1 + θ). Proof. Consider the following instance. The first job J1A of Agent A is released at time 0 and its processing time is pA 1 . The on-line algorithm schedules the job at time S. Depending on S, either no jobs are released any more or job J1B with processing time pB 1 = ε is released at time S + ε1 (ε1 is another arbitrary small A number). In the first case we get a ratio of ALG/OP T = (S + pA 1 )/p1 ; in the S+pA +θ(S+pA +ε)

S+pA +θ(S+pA )

ALG 1 1 1 1 second case we get a ratio of OP ≈ . T ≥ S+ε1 +ε+pA S+pA 1 +θ(S+ε1 +ε) 1 +θS   A A A S+p S+p +θ(S+p ) ALG 1 1 1 Hence, OP , . The algorithm may choose S so as T ≥ max pA S+pA +θS 1

1

A to minimize this expression. As (S + pA 1 )/p1 is increasing and

decreasing with S, the best choice for S is S = desired ratio of 1 + θ/(1 + θ).

θpA 1 /(1

A S+pA 1 +θ(S+p1 ) S+pA +θS 1

is

+ θ). Then we get the

Note that we can also use the example of theorem 4 to show that any online algorithm that schedules a job as soon as the machine is available has a competitive ratio of 1 + θ. If θ is a large number, then the ratio becomes terrible. A B Before giving our algorithm for 1 | online, rj | Cmax + θCmax , we introduce two conditions at a certain time t, condition 1. There exists at least one unscheduled agent A job such that A A its processing time is not more than 1+θ θ t (for any Ji ∈ J , this condition θ pA guarantee that SiA ≥ max{riA , 1+θ i }); condition 2. There exists at least one unscheduled agent B job such that its processing time is not more than (1 + θ)t (for any JiB ∈ J B , this condition 1 guarantee that SiB ≥ max{riB , 1+θ pB i }). Here, we define JaA as the current unscheduled agent A job which has the smallest release time among those whose processing times are not more than 1+θ B θ t. Similarly, we define Jb as the current unscheduled agent B job which has the smallest release time among those whose processing times are not more than (1 + θ)t.

250

G. Ding and S. Sun

Algorithm 3. Assume at a certain time t, the machine is freed, then, a. if only condition 1 is satisfied, then we start processing JaA ; if only condition 2 is satisfied, then we start processing JbB ; b. if both of the two conditions are satisfied, then compare the values of  B  i:riB ≤t pi A pi and , θ A i:ri ≤t



pB  i:rB ≤t i i then if i:rA ≤t pA ≤ , start processing JaA ; otherwise start processing i θ i JbB ; c. if none of the two conditions are satisfied, then we keep the machine idle until at least one of the two conditions is satisfied.

Note that according to algorithm 3, at a certain time t, if a job of agent A is to be started, then at least condition 1 must be satisfied; similarly, if a job of agent B is to be started then at least condition 2 must be satisfied. Theorem 6. Algorithm 3 is 2 + 1/θ-competitive for problem 1 | online, rj | A B Cmax + θCmax . It can be seen easily that this competitive ratio approaches the lower bound of theorem 5 as θ is very large. In what follows, we prove theorem 6 by the smallest counterexample strategy. If there exists one or more counterexamples of algorithm 3 whose competitive ratio is strictly larger than 2 + 1/θ, then there must exist a counterexample with the smallest total number of jobs. Here we assume that the “smallest” counterexample is composed of nA jobs of agent A and nB jobs of agent B, and denote it by I. Lemma 1. The schedule given by algorithm 3 for I includes at most two cases: 1. the schedule consists of a single block, where a block means a batch of jobs processed contiguously; 2. the schedule consists of two blocks, the first one includes the jobs of agent A and/or agent B, the second one only includes the jobs belonging to one of the two agents. Proof. If the last block of the schedule given by algorithm 3 for I includes the jobs of both agent A and agent B, then the schedule consists of only this block. This is because if there exist other blocks before it then the jobs of these blocks do not influence the schedule decision of algorithm 3 concerning the jobs of the last block, and vice versa, and the deletion of them will result in a counterexample with fewer total jobs. If the last block of the schedule given by algorithm 3 for I includes only some jobs of agent A. – There do not exist other blocks including only some jobs of agent A before the last block because the deletion of those blocks result in a counterexample with fewer total jobs.

Single-Machine Scheduling Problems

251

– Once there exists one block including the jobs of both agent A and agent B, or one block including only some jobs of agent B, immediately before this last block. There does not exist any more blocks before because the deletion of them result in a counterexample with fewer total jobs. If the last block of the schedule given by algorithm 3 for I includes only some jobs of agent B, the proof is analogous to the above. Hence, lemma 1 holds. For convenience, we use ALG3(I) and OP T (I) to denote the objective function values of I under algorithm 3 and the optimal schedule respectively, and use i i∗ Cmax (I) and Cmax (I) to denote the completion times of I corresponding to the schedule given by algorithm 3 and the optimal schedule, respectively, i = A, B. Without loss of generality we assume that there exists at least one job released at time 0. Lemma 2. Algorithm 3 is 2 + 1/θ-competitive when it acts on I and results in case 1 of lemma 1. Proof. Suppose that under algorithm 3, case 1 of lemma 1 occurs. Denote this schedule by π and denote the optimal schedule by π ∗ . In π ∗ agent A, or agent B is last completed. Case 1.1. agent A is last completed in π ∗ . In order to bound B Cmax (I) B ∗ (I) Cmax

A B Cmax (I)+θCmax (I) A∗ (I)+θC B ∗ (I) , Cmax max

we are dedicated to obtain the ratio of

firstly . If all jobs of agent A are scheduled after all jobs of agent B in π , then according to 2, we know that alljobs of agent B are not to be started condition n2 n2 B∗ B B B∗ later than j=1 pB j /(1 + θ). As Cmax ≥ j=1 pj , we get Cmax (I)/Cmax (I) ≤ 1 + 1/(1 + θ). If at least one job of agent A is scheduled before some of agent B jobs, we denote the last such a job of agent A by JsA , denote the jobs of agent B immediately after B B JsA in π by JlB , Jl+1 , . . . , Jl+u in turn, u ≥ 0, and denote a certain job(it may not A exist) before Js by Jk which belongs to either agent A or agent B. B Case 1.1.1. pB l+j ≤ (1 + θ)rl+j , j = 0, 1, . . . , u.

π has at most two subcases, as illustrated in Figure 2: d. rlB > SsA ; e. rlB ≤ this time, we might as well assume Sk ≤ rlB ≤ Ck ) .

SsA (at

rlB

? · · · JsA

rlB

B ··· JlB · · · Jl+u

? · · · Jk · · · JsA

B ··· JlB · · · Jl+u

e d B Fig. 2. Two subcases for pB ≤ (1 + θ)r l+j l+j

252

G. Ding and S. Sun

A For subcase d in Figure 2, according to condition 1, we get pA s ≤ Ss (1 + θ)/θ. B A A B B B By rl  > Ss , we have ps < rl (1 + θ)/θ. So we get Cmax (I) < rl + rlB (1 + u B B θ)/θ + j=0 pB l+j . Based on the hypothesis pl+j ≤ (1 + θ)rl+j , j = 0, 1, . . . , u and the property of algorithm 3 that the jobs satisfying condition 1 or 2 are processed in the nondecreasing order of their release times,  we obtain rlB ≤ u B B B∗ B rl+1 ≤ · · · ≤ rl+u . So, it is easy to see Cmax (I) ≥ rl + j=0 pB l+j . Thus, B Cmax (I) B ∗ (I) Cmax

<

 B rlB +rlB (1+θ)/θ+ u j=0 pl+j  u rlB + j=0 pB l+j

≤1+

1+θ θ

= 2 + 1θ .

Subcase e in Figure 2 implies that after a certain job Jk preceding JsA is B completed, although JlB has been released and it satisfies pB l ≤ (1 + θ)rl , A that is, it satisfies condition 2, algorithm 3 start processing it until Js is completed. Namely, algorithm 3 is executed according to b. Denote t = SsA , we have   nB B pB  i:rB ≤t i A i i=1 pi p ≤ ≤ . We obtain that the sum of processing times A i:ri ≤t i θ θ nB B of the jobs of agent A completed before JlB in schedule π is at most i=1 pi /θ. The machine may have a waiting time at first (if the first job released is so long that both condition 1 and 2 may not be satisfied), waiting time  condition  but the   nB B θ 1 1 A B p , p ≤ is not more than max 1+θ A B i:ri ≤t i i:ri ≤t i i=1 pi . 1+θ 1+θ  nB B   nB B B 1 B i=1 pi Thus, we get Cmax (I) ≤ 1+θ + ni=1 pB i , where the first i=1 pi + θ item of the right hand side represents the upper bound of the initial waiting time of the machine, the second item represents the upper bound of the sum the nof B B∗ (I) ≥ i=1 pB processing times of jobs of agent A completed before JlB . As Cmax i , we can get

B Cmax (I) B ∗ (I) Cmax

≤1+

1 θ

+

1 1+θ

< 2 + 1θ .

B B Case 1.1.2 There exists at least one Jl+i , 0 ≤ i ≤ u, such that pB l+i > (1 + θ)rl+i . u B B Note that at this time j=0 pl+j > (1 + θ)rl , and this indicates that job JlB u is released before time j=0 pB l+j /(1 + θ).  π also has at most two subcases, as illustrated in Figure 3: f. uj=0 pB l+j /(1 +  u A θ) > SsA ; g. j=0 pB /(1 + θ) ≤ S (at this time, we might as well assume s l+j u B Sk ≤ j=0 pl+j /(1 + θ) ≤ Ck ).

u

j=0

u

pB l+j /(1 + θ)

? · · · JsA

j=0

B ··· JlB · · · Jl+u

f

pB l+j /(1 + θ)

? · · · Jk · · · JsA

B ··· JlB · · · Jl+u

g B Fig. 3. Two subcases for pB l+i > (1 + θ)rl+i

A 1+θ For subcase f in Figure 3, according to condition 1, we have pA s ≤ Ss θ < u u B B B pl+j 1+θ u j=0 pl+j 1+θ j=0 pl+j B B B∗ + j=0 j=0 pl+j . As Cmax (I) ≥ 1+θ θ and Cmax (I) < 1+θ 1+θ θ + B u Cmax (I) 1 1 1 B j=0 pl+j , C B ∗ (I) < 1 + θ + 1+θ < 2 + θ . u

max

Single-Machine Scheduling Problems

253

B Similar to subcase e in Figure 2, for subcase g in Figure 3, we have Cmax (I) ≤  nB B B  p p n B B i=1 i i=1 i + + i=1 pi , where the first item in the right hand represents 1+θ θ the upper bound of the initial waiting time of the machine, the second item represents the upper bound of the sum of the processing times of jobs of agent A completed before JlB . nB B C B (I) 1 1 1 B∗ As Cmax (I) ≥ i=1 pi , then Cmax B ∗ (I) ≤ 1 + θ + 1+θ < 2 + θ .  nB

max

∗

B B So, for case 1.1, we obtain Cmax (I)/Cmax (I) ≤ 2 + 1/θ.

By the hypothesis of case 1.1, all the jobs of π are processed contiguously and in π ∗ agent A is last completed. Therefore, the sum of the processing times A∗ of the block is at most Cmax (I). If the first job of the block is of agent A, θ A∗ then the possible initial waiting time of the machine is at most 1+θ Cmax (I), θ A A∗ that is, Cmax (I) ≤ (1 + 1+θ )Cmax (I). Similarly, if the first job of the block is of agent B, then the possible initial waiting time of the machine is at most 1 θ A∗ A∗ 1+θ Cmax (I) ≤ 1+θ Cmax (I). Thus, no matter whether the first job of the block θ A A∗ is of agent A or agent B, we have Cmax (I) ≤ (1 + 1+θ )Cmax (I). So, for case 1.1, we get

ALG3(I) OP T (I)

∗

≤

∗

A B θ (1+ 1+θ )Cmax (I)+θ(2+ θ1 )Cmax (I) A∗ (I)+θC B ∗ (I) Cmax max

< 2 + 1θ .

Case 1.2. agent B is last completed in π ∗ . ∗

B Here the sum of the processing times of the block is at most Cmax (I). Corresponding to the case in which the first job of the block is of agent A or agent B, we θ A B B∗ A B get max{Cmax (I), Cmax (I)} ≤ (1 + 1+θ )Cmax (I) or max{Cmax (I), Cmax (I)} ≤ 1 B∗ (1 + 1+θ )Cmax (I).

So we obtain 2+

ALG3(I) OP T (I)

1 θ.

=

A B Cmax (I)+θCmax (I) A∗ (I)+θC B ∗ (I) Cmax max

∗

≤

∗

B B θ θ (1+ 1+θ )Cmax (I)+θ(1+ 1+θ )Cmax (I) B ∗ (I) θCmax

=

Lemma 3. Algorithm 3 is 2 + 1/θ-competitive when it acts on I and results in case 2 of lemma 1. Proof. For case 2 of lemma 1, π consists of two blocks, the first one includes the jobs of agent A and/or agent B, the second one only includes the jobs belonging to one of the two agents. Case 2.1. The second block only consists of some jobs of agent A. Denote them A A by JlA , Jl+1 , · · · , Jl+h , h ≥ 0. A A If pA l+j ≤ rl+j (1 + θ)/θ, j = 0, 1, . . . , h, then by algorithm 3, Cmax (I) = ∗ A Cmax (I). A that pA If there exists at least one pA l+i , 0 ≤ i ≤ h, such l+i> rl+i (1 + θ)/θ, then h A h A θ A h Cmax (I) j=0 pl+j + j=0 pl+j 1+θ θ A A h = 1 + 1+θ . j=0 pl+j > rl (1 + θ)/θ. Thus, C A∗ (I) ≤ pA max

1.1, so we have

j=0

l+j

A B Cmax (I) Cmax (I) θ A∗ (I) ≤ 1 + 1+θ . The ratio C B ∗ (I) is Cmax max B Cmax (I) ALG3(I) B ∗ (I) ≤ 2 + 1/θ, and OP T (I) ≤ 2 + 1/θ Cmax

So, for case 2.1,

the same as in case

254

G. Ding and S. Sun

Case 2.2 The second block only consists of some jobs of agent B, and denote B B them by JlB , Jl+1 , · · · , Jl+v , v ≥ 0. B If pB l+j ≤ (1 + θ)rl+j , j = 0, 1, . . . , v, then by algorithm 3, we get ∗

B B A B Cmax (I) = Cmax (I). Considering Cmax (I) < Cmax (I), we obtain B∗ B∗ Cmax (I)+θCmax (I) ∗ A B ∗ (I) Cmax (I)+θCmax

ALG3(I) OP T (I)

<

≤ 1 + 1θ .

B B If there exists at least one Jl+i , 0 ≤ i ≤ v, such that pB > (1 + θ)rl+i , u  ul+i B B 1 B u p + p Cmax (I) j=0 l+j j=0 l+j 1+θ 1 B B u then j=0 pl+j > (1 + θ)rl . So C B∗ (I) ≤ ≤ 1 + 1+θ . pB max

A B For Cmax (I) < Cmax (I), we have

1+

1 θ

+

1 1+θ

+

1 θ(1+θ)

≤2+

ALG3(I) OP T (I)

1 θ.

j=0 l+j ∗

≤

∗

B B 1 1 (1+ 1+θ )Cmax (I)+(1+ 1+θ )θCmax (I) A∗ (I)+θC B ∗ (I) Cmax max

≤

At last, through lemma 2 and lemma 3 we know that the “smallest” counterexample I does not exist. This completes the proof of theorem 6. Example 1. Consider the following instance in which agent A has a job J1A with θ B B B r1A = 0, pA 1 = 1 and agent B has a job J1 with r1 = ( 1+θ )+0 , p1 = 0. Scheduling the two jobs according to algorithm 3 results in a schedule in which J1A θ θ is completed at 1 + 1+θ and J1B is completed at 1 + 1+θ too. But it can be θ B and J1A shown easily that in an optimal solution J1 is completed at time 1+θ θ is completed at 1 + 1+θ . So, when θ is sufficiently large, the competitive ratio of algorithm 3 for this instance 1+

θ 1+θ

1+

θ 1+θ ) θ θ 1+θ

+ θ(1 +

θ 1+θ

+

≈

θ 1+θ θ 1+θ

1+

=2+

1 . θ

This particular instance implies that the algorithm 3’s competitive ratio can be achieved when θ is sufficiently large.

5

Conclusions

We consider the single-machine scheduling problems in which two distinct agents are concerned. Each agent has a set of jobs with different release times, and both of them expect to complete their respective jobs as early as possible. We take A B the makespan of each agent as its own criterion and take Cmax + θCmax as our objective function. In this paper, both off-line and on-line models are considered. For off-line model, corresponding to the preemptive case, we provide an exact algorithm. For on-line model, we construct an optimal on-line algorithm in case the jobs can be preempted. When preemption is not allowed, we give a lower bound of the competitive ratio of the problem, and construct an on-line algorithm with competitive ratio approaching this lower bound as θ large enough. There remain two future research directions. First, several different combinations of the agent’s other objective functions can be investigated thoroughly, especially for on-line case. Second, our work can be extended to include other machine environments, especially the parallel machine environment.

Single-Machine Scheduling Problems

255

References 1. Agnetis, A., Mirchandani, P.B., Pacciarelli, D., Pacifici, A.: Nondominated schedules for a job-shop with two competing users. Comput. Math. Organ. Theory 5, 191–217 (2000) 2. Agnetis, A., Mirchandani, P.B., Pacciarelli, D., Pacifici, A.: Scheduling problems with two competing agents. Oper. Res. 52, 229–242 (2004) 3. Agnetis, A., Pacciarelli, D., Pacifici, A.: Multi-agent single machine scheduling. Ann. Oper. Res. 150, 3–15 (2007) 4. Agnetis, A., Pascale, G., Pcciarelli, D.: A Lagrangian approcah to single machine scheduling problem with two competing agent. J. Scheduling 12, 401–415 (2009) 5. Baker, K.R., Smith, J.C.: A multiple-criterion model for machine scheduling. J. Scheduling 6, 7–16 (2003) 6. Cheng, T.C.E., et al.: Multi-agent scheduling on a single machine to minimize total weighted number of tardy jobs. Theoretical Computer Sci. 362, 273–281 (2006) 7. Cheng, T.C.E., Ng, C.T., Yuan, J.J.: Multi-agent scheduling on a single machine with max-form criteria. Europen J. Oper. Res. 188, 603–609 (2008) 8. Lee, K., Choi, B.-C., Leung, J.Y.-T., Pinedo, M.: Approxiation algorithms for multi-agent scheduling to minimizing total weighted completion time. Information Processing Lett. 109, 913–917 (2009) 9. Liu, P., Tang, L.: Two-agent scheduling with linear deteriorating jobs on single machine. In: Proceedings of the 14th Annual International Conference on Computing And Combinatorics, Dalian, China, pp. 642–650 10. Liu, P., Tang, L., Zhou, X.: Two-agent group scheduling with deteriorating jobs on a single machine. Int. J. Adv. Manuf. Technol. 47, 657–664 (2010) 11. Liu, P., Zhou, X., Tang, L.: Two-agent single-machine scheduling with positiondependent processing times. Int. J. Adv. Manuf. Technol. 48, 325–331 (2010) 12. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979) 13. Mor, B., Mosheiov, G.: Scheduling problems with two competing agents to minimize minamx and minsum earliness measures. Europen J. Oper. Res. (2010) doi:10.1016/j.ejor, 03003 14. Ng, C.T., Cheng, T.C.E., Yuan, J.J.: A note on the complexity of the problem of two-agent scheduling on a single machine. J. Comb. Optim. 12, 387–394 (2006) 15. Potts, C.N., Kovalyov, M.Y.: Scheduling with batching: A review. Eur. J. Oper. Res. 120, 228–349 (2000) 16. Wang, G., Vakati, R.S.R., Leund, J.Y.-T., Pinedo, M.: Scheduling two agents with controllable processing times. Europen J. Oper. Res. 205, 528–539 (2010) 17. Yang, W.H., Liao, C.J.: Survey of scheduling research involving setup times. Int. J. Sys. Sci. 30, 143–155 (1999)

Multi-Agent Asynchronous Negotiation Based on Time-Delay LiangGui Tang1 and Bo An2 1

College of Computer Science and Information Engineering, Chongqing Technology and Business University, Chongqing, P.R. China, 400067 2 Dept. of Computer Science, University of Massachusetts, Amherst, USA, MA 01003 [email protected], [email protected]

Abstract. In the Electronic Commerce applications based on MAS (MultiAgent Systems) etc., because every agent may have different negotiation strategy, reasoning mode, and that for diverse negotiation offers, the time spending in strategy computing, resource allocation and information transmission of the agent are different, so that one to many negotiation generally is asynchronous or with time-delay. This paper puts forward a mechanism with time-delay for one to many negotiation and give a negotiation control mode of multi-agents; considering the aspects of this negotiation including time dependent, opponent influence and other negotiation threads, this paper analyzes the negotiation flow for one to many negotiation with timedelay, designs sub-negotiation strategies for multi-agents; and then discusses when to offer in one to many negotiation with time-delay; brings forward a method for determining the number of negotiation opponents. The research of one to many negotiation with time-delay will improve its applicability and make agent-based automatic negotiation satisfy the needs of practical application. Experimental results validate the correctness and validity of our methods. Keywords: Multi-Agent Systems, Asynchronous Negotiation, Time-Delay.

1 Introduction Agent-based E-Commerce has greatly improved the efficiency and automation of E-Commerce [1][2][3][4]. In agent-based E-Commerce, consumer buying behavior can be divided into six stages [4]: need identification, product brokering, merchant brokering, negotiation, purchase and delivery, service and evaluation. The purpose of negotiation is to reach agreement on deal items between consumers and producers, such as price, quality, transportation, post-selling service, etc. The process of negotiation is similar to that in real life, generally including offering, offer evaluation, counteroffer generation, etc. According to the number of sellers and buyers participating in negotiation, agentbased automatic negotiation can be divided into three cases [5]: one to one negotiation (bilateral negotiation), many to many negotiation and one to many negotiation. In K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 256–269, 2010. © Springer-Verlag Berlin Heidelberg 2010

Multi-Agent Asynchronous Negotiation Based on Time-Delay

257

most Multi-Agent Systems, various auction mechanisms are adopted to realize the function of one to many negotiation. The limitations of auction mechanisms are: (1) auction mechanisms rely mostly on rigid rules and are highly structured; (2) auction mechanisms lack of flexibility; (3) it is difficult for agents to individuate their strategies; (4) agents lack of interaction during the auction process. While one to many negotiation are more flexible, for example, agents can adopt different negotiation strategies with different negotiation opponents, negotiation can be taken under different negotiation environment and negotiation protocol, etc. A negotiation cycle refers to the time spent in a round of negotiation. In existing one to many negotiation mechanisms [6][7][8] and systems (such as CSIRO’s ITA[7], MIT’ s Tete-a-Tete[9], Vrije University’s NALM[10]), the negotiation cycles are the same when agent negotiates with many opponents, which means agent’s negotiation with many opponents is synchronous. But in actual negotiation, because different agents have different strategies, reasoning mechanisms, and that for diverse negotiation offers, the time spending in strategy computing, resource allocation and information transmission of the agent are different, so that one to many negotiation generally is asynchronous or with time-delay.

2 Negotiation Frame and Control Strategy Negotiation is a dynamic interactive process for reaching agreement. During negotiation, agents exchange offers reflecting of their beliefs, desires and intentions. In order to make our discussion brief, we suppose: all agents are rational; we take the negotiation about the price of a certain product between a buyer and many sellers as background. One to many negotiation structure mainly have centralized control mode and the hierarchy structure. The centralized structure consists of only one complex and powerful agent conducting negotiation with many opponents (as in Fig.1). The centralized structure is simple and has high efficiency with small number of sellers. But when the number of sellers increases, the calculating cost of buyer increases, the processing speed slows down and the buyer’s negotiation efficiency decreases. An alternative method is using hierarchy structure, in which the buyer consists of a control agent and several sub-negotiation agents, and negotiation is composed of several sub-negotiation threads (as in Fig.2). When negotiation begins, the control agent creates several sub-agents (i.e. sub-buyers here) equal with the number of sellers and each sub-agent negotiates with unique seller separately. During negotiation, the control agent manages and coordinates the negotiation of each subagent based on certain strategy. Compared with the centralized structure, hierarchy structure can fit Asynchronous Negotiation to the diverse negotiation offers, and which has good expansibility, dynamic distribution and system stability, a failure of one sub-negotiation thread will not result in the failure of whole negotiation process. Base on the hierarchy structure, a buyer needs to make decision on two levels of negotiation strategy: negotiation control strategy and sub-negotiation strategy for each

258

L. Tang and B. An

agent. Negotiation strategy of each sub-agent is consistent with that in one to one negotiation [11][12]. Negotiation control strategy refers to rules controlling the whole negotiation process and coordinating all sub-negotiation threads.

Fig. 1. Centralized structure

Fig. 2. Hierarchy structure

Based on the analysis of Rahwan and Kowalczyk[6], we categorize the negotiation strategies for control agent into three kinds: (1) Desperate strategy: All sub-negotiation threads are independent, i.e. subagents don’t know the negotiation information of other sub-negotiation threads, and control agent is anxious to end the negotiation. Once a sub-agent finds an acceptable offer, control agent accepts it and stops other sub-negotiation threads. If several acceptable offers are found out at the same time, control agent chooses the one with the highest utility. (2) Patient strategy: All sub-negotiation threads are independent, but control agent is not anxious to end the negotiation. When one sub-agent finds an acceptable offer, other sub-negotiation threads go on. Once all sub-negotiation threads complete, control agent chooses the one with the highest utility. (3) Optimized patient strategy: All sub-negotiation threads are not independent. During negotiation, each sub-agent clearly knows the negotiation information of other sub-negotiation threads and adjusts its negotiation strategy accordingly. Optimized patient strategy regards all sub-negotiation threads as a whole. All sub-negotiation threads are interactive, and it can avoid unnecessary bargaining process. Given the condition that the control agent is not anxious to end negotiation, optimized patient strategy has the best negotiation performance.

3 One to Many Asynchronous Negotiation As former statement, one to many negotiation is generally asynchronous. The reasons causing one to many negotiation asynchronous include: (1) agents locating in different positions in network have different communication distance, and the factors affecting communication quality (QoS), e.g., bandwidth, congestion, network failure, may result in different communication delay; (2) agents with different reasoning mechanisms, processing capability, computing speed, allocating resource have

Multi-Agent Asynchronous Negotiation Based on Time-Delay

259

different processing time; (3) according to strategies, agents may decide to postpone sending offers, even though offers have already been proposed successfully. Definition 1: The response action of agent-buyer b when receives seller

i ’s offer

offer sit at time t is defined as ACTbi ,t (offer sit ) , which is denoted as follows: ⎧ reject if Bit >= Tbi . ⎪ i ACT (offer s ) = ⎨ agree if U b (offer sit ) ≥ U bi (offer bit −1 ) ⎪ offer b i otherwise b ⎩ i ,t b

t i

(1)

Definition 2: Compromising degree refers to how much agent compromises in negotiation. Suppose kit represents the compromising degree of agent

i in the tth

negotiation round. If agent is a buyer, its offer in this round is defined as follows:

offer bit = (1 + kit ) × offer bit −1 ; Else if agent is a seller, its offer in this round is defined as:

offer sit = (1 − k it ) × offer sit −1 . Fig.3 shows the flow of one to many negotiation with time-delay. Compared with the flow of one to many negotiation without time-delay, all negotiation threads are asynchronous. We also suppose the negotiation begins with seller’s offering.

Fig. 3. The flow of one to many negotiation with time-delay

260

L. Tang and B. An

3.1 The One to Many Negotiation Strategy with Time-Delay While adopting the hierarchy architecture, the negotiation between one buyer and several sellers consists of several threads of bilateral negotiation between sub-buyers and sellers. Given the condition that the control agent takes the optimized patient strategy, each sub-buyer needs to make a decision on its negotiation strategy. Different from generic bilateral negotiation, sub-agent needs to take negotiation information of other sub-agents into account. After analyzing the main factors affecting sub-agent’s decision making, we design three kinds of strategies: time, opponent and other negotiation threads dependent strategies. (1) Time dependent strategy T Strategy T takes it into account that the remaining negotiation time’s effect on negotiation strategies, i.e. agents’ sensitivity to time. Agents’ sensitivity to time embodies how the left negotiation time affects negotiation strategies, for example, if an agent has a time deadline by which an agreement should be in place, it will concede more rapidly as deadline approaches. Generally, the remaining negotiation time has a remarkable effect on negotiation strategy [13] [14]. From buyer’s perspective, formula pt = (1 + (t / Tmax ) β ) × p0 ( β > 0 ) can be used to express the remaining time’s effect on negotiation strategy, where

pt represents buyer’s offer at time t , Tmax represents the negotiation deadline, and p0 represents the buyer’s first offer. The change of p t with the remaining time has the following characteristics (as showed in Fig.4):

Fig. 4. The effect of the remaining time on negotiation strategy

① When β = 1 ,

p t linearly increases, which means the remaining negotiation

time has a consistent effect on agents’ negotiation strategy.

Multi-Agent Asynchronous Negotiation Based on Time-Delay

② When β > 1 , at the beginning of negotiation, p

t

261

changes little. Agents make

little compromise at the beginning of negotiation, but make great compromise when negotiation is to be closed. When 0 < β < 1 , the change of the slope is decreasing. Agents are eager to

③

reach an agreement as quickly as possible, and they make great compromise at the first few negotiation iterations. With negotiation’s ongoing, agents make less and less compromise. Buyer can decide the value of parameter β according to its practical situation. Similarly, when a seller negotiates with several buyers, β formula pt = (1 − (t / Tmax ) ) × p0 , ( β >0 ) can be used to express the remaining negotiation time’s effect on the seller’s negotiation strategy. Suppose kbit [T ] is the compromising degree of sub-buyer

i according to strategy

T in the tth negotiation round, and ksit [T ] is the compromising degree of sub-seller i according to strategy T in the tth negotiation round, where sub-seller

i represents the

ith sub-negotiator of seller when it negotiates with several buyers. We get: kbit [ Τ ] =

1 + ( B it / T bi ) β b

i

.

(2)

1 − ( Bit / Tsi ) β s . ks [ Τ ] = i 1 − ( Bit −1 / Tsi ) β s

(3)

1 + (B

t −1 i

i b

/T )

β bi

i

t i

Sub-buyer i and sub-seller

i make a decision on the value of parameter β bi and β s i

according to their practical situation. (2) Opponent dependent strategy O Agent decides its compromising degree according to that of its opponent. If its opponent makes great compromise, agent will do the same; otherwise agent will compromise little. In addition, with the increase of negotiation opponents, agent will make less and less compromise, because it has more and more chance to reach an agreement.

262

L. Tang and B. An

Opponent dependent strategy O takes it into account that the effect of the offers of other sub-agents’ negotiation opponent’s on agent’s compromising degree. Generally, if negotiation opponent makes little compromise, agent also makes relatively little compromise, whereas agent will make great compromise. In addition, in generic bilateral negotiation, a buyer only negotiates with one seller, and if the seller quits negotiation, the buyer would bear great loss, so the buyer will cherish the only chance of reaching an agreement, i.e. making relatively great compromise. In one to many negotiation, a buyer negotiates with several sellers at the same time. If the negotiation with a seller fails, it would be possible that buyer reaches an agreement with one of other sellers, so the buyer mostly makes relatively little compromise. Suppose kbit [O ] represents the compromising degree of sub-buyer strategy O in the

i according to

t th negotiation round, ksit [O ] represents the compromising degree

of sub-seller i according to strategy O in the

t th round of negotiation, and we get: offer s it . ) offer s it −1

(4)

offer bit −1 − 1) . offer bit − 2

(5)

kbit [O ] = γ b × (η b ) n × (1 − ks it [O ] = γ s × (η s ) n × ( Where 0 < η b ≤ 1 and 0 < η s ≤ 1 .

(3) Other negotiation threads dependent strategy P Agent also decides its compromising degree according to the negotiation information of other sub-negotiation threads. Compared with offers of other subbuyers’ opponents, if its negotiation opponent’s offer is relatively smaller, agent will make great compromise; otherwise agent will compromise little. When the control agent adopts the optimized patient strategy, all negotiation threads are not mutual independent. While making a decision on its compromising degree, in order to avoid unnecessary negotiation process, single sub-agent should take the information of other negotiation threads into account. For example, a buyer negotiates with three sellers at the same time. In a certain cycle, sub-buyer 1 receives seller 1’s offer 50, and sub-buyer 1 knows the offers of seller 2 and seller 3 are 30 and 40 respectively, i.e. the lowest offer buyers receive is 30. In order to avoid unnecessary negotiation process, sub-buyer 1’s offer to seller 1 in the next round will be not higher than 30. Similarly, sub-buyer 2’s and sub-buyer 3’s offers to seller 2 and seller 3 will also be not higher than30. In addition, the lower seller’s offer is, the more compromise its opponent makes. Because negotiation is with time-delay, when a buyer negotiates with n sellers, given the condition that buyer adopts the optimized patient strategy, each sub-buyer will not immediately calculate the counteroffer to its negotiation opponent once

Multi-Agent Asynchronous Negotiation Based on Time-Delay

263

receiving its opponent’s offer, but wait to get negotiation information of other negotiation threads, in order to avoid unnecessary negotiation process. numbt represents the number of offers buyer receives in the

t

th

negotiation round,

t represents the lowest offer buyer receives in the t th 0 < numbt ≤ n and offer smin

negotiation round. Similarly, when a seller negotiates with represents the number of offers seller receives in the

t

th

n buyers, num st negotiation round,

t represents the highest offer seller receives in the t th 0 < num st ≤ n and offer bmax

negotiation round. numbt

numst

i =1

i =1

t t offer smin = min(offer sit ) , offer bmax = max(offer bit )

Suppose kbit [ P ] represents the compromising degree of sub-buyer strategy P in the

i according to

t th negotiation round, kbit [ P ] represents the compromising degree of

sub-seller i according to strategy P in the

t th negotiation round, and we get:

t t ⎧γ 1 × ( offer smin ≥ offer bit −1 offer sit ) if offer smin ⎪ . kb [ P] = ⎨ t t −1 t −1 × − γ offer s offer b offer b otherwise ( ) ⎪⎩ 2 i i min

(6)

t t −1 ⎧γ 3 × ( offer bit offer bmax ≤ offer sit −1 if offer bmax ) ⎪ . ksit [ P ] = ⎨ t −1 t t −1 otherwise ⎪⎩γ 4 × ( offer si − offer bmax ) offer si

(7)

t i

(

)

(

Where

)

0 < γ 1 , γ 3 ≤ 1 and γ 2 , γ 4 ≥ 1 .

The selection of negotiation strategy T, O and P synthetically takes it into account that the most important factors affecting agents’ decision making in bilateral negotiation and the characteristics of one to many negotiation. Agents participating in negotiation can select and combine these strategies according to their practical situation, and then make a decision on their compromising degree. Suppose kit be agent’s compromising degree in the

t th negotiation round, and we get:

264

L. Tang and B. An

k it = wT k it [T ] + wO k it [ O ] + w P k it [ P ] .

(8)

Where 0 ≤ wT , wO , wP ≤1 and wT + wO + wP = 1 . In generic bilateral negotiation, time factor usually has a remarkable effect on negotiation strategy. We use strategy T to express the remaining negotiation time’s effect on its compromising degree. During negotiation, rational agent will pursue its greatest utility, and strategy O emphasizes negotiation opponent’s strategy’s effect on its compromising degree. Given the condition that the control agent adopts the optimized patient strategy, sub-agent needs to take the information of other negotiation threads into account, to avoid unnecessary negotiation process. 3.2 Main Factors Affecting Negotiation Performance Here we discuss factors having influence on negotiation performance in certain negotiation environment (e.g., negotiation protocols, negotiation strategies, etc.). The main factors include: MAS is a distributed environment, where agents interact with each other through network. Meanwhile negotiation highly depends on time, so communication quality and network’s QoS will affect both negotiation strategies and negotiation results. Factors affecting communication quality include: data quantity, network bandwidth, transfer delay, network failure, etc. In one to many negotiation, the control agent will perform functions such as controlling and coordinating negotiation of sub-agents: negotiation management, e.g., selection of negotiation strategies, decision making of when to offer, communication, etc.; offer evaluation; counter offer generation, etc. Given a certain negotiation environment, the main factor affecting negotiation effect is distributing process loads and the opportunities to offer. (1) The number of negotiation opponents Suppose that there are n agents participating in one to many negotiation, taking the worst situation into account (means that every two agents negotiate, i.e. many to many negotiation), negotiation between n agents forms a complete diagram. The effect of the increase of negotiation opponents on Distributing Process Capability (DPC Load, DPC_L ): the worst situation of n agents taking one to many negotiation is that agents negotiating with each other. We get: DPC_L ≈ O (n 2 ) . So that it is not the more negotiation opponents the better. In one to many negotiation, buyer needs to making a decision on how many negotiation opponents to negotiate in order to reach the best negotiation result. Fig.5 shows negotiation results’ changing with the number of negotiation opponents when taking the effect of the increase of negotiation opponent on negotiation performance into account, where we can find that: with less number of

Multi-Agent Asynchronous Negotiation Based on Time-Delay

265

opponents, agreed price decreases with the increase of negotiation opponent; with greater number of opponents, the agreed price increases with the increase of negotiation opponents.

Fig. 5. Negotiation result’s changing with the number of negotiation opponents

When buyer needs to make a decision on the number of sellers, the number of sellers can be calculated by formula (9), where N represents the maximum of the number of sellers, product),

p min represents the cost of negotiation object (bargaining

p0 represents the finally agreed price, kdpc _ l is the parameter reflecting the

influence of sellers’ increase on communication quality, and ko _ l is the parameter reflecting the other influence of sellers’ increase on buyer’s processing capability. n = arg min((1 − (

（

）

x β ) ) × ( p0 − pmin ) + pmin ) × 1 + kdpc_l × x 2 + ko _ l × x ,0 < β < 1 . N

(9)

Similarly , when a seller needs to make a decision on the number of buyers, the number of buyers can be calculated by formula (10), where N represents the maximum of the number of buyers, (bargaining product),

p min represents the cost of negotiation object

p0 represents the finally agreed price, kdpc _ l is the parameter

reflecting the influence of buyers’ increase on communication quality, and ko _ l is the parameter reflecting the influence of buyers’ increase on seller’s processing capability.

266

L. Tang and B. An

（

）

x n = arg max((1 + ( )β ) × ( p0 − pmin ) + pmin ) × 1 − kdpc _ l × x2 − ko _ l × x ,0 < β < 1 . N

(10)

(2) The decision making of when to offer In one to many negotiation with time-delay, negotiation threads are asynchronous, and sub-agents should make a decision on when to offer. In practical environment, agents don’t clearly know opponents’ negotiation strategies, processing mechanisms, and can’t forecast communication time, so the situation is greatly complex. Hence, the decision making of when to offer is a key problem for all sub-agents. We give a algorithm with the fixed ratio based negotiation plan. In a certain negotiation round, after certain sub-agent firstly receives offer from its opponent, it waits until the number of sub-agents having received offers reaches a fixed ratio, then all sub-agents having received offers begin to generate new offers. This algorithm describes the decision making process of when to offer according to the fixed ratio based plan. It uses a queue Q to store offers. Suppose the fixed ratio is R, 0 < R ≤ 1 . Begin 1) q = null ; 2) While R > Lengthq / n

insert ( offer ) ; End While 3) All the offers in queue Q, Sub-agents begin to generate new offers respectively; 4)If negotiation succeeds, turn End begin; Otherwise, turn to step 2). End begin

In this algorithm, because some negotiation threads may break up during negotiation, the sellers’ number n may constantly change. Fig.6 and Fig.7 reflect different fixed ratio’s effect on final agreed price. According to the fixed ratio based plan, in each negotiation round, after certain sub-agent firstly receives offer from its opponent, it waits until the number of subagents having received offers reaches a fixed ratio, then all sub-agents having received offers begin to generate new offers. Similarly, it is difficult for sub-buyer to choose a fixed ratio and get good negotiation result. We can find that the negotiation

Multi-Agent Asynchronous Negotiation Based on Time-Delay

267

result’s changing with fixed ratio is also not monotonic. If the greatest reflecting time difference of sellers is little, greater fixed ratio can get better result (as showed in Fig.6). If the greatest reflecting time difference of sellers is greater, less fixed ratio can get better result (as showed in Fig.7).

Fig. 6. Fixed ratio based plan with little time dependent

Fig. 7. Fixed ratio based plan with great time dependent

4 Conclusion Agent based automatic negotiation has highly promoted the intelligence of agentbased E-commerce. Compared with auction mechanisms, one to many negotiation is more flexible and interactive. Practical one to many negotiation mechanisms mostly are with time-delay, and the research of one to many negotiation with time-delay will

268

L. Tang and B. An

enlarge its applicability and make agent-based automatic negotiation meet the needs of practical application. In this paper, we analyze analyzes the structures of one to many negotiation, the strategies of control agent, and the flow of one to many negotiation without timedelay. Then the concept of one to many negotiation with time-delay is put forward, and its flow is analyzed. We design three basic negotiation strategies for sub-agents, and propose the decision making mechanisms of when to offer and how many opponents to negotiate. Experimental results validate the correctness and validity of our methods. Because most MAS work in a heterogeneous, distributed and complicated environment, our strategies and mechanisms still need to be validated in practical application. We can extend the single-attribute one to many negotiation brought out in this paper to multi-attributes many to many negotiation. The future work includes: (1) the building of a test bed for one to many negotiation; (2) studying on the characteristics of one to many negotiation with time-delay in dynamic environment;(3) research of the learning mechanisms in one to many negotiation, etc. As a work just beginning, there is still a long way to go before applying one-to-many negotiation into practical commerce application.

Acknowledgments This research is supported by the Key Science Technology Project and the Science Foundation of Chongqing under Grant No.CSTC-2005AC2090 and No.CSTC2006BB2249, supported by the Science Technology Project of Chongqing Municipal Education Commission under Grant No. KJ060704, and the Doctor Foundation of CTBU under Grant No. 09-56-12.

References 1. He, M.H., Jennings, N.R., Leung, H.F.: On Agent-Mediated Electronic Commerce. IEEE Transactions on Knowledge and Data Engineering 15(4) (July/August 2003) 2. Yarom, I., Rosenschein, J.S., Goldman, C.V.: The role of middle-agents in electronic commerce. IEEE Intelligent Systems 18(6), 15–21 (2003) 3. Youll, J.: Agent-based electronic commerce: opportunities and challenges. In: Fifth International Symposium on Autonomous Decentralized Systems, pp. 26–28 (March 2001) 4. Guttman, R.H., Moukas, A.G., Maes, P.: Agent-mediated electronic commerce: a survey. Knowledge Engineering Review 13(2), 143–152 (1998) 5. Kim, J.B., Segev, A.: A Framework for Dynamic eBusiness Negotiation Processes. In: Proceedings of the IEEE International Conference on E-Commerce (CEC’03) 6. Rahwan, I., Kowalczyk, R., Pham, H.H.: Intelligent agents for automated one-to-many ecommerce negotiation. Australian Computer Science Communications 24(1), 197–204 (2002) 7. Kowalczyk, R., Bui, V.: On Constraint-based Reasoning in e-Negotiation Agents. In: Dignum, F., Cort, U. (eds.) Agent-Mediated E-Commerce III (2003)

Multi-Agent Asynchronous Negotiation Based on Time-Delay

269

8. Brazier, F.M.T., Cornelissen, F., Gustavsson, R., Jonker, C.M., Lindeberg, O., Polak, B., Treur, J.: Compositional Design and Verification of a Multi-Agent System for One-toMany Negotiation. In: Proceedings of the Third International Conference on Multi-Agent Systems, ICMAS-98, p. 8. IEEE Computer Society Press, Los Alamitos (1998) (in press) 9. Guttman, R.H., Moukas, A.G., Maes, P.: Agent-mediated Electronic Commerce: A Survey. Knowledge Engineering Review 13(3) (1998) 10. Brazier, F.M.T., Cornelissen, F., Gustavsson, R., Jonker, C.M., Lindeberg, O., Polak, B., Treur, J.: A multi-agent system performing one-to-many negotiation for load balancing of electricity use. Electronic Commerce Research and Applications 1(2), 208–224 (2002) 11. Kraus, S.: Automated negotiation and decision making in multi-Agent environments. In: Multi-Agents systems and applications. Springer, New York (2001) 12. Faratin, P., Sierra, C., Jennings, N.R.: Negotiation Decision Functions for Autonomous Agents. Int. Journal of Robotics and Autonomous Systems 24(3-4), 159–182 (1998) 13. Sim, K.M., Choi, C.Y.: Agents that react to changing market situations. IEEE Transactions on Systems, Man, and Cybernetics, Part B 33(2), 188–201 (2003) 14. Jennings, N.R., Faratin, P., Lomuscio, A.R., Parsons, S., Sierra, C., Wooldridge, M.: Automated negotiation: Prospects, methods and challenges. International Journal of Group Decision and Negotiation 10(2), 199–215 (2001) 15. Faratin, P., Sierra, C., Jennings, N.R.: Negotiation Decision Functions for Autonomous Agents. Int. Journal of Robotics and Autonomous Systems 24(3-4), 159–182 (1998) 16. Sycara, K.: Multi-Agent Systems. Artificial Intelligence 19(2), 79–92 (1998)

Fuzzy Chance Constrained Support Vector Machine Hao Zhang1, Kang Li2, and Cheng Wu1 1

Department of Automation, Tsinghua University, Beijing 100084, China 2 Queen’s University Belfast, UK

Abstract. This paper aims to improve the performance of the widely used fuzzy support vector machine (FSVM) model. By introducing a fuzzy possibility measure, we first modify the original inequality constraints of FSVM optimization model as chance constraints. We fuzzify the distance between training data and the separating hyperplane, and use a possibility measure to compare two fuzzy numbers in forming the constraints for the FSVM model. By maximizing the confidence level we ensure that the number of misclassifications is minimized and the separation margin is maximized to guarantee the generalization. Then, the fuzzy simulation based genetic algorithm is used to solve the new optimization model. The effectiveness of the proposed model and algorithm is validated on an application to the classification of uncertainty in the hydrothermal sulfide data in the TAG region of ocean survey. The experimental results show that the new fuzzy chance constrained SVM model outperforms the original SVM model. Keywords: Fuzzy SVM, Possibility measure, Chance constrained programming, Fuzzy simulation, Genetic algorithm.

1 Introduction Support Vector Machine (SVM), originally proposed by Vapnik [1] has shown to be able to solve a wide range of classification problems due to its excellent learning performance, and has become a hotspot in the field of machine learning. However, there are still a few issues and limitations yet to be solved. For example, the effectiveness of SVM classifiers can be reduced if there exists noise or outlier in the data as SVMs are quite sensitive to noise. Lin [2-4] proposed Fuzzy Support Vector Machine (FSVM) to improve the SVM model by introducing the membership function. When training separating hyperplane, different data sample has its own contribution to the hyperplane according to the membership. A data point located at the interior of a class has a relatively higher degree of membership, while the noise or outliers have a relatively lower membership. The classification performance of SVM can be significantly improved if the impact of the noise or outliers on hyperplane of SVM can be further reduced. However, in the original FSVM model, membership is only used as the weight on the slack variables. [5] introduced the fuzzy chance constrained programming into FSVM for training K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 270–281, 2010. © Springer-Verlag Berlin Heidelberg 2010

Fuzzy Chance Constrained Support Vector Machine

271

fuzzy data. In this model, they used crisp equivalents to solve the optimization problem, which relax a support vector to a neighborhood centering around it. Using the possibility measure to frame the constraints in the optimization problem, [6] further established the following model:

r max f ( x )

r s.t. Pos {ξ | gi ( x , ξ ) ≤ 0} ≥ α i , i = 1, 2,L , p.

ξ is

a fuzzy parameter,

r gi ( x , ξ ) are constraint functions and α i are predetermined

confidence level. To solve this problem, [6] proposed two methods: crisp equivalent method and fuzzy simulation method based genetic algorithm. Although [5] introduced the chance constrained programming into the SVM model, the possibility measure was only used for the relaxation of the data points to the neighborhoods, and the optimal points were obtained in these neighborhoods to eliminate noise or outliers. In this paper, we use the basic idea of linear separability to fuzzify the distance between training data and the separating hyperplane, and then use a possibility measure to frame the constraints of FSVM model. By maximizing the confidence level we ensure that the number of misclassifications is minimized and the separation margin is maximized to guarantee the generalization. The effectiveness of the proposed model and algorithm is verified by an application to the classification of the uncertainty of hydrothermal sulfide data in ocean survey TAG region. The experiment results show that the new fuzzy SVM model outperforms the original SVM model. This paper is organized as follows. Section 2 is the preliminaries. Section 3 introduces the new fuzzy chance constrained SVM (FCSVM). In section 4 the fuzzy simulation based on genetic algorithm is used to solve the new model. Section 5 presents the numerical experiments and Section 6 concludes the paper.

2 Preliminaries We consider SVM model to classify between two classes of data, and the training data set is defined as:

r r r ( x1 , y1 ), ( x2 , y2 ),L , ( xl , yl ) ∈ℜn × {±1}, yi = +1, −1. r

with xi

(1)

= ( x1 ,L , xn )T , yi = +1 denotes positive data, yi = −1 denotes negative.

SVM is used for searching for an optimal hyperplane, separating the two classes of data correctly. If the training data set is linearly separable, there exists vector

r ( w, b) ,

272

H. Zhang, K. Li, and C. Wu

r

where w is normal vector of the optimal hyperplane, b ∈ R is the bias, holding the following inequalities:

r r w T x i + b ≥ 1, for all yi = +1.

r r wT xi + b ≤ −1, for all yi = −1. the above inequalities can be written uniformly as

(2)

r r yi ( wT xi + b) ≥ 1. The decision

function is given as:

r r r f ( x ) = sgn( wT x + b).

(3)

SVM is constructed based on the structure risk minimization principle in statistical learning theory [8], so taking misclassification into account, slack vector

r

ξ = (ξ1 , ξ 2 ,Lξl ) has been introduced to relax training data that violates constraint (2), and the training model is formulated as the following quadratic programming problem [1]: l 1 2 w + C ∑ ξi 2 i =1 rT r ⎧ y ( w xi + b) ≥ 1 − ξi s.t. ⎨ i ⎩ ξi ≥ 0, i = 1,L , l.

min

(4)

3 Fuzzy Chance Constrained Support Vector Machine Definition 1: (Dubois and Prade, 1998)[9] Let X be a nonempty set, of all subset of X , mapping Pos : the mapping satisfy: (1)

Pos(φ ) = 0.

(2)

Pos ( X ) = 1.

(3) Pos (U t∈T At ) = Sup Pos ( At ). t∈T

P( X ) is the set

P( X ) → [0,1] is called possibility measure, if

Fuzzy Chance Constrained Support Vector Machine

273

% be the fuzzy set on X . For every λ ∈ [0,1] , the necessary and Theorem 1 [8]: Let A sufficient condition for any x1 ∈ X

，x

2

∈X

{

Aλ = x | A% ( x) ≥ λ

， α ∈[0,1] , we have:

}

to be s convex set is that for

{

}

A% (α x1 + (1 − α ) x2 ) ≥ min A% ( x1 ), A% ( x2 ) .

X = ℜn is n dimension Euclidean space, A% is the fuzzy set % is convex. on X . If for all λ ∈ [0,1] Aλ is a convex set, then we call fuzzy set A

Definition 2 [8]: Let

，

Definition 3 [8]: Let (1) a% is the convex fuzzy set on X ; (2)

∃a1 , a2 ∈ X and a1 ≤ a2 , ∀x ∈ [ a1 , a2 ] , a% ( x) = 1 and lim a% ( x) = 0 , then a% is x →±∞

called a fuzzy number on X . Definition 4: (Liu, 1998) [6]. Let a% and measure [16] of

b% are two fuzzy numbers, then the possibility

a% ≤ b% is defined as:

(

)

Pos(a% ≤ b% ) = Sup{min a% ( x), b% ( y ) | x, y ∈ R, x ≤ y}. Definition 5: (Enhanced Semi-trapezoidal membership function) [8] Let a% be a fuzzy number, so its enhanced semi-trapezoidal membership function is defined as:

⎧ 0, when x ≤ a1 ⎪ ⎪ x − a1 a% ( x) = ⎨ , when a1 < x ≤ a2 ⎪ a2 − a1 ⎪⎩ 1, when a2 < x.

a1 , a2 are real numbers. (Triangular membership function) [8] Let b% is a fuzzy number, so its triangular membership function is defined as:

⎧ x + 1 − m, when m − 1 ≤ x ≤ m ⎪ b% ( x) = ⎨− x + 1 + m, when m − 1 ≤ x ≤ m ⎪ 0, other . ⎩

274

H. Zhang, K. Li, and C. Wu

m is a real number, which is the center of triangular membership function.

% has an enhanced semi-trapezoidal membership Theorem 2: Let fuzzy number 1 a1 = 0 and a2 = 1 , and b% is a fuzzy number % ≤ b% ) is equivalent to with triangular membership function, so the possibility Pos (1

function with the two parameters being

the following inequalities:

1, when m ≥ 1 ⎧ ⎪⎪1 + m Pos(1% ≤ b% ) = ⎨ ，when − 1 ≤ m < 1 2 ⎪ ⎩⎪ 0，when m < −1.

(5)

The above inequalities are also illustrated in Fig. 1.

Fig. 1. (a). When m ≥ 1

Fig. 1. (b). When −1 ≤

m <1

It is obviously that when m < −1 , by definition 4, we have Pos (1% ≤ b% ) = 0 . Classic FSVM model is defined as follows [2]: l 1 2 w + C ∑ siξi 2 i =1 rT r ⎧ y ( w xi + b) ≥ 1 − ξi s.t. ⎨ i ⎩ ξi ≥ 0, i = 1,L , l.

min

(6)

si is the degree of membership of training data. The significance of the model is that different training data is given different membership, the data belonging to the class with high level has a high membership

Fuzzy Chance Constrained Support Vector Machine

275

value, otherwise it will be low. The slack variables are weighed by membership in the objective function, thus the data with low membership has small impact on the

r w of the separating hyperplane.

objective function, so is on the normal vector

Thereby the separating hyperplane will be less affected by noise and outliers, ultimately the robustness of SVM can be improved. However, this improvement is simply based on the introduction of fuzzy membership concept, and weight on the objective function. It will be shown next that the model can further significantly improved. We propose the following fuzzy chance constrained SVM model.

r r wT xi + b is the distance between data and optimal hyperplane [10], r where xi denotes training data, we construct triangular fuzzy number d%i , and the rT r center is w xi + b . With (5) in theorem2, we have: r r ⎧ 1, when yi ( wT xi + b) ≥ 1 ⎪ r r ⎪1 + yi ( wT xi + b) r r % % Pos di ≥ 1 = ⎨ (7) ，when − 1 ≤ yi ( wT xi + b) < 1 2 ⎪ r r ⎪⎩ 0，when yi ( wT xi + b) < −1. Because

(

)

To limit the number of training data which violates (2), in other word, the number of misclassification is limited as small as possible, we need to constrain the range of fuzzy possibility measure. The larger Pos

( d% ≥ 1% ) , the smaller the slack variable ξ i

i

(refer back to Fig.1 (a) and (b)). Therefore, although we eliminate the slack variable in the original optimization model, structure risk minimization principle is still guaranteed due to the introduction of the chance constraints. In (6), the model uses the linear separability definition (2) from the traditional SVM model; however we define fuzzy linear separability as following:

r r

r

Definition 6: Let training data set {x1 , x2 ,L , xl } , for the given confidence

r

level λi , if there exists w∈ℜ , b ∈ℜ such that:

(

n

)

r r Pos d%i ≥ 1% ≥ λi , the center of d%i is yi ( wT xi + b), i = 1,L , l then the training data set is said to be fuzzy linearly separable.

276

H. Zhang, K. Li, and C. Wu

Fuzzy chance constrained SVM model is given as: l 1 2 w − C ∑ λi 2 i =1 (8) rT r ⎧⎪ Pos d%i ≥ 1% ≥ λi , i = 1,L , l the center of d%i is yi ( w xi + b) s.t. ⎨ 0 ≤ λi ≤ 1. ⎪⎩ l In order to ensure Pos d%i ≥ 1% is as high as possible, we need to make ∑ λi as

min

(

)

(

)

i =1

large as possible. From (5) we can see that it is difficult to use crisp equivalents method in [6] to solve the optimization problem, instead we choose a more general approach, fuzzy simulation based genetic algorithm.

4 Fuzzy Simulation Based Genetic Algorithm In [11], genetic algorithm has been used in chance constrained programming in a stochastic environment, and Monte Carlo stochastic simulation technique is used to check the validity of solutions. For the fuzzy chance constrained programming, we also use a Monte Carlo simulation-like method to handle the possibility measure constraints

r Pos {ξ | g i ( x , ξ ) ≤ 0} ≥ α i , i = 1, 2,L , p . This is as also called fuzzy

simulation in [12].

r r x , Pos {ξ | gi ( x, ξ ) ≤ 0} ≥ αi , i = 1,2,L, p is established if and r 0 0 0 only if there exists ξ such that g i ( x , ξ ) ≤ 0, i = 1, 2,L , p and μ (ξ ) ≥ α i . If For any given

ξ0

does not satisfy the above condition, re-generate ξ and check the constraints.

After a given number of cycles,

0

r

ξ 0 is found, x is

r

said to be feasible, otherwise x is

infeasible. Genetic algorithm is a stochastic search method which is capable of generating a global optimal solution and avoiding getting stuck at local optima [13,14,15]. The fuzzy simulation based genetic algorithm is given as follows: 1

、 Representation structure

A general genetic algorithm uses a binary vector as a chromosome to represent the real value of a decision variable, where the length of the vector depends on the required precision. An alternative approach to represent a solution is the floating decision variable, here we use vector represent a chromosome.

r r r V = ( w, b, λ ) = ( w1 ,L , wn , b, λ1 ,L , λl ) to

2

Fuzzy Chance Constrained Support Vector Machine

277

All the constraints are inequalities, so when we generate chromosome

r V , we

、 Handling constraints

r r d%i , which the center is yi ( wT xi + b) , then according to (7) we determine Pos d%i ≥ 1% , and check whether the chromosome satisfies r r Pos d%i ≥ 1% ≥ λi , if V satisfies the condition, then V is feasible, otherwise it is construct fuzzy number

(

(

)

)

infeasible, this is the process of fuzzy simulation. 3

、 Initialization process

An interior solution for the optimization problem (8) can’t be found easily, so we

λ1 = 0,L , λl = 0 , b = 0 r (0) r n vector w in ℜ until V0 satisfies the set

define integer

in

r V0(0)

.

We

generate

random

constraints in optimization problem (8). We

q as the number of chromosome, and generate q chromosomes. We

M for initialization process and mutation r (0) r (0) n operation. We randomly select a direction d in ℜ , and let V = V0 + M ⋅ d , if r (0) r (0) V satisfy the constraints, we keep V as an initial chromosome, otherwise we r (0) r (0) set M as a random number between 0 and M , until V = V0 + M ⋅ d is feasible. r (0) r (0) Repeat the above process to generate V1 ,L , Vq , finally we get a set of initial also define large positive integer

feasible chromosome. 4

、 Evaluation function r r Let

V1( k ) ,L , Vq( k ) be the set of chromosome after k iterations, and they are sorted

in an ascending order based on the fitness (values of their corresponding objective function), that is,

r r r V1'( k ) ≤ V2'( k ) , ≤ L , ≤ Vq('k ) . Evaluation function, denoted by

r eval (V ) , is to assign a probability of reproduction to each chromosome so that its

likelihood of being selected is proportional to its fitness. So we use the following equation to define evaluation function, given

a ∈ (0,1)

，the evaluation function is:

r eval (Vi ( k ) ) = a (1 − a )i −1 , i = 1', 2 ',L , q '.

(9)

So obviously we have: q'

r (k )

∑ eval (V 5

、 Selection process

i =1'

i

) ≈ 1.

(10)

278

H. Zhang, K. Li, and C. Wu

We use roulette wheel selection to select a chromosome in the process, the roulette wheel selection is as follow:

r (k )

Step 1: For every Vi

calculate the cumulative probability Pi

，

P0 = 0, i r Pi = ∑ eval (V j( k ) ), i = 1', 2 ',L , q '. j =1'

Step 2: Generate a random real number r in [0, Pq ' ] .

r (k )

Step 3: Select i th chromosome Vi

(1' ≤ i ≤ q ') such that Pi −1 ≤ r ≤ Pi

。

Step 4: Repeat step2,3 q times and obtain a new set of chromosome. 6

、 Crossover operation We define a parameter

Pc as the probability of crossover operation, for

i = 1,L, q , we generate a random number r in [0,1] . If r ≤ Pc , then we r (k ) select Vi as a parent. This gives us the expected number Pc ⋅ q of chromosome r (k) r (k) r (k) r (k) r (k) r (k) which undergoes the crossover operation. We use V '1 ,V '2 ,V '3 ,V '4 ,V '5 ,V '6 ,L every

to denote the set of selected chromosome, and divide them into the following pairs:

r

(V '

(k ) 1

r r r r r , V '(2k ) , V '3( k ) , V '(4k ) , V '5( k ) , V '(6k ) L

)(

)(

We illustrate the crossover operation by

)

r

(V '

(k ) 1

r ,V '(2k ) , at first we generate a

)

random number c in (0,1) , then the two children X , Y as follows:

r r X = c ⋅ V '1( k ) + (1 − c) ⋅ V '2( k ) r r Y = (1 − c) ⋅ V '1( k ) + c ⋅ V '(2k )

X , Y satisfy the constraints, then replace the parents by their children, otherwise regenerate c for other children until we obtain two feasible If the children

solutions .

Fuzzy Chance Constrained Support Vector Machine

7

279

、 Mutation operation We define a parameter

Pm as the probability of mutation operation, for

i = 1,L, q , we generate a random number r in [0,1] . If r ≤ Pm , then we r (k ) select Vi as a parent. This gives us the expected number Pc ⋅ q of chromosome every

which undergoes the mutation operation. We randomly select a direction d in ℜ , n

r (k )

and Vi

as a mutation parent, let V

r = Vi ( k ) + M ⋅ d . If V does not satisfy the

constraints, we set M by a random number between 0 and M until V is feasible or after a predetermined number of iterations, we set M

= 0 . We replace parent V by

its child. After above calculation evaluation function, selection process, crossover operation,

r ( k +1)

mutation operation, we get the set of chromosome V1

r ,L , Vq( k +1) for the next

iteration. Repeat above process for a given number of generations until get an optimal solution is obtained.

5 Numerical Example We carry out a numerical experiment to validate the fuzzy chance constrained SVM model and the fuzzy simulation based genetic algorithm. The model and the corresponding algorithm were coded in Matlab language and run on Intel (R) Xeon (R) CPU E5410. The method was applied to the classification problem between TAG hydrothermal sulfide vents and non-hydrothermal sulfide vents based on the marine chemical composition. The chemical compositions, including 19 chemical components such as sulfur, iron, tin and lead etc., were sampled in both the hydrothermal vent area and non-hydrothermal vent areas. The classification problem is to use the chemical composition data to train a classifier which can then be used to predict whether an unknown data is from a hydrothermal sulfide vent or a nonhydrothermal sulfide vent. Both the proposed FCSVM model and conventional SVM were trained and tested in this study. The data set includes 72 chemical data samples in TAG region as positive samples and 72 chemical samples in non-TAG region as negative samples. In every time of experience we selected half of data randomly for training and the left half of data for testing. We do experience 8 times to estimate the accuracy of classifier.

280

H. Zhang, K. Li, and C. Wu

In the experiment, all components of random direction distributed

within

d was uniformly

[ −1,1] , and q = 20, M = 5, a = 0.2, Pc = Pm = 0.7 ,

C = 0.1 . Table 1 shows the accuracy of classifier between FCSVM and lib-SVM on the test data in each time of experience. Table 1. The accuracy of FCSVM against original SVM Index of experience 1 2 3 4 5 6 7 8

The accuracy of FCSVM 97.2% 91.7% 93.1% 94.4% 91.7% 95.8% 91.7% 94.4%

The accuracy of SVM 94.4% 90.3% 93.0% 88.9% 94.4% 90.3% 86.1% 91.7%

The experimental results have clearly indicated that FCSVM is better than original SVM model.

6 Conclusion In this paper, we have analyzed the widely used FSVM model, and in order to improve the robustness of SVM, a possibility measure has been used to build constraints of the model. Then the fuzzy simulation based genetic algorithm approach has been used to solve the new FCSVM model. The proposed model and algorithm have been used to classify the uncertainty on hydrothermal sulfide data in ocean survey TAG region, and the experiment results show that the new fuzzy SVM model outperform than the original model. For further research, nonlinear decision function trained by fuzzy chance constrained SVM can be developed since most classification problems are not linearly separable and fuzzy kernel function can be very useful to achieve the projection of the data into a linear separable higher feature space.

Acknowledgment The paper is supported by UK-China Bridge in Sustainable Energy and Built Environment (EP/G042594/1), Distinguished Visiting Research Fellow Award of Royal Academy of Engineering of UK, NSFC (No.60874071) RFDP (No.20090002110035) & Oceanic Science research project of China Ocean Association (No.DYXM115-033-01).

，

Fuzzy Chance Constrained Support Vector Machine

281

References 1. Burges, C.J.C.: A Tutorial on Support Vector Machines for Pattern Recognition. J. Data Mining and Knowledge Discovery 2(2), 121–167 (1998) 2. Lin, C.F., Wang, S.H.D.: Fuzzy Support Vector Machines. J. IEEE Transactions on Neural Networks 13(2), 464–481 (2002) 3. Wang, Y.Q., Wang, S.Y., Lai, K.K.: A New Fuzzy Support Vector Machine to Evaluate Credit Risk. J. IEEE Transactions on Fuzzy Systems 13(6), 820–831 (2005) 4. Chen, Y.X., Wang, J.Z.: Support Vector Learning for Fuzzy Rule-Based Classification Systems. J. IEEE Transactions on Fuzzy Systems 11(6), 716–728 (2003) 5. Ji, A.B.: Support Vector Machine for Classification based on Fuzzy Training Data. J. Expert Systems with Applications 37, 3495–3498 (2010) 6. Liu, B.D.: Minimax chance constrained programming models for fuzzy decision systems. J. Information Sciences (112), 25–38 (1998) 7. Vapnik, V.N.: Statistical Learning Theory Translated by Xu Jianhua Zhang Xuegong, 3rd edn. Electronics Industry Press, Beijing (2009) 8. Cao, B.Y.: Fuzzy Mathematics and System. Science Press, Beijing (2005) 9. Dubois, D., Prade, H.: Possibility theory. Plenum Press, New York (1988) 10. Zhang, X.G.: Pattern Recognition, p. 293. Tsinghua University Press, Beijing (2007) 11. Iwamura, K., Liu, B.D.: A genetic algorithm for chance constrained programming. Journal of Information & Optimization Sciences 17(2), 409–422 (1996) 12. Liu, B.D., Iwamura, K.: Chance constrained programming with fuzzy parameters. Fuzzy Sets and Systems 94, 227–237 (1998) 13. Fogel, D.B.: An Introduction To Simulated Evolutionary Optimization. IEEE Transactions on Neural Networks 5(1), 3–14 (1994) 14. Goldberg, D.E.: Genetic Algorithm in Search, Optimization and Machine Learning 15. Michalewicz, Z.: Genetic Algorithm + Data Structure = Evolution Programs, 2nd edn. Springer, New York (1994) 16. Zadeh, L.A.: Fuzzy Sets as a Basis for a Theory of Possibility. Fuzzy Sets and Systems, 9– 34 (1999)

An Automatic Thresholding for Crack Segmentation Based on Convex Residual Chunhua Guo and Tongqing Wang The Key Laboratory of Optoelectronic Technology & Systems of the Ministry of Education, Chong Qing University, Chong Qing, 400030, China [email protected], [email protected]

Abstract. Automatic thresholding segmentation has been widely used in machine vision detection. From no crack defect images to crack defect images on the beam surface of straddle-type monorail, the proportion of crack is zero or small comparing to the background, so the histogram distribution is unimodal or close to unimodal. The Otsu method can be successful if the histogram is bimodal or multimodal, but it provides poor results for unimodal distribution. In this paper, a segmentation approach based on convex residual is proposed, according to the convexity and concavity of histogram of detected image, the gray value which is the maximal value of convex residual joining with betweenclass variance is the threshold, experimental results show that the segmentation performance is superior to the Otsu method and the valley-emphasis method. Keywords: Image Processing; Crack Segmentation; Convex Residual; Betweenclass Variance.

1 Introduction Thresholding is a fundamental technique for image analysis, which purpose is to identify the regions of image objects and to extract the objects from their background [1]. The basic idea of automatic thresholding is to automatically select an optimal gray threshold value for separating objects of interest from the background in an image based on their gray value distribution. In the thresholding methods of image segmentation, the selection of thresholding is a key issue, if the selected gray thresholding is too high or low, the objects and background pixels will be mistaken, then this will affect the target size and shape, or even make the target lost [2]. Automatic thresholding techniques can be roughly categorized into local thresholding and global thresholding. Local thresholding is to split the image into multiple sub-images, and use local gray-level information to choose multiple threshold values, each is optimized for the sub-image of image. After the local thresholding, the boundary of adjacent sub-images may be gray-level discontinuous, and need smooth. Although local thresholding can enhance the effect of segmentation, it still has the following shortcomings: (1) the size of each sub-image cannot be too small, otherwise the statistics result may be meaningless; (2) each sub-image segmentation is arbitrary, if there is a sub-image just lying between target and background region, then the segmentation result may be even worse; (3) local K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 282–289, 2010. © Springer-Verlag Berlin Heidelberg 2010

An Automatic Thresholding for Crack Segmentation Based on Convex Residual

283

thresholding method is complicated and slow. While global thresholding is simpler and easier to implement, it is commonly used in automated visual inspection applications. Global thresholding is to select a single threshold value from the histogram of image. Among the global thresholding techniques, the Otsu method [3] is one of the best threshold selection methods for general images. This method selects threshold values that maximize the between-class variances or minimize the withinclass variances of the histogram. The Otsu method is optimal for thresholding large objects from the background; in other words, it is good for thresholding a histogram of bimodal or multimodal distribution. The Otsu method, however, fails if the histogram is unimodal or close to unimodal. Hui-Fuang Ng [4] studied the Otsu method and proposed valley-emphasis method, which works well in most circumstances, but if the image contains noise such as particle noise, the thresholding may be over-segmentation, and the result will lose some target information. The objects of this paper is the crack defect images on the beam surface of straddle-type monorail, the crack defects mainly arise from manufacturing reasons, heavy load, changing stress and other reasons. The collected images contain many cement particle noise from no defect to small and large defects, the gray-level distributions range from unimodal to bimodal. As a result, to successfully detect defects in the automatic visual inspection application, we need to revise the Otsu method so that it will be suitable for unimodal or close to unimodal distribution images. The convex residual is introduced to Otsu method in automatic thresholding in this paper, that is to say, compute the convex hull of histogram firstly, then calculate the convex residual, when the convex residual and between-class variance maximize, the corresponding gray value will be the threshold value.

2 Automatic Threshold Selection We first analyze the limitation of thresholding based on Otsu method for crack images which have no defect or small size defect and have regular particle noise. A new thresholding algorithm is then proposed to overcome this limitation. 2.1 The Otsu Method The pixels of a given image can be represented in L gray level {0,1,..., L − 1} . The

number of pixels at level i is denoted by ni and the total number of pixels by n , the probability of occurrence of gray-level i is defined as pi = ni n .

(1)

In the case of defect detection, the threshold value is only one, the pixels of an image are divided into two classes C 0 = {0 ,1,..., k } and C1 = {k + 1,..., L − 1} (background and defects, or vice verse), where k is the threshold value. Then the probability of class occurrence and the class mean value, respectively, are given by

284

C. Guo and T. Wang

L−1

k

ω0 ( k ) = ∑ pi , ω1 ( k ) = ∑ pi . i=k +1 i =0 and

(2)

L −1

k

μ0 ( k ) = ∑ ipi ω0 , μ1 ( k ) = ∑ ipi ω1 . i =0 i = k +1

(3)

∗ Using discriminant analysis, Otsu showed that the optimal threshold value k can be determined by maximizing the between-class variance. A equivalent, but simpler formula is given as follows

∗ k = Arg

2 2 Max {ω1 ( k ) μ1 ( k ) + ω 2 ( k ) μ 2 ( k )} . 0≤ k ≤ L −1

(4)

The Otsu method is essentially to find the valley based on a statistical sense, with valley divide into two separate peaks between object and background. For the defect detection applications, histograms of images do not show clear bimodal distributions, then the valley between them may not exist; perhaps one peak of histogram may be in another one, histograms of images show unimodal distribution rather than bimodal [5], as is illustrated in the following figure. Fig.1(a) shows the test crack image, and Fig.1(b) shows the histogram of the image, because the area of crack is small proportion comparing to the background, the histogram presents a unimodal distribution, Fig.1(c) is the segmentation result of Otsu method, which fails to isolate the crack, and some background is treated as crack.

(a)

(b)

(c)

Fig. 1. Problem with Otsu threshold: (a) original crack image;(b) histogram of (a);(c)result of Otsu threshold

2.2 The Automatic Thresholding Based on Convex Residual

The aim of automatic thresholding is to find the appropriate value that separate target from background. For the case of two peaks in the histogram, the threshold value exists at the valley of two peaks, while in the case of unimodal or not obvious bimodal in the histogram, the threshold value may exist near the concave of histogram. In such cases, determining threshold value need analyze the concave and convex of histogram firstly. Specifically, the histogram of images can be treated as a

An Automatic Thresholding for Crack Segmentation Based on Convex Residual

285

plane (excluding the axis), computing the convex hull function, and calculating the convex residual, the gray-level when the convex residual and corresponding betweenclass variance reach maximum shall be the segmentation value. Fig.2 shows the convex residual of histogram. In which the thick curve is the envelope of histogram, dotted line is the convex hull function f (i ) , the gray value that the convex residual and corresponding between-class variance reach maximum shall be calculated. pi

(M1, p M1 )

f (i )

envelope

convex residual gray L-1

0

Fig. 2. Convex residual analysis of image histogram

Suppose the probabilities of histogram which gray-level from 0 to L − 1 are pi ( i = 0,1,..., L − 1 ), the steps of computing convex hull function f (t ) are as follows[6]: Step1: The maximum value of pi ( i = 0,1,..., L − 1 ) is supposed as p M 1 , which gray value is M1, then p 0 , p L −1 and p M 1 are all the points of convex hull; Step2:

p0 and p M 1 is

connected

to

form

directed

line

pM 1 p

0

,

and p L −1 with p M 1 connected to form directed line p , compute the points of L −1 p M 1 right side of the two directed line and store them in B1, B2 respectively. For example, whether the point ( i , pi ) is on the left side, right side or the extended line of p M 1 p is determined by the following third-order determinant 0 i M1 0

pi 1 pM 1 1 . p1

(5)

1

If the point is on the left side, the determinant is greater than 0, if on the right side, the determinant is less than 0, while on the extended line, the determinant is equal to 0;

286

C. Guo and T. Wang

Step3: The distance [7] between points in B1 and p is defined by the Eq.(6) M 1 p0 d =

( p M 1 − p 0 ) ⋅ i + M 1 ⋅ pi + M 1 ⋅ p 0 . 2 2 ( p M 1 − p0 ) ⋅ M 1

(6)

Then the distance between point ( i , pi ) and directed line pM 1 p can be calculated; 0 Step4: The farthest point between the points of B1 and pM 1 p is defined as pM 2 , 0

which gray value is M2. Then p M 1 and p M 2 , pM 2 and p0 are connected into lines and p respectively. Then the right side points of directed lines p M 1 p p are M2

M2 0

stored in B3 and B4 separately. Repeat Step3 and Step4 until the right side points of finally directed lines generated are empty; Step5: Repeat Step2, 3 and 4 for p L−1 p , each point found is the vertex of M1 convex hull, all these points that are the vertexes of convex hull are connected into a polygonal line together according to ascending order or descending order of gray value, this line is the convex hull function f (i ) ( i = 0,1,..., L − 1 ). After getting f (i ) , the convex residual is defined as f (i ) − pi , i = 0,1,..., L − 1 . The gray value that the convex residual and corresponding between-class variance reach maximum is selected as the threshold value for crack images, which is shown as follows

{

}

* 2 2 i = Arg Max ( f (i ) − pi )(ω1(i ) μ1 (i ) + ω2 (i ) μ2 (i)) . 0≤i≤L−1

(7)

For images that have unimodal or almost unimodal distribution, the peaks are generally corresponding to the background of crack images. The key of the convex residual formulation is the application of weight f (i ) − pi comparing to the Otsu threshold calculation. Greater convex residual can ensure that the result threshold will always reside at the larger concavity of histogram, in the same time, maximum of between-class variance can also ensure the separation of objective and background. Thus when the convex residual and between-class variance reach the maximum, the corresponding value can separate the crack defect correctly.

3 Experiments and Analysis In the experiments, we tested the performance of the convex residual method, the Otsu method and the valley-emphasis method on several crack images containing crack defects. We will illustrate our conclusions.

An Automatic Thresholding for Crack Segmentation Based on Convex Residual

287

Fig.3 shows another example of crack detection. The Otsu method was able to detect the crack but it also treated some background information as crack as seen in Fig.3 (b). The valley-emphasis method also segmented the crack but it lost weak crack, and the crack is discontinuous more or less (Fig.3 (c)), the segmentation is somewhat over-segmentation. The convex residual method successfully isolated the crack, and the crack image is more continuous, which can be seen in Fig.3 (d).

(a)

(c)

(b)

(d)

Fig. 3. Otsu method, valley-emphasis method and convex residual method:(a)original image;(b)Otsu result;(c) valley-emphasis result;(d)convex residual result

In the crack image acquisition process, due to the lights aging and other reasons, some images appear dark totally, which impact the image effects. Fig.4 is another common crack image which is darker. The difference in gray-level between the crack and the background is rather small. The Otsu method is less-segmentation as seen in Fig.4 (b), the valley-emphasis method is over-segmentation as seen in Fig.4(c). The convex residual method performed better for thresholding the crack on the test image in the case of dark light.

288

C. Guo and T. Wang

(a)

(c)

(b)

(d)

Fig.4. Otsu method,valley-emphasis method and convex residual method: (a)original image;(b)Otsu result;(c)valley-emphasis result;(d)convex residual result

From above we can see that the convex residual method is superior to the Otsu method and valley-emphasis method. The Otsu method is less-segmentation and the valley-emphasis over-segmentation somewhat when the objective is small or no proportion of the entire image. The convex residual method can resolve this problem effectively and has better robustness.

4 Conclusions In this paper we have presented the convex residual method, a revised version of the Otsu method, to automatically select the optimal threshold value for crack detection application. The method selects a threshold value that maximize the convex residual and between group variance in the gray-level histogram. Large convex residual corresponds to concave of histogram. Therefore, the convex residual method is able to select optimal threshold values for both bimodal and unimodal distributions. We have demonstrated the effectiveness of the proposed method on various crack images. The experiment results suggest that the convex residual method is effective for selecting threshold values for crack segmentation application.

An Automatic Thresholding for Crack Segmentation Based on Convex Residual

289

Acknowledgments. The authors would like to thank National Key Technology R&D Program under Grant of China (No.2007BAG06B06) for financially supporting this research.

References 1. Zhang, Y.J.: Image processing and analysis. Tsing Hua University Press, Beijing (1999) 2. Cheng, G.: The fisher criterion function method of image thresholding. Chinese Journal of Scientific Instrument. 24(6), 564–567 (2003) 3. Otsu, N.: A threshold selection method from gray-level histograms. IEEE Transactions on Systems Man Cybernet 9(1), 62–66 (1979) 4. Hui, F.: Automatic thresholding for defect detection. Pattern Recognition Letters 27, 1644–1649 (2006) 5. Zhang, Y.J.: Image segmentation. Science Press, Beijing (2001) 6. Hao, X.J.: An acceleration and improved algorithm of convex hull computing. Hebei University of Technology (2003) 7. Du, Y.Y.: An optimal algorithm for calculating convex hulls of polygons. Computer Applications and Software 15(5), 38–41 (1998)

A Combined Iteration Method for Probabilistic Load Flow Calculation Applied to Grid-Connected Induction Wind Power System Xue Li, Jianxia Pei, and Dajun Du Department of Automation, Shanghai University, Shanghai, 200072, China [email protected], [email protected], [email protected]

Abstract. Taking into account the uncertainties of load and wind power outputs, this paper proposes a combined iteration method for probabilistic load flow (PLF) applied to grid-connected induction wind power system. A probabilistic wind farm model is first developed. Then, the slip of induction generator along with the nodal voltages is corrected at each iteration of the PLF calculation, leading to a novel unified PLF iteration method for wind power generation system. Finally, Monte Carlo simulation (MCS) combined with Latin hypercube sampling (LHS) is used to perform the PLF. Simulation results show the effectiveness of the proposed method. Keywords: Combined iteration method, Grid-connected induction wind power, Probabilistic load flow, Latin hypercube sampling.

1 Introduction With the rapidly growth of wind farm capability, the wind farms will be unavoidably introduced into power grid. Due to wind power with the characters of random and interval, the probabilistic load flow (PLF) calculation for wind power integrated system has been commonly used as an efficient tool to assess the impact of wind power on grid operation. A great concern in PLF analysis was shown on the PLF calculation method for grid-connected induction wind power system. A lot of researches have been published on the deterministic load flow calculation method for grid-connected induction wind power system [1]-[3]. According to [1], the wind farms were considered as the PQ buses. The reactive power was described as the quadratic function of the real power or as the expression of the voltage magnitude and induction machine parameters. The methods above mentioned, not considering impact of the slip, are inaccurate. The induction machine was represented by means of the impedance [2]. However, the algorithm carried out to simulate the wind farm consisted of two iterative processes, which made the process yield the slow convergence. Based on the above impedance model, a unified way for the two iterative processes was presented to expedite the convergence [3]. The above mentioned researches have been carried out using deterministic wind power model disregarding the uncertainty in wind availability. However, the power K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 290–296, 2010. © Springer-Verlag Berlin Heidelberg 2010

A Combined Iteration Method for Probabilistic Load Flow Calculation

291

supplied by wind turbine behaves randomly followed the wind conditions, and probabilistic wind farm generation model seems much more appropriate for the PLF calculation [4]-[5]. In this paper, a combined iteration method for PLF, applied to grid-connected induction wind power system, is presented taking into account the uncertainties of load and wind power outputs. Firstly, a steady-state probabilistic model for wind farm is presented. Then, the PLF method for grid-connected induction wind power system is proposed. Using the Newton-Raphson (NR) algorithm, a unified iteration for the original state variables and the slip are performed. Finally, PLF using the Monte Carlo simulation (MCS) combined with Latin hypercube sampling (LHS) is performed and analyzed on the 24-bus system. The paper is organized as follows. Section 2 presents probabilistic models of main components in wind power generation system. Improved PLF method for gridconnected induction wind power system is proposed in Section 3. Section 4 gives 24bus system to illustrate the effectiveness of the proposed method. The conclusions follow in Section 5.

2 Probabilistic Models of Main Components in Wind Power Generation System Induction generator is usually used in wind farm, and its equivalent circuit can be simplified into Г-type equivalent circuit [6]. The real power and reactive power injected into grid generated by wind power generator can be expressed as

U 2 r2 / s ( r2 / s ) 2 + xk 2

(1)

r2 2 + xk ( xk + xm ) s 2 Pe r2 xm s

(2)

Pe = −

Qe = −

where xk=x1+x2. x1, r2+jx2 and xm are the stator reactance, rotor impedance and magnetizing reactance, respectively. s is the slip of the induction machine and U is the generator voltage. It can be found from (1) and (2) that the reactive power absorbed by wind power generator has the close relationship to the voltage U and the slip s when the real power Pe is certain. Moreover, note that the uncertainty of mechanical power of wind turbine induces the powers generated by wind power generator uncertain. The wind turbine model is given by [6]

Pm =

1 ρ Av 3C p 2

(3)

292

X. Li, J. Pei, and D. Du

where, ρ is the density of air (kg/m3), A is the area swept out by the turbine blades (m2), v is the wind speed (m/s), and Cp is the dimensionless power coefficient. Here, Cp can be expressed as a function of the blade tip speed ratio and be obtained by interpolation method. The wind speed are usually modeled as [6] k v c c

v c

ϕ (v ) = ( )k −1 exp[−( )k ]

(k > 0, v > 0, c > 1)

(4)

where v represents wind speed, and ϕ ( i ) is Weibull probability density function (PDF), with the shape parameter k and the scale parameter c. Further, Weibull parameters can be obtained by the mean and standard deviation of wind speed. Due to the wind speed random variation with respect to time, the corresponding power outputs of wind turbines also change.

3 Improved PLF Method for Grid-connected Induction Wind Power System 3.1 PLF Modelling for Grid-Connected Induction Wind Power System

Let N W denotes the set of the buses connected with wind farm. Based on the above analysis, suppose that wind farm is connected at ith ( i ∈ NW ) bus, the corresponding PLF equations for grid-connected induction wind power system are given by

⎧ Pei ( ei , f i , si ) − PLi − ∑ ( ei ai + f i bi ) = 0 ⎪ j∈i ⎪ ⎨Qei (ei , f i , si ) − QLi − ∑ ( f i ai − ei bi ) = 0 j∈i ⎪ ⎪ Pmi − Pei (ei , f i , si ) = 0 ⎩

(5)

where PLi and QLi are the load real and reactive powers in ith bus, respectively, ai = Gij e j − Bij f j , and bi = Gij f j + Bij e j . Pmi and Pei are the mechanical power of wind turbine and the electrical power of wind generator at the ith bus, respectively. In the paper, MCS combined with LHS [7] is used to account for the uncertainty of load and wind power outputs. It has the less computational burden than MCS combined with simple random sampling (SRS). The Newton-Raphson (NR) algorithm is used to solved (5). To retain the quadratic convergence characteristic of NR algorithm, the corresponding elements of Jacobian matrix in the correction equations of PLF must be modified [8].

A Combined Iteration Method for Probabilistic Load Flow Calculation

293

3.2 Complete Algorithm

The proposed PLF method can now be summarized as follows: Step 1: Set the initial values of the voltages and the slip. By simulating (4) with different parameters, the wind speed samples can be generated, and the corresponding wind power outputs can be calculated using (3); Step 2: Standardized daily operating curves are assigned for nodal powers. The means and standard deviations of uncertain nodal powers are then computed. To execute the MCS, random samples of nodal powers are also generated; Step 3: The mechanical power of wind turbine and the real power of wind generator are computed by (3) and (1), respectively. The equations of PLF (5) are formulated; Step 4: The Newton-Raphson algorithm is used to solved the correction equations of PLF (5). The corresponding elements of Jacobian matrix are modified; Step 5: The slip of induction machine, along with the nodal voltages, are corrected at each iteration of the PLF calculation. The correction equations are solved, and the probabilistic characteristics of nodal voltages and nodal powers are computed.

4 Numerical Example The proposed method was applied to a modified 24-bus test system as shown in Fig. 1. The numerical simulations were carried out using MATLAB. By the step-up transformer and double circuit transmission line, the equivalent wind farms were connected at the 15th node of 24-bus test system then the new nodes 25 and 26 were added. Without specification, all data were taken as per-unit value and the fiducial value was 100 MVA.

26

Fig. 1. Modified 24-bus test system

294

X. Li, J. Pei, and D. Du

There are 20 identical wind turbines in the wind farm, each of which has the nominal capacities equal to 600 kW and the rated power factor equal to 0.89. All wind turbines are assumed equipped with induction generators compensated by a capacitor bank equal to 29.7% of their nominal capacity. The parameters of wind farm are listed as follows. the density of air ρ is 1.2245kg / m 3 , the area swept out by the turbine blades A is 1840m 2 , the identical cut-in wind speeds, cut-out wind speeds and rated wind speeds of wind farm are given as 3m/s, 20 m/s and 13.5 m/s, respectively. MCS combined with LHS was adopted in PLF. 1000 wind speed samples were generated by simulating (4) with the shape parameter (k=2) and the scale parameter (c=6.7703). According to (3), 1000 wind power outputs were calculated. Standardized daily operating curves [9] were assigned for nodal powers, including eighteen load active and reactive powers and ten generation active and reactive powers. Then, the means and standard deviations of uncertain nodal powers can be derived. 1000 nodal powers random samples are generated. The wind power and nodal power samples are used to execute the MCS. In order to validate the effectiveness of the proposed method, the experiment has been carried out consisting of two iterative processes, the first PLF and the subproblem formulated for updating the slip of the induction machine. After the voltages of the buses are obtained from the result of the first PLF iteration, the real power of the wind generator can be calculated. On the other hand, the mechanical power of wind turbine can be also derived. Both powers are compared. If they are not equal, the slip is modified. With the new value of slip, the PLF analysis is repeated. The process doesn’t stop until both powers are equal or the difference between both is acceptable. Table 1. Execution Times of Different Methods Methods

Proposed Method

Execution Time (s)

50.1543

PLF consisting of two iterative processes 84.1313

Table 2. PLF Results of Different Methods

Methods

MCS(LHS) MCS(SRS)

μ σ μ σ

U18

U19

U 20

U 21

U 22

1.0921 0.0013 1.0904 0.0015

1.0743 0.0022 1.0712 0.0025

1.0000 0.0001 0.9998 0.0002

1.0000 0.0001 0.9997 0.0002

1.0380 0.0020 1.0367 0.0021

Execution Times (s) 50.1543 415.9767

A Combined Iteration Method for Probabilistic Load Flow Calculation

295

The PLF computation times generated by two methods are listed in Tables 1. From Table 1, it can be seen that the proposed method took less execution time than the PLF consisting of two iterative processes. Compared with MCS combined with LHS, MCS combined with SRS was also performed in the paper. 10000 sample sizes of MCS with SRS are assumed to be accurate. The PLF results generated by two methods are listed in Tables 2. In Table 2, Ui denotes the amplitude of voltage phasors corresponding to the ith bus, and μ and σ are the mean and standard deviation of state variable, respectively. From the Table, the proposed method obtained the similar results as the accurate method. It was also seen from Table 2 that the proposed method has less computational effort than MCS with SRS.

5 Conclusion In this paper, a combined iteration method for PLF, applied to grid-connected induction wind power system, has been presented. By introducing the slip of induction generator as the new correction, a unified iteration for the original state variables and the slip are performed using the NR algorithm. The proposed method was tested on the 24-bus system. Compared with the PLF consisting of two iterative processes, the presented method takes less computation time. The proposed method provides the accurate results and has less computational effort than MCS with SRS. The results show the effectiveness of the proposed method.

Acknowledgments This research is supported in part by the special fund for selecting and cultivating the youth teacher of Shanghai colleges and in part by the innovation fund project for Shanghai University.

References 1. Feijoo, A., Cidras, J., Dornelas, J.L.G.: Wind speed simulation in wind farms for steadystate security assessment of electrical power systems. IEEE Transactions on Energy Conversion 14(4), 1582–1588 (1999) 2. Feijoo, A., Cidras, J.: Modeling of wind farms in the load flow analysis. IEEE Transactions on Power Systems 15(1), 110–115 (2000) 3. Fuerte-Esquivel, C.R., Tovar-Hernandez, J.H., Gutierrez-Alcaraz, G., Cisneros-Torres, F.: Discussion of modeling of wind farms in the load flow analysis. IEEE Transactions on Power Systems 16(4), 951 (2001) 4. Hatziargyriou, N.D., Karakatsanis, T.S., Papadopoulos, M.: Probabilistic load flow in distribution systems containing dispersed wind power generation. IEEE Transactions on Power Systems 8(1), 159–165 (1993)

296

X. Li, J. Pei, and D. Du

5. Hatziargyriou, N.D., Karakatsanis, T.S., Lorentzou, M.I.: Voltage control settings to increase wind power based on probabilistic load flow. International Journal of Electrical Power and Energy Systems 27(9-10), 656–661 (2005) 6. Li, X., Pei, J., Zhang, S.: A probabilistic wind farm model for probabilistic load flow calculation. In: Asia-Pacific Power and Energy Engineering Conference, APPEEC 2010, pp. 1–4 (2010) 7. Yu, H., Chung, C.Y., Wong, K.P., et al.: Probabilistic load flow evaluation with Hybrid Latin hypercube sampling and Cholesky decomposition. IEEE Transactions on Power Systems 24(2), 661–668 (2009) 8. Fuerte-Esquivel, C.R., Acha, E.: A Newton-type algorithm for the control of power flow in electrical power network. IEEE Transactions on Power Systems 12(4), 1474–1480 (1997) 9. Wang, K.W., Tse, C.T., Tsang, K.M.: Algorithm for power system dynamic stability studies taking account of the variation of load power. Electric Power Systems Research 46(8), 221– 227 (1998)

Associated-Conflict Analysis Using Covering Based on Granular Computing Shuang Liu, Jiyi Wang, and Huang Lin Institute of Artificial Intelligence and Parallel Computing, Zhejiang Normal University, Jinhua, Zhejiang, 321004, P.R. China [email protected]

Abstract. Conflicts are ubiquitous phenomena in our society, and the research on them is very important both practically and theoretically. Many mathematical models and methods of conflict have been studied and proposed. In this paper, associated-conflict is introduced, and a reasonable and comprehensive approach to its analysis, using covering based on granular computing, is outlined. The model of associated-conflict analysis, given by the example of service-resource, will provide more profound insight for the conflict resolution in different fields. Keywords: associated-conflict; conflict analysis; decision-making; covering; granular computing.

1 Introduction Conflicts are widespread in our society. And some related analysis and resolution for them play an important role in business, political and lawsuits disputes, labormanagement negotiations and military operation etc. So far, many models and methods have been studied and proposed [1-7]. There are also various tools and techniques for the analysis and resolution of conflicts using graph theory, topology, differential equations, and others [4, 8-9]. However, all these researches put their emphasis on conflicts themselves such as the causes of conflict, combination of conflict relation and analysis of all kinds of conflict situations. As we all know, granular computing [10-12] is an art of science with thinking mode based on multi-layer granular structure, problem solving methods and information processing mode, which covers all theories, methods and techniques relating to granularity. Therefore, if we treat conflict as a process of granulation in different structure layers, there are some very interesting discoveries. That’s to say, there are such cases that single event or process, as a matter of fact, contains more than one conflict interest relation. Thus what the relationship among these conflict interests is and how can we solve a series of problems, such as resource shortage, hidden security risk and decision-making etc., caused by at least two kinds of conflict interest relation become crucial problem to be solved. In this paper, we are going to present the concept of associated-conflict and its analysis based on some ideas of general covering of granular computing [13]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 297–303, 2010. © Springer-Verlag Berlin Heidelberg 2010

298

S. Liu, J. Wang, and H. Lin

2 Preliminaries Formally, an information system can be defined as a pair S = (U , A) , U is a nonempty and finite set of objects, and A is nonempty and finite set of attributes. For every attribute a ∈ A , a :U →Va , where Va , called the domain of a , is the set of

∀B ⊆ A , | B |≠ 0 , for all pairs (x, y) ∈U ×U , the discernibility matrix of B in S , denoted as M (B) , is defined [14] as M ( B )= (mB ( x, y ))|U |×|U | in which mB ( x, y ) = {a ∈ B : a( x) ≠ a( y )} is the set of all attributes which can discern objects x and y . Definition: Let U be a universe, and C be its subsets family. If all subsets are not empty and UC = U , then C is called a covering of U . values of

a.

So

3 Related Work In [4], some issues of conflicts are chosen and agents taking part in the conflict dispute have to specify their standings: conflict, alliance, or neutrality. And in [15], three standings, “conflict, alliance and neutrality”, are treated as two relations: CIR and IAR (“conflict of interests” and “in ally with” relations) based on granular computing on binary relations in an aggressive Chinese wall security policy model. All these analysis about conflict and its situation is established on the equivalence relation. But it is not valid for some other cases. For example, IAR does not need to be an equivalence relation. A friend of one of our friends is not always our friend. So a covering model [13] is more reasonable to capture the conflict of interest relation since a covering holds the symmetric property intrinsically. Let U be a set of all objects, and C is a covering on U , where, ∀K ∈ C , objects in K are in conflict with each other. We call such a covering C a conflict of interest clusters to distinguish it from the conflict of interest classes in a partition model of [16]. Under this covering, the conflict set of x ∈ U is defined as conflict ( x) = U{K ∈ C | x ∈ K} − {x} . If a subject has accessed an object x , it is

x is defined as ally( x) = U − conflict ( x) . As a binary relation on U , IAR = {( x, y ) | y ∈ ally ( x )} . Remark1: Through the sets conflict ( x ) and ally ( x ) , access or supply policy is that assuming x1 ,..., xm are all objects that a subject has already accessed or demanded, a new access to or demand for object o requested from this subject is only granted if o is not in conflict ( x1 ) U... U conflict ( xm ) . In other words, o is in ally( x1) I... I ally( xm ) . not allowed to access any object in conflict ( x) . Accordingly, the ally set of

Associated-Conflict Analysis Using Covering Based on Granular Computing

299

4 Associated-Conflict in the Perspective of Granular Computing Conflict exists in different hierarchical structures and levels [17]. The cause of conflict is the interaction among objects for certain interest, and its representation just happens when there are some external factors acting on these objects. We call this kind of representation a conflict process or a conflict event.

Conflict objects

Agents Action relation

Implicated conflict objects

Implicated inherent and stationary relation

Here we introduce the associated-conflict as is shown by the flow graph above. This graph is a conflict process or event. There are two relations representing two levels in this process or event. One is action relation, an active relationship between agents and conflict objects. It can be changed by the situation of agents, such as the number of agents, the scope of agents and the sequence of agents, etc. And also agents are external factors or trigger sources. Conflict interest representation shows up immediately after agents acted on conflict objects. For all agents, the other relation is implicated relation, an inherent and stationary relationship between conflict objects and implicated conflict objects. Just like a project needs some procedures or resources, every conflict object needs some certain implicated conflict objects. It can not be changed due to the relationship of subject and object between conflict objects and implicated conflict objects. So conflict interest representation in implicated objects is always exiting. Remark2: as flow graph above shows, we call this process or event associatedconflict, which consists of two unique relations connecting one trigger sources and two conflict sources. So we can also define multiassociated-conflict which has one trigger, one conflict source and at least two implicated conflict sources.

5 An Example for the Model of Associated-Conflict Analysis We will take service-resource as an example without considering relevant application background so as to make full consideration of all kinds of situation of associatedconflict as much as possible. The purpose of this section is to illustrate the idea of associated-conflict and the model of its analysis by using covering model of conflict based granular computing. The further analysis with relevant application background will be discussed in the following section. Suppose: there are seven services (conflict objects): U1 = { A,B, C, D, E, F , G} . The conflicts of interest relation for them are as in Table1. And there are eight resources (implicated conflict objects): U 2 = {a ,b, c , d , e, f , g , h} . The conflicts of interest relation for them are as in Table2. Services and resources are conflict sources. So their relationship is inherent and stationary. We assume that the relationship between them is A = {a , c} ,

300

S. Liu, J. Wang, and H. Lin

B = {g , a, h} , C = {a, b} , D = {d, f } , E = {c, e, b, g} , F = {d } , G = {h, f } . It can not be changed. Now there are five agents (external factors or trigger sources) acting on (accessing to or demanding) these services (action relation). Assuming that the relationship between agents and services is agent (1) = { D , A} , agent (2) = { A , G } ,

agent (3) = {E , B, D} , agent (4) = {C} , agent (5) = {D, F , C , G} . It can be changed with the changing of the number of agents, the scope of agents or the sequence of agents etc. Table 1. Conflict of interest relation for service service service

A B

A C

A E

B A

B C

C A

C B

C D

C E

D C

D E

E A

E C

E D

E G

F G

G F

Table 2. Conflict of interest relation for resource resource

resource

resource

resource

a a a a b c c c d d d d d d

d f g h f d e g a c e f g h

e e e f f f f g g g g g h h

c d g a b d g a c d e f a d

In order to make this model more adaptable and catholic, we set these assumption: the priority level of each service is the same, the priority level of each agent is also the same, each resource can be used up to three times by services at the same time (the amount of each resource depending on the context specialty may be different). Since there may be some priority difference for each agent to access to or demand services, we suppose priority vector

x = (2

15

,4

15

,6

15

, 1 , 2 ) (it can be 15 15

|agent |

changed) for each agent where

∑ x( p) = 1 , 1 ≤ p ≤ |agent| and | x |=| agent | . p =1

According to access or supply policy based on covering for conflict analysis, we can construct matrix of information table for associated-conflict event or process.

Associated-Conflict Analysis Using Covering Based on Granular Computing

301

For the relationship between agents and services, combining with Table1, we have matrix X in which 1 represents that this agent has right to access to or demand this service and 0 represents this agent has no right to access to or demand this service. Accordingly, we have matrix Y , where 1 represents that this service can be supported by another resources besides some necessary resources and 0 represents that this service doesn’t need the rest of resources, for the relationship between services and resources by Table2,. Since each resource can be used up to three times by services at the same time, so |service|

we can get square matrix

Y ' in which if

∑Y(i, j) ≤3, then Y (i,) = (0,0,L,1,L,0) , '

j=1

Y (i,) = 0 , |Y |=|resource| , 1 ≤ i ≤| resource | . And we have matrix Z '' in '' which Z ( p, i ) =| service | − ∑ ( X ( p, j ) ⊕ Y ( j , i )) , 1 ≤ p ≤ |agent| , else

'

'

1≤ j ≤|service|

1 ≤ i ≤| resource | . Z can be constructed, if Z '' ( p, i) ≥ int(| service | /2) + 1 then '

Z ' ( p, i) = Z '' ( p, i) , else Z ' ( p, i ) = 0 . Then, we can get the matrix Z of information table between agents (trigger sources) and resources (implicated conflict objects) for the given Table1 and Table2 ' ' where Z = Z Y , then we have discernibility matrix M about agents in information table Z . Finally, we can get priority discernibility matrix P|M | of agents about resources from

associated-conflict

process

or

event

of

service-

resource,

where

P|M | ( p, q ) = x ( p ) × x(q)× | M | ( p, q) , 1 ≤ p, q ≤ |agent| .

6 Discussion Conflict analysis is to solve the problems that might exist in the conflict process or event, and then provides decision-making to decision-makers. Generally, there are two abnormalities caused by trigger source and implicated conflict objects respectively under the most of application background of associated-conflict. For the given example, one abnormality is resource shortage and the other is security hidden danger. The following part of this dissertation is the solution of three cases caused by two abnormal problems: (1) If the abnormality exiting in the associated-conflict process or event of serviceresource is caused by resource shortage, then we can take some measures to ensure agents as much as possible to normally access to or demand services even under the premise of services in conflict. Here are the measures: we can limit the agents in W to access to or demand services even though resources are conflicting, where agents |r |

| s|

t =1

l =1

set W = { U agent ( r (t ))} U { U agent ( s (l ))} , vector r = row(min P|M | ( p, q )) and s = column(min P| M | ( p, q )) , 1 ≤ p , q ≤ |agent| . For the example above,

302

S. Liu, J. Wang, and H. Lin

W = {agent (2), agent (4)} . If the effect is not satisfying, we can set min P|M | ( p, q )=0 to follow above step to get a new W1 , and then incorporate it into W , iteratively until the abnormality is disappeared. In the worst case, we can |r|

recycle the resources in T set where

T = U 2 − U{resource | M (r (t ), s(t ))}) , so

T = {a, b, c, d , f , g , h} for the given example.

t =1

(2) If the abnormal problem exiting in the service- resource associated-conflict process or event is the type of security hidden danger, then we can take measures to ensure normal operation of services as much as possible even under the premise of '

resources in conflict. Here are the measures: we can limit the agents in W to access to or demand services even though services are conflicting, in which agents set |r ' |

|s ' |

W = { U agent ( r (t ))} U { U agent ( s ' (l ))} , vector r ' = row(max P|M | ( p, q)) '

'

t =1

l =1

and s = column (max P|M | ( p , q )) , 1 ≤ p, q ≤ |agent| . For the example above, '

W ' = {agent (2), agent (3)} . If the effect is not satisfying, we can set max P|M | ( p, q )=0 to follow above step to get a new W1' , and then incorporate it '

into W , iteratively until the abnormality is disappeared. In the worst case, we can |r |

recycle the resources in

T ' set in which T ' = U{resource | M (r ' (t ), s ' (t ))} , so t =1

T = {e, f , g , h} for the given example. '

(3) If both simultaneously appear in the associated-conflict process or event of '

service- resource, then we can firstly limit the agents in W and then limit the agents in W − W to access to or demand the services for the consideration of security, so that we can make sure services and agents as much as possible to normally operate and access to or demand services even under the premise of both services and resources being in conflict. In the worst case, we can sequentially recycle the '

resources in T and T − T . Besides, if there is no abnormal problem existing in conflict process or event, this '

'

'

model can also help us to make decision opportunely for some purposes from W ,

W , T ' and T .

7 Conclusions This paper analyzing associated-conflict using covering based on granular computing offers deeper insights into the structure of associated-conflict, and enables the relationship between agents and service-resource to be debated. It provides us a new perspective on conflict analysis and gives us many useful clues on whether there are some useful important factors behind problems to be solved and how to use these

Associated-Conflict Analysis Using Covering Based on Granular Computing

303

implicated useful important factors, even they are in conflict, to analyze and solve problems. The presented formal model has potential applications to the security policy for information sharing, computer simulation of conflicts, privacy and so on. However how to construct the model of multiassociated-conflict analysis is our future work.

References 1. Roberts, F.: Discrete mathematical models with application to social, biological and environmental problems. Prince-Hall, Englewood Cliffs (1976) 2. Coombs, C.H., Avruin, G.S.: The structure of conflicts. Lawrence Erlbaum, London (1988) 3. Casti, J.L.: Alternative realities-mathematical models of nature and man. Wiley, New York (1989) 4. Pawlak, Z.: An inquiry into anatomy of conflicts. Journal of Information Sciences 109, 65– 68 (1998) 5. Maeda, Y., Senoo, K., Tanaka, H.: Interval density function in conflict analysis. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) New Directions in Rough Sets, Data Mining and Granular-Soft Computing, pp. 382–389. Springer, New York (1999) 6. Nakamura, A.: Conflict logic with degrees. In: Pal, S.K., Skowron, A. (eds.) Rough Fuzzy Hybridization-A New Trend in Decision-making, pp. 136–150. Springer, New York (1999) 7. Deja, R.: Conflict analysis. rough set methods and applications. In: Polkowski, L., Tsumoto, S., Lin, T.Y. (eds.) Studies in Fuzzyness and Soft Computing, Physica-Verlag, pp. 491–520. A Springer-Verlag Company, Wien (2000) 8. Pawlak, Z.: On conflicts. International Journal of Man Machine Studies 21, 127–134 (1984) 9. Pawlak, Z.: Some remarks on conflict analysis. European Journal of Operational Research 166, 649–654 (2005) 10. Yao, Y.Y.: The art of granular computing. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 101–112. Springer, Heidelberg (2007) 11. Bargiela, A., Pedrycz, W.: Granular computing: an introduction. Kluwer Acamedic Publishers, Boston (2002) 12. Yao, Y.Y.: Granular computing: basic issues and possible solutions. In: The 5th Joint Conference on Information Sciences, New Jersey, USA, vol. 1, pp. 186–189 (2000) 13. Zhu, W., Wang, F.Y.: Covering based granular computing for conflict analysis. In: Mehrotra, S., Zeng, D.D., Chen, H., Thuraisingham, B., Wang, F.-Y. (eds.) ISI 2006. LNCS, vol. 3975, pp. 566–571. Springer, Heidelberg (2006) 14. Skowron, A., Rauszer, C.: The discernibility matrices and functions in information system. In: Slowinski, R. (ed.) Intelligent Decision Support. Handbook of Applications and Advances of the Rough Set Theory, pp. 331–362. Kluwer Academic Publishers, Dordrecht (1991) 15. Lin, T.Y.: Granular computing on binary relations-analysis of conflict and chinese wall security policy. In: Alpigini, J.J., Peters, J.F., Skowron, A., Zhong, N. (eds.) RSCTC 2002. LNCS (LNAI), vol. 2475, pp. 296–299. Springer, Heidelberg (2002) 16. Brewer, D., Nash, M.: The chinese wall security policy. In: IEEE Symposium on Security and Privacy, pp. 206–214 (1989) 17. Seymour, M.L.: Consensus and conflict. New Brunswick, New Jersey (1983)

Inspection of Surface Defects in Copper Strip Based on Machine Vision Xue-Wu Zhang, Li-Zhong Xu, Yan-Qiong Ding, Xin-Nan Fan, Li-Ping Gu, and Hao Sun Computer and Information College, Hohai University, Nanjing 210098, China [email protected]

Abstract. Though copper products are important raw materials in industrial production, there is little domestic research focused on copper strip surface defects inspection based on automated visual inspection. According to the defect image characteristics on copper strips surface, a defect detection algorithm is proposed on the basis of wavelet-based multivariate statistical approach. First, the image is divided into several sub-images, and then each sub-image is further decomposed into multiple wavelet processing units. Then each wavelet processing unit is decomposed by 1-D db4 wavelet function. Then multivariate statistics of Hotelling T2 are applied to detect the defects and SVM is used as defect classifier. Finally, the defect detection performance of the proposed approach is compared with traditional method based on grayscale. Experimental results show that the proposed method has better performance on identification, especially its application in the ripple defects can achieve 96.7% accuracy, which was poor in common algorithms. Keywords: copper strip surface defects; machine vision; defect inspection; wavelet decomposition; Hotelling T2 multivariate statistics.

1

Introduction

Copper is a kind of nonferrous metal which has close relationship with human, and has been widely used in electric, light manufacturing, machine manufacturing, construction, and national defense, etc. The consumption of copper in nonferrous metal is just preceded by aluminum. As a kind of important raw material of manufacturing production, copper industry has a higher close relationship with manufacturing, even the development of whole national economy. In 2007, production of copper materials in China exceeded the level of 6,000,000 tons, corresponding to nearly seven million tons apparent consumption. The quality of copper strip surface directly affects the quality and performance of final product. However, in the rolling system of the copper strip, the surface defect detection of copper strip is a simply repetitive, fast, and highly concentrated work. The traditional methods, such as artificial visual detection and frequency flashlight detection, have the disadvantages of low random inspection rate, bad real-time capability, and low confidence in inspection, etc. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 304–312, 2010. © Springer-Verlag Berlin Heidelberg 2010

Inspection of Surface Defects in Copper Strip Based on Machine Vision

305

In 90’s, with the fast development of electrical technology and machine technology, machine vision technology has been the mainstream of surface defect inspection, and has been widely used in defect inspection of steel strip, woods, ceramics, and presswork, etc[1,2,3]. The copper strip inspection has been a difficult task for the following reasons: strongly reflective, great variety of unknown defects, and different manufacturing techniques make different defects (especially in communication and electric industry). There is little domestic mature production that uses machine vision in copper strip surface defect inspection, but foreign production costs much. The objective of this paper is to explore and research effective algorithms in automatic inspection system of copper strip surface defects. Texture surface defect inspection is one of the key factors in visual inspection. Generally, texture can be classified into two categories: statistic texture and structure texture [4]. Statistic texture follows statistic rule, but it can not be described by basic texture cells; structure texture can be made of basic texture cells through permutation and combination following certain rule. According to the texture type, surface defect inspection system can be classified into non-texture surface defect inspection and texture surface defect inspection [5]. The former mainly includes threshold method [6] and pyramid method [7], the latter mainly includes spatial domain methods and frequency domain methods [8]. The GLCM (gray level co-occurrence matrix) is one of statistic methods in the spatial domain. Reference [9] used co-occurrence matrices based on wavelet sub-image characteristics to detect defects of textile fabrics. In reference [10], the authors used GLCM and weighted Euclidian distance classifier to classify metal fracture surface image, which has good performance on classification. Techniques in the frequency domain extract textural features by conducting frequency transforms such as Fourier transforms, Gabor transforms, or wavelet transforms. Fourier transform is a kind of global method, which just depicts the spatial and frequency distribution without regarding to local information [11]. Gabor transform belongs to windowing Fourier transform and has well frequency description capability, which is similar to biological effects of human eye and can be used to extract the corresponding features from different size and different direction in frequency domain. The basic of Gabor transform is frequency function (U,V) and the parameter design of Gauss function. According to different frequency parameters and Gaussian function, Gabor transform is usually used to texture recognition. But Gabor is non-orthogonal, and redundancy exists in different feature components, which results in inefficiency in analyzing texture image [12]. Wavelet transform provides a convenient way to obtain a multi-resolution representation, from which texture features are easily extracted. It has been a popular alternative for the extraction of texture features. Since images in different scales and frequencies have inherent characteristics for the appearance of a texture, the multi-resolution, multi-channel modeling capability of wavelet is well suited for texture analysis [13, 14]. Chen et al. [15] proposed a tree-structured wavelet transform for texture classification. They used an energy criterion to select the sub-image for decomposition, and distance measures are then employed to discriminate the texture classes. Reference [16] used two-dimension wavelet transform based on different selfadaptive wavelet basis and texture pattern recognition, genetic algorithm is used to obtain different orthogonal wavelet bases from row and column, experimental results show that this method is sensitive to texture defects and can locate the position of

306

X.-W. Zhang et al.

defects effectively. Tsai and Hsiao [17] first used db4 wavelet to decompose texture image in 3 scales, and then chose a low frequency image for construction. Last they used Otsu method to segment defects, which realized automatic detection of surface defects and had a good performance on defect inspection. Many surface defect inspection methods have no consideration on the effects of gradual change in image gray level characteristics [18]. Ripple is one of the most typical defects in copper strip, which has various shapes and different sizes, and has little difference with the non-defect image. The simple defect inspection method is not effective in detecting such defect. This paper considered the effects of gradual change in image illumination, and then used statistical method based on wavelet transform to inspect the surface defects of copper strip automatically. It first used Daubechies wavelet basis to decompose image in one scale, then used Hotelling T2 to analyze wavelet coefficients statistically to inspect defects. This paper is organized as follows. Section 2 discusses the methodology, including the fundamental principle and application of DWT and Hotelling T2 statistical method in this study. Section 3 presents experimental results, which proves the effectiveness of the method proposed in this paper. The paper is concluded in Section 4, which analyzes the advantages and disadvantages of method proposed in this paper in copper strip defects inspection, and discusses the research work in late stage.

2 Methodology 2.1

Wavelet Decomposition[19]

f ( x, y ) ∈ L2 ( R 2 ) represents a two dimensional image. A standard decomposition of a two-dimensional image can be done by first applying the Φ ( x) and Ψ ( x) to each row of pixel values and then performing another Φ ( x) and Ψ ( x) transform to each column. Where Φ (⋅) and Ψ (⋅) are scale function and wavelet function, respectively. Through the first scale decomposition, four sub-images A(x,y) ,H(x,y), V(x,y), and D(x,y) have been obtained, as shown in Eqs. (1)-(4).

2.2

A( x, y ) = { f ( x, y ), Φ1k 1 ( x) Φ1k 2 ( y )}

(1)

H ( x, y ) = { f ( x, y ), Φ1k 1 ( x) Ψ1k 2 ( y )}

(2)

V ( x, y ) = { f ( x, y ), Ψ1k 1 ( x) Φ1k 2 ( y )}

(3)

D( x, y ) = { f ( x, y ), Ψ1k 1 ( x) Ψ1k 2 ( y )}

(4)

Wavelet-Based Multivariate Statistical Approach

The wavelet-based multivariate approach decomposes an image of (M×N) pixels into a set of sub-images, each of which has a size of g×h (m×n) pixels and is a multivariate processing unit, namely, g M/m, h N/n. The sub-image contains g×h

＝

＝

Inspection of Surface Defects in Copper Strip Based on Machine Vision

307

(i.e. M/m×N/n) multivariate processing units, each of which can be further decomposed into k×l (a×b) wavelet processing units, namely, k m/a, l n/b. The wavelet transform can be applied to the (m/a×n/b) pixel region of each wavelet processing unit to obtain four characteristics A(i,j), H(i,j), V(i,j), and D(i,j), whose Daubechies wavelet coefficients is (a/2×b/2), the four characteristics A(i,j), H(i,j), V(i,j), and D(i,j) are low frequency coefficients, horizontal high frequency coefficients, vertical high frequency coefficients, and diagonal high frequency coefficients, respectively. Hotelling T2 inspection [20] is one of the most often used multivariate test methods. The T2 value can be regarded as a distance value of a multivariate processing unit. The larger the T2 distance value, the more distinctive the region is from the normal area. Thus, the more easily the region can be judged as defective. To enhance the characteristics of the original image, we fuse the high frequency detail information in three orientations, signed as D( xk , yl ) . Now, the wavelet

＝

＝

decomposed characteristics of sub-image are expressed by general information A( xk , yl ) and details D( xk , yl ) , as shown in Eqs. (5)-(6).

A( xk , yl ) =

a −1 2

b −1 2

i =0

j =0

∑∑ A(i, j) a b × 2 2

a b −1 −1 2 2

D( xk , yl ) =

∑∑[δ H(i, j) +δ H

i =0

V

(5)

V(i, j) + δD D(i, j)]

j =0

a b × 2 2

(6)

Where, A( xk , yl ) is low-frequency information, D( xk , yl ) is high-frequency information, δ H , δ V , and δ D are the weights in horizontal, vertical and diagonal orientation, and δ H + δ V + δ D = 1 . Samples of wavelet-based statistics method can be expressed as Eq.(7), the total mean matrix, total covariance matrix and Hotelling T2 statistics can be obtained from Eqs.(8)-(13).

⎡ A( xk , yl ) ⎤ X W( x k , y l ) = ⎢ ⎥ ⎣ D( xk , yl ) ⎦

(7)

⎡ 1 k −1 l −1 ⎤ A( xi , y j ) ⎥ ∑∑ ⎢ ⎡ A( x, y ) ⎤ k × l i = 0 j =0 X=⎢ ⎥ ⎥=⎢ k −1 l −1 ⎣ D( x, y ) ⎦ ⎢ 1 D( xi , y j ) ⎥ ⎢⎣ k × l ∑∑ ⎥⎦ i = 0 j =0

(8)

308

X.-W. Zhang et al.

⎡ 1 g −1 h −1 ⎤ A( xi , y j ) ⎥ ⎡ A ⎤ ⎢ g × h ∑∑ i = 0 j =0 ⎥ X=⎢ ⎥=⎢ g −1 h −1 ⎢ D⎥ ⎢ 1 D( xi , y j ) ⎥⎥ ⎣ ⎦ ⎢ ∑∑ ⎣ g × h i = 0 j =0 ⎦ WA2 =

2 k −1 l −1 1 [A( x , y ) − A( x , y )] ∑∑ i j k × l − 1 i = 0 j =0

(10)

1 g −1 h −1 2 ∑∑ WA ( xi , y j ) g × h i =0 j =0

(11)

WA2 =

UCL =

(9)

2 ⎡ WA WA,D ⎤ W=⎢ 2 ⎥ ⎣ WD,A WA ⎦

(12)

T 2 = k × l[x- x]T W −1[x- x]

(13)

(m − 1)(n − 1) p Fα ( p, mn − m − p + 1) mn − m − p + 1 LCL = 0

(14)

(15)

F α ( p, mn − m − p + 1) in Eq.(14) represents the separate point of F distribution

with p and mn-m-p+1 freedom, which can get a control chart with α confidence level. In multi-unit control chart, while T 2 is larger than UCL, then the distance between sub-image and the normal area is on the large side, thus, the region would be judged as defective.

3 Experiments and Analysis To verify the feasibility of the proposed approach, we build up a machine vision system and capture pictures of copper strips from XINGRONG Copper Corporation in Changzhou, Jiangsu Province. The typical defects types are shown in Fig.1. The sample set includes 3600 256×256 images and of each class with 450 images. All of the data are divided into 2/3 (2400 images) training set and 1/3 (1200 images) testing set. In the experiments, we use following equipments: a narrow LED lighting device (model LT-191×18 of DONGGUAN technology), a CCD camera (model JAI CV-A1 of DONGGUAN technology). We select compactly orthogonal Daubechies as wavelet basis. The length of filter is 4.

Inspection of Surface Defects in Copper Strip Based on Machine Vision

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

309

Fig. 1. Copper strip samples (a) faultless (b) smearing (c) indentation (d) burrs (e) scratch (f) pit (g) hole (h) ripple

In the experiments, considering the sample training time, recognition speed of test area, recognition rate and the size of the defects and other factors, after comparing and analyzing repeated experiments, window size of sub-image, namely, multi processing unit, is 32×32, and window size of wavelet decomposition is 4 × 4, which is beneficial to improve defect recognition rate and reduces the computation time. Wavelet-based Hotelling T2 statistical method is shown in Fig.2. When T2 statistical value of the image is upper than the UCL value, it indicates that the distance from sub-image to normal image is over the range, thus it is judged as defect image. As shown in Fig.3, (a) is Hotelling T2 statistics of faultless copper strip image, (b)-(h) are Hotelling T2 statistical chart of different defects. Because characteristics change of some defects, such as ripple, scratch, etc. is a gradual process, so its Hotelling T2 statistics value is near to the statistical chart of faultless, but it can also clearly identify defects, as shown in Fig.3.

T

2

⎧ ⎪X ⎨ ⎪ ⎩C −1 2 T = a × b × [ X − X ]' C [ X − X ]

T

2

Fig. 2. Wavelet-based Hotelling T2 statistic method

310

X.-W. Zhang et al.

(a)

(e)

(b)

(f)

(c)

(g)

(d)

(h)

Fig. 3. Inspection results based on wavelet statistic (a) non-defect (b) smearing (c) indentation (d) burrs (e) scratch (f) pit (g) hole (h) ripple Table 1. Recognition results for 7 typical defects of copper strip surface using two methods Defect type

Correct detection number of Statistic method Smearing 145 Indentation 148 Burrs 148 Scratch 142 hole 147 Pit 143 ripple 145

Correct detection Statistic number of Difference method method accuracy (%) 135 96.7 149 98.7 148 98.7 132 94.7 148 98.0 141 95.3 129 96.7

Difference method accuracy (%) 90.0 99.3 98.7 88.0 98.7 94.0 86.0

A large amount of experiments show that wavelet-based statistical method has a good performance on inspecting typical defects of copper strip surface, such as smearing, gap, burrs, scratch, indentation, pit and ripple and faultless. To verify the superiority of method proposed in this paper, we used two different methods to detect defects of copper strip surface. Then support vector machine (SVM), with RBF kernel and parameter (C, γ ) = (1024,1) , was used to train the whole training set and then test it. Experimental results are shown in Table.1. The wavelet-based statistical method has a high accuracy in all kinds of defects, range from 94% to 99%. The graybased difference method has a high detection rate in detecting ‘gap’, ‘burrs’, and ‘indentation’, but a low detection rate in detecting ‘smearing’, ‘scratch,’ and ‘ripple’ defects, because feature of these defects image has little difference with non-defect image, which results in bad capacity of resisting disturbance for gray-based difference method and bad detection rate.

Inspection of Surface Defects in Copper Strip Based on Machine Vision

311

4 Conclusion In accordance with the characteristics of copper strips surface defects inspection, this paper put forwards a wavelet-based statistics method to detect defects automatically. It divided the copper strip image into several sub-images, each of which is further divided into several wavelet units. One scale db4 wavelet is used to decompose the wavelet unit, and then Hotelling T2 statistic test method is used to inspect defects, at last, support vector machine is used to classify the defects. The method has the following characteristics: (1) Wavelet analysis could use spatial and frequency information in analyzing texture; (2) Hotelling T2 statistics method judges the defect type through control chart, and has a high detection rate in detecting peeling and scratch. (3) For different lighting condition, we can set different values for UCL according to practical situation, which makes the system adjustable. (4) Support vector machine has advantages in classifying small size, non-linear and highdimension samples. It realizes easily, and has good robustness. The experimental results show that wavelet-based statistic method has a high accuracy in inspection, but it has large computation, which can not satisfy the requirements of real-time inspection. For future research, it is necessary to optimize the algorithm, improve real-time capability and improve the correct detection rate. Acknowledgments. This work is supported in part by a grant from the National Natural Science Foundation of China (No. 60872096), the Fundamental Research Funds for the Central Universities (No.BZX/09B101-21), and the Scientific Research Starting Foundation for Doctor of Hohai University (XZX/08B005-02).

References 1. Cass, R., DePietro, J.: Computational Intelligence Methods for Process Discovery. Engineering Applications of Artificial Intelligence 11, 119–138 (1998) 2. Bu, H.G., Wang, J., Huang, X.B.: Fabric Defect Detection based on Multiple Fractal Features and Support Vector Data Description. Engineering Applications of Artificial Intelligence 22, 224–235 (2009) 3. Chen, R.C., Hsieh, C.H.: Web Page Classification based on a Support Vector Machine using a Weighted Vote Schema. Expert Systems with Applications 31, 427–435 (2006) 4. Bharati, M.H., Liu, J.J., MacGregor, J.F.: Image Texture Analysis: Methods and Comparisons. Chemometrics and Intelligent Laboratory Systems 72, 57–71 (2004) 5. Zhang, Y.J.: Content-based Visual Information Search. Science Press, Beijing (2003) 6. Stojanovic, R., Mitropulos, P., Koulamas, C., Karayiannis, Y., Koubias, S., Papadopoulos, G.: Real-Time Vision-Based System for Textile Fabric Inspection. Real-Time Imaging 7, 507–518 (2001) 7. Brzakovic, D., Beck, H., Sufi, N.: An Approach to Defect Detection in Materials Characterized by Complex Textures. Pattern Recognition 23(1/2), 99–107 (1990) 8. Li, T.S.: Applying Wavelets Transform, Rough Set Theory and Support Vector Machine for Copper Clad Laminate Defects Classification. Expert systems with Applications 36(3), 5822–5829 (2008)

312

X.-W. Zhang et al.

9. Latif-Amet, A., Ertuzun, A., Ercil, A.: An Efficient Method for Texture Defect Detection: Sub-band Domain Co-occurrence Matrices. Image and Vision Computing 28, 763–774 (2002) 10. Su, J., Li, M.: Research of Metal Fracture Image Classification based on GLCM. Computer Engineering and Applications 44(9), 223–225 (2008) 11. Zhong, K.H., Ding, M.Y., Zhou, C.P.: Texture Defect Inspection Method using Difference Statistics Feature in Wavelet Domain. System Engineering and Electronics 26(5), 660–665 (2004) 12. Pichler, O., Teuner, A., Hosticka, B.J.: A Comparison of Texture Feature Extraction using Adaptive Gabor Filtering, Pyramidal and Tree Structured Wavelet Transforms. Pattern Recognition 29, 733–742 (1996) 13. Arivazhagan, S., Ganesan, L.: Texture Segmentation using Wavelet Transform. Pattern Recognition 24, 3197–3203 (2003) 14. Kim, S.C., Kang, T.J.: Texture Classification and Segmentation using Wavelet Packet Frame and Gaussian Mixture Model. Pattern Recognition 40, 1207–1221 (2007) 15. Chen, T., Kuo, C.C.J.: Texture Analysis and Classification with Tree-structured Wavelet Transform. IEEE Transactions on Image Processing 2(4), 429–441 (1993) 16. Liu, H., Mo, Y.L.: 2D Wavelet Transform with Different Adaptive Wavelet Based for Regular Texture Defect Inspection. Journal of Image and Graphics 4(4), 349–352 (1999) 17. Tsai, D.M., Hsiao, B.: Automatic Surface Inspection Using Wavelet Reconstruction. Pattern Recognition 34, 1285–1305 (2001) 18. Lin, H.D.: Computer-aided Visual Inspection of Surface Defects in Ceramic Capacitor Chips. Materials Processing Technology 189, 19–25 (2007) 19. Xu, C.F., Li, G.K.: Practical Wavelet Method. Huazhong University of Science and Technology Press, Wuhan (2001) 20. Hotelling, H.: Multivariate Quality Control. Techniques of Statistical Analysis. In: Eisenhart, Hastasy, Wallis (eds.) McGraw-Hill, New York (1947)

BIBO Stability of Spatial-temporal Fuzzy Control System Xianxia Zhang1, Meng Sun1, and Guitao Cao2 1

Shanghai Key Laboratory of Power Station Automation Technology, School of Mechatronics and Automation, Shanghai University, Shanghai 200072, China 2 Software Engineering Institute, East China Normal University, Shanghai 200062, China

Abstract. Three-dimensional fuzzy logic controller (3-D FLC) is a novel FLC developed recently for spatially-distributed systems. In this study, the BIBO stability issue of the 3-D fuzzy control system is discussed. A sufficient condition is derived and provided as a useful criterion for the controller design of the 3-D FLC. Finally, a catalytic packed-bed reactor is presented as an example of spatially-distributed process to demonstrate the effectiveness of the controller. Keywords: Fuzzy control; Three-dimensional fuzzy set; Stability analysis; BIBO stability.

1 Introduction Since the fuzzy control was first introduced and applied to a dynamic plant (steam engine and boiler combination) by Mamdani and Assilian [1], it has become one of the most active and fruitful research areas in fuzzy set theory. However, most of fuzzy controls [1],[2] in the real world are developed for lumped systems and only fewer are found for spatially-distributed systems. Actually, most of physical processes are spatially distributed systems, which are spatial-temporal systems with their states, controls, and outputs depending on the spatial position as well as the time. Such systems usually give rise to nonlinear control problems that involve the regulation of highly distributed control variables using spatially-distributed control actuators and sensors. Fuzzy control for spatially distributed systems becomes a challenging problem. Recently, a novel three-dimensional fuzzy logic controller (3-D FLC), based on three-dimensional (3-D) fuzzy sets, has been developed for spatially-distributed systems [3]. The 3-D fuzzy set is composed of a traditional 2-D fuzzy set and a third dimension for spatial information; therefore, it has the inherent capability to express spatial information. The control strategy of the 3-D FLC is similar to how human operators or experts control the temperature in a space domain. The 3-D FLC has two obvious advantages over the traditional FLC. One is that the rules will not increase as sensors increase for spatial measurement, and the other is that the controller has inherent capability to process the spatial information. Thereby, the 3-D FLC has a great potential to a wide range of engineering applications for spatially-distributed processes. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 313–323, 2010. © Springer-Verlag Berlin Heidelberg 2010

314

X. Zhang, M. Sun, and G. Cao

In this study, the stability issue of the 3-D fuzzy logic control system is discussed. Amount of efforts have been devoted to the stability analysis of conventional fuzzy control systems in the past decades [4]. The motivation of this study is the global stability analysis of conventional fuzzy two-term control system developed in [5] using the well-known small gain theorem which can result in a safe and more reliable stability region in practical application. Stimulated by it, the paper employs the small gain theorem for a global stability analysis for the 3-D fuzzy logic control system where the controlled process contains spatially distributed information. A sufficient condition is eventually derived and provided as a useful criterion for the parameter design of 3-D fuzzy two-term controller.

2 Preliminaries In this section, the analytical model of 3-D fuzzy two-term controller is presented. The analytical model can be used for the stability analysis of the 3-D fuzzy two-term controller in the following sections. Similar to the traditional two-term FLC, the 3-D two-term FLC is classified into two types: a PI type generates incremental control output Δu , and a PD type generates control output u . Contrary to its traditional counterpart, the input variables of the 3-D two-term FLC are functions of the spatial coordinates Z and represents that the input information comes from the overall space domain. For instance, the two input variables are the spatial scaled error e∗ ( z ) and the spatial scaled error change Δe∗ ( z ) . In actual applications, finite-point sensors can be used for measurement, therefore, e∗ ( z ) and Δe∗ ( z ) can be expressed in discrete form in space domain Z = {z1 ,L , zq } , i.e., e∗ ( zI ) = eI∗ and Δe∗ ( z I ) = ΔeI∗ ( I = 1,L , q ), ∗

where eI

∈ E ∗ ⊂ IR and ΔeI∗ ∈ ΔE ∗ ⊂ IR denote the scaled error and the scaled

error change at the sensing location z = zI respectively, E ∗ and ΔE ∗ denote the domains of eI∗ and ΔeI∗ respectively, the number of sensors.

IR denotes the set of all real numbers, and q is

(a) Fig. 1. Fuzzy sets for

(b)

eI∗ and ΔeI∗ (a) and for Δu (b)

Just like its traditional counterpart, the analytical model of the 3-D two-term FLC will be uniquely determined once its components are chosen. It is assumed that the

BIBO Stability of Spatial-temporal Fuzzy Control System

315

domains of both the scaled inputs are limited to [-L, L] and the domain of the output is limited to [-V, V]. Singleton fuzzifier is used for fuzzification, 2-D Mamdani linear rule base is used for control law, and triangular fuzzy sets with equal spread 2c are used for both the scaled inputs (shown in Fig. 1(a), where N fuzzy sets for positive value, N fuzzy sets for negative value, and one fuzzy set for approximate zero). Singleton fuzzy sets with center-to-center distance H are used for the output (shown in Fig. 1(b), where 2N fuzzy sets for positive value, 2N fuzzy sets for negative value, and one fuzzy set for zero). Minimum t-norm is used for “and” operation on multiple antecedents and on Mamdani implication, maximum t-conorm is used for fuzzy union operation on connecting rules, weighted aggregation approach is used for dimension reduction, and ‘center-of-sets” defuzzifier is used for defuzzification. A PI type 3-D two-term FLC has the analytical model as follows (see detailed analytical derivation in [6]).

Δu ( nT ) = H γ (ω1 s1 (nT ) + L + ωq sq (nT )) + c γ 2 H (ω1k1μ1 + ω2 k2 μ 2 + L + ωq kq μq )

(1)

sI (nT ) = eI∗ (nT ) + ΔeI∗ (nT )

(2)

γ = 1 [ω1 (1 + 2μ1 ) + ω2 (1 + 2μ 2 ) + L + ωq (1 + 2μ q )]

(3)

where

eI∗ (nT ) = GE I eI (nT ) , GE I represents the scaling gain of the actual error eI ( nT ) from z = zI ; ΔeI∗ (nT ) = GΔE I ΔeI (nT ) , ΔeI = eI (nT ) − eI (nT − T ) , GΔE I represents

the scaling gain of the actual error change ΔeI ( nT ) from z = zI ; μ I denotes a membership grade as list in Table 1, which varies in different sub-regions ( IC1 ~ IC4 ) of the inference cell Q(iI , jI ) in the rule base plane, iI and jI ( − N ≤ iI , jI ≤ N − 1 ) are integers, and ( iI c , jI c ) represents the location coordinates of Q(iI , jI ) where ( eI∗ (nT ) , ΔeI∗ (nT ) ) appears in the rule base plane; ωI denotes the spatial weight at z = z I ; k I = iI + jI + 1 ; I = 1, L , q ; T is the sampling period.

Then, its final output is given as U PI ( nT ) = U PI ( nT − T ) + GU Δu (nT ) . For a PD

type 3-D two-term FLC, its final output is given as U PD ( nT ) = GU u ( nT ) , where

u (nT ) substitutes for Δu (nT ) in (1). GU denotes the scaling gain of the output. Table 1. Value of membership grade μ I when the scaled input pair( eI∗ , ΔeI∗ ) appears in the inference cell Q (iI , jI ) Sub-region

IC1

IC2

IC3

IC4

μI

ui I = 1 + iI − e ∗ I / c

u j I = 1 + j I − Δe ∗ I / c

1- uiI

1- u j I

316

X. Zhang, M. Sun, and G. Cao

3 BIBO Stability of 3-D Fuzzy Control System In this section, the global stability analysis of the 3-D fuzzy control system is analyzed by using small gain theorem. The 3-D FLC is designed as above and the controlled process contains spatially-distributed information. The process under control is a nonlinear spatially-distributed system with one distributed control source, which exists in many real industrial environments. The nonlinear spatially-distributed system is represented by Σ′ . It is assumed that enough sensors deployed in the space domain are used to extract the spatial information of Σ′ . The nonlinear dynamic input-output relationship of Σ′ at the finite sensing points can be represented by Σ , and the nonlinear dynamic input-output relationship of Σ′ in the space domain except the finite sensing points can be represented by Σ . The 3-D FLC C and the system Σ constitute a feedback connection as shown in Fig. 2.

Fig. 2. Configuration of feedback connection

Let

e (nT ) = (e1 (nT ), L , eq (nT ))′ ∈ IR q be q -dimensional vector. And let

eI (nT ) = ydI (nT ) − yI (nT ) and ΔeI (nT ) = eI (nT ) − eI (nT − T ) be error and error change from z = zI ( I = 1,L , q ) at the n th sampling instant respectively, where ydI ( nT ) denotes the reference profile and yI ( nT ) denotes the measurement value from z = zI . The norm of e ( nT ) is given as e (nT ) assumed that e ∈ l ∞ ; i.e., q

e

l∞q

∞

= sup eI (nT ) . It is I

=sup{ e [kT ] } < ∞ , where Z + denotes nonk∈Z +

negative integers. Suppose that both the systems of the feedback connection are finitegain l ∞ stable [7] and the feedback system is well-posed [7], we have

h1 (nT ) = e (nT ) , h2 (nT ) = g1 (nT ) + u2 (nT ) ,

⎧G Δu ( nT ) for PI type g1 ( nT ) = ⎨ U , g 2 (nT ) = Σ(h2 (nT )) ⎩GU u (nT ) for PD type For the given nonlinear system Σ(h2 (nT )) , the gain or the operator norm [5] of Σ is defined by (4) over the set of admissible control signals. The system Σ is finitegain l ∞ stable; i.e., the nonlinear system has a bounded gain Σ < ∞ .

BIBO Stability of Spatial-temporal Fuzzy Control System

Σ =

Σ(v1 (nT )) − Σ(v2 (nT ))

sup

v1 (nT ) − v2 (nT )

v1 ( nT ) ≠ v2 ( nT ), n ≥ 0

∞

317

(4)

With these preparations, we state one of main results of this section as follows. Theorem 1. The sufficient condition for the nonlinear 3-D fuzzy control system to be globally BIBO stable is that the given system Σ has a bounded norm (gain) Σ < ∞

and

the

parameters

of

the

3-D

two-term

FLC

G E = {GE1 ,L, GEq } ,

GΔE = {GΔE1 ,L, GΔEq } , GU , ω1 , ω2 ,L, ωq , H , and c satisfy GU Hkmaxωmax q Σ <1 c where

kmax = max{GE1 + GΔE1 ,L, GEq + GΔEq }

ωI ′ =

ωI

ω1 + ω2 + L + ωq

;

(5)

ωmax = max{ω1′,L, ωq′ }

;

with ωI > 0 ( I = 1,L , q ).

Proof: Since the rule base plane can be decomposed into the internal region and the boundary regions as shown in Fig. 3, we will give the proof according to what regions the input pairs fall into. For simplicity, the vector norm ⋅ ∞ is written as ⋅ . The two

types of 3-D two-term FLC have similar proof procedure. For simplicity, we only give the proof procedure of the PI type 3-D two-term FLC. We assume that q1 , q2 , q3 , q4 , and q5 input pairs fall into the internal region, the up (or down) border, the right (or left) border, the border marked by ‘ o ’, and the border marked by ‘ ⊕ ’ respectively, where q1 + L + q5 = q , 0 ≤ qi ≤ q , i = 1,L ,5 . Without loss of generality, we let q2 , q3 , q4 , and q5 input pairs fall into the up border, the left border, the left border marked by ‘ o ’, and the left border marked by ‘ ⊕ ’ respectively. The input pairs falling into any boundary region will be bounded. Firstly, substituting (1) into C (e ( nT )) , we have

C (e (nT )) ≤

GU H γ [ω1s1 (nT ) + L + ωq sq (nT )] + c GU H 2γ [ω1k1μ1 + L + ωq kq μq ]

Using k I = iI + jI + 1 ≤ k = 2 N ( I = 1,L , q ) and (3), (6) becomes

C (e (nT )) ≤

GU H ω1 s1 (nT ) + L + ωq sq (nT ) + 2GU H k c ω1 + L + ωq

(6)

318

X. Zhang, M. Sun, and G. Cao

Fig. 3. Decomposition of rule base plane

Let 0 < ω I ′ =

ωI < 1 ( I = 1,L , q ), then we have ω1 + L + ωq

C (e (nT )) ≤

GU H ω1′s1 (nT ) + L + ωq′ sq (nT ) + 2GU H k c

(7 )

Substituting (2) into (7), (7) becomes

C (e ( nT )) ≤ GU H {[ω1′(GE1 + GΔE1 ) e1 (nT ) + L + ωq′1 (GE q1 + GΔE q1 ) eq1 ( nT ) ] + c [ωq′1+1GE q1+1 eq1+1 ( nT ) + L + ωq′1+ q 2GE q1+ q 2 eq1+ q 2 ( nT ) ] + [ωq′1+ q 2 +1 GΔE q1+ q 2+1 eq1+ q 2+1 ( nT ) + L + ωq′1+ q 2 + q 3GΔE q1+ q 2+ q 3 eq1+ q 2+ q 3 ( nT ) ]} GU H {[ω1′GΔE1 e1 ( nT − T ) + L + ωq′1GΔE q1 eq1 (nT − T ) ]+[ωq′1+ q 2 +1 c GΔE q1+ q 2+1 eq1+ q 2+1 ( nT − T ) + L + ωq′1+ q 2+ q 3GΔE q1+ q 2+ q 3 eq1+ q 2+ q 3 (nT − T ) ]} + +

GU HN 2(ωq′1+ q 2+ q 3+ q 4 +1 + L + ωq′1+ q 2+ q 3+ q 4 + q 5 ) + (ωq′1+ q 2+1 + L + ωq′1+ q 2+ q 3 ) − (ωq′1+1 + L + ωq′ 2 ) +2GU Hk

(8)

BIBO Stability of Spatial-temporal Fuzzy Control System

Let

319

MeJ = sup eJ (nT − T ) ( J = 1,L, q1, q1 + q 2 + 1,L , q1 + q 2 + q3 ). It is n ≥1

noted that eJ (nT − T ) can be in any region of the rule base plane. If the stability condition is hold in every region, eJ (nT − T ) is always bounded [2]. Let

Me′ = ω1′GΔE1Me1 + L + ωq′1GΔE q1Meq1 + ωq′1+ q 2+1GΔE q1+ q 2+1 Meq1+ q 2+1 + L + ωq′1+ q 2 + q 3 GΔE q1+ q 2+ q 3 Meq1+ q 2+ q 3 , then Me′ is always bounded. In the sequel, let ′ = max{ω1′,L, ωq′1+ q 2+ q 3 } and kmax ′ = max{GE1 + GΔE1 ,L, GE q1 + GΔE q1 , GE q1+1 ωmax , L , GE q1+ q 2 , GΔE q1+ q 2+1 ,L , GΔE q1+ q 2 + q 3 } , then (8) becomes

′ ωmax ′ GU Hkmax [ e1 (nT ) + L + eq1 (nT ) ] + [ eq1+1 (nT ) + L c G HMe′ + eq1+ q 2 (nT ) ] + [ eq1+ q 2+1 (nT ) + L + eq1+ q 2 + q 3 (nT ) ] + U + GU HN c 2(ωq′1+ q 2+ q 3+ q 4+1 + L + ωq′1+ q 2 + q 3+ q 4 + q 5 ) + (ωq′1+ q 2+1 + L + ωq′1+ q 2+ q 3 ) − C (e (nT )) ≤

(9)

(ωq′1+1 + L + ωq′ 2 ) +2GU Hk If at least one input pair fall into the internal region, the up border or the left border ( i.e., 1 ≤ q1 + q 2 + q3 < q ), then (9) becomes

C (e (nT )) <= λ1 e (nT ) + η1

(10)

′ ωmax ′ q GU Hkmax ; c G HMe′ η1 = U + GU HN 2(ωq′1+ q 2+ q 3+ q 4+1 + L + ωq′1+ q 2+ q 3+ q 4+ q 5 ) + c . ′ ′ ′ ′ (ωq1+ q 2+1 + L + ωq1+ q 2+ q 3 ) − (ωq1+1 + L + ωq 2 ) +2GU Hk

where λ1 =

′ = max{GE1 + GΔE1 , Let Me = Me′ = ω1′GΔE1Me1 + L + ωq′ GΔE q Meq , kmax = kmax ′ = max{ω1′,L , ωq′ } . If all the input pairs fall into L , GE q + GΔE q } , and ωmax = ωmax the internal region (i.e., q1 = q and q 2 + L + q5 = 0 ), then (9) becomes

C (e (nT )) ≤ λ2 e (nT ) + η2 where λ2 =

(11)

GU Hkmaxωmax q G HMe and η 2 = U + 2GU H k . c c

If no input pair falls into the internal region, the up border, and the left border (i.e.,

q1 + q 2 + q3 = 0 and q 4 + q5 = q ), then (9) becomes

320

X. Zhang, M. Sun, and G. Cao

C (e (nT )) ≤ λ3 e (nT ) + η3 = η3

(12)

where λ3 = 0 and η3 = 2GU HN (ωq′ 4 +1 + L + ωq′ 5 )+2GU Hk . We can obtain the inequalities similar to (10)~(12) when the down border, the right border, the right corner marked by ‘ o ’ or the right corner marked by ‘ ⊕ ’ is considered respectively. In terms of the well-known small gain theorem [8], the parameters of the 3-D twoterm FLC must satisfy the following inequalities to ensure the BIBO stability: (1) At least one input pair fall into the internal region, the up/down border or the right/left border.

′ ωmax ′ q GU Hkmax ⎧ Σ <1 ⎪λ1 Σ = c ⎪⎪ k ′ = max{GE1 + GΔE1 ,L, GE q1 + GΔE q1 , GE q1+1 ,L , GE q1+ q 2 , ⎨ max ⎪ GΔE q1+ q 2+1 ,L , GΔE q1+ q 2 + q 3 } ⎪ ′ = max{ω1′,L, ωq′1+ q 2 + q 3 } ⎪⎩ωmax

(13)

(2) All the input pairs fall into the internal region.

GU Hkmaxωmax q ⎧ Σ <1 ⎪λ2 Σ = c ⎪ ⎨kmax = max{GE1 + GΔE1 ,L, GE q + GΔE q } ⎪ ⎪ωmax = max{ω1′,L, ωq′ } ⎩

(14)

(3) No input pair falls into the internal region, the up/down border, and the right/left border.

λ3 Σ = 0 < 1

(15)

By combining (13)~(15) together, (14) is a common condition satisfying all the conditions simultaneously when the input pairs falls into any region in the rule base plane; thus, the sufficient condition for the BIBO stability of the closed-loop control system is given as (14). 

4 Example 4.1 A Catalytic Packed-Bed Reactor

The control of temperature distribution in a long and thin non-isothermal catalytic packed-bed reactor [9] is taken as an example. The dimensionless model that describes the nonlinear tubular chemical reactor is given as follows

BIBO Stability of Spatial-temporal Fuzzy Control System

εp

∂Tg ∂t

=−

∂Tg ∂z

321

+ α c (Ts − Tg ) − α g (Tg − u )

γT ∂Ts ∂ 2Ts = 2 + B0 exp( s ) − β c (Ts − Tg ) − β p (Ts − b( z )u ) 1 + Ts ∂t ∂z

(16)

subject to the boundary conditions:

z = 0, Tg = 0,

∂Ts ∂Ts = 0; z = 1, =0 ∂z ∂z

(17)

where Tg , Ts , and u denote the dimensionless temperature of the gas, the catalyst, and jacket, respectively. The values of the process parameters are given below:

ε p = 0.01, γ = 21.14, β c = 1.0, β p = 15.62, B0 = −0.003, α c = 0.5, α g = 0.5.

A uniformly distributed heat source is used with b( z ) = 1 ( 0 ≤ z ≤ 1 ). It is assumed that only catalyst temperature measurement is available. The control target is to keep the catalyst temperature uniform in the space domain with a reference value of 0.1 from zero initial condition. Five-point sensors, located along the length of the reactor with z ′ = [0 0.25 0.5 0.75 1] , are used to collect the spatial distribution of the temperature Ts ; therefore, at each sampling instant nT five spatial temperatures

Ts ( z1 , nT ), Ts ( z2 , nT ),L , Ts ( z5 , nT ) are measured. Two spatial inputs for the 3-D two-term FLC are error and error change; i.e., ez = {e1 , e2 ,L, e5 } and Δez = {Δe1 , Δe2 ,L , Δe5 } , where ei = Ts d ( zi ) − Ts ( zi , nT ) , Δei = ei (nT ) − ei (nT − T ) , and Ts d ( zi ) = 0.1 . The model represented by (16)-(17) is just used to unveil the system dynamics and is employed as the simulated model to evaluate the control scheme. For a 3-D twoterm FLC designer, the operable information is the measurement information from the multiple sensors and the fuzzy rules extracted from prior knowledge and expert experience. 4.2 3-D Two-Term FLC Design Based on BIBO Stability Condition

In order to design the parameters of the 3-D two-term FLC based on the BIBO stability condition (5), first of all, the gain or the operator norm of the given system Σ should be obtained. Since there is no available process model, it can be computed approximately by experimental data over a set of admissible control signals as in (4). In this case, the control signal is limited to [0, 0.2], and it is discretized at 501 points. Over this set of control signals, the gain of Σ is computed as 0.975 according to (15) with the sampling period T = 0.1 . The PI type 3-D two-term FLC is used as the controller with the same detailed design as in section 2. Each of eI and ΔeI is classified into five linguistic labels, i.e.

N = 2 . Triangular membership functions with the equal spread 2c ( c = 0.05 ) are

322

X. Zhang, M. Sun, and G. Cao

used for these linguistic labels. For simplicity, the same scaling gains of error and error change over the space domain are used, i.e., GE1 = GE 2 = L = GE q = GE and

GΔE1 = GΔE 2 = L = GΔE q = GΔE

;

uniform

weights

are

used,

i.e.,

ω1 = ω2 = L = ωq = 1 5 . Since the catalyst temperature of the system is expected to arrive at uniform temperature 0.1 from zero initial condition over the space domain, we set the scaling gains for each eI and ΔeI as 0.5 for a proper resolution, i.e.,

GE = 0.5 and GΔE = 0.5 ; we set the scaling gain for Δu as 0.2, i.e., GU = 0.2 . Singleton fuzzy sets with the center-center distance H = 0.05 are use for the output. With the above mentioned parameters, the inequality (5) is satisfied, which means the feedback system in Fig. 2 is BIBO stable. The controlled catalyst temperature evolution profile and the manipulated input are shown in Fig. 4 (a) and (b), and the spatially-distributed profile of catalyst temperature at the steady state is shown in Fig. 4 (c). The figures show that the catalyst temperature throughout the reactor can uniformly track the reference value at all sensing locations from the zero initial condition. It demonstrates the effectiveness of the 3-D two-term FLC. It should be noted that the reference can be tracked in the entire space domain, which validates the result derived in theorem 1.

(a)

(b)

(c)

Fig. 4. Profiles of catalyst temperature varying with time and space (a), manipulated input varying with time (b), and catalyst temperature at steady state (c)

5 Conclusions In this paper, we have presented the stability analysis of 3-D fuzzy control system. Based on the analytical mathematical model, we have analyzed the BIBO stability of the 3-D fuzzy two-term control system and derived a sufficient condition, which can be provided as a useful criterion for the design of the 3-D two-term FLC. A catalytic packed-bed reactor was presented as an example of spatially-distributed process to demonstrate the effectiveness of the controller. Acknowledgements. This work was partly supported by the National Nature Science Foundation of China under Grant 60804033, the Shanghai Municipal Natural Science Foundation (Grant: 09ZR1409600), and the Science and Technology Commission of Shanghai Municipality (Grant: 08DZ2272400).

BIBO Stability of Spatial-temporal Fuzzy Control System

323

References 1. Mamdani, E.H., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man–Mach. Stud. 7(1), 1–13 (1974) 2. Ying, H.: Fuzzy Control and Modeling: Analytical Foundations and Applications. IEEE Press, New York (2000) 3. Li, H.X., Zhang, X.X., Li, S.Y.: A Three-dimensional Fuzzy Control Methodology for a Class of Distributed Parameter System. IEEE Trans. Fuzzy Syst. 15(3), 470–481 (2007) 4. Tanaka, K., Sugeno, M.: Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems 45, 135–156 (1992) 5. Chen, G.R., Ying, H.: BIBO stability of nonlinear fuzzy PI control systems. Journal of Intelligent & Fuzzy Systems 5, 245–256 (1997) 6. Zhang, X.X., Li, H.X., Li, S.Y.: Analytical study and stability design of a 3-d fuzzy logic controller for spatially distributed dynamic systems. IEEE Trans. Fuzzy Syst. 16(6), 1613– 1624 (2008) 7. Khalil, H.K.: Nonlinear Systems, 2nd edn. Prentice-Hall, New Jersey (1996) 8. Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-Output Properties. Academic Press, New York (1975) 9. Christofides, P.D.: Robust control of parabolic PDE systems. Chemical Engineering Science 53(16), 2949–2965 (1998)

An Incremental Manifold Learning Algorithm Based on the Small World Model Lukui Shi1, Qingxin Yang2, Enhai Liu1, Jianwei Li1, and Yongfeng Dong1 1

School of Computer Science and Engineering, Hebei University of Technology, Tianjin 300401, China 2 School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin 300160, China [email protected]

Abstract. Manifold learning can perform nonlinear dimensionality reduction in the high dimensional space. ISOMAP, LLE, Laplacian Eigenmaps, LTSA and Multilayer autoencoders are representative algorithms. Most of them are only defined on the training sets and are executed as a batch mode. They don’t provide a model or a formula to map the new data into the low dimensional space. In this paper, we proposed an incremental manifold learning algorithm based on the small world model, which generalizes ISOMAP to new samples. At first, k nearest neighbors and some faraway points are selected from the training set for each new sample. Then the low dimensional embedding of the new sample is obtained by preserving the geodesic distances between it and those points. Experiments demonstrate that new samples can effectively be projected into the low dimensional space with the presented method and the algorithm has lower complexity. Keywords: manifold learning; ISOMAP; small world model; incremental algorithm.

1 Introduction Nowadays it is becoming easier and easier to get a lot of data and information with the fast development of computer techniques. However, people find that it is more and more difficult to treat the data and information because they often contain huge high dimensional data in the real world applications. Usually, these high dimensional data can be characterized by a small number of latent variations, namely, they are possibly embedded in some low dimensional nonlinear manifold. The latent variations or the low dimensional manifold can obtain by reducing the dimension of the high dimensional data. Dimensionality reduction is an important preprocessing step to deal with the high dimensional data which purpose is to discover the nonlinear manifold embedded in the high dimensional space and to gain a compact expression of the original high dimensional data. Classical linear methods, such as PCA, MDS and so on, can effectively map the linear high dimensional data into a low dimensional space. They K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 324–332, 2010. © Springer-Verlag Berlin Heidelberg 2010

An Incremental Manifold Learning Algorithm Based on the Small World Model

325

have also been widely applied in many fields. Nevertheless, these linear techniques cannot discover the nonlinear manifolds hidden in the high dimensional data. In last few years, a category of nonlinear dimensionality reduction techniques, called manifold learning, are absorbed in many researchers’ interest, which are designed to find the nonlinear manifolds hidden in the high dimensional data. The typical algorithms contain ISOMAP [1], LLE [2], Laplacian Eigenmaps [3], LTSA [4] and Multilayer autoencoders [5]. Although all these algorithms have quite different motivations and derivations, they all can execute the dimensionality reduction on the nonlinear manifolds and are applied to many real problems. Unfortunately, most of these algorithms are only defined on the training sets and are executed as a batch mode. They don’t provide a model or a formula to map the new data into the low dimensional space. When new data arrive, the only way to project them is to pool old and new points and perform these algorithms again for this pool. Namely, these methods are not generalized to treat new samples. Apparently, it is not suitable in a changing and dynamic data environment. Therefore, some researchers developed many incremental manifold learning algorithms, for example [6-11]. Nevertheless, most of these methods are complex and have high complexity. In this paper, we presented an incremental manifold learning algorithm based on the small world model, which is a better generalization of ISOMAP for new sample and has higher efficiency. According to the characteristic of the small world model, one firstly selects k nearest neighbors and some faraway points from the training set for each new sample. Then the low dimensional coordinates are discovered by preserving the geodesic distances between the new sample and those points selected. Experiments show that the presented algorithm can fast treat new samples on the base of ISOMAP.

2 Recall of ISOMAP and the Small World Model 2.1 ISOMAP ISOMAP is one of the most representative manifold learning algorithms, which generalizes MDS to the nonlinear high dimensional data. The main principle of the method is to replace Euclidean distances with the geodesic distances in MDS. Therefore, an important step of the algorithm is to compute the geodesic distances between all pairs in the original space. For the neighboring points, the input space distance provides a good approximation. For the distant points, the geodesic distances can be approximated by computing the shortest path distances in the manifold. The algorithm includes three steps: 1) Select k nearest neighbors as the neighborhood for each object. 2) Estimate the geodesic distances between all pairs by calculating the shortest path distances in the manifold. 3) Discover the low dimensional embedding by applying MDS. The details can be referred to [1]. 2.2 The Small World Model The small world phenomenon was first introduced in a social experiment by Stanley Milgram in 1960s. The principle of the small world networks is that any two individuals in the network are likely to be linked through short chains of

326

L. Shi et al.

acquaintances. To do so, each node in the small world network is directly linked its nearest neighbors as well as a small number of randomly chosen faraway nodes. It has been shown that this network structure generates graphs with small diameter and a small routing distance between any two individuals [12].

3 The Incremental Manifold Learning Algorithm 3.1 The Principle of the Algorithm To discover the low dimensional manifolds hidden in the high dimensional space, it is necessary to solve a full N×N eigenvector problem in ISOMAP, which leads to Ο(N3) complexity. For new samples, one needs to append new data into the training set as a new set and execute ISOMAP again on the new set. Apparently, the complexity is very high to project new samples into the low dimensional space for ISOMAP. Contradictorily, LLE and Laplacian eigenmaps have lower complexity because the eigenvector problem is sparse. The efficiency of mapping new data will be great improved if the eigenvector matrix is also spare in ISOMAP. In the small world network, each node is linked to its nearest neighbors and a small number of faraway nodes randomly selected. The same as the small world network, it is also feasible for a new sample to only preserve the geodesic distances between it and some points in the training set in order to gain a sparse matrix. These points contain its k nearest neighbors and some distant points randomly chosen. Let X be a high dimensional data set containing N objects and Y be its low dimensional embedding resulting from ISOMAP. Consider a new sample xnew, ynew is its low dimensional coordinates. Let dj and d *j express the geodesic distances between xnew and each point in X and corresponding Euclidean distances between ynew and each object in Y respectively. The goal of projecting xnew into the low dimensional space is to force d *j to match dj for j =1, 2, , N . In order to do that,

…

the stress function is used to measure the error between the pairwise distances in the low dimensional space and the original space in MDS. Kruskal proposed a popular form of the stress function [13]. For a new sample, Kruskal stress can be changed into

sf =

∑ (d

j

− d *j ) 2

j

∑d

2 j

j

(1)

According to the principle of the small world model, we only preserve the geodesic distances between the new sample and some points in the training set in the new algorithm. Therefore, the equation (1) can be modified as sf =

∑ (d j∈Ω

j

− d *j ) 2

∑d j∈Ω

2 j

(2)

where Ω is the set containing the nearest neighbors of the new sample and some distant points randomly chosen.

An Incremental Manifold Learning Algorithm Based on the Small World Model

327

The low dimensional embedding of the new sample can be found by minimizing the stress function. In order to do that, we can define the energy function as follows: E=

∑

1 λ ( d j − d *j ) 2 2 j∈Ω

(3)

where λ is a learning rate parameter. Here, the deepest gradient descent method is used to minimize the energy function. Let ∇E be the gradient with respect to ynew, then (d j − d *j ) Δ y new = ∇E = −λ (y j − y new ) (4) d *j j∈Ω

∑

The equation (4) tells us how to update the embedding coordinates of new data during each iterative step. The incremental manifold learning algorithm based small world model can be described as follows: Input: the new sample xnew, the training set X and its low dimensional embedding Y, the neighbor size k1, the number k2 of distant points, the embedded dimension d, the max learning steps L, the min stress value s and the learning rate λ. Output: the low dimensional coordinates ynew for xnew. Step 1 Select k1 nearest neighbors and k2 distant points for the new sample xnew based on Euclidean distances in the set X. Step 2 Estimate the geodesic distances between xnew and those points chosen in the step 1. Step 3 Randomly initialize ynew. Step 4 Set i=1. Step 5 Calculate the stress according to the equation (1). If the stress is smaller than s, go to step 9. Step 6 Compute Δ y new according to the equation (4). +1 Step 7 Set y inew = y inew + Δ y new .

Step 8 Set i=i+1. If i is smaller than L, go to step 5. Step 9 Return the low dimensional coordinates ynew of the new sample.

In the above algorithm, we suppose that the low dimensional embedding of the training set has been gained with ISOMAP before new samples are mapped. Namely, the incremental algorithm is based on ISOMAP because the new method also finds the low dimensional coordinates of new samples by preserving the geodesic distances. For other manifold learning methods, new data are not projected into the low dimensional space with the above incremental algorithm.

328

L. Shi et al.

3.2 Estimate the Geodesic Distances and Analyze the Complexity

In the above algorithm, the geodesic distances between a new sample and its k1 nearest neighbors can be approximated with Euclidean distances between them. The key is to estimate the geodesic distances between the new sample and its k2 distant points. Suppose that the training set are sufficiently sampled and that the nearest neighbors of all training objects is invariable while adding new data, the geodesic distances between the new sample and its k2 distant points can be estimated with a simple method. Let k1 nearest neighbors of the new data be x n1 , x n2 , L , x nk1 . According to Dijkstra’s algorithm, the geodesic distances between xnew and some distant point xi can be gotten with the following equation d G (x new , x i ) = min{d (x new , x n1 ) + dG (x n1 , x i ), K,

(5)

d (x new , x nk ) + d G (x nk , x i )} 1

1

where dG(i,j) and d(i,j) is respectively the geodesic distance and Euclidean distance between pair of points (i,j). In the above algorithm, the computational cost of the iterative procedure is Ο(c(k1+k2)) (c is a constant). The time cost of searching the nearest neighbors and selecting distant points of a new sample can be ignored. According to the method of estimating the geodesic distances, the cost of estimating the geodesic distances can be also ignored. Therefore, the computational complexity for mapping a new sample is Ο(c(k1+k2)). It is far below the complexity Ο(N3) of solving a full N×N eigenvector problem.

4 Experiments In this section, the presented incremental algorithm is evaluated on three data sets. They are Swiss roll data set [1], S-curve data set [2] and synthetic face images data set [1], which are often used to validate the performance of the manifold learning algorithms. We respectively pick up 2000 points from the former two data sets as sample data sets (see figure 1) and synthetic face images data set contains 698 objects described by 4096 components in different poses and lighting conditions. In experiments, the geodesic distance is used in the high dimensional space and Euclidean distance in the embedded space for all data sets. All programs were implemented in MATLAB. To check the performance of the new incremental algorithm, we divide the high dimensional data set X with n samples into two disjoint subsets X1 and X2. The former is used as the training set and the latter as the testing set. Let |X1|=m, then |X2|=n-m. The validity of the incremental algorithm is tested through the following steps. 1) ISOMAP is executed on X=X1∪X2 to obtain the corresponding low dimensional embedding Y=Y1∪Y2 firstly. Then ISOMAP is performed on the training set X1 to gain its low dimensional coordinates Y1′ . At last Y1 and Y1′ are aligned.

An Incremental Manifold Learning Algorithm Based on the Small World Model

329

2) The incremental algorithm is utilized to predict the embedded coordinates Y2′ of the testing set X2 based on the training set.. 3) Euclidean distances between each point in Y2 and all points in Y1 are computed. The distance is written as dij where i is from 1 to n-m and j is from 1 to m. 4) Euclidean distances between each point in Y2′ and all points in Y1 are calculated. The distance is written as d ij′ . 5) The relative difference between them is defined as n−m m

∑ ∑ dij − dij′

r=

(6)

i =1 j =1 n−m m

∑ ∑ dij i =1 j =1

The relative difference is computed for different m and for various k2 on three data sets. On Swiss roll data set, m=1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900. For each testing set, k1=7 and k2=5, 10, 15, 20. On S-curve data set, m=1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900. For each testing set, k1=20 and k2=5, 10, 15, 20. On face images data set, m=530, 560, 590, 620, 650, 680. For each testing set, k1=7 and k2=5, 10, 15, 20. The results on three data sets are given in table 1 to table 3. Experiments show that the proposed incremental algorithm can effectively map new samples into the low dimensional space, which is a better generalization of ISOMAP. And the method has lower complexity. As shown in experiments, the relative difference is reduced with the increasing number of selected distant points generally. Moreover, the relative difference is smaller and smaller with the increasing size of the training set. Three examples on S-curve data set are given in figure 2, figure 3 and figure 4. In figure 2, the training set and the testing set both contain 1000 samples. In figure 3, the training set includes 1500 samples and the testing set contains 500 objects. In figure 4, the training set includes 1800 objects and the testing set contains 200 samples. Here, k1=20 and k2=10 for three examples.

(a) Swiss roll data set

(b) S-curve data set

Fig. 1. Two data sets including 2000 samples

330

L. Shi et al. Table 1. Results on Swiss roll data set m 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

k2=5 0.0227 0.0201 0.0216 0.0171 0.0172 0.0148 0.0129 0.0134 0.0121 0.0100

k2=10 0.0223 0.0195 0.0215 0.0173 0.0167 0.0145 0.0132 0.0133 0.0118 0.0093

k2=15 0.0226 0.0193 0.0210 0.0170 0.0168 0.0147 0.0132 0.0133 0.0120 0.0098

k2=20 0.0221 0.0193 0.0208 0.0170 0.0167 0.0148 0.0132 0.0132 0.0117 0.0095

Table 2. Results on S-curve data set m 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

k2=5 0.0121 0.0122 0.0081 0.0111 0.0088 0.0066 0.0056 0.0044 0.0037 0.0036

k2=10 0.0121 0.0122 0.0082 0.0110 0.0088 0.0066 0.0056 0.0044 0.0037 0.0036

k2=15 0.0123 0.0122 0.0081 0.0111 0.0088 0.0066 0.0058 0.0045 0.0040 0.0039

k2=20 0.0122 0.0121 0.0081 0.0112 0.0089 0.0068 0.0062 0.0047 0.0036 0.0041

Table 3. Results on synthetic face images data set m 530 560 590 620 650 680

k2=5 0.0470 0.0480 0.0456 0.0425 0.0381 0.0343

k2=10 0.0453 0.0468 0.0462 0.0423 0.0370 0.0338

(a) The original data including 1000 objects

k2=15 0.0461 0.0466 0.0447 0.0408 0.0403 0.0318

k2=20 0.0481 0.0471 0.0460 0.0403 0.0387 0.0301

(b) The low dimensional embedding

Fig. 2. An example on S-curve data set k1=20, k2=10 and m=1000

An Incremental Manifold Learning Algorithm Based on the Small World Model

331

(a) The original data including 500 objects (b) The low dimensional embedding Fig. 3. An example on S-curve data set k1=20, k2=10 and m=1500

(a) The original data including 200 objects (b) The low dimensional embedding Fig. 4. An example on S-curve data set k1=20, k2=10 and m=1800

5 Conclusions Many applications in the real world produce lots of high dimensional data. Manifold learning algorithms can discover the nonlinear manifolds hidden in these high dimensional data. Most of manifold learning algorithms cannot project new samples into the low dimensional space because they are only defined on the training sets. In this paper, we presented an incremental manifold learning algorithm based on the small world model, which only preserves the geodesic distances between each new sample and few points. Experiments show that new samples can effectively be mapped into the low dimensional space with the incremental method and the algorithm has lower complexity. In contrast to ISOMAP, the new method has more free parameters to be selected. Among them, a very important parameter is the number of distant points, which cannot be too small or too big. Too few distant points will lead worse embedding results. And too many distant points will degrade the executing efficiency of the algorithm. The number of distant points should be far below the number of samples. Moreover, the incremental algorithm is only a generalization of ISOMAP. In the future, we will check the performance of the incremental algorithm in the real world applications.

332

L. Shi et al.

References 1. Tenenbaum, J.B., de Silva, V., Langford, J.C.: A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290, 2319–2323 (2000) 2. Roweis, S.T., Saul, L.K.: Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science 290, 2323–2326 (2000) 3. Belkin, M., Niyogi, P.: Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computation 15, 1373–1396 (2003) 4. Zhang, Z., Zha, H.: Principal Manifolds and Nonlinear Dimensionality Reduction via Local Tangent Space Alignment. SIAM Journal of Scientific Computing 26, 313–338 (2004) 5. Hinton, G.E., Salakhutdinov, R.R.: Reducing the Dimensionality of Data with Neural Networks. Science 313, 504–507 (2006) 6. Ham, J., Lee, D.D., Mika, S., Schölkopf, B.: A Kernel View of the Dimensionality Reduction of Manifolds. In: 21st International Conference on Machine Learning, pp. 369– 376. ACM Press, New York (2004) 7. Law, M.H.C., Zhang, N., Jain, A.K.: Nonlinear Manifold Learning for Data Stream. In: 4th SIAM International Conference for Data Mining, pp. 33–44. SIAM Press, Philadelphia (2004) 8. Li, H., Zhang, J., Zhou, Z.: Investigating Manifold Learning Algorithms Based on Magnification Factors and Principal Spread Directions. Chinese Journal of Computers 12, 2000–2009 (2005) 9. Shi, L., Li, J., Wu, Q., He, P., Peng, Y.: An Incremental Algorithm Based on k Nearest Neighbor Projection for Nonlinear Dimensionality Reduction. In: 5th International Conference on Machine Learning and Cybernetics, pp. 1417–1421. IEEE Press, New York (2006) 10. Zeng, X., Luo, S.: A Dynamically Incremental Manifold Learning Algorithm. Journal of Computer Research and Development 9, 1462–1468 (2007) 11. Zhan, Y., Yin, J., Zhang, G., Zhu, E.: Incremental Manifold Learning Algorithm Using PCA on Overlapping Local Neighborhoods for Dimensionality Reduction. In: Kang, L., Cai, Z., Yan, X., Liu, Y. (eds.) ISICA 2008. LNCS, vol. 5370, pp. 406–415. Springer, Heidelberg (2008) 12. Watts, D.J.: Small-worlds: The Dynamics of Networks between Order and Randomness. Princeton University Press, Princeton (1999) 13. Kruskal, J.B.: Multidimensional Scaling by Optimizing Goodness of Fit to a Nonmetric Hypothesis. Psychometrika 29, 1–27 (1964)

Crack Image Enhancement of Track Beam Surface Based on Nonsubsampled Contourlet Transform Chunhua Guo and Tongqing Wang The Key Laboratory of Optoelectronic Technology & Systems of the Ministry of Education, Chong Qing University, Chong Qing, 400030 [email protected], [email protected]

Abstract. A new algorithm of crack image enhancement of straddle-type monorail track beam surface based on nonsubsampled contourlet transform (NSCT) is presented. It is according to the characteristics of different domain in NSCT, and the fractional differentiation can enhance the mid-frequency and retain the low frequency nonlinearly. Then a new enhancement method has been proposed that the smooth sub-band texture of NSCT domain can be enhanced by the fractional differentiation; while in the high-frequency subbands of NSCT domain, each pixel of high-frequency sub-bands is divided into strong edge, weak edge and noise on the basis of direction sensitivity characteristics, then the weak edge can be enhanced nonlinearly, strong edge be retained, and noise be removed. Experimental results show that the method proposed in this paper have greatly improved visual effects and larger contrast improve index (CII) as compared with wavelet transform, and the enhancement effect is very good. Keywords: crack enhancement; nonsubsampled contourlet transform; fractional differentiation.

1 Introduction City rail transit system has the characteristics of covering less area, low-cost building, small noise, safety and so on. It is especially suitable for the region of mountain city, terrain and high concentrative building urban district. But as various reasons, the surface of straddle-type monorail appears cracks and other defects, which seriously impact the operation and safety of train. Images collected have different degree shortcomings such as edge blurring, poor contrast, etc. The property of crack is often not very apparent. Whether the images can be enhanced effectively, improving the contrast of crack and background, is not only a critical step for the following processing and analysis, but also the basis of crack segmentation and recognition. Image enhancement is an image processing method based on the psychological characteristics of human visual system. It can be classified into two categories at present [1]: global enhancement and local enhancement. Global enhancement changes overall brightness to achieve contrast enhancement in accordance with certain rules [2] [3]. In general, the global algorithms are satisfactory for the low contrast of total K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 333–341, 2010. © Springer-Verlag Berlin Heidelberg 2010

334

C. Guo and T. Wang

images, but they are poor for the local images. Local enhancement methods can enhance all images in theory. However local operator is difficult to achieve, and it is more difficult to obtain one which is suitable for all kinds of images. Multi-scale analysis of images has been widely applied in image enhancement. Wavelet transform (WT) can represent one-dimensional signals very well, but twodimensional wavelet is the kron kroncker of one-dimensional wavelet, so the direction is very limited. Donoho,Vetterli,etc [4] proposed contourlet transform (CT) in 2002, which is a new tool for the two-dimensional or multi-dimensional singular signals. However the CT is not a shift-invariant transform, which may cause pseudo-Gibbs phenomena around singular points in image processing. Arthur,etc [5] proposed an overcomplete transform that is called nonsubsampled contourlet transform (NSCT) in 2006. The NSCT is a fully shift-invariant, multiscale, and multi-directional expansion that has a fast implementation, which can effectively eliminate the phenomenon of pseudo-Gibbs. In this paper, a novel enhancement algorithm is proposed for crack images based on the NSCT. The low-frequency sub-band of NSCT is intensified using fractional differentiation, and the high-frequency sub-bands are strengthened adopting nonlinear operator. This proposed method can not only enhance the important details of images, improve the visual effects, but also suppress noise, and it has been proven to be very efficient in image enhancement as we show in this paper.

2 Methodology and Process The nonsubsampled contourlet transform is a newly developed multi-scale, and multidirectional transformation based on contourlet transform, the fully shift-invariant characteristics makes up for the limitation of contourlet transform. The NSCT can be divided into two shift-invariant parts:1) a nonsubsampled pyramid structure that ensures the multi-scale property and 2) a nonsubsampled directional filter bank (DFB) structure that gives directionality. Fig.1 displays an overview of the NSCT.

Lowpass subband

Ȧ2

Bandpass directional subbands Image

(ʌ, ʌ˅ Ȧ1

Bandpass directional subbands ˄-ʌ,-ʌ˅ (a)

(b)

Fig. 1. Nonsubsampled contourlet transform. (a) Nonsubsampled filter bank structure that implements the NSCT. (b) Idealized frequency partitioning.

Crack Image Enhancement of Track Beam Surface Based on NSCT

335

In each stage, the image is firstly decomposed into low-pass sub-band and bandpass sub-bands by means of pyramid structure, and the directional information can be obtained through directional filter bank used in band-pass sub-bands. Then the lowpass sub-band can be treated as the input image for next decomposition stage. The NSCT is a multi-resolution and multi-direction decomposition [6]. The visual characteristics of human eyes should be considered when crack images are enhanced. Physiological vision study has found that the human eyes are more sensitive to the images with details than images without details, and the visibility of noise is lower in detail images than in flat images[7][8]. Based on these, fractional differentiation is used to enhance the low frequency sub-band image of NSCT domain, and a non-linear function is employed to enhance the direction sub-bands. Thereby the visual effects of overall images are strengthened. 2.1 Low Pass Sub-band Enhancement Based on Fractional Differentiation

2 For an energy signal s (t ) ∈ L ( R ) which is square integrable, its Fourier transform of v order fractional differentiation is as following [9]: FT ∧ v d v s (t ) D s (t ) = D s (t ) = ⇔ ( D s )(ω ) v v dt v

.

(1)

∧ ∧ ∧ = (iω )v ⋅ s (ω ) = d v (ω ) s (ω ) v ∈ R +

，

v In which, v order operator Dv = D is a multiplicative operator of differential ^ v multiplier function d ( w) = (iw) , its filter function of fractional differentiation is

⎧∧d (ω ) = (iω )v = ∧a (ω ) ⋅ exp(iθ (ω )) = ∧a (ω ) ⋅ ∧p (ω ) v v v v ⎪ . ∧ ∧ ⎨ v π v a v (ω ) = ω θ v (ω ) = sgn(ω ) ⎪ ⎩ 2

，

(2)

As we can see from above, the amplitude varies with frequency and presents with fractional exponent variation, and the phase is a general Hilbert transform of frequency. Fig.2 displays an overview of amplitude characteristic of fractional differentiation. As can be seen from Fig.2, when the fractional differential order v is between 0 and 1, the high frequency of signal is significantly strengthened, middle frequency signal can also be enhanced, and very low frequency signal is non-linearly retained. The Grumwald-Letnikov(G-L) definition of fractional differentiation is starting from the classic definition of integral derivative for continuous equation, and it is obtained by extending the integral calculus to fraction calculus. The finite difference expression of fractional differentiation for one dimensional signal can be got by discretizing G-L definition, which is shown as Eq.(3)

336

C. Guo and T. Wang 3

w0.75 w

2

w

v 1

w

0.25

w0.5

0 0

2

1

3

w /Hz Fig. 2. The amplitude curve of fractional differentiation

v d f (t ) ( − v )(− v + 1) f (t − 2) v ≈ f (t ) + ( − v ) f (t − 1) + dt 2 . Γ ( −v + 1) + ... + f (t − n). n!Γ ( −v + n + 1)

(3)

We can see from [10] that the Fourier transform of high dimensional fractional order is separable, so the fractional differentiation of f ( x, y ) can also be separated in a certain conditions [11] and is equivalent to the fractional differential operation along two axes ( x and y ) respectively. For example, if the size of fractional differentiation mask is 4*4, the definitions of fractional differentiation of f ( x, y ) along two axes are as following Eq.(4) and Eq.(5).

2.2

v ∂ f ( x, y ) ( −v )(− v + 1) ≈ f ( x, y ) + ( −v ) f ( x − 1, y ) + f ( x − 2, y ) v ∂x 2 . ( −v )( −v + 1)(−v + 2) + f ( x − 3, y ) 6

(4)

v ∂ f ( x, y ) ( −v )(− v + 1) ≈ f ( x, y ) + ( − v ) f ( x, y − 1) + f ( x , y − 2) v ∂y 2 . ( −v )( −v + 1)( −v + 2) + f ( x, y − 3) 6

(5)

The Non-linear Enhancement of Direction Sub-band

2.2.1 The Structure of Threshold Study has shown that the direction sub-band coefficients of NSCT domain meet the assumption of Bayes estimation——signal shall obey general Gaussian distribution.

Crack Image Enhancement of Track Beam Surface Based on NSCT

337

Chang [12] adopted general Gaussian distribution as the priori model, and proposed a Bayes Shrink method based on Bayes criterion that the threshold of each sub-band coefficients can be obtained adaptively. The threshold value is shown as Eq.(6). 2

λ = σn σs .

(6)

Where σ n is the noise variance, σ s is the standard deviation of signal. The noise variance σ n of level l , direction k of NSCT domain is estimated by the following formula:

^

σ n = median ( fl , k ( x , y ) ) / 0 .6745 .

(7)

The corresponding standard variance of signal is estimated by maximum likelihood, where M and N are the length and width of sub-band respectively.

^2 ^2 1 M N 2 σ s = max( ∑ ∑ f ( x , y ) − σ n ,0) . MN x =1 y =1 l , k

(8)

Substitute Eq.(7) and Eq.(8) into Eq.(6), the sub-band threshold λl, k can be obtained. 2.2.2 The Design of Enhancement Function A simple enhancement function can not only enhance the edge details of direction sub-bands, but also amplify noise amplitude. To overcome this shortcoming, it is very necessary to construct a non-linear enhancement function. This enhancement function should have the feature of parameters adjustable, monotonous, fast, and satisfy the request of enhancement. The design steps are as follows: Step1. To standardize the direction sub-bands coefficients of NSCT domain of crack image

nor fl , k = fl .k / max( fl , k ) .

(9)

Where fl , k is the original direction sub-band coefficient of level l , direction k of nor NSCT domain. fl , k is the corresponding standardized coefficient. Step2. To construct the enhancement function of standardized coefficients nor nor p nor w( f ) = (1 − (1 − f ) ) ⋅ sign ( f ) . l, k l, k l, k In Eq.(10), sign ( f

(10)

nor ) is a sign function, p decides the degree of enhancement. It is l, k

easy to see that p should be greater than 1, if the value is less than 1, the edge subband is weaken rather than enhanced. Figure 3 is the curve of Eq.(10), in which p = 3 .

338

C. Guo and T. Wang

1 original values enhanced values

0.8 0.6 0.4

adjusted coefficients

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Normalized coefficients Fig. 3. The transform figure of sub-band coefficients

Step3. To calculate the enhanced sub-band coefficients nor fl .k = fl , k ⋅ max( fl , k ) .

(11)

2.2.3 Crack Image Enhancement Based on Local Statistics The sub-band coefficients of different levels and different directions in NSCT domain can be classified into strong edge, weak edge and noise according to the threshold of each sub-band coefficients. The judging criterion is as follows:

⎧ mean ( f l , k ) ≥ c λ l , k strong edge ⎪ . weak edge < λ ≥ λ mean f c f c ( ) , max( ) ⎨ l ,k l ,k l ,k l ,k ⎪ mean ( f ) < c λ , max( f ) < c λ noise l ,k l ,k l ,k l ,k ⎩

(12)

Where λl ,k is the sub-band threshold of level l , direction k of NSCT domain. c is an adjustable parameter between 1-5. The strong edge pixels of direction sub-band are retained and equal to the original pixel value. The pixels of weak edge are amplified according to the method of section 2.2.2. The pixels which are judged noise are set to 0 to suppress effectively.

3 Experimental Results and Discussion The experimental results are presented as follows, which are compared with wavelet method. The choice of experimental parameters is discussed too. 3.1 Experimental Results Comparison

Fig.4 is an enhancement experimental results of crack images, which are also compared with wavelet transformation (WT) method. In the test, nonsubsampled

Crack Image Enhancement of Track Beam Surface Based on NSCT

339

pyramid and direction filter of NSCT are chosen as “maxflat” and “dmaxflat7”, the level of decomposition is three, the number of direction is 2, 4 and 8, respectively. Among which v = 0.25 , c = 3 , p = 3 ; DB97 is selected for WT, and the decomposition level of WT is three too. From Fig.4 we can see that the proposed method is better than WT method both on enhancement details and on contrast. The results of WT method still have certain fuzzy phenomena in edge details, and the contrast is not high enough, but they are effectively improved by the method proposed in this paper. In local amplification image, the weak crack results of the proposed method show greater contrast, and the enhancement effect is better than WT, the crack edge can be distinguished from background, the texture details of enhanced crack are more obvious, and contrast is greater too.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4. Experimental results comparison: (a) original image; (b) enhanced by WT;(c) enhanced by the proposed method; (d) enlargement of part original crack;(e) enlargement of WT result; (f) enlargement of proposed method result

3.2 Comparison of Different Fractional Differentiation Operator in the Smooth Subband Coefficients

Different order of fractional differentiation in smooth subband coefficients of NSCT domain is compared in this section, in the same time the high frequency subband coefficients remain unchanged in all directions. Then the effect of image enhancement can be observed as the fractional order increasing. Figure 5 is the enhanced image for different order. It can be found that when the order increases, the details of crack edge

340

C. Guo and T. Wang

are strengthened gradually. When the order is more than 0.5, the noise of background image is significantly strengthened. Through theoretical analysis and simulation experiments, the reasonable value of fractional differential order should be between 0.1-0.3. If the order is too small, then the smooth part of image is not enhanced almost, on the contrary, the noise is noticeable.

(a)

(b)

(c)

Fig. 5. Different order of fractional differentiation for smoothing

4 Conclusions A new enhancement algorithm for crack image of track beam surface is proposed in this paper based on the nonsubsampled contourlet transform. In NSCT domain, according to the characteristics that the medium-high frequency component of signal can be enhanced and very low frequency component can be retained non-linearly utilizing fractional differentiation, the fractional differentiation is used in smooth subband to strengthen crack images. At the same time, the high frequency coefficients of direction sub-band can be classified into strong edge, weak edge and noise according to the directional sensitivity of NSCT domain, then the treatment of enhancing weak edge, retaining strong edge and noise removal would be processed, which can avoid the defects of only one enhancement method. Experimental results show that the visual effects are greatly improved comparing with the traditional wavelet algorithm, and the enhancement effect is very good.

References 1. Jang, Y.X., Wang, X.T., Xu, X.G., et al.: A method for image enhancement based on light compensation. J. Acta electronica sinica 37(4A), 151–155 (2009) 2. Pichon, E., Niethammer, M., Sapiro, G.: Color histogram equalization through mesh deformation. J. IEEE Transaction on Image Processing 2(11), 117–120 (2003) 3. Yang, S.H., Lin, D.W.: A geometry-enhanced color histogram. In: International Conference Information Technology: Research and Education, pp. 563–567. IEEE, New Jersey (2003)

Crack Image Enhancement of Track Beam Surface Based on NSCT

341

4. Do, M.N., Vetterli, M.: The contourlet transform: An efficient directional multiresolution image representation. J. IEEE Transactions on Image Processing 14(12), 2091–2106 (2005) 5. Arthur, L., Cunha Zhou, J.P., Minh, N.D.: The nonsubsampled contourlet transform: Theory, design and application. J. IEEE Transactions on Image Processing 15(10), 3089–3101 (2006) 6. Sha, Y.H., Liu, F., Jiao, L.C.: SAR image enhancement based on nonsubsampled contourlet transform. J. Journal of Electronics & Information Technology 31(7), 1716–1721 (2009) 7. Pu, T., Ni, G.Q., Li, X.Y.: Image enhancement based on the ON-OFF model of visual neurons. J. Journal of Image and Graphics. Version A 8(5), 522–526 (2003) 8. Yang, Z.Z., Zhou, J.L., Yan, X.Y., et al.: Image enhancement of based on fractional differentials. J. Journal of Computer-Aided Design & Computer Graphics 20(3), 343–348 (2008) 9. Pu, Y.F., Wang, W.X.: Fractional differential masks of digital image and their numerical implementation algorithms. J. Acta Automatica Sinica 33(11), 1128–1135 (2007) 10. Pu, Y.F.: Application of fractional differential approach to digital image processing. J. Journal of Sichuan University (Engineering Science Edition) 39(3), 124–132 (2007) 11. Pu, Y.F.: Research on application of fractional calculus to latest signal analysis and processing. Sichuan University (2006) 12. Grace, C., Bin, Y., et al.: Spatially adaptive wavelet thresholding with context modeling for image denoising. J. IEEE Transactions on Image Processing 9(9), 1522–1531 (2000)

The Class-2 Linguistic Dynamic Trajectories of the Interval Type-2 Fuzzy Sets Liang Zhao College of Electrical Engineering, Henan University of Technology, Zhengzhou 450007, China [email protected]

Abstract. The subjects of the class-2 linguistic dynamic system (LDS) are a class of complex systems. We employ interval type-2 fuzzy rules base to describe their dynamic behaviors. This paper presents the basic procedure of the class-2 LDS based on the computing with words by using interval type-2 fuzzy sets (IT2 FSs). Several numerical examples verify that our proposing method is effective. Keywords: class-2 linguistic dynamic trajectories, class-2 computing with words, in-terval type-2 fuzzy set.

1 Introduction Fuzzy theory [1] is one of the greatest achievements in the information technology field in the last century, which has given a powerful thinking way for describing the imprecise and uncertain information. With the emergency of the fuzzy theory, it has attracted many scholars’ attentions. Under their efforts, the fuzzy theory has quickly developed and obtained successful applications in the social and industrial fields. However, with the applications of fuzzy theory in the much more fields, the drawbacks of the type-1 fuzzy theory have gradually been recognized and studied. The most obvious deficiency is the determination way of the membership grade in the type-1 fuzzy theory. The crisp membership grade is not suitable for describing the uncertain world. Thinking carefully the key issue in the type-1 fuzzy theory, Literature [2] presents the type-2 fuzzy set theory in the 1990’s. Since the emergency of the type-2 fuzzy theory, it has already developed quickly. In the type-2 fuzzy theory, the membership grade of the type-2 fuzzy set (T2 FS) is not crisp number but type-1 fuzzy set (T1 FS) which makes the T2 FS be more suitable for describing the uncertain information. Literatures [3]-[5] presents the basic concepts of the linguistic dynamic system (LDS) based on computing with words (CWWs). The basic calculation unit of the LDS is the word (fuzzy set) instead of the number or symbol of the conventional dynamic system (CDS) [5]. Literatures [6]-[9] have studied the linear LDS and nonlinear LDS when T1 FSs are used as the basic calculation units. However, T2 FSs have much advantage over their T1 counterparts. Literatures [10]-[11] have studied the calculation procedures of the linear and nonlinear LDS using T2 FSs, which K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 342–349, 2010. © Springer-Verlag Berlin Heidelberg 2010

The Class-2 Linguistic Dynamic Trajectories of the Interval Type-2 Fuzzy Sets

343

represents the new direction of the LDS theory. Currently, the LDS has been applied in the complex systems modeling, analysis and control. In this paper, the trajectories of the class-2 LDS are studied based on the IT2 fuzzy theory. Firstly, we introduce the basic concepts of IT2 fuzzy theory and LDS. Then we introduce the class-2 computing with words (CWWs) based on the IT2 fuzzy sets. Finally, the trajectories of class-2 LDS are studied in detail. A few numerical examples illustrate that our proposing method is effective and valuable. The whole paper is organized as follows. Section 2 describes the basic concepts of the T2 fuzzy theory and LDS. We study the class-2 CWWs based on the IT2 FSs in section 3. Section 4 presents the trajectories of the class-2 LDS. Section 5 draws a conclusion and puts forward to the future research direction.

2 The Basic Concepts of Type-2 Fuzzy Theory and LDS 2.1 Type-2 Fuzzy Set Theory T2 FS originally is introduced by Zadeh [1], which can be very useful when such systems are used in situations where lots of uncertainties are present. Mendel etc present the formal definition as follows. Definition 1[12]: A T2 FS, denoted A , is characterized by a T2 MF μ A ( x, u ) , where

x ∈ X and u ∈ μ A ( x, u ) ⊆ [0,1] , i.e.,

A = {( x, u ) , μ A ( x, u ) | x ∈ X , u ∈ J x ⊆ [0,1]}

(1)

Meanwhile, T2 FS A can also be illustrated as A=∫

∫

u∈J x ⊆ [0,1]

x∈ X

μ ( x, u )

( x, u )

(2)

Where ∫∫ denotes union over all admissible x and u. For discrete universes of discourse, ∑ replaces ∫ . Definition 2[12]: Where all μ

( x,u ) = 1 then

A FS). All IT2 FSs A are denoted as

( )

A is interval type-2 fuzzy set (IT2

Ω( A ).

Definition 3[12]: FOU A = {( x, u ) | ∀x ∈ X , u ∈ J x = [ 0,1]} is expressed as the

footprint of the uncertainty (FOU). Obviously, FOU( A ) completely describes the IT2 FS. In the rest of this paper we will only consider the FOU( A ). Definition 4 [12]: The upper membership function (UMF) and lower membership function (LMF) of A are two T1 MFs that bound the FOU. The UMF is associated with the upper bound of FOU( A ) and is denoted μ ( x ) , ∀x ∈ X , and the LMF is associated with the lower bound of FOU( A ) and is denoted μ ( x ) , ∀x ∈ X , i.e.,

344

L. Zhao

( )

μ ( x ) = FOU A

( )

μ ( x ) = FOU A

∀x ∈ X ∀x ∈ X

(3) (4)

Note that for an IT2 FS J x = ⎡⎣ μ ( x ) , μ ( x ) ⎤⎦ , ∀x ∈ X . In this paper, the IT2 FSs are used to replace the number or symbol for the basic unit in the class-2 LDS, which makes use of the IT2 fuzzy inference method as the modeling tool for the social system. 2.2 The Class-2 Linguistic Dynamic System

The linguistic dynamic system (LDS) is proposed in [3-5], which is an effective method for the modeling and analysis of the social system based on the CWWs. In the social system, such as economical, administration and legislation systems, human employs mostly words in computing and reasoning to arrive at conclusions expressed as words from linguistic premises [6]. Therefore, the LDS has been become an important research direction. In this paper, IT2 FSs are used to set up the model of the linguistic terms. By contrast, IT2 FSs have superior to T1 FSs in handling uncertain information, which widespread exists in the social systems. An LDS is called a class1 LDS if its states are linguistic and its evolving laws are constructed based on conventional functions by using the fuzzy extension principle [6]. An LDS is called a class-2 LDS if its states are linguistic vector and its evolving laws are modeled by linguistic rules [6]. In this paper, we discuss the class-2 LDS based on class-2 CWWs, which employs the IT2 FSs as the basic calculation unit. In the class-2 LDS, IT2 fuzzy inference method is the calculation principle of the evolution process. By using it, we can obtain the trajectories of class-2 LDS.

3 The Class-2 Computing with Words Based on IT2 FS For the class-2 LDS, its key issue is the class-2 CWWs based on IT2 fuzzy inference system. If the mathematical model of the social system can not be set up, we may employ the expert knowledge and (or) experience to build their IT2 fuzzy rule models. Then based on the IT2 fuzzy inference method, we can calculate the output linguistic state vector (LSV) from the input linguistic state vectors (LSV). The whole calculation process denotes the class-2 CWWs based on IT2 FSs shown in figure 1. X ( k ) = ( X 1 ( k ) , X 2 ( k ) , , X n ( k ) ) represents the linguistic state vector (LSV) at the k time. Similarly to X ( k ) , X ( k + 1) represents the LSV at the k+1 time. The fuzzy rule base for the class-2 CWWs is illustrated as follows.

Ri: If X 1 ( k ) is A1i and X 2 ( k ) is A2i … X n ( k ) is Ani , then X 1 ( k + 1) is G1i and X 2 ( k + 1) is G2i … X n ( k + 1) is Gni .

The Class-2 Linguistic Dynamic Trajectories of the Interval Type-2 Fuzzy Sets

345

Fig. 1. The scheme diagram of the class-2 CWWs

From the above multi-input multi-output (MIMO) fuzzy rule base, we can obtain equivalent n multi-input single-output (MISO) fuzzy rule bases shown as follows.

Ri1: If X 1 ( k ) is A11 and X 2 ( k ) is A21 and…and X n ( k ) is An1 , then X 1 ( k + 1) is G1i . Ri2: If X 1 ( k ) is A12 and X 2 ( k ) is A22 and…and X n ( k ) is An2 , then X 1 ( k + 1) is G2i . Rin: If X 1 ( k ) is A1n and X 2 ( k ) is A2n and…and X n ( k ) is Ann , then X 1 ( k + 1) is Gni . From the above fuzzy bases, we can draw a conclusion that when the size of the LSV is enormous the inference process of class-2 CWWs is very complicated. Therefore, we can compute the n MISO interval type-2 fuzzy rule bases to perform class-2 CWWs. The key issue of the class-2 CWWs is the calculation of the MISO IT2 fuzzy inference process. Literature [13] presents the inference method of IT2 fuzzy system. We illustrate the process from the viewpoint of the class-2 CWWs. According to the MISO fuzzy rule base, we can get the calculation procedure of the output LSV X ( k + 1) from the input LSV X ( k ) . By using the IT2 fuzzy inference method [13], we can obtain the following formula

μB

i j

=μ

( k +1)

G ij

( x ) = ∪ ⎡⎣ μ ( ) ( x ) ∩ μ ( x, x )⎤⎦ ' j

X k

x∈ X

⎧⎡

( x ) ∩ ⎪⎨⎢ ∪ μ ' j

⎩⎪ ⎣ x1∈ X1

X1 ( k )

⎣ xn ∈X n

μX Where μ X

j

(k )

⎤

⎡

⎤

⎦

⎣ x2 ∈X 2

( x1 ) ∩ μ A ( x1 ) ⎥ ∩ ⎢ ∪ μ X (k ) ( x2 ) ∩ μ A ( x2 )⎥

⎡

∩⎢ ∪

' j

Ri

i 1

μX

( k +1)

(k )

( xn ) ∩ μ A ( xn ) ⎥ ⎪⎬ i n

⎦ ⎭⎪

(x ) = ∪μ (x )

( x ) = ⎡⎢⎣ μ X

' j

j

(k )

(5)

⎦

⎤⎫

n

M

j

i 2

2

B ij

i =1

( x), μ X

j

(k )

j = 1, 2,

' j

( x ) ⎤⎥⎦ and μ X

j

( k +1)

( x ) = ⎡⎢⎣ μ X

j

,n

( k +1)

( x), μ X

(6)

j

( k +1)

( x ) ⎤⎥⎦ .

By using the formulas, we can obtain the calculation procedure of CWWs based on IT2 FSs shown as follows.

346

L. Zhao

Step 1: Determine the IT2 fuzzy rule base. By using experts’ knowledge and (or) experience, we can decide the fuzzy rule base of the class-2 CWWs and basic linguistic terms sets of the LSVs. Then we can determinate the input LSVs. Step 2: Calculate the output LSVs from the input LSVs. By using the above formula, we can calculate the output LSVs . Several numerical examples [7-8] are illustrated for explaining the calculating procedure of the calss-2 CWWs based on IT2 FSs.

Example 1[7-8]: Under the standard atmospheric pressure, We heat the cold water, warm water and hot water, respectively. And at each t minutes we observe the evolution process in the water temperature state, which is summarized as the following three fuzzy rules. Rule 1: If the cold water is heated for about t minutes, then it will become warm water. Rule 2: If the warm water is heated for about t minutes, then it will become hot water. Rule 3: If the hot water is heated for about t minutes, then it will become vapor. For arbitrary input LSV (IT2 FSs), we can calculate its output LSV by using the IT2 inference method according to the fuzzy rules 1-3. The basic linguistic terms are illustrated as follows. 15 1 35 −0.04 x +1.6 35 −0.04 x +1.6 40 −0.04 x +1.6 1 1 1 1 Acoldwater = ∫ ∫ +∫ ∫ +∫ ∫ +∫ ∫ 0 1 x, u 15 −0.05 x +1.75 x, u 15 −0.05 x +1.75 x, u 35 0 ( ) ( ) ( ) ( x, u ) +∫

105

45

1

0

∫ ( x, u ) 0

25

Awarmwater = ∫

0

+∫

75

70

∫

−0.04 x + 3

0

Ahotwater = ∫ 105

100

∫

55

65

1

0

0

−0.04 x + 4.2

0

30

0.04 x −1

25

0

60

0.04 x − 2.2

55

0

∫ ( x, u ) + ∫ ∫

0

Avapor = ∫

0

50 0.04 x −1 70 −0.04 x + 3 1 1 1 +∫ ∫ +∫ ∫ 30 0.05 − 1.5 50 − 0.05 + 3.5 x x ( x, u ) ( x, u ) ( x, u )

105 0 1 1 +∫ ∫ 75 0 x, u x u , ( ) ( )

0

+∫

1

0

∫ ( x, u ) + ∫ ∫

0

1 ( x, u ) 1

70

0.04 x − 2.6

65

0

∫ ( x, u ) + ∫ ∫ 0

80 0.04 x − 2.2 100 −0.04 x + 4.2 1 1 1 +∫ ∫ +∫ ∫ ( x, u ) 60 0.05 x −3 ( x, u ) 80 −0.05 x + 5 ( x, u )

90 0.04 x − 2.6 105 1 1 1 1 +∫ ∫ +∫ ∫ 70 0.05 x − 3.5 x, u 90 1 x, u x u , ( ) ( ) ( )

When the input LSV X ( k ) is shown as

The Class-2 Linguistic Dynamic Trajectories of the Interval Type-2 Fuzzy Sets

X (k ) = ∫

5

0

+∫

45

40

∫

0

0

−0.05 x + 2.25

0

1

10

0.05 x − 0.25

5

0

∫ ( x, u ) + ∫ ∫

347

25 0.05 x − 0.25 40 −0.05 x + 2.25 1 1 1 +∫ ∫ +∫ ∫ 10 1/15 x − 2 / 3 25 − 1/15 x + 8 / 3 x u x u x ( , ) ( , ) ( ,u)

105 0 1 1 +∫ ∫ 45 0 x, u ( x, u ) ( )

Calculate the output LSV X ( k + 1) .

(a) The basic linguistic terms set

(b) The output LSV

Fig. 2. The result of the class-2 CWWs

In Fig. 2, figure (a) illustrates the basic linguistic terms set and initial input LSV X ( k ) and figure (b) shows the output LSV X ( k + 1) .

4 The Trajectories of Class-2 LDS Based on IT2 FSs The trajectories of class-2 LDS is the evolution law of LSV based on the class-2 CWWs when the time k elapses from 1 to n, which can be shown as

{ X ( 0 ) , X (1) , X ( 2 ) ,

}

, X ( n ) . The whole calculation procedure is illustrated as

follows. Step 1: Model the class-2 LDS based on fuzzy rule base. Step 2: Set time k = 0. The initial LSV X ( 0 ) is determined in advance. When time k adds 1, by using the class-2 CWWs we compute the LSV X ( k + 1) . Step 3: Judge k < n. If the inequality is yes, return step 2. If not, return step 4. Step 4: Get the trajectories of class-2

{

LDS X ( 0 ) , X (1) , X ( 2 ) ,

}

, X (n) .

Example 2[7-8]: Following the example 1, calculate the trajectories of class-2 LDS when the input LSV is X ( 0 ) shown in figure 3(a).

348

L. Zhao

(a) Initial LSV X ( 0 )

(b) The trajectories of class-2 LDS

Fig. 3. The result of the class-2 LDS

Example 2 illustrates that the calculation procedure of class-2 LDS is effective and valuable.

5 Conclusions In this paper, we discuss the calculation procedure of class-2 LDS based on IT2 FSs. Firstly, we discuss the basic concepts of T2 fuzzy theory and LDS. Secondly, we study the calculation procedure of class-2 CWWs based on IT2 FSs. Thirdly, we discuss the numerical procedure of class-2 LDS by using CWWs. Numerical examples illustrate that the calculation procedure in this paper is effective. The trajectories of the class-1 LDS has been studied and will be presented in the future paper. Acknowledgments. This work is supported by High-Level Talents Research Foundation of Henan University of Technology (No.2009BS060).

References 1. Zadeh, L.A.: Fuzzy Set: Information and Control 8, 338–353 (1965) 2. Karnik, N.N., Mendel, J.M., Liang, Q.: Type-2 Fuzzy Logic Systems. IEEE Trans. on Fuzzy Systems 7, 643–658 (1999) 3. Wang, F.Y.: Fundamental Issues in Research of Computing with Words and Linguistic Dynamic Systems. ACTA Automatica Sinica 31, 844–852 (2005) 4. Wang, F.Y.: Computing with Words and a Framework for Computational Linguistic Dynamic Systems. Pattern Recognition And Artificial Intelligence 14, 377–384 (2001) 5. Wang, F.Y., Lin, Y.T., James, B.P.: Linguistic Dynamic Systems and Computing with Words for Complex Systems. In: IEEE International Conference on Systems, Man, Cybernetics, pp. 2399–2404 (2000) 6. Wang, F.Y.: On the Abstraction of Conventional Dynamic Systems: from Numerical Analysis to Linguistic Analysis. Information Sciences 171, 233–259 (2005) 7. Mo, H.: Linguistic Dynamic Systems Based on Computing with Words and Their Relative Analysis. Ph.D Dissertation, Institute of Automation Chinese Academy of Sciences (2004)

The Class-2 Linguistic Dynamic Trajectories of the Interval Type-2 Fuzzy Sets

349

8. Mo, H., Wang, F.Y.: Linguistic Dynamic Systems Based on Computing with Words and Their Stabilities. Science in China Series F: Information Sciences 52, 780–796 (2009) 9. Mo, H., Wang, F.Y., Yang, T.: The Linguistic Dynamic Property of a Type of Bifurcate Mapping. Pattern Recognition And Artificial Intelligence 16, 263–270 (2003) 10. Mo, H., Wang, F.Y., Zhao, L.: On LDS trajectories under one-to-one mappings in interval type-2 fuzzy sets. Pattern Recognition And Artificial Intelligence 23, 144–147 (2010) 11. Zhao, L.: First-Order Linear Linguistic Control System and Its Stability. In: 29th Chinese Control Conference (accepted) 12. Mendel, J.M., John, R.I., Liu, F.L.: Interval Type-2 Fuzzy Logic Systems Made Simple. IEEE Trans. Fuzzy Systems 14, 808–821 (2006) 13. Liang, Q.L., Mendel, J.M.: Interval Type-2 Fuzzy Logic Systems: Theory and Design. IEEE Trans. Fuzzy Systems 8, 535–550 (2000)

Strategic Evaluation of Research and Development into Embedded Energy Storage in Wind Power Generation T.C. Yang1 and Lixiong Li2,* 1

School of Engineering and Design, University of Sussex, Brighton, BN1 9QT, UK 2 School of Mechanical Engineering and Automation, Shanghai University, Shanghai Key Laboratory of Power Station Automation Technology, Shanghai, 200072, P.R. China [email protected], [email protected]

Abstract. Embedded Energy Storage (EES) is an innovative idea presented in a previous paper. EES is associated with some major configurations of wind power generation and rechargeable batteries. Areas for further research are identified, but before resources are committed it is necessary to assess the likely benefits of ESS. In this paper, the technical and economic potential of the EES are evaluated from the viewpoint of (1) historical and future development of electrical power systems, and (2) globalisation of CO2 emission reduction and global wind energy utilisation. The evaluation uses four case studies relevant to the United Kingdom and around the world. Based on this work, a practical and low-risk methodology for research and development into this new area is proposed. Keywords: Embedded energy storage, Power system, Wind power.

1 Introduction Using variable-pitch variable-speed wind turbines has many advantages in wind energy conversion systems. These wind turbines are dominant in today’s wind power generation and their market share is increasing. There are two major electrical configurations of variable-pitch variable-speed wind turbines, both of which have a power electronic AC/DC and DC/AC frequency converter. (C-1) Configuration one (Figure 1) has a partial-scale frequency converter and a wound rotor double-fed induction generator. This is termed a DFIG scheme in [1] and other literature. The main European manufacturers for (C-1) include: Vestas, Nordex, GE (1.5 series and 3.6), Repower, Gamesa, Ecotecnia, Ingetur and Suzlon. Remark 1: The main European manufacturers for (C-1) include: Vestas, Nordex, GE (1.5 series and 3.6), Repower, Gamesa, Ecotecnia, Ingetur and Suzlon. In addition to academic literature on (C-1), for example [2], these manufacturers’ web sites also provide some useful technical information. *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 350–360, 2010. © Springer-Verlag Berlin Heidelberg 2010

Strategic Evaluation of Research and Development into EES in Wind Power Generation

351

Fig. 1. Variable-pitch variable-speed wind turbine, Configuration 1

(C-2) Configuration two (Figure 2) has a full-scale frequency converter. The generator used can be a permanent magnet synchronous generator, a wound rotor synchronous generator, or a wound rotor induction generator. Remark 2: The main European manufacturers for (C-2) include: Enercon, MEG, GE (2.x series), Zephypos, Winwind, Siemens (2.3 MW), Made, Leitner, Mtorres, and Jeumont. In both (C-1) and (C-2), there is a DC bus in the wind energy conversion system. A rechargeable battery bank can be connected to this bus as an energy storage device (not explicitly shown in Figure 1 and 2). Since the energy storage is embedded in a wind power conversion system, we call this Embedded Energy Storage (EES). More specifically, this is an embedded energy storage associated with wind power generation and batteries.

Fig. 2. Variable-pitch variable-speed wind turbine, Configuration 2

In [1], an initial study is carried out to evaluate a specific scheme of EES under (C1) configuration, addressing some technical, practical and economic issues. The study is based on the statistics of the wind speeds measured at Dunstaffnage, Scotland and the manufacturer’s data for a 2 MW turbine generator (Vestas V66). It shows many positive aspects of the proposed scheme and the overall conclusion is that the scheme studied appears attractive and worth further exploration [1]. Therefore, further research below is planned:

352

T.C. Yang and L. Li

(1) More detailed dynamic studies − a major limitation of the simulation study presented in [1] is that only steady-state models of batteries, generators, converters, transformers, and the power system are used. (2) Long-term energy management − only a short 30-minute period is studied in [1] − based on measured wind speed data (mean value and standard deviation for every 10-minute over a year). (3) Experimental studies − only simulation studies are presented in [1]. (4) Research into EES under the configuration two − only configuration one is considered in [1]. Before any detailed technical studies are initiated, it is considered necessary (1) to justify the potential of the EES since the overall cost/benefit analysis presented in [1] is not sufficiently convincing, and (2) to develop a suitable research and development (R+D) methodology − including importantly what type of battery is to be used − for this new technical area. These are therefore the two aims of this paper. The rest of this paper is organised as follows. In Section 2, we evaluate the EES from the historical and strategic view. In Section 3, we evaluate the EES from the global and power system future development viewpoint. In the mean time, subject to the completion of all necessary technical studies and validations, in Sections 2 and 3 we also propose some EES applications in a form of case studies, relevant to the United Kingdom and around the world. In section 4, we make two proposals in the EES R+D methodology aimed at low risk and seamless connections between research, development and application. Finally, some conclusions are drawn in Section 5. There are different types of rechargeable batteries with different characteristics and costs. In this paper, lead-acid batteries are considered for the EES. The reasons for this decision are explained in Section 4.

2 EES: Historical and Strategic Viewpoint, Economic Assessment of Two Case Studies 2.1 A Brief Overview of Energy Storage for Power Systems Electrical power systems in developed countries − the UK system can be considered as a typical example − can be considered technically mature before industrial-scale renewable power generation and embedded power generation took place recently. In line with ever increasing demand on electricity in the past, the fundamental technical model for investment and development was − and to large extent still is − “Generation-Transmission-Distribution- Load” (GTDL). Under this model, there is always considerable interest in energy storage for electrical power systems. For technical and economic reasons, bulk energy storage is an integrated part of many mature systems for their day-to-day operations. The Dinorwig pumped storage hydrostation is a well-know example in the UK. Around the world, between 1988 and 1995, there were five large lead-acid battery storage plants completed. In the past decade, we see increased penetration of wind power generation. Associated with electrical power generated from wind energy, there is also a great interest in exploring the application of various energy storage technologies. However,

Strategic Evaluation of Research and Development into EES in Wind Power Generation

353

all studies so far in this area are limited to an energy storage unit externally connected to a network substation. To differentiate it from the EES, we call this an Externally Connected Storage (ECS). In [1], there is a list of references (Ref 1-36 of [1]) to some recent publications on ECS research, development and applications. 2.2 Future Potential: EES for Grid-Network-Connected Wind Power Generation For its technological advantages, a DFIG scheme (Figure 1) is now commonly used for Grid-network-Connected Wind Power Generation (GCWPG) [2]. To evaluate the EES for GCWPG from a strategic point of view, we discuss three questions: (Q1) Can we envisage in the near future that there is a need for an investment in some considerable amount of capacity in energy storage for the UK power system, similar to what we did before for the Dinorwig pumped storage hydro-station? It is generally accepted that when renewable power penetration in the UK reaches near 20%, there is a requirement for some major investments in power system infrastructures − including some possible energy storage plants − to enable a further increase of renewable power generation. Recent research on “Value of bulk energy storage for managing wind power fluctuations” [3] is specifically related to this issue for the UK power system. Among the all arguments made, it points out that: (1) Due to fluctuations in wind power, if there is no energy storage to smooth it, there will be an increased requirement for the reserve of generating capacity from traditional power plants. In particular, this is to prepare for the time when the wind is low and the load is high. For a synchronized plant to provide reserve, it must run partloaded. Thermal units operate less efficiently when part-loaded, losing between 10%– 20% efficiency, although efficiency losses could be even higher, particularly, for a new gas plant. (2) An important characteristic of storage is its ability to store generation excesses during high wind and low demand, then subsequently, resupply some of this energy, and hence, reduce fuel cost. Storage can provide upward (“positive”) and downward (“negative”) reserve (while a traditional power plant, for example an open-cycle gas turbines plant, can provide only upward regulation). When generation is lower than demand, storage is discharged; when demand is lower than generation, storage is charged up. The ability of storage to provide “negative” reserve is of critical importance when low-demand conditions coincide with a high wind generation output, despite the fact that some energy will be lost through charging/discharging and storage inefficiencies. (Q2) What are the advantages of using the proposed EES for the required energy storage? (1) Battery energy storage is embedded in wind power generation, and no additional AC/DC and DC/AC converters are required, which otherwise are needed under an externally connected storage (ECS) scheme. (2) As shown in [1], for (C-1) configuration the EES is not only electrically embedded in the generation, but also can be physically embedded − in the sense that the batteries used can be placed in the hollow space inside a wind turbine tower. Therefore, in a wind power development with (C-1) configuration and an EES, there

354

T.C. Yang and L. Li

is no additional requirement for buildings, land and planning permission for the energy storage. These, however, again are needed under an ECS scheme. (3) Since an EES is embedded in wind power generation, it can modulate the output of a wind energy conversion system and therefore improve the quality of the output and reduce the costs associated with meeting the technical requirements for network connections. (4) EESs are distributed energy storages offering better reliability and security than a centralized bulk energy storage unit for an integrated power system. The systemwise control and management of distributed EESs, however, does require communication network support but this is not a problem in developed countries. Having considered the above, then: (Q3) While the 20% penetration is some years away, do we need to undertake research and development in EES now? The answer is yes. R+D and industrial adoption will take a few years. We do need to start now to gradually increase the EES capacity in the system in parallel with the increase of renewable power generations, in order to make a smooth increase of the penetration ratio up to and beyond 20%. 2.3 Future Potential: EES for Distribution-Network-Connected Wind Power Generation Distribution-network-Connected Wind Power Generation (DCWPG) is one kind of embedded power generation in the sense that power generation is embedded in distribution networks. Comparing DCWPG with GCWPG (Grid-network-Connected Wind Power Generations), power ratings of DCWPGs are lower. DCWPGs are mostly on-shore and wind speeds are normally lower. Therefore, configuration two (Figure 2) is more suitable. In the rest of this paper, we will use ENERCON’s E-33 wind turbines [4] in our hypothetical case studies. Some of the main technical data for the E-33 are:

• • • • •

Rated power: Rotor diameter: Hub height: Cut-in wind speed: Cut-out wind speed:

330 kW (maximum power 335 kW) 33.4 m 44 – 50 m 3 m/s 28 – 34 m/s

Compared with the Vestas V66 2 MW wind turbine used in a previous case study in [1], the E-33 has a wider wind-speed operational range. (For the Vestas V66, the cutin speed is 4 m/s and the cut-out speed is 25 m/s). For the all case studies in this paper, we assume a capacity factor of 30%. (The capacity factor depends on the turbine wind-speed operational range, the turbine speed-power curve, and the wind speed statistics of the site). Since an E-33 wind turbine has a full-scale frequency converter, the maximum rates of charging and discharging of the EES can be the same as the turbine rated power. Distribution systems are designed to accept bulk power at the bulk supply transformers and to distribute it to customers. Thus the flow of both real power and reactive power is always from the higher to the lower voltage levels (See all solid

Strategic Evaluation of Research and Development into EES in Wind Power Generation

355

lines in Figure 3). With a significant penetration of embedded generation (when the dashed-lined part of Figure 3 is added to the network), the power flow may become reversed and the distribution network is no longer a passive circuit supplying loads; it is an active system with power flows and voltages determined by the generation as well as the loads. This fundamental change has led to a number of technical challenges, including [5]:

• • • • • • •

Excessive fault levels in distribution networks Generator instability Inadequate reverse power capability of tap changers Interference with line drop compensation voltage control Customer exposure to steady state voltage rise Possible increased losses Disruption of fault protection equipment

Among them, the most noticeable is the voltage rise [6]. If we can have embedded wind power generation and at the same time keep one-way power flow unchanged in distribution networks, all the above problems no longer exist or can be easily solved. This can be achieved by the application of EES. The case study below is to illustrate this.

Fig. 3. Power flow in a distribution network

Case Study A. We assume that (1) there is a load centre and its 24-hour average load, also the long term average load, is 400kW, (2) the wind energy resource near this load centre is adequate for wind power generation, (3) planning permission is granted to install six E-33 wind turbines (their combined rating is 2000 kW) near the load centre, and (4) the wind turbines also have a combined EES capacity of 48 MWh (= average load 400kW × 120hours). The wind power plant is to be connected directly to the load centre to form an integrated unit GL, see Figure 4. Since the capacity factor is 30%, the average wind power generation is 600kW, 200kW more than the average load. This accounts for energy loss in battery charging/discharging and allows for some operational margin. On a broad average, GL is a self-balanced unit in power. However, although the EES can provide a reserve

356

T.C. Yang and L. Li

for an average use of five days, due to fluctuations of available wind power, it is still expected to receive a small amount of power from the distribution network as a secondary supply. On the other hand, possible surplus power − when the batteries are fully charged, the load is low and the wind is high − can be prevented by pitch angle control to limit the power received. In any event, feeding a small amount of power to the distribution network will not cause the problems listed before. Reasonably accurate short-term and long-term load and wind speed forecasts plus a good energy management control can help to achieve the goal of power self-balance [7].

Fig. 4. One-way power flow in a network having embedded generation and EES

Power balancing also means reactive power balance. While real power can be transferred over a distance at low cost, reactive power cannot be transferred easily and cheaply. Reactive power should be balanced locally. Due to the PWM (pulse-widthmodulation) control of the DC/AC part of the converter in Figure 2, the magnitude and phase angle of the currents leaving the converter can be controlled independently. Therefore, AC active/reactive powers supplied from this source to the load can be very flexible, and the reactive power demand can be met easily. Without a (C-2) type of wind power supply, an investment in some devices, for example an SVC (static Var compensator), may be necessary to meet the requirement of voltage and reactive power control. Remark 3: The issue of power quality, in particular harmonics, will be discussed further in Section 3. A wind energy conversion system normally has some filtering devices to reduce/eliminate harmful harmonics (not shown in Figure 1 and 2). For an average ten-day period, the energy consumption is 96 MWh (400kW × 240 hours). Assuming that the saving on the electrical bill is based on 80 MWh (16 MWh, about 17% of the total consumption, is still supplied from the distribution network), and that the cost of electricity is 9.5 p/kWh, the total saving in the electricity bill is £7.6k (0.095 × 80k). For a year of 365 days, the saving is £277.4k. For a five year period, the saving is £1387k. It is assumed that the lifetime of lead-acid batteries used for the EES is five years (sealed lead-acid batteries in normal ambient temperature last 5-10 years; manufacturers only give a 3 year warranty to car starting batteries due to the higher temperature in the engine compartment). It is shown in [1] that the shop price of car batteries is £50 for 0.65 kWh. Assuming that the per-kWh price for mass industrial

Strategic Evaluation of Research and Development into EES in Wind Power Generation

357

supply is one third of that, then the total cost for a 48 MWh battery bank is £1230.8k (48 × 50/0.65/3). Therefore the net operational saving over a five year period is £1387k − £1230.8k = £156.2k. For a 25-year financial planning period, the final balance sheet may look like this in Table 1. Table 1. Final balance sheet Financial Gain (g-1) operational saving over 25 years: £781k(= £156.2k × 5)

Cost (c-1) cost associated with the planning permission for the plant

(g-2) environmental benefit: an unquantified figure. (If the “pollution cost”, or an environment levy, is 1.5p per kWh, the saving over 25 years is £1095k (0.015 × 80k × 36.5 × 25.)

(c-2) the wind power plant set-up cost. (When EES is applied, ratings and costs of the equipment required for the plant-network connection can be reduced as demonstrated in [1] by a case study.)

(g-3) residual value of the wind power plant after 25 years

(c-3) the plant maintenance cost over 25 years (except for the labour, the cost of batteries changed does not need to be included)

Furthermore, the following two possibilities can be considered to improve the balance sheet: (P-1) When the wind speed is low, it is more likely that it is sunny in the daytime. If appropriate, wind power generation with EES can be combined with solar power generation, or any other forms of renewable power generation. (P-2) The EES is divided into two groups. Group A, for example has 40% of the total storage capacity, is used for frequent charging/discharging. Group B has 60% of the total storage capacity and is used for long-term reserve. Different designs of leadacid batteries can be accommodated for the different uses of the EES to reduce the overall cost and to improve the overall operational efficiency. Group A batteries should be energy-efficient in charging/discharging, and Group B batteries should be energy-efficient in long term storage. Now we consider a new case that is financially more attractive. Case Study B. We assume that a small town is to be built up to relieve housing shortage in the UK and the average consumption of power for the town is 400kW, the same as in Case Study A. The main power supply to the town will be from a wind power plant, also the same as that in Case Study A. The major difference in this case is financial: (1) A much lower-rated substation is required, with a corresponding reduction in other equipment. Consider a ten-day no-wind period; the consumption can be met by the EES’s five-day energy storage plus the supply from the network over that period, at a (constant) rate of 50% of the average load. Therefore, the required rating of the new substation is only for example 30% of the one that needs to meet the short-term high demand without the EES. The saving in this respect can be added into the lefthand side of the balance sheet above.

358

T.C. Yang and L. Li

(2) The planned wind power plant can be located near to an office block, a supermarket, or an industrial estate attached to the town but some distance away from the residential area. Since (1) such a development is very likely to use “brown field” land, and (2) the plant planning is only part of the town planning, the costly and timeconsuming process of obtaining wind power plant planning permission on its own in Case Study A may be avoided. This will remove, or greatly reduce, the value of (c-1) in the right-hand side of the balance sheet. Case Study A is less attractive not only because of the cost and the time-delay associated with the planning permission, but also because the existing distribution facilities are greatly under-used.

3 EES: From the Viewpoint of Globalisation [8] and Future Development of Power Systems 3.1 Two Case Studies for under-Developed Regions Around the World It is well-known that (1) CO2 emission reduction and environmental protection need a global effort, and (2) Developed countries get a financial reward from exporting technologies that are mainly applied in developing countries. China is now the world’s second-largest consumer of electricity after the U.S. and the demand for electricity is increasing at a fast pace. Within China, there are developed regions and under-developed regions. In developed regions of China, the transmission and distribution networks are still “weak” [9]. The large-scale integration of wind power generation into the existing system is more challenging in China than in developed countries. Therefore, the EES will play a more important role in its ability to modulate the output of a wind energy conversion system. In this subsection, we are not intending to explore it further. Instead, we use the Inner Mongolia Autonomous Province in China as an example to discuss the opportunities of wind power with EES for some under-developed regions around the world. Inner Mongolia is an inland province in China where most land is flat. Inner Mongolia is also one of the most under-developed provinces in China. However, (1) in recent years its annual GDP has increased by 18%, well above China’s average of around 10%, (2) it has rich mineral resources, and (3) it also has plentiful resources of wind energy [9]:

• • • •

At a height of 10 metres, the usable wind power in the province is about 1.01 × 108 kW. At a height of 50 metres, the usable wind power in the province is about 2.02 × 108 kW. Both of the above are about 40% of the total in China. For wind energy reserve, Inner Mongolia is therefore the No 1 province in China. The quality of the wind energy is very good; the statistical deviation of wind speeds is low. For the sites surveyed, the continuous period when the wind speed is less than 4 m/s is less than 100 hours.

Case Study C. Assuming that a remote town and the surrounding areas need an electricity supply of average 400 kW, this and the wind power plant to be built are the

Strategic Evaluation of Research and Development into EES in Wind Power Generation

359

same as those in Case Study B. Since the investment required to connect it to any existing electrical network is judged too expensive, it must be supplied by an autonomous power system (stand-alone system). Therefore, the issue for this case is how to enhance the reliability of the supply in addition to the existing capacity of “120-hour” storage. For this, the following two options can be considered: (1) similar to (P-1) in Case Study B, to consider solar power or any other forms of renewable power generation to supplement the wind power generation; and (2) to have a higher capacity of EES, similar to (P-2), this EES can be divided into two groups. Case Study D. This is the same case as Case Study C except that the main load is an industrial electrolytic aluminium plant. For such a load: (1) A considerable amount of power consumption is in the form of DC. The incoming substation/switching board of the plant can therefore have two main buses: AC and DC, each connected to the relevant bus of the wind power generation with EES. This can eliminate the loss in the AC to DC conversion required by the plant if the incoming substation only has an AC bus, and the loss in the DC to AC conversion inside the wind power energy conversion system (Figure 2). (2) For high efficiency, it is better for this kind of plant to run continuously (this also reduces the asset invisible depreciation). This leads to a more flatted load and makes the overall energy management easier. (3) This kind of load can tolerate a lower standard of power quality. From this case, one can see that some problems caused by the electricity supplied from wind power in developed countries are relatively easy to solve if wind power generation is combined with the EES and used for some industrial applications. Now consider the power quality in general. The power quality in terms of frequency deviation and voltage profile should not be a concern for wind power generation with EES, due to its fast response and flexible control of PWM DC/AC converters, so long as the energy reserve in the EES is sufficient. The main issue, however, is the harmonics caused by using power electronics. Using Case Study B as an example (Figure 4), harmonic reduction/elimination can be managed in such a way that: (a) The GL as a unit should not be a harmonic-polluting source for the network to which it is connected; and (b) Within the GL, based on the different types of loads, adequate filtering is applied to meet the required quality standard.

4 Proposals for the R+D of EES In general, R+D will involve computer simulations, industrial pilot schemes, etc. In this section, we make two specific proposals for the R+D of the proposed embedded energy storage. For energy storage applications, apart from the cost, other technical issues include:

• • • • •

Charging/discharging efficiency and storage efficiency Maximum charging/discharging rates Power density per unit size and per unit weight Energy density per unit size and per unit weight Reliability

360

T.C. Yang and L. Li

For any new technology, in addition to extensive computer simulations, studies on a scaled-downed version of a physical model are always needed. In addition to a smallscale laboratory study, a prototype of working wind power generation system with EES is the next step to follow the laboratory study.

5 Conclusions In this paper, a new idea of energy storage: Embedded Energy Storage (EES) is presented. The EES is evaluated from the historical and strategic points of view, together with some potential applications. It is believed that the EES has attractive potential. Two specific proposals for research and development into EES are also presented. Overall, the work presented here is a justification for planned future work on more detailed dynamic studies, long-term energy management, experimental studies and wide applications. Acknowledgement. This work is supported by Key Project of Science & Technology Commission of Shanghai Municipality under Grant 08160512100 and 08160705900. The author would like to thank the Chinese Government for its financial support for author’s trips to Inner Mongolia, China.

References 1. Yang, T.C.: On Embedded Energy Storage for High Penetration of Wind Power. Wind Enginpeering 32, 223–242 (2008) 2. Xu, L., CartwrightDirect, P.: Active and Reactive Power Control of DFIG for Wind Energy Generation. IEEE Transactions on Energy Conversion 21, 750–758 (2006) 3. Black, M., Strbac, G.: Value of Bulk Energy Storage for Managing Wind Power Fluctuations. IEEE Transactions on Energy Conversion 22, 197–205 (2007) 4. http://www.enercon.de/en/_home.htm 5. Kennedy, M.: Distributed Generation: Harder Than it Looks. Power Engineering, 16–19 (2003) 6. Masters, C.L.: Voltage Rise the Big Issue When Connecting Embedded Generation to Long 11 kV Overhead Lines. Power Engineering Journal, 5–12 (2002) 7. Haesen, E., Driesen, J., Belmans, R.: Robust planning methodology for integration of stochastic generators in distribution grids. In: IET Proceedings: Renewable Power Generations, vol. 1, pp. 25–32 (2007) 8. Gertmar, L., Liljestrand, L., Lendenmann, H.: Wind Energy Powers-That-Be Successor Generation in Globalization. IEEE Transactions on Energy Conversion 22, 13–28 (2007) 9. Web site of the Chinese Wind Energy Association, http://www.wffd.cn/ 10. Pudjianto, D., Ramsay, C., Strbac, G.: Virtual power plant and system integration of distributed energy resources. In: IET Proceedings: Renewable Power Generations, vol. 1, pp. 10–16 (2007) 11. Caralis, G., Zervos, A.: Analysis of the combined use of wind and pumped storage systems in autonomous Greek islands. In: IET Proceedings: Renewable Power Generations, vol. 1, pp. 49–60 (2007)

A Mixed-Integer Linear Optimization Model for Local Energy System Planning Based on Simplex and Branchand-Bound Algorithms Hongbo Ren1, Weisheng Zhou2, Weijun Gao3, and Qiong Wu3 1

Ritsumeikan Global Innovation Research Organization, Ritsumeikan University, 603-8577 Kyoto, Japan 2 College of Policy Sciences, Ritsumeikan University, 603-8577 Kyoto, Japan 3 Faculty of Environmental Engineering, The University of Kitakyushu, 808-0135 Kitakyushu, Japan [email protected]

Abstract. A Mixed-integer linear optimization model is developed to support the decision making for the sustainable use of energy in the local area. It details exploitation of primary energy sources, electrical and thermal generation, enduse sectors and emissions. The model covers both the energy demand and energy supply sides, and can provide valuable information both on the technical options, and on the possible policy measures. By aiming to realize a low-carbon energy system, the proposed optimization process provides feasible generation settlements between utility grid and distributed generations, as well as optimal diffusion of energy efficiency technologies. Moreover, the mathematical methods for solving the developed model are discussed. The focus is paid on the general solution method for mixed-integer linear optimization model including simplex algorithm and branch-and-bound algorithm. By using the suggested solution methods, the local energy system optimization problem is expected to be resolved in a reasonable time with enough precision. Keywords: Mixed-integer linear optimization, local energy system, low-carbon society, simplex algorithm, branch-and-bound algorithm.

1 Introduction The issue of global warming due to rising greenhouse gas (GHG) emissions presents a great challenge to the stability of the world’s climate, economy and population [1]. To avoid such a huge risk, the pursuit of a low-carbon society is receiving more and more attention by not only the developed but also the developing countries. As a global target, 50% reduction of GHG emissions below 1990 levels by 2050 has been confirmed by the main developed countries. However, some recent studies have recommended that worldwide emissions should be reduced by 80% by 2050 [2]. Actually, whatever the final value of the reduction goal, it is believed that numerous innovations should be realized for technology, institution and people’s behavior, towards such a bright future. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 361–371, 2010. © Springer-Verlag Berlin Heidelberg 2010

362

H. Ren et al.

Of course, a global thinking is always necessary to grasp the macro vision, the local actions can not be neglected to make the policies be realizable. It is believed that without the achievement of a local low-carbon society, the global low-carbon society will turn out to be just a dream. In order to realize a low-carbon society in the local area, the regional characteristics (e.g. natural resources, climate conditions, etc.) which may vary in many aspects should be paid enough attention. In addition, from the viewpoint of original sources of GHG emissions, it has been recognized that global climate destabilization is primarily due to the combustion of fossil fuels for energy and the resultant CO2 emissions. Therefore, it is not bombastic to say that the only way to achieve a sustainable low-carbon society is to have a sustainable lowcarbon energy system. At the same time, decrease in fossil fuels results in a considerable increase in its price and the derivatives. Actually, in all the mid-term and long-term scenarios proposed by some developed countries, the innovation of energy system is always paid the most attention. For example, according to Japanese lowcarbon society strategy, the improvement of energy intensity of end-use and the improvement of carbon intensity of end-use, which can be realized through promotion of energy efficiency and distributed renewable energy resources, contributes nearly 50% among the total CO2 emissions reduction target [3]. Thus, efficient use of energy and utilization of distributed renewable energy resources are the orders of the day when it comets to construct a low-carbon society. A wide variety of appropriate technologies is on the market now. However, a low-carbon energy system integrating energy efficiency and distributed generation technologies may illustrate a higher degree of complexity, and its economic, energetic and environmental benefits are influenced by system parameters such as variable ratios of thermal and electrical demands or intermittent supplies from renewable sources. Therefore, the optimal selection of energy technologies becomes a complex decision problem which has to consider the local energy demand and supply situation carefully. To support such decisions, an energy system optimization model will be helpful. In this study, a mixed-integer linear optimization model covering both energy demand and supply sides is developed to support the decision making for the construction of a low-carbon energy system in the local area. More specifically, the paper is structured in five more sections. Section 2 gives a definition of the local energy system planning problem. Section 3 illustrates the theoretical basis of the optimization model. In section 4, formulation of the mathematical problem relating the proposed optimization procedure is reported. Section 5 provides the solution methods for the mixed-integer linear optimization model. Finally, section 6 summarizes the conclusions of the whole study.

2 Definition of Local Energy System Planning Problem As a “no-regrets” mitigation option for CO2 emissions, energy efficiency is expected to induce more countries especially some developing countries to contribute their efforts to take moderate emission reduction target. It is recognized that the use of energy is no end in itself but is always directed to satisfy human needs and desires, which are usually realized by employing various equipments. Therefore, the potential for energy conservation can be taken into account for each component of final energy

A Mixed-Integer Linear Optimization Model for Local Energy System Planning

363

demand. Generally, the various measures that may be considered for the improvement of energy efficiency may be distinguished in the following basic categories [4]:

● Measures on architectural design: a design with green roofs or roof gardens while considering proper building orientation and shape. ● Measures on building materials: improvement of building’s envelope (roof or wall insulation), placement of heat-insulating door and window frames, introduction of thermal insulating coatings, and so on. ● Measures on energy sources: introduction of renewable (solar thermal system) or unused (ground-source heat pump, graywater utilization, etc.) energy resources. ● Measures on energy system: natural ventilation system or forced external air cooling system, advanced air-conditioner system (e.g., full-heat exchanger, zone or stratified air conditioning), ice or body storage heat system, etc. ● Measures on management and control: building energy management system, CO2-based ventilation control, and advanced sensors, etc. ● Measures on equipment efficiency: introduction of energy efficient appliance, compact fluorescent lighting, light emitting diodes, and so on. Besides the aforementioned energy efficiency actions, along with the increased awareness of global warming, more and more attentions are paid to the distributed generation technologies based on combined heat and power (CHP) and renewable energy sources, which are recognized as a “less-regrets” mitigation option. Generally, Distributed energy resources (DER) include power generation units and CHP units with nominal powers ranging from some kW to tens of MW, designed to deliver production to the utility grid or to cover, for a given amount, the load demand of single customers. For this reason, DER facilities are often located close to the load centre, which makes it convenient to be considered with the energy efficiency measures in an integrated way. In this study, the local energy system is represented as a network of energy chains, starting from the primary energy supply and ending in the end-use equipments, as shown in Fig. 1. The system is driven by an exogenous energy demand, due to the operation of various end-use equipments. The primary energy sources include fossil fuels (here, only natural gas which has been recognized as a clean fuel is considered), and local renewables (wind, solar and biomass energy, etc.). Energy convention given by several generation options comprises: CHP units (micro turbine, gas engine, fuel cell, etc.) and renewable technologies (PV system, wind turbine, etc.). Energy demand sector include direct electricity use, heating and cooling load, as well as hot water consumption. Finally, the energy end-use sector illustrates the various types of applications for the aforementioned four energy forms. This sector may include options for fuel and technology substitution, as well as various measures for energy conservation. Technology substitution is allowed for within the limits considered realistic in each demand type. Consequently, electric or district heat space heating systems may be substituted for fuel based boilers and vice versa, according to the pace at which old equipment is replaced. Heat accumulators can be used to shift the load of electric heating to night hours. New or candidate technologies in the end-use sectors can also be considered on a scenario by scenario basis. Examples of such technologies are heat pumps, solar heating systems and so on.

364

Grid Solar

H. Ren et al. Connector

Energy Efficiency

PV Heat pump

Electricity

Absorption heater

Cooling

Biomass

CHP

Natural gas

E storage

Boiler

Wind

Turbine

BEM system Green roofs Roof insulation

Heating Solar heater DER

Heat storage

High efficient appliance High efficient lighting

Hot water

Advanced air-conditioner Natural ventilation system Ventilation control system

Fig. 1. Sample layout of an integrated local energy system

In this study, among the various sectors introduced above, the focus is paid on the distributed generation and energy efficiency technologies. As distributed generation and energy efficiency measures are nowadays well known and widely spread, the main issue is to identify those that will be proven to be the more effective and reliable in the long term. With such a variety of proposed measures, the decision maker has to compensate environmental, energetic and financial factors in order to reach the best possible solution that will ensure the economically optimal combination of energy efficiency and distributed generators of a local area satisfying at the same time the final energy demands and some environmental constraints. In detail, the decisions regarding the local energy system under study concern appropriate choices for the system combination, equipment size, and running schedule of selected equipments or measures.

3 Theoretical Basis of Local Energy System Planning Local energy system planning is seen as the path towards an economic and ecological sustainable energy system while also taking into account limited financial and human resources as well as incomplete insight into the future development of economic technical and social conditions. It will be a hard work due to the innumerable alternatives concerning system structures, specifications and operational strategies. 3.1 System Interaction and Impact Boundaries Energy planning cannot be successfully achieved if the system is not adequately defined. The planner must have an overview of system’s boundaries and of all types of interactions and flows within the system or between the system and its environment [5]. As shown in Fig. 2, the economic boundaries can illustrate the cash flows between energy companies and consumers. Environmental boundaries would enable a

A Mixed-Integer Linear Optimization Model for Local Energy System Planning

365

clearer understanding of the environmental impact caused by the system being considered. Within such boundaries, the impact of different pollutants can be estimated in quantitative terms and in terms of their effects on the ecosystem, on the health of the local population, etc. Political boundaries stem from the laws, regulations and institutional frameworks that directly or indirectly affect planning decisions at the local level. These are difficult to identify and highly dependent on factors such as the national and district energy strategy, the social, economic and environmental policies. For instance, the national policy regarding the use of certain primary resources can considerably influence the energy supply options and consequently the planning at the local level. On the other hand, local actions (for example, success stories about the implementation of new energy technologies, etc.) may trigger reactions in other regions or at the national level. Fig. 2 offers a representation of the energy system within these main types of boundaries. The physical system is represented in the centre and the impact and political layers build on it. For simplicity, only the physical and measurable fluxes, between the system and its environment have been represented. The existence of different boundary types shows how complex the decision environment may be when planning. Planning problems can be of different nature and may require actions at different decision levels. Political Boundaries po

n me tal

So cia l

on vir

lic y

En

Impact Boundaries

po y lic

Physical Boundaries Generation

Costs, Emissions

Conversion

Costs, Emissions

Transport

lic y po

cp mi

Environmental Impacts

cy oli

rgy

no

Economic Impacts

En e

o Ec

Storage Consumption

Fig. 2. Impact boundaries of the local energy system

3.2 Concept of the Optimization Model The developed optimization model is a tool for analyzing competition and synergy between different alternatives of the rational use of local energy resources. For given

366

H. Ren et al.

time series of the fluctuating thermal and electrical requirements, the model calculates the optimal combination of energy conservation technologies and distributed power generators while taking into account local parameters like resource availability and climate conditions. Thus, the model provides a flexible instrument which may support decision-making when building or restructuring local energy systems. To this purpose, the model framework proposed in a previous study has been enhanced to include not only distributed generation contributions but also energy efficiency improvements [6]. In this study, the energy efficiency measures have been modeled as virtual energy supply equipments, which reduce energy demand rather than provide a more efficient supply. Actually, as is known to all, energy efficiency can provide energy and capacity to the end-users in much the same way that a power plant does. Whenever it operates, high efficiency equipment and appliances provide benefits in the form of avoided demand and, in this sense, are always available when needed. Because of this, energy efficiency measures can be designed to deliver the energy equivalent of a conventional power plant. The insulation of dwellings and the installation of energysaving light bulbs are examples of such demand-decreasing measures. In order to compare these measures with the efficient supply options and to make them accessible to optimization, they will be modeled as virtual supply processes. In the pre-developed plan and evaluation model, the objective is to minimize the overall cost of supplying energy service to a specific customer. As shown in Fig. 3, minimization of total annual energy cost is based on hourly values of energy demands, local characteristics, and energy tariffs. In addition, technical and economic data of energy conservation and distributed generation technologies should be supplied. Finally, the specific energy use, emissions, and costs for electricity imported from the utility gird have to be determined. After optimization, which yields optimal share for the different technologies considered, the total annual cost of the resulting energy system is calculated. The model does not simulate different technologies separately. Instead, it optimizes their joint operation dynamically under the time-dependent influences of intensive parameters on technology efficiencies. The flexible design of the model framework facilities easy future extensions through addition of technologies. Application of the model is no substitute for detailed engineering calculations. It serves as a preliminary stage by the use of which the multitude of technologically possible solutions can be efficiently and systematically reduced according to economic and ecological criteria defined by the decision makers. Generally, using the cost optimization criterion can be viewed as simulating the behavior of economic agents under the assumption the national priorities are always put in the first place. The optimization criteria are in accordance with the concept of lease cost planning which aims at measuring the energy supply technologies and the demand technologies by the same economic, technical and environmental parameters. In this study, in order to satisfy the preferences of different decision makers, besides the cost optimization, the model has be extended to minimize the non-renewable primary energy consumption and the CO2 emissions of the respective energy system. In this way, the model allows for dynamic energy, emission, and cost optimization of local energy systems with fluctuating energy demands and suppliers.

A Mixed-Integer Linear Optimization Model for Local Energy System Planning

367

Initial input data

INPUT

Local database

Local Information

9Energy system 9Building information 9Energy policy 9Other information

Technical Information

Optimization Model

9Climate data 9Electrical requirement 9Thermal demands 9Energy Tariffs

Energy balance

NO

YES Technical constraints

Market database 9Interest rate 9Subsides 9Capital costs 9Others

Market Information

NO

Technical database 9Power generation 9CHP 9Air-conditioner 9Lighting system 9Materials 9Others

YES 3E performances

Objective function

NO

YES Technology selection Optimal operational strategies Decision making Fig. 3. Flow chart of the optimization model

4 Mathematical Formulation of the Optimization Model To allow for the application of optimization techniques, an appropriate model should be developed in accordance to the principles mentioned above. In the following subsections, the mathematical equations of the mixed-integer linear optimization model are described. 4.1 Decision Variables As mentioned earlier, the decision making considered in this study concern three choices. For these three choices, three types of decision variables are defined respectively:

● Integer variables to reflect the alternative choices regarding system combination. ● Decision variables to reflect the equipment size based on the determined system combination. ● Decision variables to reflect the operational schedules of all system components.

368

H. Ren et al.

4.2 Objective Functions In this study, the energy system optimization model aims to determine the optimal use of resources in energy supply and conversion, and to some extent, in final energy demand. Firstly, the system is optimized by using total annual energy cost of the entire energy system over the whole study period as the objective function which is to be minimized. In a simplified form the objective function can be written as: f C = C f + C v = f ( I i , S i ) + f (Qi , j ) .

[min]

(1)

where, C f and C v indicate the fixed and variable costs, respectively. The fixed cost is usually the initial investment which should be considered from a life cycle cost viewpoint. It is a function of the equipment size S i , as well as an integer variable I i which indicate the existence or number of each equipment. The various cost is described as a function of real output Qi , j (the virtual output of energy efficiency measures as well) of the i th technology at the j th time interval. Alternatively, the environmental and energetic objective functions are described in Eqs. (2) and (3), respectively. In this study, the environmental performance is assessed based on the total CO2 emissions. The energetic performance is evaluated by the total non-renewable energy consumption.

[min]

f E = COe + CO f = f ( E j ) + f (Qi , j ) .

(2)

f P = Pe + Pf = f ( E j ) + f (Qi , j ) .

(3)

[min]

where, CO e and CO f are the CO2 emissions from utility electricity and on-site power generation, respectively. Similarly, Pe and Pf are the non-renewable primary energy consumption for utility electricity and on-site power generation, respectively. 4.3 Main Constraints Generally, one equality constraint in terms of energy balance and one inequality constraint in terms of generation capacity should be considered. Energy generation and consumption balance is formulated following the proposed system model, which includes utility grid, distributed generation, power export, and energy conservation due to energy efficiency improvement measures. Relating equality constraints have the following expressions:

¦Q i

i, j

E j  B j

Dj

j .

(4)

where, E j and B j are the imported and exported amount of utility electricity, respectively. D j indicates the real energy demands. For a stable operation, the real power output of each generator is limited in terms of maximum energy production as follows: Qi , j ≤ S i

∀i, j .

(5)

A Mixed-Integer Linear Optimization Model for Local Energy System Planning

369

5 Solution Methodology In order to optimize the local energy system with respect to the shares of the technologies that satisfy the given, fluctuating demand for heat and electricity during a representative year, the mixed-integer linear programming (MILP) approach is employed based on the simplex algorithm and a branch-and-bound (B&B) one, within individual 1-h time intervals. 5.1 Simplex Algorithm The MILP model is firstly a linear programming (LP) problem. In order to obtain an optimal solution of an LP, the research should be constrained on the set of the extreme points of its feasible region. This is because the extreme points of the feasible region of an LP correspond to the basic feasible solutions. The set of extreme points for an LP with determined technological constraints and variables can have only a finite number of extreme points; specifically, there is an upper bound for the cardinality of this set. The last observation would make one think that an approach to the problem would be to enumerate the set of extreme points, compare their corresponding objective values, and eventually select one which minimizes the objective function over this set. Such an approach would actually work for rather small formulation. But for reasonably sized LP’s, the set of extreme points, even though finite, can become extremely large. Hence, a more systematic approach to organize the search is needed. Such an approach is provided by the simplex algorithm [7]. Simplex algorithm is the general method for solving LP problems. The algorithm starts with an initial basic feasible solution and tests its optimality. If some optimality condition is verified, then the algorithm terminates. Otherwise, the algorithm identifies an adjacent basic feasible solution, with a better objective value. The optimality of this new solution is tested again, and the entire scheme is repeated, until an optimal basic feasible solution is found. Since every time a new basic feasible solution is identified the objective value is improved (except from a certain pathological case), and the set of basic feasible solution is finite, it follows that the algorithm will terminate in a finite number of iterations. 5.2 B&B Algorithm The B&B method is the basic technique for solving integer and discrete programming problems. It is based on a best-first tree search. Each node in the search tree consists of an LP problem that is formed as a linear relaxation of a corresponding MILP problem. The initial node is the LP relaxation of the original MILP problem and it is formed simply by ignoring the integrality constraints. The primal solution is calculated and served as the absolute lower limit of the objective function value and is used in the subsequent B&B algorithm to find a feasible integer solution. The main loop of the B&B algorithm manages the search tree, keeps track of the so far best MILP solution and chooses, in turn, the most promising unexplored leaf node to solve [8-9]. The general process in an iteration of B&B is shown as following:

①

Solve the LP relaxation of the problem (relaxing the integrality constraints for non-fixed integer variables).

370

② ③ ④ ① ② ③

H. Ren et al.

Let x be the optimal solution obtained, and z the optimal objective function. Bounding. If z ≥ z (the objective value of the best integer feasible solution so far), the problem can be deleted (as z is a lower bound on all integer solutions in this branch of the tree). Branching. Pick some x j whose optimal solution is fractional, and construct two problems by adding the constraints x j ≤ x *j and x j ≥ x *j in each problem. Furthermore, solving an LP problem in a node has three possible outcomes: The node is LP-infeasible. This implies that also the corresponding integer problem is infeasible. It means a dead-end in the search tree. No additional search from this node is needed. In particular, if the root node is LP-infeasible, then the original MILP problem is infeasible. The node is LP-optimal, but MILP-infeasible. This means that some of the integer constraints are not satisfied. It leads into the branch step of the algorithm. If the solution of the current node is better than the best MILP solution so far, it is necessary to continue the search from this node The node is LP-optimal and MILP-feasible. This means that the integer constraints are also satisfied. If the new solution is better than the currently best MILP solution, it replaces the best MILP solution. Then the search tree is also pruned, i.e., nodes whose solutions are not better than the new solution are removed. The solution of a child node cannot be better than the solution of the parent node, because more and more constraints are added deeper in the tree. Thus, it is not worth looking for better solutions from nodes where the LP solution is not better than the currently best MILP solution.

After the tree has been fully explored, the best found MILP solution is the optimum to the original MILP problem. If no MILP solution has been found, the original MILP problem is infeasible. The main idea in B&B is to avoid growing the whole tree as much as possible, because the entire tree is just too big in any real problem.

6 Conclusions In this study, a mathematical model has been developed for aiding in local energy system planning. The optimization model has been formulated as a MILP model with an objective function to minimize either overall energy cost, non-renewable energy consumption, or total CO2 emissions. It covers both energy demand and energy supply sides, and can provide valuable information on both the technical options and the possible policy measures. Moreover, the mathematical methods for solving the developed model are discussed. The focus is paid on the general solution method for MILP model including simplex algorithm and B&B algorithm. By using the suggested solution methods, the local energy system optimization problem is expected to be resolved in a reasonable time with enough precision. Acknowledgments. This research was supported by “Global Environment Research Fund by the Ministry of the Environment Japan” E-0804, Kansai Research Foundation for Technology Promotion, as well as Yazaki Memorial Foundation for Science and Technology.

A Mixed-Integer Linear Optimization Model for Local Energy System Planning

371

References 1. Dagoumas, A.S., Barker, T.S.: Pathways to a low-carbon economy for the UK with the macro-econometric E3MG model. Energy Policy 38(6), 3067–3077 (2010) 2. Parry, M., Palutikof, J., Hanson, C., Lowe, J.: Squaring up to reality. Nature Reports Climate Change 2, 68–70 (2008) 3. National Institute for Environmental Studies: Japan Scenarios and Actions towards LowCarbon Societies (LCSs), Tokyo (2008) 4. Diakaki, C., Grigoroudis, E., Kolokotsa, D.: Towards a multi-objective optimization approach for improving energy efficiency in buildings. Energy and Buildings 40(9), 1747– 1754 (2008) 5. Steidle, T., Schlenzig, C., Cuomo, V., Macchiato, M., Lavagno, E., Ryden, B., Willemsen, S., Grevers, W., Jank, R.: Advanced Local Energy Planning (ALEP) — a Guidebook, Program Energy Conservation in Buildings and Community Systems, Annex 33. International Energy Agency (2000) 6. Ren, H., Gao, W.: A MILP model for integrated plan and evaluation of distributed energy systems. Applied Energy 87(3), 1001–1014 (2010) 7. Rong, A., Lahdelma, R.: An efficient linear programming model and optimization algorithm for trigeneration. Applied Energy 82(1), 40–63 (2010) 8. Lin, Y., Austin, L.M., Burns, J.R.: An implicit branch-and-bound algorithm for mixedinteger-linear programming. Computers and Operations Research 17(5), 457–464 (1990) 9. Chan, D., Ku, C., Li, M.: A method to improve integer linear programming problem with branch-and-branch procedure. Applied Mathematics and Computation 179(2), 484–493 (2006)

IEC 61400-25 Protocol Based Monitoring and Control Protocol for Tidal Current Power Plant Jung Woo Kim and Hong Hee Lee School of Electrical Engineering, University of Ulsan, Nam-Gu, 680-749 Ulsan, South Korea {onethingtrue,hhlee}@mail.ulsan.ac.kr

Abstract. Wind energy and tidal current power have a common operation principle. Tidal current power converts kinetic energy of fluid to electric power. The communication infrastructure is very important to control the system and to monitor the working conditions of tidal current power plant. In this paper, the monitoring using international standard IEC 61400-25, which successfully operate for wind power system, is consequently applied to the tidal current power system. MMS service for communication between the tidal current power system and the remote control center is implemented to verify the performance of the monitoring system. Keywords: IEC 61400-25, Tidal current power, Monitoring, Remote control.

1 Introduction In recent years, the increasing of environmental concerns about global warming and the harmful effects of carbon emissions have created a new demand for clean and sustainable energy sources, such as wind, tidal, photovoltaic, biomass and geothermal power. Among these energy sources, tidal current power energy has more advantages of power per unit size and a smaller effect of changes in the weather than other renewable resources. Generally, each tidal current power generator is separately installed with long distance within the same tidal current farm. And the control center is far away from the tidal current plant, which is difficult to access due to nature of ocean. Therefore, the remote control and monitoring system are very important. In case of wind power plants, IEC (International Electrotechnical Commission) was established as an international standard of Communications for monitoring and control of wind power plants. Most of the communication compatibility to operating system was solved successfully using the IEC 61850 standard, which includes parts IEC618050-7-420 to monitor and control for DER (Distributed Energy Resource). However, there is no international standard of communication protocol for monitoring and control of tidal current power plants. In this paper, we suggest IEC61400-25 protocol to apply for tidal current power. The tidal current power is modeled into information model using IEC61400-25. The implementation of MMS (Manufacturing Message Specification, ISO 9506) Service K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 372–379, 2010. © Springer-Verlag Berlin Heidelberg 2010

IEC 61400-25 Protocol Based Monitoring and Control Protocol

373

and the communication systems for tidal power plant are constructed. The experimental result is shown to verify the effectiveness of the proposed communication system.

2 Overview IEC 61400-25 The IEC 61400-25 standard is a basis for simplifying the roles that the wind turbine and SCADA systems have to play. The crucial part of the wind power plants information, information exchange methods, and communication stacks are standardized by IEC 61400-25.

Fig. 1. Conceptual model of IEC 61400-25 series communication

Fig. 1 shows the communication model of IEC 61400-25 which defines all details required to connect the components of wind power plants in a multi-vendor environment and to exchange the information available by components. This is done by definitions in the document or by reference to other commonly used standards such as IEC 61850. IEC 61400-25 consists of six parts: IEC 61400-25 Part 25-1: Overall description of principles and models IEC 61400-25 Part 25-2: Information models IEC 61400-25 Part 25-3: Information exchange models IEC 61400-25 Part 25-4: Mapping to communication profiles IEC 61400-25 Part 25-5: Conformance testing IEC 61400-25 Part 25-6: LN classes and Data classes for Condition Monitoring

3 Information Model in Tidal Current Power System Fig. 2 shows a general tidal current generator. The generator using in the tidal power is almost same as the one in the wind power system. The orders of the energy conversion are fluid, blade, generator and converter. We propose an information model for logical nodes in the tidal current power plants as shown Fig. 3.

374

J.W. Kim and H.H. Lee

Fig. 2. Tidal current power tower

The components of the tidal current power plants correspond to the logical node. Compared to wind power plant’s logical nodes, pitch information node is added in tidal current power plant’s logical nodes. And the items, which are used in wind power, such as velocity, direction, air temperature, air density, etc., will be redefined in the tidal current power plant’s items. Table 1 shows the logical nodes in the tidal current power plants.

Fig. 3. Logical node for tidal current power plant

Service subset of ACSI(Abstract Communication Service Interface) needs to carry out all information exchanges within the WPPs. As defined in IEC 61400-25 Part 3, it describes client certification, transmission of control command, protocol and mechanism of data exchange.

IEC 61400-25 Protocol Based Monitoring and Control Protocol

375

Table 1. Proposed Logical Node List (M: Mandatory, O: Optional)

LN classes

TTUR TROT TTRM TGEN TCNV TTRF TNAC TYAW TTOW TMET TALM TPCH TREP TAPC TRPC

Description

Tidal current turbine general information Tidal current turbine rotor information Tidal current turbine transmission information Tidal current turbine generator information Tidal current turbine converter information Tidal current turbine transform information Tidal current turbine nacelle information Tidal current turbine yawing information Tidal current turbine tower information Tidal current power plant meteorological information Tidal current power plant alarm information Tidal current turbine pitching information Tidal current turbine report information Tidal current power plant active power control information Tidal current power plant reactive power control information

M/O

M M O M O O O M O M M O O O O

4 Mapping of Communication Profile Mapping of communication profile is described in IEC 61400-25 Part 25-4. According to Fig. 4, the SCSM (Specific Communication Service Mapping) maps the abstract communication services, objects and parameters to the specific application layers. The wind power plant information models and the information exchange models need to be mapped to appropriate protocols. The protocol TCP/IP is the basic lower layer protocols provided by all mappings. Specific data links and physical layers are beyond the scope of this standard. Depending on the technology of the communication network, these mappings may have different complexities, and some ACSI services may not be supported directly in all mappings but the equivalent services are provided [7].

Fig. 4. Mapping and communication profiles

376

J.W. Kim and H.H. Lee

In order to decide the solution which is suitable for monitoring and control application, the mapping which can satisfy supplier and client firstly has to be selected. Moreover, the selected mapping should have conformance fitted for the international standard to be at least selected. Recently, the IEC 61400-25-4 standard has been published with the new proposal, which includes the following 5 mapping methods: - SOAP based Web Services - OPC/XML-DA - IEC 61850-8-1 MMS - IEC 60870-5-104 - DNP 3(Distributed Network Protocol 3)

5 Implementation of MMS Service Fig. 5 shows a flow chart to implement MMS service. In the select mode, it is determined if MMS service is working as MMS server or MMS client before declaring variables. After initializing memory of each section, the logging parameters of WPP are defined. We set 4 lower layers from OSI 7 layer for MMS service of WPP and MMS object is added. MMS server remains in ready-state response until the request of MMS client [8].

Fig. 5. Service flow chart of MMS server

IEC 61400-25 Protocol Based Monitoring and Control Protocol

377

MMS service is implemented using SISCO’s MMS-EASE Lite tool. It is used to develop communication services in IEC 61850 based on Substation Automation System. But MMS service used in IEC 61400-25 is the same as in IEC 61850 on basic communication method and protocol. Therefore, this work develops MMS service using modified information model and functions with SISCO’s tool to apply IEC 61400-25 service. Abstraction model which is defined in IEC 61850-7 should be mapped into the application layer. ACSI object of IEC 61850-7-2 should be mapped into IEC 61400-25 MMS object. The VMD, Domain, Named Variable, Named Variable List, Journal, and File management included in information model and the various control blocks are also mapped into the corresponding MMS services [8]. Fig. 6 shows operation window of MMS communication service in IEC 61400-25. This work describes transmission service of various information (frequency, speed and temperature, etc.) between MMS server and client using logical node, regarding rotator, generator and converter which are already defined.

Fig. 6. MMS communication between server and client

6 Experiments and Discussion In this paper, MMS server and client are developed by PC by using Visual C++ and MMS-EASE lite supplied by SISCO. And a state of communication is displayed by Ethereal (Network protocol analyzer). Fig. 7 shows network construction of control and monitoring in tidal current power plants. Communication between server and client is achieved by using MMS service.

378

J.W. Kim and H.H. Lee

Fig. 7. Network based tidal current power plant system

MMS server receives information from the tidal current power plants. MMS client takes information from MMS server using MMS service. The Ethereal can be visualized by MMS service as shown in Fig. 8.

Fig. 8. Analysis of MMS using Ethereal tool

5 Conclusions This paper proposed an IEC 61400-25 communication protocol applied to the tidal current power plant, for monitoring and control. And this work developed a test bed system. The information model of tidal current power plant and the MMS

IEC 61400-25 Protocol Based Monitoring and Control Protocol

379

communication were successfully implemented. The MMS message was monitored and analyzed through the Ethereal network analyzer to verify the performances of the proposed system. And the stable operation of network system, which was constructed using PC, was finally approved. Acknowledgment. This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning(KETEP) grant funded by the Korea government Ministry of Knowledge Economy. (No. 2007-P-EPHM-E-04-0000).

References 1. International Standard IEC 61400-25: Wind Turbine Generator Systems part 25: Communications for Monitoring and Control of Wind Power Plants, WD edn. (2001) 2. Xie, Z., Manimaran, G., Vittal, V.: An Information Architecture for Funiture Power Systems and Its Reliability Analysis. IEEE Trans. on Power Systems, 857–863 (2002) 3. Qiu, B., Liu, Y., Phadke, A.G.: Communication Infrastructure Design for Strategic Power Infrastructure Defence (SPID) System. IEEE Power Engineering Society Winter Meeting, 672–677 (2002) 4. Tomsovic, K., Bakken, D.E., Bose, A.: Designing the Next Generation of Real-Time Control, Communication, and Computations for Large Power Systems. Proceeding of the IEEE 93, 965–979 (2005) 5. Khatib, A., Dong, X., Qiu, B., Liu, Y.: Thoughts on Future Internet Based Power System Information Network Architecture. In: Power Engineering Society Summer Meeting, vol. 1, pp. 155–160. IEEE, Los Alamitos (2000) 6. International standard: WD IEC 61400-25 Wind Turbine Generator Systems Part 25: Communications for Monitoring and Control of Wind Power Plants. First edn. (2003) 7. International standard: IEC 61850-7-1: Basic Communication Structure for Substation and Feeder Equipment – Principles and Models. First edn. (2003) 8. Kim, T.O., Jung, W.K., Lee, H.H.: Network Construction using IEC 61400-25 Protocol in Wind Power Plants. In: Huang, D.-S., Jo, K.-H., Lee, H.-H., Kang, H.-J., Bevilacqua, V. (eds.) ICIC 2009. LNCS (LNAI), vol. 5755, pp. 1049–1058. Springer, Heidelberg (2009) 9. Lee, J.H., Seo, M.J., Kim, G.S., Lee, H.H.: IEC 61400-25 Interface Using MMS and Web Service for Remote Supervisory Control at Wind Power Plants. In: International Conference on Control, Automation, and Systems (2008)

Adaptive Maximum Power Point Tracking Algorithm for Variable Speed Wind Power Systems Moo-Kyoung Hong and Hong-Hee Lee School of Electrical Engineering, University of Ulsan, Nam-Gu, 680-749 Ulsan, South of Korea {nice1204,hhlee}@mail.ulsan.ac.kr

Abstract. This paper proposes an adaptive maximum wind power extraction algorithm for variable speed wind power generation systems. The proposed control algorithm is analyzed with the air density effects on the maximum power point tracking (MPPT) method, which is based on the conventional hill climb searching (HCS). To implement this algorithm, the memory features and the initial tip speed ratio (TSR) are taken into account. The control scheme is performed in three separate operation steps according to the power curve of a generator to improve the tracking performance. The feasibility and effectiveness of the proposed MPPT strategy have been verified through simulations and experimental results. Keywords: Hill Climb Searching (HCS), Maximum Power Point Tracking (MPPT), Permanent Magnet Synchronous Generator (PMSG).

1 Introduction Nowadays, wind energy has become one of the most important renewable energy resources. In a wind power system, wind speed is varying in magnitude and direction at any time. To extract the maximum power under given wind conditions, the performance of maximum power point tracking (MPPT) algorithm is very important. There are three types of well-known maximum wind power extraction methods, i.e., Tip Speed Ratio (TSR), Power Signal Feedback (PSF), and Hill-Climb Searching (HCS)[1]. In these methods, the HCS algorithm is more robust and easier to develop in a wind power system than other methods. However, there are still some disadvantages associated with the HCS algorithm, which can significantly deteriorate the performances, especially under abrupt wind changing conditions. Accordingly, an optima solution is definitely required to extract maximum power stably under varying wind speed and air density. [3] In this paper, we propose a new strategy taking into account the air density effects for the MPPT method based on HCS algorithm. Due to temperature changing and air pressure, the air density is also varying. To adapt and compensate this effect, memory features and initial TSR value are used in the MPPT control algorithm. The feasibility and effectiveness of the proposed MPPT strategy have been verified through simulations and experimental results. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 380–388, 2010. © Springer-Verlag Berlin Heidelberg 2010

Adaptive MPPT Algorithm for Variable Speed Wind Power Systems

381

2 Wind Power Generation System 2.1 System Configuration Fig 1 shows the block diagram of entire wind power generation system used for control method. The wind power is emulated with DC motor, which is controlled according to the real wind characteristic profile, and PMSG is operated as the wind generator.

Fig. 1. Block diagram of entire wind power generation system

2.2 Wind Turbine Characteristics The wind power is expressed: Po =

where

1 ρ AV 3CP 2

Po : Energy extracted from rotor [W] 2

A : Rotary area of blade [ m ] CP: Power Coefficient.

(1) 3 ρ : Air density [ kg / m ]

V : Wind velocity [ m / s ]

Tip speed ratio is defined in (2), and the torque which is produced by the wind turbine can be given like (3) from (1) and (2).

λ=

ωrotor Rblade

(2)

Vwind 1 2

τ m = ρ AV 2Cτ

(3)

where Cτ is torque coefficient: . Cτ =

CP

λ

(4)

382

M.-K. Hong and H.-H. Lee

3 Maximum Power Point Tracking 3.1 Disadvantages of the Conventional HCS Algorithms As shown in Fig 2(a), if the tracking step size becomes larger, the speed of maximum power tracking can be faster. But, it deteriorates the efficiency of the MPPT control. On the contrary, if we make the step size smaller, the efficiency can be improved. However, the tracking speed becomes much slower.

Fig. 2. (a) Trade-off between step size and speed-efficiency (b) Misleading decision of conventional HCS under changing wind conditions

When wind speed changes, the misleading decision of conventional HCS method causes the failure in keeping maximum power and HCS algorithm moves downhill as illustrated in Fig 2(b). 3.2 Air Density Effect The wind power linearly varies with the air density sweeping on the blades. The air density ρ varies with pressure and temperature in accordance with the gas law: p R ⋅T where p : Air pressure, T : Temperature on the absolute scale, R : Gas constant.

ρ =

(5)

Fig. 3. (a) Variation of temperature (b) Variation of Air Density from 2004 to 2009 in Daegwallyeong area

Adaptive MPPT Algorithm for Variable Speed Wind Power Systems

383

Fig.3 (a) shows the temperature variation of six years of data from 2004 to 2009 in Daegwallyeong, Korea. As the temperature changes, the air density changes from 3 3 1.11 kg/m to 1.197 kg/m .

Fig. 4. Maximum power curves for different air densities

Fig. 4 shows the effect of air density on the power curve of the generator under various wind speeds. When the air density is varying, the corresponding power curves are also changed according to (1). More clearly, the maximum power curves are shifted, which can be realized in three different colors in Fig. 4. Therefore, it is necessary to consider this change to effectively improve the MPPT algorithm. The following subsection explains the proposed control method to overcome the mentioned problem in detail. 3.2 Proposed MPPT Algorithm In this paper, three steps are analyzed according to the operation of a generator. The tracking process to search maximum power point can be divided into three sections. These sections are defined by the output variations: increases, maintenance, or decreases as the generator rotating speed is increased. The flowchart of the proposed algorithm is shown in Fig. 5. In the flow chart, there are three steps ΔP is power variation, V W is the wind speed, and λinitial is the initial TSR value. The three steps are explained as follows: Step 0: Run the conventional HCS algorithm to search for approximate maximum power point using the initial TSR. After that, new TSR value is calculated. Step 1: The rotor speed reference is saved according to wind speed. The optimal TSR is updated for next step use. Step 2: In this step, the rotor speed reference and wind speed are extracted from step 1 for control algorithm calculation. By adopting the memory feature, the accurate rotor speed reference can be achieved for extracting maximum power point. Likewise, the step 0 and step 1 are done again continuously.

384

M.-K. Hong and H.-H. Lee

Fig. 5. Maximum Power Point Tracking Algorithm Flow Chart

4. Simulation 4.1 Simulation Model The wind energy conversion system was simulated using MATLAB SIMULINK. The system consists of four parts: air density and wind condition, wind turbine model, Table 1. Parameter of Cp

Table 2. Wind Turbine Model Parameters

controller, generator, and power system. The wind turbine emulator (WTE) must exactly produce torque – speed curve for real wind turbine when the wind speed changes. In order to generate the torque-speed curve as real wind turbine, Cτ − λ must be described at least up to the 6th order polynomial:

Adaptive MPPT Algorithm for Variable Speed Wind Power Systems

385

i =6

Cτ (λ ) = a0 + ∑ ai λ i

(6)

i =1

Table 1 shows the coefficients ai . Parameters of Cp and Wind Turbine Model parameters which are used in WTE are shown in Table 1 and Table 2, respectively. 4.2 Simulation Results Fig. 7 (a) and (b) show the simulation results with changing of wind speed from 6m/s to 8m/s at t = 2.5s and from 8m/s to 10m/s at t = 12.5s. WTE gives torque reference according to changing wind speed and air density. MPPT algorithm is activated, and the controller starts to track the maximum power point (MPP) by adjusting the DC/DC converter duty cycle.

Fig. 6. Simulation result of proposed MPPT algorithm (a) Wind turbine power and DC power (b) Power coefficient

As shown in Fig. 7(a), the extracted DC power is close to the maximum wind power, and the power coefficient reaches its maximum value in Fig. 7(b). Simulation results show that the proposed MPPT algorithm has good dynamic as well as the steady state performances throughout the MPPT process. The proposed MPPT algorithm can exactly extract the maximum wind turbine power at any given wind condition.

5 Experimental Results The WTE was implemented by 2.2kW, 1750rpm DC motor with torque control as shown in Fig. 8. The control algorithm for the WTE was implemented using a floating-point DSP TMS320F28335 with a clock frequency of 150 MHz. The wind speed profile is sent from PC to processor via RS-232, and the WTE can be controlled and supervised by a monitoring program on PC. In the prototype shown in Fig. 9, the DC motor is used to emulate as a wind turbine and rotates a 3kW PMSG. The generator stator output voltage is connected to an uncontrolled rectifier. A 10Ω resistor is used as load to consume the power which is generated by PMSG. The parameters of DC motor and PMSG which were used in experiments are same as simulation conditions.

386

M.-K. Hong and H.-H. Lee

Tref dn=

n

Tdt 2πJ

Nref

Iref

i

v n

Fig. 7. Wind turbine emulator

Fig. 8. System Configuration of WTE and PMSG controller

Fig. 10 and 11 show the experimental results of conventional HCS MPPT algorithm and proposed MPPT algorithm, respectively. In Fig 10 (a) and 11 (a), the blue line, the red line line represent for the maximum wind power and the DC power, respectively.

Fig. 9. Experimental results of conventional HCS algorithm (a) Wind Turbine Power and DC power (b) Power coefficient

In Fig. 10, the extracted power of conventional HCS algorithm is lower than the derived power obtained in the proposed MPPT algorithm, and the power coefficient is unstable than its maximum value. The proposed algorithm illustrates the fast response of tracking maximum power as compared to the conventional HCS algorithm. As a result, the proposed algorithm gives the effectiveness of power tracking in the dynamic conditions as the wind speed and air density changes. As seen in Fig. 11, the extracted DC power is close to the maximum wind power, and the power coefficient

Adaptive MPPT Algorithm for Variable Speed Wind Power Systems

387

Fig. 10. Experiment result of proposed MPPT algorithm (a) Wind Turbine Power and DC power (b) Power coefficient

is stable at its maximum value, 0.475. When wind speed changes from 6m/s to 8m/s and from 8m/s to 10m/s, the DC power is quickly adjusted to reach the new maximum wind power.

6 Conclusions This paper proposed a MPPT algorithm based on utilizing the memory features and the initial TSR value. The proposed algorithm effectively overcomes the drawbacks existed in the conventional HCS algorithm such as misleading tracking power curve and trade-off between step size and speed-efficiency. Based on the operation characteristic of generator, the proposed algorithm adapts three steps and accurately performs self tuning to update reference speed data as the wind speed changes. Furthermore, this algorithm does not require the knowledge of intangible turbine mechanical characteristic such as its power coefficient curve, power or torque characteristic, or torque characteristics. The feasibility and effectiveness of the proposed MPPT strategy are entirely verified by the simulations and experimental results. It can be seen from the results that the proposed MPPT algorithm is an adaptive algorithm to be a superior strategy to be used in wind power system. Acknowledgment. The authors would like to thank Ministry of Knowledge Economy which partly supported this research through the Network-based Automation Research Center (NARC) at University of Ulsan.

References 1. Wind energy for the eighties’. Peter Peregrinus Ltd., Stevenage, UK (1982) 2. Kazmi, S.M.R.: A novel algorithm for fast and efficient maximum power point tracking of wind energy conversion system. In: Proceedings of the 2008 International Conference on Electrical Machine (2008) 3. Wang, Q.: An intelligent maximum power extraction algorithm for inverter-based variable speed wind turbine systems. Proceedings of IEEE 19, 1242–1249 (2004) 4. Bhadra, S.N., Kastha, D., Banerjee, S.: Wind Electrical System. Oxford University Press, Oxford (2005)

388

M.-K. Hong and H.-H. Lee

5. Mukund, R.P.: Wind and Solar Power System, pp. 68–79. CRC Press, Boca Raton (1999) 6. Thiringer, T., Linders, J.: Control by variable rotor speed of a fixedpitch wind turbine operating in a wide speed range. IEEE Trans. Energy Conv. EC-8, 520–526 (1993) 7. Ermis, M., Ertan, H.B., Akpinar, E., Ulgut, F.: Autonomous wind energy conversion systems with a simple controller for maximum-power transfer. Proc. Inst. Elect. Eng. B 139, 421–428 (1992)

Modeling and Simulation of Two-Leaf Semi-rotary VAWT Qian Zhang, Haifeng Chen, and Binbin Wang School of Electronic and Information Engineering Zhongyuan Institute of Technology Zhongyuan Middle Road 41, Zhengzhou, China [email protected], [email protected], [email protected]

Abstract. In this paper, according to the structural characteristics of two-leaf semi-rotary VAWT(vertical axis wind turbine), the microelement method and the coordinate system rotation method are used to establish the mathematical model of wind turbine and, the mathematical model is simulated in the MATLAB platform by using the PID control method based RBF neural network tuning. The simulation result shows the revolution speed of VAWT can converge to rated value in a relative short period time after wind speed increasing.

1

Introduction

With the development of society, fossil fuels become increasingly exhausted. In the process of seeking new energy, wind energy reflects strong competitiveness. It is clean, safe, and renewable, so it has been very popular in recent years [1]. The wind is rich in energy. It is estimated that the total of wind energy resource around the earth can reach 2 trillion KW per year [2]. To make full use of wind resource, many countries are constantly increasing investment. A worldwide increase in the total installed capacity of wind power plants is expected to 20% each year and to reach a total installed capacity of 90 GW[3]. The wind turbines transform wind energy into mechanical energy. According to the wheel spindle, the wind turbines can be divided into HAWT and VAWT. A VAWT has good symmetry of stress and environmental noiselessness features. It has nothing to do with the wind direction, so there is no “wind damage” phenomenon [4]. Semi-rotary structure increases the blade area by wind to some extent, so semi-rotary VAWT has better aerodynamic characteristics than conventional straight board VAWT. In this paper, a two-leaf semi-rotary VAWT is introduced, whose blades mounted on the both side of spindle is 90-degree difference. Therefore, when the turbine works, it will make the blades produce non-uniform “wind movement” [5]. This characteristic makes the wind turbine have higher wind energy utilization efficiency and self-starting ability. The twoleaf semi-rotary VAWT has simple structure, convenient maintenance and installment, which makes it suitable to remote mountain areas and other off-grid power supply. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 389–398, 2010. c Springer-Verlag Berlin Heidelberg 2010 

390

Q. Zhang, H. Chen, and B. Wang

VSCF (Variable Speed Constant Frequency) control technology is common used in HAWT and the control effect is quite well. VAWT has low wind energy utilization efficiency, if VAWT can be combined with VSCF control technology, its efficiency will increase to some extent and, VAWT will be accepted by more people. RBF is a local approximation neural network. It can train the network by time slice and approximate to any degree of accuracy in any continuous function [6], [7]. In industrial production process, PID control is the most common control technology, which is mature and flexible to use. Three parameters are only required when PID controller is used, but they are difficult to adjust. In this paper, a PID control technology based on RBF neural network tuning is adopted to simulate the mathematical model of wind turbine in the MATLAB platform.

2

Wind Turbine Dynamic Model

For any control system, the mathematical model is the most fundamental basis of its control. For wind turbines, the uncertainty of aerodynamics and complexity of power electronics make the dynamic model difficult to establish [8]. The technology of HAWT is matured and its model establishing method is quite fixed. While VAWTs are diverse, so there is no uniform method for them. And the variety of VAWT makes modeling quite difficult. In addition, there are less references and theory, which produces more uncertainty in modeling. Therefore, the model about VAWT is quite difficult in establishing. For the purpose of establishing an accurate model, it is necessary to know the structure of two-leaf semi-rotary VAWT. The structural diagram is as shown in Fig .1:

Fig. 1. The structure of two-leaf semi-rotary VAWT

From Fig .1, we know that this VAWT belongs to lifting-dragging mixed type wind turbine. Due to the dominating dragging force, we still consider it as resistance-type wind turbine. In order to simplify the structure of wind turbine in this paper, we can just analyze one blade working situation. And the movement trace of single blade is as shown in Fig .2:

Modeling and Simulation of Two-Leaf Semi-rotary VAWT

391

Fig. 2. The structure of two-leaf semi-rotary VAWT

Fig. 2 is the top view of single blade operation, where O is wind turbine spindle, OA = R, p is any point in blade. Axis-l is established along the blade direction which is H to A and A is the origin of axis-l. According to semi-rotary characteristic, the line where the blade leaves and the circle intersect at point H which is a fixed point and, the angular velocity of blade revolving around point O is twice as much as that revolving around point H [9]. If blade width is a and the angular velocity of blade revolving O is ω, then the speed of point p at the normal direction of blade can be expressed as: vn = ω/2 × HP = (2R cos θ + l)ω/2 .

(1)

The wind speed at the normal and tangential direction of blade can be respectively expressed as: v⊥ = v cos θ, vr = v sin θ . Thus, the relative wind speed at the normal direction of blade is: vrel = v⊥ − vn = v cos θ − (2R cos θ + l)ω/2 .

(2)

According to the aerodynamics, the aerodynamic acting on object surface can be expressed as: 2 /2 . (3) F = ρSvrel Where, ρ is air density (usually is 1.25kg/m3), S is the area that wind acts on object, vrel is relative wind speed. According to (3), the torque that the wind turbine gained at point p is: 2 dTdr = ρhvrel (R cos θ + l)dl/2 .

(4)

Where, the torque arm is equal to (R cos θ + l) and dl is the micro-element at axis-l division. vr leaves along the blade tangent direction so that there is no area that wind acts on object, thus it contributes nothing to blade. The movement

392

Q. Zhang, H. Chen, and B. Wang

of blade is symmetrical on axis-x, so according to (4), the average torque gained by a whole blade can be expressed as:  π2  a2 2 1 2 Tdr = [ ρhvrel (R cos θ + l) dl ] dθ π 0 − a2 2  π2 2 a3 5Rω 2 1 = ρh[Ra(v − Rω)2 cos2 θ + ( − vω)] cos θdθ π 0 2 12 4 2ρhRa ρha3 5Rω 2 = (v − Rω)2 + ( − vω) . 3π 12π 4 Because the value of the latter is very small, it can be ignored. The expression above can be expressed as: Tdr = 2ρha(v − Rω)2 /3π .

(5)

In this paper, for the two-leaf semi-rotary VAWT, the total torque gained by wind turbine can be regarded as 2Tdr because of the little influence caused by two blades mutual covering. Assumed that the resisting torque of wind turbine is Tf , and then the wind turbine dynamic equation can be expressed as: 2Tdr − Tf = Jdω/dt . If v0 is rated wind speed and ω0 is rated angular velocity, the following equation will be gained: 2Tdr (v0 , ω0 ) − Tf = 0 . (6) Only steady revolution, the wind turbine would have higher quality power output. So the constant revolution speed control must be guaranteed. When the wind speed is higher than rated wind speed, a step motor can be used to keep wind turbine working at rated revolution speed. The process of adjustment is as shown in Fig. 3 and the wind turbine control system block diagram is as shown in Fig. 4:

Fig. 3. Graph of position adjustment

In Fig. 3, α is defined as blade power angle and it is adjusted through a step motor in Fig. 4. When the power angle is equal to α, the fixed point H

Modeling and Simulation of Two-Leaf Semi-rotary VAWT

393

Fig. 4. Wind turbine control system block diagram

Fig. 5. Coordinate system after rotating

will move to point H  . At this time, the torque of wind turbine is no longer symmetrical about the horizontal axis. For convenient analysis and calculation, a new coordinate system will be gained through counterclockwise rotating original coordinate system by 2α degrees. As shown in Fig. 5: In the coordinate system x oy  , the wind speed can be divided into two parts: v1 and v2 . Their value can be expressed as: v1 = v cos(2α), v2 = v sin(2α) . According to Fig. 4 and (5), under the effect of v1 , the average driving torque gained by wind turbine can be expressed as: Tv1 = 2ρha(v1 − Rω)2 /3π . When v2 acts on the wind turbine alone, the wind plays a negative role on wind turbine while the blade leaves above axis-x ; and it plays a positive role on wind turbine while the blade leaves below axis-x . It is easy to know that v2 does not work on wind turbine in a period for the symmetry of action. Therefore, the average driving torque gained by wind turbine after power angle adjustment is: Tα = Tv1 = 2ρha[v cos (α) − Rω]2 /3π .

(7)

From (6) and (7), the kinematic equation of wind turbine after power angle adjustment can be expressed as: q 2 − P 2 = −(3πJ/4ρhRa) · dq/dt .

(8)

394

Q. Zhang, H. Chen, and B. Wang

Where, q = v cos(2α), p = v0 − Rω. From (8), we can gain a discrete equation as shown below: q(k) =

emTs [q(k

2pq[q(k − 1) − p +p, − 1) + p] − [q(k − 1) − p]

ω(k) = (v cos(2α) − q(k))/R .

(9)

Where, m = 3ρhRap/3πJ, Ts is sampling period and ω(k) is the angular speed of wind turbine at k·Ts .

3

MATLAB Simulation of Wind Turbine Model

In the last few decades, extensive research has been carried out in developing the theory and the application of ANN and it has emerged to be a powerful mathematical tool for solving various practical problems like pattern classification and recognition, medical imagine, speech recognition, and control [10],[11]. ANN is formed by the artificial neurons interconnection [12]. The so-called neural network model is a network composed of some simple unit that interconnecting in parallel. This network can simulate the interaction made by species living in the real world [13]. RBF neural network is three-layer feed forward network. The mapping from input to output is nonlinear while that from hidden layer space to output layer space is linear. In this paper, PID control based on RBF neural network tuning is used and the system input is error signal Δu(k), system output is ω(k)and ω(k − 1). The middle layer contains 6 neurons. The output of network has only one neuron. So the structure of RBF neural network is 3-6-1, as shown in Fig. 6. And the wind turbine control system block diagram is shown in Fig. 7. In Fig. 6, the input layer acts as signal transmission layer, which directly sends the input signal to hidden layer. The hidden layer function is Gaussian function as shown below: hj = exp(−  X − Cj 2 /2b2j )

(j = 1, 2, · · · , 6).

The center vector of note j is: Cj = [cj1 , cj2 , cj3 ]T . The base width vector is: B = [b1 , b2 , · · · , b6 ]T . Where bj is the base width parameter of note j, which is greater than 0. The network weight vector is: W = [w1 , w2 , · · · , w6 ]T .

Modeling and Simulation of Two-Leaf Semi-rotary VAWT

395

Fig. 6. RBF neural network structure

Fig. 7. The control system block diagram including RBF

Because the output function of identification network is linear summation function, the output of identification network is: ωm (k) = w1 h1 + w2 h2 + w3 h3 + · · · + w6 h6 . And the identifier performance index function is : JI = (ω(k) − ωm (k))2 /2 . According to gradient descent method, the iterative algorithm of output weights, the iterative algorithm of node center and that of node base width parameters are: ωj (k) = ωj (k − 1) + η(ω(k) − ωm (k))hj + α(ωj (k − 1) − ωj (k − 2)) , Δbj = (ω(k) − ωm (k))ωj hj  X − Cj 2 /b3j , bj (k) = bj (k − 1) + ηΔbj + α(bj (k − 1) − bj (k − 2)) , Δcji = (ω(k) − ωm (k))ωj (xj − cji )/b2j , cji = cji (k − 1) + ηΔcji + α(cji (k − 1) − cji (k − 2)) , Where, η is learning rate and α is momentum factor. The sensitivity algorithm of system output to control input change is : ∂ωm (k)  cji − x1 ∂ω(k) ≈ = ω j hj , ∂Δu(k) ∂Δu(k) j=1 b2j 6

396

Q. Zhang, H. Chen, and B. Wang Table 1. Wind Turbine Parameters Quantity Symbol Value Unit Wind turbine radius R 0.6 m Moment of inertia J 7.4 Kgm2 Friction resistance moment Tf 2.4 Nm Blade height h 0.65 m Blade width a 0.54 m

Where, x1 = Δu(k). In this paper, the controller uses digital incremental PID controller. And the indicator of neural network tuning is chosen as: E(k) = e(k)2 /2 . Where e(k) is the deviation of input and output. The tuning algorithms of kp , ki and kd adopt gradient descent method: Δkp = −η

∂E ∂E ∂ω ∂Δu ∂ω (e(k) − e(k − 1)) , = −η = ηe(k) ∂kp ∂ω ∂Δu ∂kp ∂Δu

∂E ∂E ∂ω ∂Δu ∂ω e(k) , = −η = ηe(k) ∂ki ∂ω ∂Δu ∂ki ∂Δu ∂E ∂E ∂ω ∂Δu ∂ω Δkd = −η [e(k) − 2e(k − 1) + e(k − 2)] . = −η = ηe(k) ∂kd ∂ω ∂Δu ∂kd ∂Δu Where, ∂ω/∂Δu can be obtained through the neural network identification [14]. The parameters of two-leaf semi-rotary are as shown in Table 1: Δki = −η

Fig. 8. Step response of wind turbine angular velocity

The rated angular of wind turbine can be drawn as ω0 = 5.56rad/s according to v0 =7m/s. In this paper, when the wind speed suddenly increases from v0 to 1.2v0 , the method of PID control based RBF neural network tuning will be adopted to ensure wind turbine constant revolution. Simulating the mathematical model of two-leaf semi-rotary wind turbine in platform of MATLAB, the simulation result is as shown in Fig. 8:

Modeling and Simulation of Two-Leaf Semi-rotary VAWT

4

397

Conclusion

In this paper, the aerodynamic model of two-leaf semi-rotary VAWT is established through micro-element method and coordinate rotation method according to aerodynamics. Using coordinate rotation method is easy to analyze the operation situation of VAWT when the power angle is nonzero. But in simulation, a simple step response is just used, which couldn’t fully reflect the complex and random wind movement. From simulation, the PID control based RBF neural network tuning can achieve constant speed control in the platform of MATLAB and the result shows: 1. Sampling results of the first time has overshoot phenomenon, which is because the entire wind turbine model is not established. Actually, when the driving torque of wind turbine overcomes the friction resistance moment and electromagnetic resistance moment of generators, it also overcomes the friction resistance moment of itself. When the wind suddenly increases, the rotation of generator also has sudden increase in the trend which can produce a sudden increase of electromagnetic torque that can greatly limit the wind turbine revolution speed. 2. Under the regulation of PID control based RBF neural network tuning, the wind turbine can reach rated revolution speed in a relatively short period of time.

References 1. Kong, Y., Wang, Z.: Layered Multi-mode Optimal Control Strategy for Multi-MW Wind Turbine. J. Journal of Donghua University (Eng. Ed.) 25(3), 317–323 (2008) 2. Bao, E.: Development Of Wind Power Generation Technology. J. Renewable Energy 114(2), 53–55 (2004) 3. Antipov, V.N., Danilevich, Y.B.: Analysis and Investigation of Size Spectrum of Synchronous Machines Operating as Wind-Power-Generators in the Range of Rotation Frequencies of 75-300 rpm. J. Russian Electrical Engineering 80(1), 23–28 (2009) 4. Yang, F., Wu, J., Yang, J.: Analysis of Structural Mechanics for Lattice VAWT. J. Electric Power Construction 29(11), 67–70 (2008) 5. Two-leaf Vertical Axis Wind Turbine. P. Prospectus, http://www.qianyan.biz/Patent-Display/200810024858.html 6. Li, G.: Intelligent Control and MATLAB Implementation, p. 31. Electronic Industry Press, Beijing (2005) 7. Xu, N.: Neural Network Control, 3rd edn., p. 25. Electronic Industry Press, Beijing (2009) 8. Ye, H.: Technology of Wind Turbine Control, p. 193. Machine Industry Press, Beijing (2002) 9. Tao, Z., Qiu, Z.: Stability and Maximum Speed of Revolution of Two-leaf Semirotary VAWT. J. Journal of Anhui University of Technology 23(4), 421–424 (2006) 10. Shin, M., Goel, A.L.: The range of fields in which RBF methods have been employed is very impressive, including geophysics, signal processing, meteorology, orthopedics, pattern recognition, and computational fluid dynamics. J. IEEE Trans. Software Engineering 26(6) (2000)

398

Q. Zhang, H. Chen, and B. Wang

11. Suresh, S., Omkar, S.N., Mani, V.: Parallel Implementation of Back-Propagation Algorithm in Networks of Workstation. J. IEEE Trans. 16(1), 24–33 (2005) 12. Knoblock, C.: Neural networks are learning devices inspired by the workings of the brain. J. IEEE Computer Society 11(4), 4 (1996) 13. Chen, M.: Neural Network Model, p. 4. Dalian University of Technology Press, Dalian (1995) 14. Liu, J.: Advanced PID Control MATLAB Simulation, 2nd edn., pp. 162–164. Electronics Industry Press, Beijing (2004)

Identification of Chiller Model in HVAC System Using Fuzzy Inference Rules with Zadeh’s Implication Operator Yukui Zhang1, Shiji Song1, Cheng Wu1, and Kang Li2 1

2

Department of Automation, Tsinghua University, Beijing 100084 School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast BT9 5AH, UK {zyk08,shijis}@mail.tsinghua.edu.cn

Abstract. In the heating, ventilating, and air-conditioning (HVAC) system, chiller is the central part and one of the primary energy consumers. For the purpose of saving energy, the identification of the chiller model is of great significance. In this paper, based on fuzzy inference rules with Zadeh’s implication operator, the model of chiller in HVAC is identified. The mean square error (MSE) is employed to evaluate the approximating capability of the fuzzy inference system. The objective of the problem is to minimize MSE. Since the Zadeh’s implication operator is adopted in the fuzzy inference, the output of the system becomes a continuous but non-smooth function. In addition, the objective function contains many parameters that need to be optimized, consequently, traditional optimization algorithms based on gradient descent method fail to work. Therefore, an improved genetic algorithm (GA) is applied to minimize the MSE. Actual operational data of a chiller in HVAC are gathered to train the fuzzy inference system. Numerical experiment results validate the accuracy and efficiency of proposed fuzzy model and the improved GA algorithm. Keywords: Chiller, fuzzy inference system, implication operator, improved genetic algorithm.

1 Introduction As well known, heating, ventilating, and air-conditioning (HVAC) system plays an important role in modern life. It’s a must in large buildings such as hotels, workshops, hospitals, etc. In terms of energy consumption, the energy consumed in HVAC system accounts for more than 50% of total building energy [1]. As a convention, HVAC system is run under maximum load, however, in most cases, the real load at the operational time is lower than full load. Obviously, these conditions cause large sum of waste. Consequently, in order to save energy, the efficiency of HVAC system needs to be improved. A typical HVAC system consists of an indoor air loop, chilled water loop, refrigerant loop, condenser water loop and outdoor air loop. Researchers concentrated K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 399–408, 2010. © Springer-Verlag Berlin Heidelberg 2010

400

Y. Zhang et al.

on different constituents, some on chilled water loop[2] , some on condenser water loop[3]. Certainly, more and more scholars considered the optimization of HVAC systems as a whole [4-5], but their results still have a lot of space for improvement. In order to control the whole system, the models of HVAC need to be identified. Among the five procedures mentioned above, two water loops and the refrigerant loop are of most importance, because the energy consumption of the three parts account for overwhelming proportion of the total power use. The primary energy consumers of the water loops and refrigerant loop are pumps and chiller respective. Since the mathematical model of pumps can be described as power functions [6], the main task now is to identify the model, namely the input-output relation, of chiller. In terms of HVAC models, several popular algorithms are adopted to deal with the modeling or forecasting issues on HVAC system. Hao Peng and Xiang Ling[7] predicted the pressure drop and heat transfer characteristics in the plate-fin heat exchangers (PFHEs) based on Artificial Neural Networks (ANN). Mahendra Kumar and I.N. Kar[8] made use of support vector machines (SVM) to model two HVAC relationships. Rafael Alcala et al developed accurate fuzzy logic controllers for HVAC by proposing a fuzzy system[9]. Servet Soyguder and Hasan Alli modeled and designed the HVAC systems and controlled the fan speed by virtue of Adaptive Neuro-Fuzzy Inference System(ANFIS)[10-11]. The aim of this paper is to model the input-output functional mapping of chiller by using the fuzzy inference system (FIS). In 1965, Zadeh[12] firstly published an article which initiated the study of FIS. At present, there are two popular types of FIS which are proposed by Mamdani et al. [13] in 1975 and Wang et al. [14] in 1992. In this paper, we adopt the former one to identify the chiller’s model. In 1973, Zadeh proposed an implication operator Rz , which connects the antecedent part and consequent part of fuzzy rules and yields an numeric result[15]. In the Mamdani’s FIS, a weighted averaged method is usually adopted, wherein operator Rz is used to calculate the weights. The fuzzy sets of the rules are described by Gaussian Membership Function (GMF), and the essential parameters of GMF are its center and variance. We adopt the mean square error (MSE) to evaluate the difference between the real output and the output of our Mamdani’s FIS. Obviously, the MSE varies with the parameters, and we want to decrease the MSE to improve the approximating capability of the fuzzy inference system. So, the task now is to minimize MSE by adjusting the centers and variances, a continuous but non-smooth optimization problem is then formulated, and many parameters of the model need to be optimized. Traditional variously optimization algorithms based on gradient descent methods can’t be used to solve this problem. So the improved Genetic Algorithms (GA) is proposed and used to minimize the MSE. As we know, crossover is an important operation in the procedures of GA, during this procedure pairs of parent chromosomes produce their offspring. The traditional GA method is designed by one-point crossover approach[16], that is, only one crossover point occurs in a chromosome pair, so their offspring are too similar to themselves, i.e., the search capability is not enough powerful for multi-parameter optimization problem. Subsequently, a uniform crossover approach was proposed by researchers [17], which involves on the average L / 2 crossover points for strings of

Identification of Chiller Model in HVAC System Using Fuzzy Inference Rules

401

length L , and this method causes the offspring retaining little similarity to their parents. In this paper, for the design of GA, we proposed an improved crossover approach which involves as many crossover points as the parameters to be optimized. We apply the actual operating data of Carrier 19XR6767467DJS cold water chiller to identify the chiller model. Some numeric experiment results validate the accuracy and efficiency of the identified model by FIS and improved GA algorithm by crossover. This paper is divided into six parts and organized as follows. Section II introduces the necessary notations and preliminaries. Section III models the chiller based on the fuzzy inference system with implication operator. Section IV proposed the optimization algorithm based on GA with improved crossover method. In the fifth section, numerical results are yields by simulation. Section VI concludes this paper.

2 Notations and Preliminaries In this paper, we consider a Multiple-Input-Single-Output (MISO) system. The basic notations are listed as follows:

x = ( x1 , x2 , L xM ) ∈ R M : the system input. y : the system output. In this paper, we will apply Mamdani’s fuzzy inference system, main notations related to this paper are denoted as follows: N i : the number of fuzzy sets of xi . Ai j : the jth fuzzy set of xi . u A j : the membership function of Ai j . i

M

N = ∏ N i : the number of rules. i =1

Ai , j : the fuzzy set chosen from { Ai1 , Ai2 , L AiN i } in the jth rule. u Ai , j : the membership function of Ai , j . B j : the fuzzy set of y in the jth rule. uB j : the membership function of B j . y j : the center of gravity of B j . f ( x ) : the output yielded by the FIS with input x.

Notations related to sample data are listed as follows: K : the number of sample data. x( k ) = ( x1 (k ), x2 (k ),L xM (k )) ∈ R M : the kth sample input. y ( k ) : the kth sample output. e(k ) = f ( x (k )) − y ( k ) : the error between the output of FIS and the real output.

402

Y. Zhang et al.

As we employ the Zadah implication operator in this paper, two mathematical operators need to be introduced: a ∧ b : [0,1] 2 → [0,1] : a ∧ b = min(a, b). a ∨ b : [0,1] 2 → [0,1] : a ∨ b = max(a, b). Because we adopt the GMF to represent the fuzzy sets, the following parameters need to be defined: c X : the center of X .

σ X : the variance of X .

Where X is a fuzzy set with Gaussian membership function.

3 Mathematical Model of Chiller Usually, a typical HVAC system consists of an indoor air loop, chilled water loop, refrigerant loop, condenser water loop and outdoor air loop. Since the chiller is the primary energy consumer of the refrigerant loop and accounts for a large proportion of the total power use of the whole HVAC system, in order to save its energy, we need to identify the mathematical model of chiller to facilitate the control of a whole system. The typical Mamdani’s fuzzy inference rules are adopted and described as follows: R i : IF x1 is A1i and x 2 is A2i and L and x m is Ami , THEN y is B i , i = 1,2, L n.

(1)

Zadeh’s implication operator R z is defined as follows [15]: Rz : [0,1]2 → [0,1] : Rz (a, b) = (1 − a ) ∨ (a ∧ b).

(2)

Using the operator R z , we can calculate the output f (x ) of FIS by using a popular fuzzy weighted-average method, f (x ) is then determined as: M

N

f ( x) =

∑y j =1

j

• Rz (∏ u Ai , j ( xi ), uB j ( y )) i =1

N

M

j =1

i =1

∑ Rz (∏ u Ai , j ( xi ), uB j ( y))

.

(3)

In the following, GMF is used to represent a fuzzy set. Suppose X is an arbitrary fuzzy set with GMF, then we have −( y −c X ) 2

u X ( x) = e

2σ X 2

.

(4)

where u X ( x) is the membership function of X . Since the center of gravity of a fuzzy set X with GMF is just its center c X , from the construction of the jth fuzzy inference rule above, we have

Identification of Chiller Model in HVAC System Using Fuzzy Inference Rules

y j = cB j .

403

(5)

According to (4) and (5), we can rewrite (3) as the following form: − ( y − c Ai , j ) 2

M

N

f ( x) =

∑c j =1

Bj

• Rz ( ∏ e

,e

2σ B j 2

i =1

M

N

∑ R (∏ e j =1

2σ Ai , j

−( y −cB j ) 2

2

− ( y − c Ai , j ) 2

−( y − cB j ) 2

2σ Ai , j

2σ B j

2

z

,e

) .

(6)

2

)

i =1

To evaluate the approximating ability of the above FIS, we now apply the Mean Square Error (MSE) as the objective function of the optimization problem. The MSE is defined as MSE =

1 K

K

∑ (e(k )) . 2

(7)

k =1

We fixed the parameters of the fuzzy sets in the antecedent part of the fuzzy rule, while adjusting the other ones in the consequent part. So we will minimize the MSE optimizing c B and σ B , j = 1,2,L N . This investigated in the next section. via

j

optimization

problem

will

be

j

4 Optimization via Improved GA Since the Zadeh’s implication operator is adopted in this paper, the objective of the problem becomes continuous but non-smooth function. In addition, there are many parameters to be optimized which result in large search spaces of the optimization problem. Consequently, traditional variously optimization algorithms based on gradient descent method fail to work. So, in this section we adopt the improved Genetic Algorithms (GA) to minimize the MSE. As a heuristic algorithm, GA is suit for solving complicated problems with large search spaces, and able to provide optimal or near optimal solutions. In this paper, we adopt binary codes for GA, that is, a chromosome is represented by a 0-1 string with length L . It is composed of 2N sub-chromosomes, and the length of a sub-chromosome is l .Thus we have L=2·l·N. Each sub-chromosome represents a parameter to be optimized, that is, all the centers and variances of the consequent part of the fuzzy rules can be assigned according to a single chromosome. Among the design procedures of GA, crossover is an important one. During this procedure pairs of parent chromosomes produce their offspring. The traditional GA method is designed by one-point crossover approach[16], that is, only one crossover point occurs in a chromosome pair, so their offspring are too similar to themselves, i.e., the search capability is not enough powerful for multi-parameter optimization problem. Subsequently, a uniform crossover approach was proposed by researchers

404

Y. Zhang et al.

[17], which involves on the average L/2 crossover points for strings of length L, and this method causes the offspring retaining little similarity to their parents. Now we present an improved crossover approach which involves as many crossover points as the parameters to be optimized, which will be described in step 5 of the improved GA algorithm. Let G and S denote the number of total generations and the number of chromosomes in each generation respectively. Define crossover probability and mutation probability as pc and pm. Denote the chromosomes in one generation as Ci, i=1,2…S. The improved GA algorithm is designed in detail as follows: Step1: Initiation Let g=0, which denotes current generation number. Generate a set of S random 0-1 strings with length L. Step2: Decoding and Evaluation 1) For a certain sub-chromosome of some chromosome C i , suppose the corresponding parameter ranges from Pmin to Pmax. Now we consider this subchromosome as a binary number, and suppose the value of this binary number is V, e.g. the V of a sub-chromosome which is 1-1-0-0-1 equals to 25. Then assign

V • ( Pmax − Pmin ) + Pmin to the corresponding parameter. 2l − 1 2) For all of the other sub-chromosomes of this chromosome, assign the corresponding parameters according to stage 1). After this stage, all of the parameters to be optimized are assigned values based on this chromosome. 3) Substituting the parameters to formula (7) and calculating the corresponding MSE and fitness, where the fitness is defined as Fi =1/(1+ MSEi). 4) For all of the other chromosomes, the corresponding MSEi and Fi are then given according to stage 1) to 3) respectively. After this stage, all of the chromosomes are decoded and evaluated. Step3: Stopping Criterion If g≥G, go to step 7. Otherwise, go on to step 4. Step4: Selection S

1) Finding the total fitness of the generation: TF = F . ∑ i i =1

2) Getting the probability pi of selection for each chromosome by pi=Fi/TF. i

3) Calculating the cumulative probability P = ∑ p j and letting P0 = 0 . i j =1

4) Generating a random number r in (0, 1]. Finding the proper integer n satisfying Pn
Identification of Chiller Model in HVAC System Using Fuzzy Inference Rules

405

5) Repeat stage 4) S times. After these 5 stages, a new set of chromosome is generated. Step 5: Crossover As mentioned above, we propose an improved crossover approach to adequately mix up two chromosomes, while ensuring the offspring not to differ from their parents excessively. The stages are described as follows: 1) Generating a random number rc in [0, 1] for each chromosome of the code set generated in step 4. 2) If rc is smaller than pc, select the corresponding chromosome for crossover. If the last chromosome, i.e. Cs, is processed, stop the crossover step. 3) If there have been two chromosomes ready to produce their offspring, continue to stage 4). Otherwise, return to 1). 4) Generating 2N random integers ni , i=1,2,…2N ranging in [1, l ] . For the pair of ith sub-chromosomes of the two selected chromosomes, we applied the one-point crossover method [16]. This method involves totally 2N crossover points, which equals to the parameter number. 5) Return to 1). After these 5 stages, we get a new chromosome set. Step 6: Mutation Mutation is relatively simple among all the steps of the algorithm. Traversing s from 1 to S, for each binary bit of Cs, generating a random number r in [0,1], and if r is smaller than pm, change this bit to its opposite number, i.e. if the bit is b, change it to 1-b. Finally, let this new chromosome set be the next generation. Let g=g+1 and go back to step 2. Step7: Termination Output the results such as the optimized MSE, the optimized Mean Absolute Error (MAE), etc. generate the relevant figures.

5 Simulation Results In order to establish the chiller model base on the FIS with Zadeh’s implication operator mentioned above, we need relevant operating data of chiller. In this paper, we select the actual operating data of Carrier 19XR6767467DJS cold water chiller. 100 items of data in all are gathered to construct the fuzzy inference system. Three typical records are listed in table 1. Since the difference between inlet and outlet temperature of chilled water are maintained at 5 , it is reasonable to take one as an input variable and ignore the other. Here we choose inlet temperature and discard the outlet one. For the same

℃

406

Y. Zhang et al.

reason, we select cooling water inlet temperature and ignore the outlet counterpart. So the chiller is considered as a 3-input-single-output system. The input variables composite of Quantity of Cold x1, Inlet Temperature of Chilled Water x2, Inlet Temperature of Cooling Water x3. The output variable is Input Power y. Table 1. Typical records of Chiller operation data Chilled Quantity of Water Inlet Cold (KW) Temperature ( ) 2830 10 2650 11 2946 13

Record Num

℃

1 2 3

Chilled Water Outlet Temperature ( ) 5 6 8

℃

Cooling Water Inlet Temperature ( ) 26 32 28

℃

Cooling Water Outlet Input Temperature Power (KW) ( ) 31 527 37 550 33 521

℃

Before simulating, values of some essential parameters must be specified as follows: N1=7, N2=5, N3=5, N=N1·N2·N3=175, K=100.

c A j j = 1,2,... N j are assigned uniformly between min{xi ( k ), k = 1,2, L K } and i

max{xi (k ), k = 1,2,L K }

.

⎧5, if i = 1, ⎩1, if i = 2 or 3.

σA = ⎨ j

i

S = 60, G = 50, pc = 0.5, pm = 0.2, N = 175, M = 3. The range of c B is j

[min{ y ( k ), k = 1,2, L K }, max{ y ( k ), k = 1,2, L K }] , j = 1,2, L N . The range of σ B is [1, 10] , j = 1,2, L N . j Figure 1 and 2 illustrate two distinct optimization processes respectively.

Fig. 1. Optimizing process 1

Fig. 2. Optimizing process 2

Identification of Chiller Model in HVAC System Using Fuzzy Inference Rules

407

The final optimal numerical results are listed in table 2. In addition to MSE, we adopt the Mean Absolute Error (MAE) and Mean Relative Absolute Error (MRAE) to evaluate the effectiveness of the applied fuzzy model and GA with improved crossover method. Table 2. Final optimal results

simulation num

MSE(KW2)

MAE(KW)

MRAE(%)

Simulation 1

182.77

10.89

2.06

Simulation 2

181.87

10.86

2.05

According to the numerical results, the MRAEs are about 2%, which indicates the accuracy of our model. Comparing to the initial MSEs 211.2 and 221.5 respectively, the optimized MSEs decreased by 13.5% and 17.9% which shows the efficiency of the applied GA with improved crossover approach. Finally, we obtain the optimal f * ( x ) with the optimized parameters c *B j and σ B* j , j = 1,2,L N . Consequently, the chiller model is then identified.

6 Conclusions The chiller is one of critical components in terms of saving energy consumed in HVAC. Based on the Zadeh’s implication operator Rz, a Mamdani’s fuzzy inference system is adopted to identify the model of chiller. We apply the mean square error (MSE) to evaluate the approximating ability. The objective of the problem is to minimize MSE. The traditional variously optimization algorithms based on gradient descent method can’t used to solve this problem because the objective of the problem is a continuous but non-smooth function, and contains many parameters to be optimized. An improved GA algorithm by new crossover procedures is presented. Numeric experiment results show the accuracy and efficiency of proposed fuzzy model and improved GA algorithm.

Acknowledgements The paper is supported by UK-China Bridge in Sustainable Energy and Built Environment (EP/G042594/1), Distinguished Visiting Research Fellow Award of Royal Academy of Engineering of UK, NSFC (No.60874071) & RFDP (No.20090002110035).

References 1. Hodge, B.K., Taylor, R.P.: Analysis and design of energy systems. Prentice Hall, New Jersey (1999) 2. Kirsner, W.: Designing for 42°F chilled water supply temperature-—does it save energy? ASHRAE J. 40(1), 37–42 (1998)

408

Y. Zhang et al.

3. Shelton, S.V., Joyce, C.T.: Cooling tower optimization for centrifugal chillers. ASHRAE J. 33(6), 28–36 (1991) 4. Hartman, T.B.: Global optimization strategies for high-performance controls. ASHRAE Trans. 101(2), 679–687 (1995) 5. House, J.M., Smith, T.F.: A system approach to optimal control for HVAC and building systems. ASHRAE Trans. 101(2), 647–660 (1995) 6. Li, Y.X., Cai, X.B., et al.: Fuzzy control technology for energy saving and its application on HVAC system. China Construction Press 7. Hao, P., Xiang, L.: Neural networks analysis of thermal characteristics on plate-fin heat exchangers with limited experimental data. Applied Thermal Engineering 29, 2251–2256 (2009) 8. Kumar, M., Kar, I.N.: Non-linear HVAC computations using least square support vector machines. Energy Conversion and Management 50, 1411–1418 (2009) 9. Alcalá, R., Casillas, J., et al.: A genetic rule weighting and selection process for fuzzy control of heating, ventilating and air conditioning systems. Engineering Applications of Artificial Intelligence 18, 279–296 (2005) 10. Soyguder, S., Alli, H.: Predicting of fan speed for energy saving in HVAC system based on adaptive network based fuzzy inference system. Expert Systems with Applications 36, 8631–8638 (2009) 11. Soyguder, S., Alli, H.: An expert system for the humidity and temperature control in HVAC systems using ANFIS and optimization with Fuzzy Modeling Approach. Energy and Buildings 41, 814–822 (2009) 12. Zadeh, L.A.: Fuzzy Sets. Information and Control 8(3), 338–353 (1965) 13. Mamdani, E.H., Assilian, S.: An experiment in linguistic synthesis with a fuzzy logic controller. Int. J. Man-Mach. Stud. 7(1), 1–13 (1975) 14. Wang, L.X., Mendel, J.M.: Generating fuzzy rules by learning from examples. IEEE Trans. Systems, Man, and Cybernetics 22(6), 1414–1427 (1992) 15. Zadeh, L.A.: Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Systems Man Cybernet. 3, 28–34 (1973) 16. Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975) 17. Spears, W.M., De Jong, K.A.: On the Virtues of Parameterized Uniform Crossover. In: Proc. of the Fourth International Conference on Genetic Algorithms, pp. 230–236 (1991)

An Improved Control Strategy for Ball Mill Grinding Circuits Xisong Chen, Jun Yang, Shihua Li, and Qi Li School of Automation, Southeast University, Nanjing, Jiangsu Province, 210096, China [email protected]

Abstract. An improved control strategy is proposed to control ball mill grinding circuits for energy saving and pollution reduction. A two-layer optimization architecture combined by particle size optimization layer and energy optimization layer is developed, where the optimal particle size set-point is calculated first, followed by the energy optimization step. A control method adaptive to the wear of liner in ball mills is also proposed. Simulation studies demonstrate that the fresh ore feed rate has been increased and the energy efficiency has been increased. Keywords: Optimization; Energy efficiency; Expert system; Grinding circuit.

1 Introduction The function of grinding circuits is to reduce particle size of ore to a size called the liberation size such that the valuable mineral constituent is exposed and can be recovered in the subsequent operation. The grinding circuits typically represent about 50% of the total expenditure of a concentrator plant, only 10% of the electrical power consumed contributing to grinding of ore. Grinding to a size much smaller than the liberation size may also bring some disadvantages. Besides the higher power consumed in grinding, too fine particles or slimes are difficult to be recovered, which will also produce larger volume of tailing discharge and make pollution control more serious. The general objective of grinding circuit is to maximize the fresh ore feed rate to achieve high energy efficiency. The control schemes mainly applied in industrial fields includes PID control [1], model predictive control (MPC) [1-3], expert control [4-6], etc. Although these schemes have made some success in grinding circuits, there still exits space to improve the control strategies for higher energy efficiency and pollution control. First, most of the control schemes keep the particle size unchanged no matter whether the properties of ore have changed or not. In fact, for the ore with lower sulfur, for example, the ore can be ground to a coarser size. In the presence of a coarser size, the fresh ore feed rate can be increased and the energy can be then saved. Second, a mill needs to replace liner every half to one year. During the early days after the liner has been replaced, the effective grinding volume is relatively smaller, the feed rate needs to be controlled carefully to avoid overload. While in the later stage, the feed rate are allowed to increase since the effective grinding volume has enlarged due to wear and tear of the liner. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 409–415, 2010. © Springer-Verlag Berlin Heidelberg 2010

410

X. Chen et al.

This paper introduces an improved control strategy for ball mill grinding circuits. A two-layer optimization architecture including particle size optimization and energy optimization is introduced. Different operation conditions are considered to maximize fresh ore feed rate and to reduce the tailing discharge. The paper is organized as follows: it starts with a brief description of the ball mill grinding circuit. An improved control strategy combined by a two-layer optimization architecture is then formulated. Simulation study is constructed and analyzed, and conclusions are given finally.

2 Ball Mill Grinding Circuits Description The typical grinding circuits operate in a closed loop as shown in Fig. 1, which consists of a ball mill, a pump sump, hydrocyclones and associated pumps and solids feeding conveyors. product vibratory conveyor

ore

Hydrocyclones

Circulating load

water

Ball mill water Pump sump

pump

Fig. 1. Process diagram of grinding process

A ball mill consists of a cylindrical shell rotated about its axis by a motor. Heavy metallic balls called grinding media are loaded into the cylinder. The coarse ore (usually from primary crusher) is fed into the mill together with water in wet grinding. The tumbling action of the balls within the revolving mill crushes the feed to finer sizes. The slurry containing the fine product is discharged from the mill to a pump sump, and then pumped to hydrocyclones for classification. The slurry is separated into two streams: an overflow stream containing the finer particles and the underflow stream containing the larger particles as circulating load. The overflow is the desired product. The circulating load is recycled back to the ball mill for further grinding [3].

3 Improved Control Strategy The schematic diagram of the improved control strategy for ball mill grinding circuits is shown in Fig. 2.

An Improved Control Strategy for Ball Mill Grinding Circuits

411

Setpoints

Particle size optimization layer Optimization

Constraints

Subsequent measured variable: recovery rate , grade,tailing discharge , etc. Setpoints for base loops

Energy optimization layer

Disturbance observer based multi-loop control (DOMC)

Base regulatory loops

Measured variables

Dynamic modeling

Manipulated variables Grinding circuits

Grinding circuits

Fig. 2. Schematic diagram of the improved control strategy for grinding circuits

3.1 Particle Size Optimization Layer On the top of the architecture in Fig.2 is the optimization layer in which there are two sub-layers: the particle size optimization layer and energy optimization layer. The particle size optimization layer is to find the optimal particle size set-point since the Prior operation

Subsequent operation

Crusher circuits

Grinding circuits

Floatation

Magnetic concentration

Sp

Setpoints for crusher

Particle size optimization

Setpoints for floatation , magnetic concentration , etc.

Production setpoints optimization

Assay data, G fo , Gdo , Rr , etc. Fig. 3. Particle size optimization layer for grinding circuits

412

X. Chen et al.

ore properties shift from time to time in real practice. A too fine particle size of grinding circuit not only consumes too much energy but also causes more tailing discharge which leads to lower metal recovery rate (Rr). From the standpoint of energy saving and pollution reduction, it is necessary to regulate the particle size set-points with the variation of ore properties. In fact, experienced operators can regulate the particle size set-points and other set-points according to the laboratory assay data of fine ore grade (Gfo), discharged ore grade (Gdo) and Rr, etc, as shown in Fig. 3. The prior process mainly refers to crusher operation, while the subsequent operations usually include floatation process and magnetic process, etc. The optimal particle size set-point for a grinding circuit is based on Gfo, Gdo and Rr and so on according to library assay. Variables are fuzzifyed as shown in Fig 4, where H, M and L represents high, medium and low respectively.

Gfo Gdo Rr

M

L

-3

-2

-1

0

1

H

2

3

Fig. 4. Process diagram of grinding process

An expert system can be constructed here according to our early paper [6]. The knowledge base is established by production rules whose basic pattern is “IF conditions THEN results”, where the condition is a characteristic state of the process or a logic combination of characteristic states and the result is an operation. Some main rules include: R1: IF G fo = H AND Gdo = H AND Rr =L THEN DEC S *p

R 2 : IF G fo = H AND Gdo = M AND Rr =L THEN DEC S *p R 3 : IF G fo = L AND Gdo = H AND Rr = L THEN INC S *p R 4 : IF G fo = L AND G do = M AND Rr = L THEN INC S *p

where S *p represent the set-point of particle size, while INC and DEC represent increase and decrease respectively. Other rules to regulate S *p can also be integrated in the expert system according to process knowledge. 3.2 Energy Optimization Layer The energy optimization layer is the second sub-layer, and the detailed illustration of this layer can be found in paper [6]. Here we introduce a modified control strategy adaptive to the wear of the liner in the mill to maximize energy efficiency.

An Improved Control Strategy for Ball Mill Grinding Circuits

413

Normally, the liner in a ball mill should be replaced every half to one year. The effective grinding volume is quite small immediately after the liner has been replaced. It is desired during this stage to decrease fresh ore feed rate to avoid overload. On the contrary, fresh ore feed rate can be increased during the later stage since the liner has been worn greatly and the effective grinding volume has been enlarged. For simplicity, three working modes have been divided here according to different stages.

⎧ Mode I ⎪ Mode = ⎨ Mode II ⎪⎩ Mode III

t ≤ tsp1 t sp1 < t ≤ tsp 2 tsp 2 < t

(1)

where tspi (i=1,2) is different time to divide the mill into 3 working stages. The control strategy is illustrated as the following

⎧ Mode I ⎪ ⎨ Mode II ⎪⎩ Mode III

F f 1 = Ψ1 ( F f 0 ) Ff 2 = F f 0 Ff 3 = Ψ 2 ( Ff 0 )

(2)

where Ffi (i=1,2,3)is the ideal resetting value (IRV) of fresh ore feed rate for the early, medium and later stage after a liner has been replaced for the mill; Ff0 is the nominal fresh ore feed rate, and Ff1, Ff2 are linear functions of Ff0. It should be pointed out that the working stages can be more divided which will be more adaptive to the real industrial practice of course.

4 Simulation Study PID or MPC control strategies demonstrate limited abilities in control ball mill grinding circuits in the presence of strong disturbances, so we integrate disturbance observer based multi-variable control (DOMC) here proposed by us before[7,8]. Consider the following process model [7]

⎡ S p ( s ) ⎤ ⎡ G11 ( s ) G12 ( s ) ⎤ ⎡ Ff ( s ) ⎤ ⎢ F ( s ) ⎥ = ⎢G ( s ) G ( s ) ⎥ ⎢ F ( s ) ⎥ ⎣ c ⎦ ⎣ 21 ⎦⎣ d ⎦ 22

(3)

where Sp, Fc, Ff , Fd is the particle size, circulating load, fresh ore feed rate and dilution water flow rate respectively, and

G11 ( s ) = g11 ( s )e −τ11n s =

−0.58 −0.68 s e 2.5s + 1

G12 ( s ) = g12 ( s )e−τ12n s =

4(1 − 0.9938e−0.47 s ) −0.2 s e (2s + 1)(6s + 1)

G21 ( s ) = g 21 ( s )e −τ 21n s =

2.2 −0.6 s e 6s + 1

G22 ( s ) = g 22 ( s )e −τ 22 n s =

2.83 −0.13s e 3.5s + 1

414

X. Chen et al.

Set the objective function k +P

k + M −1

j = k +1

j =k

min Φ k = ∑ [Y ( j | k ) −Y * ]Q[Y ( j | k ) − Y * ] + ∑ {[U ( j | k ) −U * ]R[U ( j | k ) + ΔU T ( j | k ) S ΔU ( j | k )} s.t. U min ≤ U ( j | k ) ≤ U max , j = k ,...k + M − 1

(4)

ΔU min ≤ ΔU ( j | k ) ≤ ΔU max , j = k ,...k + M − 1 where Y

= [ y1 , y2 ] are the controlled variables of Sp, Fc, and Y * = [ y1* , y2* ] are the

set-points of Sp and Fc,.

U = [u1 , u2 ] are the manipulated variables, and

U * = [u1* , u2* ] are the IRVs of Ff and Fd. Q is the error weighting factor vector , and

Particle size (%-200mesh)

R is the input weighting factor vector.

72

71.5

71

Fresh ore feed rate (t/h)

0

20

40 60 Time (min) (1)

80

100

80

100

76 74 72 70 68 0

20

40 60 Time (min)

(2) Fig. 5. Effect of particle size (1) and ore feed rate (2) after the implementation of control strategy

An Improved Control Strategy for Ball Mill Grinding Circuits

Assume the current working stage of the ball mill is Mode II, and

415

u1* = F f 0 is 75t/h,

y1* = S *p is changed from 72% (-200mesh) to 71% (-200mesh) after the proposed optimization has been put into operation. Fig. 5 shows the particle size and fresh ore feed rate curves respectively. As shown in Fig.5, the fresh ore feed rate has increased from the original 75t/h to more than 76t/h. When the working mode has changed from Mode II to Mode III, fresh ore feed rate will be observed also increased similar to Fig.5 which is omitted here for space. It is obvious that the fresh ore feed rate has been improved and then the energy efficiency has been increased.

5 Conclusions An improved control strategy is proposed to control ball mill grinding circuits for energy saving and pollution reduction. A two-layer optimization architecture including particle size optimization layer and energy optimization layer is put forward, in which the optimal particle size is calculated first, and then the energy optimization is carried out. A control method adaptive to the wear of liner in ball mills is also proposed. Simulation studies demonstrate that the tonnage has been increased and so the energy efficiency has been increased. However, compared with other control and optimization strategies, the proposed method may need a long-term data collection and processing from experienced experts or operators. Acknowledgments. This work is supported by National High-Tech Research and Development Program of China (Project 2009AA04Z140), Provincial Natural Science Foundation of Jiangsu (BK2009270) and a grant from the PhD Programs Foundation of Ministry of Education of China (20070286040).

References 1. Pomerleau, A., Hodouin, D., Desbiens, A., Gagnon, E.: A survey of grinding circuit control methods: from decentralized PID controllers to multivariable predictive controllers. Powder Technology 108(2), 103–115 (2000) 2. Ramasamy, M., Narayanan, S.S., Rao, C.D.P.: Control of ball mill grinding circuit using model predictive control scheme. Journal of Process Control 15, 273–283 (2005) 3. Xisong, C., Junyong, Z., Shihua, L., Qi, L.: Application of model predictive control in ball mill grinding circuit. Minerals Engineering 20(11), 1099–1108 (2007) 4. Yianatos, J.B., Lisboa, M.A., Baeza, D.R.: Grinding capacity enhancement by solids concentration control of hydrocyclone underflow. Minerals Engineering 15, 317–323 (2002) 5. Christophe, B., Carol, B., Alain, B., van Wayne, D.: Advanced control of gold ore grinding plants in South Africa. Minerals Engineering 18, 866–876 (2005) 6. Xisong, C., Qi, L., Shumin, F.: Supervisory expert control for ball mill grinding circuits. Expert Systems with Applications 34(3), 1877–1885 (2008) 7. Xisong, C., Jun, Y., Shihua, L., Qi, L.: Disturbance observer based multivariable control of ball mill grinding circuits. Journal of Process Control 19(7), 1025–1031 (2009) 8. Jun, Y., Shihua, L., Xisong, C., Qi, L.: Disturbance rejection of ball mill grinding circuits using DOB and MPC. Powder Technology 198(2), 219–228 (2010)

Sliding Mode Controller for Switching Mode Power Supply Yue Niu1, Yanxia Gao1,*, Shuibao Guo1,2, Xuefang Lin-Shi2, and Bruno Allard2 1

School of Mechatronical Engineering and Automation, Shanghai University 200072 Shanghai, China {jerriler,gaoyanxia}@shu.edu.cn, [email protected] 2 Lab. AMPERE (CNRS UMR 5005)-INSA-Lyon, Villeurbanne Cedex – France {xuefang.shi,bruno.allard}@insa-lyon.fr

Abstract. Switching Mode Power Supply (SMPS) is a nonlinear system in nature, for which the performance of linear control strategy is constrained. Therefore, this paper presents a sliding mode controller for SMPS. Due to its simplicity and robustness against parameter variations and disturbances, Sliding Mode Control (SMC) has been widely used in DC-DC power conversion applications. This paper takes a Buck converter as an example to analyse the SMC application in SMPS. The simulation and experimental results are presented. Keywords: Sliding Mode Control, Buck Converter, DPWM.

1 Introduction In order to reduce the size of passive components and improve system performance, recently digital controller has become an attractive research topic in SMPS for high performance applications. The digital PID control is simple and easy to implement at low cost and calculation time. However, due to the nonlinearity characteristics of SMPSs, it is difficult for a linear PID controller to offer high dynamic performance over wide operating conditions and parameter range. Some auto-tuning process of the digital PID controller are proposed [1-2]. Unfortunately on-line tuning solutions increase algorithm complexity and thus the silicon area of the IC controller and power consumption as well. As SMPSs represent a particular class of Variable Structure Systems (VSS), nonlinear control methods, especially a Sliding-Mode-Control (SMC) [3] can enhance the dynamic performances of SMPSs. One feature of SMC is its robustness against parameter uncertainties. Moreover it is particularly suitable for high-frequency SMPS application as the control behavior approaches the ideal SMC with the increase in the switching frequency [3]. To acquire high performance for VSS, an ideal SMC must be operated at high enough switching frequency on the sliding surface. Thus the nature of the SMC is to ideally operate at an infinite switching frequency such that the controlled variables can track a certain reference path to achieve the desired dynamic response and steady-state *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 416–424, 2010. © Springer-Verlag Berlin Heidelberg 2010

Sliding Mode Controller for Switching Mode Power Supply

417

operation [4-6]. Hence for SMC to be applicable in power converters, their switching frequencies must be constricted within a practical range to meet practical application. The most popular way to implement SMC for switching power converters is Hysteresis-Modulation (HM) implementation, which requires a bang–bang type controller to perform the switching control [4-6]. The introduction of a hysteresis band with the boundary conditions S(x) =σ and S(x) = −σ provides a form to limit the infinite high switching frequency. However, due to the lack of systematic design methods and implementation criteria, the implementation of HM-based SMC for SMPS still relies on the trial-and-error tuning of the σ value to achieve the desired control performance.. Moreover, the HM-based SMC operating switching frequency is variable, which cases the difficulties to design a filter. By contrast, the PWM-based SMC presented for SMPS application in this paper, can limit the finite high switching frequency on one hand, and can fix the variable switching frequency on other hand.. This paper details with the design of PWM-based SMC for a Buck converter. Section 2 outlines the procedure of application SMC to DC-DC converters. Section 3 surveys the sliding-mode controller based on Buck converter. Section 4 presents the MATLAB simulation results, the test platform and the experimental results.

2 Digital Sliding Mode Control for DC-DC Converters To fix the variable switching frequency, most of existing sliding mode control is realized by analog PWM-based controller. Unfortunately analog PWM-based SM controller is highly sensitive to analog component variations, and it can be mixed only with analog signals in the converter and not with signals inside a high-level digital power management unit. By contrast, digital PWM-based SM controller is insensitive to analog component variations and offers more flexibility. 2.1 The Switching Control Law The basic principle of SMC is to force the state to move toward and to reach a chosen sliding surface S in finite time. During the sliding mode, the trajectory tends asymptotically to the equilibrium point. The system dynamic behavior is controlled by the location of the sliding surface in the state space. Considering a DC-DC converter is a general SISO (single-input-single-output) autonomous nonlinear system, the sliding mode controlled system can be modeled by:

⎧ xٛ = f ( x ) + g ( x )u ⎨ T ⎩ S ( x) = K x

∈

(1)

Where x Rn is the variable vector of SMC, f and g are state vectors defined on Rn , S is the sliding surface, K = [ K1 , K2 , , Km ] ∈ R is the sliding gains parameters. In order that the sliding motion can exist, it is necessary to determine the switching control law. Since there are only two available choices “ON” ( u+ =1) and “OFF” (u= 0 ) for switch action in DC-DC converters, then the method can be only selected in equations (2) and (3) as shown: T

T

T

n× m

418

Y. Niu et al.

⎧0, S < 0 u=⎨ ⎩1, S > 0

(2)

⎧1, S < 0 u=⎨ ⎩0, S > 0

(3)

2.2 Equivalent Control

In [3], a relationship between HM-based SM controller and PWM-based SM controller has been established. Two key results are useful: First, in the SMC, the discrete control input u can be theoretically replaced by a smooth function known as the equivalent control signal ueq. Second, at a high switching frequency, the equivalent control is effectively a duty-cycle control. Based on this useful establishment, the control input u can be modified into a pulse width modulation as ueq during every duty cycle with two switching input u+ and u−. An equivalent control law which forces the system state variables trajectory to the sliding surface can represent the duty ratio control. When the state variable phase trajectory is on the sliding surface, the following conditions must be satisfied: ٛ

S ( x ) = 0, S ( x ) = 0

(4)

Thus the equivalent control signal ueq can be described as: −1

ueq = − ⎡⎣ K T g ( x ) ⎤⎦ K T f ( x )

(5)

2.3 PWM-Based SMC

The conventional PWM control is widely used in DC-DC converters, in which the control input u is switched between ‘1’ and ‘0’ once per switching cycle for a fixed small duration Δ . The time instance where the switching occurs is determined by the sampled value of the state variables at the beginning of each switching cycle. Duty ratio d is then the fraction of the switching cycle in which the control holds the value 1. It is normally a smooth function of the state vector x, and it is denoted by d(x) , where 0 < d(x) <1. Then for each switching cycle interval Δ during the time (τ, τ + Δ), the control input u can be written as

⎧ u + = 1 τ ≤ t < τ + d ( x)Δ u=⎨ − ⎩u = 0 τ + d ( x)Δ ≤ t < τ + Δ

(6)

Hence, in the ideal PWM control,

ueq = d ( x)

(7)

Sliding Mode Controller for Switching Mode Power Supply

419

2.4 Existence Conditions

To achieve the design specifications, the SMC must maintain the variables’ state trajectory on the surface for all subsequent time and the state trajectory slides along this surface. Thus it is necessary to determine the sliding motion range for the SM controlled power converter, i.e. to ensure the existence conditions for sliding surface. This task is performed using the Lyapunov’s second method, where the Lyapunov’s function is generally defined as V = (1/ 2) S ( x) 2 . The existence conditions must be satisfied: S ( x ) S ( x ) < 0 . Thus the existence conditions of SMC can be written as: i ⎧ S ( x) > 0 S i S < 0, i.e,. ⎨ ⎩ S ( x) < 0

if S ( x) < 0 if S ( x) > 0

(8)

3 Sliding Mode Control for a Buck Converter The Buck converter is one type of the basic DC-DC converters used in industries. We present a practical design of PWM-based SMC application -for a buck converter. Our focus in this design is the application of SMC to converter operating in continuous current mode (CCM), and the discontinuous current mode (DCM) is not discussed here.

Fig. 1. The buck converter

Taking the output voltage error and its differential and integral portions as the state variables, the Buck converter shown in Fig.1 can be described as a form of state-space description:

xٛ = Ax + Bu + D Assuming that

(9)

Vref is constant and capacitor ESR is zero, then

⎡0⎤ ⎡ 0 ⎡ 0 ⎤ 1 0⎤ ⎢Vref ⎥ ⎢ 1 ⎢ Vin ⎥ 1 ⎥ A= ⎢− − 0⎥, B = ⎢− ⎥, D= ⎢ ⎥, u∈{0,1} ⎢ LC⎥ ⎢ LC RC ⎥ ⎢ LC⎥ 0 0⎦ ⎣ 1 ⎣ 0 ⎦ ⎣0⎦

(10)

420

Y. Niu et al.

Where L is the inductance value, R is the load resistance, C is the capacitance value, and Vin represents the input voltage. 3.1 System Modeling for Sliding Mode Controller

The surface of the SMC at the stable equilibrium point, S (x) = 0 can be detailed as: S ( x ) = K T xٛ

(11)

S( x ) = k1 x1 + k2 x2 + k3 x3 = k1 x1 + k2 xٛ1 + k3 ∫ x1dt = 0

(12)

Or

Where, x1, x2 and x3 are respectively the voltage error and its differential and integral portions. As mentioned above, the equivalent control signal ueq has been proven equivalent to PWM duty ratio d for SMC. Then the derivation of the PWM-based SMC can be achieved by the equivalent control on the sliding surface S. Substituting x = V − V and x = x = − dV / dt into equation (5), then: 1

ueq =

1 Vin

ref

out

2

1

out

⎡ ⎤ ⎛ k1 1 ⎞ dVout ⎛k 1 ⎞ + LC ⎜ 3 − ⎢Vref − LC ⎜ − ⎟ ⎟ (Vref − VOUT ) ⎥ ⎢⎣ ⎥⎦ ⎝ k2 RC ⎠ dt ⎝ k2 LC ⎠

(13)

Where ueq is continuous and equals to PWM duty ratio d , and 0 < ueq =d <1 , parameters k1/k2 and k3/k4 are to be determined which corresponds to the desired SMC dynamics that will be discussed. 3.2 Determination of SMC Parameters

The selection of sliding parameters should be based on the desired dynamic properties. In this way, the sliding motion ensures that the state trajectory of the system under SMC operation will always reach a stable equilibrium point. According to Ackermann’s Formula for a standard second-order system dynamics, equation (12) can be transformed into: xٛ1 + 2ζωn xٛ1 + ωn 2 x1 = 0

(14)

Where, ω = k / k is the undamped natural frequency and ζ = k / 2 k k is the damping ratio. Thus the sliding parameters now depend on the desired dynamic performance that can be chosen by user. n

3

2

1

2

3

3.3 Derivation of Existence Conditions

Recalling of the existence conditions in equation (8), the existence conditions must be satisfied. It is obvious that x1, x2 and S(x) construct the sliding surface in a 3-D space, which is the existence range of sliding motion for SMC. In order to simplify the analysis of the existence, the 3-D space can be mapped onto the 2-phase-plane (x1, x2).

Sliding Mode Controller for Switching Mode Power Supply

421

Assuming K2 is positive, the existence range can be represented with the two lines in x1, x2 phase-plane: Vref −Vin ⎧ ⎛ K3 1 ⎞ K1 1 <0 ⎪λ1 (x) = ( − )x2 + ⎜ − ⎟ x1 + K2 RC LC ⎪ ⎝ K2 LC ⎠ ⎨ ⎪ λ (x) = ( K1 − 1 )x + ⎛ K3 − 1 ⎞ x + Vref > 0 ⎟ 1 2 ⎜ ⎪ 2 K2 RC LC ⎝ K2 LC ⎠ ⎩

(15)

Thus the two lines λ ( x) =0 and λ ( x) =0 respectively determine the two boundaries of sliding surface existence range. Detailing equation (8) leads to: Case: S → 0 ,u = 1 then: S < 0 1

+

2

+

K ⎞ K ⎞ K ٛ ⎛ ⎛ S ( x ) S →0+ = ⎜ K1 − 2 ⎟ x2 + ⎜ K3 − 2 ⎟ x1 + 2 (Vref − Vin ) < 0 RC ⎠ LC ⎠ LC ⎝ ⎝ Thus, the parameters are K / K < 1/ ( RC) and 1

2

(16)

K 3 / K 2 < 1 / ( LC ) .

3.4 Stability of SMC

The stability is an important item which is to ensure that the sliding surface will direct the state trajectory toward the stable equilibrium points in existence regions. According to equation (12), the S (x) = 0 can be rewritten in Laplace form as:

s2 +

K1 K s+ 3 =0 K2 K2

(17)

Firstly, using the Routh’s stability criterion to this second order linear polynomial Δ to determine the stability conditions, s2 Δ= s

1

s

0

1

K3 / K2

K1 / K 2 K1 K 3 / K 2

(18)

0 2

0

The condition for the stability must meet Δ > 0, which means all the coefficients x1, x2, x3must be with the same sign. Secondly, extracting the time differential, the S (x) = 0 can be rearranged into a stand second-order system form:

s 2 + 2ζωn s + ωn 2 = 0

(19)

Where the values of damping ratio ζ is:

ζ = K1 / 2 K 3 K 2

(20)

Since ζ > 0 then K1 must be positive. Recalling that K1, K2, K3 must be with the same sign, thus finally all the coefficients should be positive for stable operation.

422

Y. Niu et al.

4 Simulation and Experimental Results The time domain behavior of the proposed SMC is verified on a buck converter using Matlab/simulink, where the Buck converter circuit elements are: Vin=3V, L=4.7uH, C=22uF, R=5Ω and switching frequency fs=1MHz. The simulation mode is shown as Fig.2. To comply with the design equations regarding with the stability and the transient response, the choice for system dynamic performance should be considered with the trade-off between ωn and ξ , where large of simulations have been carried out. Finally we set the bandwidth of the SMC response fω at one-fifteenth of the switching frequency f s , i.e., f s = 15 × ωn ( 2π ) , and choose the damping ratio ξ = 1 . Thus the sliding parameters are determined as: K1 K 2 = 4π f s 15 and K3 K 2 = 4π 2 f s 2 152

Fig. 2. The model of the SMC for a buck converter

Fig.3 shows that the output voltage follows the reference by a slope function until steady state. Then the load suddenly changes from 0.3A to 0.46A (R: 5Ω→3.3Ω), Fig. 4 shows the transient response of the SMC at switching frequency 1MHz. It can be seen that the dynamic response of SMC is satisfying, where the transient response is very fast. Vout

1.5058

2.5

transient output voltage at 4MHz

1.5037 1.5012 1.4987 output Voltage: V / div

2

1.5

1

1.4962 1.4937 1.4912 1.4887 1.4862 1.4837 1.4812

0.5

1.4787 1.4762

0 0

1

2

3

Fig. 3. Output voltage Vout

Time

4

1.4737

2.5

2.55

2.6

2.65

2.7 2.75 Time: s / div

2.8

2.85

2.9 -4

x 10

Fig.4. Dynamic response of SMC when load changes from 0.3A to 0.46A

The functionality of the controller designed and proposed DPWM are experimentally verified using a discrete buck converter with 3.0V input and 1.5V output voltage. The voltage feedback is performed by a 10-bit A/D converter AD9203, and the digital PWM is realized in an 11-bit hybrid DPWM architecture [10]. The implementation of the proposed digital controller DPWM-based is performed on a Xilinx XC2VP30 FPGA with an external 32MHz system clock. A

Sliding Mode Controller for Switching Mode Power Supply

423

VHDL design approach is used to synthesize the controller with Xilinx ISE development software. The test platform is pictured in Fig.5.

Fig. 5. Experimental test platform

Fig.6 shows the steady-state output voltage and the corresponding PWM signal in steady state operation at 4MHz. It can be seen that the system has favorable steadystate performance. Fig.7 shows the transient output voltage of SMC

when load

varies from 0.3A to 0.46A. These results prove that the dynamic response of SMC is very fast (<8.6µs) and the offset on the output voltage is also very small (<19mV). Clearly, the proposed DPWM-based SM controller is with good disturbance rejection and fast speed of dynamic response.

Fig. 6. Output voltage and the corresponding PWM signal

Fig. 7. Transient output voltage of SMC

5 Conclusion This paper presents a fully synthesizable PWM-based digital SMC for low-power (some watts) and high-frequency (MHz and over) SMPS. By employing the proposed hybrid DPWM, the digital SM controller can accommodate SMPS with high control performance at high constant switching frequency. The FPGA implementation is tested inside a buck converter (step-down SMPS) at the switching frequency 1MHz. The experimental results validate the functionality of the proposed PWM-based digital SM controller.

424

Y. Niu et al.

Acknowledgments This research is co-supported by Shanghai University “11th Five-Year Plan” 211 Construction Project, Shanghai Key Laboratory of Power Station Automation Technology and French Government Scholarship (No.K06D20).

References 1. Stefanutti, W., Mattavelli, P., Saggini, S., Ghioni, M.: Auto-tuning of Digitally Controlled Buck Converters Based on Relay Feedback. In: Proc. IEEE PESC Conf., pp. 2140–2145. IEEE Press, Los Alamitos (2005) 2. Zhao, Z., Lee, H., Feizmohammad, A.: Limit-cycle Based Auto-tuning System for Digitally Controlled Low-power SMPS. In: Proc. IEEE APEC Conf., pp. 1143–1147. IEEE Press, Los Alamitos (2006) 3. Tan, S., Lai, Y., Tse, C.: General Design Issues of Sliding-mode Controllers in DC-DC Converters. IEEE Transactions on Industrial Electronics 55(3), 1160–1174 (2008) 4. Castilla, M., de Vicuna, L.C., Lopez, M., Lopez, O., Matas, J.: On the Design of Sliding Mode Control Schemes for Quantum Resonant Converters. IEEE Transactions on Power Electronics 15(6), 960–973 (2000) 5. Ackermann, J., Utkin, V.: Sliding Mode Control Design Based on Ackermann’s Formula. IEEE Transactions on Automatic Control 43(2), 234–237 (1998) 6. Utkin, V., Guldner, J., Shi, J.X.: Sliding Mode Control in Electromechanical System. Taylor and Francis, London (1999) 7. Perruquetti, W., Barbot, J.P.: Sliding Mode Control in Engineering. Marcel Dekker, New York (2002) 8. Tan, S.C., Lai, Y.M., Tse, C.K., Cheung, M.K.H.: A Fixed-Frequency Pulse width Modulation Based Quasi-Sliding-Mode Controller for Buck Converters. IEEE Transactions on Power Electronics 20(6), 1379–1392 (2005) 9. Castilla, M., de Vicuna, L.C., Lopez, M., Lopez, O., Matas, J.: On the Design of Sliding Mode Control Schemes for Quantum Resonant Converters. IEEE Transactions on Power Electronics 15(6), 960–973 (2000) 10. Shuibao, G., Xue, F., Lin, S., Allard, B.: High-resolution digital PWM controller for highfrequency low-power SMPS. In: 13th European Conference on Power Electronics and Applications, EPE’09, pp. 1–9 (September 2009)

Expression of Design Problem by Design Space Model to Support Collaborative Design in Basic Plan of Architectural Design Yoshiaki Tegoshi1, Zhihua Zhang2, and Zhou Su3 1

Faculty of Environmental Studies, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku, Hiroshima 731-5193 Japan 2 Department of Human Life Science, Sanyo Women’ College, Sagatahonmati 1-1, Hatsukaichi, Hiroshima 738-8504, Japan 3 Faculty of Science and Engineering, Waseda University, Ohkubo3-4-1, Shinjyuku, Tokyo 169-8555, Japan [email protected]

Abstract. In collaborative design, the differences between designers are likely to occur in viewpoint and recognition gaps of design problems. It will spend time and labor by the time for designers to acquire the common awareness of the issues. In order to solve the problem and support collaborative design, we propose a structure with three features as an interaction process model: The features are the common view of design space, grasp of mutual search domain, and mutual evaluation of design constraints. Based on the interaction process model, collaboration design of zone arrangement of an elementary school was performed as the verification case, the conversation record and the design history which were accumulated on a server were analyzed and the usefulness of the model was verified. Keywords: Design Space, Knowledge Representation, Collaborative Design, Design Support Systems.

1 Introduction An architectural design is often socialized because of the complexity in general. It consists of designers or experts from various specialized fields, gathering to make a team and do the joint work. On the other hand, the inventiveness at early stage in design exerts a big influence on the following design. The collaborative design if introduced as two or more designers cooperate at this stage, various design proposals to which each designer's individuality taken are contrived and possible to be examined. Moreover, the possibility of design can be expected on the design proposal by designers who act each other on a creative impulse and development into new creative endeavors more than individually case. However, it comes to be going to spend labor and time by the time to obtain a common awareness of the issues because the difference and the perception gap of aspects to the design problems are caused easily between designers. In this cause, the individual variation is enumerated in the K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 425–433, 2010. © Springer-Verlag Berlin Heidelberg 2010

426

Y. Tegoshi, Z. Zhang, and Z. Su

idea process until the limiting condition and the design solution that there was a difference in the place of the design that each designer considers in the process of searching for the design solution, and each designer assumed are selected. From such a background in the collaborative design of the architectural initial designing stage, to cancel the cause that is the obstruction that designers obtain a common awareness of the issues and do the interaction effectively, it is necessary to discuss the mechanism for designers that can recognize the design space on the same design problem, to confirm the search domain and evaluate design constrains each other. To support the collaborative design in a basic plan of an architectural design, in this paper, after recognizing the place of the design together among designers, and understanding a search area each other, we will proposed a mechanism that can evaluate design constrains for designers each other by the interaction process. This is based-on the model of design space which is proposed for design problem and design knowledge representation. The interaction process in the design idea generation when the zone arrangement of the elementary school design cooperates and is designed based on this is considered.

2 Related Works In the field of architectural design, as for the frame that systematically makes to the division of labor and does the design work, it is put to practical use, and the frame that supports the coordinated design is requested. The attempt to support the coordinated design is done from various viewpoints like modeling the action analysis of the technical side of LAN construction technology for the coordinated design etc. and the group and design information and decision makings, etc. and achieves an effect. For instance, LAN construction technologies of both corners for coordinated design and operation technologies of network environmental setting and the attempt of web-based group work that use the Internet and information exchange technology [1] is introduced. And Sasada and Kaga et al. try to proposed method of supporting communications of the client with the designer between remote places which the design act was caught and adopted with a part of social process by the method [2], [3]. Moreover, Kitao et al. analyze adjustment person's adjustment act when two or more designers’ center on adjustment person's idea and the cooperation of labor is designed, and are proposing the design process where the design methodology is treated from a behavioral science aspect as a cooperation of labor design technique [4]. The support environment in which design information with different expression form can be smoothly exchanged is constructed to the exchange technology of design information corresponding to both of an asynchronous and synchronous work. Toizumi et al. constructed an environment that was computer-aided of the design conference to improve the inefficiency of the design information processing in the design team that did with synchronization [5]. H. Yamaguchi et al. [6] proposed modeling techniques of design knowledge in basic design of power plant equipment and represent a design problem firstly by concept of design space, which describes a search scope of design plans and design conditions. Z. Zhang et al. [7] proposed a model for knowledge and object management in manufacturing design of welding vessels based-on concept of assembly structure which is described as the combination

Expression of Design Problem by Design Space Model to Support Collaborative Design 427

of functional elements and their dependencies defined, and shows the possibilities of integrated representation and management about design objects, design knowledge, and design information for a knowledge-based manufacturing design support systems. X.F. Zha and H. Du [8] [9] developed a knowledge-intensive distributed design model and a framework for collaborative design modeling and decision support and test it on modular micro-robotic systems design. These researches are related to our object in the point of modeling the design problem, the coordinated activity between designers and the operation of design information etc. However, it is different from the generation case with the design solution by the joint design work based on the design space in our research.

3 Basic Concept 3.1 Design Space The concept of design space can be defined as following. A design proposal is treated as sets of attributes which concerned the performance, strength, and the structure, etc. about a design object in a concrete instance. The whole design proposal expressed by sets of these design attributes together with the restriction that it had to satisfy is called a Design Space [6]. A Design Space (Ds) can be formally represented by a quartet of Ds = {P, D, R, and C}. Where, P is a set of the design attributes or the design parameters; D is a set of the domain constraints on the value of the design variables; R is a set of the relations exist in the design attributes; and C is a set of limiting conditions that some of the design attribute should satisfy in design, called as design restriction. And the R is redefined by the functions F of design parameters when the modelling of a concrete design problem is necessary. In the architectural design field, it is necessary to think about the situation and the scene, etc., when the spatial domain is fixed for a design. Here, the range that design activity reaches or the design phase that a designer considered for a certain design problem is called as design place. The design place is a gathering of set of design spaces about a design problem. 3.2 View of Interaction Process in Collaborative Design Generally, a design process is a knowledge-intensive and collaborative activity, especially for some complex design problems which cannot be performed by a single designer work alone. This leads the increasing needs for the research of collaborative design. The continuing growth of the design knowledge and supporting information and the ever-increasing complexity of design problems have led to the increasing specialization. Integrated design requires skills of many designers/users and experts; each participant creates models and tools to provide information or simulation services to other participants given appropriate input information [8]. On the other hand, in our research group, when a designer teaches the beginners, it pays attention to the individual variation concerning the attitude of the design proposal to the same design problem, and the method of searching for a common idea is needed in the collaborative design.

428

Y. Tegoshi, Z. Zhang, and Z. Su

This research aims to develop a methodology based-on the concept of design space described previously, and provide a framework for collaborative design modeling and design support. Based on the model of design space, a design problem is expressed by using a quartet of {P, D, R, and C}, this is basically means the space that expresses one designer's design consideration when he/she think about and modeling a design problem. In the collaborative design, two or more designers join together to do the design problem modeling and there is a possibility of putting out a different design proposal from each standpoint. At this time, it is thought that the following individual variations are caused. Firstly, the individual variation concerning the domain modeling of the design problem exists. This controls the difference on the definition of design space about the design problem. Secondly, there exists the individual variation concerning the base attributes concerning the design. This influences the method of searching for the design solution. Thirdly, there exists the individual variation on the recognition concerning the design restriction by different design knowledge or design experience. This influences the accuracy of the design solution and the search efficiency of the design solution. Based on the analysis, we describe the concept of interaction design space (IDS, following) for collaborative design in Fig. 1. It shows a view of interaction process in collaborative design. IDS images the process of two or more designers cooperate and earning the common design solutions. Here, designer A, designer B and designer C has his own design plan individually for a given problem, and aim to get a common area of design plan as the design solution for the design problem. It is useful also for the student education that makes this an architectural design specialty. Design Place Joint Design Space

Designer B

1. 2.

Designer A Common area

3.

Recognize the place of the same design. Confirm a search area each other. Evaluate the limiting condition mutually.

Designer C

Fig. 1. View of Interaction Process in Collaborative Design (IDS)

In Fig.1, about each design problem, generally designers advance the collaborative design synchronously to get the proposal by steps as following from step 1 to step 3 by cooperating. Here, the advancing designing order is an overarching point. In general, the designers are often advancing the design in order of 1→3→2. For our model, the mechanism of the interaction of the design proposal is achieved by advancing the collaborative design in order of 1→2→3.

Expression of Design Problem by Design Space Model to Support Collaborative Design 429

3.3 Expression Example of Design Space An architectural design can express the search area by using the design attribute and the limiting condition in which the function and shape are shown. Figure 2 shows a zone generating problem and the representation of design space for the design of an elementary school construction, which own the parameter f in width and d in depth of the arranged zone area.

Fig. 2. A Representation of Design Space on the Zone Generating Problem

Here, to simplify the problem, the shape of zone is treated as a rectangle. The zone should fill area by consideration of design parameters that composes ap, the area of room, u, the kind of usage of the zone, hn, the numbers of people for one class, n, the number of classes, h, the number of users, and as, the area that can be accommodated. Figure 2(b) shows the IDS that designers agreed on the relational expression and the limiting condition between the kinds, the number of design attributes. To make the design solutions to be obtained easily, the viewpoint of thought of a designer change it based on his experience. For instance, in modeling the design problem by design space, he will pay attention to decide the main design attributes which are considered as first and important design parameters (following, called as base attribute) to decide the other design attributes. This is a process of the design attributes from which other design attributes is requested according to relations R between attributes. In the collaborative design, the perception gap to the search route is adjusted by mutually understanding the class of the base attribute. Therefore, cooperation by the progress process of the design becomes possible by understanding a search route among designers and the effect of the interaction that causes each proposal of a design is achieved. In addition, it becomes possible to make the search area narrow efficiently. For instance, in the zone generating problem mentioned above, the class of the base attributes that three designers assumed is shown as Figure 3 in (a), (b), and (c) respectively as the three group attributes of {u,h,n}, {u,hn,n} and {u,h,hn}. In this case, it is the situation of individual variation concerning the domain modeling on design problem. Based on IDS and the discussion of designers, the common one has to be selected cooperatively from the class. In this problem, the class in Figure 3(c) was selected. Based on this decision, next step in our systems, the design space will be described by the DSP knowledge representation language. This is

430

Y. Tegoshi, Z. Zhang, and Z. Su

Fig. 3. Class of the Base Attribute and Relations of the Attributes

a non-procedure type language that developed for the construction of design support systems. It corresponds to a chain of flows of the generation and verification method with a multivalent function (assumption generator) that generates the design solution definitely. Moreover, it has the development setting where the edit function of a catalog and numerical table to express the calculation function and the design material such as the development setting, the generation verifications, and reflexives of the knowledge base and the display functions of the design solution were organically united.

4 Application Example to Collaborative Design and Simulation It is applied the collaborative design to an elementary school construction problem. It went by the composition of three people a group by using the above-mentioned technique, and using the NC construction design system. Here, a basic plan of an architectural design is divided into three stages as to be the zone arrangement; the arrangement in space and the room arrangement. The interaction process records the search area of the conversation record and the design solution that the designer on the way did mutually in the data base on the server at each design stage. In the zone arrangement, usages of kind u and number of users h are assumed as the design condition which are arranged on the site of width fl=160m and dl=110m in depth. Next, the kind of the arranged zone is made the outdoor zone, the movement zone, the management zone, and a study zone. When the zone is arranged, it is assumed to be an arrangement condition that each zone doesn't come in succession mutually. Moreover, the position where the front gate is arranged is adjusted to the east side of the site with the frontal road. The passage and the stairs, etc. are made common space. A lot of design information is included in the conversation done among designers in the collaborative design, and it influences the problem solving. It is necessary to analyze the conversation record to analyze this design information, and to clarify the process of the problem solving done among designers. To analyze the conversation record, we divide the unit of the utterance that shows the meaning of the content as the same. The code to understand the interaction of the mutual act according to the type is made and the method of the protocol analysis was used. Here in this research,

Expression of Design Problem by Design Space Model to Support Collaborative Design 431

the attribute that composes IDS and each kind of the relational expression are encoded and the conversation record in the collaborative design is classified. The classification method is shown as Table 1. Utterance that shows feature and character of design object; Utterance that shows design domain by expertise; Utterance of common points that recognize place like progress and question; Utterance concerning design restriction that satisfies design objective; Remark that requests agreement to proposal of design formulas; are made to correspond to the design attributes of A, D, P, C, and R and the relational expression between them respectively. Table 1. Method of classifying conversation record Design protocol Type of design attributes

Code_A

Design domain

Code_D

Design phase

Code_P

Design restriction

Code_C

Relations between design Code_R attributes

Content Utterance that shows feature and character of design object Utterance that shows design domain by expertise Utterance of common points that recognize place like progress and question, etc. Utterance concerning design restriction that satisfies design objective Remark that requests agreement to proposal of design formulas

The conversation record of each phase of the design can be divided by arranging it by classifying according to the stepwise refinement of the design, and making them correspond to the design attributes that composes IDS. Here, the zone arrangement is assumed to be a configuration with IDS that generates each zone in the outdoor zone, the movement zone, the management zone, and the study zone and IDS that synthesizes the generated each zone.

Fig. 4. Result of Application Example to Collaborative Design and Simulation

432

Y. Tegoshi, Z. Zhang, and Z. Su

By the analysis of conversation record, designers do the collaborative design and simulation on real design problem. Figure 4 shows the result of the application example and the simulation to the elementary school zone design problem. Aspect ratios f/d of designer A in Fig. 4 are assumed from 1/6 to 6/1. 2/1 and designers C are from 1/3 to 3/1 in the aspect ratio that designer B assumed from 1/2. Designers evaluate an aspect ratio each other mutually, and obtain 3/1 from aspect ratio 1/6, and the design proposal is generating cooperatively verified, and IDS of (i) in Fig.4 .

5 Conclusion In this paper, for support the collaborative design of basic plan of Architectural Design, to cancel the cause of becoming an obstruction that obtains a common awareness of design solutions, we firstly represented design problem by design space model, and then by understanding the search area and the mechanism that is evaluated by designers each other, and proposed as an interaction process model. It was shown that IDS that satisfied the restriction that two or more designers had assumed as a result of analyzing the generation process of the design solution in the collaborative design existed. Moreover, designers confirmed IDS was synchronously recognized together as a result of having it the correspondence of IDS to the conversation record by the adjustment of the example. Designers were able to acknowledge the difference of the idea to the design proposal mutually. It was confirmed to be able to shorten time to be going to require the design proposal by generation by cooperating acknowledging the perception gap caused between designers. In addition, it is necessary to confirm the effectiveness of this method by adjusting to the example of design problem with the complexity or more as future tasks.

References 1. Morozumi, M., et al.: Construction of cooperated remoteness design support environment that uses gigabit network and the operated evaluation. In: 25th Symposium on Information, System, Use and Technology, Architectural Institute of Japan, pp. 85–90 (2002) 2. Kaga, A., et al.: Research of network type cooperation design support system, Idea of system construction. In: 19th Symposium on Information, System, Use and Technology, Architectural Institute of Japan, pp. 229–234 (1996) 3. Tomohiro, F., et al.: Research of network type cooperation design support system, Operation of system by World Wide Web. In: 19th Symposium on Information, System, Use and Technology, Architectural Institute of Japan, pp. 229–234 (1996) 4. Kitao, Y., et al.: Research of cooperation of labor design technique of new town by mastering architect method, –Adjustment act and role in environmental design process of mastering architect. Architectural Institute of Japan Plan System Thesis Collection 512, 183–190 (1998) 5. Toizumi, K., et al.: Research on support environment by computer of design conference, – Analysis of conference in architectural design and development of supporting system. Architectural Institute of Japan Plan System Thesis Collection 512, 293–299 (1999)

Expression of Design Problem by Design Space Model to Support Collaborative Design 433 6. Yamaguchi, H., Zhang, Z.H., et al.: Knowledge Representation Model for Basic Design of Power Plant Equipment. Transactions of Information Processing Society of Japan 41, 3180– 3192 (2000) 7. Zhang, Z.H., Nagasawa, I., et al.: A Design Object Model for Manufacturing Design of Welding Vessel Objects. Transactions of Information Processing Society of Japan 41, 123– 135 (2000) 8. Zha, Y.F., Du, H.: Knowledge-intensive collaborative design modeling and support: Part I: Review, distributed models and framework. Computers in Industry 57, 39–55 (2006) 9. Zha, Y.F., Du, H.: Knowledge-intensive collaborative design modeling and support: Part II: System implementation and application. Computers in Industry 57, 56–71 (2006)

Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles Behnam Ganji, Abbas Z. Kouzani, and H.M. Trinh School of Engineering, Deakin University, Geelong, Victoria 3217, Australia {bganj,kouzani,hmt}@deakin.edu.au

Abstract. This paper presents a drive cycle analysis of hybrid electric vehicle power train configurations. Based on fuel economy and emissions factors, a tradeoff between conventional, series hybrid, parallel hybrid, and a parallelseries hybrid is drawn. The operational characteristics of conventional and hybrid electric vehicles are evaluated from the standpoint of fuel economy and emissions. First, models are formed for conventional, series hybrid, parallel hybrid, and series-parallel hybrid vehicles. Then, the models are simulated for fuel consumption, powertrain average efficiency, and emissions using drive cycles containing target speeds in predefined patterns. The simulation results signify that hybrid electric vehicles achieve lower fuel consumption and emissions. Finally, the powertrain configurations that are suitable for city and highway driving are determined. Keywords: Hybrid electric vehicles, modeling and simulation, fuel consumption, emissions.

1 Introduction With the population now growing, the demand for petroleum continues to increase whilst oil supplies remain finite. At some future date, conventional oil supplies will no longer be able to satisfy the global demands. The oil production will peak and then commence to decline. There is no certainty when the peaking will occur, but it has been forecasted that it could happen soon [1]. According to a report in 2006, 80 million barrel petrol is used per day. The transportation section consumes 66% of this petrol and the portion of the land transportation is 56% of this quantity [2]. Consuming this amount of fuel continues to aggravate the environmental issues. Consequently, when it comes to energy security and climate change concerns, cars and trucks are considered one of the principal problems. They consume relatively a big portion of the oil in the world and emit a large amount of greenhouse gas (GHG). To overcome these problems, policymakers try to find solutions so that energy efficiency and emissions could be brought into focus at the United Nations Climate Change Conference in Copenhagen [3]. In response to the increase in the impacts of the vehicles on the environment and energy security, improving fuel efficiency can play a massive role. There are many techniques to improve the efficiency in vehicles. Over the past few decades, there has K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 434–444, 2010. © Springer-Verlag Berlin Heidelberg 2010

Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles

435

been significant development in automobile engine and body technology. Thus, there are little gains achievable in fuel economy through vehicle engine and body. This has resulted in the emergence of innovative technologies for automotive industry. Since pure electric vehicles do not have adequate source of energy for propulsion in relatively long distance ranges, and because their recharging process requires several hours, hybrid electric vehicles (HEVs) can be considered as alternative means of transportation. In comparison with conventional vehicles, fuel economy and reduced emissions have improved in HEVs [4-6]. As the HEV technology is considered as a successful solution to address some energy efficiency and emissions concerns about vehicles, by 2035 almost 80% of all vehicles introduced to the market will be hybrids, diesels, or turbocharged gasoline engines [3, 7-9]. An HEV enjoys a combination of at least two power sources whilst a conventional car employs only one source of propulsion. Although HEVs have made it possible to improve fuel efficiency and reduce emissions, considerable energy effectiveness can be achieved also with driving profile, environmental effects, driving behavior, control strategy as well as drive train configuration. The energy efficiency in vehicles can be investigated in two parts: (i) efficiency of fuel making process and fuel distribution system (well-to-tank efficiency-WTTE), and (ii) efficiency of the vehicle itself (tank-to-wheel efficiency-TTWE). Without considering the plug-in HEV option, improvements in “tank-to-wheels efficiency directly correspond to improvements in well-to-wheels efficiency” [10]. Table 1 shows the efficiencies in two categories of vehicles. It reveals the importance of TTWE and its impact on total efficiency. Table 1. Fuel efficiency of different vehicle configurations [11] Conventional car HEV

WTTE% 88 88

TTWE% 16 37

Total Efficiency 14 32

Simulation and computer modeling can be implemented to decrease the expense of a product from design to prototype and mass production. The interest in HEVs has generated a number of simulation software programs such as simple electric vehicle simulation (SIMPLEV) [12] from the DOE’s Idaho National Laboratory, MARVEL and PSAT from Argonne National Laboratory, CarSim from AeroVironment Inc., JANUS from Durham University, ADVISOR from the DOE’s National Renewable Energy Laboratory, Vehicle Mission Simulator [13], and others [14]. Among these software tools, PSAT and ADVISOR are more popular than the others. PSAT is a look forward simulator. It allows the user to model more than 200 predefined powertrains including conventional, pure electric, fuel cell and all the types of hybrid. The features such as control strategies including the propelling, breaking, and shifting strategies make it more accurate to predict the fuel economy and vehicles performance. In this study, PSAT is used to model and simulate the devised configurations. The contribution of this paper is the determination of the configuration of HEVs that is more appropriate for city driving and the configuration of HEVs that is more suitable for highway driving.

436

B. Ganji, A.Z. Kouzani, and H.M. Trinh

2 Hybrid Electric Vehicles An HEV is a complex system consists of many components. The necessary propulsion energy is generated and converted in some of these components which are known as powertrain or propulsion system. The ways of arrangement of these components determines the configuration of a powertrain. There are a vast range of feasible configurations for an HEV. However, based on the configuration of powertrain it can be classified by series hybrid, parallel and series parallel hybrid. 2.1 Series Hybrid In this type, which has the simplest topology, the vehicle acts like an electric car with an on-board generator as a battery charger. When driving at slow speeds, the controller draws power from the batteries to drive EM. In this case, the vehicle acts like a pure electric car. During acceleration, the internal combustion engine (ICE) drives the generator to compensate the power being drawn from the batteries. The generator provides power to run EM, and if necessary, additional power may be drawn from the generator to recharge the batteries. The energy from regenerative braking is converted into electricity and stored in the batteries. Since ICE is not coupled to the wheels, it operates in a narrow region at near optimum efficiency [15]. In this type, clutch and multi-speed transmission is omitted, so in comparison with a conventional vehicle, the fuel economy is increased and the emissions are decreased. Fuel economy of the series hybrid is dependent on road load and is investigated in this study. There are two stages of energy conversion during power flow to wheels, ICE/generator and battery/motor. These conversions cause a loss of energy. This issue is considered as a drawback for the series hybrid in comparison with parallel hybrid, thus making it more suitable for city driving. 2.2 Parallel Hybrid Depending on how the electric motor contributes in the vehicle propulsion, parallel hybrid can be divided to three groups: micro hybrid, mild hybrid and full hybrid. In micro hybrid, the vehicle has an electric motor but it does not supply additional torque when ICE runs. In other words, the electric motor does not contribute to propulsion. The electric motor provides functions such as starter/alternator, managing engine on/off, auxiliary loads, and the use of regenerative braking in order to charge the battery. Fuel economy of the micro hybrid is reported in the range of 5% to 15% [16]. In mild hybrid, EM contributes to providing supplementary torque to ICE, but it is not the only source of propulsion. This system also supports features such as starter/alternator, managing engine on/off, auxiliary loads, and the use of regenerative braking in order to charge the battery. Fuel economy of the mild hybrid is reported in the range of 15% to 25% [16]. In full hybrid, which is also called parallel hybrid, both ICE and EM are capable of producing the force that is need for the propulsion system. This approach eliminates the generator that is used in the series HEVs. In parallel HEVs, there are different

Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles

437

ways to configure the transmission system. When ICE is on, the controller distributes the energy between the propulsion and the energy storage system. The split of energy between the two is determined by the speed and the driving pattern. For example, under acceleration, more power is allocated to the drivetrain than to the energy storage system. During the periods of idle or low speed, more power is allocated to the batteries than the propulsion. When ICE is off, the parallel hybrid can run like a pure electric vehicle. The batteries provide additional power to the transmission when ICE is unable to produce enough energy to run auxiliary systems such as the air conditioner and heater. Similar to other types of HEVs, energy can be saved during regenerating breaking. The significant advantage of the hybrid is that less energy is wasted during conversion stages. It has been shown that fuel economy in the parallel hybrid is 4% better than that of the series hybrid [4]. However, fuel economy is dependent on road loads. Fuel economy of the parallel hybrid is reported around 40% [16]. The main drawback of the parallel hybrid is the complexity of its transmission and control system. 2.3 Series-Parallel Hybrid In this type, the advantages and complications of both series and parallel hybrids are combined. The engine can both drive the transmission directly (as in the parallel HEV) and be effectively disconnected from it (as in the series HEV). In this configuration, the engine can often operate at near optimum efficiency. At low speeds, the vehicle operates as a series vehicle, while at high speeds, where the series hybrid is less efficient, the vehicle acts like a parallel one. Since the system needs a generator, a larger storage system, and a complicated power flow control system, it has a higher cost than the other two hybrid types. However, the seriesparallel HEV has the potential to perform better than each of the two other systems alone. Fuel economy of the series-parallel hybrid is investigated in this study. In summary, considering the stated review of the popular hybrid types and their associated attributes, an outline of different power train configuration and the classification of hybrid vehicles is graphically presented in Fig. 1 and Fig. 2.

3 Modeling of Conventional and Hybrid Electric Vehicles In a car, it is difficult to predict the relations among its internal components and systems because of dynamic interactions among the components and also the components’ complex nature. Usually, prototyping and testing each design combination is expensive and time intense. Therefore, modeling and simulation are crucial for concept assessment, prototyping, and analyzing conventional and hybrid vehicles. Simulation and modeling tools model vehicle components in different level of details [14, 17]. Modeling based on time scale can be divided into static, quasi-static, low-frequency dynamic, high-frequency dynamic and detailed physics. PSAT, which is used in this study, is considered as a low frequency dynamic vehicle modeling software. It can be used to evaluate fuel economy, performance and emissions of conventional, pure electric, hybrid electric, and hybrid electric FC. Furthermore,

438

B. Ganji, A.Z. Kouzaani, and H.M. Trinh

PSAT is designed to connecct to other simulation tools with the aid of PSAT-PRO [[18] in order to be used as a hard dware-in-the-loop/software-in-the-loop testing program..

(a)

(b)

(c) Fig. 1. Power train configurattion. (a) Series HEV. (b) Parallel HEV. (c) Series-Parallel HE EV.

Fig. 2. Veehicle classification and hybridization concept

Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles

439

We have created models in PSAT for the following vehicles: conventional, series hybrid, parallel hybrid, and series-parallel hybrid. Each component corresponds to an actual vehicle. The initial values and other assumptions for the modeled conventional and hybrid vehicles are listed in Table 2. As an example, the block diagram description of the model for the parallel hybrid is shown in Fig. 3. The only traction power source in the conventional vehicle is ICE, but the electric motor provides an extra source of power. This feature makes HEVs more flexible and hence more efficient. In conventional vehicles based on the driver demand, ICE provides the necessary torque from chemical energy. It then flows through the powertrain in a forward direction to provide the traction power. Similarly in HEVs, the two sources of energy, chemical and electrical are used to supply traction demand. The traction force in the all models (Conventional and HEVs) is derived from the equation of solid body motion in the following form. 1 2

(1)

The modularity in the model is to facilitate the consequence between the vehicle components. There is a common format for the components. Each component has the same input and output parameters. For each module, there is an input from the controller, for instance, engine on/off command or gear shift number in transmission block. There is an output from each module. This output is used in powertrain controller as well as for post-processing. The second port (input and output) carry the effort such as voltage in electrical parts and torque in mechanical ones. The last port carries the flow for instance current or speed. Table 2. The initial values and assumptions Description Engine type IC Engine Power – kW Electric Motor #1 Peak Power-kW Electric Motor#1Cont Power – kW Electric Motor #2 Peak Power-kW Electric Motor#2Cont Power – kW Engine Torque Max – Nm Transmission type Final Drive Ratio Vehicle Mass (m) Fuel Heating Value Fuel Density Vehicle Frontal area (A)-m2 Vehicle Drag Coeff. (CD) Tire Rolling Res. (Crr) Wheel Radius (m) Engine Max Eff.

Conventional Si 100.52 173.2 Ct 4.07 1553.59 43000000 0.749 2.18 0.3 0.008 0.292 0.359

Series Si 100.52 170 81.32 173.2 No Trns. 3.55 1844.5312 43000000 0.749 2.18 0.3 0.009 0.348 0.365

Parallel Si 85.72 32.26 16.13 140.5 dm 4.44 1678 43000000 0.749 2.25 0.3 0.008 0.365 0.365

SeriesParallel Si 100.5 52.35 26 15.39 7.695 173.2 dm 3.77 1500 43000000 0.749 2.06 0.31 0.007 0.290 0.365

The driver block is used to model the accelerator and brake pedals. Through the driver block, power traction demand is imposed to the model. The desired vehicle

440

B. Ganji, A.Z. Kouzani, and H.M. Trinh

speed is compared with the actual speed, and a PI controller is used to request more or less torque to the vehicle. The control block supervises the components such that the demand can be guaranteed by the entire powertrain. The driver demand in vehicle simulation tools is determined by “drive cycle”. Drive cycle specifies the speed in a predefined pattern. Before prototyping, it is important to simulate the vehicle under different driving environments [19]. In recent years, several standard driving cycles have been developed. They are used to describe various driving modes in different regions. Some of them were developed to evaluate the statistical flow of traffic in urban areas. In contrast, there exist some which consider rural or cross-country driving. In order to investigate the fuel economy and emissions factors among the conventional powertrain and HEV configurations, two frequent drive cycles are used in this work - a highway (HWFET) and an urban drive cycle (UDDS). These drive cycles are shown in Fig. 4.

Fig. 3. The modeled parallel hybrid powertrain configuration

Fig. 4. Drive cycles used in our simulations. (left) Highway fuel economy driving schedule (HWFET). (right) Urban dynamometer driving cycle (UDDS).

Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles

441

Table 3. Simulation results

Description

Unit

HW FET

UDD S

HWF ET

UD DS

HW FET

UDD S

SeriesParallel HEV HWF UD ET DS

Cycle distance THERMAL INFORMATION Fuel economy

km

16.5

11.97

16.5

11.97

16.5

11.97

16.5

11.97

liter/10 0km liter/10 0km kg g/km g/km g/km g/km

6.13

8.59

4.74

3.42

4.54

5.459

4.475

3.885

6.18

8.67

4.79

3.45

4.61

5.601

4.619

3.925

0.73 0.06 1.06 0.04 120.7

0.78 0.17 3.67 0.09 175.1

0.59 0.07 0.98 0.09 111.51

0.31 0.11 1.85 0.07 78.35

0.56 0.035 0.435 0.037 118.6

0.601 0.118 2.360 0.067 138.9

0.565 0.062 1.058 0.039 107.5

0.347 0.162 3.210 0.069 87.56

% %

70 69.6

70 69.55

70 60.68

70 59.62

70 63.91

70 61.12

70 70.12

70 68.59

%

28.77

25.83

33.35

31.34

34.23

28.29

32.59

30.97

%

85

85

91.17

90.45

-

-

89.64

89.10

%

96.75

93.13

-

-

93.12

90.16

94.55

92.17

-

-

83.89

87.08

86.11

84.96

87.58

92.77

25.39

18.97

28.15

38.13

30.41

27.02

31.02

35.22

Conventional Vehicle

Fuel economy gasoline equivalent Fuel mass HC emissions CO emissions NOx emissions CO2 emissions ELECTRICAL INFORMATION Initial ESS SOC Final ESS SOC COMPONENT AVERAGE EFFICIENCIES Engine Bidirectional Efficiency Generator Bidirectional Eff. Transmission Bidirectional Eff. Motor Bidirectional Efficiency Powertrain Bidirectional Eff.

%

Series HEV

Parallel HEV

4 Simulation Two drive cycles are selected to study the behavior of different powertrain configurations in city and highway driving (see Fig. 4). From the point of view of efficiency and fuel economy, engine efficiency is usually less than 36% in all types. In the conventional vehicle, considering the overall losses in powertrain, total efficiency is less than 20% in cities. In this type, the produced torque and speed by ICE is dependent on the road load. Therefore, the engine cannot work in an efficient region at all times. However, in higher speeds and around highways speeds, the engine and powertrain run more efficiently. This matter is demonstrated in Fig. 5(a) by the density of operating points on the torque-speed map with the contours of efficiencies for the two tested drive cycles. The average efficiency shows more than 6% improvement in highway driving. Although emissions in the conventional vehicle are controlled effectively by using a fuel-injection system and catalysts in exhaust system, still they are significant

442

B. Ganji, A.Z. Kouzani, and H.M. Trinh

(a)

(b)

(c)

(d)

Fig. 5. Engine operating points (left: HWFET, right: UDDS). (a) Conventional. (b) Series HEV. (c) Parallel HEV. (d) Series-Parallel HEV.

Drive Cycle Analysis of the Performance of Hybrid Electric Vehicles

443

contributors to pollutant emissions. The simulation results reveal that the road load has an enormous effect on increasing emissions. Accordingly, HEVs can be considered as the appropriate alternative to conventional vehicles in order to reduce emissions and increase fuel economy. Due to the existence of electric motor in HEVs, the load can be balanced and ICE can be commanded to work near its optimal operating region. The density of operating points on torque-speed map with the contours of efficiencies for two drive cycles are shown in Fig. 5(b-d). In all HEV configurations, engine runs near its maximum efficiency to some extent (graphs reveal the dense points around the efficient regions on the map), thus comparing to the conventional vehicle they are more efficient. The obtained results in Table 3 and Fig. 5 approve this observation. However, selecting the appropriate powertrain configuration can influence the fuel economy and the amount of emissions. From the summarized results in Table 3, we can conclude that the series hybrid is more appropriate for city driving especially with a plug-in feature and for commuter drivers lower emissions can obtain [20]. In contrast, parallel HEV is less efficient than series type in stop-and-go driving. Therefore, this type is more suitable for highway driving; hence it can be a proper substitution for conventional vehicles in long distance and out of cities driving. With the expense of complexity, series-parallel brings together the advantages of both HEVs. From the obtained results this observation can be confirmed.

5 Conclusion This paper presented an overview of vehicles from conventional to hybrid and classified them according to the use of electricity in the propulsion system. Using two sources of energy in the propulsion system allows very diverse set of powertrain configurations. A conventional vehicle and three types of HEVs were modeled in this study. The technical characteristics of the models are comparable providing the ability to compare the simulation results. Two frequent standard drive cycles, HWFED and UDDS, were employed to simulate the models under urban and highway driving conditions. The simulation results demonstrate better fuel economy and lower emissions for all types of hybrid vehicles. The operating points of engines on efficiency maps for two drive cycles have been extracted. The graphs and calculated average efficiencies imply that the series hybrid performs better in city driving. In contrast, the parallel hybrid achieves lower emissions and fuel consumption in highway driving. This reveals that series HEVs are a proper substitution for conventional cars in city driving, and parallel HEVs for the speedy cars in highway driving. In series-parallel HEVs, we can achieve the advantages of both series and parallel configurations. In this type, the simulation results from the two drive cycles showed lower emissions and better efficiencies. However, this configuration is more complex; hence, its complicated control system and additional equipments augment its cost. Acknowledgment. The authors would like to thank AUTOCRC which provides financial support for this research.

444

B. Ganji, A.Z. Kouzani, and H.M. Trinh

References 1. http://www.issues.org/21.3/hirsch.html 2. Fuhs, A.: Hybrid vehicles and the future of personal transportation. CRC Press, Boca Raton 3. United Nations Climate Change Conference (December 2009), http://en.cop15.dk/ 4. Cuddy, M.R., Wipke, K.B.: Analysis of the fuel economy benefit of drivetrain hybridization, vol. 1243, pp. 101–111. SAE Special Publications (1997) 5. Riezenman: Engineering the EV future. IEEE Spectrum 35(11), 18–20 (1998) 6. Wouk, V.: Hybrids: then and now. IEEE Spectrum 32(7), 16–21 (1995) 7. Heywood, J., et al.: An Action Plan for Cars The Policies Needed to Reduce U.S. Petroleum Consumption and Greenhouse Gas Emissions (2009) 8. Bandivadekar, A., et al.: On The Roa. 2035: Reducing Transportation’S Petroleum Consumption and Ghg Emissions. MIT Laboratory for Energy and the Environment, Cambridge (July 2008) 9. Stauffer, N.: Doubling vehicle fuel economy by 2035: Technically feasible, challenging in scope. Energy Futures. MIT Energy Initiative (2008) 10. Katrašnik, T.: Analytical framework for analyzing the energy conversion efficiency of different hybrid electric vehicle topologies. Energy Conversion and Management 50, 1924–1938 (2009) 11. Toyota Motor Corporation.: Toyota hybrid system. Technical report (May 2003), http://www.toyota.co.jp 12. Cole, G.: Simple electric vehicle simulation (SIMPLEV). v3.1, DOE Idaho National Eng. Lab 13. Noons, R., Swann, J., Green, A.: The use of simulation software to assess advanced powertrains and new technology vehicles. In: Proc. Electric. Vehicle Symp., Brussels, Belgium, vol. 15 (October 1998) 14. Gao, D.W., Mi, C., Emadi, A.: Modeling and simulation of electric and hybrid vehicles. Proceedings of the IEEE 95(4), 729–745 (2007) 15. Miller, J.M., Nicastri, P.R., Zarei, S.: The automobile power supply in the 21st century. In: Proc. Power Systems World Conf. Expo. (1998) 16. Green Car Congress (August 2004), http://www.greencarcongress.com/ 17. Cheng, Y., et al.: Global modeling and control strategy simulation. IEEE Vehicular Technology Magazine 4(2), 73–79 (2009) 18. http://www.transportation.anl.gov/software/PSAT/ 19. Onoda, S., Emadi, A.: PSIM-Based Modeling of Automotive Power Systen: Conventional. Electric and Hybrid Electric Vehicles, IEEE Transactions on Vehicular Technology 53(2) (March 2004) 20. Gong, Q., Li, Y., Peng, Z.R.: Trip-based optimal power management of plug-in hybrid electric vehicles. IEEE Transactions on Vehicular Technology 57(6), 3393–3401 (2008)

Supply Chain Network Equilibrium with Profit Sharing Contract Responding to Emergencies Ating Yang and Lindu Zhao* Institute of Systems Engineering, Southeast University, Nanjing, Jiangsu 210096, P.R. China [email protected], [email protected]

Abstract. In the real market competition, the supply chain network equilibrium state is too ideal to obtain and contracts can be used to coordinate the supply chain network. In this paper, we establish a supply chain network equilibrium model with random demands and introduce profit sharing contract to the supply chain network model in order to be equilibrium. Then analyze the impacts that emergencies have on this equilibrium state. Through numeral examples we prove that the manufacturers and retailers can adjust the contract parameters to achieve a new supply chain network coordination state through bargaining when the demands increase suddenly as a result of emergent events. Keywords: supply chain network equilibrium, profit sharing contract, emergency, coordination.

1 Introduction With the development of science and technology, we are entering an era, during which individual businesses no longer compete as stand-alone entities, but rather as supply chains. In this situation, supply chain network is becoming a growing research topic nowadays. Supply chain network is consisted of multiple manufacturers, multiple suppliers, multiple retailers and multiple customers. Nagurney et al. [1] first brought forward the concept “supply chain network equilibrium”. They initiated a novel supply chain network equilibrium model in the case of the consumers’ demand for the product which can be expressed as a deterministic function and further showed that the supply chain network equilibrium model can be formulated as a variational inequality. Dong et al. [2] extended the model to the case where the consumers’ demand is assumed to be a random variable with a known probability distribution, and they also presented a variational inequality formulation. However, the equilibrium state of supply chain network is an ideal and optimal state to achieve. Chee et al. [3] indicated that the market could hardly be in equilibrium state, due to the private or imperfect information, decentralized decisionmaking, convert behaviors and so on. So, we can use contracts to coordination the *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 445–455, 2010. © Springer-Verlag Berlin Heidelberg 2010

446

A.T. Yang and L.D. Zhao

supply chain network. Cachon [4] described the variety contracts in the newsvendor model, and proved that the buyback contract and profit sharing contract can coordinate a single supply chain. Giannoccaro et al. [5] proposed a model of profit sharing contract aimed at coordinating a three-stage supply chain. Teng and Hu [6] established a supply chain network equilibrium model with random demands and introduced buy back contract and profit sharing contract into the supply chain network equilibrium model. Supply chain emergency management is a new research area in recent years and has attracted a widespread interest of scholars [7-8]. In the past few years, emergencies (e.g. “5.12 Sichuan Earthquake”, “H1N1 Influenza”, “Global financial crises” and so on) all have brought disadvantageous impacts on supply chain system. Therefore, it is very essential and important to research how the normal supply chain network responds to unexpected events. Yu et al [9] investigated the supply chain coordination problem under demand disruptions. By using the quantity discount contract, the two-stage newsvendor supply chain can be coordinated. Cao and Lai [10] and Sun and Ma [11] described a revenue sharing contract model for a two-stage supply chain that faced stochastic market demand in response to an emergent event. Most of the studies, however, still focus on the impacts of emergencies on a single supply chain and few have expanded it to supply chain network. So based on Teng [12], in this paper, we introduce profit sharing contract into the supply chain network equilibrium model and analyze the impacts of emergent events have on this equilibrium model. Then prove that manufacturers and retailers need to adjust the contracts parameters to achieve a new supply chain network equilibrium state so that all of them can be satisfied with the profit allocation.

2 Basic Supply Chain Network Model with Random Demand In this section, we develop a supply chain network equilibrium model with random demands at the retailer level. The representative decision makers in our framework are manufacturers and retailers, with the consumers being represented through the random demands for product at the different retail outlets. The supply chain network structure is as depicted in Fig. 1.













C   * 









& (

Fig. 1. Network structure of supply chain

SCN Equilibrium with Profit Sharing Contract Responding to Emergencies

447

In particular, we consider m manufacturers involved in the production of a homogeneous product of short life cycle, which can then be purchased by n retailers, who, in turn, respond to the consumers via the random demand functions. The links in the network denote transaction links. In this model, manufacturers must satisfy all of the retailers’ orders. Meanwhile, assume that all information is symmetrical. Retailers can bargain with manufacturers about the parameters of profit sharing contract. The retailers must choose order quantities and manufacturers before the start of a single selling season that has random demands. Let D j be demand at retailer

j and u j be its expectation. Let Pj be the demand distribution function of the

retailer

j and p j be its density function. Pj ( j = 1, 2, ⋅⋅⋅, n) is differentiable, strictly

increasing and Pj (0) = 0 , Pj ( x) = 1 − Pj ( x) . Let ρ ij denote the wholesale price charged for the product by manufacturer i to retailer j and ρ j denote retail price of retailer j . The transaction quantity between manufacturer i and retailer j is denoted by qij , then group all the qij into the mn -dimensional column vector Q . So, according to conservation of flow equation, we have qim =

n

m

j =1

i =1

∑ qij and q rj = ∑ qij .

Let S j ( q rj ) be expected sales at the retailer j ,

q rj

S j (q rj ) = E ⎡⎣ min( q rj , D j ) ⎤⎦ = q rj − ∫ Pj ( x )dx

(1)

0

Let I j ( q rj ) be expected left over inventory at the retailer

j . The retailer

j earns v j per unit unsold at the end of season, where is net of any salvage expenses. I j ( q rj ) = E ⎡⎣ ( q rj − D j ) + ⎤⎦ = q rj − S j ( q rj )

(2)

Let L j ( q rj ) be the lost sales function at the retailer j . For each unsatisfied demand retailer j incurs a punishment cost g j , including goodwill cost and customer lose, and manufacturer i also incurs an analogous cost g ij .

L j ( q rj ) = E ⎡⎣ ( D j − q rj ) + ⎤⎦ = u j − S j ( q rj )

(3)

448

A.T. Yang and L.D. Zhao

3 Supply Chain Network Equilibrium Model with Profit Sharing Contract In this section, we introduce profit sharing contract in the basic supply chain network model. Let φij be the fraction of the link ( i ; j ) profit the retailer

j keeps, so 1 − φij is

the fraction the manufacturer i keeps. φij is an array of constants, which is determined through bargaining between manufacturer i and retailer

j . Hence, the additional

transfer payment from retailer j to manufacturer i is expressed as:

Tij ( ρ j , qij ) = qij ρ ij +

qij q

r j

(1 − φij ) ⎡⎣ q rj − S j ( q rj ) ⎤⎦ v j +

qij q rj

(1 − φij ) ρ j S j ( q rj )

(4)

3.1 Optimality Condition of Manufacturers Group all the qim into the column vector q∈R+m . Assume that each manufacturer i is faced with a production cost function

f i , which depends on the entire vector of

production outputs. The transaction cost between each manufacturer and retailer is denoted by cij . Hence, the optimization problem of manufacturer i can be expressed as: n

n

n

j =1

j =1

j =1

max ∏ im = ∑ Tij ( ρ j , qij ) − fi (q ) − ∑ cij (qij ) − ∑ g ij ⎡⎣u j − S j ( q rj ) ⎤⎦

(5)

subject to qij ≥ 0 , for all i , j . After substituting Tij ( ρ j , q ij ) , (5) can be expressed as: n

n

max ∏im = ∑ qij ρij + ∑

qij

j =1 q

j =1

n

r j

n

(1 − φij ) ⎡⎣q rj − S j (qrj )⎤⎦ v j + ∑

qij

r j =1 q j

n

(1 − φij ) ρ j S j (q rj ) (6)

− fi (q) − ∑ cij (qij ) − ∑ gij ⎡⎣u j − S j (q ) ⎤⎦ j =1

j =1

r j

We assume that manufacturers compete in a non-cooperative fashion and the cost functions for each manufacturer are continuous and convex, and then the optimality condition for all manufacturers satisfies the following variational inequality:

SCN Equilibrium with Profit Sharing Contract Responding to Emergencies

q ⎡ ∂ ijr S j ( q rj ) ⎢ ⎢ − ρ − (1 − φ ) v + (1 − φ )( v − ρ ) q j − g ij Pj ( q rj ) ∑ ∑ ij ij j ij j j ∂ q ij i =1 j =1 ⎢ ⎢ ⎣ ∂ f ( q ) ∂ cij ( q ij ) ⎤ ∗ + i + ⎥ × ⎡ q ij − q ij ⎤⎦ ≥ 0 ∂ q ij ∂ q ij ⎥⎦ ⎣ m

449

n

(7)

3.2 Optimality Condition of Retailers A retailer j is faced with what we term a handling cost c j , which may include the display and storage cost associated with the product. For the sake of generality and to enhance the modeling of competition, we write c j = c j (Q) . Hence, the optimization problem of retailer

j can be expressed as: m

max ∏rj = ρ j S j (qrj ) + v j ⎡⎣qrj − S j (qrj )⎤⎦ − g j ⎡⎣u j − S j (qrj )⎤⎦ − c j (Q) − ∑Tij (ρ j , qij )

(8)

i =1

subject to qij ≥ 0 , for all i , j . After substituting Tij ( ρ j , q ij ) , (8) can be expressed as: m

max ∏ rj = ρ j S j (q rj ) + v j ⎡⎣ q rj − S j (q rj ) ⎤⎦ − g j ⎡⎣u j − S j (q rj ) ⎤⎦ − c j (Q) − ∑ qij ρij i =1

m

+∑ i =1

qij q rj

m

qij

(1 − φij ) ⎡⎣ q − S j (q ) ⎤⎦ v j + ∑ r (1 − φij ) ρ j S j (q ) q r j

r j

i =1

(9)

r j

j

Assuming that the handling cost for each retailer is continuous and convex, the optimality condition for all the retailers satisfies the following variational inequality:

q ⎡ ∂ ijr S j ⎢ ⎢(v − ρ − g ) P (q r ) + ∂c j (Q) + ρ − φ v + (1 − φ )( ρ − v ) q j ∑∑ j j j j j ij ij j ij j j ∂qij ∂qij (10) i =1 j =1 ⎢ ⎢ ⎣ × ⎡⎣ qij − qij∗ ⎤⎦ ≥ 0 m

n

450

A.T. Yang and L.D. Zhao

3.3 Optimality Condition of the Supply Chain Network The summation of inequalities (7) and (10), after algebraic simplification, yields the equilibrium condition of the supply chain network. m

n

⎡

∑∑⎢(v i =1 j =1

⎣⎢

j

− ρ j − g j − gij )Pj (qrj ) +

∂c (Q) ⎤ ∂fi (q) ∂cij (qij ) ∗ + − vj + j ⎥ × ⎡qij − qij ⎤⎦ ≥ 0 ∂qij ∂qij ∂qij ⎦⎥ ⎣

(11)

Then optimal condition of wholesale price charged for the product by manufacturer i to retailer j is:

ρij* =

∂

qij

∂

qij

S j (q rj )

q ∂fi (q) ∂cij (qij ) + − gij Pj (q rj ) − (1 − φij )v j + (1 − φij )(v j − ρ j ) ∂qij ∂qij ∂qij r j

(12)

Equals to

ρ ij* = ( ρ j + g j − v j ) Pj ( q rj ) −

∂c j (Q ) ∂qij

+ φij v j − (1 − φij )( ρ j − v j )

q rj

S j ( q rj )

(13)

∂qij

subject to qij ≥ 0 , for all i , j . With profit sharing contract, the manufacturers and retailers determinate the parameters based on the status of each other. Thus, the supply chain network will be equilibrium, and all decision-makers will satisfy with the profit allocation.

4 Impacts of Emergencies on Supply Chain Network Equilibrium For the above supply chain network with profit sharing contract, after determining the contract parameter φij with retailer

j , manufacturer i will obtain an optimal order

quantity qij* according to the equilibrium state and then arrange his production plan. But when emergencies happen the demand distribution function of retailer

j may

change to G j , which is also differentiable, strictly increasing and G j (0) = 0 . Let Gj ( x) = 1− Gj ( x) and its expectation u 'j = E ⎡ DJ' ⎤ . ⎣ ⎦ Hence, the optimization problem of manufacturer i turns to be:

SCN Equilibrium with Profit Sharing Contract Responding to Emergencies

n

n

max ∏im = ∑ qij ρij + ∑

qij

j =1 q

j =1

r j

n

(1 − φij ) ⎡⎣qrj − S 'j (qrj ) ⎤⎦ v j + ∑

qij

r j =1 q j

n

j =1

n

' j

j =1

r j

j =1

+ ij

(1 − φij ) ρ j S 'j (qrj ) − fi (q) +

−∑ cij (qij ) − ∑ gij ⎡⎣u − S j (q ) ⎤⎦ − ∑ λ ( qij − q n

∗ ij

451

) − ∑λ ( q n

− ij

j =1

∗ ij

− qij )

+

(14)

The optimization problem of retailer j turns to be: m

max ∏rj = ρ j S 'j (q rj ) + v j ⎡⎣ q rj − S 'j (q rj )⎤⎦ − g j ⎡⎣u 'j − S 'j (q rj )⎤⎦ − c j (Q) − ∑ qij ρij i =1

m

+∑ i =1

qij q rj

m

qij

(1 − φij ) ⎡⎣q − S (q ) ⎤⎦ v j + ∑ r (1 − φij ) ρ j S (q ) q r j

' j

r j

i =1

(15)

r j

' j

j

*

After the emergency, if retailer j adjusts his order quantity qij to a new one, then it will add extra production cost to manufacturer i . That is, when the new order * quantity qij > qij , we need to add new production cost λij for the increased order

+

* quantity qij − qij because the original production plan has been changed; * when qij < qij , then we need to spend new disposal expense λij for the left over

−

* products qij − qij .

5 Numerical Examples In this section, we apply the two-stage prediction–correction method [13] to a numerical example, which consists of two manufacturers and two retailers. In the example, we assume that the demands associated with the retail outlets follow a uniform distribution. Hence, we assume that the random demand Pj of retailer distributed in [0, b j ] , j = 1, 2 . The distribution function of Pj is expressed as: ⎧ 0, ⎪ Pj ( x ) = ⎨ x ⎪b ⎩ j

otherw ise x ∈ [0, b j ]

j = 1, 2 .

The other functions are given by:

cij (qij ) = 0.5qij2 + 3.5qij , i = 1, 2; j = 1, 2 .

j is

452

A.T. Yang and L.D. Zhao

f1 (q) = 2.5q1m2 + q1m q2m + 2q1m ; f 2 (q) = 2.5q2m 2 + q1m q2m + 2q2m .

c1 (Q ) = 0.5 ( q11 + q21 ) ; c2 (Q ) = 0.5 ( q12 + q22 ) . 2

2

Assume ρ j = 100 ; v j = 10 ; g j = 10 , gij = 10 ; φij = 0.8 . We set bj = 10 when no emergencies happen. The algorithm is implemented in MATLAB and converges well, (see Fig. 2). The convergence criterion used is X k − max[ X k − F ( X k ), 0] ≤ 0.001 and * * * yields the following equilibrium state pattern: q11 = q12* = q21 = q22 = 3.0935 ; * ρ11* = ρ12* = ρ21* = ρ22 = 23.0955 . Then we have the profits of manufacturers and

retailers: ∏1m = ∏ 2m = 64.8152 , ∏1r = ∏ 2r = 366.4327 .

Fig. 2. The convergence of the simulation

We respectively set different φij , the results are given by Table 1. Through bargaining, manufacturers and retailers can determine the parameter of the profit sharing contract. So, the supply chain network can achieve coordination by adjusting the wholesale price from manufacturer i to retailer j . Suppose that the demand scale increased suddenly because of emergencies. Suppose that the demands still follow a uniform distribution, while just the parameter b j has altered. Set bj = 15 , λij+ = 1, i = 1, 2; j = 1, 2 and the profit sharing contract parameter φij = 0.8 . If the retailers still hold the order quantities as bj = 10 ,

SCN Equilibrium with Profit Sharing Contract Responding to Emergencies

453

then their profits will decline while the profits of manufacturers increase (see column 3 in Table 2). So, the retailers prefer to order as the new equilibrium state when

bj =15 . By using the two-stage prediction–correction algorithm, we can obtain the * * following equilibrium state pattern: q11* = q12* = q21 = q22 = 3.8921 ; * * * * ρ11 = ρ12 = ρ21 = ρ22 = 30.1269 . Table 1. The profit sharing contract with different parameters

φij

qij*

ρij*

∏ im

1 0.95 0.90 0.85 0.80 0.75

3.0935 3.0935 3.0935 3.0935 3.0935 3.0935

32.9430 30.4811 28.0192 25.5573 23.0955 20.6336

36.4527 43.5432 50.6337 57.7241 64.8152 71.9057

However, when the equilibrium state has changed, the profits of manufacturers decline compared to no emergencies situation. That is, the supply chain network equilibrium state is broken because of the emergent events. Hence, the manufacturers and retailers have to adjust the profit sharing contract parameter to achieve a new supply chain network coordination state. In this example, we set φij = 0.7 , under this circumstance both the profits of manufacturers and retailers have increased (see column 5 in Table 2). Therefore, we can conclude that the profit sharing contract can coordinate the supply chain network when emergences happen. Table 2. The order plan with emergencies parameters

bj

10

15

15

15

φij

0.8

0.8

0.8

0.7

qij

3.0935

3.0935

3.8921

3.8921

ρij

23.0955

40.9346

30.1269

24.5298

∏

64.8152

124.6811

47.4936

99.9440

366.4327

306.3442

902.8282

537.0326

m i

∏

r j

454

A.T. Yang and L.D. Zhao

6 Conclusions This paper studies the supply chain network equilibrium model with the profit sharing contract under random demand. Contracts in the supply chain network are tools which can coordinate and improve the profit allocation of decision makers. However, when emergencies happen, the demands may change suddenly and the equilibrium state will be broken. In this situation, through numerical examples we show that the manufacturers and retailers can adjust the contracts parameters together to achieve a new supply chain network equilibrium state. That is to say, the profit sharing contract has an anti-emergency ability. We only introduce the profit sharing contract to coordinate the supply chain network in this paper. There are some other contracts that may have this ability, such as quantity discount, buy back, option contract and so on. So, we can try to study and compare these contracts in future work.

Acknowledgment This work was supported by the National Key Technology R&D Program of China (No. 2006BAH 02A06).

References 1. Nagurney, A., Dong, J., Zhang, D.: A supply chain network equilibrium model. Transp. Res. Part E: Logist. Trans. Rev. 38, 281–303 (2002) 2. Dong, J., Zhang, D., Nagurney, A.: A Supply Chain Network Equilibrium Model with Random Demand. Eur. J. Oper. Res. 56, 194–212 (2004) 3. Chee, Y.W., John, J., Hans, H.H.: Supply chain coordination problems: literature review. Working Paper No. 08–04 (2004) 4. Cachon, G.: Supply chain coordination with contracts. In: Handbooks in Operations Research and Management Science: Supply Chain Management. Kluwer Academic Publishers, Dordrecht (2003) 5. Giannoccaro, I., Pontrandolfo, P.: Supply chain coordination by revenue sharing contracts. Int. J. Prod. Econ. 89, 131–139 (2004) 6. Teng, C.X., Hu, Y.X.: The study of supply chain network equilibrium model with contracts. In: Second International Conference on Innovative Computing, Information and Control, ICICIC, Kumamoto, p. 410 (2007) 7. Ritchie, B., Brindley, C.: An emergent framework for supply chain risk management and performance measurement. J. Oper. Res. Soc. 58, 1398–1411 (2007) 8. Zhu, H.J., Tian, Y.: Research on the classification and grading system of emergencies in supply chain emergency management. In: International Conference on Management Science and Engineering-16th Annual Conference Proceedings, ICMSE, Moscow, pp. 513–517 (2009) 9. Yu, H., Chen, J., Yu, G.: How to coordinate supply chain under disruptions. Systems Engineering- Theory & Practice 25, 9–16 (2005)

SCN Equilibrium with Profit Sharing Contract Responding to Emergencies

455

10. Cao, E.B., Lai, M.Y.: Supply chain coordination under disruptions with revenue sharing contract. J. of Wuhan Uni. of Sci. & Tech. (Natural Science Edition) 30, 557–560 (2007) (in Chinese) 11. Sun, L., Ma, Y.H.: Supply chain coordination in emergent events using a revenue-sharing contract. J. of Beijing Uni. of Chem. Tech. 35, 97–99(2008) (in Chinese) 12. Teng, C.X., Hu, Y.X., Zhou, Y.S.: Supply chain network equilibrium with stochastic demand under disruptions. Systems Engineering- Theory & Practice 29, 16–20 (2009) (in Chinese) 13. Fu, X.L.: A two-stage prediction-correction method for solving monotone variational inequalities. J. Comput. Appl. Math. 214, 345–355 (2008)

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow: A Review Yijuan Di1, Minrui Fei1,*, Xin Sun1, and T.C. Yang2 1

Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, Shanghai, 200072, China 2 University of Sussex, UK [email protected], [email protected], [email protected], [email protected]

Abstract. Human bronchial tree is breath passage between pulmonary capillary vessel and fresh air and it is a key part of the human respiratory system. Clinical experience shows that abnormal or diseases of the respiratory system often occur at some bifurcated segments of the bronchial tree. Therefore, for better diagnosis and treatment of such diseases, it is necessary to investigate morphology structure of human bronchial tree and its internal airflow dynamics. In this paper, approaches for the modeling of bronchial tree are outlined based on the morphology structure data obtained from the two sources, i.e. anatomical experiments and CT/MRI. In addition, the simulation of airflow in bifurcating airways is summarized from the aspect of simulation methods and CFD. Finally, possible future research is suggested to develop: (1) a universal dynamic model of the airflow in the bronchial tree; and (2) a user-friendly software package for modeling and simulation. Keywords: Bronchial Tree, Airflow, Modeling, Simulation, Anatomical Experiments, CT/MRI, CFD.

1 Introduction In human breathing physiology, the bronchial tree is breath passage between pulmonary capillary vessel and fresh air and it is a key part of the respiratory system. Some diseases, e.g. tumor, chronic obstructive pulmonary disease (COPD), obstructive sleep apnea syndrome (OSAS), commonly occur in the airways of bronchial tree and seriously affects human body health [1-4]. Therefore, it is necessary to study human bronchial tree for better diagnosis and treatment these diseases. The two major points in the research of human bronchial tree are (1) morphological modeling of bronchial tree; and (2) the airflow simulation in bifurcating airways. These two points have become the most basic and revealing problems in the general case of breathing physiology, which can provide useful information for operation plan, drug particle delivery pattern, pollution dispersion, etc. *

Corresponding author.

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 456–465, 2010. © Springer-Verlag Berlin Heidelberg 2010

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow

457

As the medical ethics, experiments data of bronchial tree can not be directly measured from human living body and direct airflow simulation in human bifurcating airways can not be made. Thus, anatomic experiment and two advanced medical imaging, i.e. X-ray computerized tomography (CT) and magnetic resonance imaging (MRI), has become main approaches to extract the data from human bronchial tree. And fluid characterization on the airflow in bifurcating airways was often investigated in animals. In the 1960s, anatomic experiment firstly became the method of extraction structure data from bronchial tree for generating morphology structure models of bronchial tree [5-18]. However, the data obtained has deviated from actual structure data of human bronchial tree, which may result in incorrect conclusion and limit the research on bronchial tree. With the development of computational capability and advanced medical imaging, CT/MRI makes it possible to generate real 3D bronchial tree models for internal airflow dynamics simulation [19-26]. In the following, the modeling approaches of human bronchial tree are categorized based on two data sources, i.e. anatomical experiments and CT/MRI. The simulation of airflow in bifurcating airways is then reviewed from the view of simulation methods and computational fluid dynamics (CFD). Finally, the problems and prospects in modeling of bronchial tree and airflow simulation are briefly illustrated.

2 Modeling of Human Bronchial Tree According to the sources of morphology structure data, modeling of human bronchial tree can be classified into two categories, as shown in Fig.1.

Fig. 1. Modeling of human bronchial tree

2.1 Modeling Based on Anatomical Experiments In 1963, by measuring the length and the diameter of each bronchium in bronchial tree of five normal volunteers, Weibel [5] proposed a symmetric model, which took the human lung as 23 generations regularly bifurcated airways, while the effect of airway branching asymmetry was neglected. Horsfield et al. [6, 7] presented an asymmetric model with the spatial location of airways neglected. Although these two classical models were developed based on careful measurements, none of them included information on the angle between two successive bifurcation planes. In 1980, to account for the 3-dimensional (3D) nature of the airway system, an asymmetrical model of the upper trachea-bronchial tree was proposed by WenJia [8]. In this model, the orientation angle of each tube relative to its parent tube was chosen

458

Y. Di et al.

from the two solutions computed from the measured gravitational inclination and the branching angle. In later research, numerous studies on bifurcating flow have been performed to investigate the effect of bifurcation on airflow pattern and particle deposition et al. They considered the human lung as either symmetric bifurcation airways [3, 9-12], or asymmetric airways [13-18], and most of these model were based on the Weibel model or Horsfield model. However, actual human bronchial tree airways show distinct irregular features, and the regular airway models may not reflect the realistic flow characteristics in the human bronchial tree. 2.2 Modeling Based on CT/MRI As the emergence of CT/MRI, numerous studies on bronchial tree no longer simply depended on anatomical experiments. CT/MRI, as an in-vivo and non-invasive tool, showed the significant improvements and accuracy increase in morphological modeling of human bronchial tree [19-26]. The 3D mathematical morphology (3DMM), diffusion limited aggregation (DLA), energy-based modeling and fractal representations were used to develop a 3D model of the human airway tree [19]. Results showed morphometric characteristics of the bronchial tree were in good agreement with those reported. Using a semi-transparent volume rendering technique, a realistic 3-D model of the bronchial tree structure was developed [20] based on CT scans. In this study, the tree segmentation was performed by a DLA-based propagation. Merryn et al. [21] modeled entire conducting airway system in human bronchial tree by a volume-filling algorithm to generate airway centerline locations within detailed volume descriptions of the lungs. Curvilinear airway centerline and diameter models had been fitted to human bronchial tree extracted data from CT scanners. Subsequently, MRI was combined with CT together to extract morphology structure data from human bronchial tree, e.g. Montesantos et al. [22] generated a hybrid conceptual model based on MRI and high-resolution CT. A novel layer-bylayer searching algorithm was presented to reconstruct branches of human airway tree from point clouds obtained from CT/MRI [23]. Once the searching direction was fixed in each lay, the topological relationship between branches and the relative location of each branch was identified. Furthermore, in 2001, a 4-dimensional (4-D) NURBS-based cardiac-torso (NCAT) phantom was developed by Segars [24] based on 4-D tagged MRI and respiratory-gated CT. By this NCAT phantom, few model of the airway tree was built. For example, Garrity et al. [25] combined a segmentation technique with an algorithmic technique formulated by Kitaoka et al. [19], and developed a dynamic respiratory model for the individual lobes of the lungs and for the bronchial tree.

3 Simulation of Airflow in Bifurcating Airways In this section, the simulation of airflow in bifurcating airways is firstly summarized from the aspect of simulation methods and CFD. Then, the simulation methods mainly are elaborated from experimental simulation and numerical simulation.

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow

459

3.1 Simulation Based on Methods Over the last few decades, airflow simulation in bifurcating airways of human bronchial tree has been largely studied base on mechanical properties and dynamics properties of lung. The simulation methods of airflow in bifurcating airways are shown in Fig. 2.

Fig. 2. The simulation method of airflow in Bifurcating airways

Experimental Simulation Methods. Numerous experimental investigations have been performed in the past few decades to estimate the airflow in the human respiratory airways. In 1978, based on a multialveolar model described by a graph, a method was proposed by Chauvet [26] to deal with ventilator inequalities to investigate the variation of press and velocity. Through a Y-shaped tube bifurcation, a stead streaming displacement of fluid elements was observed on the oscillatory velocity vector field [27]. In this research, the relationship among displacement, Reynolds (Re) and alpha was studied. Ben [28] analyzed the gaseous dispersion of gas mixing in the human pulmonary airway by bolus response methods and found that gaseous dispersion was strongly related to flow velocity and it matched well with the theory of turbulent or disturbed dispersion. Snyder et al. [29] investigated the flow energy loss, and found that the flow energy loss was nearly independent of flow direction. An analysis of these observations suggested that kinetic energy factors, not shear stresses, accounted for most of the energy dissipated in central airways and in simple bifurcating sections. The pressure-flow relationship at high-frequency and low stroke volume was measured in human upper and central airways by a respiratory tract model [30]. Plotted against Re/β (ratio between Reynolds and Womersley numbers), the normalized instantaneous pressure drop was generally smooth throughout the range of Re/β except for the appearance of a transition at the lowest frequency, suggesting the onset of turbulent flow. By 3-D laminar flow profiles measured, the secondary flow retarded the onset of local flow separation by convecting axial

460

Y. Di et al.

momentum into the boundary layer, which was adjacent to the diverging wall of the bronchial tube [31]. Kulkarni et al. [32] studied the airflow in first four generations of the bronchial tree based on a 2-D, fluid mechanical model. The computation results for an asymmetric breathing cycle showed that preferred flow was proximal to the trachea, areas of recirculation, and time-averaged expiratory shear stresses at the airmucus interface. In Popwell’s study [33], the simulation of airflow was carried out under different frequency, tidal volume and Reynolds through a model casted in a transparent silicone elastomer. In recent years, the oscillatory flow in human lung airways down to the 6- generation through the branching network was studied by digital particle image velocimetry (DPIV) [34]. It was used to analyze lung ventilation during breathing under rest conditions and for high frequency ventilation (HFV). In additional, a distributed model of bronchial tree based on Weibel model was proposed by Ginzburg et al. [9] to simulate the global dynamic characteristics of human lung. In the resulting model, local mechanical characteristics of each airway were represented by RCL circuits, and parameters of the electrical components were determined from local physiological data. The global resistance to airflow and the dynamic compliance of bronchi decreased as the forced oscillation frequency increased. Numerical Simulation Methods. While current in-vivo experimental techniques can only reliably provide total or regional airflow features, numerical simulations can often provide much more detailed information in both space and time and it has become the effective ways to study airflow dynamics in bifurcating airways [11-12, 15-16, 19, 35-40]. The methods of numerical simulations for airflow can be approximately classified into six categories: The control volume method (CVM): The fluid flow in bifurcations of a symmetric tree was investigated by Andrade et al. [15] by the simulation of a 2D Navier-

Stokes and continuity equations, which was solved for the velocity and pressure fields by CVM. Flow distribution was significantly heterogeneous at an increased Reynolds number. At low Reynolds number, the flow distribution throughout the airways was quite uniform while the nonlinear contribution from the inertial terms became relevant at high Reynolds number. Liu et al. [11, 16] studied the inspiratory flow characteristic in symmetric and asymmetric three-generation lung airway respectively. By the simulation of three-dimensional laminar Navier-Stokes equations based on CVM, relations between Reynolds number and flow characteristics, including flow patterns and pressure drop, was established. The finite element method (FEM): FEM was applied to numerically study steady inspiratory and expiratory flow in a symmetrical bifurcating airway model [35]. In this research, most of the important flow features observed in experiments were captured in the numerical simulation. The numerical simulation of fluid flow with high-frequency and low-tidal-volume ventilation was carried out by FEM in the first 6-generations of human bronchial tree [36]. Results indicated preferred flow proximal to the trachea, areas of recirculation, and time-averaged expiratory shear stresses at the air-mucus interface. Yang et al. used FEM to calculate the nonlinear deformation of the thin-wall structure of 3-generation airway [37]. The deformation of the

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow

461

collapsible tube increased the resistance to the fluid flow and formed a jet that altered the downstream flow pattern significantly. The finite difference method (FDM): Airflow fields within bronchial airway bifurcations were computed by solving the three-dimensional Navier-Stokes equation with FDM [38]. The large eddy simulation (LES): LES resolves large-scale energy-containing turbulent eddies and parameterizes small-scale unresolved eddies and has become an important numerical simulation methods of fluid mechanics. Luo [39] employed LES method to study the transitional and turbulent airway flow during inspiration in a single asymmetric bifurcation model. The influence of the non-laminar flow on the patterns and the particle paths was investigated in both 2D and 3D models. Results indicated that the differences in geometry can heavily influence the nature of the flow. Transitional and turbulent flow within the trachea and bronchial of asymmetric upper airways during high frequency oscillatory ventilation (HFOV) were also investigated by using LES [40]. The finite volume method (FVM): By the use of FVM, transient laminar inhalation with a realistic inlet waveform was simulated in an asymmetric out-of-plane human upper airway model. Results showed that the flow fields are similar to those obtained with equivalent Reynolds numbers at steady state. Stronger axial and secondary velocities occurred at all upper branch locations during flow deceleration because of the dynamic lingering effect [19]. The fluid network theory: Based on the fluid network theory, Lai et al. [12] investigated a three-element model of lumped parameter. The pressure and flow rate distributions in different positions of the lungs during normal respiration and partial bronchial obstruction were compared. 3.2 Simulation Based on CFD For the limitations [3] of experimental method and simulation analysis, i.e. complexity of questions and high-cost, computational fluid dynamics (CFD) simulation has become a powerful and often indispensible tool for studying airflow in realistic and complicated airway geometry[3, 37, 41-45]. CFD simulations can provide not only 3D flow pattern within the airways, but also detailed particle deposition patterns. Lin et al. [41] studied the upper and intra-thoracic airways and their role in determining central airflow patterns by CFD technique. The presence of turbulent laryngeal jet on air flow in the human intra-thoracic airways was found to increase the maximum localized wall shear stress by three-folds. It was found that turbulence could significantly affect airway flow patterns as well as tracheal wall shear stress. Some airflow simulations in human bronchial airways operated by CFD were compared with experimental measurements and good agreements have been obtained. For example, Rochefort et al. [42] carried out both experimental study and CFD simulation on the same human proximal airways, and they found the velocity agreement between measurement and calculation was within 3%. Main and secondary

462

Y. Di et al.

flow velocity intensities were similar, as were velocity convective patterns. This combined numerical and experimental approach of flow assessment in realistic human airway geometries confirmed the strong dependence of airway flow patterns on local and global geometrical factors. The CFD-simulated flow features qualitatively had a good agreement with the PIV-measured. More recently Ma et al. [43] investigated the CFD predictions of flow field and particle trajectory with experiments within a scaled-up model of alveolated airways. At 12 selected cross section, the velocity profiles obtained by CFD was in agreement with those by particle image velocimetry (within 1.7% on average), and the CFD predicted trajectories also matched well with particle tracking velocimetry experiments. Different from above mentioned studies only considering a straight smooth airway, Yang et al. [3] investigated the effect of COPD on respiratory flow (airflow pattern, airflow rate and pressure drop) on four different four-generation models based CFDsimulation Laminar Navier-Stokes equation. The calculation showed that the flow separation existed in the conjunction region. In a bifurcation airway, the re-circulation cells generated both upstream and downstream blocked the air from entering the downstream branches. Additional, some CFD commercial software programs in the market has got application in airflow investigation in human bronchial tree, such as, FIDAP and FLUENT. McHugh et al. [36] studied the fluid flow from high-frequency and lowtidal-volume ventilation in bronchial tree based on FIDAP CFD package. The air flow pattern and flow rate distribution in human lung model were numerically studied using low Reynolds number (LRN) k-ω turbulent model [44] and the CFD solver FLUENT. Compared with the laminar flow, the secondary flow of turbulent flow was weaker than that of the laminar flow, and the turbulent axial flow velocity profile was more flat in the center of the tube. The ratio of the air flow rate between left and right lobes was insensitive to Reynolds number.

4 Problems and Prospects From all above mentioned, a large number works on the modeling of human bronchial tree and the simulation of airflow in bifurcation airways has been generally performed, but there still exist some problems on them: (1) It can be found that these models or simulation methods was special and individual, lack of the universal property model or method between them. (2) The modeling of human bronchial tree and internal airflow simulation were studied from the viewpoint of microscopic levels (such as organ levels) while the respiration investigation from molecular levels in the bronchial tree also has made [45, 46]. However, there hasn’t a synthetic study both from the microscopic levels and from the molecular levels. (3) The modeling and simulation works are far from practical use. For these issues, it is necessary to further develop a universal variable-parameter dynamic model for better study realistic dynamical respiratory motion. This model can be based on advanced medical imaging, biomechanics and mathematical methods.

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow

463

Currently, there are some commercial software programs in the global market for research and clinical applications [47, 48]. However, by using the published algorithms or the commercial software programs in the market, many flaws occur in the generated geometries, which impair the use of the models directly. Therefore, for improving surgical operation, recovery of postoperative respiratory function and particle disposition, a user-friendly software package combined the geometries with workable numerical models for modeling of bronchial tree and internal airflow dynamics simulation is suggested to develop. Acknowledgments. This research is supported by Mechatronics Engineering Innovation Group project from Shanghai Education Commission, The graduate student innovation fund of Shanghai University under Grand SHUCX101015.

References 1. Barber, H.B., Woolfenden, J.M., Donahue, D.J., Nevin, W.S.: Small Radiation Detectors For Bronchoscopic Tumor Localization. IEEE Transactions on Nuclear Science NS-27, 496–502 (1979) 2. Ye, X., Sonka, M., McLennan, G., Junfeng, G., Hoffman, E.A.: MDCT-based 3-D texture classification of emphysema and early smoking related lung pathologies. IEEE Transactions on Medical Imaging 25, 464–475 (2006) 3. Yang, X.L., Liu, Y., Luo, H.Y.: Respiratory flow in obstructed airways. Journal of Biomechanics 39, 2743–2751 (2006) 4. Shome, B., Wang, L.P., Santare, M.H., et al.: Modeling of Airflow in the Pharynx with Application to Sleep Apnea. Journal of Biomechanical Engineering 120, 416–422 (1998) 5. Weibel, E.R.: Morphometry of the Human Lung, p. 19. Academic Press, New York (1963) 6. Horsefield, K., Cumming, G.: Morphology of the bronchial tree in man. Journal of Applied Physiology 24, 373–383 (1968) 7. Horsfield, K., Dart, G., Olson, D.E., Filley, G.F., Cumming, G.: Models of the human bronchial tree. Journal of Applied Physiology 31, 207–217 (1971) 8. Wen, J., David, S.S., Wang, C.S.: A three-dimensional model of the upper tracheobronchial tree. Bulletin of Mathematic Biology 42, 847–859 (1980) 9. Ginzburg, I., Elad, D.: Dynamic model of the bronchial tree. Journal of Biomedical Engineering 15, 283–288 (1993) 10. Zhang, Z., Kleinstreuer, C., Kim, C.S.: Aerosol deposition efficiencies and upstream release positions for different inhalation modes in an upper bronchial airway model. Aerosol Science and Technology 36, 828–844 (2002) 11. Liu, Y., Soa, R.M.C., Zhang, C.H.: Modeling the bifurcating flow in a human lung airway. Journal of Biome-chanics 35, 465–473 (2002) 12. Lai, W., Tan, X.P., Pei, J.: Air distribution in the bronchial tree of human lungs. In: Wu, S., Xue, Y., Xue, G.C., Zhi, Z. (eds.), vol. 21, pp. 229–232 (2004) 13. Tanaka, G., Ito, J., Oka, K., Tanishita, K.: Local gas transport rate in oscillatory flow in a model human central airways. American Society of Mechanical Engineers, Bioengineering Division (Publication) BED 33, 179–180 (1996) 14. Kurujareon, J., Holdo, A., Calay, R.K.: Effects of boundary conditions for the CFD modelling of respiratory flow in an asymmetric bifurcation. American Society of Mechanical Engineers. Bioengineering Division (Publication) BED 39, 103–104 (1998)

464

Y. Di et al.

15. Andrade Jr., J.S., Alencar, A.M., Almeida, M.P., et al.: Asymmetric Flow in Symmetric Branched Structures. Physical Review Letters 81, 926–929 (1998) 16. Liu, Y., So, R.M.C., Zhang, C.H.: Modeling the bifurcating flow in an asymmetric human lung airway. Journal of Biomechanics 36, 951–959 (2003) 17. Li, Z., Kleinstreuer, C., Zhang, Z.: Simulation of airflow fields and microparticle deposition in realistic human lung airway models. Part I: Airflow patterns. European Journal of Mechanics B/Fluids 26, 650–668 (2007) 18. Van Ertbruggen, C., Hirsch, C., Paiva, M.: Anatomically based three-dimensional model of airways to simulate flow and particle transport using computational fluid dynamics. Journal of Applied Physiology 98, 970–980 (2005) 19. Kitaoka, H., Ryuji, T., Suki, B.: A three-dimensional model of the human airway tree. Journal of Applied Physi-ology 87, 2207–2217 (1999) 20. Fetita, C.I., Preteux, F.: Bronchial tree modeling and 3D reconstruction. In: Proceedings of SPIE - The International Society for Optical Engineering, vol. 4121, pp. 16–29 (2000) 21. Merryn, H., Tawhai, P.H., et al.: CT-based geometry analysis and finite element models of the human and ovine bronchial tree. J. Appl. Physiol. 97, 2310–2321 (2004) 22. Montesantos, S., Fleming, J.S., Tossici-Bolt, L.: A Spatial Model of the Human Airway Tree: The Hybrid Conceptual Model. Journal of Aerosol Medicine and Pulmonary Drug Delivery 23, 59–68 (2010) 23. Ding, S.L., Ye, Y., Tu, J.Y., Subic, A.: Region-based geometric modelling of human airways and arterial vessels. Computerized Medical Imaging and Graphics 34, 114–121 (2010) 24. Segars, W.P.: Development of a new dynamic NURBS-based cardiac-torso (NCAT) phantom, PhD dissertation, The University of North Carolina (May 2001) 25. Garrity, J.M., Segars, W.P., et al.: Development of a Dynamic Model for the Lung Lobes and Airway Tree in the NCAT Phantom. In: Nuclear Science Symposium Conference Record, vol. 3, pp. 1858–1862. IEEE, Los Alamitos (2002) 26. Chauvet, G.: Mathematical study of flow in a non-symmetrical bronchial tree: Application to asynchronous ventilation. Mathematical Biosciences 42, 73–91 (1978) 27. Haselton, F.R., Seherer, P.: Flow visualization of steady streaming in oscillatory flow through a bifurcation tube. Journal of Fluid Mechanics 123, 315–333 (1982) 28. Ben Jebria, A.: A mechanical approach to the longitudinal dispersion of gas flowing in human airways. Journal of Biomechanics 18, 399–405 (1985) 29. Snyder, B., Olson, D.E., et al.: Reversible and irreversible components of central-airway flow resistance. Journal of Biomechanical Engineering 109, 154–159 (1987) 30. Jebria, A.B., Ultman, J.: Pressure-flow characteristics of the upper and central airways during high-frequency oscillation. In: IEEE/Engineering in Medicine and Biology Society Annual Conference, vol. 10, pp. 692–693 (1988) 31. Snyder, B.: Flow development in a model airway bronchus. Journal of Fluid Mechanics 207, 379–392 (1989) 32. Kulkarni, A.K., Meyers, J., MeHugh, P.: A numerical simulation of airflow in human bronchial passages during high-frequency ventilation. In: Proceedings of the Northeast Conference on Bioengineering, pp. 113–114 (1990) 33. Popwell, R.E., Peattie, R.A.: Transition to turbulence in oscillating flows through a model bronchial bifurcation. American Society of Mechanical Engineers, Bioengineering Division (Publication) BED 26, 215–218 (1993) 34. Katrin, Adler, Brücker, C.: Dynamic flow in a realistic model of the upper human lung airways. Experiments in Fluids 43, 411–423 (2007)

Modeling of the Human Bronchial Tree and Simulation of Internal Airflow

465

35. Zhao, Y., Brunskill, C.T., Lieber, B.B.: Inspiratory and expiratory steady flow analysis in a model symmetrically bifurcating airway. Journal of Biomechanical Engineering 119, 52– 58 (1997) 36. McHugh, P.M., Kulkarni, A.K.: Numerical simulation of flow from asymmetrical highfrequency ventilation in the bronchial tubes. American Society of Mechanical Engineers, Bioengineering Division (Publication) BED 17, 63–66 (1990) 37. Yang, X.L., Liu, Y., Yang, J.M.: Expiratory flow in a rigid three-generation airway with one collapsible segment. Communications in Numerical Methods in Engineering 25, 553– 563 (2009) 38. Balashazy, I., Hofmann, W.: Particle deposition in airway bifurcations - I. Inspiratory flow. Journal of Aerosol Science 24, 745–772 (1993) 39. Luo, X.Y., Hinton, J.S., Liew, T.T., Tan, K.K.: LES modelling of flow in a simple airway model. Medical Engineering & Physics 26, 403–413 (2004) 40. Martin, A.N., John, E.C.: Large eddy simulation of high frequency oscillating flow in an asymmetric branching airway model. Medical Engineering & Physics 31, 1148–1153 (2009) 41. Lin, C.L., Tawhai, M.H., McLennan, G., Hoffman, E.A.: Characteristics of the turbulent laryngeal jet and its effect on airflow in the human intra-thoracic airways. Respiratory Physiology and Neurobiology 157, 295–309 (2007) 42. Rochefort, L., Vial, L., Fodil, R., Maıtre, X., et al.: In vitro validation of computational fluid dynamic simulation in human proximal airways with hyperpolarized 3He magnetic resonance phase-contrast velocimetry. Journal of Applied Physiology 102, 2012–2023 (2007) 43. Ma, B., Ruwet, V., Corieri, P.: CFD simulation and experimental validation of fluid flow and particle transport in a model of alveolated airways. Journal of Aerosol Science 40, 403–414 (2009) 44. Luo, H.Y., Liu, Y.: Modeling the bifurcating flow in a CT-scanned human lung airway. Journal of Biomechanics 41, 2681–2688 (2008) 45. Gray, N.F., O’Neill, C.: Acid mine-drainage toxicity testing. Environmental Geochemistry and Health 19, 165–171 (1997) 46. Yu Svirezhev, M., Svirejeva-Hopkins, A.: The model of long-term evolution of the carbon cycle. Ecological Modelling 216, 114–126 (2008) 47. Vida, Diagnostics: Home, http://www.vidadiagnostics.com 48. Mimics, http://www.materialise.com/mimics

Robust Semi-supervised Learning for Biometrics Nanhai Yang, Mingming Huang, Ran He, and Xiukun Wang Department of Computer Science and Technology, Dalian University of Technology, 116024 Dalian, China

Abstract. To deal with the problem of sensitivity to noise in semi-supervised learning for biometrics, this paper proposes a robust Gaussian-Laplacian Regularized (GLR) framework based on maximum correntropy criterion (MCC), called GLR-MCC, along with its convergence analysis. The half quadratic (HQ) optimization technique is used to simplify the correntropy optimization problem to a standard semi-supervised problem in each iteration. Experimental results show that the proposed GRL-MCC can effectively improve the semi-supervised learning performance and is robust to mislabeling noise and occlusion as compared with GLR. Keywords: Biometrics, Semi-supervised Learning, Robust, Correntropy.

1 Introduction With the advent of information age, data collection and storage competency is greatly improved. Discovery and extraction of useful information from the mass data becomes common requirement of bioinformatics and biometrics field. Since labeling often requires expensive human labor, one often faces a lack of sufficient labeled data. However, large numbers of unlabeled data can be far easier to obtain. In 1992, the concept of “semi-supervised” is proposed for the first time, which uses a large number of unlabeled data to assist insufficient labeled data for learning. In the recent decades, graph-based semi-supervised learning (SSL) have been becoming one of the hottest research area in SSL field, which models the whole data set as an undirected weighted graph, whose vertices correspond to the data set, and edges reflect the relationships between pair wise data points. In practice, during collecting data, noise may occur. Taking image data as an example, the training data may contain undesirable artifacts due to image occlusion, illumination, or image noise. At the same time, for supervised learning, mislabeling of training data may occur and deteriorate the performance of the learning model. Therefore, noisy problem has been received considerable interests from the machine learning and data mining communities. In graph-based SSL, noise problem has not been covered thoroughly. [1] proposed a heuristic method to remove the bridge points that connect different classes; [2] proposed a robust graph-based SSL method which can determine the similarities between pairwise data points automatically and is not K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 466–476, 2010. © Springer-Verlag Berlin Heidelberg 2010

Robust Semi-supervised Learning for Biometrics

467

sensitive to outliers. Most existing methods adopt heuristic strategy to resolve the noise problem and hence lack of theory foundation. Gaussian-Laplacian Regularized (GLR) framework is a classical framework, most of traditional graph-based SSL algorithms are based on it [3][4][5]. This paper analyzes the effect of noise on GLR, which is based on least square criterion and is sensitive to noise [6]. To solve this problem, we propose a robust semi-supervised learning framework based on maximum correntropy criterion (MCC) [6][7], called GLR-MCC, along with its convergence analysis. The half quadratic (HQ) optimization technique is used to simplify the correntropy optimization problem to a standard semi-supervised problem in each iteration, then a greedy algorithm is proposed to increase the objective function until convergence. Promising experimental results on the UCI databases and face databases demonstrate the effectiveness of our method. The remainder of this paper is organized as follows. In section 2, we briefly review GLR method. In section 3, we discuss robust GLR-MCC framework and present an algorithm based on half quadratic to optimize it. We give the experimental results in section 4 and summarize this work in section 5.

2 GLR In SSL, we are given a set of data points l

Χl = { x i }i =1 are

labeled, and Χu

n

= { x i }i =l +1 are

Χ = {x 1,...xl , xl +1,..., x n } ∈ Rd

unlabeled. Each xi

∈X

where

is drawn from a

fixed but usually unknown distribution p(x ) . In the graph-based SSL methods, model the whole data set as a graph G = (V , E ) .V is the vertex set, which is equivalent to the data set, and E is the edge set. Associated with each edge eij ∈ E is a nonnegative, weight wij

≥ 0 reflecting

how similar datum i is to datum j. If there is no edge present

between samples xi and xj , w ij = 0 . The basic assumption behind graph-based SLL is the cluster assumption [8][9], which states that two points are likely to have the same class label if there is a path connecting them that passed through the regions of high density only. Zhou et al. [10] further explored the geometric intuition behind this assumption: 1) nearby points are likely to have the same label, and 2) points on the same structure (such as a cluster or a submanifold) are likely to have the same label. Note that the first assumption is local, whereas the second one is global. The cluster assumption tells us to consider both the local and global information contained in the data set during learning. According to the cluster assumption, the general geometric framework for SSL seeks an optimal classification function f by minimizing the following objective Η( f ) = ηC (x L ) + βS ( f )

(1)

468

N. Yang et al.

where C(.) denotes the loss function, which measures loss between the predictive labels and original labels, and S (f ) reflects the intrinsic geometric information of the marginal distribution p(x ) . The reason why we should punish the geometrical information of f is that in SSL, we only have a small portion of labeled data (i.e. l is small), which are not enough to train a good learner by purely minimizing the structural loss of f . Therefore, we need some prior knowledge to guide us to learn a good f . What p(x ) reflects is just such type of prior information. Moreover, it is usually assumed [20] that there is a direct relationship between p(x ) and p(y | x ) , i.e. if two points xi and xj are close in the intrinsic geometry of p(x ) , then the conditional distributions p(y | x 1 ) and p(y | x 2 ) should be similar. In other words, the conditional probability distribution p(y | x ) should vary smoothly along the geodesics in the intrinsic geometry of p(x ) . GLR framework aims to solve the following optimization problem: l

n

n

f = arg min ∑ (fi − yi )2 + ζ ∑ ∑ (fi − f j )2 wij f

i =1

(2)

i =1 j =1

where the former is the loss item, the latter is the smoothness item. T f = ⎡⎣ f1, f2,..., fl ,..., fn ⎤⎦

,

fi

is the classifier.

y = [y1, y2 ,...yl ,...yn ]T

,

yi

is the original label.

The smoothness item can be approximated by n

n

∑ ∑ (fi − fj )2 wij

(3)

= f T Lf

i =1 j =1

where L is the graph Laplacian L = D − W ∈ \n×n , adjacency graph, D

= diag (∑ i w1i ,...,∑ i wni ) .Then

wij

are the edge weights in the data

the total loss we want to minimize

becomes l

f = arg min ∑ ( fi − yi )2 + ζ f T Lf f

(4)

i =1

By setting ∂J / ∂f = 0 we can get that f = (J + ζ L)−1Jy

(5)

where J ∈ \n×n is a diagonal matrix with its ( i, i ) -th entry, y is a n × 1 column vector with its i -th entry ⎧ ⎪ 1, if x i is labeled; J (i, i ) = ⎪ , y(i ) = ⎨ ⎪ 0, otherwise, ⎪ ⎩

⎧⎪ yi , if x i is labeled; ⎪⎨ ⎪⎪ 0, otherwise ⎩

Robust Semi-supervised Learning for Biometrics

469

Since (2) is based on l2 norm, noise will significantly impair objective function. GLR’s performance will decrease rapidly when noise occurs [6].

3 Robust SSL Based on MCC Theoretical analysis and experiment results [6] demonstrate that correntropy criterion can be very useful in non-Gaussian noise, especially in the impulsive noise environment. 3.1 Problem Formulation Recently, the concept of Maximum Correntropy Criterion (MCC) was proposed for ITL [11][19]. MCC is a local criterion of similarity and is very useful for cases when the noise is nonzero mean, non-Gaussian, with large outliers [6]. The correntropy is a generalized similarity measure between two arbitrary scalar random variables A and B, defined by Vσ (A, B ) = E [kσ (A − B)]

(6)

where kσ (.) is a kernel function that satisfies Mercer’s theory [12] and E[.] is the expectation. In practice, the joint probability density function is often unknown and only a finite number of data

{(ai , bi )}ni =1

are available, which leads to the sample

estimator of the correntropy: n  1 Vσ (A, B ) = ∑ kσ (ai − bi ) n i =1

(7)

The correntropy has a clear theoretic foundation and some important properties [13], such as symmetric, positive and bounded. Substituting Gaussian kernel function g(x ) = exp(−

||x ||2

σ2

) into

(7) and combing (4), we get following maximum correntropy

problem: l

E (f ) =

n

i =1

l

=

∑ exp(− i =1

T

n

∑ g(fi − yi ) + ζ ∑ ∑ g(fi − fj )wij i =1 j =1

2

( fi −yi ) σ2

n

n

) + ζ ∑ ∑ exp(− i =1 j =1

T

where f = ⎡⎣ f1, f2,..., fl ,..., fn ⎤⎦ , y = ⎡⎣ y1, y2,..., yl ,..., yn ⎤⎦ .

(8) ( fi − fj )2 σ2

)wij

470

N. Yang et al.

The optimal solution f is obtained by maximizing (8) according to Karush-KuhnTucker (KKT) conditions. As observed in (8), the former is a loss item: if yi is mislabeled at initial and fi is labeled correctly, fi ≠ yi , then we regard yi as outlier and expect it has small contribution to the objective; if yi is labeled correctly at initial and its corresponding data is the outlier, we can’t correctly evaluate fi, and hence we also expect fi has small contribution to the objective. The latter is a smoothness item, note that we hope the same label points are likely nearby according to the cluster assumption. If fi and fj have the same label, then xi and xj are potentially in the same cluster, and wij is may have a high value. If fi and fj belong to different classes, then xi and xj are potentially in different clusters, and wij should be low. We also recognize that it is hard to find a closed form for (8) due to its nonlinear. Hence the objective of (8) is difficult to be directly optimized. In the next section, we introduce an approximate strategy to optimize it. 3.2 Optimization Algorithm via Half Quadratic In ITL, the half-quadratic technique [11][14][15], expectation maximization (EM) method [16] and conjugate gradient algorithm [17] can be utilized to solve nonlinear ITL optimization problem. Here, we follow the lines of [18] and derive an algorithm to solve (8) based on the half-quadratic technique. Based on the theory of convex conjugated functions [13], we can easily derive the following proposition Proposition 1: There exists a convex conjugated function of g(x ) , such that, g(x , σ) = exp(−

||x ||2 σ

2

) = sup (p

||x ||2

p ∈R−

σ2

− ϕ(p))

(9)

and for a fixed x , the maximum is reached at p = −g(x , σ) . Substitute (9) into (8), we have the augmented objective function in an enlarged parameter space Eˆ(f , P,Q) =

l

∑ ( pi i =1

( fi −yi )2 σ2

)

n

n

⎛ − ϕi (pi ) + ζ ∑ ∑ ⎜⎜⎜qij ⎝ i =1 j =1

( fi − fj )2 σ2

⎞ − φij (qij ) ⎟⎟⎟ wij ⎠

(10)

where P is diagonal matrix, and P (i, i) = pi and Q = [qij ] ∈ \n×n store the auxiliary variables introduced in half quadratic optimization. According to the Proposition 1, we have that for a fixed f , the following equation holds. E ( f ) = sup Eˆ( f , P ,Q ) P ,Q

(11)

Then we can conclude that maximizing E (f ) is identical to maximizing the augmented function Eˆ(f , P,Q ) .

Robust Semi-supervised Learning for Biometrics

max E (f ) = max Eˆ(f , P,Q ) f

471

(12)

f ,P ,Q

Then we can divide into two steps to solve this problem: 1) Calculate auxiliary variations P and Q : pi = exp(−

( fi −yi )2 σ

2

) , qij = exp(−

( fi − f j )2 σ2

)

2) Substitute P and Q into (11), we get: l

arg max ∑ exp(− f

( fi −yi )2 ( fi −yi )2 σ2

i =1

)

σ2

n

n

+ ζ ∑ ∑ exp(−

( fi − fj )2 ( fi − fj )2 σ2

i =1 j =1

)

σ2

wij

(13)

By transforming smoothness item, we can easily get: n

n

∑ ∑ exp(− i =1 j =1

( fi − fj )2 ( fi − fj )2 σ2

)

σ2

wij = f T L ′f

where L ′ = D ′ − W ′ ,W ′ is the new weight matrix, wij ′

(14)

= exp(−

( fi − f j )2 σ2

)wij , D ′ is

a diagonal

matrix D ′(i, i ) = ∑ j Wij ′ . By derivation, we obtain the solution of (12) f = (J ′ + ζ L ′)−1J ′y

(15)

where J ′ = diag([p1,..., pn ]) , y is the a n × 1 column vector ⎧⎪ y , if x i is labeled; y(i ) = ⎪⎨ i ⎪⎪ 0, otherwise ⎩

All above is about two-class case, it is easy to extend GLR-MCC algorithm to multiclass classification problems. Suppose there are c classes, we can obtain the solution F = (J ′ + ζ L ′)−1J ′Y

(16)

where both F and Y are matrixes, ⎡f ′ ⎢ 11 ⎢ ⎢f ′ F = ⎢ 21 ⎢ ... ⎢ ⎢f ′ ⎢⎣ n 1

⎡y ′ y ′ ... f1k ′ ⎤⎥ ⎢ 11 12 ⎥ ⎢ ⎢ y21′ y22′ f22′ ... f2k ′ ⎥ ⎥,Y = ⎢ ⎢ ... ... ... ... ⎥ ... ⎥ ⎢ ⎥ ⎢y ′ y ′ fn 2′ ... fnk ′ ⎥ ⎢⎣ n 1 n2 ⎦ f12′

... y1k ′ ⎤⎥ ⎥ ... y2k ′ ⎥ ⎥, k =c. ... ... ⎥ ⎥ ... ynk ′ ⎥⎥ ⎦

Intuitively, (16) is not the standard labels, so we propose a greed algorithm to seek a local optima. fi ′ = [ fi1′ fi 2′ ... fik ′ ] ∈ \d , we should discrete it to get the standard labels at first which can’t guarantee to increase the value of (13); then we adopt the greed algorithm to calculate a optimal solution, i.e., if the solution after discreting can’t increase the value of (13), we retain original value, otherwise replace it; finally

472

N. Yang et al.

we make use of the F value as Y in the next iteration, and recompute (16) until convergence. The main procedure of GLR-MCC is summarized as follows: Algorithm 1: GLR-MCC l

n

Input: Χ = { x1,...xl , xl +1,..., xn } ∈ Rd , Χl = { x i }i =1 are labeled, Χu = { x i }i =l +1 are unlabeled. Output: The labels of all the data points. Step 1. Calculate W ; Step 2. Repeat (a)Calculate W ′ by wij ′ = exp(−

( fi − f j )2

σ2

)wij ,

and calculate L ′ by L ′ = D − W ′ (b) Calculate J ′ by (15) and F by (16) (d) Adopt the greed algorithm to calculate the labels (discrete F ) Until all the labels are convergence to a fixed point. 



Proposition 2. Denote E t  E (f t , P t ,Qt ) is the function in the t-th iteration, then the 

sequence {E t }t =1,2,... generated by GLR-MCC converges. Proof: According to [11] and the step 2 of Algotithm1, we have     E ( f t , P t ,Q t ) ≤ E ( f t , P t +1,Q t ) ≤ E (f t , P t +1,Q t +1 ) ≤ E (f t +1, P t +1,Q t +1 )

The function increases at each iteration maximization step. Therefore, the sequence  {E t }t =1,2,... is non-decreasing. We can easily verify that E (f ) is bounded (property of 

correntropy [6]), and by (12) we get that E (f t , P t ,Q t ) is also bounded. Consequently we can conclude that GLR-MCC will converge.

4 Experiments

、

In this section, we verify the robustness of our proposed GLR-MCC algorithm and compare the performance with GLR method. Three UCI Databases ORL and Yale face databases are selected for performance evaluation. Details of these datasets are shown in Table 1. Table 1. Datasets used in experiments Datasets Balance-scale Sonar Tic-Tac-Toe(TTT) ORL Yale

Dimension 4 60 9 1024 1024

Classes 3 2 2 40 15

Instances 625 208 958 400 163

Feature real number real number real number non-negative non-negative

Robust Semi-supervised Learning for Biometrics

473

In all the experiments, training and testing samples are randomly selected from each dataset and semi-supervised learning accuracy is reported. In all the figures, the x-axis represents the percentage of randomly labeled samples, and the y-axis is the average classification accuracy over 50 independent runs. Traditional graph construction methods typically decompose the graph construction process into two steps, graph adjacency construction and graph weight calculation. For graph adjacency construction, there exist two widely used methods: ε -ball neighborhood and k-nearest neighbors. For graph weight calculation, there exist three frequently used approaches•1) Gaussian Kernel 2) Inverse Euclidean Distance 3) Local Linear Reconstruction Coefficient. In our experiments, we adopt k-nearest neighbors and Gaussian Kernel to calculate W . K-nearest neighbors: The samples x i and x j are considered as neighbors if x i is among the k-nearest neighbors of x j or x j is among the k-nearest neighbors of x i where k is a positive integer and the knearest neighbors are measured by the usual Euclidean distance. Gaussian Kernel: ⎧ − || xi − x j || ⎪ δ Wij = ⎨e ⎪⎩ 0

2

i is j ' s neighbor i isn ' t j ' s neighbor

In all of the experiments, k is set to 20, σ is set to 0.1. 4.1 UCI Datasets We applied GLR-MCC and GLR to three UCI data sets in UCI machine learning repositories: Balance–scale Sonar and TTT. Balance-scale database was generated to model psychological experimental results. Each example is classified as having the balance scale tip to the right, tip to the left, or be balanced. The attributes are the left weight, the left distance, the right weight, and the right distance. The correct way to find the class is the greater of (left-distance × left-weight) and (right-distance × rightweight). If they are equal, it is balanced. Sonar data set contains signals obtained from a variety of different aspect angles, spanning 90 degrees for the cylinder and 180 degrees for the rock. Tic-Tac-Toe (TTT) database encodes the complete set of possible board configurations at the end of tic-tac-toe games, where "x" is assumed to have played first. We randomly select 5% of training samples as mislabeling noise. Table 2. Classification accuracy on mislabeling noise

Datasets Balancescale Sonar TTT

40% 89.70 77.45 81.77

GLR 60% 90.40 80.32 82.82

70% 90.38 83.91 82.87

40% 91.66 79.34 82.27

GLR-MCC 60% 92.54 82.88 83.40

70% 93.93 85.84 84.37

Table 2 shows the results of different methods under mislabeling noise. GLR-MCC achieves the highest average correct classification rate on all testing datasets. Since

474

N. Yang et al.

the mislabeling noise changes the relationship of neighborhood near the boundary of different classes, GLR’s performance decreases rapidly as number of training samples becomes smaller. Since GLR-MCC tries to reduce the contribution of noise points, it can efficiently deal with the mislabeling noise so that it performs better than GLR. 4.2 ORL and Yale Face Datasets In real face recognition system, it is necessary to label large amount of training facial images to learn a robust classifier. But during collection of facial images, two types of noise maybe occur. The one is mislabeling noise and the other is image noise. The mislabeling noise occurs when we mislabel one person’s facial image to another; and the image noise occurs when the person’s face is close to border of the camera so that we can only crop part of the face. In this experiment, we use ORL and Yale face databases to validate different methods under above two noises. The ORL database contains ten different images of each of 40 distinct subjects. The images were taken at different times, varying the lighting, facial expressions and facial details. The Yale face database consists of 165 gray-scale images of 15 individuals. There are 11 images per subject, with variations of facial expressions or different configurations. Each facial image is in 256 grey scales per pixel and cropped into size of 32 × 32 pixels by fixing the positions of two eyes. For each individual images are randomly selected for training and the rest are used for testing. And 5% facial images per person is randomly selected as noisy samples.

95 average classification accuracy(%)

average classification accuracy (%)

Fig. 1. Cropped facial images and their corresponding noisy images on ORL database

90 85 80 75 GLR-MCC GLR

70 65 40

50

60

70

80

percentage of randomly training points

(a) ORL

90

75

70

65 GLR-MCC GLR

60

55 40

50

60

70

80

percentage of randomly training points

(b) Yale

Fig. 2. Average classification accuracy on mislabeling noise

90

average classification accuracy (%)

average classification accuracy(%)

Robust Semi-supervised Learning for Biometrics

90 85 80 75

GLR-MCC GLR

70 40

50

60

70

80

percentage of randomly training points

(a) ORL

90

475

72 70 68 66 64 62

GLR-MCC

60

GLR

58 40

50

60

70

80

90

percentage of randomly training points

(b) Yale

Fig. 3. Average classification accuracy on image noise

Fig.2 shows the results of different methods under mislabeling noise. The average correct classification rate of GLR-MCC is clearly higher than that of GLR, at 60% the correct classification rate of GLR-MCC is significantly better than GLR. The results demonstrate that GLR-MCC provides an efficient method under mislabeling noise. Fig.3 shows the results of different methods under noisy images. Similar to mislabeling noise, the average correct classification rate of GLR-MCC is higher than that of GLR, GLR-MCC rises noticeably at 60%. When training samples are small, noise has significant effect on the results. Since GLR-MCC can remove the noise images, the results are better than GLR. Experimental results illustrate that GLRMCC is robust against training noise for both images and labels.

5 Conclusion In this paper, we propose a novel robust GLR-MCC framework based on MCC. It utilizes MCC as objective to replace GLR’s least squares criterion, and hence is not sensitive to outliers theoretically. This method firstly calculates graph weight, then calculates labels according to our objective. Lastly a greedy strategy is proposed for optimizing the problem based on the half quadratic technique, along with its convergence analysis. Experiments on recognition tasks have demonstrated GLRMCC is robust against noise comparing with GLR. In our future work, we will focus on the theoretical analysis and accelerating issues of our algorithm.

Acknowledgments This work was supported by DUT R & D Start-up costs.

References 1. Wang, F., Zhang, C.S.: Label propagation through linear neighborhoods. IEEE Transactions on Knowledge and Data Engineering 20(1), 55–67 (2008) 2. Wang, F., Zhang, C.S.: Robust self-tuning semi-supervised learning. Neurocomputing 70, 2931–2939 (2007)

476

N. Yang et al.

3. Belkin, M., Matveeva, I., Niyogi, P.: Regularization and Semi-supervised Learning on Large Graphs. In: Shawe-Taylor, J., Singer, Y. (eds.) COLT 2004. LNCS (LNAI), vol. 3120, pp. 624–638. Springer, Heidelberg (2004) 4. Zhou, D., Bousquet, O., Lal, T.N., Weston, J., Scholkopf, B.: Learning with Local and Global Consistency. In: NIPS 16 (2004) 5. Zhu, X., Ghahramani, Z., Lafferty, Z.: Semi-Supervised Learning Using Gaussian Fields and Harmonic Functions. In: International Conference on Machine Learning, ICML (2003) 6. Liu, W., Pokharel, P.P., Principe, J.C.: Correntropy: Properties and application in nongaussian signal processing. IEEE Transactions on Signal Processing 55(11), 5286–5298 (2007) 7. He, R., Hu, B.G., Yuan, X.T., Zheng, W.S.: Principal Component Analysis Based on Nonparametric Maximum Entropy. Neurocomputing (2009) 8. Chapelle, O., Weston, J., Scholkopf, B.: Cluster kernels for semi-supervised learning. Advances in Neural Information Processing System 15, 585–592 (2003) 9. Chapelle, O., Weston, J., Scholkopf, B.: Cluster kernels for semi-supervised learning. In: NIPS 15 (2003) 10. Zhou, D., Bousquet, O., Lal, T.N., Weston, J., Scholkopf, B.: Learning with Local and Global Consistency. In: NIPS 16 (2004) 11. Liu, W.F., Pokharel, P.P., Principe, J.C.: Correntropy: Properties and Applications in NonGaussian Signal Processing. IEEE Transactions on Signal Processing 55(11), 5286–5298 (2007) 12. Vapnik, V.: The Nature of Statistical Learning Theory. Springer, New York (1995) 13. Rockfellar, R.: Convex analysis. Princeton Press, Upper Saddle River (1970) 14. He, R., Hu, B.G., Yuan, X.T.: Robust discriminant analysis based on nonparametric maximum entropy. In: Asian Conference on Machine Learning (ACML), NanJing, China (2009) 15. Nikolova, M., Ng, M.K.: Analysis of half-quadratic minimization methods for signal and image recovery. Society for Industrial and Applied Mathematics 27(3), 937–966 (2005) 16. Yang, S.H., Zha, H.Y., Zhou, S.K., Hu, B.G.: Variational graph embedding for globally and locally consistent feature extraction. In: ECML PKDD, pp. 538–553 (2009) 17. Grandvalet, Y., Bengio, Y.: Entropy Regularization (2006) 18. Yuan, X.T., Hu, B.G.: Robust feature extraction via information theoretic learning. In: International Conference on Machine Learning, ICML (2009) 19. He, R., Hu, B.G., Zheng, W.S., Guo, Y.Q.: Two-stage Sparse Representation for Robust Recognition on Large-scale Database. In: Twenty-Fourth AAAI Conference on Artificial Intelligence, AAAI-10 (2010) 20. Belkin, M., Niyogi, P., Sindhwani, V.: Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples. Journal of Machine Learning Research 7(11), 2399–2434 (2006)

Research on Virtual Assembly of Supercritical Boiler Pi-guang Wei, Wen-hua Zhu, and Hao Zhou CIMS and Robot Center, Shanghai University, Shanghai 200072, China [email protected]

Abstract. Supercritical boiler is an important measure to solve problems like electricity shortage or energy intensity, with its high combustion efficiency. As supercritical boiler is a large and complex product, it may appear some problems of collision, location error, or assembly order reverse in assembling process. In order to solve these problems, this paper studies the virtual assembly technology of supercritical boiler. This includes the establishment of model of supercritical boiler, the transformation technology of the model and the Lightweight model technology. Taking 600MW supercritical boiler as an example, a virtual assembly simulation is established in virtools environment. Although the supercritical boiler product contains about 100,000 parts, this technique can achieve fluency of virtual assembly process. It also realizes higher humanmachine interaction, and guilds the assembly process of supercritical boiler. Keywords: Virtual Assembly; Supercritical Boiler; Virtual Reality; Virtools; Lightweight model; Interaction.

1 Introduction On February 16, 2005, "Kyoto Protocol" signed by Parties of "United Nations Framework Convention on Climate Change" provides that developed countries reduce the emission of greenhouse gas by 5.2% based on 1990, between 2008 to 2012[1]. In response to this request, to cope with the problems such as increasingly serious energy intensity, power shortage, low energy utilization, environmental pollution, we need to research and develop the technology of supercritical boiler with its high combustion efficiency. As supercritical boiler is a large and complex product, the assembly process of supercritical boiler is also a complex process. In the process of supercritical boiler assembly, there may be some series problems such as collision occurred, spatial error, reverse sequence of assembly because of the incomprehension of previous design and plans. While most way of boiler assembly process is using welding, the boiler assembly is an irreversible process at the same time. Therefore, the assembly process problems, will greatly influence the whole project, or even result in a huge waste. Address the issues raised above, many people have done the correlative research. Xiangping Zhou puts forward some management measures, to ensure the normal operation of the supercritical boiler assembly[2]. Wanlong Shi an engineer from the Third Engineering Company of Heilongjiang Province studies the Discussion of installation procedure for 660 MW tower type supercritical boiler in northern part of K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 477–485, 2010. © Springer-Verlag Berlin Heidelberg 2010

478

P.-g. Wei, W.-h. Zhu, and H. Zhou

China[3]. These measures made a significant contribution to solve the problems in the supercritical boiler assembly. But now, as the greatly development of virtual reality technology, can we use this technology of virtual assembly to solve the problem. The answer is yes. Today's virtual assembly technology has greatly developed. Washington University and The American National Standards Institute of Technology together develop the VADE(Virtual Assembly Development Environment)[4]; Jung from Bielefeld University developed such virtual assembly system CODY based on the knowledge [5], this system has an important characteristic which supports natural language to take simple assembling orders to achieve more natural human-machine interaction; Zhuhua Liao form Institute of Plasma Physics tells about assembling characteristic of EAST, presents movement model of overhead crane, user interface model and structure of multi-3D scene[6]; Jianhua LIU from Beijing Institute of Technology proposed a virtual assembly-oriented multi-layer exact collision detection algorithm based on the analysis of the problem in the virtual assembly environment[7]. CIMS/ERC, Tsinghua University develops a virtual assembly supported system (VASS), with which a product can be analyzed, simulated and pre-assembled virtually[8]. Based on the studies of virtual assembly above, in order to solve the problems in the assembly of supercritical boiler, this paper puts forward the concept of virtual assembly of supercritical boiler. A virtual assembly simulation is established in virtools environment after studying the structure of supercritical boiler, making full use of lightweight technology. The technology of virtual assembly can realize the virtual assembly process of products with more than 10,000 parts. The hardware requirements also are not very high, and it’s no need to develop special program. The only thing need is to define the simple chart to realize the complex process of products virtual assembly, which can highly realize human-machine interaction.

2 The Technologies of Virtual Assembly 2.1 The Analysis and Planning of Supercritical Boiler Model Supercritical boiler is one of the main three components, which compose the thermal power plant. A supercritical boiler generally contains about 100,000 parts, while the cost of a supercritical boiler can reach 3 to 4 billion and its size is very large. Because of the complexity of its structure, the supercritical boiler assembly process is a huge and complex process. However, in this complex process we can still find some of the regular rules. Generally, the process of steel frame assembly is before the process of the inside parts assembly such as the heating surface assembly, the pressure parts assembly and so on. Boiler assembly mainly takes the bulk approach. To reduce the amount of high-altitude operations and to accelerate the construction schedule, assemble local components in the ground at first which will be installed on the whole boiler. Therefore, we can grasp the main part of the virtual assembly during the boiler assembly process, ignore the small details and the partial assembly. In summary of boiler assembly process, the sequence of boiler virtual assembly must meet the following requirements: Steel installation must start from front furnace to back

Research on Virtual Assembly of Supercritical Boiler

479

furnace, while installations of left and right side start at the same time and install the steel on each layer separately from bottom to top. The heating surface and pressure parts are assembled by division, which divided into the vertical section of wall, spiral section wall, ash bucket, superheater, reheater, after the smoke well, rigid beams, roof piping, air-type expansion device, start the separator and other components. 2.2 Lightweight Model Technologies Because the model is very large and complex, we need to lightweight the model, if we want to accomplish the virtual assembly of the supercritical boiler with only one common computer. Based on the research of modeling character and virtual environment as well as the lightweight mode, we take methods as follows. Simplifying the Meshes. The 3d models depend on the meshes, which contains material, texture and animation. We can see the relationships between them in fig 1. In the test of the virtual environment, through the precision of the model meshes, the spatial capacity can be greatly reduced. Also during the test, it is definitely clear that the description of the meshes take an 80% percentage of the all the data. For the boiler product, it contained huge amount of parts, however, in the virtual environment the same models with different serial have the same meshes. As to these kinds of models, they could share the same meshes rather than corresponding to the mesh of their own. Meanwhile, for some parts with simple structure but fine mesh, we can use the special plug-in to simplify the mesh. Comparing the parts before simplification, the spatial capacity of the model reduced to 60%.

Fig. 1. The figure tells about the relationships of mesh, model, material, texture and animation

3DXML Lightweight Technology. 3DXML is one kind of technology which is used to describe the 3d data in a simple and agile way, which is launched by Dassault System in June 2005. LOD Technology. LOD (Level Of Detail) is one kind of lightweight technology that was created especially for the improvement of the efficiency of the modeling. It can dynamically describe the complexity of the model meshes according to the screen percentage of the model. For instance, when the model only covers a small part of the screen, we do not need to have a detailed description of its structure and the overall description of the object can meet the demands. The relationship is shown in the fig 2.

480

P.-g. Wei, W.-h. Zhu, and H. Zhou

Fig. 2. The fig instructs the mechanism of LOD

2.3 The Introduction of Virtools Virtools technology is a powerful solution for the 3d visualization and interactivity, which is mainly used in 3D timely interaction. It also can make an integration of the text, graphics, video and sound. The particular character of virtools is that it builds in hundreds of BB (Building Blocks). Though simple operations and settings, we can define the complex interactive behaviors. So it is a powerful, easy to use, and very effective development tool.

3 The Realize of Virtual Assembly of Supercritical Boiler 3.1 The Establishment of Boiler Model The Model of Heating Parts and Pressure Parts. The heating parts and the pressure parts are the main parts of the boiler design. The 3D models of these parts have already been found at the beginning of the boiler design, and we can inherit and use 3D models of these parts. The designed models of the heating parts and the pressure parts in NX can be transformed to the environment of Virtools by some intermediate format. There are two methods specifically. One way is to use VRML format in NX, and the models can be transformed to the format files with an extension of .wrl, which can be opened in Virtools directly; Another way is to use 3DXML format. As NX does not support the format, and CATIA has very good interface with 3DXML, the first step is to export the NX models into CATIA. In the process we should use STEP format file, which is a neutral file and is playing a role of a bridge in the process of transforming one format file into another. The models exported into CATIA by STEP format file can be transformed to format files of 3DXML directly, which can be loaded in Virtools directly. The Model of Steel Frame. The steel models used in the virtual simulation are also using the initial designing models such as the heating parts and the pressure parts, but the designing environment of the steel frame models are different, so all the methods

Research on Virtual Assembly of Supercritical Boiler

481

above are not practical. The steel frame models are 3D models in AUTOCAD. Because AutoCAD and 3DS MAX all belong to Autodesk Company, and the plug-in interfaces have already been developed between 3DS MAX and Virtools, we can transform models into 3DS MAX. 3DS MAX has a very good compatibility with AutoCAD and can open AutoCAD models directly. By Virtools Export, we can export the models into files with an extension of .nmo, which can be loaded in Virtools directly. The Establishment of Scene Model. Boiler’s scene model plays an emphasized role in the virtual assembly, which aim is to enhance the realism of virtual assembly. While boiler scene design does not belong to the contents of the boiler, there is no ready-made model of the scene that can be borrowed. As the boiler scene is to enhance the effect of rendering type, we do not need to over-build scenario models seriously. We build the scene model using the SketchUp environment, which is a very simple software. This software is very suitable for building models in threedimensional design, and it is very efficiency. The established three-dimensional model can be exported as 3DS format, which can be loaded directly in the Virtools environment. 3.2 Lightweight Model In the process of converting models above, there are a lot of techniques to lightweight the model. Taking the heating parts and pressure parts for example, we can use the way of conversion into CATIA 3D XML technology to lightweight the model, while during the conversion of VRML format we use the way of the X3D(Extensible 3D) to lightweight the model. Comparing the two conversion methods, the results show that the lightweight model by adoption of 3D XML technology is smaller. As to the steel model, we use the streamlining plug of Polygon Cruncher from3D MAX to simplify the model. In the establishment of scene model, we use the 3DS format to simplify the model. All the conversion methods and lightening techniques are shown in Fig. 3.

Fig. 3. This fig tells about the conversion methods and lightening techniques

482

P.-g. Wei, W.-h. Zhu, and H. Zhou

3.3 Design of Virtual Assembly Virtual assembly of supercritical boiler is realized in the environment of virtools. It is easy to define the movements, the route, the statement of the model and the visual angle of the camera coupled with the instruction of words, sound or video. The virtual assembly includes the assembly of steel frame and internal model. Camera at the appropriate point as well as some instructions is also necessary. And in order to exhibit the virtual assembly better, higher human-machine interaction is needed as well. The Virtual Assembly of Steel Frame. In reality, the assembly is carried out from the boiler front to the boiler back as well as the left and right parts assembly at the same time. According this procedure, we simulate virtual assembly of steel frame from the front to the back, apply the presentation form layer by layer. The fig. 4. followed shows the steel assembly process. It is achieved through BB (Building Blocks) like Group Iterator Show Delay and so on to carry out the effect level by level.

、

、

Fig. 4. This fig tells about the process of steel assembly

Fig. 5. This fig tells about the procedure of internal parts assembly

Research on Virtual Assembly of Supercritical Boiler

483

The Virtual Assembly of Inside Parts. The models inside the boiler are assembled after completing the assembly of boiler steel frame. For presenting the internal model much better, we disguise the steel frame model when it is being assembled. The assembly of internal model is mainly presented by putting all the components together. We apply the form of the assembly from top to bottom. Fig.5 shows the procedure of internal parts assembly. It is realized by BB like Show Move To and Delayer.

、

3.3.3 Design of Shifting Camera Angle. As the huge boiler model and the complexity of the boiler assembly, the boiler assembly process can not be present only by one perspective. Therefore, shifting camera angle constantly during the assembly for expressing the objects to be described is essential. In the virtools, camera is defined as an object whose position and orientation can be set by BB like Bezier Progression Position On Curve Set Position Set as Active Camera Look At and so on. It’s schema can be seen in Fig.6.

、

、

、

、

Fig. 6. The specific flowchart of Shifting Camera Angle

The Embedded Explanation. Explanations such as text or sound play a supporting role in the assembly process of boiler. A complete virtual assembly should have these factors. In the virtools environment, you can show the text by the use of Text Display, and play music by Wave Player. It is important to note that explanations must correspond to model assembly process. In order to meet this requirement we use the Sequencer and Parameter Selector. The combination of these two BBs can achieve to trigger different parameters in different times. The sequence of model assembly determines the trigger time of Sequencer, and output parameters have been pre-set in Parameter Selector. The two BBs together achieve to output the corresponding parameters at the right time. The Setting of Interaction of virtual assembly. In the virtual assembly of model, the interaction is a very important factor. Through human-computer interaction, people's initiative can be reflected. The virtual assembly process is showed in a better way and meet the people's will much more. In the virtools, it is easy to implement interaction. There are special BBs to achieve interaction between people and the computer by mouse, keyboards, joystick or other equipment. These BBs are Key Event, Push Button, Joystick Controller, Switch On Key, Mouse Waiter and so on. In the virtual assembly process of boiler, there are two virtual assembly modes. One is the automatic assembly mode, and the other is the free assembly mode. In free assembly mode, people can control perspective by mouse, keyboard and other devices. The

484

P.-g. Wei, W.-h. Zhu, and H. Zhou

switch of automatic mode and free mode is realized by BB such as Key Event, Get Current Camera, Set As Active Camera and so on. The specific flowchart is shown in fig 7.

Fig. 7. This fig tells about the specific flowchart of switching mode

4 Conclusion The virtual assembly takes a very good guidance on supercritical boiler assembly process, and is beneficial to avoid the problems of supercritical boiler assembly process. Based on the virtools environment, the virtual assembly technology is a new kind of technology. This virtual assembly technology can achieve more than 100,000 parts of virtual assembly process of products. In the establishment of virtual assembly process, it can easily integrate such factors as image, sound, video, etc. It also can achieve higher level man-machine interaction. The model lightweight process can use the common lightweight technology such as 3DXML, to expression more complex product with smaller model. Establishing the virtual assembly process doesn't need programming, and the only thing need is to define standard flowcharts, so the operation is convenient as well as easy. Acknowledgment. The project is supported by Shanghai informationization special project from Shanghai Municipal Science and Technology Committee (Project No.08DZ1124300), partially supported by the Municipal Program of Science and Technology Talents, Shanghai China. (Project No. 09QT1401300), and Shanghai Leading Academic Discipline Construction (Project No. Y0102). The authors gratefully acknowledge the encourages from Prof. Ming-lun FANG et al, and acknowledge the support of colleague in CIMS Center of Shanghai University.

References 1. Wang, D.B.: The Research On Key Technologies in Supercritical Pressure Boiler Design Platform. Shanghai University 147, 1–2 (2010) 2. Zhou, X.P.: Research On Quality Control Measures of Boiler Assembly. Guangdong Science and Technology 202, 207–208 (2008) 3. Shi, W.L.: Discussion of installation procedure for 660 MW tower type supercritical boiler in northern part of our country. Heilongjiang Electric Power 30, 311–314 (2008) 4. Jayamam, S., Jayaram, U., Wang, Y.: A virtual assembly design environment. IEEE Computer Graphics and Applications (S0272-1716) 19, 44–50 (1999)

Research on Virtual Assembly of Supercritical Boiler

485

5. Jung, B., Hoffhenke, M., Wachsmuth, I.: Virtual Assembly with Construction Kits. In: Proceedings of DETC, ASME Design for Engineering Technical Conferences, pp. 1–7 (1997) 6. Liao, Z.H., Liu, X.P.: Design of 3D Interaction and User Interface for EAST Assembly Simulation. Journal of System Simulation 16, 2329–2343 (2004) 7. Liu, J.H., Yao, J., Ning, R.X.: Research and Realization of Collision Detection Algorithm in Virtual Assembly Environment. Journal of System Simulation 16, 1775–1778 (2004) 8. Li, Z., Zhang, L.X., Xiao, T.Y.: Introduction of a Virtual Assembly Supported System. Journal of System Simulation 14, 1150–1153 (2004)

Validation of Veracity on Simulating the Indoor Temperature in PCM Light Weight Building by EnergyPlus Chun-long Zhuang1,2, An-zhong Deng1, Yong Chen1, Sheng-bo Li1, Hong-yu Zhang1, and Guo-zhi Fan1 1

Department of Barracks’ Management and Environmental Engineering, PLA Logistic Engineering University, Chongqing 400016, China 2 College of Urban Construction and Environmental Engineering, Chongqing University, Chongqing 400030, China

Abstract. This article surveys the EnergyPlus constructions solution algorithm and heat balance method in EnergyPlus, presents the new conduction finite difference solution algorithm and enthalpy-temperature function features, describes the implementation of the module, and validates the veracity of the possible applications for the capabilities in EnergyPlus by PCM experiment dates. The most relativity difference is 12.41% and the least is 0.71% between the simulation and testing value in sequential 36h on the A envelope condition, wheras on the B envelope condition the most relativity difference is 8.33% and the least is 0.33% in sequential 72h.The results to date has shown good agreement with well established in this simulation tools and shown that the algorithm incorporated into EnergyPlus can simulate the PCM in building construction.

（PCMs）, Indoor

Keywords: Energyplus, Validation, Phase change materials temperature.

1 Introduction The use of PCMs in buildings, especially passive and active solar buildings, has received considerable interest since the first published application in 1940. The energy storage by wallboards is of particular importance to store and recover solar energy, many applications have been considered and some can be found in Refs.[1–3].Up to now, using PCMs in building materials was limited by many difficulty such as the engineering algorithm to confirm the PCMs dosage and phase change temperature in different climate area and incorporating the PCM into the wallboard ect. EnergyPlus —new programs of simulating this type of material in building fabric may allow us an easier handling and the use of PCM in a more handy way. The EnergyPlus project began in the late 1990s as an effort to combine the best features of BLAST and DOE-2, two popular energy analysis programs used internationally, into a single program [4-6]. The goal was to provide the simulation community with a single platform for model development, saving costs and time in the never K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 486–496, 2010. © Springer-Verlag Berlin Heidelberg 2010

Validation of Veracity on Simulating the Indoor Temperature

487

ending process of keeping the programs up to speed with the rapid changes in the HVAC industry and maximizing the impact that a new program might have on the architectural and engineering community. For heat transfer, EnergyPlus adopted the Conduction Transfer Function CTF solution algorithm to calculate the conduction through the common walls. For high performance building designs, for example the advanced constructions—PCM, this had been done by addition a new solution algorithm that utilized an implicit finite difference procedure in EnergyPlus. This paper surveyed the EnergyPlus constructions solution algorithm and heat balance method in EnergyPlus, presented the conduction finite difference solution algorithm features, described the implementation of the module, and validated the veracity of the possible applications for the capabilities in EnergyPlus by PCM house experiment dates.

（ ）

2 EnergyPlus Solution Algorithm 2.1 Conduction Finite Difference Solution Algorithm [4] EnergyPlus models followed fundamental heat balance principles very closely in almost all aspects of the program. However, the simulation of building surface conventional constructions had relied on a transfer function transformation carried over from BLAST. This had all the usual restrictions of a transformation-based solution: constant properties, and fixed values of some parameters. As the energy analysis field moved toward simulating more advanced constructions, such as phase changed materials (PCM), it becomes necessary to step back from transformations to more fundamental forms. Accordingly, a conduction finite difference solution algorithm has been incorporated into EnergyPlus. This did not replace the CTF solution algorithm, but complemented it for cases where the user needed to simulate phase change materials or variable thermal conductivity. It was also possible to use the finite difference algorithm for zone time steps as short as one minute, corresponding to the system time step. The algorithm used an implicit finite difference scheme coupled with an enthalpy-temperature function to account for phase change energy accurately. The One dimensional steady heat conduction process through the wall was shown as the formula (1).The implicit formulation for an internal node was shown in equation (2).

ρ

∂(c p t ) ∂τ

∂ 2t =k 2 ∂x

(1)

ρcpΔx(Ti,new −Ti,old ) k(Ti−1,new −Ti,new) k(Ti+1,new −Ti,new) Δt

=

；

Δx

+

Where: T is node temperature

i =node being modeled;

i + 1 =adjacent node to interior of construction;

Δx

(2)

488

C.-l. Zhuang et al.

i − 1 =adjacent node to exterior of construction; new =new temperature at end of time step; old =temperature at end of previous time step;

Δt =calculation time step; Δx =finite difference layer thickness (always less than construction layer thickness);

Cp

=specific heat of material;

ρ = density of material.

Then, this equation is accompanied by a other equation—enthalpy-temperature function 3 that relates enthalpy and temperature.

hi = HTF (Ti )

(3)

Where HTF was an enthalpy-temperature function that uses user input data. The enthalpy-temperature function [7-10] is showed as the formula (4) and 5 .

where

， H = enthalpy,

⎧ 2 ⎪ ρ ∂H = k ∂ T ⎪ ∂t ∂y 2 ⎪ ⎨T y =0 = To ⎪ ⎪−k ∂T = h ⎡⎣T ( yo ) − T f ⎤⎦ ⎪ ∂y y = yl ⎩

⎪ ⎪ H / c p,s ⎪⎪ H + (T − ε )r / 2ε m T =⎨ c p , s + r / 2ε ⎪ ⎪ H − Hl ⎪Tm + ε + c p ,l ⎪⎩ Where,

c p ,s

and

c p ,l

(4)

H < Hs H s ≤ H ≤ Hl

(5)

H > Hl

;

=respectively solid state and liquid state specific heat of material

ε =the half of phase change temperaturerange ΔTpc ( K ) ; r =phase change latent heat (J·kg )；

H s and H l =respectively solid state and liquid state enthalpy of material, which

was showed as follow:

H s = c p , s (Tm − ε )

(6)

Validation of Veracity on Simulating the Indoor Temperature

489

H l = c p ,s ( Tm + ε ) + r

(7)

It turned out that there were four node types: external surface nodes, internal surface nodes, internal nodes such as shown above, and then nodes occurring at the material interfaces. The formulation of all node types was basically the same. So, equations such as (2) and (3) were formed for all nodes in a construction. Because the solution was implicit, a Gauss-Seidell iteration scheme was used to update to the new node temperatures in the construction. Because of this, the node enthalpies get updated each iteration, and then they were used to develop a variable C p if a phase change material was being

（）

simulated. This was done by including a third equation 8 for C p . Cp =

(h

i , new

− hi ,old )

(8)

Ti ,new − Ti ,old

The iteration scheme assured that the correct enthalpy, and therefore the correct

C p was used in each time step, and the enthalpy of the material was accounted for accurately. Of course, if the material was regular, the user input constant

C p was

used. The algorithm also had a provision for including a temperature coefficient to modify the thermal conductivity. 2.2

Zone Air Temperature Solution Algorithm

The zone air heat balance was the accurate method that coupled with the indoor air, outside surface heat balance and inside surface heat balance on the building fabrics, as were showed in Figure1, 2 and 3.By this method a variable adaptive time step shorter than one hour could be used for updating the system conditions. For stability reasons it was necessary to derive an equation for the zone temperature that included the unsteady zone capacitance term and to identify methods for determining the zone conditions and system response at successive time steps. From Figure 3, it could be seen that for energy storage in the zone air, a heat bal-

（9）[4],[11]:

ance equation would be similar to the following formula

CZ

dTz Nsl − = ∑QQ i + dt i=1

Nsurfaces

Nzones −

−

i ∑ hA i i ( Tsi −Tz ) + ∑ mC p ( Tzi −Tz ) + minf Cp (T∞ −Tz ) +Qsys

i=1

i=1

(9)

490

C.-l. Zhuang et al.

Fig. 1. Outside surface heat balance

Fig. 2. Inside surface heat balance

Fig. 3. Zone air heat balance

Fig. 4. Building - Isometric View

Where: CZ

dTz = energy stored in zone air dt

N surfaces

；∑

loads

i =1

hi Ai (Tsi − Tz )

N zones −

∑ m C (T i

i =1

p

zi

； ∑Q N sl

i =1

−

i

= sum of the convective internal

；

= convective heat transfer from the zone surfaces

− Tz )

−

= heat transfer due to inter zone air mixing;

；

heat transfer due to infiltration of outside air

Qsys

m inf C p (T∞ − Tz )

—

一system output.

Where, CZ was the product of the mass of the zone air and the specific heat of air.

（10）for the

Used in conjunction with a third order finite difference approximation

derivative term and time steps between 0.1 and 0.25 hours, this method of calculating the zone air heat balance had been shown to be stable [12]. Thus, the basic heat balance approach was to set up energy balance equations using control volumes at the inside and outside faces of each surface in a particular zone as well as a control volume around the zone air.

Validation of Veracity on Simulating the Indoor Temperature

491

While a solution could be found for this system of equations, the actual solution process was the third order finite difference approximation of the derivative term, dTz 3 1 −1 ⎛ 11 ⎞ ≈ (δ t ) ⎜ Tzt − 3Tzt −δ t + Tzt−2δt − Tzt −3δt ⎟ + O(δ t 3 ) . dt t 2 3 ⎝6 ⎠

(10)

One final rearrangement was to move the lagged temperature in the derivative approximation to the right side of the equation. The explicit appearance of the zone air temperature was thus eliminated from one side of the equation. An energy balance equation that included the effects of zone capacitance was then obtained by dividing both sides by the coefficient of Tz, as was showed in formula

（11）: This was the

form currently used in EnergyPlus. Nsurfaces

Nsl .

T= t z

Nzones .

.

.

∑Q+ ∑hAT + ∑mCT +m CT +m CT i=1

i

i=1

i i si

i

i=1

p zi

inf

p∞

sys

p supply

3 1 ⎞ ⎛C ⎞⎛ −⎜ z ⎟⎜−3Tzt−δt + Tzt−2δt − Tzt−3δt ⎟ . 2 3 ⎠ ⎝δt ⎠⎝

N

Nzones . . . 11Cz surfaces + ∑hA i i + ∑mC p +minf Cp +msys C 6 δt i=1 i=1

(11)

3 Method or Experimental The most new EnergyPlus Version 3.1 was used to model a range of building specifications as specified in ANSI/ASHRAE Standard 140-2007- Standard Method of Test for the Evaluation of Building Energy Analysis Computer Programs and in the Building Energy Simulation Test (BESTEST) and Diagnostic Method. The ability of EnergyPlus to predict thermal loads was tested using a test suite of 18 cases which included buildings with both low mass and high mass construction, without windows and with windows on various exposures, with and without exterior window shading, with and without temperature setback, with and without night ventilation, and with and without free floating space temperatures. The annual heating and cooling and peak heating and cooling results predicted by EnergyPlus for 13 different cases were compared to results from 8 other whole building energy simulation programs. Based on 62 separate possible comparisons of results for the BASIC test cases, EnergyPlus was within the range of spread of results for the other 8 programs for 54 of the comparisons [13-14]. But it had not been reported on the condition for PCM building envelope in ANSI/ASHRAE Standard 140-2007 - Standard Method of Test. 3.1 Simulation Building Size and Envelope

Figure 4 showed the Building - Isometric View of southeast corner with Windows on South and north Wall. The PCM room dimension length×width×high = 1.5m×1.5m×2.0m door and windows the door was in the east wall and dimension 1.5m×0.6m window dimension:0.3m×0.4m. The wall, roof and floor adopted

； ；

492

C.-l. Zhuang et al.

the same envelope structure, namely, in Energyplus the MATERIAL: Regular object, Material Property: Temperature Dep: CondFD, the material thermal characteristics and envelope structure were showed in table 1, 2. For the validation of using EnergyPlus to simulate the indoor air temperature in PCM house, and according to the prophase experiment projects, in Group – Surface Construction Elements: Material Property: Temperature Dep: CondFD, we adopted two kinds of phase change temperature materials, the temperature of phase change was 40 and 33 (40 phase change material, recorded as 40# PCM, the latent heat was about 80000 J/kg, 33 phase change material, recorded as 33# PCM, the latent heat was about 70000 J/kg),so two kinds of envelope was adopted in simulation, namely A and B ,which were showed as table 2. Created the 6mm thick plank materials in term of the 5kg/m 2 surface densities, and were installed between the wood board and EPS foam board according to the different design. EnergyPlus now automatically calculates the local outdoor air temperature and wind speed separately for the zone and surface exposed to the outdoor environment. The zone centric or surface centric are used to determine the height above ground. Only local outdoor air temperature and wind speed are currently calculated because they are important factors for the exterior convection calculation for surfaces and can also be factors in the zone infiltration and ventilation calculations. The Int.Surface Coeff. and Ext. Surface Coeff was 8.7 and 23 W / m2.K separately.

℃

℃ ℃

℃

3.2 Weather Data

The Solar and Wind Energy Resource Assessment (SWERA) was adopted in this simulation. The SWERA was funded by the United Nations Environment Program, and was developing high quality information on solar and wind energy resources in 14 developing countries. Typical year hourly data were available for 156 locations in Belize, Brazil, China, Cuba, El Salvador, Ethiopia, Ghana, Guatemala, Honduras, Kenya, Maldives, Nicaragua, and Sri Lanka. The data were available from the SWERA project web site.In this paper we changed the SWERA file of Fig. 5. Experiment house Chongqing area fine and put the testing outdoor dry bulb temperature into the Chongqing SWERA weather data from 8/13/2007 to 14/8/2007and 8/23/2007 to 26/8/2007. 3.3 Experiment Set-Up

The PCM light weight experiment house was showed in Figure5,the test condition was the same as the simulation condition from the aspect of envelope structure, as was expatiated before, The experiment process and method might obtain from the reference [15-16].

Validation of Veracity on Simulating the Indoor Temperature

493

Table 1. Material thermal characteristics Element Thermal conductivity (w / m·k) Thickness Thermal resistance (m2.k/w) coefficient (w / m2.k) Thermal inertia index

veneer 0.17 5mm 0.03 4.57 0.13

EPS 0.042 50mm 1.19 0.36 0.43

33#PCM 0.30 10mm 0.033 28.65 0.95

40#PCM 0.30 10mm 0.033 28.65 0.95

Table 2. Envelope structure Element A envelope out to in B envelope out to in

veneer 5mm 5mm

40#PCM 10mm 10mm

EPS 50mm 50mm

33#PCM 10mm

veneer 5mm 5mm

4 Results and Discussion 4.1 Comparison of Indoor Air Temperature

(1) Comparison of A envelope simulation and testing Fig.6 is the Comparison of the indoor temperature between the simulation and testing value in sequential 36h on the A envelope 40#PCM condition. The most relativity difference is 12.41% and the least is 0.71% between the simulation and testing value in 40#PCM condition, the simulation and testing value are consistency basically. It may be the reasons that created the error, firstly we only changed the outdoor dry bulb temperature in SWERA file of Chongqing area, and that solar radiation and wind speed condition would great effect on the indoor air temperature, secondly we do not change the weather data before 8 /13, the weather in 8/12 will influence the indoor air environment especially in summer because of the PCM building envelope thermal inertia, lastly there are some difference in the material thermal characteristics between the simulation input value and actual testing process, which will affect the validity results.

（

）

Fig. 6. Comparison of 40#PCM simulation and testing in sequential 36h

Fig. 7. Comparison of 40#+33#PCM simulation and testing in sequential 72h

494

C.-l. Zhuang et al.

Fig. 8. Comparison of A fabric south wall interior surface temperature

Fig. 9. Comparison of B fabric south wall interior surface temperature

4.2 Comparison of B envelope Simulation and Testing

Fig.7 showed the Comparison of the indoor temperature between the simulation and testing value in sequential 72h on the B envelope 40#+33#PCM condition. The most relativity difference is 8.33%, and the least is 0.33% between the simulation and testing value on the B envelope condition, the simulation and testing value are more consistency than the A envelope. It may be the reasons that we used two PCM layers and increased the PCM dosage, which leaded to the improvement on the building envelope thermal inertia, and it weakened the influence of the outdoor air environment to the indoor. we change the outdoor dry bulb temperature from 8 /22 to 8 /27 in the weather data, this cam weaken the influence of the 8 /22 weather condition on the indoor air environment in 8 /23. So, it is very important that using feasible weather data when simulating in Energyplus. Comparison results of A and B envelope condition have shown good agreement with well established in this simulation tools and shown that the algorithm incorporated into EnergyPlus can simulate the conduction through the PCM walls and the indoor air temperature in PCM Light Weight Building 4.3 Comparison of Fabric Interior Surface Temperature (1) South Wall Interior Surface Temperature Concerning the evolutions of other walls interior temperatures being similar, we only take the south walls interior temperatures to analyse.Fig.8,9 shows the curves of mean temperatures for the interior surfaces of A and B fabric south walls for the cases with PCM material. The most relativity difference is 8.18% and 7.16% and the least is 0.54% and 0.65% between the simulation and testing value in two kinds of fabrics condition, the simulation and testing value are consistency basically. The reason that created the error was the same as the before-mentioned Comparison of indoor air temperature condition.

，

Validation of Veracity on Simulating the Indoor Temperature





495

!

Fig. 10. Comparison of A fabric roof interior surface temperature

Fig. 11. Comparison of B fabric roof interior surface temperature

(2)Roof Interior Surface Temperature Fig.10, 11 shows the curves of mean temperatures for the interior surfaces of A and B fabric south walls for the cases with PCM material. The most relativity difference is 6.65% and 6.03% and the least is 0.22% and 0.38% between the simulation and testing value.

5 Conclusions This paper validates the veracity of the possible applications for the PCM fabrics in EnergyPlus by PCM experiment dates. The most relativity difference is 12.41% and the least is 0.71% between the simulation and testing value in sequential 36h on the A envelope condition, the most relativity difference is 8.33% and the least is 0.33% in sequential 72h on the B envelope condition. The most relativity difference for south wall interior surface temperature is 8.18% and 7.16% and the least is 0.54% and 0.65% between the simulation and testing. The most relativity difference for roof interior surface temperature is 6.65% and 6.03% and the least is 0.22% and 0.38% between the simulation and testing value. The difference in the weather dates and the material thermal characteristics will bring great influence on the error between the simulation and testing value. So, it is very important that using feasible weather data and material thermal characteristics value when simulating in Energyplus. EnergyPlus can simulate the conduction through the PCM walls and the indoor air temperature in PCM Light Weight Building while maintaining all the other aspects of a detailed energy simulation. The scheme of carrying the node enthalpy along with the simulation has proven to be very robust, and provides a completely accurate accounting for the phase change enthalpy.

，

496

C.-l. Zhuang et al.

References 1. Ismail, K.A.R., Henriquez, J.R.: Thermally effective windows with moving phase change material curtains. Applied Thermal Engineering 21, 1909–1923 (2001) 2. Stritih, U.: Heat transfer enhancement in latent heat thermal storage system for buildings. Energy and Buildings 35, 1097–1104 (2003) 3. Khudhair, A.M., Farid, M.M.: A review on energy conservation in building. Applications with thermal srorage by latent heat using phase change materials. Energy Conversion and Management 45, 263–275 (2004) 4. Berkely, L.: National Laboratory, University of Illinois, University of California. EnergyPlus 3.1Manual. LBNL in Berkeley, California (2009) 5. Jacobs, P., Henderson, H.: State-of-the-Art Review of Whole Building. Building Envelope, and HVAC Component and System Simulation and Design Tools, ARTI-21CR-60530010–30020-01 (2001) 6. Crawley, D.B., et al.: EnergyPlus: Creating a New-Generation Building Energy Simulation Program. Energy & Buildings 33(4), 443–457 (2001) 7. Guo, X.L., Kong, X.Q., Chen, S.N.: Numerical Heat Transfer. China Sciencete and Chnology Press (1988) 8. Zhou, J., Huang, S.Y.: Caculation for Phase Change Thermal Tranfer. Energy Technology (1), 11–14 (2000) 9. Zhang, Y.P., Hu, H.P., Kong, X.D., et al.: Phase Change Energy Storage-—Theory and Application. China Sciencete and Chnology Press (1996) 10. Ye, H., He, H.F., Ge, X.S.: The Comparative Numerical Investigations on the Melting Process of Form-Stable Phase Change Material Using Enthalpy Formulation Method and Effective Heat Capacity Formulation Method. Acta Energ1ae Solaris Sinica 25(4), 488– 491 (2004) 11. Richard, K., Curtis, O.: Modularization and Simulation Techniques for Heat Balance Based Energy and Load Calculation Programs: The Experience of The Ashrae Loads Toolkit and Energyplus. Building Simulation, 43–50 (2001) 12. Taylor, R.D., Pedersen, C.O., Fisher, D.E.: Elta: Impact of Simultaneous Simulation of Buildings and Mechanical Systems in Heat Balance Based Energy Analysis Programs on System Response and Control. In: Conference Proceedings IBPSA Building Simulation, Nice, France, pp. 20–22 (August 1991) 13. Energy Efficiency and Renewable Energy Office of Building Technologies, U.S. Department of Energy. EnergyPlus Testing with Building Thermal Envelope and Fabric Load Tests from ANSI/ASHRAE Standard 140-2007 Energy Plus Version 3.1.0.027 (April 2009) 14. IEA 1995. Building Energy Simulation Test (BESTEST) and Diagnostic Method. National Renewable Energy Laboratory, Golden, Colorado (February 1995) 15. Li, B.Z., Zhuang, C.L., Deng, A.: Study on improving the indoor thermal environment in light weight building combining pcm wall and nighttime ventilation. Journal of Civil, Architectural & Environmental Engineering 31(3) (2009) 16. Zhuang, C.L., Deng, A.Z., Li, S.B., et al.: The field study of temperature distribution in the floor radiation heating room with new phase change material. Journal of Center South University of Technology (14), 91–94 (2007)

Positive Periodic Solutions of Nonautonomous Lotka-Volterra Dispersal System with Delays Ting Zhang, Minghui Jiang, and Bin Huang Institute of Nonlinear and Complex System, China Three Gorges University, YiChang, Hubei 443002, China [email protected]

Abstract. In this paper, a general class of nonautonomous LotkaVolterra dispersal system with discrete and continuous infinite delays is investigated. This class of Lotka-Volterra systems model the diffusion of a single species into n patches by discrete dispersal. By using Schauder’s fixed point theorem, we prove the existence of positive periodic solutions of system. The global asymptotical stability of positive periodic solution is discussed and the sufficient conditions for exponential stability are also given. we give an example to illustrate the validity of the results in the end. The conditions we obtained are more general and it can be extended to several special systems. Keywords: Lotka-Volterra dispersal system, Positive periodic solutions, Schauder’s fixed point theorem, Global asymptotical stability, Global exponential stability.

1

Introduction

The Lotka-Volterra system forms a significant component of the models of population dynamics. In recent years, it has been usefully used in population dynamics, epidemiology, biology and any other areas in general. Because of its theoretical and practical significance, the Lotka-Volterra system has been extensively and intensively studied[1-10]. There have many papers on the study of the Lotka-Volterra type systems and it have been developed in[4,6,7,9,11]. Now the basic questions of this model are the persistence, extinction, global asymptotic behaviors, global exponential behaviors, existence of positive solutions and periodic solutions and so on. In this paper, we considered the following case of combined effects: dispersion, time delays, periodicity of environment. We study the following general nonautonomous Lotka-Volterra type dispersal system with discrete and continuous infinite time delays: x˙ i (t) = xi (t)[ri (t) − −

n   0 j=1

n  j=1

aij (t)xj (t) −

c (t, s)xj (t −∞ ij

n  j=1

+ s)ds] +

bij (t)xj (t − τij (t)) n  j=1

Dij (t)(xj (t) − xi (t)),

K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 497–505, 2010. c Springer-Verlag Berlin Heidelberg 2010 

(1)

498

T. Zhang, M. Jiang, and B. Huang

with the initial condition xi (s) = φi (s) f or s ∈ (−∞, 0], i = 1, 2, · · · , n.

(2)

where xi (t) represents the density of species in ith patch; ri (t) denotes the growth rate of species in patch i; aij (t) represents the effect of intraspecific (if i = j) or interspecific (i  = j)interaction at time t, bij (t) represents the effect of intraspecific (if i = j) or interspecific (i  = j) interaction at time t − τij (t); 0 c (t, s)x (t + s)ds represents the effect of all the past life history of the ij j −∞ species on its present birth rate; Dij (t) is the dispersion rate of the species from patch j to patch i. The model in this work can be extended to the models of [1-6]. In this paper, we assume the following assumptions hold for system (1): (H1 ) The bounded functions cij (t, s) are defined on R×(−∞, 0], and nonnegative continuous with respect to t ∈ R and integrable with respect to s on (−∞, 0] 0 such that −∞ cij (t, s)ds is continuous and bounded with respect to t ∈ R; (H2 ) τij (t) is continuous and differentiable bounded on R, τ˙ij (t) is uniformly continuous with respect to t on R and inf {1 − τ˙ij (t)} > 0; t∈R

(H3 ) ri (t), aij (t), bij (t), cij (t, s) are continuous periodic functions with period ω > 0, and for any ε > 0, there is a constant l = l(ε) > 0, such that in any interval [t, t + l], there exists ω such that the following inequalities hold: |ri (t + ω) − ri (t)| < ε, |aij (t + ω) − aij (t)| < ε,  |bij (t + ω) − bij (t)| < ε,

2

(3)

0

−∞

|cij (t + ω, s) − cij (t, s)|ds < ε.

(4)

Preliminaries

Throughout from this paper, we suppose that h(t) is a bounded function defined on R. Define hu = lim sup h(t), hl = lim inf h(t). t→∞

t→∞

Definition 1. An ω-periodic solution x∗ (t) of system (1) is said to be globally asymptotically stable, if for any positive solution x(t) of system (1), we have that lim x(t) − x∗ (t) = 0.

t→∞

Definition 2. An ω-periodic solution x∗ (t) of system (1) is said to be globally exponentially stable with convergence rate ε > 0, if for any positive solution x(t) of system (1), we have that lim x(t) − x∗ (t) = 0(e−εt ). t→∞

Definition 3. The {ζ, 1}-norm of a vector x ∈ Rn is defined by x{ζ,1} = n  ςi |xi |, ζi > 0 for i = 1, 2, · · · , n. i=1

Positive Periodic Solutions

499

Lemma 1. The solution x(t) of system (1) is said to be a positive solution if it satisfies the initial condition (2). Proof. From Eq.(1), we have that: ln xi (t) − ln xi (0) =

t

0 [ri (t) −

−

n 

0

n  j=1

aij xj (t) −

c (t, s)xj (t −∞ ij

j=1

n 

+ xi1(t)

j=1

n  j=1

bij (t)xj (t − τij (t))

+ s)ds

(5)

Dij (t)(xj (t) − xi (t))]dt.

When t is finite, then the right-hand side of Eq.(5) is also finite, thus ln xi (t) > −∞, which implies xi (t) > 0 for any finite time t > 0, i = 1, 2, · · · , n. Then this lemma can be proved.

3

The Existence of Positive Periodic Solutions

In this section, the conditions about the existence of positive periodic solutions of system (1) are derived. Theorem 1. For the system (1) of initial condition (2), if alii > 0, and there exist positive constants η, ξ such that for all i = 1, 2, · · · , n, the following inequalities hold: η ≤ lim inf xi (t) ≤ lim sup xi (t) ≤ ξ, t→∞

n 

i = ril − ( −

where ξ =

n  j=1

j=1,j=i

(6)

t→∞

auij +

n  j=1

buij +

n   0

−∞ cij (t, s)xj (t

j=1

+ s)ds)ξ (7)

u Dij > 0,

max {

riu +

i=1,2,··· ,n

n  j=1 alii

u Dij

}, η =

one positive ω-periodic solution.

min { ui }, i=1,2,··· ,n aii

then system (1) has at least

Proof. Assume x(t) = (x1 (t), x2 (t), · · · , xn (t)) is any positive solution of system (1) with initial condition (2). Define V (t) = max{xj (t)} for t ≥ 0, j = 1, 2, · · · , n. Suppose V (t) = xi (t), so xi (t) ≥ xj (t)(j = 1, 2, · · · , n). The upper right derivative of V (t)along the solution of system (1) is: D+ V (t) = x˙ i (t) = xi (t)[ri (t) − −

n   0 j=1

n  j=1

aij (t)xj (t) −

−∞ cij (t, s)xj (t

j=1

+ s)ds] +

≤ xi (t)[ri (t) − aii (t)xi (t)] + ≤ xi (t)[riu − alii xi (t) +

n 

n  j=1

n  j=1

u Dij ].

bij (t)xj (t − τij (t)) n  j=1

Dij (t)(xj (t) − xi (t))

Dij (t)xi (t)

500

T. Zhang, M. Jiang, and B. Huang

By using comparability theorem, we have lim sup xi (t) ≤ ξi , i = 1, 2, · · · , n, where ξi = {

riu +

n  j=1 alii

t→∞

u Dij

}, taken ξ =

max {ξi }, then lim sup xi (t) ≤ ξ. t→∞

i=1,2,··· ,n

On the other hand, there exists a positive constant T for t ≥ T such that x˙ i (t) ≥ xi (t)[ril − auii xi (t) −

n  j=1,j=i

auij ξ −

n  j=1

u Dij −(

n 

j=1

buij +

n   0 j=1

cu (s)ds)ξ]. −∞ ij

By using comparability theorem, we can get that lim inf xi (t) ≥ ηi , where ηi = ril −(

n  j=1,j=i

au ij +

n  j=1

bu ij +

n   j=1

0 −∞

cu ij (s)ds)ξ−

au ii

n  j=1

t→∞

u Dij

> 0. Taken η =

min

{ηi }, then

i=1,2,··· ,n

lim inf xi (t) ≥ η. Thus we can get that η ≤ lim inf xi (t) ≤ lim sup xi (t) ≤ ξ.

t→∞

t→∞

t→∞

Then (6) is proved. Then let C = C((−∞, 0], Rn ) be a banach space with the norm φ = max sup |φi (s)|. i

−∞≤s≤ω

˙ ≤ L, ξ = Let Ω = {x(s) ∈ C : ηi ≤ xi (s) ≤ ξi , i = 1, 2, · · · , n, x(s) max {ξi }}, then Ω is a convex compact set in C. Define a map F from Ω to

i=1,2,··· ,n

C:

F : φ(s) → x(s + ω, φ).

x(t) = x(t, φ) is the solution of system(1) with the initial condition xi (s) = φi (s) for s ∈ (−∞, 0). It is easy to prove that x(s ˙ + ω) ≤ L, then if φ ∈ Ω, we have x ∈ Ω, it means that F Ω ⊂ Ω. By using Schauder’s fixed point theorem, there exists φ∗ ∈ Ω such that F φ∗ = φ∗ . Thus x(t, φ∗ ) = x(t, F φ∗ ) i.e. x(t, φ∗ ) = x(t + ω, F φ∗ ). It means that x(t, φ∗ ) is a positive periodic solution of system (1). Then the proof is complete. Remark 1. From Theorem 1, we can know that the upper and lower bounds of x∗ (t) are ηi ≤ x∗i (t) ≤ ξi (i = 1, 2, · · · , n). It may be useful in the other applicated areas.

4

The Globally Asymptotical Stability

In this part, we discuss about the global asymptotical stability of positive peri+∞  odic solutions. We assume that s|cij (t, s)|ds < +∞ and τij (t) is continuously 0

−1 (t) is an inverse differentiable with τ˙ij (t) < 1. Define σij (t) = t − τij (t), then σij function of σij (t).

Positive Periodic Solutions

501

Theorem 2. Assume that system(1) has a positive ω-periodic solution x∗ (t) = (x∗1 (t), x∗2 (t), · · · , x∗n (t)), and there exist positive constants λ1 , λ2 , · · · , λn such that: n 

λi aii (t) + −

n  j=1

|aji (t)|λj −

j=1,j=i 0 λj −∞ |cji (t

− s, s)|ds

n 

−1 (t))| |bji (σji

λ −1 1−τ˙ji (σji (t)) j j=1 n  Dji (t) − λj η ≥ δi j=1

(8)

> 0,

for all i = 1, 2, · · · , n, δi > 0, λi > 0 hold, then x∗ (t) is globally asymptotically stable. Proof. Construct a Lyapunov function: V (t) =

n  i=1

n 

λi | ln xi (t) − ln x∗i (t)| +

−x∗i (s)|ds

+

n  i,j=1

λi

0

i,j=1

t

−∞ t+s

λi

t

−1 (s))| |bij (σij |x (s) −1 t−τij (t) 1−τ˙ij (σij (s)) i

(9)

|cij (θ − s, s)||xi (θ) − x∗i (θ)|dθds.

The derivative of V (t) along the solution of system (1) is: dV (t) dt

=

n  i=1

−

λi sign|xi (t) − x∗i (t)|[−aii (t)|xi (t) − x∗i (t)| n 

j=1,j=i

aij (t)|xi (t) − x∗i (t)| −

−x∗j (t − τij (t))| − +

n  i,j=1

n   0

j=1 −1 |bij (σij (t))|

λi [ 1−τ˙

−1 ij (σij (t))

−∞

n  j=1

bij (t)|xj (t − τij (t))

|cij (t, s)||xj (t + s) − x∗j (t + s)|ds]

|xi (t) − x∗i (t)|

(10)

x∗i (t

−|bij (t)||xi (t − τij (t)) − − τij (t))|] n 0  + λi [ −∞ |cij (t − s, s)||xi (t) − x∗i (t)|ds i,j=1 0 − −∞ |cij (t, s)||xi (t + s) − x∗i (t + s)|ds] n ∗  xj (t)x∗ i (t)−xi (t)xj (t) + λi Dij (t) sign|xi (t) − x∗i (t)|. xi (t)x∗ (t) i,j=1

i

And we have n  j=1

Dij (t)

∗ xj (t)x∗ i (t)−xi (t)xj (t) sign|xi (t) xi (t)x∗ (t) i

− x∗i (t)| ≤

n  j=1

Dij (t) η |xi (t)

− x∗i (t)|. (11)

Now we prove (11) in four cases: Case(1): If x∗i < xi , x∗j < xj , then we have: n  j=1

Dij (t)

∗ xj (t)x∗ i (t)−xi (t)xj (t) sign|xi (t) xi (t)x∗ (t) i

− x∗i (t)| ≤ ≤

n  j=1 n  j=1

Dij (t)|

xi (t)(xj (t)−x∗ j (t)) | xi (t)x∗ i (t)

Dij (t) |xi (t) η

− x∗i (t)|;

502

T. Zhang, M. Jiang, and B. Huang

Case(2): If x∗i < xi , x∗j > xj , then we have: n 

Dij (t)

j=1

xj (t)x∗i (t) − xi (t)x∗j (t) sign|xi (t) − x∗i (t)| ≤ 0 xi (t)x∗i (t)

Then n  j=1

Dij (t)

∗ xj (t)x∗ i (t)−xi (t)xj (t) sign|xi (t) xi (t)x∗ i (t)

− x∗i (t)| ≤

n  j=1

Dij (t) η |xi (t)

− x∗i (t)|;

Case(3): If x∗i > xi , x∗j < xj , then the proof is the same as case(1); Case(4): If x∗i > xi , x∗j > xj , then the proof is the same as case(2). Then (11) hold. Combining with (8)-(11), we have that dV (t) dt

≤− − ≤−

n 

[λi aii (t) +

i=1 n 

j=1 n  i=1

λj

0

n  j=1,j=i

n 

|aji (t)|λj −

|c (t − s, s)|ds − −∞ ji

n  j=1

j=1

λj

−1 (t))| |bji (σji

λ −1 1−τ˙ji (σji (t)) j

Dji (t) η ]|xi (t)

− x∗i (t)|

(12)

δi |xi (t) − x∗i (t)|.

Thus, we we can get that: 

n +∞ 

0

δi |xi (t) − x∗i (t)|ds ≤ V (0) − V (+∞) ≤ V (0) < +∞.

(13)

i=1

According to Theorem 1, we can know that x(t) and x∗ (t) are bounded on [0, +∞). Hence |xi (t) − x∗i (t)|(i = 1, 2, · · · , n) are bounded and uniformly continuous on [0, +∞). Then from (13) we can get that lim xi (t) − x∗i (t) = 0. t→∞

Therefore, x∗ (t) is globally asymptotically stable.

5

The Globally Exponential Stability

Theorem 3. If there exist constants ε > 0, ρ > 0, ζ1 , ζ2 , · · · , ζn such that ζi (aii − ερ) + −

n  j=1

ζj

0

n  j=1,j=i

|aji (t)|ζj −

|c (t − s, s)|e −∞ ji

−εs

n  j=1

ds −

−1 (t))| |bji (σji

ζ e −1 1−τ˙ji (σji (t)) j

n 

j=1

ζj

Dji (t) η

≥ 0,

| ln xi (t) − ln x∗i (t)| ≤ ρ|xi (t) − x∗i (t)|,

−1 ε(σji (t)−t)

(14)

(15)

for all t > 0, i = 1, 2, · · · , n and ζi > 0 hold, then x∗ (t) is globally exponentially stable with convergence rate ε.

Positive Periodic Solutions

503

Proof. Construct a Lyapunov function: n 

V1 (t) =

i=1

+ +

ζi | ln xi (t) − ln x∗i (t)|eεt n 

i,j=1 n  i,j=1

ζi ζi

t

−1 |bij (σij (s))|

|x (s) −1 t−τij (t) 1−τ˙ij (σij (s)) i

0

t −∞ t+s

−1

− x∗i (s)|eεσij

(s)

ds

(16)

|cij (θ − s, s)||xi (θ) − x∗i (θ)|eε(θ−s) dθds.

The derivative of V1 (t) along the solution of system (1) is: dV1 (t) dt

=

n  i=1

−

ζi sign|xi (t) − x∗i (t)eεt |[ε| ln xi (t) − ln x∗i (t)| − aii (t)|xi (t) − x∗i (t)| n 

aij (t)|xi (t) − x∗i (t)| −

n 

bij (t)|xj (t − τij (t)) − j=1 j=1,j=i n   0 ∗ − −∞ |cij (t, s)||xj (t + s) − xj (t + s)|ds] j=1 −1 n  −1 |bij (σij (t))| ε(σij (t)−t) ∗ + ζi eεt [ 1−τ˙ (σ |x (t) − x (t)|e −1 i i (t)) ij ij i,j=1 −|bij (t)||xi (t − τij (t)) − x∗i (t − τij (t))|] n 0  ζi eεt [ −∞ |cij (t − s, s)||xi (t) − x∗i (t)|e−εs ds + i,j=1 0 − −∞ |cij (t, s)||xi (t + s) − x∗i (t + s)|ds] n ∗  xj (t)x∗ i (t)−xi (t)xj (t) ζi eεt Dij (t) sign|xi (t) − x∗i (t)|. + xi (t)x∗ (t) i i,j=1

x∗j (t − τij (t))|

Combining with (14)-(16), we have that dV1 (t) dt

≤ −eεt −

n  i−1

n 

j=1

ζj

[ζi (aii − ερ) + 0

n  j=1,j=i

−∞ |cji (t − s, s)|e

And from (17), we know

n  i=1

n 

|aji (t)|ζj −

−εs

ds −

n  j=1

n  j=1

−1 (t))| |bji (σji

ζ e −1 1−τ˙ji (σji (t)) j

D (t) ζj jiη ]|xi (t)

−1 ε(σji (t)−t)

− x∗i (t)| ≤ 0.

(17)

ζi | ln xi (t) − ln x∗i (t)|eεt ≤ V1 (t) ≤ V1 (0), so

ζi | ln xi (t) − ln x∗i (t)| ≤ V1 (0)e−εt .

i=1

From Theorem 2, we can know that x∗ (t) is globally asymptotically stable. So there exist positive constants m1 , m2 such that m1 ≤ xi (t) ≤ m2 for all t ≥ 0, i = 1, 2, · · · , n. Thus ,there exists a constant μ such that |xi (t) − x∗i (t)| ≤ n  μ| ln xi (t) − ln x∗i (t)| for all t ≥ 0, i = 1, 2, · · · , n. Then it get that ζi |xi (t) − i=1

x∗i (t)| ≤ μV1 (0)e−εt . It means that xi (t) − x∗i (t){ζ,1} = o(e−εt ). So x∗ (t) is globally exponentially stable with convergence rate ε by Definition 2. Then this proof is complete.

504

6

T. Zhang, M. Jiang, and B. Huang

Numerical Example

In this part, we set an example to illustrate the validity of the results in our theorems. Example 1. Consider the following 3-species system which is a particular case of system (1): ⎧ dx1 (t) ⎪ ⎪ = (8 + sin t)x1 (t) − (4 + cos t)x21 (t) − (1.3 + sin 2t)x1 (t)x2 (t − π) ⎪ ⎪ dt ⎪ ⎪ ⎪ +(7 − sin t)x1 (t)x3 (t − π) + (1 + sin t)(x2 (t) − x1 (t)) ⎪ ⎪ ⎪ ⎪ +(3 + cos t)(x3 (t) − x1 (t)), ⎪ ⎪ ⎪ dx (t) ⎪ ⎪ ⎨ 2 = (9 + cos t)x2 (t) − (1 + cos t)x1 (t − π)x2 (t) − (6 + cos t)x22 (t) dt +(3.5 + cos t)x2 (t)x3 (t − π) + (4 + cos t)(x1 (t) − x2 (t)) ⎪ ⎪ ⎪ ⎪ +(1 + sin t)(x3 (t) − x2 (t)), ⎪ ⎪ ⎪ ⎪ dx3 (t) ⎪ ⎪ ⎪ dt = (5.5 + 2 sin t)x3 (t) − (0.2 + sin t)x1 (t − π)x3 (t) ⎪ ⎪ ⎪ ⎪ +(3 + cos t)x2 (t − π)x3 (t) − (7.5 + sin t)x23 (t) ⎪ ⎩ +(1 + sin t)(x1 (t) − x3 (t)) + (4 + cos t)(x2 (t) − x3 (t)). 8 x1(t) x2(t) x3(t) 7

x1(t),x2(t),x3(t)

6

5

4

3

2

1

0

5

10

15

20

25 time t

30

35

40

45

50

Fig. 1. The system converges to a positive 2π -periodic solution

with the initial condition x1 (s) = 2, x2 (s) = 3.8 + 0.7s, x3 (s) = 1.7 + 0.3s2 , s ∈ 14 14 17 32 . [−π, 0]. Then we get ξ = max { , , } = i=1,2,3 3 5 13 3

Positive Periodic Solutions

505

It is easy to verify that all the conditions required in Theorem 1 are satisfied. Then this system exists a positive solution. Fig.1 shows that the system converges to a positive 2π-periodic solution.

7

Conclusion

In this paper, some new results of the positive periodic solutions of a nonautonomous Lotka-Volterra dispersal system with discrete and continuous infinite delays are obtained. We discuss about the global asymptotical stability and global exponential stability and the existence of positive periodic solutions in this Lotka-Volterra system. The results are typical and more general, and it can be reduced in the other systems.

Acknowledgments This work is supported by the Graduate Scientific Research Creative Foundation of Three Gorges University (200945) (200946) and the Scientific Innovation TeamProject of Hubei Provincial Department of Education (T200809) and Hubei Natural Science Foundation (No.D20101202).

References 1. Meng, X., Chen, L.: Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay. J. Math. Anal. Appl. 05, 84 (2007) 2. Lin, W., Chen, T.: Positive periodic solutions of delayed periodic Lotka-Volterra systems. Physics letters A 334, 273–287 (2005) 3. Meng, X., Jiao, J., Chen, L.: Global dynamics behaviors for a nonautonomous Lotka-Volterra almost periodic dispersal system with delays. Nonlinear Analysis 68, 3633–3645 (2008) 4. Liang, Y., Li, L., Chen, L.: Almost periodic solutions for Lotka-Volterra systems with delay. Commun. Nonlinear Sci. Numer. Simulat. 14, 3660–3669 (2009) 5. Teng, Z.: Nonautonomous Lotka-Volterra systems with delays. Journal of Differerntial Equations 179, 538–561 (2002) 6. Zhu, C., Yin, G.: On competitive Lotka-Volterra model in random environments. J. Math. Anal. Appl. 170, 154–170 (2009) 7. Li, Y.: Positive periodic solutions of periodic neutral Lotka-Volterra system with distributed delays. Chaos, Solitons and Fractals 37, 288–298 (2008) 8. Luo, H., Li, W.: Periodic solutions of a periodic Lotka-Volterra system with delays. Applied Mathematics and Computation 156, 787–803 (2004) 9. Faria, T.: Sharp conditions for global stability of Lotka-Volterra systems with distributed delays. J. Differential Equations 246, 4391–4404 (2009) 10. Meng, X., Chen, L.: Almost periodic solution of nonautonomous Lotka-Volterra predator-prey dispersal system with delays. Journal of Theoretical Biology 243, 562–574 (2006) 11. Lu, G., Lu, Z., Lian, X.: Delay effect on the permanence for Lotka-Volterra cooperative systems. Nonlinear Analysis. Real Word Application 10, 005 (2009)

An Algorithm of Sphere-Structure Support Vector Machine Multi-classification Recognition on the Basis of Weighted Relative Distances Shiwei Yun,Yunxing Shu, and Bo Ge Luoyang Institute of Science and Technology Henan, Luoyan, 471003, China [email protected]

Abstract. Theories on sphere-structure support vector machine (SVM) and multi-classification recognition algorithms were studied in the first place, and on this basis, in view of the issue of the difference in the hypersphere radiuses resulted from the difference in the quantity of the training samples and the discrepancy in their distributions, the concepts of relative distance and weight were introduced, and subsequently a new algorithm of sphere-structure SVM multi-classification recognition was proposed on the basis of weighted relative distances. Accordingly, the data from the UCI database were used to conduct simulation experiments, and the results verified the validity of the algorithm propose. Keywords: pattern recognition; sphere-structure SVM; relative distance; weighting.

1 Introduction Pattern classification is a kind of pattern recognition technology, which makes use of the machine system with the computer as its center to create the decision-making boundaries between the various pattern classes on the basis of decision-making theories and to recognize the class of the unknown sample through certain algorithms. In past few decades, pattern recognition technologies have been successfully applied in state inspection, fault diagnosis, character recognition, speech recognition, and image recognition, etc., which has promoted the development of artificial intelligence [1]. Support vector machine adopts the structure risk principle in statistical learning theories [2] and has realized the training of small samples and demonstrated a good generalization capability, thus to a certain extent having solved the “over-fitting problem” and the “generalization capability problem” in neural networks. Meanwhile, the adoption of the kernel technology has remarkably enhanced the processing capacity of nonlinear data, and this, to some extent, also avoids the “dimension disaster problem” and the “network scale explosion problem” in neural networks [3]. Therefore, SVM has made a new international hot spot in the research in the field of machine learning [4,5]. K. Li et al. (Eds.): LSMS/ICSEE 2010, Part I, LNCS 6328, pp. 506–514, 2010. © Springer-Verlag Berlin Heidelberg 2010

An Algorithm of Sphere-Structure SVM Multi-classification Recognition

507

SVM was proposed on the basis of the problem of dichotomy. As for the multiclassification problem, the main algorithms at present are as follows: the one-againstrest method [6], the one-against-one method [7], and the tree-based SVM method [89], etc. The basic idea of these algorithms is to use multiple hyperplanes to divide the data space, with each kind of data enclosed within an area formed by several hyperplanes, and all the kinds of data are not off the optimal hyperplanes of the SVMs. The idea of the sphere-structure SVM algorithm is as follows: in a high-dimension feature space, a hypersphere is to be created for every pattern class which can cover its training samples and has the minimum volume, so as to realize the division of the training sample space. Reference [5] proposed a sphere-structure-based SVM algorithm its multi-classification decision rules. References [6-9] proposed two kinds of improved algorithms on the basis of sphere-structure SVMs. Compared with the standard SVMs, the multi-classification recognition algorithm on the basis of spherestructure SVMs is more suitable for the multi-classification recognition problem involving more samples and is easier to extend new classes. This study proposed a new algorithm of multi-classification recognition on the basis of sphere-structure SVMs. First, for each pattern class, a hypersphere was created, which divides the sample space; for the samples to be tested, considering the quantity and distribution of the training samples, the relative distances from the samples to be tested to the spherical centers of various kinds and the corresponding weights were introduced; according to the weighted relative distances, a membership function was established, and pattern recognition was realized through the extent of the membership. Finally, simulation experiments verified the validity of the method proposed.

2 Basic Theories of Sphere-Structure SVMs Suppose X = { x1 , x2 ,", xn } as the set of the training samples for k kinds in a d d

dimension space R ; X

m

= { x1 , x2 , " , xnm } and m = (1, 2, " , k ) as the sample set m

m

m

for the m -th kind in X ; nm as the total number of m -th kind of training sample, k

∑n

m

= n . A sphere-structure SVM is to create a minimum hypersphere for each

m =1

kind of training sample and to make the hypersphere include all the training samples belonging to this kind. This question can be transformed into the following quadratic programming questions:

min Rm 2

(1)

s.t. ( xim − am )T ( xim − am ) ≤ Rm 2 , i = 1, 2," , nm

(2)

in which am and Rm respectively refers to the spherical center and the spherical radius derived from the training of

m -th sample.

508

S. Yun,Y. Shu, and B. Ge

Then the slack variable

ξim

is introduced, and the quadratic programming

questions (1) and (2) can be changed into: nm

min Rm + Cm ∑ ξim

(3)

⎧⎪( xim − am )T ( xim − am ) ≤ Rm 2 + ξim s.t. ⎨ m , i = 1, 2," , nm ⎪⎩ξi ≥ 0

(4)

2

i =1

in which Cm > 0 is a regularization parameter, reflecting the relation between the spherical radius and the quantity of the samples outside the sphere. According to the KKT condition, the Lagrange Multiplier Method is used to realize the dual programs of the quadratic programming questions, just as follows: nm

nm

max L = ∑ α im ( xim ⋅xim ) − ∑ α imα mj ( xim ⋅ x mj )

(5)

⎧ nm m ⎪∑ α i = 1 s.t. ⎨ i =1 , i = 1, 2," , nm ⎪0 ≤ α m ≤ C i m ⎩

(6)

i =1

i , j =1

Solve the quadratic programming questions and get the spherical center nm

am = ∑ α im xim . i =1

Among the solutions, only a small portion satisfies α i > 0 , and the corresponding sample points are called support vectors, in which the sample points corresponding to m

0 < α i < Cm are called boundary support vectors, and these points are on the m

boundaries of the hypersphere; the sample points corresponding to α i = Cm are m

called outside support vectors, and these points are off the hypersphere. If any boundary support vector

xkm is used, the square of the spherical radius can be

calculated as:

Rm 2 = ( xkm − am )T ( xkm − am ) nm

nm

i =1

i , j =1

= ( xkm ⋅ xkm ) − 2∑ αim ( xkm ⋅ xim ) + ∑ αimα mj ( xim ⋅ x mj ) For any unknown sample point spherical center is:

(7)

x , the square of the distance from the point to the

An Algorithm of Sphere-Structure SVM Multi-classification Recognition

509

Dm 2 = ( x − am )T ( x − am ) nm

nm

i =1

i , j =1

= ( x ⋅ x ) − 2∑ αim ( x ⋅ xim ) + ∑ αimα mj ( xim ⋅ x mj )

(8)

In reality, sample points generally will not be distributed spherically. In order to make this method have more extensive practicality, just as the SVMs method, so the kernel method can be used, that is, find a nonlinear mapping ϕ , and map the sample into the high-dimension feature space. The inner product operation in the feature space can be replaced by the kernel function, therefore it is unnecessary to know the specific form of the nonlinear mapping, and thus the operation complexity will not be increased. Theories have proved that as long as the kernel function K ( x , y ) satisfies the Mercer theorem, the kernel function can represent the inner product of the two vectors within the high-dimension space [2]. At this point the quadratic programming questions (5) and (6) can be transformed into: nm

nm

max L = ∑ α K ( x ,x ) − ∑ α imα mj K ( xim , x mj )

(9)

⎧ nm ⎪ αi = 1 , i = 1, 2," , nm s.t. ⎨∑ i =1 ⎪0 ≤ α ≤ C i ⎩

(10)

i =1

m i

m i

m i

i , j =1

Solve the quadratic programming questions and get the spherical center nm

am =

∑α i =1

m m i ϕ ( xi ) . m

If any boundary support vector xk is used, the square of the spherical radius can be calculated as: Rm 2 = (ϕ ( xkm ) − am )T (ϕ ( xkm ) − am ) nm

nm

i =1

i , j =1

= K ( xkm ⋅ xkm ) − 2∑ αim K ( xkm ⋅ xim ) + ∑ αimα mj K ( xim ⋅ x mj )

(11)

For any testing sample point x , the square of the distance from x to the spherical center of the m -th sphere is:

Dm 2 ( x ) = (ϕ ( x ) − am )T (ϕ ( x ) − am ) nm

nm

i =1

i , j =1

= K ( x ⋅ x ) − 2∑ αim K ( x ⋅ xim ) + ∑ α imα mj K ( xim ⋅ x mj )

(12)

510

S. Yun,Y. Shu, and B. Ge

3 Multi-classification Fuzzy Recognition on the Basis of SphereStructure SVMs 3.1 Weighted Relative Distances

The basic procedures of using the sphere-structure SVMs to conduct multiclassification recognition are as follows: first, train the various kinds of training samples and construct multiple hyperspheres to divide the sample space; then, test the testing samples and classify in accordance with certain rules. Generally, testing samples fall into three categories: (1) enclosed by only one hypersphere (as in Figure 1(a)); (2) not enclosed by any hypersphere (as in Figure 1(b)); (3) enclosed by multiple hyperspheres (as in Figure 1(c)).

Fig. 1.

Under general circumstances, because of the differences in the quantity and distribution of the training samples, the sizes of the radius of the training hyperspheres will also be different, and the longer the spherical radius, the more dispersed the samples; the more the sample points, the more concentrated the samples. The multi-classification recognition algorithms proposed in references [5-7] do not take these circumstances into account. Therefore, this study proposed a fuzzy recognition algorithm on the basis of weighted relative distances. Suppose nm as the number of the m -th kind of training sample, Rm as the m kind of spherical radius, and Dm ( x ) as the distance from the testing sample point x to the m -th spherical center, define

d m ( x ) = ωm

Dm ( x) Rm

(13)

as the weighted relative distance from point x to the m -th spherical center, in which ωm is the weight value. Define

μm ( x) =

1/ d m ( x) k

∑1/ d i =1

m

( x)

(14)

An Algorithm of Sphere-Structure SVM Multi-classification Recognition

511

as the degree of membership of point x the m -kind. Lastly, conduct pattern recognition according to the size of the membership function. The testing samples are affiliated to the kinds whose degree of membership is larger. Because the number of the sample points and the selection of the kernel function both will influence the distribution of the sample points, therefore the selection of the weight function ωm is related to the number of the training sample points and to the selection of the kernel function, namely, the weight function ωm = f ( nm , σ ) , in which σ is the parameter in the kernel function. Because the number of sample points will influence their distribution, and the size of σ will influence the compactness of the boundaries of the hyperspheres, the weight function shall bear the following properties: (1) ωm = f ( nm , σ ) is the increasing function of nm . Because the more the sample points are, the more concentrated the samples are, under the circumstance that the relative distances Dm ( x ) / Rm are identical, the probability for an unknown sample to be affiliated to the kind whose sample points are more dispersed is larger, that is, the weighted relative distances to those kinds whose sample points are fewer shall be shorter, namely, the fewer the sample points, the smaller the weight values. (2) With the strengthening of the nonlinearity of the kernel function, ωm = f ( nm , σ ) reduces the degree of differences in weight values between different kinds. Because the larger the nonlinearity of the kernel functions is, the more compact the hyperspheres will be, and correspondingly the smaller the influence of the sample number and distribution will be. Therefore, the extent of the relative changes among the weight values of different kinds shall become smaller. The specific function form of ωm is generally determined by experience or experiments. 3.2 Procedures of the Pattern Recognition Algorithm

(1) For every kind of training sample, construct respectively the corresponding minimum hyperspheres, and use equation (11) to calculate the radius of various kinds of hyperspheres; (2) Use equation (12) to calculate the distances from the samples to be tested to the spherical centers of various kinds of hyperspheres; (3) Determine the weight function, and use equation (14) to calculate the degree of membership of the samples to be tested; (4) Conduct decision-making classification according to the principle of maximum degree of membership.

4 Experimental Analysis 4.1 Simulation Experiments

In order to verify the validity of the proposed algorithm, this study adopted the Wine sample set and the Vehicle sample set from the UCI database to conduct simulation

512

S. Yun,Y. Shu, and B. Ge

experiments, in which the kernel function was the same, that is, the radial basis kernel function. For the Wine sample set, we selected three kinds of training samples, whose numbers were 20, 30, and 20 respectively, and the remaining ones were used as testing samples. The weight function adopted was ωm = f ( nm , σ ) = exp( −σ / nm

0.9 + 0.02σ

) , and

the parameter σ in the kernel function ranged from 1 to 20 with an interval of 1. Altogether 20 experiments were conducted. For the Vehicle sample set, we selected four kinds of training samples, whose numbers were 100, 108, 109, and 106 respectively, and the remaining ones were used as testing samples. The weight function adopted was ωm = f ( nm , σ ) = exp( −σ / nm

0.5+ 0.05σ

) , and the parameter σ in the

kernel function ranged from 1 to 4 with an interval of 0.2. Altogether 16 experiments were conducted. For the Ionosphere sample set, we selected two kinds of training samples, whose numbers were 42 and 75 respectively, and the remaining ones were used as testing samples. The weight function adopted was 2 +σ

ωm = f ( nm , σ ) = exp( −σ / nm ) , and the parameter σ in the kernel function ranged from 1 to 2 with an interval of 0.05. Altogether 21 experiments were conducted. For the Iris sample set, we selected three kinds of training samples, whose numbers were 12, 20 and 30 respectively, and the remaining ones were used as testing samples. 0.5 + 0.05σ The weight function adopted was ωm = f ( nm , σ ) = exp( −σ / nm ) , and the parameter σ in the kernel function ranged from 1 to 10 with an interval of 1. Altogether 10 experiments were conducted. For the Waveform sample set, we selected three kinds of training samples, whose numbers were100 200 and 300 respectively, and the remaining ones were used as testing samples. The weight function adopted was 1+ 0.05σ ωm = f ( nm , σ ) = exp( −σ / nm ) , and the parameter σ in the kernel function ranged from 2 to 6 with an interval of 0.5. Altogether 9 experiments were conducted. All the testing correctness rate and testing time of the algorithm were compared with those of the algorithm proposed in reference [5]. The results of the experiments are shown in Table 1. 4.2 Result Analysis

The experiment results suggest that testing correctness rate is mainly related to the selection of the kernel function parameters. Under the same kernel function parameter, because the algorithm in the present study took sample number and distribution into full consideration, the correctness rate of the algorithm in this study was higher than that in reference [5]. As for testing time, with the increase of the parameter of the radial basis kernel function, the testing time of the algorithm in the present study was less than that in reference [5], and the reason is as follows: with the increase of the parameter of the radial basis kernel function, there are relatively more overlapping among the hyperspheres, and for the decision function for the sample points in the overlapping space, the degree of complexity of the algorithm in reference [5] was higher, thus the algorithm used in the present study is better as far as testing time is concerned.

An Algorithm of Sphere-Structure SVM Multi-classification Recognition

513

Table 1. Comparison between algorithm in this study and algorithm in reference [5] Data sets

Wine

Vehicle

Iris

Ionosphere

Waveform

Method Method in this study Method in reference [5] Method in this study Method in reference [5] Method in this study Method in reference [5] Method in this study Method in reference [5] Method in this study Method in reference [5]

average correctness rate

highest correctness rate

testing time(s)

0.8911

0.9533

0.0766

0.8014

0.8598

0.0781

0.7305

0.7057

0.8642

0.7116

0.6918

0.9023

0.9589

0.9667

0.0305

0.9133

0.9556

0.0310

0.6461

0.7650

0.0930

0.6416

0.6923

0.0967

0.8262

0.8352

6.1911

0.8048

0.8191

6.5276

5 Conclusion On the basis of studying the multi-classification pattern recognition based on spherestructure SVMs, a kind of fuzzy multi-classification recognition algorithm was proposed on the basis of weighted relative distances. Because the decision function of this algorithm takes into full consideration the sample number and the influence of the kernel function on the distribution of the training samples, the testing correctness rate is higher. Meanwhile, because the decision function of the present algorithm only used weighted relative distances, compared with the decision function used in reference [5], its degree of complexity in calculation is smaller. Therefore, in the aspect of testing time the present algorithm is better than that in reference [5], and the present algorithm is easier to be realized. However, as for the determination of the weight function, so far there have been no unified theories to be guidance, and this is a further research direction for the present study.

References 1. Bishop, Christopher, M.: Pattern Recognition and Machine Learning. Springer, Heidelberg (2006) 2. Vapnik, V.N.: The Nature of Statistical Learning Theory. Springer, Heidelberg (1995)

514

S. Yun,Y. Shu, and B. Ge

3. Zhang, X.: Introduction to Statistical Learning and Support Vector Machines. Acta Automatica Sinica 26, 32–42 (2000) 4. Gestel, T.V., Suykens, A.K., Brabanter, J.D., Moor, B.D., Vandewalle, J.: Kernel canonical correlation analysis and least squares support vector machines. In: Proceedings of the International Conference on Artificial Neural Networks (2001) 5. Zhu, M., Liu, X., Chen, S.: Solving the Problem of Multi-class Pattern Recognition with Sphere-structured Support Vector Machines. Journal of NanJing University Natural Sciences 39, 153–158 (2003) 6. Li, H., Lin, J.: Multiclass Classification Based on the One-class Classification and its Application. Journal of South China University of Technology (Natural Science Edition) 32, 82–88 (2004) 7. Wang, J., Neskovic, P., Cooper, L.N.: Bayes classification based on minimum bounding spheres. Neurocomputing 4-6(70), 801–808 (2007) 8. Wu, Q., Jia, C., Zhang, A., et al.: Improved Algorithm Based on Sphere Structure SVMs and Simulation. Journal of System Simulation 20, 345–348 (2008) 9. Gu, L., Wu, H., Xiao, L.: Sphere-structured Support Vector Machines Classification Algorithm Based on New Decision Rule. Journal of System Simulation 20, 2901–2904 (2008)

Author Index

Abeykoon, Chamil II-9 Affenzeller, Michael II-368 Alla, Ahmed N. Abd II-190 Allard, Bruno I-416 An, Bo I-256 Anqi, Wang II-351 Auinger, Franz II-368 Banghua, Yang

Fan, Guo-zhi I-486 Fan, Jian I-147 Fan, Xin-Nan I-304 Fan, Zhao-yong I-104 Fei, Minrui I-1, I-62, I-87, I-147, I-221, I-456, II-40, II-49, II-450, II-475 Fu, Liusun II-505 Fu, Xiping II-49

II-250

Cai, Zongjun I-36 Cao, Guitao I-313 Cao, Longhan II-93 Cao, Lu II-58 Cao, Luming I-36, I-112, II-493 Chen, Haifeng I-389 Chen, Hui II-333, II-505 Chen, Jindong II-207 Chen, Jing I-136 Chen, LiXue II-138 Chen, Min-you I-104, II-342 Chen, Qigong I-87 Chen, Rui II-234 Chen, Wang II-148 Chen, Xiaoming II-120 Chen, Xisong I-409 Chen, Yong I-486 Cheng, Dashuai I-36, I-112, II-493 Cheng, Liangcheng I-45 Chiu, Min-Sen I-36, I-112, II-493 Dai, Rui II-93 Deng, An-zhong I-486 Deng, Jing I-7, II-9, II-40 Deng, Weihua I-1, II-450 Dengji, Hu II-351 Di, Yijuan I-456 Ding, Guosheng I-244 Ding, Yan-Qiong I-304 Dong, Yongfeng I-324 Du, Dajun I-62, I-290 Du, Yanke II-266 Duan, Ying-hong II-259

Ganji, Behnam I-434 Gao, Lixin I-157 Gao, Weijun I-361 Gao, Yanxia I-416 Ge, Bo I-506 Geng, Zhiqiang II-84 Gou, Xiaowei II-69 Gu, Hong I-52 Gu, Li-Ping I-304 Guanghua, Chen II-351 Guo, Chunhua I-282, I-333 Guo, Fengjuan I-205 Guo, Huaicheng I-120 Guo, Shuibao I-416 Guo, Xingzhong II-433 Guo, Yi-nan I-213 Hadow, Mohammoud M. II-190 Hailong, Xue II-400 Ha, Minghu II-241 Han, Yongming II-84 Hao, Zhonghua II-333, II-505 He, Qingbo II-218 He, Ran I-466 He, Tian-chi II-1 He, Wei II-342 Hong, Moo-Kyoung I-380 Hou, Weiyan I-129 Hu, Gang II-342 Hu, Huosheng II-450 Huang, Bin I-497 Huang, Hong II-390 Huang, Lailei II-277 Huang, Mingming I-466 Huang, Quanzhen I-95

516

Author Index

Huang, Zaitang II-166 Hutterer, Stephan II-368 Irwin, George W.

II-40, II-456

Jia, Fang II-158 Jia, Li I-36, I-112, II-69, II-493 Jiang, Enyu I-95 Jiang, Jing-ping I-77 Jiang, Lijun II-225 Jiang, Ming I-87 Jiang, Minghui I-497 Jiang, Xuelin I-185 Jiao, Yunqiang II-360 Jing, Chunwei I-233 Jingqi, Fu II-128 Kang, Lei II-467 Karim, Sazali P. Abdul II-190 Kelly, Adrian L. II-9 Kim, Jung Woo I-372 Kong, Zhaowen I-185 Koshigoe, Hideyuki II-288 Kouzani, Abbas Z. I-434 Lang, Zi-qiang I-104 Lee, Hong-Hee I-372, I-380 Liao, Ju-cheng I-104 Li, Bo II-148 Li, Jianwei I-324 Li, Kang I-1, I-7, I-62, I-270, I-399, II-9, II-40, II-379, II-456 Li, Lixiong I-350, II-475 Li, Nan I-196 Li, Qi I-409 Li, Ran II-199 Li, Sheng-bo I-486 Li, Shihua I-409 Li, Xiang II-180 Li, Xin II-180 Li, Xue I-290 Li, Yaqin I-45 Li, Yonghong I-26 Li, Yongzhong I-233 Li, Zhichao II-467 Liang, Chaozu I-185 Lin, Huang I-297 Linchuan, Li II-400 Lin-Shi, Xuefang I-416 Littler, Tim II-421

Liu, Enhai I-324 Liu, Fei II-442 Liu, Jiming II-277, II-410 Liu, Li-lan I-166, II-101 Liu, Qiming II-266 Liu, Shuang I-297 Liu, Xiaoli II-93 Liu, Xueqin I-7 Liu, Yi II-390 Lu, Da II-379 Lu, Huacai II-433 Luojun, II-218 Luo, Yanping I-157 Ma, Shiwei I-185, II-69, II-333, II-505 Martin, Peter J. II-9 McAfee, Marion I-7, II-9 McSwiggan, Daniel II-421 Meng, Ying II-433 Menhas, Muhammad Ilyas II-49 Ming, Rui II-475 Niu, Qun Niu, Yue

II-21, II-456 I-416

Pan, Feng II-207 Pang, Shunan II-30 Pedrycz, Witold II-241 Pei, Jianxia I-290 Peng, LingXi II-138 Qi, Jie II-30 Qian, Junfeng II-333 Qin, Xuewen II-166 Qin, Zhaohui I-129 Qu, Fu-Zhen II-148 Qu, Gang II-297 Ren, Hongbo

I-361

Shao, Yong I-95 Shi, Benyun II-410 Shi, Jiping I-112, II-493 Shi, Lukui I-324 Shi, Xiaoyun II-84 Shi, Yan-jun II-148 Shiwei, Ma II-351 Shu, Yunxing I-506 Shu, Zhi-song I-166, II-101 Song, Shiji I-399, II-484

Author Index Song, Yang I-1, I-129 Steinmaurer, Gerald II-368 Su, Hongye II-360 Su, Zhou I-425 Sun, Bo II-505 Sun, Hao I-304 Sun, Meng I-313 Sun, Shijie I-244 Sun, Sizhou II-433 Sun, Xin I-456 Sun, Xue-hua I-166, II-101 Tan, Weiming II-166 Tang, Jiafu II-297, II-312 Tang, LiangGui I-256 Tang, Zhijie II-218 Tao, Jili II-120 Tegoshi, Yoshiaki I-425 Tian, Shuai I-166 Trinh, H.M. I-434 Wang, Binbin I-389 Wang, Chao II-241 Wang, Fang I-157 Wang, Haikuan I-129 Wang, Hongbo I-196 Wang, Hongshu II-305 Wang, Jia II-128 Wang, Jianguo II-69 Wang, Jiyi I-297 Wang, Junsong I-69 Wang, Ling II-49, II-69 Wang, Lingzhi II-110 Wang, Linqing II-312 Wang, Nam Sun I-45 Wang, Qiang II-58 Wang, Shentao II-93 Wang, Shicheng II-234 Wang, Shujuan II-234, II-467 Wang, Tongqing I-282, I-333 Wang, Xihong I-147 Wang, Xin II-499 Wang, Xiukun I-466 Wang, Xiuping I-136 Wang, Yu II-297 Wei, Jia II-225 Wei, Lisheng I-87 Wei, Pi-guang I-477 Weimin, Zeng II-351 Wen, Guangrui I-16

Wen, Guihua II-225 Wu, Cheng I-270, I-399, II-484 Wu, Jue II-138 Wu, Jun II-158 Wu, Liqiao II-305 Wu, Mingliang II-93 Wu, Qiong I-361 Xiao, Da-wei I-213 Xiaoming, Zheng II-250 Xiaozhi, Gao II-400 Xia, Qingjun I-205 Xu, Jing I-233 Xu, Lijun I-221 Xu, Li-Zhong I-304 Xu, Rui II-266 Xu, Zhi-yu I-176 Yang, Aolei II-456 Yang, Ating I-445 Yang, Bo II-75 Yang, Huizhong I-45 Yang, Jun I-409 Yang, Lei II-138 Yang, Mei I-213 Yang, Nanhai I-466 Yang, Qingxin I-324 Yang, T.C. I-221, I-350, I-456 Yang, Yan-li II-342 Ye, Longhao I-120 Yin, Wenjun II-484 Yin, Xueyan II-442 Yonghuai, Zhang II-250 Yoshii, Takako II-288 Yuan, Ruixi I-69 Yu, Chunyan II-305 Yu, Shuxia I-120 Yu, Tao I-166, II-101 Yu, Wei I-221 Yu, Xiao-ming I-77 Yu, Yajuan I-120 Yun, Shiwei I-506 Zeng, Xiangqiang I-95 Zhai, Guofu II-467 Zhai, Jin-qian I-104, II-342 Zhang, An I-205, II-58 Zhang, Changjiang II-75 Zhang, Chao II-390 Zhang, Dexing II-180

517

518 Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang, Zhang,

Author Index Hao I-270, II-333 Hong-yu I-486 Hui II-390 Jie I-176 Jun I-52 Long II-234, II-467 Qian I-389 Qingling II-321 Tiezhi II-321 Ting I-497 Xianxia I-313 Xining I-16 Xue-Wu I-304 Yaozhong I-205 Yuanyuan II-84 Yue II-321 Yukui I-399 Yuli II-484 Zhihua I-425

Zhao, Guangzhou I-52, I-196, II-199, II-379 Zhao, Liang I-342 Zhao, Lindu I-26, I-445 Zhao, Ming I-16 Zhao, Ming-sheng II-1 Zhao, Shian II-110 Zhao, Wanqing II-456 Zheng, Dong II-250 Zheng, Xiaoming II-180 Zhou, Caihui II-225 Zhou, Fang II-21 Zhou, Hao I-477 Zhou, Taijin II-21 Zhou, Weisheng I-361 Zhuang, Chun-long I-486 Zhu, Peng II-1 Zhu, Wei II-390 Zhu, Wen-hua I-477 Zhu, Yong II-120