Advances in Linear Matrix Inequality Methods in Control (Advances in Design and Control)

Advances in Linear Matrix Inequality Methods in Control

Advances in Design and Control SIAM's Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines. Editor-in-Chief John A. Burns, Virginia Polytechnic Institute and State University Editorial Board H. Thomas Banks, North Carolina State University Stephen L. Campbell, North Carolina State University Eugene M. Cliff, Virginia Polytechnic Institute and State University Ruth Curtain, University of Groningen Michel C. Delfour, University of Montreal John Doyle, California Institute of Technology Max D. Gunzburger, Iowa State University Rafael Haftka, University of Florida Jaroslav Haslinger, Charles University J. William Helton, University of California at San Diego Art Krener, University of California at Davis Alan Laub, University of California at Davis Steven I. Marcus, University of Maryland Harris McClamroch, University of Michigan Richard Murray, California Institute of Technology Anthony Patera, Massachusetts Institute of Technology H. Mete Soner, Carnegie Mellon University Jason Speyer, University of California at Los Angeles Hector Sussmann, Rutgers University Allen Tannenbaum, University of Minnesota Virginia Torczon, William and Mary University Series Volumes El Ghaoui, Laurent and Niculescu, Silviu-lulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J. William and James, Matthew R., Extending H°° Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

Advances in Linear Matrix Inequality Methods in Control

Edited by Laurent El Ghaoui University of California, Berkeley Silviu-lulian Niculescu Universite de Technologie de Compiegne Compiegne, France

slam.

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2000 by the Society for Industrial and Applied Mathematics. 1098765432 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Cataloging-irvPublication Data Advances in linear matrix inequality methods in control / edited by Laurent El Ghaoui, Silviu-lulian Niculescu. p.cm. — (Advances in design and control) Includes bibliographical references. ISBN 0-89871-438-9 (pbk.) 1. Control theory. 2. Matrix inequalities. 3. Mathematical optimization. I. El Ghaoui, Laurent. II. Niculescu, Silviu-lulian. III. Series. QA402.3.A364 2000 629.8—dc21 99-39162

,•«

ELlcUTL is a registered trademark.

Contributors R. J. Adams C. Andriot P. Apkarian V. Balakrishnan E. B. Beran

S. Boyd

S. Dasgupta C. E. de Souza

S. Dussy

L. El Ghaoui E. Feron

Electrical Engineering Department, University of Washington, Seattle, WA 98195 Internet: rj adamsQuwashington. edu Service Teleoperation et Robotique, CEA, route du Panorama, 92 265 Fontenay aux Roses, France Internet: andriotQCYBORG. cea. fr CERT/DERA, 2 Av. Ed. Belin, 31055 Toulouse, France Internet: apkarianflcert.fr School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285 Internet: raguQecn. purdue. edu Axapta Financials Team, Damgaard International A/S Bregneroedvej 133, DK-3460 Birkeroed, Denmark Internet: ebbQdk. damgaard. c om Information Systems Laboratory, Electrical Engineering Department, Stanford University, Stanford, CA 94305 Internet: boydQst anf ord. edu Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242 Internet: soura-dasguptaQuiowa.edu Laboratorio Nacional de Computagao Cientffica-LNCC, Department of Systems and Control, Av. Getulio Vargas, 333, 25651-070 Petropolis, RJ, Brazil Internet: csouzaQlncc.br Aerospatiale-Matra Lanceurs 66, route de Verneuil, 78133 Les Mureaux Cedex, France Internet: stephane. dussyQespace. aerospatiale. f r Laboratoire de Mathematiques Appliquees, ENSTA, 32, Bvd. Victor, 75739 Paris Cedex 15, France Internet: elghaouiOensta.fr Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 Internet: f erondtait. edu v

VI

J. P. Folcher M. Fu

K. M. Grigoriadis J.-P. A. Haeberly S. R. Hall T. Iwasaki

U. Jonsson

C. Lemarechal

M. Mesbahi

M. V. Nayakkankuppam

S.-I. Niculescu J. Oishi

F. Oustry

M. L. Overton

Contributors Laboratoire de Mathematiques Appliquees, ENSTA, 32, Bvd. Victor, 75739 Paris Cedex 15, France Internet: folcherQensta.fr Department of Electrical and Computer Engineering, The University of Newcastle, Newcastle, NSW 2308, Australia Internet: eemfSee. newcastle. edu. au Department of Mechanical Engineering, University of Houston, Houston, TX 77204 Internet: karolosfluh.edu Mathematics Department, Fordham University, Bronx, NY 10458 Internet: haeberlyOmurray. f ordham. edu Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 Internet: hallQmit.edu Department of Control Systems Engineering, Tokyo Institute of Technology, 2-12-1 Oookayama, Meguro-ku, Tokyo 152, Japan Internet: iwasakiQctrl. titech.ac.jp Division of Optimization and Systems Theory, Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden Internet: ulf j Qmath .kth.se INRIA Rhone-Alpes, ZIRST - 655 avenue de 1'Europe 38330 Montbonnot Saint-Martin, France Internet: Claude. LemarechalQinria. fr Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 Internet: mesbahiQhafez. jpl. nasa. gov Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, MD 21250 Internet: madhuQ c s. nyu. edu HEUDIASYC (UMR CNRS 6599), UTC-Compiegne, BP 20529, 60205, Compiegne, Cedex, France Internet: silviuQhds .utc.fr Production Engineering Department Laboratory, NEC Corporation, Tomagawa Plant 1753 Shimonumabe, Nakahara-ku Kawasaki, 211 Japan Internet: bigstoneQmsa. biglobe .ne.jp INRIA Rhone-Alpes, ZIRST - 655 avenue de 1'Europe 38330 Montbonnot Saint-Martin, France Internet: Francois. OustryQinria. f r Computer Science Department, Courant Institute, New York University, New York, NY 10012 Internet: overtonQcs. nyu. edu

Contributors F. Paganini G. P. Papavassilopoulos A. Rantzer M. G. Safonov C. W. Scherer

A. Trofino

S.-P. Wu K. Y. Yang

vn

Department of Electrical Engineering, UCLA, Los Angeles, CA 90095 Internet: paganiniQee. ucla. edu Department of EE—Systems, University of Southern California, Los Angeles, CA 90089 Internet: yorgosQbode. use. edu Lund Institute of Technology, Box 118, s-221 00 Lund, Sweden Internet: rantzer®control. 1th. se Department of EE—Systems, University of Southern California, Los Angeles, CA 90089 Internet: msafanovQusc.edu Mechanical Engineering Systems and Control Group, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands Internet: c. w. schererQwbmt. tudelft. nl Department of Automation and Systems, Universidade Federal de Santa Catarina, 88040-900 Florianopolis, SC, Brazil Internet: trof inoQlcmi. uf sc. br Numerical Technology, Inc., Santa Clara, CA 95051 Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA 02420 Internet: kyangQll. mit. edu

This page intentionally left blank

Contents Preface

xvii

Notation

xxv

I

Introduction

1

1 Robust Decision Problems in Engineering: A Linear Matrix Inequality Approach 3 L. El Ghaoui and S.-I. Niculescu 1.1 Introduction 1.1.1 Basic idea 1.1.2 Approximation and complexity 1.1.3 Purpose of the chapter 1.1.4 Outline 1.2 Decision problems with uncertain data 1.2.1 Decision problems 1.2.2 Using uncertainty models 1.2.3 Robust decision problems 1.2.4 Lagrange relaxations for min-max problems 1.3 Uncertainty models 1.3.1 Linear-fractional models 1.3.2 Operator point of view 1.3.3 Examples 1.3.4 Toward other models 1.4 Robust SDP 1.4.1 Definition 1.4.2 Lagrange relaxation of a robust SDP 1.4.3 A dual view: Rank relaxation 1.4.4 Extension to nonconvex nominal problems 1.4.5 On quality of relaxations 1.4.6 Link with Lyapunov functions 1.5 Algorithms and software for SDP 1.5.1 Facts about SDP and related problems 1.5.2 Interior-point methods 1.5.3 Nondifferentiable optimization methods 1.5.4 Software 1.6 Illustrations in control 1.6.1 Optimal control and SDP duality ix

3 3 4 4 4 5 5 5 6 7 8 9 11 12 13 14 14 15 16 17 18 18 21 21 22 24 24 25 25

x

Contents 1.6.2 Impulse differential systems 1.6.3 Delay differential systems 1.6.4 Open-loop and predictive robust control 1.7 Robustness out of control 1.7.1 LMIs in combinatorial optimization 1.7.2 Interval calculus 1.7.3 Structured linear algebra 1.7.4 Robust portfolio optimization 1.8 Perspectives and challenges 1.8.1 Algorithms and software 1.8.2 Nonconvex LMI-constrained problems 1.8.3 Dynamical systems over cones 1.8.4 Quality of relaxations for uncertain systems 1.9 Concluding remarks

II

Algorithms and Software

26 27 28 29 29 30 31 33 33 33 34 35 36 37

39

2 Mixed Semidefinite-Quadratic-Linear Programs

41

Jean-Pierre A. Haeberly, Madhu V. Nayakkankuppam, and Michael L. Overton 2.1 Introduction 2.2 A primal-dual interior-point method 2.3 Software 2.3.1 Some implementation issues 2.3.2 Using SDPpack 2.3.3 Miscellaneous features 2.4 Applications 2.5 Numerical results 2.6 Other computational issues 2.7 Conclusions

41 44 47 47 48 50 51 51 53 54

3 Nonsmooth Algorithms to Solve Semidefinite Programs Claude Lemarechal and Francois Oustry 3.1 Introduction 3.2 Classical methods 3.3 A short description of standard bundle methods 3.4 Specialized bundle methods 3.4.1 Rich oracle: Dual methods 3.4.2 Rich oracle: A primal approach 3.4.3 A large class of bundle methods 3.5 Second-order schemes 3.5.1 Local methods 3.5.2 Decomposition of the space 3.5.3 Second-order developments 3.5.4 Quadratic step 3.5.5 The dual metric 3.5.6 A second-order bundle method 3.6 Numerical results 3.6.1 Influence of e 3.6.2 Large-scale problems

57 57 60 61 64 64 65 67 70 70 70 71 71 72 73 74 74 75

Contents 3.6.3 Superlinear convergence 3.7 Perspectives

xi 76 76

4 sdpsol: A Parser/Solver for Semidefinite Programs with Matrix Structure 79 Shao-Po Wu and Stephen Boyd 4.1 Introduction 79 4.1.1 Max-det problems and SDPs 79 4.1.2 Matrix structure 80 4.1.3 Implications of the matrix structure 81 4.1.4 sdpsol and related work 81 4.2 Language design 82 4.2.1 Matrix variables and affine expressions 82 4.2.2 Constraints and objective 83 4.2.3 Duality 83 4.3 Implementation 84 4.3.1 Handling equality constraints 85 4.4 Examples 86 4.4.1 Lyapunov inequality 86 4.4.2 Popov analysis 86 4.4.3 Z)-optimal experiment design 88 4.4.4 Lyapunov exponent analysis 90 4.5 Summary 91

III

Analysis

93

5 Parametric Lyapunov Functions for Uncertain Systems: The Multiplier Approach 95 Minyue Fu and Soura Dasgupta 5.1 Introduction 95 5.2 Multipliers and robust stability 97 5.3 Parametric KYP lemma 98 5.4 Main results on parametric Lyapunov functions 100 5.5 Comparison with other schemes 103 5.5.1 Generalized Popov criterion 103 5.5.2 The AQS test 105 5.6 Generalization to discrete-time systems 106 5.7 Conclusions 107 6 Optimization of Integral Quadratic Constraints Ulf Jonsson and Anders Rantzer 6.1 Introduction 6.2 IQC-based robustness analysis 6.2.1 Robust stability analysis based on IQCs 6.2.2 Robust performance analysis based on IQCs 6.3 Signal specifications 6.4 A robust performance example 6.5 LMI formulations of robustness tests 6.6 Numerical examples 6.7 Conclusions

109 109 Ill 112 113 115 116 117 121 125

xii

Contents 6.8 Appendix 6.8.1 Proof of Theorem 6.2 and Remark 6.4 6.8.2 Multiplier sets for Example 6.3

126 126 127

7 Linear Matrix Inequality Methods for Robust H2 Analysis: A Survey with Comparisons 129 Fernando Paganini and Eric Feron 7.1 Introduction 129 7.2 Problem formulation 130 7.2.1 The H2 norm 130 7.2.2 Robust H2-performance 131 7.3 State-space bounds involving causality 133 7.3.1 The impulse response interpretation 133 7.3.2 Stochastic white noise interpretation 135 7.3.3 Refinements for LTI uncertainty 137 7.3.4 A dual state-space bound 137 7.4 Frequency domain methods and their interpretation 139 7.4.1 The frequency domain bound and LTI uncertainty 139 7.4.2 Set descriptions of white noise and losslessness results 141 7.4.3 Computational aspects 143 7.5 Discussion and comparisons 143 7.5.1 NLTV uncertainty 143 7.5.2 LTI uncertainty 145 7.5.3 A common limitation: Multivariable noise 146 7.6 Numerical examples 147 7.6.1 Comparing Js and Js 147 7.6.2 NLTV case: Gap between stochastic and worst-case white noise . . 148 7.6.3 LTI case: Tighter bounds due to causality 148 7.6.4 LTI case: Tighter bounds due to frequency-dependent multipliers . 148 7.6.5 LTI case: Using both causality and dynamic multipliers 148 7.6.6 A large-scale engineering example 149 7.7 Conclusion 150

IV

Synthesis

8 Robust H2 Control Kyle Y. Yang, Steven R. Hall, and Eric Feron 8.1 Introduction 8.2 Notation 8.3 Analysis problem statement and approach 8.4 Upper bound computation via families of linear multipliers 8.4.1 Choosing the right multipliers 8.4.2 Computing upper bounds on robust H2-performance 8.5 Robust H2 synthesis 8.5.1 Synthesis problem statement 8.5.2 Synthesis via LMIs 8.5.3 Synthesis via a separation principle 8.6 Implementation of a practical design scheme 8.6.1 Analysis step 8.6.2 Synthesis step

153 155 155 157 158 160 160 160 164 164 165 166 167 168 168

Contents 8.6.3 Design examples 8.7 Summary

xiii 169 173

9 A Linear Matrix Inequality Approach to the Design of Robust H2 Filters 175 Carlos E. de Souza and Alexandre Trofino 9.1 Introduction 175 9.2 The filtering problem 176 9.3 Full-order robust filter design 177 9.4 Reduced-order robust filter design 183 9.5 Conclusions 185 10 Robust Mixed Control and Linear Parameter-Varying Control with Full Block Scalings 187 Carsten W. Scherer 10.1 Introduction 187 10.2 System description 188 10.3 Robust performance analysis with constant Lyapunov matrices 189 10.3.1 Well-posedness and robust stability 189 10.3.2 Robust quadratic performance 191 10.3.3 Robust H2 performance 193 10.3.4 Robust generalized H2 performance 194 10.3.5 Robust bound on peak-to-peak gain 195 10.4 Dualization 196 10.5 How to verify the robust performance tests 197 10.6 Mixed robust controller design % 199 10.6.1 Synthesis inequalities for output feedback control 199 10.6.2 Synthesis inequalities for state-feedback control 201 10.6.3 Elimination of controller parameters 201 10.7 LPV design with full scalings 202 10.8 Conclusions 206 10.9 Appendix: Auxiliary results on quadratic forms 206 10.9.1 A full block 5-procedure 206 10.9.2 Dualization 206 10.9.3 Solvability test for a quadratic inequality 207 11 Advanced Gain-Scheduling Techniques for Uncertain Systems Pierre Apkarian and Richard J. Adams 11.1 Introduction 11.2 Output-feedback synthesis with guaranteed Z/2-gain performance 11.2.1 Extensions to multiobjective problems 11.3 Practical validity of gain-scheduled controllers 11.4 Reduction to finite-dimensional problems 11.4.1 Overpassing the gridding phase 11.5 Reducing conservatism by scaling 11.6 Control of a two-link flexible manipulator 11.6.1 Problem description 11.6.2 Numerical examples 11.6.3 Frequency and time-domain validations 11.7 Conclusions

209 209 210 214 214 215 218 218 221 221 224 226 228

xiv

Contents

12 Control Synthesis for Well-Posedness of Feedback Systems Tetsuya Iwasaki 12.1 Introduction 12.2 Well-posedness analysis 12.2.1 Exact condition 12.2.2 P-separator 12.3 Synthesis for well-posedness 12.3.1 Problemion3mulation 12.3.2 Main results 12.3.3 Proof of the main results 12.4 Applications to control problems 12.4.1 Fixed-order stabilization 12.4.2 Robust control 12.4.3 Gain-scheduled control 12.5 Concluding remarks

V Nonconvex Problems

229 229 230 230 231 232 232

235 237 240 240 242 244 246

249

13 Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints 251 Karolos M. Grigoriadis and Eric B. Beran 13.1 Introduction 251 13.2 LMI-based control design with rank constraints 252 13.2.1 Control problem formulation 252 13.2.2 Stabilization with prescribed degree of stability 253 13.2.3 HOO control 255 13.3 Alternating projecting schemes 257 13.3.1 The standard alternating projection method 257 13.3.2 The directional alternating projection method 258 13.4 Mixed alternating projection/SP (AP/SP) design for low-order control . . 259 13.5 Numerical experiments 263 13.5.1 Randomly generated stabilizable systems 263 13.5.2 Helicopter example 264 13.5.3 Spring-mass systems 265 13.6 Conclusions 267 14 Bilinearity and Complementarity in Robust Control Mehran Mesbahi, Michael G. Safonov, and George P. Papavassilopoulos 14.1 Introduction 14.1.1 Preliminaries on convex analysis 14.1.2 Background on control theory 14.2 Methodological aspects of the BMI 14.2.1 Limitations of the LMI approach 14.2.2 Proof of Theorem 14.1 14.2.3 Why is the BMI formulation important? 14.3 Structural aspects of the BMI 14.3.1 Concave programming approach 14.3.2 Cone programming approach 14.4 Computational aspects of the BMI 14.5 Conclusion

269 269 270 271 274 275 276 278 279 279 282 289 291

Contents

VI

Applications

xv

293

15 Linear Controller Design for the NEC Laser Bonder via Linear Matrix Inequality Optimization 295 Junichiro Oishi and Venkataramanan Balakrishnan 15.1 Introduction 15.2 Controller design using the Youla parametrization and LMIs 15.2.1 Youla parametrization 15.2.2 Controller design specifications as LMI constraints 15.3 Design of an LTI controller for the NEC Laser Bonder 15.3.1 Modeling of NEC Laser Bonder 15.3.2 Design specifications 15.3.3 Setting up the LMI problem 15.3.4 Numerical results 15.4 Conclusions

295 296 297 299 301 301 303 304 305 306

16 Multiobjective Robust Control Toolbox for Linear-Matrix-InequalityBased Control 309 Stephane Dussy 16.1 Introduction 309 16.2 Theoretical framework 310 16.2.1 Specifications 310 16.2.2 LFR of the open-loop system 311 16.2.3 LFR of the controller 311 16.2.4 LMI conditions 311 16.2.5 Main motivations for the toolbox 313 16.3 Toolbox contents 314 16.3.1 General structure of the toolbox 314 16.3.2 Description of the driver mrct 314 16.3.3 Description of the main specific functions 315 16.4 Numerical examples 316 16.4.1 An inverted pendulum benchmark 316 16.4.2 The rotational/translational proof mass actuator 318 16.5 Concluding remarks 320 17 Multiobjective Control for Robot Telemanipulators Jean Pierre Folcher and Claude Andriot 17.1 Introduction 17.2 Control of robot telemanipulators 17.2.1 Teleoperation systems 17.2.2 Background on network theory 17.2.3 Toward an ideal scaled teleoperator 17.3 Multiobjective robust control 17.3.1 Problem statement 17.3.2 Control objectives 17.3.3 Conditions for robust synthesis 17.3.4 Reconstruction of controller parameters 17.3.5 Checking the synthesis conditions 17.4 Force control of manipulator RD500 17.4.1 Single-joint slave manipulator model 17.4.2 Controller design

321 321 322 322 322 324 326 326 326 328 330 331 332 332 334

xvi

Contents 17.4.3 Results 17.5 Conclusion and perspectives

336 338

Bibliography

341

Index

369

Preface Linear matrix inequalities (LMIs) have emerged recently as a useful tool for solving a number of control problems. The basic idea of the LMI method in control is to interpret a given control problem as a semidefinite programming (SDP) problem, i.e., an optimization problem with linear objective and positive semidefinite constraints involving symmetric matrices that are affine in the decision variables. The LMI formalism is relevant for many reasons. First, writing a given problem in this form brings an efficient, numerical solution. Also, the approach is particularly suited to problems with "uncertain" data and multiple (possibly conflicting) specifications. Finally, this approach seems to be widely applicable, not only in control, but also in other areas where uncertainty arises.

Purpose and intended audience Since the early 1990s, with the developement of interior-point methods for solving SDP problems, the LMI approach has witnessed considerable attention in the control area (see the regularity of the invited sessions in the control conferences and workshops). Up to now, two self-contained books related to this subject have appeared. The book Interior Point Polynomial Methods in Convex Programming: Theory and Applications, by Nesterov and Nemirovskii, revolutionarized the field of optimization by showing that a large class of nonlinear convex programming problems (including SDP) can be solved very efficiently. A second book, also published by SIAM in 1994, Linear Matrix Inequalities in System and Control Theory, by Boyd, El Ghaoui, Feron, and Balakrishnan, shows that the advances in convex optimization can be successfully applied to a wide variety of difficult control problems. At this point, a natural question arises: Why another book on LMIs? One aim of this book is to describe, for the researcher in the control area, several important advances made both in algorithms and software and in the important issues in LMI control pertaining to analysis, design, and applications. Another aim is to identify several important issues, both in control and optimization, that need to be addressed in the future. We feel that these challenging issues require an interdisciplinary research effort, which we sought to foster. For example, Chapter 1 uses an optimization formalism, in the hope of encouraging researchers in optimization to look at some of the important ideas in LMI control (e.g., deterministic uncertainty, robustness) and seek nonclassical applications and challenges in the control area. Bridges go both ways, of course: for example, the "primal-dual" point of view that is so successful in optimization is also important in control. xvii

xviii

Preface

Book outline In our chapter classification, we sought to provide a continuum from numerical methods to applications, via some theoretical problems involving analysis and synthesis for uncertain systems. Basic notation and acronyms are listed after the preface. After this outline, we provide some alternative keys for reading this book. Part I: Introduction Chapter 1: Robust Decision Problems in Engineering: A Linear Matrix Inequality Approach, by L. El Ghaoui and S.-I. Niculescu. This chapter is an introduction to the "LMI method" in robust control theory and related areas. A large class of engineering problems with uncertainty can be expressed as optimization problems, where the objective and constraints are perturbed by unknownbut-bounded parameters. The authors present a robust decision framework for a large class of such problems, for which (approximate) solutions are computed by solving optimization problems with LMI constraints. The authors emphasize the wide scope of the method, and the anticipated interplay between the tools developed in LMI robust control, and related areas where uncertain decision problems arise. Part II: Algorithms and software Chapter 2: Mixed Semidefinite-Quadratic-Linear Programs, by J.-P. A. Haeberly, M. V. Nayakkankuppam, and M. L. Overton. The authors consider mixed semidefinite-quadratic-linear programs. These are linear optimization problems with three kinds of cone constraints, namely, the semidefinite cone, the quadratic cone, and the nonnegative orthant. The chapter outlines a primaldual path-following method to solve these problems and highlights the main features of SDPpack, a Matlab package that solves such programs. Furthermore, the authors give some examples where such mixed programs arise and provide numerical results on benchmark problems. Chapter 3: Nonsmooth Algorithms to Solve Semidefinite Programs, by C. Lemarechal and F. Oustry. Today, SDP problems are usually solved by interior-point methods, which are elegant, efficient, and well suited. However, they have limitations, particularly in large-scale or ill-conditioned cases. On the other hand, SDP is an instance of nonsmooth optimization (NSO), which enjoys some particular structure. This chapter briefly reviews the works that have been devoted to solving SDP with NSO tools and presents some recent results on bundle methods for SDP. Finally, the authors outline some possibilities for future work. Chapter 4: sdpsol: A Parser/Solver for Semidefinite Programs with Matrix Structure, by S.-P. Wu and S. Boyd. This chapter describes a parser/solver for a class of LMI problems, the so-called maxdet problems, which arise in a wide variety of engineering problems. These problems often have matrix structure, which has two important practical ramifications: first, it makes the job of translating the problem into a standard SDP or max-det format tedious, and, second, it opens the possibility of exploiting the structure to speed up the computation. In this chapter the authors describe the design and implementation of sdpsol, a parser/solver for SDPs and max-det problems, sdpsol allows problems with matrix structure to be described in a simple, natural, and convenient way.

Preface

xix

Part III: Analysis Chapter 5: Parametric Lyapunov Functions for Uncertain Systems: The Multiplier Approach, by M. Fu and S. Dasgupta. This chapter proposes a parametric multiplier approach to deriving parametric Lyapunov functions for robust stability analysis of linear systems involving uncertain parameters. This new approach generalizes the traditional multiplier approach used in the absolute stability literature where the multiplier is independent of the uncertain parameters. The main result provides a general framework for studying multiaffine Lyapunov functions. It is shown that these Lyapunov functions can be found using LMI techniques. Several known results on parametric Lyapunov functions are shown to be special cases. Chapter 6: Optimization of Integral Quadratic Constraints, by U. Jonsson and A. Rantzer. A large number of performance criteria for uncertain and nonlinear systems can be unified in terms of integral quadratic constraints. This makes it possible to systematically combine and evaluate such criteria in terms of LMI optimization. A given combination of nonlinear and uncertain components usually satisfies infinitely many integral quadratic constraints. The problem to identify the most appropriate constraint for a given analysis problem is convex but infinite dimensional. A systematic approach, based on finite-dimensional LMI optimization, is suggested in this chapter. Numerical examples are included. Chapter 7: Linear Matrix Inequality Methods for Robust H2 Analysis: A Survey with Comparisons, by F. Paganini and E. Feron. This chapter provides a survey of different approaches for the evaluation of H2performance in the worst case over structured system uncertainty, all of which rely on LMI computation. These methods apply to various categories of parametric or dynamic uncertainty (linear time-invariant (LTI), linear time-varying (LTV), or nonlinear timevarying (NLTV)) and build on different interpretations of the H2 criterion. It is shown nevertheless how they can be related by using the language of LMIs and the so-called 5-procedure for quadratic signal constraints. Mathematical comparisons and examples are provided to illustrate the relative merits of these approaches as well as a common limitation. Part IV: Synthesis Chapter 8: Robust H2 Control, by K. Y. Yang, S. R. Hall, and E. Feron. In this chapter, the problem of analyzing and synthesizing controllers that optimize the H2 performance of a system subject to LTI uncertainties is considered. A set of upper bounds on the system performance is derived, based on the theory of stability multipliers and the solution of an original optimal control problem. A Gauss-Seidel-like algorithm is proposed to design robust and efficient controllers via LMIs. An efficient solution procedure involves the iterative solution of Riccati equations both for analysis and synthesis purposes. The procedure is used to build robust and efficient controllers for a space-borne active structural control testbed, the Middeck Active Control Experiment (MACE). Controller design cycles are now short enough for experimental investigation. Chapter 9: A Linear Matrix Inequality Approach to the Design of Robust H2 Filters, by C. E. de Souza and A. Trofino. This chapter is concerned with the robust minimum variance filtering problem for linear continuous-time systems with parameter uncertainty in all the matrices of the system state-space model, including the coefficient matrices of the noise signals. The

xx

Preface

admissible parameter uncertainty is assumed to belong to a given convex bounded polyhedral domain. The problem addressed here is the design of a linear stationary filter that ensures a guaranteed optimal upper bound for the asymptotic estimation error variance, irrespective of the parameter uncertainty. This filter can be regarded as a robust version of the celebrated Kalman filter for dealing with systems subject to convex bounded parameter uncertainty. Both the design of full-order and reduced-order robust filters are analyzed. We develop LMI-based methodologies for the design of such robust filters. The proposed methodologies have the advantage that they can easily handle additional design constraints that keep the problem convex. Chapter 10: Robust Mixed Control and Linear Parameter-Varying Control with Full Block Scalings, by C. W. Scherer. This chapter considers systems affected by time-varying parametric uncertainties. The author devises a technique that allows us to equivalently translate robust performance analysis specifications characterized through a single Lyapunov function into the corresponding analysis test with multipliers. Out of the multitude of possible applications of this so-called full block <S-procedure, the chapter concentrates on a discussion of robust mixed control and of designing linear parameter-varying (LPV) controllers. Chapter 11: Advanced Gain-Scheduling Techniques for Uncertain Systems, by P. Apkarian and R. J. Adams. This chapter is concerned with the design of gain-scheduled controllers for uncertain LPV systems. Two alternative design techniques for constructing such controllers are discussed. Both techniques are amenable to LMI problems via a gridding of the parameter space and a selection of basis functions. The problem of synthesis for robust performance is then addressed by a new scaling approach for gain-scheduled control. The validity of the theoretical results is demonstrated through a two-link flexible manipulator design example. This is a challenging problem that requires scheduling of the controller in the manipulator geometry and robustness in the face of uncertainty in the high frequency range. Chapter 12: Control Synthesis for Well-Posedness of Feedback Systems, by T. Iwasaki. A wide variety of control synthesis problems reduces to the same type of computational problems involving LMIs. The main objective of this chapter is to show explicitly a general class of control problems that reduce to the same computational problem. The class is defined by the closed-loop specifications given in terms of the well-posedness of feedback systems. The author shows that this class includes many existing control problems as special cases, and such problems can be reduced to a search for structured positive definite matrices satisfying LMIs and possibly rank constraints.

Part V: Nonconvex problems Chapter 13: Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints, by K. M. Grigoriadis and E. B. Beran. Recently a large class of fixed-order control design problems has been formulated in a unified way as LMI problems with additional coupling matrix rank constraints. Because of the nonconvexity of the rank constraints, efficient convex programming algorithms cannot be used to find a solution. In this chapter, the method of alternating projections along with efficient SDP algorithms are proposed to address these problems of combined LMIs and coupling matrix rank constraints. Alternating projection algorithms exploit the geometry of these problems to obtain feasible solutions. Directional alternating projection methods that enhance the computational efficiency of the algorithms

Preface

xxi

are also described. Extensive computational experiments are provided to indicate the effectiveness and complexity of the proposed algorithms. Chapter 14: Bilinearity and Complementarity in Robust Control, by M. Mesbahi, M. G. Safonov, and G. P. Papavassilopoulos. The authors present an overview of the key developments in the methodological, structural, and computational aspects of the bilinear matrix inequality (BMI) feasibility problem. In this direction, the chapter presents the connections of the BMI with robust control theory and its geometric properties, including interpretations of the BMI as a rank-constrained LMI, as an extreme form problem (EFP), and as a semidefinite complementarity problem (SDCP). Computational implications and algorithms are also discussed.

Part VI: Applications Chapter 15: Linear Controller Design for the NEC Laser Bonder via Linear Matrix Inequality Optimization, by J. Oishi and V. Balakrishnan. The authors describe a computer-aided control system design method for designing an LTI controller for the NEC Laser Bonding machine. The procedure consists of numerically searching over the set of Youla parameters that describe the set of stabilizing LTI controllers for an LTI model of the Laser Bonder; this search is conducted using convex optimization techniques involving LMIs. The design specifications include constraints on the transient and steady-state tracking of a specific input, as well as bounds on the norms of certain closed-loop maps of interest. The design procedure is shown to yield a controller that optimally satisfies the various performance specifications. Chapter 16: Multiobjective Robust Control Toolbox for Linear-Matrix-InequalityBased Control, by S. Dussy. This chapter presents a collection of LMI-based synthesis tools for a large class of nonlinear uncertain systems, packaged in a set of Matlab functions. Most of them are based on a cone complementarity problem and provide solutions for the robust state- and output-feedback controller synthesis under various specifications. These specifications include performance requirements via a-stability or C-2-ga.m bounds and command input and outputs bounds. Two numerical examples, namely, an inverted pendulum and a nonlinear benchmark, are provided. Chapter 17: Multiobjective Control for Robot Telemanipulators, by J. P. Folcher and C. Andriot. This chapter addresses the control of robot telemanipulators in the presence of uncertainties, disturbance, and measurement noise. The controller design problem can be divided into several multiobjective robust controller synthesis problems for LTI systems subject to dissipative perturbations. Synthesis specifications include robust stability, robust performance (H2 norm) bounds, and time-domain bounds (output and command input peak). Sufficient conditions for the existence of an LTI controller such that the closed-loop system satisfies all specifications simultaneously are derived. An efficient cone complementarity linearization algorithm enables us to solve numerically the associate optimization problem. This multiobjective synthesis approach is used to design the force controller of a single joint of a slave manipulator.

How to read the book The book is the result of the work of many different authors, each with a different point of view. We have sought to provide a coherent and unitary text, without sacrificing the

xxii

Preface

individuality of each contribution. Some redundancy might result from this; however, we feel it is always beneficial to provide different views on similar problems. There are several themes, emphasized as key words in the list below, that come across the sequential order of the book. An interested reader might read the corresponding chapters independently. • Several chapters deal with the so-called <S-procedure and its applications in control: Chapter 1 puts it in the context of Lagrange relaxations, Chapters 8 and 10 generalize the result to different settings. Also, Lyapunov functions (a theme recurrent in the book) can be interpreted in the context of Lagrange multipliers, as explained in Chapter 1. • The robust 13.2 problem is evoked first in its analysis aspect in Chapter 7. Chapters 8 and 10 deal with the corresponding control synthesis problem in different ways; Chapter 9 addresses the issues of filter design in this context. • Nonconvex optimization problems arise mainly in three forms in control, as LMI problems with rank constraints, as BMI problems, or as cone complementarity problems. The similarity between these points of view is outlined in Chapter 1 and are explored in more detail in Part V. The so-called cone complementarity formulation, mentioned in Chapters 1 and 14 (under the name SCDP) is used in a practical context in Chapters 16 and 17. • Linear parameter-varying (LPV) systems are considered in Chapters 10 and (to a lesser extent) 12, where a general framework, encompassing robust control, is outlined. Chapter 11 is devoted to techniques improving the possible conservatism of the method. • The search for multipliers in the context of robustness analysis is addressed in Chapter 5 under a "parametric Lyapunov function" point of view, while Chapter 6 develops the "integral quadratic constraint" (IQC) approach; Chapter 17 uses a very similar approach in a practical application. Note that Chapter 1 makes some connections between Lyapunov functions, multipliers (in the above "control" sense), and Lagrange relaxations in optimization. • Algorithms and software issues are broadly mentioned in Chapter 1. Chapters 2 and 3 describe algorithms, and Chapter 4 describes a parser/solver for a general class of LMI problems. Chapter 16 uses a specialized Matlab toolbox for some robust control problems. • Some numerical results and applications are mentioned throughout the book, coming across as illustrations of software performance (Part II), applications of system analysis (Chapters 6 and 7), or control synthesis methods (Chapters 11 and 13). Part VI is dedicated to realistic design examples.

Acknowledgments This project began a long time ago (in October 1996!). During this long process, the book has benefited from the enthusiastic participation and feedback from the authors, whose patience is remarkable. We express our thanks to all of them. Some of the authors have particularly helped our editorial job, improving the overall message. For example, Michael Overton and Francois Oustry have completely rewritten a section on LMI algorithms in the introductory chapter; Fernando Paganini and Eric Feron decided to merge their contribution on robust H2 analysis in a single survey chapter.

Preface

xxiii

We also would like to thank the anonymous reviewers, who have carefully read the manuscript and provided us with very useful suggestions for improvement. We are grateful to John Burns, editor-in-chief of this series, for his visionary support, making this book a reality. Our SIAM correspondents, Vickie Kearn and Marianne Will, offered us precious help at all stages of this project. This book would not have been possible without the support of EDF (Electricite de France), under contract P33L74/2M7570/EP777; special thanks goes to Clement-Marc Falinower, head of the "Service Automatique" at EDF. His support partially funded, in particular, several key visits of contributors to ENSTA during the project, including those of Silviu-Iulian Niculescu, Michael Overton, Stephen Boyd, and Eric Feron. Special thanks go to Jean-Luc Commeau, who helped us with various computer problems. Finally, we would like to thank our home institutions, ENSTA (Paris) and CNRSHEUDIASYC (Compiegne), for their support. Chapter 1 was partially funded by a support from EDF under contract P33L74/ 2M7570/EP777. Section 1.5 was written with precious help from Michael Overton and Frangois Oustry. Chapter 2 was supported in part by National Science Foundation grant CCR-9731777 and in part by U.S. Department of Energy grant DE-FG02-98ER25352. Chapter 4 was partially supported by AFOSR contract F49620-95-1-0318, NFS contracts ECS-9222391 and EEC-9420565, and MURI contract F49620-95-1-0525. Chapter 15 was supported in part by the NEC Faculty Fellowship and a General Motors Faculty Fellowship.

LAURENT EL GHAOUI SILVIU-IULIAN NICULESCU

This page intentionally left blank

Notation

C C° C + , C , C~ a*, 3ft(a) Ex Hp L p (R,R n X m ) 5 int5 S*

The real numbers, real k-vectors, real m x n matrices. R+ , onn j R Th real enumbers nonnegativ and the e imaginary numbers, respectively. The complex numbers. Unit circle of the complex plane. The open right half, closed right half, and open left half complex planes, respectively. The complex conjugate and the real part of Expected value of (the random variable) x. pth Hardy space (in this book, p is either 2 or oo). pth Hilbert space of functions mapping R to R n x m . (p - 2 in this book; also, the abbreviation Li2 is used.) The closure of a set 5. The interior of a set 5. The dual cone of a set 5 C Sn is defined to be

5n Space of real, symmetric matrices of order n. <S" Space of real, positive semidefinite, symmetric matrices of order n. CoS Convex hull of the set 5 C R n , given by

Ik MT M* Tr(M) Tr(ATB) KerM

(Without loss of generality, we can take p = n + I here.) The k x k identity matrix. The subscript is omitted when k is not relevant or can be determined from context. Transpose of a matrix Complex-conjugate transpose of a matrix where a* denotes the complex conjugate of Trace of M e R n x n ; i.e., £

Notation

XXVI

IrnM Range of a matrix M. RankM Rank of a matrix M. M1- Orthogonal complement of M, a matrix of maximal rank such that MTM-L — 0; the rows of ML form a basis for the nullspace ofM T . Aft Moore-Penrose pseudoinverse of M. Af > 0 M is symmetric and positive semidefmite; i.e., M = MT and for all Af > 0 M is symmetric and positive definite; i.e., zTMz > 0 for all nonzero z € Rn. Af > N M and N are symmetric and M1/2 For M > 0, M1/2 is the unique Z = ZT such that sectX Bilinear sector transformation of a square matrix, defined (when applicable) a The maximum eigenvalue of the matrix For P = PT, Q = QT > 0, Amax(P, Q) denotes the maximum eigenvalue of the symmetric pencil The minimum eigenvalue of ||M|| The spectral norm of a matrix or vector It reduces to the Euclidean norm, i.e., ||x|| = VxTx, for a vector x. \\M\\F The Frobenius norm of a matrix or vector It reduces to the Euclidean norm, i.e., ||x|| = VxTx, for a vector x. diag(- • •) Block-diagonal matrix formed from the arguments, i.e.,

Vec(Af) vector obtained by stacking up the columns of matrix M. A <8> B The Kronecker product of matrices A, B. Herm(M) The Hermitian part of matrix M, i.e., the matrix Partitioned matrix notation for transfer functions:

The upper and lower linear fractional transformations of a matrix A and a partitioned matrix M, defined as

and

votationn

xxvii The linear fractional transformation (LFT) or star product of two partioned matrices M, JV, defined as

A Uncertainty matrix. A Uncertainty set, including the zero matrix in general.

This page intentionally left blank

Part I

Introduction

This page intentionally left blank

Chapter 1

Robust Decision Problems in Engineering: A Linear Matrix Inequality Approach L. El Ghaoui and S.-I. Niculescu 1.1

Introduction

1.1.1

Basic idea

The basic idea of the LMI method is to formulate a given problem as an optimization problem with linear objective and linear matrix inequality (LMI) constraints. An LMI constraint on a vector x e Rm is one of the form

where the symmetric matrices Fj = F? G R JVxJV , i = 0,... ,m, are given. The minimization problem where c € R m , and F > 0 means the matrix F is symmetric and positive semidefinite, is called a semidefinite program (SDP). The above framework is particularly attractive for the following reasons. Efficient numerical solution. SDP optimization problems can be solved very efficiently using recent interior-point methods for convex optimization (the global optimum is found in polynomial time). This brings a numerical solution to problems when no analytical or closed-form solution is known. Robustness against uncertainty. The approach is very well suited for problems with uncertainty in the data. Based on a deterministic description of uncertainty (with detailed structure and hard bounds), a systematic procedure enables us to formulate an SDP optimization problem that yields a robust solution. This statement has implications for a wide scope of engineering problems, where measurements, modelling errors, etc., are often present. 3

4

El Ghaoui and Niculescu

Multicriteria problems. The approach enables us to impose many different (possibly conflicting) specifications in the design process, allowing us to explore trade-offs and analyze limits of performance and feasibility. This offers a drastic advantage over design methods that rely on a single criterion deemed to reflect all design constraints; the choice of a relevant criterium is sometimes a nontrivial task. Wide applicability. The techniques used in the approach are relevant far beyond control and estimation. This opens exciting avenues of research where seemingly very different problems are analyzed and solved in a unified framework. For example, the method known in LMI-based control as the S-procedure can be successfully applied in combinatorial optimization, leading to efficient relaxations of hard problems.

1.1.2

Approximation and complexity

At this point we should comment on a very important aspect of the LMI approach, which puts in perspective the above. The LMI approach is deeply related to the branch of theoretical computer science that seeks to classify problems in terms of their computational complexity [323]. In a sense, the approach is an approximation technique for a class of (NP-) hard problems, via polynomial-time algorithms, and thus hinges on a dividing line between "hard" and "easy" in the sense of computational complexity. This point of view shows that the LMI approach is not only a numerical approach to practical problems. It also requires one to consider the complexity analysis of a given problem and to seek an approximation in the case of a "hard" problem. This concern is quite new in control; see [53] for a survey and references on complexity analysis of control problems.

1.1.3 Purpose of the chapter In this chapter, we describe a general LMI-based method to cope with engineering decision problems with uncertainty and illustrate its relevance, both in control and in other areas. This chapter is intended to serve two kinds of objectives. The first is to introduce the reader to the LMI method for control and help understand some of the issues treated in more detail in the subsequent chapters. References that would usefully complement this reading include a book on the LMI approach in system and control theory by Boyd, El Ghaoui, Feron, and Balakrishnan [64], a book on algorithms for LMI problems by Nesterov and Nemirovskii [296], and [404]. Also, the course notes [66] contain many results on convex optimization and applications. Another purpose of this chapter is to open the method to other areas. In our presentation, we have tried to give an "optimization point of view" of the method in an effort to put it in perspective with classical relaxation methods. This effort is mainly motivated by the belief that the approach could be useful in many other fields, especially regarding robustness issues. This topic of robustness is receiving renewed attention in the field of optimization, as demonstrated by a series of recent papers [42, 127, 39].

1.1.4

Outline

Section 1.2 defines robust solutions to problems with uncertain data and outlines the general ideas involved in Lagrange relaxations for these problems. Section 1.3 is devoted to models of uncertain systems. In section 1.4, we concentrate on SDP problems with uncertain data matrices and detail a general methodology for computing appropriate bounds on these problems. Section 1.5 briefly describes algorithms, software, and related

1.2. Decision problems with uncertain data

5

issues for solving general SDPs. (This section, written with the help of Michael Overton and Frangois Oustry, can be read independently of the rest of this chapter.) Some illustrative examples taken from the control area, which complement those put forth in the other chapters, are given in section 1.6. Section 1.7 presents a number of examples taken out of control. Finally, section 1.8 explores some perspectives and challenges that lie ahead.

1.2 1.2.1

Decision problems with uncertain data Decision problems

Many engineering analysis and design problems can be seen as decision problems. In control engineering, one must decide which controller gains to choose in order to satisfy the desired specifications. This decision involves several trade-offs. In this context, decision problems are based on two "objects": a set of decision variables that reflect the engineer's choices (controller gains, actuator location, etc.) and a set of constraints on these decision variables that reflect the specifications imposed by the problem (desired closed-loop behavior, etc.). Such problems can be sometimes translated in terms of mathematical programming problems of the form (1.3)

minimize /o(x) subject to x € #,

fi(x) < 0,

i = 1,... ,p,

where /o, /i,..., fp are given scalar-valued functions of the decision vector x G Rm, and X is a subset of R m . In some problems, X is infinite dimensional, and the decision vector x is a function. In the language of control theory, we can interpret the above as a multispecification control problem, where several (possibly conflicting) constraints have to be satisfied. Let us comment on the limitations of the above mathematical framework. The desired specifications are sometimes easy to describe mathematically as in the above model (for example, when we seek to ensure a desired maximum overshoot). Others are more difficult to handle and not less important, such as economic constraints (cost of design), reliability and ease of implementation (time needed to obtain a successful design), etc. These "soft" constraints can be neglected or taken into account implicitly (for example, choosing a PID controller structure, as opposed to a more complicated one). A classical way to handle these constraints is by hierarchical decision, where soft constraints are taken into account at a higher level of managerial decision (e.g., allowing a constrained budget for the whole control problem).

1.2.2

Using uncertainty models

In some applications, the "data" in problem (1.3) is not well known; the previous "optimization model" falls short of handling this case, so it would be useful to consider decision problems with uncertainty. For this, it is necessary to introduce models for uncertain systems; this is the purpose of section 1.3. There are other motivations for us to use such models. Real-world phenomena are very complex (nonlinear, time-varying, infinite-dimensional, etc.). To cope with analysis and design problems for such systems, we almost always have to use (finite-dimensional, linear, stationary) approximations. We believe that a finer way to handle complexity is to use approximations with uncertainty. We thus make the distinction between "physical" and "engineering" uncertainties. The first are due to ignorance of the precise physical behavior; the second come from a voluntary simplifying assumption. (This technique can be referred to as embedding.)

6

El Ghaoui and Niculescu

Assume, for instance, that in a complex system two variables p, q are linked by a nonlinear relation p = 6(q). The function 6 may be very difficult to identify (this is a physical uncertainty). Moreover, we may choose to "embed" this nonlinearity in a family of linear relations of the form p = 6q, where the scalar 8 is inside a known sector. (The latter may be inferred from physical knowledge.) We thus obtain a linear model with (engineering) uncertainty that is more tractable than the nonlinear one. The technique of embedding may be conservative, but it usually gives rise to tractable problems.

1.2.3 Robust decision problems In some applications, the "data" in problem (1.3) is not well known; that is, every /j depends not only on x but also on a real "uncertainty matrix"1 A. The latter is only known to belong to a set of admissible perturbations A. We may then define a variety of robust decision problems and associated solutions. These problems are based on a worst-case analysis of the effect of perturbations. Robustness analysis We define the robust feasibility set

The analysis problem is to check if a given candidate solution x is robustly feasible. Robust synthesis A robust synthesis problem is to find a vector x that satisfies robustness constraints. We may distinguish the following: • A robust feasibility problem, where we seek a vector x that belongs to • A robust optimality problem, defined as

Here, we seek to minimize the worst-case value of the objective fo(x, A) over the set of robustly feasible solutions. (We note that the objective function /o may be assumed independent of the perturbation without loss of generality.) Parameter-scheduled synthesis. In such a problem, we allow the decision vector to be a function of the perturbation. In this case, we assume that the decision vector is not finite dimensional but evolves in a set X of functions (of the perturbation A). The problem reads formally as the robust synthesis ones, but with X a set of functions. Mixed parameter- scheduled/robust problems. In some problems, the decision vector is a mixture of a set of real variables and of functions of some parameters. In control, this kind of problem occurs when only some parameters are available for measurement. 1

We take the uncertainty to be a matrix instead of a vector for reasons to be made clear later.

1.2. Decision problems with uncertain data

7

In control theory, we interpret the above problems as robust multispecification control problems, where several (possibly conflicting) constraints have to be satisfied in a robust manner. We note that these problems are in general very hard to solve (noncomputationally tractable). References on complexity of such problems in control include [86, 291, 72, 52, 393, 144]. Robust decision problems first arose (to our knowledge) in the field of automatic control. In fact, robustness is presented in the classical control textbooks as the main reason for using feedback. Good references on robust control include the book by Zhou, Doyle, and Glover [446]; Dahleh and Diaz-Bobillo [88]; and Green and Limebeer [175]. In other areas of engineering, uncertainty is generally dealt with using stochastic models for uncertainty. In stochastic optimization, the uncertainty is assumed to have a known distribution, and a typical problem is to evaluate the distribution of the optimal value, of the solution, etc.; a good reference on this subject is the book by Dempster [97]. Models with deterministic uncertainty are less classical in engineering optimization. Ben Tal and Nemirovski consider a truss topology design problem with uncertainty on the loading forces [41, 40]. In a more recent work [42], they introduce and study the complexity of robust optimality problems in the sense defined above in the context of convex optimization. Their approach is based on ellipsoidal bounds for the perturbation. The paper [127] addresses the computation of bounds for a class of robust problems and uses a control-based approach to solve for these bounds via linear-fractional representations (LFRs) and SDR

1.2.4

Lagrange relaxations for min-max problems

Robust decision problems such as problem (1.5) belong to the class of min-max problems. To attack them, we can thus use a versatile technique, called Lagrange relaxation, that enables us to approximate a set of complicated constraints by a "more tractable" set. To understand the technique, let us assume that the uncertainty set A is a subset of R' that can be described by a finite number of (nonlinear) constraints

where the g^s are given scalar-valued functions of the perturbation vector 6. Now, observe that the property

holds if there exist nonnegative scalars TI, . . . , rq such that

The above condition is rewritten

The above idea can be used for general min-max problems of the form (1.5). The basic idea is to approximate a min-max problem by a (hopefully easier) minimization problem. To motivate this idea, we take an example of the robust optimality problem (1.5) with no constraints on x (p = 0). That is, we consider

8

El Ghaoui and Niculescu The Lagrange relaxation technique yields an upper bound on the above problem:

In some cases, computing the (unconstrained) maximum in the above problem is very easy; then the min-max problem is approximated (via an upper bound) by a minimization problem (the minimization variables are x and r). (The scalars TJ appearing in the above are often referred to as Lagrange multipliers.) Note that, for x fixed, the above problem is convex in r ; if /o is convex in rr, then we may use (subgradient-based) convex optimization to solve the relaxed problem. The technique thus requires that computing values and subgradient of the "max" part (a function of x, r) is computationally tractable. The upper bounds found via Lagrange relaxations can be improved as long as we are able to solve and compute corresponding subgradients for a constrained "max" problem in (1.8). For example, we may choose to relax only a subset of the constraints &(£) < 0 and keep the remaining ones as hard constraints in the maximization problem. Different choices of "relaxed" versus "nonrelaxed" constraints lead to different Lagrange relaxations (and upper bounds). A special case: The 5-procedure. The well-known <S-procedure is an application of Lagrange relaxation in the case when the g^s are quadratic functions of 6 e R'. In this case, checking (1.6) yields a simple LMI in the multipliers T^ The 5-procedure lemma, which we recall below, is widely used, not only in control theory [258, 140, 64] but also in connection with trust region methods in optimization [376, 381]. Lemma 1.1 (5-procedure). Let go, . . . ,gp be quadratic functions of the variable 6 € Rm:

where F» = F?'. The following condition on po, • • • > 9p

holds if there exist scalars T» > 0, i = 1, . . . ,p, such that

When p=l, the converse holds, provided that there is some 60 such that gi(6o) > 0.

1.3 Uncertainty models A typical decision problem is defined via some data, which we collect for convenience in a matrix M. For example, if the problem is defined in terms of the transfer function of an LTI system M(s) — D + C(sl — A)~1B, the matrix M contains the four matrices A, B, C, D. When the data is uncertain, we need to describe as compactly as possible the way the perturbation A affects the data matrix and also the structure of the perturbation

1.3. Uncertainty models

9

set A. This will result in an uncertainty model for the data. Formally, an uncertainty model for the data is defined as a matrix set

where M is a (possibly nonlinear) matrix-valued function of the "perturbation matrix" A, and A is a matrix set.

1.3.1

Linear- fractional models

Definition A linear- fractional model is a set of the form (1.11), where the following three assumptions are made. Linear-fractional assumption. We assume that the matrix-valued function can be written, for almost every A, as

for appropriate constant matrices M,L,R and Uncertainty set assumption. In addition, the uncertainty set A is required to satisfy the following. For every vector pair p, g, the condition

can be characterized as a linear matrix inequality on a rank-one matrix:

where $A is a given linear operator "characterizing" A. Well-posedness assumption. To simplify our exposition, we make a last assumption from now on that the above model is well posed over A, meaning that

Checking well-posedness is very difficult in general; using the theory developed in section 1.4, we can derive sufficient LMI conditions for well-posedness. Chapter 12 discusses the issue of well-posedness in more detail.

Motivations The above uncertainty models for the data seem very specialized. However, they can cover a wide variety of uncertain matrices, as we now show. First, the linear-fractional assumption made on the matrix function M(A) is motivated by the following lemma, a constructive proof of which appears in, e.g., [56, 446].

10

El Ghaoui and Niculescu

Lemma 1.2. Every (matrix-valued) rational function M(<5) of 6 € Rp that is well defined for 6 = 0 admits an LFR of the form

valid for whenever det(/ — DA) ^ 0 for appropriate matrices M, L, R, D and integers We note that, when I = 1 (that is, for a monovariable rational matrix function), the matrices M, L, R, D are simply a state-space realization of the transfer matrix M+L(sI— D)~1R, where s = 1/6. An LFR of M can then always be constructed in a "minimal" way, such that the size of matrix A is exactly the order of M in s. In the multivariable case, the integers TI are greater than the degrees of the corresponding variable Si in M, and the issue of minimality is much more subtle. Finally, we note that when M is polynomial in its argument, an LFR can always be constructed such that D is strictly upper triangular, so that this LFR is everywhere well posed, as is M. The LFR generalizes the state-space representation known for (monovariable) transfer matrices to the multivariable case.2 For other comments on the LFR formalism, see, for example, [108, 254]. The above lemma shows that we can handle almost arbitrary algebraic functions of perturbation parameters, provided we define the perturbation matrix A as a diagonal matrix, with repeated elements. (In the following, we pay special attention to such diagonal perturbation structures.) The uncertainty set assumption made on the set A can appear also very specialized. In fact, it can handle a wide array of uncertainty bounds. The following is a short list of examples. Unstructured case. Assume

This case is referred to as the "unstructured perturbations" case and is a classic since the development of HQQ control. The corresponding characterization is

Euclidean-norm bounds. Assume

The corresponding characterization is

(In the above,
1.3. Uncertainty models

11

Maximum-norm bounds. Assume

The corresponding characterization is

(Again, pi (resp., qi) denotes the ith Ti x 1 block in vector q (resp., p).) Sector bounds. Assume A = {s • I \ s G C, characterization is

s + s* > 0}. The corresponding

Of course, it is possible to mix different bounds to cover more complicated cases.

1.3.2

Operator point of view

The above uncertainty models for data matrices can be interpreted as input-output maps, as follows. Given A e A, the input-output relationship between two vectors u and t/, can be rewritten, if det(7 — DA) ^ 0, as a quadratic one:

The above representation is not surprising: It shows that a nonlinear (algebraic) constraint can always be seen as a quadratic one, provided we introduce enough "dummy" variables (here, the vectors p, q). In view of our assumption on the uncertainty set A, the "uncertain constraint" can be rewritten in a quadratic matrix inequality form

The operator point of view allows us to consider (linear) dynamical relationships. For example, the LTI system where A is a constant matrix, can be represented in the frequency domain as

When we need to analyze stability, we restrict ourselves to all complex s such that 5Rs > 0. In this case, the uncertain system to be considered is

In section 1.4.6, we follow up this idea in connection with Lyapunov theory.

12

1.3.3

El Ghaoui and Niculescu

Examples

Let us now show how the above tools can be used to describe compactly a wide array of uncertain matrices and dynamical systems. To simplify our description, we take the operator point of view mentioned previously. Matrices The above framework covers the case when some (matrix) data occurring in the robust decision problem is affected by a perturbation vector <5, in an algebraic manner, and the vector 6 is unknown-but-bounded. Consider, for example, an "uncertain matrix" M(£), where M is an algebraic function of vector <5, and 6 is unknown-but-bounded in the maximum-norm sense. This matrix can be represented as an input-output operator u —> y := M(<5)u in the form

The above form consitutes a linear-fractional model for the uncertain linear constraint y = M(6)u, \\6\\oo < 1. Model (1.18) can also be represented in the quadratic matrix inequality form (1.17), where $A is defined in (1.16). Dynamical systems The input-output relationship y = M(A)w may involve a dynamical system; in this case also, LFR models can be used. For example, the uncertain dynamical system

can be represented, with the assumptions made previously in force, as

It is possible to consider nonlinear systems with the above representations, using an "embedding" technique. Roughly speaking, if we know bounds on the trajectories of the above uncertain system, then these bounds will also hold for the nonlinear system

if A(y) e A for every y. (This technique, which is more powerful than the classical linearization technique, is related to the comparison principle in differential equations; see [238] for more details.) We take another example arising in the control of linear systems with delayed input, where we assume only time-delay interval uncertainty. Consider the following (frequencydomain) constraint:

where HQ is a transfer function that represents the nominal system of the form

1.3. Uncertainty models

13

with Hnd free of delay and 6n as the nominal delay. The uncertainty 6 is of the form where A is an unknown-but-bounded real scalar (— 1 < A < 1). A way to handle this class of systems is to aproximate the delay uncertainty term using a first-order real Fade approximant. An LFR of the approximation is

Using this LFR, we may obtain an LFR for the delayed system. (An alternate representation of this kind of uncertainty is given below.) Analysis and comparisons between several approximations of the delay uncertainty can be found in [415].

1.3.4

Toward other models

We now briefly mention other possible models. (For other remarks and comments on uncertainty modeling, we refer to [378].) Integral quadratic constraint models An alternative to the class of the above parameter-dependent models is one where we assume that some signals in a known linear time-invariant (LTI) system are subject to integral quadratic constraints (IQCs) that are usually written in the frequency domain. Thus, the unknown part of the system is only subject to energy-type constraints. For other comments on this formalism, see, for example, [274, 341]. This class of systems is perhaps more versatile than the class of parameter-dependent systems seen above. We can use this models to "embed" a large class of complicated systems, such as systems with delays, parameter-dependent systems with bounds on the parameter's rate of variation, infinite-dimensional systems, etc. Consider for example the following system:

where the uncertainty block A(£) satisfies an IQC. This IQC is defined as follows: let II : jH »—> C r x r be a bounded measurable function, with appropriate r. The corresponding IQC is

where p, q are the Fourier transforms of p, q e L2[0, oo). As an example, we consider again the delayed system (1.19) with 6n = 0, and the uncertainty delay interval of the form [0, p\. A different way to handle the uncertainty on the delay is to suppose that the uncertain "quantity" is Aw(t) = u(t — h) — u(t), h < p (here, u(-) is the input signal). This signal satisfies the IQC above, with (see [341]):

where a and (3 are bounded real-valued functions on the imaginary axis such that

IQCs are examined in more detail in Chapter 6 of this book.

14

El Ghaoui and Niculescu

Polytopic models In a polytopic model [64], we assume that the state-space matrices of the system are only known to lie in a given polytope. For example,

Here, the set of admissible perturbations of T> is the polytope {A A; > 0, ^ A* = 1}, and the mapping assigns the linear combination ]T) \Ai to the vector A. Polytopic systems are considered in Chapter 9 of this book. Refined perturbation sets The above models are very rough in that they impose no restriction on the rate of change of the perturbed elements. For example, the polytopic model (1.22) could be used for a system subject to possible failures; each matrix Ai in (1.22) represents a specific operating mode for the system. In practice, this model will be very conservative, as it does not prevent the system from jumping arbitrarily fast from a mode to any other. Some refined models simply impose a bound on the rate of variation of the perturbation parameters and make use of LMI analysis, which takes into account this bound. Stochastic models These models are often used in engineering to describe uncertainty and could be interpreted as more refined than deterministic ones. For example, for a system with several failure modes, a model that is more refined than the polytopic model could be based on systems with Markovian jumps. With the matrices Ai of the polytopic model, we associate a "transition probability matrix" that describes the probability of change from one mode to the other. Similarly, it is possible to consider a linear-fractional model, where the uncertain matrix A obeys a known white-noise-type distribution; see [118] for an example. The reader should not make a conflict out of the distinction deterministic/stochastic: it is possible to consider stochastic models with deterministic uncertainty. For example, we may consider a linear system with Markovian jumps, where the transition probability matrix is unknown-but-bounded (see [121]).

1.4

Robust SDP

Robust decision problems are defined in terms of (a) a "nominal" problem of the form (1.3) and (b) an uncertainty model for the data. Equipped with (matrix-based) uncertainty models for the data described previously, we need to define more precisely the form of the nominal problem. It turns out that a class of problems very rich in applications is precisely an SDP. This motivates the study of robust SDP problems, which correspond to an SDP with uncertain data.

1.4.1

Definition

Consider the case when the constraints /*(#, 6) < 0, i = 1,... ,p, can be written as

where F is affine in the decision variable x and takes its values in the set of symmetric matrices.

1.4. Robust SDP

15

We make the assumption that F is a linear-fractional function of A, with LFR where F(x) = F(x)T and L(x) are affine matrix-valued functions of x, and R, D are given matrices, while the uncertainty matrix A is restricted to a set A. We assume that the above LFR is well posed over A, and, in addition, we assume that the set A satisfies the uncertainty set assumption defined in section 1.3.1. The problem is called a robust SDP problem. Note that we assumed that the "objective vector" c is independent of perturbation; this is done without loss of generality. The above problem is convex; however, it is in general very difficult to solve (NP-hard; see [154, 323]). Checking if a given candidate solution x is robustly feasible is already very difficult in general. Our objective is to find lower bounds on this problem in the form of SDP.

1.4.2

Lagrange relaxation of a robust SDP

Let us fix the decision variable x and seek a sufficient condition for Using the LFR, and using the well-posedness assumption, we rewrite the above as

With the previous characterization p = Ag, the above is seen to be equivalent to

such that

We are ready now to use Lagrange relaxation, following the general ideas evoked in section 1.2.4. The previous condition holds if there exists a positive semidefinite matrix S such that V

or, equivalently,

where $^ is the dual map of $ A • Hence, the Lagrange relaxation of robust SDP is an SDP of the form min CTX subject to 5 > 0 and (1.24). The variables in this SDP are x (the original decision variable) and 5 (the "multiplier" matrix). The SDP provides an approximation of the robust SDP in the following senses: (a) it yields an upper bound on the robust SDP; (b) any optimal x is robust (i.e., it is feasible V A e A); (c) in the unstructured case (A defined by (1.14)), the approximation is exact.

16

El Ghaoui and Niculescu

A special case: Robust linear programming (LP) Consider the LP

Assume that the a^'s and 6i's are subject to unstructured perturbations. That is, th e perturbed value of [af bi]T is [af 6j]T + <§i, where \\6i\\2 < p, i = 1,..., L, and p > 0 is a given measure of the perturbation level. We seek a robust solution to our problem; that is, we seek x that minimizes CTX subject to the robustness constraints

This is a special case of a robust optimality problem, as defined in section 1.2.3. Using the previous theory, it is straightforward to show that the robust LP is equivalent to the problem

The above problem is a second-order cone program (SOCP) for which efficient specialpurpose interior-point methods are available (see section 1.5.1). We note that for every p > 0, the above problem yields a unique solution, that is, a regular (in particular, continuous) function of the problem's data (the o^'s and the &j's). Thus, the approach provides a regularization procedure for the LP (1.25). A simple variation of the robust LP is LP with implementation constraints, where we seek a solution x such that for every 8x, \\6x\\ < p, the modified vector x + 6x still satisfies the constraint of the original LP. This problem arises for example in the area of optimal filter design, where the filter's coefficients can be implemented only with finite precision.

1.4.3

A dual view: Rank relaxation

Lagrange relaxation can be interpreted in a dual way, in terms of relaxation of a rank constraint of the rank-one, positive semidefinite matrix

which appears in (1.23). Define

Define also the convex set

Let us fix the decision variable x again. Condition (1.23) holds if and only if

The robust SDP problem writes as the max-min problem

1.4. Robust SDP

17

Relaxing the rank constraint yields lower bound

which, by virtue of the saddle-point theorem (recall X is convex), is equivalent to the SDP in variable X

The above is the dual of the SDP obtained via Lagrange relaxation. We will see in section 1.7.1 how this connection between Lagrange and rank relaxation has been recognized earlier in the special case of combinatorial optimization problems. For a related point of view on the geometrical properties of rank-constrained symmetric matrices, see [192] and references therein.

1.4.4

Extension to nonconvex nominal problems

In many cases, notably in control synthesis, we encounter problems where the nominal problem is nonconvex. Most of these problems can be turned into cone complementarity problems (see section 1.5.2 for more details). We have seen how to handle uncertainty in (convex) SDPs, which are symmetric eigenvalue problems. To illustrate the ideas, let us consider a (nonconvex) unsymmetric eigenvalue problem. For example, consider the following robust synthesis problem:

where p(-) denotes the spectral radius, and M(x, A) is affine in x and rational in A € A, given in an LFR (We note that the fact that M is affine in x is done without much loss of generality.) Robustness analysis Let us fix A, x, and seek sufficient condition for

Let us first "forget" uncertainty and apply Lagrange relaxation to the problem of checking if We obtain that A is an upper bound on spectral radius if and only if there exist X > 0 such that The above condition is an LMI in X, which implies that we can apply the robust SDP methodology to handle the case when M is uncertain. We obtain that A is an upper bound on maximal spectral radius if there exist P > 0 and 5 such that

For fixed x, minimizing A a (quasi-) convex problem (over S and P) results in an upper bound on the worst-case spectral radius.

18

El Ghaoui and Niculescu

Robust synthesis We now seek a sufficient condition, A being fixed, so that holds. Now x becomes a variable. Due to possible cross products between x and P in the previous sufficient condition, the problem is not convex, contrary to what happens in the robust SDP case. Let X = A2P, Y = P"1, and write problem as a conic complementarity problem

Condition (1.27) is true if and only if the optimum is A 2 n. The problem we obtain is a generalization of the SDP, in the sense that the SDP is a special complementarity problem (see section 1.5). In general, the above problem is not convex; however, several efficient techniques can be used (see Part V). One of these techniques, described in Chapters 16 and 17, relies on a simple linearization of the quadratic objective and leads to a sequence of SDPs. In some cases, this approximation technique is guaranteed to yield the global optimum of the original problem (see [128]).

1.4.5

On quality of relaxations

Associated with the relaxation methods comes the need to evaluate the quality of the approximations involved. In the context of a robust SDP, the first results are due to Nemirovski [39]. To illustrate the ideas involved in this analysis, consider the following robustness analysis problem: Given I symmetric n x n FI, ..., F/, and a scalar p, check if

This seemingly simple problem is NP-hard. If we apply Lagrange relaxation, we obtain that a sufficient condition for property PI (i.e., p = I in (1.28)) to hold is

To evaluate the quality of the above approximation, we seek the smallest p such that condition (1.29) implies Pp holds. Clearly, such a p satisfies p > 1; the closer to 1, the better the approximation is. Nemirovski has proved that p < min(v/n, %//)• In terms of control, we interpret this result as a bound on the quality of the approximation to a kind of "/z-analysis" problem [108], with affine dependences and Euclidean-norm bounds. More refined results, including for other norm bounds, are underway [290].

1.4.6

Link with Lyapunov functions

There is more than one connection between the approach proposed here and Lyapunov theory; we will briefly hint at these connections here.

1.4. Robust SDP

19

Lyapunov functions and Lagrange multipliers Lyapunov functions can be interpreted in the context of Lagrange relaxations, which establish an interesting link between two "central" methods in control and optimization. For the sake of simplicity, we first consider the problem of establishing stability of the LTI system where x 6 Rn is the state, and A 6 R n x n is a (constant) square matrix. It is well known that the system is stable if and only if This shows that the stability problem is a robustness analysis problem (with respect to "uncertain" parameter s). To attack it using Lagrange relaxation, we first use an LFR of the problem. Introduce p = sx] our robustness analysis problem is equivalent to checking if Let us eliminate the "uncertain parameter" s by noting that Let us apply a Lagrange relaxation idea now. A sufficient condition for stability is that there exists a "multiplier" matrix S > 0 such that A simple computation leads to the classical Lyapunov condition which corresponds to the Lyapunov function candidate V(x) = xTSx. Let us give another example of the "Lagrange relaxation" method for dynamical systems. Consider a system with delayed state The system is stable independent of the delay r > 0 if An equivalent stability condition is that

This time, we may relax the two matrix inequality constraints, which after some matrix calculations yields the sufficient (possibly conservative) condition for stability: There exists symmetric matrices 5, Z such that 5 > 0, Z > 0, and

To this condition, we can associate a "Lyapunov function candidate," termed as a Lyapunov-Krasovskii functional, defined as

20

El Ghaoui and Niculescu

Quadratic stability The robust decision methodology can also be applied in the quadratic stability analysis for an uncertain system subject to unknown-but-bounded uncertainty:

where A satisfies the assumptions of section 1.3.1 and A is defined via the LFR

(Recall that we assume that this LFR is well posed over A.) To analyze stability, the idea is to apply Lagrange relaxation twice. In a first step, we "forget" uncertainty and apply Lagrange relaxation to the "frozen" system. This results in a sufficient condition for stability: There exists a matrix 5 such that

This condition is known as a quadratic stability condition, since it requires the existence of a quadratic Lyapunov function proving stability of the uncertain system. In general, the above condition is still not tractable, despite the fact it is a convex condition on S. However, the relaxed problem above now belongs to the realm of robust SDP. We can apply Lagrange relaxation again. We obtain that if there exist matrices S and W such that

holds, where $A is the linear operator that characterizes the uncertainty set A, then the system is quadratically stable. The above problem is an SDP, with decision variable S (linked to the quadratic Lyapunov function proving quadratic stability), and W. When A is the set defined by (1.14), the above condition is the well-known small gain theorem (see [64] and references therein). The above idea can be extended in various ways, as illustrated in some other chapters (Parts III and IV) of this book. For example, we may consider parameter-dependent, or frequency-dependent Lyapunov functions, based on so-called IQCs. Stochastic Lyapunov functions Stochastic Lyapunov functions have been introduced by Kushner [237], and their use in the context of LMI optimization has been introduced in [64, 120, 121]. Consider stochastic system of the form

where w is a scalar standard Wiener process (the "derivative" of this process can be interpreted as a random perturbation on the state-space matrix, hence the name multiplicative noise given to such perturbations). The above systems are useful models in a number of areas, most notably finance [256]. The second moment X = E(xxT) obeys the deterministic equation

1.5. Algorithms and software for SDP

21

The stability of this system hinges on the existence of a linear Lyapunov function

where 5 > 0, which proves stability of the deterministic system (1.33) (evolving over the positive semidefinite cone). It can be shown that the (second-moment) stability is guaranteed if and only if

A stochastic Lyapunov function (in the sense of Kushner), namely corresponds to this Lyapunov function. One of the advantages of LMI formulations to stochastic control problems is that once an LMI formulation is known, it is possible to apply Lagrange relaxation in the case when the "data" (for example, the "nominal" matrix A in the above) is uncertain.

1.5

Algorithms and software for SDP

Algorithms for SDPs are now the subject of intense research in the field of optimization, especially since 1995. This progress should have a great impact on LMI-based control as algorithms become more efficient. Recent methods to solve a general SDP can be roughly divided in two classes: interior-point methods and nondifferentiable optimization methods. These algorithms have been implemented, and several interior-point software packages are now generally available.

1.5.1

Facts about SDP and related problems

We briefly recall some important results on SDP and related problems.

Duality The problem dual to problem (1.37) is

where Z is a symmetric NxN matrix and Ci is the ith coordinate of vector c. When both problems are strictly feasible (that is, when there exist x, Z, which satisfy the constraints strictly), the existence of optimal points is guaranteed [296, Theorem 4.2.1], and both problems have equal optimal objectives.

Related convex problems We list several problems that are related to SDPs. Second-order cone programming problems are special cases of SDPs. They take the form

where Ci € R n i X m , di € R n< , &i € R m , /» 6 R, i = 1,..., L. The special structure of SOCPs can be exploited in algorithms to yield much faster methods than using generalpurpose SDP methods; see, e.g., [296, 240, 9], as well as Chapter 3 of this book.

22

El Ghaoui and Niculescu Determinant maximization (max-det) problems are of the form

where G is an affine function taking values in the set of symmetric matrices of order p. Max-det problems are convex optimization problems and can be solved in polynomial time with interior-point methods [296, 402]. Chapter 2 of this book provides more details on max-det problems. Other interesting applications of max-det problems are reviewed in [406]. Generalized eigenvalue problems (GEVPs) are generalizations of SDPs, of the form minimize A subject to Here, A, B, and C are symmetric matrices that depend affinely on x e Rm. GEVPs are not convex but are quasi convex and amenable to interior-point methods similar to those used for SDPs; see [63, 295].

1.5.2

Interior-point methods

One way to view interior-point methods is as a reduction of the original problem to a sequence of differentiable, unconstrained minimization subproblems. Each subproblem can then be efficiently solved, perhaps approximately. These methods can be described in the context of self-concordant barrier and potential functions; this is the point of view taken in [296, 404]. Primal-dual interior-point methods Without doubt, the most efficient interior-point methods for LP are the primal-dual interior-point methods [423]. When a primal-dual interior-point method is used, each subproblem can be described by a linear least squares problem or, equivalently, a system of linear equations. These methods are very efficient for four reasons: (1) they enjoy robust convergence from infeasible starting points, with guaranteed complexity when implemented accordingly (though typically this is not done in practice); (2) they converge rapidly, even if the solutions are not unique; (3) the linear systems to be solved are sparse (or can be made sparse by, for example, using standard techniques to handle dense columns in the constraint matrix); consequently, very large problems can be solved; (4) the inherent ill conditioning in the linear systems to be solved does not, in practice, greatly limit achievable accuracy. However, this ill-conditioning does mean that iterative methods (e.g., conjugate gradient) are not generally practical to solve the linear systems: the standard workhorse is sparse Cholesky. Since 1995, there has been a flurry of papers on primal-dual interior-point methods for SDPs; a complete bibliography on this subject is outside the scope of this chapter. Relevant references include [297, 8, 233, 286, 338, 391,190, 404, 5, 219]. In these methods, the subproblem to be solved is again simply a system of linear equations, although this is not always equivalent to a least-squares problem. As in LP, these methods converge very rapidly; various complexity results are now available; software packages have also recently become available. However, primal-dual methods are not necessarily the best at exploiting sparsity in the data. The linear system that must be solved at every iteration invariably has dimension m, the number of variables in the primal form of the SDP. Therefore, with present technology, primal-dual methods are highly efficient if m is not too large (say, less than 1000), but if it is much larger, they are not practical unless sparsity can be exploited. See [38] for a primal alternative to primal-dual methods that

1.5. Algorithms and software for SDP

23

quite effectively exploits sparsity in SDPs arising in applications to graph theory. Even this approach, however, is limited by the size of linear systems that can be solved. In what follows, we briefly describe primal-dual path-following methods, following Kojima et al. [233]. Optimality conditions and central path When the primal problem (1.2) and its dual (1.35) are both strictly feasible, then we may write the optimality conditions

where £ is an affine subspace:

We can interpret the above conditions as complementarity conditions over the positive semidefinite cone, similar to those arising in LP [233]. The convexity of the original SDP is transported here into a property of monotonicity of £, which is crucial for the algorithm outlined below to converge globally. We introduce a parameter p, that modifies the above conditions, as follows:

Under strict feasibility conditions, the above system defines a unique primal-dual pair (F M ,Z M ). Solving (1.40) for decreasing values of the parameter \JL leads to an optimal solution. The path followed by (F M ,Z M ) as /u —» 0 is referred to as the central path, which explains the qualification "path following." For each value of /^, the conditions (1.40) can be interpreted as the optimality conditions for a differentiable problem involving a barrier function. Thus, the above description of interior-point methods can be interpreted in the context of barrier functions [296, 404]. Predictor and corrector steps Assume that we have found a pair (Ffc, Z<-) that is close to the central path (i.e., approximately solve (1.40)). To find the new iterate, two steps are taken. In the predictor step, we seek to approach the optimum, that is, to satisfy ZF = 0. In the corrector step, we seek to return close to the central path by ensuring ZF = fj,I. The search directions 6x, 6Z for each step (predictor and corrector) are computed by linearization of the constraint ZF = 0 (predictor) or ZF = p,I (corrector) around the current value of (x,Z). Each step thus gives rise to a linear system in the elements of 6x, 6Z. Note that ZF can be linearized in a number of ways, depending on the specific method used. The step lengths may be chosen small enough so that the new iterates will be guaranteed to be close to the central path again (and thus, strictly feasible). Here, closeness is measured using a proximity function such as

(Many alternatives are possible; see [233].) Depending on the proximity function used, we may define long-step and short-step strategies (see, for example, [286]). There are two different approaches to initialization that have become standard. One approach uses an "infeasible start," i.e., points that satisfy the cone inequality constraints

24

El Ghaoui and Niculescu

but not the linear equality constraints. If the original problem turns out to be either primal or dual infeasible, iterates with large norms will typically be generated, alerting the user that the problem is most likely either primal or dual infeasible. The other approach is to construct a modified problem for which feasible starting points are known, which then guarantees that either the original problem will be solved or a certificate of infeasibility will be found [298]. Although this approach is very attractive theoretically, it remains unclear whether it is better than the infeasible-start approach in practice. In fact, the whole issue of feasibility is much more subtle for SDP than it is for LP. An interesting discussion on this subject can be found in [257].

1.5.3

Nondifferentiable optimization methods

When the number of variables is sufficiently large, and sparsity cannot be effectively exploited, it may be advantageous to consider alternative methods to solve SDPs. For simplicity, we only discuss an LMI feasibility problem, where a solution to a set of LMIs is sought; such problems appear routinely in control, e.g., in stability analysis. An LMI feasibility problem is easily formulated as an eigenvalue minimization problem of the form

where F(x) is an affine function taking values in the set of real symmetric matrices and Amax denotes largest eigenvalue. (To see this, note that there exists an x such that F(x) < 0 if and only if the minimum of Amax(F(a;)) is below zero.) The function f ( x ) = Amax(F(a;)) is a nondifferentiable function; yet it is convex and all theoretical tools from convex analysis apply to this function: In particular, first-order and secondorder generalized derivatives can be explicitly quantified and numerically implemented for this class of functions [305, 397, 373, 310, 137]. In Chapter 3 of this book, an extensive review of existing nondifferentiable optimization methods (NDOMs) is proposed ranging from early subgradient methods to the modern bundle methods. Chapter 3 also includes some very recent developments on second-order bundle methods as well as numerical results of a new type: a mixed polyhedral-semidefinite bundle method enables us to check the stability of polytopic linear differential inclusions with 1000 state variables (which implies more than m = 500,000 variables). A second-order bundle method is shown to bring in practice an asymptotic superlinear rate of convergence for smaller problems. Some perspectives are sketched to see how the latter methods could be made efficient on large-scale problems.

1.5.4

Software

Software for solving SDPs is generally available, both freely on the Internet and commercially. (To our knowledge, most of these SDP algorithms use interior-point methods.) Much of this has become available quite recently. Here is a (probably incomplete) list: • The Matlab control toolbox, developed by Cabinet and others [153] uses Nesterov and Nemirovski's projective algorithm [296, 293]. • The code sdpsol, described in Chapter 2 of this book, is a parser/solver for SDPs and max-det problems, sdpsol uses the code SP, developed by Vandenberghe and Boyd [402]. http: //www. Stanford. edu/"boyd/group_index. html The toolbox Imitool, developed by El Ghaoui, Nikoukha, and Delebecque [126],

1.6. Illustrations in control

25

and later by Commeau, is an interface to the codes SP, SDPpack, and SDPHA. http : //www . ensta . f r/~gropco/ • The code SDPpack, outlined in Chapter 3 of this book, implements a primal-dual algorithm due to Alizadeh, Haeberly, and Overton [8] and Alizadeh et al. [6]. http : //www . cs . nyu . edu/cs/f aculty/overton/sdppack/sdppack . html • The code SDPA: developed by Fujisawa, Kojima, and Nakata; the underlying algorithm is described in [233]. ftp : //ftp . is . t itech . ac . jp/pub/OpRes/sof tware/SDPA

• The code SDPHA: developed by Brixius, Sheng, and Potra; for a description of the underlying algorithm, see the web page. http : //www . cs . uiowa . edu/brixius/SDPHA

• The code SDPT3: uses the algoritm of Todd, Toh, and Tiituncii [391]. http : //www . math . nus . sg/"mattohkc/SDPT3 . tar . Z • The code SeDuMi: developed by Sturm. http : //www . crl . mcmaster . ca/ASPC/people/sturm/sof tware/

1.6

Illustrations in control

This book abounds with examples of applications of the LMI framework to control problems. In this section, we seek to provide complementary examples.

1.6.1

Optimal control and SDP duality

Notions central to SDP theory, such as duality, optimality conditions, sensitivity analysis, etc., have usually interesting interpretations for the analysis of (optimal) control problems. For example, the perturbation theory for the general SDP recently developed by Shapiro [372, 373] and Bonnans, Cominetti, and Shapiro [55] can be used for the analysis of Lyapunov and Riccati inequalities and equations encountered in optimal control. The following example shows how SDP duality and optimality theory can be used in an optimal control problem. In a variety of stochastic control problems, a Riccati equation of the following form [422] arises: where Q = QT > 0, .R > 0, and A, B, G are matrices of appropriate sizes. When G = 0, we recover a standard Riccati equation (which can be solved via the eigenvectors of a Hamiltonian matrix). In the general case G ^ 0, the problem of computing the solution is nontrivial. Under hypotheses that are physically reasonable, detailed below, we can solve the above equation using an SDP as follows. Consider the SDP

We form the problem dual to (1.43) (see section 1.5):

26

El Ghaoui and Niculescu

The above problem is strictly feasible if and only if the underlying stochastic system is stabilizable in a stochastic sense (with static state-feedback control law u = Kx, K — UX~1}. Likewise, it is easy to show that the primal problem (1.43) is strictly feasible when this system is detectable. Let us make the hypothesis of primal and dual strict feasibility. Since both primal and dual problems are strictly feasible, it follows that a set of feasible primal-dual variables (P, X, U, S) are optimal if and only if

from which we get

With R > 0, we obtain U = KX, where K = -R~1BTP. The first equation of (1.45) then writes Finally, we observe that X > 0: if Xz = 0 for some z, then Uz = 0 (from U = KX). Since U, X are feasible for (1.44), we obtain

which implies z = 0. The conclusion is that P satisfies the Riccati equation; furthermore, the control law u = Kx is stabilizing, since X > 0, and

For other references where duality theory and control links are exploited, see, for example, [413]. The above approach can be extended to some other classes of nonstandard Riccati equations, involving, for example, jump linear systems [3], problems with indefinite control weighting matrix R [324], etc.

1.6.2

Impulse differential systems

Impulse differential systems are systems subject to state discontinuities (or jumps). These systems are hybrid, in the sense that they obey to a mixture of discrete- and continuoustime behaviors. See [27, 358] for references and also [439] for an application in the context of telecommunications. In classical control, hybrid systems are very common: they arise when one controls a (continuous-time) system with a discrete-time controller. A typical impulse linear differential system is of the form

where A, M are given square n x n matrices, and r», i = 0, 1, . . . , is a strictly increasing infinite sequence of time instants, which determine when the (otherwise smooth) trajectory suddenly jumps from x(r~) to X(T*). By proper choice of the impulse matrix M, one seeks to guarantee stability, using small impulses at properly chosen time instants. (In the following, we assume the jumps are separated by a fixed interval T.)

1.6. Illustrations in control

27

The stability analysis of impulse systems is nontrivial in general. In [439], Yang and Chua have devised (based on an earlier result by [358]) a stability criterion, using quadratic Lyapunov functions. It turns out that it is easy to rewrite this criterion in the form of an LMI condition, as follows. If there exists S > 0 such that

then the system is asymptotically stable. The above conditions can be interpreted in terms of the ellipsoid S — {£ | £TS£ < 1}: the first LMI says that, during the "continuous-time behavior," the trajectory never escapes the (possibly bigger) ellipsoid {£ | £TS£ < e~2aT}; the last two conditions ensure that the jump brings the state back to £, which implies some kind of invariance for S. Note that we do not assume the continuous system to be stable, since the decay rate a may be negative; in some sense, we are requiring that the "degree of stability" of the discrete-time behavior dominates that of the continuous-time one. The above condition is an LMI in 5 for fixed A, a, which shows that the problem of checking stability (based on quadratic stability) is tractable, provided a plane search (on A, a) is used. The main advantage of the LMI formulation is that it allows for designing an appropriate impulse M: with the change of variables U = SM, the above conditions become convex in both S, U. With an LMI (sufficient) condition stability in hand, we can address a variety of control problems. For example, we can extend the results to nonlinear and/or uncertain systems; for example, we may allow the matrix A to be unknown-but-bounded, or consider the case when the system without impulses is LTI with a sector-bounded nonlinearity, and apply Lur'e stability conditions [64]. Likewise, it is possible to deal with optimal (state-feedback) control problems using the framework used in section 1.6.1.

1.6.3

Delay differential systems

Delay systems arise in many applications, especially when a physical or biological transport phenomenon occurs (see, e.g., Murray [288] for interesting examples in biology). A typical problem in control of delay system is to analyze stability as a function of delay, where the delay is viewed as a parameter of the system. The problem is still largely open [100]; even in the linear case, and with multiple (noncommensurable) delays, the problem is NP-hard (see Toker and Ozbay [395] for a precise statement and results). We may distinguish delay-independent stability analysis (i.e., no information on the delay size is used) and delay-dependent stability analysis (based on this information). Sufficient stability conditions, both delay independent or not, have been devised using LMIs; see, e.g., [428, 248, 143]. Important tools in stability analysis are based on extensions of classical Lyapunov theory, leading to Krasovskii functionals or Razumikhin functions [186]. The Razumikhin theorem has been linked to the S-procedure of section 1.2.4 in [301]. A survey of results and further comments are given in [301]. In section 1.4.6, we have given a delay-independent stability result, based on Lagrange relaxations, which is related to a particular choice of Lyapunov-Krasovskii functional candidate. This approach can be extended to a delay-dependent result, using an embedding technique (comparison principle) as follows. Consider the stability analysis problem for the delay-differential system (1.30). We introduce the distributed delay system

28

El Ghaoui and Niculescu

and argue that the above system is an embedding of the original system (1.30), in the sense that stability of the above guarantees that of (1.30) (in fact, the above is an integration of the original equations over one delay interval). Using Laplace transforms and straightforward computations, we obtain a sufficient condition for stability:

It is then straightforward to apply the Lagrange relaxation technique (see section 1.4.6) and obtain the following delay-dependent relaxed LMI condition:

Computing a lower bound on the largest allowable delay is a generalized eigenvalue optimization problem (GEVP). Refinements of these results can be found in [300]. To the above LMI condition corresponds a Lyapunov-Krasovskii functional candidate of the form

The above ideas can be extended to multiple delays (seen as independent uncertain parameters). A different idea, leading to similar results, is based on the IQC theorem, combined with the Kalman-Yakubovich-Popov lemma [143] (see Chapter 6 for more on IQCs). The basic idea here is to view the delay as an uncertainty that satisfies a given IQC (see section 1.3.1) and then apply IQC analysis. It is possible to refine the results using a more computationally intensive LMI condition, using an idea similar in some sense to the one exploited in the context of robustness analysis with multipliers (see Chapters 6 and 11, for example). Indeed, for the system (1.30), we may exhibit a Lyapunov-Krasovskii functional that yields a necessary and sufficient condition for stability [207]. Unfortunately, this functional cannot be parametrized using a finite-dimensional function (as is the case when working with LTI systems and quadratic Lyapunov functions); rather, it depends on several matrix functions (defined over an interval [—r 0]). It is possible to discretize this parametrization [179] and obtain LMIs whose degree of conservativeness is arbitrarily low, with an inversely proportional computational effort.

1.6.4

Open- loop and predictive robust control

Robust optimization opens up the possibility of designing robust open-loop controllers for uncertain systems. This idea is suggested in [66], and we briefly elaborate on it now. Consider, for example, a dynamical system

where A*, B* € R nxn ", Cfc 6 R n » x n , and D* are unknown-but-bounded time- varying matrices. We are given a time horizon T > 0 and denote by u and y the vectors (UQ, . . . ,WT-I) € RTn" and (yo> • • • >2/T-i) € RTn". We also define three polytopes (or ellipsoids) U C RTriu, AQ, and y € Rrn", sets of admissible controls, initial conditions, and outputs constraints, respectively.

1.7. Robustness out of control

29

The problem is to find a u e U, if any, such that for every XQ € AQ, the resulting output satisfies y £ y. If the state-space matrices are exactly known, the problem is equivalent to a simple linear program. In the case when the state-space matrices are uncertain, the robust counterpart can be approximated using the robust decision methodology. In terms of optimization, this versatile framework gives rise to challenging large-scale problems. One way to avoid the curse of dimensionality is to construct ellipsoids of confidence for the future state and optimize (over a few time steps) the control vector. This idea is similar to the one used in model predictive control. The construction of the corresponding ellipsoids of confidence can be done using the techniques proposed shortly in section 1.7.2, in a recursive manner.

1.7

Robustness out of control

The title of this section is borrowed from that of a talk by J. C. Doyle [105]. The LMI formalism provides a unitary framework, not only for various control problems, but also in other fields, where LMIs have recently emerged as a useful tool. We give some examples of this interplay, concentrating on areas where control-related notions (e.g., robustness) seem to play a role that goes sometimes unrecognized yet, such as interval linear algebra.

1.7.1

LMIs in combinatorial optimization

SDPs are now recognized as yielding very efficient relaxations for (hard) combinatorial problems [161]. It turns out that these SDP relaxations for combinatorial problems can be obtained as special cases of the methodology outlined in section 1.4. (The connection between 5-procedure and such relaxations is presented in [66].) In turn, the SDP relaxations open up interesting perspectives for handling combinatorial problems with uncertain data, in view of the tools developed for the robust SDP. We illustrate this via a simple example. Consider the NP-hard problem

where W is a given symmetric matrix. When W assumes some special structure, the above problem is known in the combinatorial optimization literature as "the maximum cut" (MAX-CUT) problem [161, 154]. This problem is a robustness analysis problem, as defined in section 1.2.3. First we note that, without loss of generality, we may assume W > 0 and rewrite the problem as a robust synthesis problem:

Let us now apply our results, with

for appropriate matrices F, L, R, and with A = diag(<$i,..., <5j). The uncertainty set is

The above set satisfies the assumptions made in section 1.3.1.

30

El Ghaoui and Niculescu

We obtain easily the following SDP approximation, yielding an upper bound on the original problem: (1.47)

minTrS subject to W < 5, 5 diagonal.

In [161], Goemans and Williamson have also proposed an SDP relaxation (i.e., an upper bound) of the problem. The technique is actually a special case of the general technique described in section 1.4.3. Introduce the variable X = 68T and rewrite problem (1.46) in the equivalent form

Relaxing the nonconvex constraint rankX = 1, we obtain an SDP that is exactly the "dual" (in the sense of SDP; see section 1.5.1) of (1.47). As shown in [161], the above relaxation is the most efficient currently available (in terms of closeness to the actual optimum in a certain stochastic sense). Once an SDP formulation of the original problem is known, we can consider problems in which the "data matrix" W is uncertain, starting from the "nominal problem" (1.47). (In practice, such problems often arise in telecommunications, and the matrix W is estimated from statistics.)

1.7.2

Interval calculus

A basic problem in interval computations is the following. We are given a function f from R/ to Rm and a set confidence V for 6 € R* in the form of a product of intervals. We seek to estimate intervals of confidence for the components of x = f (6) when 6 ranges to V. (Sometimes, f is given in implicit form, as in the interval linear algebra problem; we return to this case in the next section.) Obtaining exact estimates for intervals of confidence for the elements of solutions x is in general NP-hard (see [345, 346] for the interval linear algebra case). One classical approach to this problem resorts to interval calculus, introduced by Moore [287], where each one of the basic operations (+, —, x, /) is replaced by an "interval counterpart," and standard (e.g., LU) linear algebra algorithms are adapted to this new "algebra." Many refinements of this basic idea have been proposed, but the algorithms based on this idea have in general exponential complexity. (For control applications of interval calculus, see, e.g., [99].) Robust SDP can be used for this problem as follows. Assume we can describe f explicitely as a rational function of its arguments; from Lemma 1.2, we can construct (in polynomial time) an LFR of f in the form Assume first that we seek an ellipsoid of confidence for the solution in the form where XQ € Rn is the center and P > 0 defines the "shape" (our parametrization allows for degenerate, "flat" ellipsoids to handle cases when some components of the solution are certain). We seek to minimize the "size" of £ subject to f(<$) € £ for every 8 € *D. Measuring the size of £ by TrP (other measures are possible), we obtain the following equivalent formulation of the problem: minimize TrP subject to

1.7.

Robustness out of control

31

The above is obviously a robust SDP problem (in variables xo,P) for which an explicit SDP counterpart (approximation) can be devised, provided V takes_the form of ajgeneral) ellipsoidal set. (A typical set is a product of intervals II [^ <5i], where £i; 8i are given.) The above method finds ellipsoids of confidence, but it is also possible to find intervals of confidence for the components of f (6) by modifying the objective of the above robust SDP suitably (for example, if we minimize the (1,1) component of the matrix variable P instead of its trace, we will obtain an interval of confidence for the first component of when 6 ranges to T>).

1.7.3

Structured linear algebra

Many linear algebra problems involve matrices with a given structure (in identification, for instance, we obtain least squares problems in which the coefficient matrix has a Toeplitz structure). One of the most active research areas in numerical linear algebra pertains to exploiting perturbation structure, both for error analysis and for computing the solution. Consider, for example, a polynomial interpolation problem leading to a linear system of the form Ax = 6, where x contains the polynomial coefficients and A is a matrix with a Vandermonde structure. In some applications, the interpolation points are not precisely known. This results in an "uncertain Vandermonde system" of the form A(6)x = b(6), where A(<5), b(£) are matrix- valued (resp., vector- valued) polynomial functions of a perturbation vector 6. A structured linear equation is a model of the form

where A(6), b(<5) are (affine) functions of a parameter vector 6 € R', and x is the variable. We assume that 6 is unknown-but-bounded; for example, ||£||oo < 1- (Note that (1.50) is not an equation but a model of the way an equation is subject to perturbations.) The above model can be represented in the linear-fractional form:

A number of problems pertain to the structured equation (1.50). Robust least squares One such problem is robust least squares, where we seek a vector x that minimizes the worst-case residual

Such a vector can be termed robust in the sense that it minimizes the effect of perturbations on the "performance" (measured by the residual). The above problem has many implications in control, notably in robust identification. The robust least-squares problem is a robust SDP problem, since the nominal problem, i.e., the classical least-squares problem, is a special case of an SDP. Details on this approach are given in [125].

32

El Ghaoui and Niculescu

Condition estimates and accurate solutions Another problem is to measure the sensitivity of a candidate solution x. A way to quantify this sensitivity (or backward error) is to compute

It is also interesting to compute a vector x that minimizes (an upper bound on) the above quantity. Such a vector is a solution for (1.50) with minimum sensitivity with respect to the perturbation vector 6; in this sense, it is accurate (with respect to the given perturbation structure and bounds). As seen in the next paragraph, the above problems are robust SDP problems, which can be attacked using the methodology described previously. Interval linear algebra The basic problem of interval linear algebra is a slight variation on the problem considered in section 1.7.2 and is as follows. We are given matrices A e R nxn , b € Rn, the elements of which are only known within intervals; in other words, [A b] is only known to belong to an "interval matrix set" U. we seek to compute intervals of confidence for the set of solutions, if any, to the equation Ax = b. To attack this problem we seek an ellipsoid of confidence in the form (1.48), where P > 0. For simplicity, we assume first that A, b are given; using the S-procedure, we find that a necessary and sufficient condition for £ to be an ellipsoid of confidence is that there exists a scalar multiplier r such that

This condition is easily converted in an LMI in XQ,P. Prom then on, we can use the robust SDP methodology to handle the case when A, b are uncertain. For further reference, we detail the result in the special case when [A b] is an uncertain matrix subject to "unstructured additive perturbations." Assume

where [A b] € R nx ( m + 1 ) and p > 0 are given. In this case, our results are exact and yield an ellipsoid center of the form

(We assume that crmin([^4 6]) > p; otherwise the ellipsoid of confidence is unbounded. Except in degenerate cases, this guarantees the existence of the inverse in the above.) Structured total least squares The so-called structured total least squares (STLS) problem, which arises in many areas, e.g., in identification and other inverse problems in engineering [96], is as follows. We are given A(6) e R mxn , b(<$) € Rm, two (algebraic) functions of a parameter 6 € R'. The STLS problem is Here, we are interested in finding one value of the "perturbation" vector 6 that makes the vector x feasible. In this sense, the problem belongs to the class of "backward error

1.8. Perspectives and challenges

33

analysis" problems: A perturbation is sought such that the linear equation above becomes consistent. The STLS problem can be solved (at least approximately) by resorting to robust optimization as follows. A scalar p is an upper bound on the objective in (1.53) if there exists x € Rm such that

Assume that we have computed an ellipsoid of confidence of center XQ and shape matrix P, denoted by £p, for the set of solutions of the above constraint. The quantity min p subject to £p is not empty is an upper bound on the STLS problem. The above problem turns out to be a GEVP, with variables p, XQ, PIn the unstructured (additive) perturbations case, we recover the classical notion of the total least-squares solution developed by Golub and Van Loan [171]. The upper bound computed via GEVP is exact; the ellipsoid of confidence can be shown to be reduced by the singleton, and the optimal center is given by (1.52), where p = crmin([A &]).

1.7.4

Robust portfolio optimization

A portfolio is a collection of financial assets, say, pi,... ,pn. Each asset Pi has a (random) return over a given horizon, denoted TJ. If x» is the proportion of the total value of an asset p invested in the ith asset, then the return of the portfolio over the given horizon is p = XiTi H V xnrn = rTx. The return of each asset is, of course, unknown a priori. However, it is possible to deduce from past measurements the values of the mean // = E(r) and covariance matrix V = E(rrT). The standard mean-variance portfolio choice problem, introduced in the fifties by Markowitz [263], is to minimize the variance of the asset's return, xTFx, subject to a given level a of mean return, a = HTX. This problem can be written

(In the above, e stands for the n-vector with every component equal to one.) The problem above belongs to the class of a convex quadratic problem (a subset of SDP problems), and, as such, can be very efficiently solved using interior-point methods. One may then consider variations of this problem and solve them using the LMI framework. We obtain one interesting example when we assume that the mean and covariance matrices are unknown-but-bounded. This leads to portfolio choices that are robust with respect to uncertainty in mean and covariance. Another interesting perspective is to formulate the problem in a dynamic (multistage) setting and apply robust control techniques to devise a robust portfolio update.

1.8 Perspectives and challenges 1.8.1

Algorithms and software

Efficient algorithms and software should exploit the structure of the given problem in the most systematic way. For example, many problems in control lead to SDPs of the form (1.2), where the coefficient matrices are sparse. A sparse SDP solver would thus be

34

El Ghaoui and Niculescu

extremely useful. Interior-point methods seem difficult to adapt directly to sparse LMI problems. To understand why, recall from section 1.5.2 that at each step we have to find (approximate) solutions to a complementarity equation of the form

where /z is a parameter, and F (resp., Z) is the primal (resp., dual) variable. Sparsity of F does not imply that of Z, and this makes the use of sparse computations uneasy to propagate throughout the iterations. In contrast, the nondifferentiable methods, described in section 1.5.3, and in more detail in Chapter 3, seem promising for sparse SDPs. In particular, first-order (bundle) methods only require the computation of a (few) subgradient(s), and this can be done with sparse calculus. One problem with such methods is slow convergence, but we believe that using second-order information (such as t/-Lagrangians) , it will be possible to increase convergence speed at a controlled increased cost per iteration (recall that with the C/-Lagrangian method, we work on a subspace of dimension equal to the multiplicity at the optimum, and this number is often much smaller than the size of the problem). More theoretical studies are needed to make this statement anything else than a belief. In particular, more work is needed on generic multiplicity at the optimum for SDPs, along the lines of [310, 326, 7]. Sparsity is not the only structure of which we may take advantage. SDPs that pertain to matrices with Toeplitz or Lyapunov, etc., structure can often be solved much faster than general SDPs. (Structure is also crucial information to use for solving linear equations.) This issue is raised in detail in Chapter 4 of this book.

1.8.2

Nonconvex LMI-constrained problems

A number of problems arising in control and other fields are not convex. We encounter the following classes. (We only discuss feasibility problems, to simplify; see Part V of this book for more on these problems.) Bilinear -matrix inequality (BMI) problems are of the following form: Find x such that where F,, Gij are given symmetric matrices. This class contains much of combinatorial optimization; a typical example of such a problem is the MAX-CUT problem mentioned in section 1.7.1. Rank-constrained problems (RCPs) are of the following form: Find x such that where F(-),G(-) are two affine matrix functions (with F(-) symmetric), and k is a given integer. Rank-constrained problems arise in, e.g., reduced-order feedback control (see [108, 124, 128]). Conic complementarity problems (CCPs) consist of solving

where K, is a convex subset of the cone of semidefinite matrices, and F, Z are variables. When the problem is "monotone" (that is, when the set /C is a "self-dual" cone), then the above problem can be solved using primal-dual interior-point methods for SDPs. For

1.8. Perspectives and challenges

35

general cones /C, however, the above problem is hard. Conic complementarity problems are generalizations of the famous linear complementarity problems and arise in many situations, including robust control (see [128]); see section 1.4.4 for an example. The above problems are in general NP-hard. As we will see, they are polynomially equivalent, in the sense that we may transform one into the other in polynomial time. This equivalence can be used to devise efficient relaxation algorithms. First, we may rewrite any BMI in the form of an RCP, as follows. Define and rewrite the constraint in (1.54) as

We obtain a rank-constrained problem. Dropping the rank constraint yields an LMI condition, which provides us with a relaxation of the original BMI problems, very similar to the one used in combinatorial optimization (see section 1.7.1). Conversely, any rank-constrained problem is a BMI and a CCP. First we may assume without loss of generality that the matrix G(x) is square (UG x UG] and symmetric. Next, the constraint RankG(:r) < k is equivalent to saying that TrZF = 0, F > 0, Z > 0, where

for some M € R n c X f e , 5 e R n G X n °, S > I, U, V of appropriate size. Note that the constraint TrZF = 0 can be written as a BMI, which shows the polynomial equivalence between BMIs and RCPs. By solving the CCP

where /C denotes the cone of positive semidefinite pairs (F,Z) of the form (1.57), with 5 > /, we solve the original RCP. The conic complementarity formulation seems very similar to the problem we encounter in primal-dual interior-point methods for solving SDPs. This suggests an algorithmic approach to such problems, where we directly apply a primal-dual interior-point method to the problem. (Of course, this method will converge globally when the complementarity problem is monotone.) For nonmonotone problems, the method will at least provide a local minimum.

1.8.3

Dynamical systems over cones

The developments made in section 1.4.6 seem to indicate that to analyze a (possibly stochastic) dynamical system with a vector state x € Rn in the context of LMIs, it is fruitful to consider the (equivalent) system describing the evolution of the matrix X = XXT (or its expected value, if a stochastic system is at hand). By definition, the resulting (deterministic) dynamical system leaves the positive semidefinite cone invariant. This motivates a closer study of dynamical systems of the form

where is a deterministic map, with the property that the positive semidefinite cone is invariant for (1.58). There seem to be a number of (apparently unexplored) interesting

36

El Ghaoui and Niculescu

connections between several optimization and control notions, all related to systems of the form above. For example, for the LTI system x = Ax, the resulting cone-invariant system takes the form (1.58), with (f>(X) = ATX + XAT. A (linear) Lyapunov candidate of the form V(X) = TrXS (with S > 0 given) proving stability of system (1.58) can be interpreted as a primal-dual gap (X is the "primal" variable, S the "dual" one) and is related to a quadratic Lyapunov function proving stability of the original (state-vector) system. The complementarity condition of the form (1.40) encountered in SDP, where a product of primal and dual variables are inverse of each other, can be related to the notion of adjoint system. Indeed, we note that if X(t) is a trajectory of (1.58), with X(t) invertible for every t > 0, then the inverse matrix S(t) = X(i)"1 satifies the adjoint system

It would be interesting to see how the above connections can be exploited in control and optimization problems and if the study of cone-invariant systems (1.58) can be made for cones other than the positive semidefinite cone.

1.8.4

Quality of relaxations for uncertain systems

As seen in section 1.4.5, the problem of estimating the quality of relaxations is crucial in robust SDP, and this opens several challenging new problems. Good quality estimates would be of great importance for robust control, as they would enable us to bound the conservativeness of well-known tools such as //-analysis. Consider for example the uncertain dynamical system

where the n x n matrix A is an algebraic function of its argument 6 € R*. A sufficient condition for stability is quadratic stability (see section 1.4.6):

In turn, the above condition can be relaxed using robust SDP techniques, leading to a (tractable, sufficient) LMI condition of the form (1.32). In some cases, it is possible to assess the possible conservativeness of this LMI condition over the quadratic stability one. For example, if A is affine in 6, then Nemirovski's results apply (see section 1.4.5), leading to a quality bound (in the sense defined in section 1.4.5) of min(y/n, %//)• The more general case when the matrix A is nonlinear (algebraic) in the parameter vector 6 can perhaps be attacked using a singular perturbation technique, as follows. Represent A in the LFR format

As usual, we assume this LFR to be well posed over A. Now let c > 0, and associate to system (1.59) the following "augmented" system:

where A is the set defined in (1.15).

1.9. Concluding remarks

37

Under certain conditions classical in singular perturbations theory, involving (robust) stability of the matrix DgpA — / for A e A,3 the above system can serve as an approximation to system (1.59). For every e > 0, the above system is equivalent to an uncertain linear system, with affine dependence on the uncertain parameters 6. For this system, the conservativeness estimate for quadratic stability is mm(\/n -f AT, >//), where N = 7*1 H hrj. Since by definition TJ > 1, we obtain the conservativeness estimate \/ The challenge here is to see how this analysis can be extended to the case of arbitrarily small e in order to give a rigorous conservativeness estimate.

1.9

Concluding remarks

In this chapter, we sought to give an overview of the LMI method. We proposed a unitary framework based on optimization ideas for addressing a large class of decision problems with uncertainty. We have indicated some possible research directions where this method could be applied. In control theory, to borrow words from Doyle, Packard, and Zhou [108], LMIs play the same central role in the postmodern theory as Lyapunov function and Riccati equations played in the modern, and in turn various graphical techniques such as Bode, Nyquist and Nichols plots played in the classical. Thus, the LMI approach constitutes a basis for a "postmodern control theory" [108], which allows robust, multicriteria synthesis. Benefits of both classical (PID) or modern (LQG, HOQ) approaches are combined: reduced number and clear physical interpretation of design parameters and simplicity of numerical solution, even for multivariable problems. In addition, this approach provides guarantees for the design and allows (possibly competing) specifications. In our opinion, LMI-based control methods have reached a certain degree of "academic" maturity, even though many control areas are yet unexplored. Both theoretical grounds and efficient algorithms are now available, and this should lead to more and more industrial applications, some of which are in progress. We strongly believe that the LMI approach could be applied to many other engineering problems, especially design problems with robustness and multicriteria issues. We presented some of these problems with an optimization point of view. The reason is that the language of optimization seems the best suited as a common framework for these problems. In some ways, we could widen the scope of the above quote from [108] by saying that LMIs (and convex optimization ideas) should play the same central role in "postmodern" engineering as numerical linear algebra (least squares, eigenvalue problems, etc.) played in the past 20 years.

Acknowledgments We are grateful to Michael Overton and Francois Oustry, who completely rewrote the section on algorithms (section 1.5). The first author is greatly indebted to Arkadi Nemirovski for many stimulating discussions on the subject of robust optimization.

3

If A is polynomial in 6, then Dqp can be constructed as a strictly upper triangular matrix, and jpA — / is stable for every A € A.

This page intentionally left blank

Part II

Algorithms and Software

This page intentionally left blank

Chapter 2

Mixed Semidefinite-Quadratic-Linear Programs Jean-Pierre A. Haeberly, Madhu V. Nayakkankuppam, and Michael L. Overt on 2.1 Introduction In this chapter, we consider mixed semidefinite-quadratic-linear programs (SQLPs). These are linear optimization problems with three kinds of cone constraints, namely, the semidefinite cone, the quadratic cone, and the nonnegative orthant. This chapter outlines a primal-dual path-following method to solve these problems and highlights the main features of SDPpack, a Matlab package that solves such programs. Furthermore, we give some examples where such mixed programs arise and provide numerical results on benchmark problems. A mixed SQLP has the form

where y € Rm and

The first set of constraints (2.2) consists of L semidefinite cone constraints of size n i , . . . , r i L , respectively. The second set of constraints (2.3) consists of M quadratic

41

42

Haeberly, Nayakkankuppam, and Overton

cone constraints of size pi, . . . ,PM, respectively. The final constraint (2.4) consists of po ordinary linear inequality constraints. We use y instead of x as the optimization variable, with maximization objective bTy instead of minimization objective CTX as used elsewhere in this book in order to make the notation used here more compatible with the SDPpack software to be described later in this chapter. It is convenient to rewrite the problem as follows:

where We write

and

The dual optimization problem is

where the dual variables are Z (fc) € S"*, k = 1,...,L, z(fc) € R pfc , and £<*> € R, fc = 1,...,M, and 2;^°^ € Rp°. Note that the constraints Z^ > 0 are semidefinite constraints. We write and

Let us denote the primal objective (2.1) by P(y) and the dual objective (2.9) by ,z^). If y and (Z,z,z^}, respectively, satisfy the primal and dual feasibility constraints, it immediately follows that

Since Z and F are both positive semidefinite, TrZF > 0; this follows using the Cholesky factorizations Z — RTR and F = SPS as

2.1. Introduction

43

where || • \\F stands for the Probenius norm. Likewise, Cauchy-Schwarz shows that zTf > 0, since

Also, of course, (z^}Tf^

> 0 is implied by z^ > 0, /<°> > 0. Thus we have

with equality (a zero duality gap) if and only if TrZF = 0, zTf = 0, and These three conditions are called complementarity conditions and may be summarized as

Furthermore, again using Cholesky, we see that if TrZF = 0, then the matrix product ZF must itself be zero (since SRT = 0), and therefore (since Z and F are symmetric) that ZF + FZ = 0. (Recall that this is a block-diagonal matrix equation.) Likewise, we can also elaborate on the second complementarity condition. The condition for equality in Cauchy-Schwarz shows that if zTf = 0, then

Following [2], this condition may be conveniently expressed as

where eq is a block vector whose fcth block is and the block-diagonal matrix

has blocks defined, for a e R, a e Rp, by the "arrow matrix"

Finally, the third complementarity condition ( z ) T f = 0 implies the componentwise condition (z (0) )fc(/ (0) )fc = 0, k = 1, . . . ,po, since z(0) > 0 and /(0) > 0. Letting ei = [1, . . . , 1]T, we write this as

This much is just linear algebra. To go further we make use of a well-known result from optimization theory [296] stating that if strictly feasible primal and dual variables exist (points that satisfy all the inequalities strictly), then the optimal primal and dual objective values are equal and that there exist primal and dual variables achieving these optimal values. Thus, the complementarity conditions must hold at the (optimal) solutions. In what follows we make this strict feasibility assumption and we also assume, mainly for convenience, that there are no redundant equality constraints in the dual formulation of the problem.

44

2.2

Haeberly, Nayakkankuppam, and Overton

A primal-dual interior-point method

The SQLP described in section 1 can be efficiently solved with interior-point methods. Of the many flavors of interior-point methods [297, 423] , we briefly describe the algorithm implemented in the public domain package SDPpack [6]. This is a primal-dual pathfollowing algorithm based on the "XZ+ZX" direction of [8] (to be translated here as "ZF+FZ" given the notational differences), using a Mehrotra predictor-corrector scheme. The optimality conditions are completely determined by the three complementarity conditions together with the primal and the dual constraints. Let svec denote a map from symmetric, block diagonal matrices (with block sizes ni, . . . , UL) to column vectors, so that svec(F) G R N , where

and such that TrZF = svec(^)Tsvec(F) for all symmetric block-diagonal matrices Z, F. Then we may abbreviate the primal slack linear constraints (2.5)-(2.7) by

where

and

The dimensions of As, A9, and A^ are, respectively, m x AT, m x m x PQ. The dual equality constraints (2.10) are then written

Furthermore, the complementarity conditions can be written

where eq and e^ were defined in section 2.1. We collect the primal and dual equality constraints and a relaxed form of the complementarity condition, called a "centering"

2.2. A primal-dual interior-point method

45

condition, in the nonlinear system of equations

where p, > 0 is a parameter. It is well known that, for all p, > 0, this nonlinear system of equations has a unique solution satisfying the cone inequality constraints, and the set of such solutions is called the central path [296]. As /u —> 0, this central solution converges to a (not necessarily unique) primal-dual solution of the SQLP. The algorithm we now outline uses Newton's method to approximately follow the central path as n —» 0. For a given approximate primal-dual solution (y, F, /, /(°\ Z, z, z^) and central path parameter n, Newton's method defines a "search direction" by the linear system of equations

where the Jacobian J of H is the matrix

Here the symbol * is the "symmetrized Kronecker product" [8]: For symmetric blockdiagonal matrices ^4, B, X (with block sizes

This linear system is not solved directly but via an efficient method exploiting block Gaussian elimination that is outlined in the next section. Once primal and dual "search directions" are computed, it is necessary to compute steps along these to the relevant cone boundaries, so that subsequent iterates will respect the inequality constraints. This requires an eigenvalue computation of a block-diagonal symmetric matrix to determine the step to the boundary of the semidefinite cone, the solution of M quadratic equations to determine the step to the boundary of the quadratic cone, and an ordinary ratio test to determine the step to the boundary of the positive

46

Haeberly, Nayakkankuppam, and Overtoil

orthant. The minimum of these step lengths determines the maximum step that can be taken. This procedure is used to determine one step length for the primal variables and another for the dual variables. At a more detailed level, the algorithm uses a predictor-corrector scheme. This requires the following steps, which constitute one interior-point iteration: • Given a current iterate (y,F, /, / ° Z , z, z ) , compute a predictor search direction by solving the linear system (2.13), with /u = 0, for (Aypred, AFpred, A/pred, (A/(°)) pred , AZ pred , Az pred , (Az (0) ) pred ). • Compute the primal and dual step lengths to the cone boundaries along this direction, apred and /?pred, respectively, so that

and

are each on the boundary of the product of the semidefinite cone, the quadratic cone, and the positive orthant. • Compute 3

where v = 53*11 nk + Y^k=i(Pk + 1) + Po- This expression for p, is known as Mehrotra's formula. • Recompute the relaxed complementarity conditions in the right-hand side of (2.13) using the Mehrotra formula for /z and including second-order corrections, by substituting

for the last three components in the right-hand side of (2.13). Solve the linear system again for a corrector search direction. • Compute primal and dual step lengths a and 0 along the corrector search direction and update the primal and dual variables accordingly. The step lengths are chosen so that the new iterates satisfy the inequality constraints. Furthermore, they are never set to a value larger than unity. This can be done by taking the step length to be the smaller of unity and a fixed fraction (say, 0.999) times the step to the boundary of the cone. If a unit primal (dual) step is taken, the new iterate also satisfies the primal (dual) equality constraints. The choice of the sequence of values for p, greatly influences the performance of the algorithm. The method described above is Mehrotra's predictor-corrector scheme [276], originally proposed for linear programming, which significantly reduces the number of

2.3. Software

47

interior-point iterations needed to compute a solution of given accuracy. The predictor step helps calculate a target value for /z, while the corrector step includes some second-order terms lost in the linearization of H in the predictor step. For a detailed interpretation of Mehrotra's predictor-corrector scheme, see [423, pp. 193-198].

2.3

Software

In this section, we first discuss some implementation details involved in solving the linear system arising in each interior-point iteration. Then we highlight the main features of SDPpack (Version 0.9 Beta),4 a package (based on Matlab 5.0) for solving SQLP, written by the authors together with F. Alizadeh and S. Schmieta.

2.3.1

Some implementation issues

The Newton system (2.13) admits a reduction to a compact Schur complement form via block Gauss elimination. Let

so that

Eliminating (svec(AF), A/, A/°) in (2.13), the matrix J transforms to

and on further elimination of (svec(AZ), Az, Az 0 ), we arrive at the m x m Schur complement matrix in a linear system involving only the variable Ay. The Schur complement matrix M can be computed as the sum of the individual Schur complements for the semidefinite (Ms), the quadratic (M9), and the linear (M*) parts. Moreover, Ms and Mq can be computed blockwise. Writing A5fc' (likewise A^fc' ) to mean the columns of Aa (likewise A q ) corresponding to the fcth block, we can verify that M = Ma + Mq + Me, where

The contribution to the (i, j ) entry of Ma from the fcth semidefinite block can be computed as F/ ' • Vj ' , where the n^ x njt symmetric matrix V ' is the solution to the 4

Available from the SDPpack web page: http://www.cs.nyu.edu/overton/sdppack.

48

Haeberly, Nayakkankuppam, and Overtoil

Lyapunov equation where #Jfc) = Z^F^ + F^ Z^ . The standard method to solve this equation is via the spectral factorization F^ = Pdiag(A)P:r, where P is an orthonormal matrix of eigenvectors and A = [Ai, . . . , A n J T , a vector of eigenvalues. The Lyapunov equation is solved by computing

where the symbol o in (2.17) denotes the Hadamard (or componentwise) product. The contribution from the fcth quadratic block to Mq can be computed efficiently by observing that the arrow matrix arw(^*\^^) is the sum of a diagonal matrix and a matrix of rank two.

2.3.2 Using SDPpack We now describe how to use the main routines of SDPpack (Version 0.9 beta); in what follows, » denotes the Matlab prompt. The problem data and an initial starting point are stored using structures: Matlab variables A.s, A.q, A.I b C.s, C.q, C.I blk.s, blk.q, blk.l X.s, X.q, X.I

Data 6

y

Z.8,Z.q,Z.l

For example, the following lines of Matlab code set up a randomly generated SQLP: » » » »

setpath; '/. add SDPpack subdirectories to Matlab path blk.s = [ 5 5 5 ] ; blk.q = [10 10]; blk.l = 30; m = 10; sqlrnd; 7. set up a random, strictly feasible SQLP

The constraint matrix A . s may be set up using the routine makeA, which assumes that the matrices F» are stored using Matlab's cell arrays. Here is an example with two semidefinite blocks and one quadratic block: » » » » » » » »

clear blk.s = [33]; blk.q = 3; m = 2; b = [1 2] ' ; negF{l} « speye(6); negF{2}(l:3,l:3) = sparse([l 2 3 ; 2 4 5 ; 3 5 6 ] ) ; negF{2}(4:6,4:6) = sparse([l 0 0 ; 0 2 2 ; 0 2 3 ] ) ; A.s = makeACnegF, blk.s); '/. creates the 2 x 12 matrix A.s C.s(l:3,l:3) = sparse (ones (3, 3) );

» C.s(4:6,4:6) = sparse (zeros (3, 3) );

» A.q - [1 2 3; 4 5 6]; » C.q = [1 -1 -1]';

2.3. Software

49

Note that when there is more than one semidefinite block (as in the example above with two blocks), the constraint matrices negF{i} containing —Fi (i = 1,... , m) and the cost matrix C. s containing FQ must be stored in sparse format. The routines import (export) can be used to load (save) ASCII data from (to) a file. Once the problem data are set up, the optimization parameters, which are stored in the structure opt, must be assigned appropriate values; these may be initialized to their default values with the routine » setopt; The next step is to provide a starting point, i.e., the variables X, y, and Z. Generally, it suffices to provide an infeasible starting point, i.e., points for which (2.5)-(2.7) and (2.10) need not necessarily hold, although the cone inequalities (2.8) and (2.11) must be satisfied. A default starting point (usually infeasible) may be generated with the routine » init; which initializes the primal variable y to zero, the primal slack variables Z.s, Z.q, Z.I to scalefac * (7,e q ,e^), and the dual variables X.s, X.q, X.I to scalefac * (I,e 9 ,e^), where scalefac is a user-defined scale factor set to a default value by setopt. Again, if there is more than one semidefinite block, the matrices X. s and Z. s must be stored in the sparse format. The solver can then be invoked with the Matlab function call >>

[X, y, Z, iter, compval, feasval, objval, termflag] = ... fsql(A, b, C, blk, X, y, Z, opt);

or by the driver script » sql;

which optionally prints out additional summary information5 about the accuracy to which the problem was solved and the CPU time consumed. In the later iterations, it is common to see Matlab warnings about ill-conditioned systems of linear systems. This ill conditioning is a common phenomenon in interior-point methods. The warnings may be suppressed by the Matlab command warning off. When the algorithm terminates, a termination flag indicates the cause for termination, which is usually one of the following: • A solution of the requested accuracy has been found. In this case, the sum of the primal infeasibility, the dual infeasibility and complementarity at the computed solution satisfies both an absolute and a relative tolerance, specified in the opt structure. • An iterate (F, /, /^) or (Z,z,z^) violates a cone constraint—a normal phenomenon that usually happens due to numerical rounding errors in the last stages of the algorithm. In this case, the previous iterate, which does satisfy the cone constraints, is returned as the computed solution. • The Schur complement matrix is numerically singular. If this occurs in the early stages of the algorithm, the most likely cause is linear dependence in the rows of the constraint matrix A. Preprocessing can be done (cf. section 2.3.3) to detect and rectify this situation. 5

References to "primal" and "dual" in the output (and in the User Guide) actually refer to the "dual" and "primal" problems in the notation used here.

50

Haeberly, Nayakkankuppam, and Overton • The algorithm fails to make sufficient progress in an iteration. The progress made in an iteration is deemed insufficient (based on parameters in the opt structure) if the step causes an inordinate increase in the primal or the dual infeasibilities without a commensurate decrease in complementarity. Again, the previous iterate is returned as the computed solution. • The norm of (F, /, /^) or of (Z, z, z^) exceeds a threshold (defined by the user in the opt structure). Primal (dual) unboundedness is an indication of dual (primal) infeasibility.

For full details, we refer the reader to the SDPpack User Manual [6]; see also Table 2.1. Due to the loop overhead in Matlab, each block diagonal matrix is stored as one large sparse matrix, instead of as individual blocks. This has the advantage that a sparse block in such a matrix is efficiently handled using Matlab's sparse matrix operations. On the other hand, a dense block suffers the penalty involved in sparse matrix access. The loop overhead in Matlab also hinders the effective use of sparsity in the formation of the Schur complement matrix. For instance, in the truss topology design problems, most of the blocks in every constraint matrix are completely zero, and hence do not contribute to the Schur complement at all (see (2.15)), but this could not be exploited effectively. A stand-alone C code currently under development incorporates some techniques to improve performance on sparse problems.

2.3.3

Miscellaneous features

The package provides several support routines that include the following: • The routines scomp, pcond, and dcond to examine complementarity and degeneracy conditions [7, 132]. These may be used, in particular, to check whether the computed primal and dual solutions are unique. • The routine makesql to generate random SQLPs with solutions having prescribed "ranks." This routine lets the user specify — for each semidefmite block k, the rank of F^ (likewise — for each quadratic block k, whether f^ (likewise z^) lies in the interior of the quadratic cone, or on its boundary, or is identically zero; and — for the linear block, the number of components of /^ (likewise z^) that are positive. Since nondegeneracy automatically implies bounds on ranks of the solution [7], this routine is particularly useful in generating test problems whose solutions have specified degeneracy and complementarity properties. The routine preproc to examine the consistency of the constraints. If they are consistent, the redundant constraints, if any, are eliminated via a QR factorization of the constraint matrix A (see (2.14)). The specialized solvers dsdp and Isdp (based on the "XZ method" [190, 235]) that exploit the special structure of problems arising in combinatorial optimization (for example, MAX-CUT relaxations and the Lovasz ^-function). In these applications, the number of semidefinite blocks L = 1 and the matrices F^\ . . . , F$ are all of the form e»ef or CjcJ + e^ef . Here e, stands for the unit vector with 1 in the ith location and zeros everywhere else.

2.4. Applications

51

More information about any routine is available by typing help from the Matlab prompt.

2.4

Applications

We consider two examples of SQLP. Robust least squares. This problem is taken from [125]. Given B, AB 6 R r x s (r > s] and d, Ad G R r , we want to compute the unstructured robust least squares solution x G Rs that minimizes the worst-case residual

where the perturbation [AB Ad] is unknown, but the bound p on its norm is known. It is shown in [125] that this problem can be formulated using quadratic cone constraints. Adding componentwise bound constraints, say, x m j n < x < xmax, converts this into an SQLP with a quadratic and a linear component. Clock mesh design. This circuit design problem, taken from [405], deals with the propagation of a clock signal in integrated circuits. A rectangular mesh with fixed grid points is given. The interconnecting wire segment between grid points i and j is assumed to be a rectangular strip of a fixed length, but variable width dij. The electrical characteristics of each wire segment are modeled as a 7r-segment, i.e., a grounded input capacitance, a grounded output capacitance, and a connecting conductance, all of which are assumed to depend affinely on the design parameters d^. A synchronous clock signal (of fixed frequency) applied to a fixed subset of the grid points has to be propagated to the entire mesh with minimal delay. There is an inherent trade-off between the dominant delay (see [405] for the definition) and the power consumption of the circuit. The goal is to optimally choose the widths d^ of the rectangular wire segments so that the power consumption of the circuit is minimized for a fixed value of the dominant delay that we are willing to tolerate. It is shown in [405] that this can be formulated as a semidefinite program (SDP). Physical limitations on the width of the segments further impose bound constraints of the form dmin < d^ < dmax, which results in an SQLP with a semidefinite and a linear block. Now consider a slight variation of this problem: The objective is the same as before, but now we have a rectangular mesh whose grid spacing is not fixed. Here the wire widths are fixed, but the spacing li between successive columns and TJ between successive rows of grid points are variables with /min < li < lmax and 7"min < TI < rmax. Let us further impose a restriction on the maximum allowable size of the circuit by imposing a bound on the length of the diagonal of the rectangular mesh. This last bound constraint is a convex quadratic constraint, thus resulting in an SQLP with semidefinite, quadratic, and linear components.

2.5

Numerical results

In this section, we present numerical results on a variety of test problems, both randomly generated and from applications. The latter come from control theory (LMI problems) and truss topology design. The randomly generated problems include general dense problems, the unstructured robust least squares problem, and problems with specific sparsity structure such as those arising from the clock mesh design application.

Haeberly, Nayakkankuppam, and Overtoil

52

Table 2.1: Legend for termination flag tf. tf -2 -1 0 1 2 3 4 5 6 7

Description || (Z, z , z ( ° ) ) \ \ exceeds user-defined limit. ||(F, /, /(°))|| exceeds user-defined limit. A solution of requested accuracy has been found. (Z, z, 2^°)) or (F, /, /(°)) is outside the cone or on its boundary. F has a nonpositive eigenvalue. Schur complement matrix is numerically singular. Insufficient progress. Primal or dual step length is too small. Exceeded maximum number of iterations allowed. Failed data validation checks.

In the tables, the first column (#) is the problem number, the second column is the dimension of the primal slack variable

m is the dimension of y, iter is the number of interior-point iterations, r^, rp, and rc are, respectively, the dual infeasibility norm (see (2.10)), the primal infeasibility norm (see (2.5)-(2.7)), and the complementarity Tr(ZF) + zTf + (z(0))T/(°) evaluated at the computed solution. The quantity scalef ac is the factor by which the default starting point ((/, e q ,e£),0, (7,e 9 ,e^)) is scaled. When scalef ac is omitted, it means that the default value of 100 was used. The quantities rp, r(Z, z, z^) in the tables, are provided only for the LMI and the truss topology design problems,6 and not for any of the randomly generated ones. All the runs were performed on a Sun Spare Ultra II workstation (200 MHz CPU with 192 MB of memory). Unless explicitly noted otherwise in the tables, all the problems were solved with SDPpack's options set to their default values. Random dense problems. Table 2.2 shows results for problems with only semidefinite constraints. Table 2.3 shows results for problems with only quadratic cone constraints. In Table 2.2, each problem has three blocks of equal size; the five runs tabulated there correspond to block sizes of 20, 40, 60, 80, and 100. Each problem in Table 2.3 also has three blocks of equal size; the block sizes for the five runs are 50, 100, 150, 200, and 250. Clock mesh design. This problem from [405] was described in section 2.4. These are problems with one semidefinite block and a linear block. The constraint matrices 6

Available from the SDPpack web page.

2.6. Other computational issues

53

Table 2.2: Randomly generated with semidefinite blocks only.

# II dim 1 630 2 2460 5490 3 9720 4 5 15150

m 60 120 180 240 300

iter 14 13 15 13 14

Td

-13 -13 -12 -12 -10

rp -14 -13 -13 -13 -13

rc -14 -12 -11 -12 -07

CPU sees 1.88e+01 1.57e+02 8.45e+02 2.13e+03 5.72e+03

tf 0 0 0 0 4

Table 2.3: Randomly generated with quadratic blocks only.

# II dim 1 150 2 300 3 450 4 600 5 750

m 50 100 150 200 250

iter 8 8 8 8 8

T
rp -15 -14 -13 -14 -13

Tc

-10 -13 -10 -11 -10

CPU sees 5.40e-01 1.29e+00 3.48e+00 7.00e+00 1.28e+01

Table 2.4: Clock mesh design problems (scalefac = 1000): #2 a 12 x 12 mesh.

# II dim 1 3609 2 14989

m 144 312

iter 16 21

Td

-13 -12

rp -16 -16

rc -09 -09

tf 0 0 0 0 0

is an 8 x 8 mesh, and

CPU sees 1.92e+02 2.84e+03

tf 0 0

are extremely sparse, and the spatial mesh adjacency relation they describe gives them a banded structure; the results are shown in Table 2.4. Robust least squares. This problem from [125] was described in section 2.4. Numerical results for this problem class are shown in Table 2.5. Truss topology design. These problems are pure SDPs that arise in truss topology design. The constraint matrices here are extremely sparse even for the moderately sized problems. The results are summarized in Table 2.6. LMI problems. These problems are pure SDPs that arise in control applications. On the problems where the final complementarity residual is unsatisfactory (see Table 2.7), the dual iterates indeed get large, indicating that the primal problem may be infeasible or weakly infeasible [257].

2.6

Other computational issues

Several enhancements to the basic interior-point iteration of section 2.2 have been proposed, and some used with success, in linear programming. We are concerned with extending some of these techniques to SQLP, namely, • Gondzio's multiple centrality corrections [172], and • Mehrotra's higher-order corrections [275].

Haeberly, Nayakkankuppam, and Overton

54

Table 2.5: Unstructured robust-least squares problems (scalef ac = 1000). # II dim 1 1103 2 2203 3 3303 4 4403 5 5503

m 102 202 302 402 502

iter 7 7 7 7 7

Td

-13 -12 -12 -12 -12

rp -13 -13 -13 -12 -12

rc -09 -09 -09 -09 -09

CPU sees 3.32e+00 1.74e+01 4.93e+01 1.02e+02 1.86e+02

tf 0 0 0 0 0

Table 2.6: Truss topology design. # || dim 1 19 2 331 3 91 4 37 5 1816 6 901 7 451 8 6271

m 6 58 27 12 208 172 86 496

iter 10 12 15 11 15 28 29 18

rd -14 -13 -14 -13 -10 -07 -06 -10

rp -15 -14 -15 -16 -14 -13 -13 -14

Tc

-11 -11 -08 -11 -09 -07 -08 -11

2>(Z,*,*W) 9.00e+00 1.23e+02 9.11e+00 9.01e+00 1.33e+02 9.01e+02 9.00e+02 1.33e+02

?(y)

9.00e+00 1.23e+02 9.11e+00 9.01e+00 1.33e+02 9.01e+02 9.00e+02 1.33e+02

CPU sees 4.20e-01 4.81e+00 1.86e+00 6.10e-01 7.28e+01 5.85e+01 2.40e+01 1.13e+03

tf 0 0 4 0 0 4 4 0

The main motivation behind considering these enhancements for SQLP is the ratio of the cost of forming and factorizing the Schur complement matrix relative to the cost of a backsolve. For SQLP, this ratio is typically much larger than for LP. Thus, having expended the effort involved in forming and factorizing the Schur complement matrix, it is worthwhile to improve the quality of the search direction by solving multiple linear systems with the same coefficient matrix but different right-hand sides. Both the above schemes can be extended to SQLP and are described in detail elsewhere [184]. In our preliminary experiments, adapting Gondzio's multiple centrality corrections to SQLP reduces iteration count and CPU time roughly to the same extent as reported for LP. On the other hand, Mehrotra's higher-order corrections reduce the number of iterations and the overall CPU time significantly; this observation is different from LP experience, where the overall CPU time is typically not reduced.

2.7

Conclusions

In this chapter, we have described mixed SQLPs. These are linear optimization problems with mixed cone constraints, namely, the semidefinite cone, the quadratic cone, and the nonnegative orthant. This problem class includes pure semidefinite programming, convex quadratically constrained quadratic programming, and linear programming as special cases. We have presented a primal-dual interior-point algorithm for such problems and have described the salient features of SDPpack (Version 0.9 beta), a software package that implements this algorithm. A couple of examples have been given to illustrate how such mixed programs arise in practice, and the numerical results shown here demonstrate the performance of SDPpack on several classes of test problems. Finally, we have briefly listed some enhancements of interior-point algorithms for LP; their extensions to SQLP have been implemented in a new stand-alone C code currently under development.

55

2.7. Conclusions Table 2.7: LMI problems from HQQ control (scalefac = 1000).

# II d I 41 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

51 51 51 51 51 51 51 51 66 97 120 178 227 273 51

m 13 13 13 13 13 13 13 13 13 21 31 43 57 73 91 13

iter 12 14 16 15 17 33 11 15 17 27 23 25 29 30 18 17

Td

-08 -09 -05 -06 -04 -02 -05 -05 -07 -05 -07 -08 -04 -07 -06 -07

rp -12 -11 -12 -10 -12 -11 -11 -11 -13 -07 -07 -07 -09 -09 -08 -13

rc -10 -08 -10 -11 -09 -08 -07 -09 -11 -05 -04 -01 -02 -03 -02 -11

V(Z,z,zW) -2.03e+00 -l.lOe-fOl -5.70e+01 -2.75e+02 -3.63e+02 -4.49e+02 -3.91e+02 -1.16e+02 -2.36e+02 -1.09e+02 -6.60e+01 -2.00e-01 -4.44e+01 -1.30e+01 -2.39e+01 -2.36e-f02

P(y)

-2.03e+00 -1.10e+01 -5.69e+01 -2.75e+02 -3.63e+02 -4.49e+02 -3.91e+02 -1.16e+02 -2.36e+02 -1.09e+02 -6.59e+01 -1.78e-01 -4.44e+01 -1.30e+01 -2.40e+01 -2.36e+02

CPU sees 7.60e-01 9.20e-01 9.90e-01 9.10e-01 1.04e+00 2.00e+00 6.90e-01 9.10e-01 1.05e+00 2.30e+00 3.23e+00 5.31e+00 8.73e+00 1.39e+01 2.23e-f01 1.08e+00

tf 2 4 1 2 1 2 1 1 2 -1 -1 -1 2 5 -1* 2

Acknowledgments We thank Pascal Gahinet, Arkadii Nemirovskii, and Lieven Vandenberghe for supplying us with the LMI, the truss, and the clock mesh design problems, respectively.

This page intentionally left blank

Chapter 3

Nonsmooth Algorithms to Solve Semidefinite Programs Claude Lemarechal and Frangois Oustry 3.1 Introduction Many practical applications of LMI methods require the solutions of some large-scale semidefinite programs (SDPs). In this chapter, we describe nonsmooth algorithms to solve SDPs, with two motivations in mind: - to increase the size of tractable eigenvalue optimization problems, - to improve asymptotic properties of the algorithms. Both properties cannot be achieved simultaneously. Yet a good trade-off must be identified between problem size and speed of convergence; and this happens to be possible with algorithms for nonsmooth optimization, which in addition offer a lot of flexibility. After a short introduction of the objects necessary for later understanding, we outline in section 3.2 the existing nonsmooth algorithms that can be used "off the shelf" in the present framework. Among them are bundle methods, which have recently been specialized to exploit the particular structure of the problem; section 3.3 is devoted to them, while their specialized versions form section 3.4. Finally, even though the problem belongs to the realm of nonsmooth optimization, it lends itself to second-order analysis. Accordingly, second-order algorithms can be applied, which bring good asymptotic properties. This is the subject of section 3.5, which contains a condensed introduction to the necessary theory, together with corresponding implementations. Comparative numerical illustrations are given in section 3.6, and we conclude with a section on possible future developments. Original structure: Convexity. Let us write an SDP as minimize CTX

subject to

where c G Rm is the cost vector and F is an affine mapping from 57

58

Lemarechal and Oustry

The key characteristic of the problem is the constraint-function / := Amax o F, more precisely, the function X i-> Xmax(X). This latter function is convex: it is the support function of the compact convex set

This can be seen from the Rayleigh variational formulation

where • is the standard scalar product in «Sn. Interior-point methods transform the original problem (3.1) to a smooth one by adding some structure, thereby obtaining polynomiality; but some information is then lost, which is useful for local convergence. By contrast, nonsmooth analysis directly handles the problem with its "pure structure." The role of convex analysis was first emphasized by R. Bellman and K. Fan in [37] and developed further in [195, 247], among others. A first step is to express the subdifferential dAmax(X), and this comes easily from elementary convex analysis: indeed (see, for example, Proposition VI. 2. 1.5 in [194]) (3.2) shows that this subdifferential is the face of Cn exposed by X. In other words, let r be the multiplicity of A max (X) and let Q be an n x r matrix whose columns form an orthonormal basis of the corresponding eigenspace. Then This formulation will be used in the so-called rich-oracle methods. For poor-oracle methods we will rather refer to an equivalent form:

Once dAmax is thus characterized, df is derived by simple chain rules. In fact, denote by F* : Sn -* Rm the adjoint of T\

Then the subdifferential of / at x is just the image by F* of the subdifferential of Amax at X = F(x):

Remark 3.1. A further advantage of this approach via nonsmooth analysis is that it does not rely on affinity of the mapping A: smoothness suffices. When A is not affine, f is no longer convex, but its essential composite structure is preserved. Namely, (3.5) can be applied by replacing the linear operator F* by F*:

Note that the subdifferential (of Amax, and hence of /) is highly unstable: it changes drastically when the multiplicity of Amax changes. As will be seen in section 3.3, the whole idea of so-called bundle methods can be seen as constructing good approximations of a much more stable object than the subdifferential of a convex function, the approximate subdifferential. For e > 0, the £-subdifferential d£f(x) is defined by

3.1. Introduction

59

It is not hard to see that, for the maximum eigenvalue function,

and for / = Amax o A the same chain rule still applies:

From (3.6) we see that d£Xmax(X) is no longer a face of Cn: it is the intersection of Cn with the half-space {V € Sn : V • X > Xmax(X) — e}. In particular, almost all matrices in d£Xmax(X) have rank n (for e > 0), whereas the rank of matrices in d\max(X) is at most the multiplicity r, which is 1 for almost all X; this gives an idea of the big gap existing between these two convex sets. SDP and maximum eigenvalue minimization. For the sake of simplicity, most of the algorithms we present here are designed to solve the unconstrained problem

Note that nonsmooth algorithms can be used to solve (3.1) as well, using exact penalty or more sophisticated techniques as in [244]. Remark 3.2. In some cases, for example, when the diagonal of F(x) is fixed [189] or free [307], transforming (3.1) to (3.8) can be done efficiently because an efficient value for the penalty coefficient is known. Such is often the case when (3.1) is the relaxation of a graph problem; see [178] or [160] for more on the origins of these problems. We also mention that algorithms for nonsmooth optimization extend in a straightforward way to the more general problem

where Amax = AI > A2 > • • • > An are the eigenvalues in decreasing order and r < n is a given integer. In fact, the above function is convex as well. Poor, rich, and intermediate oracles. For a resolution algorithm to work, it is necessary to compute information regarding (3.1) or (3.8) for given matrices X = F(x): Typically, it can be the spectrum of X, or its QR decomposition, etc. We find it convenient to consider this computation as done by an oracle that, having X as input, answers the required information. A distinguished feature of nonsmooth optimization is that it can be satisfied with "poor oracles," which compute only the maximal eigenvalue, together with just one associated eigenvector. In view of (3.5) and (3.4), this provides a subgradient of Amax or of /. Needless to say, poor oracles are advantageous for problems with large n. The price to pay is that the corresponding algorithms usually have modest speed of convergence. On the other hand, nonsmooth optimization accommodates "intermediate" oracles, computing a number r € {1, . . . , n} of largest eigenvalues and associated eigenvectors (with r possibly varying along the iterations). This supposedly improves performances; it is thus possible to balance sophistication of the oracle with convergence speed of the minimization algorithm.

60

3.2

Lemarechal and Oustry

Classical methods

The nonsmooth optimization problem (3.8) can be solved by any existing algorithm for nonsmooth optimization, ignoring the particular structure enjoyed by the function Amax. Such methods use a poor oracle and are fairly classical. They can be divided into two classes, based on totally different motivations. Subgradient methods. The methods of the first class have developed in the 1960s, mainly in Russia and Ukraine (see the review [335]). Let the oracle have computed at Xk a subgradient gk of the form F*vvT for some eigenvector v; see (3.5) and (3.4). Then the basic subgradient method computes

For the moment, Wk is the identity Im in Rm. As for the step size tk > 0, known formulas are

(the second formula is valid only if the optimal value / of (3.8) is known). The above method is purely "Markovian," in the sense that iteration k depends only on the differential information of / at xk. In the 1970s, N. Z. Shor inserted a memory mechanism to update the preconditioner Wk in (3.9). As explained in his monograph [375], the basic object for this is space dilation along a vector £ = & € Rm, i.e., the operator that multiplies by a = ak > 0 the component along £ of each vector in Rm. The well-known ellipsoid algorithm is obtained with & = 9k and specific values for a*, tk. On the other hand, taking & = 9k — 9k-i results in the so-called r-algorithm, lesser known but rather efficient in many cases. Pure cutting-plane method. The methods of the second class are all based on approximating / by the polyhedral (or cutting-plane) "model"

Here, f ( x i ) = A max (F(xj)) and The next iterate Xfc+i is computed with the help of this information, and the whole success of the method will rely on the optimizer's ability to appropriately weight the elements of the bundle. Now, the pure cutting-plane method uses the simplest strategy: take for xk+i a minimizer of /&; this amounts to solving the linear program with an additional variable r:

A simple example, due to A. S. Nemirovski and reported in [194, section XV.1.1], illustrates the instability of this scheme: in some situations, billions of iterations may be needed to obtain just one-digit accuracy.

3.3. A short description of standard bundle methods

61

Method of analytic centers. A recent strategy to cope with the above instability is as follows. In the graph-space Rm x R, consider the set Pk defined by

here Ik := mini/i(zi) is the best function value returned by the oracle up to the fcth iteration. Because fk is polyhedral, Pk is a convex polyhedron; write it in terms of abstract affine inequalities as

In the pure cutting-plane method, Zk+i = (xk+i, fkfak+i}) is the lowest point in Pk. A more stable point was defined in [379]: the analytic center of P^, i.e., is the unique solution of the optimization problem

Here, we recognize the usual barrier associated with a convex polyhedron. Naturally, the above optimization problem cannot be solved exactly (this would imply an infinite subalgorithm at each iteration k of the outer nonsmooth optimization algorithm). On the other hand, the numerous works in interior-point methods and complexity theory can be used as a guide to control the number of Newton iterations to be performed for each /c, while preserving the main convergence properties of the outer algorithm. An algorithm based on these ideas, ACCPM, is described in [162]. Appropriate implementations of this algorithm have been used successfully for some applications; as for its theoretical properties, see [294], [163] for some first complexity results. Bundle methods. Another stabilization process is achieved by bundle methods. In the framework or eigenvalue optimization, this was done, for example, in [231, 278, 366] (t method of [231] is best explained in [232]). Actually, these works deal with the nonconvex case (F(-) nonaffine). Numerical experiments are reported in [366] for relaxations of combinatorial problems with n ~ 100 and m ~ 1000. Actually, these methods have recently received increased attention and rather sophisticated variants, exploiting the structure of the present problem, have been developed. For future reference, we find it convenient to study this class of methods more deeply.

3.3

A short description of standard bundle methods

For a study of bundle methods, we recommend [194, Chapter XV]; see also [230] for their application to nonconvex problems, and the direct resolution of constrained problems such as (3.1), without a detour via (3.8). We now demonstrate the principles underlying these methods, to minimize a general convex function. At the fcth iteration, we have a stability center, call it Xk, and we perform the following operations: (i) we choose a mode/, call it (pk, supposed to represent /; (ii) we choose a norming, call it || • ||fc; (iii) we compute a candidate yk+i realizing a compromise between diminishing the model

62

Lemarechal and Oustry and keeping close to the stability center

(iv) the candidate is then assessed in terms of the true function /, and the stability center xk is - moved to y^+i if it is deemed good enough (descent step), - or left as it is (null step). Thus, two sequences are involved in this description, rather than one: The sequence of iterates is now denoted by {yfc}, while {xk} is rather a subsequence of selected iterates. Depending on the specific choices made for the above operations, a whole family of bundle methods can be constructed. We now describe probably the most efficient of them: cutting planes with stabilization by penalty. Such a method has the following features: (i) The model taken for the kth iteration is still the cutting-plane function 0. (iii) The candidate is then the minimizer of 1 2 the function (pk(-} + % p>k\\ • — xk\\kRemark 3.3. This last function is related to the Moreau-Yosida regularization of f:

Call yk+i the unique solution of this last problem when x = xk. Then, insofar as (pk ~ /, yfc+i approximates the so-called proximal point

The present stabilization of cutting planes is therefore also called the proximal bundle method. Then the resulting algorithm has the following form. Algorithm 3.1 (cutting planes with stabilization by penalty). The initial point x\ is given, together with a stopping tolerance 5 > 0. Choose a spring strength u.\ > 0, a positive definite matrix MI £ R m x m , and a descent coefficient u> e ]0,1[. Initialize the iteration counter k = I and yi = xi; compute /(yi) and g\ € df(y\)• STEP 1. Compute the solution yk+i of

and set the decrement of the model

STEP 2. If6k < 6 stop.

3.3. A short description of standard bundle methods

63

• STEP 3. Call the oracle to obtain f ( y k + i ) and gk+i e d f ( y k + i ) . If

set Zfc+i = yk+i, choose Hk+i > 0 and Mfc+i >- 0. Otherwise set £fc+i = Xfc (nuJJ sep. step). Update the model Replace k by k 4- 1 and loop to Step 1. Naturally, the "master program" (3.12) is nothing more than a linear-quadratic program resembling (3.11):

This form makes it easier to write the dual of (3.12), a crucial point to fully understand the different variants of bundle methods. Let us denote by the unit simplex of Rfc and by &\ the linearization errors between the yi's and Xk'

In the result below we also introduce Wk := M.. l and the dual norm Lemma 3.1 (see [194, LemmaXV.2.4.1]). For /Zfc > 0 and Mfc >- 0, the unique solution of the penalized problem (3.12) = (3.14) is

where <5 e Rfc solves

Furthermore, with 6k of (3.13) and gk := Z^i=i ^iPi>

we

have the primaJ-duaJ relation

We also give the following result, relating the subdifferentials at previous iterates with the approximate subdifferential at the current stability center. Proposition 3.1 (see [194, Proposition XIV.2.1.1]). For all y € Rm and a e Afc, we have

where efc is the vector of linearization errors defined in (3.15). In particular, for e > mint elk,

64

Lemarechal and Oustry

Thus, for a solving (3.17), consider £k and g~k of Lemma 3.1. We see that g^ € d £ k f ( x k ) . Indeed we have A nice interpretation of Lemma 3.1 in terms of the subgradient algorithm (3.9) ensues; this interpretation will be important for our further development. If we compare (3.16) with (3.9), we see that (l/fJ>k) can be interpreted as a step length and that the chosen subgradient is now the projection (for the current dual norm) of the origin onto the approximation Sk(£k) C d £ k f ( x k ) . In what follows, we denote by Algorithm 3.1* the (dual) version of Algorithm 3.1, where we solve (3.17) instead of (3.12) and yk+i is updated by (3.16). Remark 3.4. The relation gk G d£kf(xk) implies As a result, Xk satisfies an approximate minimality condition if both \\gk\\ and £k are small. In view of (3.18), this will be the case when 6k is small, providing that p-k^max(Mk) is not large. This explains the stopping criterion in Step 2 of the algorithm. To conclude this section, we mention another important point: It is actually possible to "clean the bundle" so as to avoid an intolerably growing number of affine functions in the model
3.4

Specialized bundle methods

It is observed in [366] that dramatic improvements can be obtained with a slightly enriched oracle, which computes at Xk not only f ( x k ) and some eigenvector Vk, but also a few more eigenvectors corresponding to "almost maximal" eigenvalues. This extra information can be inserted into the the definition of /t, which will approximate / more accurately while staying polyhedral. This is the basis for the methods described in this section.

3.4.1

Rich oracle: Dual methods

The first instance of a nonsmooth algorithm to solve (3.8) is probably the pioneering work [87], which definitely adopts a dual approach. As mentioned above, the idea there is to realize that the eigenvectors associated with eigenvalues of F(x) falling within e of Amax(jP(x)) yield an enlarged df(x) (denoted by S£f(x) below), which turns out to produce good descent directions. A quantitative analysis of this enlargement is given in [307]. At X e Sn and for e > 0, we use the following notation: matrix whose columns form an orthonormal basis of the re(X)-dimensional space spanned by the corresponding eigenvectors. Then, by analogy with (3.3) and (3.5), we consider the following enlarged subdifferentials S£Xmax(X) and 6£f(x):

3.4.

Specialized bundle methods

65

Since the columns of Q£(X) span in particular the first eigenspace of X, it is clear that 6£f(x) contains the exact subdifferential. To see that this enlargement is included in the e-subdifferential d £ f ( x ) , it suffices to note that any V = Q£(X)ZQ£(X}T € <5 £ A max (X), which obviously lies in Cn, also lies in the half-space {V e Sn : V • X > Xm&x(X) - £}'• In fact,

since (3.6) and (3.7), we do have

Together with

The whole purpose of the first-order analysis in [307] then consists in quantifying the distance between 6£f(x) and def(x) (knowing that we would like to approximate the latter set). Clearly, 6£Xmax(X) does not contain any full-rank matrix if r£(X) < n; this seems to imply that this set is a poor approximation of the (full-dimensional) esubdifferential. Nevertheless, it is proved in [307] that the shortest element

does provide a good descent direction de — —g£. Specifically, / is guaranteed to decrease along d£ by a quantity proportional to the gap defined at X — F(x) by &£(X) := Prom there, a minimization algorithm can be constructed. At each iterate x = Xk, one chooses e = £fc, one constructs the inner approximation <5 e /(zfc), one computes its corresponding shortest element to obtain a direction dk , along which a line search is performed. With a choice of £k careful enough to appropriately control the gap A£fc (Xfc), a globally convergent method is obtained [307].

3.4.2

Rich oracle: A primal approach

The algorithm outlined above implies a delicate choice of e. In fact, this parameter plays an ambivalent role: (i) It is the tolerance on the largest eigenvalue, driving the quality of the direction of search; as such, e should be "large." (ii) It is also the final accuracy driving the quality of the solution: The algorithm stops when 6£f(xk) contains a small vector (see Remark 3.4); as such, e should be "small." In the present section, we combine the two subdifferential enlargements of sections 3.3, 3.4.1. This allows us to restrict the e of the latter section to its role (ii) of a stopping tolerance. We therefore fix this e to a small tolerance e. The resulting 8sf becomes a slim enlargement of d/, but it is enriched with the bundle Sfc(£fc) of Proposition 3.1. Alternatively, we can consider Sgf as enriching the bundle Sfc(£fc). In addition to this mutual enrichment, we also transform the dual formulation of section 3.4.1 to a primal one as in section 3.3. Since we are dealing now with f-subgradients, in particular those belonging to at iteration k, we need to adapt Proposition 3.1. At iteration fc, we have k — I previous e-subgradients, & € 6ef(yi) for i = 1, . . . , fc — 1, and at the current point

66

Lemarechal and Oustry

which will be considered as a local dual variable (together with the vector a € A&). We define now the ^-linearization errors at iteration yk:

We see here that the kih linearization error plays a particular role since it becomes a function of the dual variable g. Proposition 3.2. For all y € Rm, a € Afc, and gk € S s f ( y k ) , we have

In particular, for

Proof. Follow line by line the proof of [194, Proposition XIV.2.1.1], starting with the e-subgradient inequalities

instead of the subgradient inequalities. Our (inner) approximation o f d £ f ( x k ) is now SktS(e) of (3.26). Then the dual program (3.17) is no longer quadratic and becomes

where Qk and rk stand, respectively, for Qg(F(yk)) and re(F(yk)) of (3.21). In section 3.3 we presented a primal formulation and interpreted it in the dual space. Here we do a reverse presentation by introducing the Lagrange multiplier fj,k associated with the constraint (*) above: (3.27) can be written in a penalized form

Prom now on, we assume nk > 0. Going further into our primalization process, we construct the dual of (3.28) and obtain the following result. Proposition 3.3. Up to the multiplicative factor p,k > 0, the dual of (3.28) is

3.4.

Specialized bundle methods

67

If (tk+i,yk+i) and (a, Z) solve, respectively, (3.29) and (3.28), we have the following primal-dual relations wher

where

Proof. Problems (3.28) and (3.29) are two convex optimization problems: a convex quadratic objective is minimized under LMI constraints; we apply duality theory (see [373], for example). Consider (3.29); for fixed y G Rm and t large enough, the constraints are strictly satisfied; i.e., the Slater condition clearly holds. Then there is no duality gap and it suffices to prove that the dual of (3.29) is (3.28). For this let us introduce a (k — l)-uple a G R/0"1 of nonnegative scalar multipliers (associated with the k — 1 scalar constaints) and a positive semidefinite matrix multiplier U € <Srfe (associated with the LMI constraint). The associated Lagrangian is

The dual of (3.29) is then

For the minimization in (3.32) to have a finite value we need the multipliers to satisfy the following hidden constraint:

On the other hand, the Lagrangian is strongly convex in y and the optimality condition for the minimization part gives us

Plugging this relation into the Lagrangian, setting ak = Trt/, we obtain (3.28) after multiplication by

3.4.3 3

and

A large class of bundle methods

Starting from the above development, several options can be made to obtain implementable bundle methods, which will combine the global convergence properties of a bundling mechanism with the fast termination allowed by a rich oracle.

68

Lemarechal and Oustry

Models. Consider the function (f>k defined in (3.31). Prom (3.23) and (3.6), we have for all gi € 6sf(yi), * = 1, . . . , A; - 1, For these Vj's, we therefore have On the other hand,

Then introduce the set Recall (3.2); the following inequalities hold:

We see that, if based on the intermediate model above, a bundle method can be seen as a process to update approximations Ck of the big convex set Cn. Furthermore, observe that the first inequality becomes an equality when e = 0. Then such an approach resembles that of [189], which consists in approximating Cn by

here V G Cn is an aggregate matrix that contains information from the previous iterates, and the columns of Pk are orthonormal vectors storing past and current information. Implementations. Consider now the effective resolution of the quadratic-sdp problem (3.29) or (3.28). In view of its nonlinear constraints, it cannot be solved exactly; and since it must be solved at each iteration of the outer minimization algorithm, a fast method is in order. In [189] (where 5 = 0), this subproblem is solved using interior-point methods; the bundle scheme is thereby assigned the role of a master program controlling the size of small quadratic-sdp problems. Here, we propose the approximate resolution of (3.28) in two steps: (qpi) compute an approximate solution gk of

(qp2) then solve the quadratic program

where

3.4. Specialized bundle methods

69

Four main reasons lead us to this choice: (i) The support function of the set fizf(yk] is known explicitly; therefore, an efficient method, called the support black-box function in [307, section 3.1], can be used to obtain an approximation of (3.34). (ii) Step (qpi) can be viewed as a refinement of the oracle giving f(yk) and gk'. Now gk coming from (qpi) is a carefully selected element of &sf(yk), possibly more efficient than any exact subgradient at yk. (iii) By decomposing (3.28) in two steps, local and global work are cleanly separated: While (qp2) guarantees global convergence, the solution gk of (3.34) is exactly the local information needed to form the so-called W-Hessian of / and get asymptotic quadratic convergence (see section 3.5 below). (iv) Step (qp2) is similar to (3.17), with approximate instead of exact subgradients: Then the two steps can be naturally inserted in the general scheme of (3.3) (see Algorithm 3.2 below); the resulting method can be seen as a polyhedral-semidefinite proximal bundle method. In view of this last point, the notation of (3.10) can still be used (even though the #i's have a different origin), keeping in mind that it is now f — e that underapproximates /. Indeed, we can see from (3.31), (3.33) that the models are ordered as follows: For all

yeRm,

In a way, the present approach consists in replacing the model (f>k of (3.31) by fk — £. Thus, the analysis of (3.17) can be reproduced to obtain the analogue of Lemma 3.1. Lemma 3.2. For fj,k > 0 and Mk > 0, the unique solution of

Furthermore, setting we have

Algorithm 3.2 (polyhedral-sdp proximal bundle method). The initial point x\ is given, together with a stopping tolerance 6 > 0. Choose a spring strength u.\ > 0, the tolerance e > 0, a positive definite matrix M, and a descent coefficient u e ]0,1[. Initialize the iteration counter k = 1 and y\ = x\; compute f ( y \ ) and g\ G d f ( y \ ) . • STEP 1. Compute a solution a of (qp2) and set

• STEP 2. If6k < 8 stop.

70

Lemarechal and Oustry • STEP 3. Call the oracle. Compute an approximate solution gk+i € 6sf(yk+i) of (3.34) and ek according to (3.36). If set Xk+i = yjt+i, and choose fj,k+i > 0 and Mk+i >- 0. Otherwise set (null step). Replace k with k + l and loop to Step 1.

When £ = 0, Algorithm 3.2 is exactly Algorithm 3.1*. Numerical experiments reported in section 3.6 will show that the introduction of £ together with (qpi) significantly improves the speed of convergence. As for theoretical properties, the global convergence of Algorithm 3.2 is analyzed just as in [194, section XV. 3. 2]. Via an appropriate control of the sequence {/ifcAmax(.Mfc)}, and assuming 6 > 0, the algorithm stops at some iteration k; then the last iterate satisfies an approximate minimality condition, alluded to in Remark 3.4.

3.5

Second-order schemes

The rich structure enjoyed by Amax allows an appropriate second-order analysis, even though VAmax does not exist. This can be performed with tools from differential geometry: [24, 310, 311, 374] or from convex analysis via the recent ZY-Lagrangian theory of [245].

3.5.1

Local methods

With an appropriate second-order approximation of Amax on hand, Newton-type methods can be developed for (3.1) or (3.8). This is done by M. L. Overton in [308], inspired by the work of R. Fletcher [137]. The approach is further developed in [309, 311, 310, 374, 130], and revisited in the light of ZY-Lagrangians in [305]. These methods are local, in the sense that convergence can be guaranteed only in a neighborhood of a solution x*. Furthermore the multiplicity r* of Amax(F(x*)) must be known. It is then natural to combine the globally convergent bundle methods of section 3.4 with the present fast local methods by inserting in the former the appropriate second-order information used by the latter. For methods of section 3.4.1, this is done in [307], Without any assumption on (3.8), the resulting algorithm generates a minimizing sequence. Its quadratic convergence needs some strict complementarity and nondegeneracy assumptions. Here, we proceed to demonstrate how the same metric can be inserted in a primal formulation like section 3.4.3. For this, we first need to explain what such a metric is; we do this via the W-Lagrangian theory.

3.5.2

Decomposition of the space

Notationally, a hat on a matrix or vector (say, x) will suggest some intermediate variable, rather than the current point of an algorithm. We denote by Air the set of matrices whose maximum eigenvalue has a fixed multiplicity r 6 {1, . . . , n}. At a point X € Mr, we decompose the space <Sn as the direct sum of U\mM(X) and V\max(X], the tangent and normal spaces to M.r at X. These two subspaces have an explicit description:

3.5. Second-order schemes

71

where r and Q are the r0(X) and QQ(X) of (3.21). To this decomposition of <Sn, there corresponds the following decomposition of the space of variables Rm: At a point x e Rm such that F(x) = X e Mr, i.e., x e A~lMr, we write where W/(x) := /"^A™^) and Vf(x) := Uf(x}JL. It is shown in [305] that Uf (x) is the largest subspace where the directional derivative /'(x, •) of / at x is linear, and Vf(x) is the linear subspace parallel to the affine hull of df(x).

3.5.3

Second-order developments

Take x € A~lMr and G € Cn such that f*G e df(x). we define the following symmetric operator H(X, G):

At the primal-dual pair (X,G)

where The operator induced by H(X,G) in the subspace U\maiX(X) is called the ZY-Hessian of Amax at (X,G). It collects the relevant second-order information on Amax along M.r as follows. Theorem 3.1. For all D € Sn such that X + D e Mr, we have

In the space of decision variables, (3.44) is written as the following: For all d e Rm such thatF(x) + FdeMr,

Proof. The first development (3.44) is given in [305, Theorem 4.12, Corollary 4.13]. The second development is obtained by taking D = F d and by noticing that

3.5.4

Quadratic step

Now we come back to our algorithmical motivations. So far, the viscosity parameter I had the role (ii) of a stopping criterion (see the beginning of section 3.4.2). At this stage we assign it an additional role: (iii) e enables us to "stabilize" the multiplicity of At iteration fc, assume that yk = Xk (i.e., the previous step was a descent step). As in section 3.4.1, we denote by r^ and Qk the approximate multiplicity rs(F(xk)) and the associated matrix Qs(F(xk)) of (3.21). The rationale for our quadratic step is to minimize the quadratic approximation of (3.45) (or an "e-regularization" of it) on the linearized constraint F(xk + d) € Mrk (knowing that MTk is supposed to contain the optimal solution). For this we introduce a dual matrix Gk € Cn such that T* Gk = 9k,

72

Lemarechal and Oustry

where gk € Sg\maLX(F(xk)} is the output of (qpi) in section 3.4.3. We also introduce the symmetric operator

where F* := F(xk)In [305, section 5.4], the constraint requiring F(xk+d) € M.ric is shown to be linearized as F(xk +d) € ^AmaxC-^fc)- This is a linear equation, which can be written, using (3.42),

We therefore obtain the following quadratic step:

We define Uk and Vfc, the e-regularizations ofltf(xk)

and

We assume that e-transversality holds at xk'- [D<j>]k is onto (or, equivalently, [D(f>]^ is injective). If in addition the Hessian matrix Hk is positive definite on Uk, then (3.47) has a unique solution dk. Under these assumptions (3.47) can be solved in two steps. Normal step. Since [D]k is onto, it has a right inverse [D(f>]*k)-1. Define Then we have

Tangent step. The set of directions d satisfying the linear equation of (3.47) is vk +Uk . Then, setting u — d — vk, (3.47) is the simple problem

whose unique solution is Here Puk is the orthogonal projection onto Uk\ Puk is symmetric, Pfik Range Puk =UkThe unique solution of (3.47) is then

3.5.5

= Im and

The dual metric

The previous section concerned a quadratic model of / yielding fast asymptotic convergence of an algorithm. Now we suggest a possibility to choose the metric in (qp2). We look for a matrix Wk of the form Wk = Wuk + Wyfc such that Wuk and W"vfc are symmetric , Wyk + Wvk is positive definite , Range and Range

3.5. Second-order schemes

73

Prom the previous paragraph, it is rather natural to take

Now we want to choose Wyfe so that the dual problem (3.34) is well conditioned. Notice that the enlargement 6 e f ( x k } can be described by

where gc := T* . Then the metric induced by || • \\k^ in Srit is [D4>}koWVk o [£>]£. Thus a natural choice for W\;k is

since in this case the induced metric in 5rfc is the standard Probenius norm.

3.5.6

A second-order bundle method

Now we can state an algorithm piecing together the objects defined in the preceding sections. When Algorithm 3.2 has just performed a descent step, the model seems good and a Newton step can be tried. On the other hand, a null step seems to indicate that the bundle is not rich enough (at least locally), and a more conservative strategy is advisable to enrich the bundle. Algorithm 3.3 (second-order proximal bundle method). The initial point x\ is given, together with a stopping tolerance 6 > 0. Choose a spring strength Hi > 0, the tolerance E > 0, a positive definite matrix MI, and a descent coefficient m €]0, 1[. Initialize the iteration counter k = 1 and y\ = x\; compute f ( y \ ) and g\ € df(y\). • STEP 1. If yk = xk goto Step 1.1 else goto Step 1.2. STEP 1.1. Compute dk '.= vk + uk, where vk and uk, respectively, are given by (3.49) and (3.51). Set

STEP 1.2. Compute a solving (qp2) and set

STEP 2. If6k < 6 stop. STEP 3. Call the oracle. Compute an approximate solution gk+i € Sef(yk+i) of (3.34) and ek according to (3.36). If

Replace k with k + 1 and loop to Step 1.

Lemarechal and Oustry

74

Figure 3.1: Influence of e. As far as the bundling mechanism is concerned, this algorithm is formally the same as Algorithm 3.2. Its global convergence is therefore established likewise, via appropriate safeguards on the various metrics used (see the end of section 3.5.1). Its local convergence is also easy to analyze: When /ifc = 1 is accepted for the descent test, the algorithm becomes the second-order bundle method of [307] and therefore enjoys the same asymptotic properties.

3.6

Numerical results

3.6.1 Influence of e This first example is aimed at illustrating an observation made by Schramm and Zowe and reported by Overton in [309]. We consider the Lovasz problem for a class of undirected graphs given in [366]: given integers a > I and a> > 3, set n = auj + 1 and define the graph C%~1 to have the property that vertices i and j (in {!,...,n}) are adjacent if j - i
where e is the n-vector on all ones. Figure 3.1 shows the influence of e on a small instance of (3.54), where a = 3 and o> = 4: this gives m = 39 variables in the 13 x 13 symmetric matrix M. We observe here not only the improvement of the algorithm for e > 0 but also a super-linear behavior

3.6. Numerical results

75

Table 3.1: Lovasz problem. a 10 20 20

(jj 6 10 50

n 61 201 1001

m 1891 1809 49049

r

10.120845 20.106 99.2

\\9\\ 6.1 10~7 4.2 10~3 1.1 KT1

# iter 129 481 96

CPU time 20s 16 min 5h

without the use of second-order information. This might be explained by the peculiar structure of these problems: The multiplicity at the solution is very high (r* = 7 to precision 10~8) and V*, the affine hull of df(x*}, is therefore very large. Moreover, first-order information is enough to get a second-order approximation of / in V* [245].

3.6.2

Large-scale problems

Lovasz problem. In order to be able to solve (3.54) when n is large, we exploit in the oracle the sparse structure of the adjacency matrices M. In Table 3.1, we report for several values of a and a;, the final value of the function, the norm of the corresponding aggregate subgradient, the number of total iterations (descent or null steps), and the total CPU time used by Algorithm 3.2 on a Sun Spare Ultra 1 with 196 MB RAM. Quadratic stability of polytopic linear differential inclusions (PLDIs). In this paragraph, along the lines of [64, Chapter 5], we analyze the stability of a PLDI

where the A^s are given stable n x n real matrices. Then we define the structured mapping:

a sufficient condition for (3.55) to be quadratically stable is

Then an equivalent condition for (3.56) to hold is to require that the value of the eigenvalue optimization problem

is negative. Actually, using a generalization of Lyapunov's theorem [199, Theorem 2.4.10], we can drop the constraint P > 0; the equality constraint can also be removed by eliminating one variable. Yet for the implementation of the oracle presented in section 3.4.3, we do keep the matricial structure of the decision variable P instead of decomposing P on a basis of <Sn; the minimized functions have the form

where E is the symmetric matrix with zeros everywhere except Enn = n and P equals P everywhere except Pnn := — Y%=i PH- The adjoints of P •-> F-P have a similar matricial structure. This formulation eases the resolution of (3.57) when n and m = "H"1) - l are large numbers. In Table 3.2, we take L = 2 and we generate stable random matrices

Lemarechal and Oustry

76

Table 3.2: PLDI, n > 500.

n 500 1000

m 125249 500499

/*2

io-

5.110-2

\\9\\2

io-

6.3 10~2

#iter 80 47

CPU time 7h 20 h

AI and AI of various sizes. The most difficult situations are when the value of (3.57) is nonnegative. We report for such situations the final value of the function, the norm of the corresponding aggregate subgradient, the number of total iterations (descent or null steps), and the total CPU time. Algorithm 3.2 is run with the viscosity parameter e = 10~3 and the stopping tolerance 6 = IQ~2. We note that the observed multiplicity to the precision 10 * e is, in these situations, r* = n.

3.6.3

Superlinear convergence

In this paragraph we focus our attention on a small instance of (3.57): n = 5 and the matrices A\ and AI are given by

Although both A\ and A2 are stable, their respective spectral abscissa are a(Ai) = —2.93e — 05 and 01(^2) = — 3.25e — 06; there is no quadratic Lyapunov function proving the stability of (3.55). Indeed, using the second-order proximal bundle method, we solve (3.57) to machine precision: At the final point P* generated by the algorithm we have g € df(P*) with \\g\\ < 2e - 15. The final value is /* := /(P*) = 0.01149788039455 with multiplicity 4 to the machine precision; the next eigenvalue is AS(^" • E + F - P*) = —0.00670989012524. The corresponding solution is positive definite,

Figure 3.2 shows us the acceleration produced by the introduction of second-order information. In particular, we observe the asymptotic superlinear convergence. Algorithm 3.2 is run with e — 10~8. Algorithm 3.3 is initialized with e = 10~~3, and a dynamic update of the form lk+i '•= 0-9 * £* is used when e~k is too large compared wit the £k of (3.39). This management of e enables us to enter more rapidly the region of superlinear convergence.

3.7 Perspectives Depending on the type of problem to be solved, the need for further work can be anticipated in the following directions. Poor oracles. Despite some isolated experiments reported in section 3.3, the re capabilities of standard bundle methods have not been really investigated so far. Can they solve problems for which other alternatives are impractical or inefficient? How

3.7.

Perspectives

77

Figure 3.2: Acceleration with second-order information. large of problems can be solved in practice? To what extent can intermediate oracles be helpful, and how can their additional information be used most efficiently? These questions are not clearly answered yet. Rich oracles. The real issue underlying the ideas in section 3.4.1 is not completely understood. Are there connections between the dual and primal approaches adopted there? Do these approaches come from a general approximating scheme, or are they just isolated tricks without a firm ground? These questions are important for further developments, for an adequate link with section 3.5, and also for a direct resolution of (3.1). Another issue is the bulk of work necessary for methods of section 3.5: Computing the ZY-Hessian appearing in (3.43) really involves heavy computations. The question of approximating this operator by quasi-Newton updates is therefore important for numerical implementations. On the theoretical side, we mention the question of relating the present nonsmooth analysis with interior points. For example, it should be interesting to relate the W-Hessian with the Hessian of the potential function associated with

This page intentionally left blank

Chapter 4

sdpsol: A Parser/Solver for Semidefinite Programs with Matrix Structure Shao-Po Wu and Stephen Boyd 4.1 Introduction This chapter describes a parser /solver for a class of linear matrix inequality (LMI) problems, the so-called max-det problems. A variety of analysis and design problems in control, communication and information theory, statistics, combinatorial optimization, computational geometry, circuit design, and other fields can be expressed as semidefinite program (SDP) problems or determinant maximization problems (max-det problems). These problems often have matrix structure, which has two important practical ramifications: First, it makes the job of translating the problem into a standard SDP or max-det format tedious, and, second, it opens the possibility of exploiting the structure to speed up the computation. In this chapter, we describe the design and implementation of sdpsol, a parser /solver for SDPs and max-det problems, sdpsol allows problems with matrix structure to be described in a simple, natural, and convenient way. Moreover, it allows algorithms to exploit problem structure to gain efficiency. Although the current implementation of sdpsol exploits only block-diagonal structure in the solution algorithm, the language and parser were designed with exploiting general matrix structure in mind.

4.1.1

Max-det problems and SDPs

A max-det problem has the form minimize subject to

79

80

Wu and Boyd

where the optimization variable is the vector x € R m . The matrix functions G^ : Rm —>• RliXli and F<*> : Rm -* R«« xri « are affine:

where G^*' and F^ are symmetric for j = 0, . . . , m. The inequality signs in (4.1) denote matrix inequalities. We call G^(x) > 0 and F^(x) > 0 LMIs in the variable x. Of course the LMI constraints in (4.1) can be combined into one large block-diagonal LMI with diagonal blocks G (i >(z) and F<*>(x). When K = 1 and G^(x) = 1, the max-det problem (4.1) reduces to the well-known SDP problem:

The max-det problem (4.1) and the SDP (4.2) are convex optimization problems. In fact, LMI constraints can represent many common convex constraints, including linear inequalities, convex quadratic inequalities, and matrix norm and eigenvalue constraints. SDPs and max-det problems arise in many applications; see Alizadeh [5], Boyd, El Ghaoui, Feron, and Balakrishnan [64], Lewis and Overton [247], Nesterov and Nemirovsky [296, section 6.4], Vandenberghe and Boyd [404], and Vandenberghe, Boyd, and Wu [407] for many examples. SDPs and max-det problems can be solved very efficiently, both in worst-case complexity theory and in practice, using interior-point methods (see [404] and [407]). Current general-purpose solvers such as SP [402], MAXDET [427], SDPT3 [392], SDPA [145], SDPMATH [74], and SDPpack [6] use the standard format (4.2) (and (4.1)) or a simple variation (e.g., its dual form).

4.1.2

Matrix structure

In many SDPs and max-det problems the optimization variables are matrices of various dimensions and structure, e.g., row or column vectors, or symmetric or diagonal matrices. In general, the optimization variables can be collected and expressed as (X^), . . . , X^ M ^), where X^ e RpiX9< and X^ may have structure (e.g., symmetric, diagonal, upper triangular, etc.). These matrix variables can be vectorized into the single vector variable x that appears in the standard format problems (4.1) and (4.2). To vectorize X^t we find a basis E^\ . . . , E£] such that

where x^ € Rmi denotes the vectorized X^. For example, if X^ € RpiXqi has no structure, we have x^ = vec(X^) and mi = p^; if X^ € R piX9< is diagonal (pi = ft), we have rpW = diag(X^) and m* = p,. Doing this for i = 1,...,M, we obtain the vectorized variable x € Rm, m = mi + • • • -f- mM> and

Note that each variable X^ of the problem corresponds to part of the vectorized variable x. With this correspondence, we can convert the problem to the standard form (4.1) or (4.2), and vice versa.

4.1. Introduction

81

The following example illustrates this vectorization process. We consider the problem

where A, B are given matrices (i.e., problem data), and the symmetric P G R n x n and diagonal R G R fcxfc are the optimization variables. We vectorize P and R as

where Pij denotes an nx n zero matrix except the (i,j) and (j, i) entries are 1; R± denotes a k x k zero matrix except the (i,i) entry is 1. Substituting (4.4) and (4.5) for P and R everywhere in the SDP (4.3), we obtain the problem of the form (4.2) in the optimization variable

4.1.3

Implications of the matrix structure

Clearly it is straightforward but inconvenient to convert an SDP or a max-det problem with matrix structure into the standard form (4.2) or (4.1). This conversion obscures the problem structure and the correspondence between the variables X^ and the vectorized variable x, which makes it hard to interpret the results after the problem is solved. Moreover, the problem structure can be exploited by interior-point methods for substantial efficiency gain (see [403] and [67] for examples). To illustrate the idea with a simple example, consider the operation

that evaluates the —ATP — PA term in the first matrix inequality of (4.3). C(P) is an O(n3) operation because it involves matrix multiplications of n x n matrices. However, if we vectorize P as shown in (4.4), then C(P] becomes

which is an O(n4) operation.

4.1.4

sdpsol and related work

In this paper, we describe the design and implementation of sdpsol, a parser/solver for SDPs and max-det problems with matrix structure. Problems can be specified in the sdpsol language in a form very close to their natural mathematical descriptions. As an

82

Wu and Boyd

example, the SDP (4.3) can be specified as variable P(n,n) symmetric; variable R(k,k) diagonal;

minimize objective = Tr(P); The problem specification is then compiled by the sdpsol parser into the standard form and passed to the solution algorithm to be solved. The solution (optimal variables) will again be interpreted by the parser and returned in their specified structure. Evidently, the sdpsol parser/solver automates the tedious and inconvenient step of converting an SDP or a max-det problem into the standard form. It also allows a structural description of the problem such that sophisticated solution algorithms can exploit the problem structure to gain efficiency. There exist several similar tools that use a specification language to describe and solve certain mathematical programming problems. A well-known example is AMPL [139], which handles linear and integer programming problems. LMITOOL [126] and the LMI Control Toolbox [151] provide convenient Matlab interfaces that allow the user to specify SDPs with matrix structure (specifically the ones arising in control), using a different approach from the parser/solver described in this paper. In section 4.2, we describe the design of the specification language. A preliminary implementation of sdpsol (version beta) is described in section 4.3. In section 4.4, we give several examples to illustrate various features of sdpsol.

4.2 4.2.1

Language design Matrix variables and affine expressions

The fundamental data structure of the sdpsol language is, as in Matlab [266], the matrix. The language provides a Matlab-like grammar, including various facilities to manipulate matrix expressions. For example, it provides commands that construct matrices, operations such as matrix multiplication and transpose, and matrix functions such as trace and inner product. The language also supports assignments; i.e., expressions can be assigned to internal variables that can later be used in the problem specification. Various algorithm parameters, such as tolerance or maximum number of iterations, can be initialized using assignments. An important extension beyond Matlab is that the sdpsol language provides matrix variables and, by extension, matrix expressions formed from matrix variables and matrix constants. Matrix variables of various dimensions and structure can be declared using variable declaration statements. For example, the statements variable P(7,7) symmetric; variable x(k,l), y; declare a 7 x 7 symmetric variable P, a k x 1 variable x, and a scalar variable y. Variables can be used to form affine expressions, i.e., expressions that depend affmely on the optimization variables. For example, the expression (assuming P is a variable and

4.2. Language design

83

A, D are constant square matrices) is an affine expression that depends affinely on P. Affine expressions can be used to construct LMI constraints and the objective function of SDPs and max-det problems. As each affine expression is parsed, sdpsol keeps track of its dependence on the variables, adding the dependency information to an internal table for later reference, e.g., by a solver.

4.2.2

Constraints and objective

Various types of (affine) constraints are supported by the language, including matrix inequalities, componentwise inequalities, and equality constraints. Constraints are formed with affine expressions and relation operators, as in

which specify the matrix inequality ATP + PA + D < — /, the componentwise inequality diagP > 0, and the equality TrP = 1. The objective function is specified via an assignment, as in maximize objective = Tr(P) ; which assigns TrP to the internal variable objective and makes maximizing it the objective. A feasibility problem (i.e., a problem that determines whether a set of constraints is feasible or not) is specified without an objective statement.

4.2.3

Duality

We associate with the max-det problem (4.1) the dual problem (see [407] for details):

where the dual variables are symmetric matrices W^ G R iiXii for i = 1, . . . , K, symmetric matrices Z(i) € R n i X U i for i = 1, . . . , L, and a vector z e Rp (assuming A € R pxm and 6 e Rp in (4.1)). We will refer to (4.1) as the primal problem of (4.6). Note that (4.6) is itself a max-det problem. Similarly, we can associate with the SDP (4.2) the dual

which is also an SDP.

84

Wu and Boyd

The dual problem has several practical uses; e.g., the optimal dual variables give a sensitivity analysis of the primal problem. The most efficient SDP and max-det solvers are primal dual; i.e., they simultaneously solve the primal and dual problems, sdpsol automatically forms the dual from the primal problem specification. The dual problem is used by the solver and also, if requested, returns to the user optimal dual variables. There is a simple relation between the form (size and structure) of the primal constraints and the dual variables (and vice versa). Observe that for each constraint in (4.1) (or (4.2)), there is a dual variable of the same dimensions and structure associated with it. The variables associated with the primal LMI constraints are constrained to be positive definite in the dual problem, while the variables associated with the equality constraints are unconstrained. Taking the SDP (4.3) as an example, we have that sdpsol associates the dual variables

with the three matrix inequalities, respectively, and z € R with the equality constraint. After some manipulation (which can be easily performed by sdpsol), we have the dual problem

where P^, Ri are given by (4.4) and (4.5), 6ij = I if i = j, otherwise zero. sdpsol provides an alternate form of constraint specifications, in which the user associates a name with each constraint, which is then used by sdpsol to refer to the dual variables. For example, the statements (assuming P,A,D£ R n x n ) constraint lyap constraint equ specify (and name) the constraints ATP+PA+D < —I and TrP = 1. sdpsol associates with these constraints the dual variables with names lyap -dual G R nxn , lyap .dual symmetric, equjdual € R. After the solution phase, sdpsol returns the optimal values of these dual variables.

4.3 Implementation We have implemented a preliminary version of the parser/solver, sdpsol version beta. The parser is implemented using BISON and FLEX. Two solvers, SP [402] and MAXDET [427], are used to solve the resulting SDPs and max-det problems. Both solvers exploit only block-diagonal structure, so the current version of sdpsol is not particularly efficient. Both SP and MAXDET are primal-dual methods that require a feasible starting primal and dual point, sdpsol uses the method described in [404, section 6] to handle

4.3. Implementation

85

the feasibility phase of the problem; that is, sdpsol either finds a feasible starting point (to start the optimization phase) or proves that the problem is infeasible. If there is no objective in the problem specification, sdpsol simply solves the feasibility problem only. A Matlab interface is provided by sdpsol to import problem data and export the results, sdpsol can also be invoked from within Matlab interactively.

4.3.1

Handling equality constraints

Neither SP nor MAXDET handles problems with equality constraints, so the current implementation of sdpsol eliminates them, uses SP or MAXDET to solve the resulting problem (which has no equality constraints), and finally reassembles the variables before reporting the results. The equality constraints given in the problem specification can be collected and put into the form of Ax = b (A € R p x m ,6 e Rp) as given in (4.1) and (4.2). We assume that the rank of A, say, r, is strictly less than m (otherwise the problem has a trivial solution). The matrix A is often not full rank, because sdpsol does not detect and remove redundancy in the equality constraints specified. For example, a 2 x 2 matrix variable X can be constrained to be symmetric by the constraint specification X == X ' ;

sdpsol interprets the above statement as the following four equality constraints:

Evidently one constraint X^\ = X\<2 is necessary. This will show up as two zero rows in A and two identical rows of A; in particular, A will not be full rank. Instead of keeping track of such situations, sdpsol builds up the matrix A paying no attention to rank, sdpsol then performs a complete orthogonal decomposition (see [171]) on A such that

where P is a column permutation matrix (for column pivoting), R\\ G R r x r is upper triangular, and Q, U € R pxp are orthogonal matrices. All x € Rm satisfying the equality constraints can be expressed as

we can express the max-det problem (4.1) as

86

Wu and Boyd

where x 6 Rm is the optimization variables and Bkj denotes the ( k , j ) entry of B. Evidently, the above process reduces the optimization problem (4.1) in Rm with equality constraints to (4.10) in Rm without any equality constraint. Similar methods apply to the SDP (4.2). We should add that eliminating variables destroys the matrix structure that could be exploited by a solver. A future version of sdpsol would directly pass the equality constraints to an SDP/max-det solver that handles equality constraints.

4.4

Examples

In this section, we give several examples to illustrate various features of sdpsol.

4.4.1

Lyapunov inequality

As a very simple example, consider a linear system described by the differential equation

where A € R nxn is given. The linear system is stable (i.e., all solutions of (4.11) converge to zero), if and only if there exists a symmetric, positive definite P such that the Lyapunov inequality is satisfied. The problem of finding such a P (if one exists) can be specified in the sdpsol language as follows: '/. Lyapunov inequality variable P(n,n) symmetric;

In the problem specification, an n x n symmetric variable P is declared. An affine expression ATP + PA is formed and is used to specify the Lyapunov inequality. Another LMI, P > 0, constrains P to be positive definite. Since no objective is given, sdpsol solves the feasibility problem that either finds a feasible P or shows that the problem is infeasible. Note that this problem can be solved analytically. Indeed, we can solve the linear equation ATP + PA + / = 0 for the matrix P, which is positive definite if and only if (4.11) is stable.

4.4.2

Popov analysis

Consider the mass-spring-damper system with a vibrating base shown in Figure 4.1. The masses are 1kg each, the dampers have the damper constant 0.5nt/m/s, and the springs are nonlinear: Fi = 0i(£i), where

The base vibration is described by

4.4.

Examples

87

Figure 4.1: Mass-spring-damper system. where RMS(to) < 1 but is otherwise unknown. Our goal is to find an upper bound on RMS(z). The mass-spring-damper system can be described by the Lur'e system (see [64, section 8.1]) with 7 states and 3 nonlinearities

where x e R7, p € R3 and the functions 4>i satisfy the [0,1] sector condition; i.e., for i — 1,2,3. One method to find an upper bound is to find a 72 and a Lyapunov function of the form

where Cqi denotes the ith row of Cq such that

for all w,z satisfying (4.12). The square root of 72 yields an upper bound on RMS(z). The problem of finding 72 and the Lyapunov function (4.13) can be further relaxed using the «S-procedure to the following SDP (see [64, section 8.1.4]):

Wu and Boyd

88

Figure 4.2: RMS(2) of mass-spring-damper system and upper bound. where P = PT e R 7x7 , L,T e R 3x3 diagonal, and 72 € R+ are the optimization variables. The square root of the optimal 72 of (4.14) is an upper bound of RMS(z) if there exists P, L, T that satisfy the constraints. The SDP (4.14) can be described using the sdpsol language as follows: '/. Popov analysis of a mass-spring-damper system variable P(7,7) symmetric; variable L(3,3), T(3,3) diagonal; variable gamma_sqr;

Again, the specification in the sdpsol language is very close to the mathematical description (4.14). The objective of the problem is to minimize RMS_bound_sqr, the square of the RMS bound, which is equal to the variable gamma_sqr. Unlike the previous example, this problem of finding an upper bound on RMS(z) has no analytical solution. However, one can easily specify and solve the problem using sdpsol. We show in Figure 4.2 an envelope of possible RMS(z) obtained via Monte-Carlo simulation and the upper bound obtained via solving (4.14). RMS(z) of the nominal system is shown in the dashed line.

4.4.3 .D-optimal experiment design Consider the problem of estimating a vector x from a measurement y = Ax + w, where w ~ AT(0, /) is the measurement noise. The error covariance of the minimum-variance estimator is equal to A^(A^}T = (ATA)~l. We suppose that the rows of the matrix

89

4.4. Examples

Figure 4.3: D-optimal experiment design. can be chosen among M possible test vectors

The goal of experiment design is to choose the vectors a» so that the determinant of the error covariance (ATA)~l is minimized. This is called the D-optimal experiment design. We can write ATA = Y ^ - , AtT/*M*) , where A» is the fraction of rows a/c equal to * ** A ^~ ...v,^ ^ (l'. We ignore the fact that the numbers Aj are integer multiples of l/q and instead treat them as continuous variables, which is justified in practice when q is large. In D-optimal experiment design, A» are chosen to minimize the determinant of the error covariance matrix, which can be cast as the max-det problem

This problem can be described in the sdpsol language as 7. D-optimal experiment design variable lambda(M,l); lambda .> 0; sum(lambda) == 1; Cov_inv = zeros(p,p); for i=l:M; Cov_inv = Cov_inv + lambda(i,l)*v(:,i)*v(:,i)';

end;

minimize log_det_Cov = -logdet(Cov_inv); In the specification, an M-vector lambda is declared to be the optimization variable. The

90

Wu and Boyd

Figure 4.4: Mass-spring system. (componentwise) inequality constraint specifies that each entry of lambda is positive, and the equality constraint says the summation of all entries of lambda is 1. A for-loop is used to construct Cov_inv, the inverse of the covariance matrix, to be ]T)i=i Aj7/*M*' . The objective of the optimization problem is to minimize the log determinant of the inverse of Cov_inv. An implicit LMI constraint, Cov_inv > 0 is added to the problem as soon as the objective is specified. This LMI corresponds to the G(x) > 0 term in (4.1), and it ensures that the log-determinant term is well defined. A numerical example of /^-optimal experiment design involving 50 test vectors in R2 is shown in Figure 4.3. The circle is the origin; the dots are the test vectors that are not used in the experiment; the x's are the vectors that are used. The optimal experiment allocates all measurements to only two test vectors. • r

4.4.4

*jP

Lyapunov exponent analysis

Consider the mass-spring system shown in Figure 4.4 [63, section 7]: Two unit masse and a wall (infinite mass) are connected by nonlinear springs with spring constants that can change instantly over the range of [1,2]. We would like to find an upper bound on the rate of extracting energy from (or pumping energy into) the system by loosening and stiffening the springs. The system can be described by the differential inclusion

where the four extreme cases (i.e., ki = 1 or 2, i = 1,2) are denoted by y = Ajy, j = 1,..., 4. It is shown in [63, 64] that an upper bound is given by the solution of the generalized eigenvalue problem:

The above problem is a quasi-convex optimization problem and cannot be posed as an SDP or a max-det problem. However, we can solve (4.16) via bisecting an interval of a

4.5. Summary

91

and solve, for each a, the SDP feasibility problem

until the minimum feasible a is found. We can specify (4.17) in the sdpsol language as '/, Lyapunov exponent analysis via bisection variable p(4,4) symmetric; A1'*P+P*a1-2-alpha*p<0; A2'*P+P*a2-2-alpha*p<0; A3'*P+P*a3-2-alpha*p<0; A4'*P+P*a4-2-alpha*p<0; Tr(P) == 1; P>0;

With a simple Matlab script that performs the bisection and checks the feasibility results returned by sdpsol, we can solve the problem easily. In fact, since a > 0 and

we can use bisection over the interval [0,2.12] and get the optimal bound w 0.33.

4.5

Summary

Using a parser/solver (such as sdpsol described in this paper) to solve SDPs and max-det problems has the following advantages: • Problems with matrix structure can be conveniently specified. The process of converting problems into the standard forms is automatized. • Problem structure is kept during compilation and can be exploited fully by sophisticated solvers to gain efficiency. • The solver can be easily updated or changed by modifying its interface with the parser. There is no need to change the problem specifications.

Acknowledgments The authors would like to thank Lieven Vandenberghe, Laurent El Ghaoui, Paul Dankoski, and Michael Grant for very helpful discussions, comments, and suggestions.

This page intentionally left blank

Part III

Analysis

This page intentionally left blank

Chapter 5

Parametric Lyapunov Functions for Uncertain Systems: The Multiplier Approach Minyue Fu and Soura Dasgupta 5.1

Introduction

This chapter proposes a parametric multiplier approach to deriving parametric Lyapunov functions for robust stability analysis of linear time-invariant (LTI) systems involving uncertain parameters. This new approach generalizes the traditional multiplier approach used in the absolute stability literature, where the multiplier is independent of the uncertain parameters. The main result provides a general framework for studying multiaffine Lyapunov functions. It is shown that these Lyapunov functions can be found using LMI techniques. Several known results on parametric Lyapunov functions are shown to be special cases. Our focus is on the existence of parametric Lyapunov functions. Our motivation for considering parametric Lyapunov functions stems from two facts. First, they can offer less conservative robust stability conditions than parameter-independent Lyapunov functions that are used in the quadratic stability theory. Second, they can be applied to stability analysis of systems with time-varying parameters. In the latter case, one possible approach is to find a parametric Lyapunov function that ensures that the "frozen" uncertain system (i.e., the uncertain system with "frozen" parameters) is robustly stable with certain stability margin. This margin can then be used to determine the "average" time variation for the parameters that could be withstood without losing stability. The use of parametric Lyapunov functions can be traced back to the work of Parks [325], who proves the Routh-Hurwitz stability condition for polynomials directly using a Lyapunov matrix called Hermite matrix. The unique feature of the Hermite matrix is that it is bilinear in the coefficients of the polynomial. The well-known Popov criterion [337], when specialized to dealing with a single constant uncertain parameter rather than sector-bounded nonlinearity, yields a Lyapunov function that depends on the un95

96

Fu and Dasgupta

certain parameter affinely. In fact, most absolute stability criteria given in the literature give, explicitly or implicitly, a parametric Lyapunov function when the uncertainty or nonlinearity they consider reduces to constant uncertain parameters. Recently, Dasgupta et al. [92] studied the existence of parametric Lyapunov functions for the following family of systems:

and h(q) is affine in the elements of q. This type of uncertainty is called rank-one uncertainty. It is shown in [92] that the uncertain system above is robustly stable, i.e., stable V g e Q, if and only if there exists some constant stable matrix F and a compatibly dimensioned vector w such that the augmented uncertain system

admits a parametric Lyapunov function P(q) that depends on q multiaffinely. A procedure for constructing F, iy, and P(q) is given in [92]. It is also shown how this result is used in robust stability analysis for linear systems with time-varying parameters. In Feron, Apkarian, and Gahinet [135] and Haddad and Bernstein [183], a more general class of uncertain matrices

are considered along with the so-called generalized Popov criterion to generate parametric Lyapunov functions. This kind of uncertain system is also studied by Gahinet et al. [149] using an affine Lyapunov matrix

where P», i = 0,1,...,p, are symmetric. In contrast to [92], the robust stability tests given by the generalized Popov criterion and an affine Lyapunov matrix above are conservative in general. In this chapter, we study a more general family of uncertain systems described by

where the nominal matrix AQ € 7£nxn is assumed to be stable, B € 72.nxm is a constant full-rank matrix, m < n, C(q] 6 7£mxn and D(q) € ftmxm are affine in q as follows:

and D(q) is invertible V q € Q. Also, we seek a multiaffine Lyapunov matrix P(q) PT(q) > 0 of the form

5.2. Multipliers and robust stability

97

such that

The tool we use for establishing parametric Lyapunov functions is the multiplier approach, which is popularly used in the absolute stability literature. The basic idea of this approach is to find a transfer matrix of certain structure, called multiplier, such that the cascade of the multiplier and some transfer matrix related to the uncertain system is strictly positive real. What makes our approach unique is the use of parametric multipliers, whereas in the literature only constant multipliers are used. Our main result gives a sufficient condition for the existence of a multiaffine Lyapunov function in terms of the existence of an affine multiplier with a special structure. It turns out that this multiplier can be found by solving a set of linear matrix inequalities (LMIs). The solution to the LMIs automatically provides a multiaffine Lyapunov function for the given uncertain system. We then focus on the analysis of the conservatism of the method we propose. Two schemes have been analyzed in detail, namely, the so-called generalized Popov criterion for the multiparameter case, which has been studied by many authors, and the so-called affine quadratic stability (AQS) test for the single parameter case. It is found that the former renders a constant multiplier, and the latter results in a constant multiplier in the single parameter case and an affine multiplier in general. Subsequently, a frequency domain interpretation of these two schemes is given.

5.2

Multipliers and robust stability

To motivate the multiplier approach, we consider the following uncertain transfer matrix associated with (5.6):

Lemma 5.1. The system (5.6) is stable if and only ifG~1(s^q) is stable. Proof 5.1. The characteristic polynomial of the system (5.6) is given by

Hence, the eigenvalues of the system (5.6) coincide with the poles of G~l(s,q) modulo the stable eigenvalues of AQ. In view of the result above, we are interested in finding an affine transfer matrix, called affine multiplier, of the form

such that the transfer matrix below

is strictly positive real (SPR) V q € Q. A few remarks are in order.

98

Fu and Dasgupta 1. A direct consequence of the existence of such a multiplier is that G~1(s,q) and hence the system (5.6) are robustly stable. This follows from the trivial fact that an SPR function is stable. 2. If B and C(q) are rank one, which is slightly more general than the uncertain system (5.1) where D(q) = 1, the corresponding G(s,q) is a single-input singleoutput transfer function depending on q affinely. That is, the numerator coefficients of G(s,q) are affine in q and the denominator of G(s,q) is parameter independent. In this case, it is known that G~l(s,q) is robustly stable if and only if a parameter-independent multiplier K(s) (possibly with a high order) exists to render K(s)G~1(s,q) robustly SPR; see Anderson et al. [12]. The parametric Lyapunov functions given in [92] are actually based on the existence of such a multiplier. The multipliers we allow in this chapter are more general than [92] in the sense that they are paremeter dependent but more restrictive in the sense that they share the same AQ as G(s,q). As we will see later, this restriction is used to assure that the multiplier will yield a multiaffine Lyapunov matrix (5.8). 3. Although the multiplier approach is a frequency domain method, it has a natural state-space domain interpretation. This interpretation is achieved using a version of the well-known Kalman-Yakubovich-Popov (KYP) lemma called the parametric KYP lemma that is to be introduced in the next section. This result is the key device for deriving a multiaffine Lyapunov function from an affine multiplier. It is this lemma that motivates the use of parametric multipliers.

5.3

Parametric KYP lemma

This section presents a version of the well-known KYP lemma, which is instrumental in dealing with systems with uncertain parameters. To prepare for the result, we first introduce a known generalized KYP lemma by Willems [419]. Lemma 5.2 (generalized KYP lemma). Given A € R nxn , B € R nxfc , and symmetric fi e R("+fc) *("+*), there exists a symmetric matrix P € R nxn such that

if and only if there exists some e > 0 such that

Rirther, if A is Hurwitz stable and the upper left nxn block of ft is positive semidefinite, then P above, when it exists, is positive definite. Our desired result is given as follows in Lemma 5.3. Lemma 5.3 (parametric KYP lemma). Given matrices A € R n x n ,5 e R nxm , ra < n, a hyperrectangular set Q C Rp, a parametric matrix £l(q) € R( n + m ) x (n+m) described by

5.3. Parametric KYP lemma where OM(
JS

99

multiaffine in q and

Then the following conditions are equivalent: (i) There exists e > 0 such that

(ii) There exists a multiaffine matrix

such that

(iii) The inequality (5.17) hoJds at all vertices of Q. (iv) The inequality (5.19) holds at all vertices of Q. Proof 5.2. The equivalences (i)<=>(iii) and (ii)<=>(iv) are obvious due to the assumption in (5.16). The implication (ii)=>(i) follows directly from Lemma 5.2. More precisely, for each fixed q e Q, there exists e(q] > 0 such that (5.17) holds V Re[s] > — e(q). Since Q is a compact set, exists and is positive. Subsequently, (5.17) holds for Re[s] > — e. To show (iii)=>(iv), we apply Lemma 5.2 and induction. Without loss of generality, we assume Q = [0, l]p. Suppose (iii) holds for p = 1; i.e., Q = [0, 1]. Let PQ and PI be any solutions to (5.19) at q = 0 and
which is affine in q, symmetric, and positive definite. It is obvious that the corresponding U(q) is negative definite at q = 0 or 1. That is, (iv) holds for p = 1. Suppose (iii)=>(iv) for p = k; we need to prove it for p = k + 1. Assume (iii) holds for p = k + 1 and write q = (qk,qk+i). Then, (5.17) holds V qk G [0, I]k,qk+i = 0, and qis+i = 1. By the assumption, there exists Po(qk) and Pi(qk), both multiaffine, symmetric, and positive definite, such that the corresponding Il(g), denoted by HO^) and IIi(g fc ), are negative definite V qk 6 [0, l] fc . Now we apply the same "trick" for p = I again. That is, define

Then P(q) is multiaffine, symmetric, and positive definite. Also, it is straightforward to verify (5.19) at qk € [0, l]k, qk+i = 0, and 1, when the above P(q) is applied. That is, (iv) holds for p = k + 1.

100

Fu and Dasgupta

Remark 5.1. It is important to .know that the proof above gives an algorithm for constructing the multiaffine Lyapunov matrix P(q) required for the parametric KYP lemma. Alternatively, P(q) can be searched using SDP algorithms because (5.19) is an LMI at each vertex of Q. The latter is efficient especially when only a few terms of Pj, P^,..., are sought.

5.4

Main results on parametric Lyapunov functions

Under a "convexity condition," our main result below provides a necessary and sufficient condition for the existence of a multiplier of the form (5.11). This condition automatically renders a multiaffine Lyapunov matrix. Theorem 5.1. Given the uncertain system in (5.6), suppose there exists an a/fine multiplier K(s,q) of the form (5.11) such that the transfer matrix H(s,q) in (5.15) is SPR at all vertices of Q. In addition, the convexity condition below is satisfied:

Then, the following properties hold: (i) H(s,q) is SPR V q E Q.

(ii) H(s, q) has the following nth-order realization

(iii) There exists a multiafRne P(q) = PT(q) of (5.8) to establish the robust SPR property of H(s,q), i.e.,

holds V q € Q, where

(iv) (5.24) holds V q € Q if and only if it holds at all corners ofQ. (v) The same P(q) above is a Lyapunov matrix for establishing the robust stability of (5.6). Conversely, if there exists P(g) of the form (5.8) and a multiplier K(s, q) of the form (5.11) such that the convexity condition (5.22) is satisfied and that the LMI (5.24) holds at all vertices of Q, then K(s, q)G~1(s, q) is SPR V q € Q. Remark 5.2. The property (iv) shows that the multiaffine Lyapunov matrix P(q) can be searched using LMI techniques. Indeed, the matrix in (5.24) is linear in PO, P», Py, . . . , (defined in (5.8)). Subsequently, the existence of such P(q) is equivalent so that the 2P LMIs (5.24), one for each corner of Q, are feasible.

5.4.

Main results on parametric Lyapunov functions

101

Now the proof of Theorem 5.1 follows. Proof 5.3. Define

Then, K(s,q)G~1(s,q) being SPR V q e Q is equivalent to the existence of e > 0 such that

V Re[s] > —e and
Using Lemma 5.3, the SPR condition above is equivalent to the existence of a multiaffine P(q) such that the following holds at all vertices of Q:

Pre- and postmultiplying both sides by

and its transpose, respectively, will yield (5.24). Remark 5.3 (LMI solution). The inequalities (5.22) and (5.24) represent a finite set of LMIs (p from (5.22) and 2P from (5.24)). These LMIs are jointly convex in PQ, Pi, Pij..., CM, and Dki- Thus, efficient LMI algorithms, such as interior-point algorithms, can be applied to compute the affine multiplier and and the associated multiaffine Lyapunov matrix. The following result shows that the form of the affine multiplier can be simplified to have one fewer pair of Cki,Dki- In particular, for single parameter uncertain systems, the existence of affine multipliers is equivalent to the existence of a constant multiplier, provided that the convexity condition mentioned above is satisfied. Theorem 5.2. Given the uncertain system in (5.6), suppose there exists an affine multiplier K(s,q) of the form (5.11) such that the transfer matrix H(s,q) in (5.15) is SPR at all vertices ofQ and the convexity condition in (5.22) is satisfied. In addition, assume that [Cp, Dp] has rank m (full rank). Then there exists Ep^nmxm such that Further, Jet and

Then K(s^q)G~1 (s,q) is SPR V q e Q. In particular, ifp = 1 then the resulting multiplier K(s, q) is parameter independent.

102

Fu and Dasgupta

The proof of the theorem above hinges upon the following lemma. Lemma 5.4. Suppose A,B ennxk,k
Then there exists some H € nnxk such that

Proof 5.4. Let T e 7£nxn be a nonsingular transformation matrix such that

and denote Then (5.28) implies

This results in BI =0. Consequently,

So, H = BI and B = AH and rank(B) = rank(#). Finally, H + HT > 0 follows from (5.28) and the full-rank property of A. So (5.29) holds. Now the proof of Theorem 5.2 follows.

Proof 5.5. Using Lemma 5.4, the convexity condition (5.22) and the full-rank property of [Cp, Dp] imply that Ep exists to satisfy (5.25). Since K(s, q)G~1(s, q) is SPR V q G Q, there exists e > 0 such that

It follows that

Remark 5.4. The assumption that [Cp, Dp] is full rank is always satisfied for the single parameter case. This condition, however, may not be satisfied in general. Whenever this condition holds, the result above shows that the LMIs (5.22) and (5.24) are simplified without any harm by setting [Ckp Dkp] = 0. It is, however, difficult to simplify further because the resulting [Cki, Afci] may not meet the convexity condition. Remark 5.5. The result above shows how to simplify the multiplier. But it makes no implication that similar simplification can be made for the Lyapunov matrix. For example, it is not claimed that a constant multiplier renders a constant Lyapunov matrix.

5.5. Comparison with other schemes

5.5

103

Comparison with other schemes

Theorem 5.1 gives a sufficient condition for the existence of a multiaffine Lyapunov matrix that is of the same order as the plant. In this section, we show that two other known schemes for obtaining parametric Lyapunov functions are special cases of Theorem 5.1. One of these two schemes is a generalized Popov criterion derived based on the so-called 5-procedure [135, 149, 203, 183, 380], and the other is called the AQS test by Cabinet et al. [149]. We show that the former gives a parameter-independent multiplier, while the latter gives a parameter-independent multiplier in the single parameter case and an affine multiplier in general.

5.5.1

Generalized Popov criterion

Consider the following system:

Or, equivalently,

where q = (gi, . . . ,qp) € Q = [— 1, l]p (without loss of generality), A0,Ai £ R n , i = 1, . . . ,p, and AQ is asymptotically stable. The generalized Popov criterion seeks a Lyapunov function of the following form [149]:

Note that this function appears to be more general than an affine quadratic Lyapunov function due to the integral terms. The following sufficient condition is established [135, 149], which is generalized from [183, 203, 380]. Lemma 5.5. The system (5.31) is robustly stable with the Lyapunov function (5.33) if there exists symmetric matrices PQ, Pd = diag{Pj}, S = diag{S"i} > 0 and skew symmetric matrix T = diag{Ti} such that the following LMI is satisfied:

What we intend to show here is that the condition in Lemma 5.5, if satisfied, will lead to a parameter-independent multiplier A"(s), which in turn leads to a multiaffine Lyapunov function without an integral term (cf. (5.33)). Theorem 5.3 is the result.

104

Fu and Dasgupta

Theorem 5.3. Consider the uncertain system (5.32) and suppose that the condition in Lemma 5.5 is satisfied. Define A = diag{gi/n}, C(q) = AC,

Then K(s) is invertible and K(s)G-1(s,q) is SPR V q E Q.

Proof 5.6. Suppose the condition in Lemma 5.5 holds. A straightforward application of Lemma 5.3 yields

for some e > 0, where Define

It is straightforward to verify that

That is,

holds V A. The idea above is in fact borrowed from [277]. It is straightforward to verify that an alternative representation of K(s) in (5.36) is given by

Noting that S and Pd are symmetric and T is skew symmetric, we have that (5.37) is equivalent to

Remark 5.6. Note that the Lyapunov function in (5.33) involves integral terms. However, the existence of a multiplier K(s) in (5.36) implies that an alternative Lyapunov function can be found that does not involve integral terms. This claim follows from property (iv) in Theorem 5.1. The trade-off is that the new Lyapunov function may involve multiafRne terms. But in the special case where p= I, we can conclude that the Lyapunov function obtained using Theorems 5.3 and 5.1 is affine.

5.5. Comparison with other schemes

5.5.2

105

The AQS test

Consider the uncertain system (5.31) and the affine Lyapunov matrix

The AQS test proposed in Gahinet et al. [149] amounts to finding such P(q] that the following two conditions are satisfied:

The use of the constraint (5.40), which adds to the conservatism of the method, is to assure that (5.39) holds V q € Q. The AQS test requires us to solve the set of LMIs above. The result below gives an interpretation of the AQS in terms of the multiplier approach. Theorem 5.4. Given the uncertain system (5.31), suppose there exists an affine Lyapunov matrix of the form (5.38) such that (5.39) holds. Define

Then, K(s, q)G~l(s, q) is SPR V q e Q. The convexity condition for K(s, q) as stated in Theorem 5.1 hoJds when (5.40) is satisfied. Further, for the single parameter case, let AI be decomposed into

where B € R nxm , C e R,mxn are full-rank matrices and m < n. Then there exists some such that

and

Proof 5.7. The properties about_K(sLq)G~l(s, q) are trivial. The existence of H follows from Lemma 5.4. To show that K(s)G~l(s,q) is SPR V q e [-1, 1], we take p = 1 and note that (5.39) implies the existence of e > 0 such that

106

Fu and Dasgupta

Pre- and postmultiplying the above by the full-rank matrix (si — Ao)B and its Hermitian gives Using (5.44), the above becomes

5.6

Generalization to discrete-time systems

Consider the uncertain discrete-time system where all the entries are the same as in the continuous-time case. Searching for parametric Lyapunov functions in the discrete-time case seems more involved than in the continuoustime case. The reason is that the stability requirement in the discrete-time case becomes finding P(q) = PT(q) > 0 such that Even when A(q] and P(q) are both affine, the inequality above involves cubic terms. Alternatively, the following equivalent condition to (5.48) can be analyzed:

This condition is often used for quadratic stability analysis when P(q) is restricted to be constant. When parametric Lyapunov matrices are considered, condition (5.49) again seems to be powerless because P~l(q) is nonlinear in q. In this section, we show that the multiplier idea studied in previous sections can be generalized to discrete-time systems with ease. We first introduce a counterpart of the parametric KYP lemma for discrete time. Lemma 5.6 (discrete-time parametric KYP lemma). Given matrices A, B, £t(q),Q as in Lemma 5.2, the following two conditions are equivalent: (i) There exists 0 < e < 1 such that

(ii) There exists a multiaffine matrix

such that

5.7.

Conclusions

107

(iii) The inequality (5.50) holds at all vertices of Q. (iv) The inequality (5.52) holds at all vertices of Q. Proof 5.8. The proof is essentially the same and is based on a discrete-time version of Lemma 5.2. So the details are omitted. Applying the lemma above, the counterpart of Theorem 5.1 is easily obtained. Theorem 5.5. Given the uncertain system in (5.47), suppose there exists an affine multiplier K(z,q) of the form (5.11) such that the transfer matrix H ( z , q ) in (5.12) is SPR at all vertices of Q. In addition, the convexity condition in (5.22) is satisfied. Then the following properties hold: (i) H(z,q) is SPR V q E Q.

(ii) H ( z , q ) has the nth-order realization

(iii) There exists a multiaffine P(q) = PT(q] to establish the robust SPR property of H(z,q); i.e.,

hoids V q e Q, where

(iv) The same P(q) above is a Lyapunov matrix for establishing the robust stability of (5.47). Conversely, suppose there exists P(q) of the form (5.8) and a multiplier K(z,q) of the form (5.11) such that the convexity condition (5.22) is satisfied and that the LMI (5.54) holds at all vertices ofQ. Then K(z, q)G~l(z, q) is SPR V q € Q. Proof 5.9. The proof is virtually identical to the continuous-time case. The details are thus omitted. D Remark 5.7. Note that, as in the continuous-time case, (5.54) is affine in PQ> PI, Pij, ••••> C'fci? Dki. Hence, finding a multiaffine Lyapunov matrix P(q) amounts to solving a finite number ofLMIs (p for (5.22) and 2P for (5.54) at the vertices of Q).

5.7

Conclusions

In this chapter, we have studied the use of the multiplier approach to generate parametric Lyapunov functions for linear systems with parameter uncertainty. In the process of doing so, we have derived an extended version of the KYP lemma, the parametric KYP lemma, as a general tool to study the robust stability with parameter uncertainty. Using this lemma, we have provided conditions under which an affinely parameterized multiplier

108

Fu and Dasgupta

exists to establish the robust stability of the uncertain system. This type of parametric multiplier then naturally leads to a multiaffine Lyapunov function that can be used to establish robust stability in the state-space domain. Although not studied in this chapter, we point out that parametric Lyapunov functions can also be used in dealing with timevarying parameters; see [92]. Also shown in this chapter is that some previous results in the literature on parametric Lyapunov functions lead to special multipliers. This analysis confirms the generality of our approach. We have also demonstrated the ease of adapting our approach to discrete-time systems despite the observation that uncertain parameters in the discrete-time case appear to be harder to deal with.

Chapter 6

Optimization of Integral Quadratic Constraints Ulf Jonsson and Anders Rantzer 6.1 Introduction Computational issues in stability and performance analysis have been studied for a long time. In particular, it was shown in a classical paper by Zames [444] that a stable linear system G in negative feedback interconnection with a slope restricted nonlinearity is stable if there exists a "multiplier" M such that

The multiplier can be selected from a specific class, and Zames noted that the main difficulty in applying this stability condition was to find a suitable multiplier from the class. Several more sophisticated multiplier-based stability results were developed during the period 1965-1975. For example, a criterion for slope restricted nonlinearities was derived in [445] and a criterion for slowly time-varying parameters in [385]. See also the books [98, 419]. All these results had their main limitation in the computability of the multipliers. The difficulty was mainly due to the lack of powerful computers and suitable numerical software. Since the early 1980s a lot of work has been focused on robustness analysis of uncertain systems. In particular, multiplier-based methods for diagonally perturbed linear timeinvariant (LTI) systems received attention through the work in [102, 350]. The theory was later extended to treat also parametric uncertainty [131, 441]. Since both the system and the uncertainty are time invariant, it is possible to do the analysis frequency by frequency; see [31, 314, 443]. This is not the case for systems that contain time-varying or nonlinear components. The development and implementation of efficient algorithms for solution of convex optimization problems involving linear matrix inequalities (LMIs) has greatly increased the possibilities for efficient computations in system analysis. For references, see [64,119, 153, 296]. Multiplier computation in analysis of systems with various forms of dynamic and parametric uncertainty was treated in [28, 29, 191, 262]. The idea in these papers is 109

110

Jonsson and Rantzer

to find a finite-dimensional affine parameterization of a subset of the multipliers. The stability criterion restricted to this subset is convex, but still includes an infinite number of constraints corresponding to different frequencies. However, using the KalmanYakubovich-Popov (KYP) lemma, it is possible to replace the frequency dependence with an additional matrix variable. In this way, the problem is converted to a finite-dimensional LMI. Characteristic for all the LMI-based methods for multiplier optimization mentioned so far is that each approach only considers a restricted class of problems. An important reason for this is the appearance of nonconvex side conditions in the traditional frameworks for stability analysis based on passivity, dissipativity, and parameterized Lyapunov functions. For example, in multiplier-based passivity analysis there are generally nonconvex invertibility and factorizability conditions on the multipliers. A unified approach for multiplier-based robustness analysis was suggested in [268]. The idea is to use integral quadratic constraints (IQCs) to describe uncertainty, timevarying parameters, performance criteria, and frequency characteristics for signals. The methodology was later extended in [273] to give a theory that allows us to combine multipliers according to various robustness criteria without any need for invertibility or factorizability conditions. In this chapter we suggest a unifying format for multiplier computation by LMI optimization. The format is based on the IQC paradigm. The main idea is to find finite-dimensional parameterizations of the multipliers in terms of a basis. The stability condition is then transformed to an equivalent finite-dimensional convex feasibility test that can be solved by LMI methods. All the examples mentioned above can be treated in our format. We discuss robustness analysis with IQCs and in particular robust performance analysis with IQCs in the first three sections.

Preliminaries We let RL™ xm denote the proper real rational matrix functions with no poles on the imaginary axis. The subspace RHoo C RL™ xm consists of functions with no poles in the closed right half plane. The truncation operator PT is defined as

for any function /. We let L™[0, oo) denote the space of Rm-valued square integrable functions with norm defined by

The extended space L^[0, oo) is the vector space of functions / satisfying Ppf 6 L^fO, oo) for all T > 0. An operator H : L£[0, oo) -* L£[0, oo) is said to be causal if PTHPT = PTH for all T > 0. This means that the value at a certain time instant does not depend on future values of the argument. A causal operator H on LJ£[0, oo) is bounded if H (0) = 0 and if the gain defined as

is finite; see [419] for a more detailed discussion.

6.2. IQC-based robustness analysis

111

Figure 6.1: System consisting of a nominal plant G and a bounded causal perturbation A.

6.2

IQC-based robustness analysis

We consider systems consisting of a nominal transfer function G in a positive feedback interconnection with a bounded causal operator A; see Figure 6.1. We will for simplicity of notation consider only square systems. Extension to the nonsquare case is immediate. The basic idea behind the IQC approach for robustness analysis is to find a description of A in terms of bounded and Hermitian- valued multipliers, II, that satisfy

V square integrables y and v = A(y). Here y and v denote the Fourier transforms of y and v, respectively. All multipliers used in this chapter are assumed to be rational transfer functions. The reason for this becomes apparent when we discuss multiplier computation in section 6.5. We will generally consider multipliers consisting of a proper part H# = H^ 6 and a nonproper Popov multiplier of the formS

where A G R m x m . The Popov multiplier gives useful descriptions of static nonlinear ities, parametric uncertainties, and slowly varying parameters; see, for example, [221]. We use the Popov multipliers in (6.3) to define constraints of the form

where 7 > 0. We note that in order to use the Popov multiplier we must ensure differentiability of y. This enforces a condition on strict properness of the nominal transfer function G(s) that would be overly restrictive in applications when A is a sparse matrix. This problem can be overcome. If we introduce PA : Rm —* Rm as the orthogonal projection onto the range of AT, then we can use the derivative (P^y)' = ^(P\y} m (6-4) instead of y. This works since Im^A7")-1 = Ker(A). Hence, P\y _L Ker(A) and only the part of y that contributes to the integral in (6.4) needs to be differentiated. The following definition of IQC will be used in the chapter.

Definition 6.1 (IQC). Let UB = U*B e RL^x2m and let UP be the Popov multiplier in (6.3). We say that A satisfies the IQC defined by H = UB + HP if there exists a positive constant 7 such that

V y and v = A(y) such that y, (P^y)',v € L^O, oo). We use the notation yo = y(0).

112

Jonsson and Rantzer

In order to obtain the most accurate robustness condition we need to find as many multipliers for the perturbation A as possible. The next two properties can be used to combine multipliers for this purpose. Property 1. Assume that A satisfies the IQCs defined by HI, ... ,IIn. Then A also satisfies the IQC defined by the conic combination Y^=i aiH-ii where a* > 0, i = 1,..., n. Property 2. Assume that A = diag(Ai,..., A n ), and that, for i = 1,..., n, Aj satisfies the IQC defined by

where the block structures are consistent with the size of the perturbations A;. Then A satisfies the IQC defined by

This is easily seen by writing out the expression for the IQC in Definition 6.1. It follows from the first property that we always can characterize A with a convex cone of multipliers. We use the notation HA for any such convex cone of multipliers that describe A in the sense that A satisfies the IQC defined by II for every II G HA- We note that such a multiplier description in general only is a partial characterization of A and the corresponding robustness criterion will only be a sufficient condition. In applications it is important to find a description of A that is as accurate as possible. A description of A can be improved by addition and augmentation of convex cones of multipliers. Addition. If A is described by the convex cones HJA? i = l , . . . , n , then A is also described by

Diagonal augmentation. If, for i = l,...,n, A* is described by the convex cone HA;, then A is described by the convex cone

6.2.1 Robust stability analysis based on IQCs We consider stability of the system in Figure 6.1. The system equations are given by

We assume that G is a bounded causal operator with transfer function G € RH™ xm . The perturbation A is a bounded causal operator on L^O, oo). The input g is arbitrary in L^O, oo) and / € L^fO, oo) is assumed to satisfy the differentiability condition (PA/)' € L^O, oo). We think of / as the response due to the initial conditions. Stability and well-posedness for the system in (6.5) is defined as follows.

6.2. IQC-based robustness analysis

113

Definition 6.2. The feedback interconnection of G and A is well posed if there exists a solution (y,(P A y)» € L£[0,oo) x Lg[0,oo) x L£[0,oo) for any (/,(P A /)',0) € L™[0, oo ) x L™[0,oo) x L^[0, oo) and if the map (/,-» (y,w) is causal. The feedback interconnection is stable if in addition there are positive constants ci, 02, and ca such that

The next theorem gives conditions for stability of the system in (6.5). The case when HP = 0 follows from [273]. The more general situation with Popov multipliers was proved in [220, 221]. Similar results can be obtained by using the ideas in Corollary 4 in [273]. Theorem 6.1. Let G £ RH^xm and let A be a bounded causa] operator on I/£[0, oo). Assume that (i) for ail T € [0, 1], the interconnection of G and rA is we]] posed; (ii) for all r € [0, 1], rA satisfies the IQC denned by U = UB + HP; (iii) AG(oo) = 0, where A is the Popov parameter in HP; (iv) the inequah'ty

Then the feedback interconnection of G and A is stable. Remark 6.1. It is sometimes usefu] to consider more general parameterizations than rA. This is needed, for example, for the IQC that describes an uncertain time delay in an example later in this chapter. More generalized statements of the theorem that provides suitable conditions on such a parameterization are given in [220, 342] and [271]. Remark 6.2. The third condition of the theorem statement ensures that P\y is differentiable. It can be omitted when Up = 0.

6.2.2

Robust performance analysis based on IQCs

For robust performance analysis we consider the system described by the equations

where A is a bounded causal operator on L^[0, oo) and where G is a bounded causal operator with rational transfer function. We assume that G is block partitioned according to the size of the signals, i.e.,

System (6.6) can be illustrated with the block diagram in Figure 6.2. Performance of the system in (6.6) is generally measured in terms of disturbance attenuation. A measure of this attenuation can, for example, be obtained in terms of the

Jonsson and Rantzer

114

Figure 6.2: System setup for robust performance analysis. energy ratio of the error signal z and the disturbance w. It is important to exploit the spectral characteristics of the disturbance w when studying such a performance measure. We will—as has been suggested in, for example, [269, 268] and [317]—study performance criteria in terms of signals from a set Winp C L^O, oo). This set is assumed to be defined in terms of a convex cone of self-adjoint multipliers, Tinp C RL<£9, in the following way:

We note that Winp = L|[0, oo) if T,np = {0}. If in a certain application we have a set of signals W, then we need to find a convex cone Tjnp C RL[0, oo) defined by the convex cone Tinp. The system in (6.6) is said to have robust performance according to the performance specification nperf = n*erf € RL^X 9 if (i) the feedback interconnection of GU and A is stable; (ii) the inequality

The next theorem gives conditions that ensure robust performance for the system in (6.6). Theorem 6.2. Assume that (i) the interconnection ofGu and A is stable; (ii) A satisfies the IQC defined by (iii) A [ GH(OO) £12(00) ] = 0, where A is the Popov parameter in Up; (iv) there exists T G Tinp such that

Then the system in (6.6) has robust performance.

6.3. Signal specifications

115

Proof 6.1. The proof is given in Appendix 6.8.1. Remark 6.3. The third condition of the theorem can be omitted if Up = 0. Remark 6.4. The stability assumption (i) in Theorem 6.2 can be replaced by the following conditions: (ia) for any T e [0, 1], the interconnection of G\\ and rA is we]] posed; (ib (ii)' for any T € [0, 1], rA satisfies the IQC denned by This is also proved in Appendix 6.8.1.

6.3

Signal specifications

Spectral information on the disturbance should be used to reduce conservatism in robust performance analysis. The multiplier set Tinp, which defines the input signal to be in a conic subset Winp C L^O, oo), can be used to describe various types of spectral information. In order to get an idea of how Tinp should be chosen we consider the case when L2-performance is studied, and we know that the input signal is constrained to be in the set Winp defined by Tj np . For any T 6 Tinp we obtain the combined multiplier

We can see from the frequency domain criterion in (6.7) that T is only contributing in a useful way at frequencies where it is not positive definite. We give two examples that show how Tinp can be chosen for two important signal classes. The first was suggested in [268]. Dominant harmonics Let w e 112(0,00) be a bandpass signal with supp w € [— 6, — o] U [a, 6], where supp w denotes the support of the Fourier transform of w. Then w is in the set Wjnp defined by

It is clear from the robust performance condition in (6.7) that we should choose a multiplier in Tinp that is as close as possible to

Signals with a given spectral characteristic Noise signals in a system are often modeled as stochastic white noise or filtered stochastic white noise. In the IQC framework it is more convenient to consider so-called deterministic white signals. A realization of a stochastic white signal has infinite energy and it would for that reason be appropriate to consider the deterministic white signals to be in L^O, oo). This was the approach in [269]. A recent approach for deterministic white signals with finite energy was developed by Paganini in [317, 320] and Chapter 7 of this

116

Jonsson and Rantzer

book. We will consider a class of signals with given spectral characteristics that gives analysis results that are similar to Paganini's. For simplicity we consider only scalar signals. Let

where H e RH^ 1 is strictly proper and normalized such that

We can regard W as a set of filtered deterministic noise signals. We have the following lemma. Lemma 6.1. We have W# = VVinp when

Proof 6.2. Let w € WH and T e Tinp; then

which proves that W# C Winp. For the opposite direction we first note that Winp is unchanged if we use

where C(jR) denotes the continuous functions on the imaginary axis that are real valued, even, and have a limit at u; = oo. This follows since the set of T = T* € RL^1 is dense in C(jR). The claim follows if we show that no w & WH can be in Winp. For this we define W0(u) = f£ \H(jv)\*dv. This is a function of bounded variation. It belongs to the dual space of C(jR), and it defines a linear functional in terms of the following Stieltjes integral:

It follows that Tinp is the half-space

Every nonzero w & WH defines an element W(u) = f£ |ty(jV)|2 dv of bounded variation, which defines another half-space. It follows that there exists T € T}np such that (T, W) < 0. This proves that W// = Winp.

6.4

A robust performance example

We will here illustrate the ideas from the last two sections with a simple example. Consider the system in Figure 6.3. The perturbation A is assumed to be a dynamic uncertainty with || A || oo < 1, where || • ||oo denotes the HOQ norm. We assume that w belong to the set in WH in (6.9) for some strictly proper filter H. We want to compute th

6.5. LMI formulations of robustness tests

117

Figure 6.3: System for the example in section 6.4. performance measure

where the closed-loop system operator is defined as

If we assume that the nominal system is stable, then the system in Figure 6.3 can be put into the form in Figure 6.2 with

We will now use Theorem 6.2 to formulate an optimization problem for the computation of an upper bound of (6.10). The spectral characteristic of w can be described by Tjnp in Lemma 6.1 and A is described by the multipliers

We can ensure that the optimal value of (6.10) is less than 71/2 if the condition (iv) in Theorem 6.2 holds for some II € HA and T e Tinp and with

The best upper bound is obtained by solving the optimization problem

This is an infinite-dimensional optimization problem. We will in the next section formulate a methodology that can be used to obtain suboptimal solutions to this optimization problem. Numerical computations for this example will be given in section 6.6

6.5

LMI formulations of robustness tests

In this section we derive a method for practical application of the robust stability and performance results in section 6.2. The first step in the analysis is to derive a multiplier

118

Jonsson and Rantzer

description of the perturbation A. The multipliers are collected in a set HA that can be assumed to be a convex cone. In case of performance analysis we also need multipliers for the performance criterion and multipliers that constrain the range of the input to the system. In order to keep the presentation as simple as possible we treat only the case of robust stability analysis. Extension to robust performance analysis is straightforward. Robust stability analysis based on IQCs can now be formulated as the following convex feasibility test. Robustness Test 1 (infinite-dimensional feasibility test). Find II € HA such that

V u ; € [0,oo]. The convex cone HA is generally infinite dimensional. It consists of multipliers on the form II = II B 4- HP, where UB G RL^x2m and where Up is a Popov multiplier. The Popov term lip involves only a finite number of parameters, namely, the parameters in A. However, HB can generally be chosen from a set of rational transfer functions of arbitrary order. The infinite dimensionality of the robustness test makes it hard to implement directly as a numerical algorithm. We will introduce a format for finite-dimensional parameterizations of convex subcones of HA- If the robustness test is restricted to this subcone, we obtain a finitedimensional convex feasibility test that generally can be transformed to an equivalent feasibility problem for LMIs. The subcone is defined in terms of a basis multiplier and a convex cone of parameters according to the following definition. Definition 6.4. Let \£ = [ ^a ^6 ], where \£a and ^/b are N x m rational transfer functions, and let MA C R Nx7V be a convex cone of symmetric matrices. Then > MA) C EA denotes the subcone defined as

We call ^ a basis multiplier, and the set MA is called the parameter set. We note that \£ in general is nonproper due to the Popov multipliers. If we for a given M € MA write out the expression for the corresponding multiplier we see that it has the structure

Before we give an example we notice that addition and diagonal augmentation of subcones on the the form in Definition 6.4 can be done in a simple way. Addition. Let

where Augmentation. Let P

then

6.5. LMI formulations of robustness tests

119

where diag(MA 1 ,MA 2 ) is defined analogously with the same operation in the case of addition. We next give an example that illustrates how parameterizations on the form above can be obtained from well-known multiplier descriptions for a repeated uncertain parameter. Example 6.1. An uncertain real parameter 61, where 6 € [—1,1] can be described by the following sets of multipliers:

We can use the parameterizations

and

Here R 6 RH^ lXm ,5 0 € RH%xm, and7 U e 5Nl, V € R m x N 2 . It is easy to show that any X and Y satisfying the stated properties can be factorized in this way. We can now use the finite-dimensional convex cones Pi&(y\,Mi&,) C HIA, ^2A(^2» A^2A) C , where

where U € SNl, and V e Rm*N*. We note that the constraint that specifies the parameter set MIA is given in terms of a nonstrict LMI. Assume that we have obtained a finite-dimensional cone P&.($!, M&) C HA- If we restrict Robustness Test 1 to PA we get the following finite-dimensional test. Robustness Test 2 (finite-dimensional feasibility test). Find M € MA such that where

is a rational transfer function. 7

«Sn C R nxn denotes the real symmetric matrices.

120

Jonsson and Rantzer

We note that assumption (iii) in Theorem 6.1 implies that $ is a proper transfer function. We will next show how the frequency domain inequality in (6.13) can be transformed to an equivalent LMI. The means for doing this is to obtain a state-space realization of $ and then invoke the KYP lemma, which we state next. Lemma 6.2 (KYP). Let M be a real symmetric matrix. For the system $(s) = C(sI— A)~1B + D, where det(jul — A) ^ 0 V w, the following inequalities are equivalent: (a)

(b) There exists a matrix P = PT £ R n x n , where n = dim(A), such that

The equivalence holds for nonstrict inequalities if in addition (A, B) is controllable. Proof 6.3. The lemma follows from [420] in the case when (.A, B) is controllable. The formulation of the case with strict inequalities does not require controllability. A proof of this is given in [340]. Introduce a realization $(s) = C$(sl — A$)~1B$ + D$. It is by the assumptions on G and \P no restriction to assume that A$ has no eigenvalues on the imaginary axis. Hence, it follows by the KYP lemma that Robustness Test 2 is equivalent to the following LMI test. Robustness Test 3 (feasibility test with LMIs). Find P = PT and M € MA such that

The last constraint is an LMI condition. The parameter set MA can often can be specified in terms of strict or nonstrict LMIs; see, for example, Example 6.1. Strict LMIs are easier to solve numerically. We will here formulate conditions that allow us to solve the nonstrict LMIs as strict LMIs. Since MA is a convex cone we have riMA D MA, where ri MA denotes the closure of the relative interior of MA- This proves the equivalence of the following two problems. 1. Find M € MA such that 2. Find M € riMA such that This means that every constraint in MA that is formulated in terms of a strictly feasible nonstrict LMI can be solved in terms of its strict version.

6.6.

6.6

Numerical examples

121

Numerical examples

In this section we will give two numerical examples that verify the applicability of the suggested method for multiplier computation. All numerical computations are done with LMI-Lab [153]. In the first example we treat the robust performance problem in section 6.4. Example 6.2. We will here briefly show how the robust performance problem in section 6.4 can be transformed to a suboptimal eigenvalue problem. We first find finitedimensional restrictions of the cones HA and Tinp. The most difficult part is to find a finite-dimensional description of the cone Tinp that describes WH in terms oflQCs. It was shown in Lemma 6.1 that WH = VVjnp when

In order to find a finite-dimensional parameterization of Tinp, let T = ^[npU-mp^f-mp, where #inp e RH^xl and where U-mp G SN . Assume that we have the realization $(s) = ^?inp(s)H(s) = C$(sl - A$)~1B$. Then the condition T € Tj n p becomes

A weh'-Jcnowa argument shows that P$ = P^ satisfies the Lyapunov equation

ft follows that we can use "Pi

We can now use the following finite-dimensional cones: HA- We can use ?>&.($, MA) C HA, where

and where R 6 RH^xl and U e SN . nperf • We could use the convex cone Pperf (\&, Mperf (7)) = IIperf (7), where

It is, however, advantageous to fix the value of x to be 1. In other words we Jet

This wi]] aliow us to compute a suboptimaJ vaiue 0/7 in the optimization problem (6.11) as an eigenvalue problem for LMIs.

Jonsson and Rantzer

122

Table 6.1: Numerical results for Example 6.2. Here njec denotes the number of unknown (decision) variables in the problem. It gives a measure of the complexity of the computations. ftdec 80

140 175 215

R 1 1 Ritz(10, 1) Ritz(10,2)

#inp

0

V&O mp V&O *inp X&O x mp

7opt

6.96 0.59 0.54 0.54

. We use

where ^f-mp €

, and where Mjnp is defined as in (6.15) above.

We obtain a finite-dimensional parameterization 7>Tot(^rTot)-Wrot(7)) for the total multiplier in (6.11) as

Hence, we get

and

The optimization problem in (6.11) restricted to the cone
and that the filter H defining the spectral characterestic of the input signal is

We obtain the results in Table 6.1, where

We wih" next show that the numerical results in Table 6.1 are reasonable. First we notice that the optimization problem in (6.10) can be formulated equivalently as

123

6.6. Numerical examples

Figure 6.4: PI control with antiwindup compensation. where

Since the spectral characteristic denned by H is of low-pass form with a sharp resonance around uj = I and since the weighting filter W is of high-pass form with low gain up to uj = 15, it is resonable to expect that the objective of (6.16) changes only a little if we disregard the unmodeled dynamics, i.e., ifW(s) = 0. In this case the optimal value of (6.16) becomes

which is not far from the best value 7opt = 0.54 obtained in Table 6.1. The next example considers stability of an antireset windup compensator for a PI controller. We will obtain stability bounds that cannot be obtained with classical methods such as, for example, the Popov criterion and the off-axis circle criterion. Example 6.3. Consider the control system in Figure 6.4. A process GQC~ST is controlled with a PI controller. There is also an antireset windup compensator to reduce the effect of integrator windup due to saturation in the actuator. The particular antireset windup compensator in this example is of tracking type; see [23] for further details. The idea is to feed the error signal ea to the integrator when the actuator saturates. With a suitably chosen tracking time Tt this device prevents the integrator from integrating up to a large value during saturation. The filter W corresponds to the sensor dynamics. We will in this example study stability of the system in Figure 6.4. In particular we study how large the time delay, T, can be before the system becomes unstable. For this purpose we assume that the time delay is uncertain but restricted to be in the interval [0, TO]. We will search for a bound on TO such that stability is ensured. The system can be transformed to the nominal form in Figure 6.1, with

where (p corresponds to the saturation nonlinearity, AT = e K + K/(sTi).

sT

— 1, and where GPI(S) =

124

Jonsson and Rantzer

We need to find a multiplier description II A for the diagonal operator A. This is done by augmenting multiplier descriptions for the saturation nonlinearity (p and for Ay. The saturation nonlinearity is assumed to be the function

It is an odd nonlinearity with slope restricted to the interval [0,1]. It can thus be described by the multipliers that follow from [445], i.e.,

where h$ > 0, and H is a strictly proper rational transfer function with a corresponding weighting function h that satisfies the the LI-norm condition

In the case that the first row of the system matrix G in (6.17) is strictly proper, then we also use the Popov multiplier

where A e R. The operator AT = e~sT — 1, where T e [0,To], is described by the multipliers

see [273]. We will for the example assume that K = 0.07, T» = Tt = 0.11, and

The strict properness of W implies that the first row of G(s) (6.17) is strictly proper and we can use the Popov multipliers in (6.20) in the anaJysis. In Appendix 6.8.2, we derive the sets Pi^iy, Mlv?), p2 MA T ) ^or tfle multipliers in (6.19), (6.20), and (6.21), respectiveJy. A finite-dimensional multiplier description of the diagonal operator A in (6.18) is then obtained as

It turns out that we can obtain good results with very simple basis multipliers for this particular example. Table 6.2 shows numerical results obtained with LMI-Lab. The obtained bound on TO is very close to optimal. To see this consider the transfer function

6.7. Conclusions

125

Table 6.2: Numerical results for the antireset windup compensator example. The result shows stability for time delays up to T0 = 0.01.

Figure 6.5: The diagram shows the Nyquist curve for Gci- The stability condition in (6.22) can be satisfied with proper choice of the Popov parameter A and the multiplier H.

where the time delay has been included. If the time delay T is given (not uncertain but bounded as before), then stability of the control system in Figure 6.4 can be investigated by considering the inequality

where A 6 R and where H satisfies the same conditions as above. The Nyquist curve of Gci is given in Figure 6.5 for the case when T = 0.01. The system is very close to instability for this choice of time delay. In fact, the high frequency part of the Nyquist curve is very close to pass through the point (—1,0) in which case the system would be unstable for the case with unity gain feedback. However, from the numerical results in Table 6.2 it follows that we can satisfy the condition in (6.22) by using an appropriate Popov paramater A and with a suitably chosen multiplier H.

6.7

Conclusions

We have introduced a format for multiplier computation in robustness analysis. It is applicable to a large class of systems. The resulting computational problem can be formulated as a convex optimization problem in terms of linear matrix inequalities. It is easy to implement the ideas in terms of a set of subroutines that interfaces any of the existing software packages for solving convex optimization problems in terms of LMIs. All manipulation of transfer functions is then done in terms of corresponding state-space realizations. A Matlab 4 version of such a toolbox has been used for the examples in this chapter [224]. A new toolbox for IQC analysis is currently under development by Megretski and coworkers; see [272], The new toolbox uses the new data structures and the graphical user interface builder of Matlab 5.0 to obtain a user friendly environment for system analysis. There are many interesting problems that remain to be solved. It would, for example, be useful to develop efficient mathematical methods for choosing basis multipliers, in other words, methods that can give guidelines for improving a particular basis multiplier.

Jonsson and Rantzer

126

It is also useful in this perspective to have methods for obtaining bounds on what can be achieved with a particular infinite-dimensional cone of multipliers. This gives an indication of the quality of a particular basis. We have used results from duality theory for infinite-dimensional convex programs to obtain such bounds for general classes of problems in [220, 222, 223].

6.8

Appendix

6.8.1

Proof of Theorem 6.2 and Remark 6.4

Let w €. Ii2[0,oo) be in Winp. The stability assumption implies that A(y) € L™[0, oo). Consider the operator

We note that the condition A [ GH(OO)

£12(00) ] = 0 implies that

Hence, H is a bounded and Hermitian- valued operator on L g * (—00,00). Evaluation of the corresponding quadratic functional for [ A(y) w* ]* gives

The equality follows from the observations

and

where T 1 denotes the inverse Fourier transform. We note that assumed that the system is initially at rest. Hence, we get

= 0, since it is

This proves the theorem. The statement of Remark 6.4 follows, since the upper left block of the inequality (6.7) implies that

V u € [0,oo]. By assumption (ib) the second term is nonnegative. Hence, the stability follows from Theorem 6.1.

127

6.8. Appendix

6.8.2

Multiplier sets for Example 6.3

To obtain a finite-dimensional parameterization of the multipliers in (6.19) we introduce strictly proper basis functions Hi G RL^1, i = 1, . . . , N, for a finite-dimensional parameterization of H . We can express H as

where A+, A{ > 0. We can assume that Hi = HiC + Hiac, where the causal part has the realization HiC = CiC(sI — AiC}~lBiC and where the anticausal part has the realization Hiac = Ciac(sl — Aiac}"1 Biac. The Lx-norm of the weighting function corresponding to Hi can be computed as

Then the constraint ensures that \\h\\i < /IQ. To obtain a parameterization in the format ^(^A/A), we define

We can now use

v(^fiv^

MI^,), where ^i^, is as above and where

The Popov multiplier in (6.20) can in our format be represented as PI^\

,), where

To obtain a parameterization on the form P&T(ty, MA) in equation (6.21), we let x(ju) = R(juj)*UR(ju>), where R € RH^,xl is a basis multiplier and where U = UT > 0 are the corresponding coordinates. We note that we have the factorization ^Q(S) = H*(s)H(s), where

We now get the parameterization / PA T (^A T >MA T ),

This page intentionally left blank

Chapter 7

Linear Matrix Inequality Methods for Robust H2 Analysis: A Survey with Comparisons Fernando Paganini and Eric Feron 7.1 Introduction The robust H2 problem is rooted in the efforts of the 1970s to provide stability margins to the H2-optimal (LQG) regulator. The difficulties encountered in this combination of classical and modern control [348, 101, 109] led to a shift in focus to other performance criteria (Hoc, LI), which by means of the small gain theorem are directly linked to robust stability guarantees. Robust control based on the HOC and LI measures is now a mature field [446, 88]. The impression has remained, however, that such methods lean too heavily on robustness and sacrifice an adequate view of performance; the latter is often more naturally described by an H2 criterion, which can be used to capture both the transient response of the system and the response to stationary noise. These interpretations are reviewed in section 7.2. The promise of a successful combination of robustness and H2-performance was renewed in the late 1980s by the close ties between the state-space methods of H2 and HQQ control (see [107]). An abundant literature has pursued combinations of H2 and HQQ in various forms (see [48, 228, 447, 110, 384, 330, 334, 386, 187, 210, 134] and references therein), including mixed K^/Hoo control (multiobjective design) and the related but separate question of robust H2-performance analysis. Focusing on the latter question, in section 7.3 we provide a review of these methods, aiming at the most general results and a streamlined presentation, which is difficult to find in this large literature. Central to this task are linear matrix inequalities (LMIs) and the so-called «S-procedure, which refers (see Chapter 1 or [433]) to the use of multipliers to bound an optimization under nonconvex quadratic constraints; the procedure is called "lossless" when the bound is nonconservative. The above state-space theory has the limitation that it is not accompanied by ex129

130

Paganini and Feron

ternal, input-output descriptions or necessary and sufficient characterizations, as are available for robust stability and HQQ- or Li-performance. In pursuit of these characterizations, an operator version of robust H2-performance was proposed in [318, 319, 320], which views the H2 norm is the worst-case response under a suitable class of "typical" white noise signals. This method also brings in an infinite-dimensional «S-procedure, which in this case is proven to be lossless. In section 7.4 we will review this alternative diiv' the resulting characterizations, both in the frequency domain and in the state space. Ihe surveyed methods for robust H2-performance evaluation will be expressed in terms ^f a series of SDPs. Subsequently, in sections 7.5 and 7.6 we will proceed to compp/e these approaches in mathematical terms, in terms of computational cost, and by illustrative examples, showing their relative strengths and limitations. The common language will be key to providing transparent connections. We conclude in section 7.7 v;dth a few final remarks on the state of this area of research.

7.2

Problem formulation

The chapter will focus on continuous-time signals and systems, but analogous methods can be developed for the discrete time counterpart, which is in fact somewhat easier since some objects (impulses, white noise) are more easily formalized in that setting.

7.2.1

The H2 norm

We begin by defining the H2 norm of a linear time-invariant (LTI) system and remarking on its motivation as a performance measure. Let H be a stable LTI system of transfer function H(ju); the H^ norm of H is defined as

If H is finite dimensional, we can compute this norm using a state-space realization

which must have no feed-through term for the Hfe norm to be finite. Then we have

The main motivations of the H2 norm as a performance measure are the following: 1. \\H\\2 is the steady-state response of the system to white noise; this idealization of signals with flat spectrum over all frequencies is a common modeling tool for disturbances. Both deterministic (see section 7.4) and stochastic formalizations of this notion are possible. The stochastic version can be related to (7.3) by noting that Q satisfying AQ + QAT 4- BBT = 0 is the asymptotic covariance of the state x when it is driven by continuous-time white noise (formally treated in section 7.3.2); thus the steadystate variance of the output z = Cx is Tr(CQCT) = \\H\\%. 2. For scalar inputs, \\H\\% is the energy of the system impulse response; thus this quantity can be used to measure the transient response of an error variable in

7.2.

Problem formulation

131

response to known inputs or initial conditions (which may be generated by an impulse). In the multivariable case, we can write

where e^ denotes the fcth coordinate vector in input space. In other words, the norm measures the sum of impulse response norms over every input channel. Equivalently, \H.^ is equal to JEH-Hvc^)!^, the expected response for an impulse of random direction VQ with covariance E(VQV^} = / (which generates an initial condition of covariance E(XQXQ) = BBT). This interpretation is closely related to (7.2); if P satisfies ATP + PA + CTC = 0, XQPXQ is the energy of the transient response starting from x(0) = XQ; Tr(BTPB) adds this energy over initial conditions obtained by applying an impulse at each channel. 3. For scalar outputs, \\H\\z is the I<2 to LQQ induced norm of the system

and thus measures amplitude deviations for finite energy disturbances. The above motivations, mainly the first two, strongly motivate the use of this norm as a measure of performance. In particular, notice that by adding an input filter or weighting function, we can use the impulse response to study the response to given, known inputs (e.g., steps) and the white noise response to study the response to disturbances of known (colored) spectrum. The common feature is thus the following: when there is information on the spectral content of the inputs, H2 is an adequate performance measure. In contrast, other methods such as H^ treat disturbances or commands as being the worst in a very broad class, which is often unrealistic. Optimal control based on the H^-performance measure has an elegant solution, developed in the 1960s under the name LQG control (see [11]).

7.2.2

Robust H2-performance

Figure 7.1: Uncertain system (M, A). The robust ££2 problem arises when model uncertainty is added to an H2-performance specification; a standard formulation is depicted in Figure 7.1. The nominal map M is

132

Paganini and Feron

taken to be a finite-dimensional LTI system with state-space realization

Notice that for the problem to be meaningful there can be no feed-through term between v and z. In addition, we will assume here that Dqv = 0; i.e., there is no unfiltered input seen from the uncertainty A. This assumption is required in all the approaches except possibly for that of section 7.3.4, so we will adopt it from here onward. The uncertainty A is modeled as an unknown operator on the space Li2 of squareintegrable, vector-valued functions, restricted to the spatial structure

This standard form accommodates a variety of uncertainty models, including real parameter variations and unmodeled dynamics, which can be LTI, linear time-varying (LTV) or nonlinear (NL). These choices are reflected on the dynamic nature of the perturbation blocks, which are normalized to size 1 in the Ii2-induced norm. BA denotes the unit ball of structured perturbations under consideration. Remark. It is customary to restrict A to be a causal operator (past outputs depend only on past inputs). This restriction deserves some comment since A does not represent a physical system, it merely provides a parameterization for a class of systems. The convenience of a causal parameterization is that it allows for the fictitious feedback loop of Figure 7.1 to be treated like a "physical" loop and to consider questions of stability (which go beyond the signal space L2) within this framework. As we shall see, the causality restriction plays an important role in the bounds for robust H2-performance. The system (M, A) is robustly stable if M is stable and if / — AMn has a causal, bounded inverse for every A G BA- The question of robust stability under structured uncertainty has been extensively studied (e.g., [314, 441, 370, 274]). We will assume in what follows that the system is robustly stable. With this assumption, the closed-loop map T ZU (A) from v to z is a bounded operator for all A e BA; we wish to impose an H2-performance specification on this map. Now the only case in which we have an unequivocal definition of ||TZV(A)||2 (analogous to (7.1)) is when this map is LTI, i.e., when the blocks of A are LTI or real parametric. For this case, the question at hand of how to evaluate supA ||T2V(A)||2 is purely computational; unfortunately, in this case robust performance evaluation is necessarily hard, because already the robust stability problem is known to be NP-hard [72]; so the aim here is to obtain tight, tractable bounds. As is standard in robust stability and HQQ- or LI-performance problems, these bounds will involve uncertainty "scalings" or "multipliers" of the form

which commute with the structure (7.7); we denote by A the set of positive matrices of the form (7.8). With these other performance measures, the resulting bounds have also an interpretation in their own right as tests for LTV or NL uncertainty structures [227, 370, 274, 336]; we wish to emulate this in the H2 context. For NLTV perturbations, however, the right definition of the H2 norm is not so clear, a difficulty that was pointed out in [384]; in particular, one can build on various

7.3. State-space bounds involving causality

133

approaches: impulse response measure, stochastic white noise rejection, or deterministic versions of white noise. Sections 7.3 and 7.4 will provide an overview of these approaches and cast them in a common language, by exploiting the quadratic structure of uncertainty (described by quadratic signal constraints), noise (in terms of its spectrum, or second-order statistics), and performance (I<2 norms, variance). This structure allows for the use of state-space tools of linear-quadratic optimization, as well as the «S-procedure [433, 274] in which quadratic constraints are incorporated into a cost function by means of Lagrange multipliers. These tools come together to yield convex methods for analysis involving LMIs.

7.3

State-space bounds involving causality

We begin by stating an SDP problem, where the unknowns involve a state-space matrix and a matrix of constant uncertainty multipliers. To simplify the formulas we will assume that Dqp and Dzp are zero. Problem 1. Find Js,nitv := inf Tr(B^PBv) subject to P > 0, A € A, and

Remark. This problem can also be stated with (7.9) replaced by a Riccati equation ((7.13) below), which depends nonlinearly on A. We prefer the LMI version due to its convexity, but some authors [384, 334] have pursued nonconvex computation with the Riccati version; in the case of a scalar multiplier, nonconvexity can be handled easily by a linear search. We will show that Js,nitv provides a bound for robust H2-performance under structured, causal, otherwise arbitrary perturbations A. This holds both for the impulse response and the stationary noise interpretations. For simplicity we take A = diag[Ai , . . . , and the multiplier structure A = diag[Ai/, • • • , A.F/].

7.3.1

The impulse response interpretation

As remarked in section 7.2, the H2 criterion can be interpreted as a measure of the impulse response energy, added over the input channels. For the closed-loop map we write

If T ZW (A) is LTI, this coincides with the standard H2 criterion mentioned before, but the definition also applies to nonlinear time-varying (NLTV) systems as well. Remark. Alternatively, one could use the average response to an impulse of random direction where VQ is a random vector of covariance E(VQVQ) = I. While both notions need not be equivalent for a given NL system, the same bounds apply in the worst case over A. Many authors [384, 334, 210, 64, 134, 187] have applied this approach for robust H2 performance under a variety of uncertainty structures and obtained bounds of the form of Problem 1 or its Riccati equation counterpart. The following is a derivation based on the «S-procedure.

134

Paganini and Feron

Theorem 7.1. Suppose that the system of Figure 7.1 is robustly stable against the ball of structured NLTV perturbations. Then

Proof. To begin with, we observe that the effect of the impulse in the fcth channel is to "load" an initial condition XQ = Bve^ in the system, which subsequently responds autonomously. For this reason we first focus on the problem for fixed initial condition and no input, We now write the bounds

In the first step the iih uncertainty block is replaced by the integral quadratic constraint (IQC) || Ibilli? where qi and pi are the corresponding portions of q, p; this constraint would characterize the class of contractive (possibly noncausal) NLTV operators. However, by requiring p € L»2[0, oo), we are imposing some causality in the problem by not allowing p to anticipate the impulse. This does not, however, impose full causality in the map from q to p, hence the inequality. The second step is an application of the 5-procedure: the constraints are incorporated into the cost function via Lagrange multipliers A* > 0, and it is straightforward to show the stated inequality. What is not obvious is that in this case of IQCs the «S-procedure is lossless, and in fact [274, 433] we have equality in (7.11). This property will be relevant to the comparisons of section 7.5. To compute (7.11), observe that for fixed A* we have a classical linear-quadratic optimization problem

Following, e.g., [420], it is known that the optimal value of (7.12) is x^Pxo, where P is the stabilizing solution of the algebraic Riccati equation

For such a solution to exist, the HQQ norm condition

must hold, which happens to be feasible given the robust stability assumption. To see this, notice that robust stability under NLTV uncertainty implies [370] that || A^MnA'i ||c can be made less than 1, and this implies (7.14) by scaling up A. To derive an LMI version, we use the fact (see [420]) that the Riccati solution is the minimizing solution of the LMI (7.9), which is jointly affine in A and P. Thus we have

7.3. State-space bounds involving causality

135

which is a convex optimization problem. Now to consider the sum over impulse channel, we write

Here (7.17) results from (7.15). Notice that we have commuted the supremum with the sum in (7.16) and the infimum with the sum in (7.18), both potentially conservative steps (see section 7.5.3) for the multivariable case. In the scalar v case, the only source of conservatism is (7.10). D A final comment on the preceding result is that for NLTV uncertainty the response to an impulse may vary in time; however, exactly the same argument shows that the bound applies to the response to an impulse 6t applied at time t. Furthermore, it follows that the bound applies to the impulse response averaged over time, which we denote

for the single input case (and with a spatial sum for the multi-input case).

7.3.2

Stochastic white noise interpretation

We now inquire to what degree the preceding bound can be applied to questions of stationary white noise rejection. Of course, in the LTI uncertainty case the implication is direct, since we are bounding the standard H2 norm. The LTV and NLTV cases require more care; in particular, since time variation is allowed the output to stationary noise will not be stationary, and any performance measure must resolve this by taking either the average or the worst case over these variations. We consider here the averaging approach; the worst-case measure is considered in section 7.4. Define the average output variance norm of a system H as

where z is the response of H to stochastic white noise (formalized below). Does Problem 1 provide a bound for supA ||Tzv(A)||2)aoi,? The answer is affirmative for LTV uncertainty, since for LTV systems \\H\\2,avgimp = ||#H2,aotM as shown in [32]. However, this identity relies on superposition and hence does not extend to NL systems. Nevertheless, a stochastic argument can be given to show that the bound still applies for NLTV, causal uncertainty. This method was proposed in [330]; we extend it here to the structured uncertainty problem. Theorem 7.2. Suppose that the system of Figure 7.1 is robustly stable against the ball BA. of structured NLTV perturbations. Then

136

Paganini and Feron

Proof. As before, robust stability implies the feasibility of (7.9). To formalize the notion of continuous time white noise applied in v, (7.4) is replaced by the stochastic differential equation

in the sense of Ito calculus, where V(t) is multidimensional Brownian motion. For an introduction to stochastic calculus, including the Ito formula used below, see [304]. The white noise v(t) would be the derivative of V(t), except that this process is not differentiate, hence the need for this alternative calculus. For (7.19) to be well defined, the stochastic process p(t) must be as follows: • adapted to V(t), meaning that it is a function of past values of V(t): this in fact holds when p(t) is obtained through the configuration of Figure 7.1 with a causal

A;

• locally square integrable, i.e., E(f*2 \p(t)\2dt) < oo, which follows from robust stability. Remark. Additional technical assumptions are needed for stochastic differential equations that may possibly restrict the allowable class of A; these are outside our scope. Now we use Ito calculus to compute stochastic differentials; the relevant rules are dt2 = 0, dtdV = 0, (dV)(dV)T = Idt, and the Ito formula

Applying these to f(x) = xTPx for a symmetric matrix P, we have

Now take P, A satisfying (7.9); pre- and postmultiplying by (xT,pT) and (xT,pT)T we obtain

Integrating in [0,r] and exploiting |bi||L2[o,r] < llgtltaflVr] (A< is causal, contractive) gives

Now integrating (7.20) on the interval [0,r] for zero initial conditions, and using (7.21), we find

7.3.

State-space bounds involving causality

137

Taking expectations, the last term (a stochastic integral of zero mean) disappears, and we are led to which implies that Taking supremum over A and infimum over P, A completes the proof.

7.3.3 Refinements for LTI uncertainty We have described the constant multiplier method, suitable for arbitrary contractive perturbations, but additional properties of the uncertainty can also be exploited. For LTI uncertainty dynamic multipliers can be used; using, e.g., the impulse response interpretation (they are all equivalent for LTI systems), (7.10)-(7.11) can be replaced by

where \i(ju) are in principle any positive transfer functions. However, at this level of generality the restriction p € L2[0, oo) (related to causality) is not easily handled. To obtain a computable bound of the style of Problem 1, \i(ju) must be restricted a priori to the span of a finite set of rational basis functions, as is done in [134].8 This means that the multipliers A(JCJ) would be restricted to a certain subspace SN of rational functions, N denoting the dimensionality of this space, leading to Problem 2. Problem 2. Find

The preceding optimization bounds (7.23) from above and can be reduced to SDP computation by state-space methods. For details on this procedure—in particular, how to handle noncausal multipliers—we refer to [134]. Further refinements ("G-scalings," Popov multipliers [187, 437, 57]) apply to the real parametric uncertainty case; we refer to Chapter 9 in this book for some of these methods.

7.3.4 A dual state-space bound We briefly discuss a bound that is dual to Problem 1; this bound uses a different restriction on feed-through terms in the realization (7.4)-(7.6): The terms with output z must be zero but not necessarily Dqv. To set things in a common ground we will assume, as before, that all the D terms are zero. Problem 3. Find 3'

8 [134] uses passive rather than contractive uncertainty blocks; the methods are easily adapted to the present case.

138

Paganini and Feron

Note that (7.24) is feasible in Q provided that (compare with (7.14))

whose feasibility in A is once again equivalent to robust stability under NLTV perturbations. A first question is whether J8,nitv = JB: ^ is shown in section 7.6 via examples that the answer is negative and also that there is no fixed ordering between them. In terms of interpretation, J'a has not been studied as much, but the following can be shown. Theorem 7.3. Suppose that the LMI (7.24) is feasible. Then with A varying in the class of real constant matrices of the structure (7.7),

Proof. The main idea, which can be traced back to [50] in the context of Riccati equation conditions, is to bound the controllability Gramian (asymptotic state covariance) QA of the stable closed-loop

In other words, QA is the solution to the Lyapunov equation (analogous to (7.3))

We will show that if Q, A e A, satisfy (7.24), then

by deriving a Lyapunov equation for the difference Q — QA- To do this, first rewrite (7.24) by Schur complement as

Subtracting (7.25) from it gives

which can be rewritten after some algebra as

Since the last two terms are nonnegative (note that ||A|| < 1), we conclude that

that leads to (7.26) using the stability of A + Bp&Cq. Having shown this, we complete the proof by writing

and taking supremum over A, infimum over Q, A.

7.4. Frequency domain methods and their interpretation

139

Extensions and open questions • A similar procedure shows (see [389]) that the bound applies to the case of a memoryless time-varying uncertainty A(t), under the || • ||2,aov-norm. • However, we are not aware of analogous studies with a stochastic interpretation for the case of perturbations with memory (even LTI). Given (7.26), one can conjecture that if Q, A satisfy (7.24), Q should bound the time-averaged covariance of the system state when driven by continuous-time white noise, and when the uncertainty is structured, causal NLTV. This conjecture might follow if instead of this algebraic argument one attempts a signal-based 5-procedure argument similar to the one in Theorem 7.2; the extension is not, however, immediate and the conjecture is, to our knowledge, open. • A case where the extension to perturbations with memory has been pursued is in the Ii2 to LOQ induced norm interpretation (also studied in [50] for static matrix A). Notice that only for scalar outputs does this correspond to the H2 norm, so we focus on this case (see [210, 50]) for generalizations). Here Tr(C2QCj) = \CZQC^\, and it is shown in [210, 377] that Problem 3 gives a bound on the Li2 to LOO induced norm of T 2U (A) under arbitrary NLTV, causal uncertainty. In other words

The proof [210, 377] follows an e>-procedure argument similar to the one in section 7.3.1, plus a more extensive use of LMI properties. Another open question is the refinement of this bound by dynamic multipliers for LTI uncertainty, analogous to section 7.3.3.

7.4

Frequency domain methods and their interpretation

A separate line of research [318, 319, 320] has sought "external" characterizations of robust H2-performance, paralleling the frequency domain theory available for robust stability or Hoo-performance. This includes the structured singular value methods [314] for LTI uncertainty, as well as recent strong results that have shown the losslessness of the convex upper bound over classes of NLTV uncertainty [370, 274, 336]. These results are replicated in the ^-performance setting by applying an operator point of view to stationary white noise rejection.

7.4.1

The frequency domain bound and LTI uncertainty

Consider the following optimization. Problem 4. Find J f) i ti := inf /!^,Tt(y(a;))^ subject to A(w) € A and

140

Paganini and Feron

To relate this condition to the structured singular value methods [314], notice that replacing Y(w) by 72/ in (7.27), imposes the robust Hoo-performance specification

This means that the worst-case gain of the system across frequency and spatial direction is bounded by 7. In (7.27) the "slack variable" Y(u) allows such gain to vary, provided that the accumulated effect over frequency and spatial direction is minimized, which is indicative of an H2-type specification. For LTI uncertainty, this argument is easily formalized as follows. Theorem 7.4. Suppose that the family ofLMIs (7.27) is feasible. Then with A varying in the class of LTI perturbations of the structure (7.7),

Proof. Concentrating for simplicity on the full block case A = diag[Ai, . . . , consider a fixed frequency u. Multiplying (7.27) by (p(ju)*,v(ju)*) on the left and its adjoint on the right yields (refer to Figure 7.1)

Since A is time invariant and contractive we have ^(j'a;)) > jpiO'^)), so \i(ju) > 0 implies

follows. Using definition (7.1), we have

for every A, which proves the desired upper bound. The preceding proof is close in spirit to the methods of the structured singular value theory, by which the above-mentioned HQQ condition is proved. Analogously, one can also refine the bound for the case of real perturbations by adding to the left-hand side of (7.27) a "G-scaling" term of the form

where the additional scaling G(u) has the structure G = diag[Gi, • • • » G r , 0,..., 0], with blocks Gi = G* in the locations of the real uncertainty (see [441]). Proof of sufficiency follows similar lines. The next question we discuss is the conservatism of this bound, analogously to the study of "/x-simple" structures pursued in [314]. We state the following result. Proposition 7.1. Assume that v is scalar, and that 2L + F <2 in the structure (7.7). Let BA denote the ball of LTI, possible noncausal perturbations of this structure. Then

7.4. Frequency domain methods and their interpretation

141

This characterization is a direct consequence of the structured singular value theory, since for scalar v, Y(u) measures the worst-case gain of the system at that frequency, as follows from the characterization of /^-simple structures [314], adding the usual "performance block." For noncausal uncertainty this worst-case gain is achievable at every frequency by a suitably chosen A(ju;), which thus gives an H2 norm equal to the stated bound. Having identified the nonconservative case, we will now comment on the sources of conservatism in the general case: • Non-/i-simple structures. Clearly, as the number of blocks increases we expect a similar conservatism as the one we have in the structured singular value. • Multivariable noise. As we see in the preceding proof, we are imposing the matrix bound Tzv(&)(juj)*Tzv(A.)(j(jj) < Y(u>) as a way of constraining their traces. We will see in section 7.5.3 that this can be conservative. • Causal uncertainty. Imposing this, A(jfo;) would not be allowed to interpolate any frequency function; we will return to this issue in section 7.5.

7.4.2

Set descriptions of white noise and losslessness results

In view of these limitations, it is natural to inquire what exactly is the bound Jf,iti computing in regard to this problem. It turns out that frequency domain conditions correspond to treating white noise, as well as the uncertainty, from a worst-case perspective, in a sense explained below. This approach will apply as well to the case of NLTV uncertainty; in that case the conditions will involve the use of constant uncertainty scalings A G A, as the ones considered in section 7.3. In particular, we will consider the following modified problem. Problem 5. Find J

We now outline the set-based approach to white noise rejection [318, 320] that is directly tailored to the robustness analysis problem. By treating both noise and uncertainty from a worst-case perspective, exact characterizations can be obtained. At first sight, this proposition may seem strange to readers accustomed to a stochastic treatment of noise; notice, however, that a stochastic process is merely a model for the generation of signals with the required statistical spectrum, and other models (e.g., deterministic chaos) are possible. Here we take the standpoint that rather than model the generating mechanism, we can directly characterize a set of signals of white spectrum, defined by suitable constraints, and subsequently pursue worst-case analysis over such set. One way to do this, inspired in statistical tests for white noise commonly used in time series analysis, is to constrain the cumulative spectrum. We focus for the moment on scalar Li2 signals and define the set

The imposed constraints are represented in Figure 7.2. The signals in W^^B have approximately flat spectrum (controlled by the accuracy 77 > 0) up to bandwidth B, since the integrated spectrum must exhibit approximately linear growth in this region.

142

Paganini and Feron

Figure 7.2: Constraints on the accumulated spectrum. Notice that this integrated spectrum will have a finite limit as ft —»• oo, for Ii2 signals, so we only impose a sublinear upper bound for frequencies above B. Having defined such approximate sets, the white noise rejection measure for a given system H will be based on the worst-case rejection of signals in W^.B in the limit as

For an LTI system under some regularity assumptions, || • ||2,um can be shown to coincide with the standard H^ norm; see [320] for details. Notice that the preceding definition can apply to any bounded operator, even nonlinear. We are now ready to state a characterization of the frequency domain test. Theorem 7.5. Suppose that the system of Figure 7.1 is robustly stable against the ball B A of structured NLTV perturbations. Then

This result is proved in [320] and is based on the method of IQCs, which are used as in [274] to characterize the uncertainty but here are used also for the set W^B- In particular (7.29) imposes a parametric (in (3) family of constraints on v(ju), which are quadratic and shift invariant. The 5-procedure in this case generalizes to include an infinite-dimensional Lagrange multiplier corresponding to these constraints, in addition to the uncertainty multipliers. It is precisely this multiplier that gives rise to Y(u) in Problems 4 and 5. It is shown in [320] that such «5-procedure is lossless; the key step in proving Theorem 7.5. We remark that a similar interpretation can be given to the cost function Jf,iti, as a test for robust performance over white noise sets in the sense defined above, for uncertainty of arbitrarily slow time variation, in the sense of [336]. Thus this method provides a parallel theory to the analysis with an Hoo-performance measure. An important comment in regard to Theorem 7.5 is that it computes the limit a posteriori to the supremum, instead of computing supA ||T2w(A)||2)U,n. In other words

7.5. Discussion and comparisons

143

it is a uniform limit across A that makes the test exact. In fact this interchange can be conservative when the method is extended to multivariable noise (by imposing crosscorrelation constraints on the components), as was recently pointed out in [387]; we return to this in section 7.5.3.

7.4.3 Computational aspects In terms of computation, Problem 4 is infinite dimensional as posed. Note, however, that with A (jo;) free to vary in frequency, the problem decouples over frequency, and a gridding method analogous to the /u-analysis tools [31] can be used. At a given frequency point uji we can solve the low-dimensional SDP as follows.

Problem 6. Find inf Tr(Y(ui)) subject to

We obtain a value that has the intuitive appeal of bounding the worst-case gain at each frequency and can subsequently be accumulated over the frequency grid. Other computational methods based on this condition include basis functions and the use of a dual form [170]. If A is constant, a gridding method is not attractive because it would give a large, coupled problem. However, in this case the problem can be reduced exactly to the following finite-dimensional state-space problem, which is of the LMI type in the unknowns Z, P_, P+, and A. For details on this equivalence see [320]. Problem 7. Jf, n itv = inf Tr(Z) subject to

7.5

Discussion and comparisons

We have presented a number of approaches for the evaluation of robust H2-performance. As we have seen many different performance criteria fall under the umbrella of "H2," since they all coincide in the case of LTI uncertainty. In this case all upper bounds can be directly compared. In problems involving LTV or NL perturbations, the comparison is more involved since we are using different interpretations; nevertheless, some direct comparisons can be pointed out, as explained in this section. We will focus on the bounds Js and Jf of sections 7.3.1-7.3.3 and section 7.4, respectively.

7.5.1

NLTV uncertainty

For arbitrary NLTV perturbations, we have seen that Problem 1 provides a bound on the impulse response as well as the average response to stochastic noise in the worst case

144

Paganini and Feron

over A. In contrast, Problem 5 applies to the worst-case white noise response, which may be a larger quantity. Concretely, for scalar noise we clearly have the relationship

since the impulse (or, more exactly, an Ii2 approximation) is always an element of the "white" set W^B- The following questions arise: • Can the inequality be strict? • Do the answers to Problem 1 and Problems 5 and 7 reflect an analogous inequality? This is not obvious since since we only know that J8,nitv is an upper bound for the first quantity in (7.30). These questions have an affirmative answer. We first discuss the inequality between the LMI bounds, which can be proved (in general, including also multivariable noise) by transforming Problem 1 to the equivalent Problem 8. Problem 8. Js,nitv = inf Tr(Z) subject to

This transformation can be done by standard matrix operations, the variables P and A being the inverses of P and A in Problem 1. It is not hard to observe that the constraints on Z are weaker in Problem 8 than in Problem 7; hence the minimization will give a smaller number. This shows that Proposition 7.2 holds. Proposition 7.2. Js,nitv < Jf.nitv In the next section we will provide an example where Js,nitv < Jf.nitv This implies, in particular, that the inequality in (7.30) can be strict, answering also the first question above. The previous derivation does not, however, provide much insight. We now give an alternative argument, based on the frequency domain, focusing for simplicity on the case of scalar noise. A frequency domain version of the optimization (7.11) is

Now introducing the slack variable Y(u) to bound the integrand in (7.31) we can rewrite this problem as the minimization of /^ Y(U>)T£% subject to

V p(ju) in the Fourier image of L2JO, oo). Since z and q are the response of M to v(t) = 6(t),

7.5. Discussion and comparisons

145

which translates (7.32) to

Now we have an expression that closely resembles Problem 5; in fact, if p(ju] were allowed to be any frequency function, (7.33) would reduce to (7.27) and the two problems would be equivalent. However, the constraint p(t) € L2[0, oo), which embeds some causality in Problem 1, implies that p(juj) is not free but must have an analytic continuation to the right half plane. This restriction will in general lead to a smaller supremum, and thus we prove Proposition 7.2. Remark. A consequence of the previous argument is that in the scalar noise case, if we remove the causality restriction on A, then the worst-case impulse response is Jf, n itv, so the impulse is the worst-case white noise signal in that case. The situation is different if causality is enforced and may lead to a smaller result. A clarification is in order here: We are not saying that Theorem 7.5 fails with causal perturbations (indeed it holds; see [317]); what happens is that the impulse ceases to be the worst-case signal. Also, a gap appears between treating white noise in a worst-case perspective or in an average case perspective.

7.5.2

LTI uncertainty

We can also extend the previous argument to the case of LTI uncertainty. Notice that here we have an unambiguous H2 norm we are trying to compute, for which both approaches provide bounds. In this regard, once again we find that removing the restriction p G ^[0, oo) from (7.23) will lead to the result Jf,iti but that there is a gap between the two. This would mean that for causal, LTI uncertainty, the state-space approach could in principle give a tighter bound. Notice, however, that we do not have a Js,iti bound, only a family J^iti obtained by basis expansions of order N for the frequency-varying scalings. This means that while

we know nothing about the situation with a given, finite N. This is particularly relevant since the computational cost of state-space LMIs grows dramatically with the state dimension, in contrast with a more tractable growth rate for computation based on Problem 4, as is now discussed. Computation time for a standard LMI solver such as [153] grows approximately as n^ec, where n^c is the number of decision variables. Now for state-space conditions such as Problem 1, ridec includes the variables of the P matrix, which are of the order of the square of the number of states; roughly speaking, then, computation time grows as the tenth power of the state dimension, which indicates that the cost will be prohibitive for high-dimensional models. This also sets a limit on the applicability of dynamic multipliers for Problem 2, since they increase the state dimension. In contrast, the decoupled frequency-domain bound of Problem 4 for LTI uncertainty is not very sensitive to this problem: While of course a high-order system calls for the use of a fine frequency grid, this effect is milder since computation time grows linearly with the number of gridpoints. Also, there is no limitation on the order of multipliers. We will return to this issue in section 7.6.

Paganini and Feron

146

7.5.3 A common limitation: Multivariable noise Throughout this chapter we have focused on upper bounds for robust ^-performance, and only in Theorem 7.5 and Proposition 7.1 have we exhibited characterizations of some of these bounds. The conservatism inherent in, e.g., Theorems 7.1 or 7.4 has not been quantified. Recently, [387] has pointed out a source of conservatism in these approaches in the treatment of multivariable noise, which we now explain. We focus the discussion on the LTI uncertainty case, but in principle analogous limitations hold in general. The following example is adapted from [387]. Consider the system with A being a full 2 x 2 LTI uncertainty block, v G R2, p € R2, q G R2, and z e R, and state-space matrices

From an input-output perspective, the uncertain system is of the form

where W is a system of transfer function

In this case the worst-case

norm can be computed exactly:

Now let us consider the bound of Problem 4. It is not difficult to show that the LMI (7.27) reduces to

aand therefore the cost is

which shows a conservatism of a factor of \/2 with respect to the true worst-case H2 norm. More insight is gained by looking at the bound

that corresponds to (7.28) from the proof of Theorem 7.4. For this bound to hold for every A, Y(u) is forced to satisfy (7.34); however, each A can only produce gain in a one-dimensional direction, which means that the H2 norm will never be that large.

7.6. Numerical examples

147

We now look at this example from the point of view of the state-space bounds of section 7.3.1 and its refinements for LTI uncertainty of section 7.3.3. Instead of computing the state-space LMIs, however, it is easier to directly evaluate the sum

which is upper bounded by Js,nitv (and also by J^lti) as follows from the proof of Theorem 7.1. In this problem,

and the same happens for 626(1). Therefore, the sum in (7.35) is once again 2||W||2, which indicates that we will have at least a factor of \/2 of conservatism, like we had with Problem 4. The key difficulty is in the interchange of the supremum with the sum: Different perturbations A achieve the worst case for each input channel. It is clear that an analogous example can be constructed for noise dimension m, giving a conservatism of \/m. Thus we find that the need for a convex characterization is ill suited to treat the decoupling of uncertainty with input channel direction in both methods. At this time, this appears to be inherent to any convex method for this problem.

7.6

Numerical examples

We consider in this section several examples of robust H2-performance evaluation to compare the different approaches. The examples all involve scalar noise, so that the preceding limitation does not come into play. In all cases, the convex optimization problems were solved using the LMI Control Toolbox [153].

7.6.1

Comparing Js and J's

As mentioned before, the two dual state-space bounds of section 7.3 are not equivalent. This is now illustrated. For

it can be shown by LMI computation that J8,nitv < Jg> but the opposite inequality holds for

Paganini and Feron

148

7.6.2 NLTV case: Gap between stochastic and worst-case white noise This example exhibits a case where the inequality in Proposition 7.2 is strict. For the system

we obtain confirming that the gap can be indeed significant.

7.6.3

LTI case: Tighter bounds due to causality

We can also use the same example for the case of LTI uncertainty and solve Problem 4 with frequency-dependent multipliers, which gives Jf,iti = 3.434. Notice that this only reduces modestly the constant multiplier result Jf,nitv and is still larger than J8,nltvSo here the benefit of frequency-dependent multipliers is outweighed by the causality constraint embedded in Problem 1.

7.6.4

LTI case: Tighter bounds due to frequency-dependent multipliers

We now show an example in the extreme opposite end, where Jf,iti provides a nonconservative estimate of robust performance, tighter than the state-space bounds. Consider the second-order system with uncertain damping

Here 6 € [—1,1] is parametric uncertainty in the damping coefficient. In this case, the H2 norm of the system can be computed analytically and is equal to 1/^2(2 + 6), which has a maximum of 1/V2 = .707 over the range of uncertainty. Rewriting the problem in standard form, we compute the different bounds. Problem 4 with frequency-dependent multipliers gives 0.787 when the real parametric nature of the uncertainty is not exploited and tightens to 0.707 when we add the additional G-scaling. Problem 1 with constant multipliers gives a value of 0.883; we also studied the bound when a Popov multiplier was included as in [437], but the bound did not improve. So in this case the use of frequency-dependent multipliers is essential, while the causality restriction does not come into play. For the state-space method to become tighter would require the use of frequency-dependent multipliers as in Problem 2.

7.6.5

LTI case: Using both causality and dynamic multipliers

Next, we study an example used in [134] to illustrate the improvement due to dynamic multipliers and analyze the role of causality. Table 7.1, adapted from [134], gives values for J^lti from Problem 2 as a function of the order N of the dynamic multipliers. In comparison, the frequency-domain bound from Problem 4 on the same problem with dynamic multipliers gives a value of Jf,iti = 314.15, similar to the bound obtained from Problem 1 and constant multipliers but significantly worse than the bounds that

149

7.6. Numerical examples Table 7.1

Figure 7.3: Frequency responses and bound.

incorporate both causality and dynamic multipliers. More insight is gained by looking at the frequency plots below. In Figure 7.3, the envelope curve depicts the bound Y(u>) on the worst-case gain at each frequency, obtained from Problem 4. Since this problem has single-block uncertainty and scalar noise, it follows from the //-theory that this bound can be achieved at every frequency: i.e., for each uj we can find some A (a;) such that the system gain is in the envelope. The question of causality comes in to play when we want to interpolate such family of perturbations with a causal LTI system. To illustrate this, consider the other curves in the figure, which represent the gains |Tzv(Ai)(ju;)|2, with specific choices of constant, complex Ai. The central curve is for A = — 1; the others are for different phases. Each curve touches the envelope at a certain frequency for which the selected phase is the worst case. Now, while it is possible to interpolate a finite set of phases with a causal, contractive perturbation (this would give a response with all these peaks and "valleys" between them), it is in general impossible to do it for all u, hence the conservatism of the method of Problem 4.

7.6.6

A large-scale engineering example

As a final example with engineering content, we consider the application of these techniques to the Middeck Active Control Experiment (MACE) [285], which studies the control of lightly damped mechanical structures for space applications.9 The disturbance rejection specifications make it a natural testbed for studies in robust H2-performance [204, 437]. A feature of these problems is that a large number of modes fall in the frequency 9

We thank Kyle Yang for providing the MACE models.

150

Paganini and Feron

range of interest, leading to high-order models. For example, the simplest MACE model used for SISO control in [204, 437] has 24 states (which doubles to 48 once a controller is included) and 4 real uncertain parameters. This gives 1176 variables for the matrix P in Problem 1, which shows the high cost of state-space analysis, especially for LMI computation, which has growth rate n^ec. In addition, due to memory requirements this problem is beyond the reach (see [437]) of the standard LMI software [153] on workstation platforms. For this reason, [204, 437] have had to rely on nonconvex techniques based, for example, on the Riccati equation version (7.13). These have no convergence guarantees and actually have difficulty handling a more detailed, 59-state model required for MIMO (three-axis) control. In comparison, the frequency-domain bound of Problem 4 applied to the SISO model has 12 decision variables for A(o>), G(u>), and Y(UJ) at each u>. Using an n^ec growth rate for the LMI solver, the computational cost per frequency gridpoint is about 1010 times cheaper than state-space computation; even using a very fine frequency grid (e.g., 10,000 points), the overall cost is many orders of magnitude below. Also, memory constraints are not an issue here. We have computed the frequency-domain bounds of Problem 4 for both MACE models on a SUN workstation, in reasonable computation time and with no convergence difficulties.

7.7

Conclusion

We have reviewed different approaches for the evaluation of robust H2-performance under structured uncertainty, all of which are based on multipliers for the uncertainty and SDP computation. However, the approaches build on different interpretations of the H^ metric and do not give equivalent results. For arbitrary NLTV uncertainty (constant uncertainty multipliers), we have the following problems: (i) Problem 1, which bounds the impulse response norm, and stochastic white noise response under causal uncertainty; (ii) Problem 3 that bounds the LZ —> L^ induced norm; and (iii) Problem 5 (equivalent to Problem 7) that can be characterized as lossless under a worst-case treatment of white noise rejection. The three problems give different numerical values, which is not surprising due to the different nature of the interpretations. The main open question here is whether a stochastic interpretation can be developed for Problem 3, more general than the very limited memoryless case that was discussed. For the LTI uncertainty case, all approaches are attempting to compute the same quantity. In this case refinements are available only for (i) and (iii) in the above list (Problems 2 and 4, respectively) using frequency-dependent multipliers. We have seen that by using a multiplier of high enough order, Problem 2 will give a tighter bound; this advantage should be taken with a grain of salt, however, given the high cost of state-space LMI computation as state dimension is increased. In comparison, the frequency-domain method is always tractable: It essentially comes for free given a // analysis for robust stability and would appear to be a good first-cut analysis. A competing first-cut test is the constant multiplier bound of Problem 1 (or its extension to Popov multipliers) provided that the state dimension is not too large. Only in rare cases it would probably be justified to make an expensive dynamic multiplier computation in state space. In this case, a reasonable combination of the approaches is to find the multipliers A.(ju) from Problem 4 and based on this information select an appropriate basis function to use in Problem 2.

7.7. Conclusion

151

However, we have seen that both approaches are potentially conservative when dealing with multivariable noise signals, with a gap increasing with the noise dimension. This difficulty appears at present to be inherent in a convex analysis for this problem, but further research is required to settle this question, which also highlights the need for lower bounds to assess these gaps, analogously to those available [314, 441] for other performance measures. Some work in this direction for the state-space bounds is reported in [57]. We have concentrated on the question of robust Eb analysis and have not attempted to cover the (more difficult) problem of controller synthesis. We mention, however, that the approach of section 7.3 has led to a family of synthesis methods combining mixed H2/Hoo control with a search over multipliers (see, e.g., [384, 334] or Chapter 9 in this book). Also synthesis methods based on a set-based approach for noise as in section 7.4 have been developed in [89]. A rare feature of HQQ theory has been the complete duality between two ways of thinking, one based on state-space and linear-quadratic optimization, the other on operator theory and the frequency domain. After two decades of research, the robust H2 problem has found these two faces but has not achieved the same unity or a satisfactory treatment of multivariable white noise: LQG has not quite adapted to the world of robustness.

This page intentionally left blank

Part IV

Synthesis

This page intentionally left blank

Chapter 8

Robust H2 Control Kyle Y. Yang, Steven R. Hall, and Eric Feron 8.1

Introduction

Among all performance indices known to control engineering, the H2-performance index holds a special place for historical and practical reasons. The historical reasons are that minimizing the H2 norm of a linear control system via feedback, better known as the LQG problem, is among the first optimal control problems to have been solved analytically (for an extensive presentation and bibliography, see [11]). The practical reason is that this problem can be solved using reliable and fast computational procedures [25, 239, 409]. It is, however, well known that the performance of the LQG-optimal controller can be very sensitive to perturbations on the nominal system [101]. In view of this fact, devising analysis and synthesis tools that will, respectively, evaluate and minimize worst-case H2 norms of control systems is especially relevant. In this chapter, we consider the following problems: Given a linear system perturbed by linear time-invariant (LTI) perturbations, what is its worst-case H2 norm? Second, given such a perturbed system, what LTI control system will minimize the closed-loop worst-case H2 norm? The first question is the subject of a number of works. Packard and Doyle [313], and Bernstein and Haddad [47, 48, 49] are among the first to consider the problem of robust H2-performance in the face of dynamic and parametric uncertainty. Stoorvogel [383, 384], Petersen and McFarlane [332], and Petersen, McFarlane, and Rotea [334] find bounds on the worst-case H2 norm of a system subject to norm-bounded, noncausal, possibly nonlinear and time-varying uncertainties. Peres and Geromel [328] and Peres, Geromel, and Souza [329] find upper bounds on the H2 norm of linear time-varying (LTV) and uncertain LTI systems based on quadratic Lyapunov functions. The book and the papers by Boyd, El Ghaoui, Feron, and Balakrishnan [64, 136, 60] and Feron [133] show that the computation of all these bounds on H2-performance can be reduced to convex optimization problems involving linear matrix inequalities (LMIs), which can be solved via efficient convex optimization techniques. In [64,133], attempts are made to refine the upper bounds on H2-performance when dealing with particular classes of perturbations such as static nonlinearities and parametric uncertainties, using Lur'e-Lyapunov functions and causal multipliers. Other attempts at obtaining reliable upper bounds on robust H2-performance include the recent paper by Paganini, D'Andrea, and Doyle [321]. 155

156

Yang, Hall, and Feron

This chapter presents a general account of the recent research on the use of multipliers to evaluate and design controllers that optimize the worst-case H2 norm of linear systems perturbed by LTI perturbations. Using multipliers is a well-known technique to determine the stability of uncertain systems (see [98, 419] and the references therein) and has proven to yield effective computational procedures [357, 355, 102]. A variety of other methods have been used in practice to try and make H2 controllers less sensitive to parameter variations in structural systems. For a survey of the most promising of these techniques, the reader is referred to [177]. However, these methods (such as the sensitivity-weighted LQG method or the multiple-model method) do not provide a priori guarantees that the system will be robust nor can they provide a bound on the performance of the system. This chapter uses an iterative scheme, commonly referred to as a "D-K iteration," [446] to design robust controllers. A D-K iteration consists of alternating synthesis and analysis steps. In the analysis, or D, step, the controller is held fixed, and optimal weights (stability multipliers) are found to calculate the performance of the closed-loop system. In the synthesis, or K step, the weights are held fixed and a controller is synthesized. One advantage of this design methodology is that at each step, the closed-loop cost should decrease monotonically. Unfortunately, the overall D-K iteration is not convex. It can converge to a local minimum and can even fail to converge. To be clear, note that we refer to a D-K iteration as a "design" (not a "synthesis") method. Several methods exist to synthesize robust, dynamic H2 controllers. Synthesis performed using coupled Riccati equations is presented in [48]. In contrast, a robust synthesis performed using an optimization subject to LMI constraints is discussed in [438]. Not all robust control design techniques rely upon a D-K iteration. The great difficulty with trying to simultaneously optimize all the unknowns in a robust control design is that the cost function and constraints are bilinear. Some work has been conducted into the solution of bilinear matrix inequalities, but such routines typically require the solution of alternating LMIs [165]. Later in this chapter, we restrict our attention to designing H2 controllers for systems described by a particular multiplier, namely, the Popov multiplier. A design scheme for these controllers, the so-called H2/Popov controllers, is investigated by How [201]. How sets up an augmented cost function, composed of a bound on the H2 cost and an associated, nonlinear, matrix equality constraint on the performance. This augmented cost function is then minimized using a gradient search algorithm. This technique simultaneously solves for both the controller and weighting functions (multipliers). In practice, this can be a problematic optimization, as it can be difficult to find an initial guess for the unknown parameters. The multipliers are not motivated by any physical or heuristic rules. This chapter is organized as follows. First, the general robust H2 analysis problem is formulated and upper bounds on robust H2-performance are proposed via a multiplier formulation. Then, the problem specified in the first section is specialized for two classes of uncertainties and a complete solution procedure is proposed for both cases that relies on either LMIs or Riccati equations. In the third part of this chapter, the problem of synthesizing robust controllers that minimize a worst-case ^-performance index is considered and solved. In the fourth and last part of this chapter, numerical issues associated with the design of H2/Popov controllers are considered. Numerical results are provided including those for a flexible space structure, the Middeck Active Control Experiment (MACE).

8.2. Notation

157

8.2 Notation L 2 (R) denotes the Hilbert space of functions h mapping R into R n x p that satisfy

It is equipped with the standard scalar product

and, for all h 6 L2(R), the Euclidean norm {/i, /i)1/2 of h is noted \\h\\2. L 2 (R+) denotes the subspace of L 2 (R) made of the functions h satisfying h(t) = 0 when t < 0. L 2e denotes the space of functions h mapping R into R n x p satisfying h(t) = 0 for t < 0 and

Following the usage of Francis [142], we suppress the dependence of these spaces on the integers n and p. For a given operator A and a function p mapping R into R n , (Ap)(£) denotes the value taken by the image function Ap at time t. An operator A is causa/ if for any function p and any time t, (Ap)(t) depends only on the past values of p up to time t. It is anticausal if (Ap)(t) only depends on the future values of p from time t. In any other case, A is noncausal. Given a set of operators AI, . . . , A^, diag(Ai, . . . , AX,) stands for the operator that maps the function taking the value \u\ (t)T ... UL (t)T] at time t to the function taking the value [(AiUi)(t) T . . . (AI,UL)(*)T] at time t. Let G map L2(R) into L2(R) and be linear. The adjoint of G, noted G*, is the unique linear operator satisfying rrt

rr\

Defining s as the usual Laplace variable, the transfer function of G is denoted by G(s), if it exists. Let us now introduce the notions of finite-gain positivity and passivity that we will use throughout this chapter. Definition (see [343]). An operator F mapping L 2 (R) into L2(R) has finite gain (or, equivalently, is bounded) if there exists a positive k such that for any u € L2, The smallest such k is called the gain of F and noted Definition (see [98]). A linear operator G mapping L2(R) into L 2 (R) is said to be positive if for any u 6 L2(R), G is said to be strictly positive if there exists k > 0 such that for any u € L2(R), Definition (see [98]). A linear, causal operator G mapping L2e into L2e is said to be passive if for any u e L2e and T > 0,

158

8.3

Yang, Hall, and Feron

Analysis problem statement and approach

Consider the system

where a: : R -> R n , w : R -> R n ™, z : R -> Rn", e : R -+ R n «, and d : R -» Rn" and all quantities are equal to 0 for t < 0. Assume that the matrix A is stable. A is a perturbation that satisfies the following set of assumptions:

In addition, A is bounded in some form: For example, A may be bounded above and below by the identity matrix, or A may be only known to be a passive system. A may be known to be a static gain or be a dynamic operator or any combination of the above. Many of these formats and the equivalences among them are described in detail in [102, 98], for example. Much of the existing literature is devoted to studying the robust stability of system (8.1) against an uncertainty, A. The system is robustly stable if, for w = 0 and any initial condition XQ, the signals x, w, z, and e belong to L2(R+) for all allowable instances of A. In the analysis problem, we assume system (8.1) to be robustly stable, and we are interested in evaluating its worst-case fL^-performance against the uncertainty A: Let

where

The H2 norm of system (8.1) is defined as

Equivalently, using Parseval's theorem, ||G ||2 may also be expressed as ||0A||2», where PA is the impulse matrix of G'A- In the subsequent developments of this chapter, it will also be convenient to express it as

where e is the output of system (8.1) with the following assumptions: The input d is identically 0 and the initial condition XQ is equal to Bdv, where v is a random variable

8.3. Analysis problem statement and approach

159

satisfying EvvT = /. (The expectation appearing in (8.4) is therefore to be taken with respect to v.) The robust H2 analysis problem is to compute the worst-case H^ norm of system (8.1) over all possible values of A that satisfy (8.2) and the additional boundedness conditions. This computation is, in general, a challenging problem. Thus, we propose to replace it by the computation of upper bounds on the robust H2 norm, using a technique similar to the classical technique of Lagrange multipliers: Consider any family, M, of operators, M, mapping L2(R) into La(R). For an appropriate choice of Ai, the following lemma gives an upper bound on the worst-case H^ norm of system (8.1). Lemma 8.1. The following inequality holds:

where w, z, and e are the inputs and outputs of the system

where all variables belong to L2(R+), and v is a random variable satisfying EvvT = /. For this lemma to hold, it is sufficient that

for any admissible perturbation A and any z and w such that z = Aty. Note that the multiplier M can indeed be seen as a Lagrange multiplier. Such an approach is not unlike the one encountered in the papers by Yakubovich [433, 434], Fradkov and Yakubovich [141], and Megretsky [269, 267, 270], where it is called the 5-procedure. In the remainder of this chapter, we will show that a suitable choice of the family of multipliers M allows for the right-hand side of (8.5) to be easily computed using state-space computations. Note that, using Parseval's theorem, the computation of the upper bound in Lemma 8.1 may be done in the frequency domain as well. The essential difficulty with frequency-domain approaches, however, is their inability to easily discriminate signals belonging to L2(R-t-) from signals that belong to L2(R). The result is that a frequency-domain approach may lead to perturbations w that may anticipate the impulse Bjv sent in the system, which is impossible in physically significant systems and may result in conservative estimates. For that reason, the analysis of robust H2-performance distinguishes itself from the analysis of robust stability or robust Hoc-performance. A detailed discussion of this issue together with numerical comparisons is provided in Chapter 7 of this book, by Paganini and Feron, and in [322].

Yang, Hall, and Feron

160

8.4 8.4.1

Upper bound computation via families of linear multipliers Choosing the right multipliers

Depending on the perturbation class under consideration, the choice of multipliers M will vary. There is a significant body of literature devoted to determining the appropriate choice of multipliers for the families of uncertainties under consideration. An extensive presentation of the techniques to obtain large classes of multipliers is presented in [273], for example. The following are examples of such multipliers. If A is a diagonal matrix of positive operators, then the largest known family of corresponding multipliers is written as

where M\i is any diagonal, positive, and self-adjoint operator. If, in addition, A is a static gain, then M\i need only be a diagonal and positive operator. Likewise, if A is a matrix of numbers that are only known to lie between —1 and 1, then the largest known useful family of multipliers is given by

where Mn is self-adjoint and M\i is a skew-symmetric operator. If A is any diagonal, LTI operator whose gain is less than 1, then the largest known family of multipliers is

where Mn is any positive, self-adjoint operator.

8.4.2

Computing upper bounds on robust H2-performance

In this section, we consider numerical schemes to solve the right-hand side of (8.5). Two specific uncertainty instances will be considered. First, A will be assumed to be an unknown but diagonal, passive, and LTI operator. Second, A will be assumed to be a diagonal matrix whose diagonal elements all have absolute value less than 1. In this second case, a special class of multipliers, named Popov multipliers, will be considered. It will be shown that they deserve special treatment. LTI and positive uncertain operators Following an idea arising in [61, 353, 64], we consider the family of finite-dimension operators

8.4. Upper bound computation via families of linear multipliers

161

(We refer the reader to [142] for a complete discussion of the representation of self-adjoint and general noncausal operators via transfer functions with unstable poles.) Thus, the set M is parameterized linearly by the real numbers mij, i = 1,..., np, j = 0,..., N. Each transfer function Mi(s) may alternatively be written as Mj(s) = CMi(sI — ^Mi)~1-5M< + Dmi, where

and

Likewise, the transfer function M(s) may also be written as M(s) = CM(S!-AM)' DM, with

and

Positivity of M is ensured by the inequality Mi(ju) -f- Mi(ju>)* > 0 for all u; e R and can be expressed in a convenient form via a straightforward application of Theorems 3 and 4 of Willems [420], subsequently corrected in [421]. Lemma 8.2. The inequality

is satisfied if and only if there exists a symmetric matrix Pi satisfying

Note that this lemma requires controllability of assumption is indeed satisfied.

^ to hold and that this

162

Yang, Hall, and Feron

In order to calculate the upper bound on the worst-case H2 norm given in Lemma 8.1 we introduce the augmented system

where AMH , BMH, CMH, DMH, CM He are given by

We now summarize the computation of the upper bound in the following theorem. Theorem 8.1. Consider system (8.6). The quantity

where e, w, and z satisfy (8.6), is computed as the minimum of TrF over the variables F, P, PI, ..., Pnw, rriij, i = 1,...,n w , j = 0,..., N, satisfying the constraints

and

Where

has been partitioned conformally with the dimensions of A, Aca, and AM (where AC& and AM are given by (8.10)). A detailed proof of this theorem may be found in [134]. Popov multipliers for real parametric uncertainties Consider the case of system (8.1) with Dzw = 0. When real uncertainties are present and when the system is scaled such that these uncertainties lie between —1 and 1, a popular class of multipliers known as Popov multipliers can be employed. A Popov multiplier is described by

8.4. Upper bound computation via families of linear multipliers

163

where H and N are diagonal and the coefficients of H are nonnegative. Popov's multiplier does not lend itself to exactly the same calculations as other multipliers, because of the derivative term present in it. More precisely, given the impulsive input d = 6(t)v we compute

Thus, Lemma 8.1 needs to be adapted and becomes Lemma 8.3. Lemma 8.3. The following inequality holds:

where w, z, and e are the inputs and outputs of the system (8.6). Under these conditions, the right-hand side of (8.5) can be computed using state-space techniques given by the following theorem. Theorem 8.2. The robust H2 -performance of system (8.1) is bounded above by the minimum of maxA TrjBj(P + Cf(A + I)NCz)Bd over the variables P, H, and N that satisfy the constraints H > 0 and

If, in addition, the multiplier simplifies to

is required to be positive, then the objective function

and computing the upper bound on robust ^-performance reduces to optimizing a linear objective subject to LMI constraints [438]. If N is not required to be positive, then a more accurate upper bound as well as lower bounds on the worst-case ^-performance may be obtained by solving another set of LMIs, as described in detail in [58]. The LMI condition (8.18) may also be written as the equivalent Riccati equality constraint

It can be shown that the solution to (8.20) is the smallest (in the sense of the partial ordering of symmetric matrices) of all matrices satisfying (8.18).

Yang, Hall, and Feron

164

Figure 8.1: The controller design problem.

8.5

Robust H2 synthesis

We turn now to the question of robust H^ controller synthesis. For a fixed set of multipliers, we wish to design a full-order LTI controller that minimizes the cost function. Because the form of the resulting synthesis equations (and inequalities) is highly dependent on our choice of multipliers, we choose to specify a priori that we will use the Popov multipliers discussed previously. Popov multipliers are convenient, because they do not increase the number of states in the controller beyond those of the plant. Furthermore, H2/Popov controllers have been used succesfully in structural control problems [204], and we can compare the utility of our synthesis routines with those used in previous designs.

8.5.1 Synthesis problem statement Consider the system in Figure 8.1. The LTI plant, G, in the system is given by

with all signals as defined for (8.1), and u : R —* Rn". Note that A is no longer assumed to be stable. The realization is assumed to be minimal. Also, we assume that (A, Bu) is stabilizable and that (A, Cy) is detectable. The exogenous disturbance, d, is assumed to be stationary and ergodic. As in the analysis problem, the goal is to minimize (E \\e\\%). However, for simplicity, in this case we assume that the control cost is decoupled from the state cost such that C^Deu = 0. The uncertainty block, A, is a diagonal matrix whose diagonal entries are nw static, independent, real, uncertain parameters. For convenience, we assume that all the entries of A have magnitude less than unity. Note that the dynamics matrix of a perturbed system is not the nominal A, but A + A.A, where A.A = BWACZ and —/ < A < /. Our goal is to find the LTI controller, K, given by

with xc : R —> Rn, that minimizes the H2 norm of the system, as measured at the output error signal, e. For the nominal system, the controller that minimizes the H2 norm is given by the celebrated LQG controller. However, it is also known that the problem of determining whether a system that is subject to real parametric uncertainties is robustly

8.5. Robust H2 synthesis

165

stable is NP-hard [71, 442]. Therefore, finding the extremal values of the H2 norm is not generally possible at low computational cost. For a closed-loop system with a known controller, the performance can be analyzed using (8.19) if the given matrices are replaced by their closed-loop counterparts. That is, A «— A, where

Ce <— Ce, and so on. The analysis Riccati equation (8.20) is then of size In rather than n. The closed-loop cost function that bounds the system H2 norm is therefore given by

8.5.2

Synthesis via LMIs

Robust H2 synthesis appears complicated because, ostensibly, we must minimize a nonlinear cost function subject to nonlinear constraints. The closed-loop cost equation (8.27) cannot be solved using standard Riccati equation solvers because it contains unknown controller parameters. Even if the constraint is written as a matrix inequality, we find that it is bilinear in the unknowns. Similarly, writing out (8.27) reveals that the cost is quadratic in the unknown controller input matrix, Bc. Fortunately, we do not have to solve a nonlinear optimization. The following theorem, from [251], shows that the cost function and constraint can be linearized, making the problem convex. The proof of this theorem relies heavily upon the elimination lemma of [315]. Theorem 8.3. For the plant, G, with controller, K, and fixed stability multipliers, H and N, define a matrix Bc e R nxn « that is full column rank. Furthermore, define

Tien an H2/Popov controller exists if and only if a positive definite, symmetric matrix P 6 R2n exists and minimizes

while satisfying

and

166

Yang, Hall, and Feron * A

A

T1

for some symmetric matrix Qu € Rn, where A = A — BWCZ and Eu = D^uDeu. Furthermore, the closed-loop cost is equal to the constrained minimum of (8.29). Given a solution to the optimization problem for P and Qu, the resulting controller can be found via an algebraic solution of the closed-loop LMI robustness constraint. Interestingly, for a case without an uncertainty block, this theorem yields the solution for the nonrobust LQG controller. Furthermore, in the case where the N scaling function is set to zero, this formulation produces a closed-form realization for the controller in terms of the Lyapunov matrix [438].

8.5.3

Synthesis via a separation principle

We now discuss the separation structure of the H2/Popov controller. It is well known that for a given, fixed LTI system, any output feedback controller that stabilizes the system could be built by combining the solutions of two separate problems—a state feedback problem and an output estimation problem (see, for instance, [446]). This can be discerned via a Youla parameterization of the controller. However, this separation structure is not always exploited for controller synthesis. The separation structure exists but is not considered useful. Two obvious examples where the separation principle plays an explicit and significant role in controller synthesis are the E^ (LQG) controller and the HQO controller [107]. The separation principle requires us to solve two subsidiary problems—a robust fullinformation (FI) problem and a robust output estimation (OE) problem. In the FI problem, we desire to find the static feedback gain that minimizes the cost for the uncertain system. This can be found by solving a single Riccati equation. In the OE problem, we design a dynamic filter to track a system input with minimum error on average, leading to a solution via a pair of coupled Riccati equations. The final H2/Popov control signal is equal to the optimal state estimate from the OE problem, multiplied by the gain from the FI problem. This is summarized in the following theorem from [437, 436]. Theorem 8.4. Consider the plant, G, with controller, K. Assume that we are given fixed stability multipliers, H and N. Define the following constant matrices

where Eu = D^uDeu. Furthermore, assume that both Z and £ are positive definite. If a symmetric matrix R e Rn exists such that it is the unique, positive semidefinite, stabilizing solution to the related full-state feedback Riccati equation

*

where A = A — BWCZ, and if positive semidefinite, symmetric matrices X € Rn and Y € Rn exist that are the stabilizing solutions to the related output estimation coupled

8.6. Implementation of a practical design scheme

167

Riccati equations

and

where Ed = DydD^d, and

then a controller given by

will stabilize the system for all allowable A. Furthermore, for all allowable A, the closedloop performance of the system, J, will be bounded by

These results are effectively a stochastic version of principles found in [107] and [110]. The coupled Riccati equations are also related to those of [48] . In [436], the cost function, JOY, is shown to overbound the H2 cost of the system. It is believed to be the case, though it has not been proven, that the controller resulting from the separation-based solution is identical to a controller derived from the LMI-based solution of Theorem 8.3.

8.6

Implementation of a practical design scheme

In this section, we will compare three different methods used to design H2 /Popov controllers. They are the following: (1) a D-K iteration in which both the analysis and synthesis steps are solved as LMI problems (using Theorems 8.3 and 8.2), (2) the augmented cost function approach of How [204], and (3) a novel D-K iteration that solves the synthesis problem using the separation principle Riccati equations and solves the analysis problem with a gradient optimization. The steps in the final method will next be described in detail, and then two design problems will be presented.

168

8.6.1

Yang, Hall, and Feron

Analysis step

The H2/Popov analysis problem requires that, for a fixed controller, we find the (2n)2 parameters of the Lyapunov matrix, JP, and the 2nw parameters in the scalings, H and JV, that minimize the closed-loop cost function (8.27). Since the problem is convex, there are no local minima. In fact, because it is convex, there are quite a few different algorithms that could be chosen to solve the problem. Even algorithms that are not specifically designed for convex problems, such as a gradient search routine, should converge to the unique minimum. Regardless of the solution technique, however, we need to make the routine as efficient as possible, both in terms of speed as well as in terms of memory requirements, so that it can be used for problems with greater than, say, 30 states. We can reduce the size of the analysis problem by explicitly minimizing over just the stability multipliers, rather than both the multipliers and the Lyapunov matrix. Consider the closed-loop cost, J7, given in (8.27). The gradient of J', with respect to H alone, is a size nw vector, |^ = [ • • • ^ • • -]T. Each of the scalar derivatives, $£, is equal to Tr(BdBdlHr)> wnere ^t is *ne **h diagonal element in the H matrix. However, we wish to consider only gradient directions that keep the optimization trajectory in the space specified by the closed-loop form of the Riccati constraint (8.20). By examining the first-order variations of the Riccati equation with respect to P and /i», we find that the change in P due to a change in H can be calculated using a Lyapunov equation. After defining

where Q indicates that a matrix is closed loop, we can appeal to duality to show that

where Qu is the solution to the Lyapunov equation

and A#i is a matrix of zeros except for the ith diagonal element, which is equal to unity. Therefore, to calculate the constrained gradient of P with respect to H, we need only evaluate a single Lyapunov equation, (8.50), and then calculate nw scalar values using (8.49). The constrained gradient with respect to N is calculated similarly. The analysis routine is implemented using the sequential quadratic programming (SQP) algorithm found in Matlab [174] with a slightly modified line search routine. SQP routines do not rely upon convexity. Constraints are imposed that hi > 0, rij > 0 for

8.6.2 Synthesis step Given fixed scalings, H and N, we wish to solve equations (8.36)-(8.38) for R, X, and Y to synthesize an H2/Popov controller. In principle, solving these three size-n, nonlinear equations poses no serious memory problems for practical systems. The important question is whether or not these equations can be solved quickly. One method that is used to solve this type of coupled equations is a homotopy algorithm [83]. The problem with such methods is that in order to calculate the derivative

8.6. Implementation of a practical design scheme

169

of the solution versus the homotopy parameters, they require the solution of two matrix equations of size n2 x n2 at each step. Thus, this method is not attractive for large systems. On the other hand, a straightforward method to try and solve the coupled equations is to iterate between solutions of standard Riccati equations; i.e., first solve (8.36) for R, then to fix X = Xk and solve (8.38) for y fc+ i, then to fix Y = Yk+i and solve (8.37) for Xk+2 using standard Riccati solvers. This sequence is repeated until the solutions appear to converge. We refer to this procedure as the "standard" iteration. The main problem with the standard iteration is that (8.37) does not necessarily have a stabilizing solution. Often, the procedure can iterate for some time, but eventually Yk enters a region where there is no solution for Xk+i. This is because the sign of the quadratic X term is positive. It is akin to the form of a typical HQQ Riccati equation. In contrast, (8.38) has a negative quadratic term, so, given detectability assumptions, it always has a stabilizing solution for Y. Reference [438] proposes two alternative iterative methods that eliminate the problem of not finding a solution for X. The first of the iterative methods, called the control gain (CG) iteration, solves the following equation in lieu of solving (8.37):

where (• • •) indicates a repeat of the previous term in parentheses. It can be seen that, if the iteration is on a stationary point (Xk — Xk+i), then this equation is the same as (8.37). The primary difference is that the quadratic term is now negative, so that a stabilizing Xk+i always exists. The CG iteration is akin to a perturbation method because it always has a solution at the initial step. A valid initial condition is Xk=Q = X = 0. The second iterative scheme described in [438] is the CG with relaxation (CGR) iteration. In this scheme, (8.38) is effectively solved twice for every new solution of (8.51). Unfortunately, because these are matrix rather than scalar equations, there is no known way to determine if these iterative techniques will converge from any given starting point. However, [438] contains a comparison of the CG, CGR, and standard iterations, together with a synthesis algorithm based on the LMI method of Theorem 8.3. It is shown on an 8-state benchmark example that the CG and CGR iterations could synthesize controllers for a much wider range of uncertainties than could the standard iteration. The LMI-based synthesis algorithm synthesized controllers for all desired levels of uncertainty. However, in the ranges where the CG and CGR iterations converged, they were significantly faster than the LMI-based algorithm.

8.6.3

Design examples

We now present H2/Popov controller designs that were performed using the three different design methodologies. In the cases using a D-K iteration, algorithms were initialized using a synthesis step with H = I and N = 0. The routines iterated between analysis and synthesis steps until the cost function (evaluated after every synthesis step) converged. All results were performed in Matlab on a Sun Sparc-20 workstation. For more details, the reader is referred to [437, 436].

Yang, Hall, and Feron

170

Figure 8.2: The MACE system.

Four mass system This problem consists of 4 masses connected by 3 springs, corresponding to 8 states. Using a noncollocated force input, the goal is to control the position of one of the masses, which is measured. The stiffness of two of the springs is uncertain (see [438] for details). The system is small enough that a D-K iteration that utilizes LMIs for both the analysis and synthesis steps can be run [438]. Equivalent designs for H2/Popov controllers for this system also appear in [202]. The proposed algorithm converged smoothly and quickly to a design for this system. The CG iteration was used for all synthesis steps. Overall, the design required 3.5 minutes to converge (w22 sec/iteration). In contrast, the LMI-based design required 129 seconds per iteration [438]. Furthermore, as expected, the controller correctly matched the designs found in [438] and [202].

MACE system The MACE development model is discussed in more detail in [204]. The structure, pictured in Figure 8.2, consists of four Lexan tubes, connected by aluminum nodes. It is suspended by wires from a sophisticated air-spring suspension system. In the center of the structure are three torque wheels, and at one end is an actuated payload—the pointing/scanning payload. The goal of the controller is to keep the pointing/scanning payload as still as possible, on average. For this configuration of the MACE system, broadband disturbances are injected into the system through the actuators. These actuators also serve to control the structure. Furthermore, the performance and sensed signals for the structure are the same. We design controllers for the same two models discussed in [204]: a 24-state SISO model and a 59-state 3-input/3-output model. The SISO model has 4 uncertain frequencies in its modes, and the MIMO model has 11 uncertain frequencies.

171

8.6. Implementation of a practical design scheme Total solution time 12:00

#of synthesis steps 4

Avg. synthesis time 0:44

#of analysis steps 3

Avg. analysis time 3:16

Time per full iteration 4:00

Table 8.1: SISO MACE problem: Solution time (min:sec). Data do not include the iteration run as a check on convergence or the initial controller synthesis.

Figure 8.3: Cost function of the SISO MACE problem vs. time. Cost after a synthesis step denoted by o; cost after an analysis step denoted by *. Open loop cost is 17.1. Plot includes extra iteration. These models were found to be too large to be run using the LMI-based D-K iteration used on the previous example. The results, however, can be compared with the H2/Popov controllers designed using the augmented cost function optimization of How [204]. The new D-K iteration performed very well for the SISO MACE system. All synthesis steps were performed using a CG iteration. The overall D-K iteration converged in only three iterations (4 synthesis steps and 3 analysis steps). However, to double check that the solution had actually converged to the correct solution, the iteration was forced to restart from the solution point with tighter tolerances. As shown in Table 8.1, the total solution time for the problem was only 12 minutes. The cost function is plotted as a function of time in Figure 8.3. The plot graphically demonstrates that the analysis steps required more time to solve than the synthesis steps. This design can be compared with the SISO controller design of [204]. It is estimated that, after accounting for differences in computational speed, the new routine is at least three times faster than the older routine. With a nominal plant, the new controller guarantees a cost no higher than 0.103 times the open-loop cost, while the older design only guarantees 0.11 times the open-loop cost. To check a measure of the conservatism of our control designs, we examine cases when the stiffness changes in the four modes are exactly correlated; i.e., they are either all 1%

172

Yang, Hall, and Feron

Figure 8.4: SISO MACE: H2/Popov robustness guaranteed to ±0.02 (dotted line). Cost normalized by open loop cost. LQG performance=>solid line. H^/Popov performance=>dashed line. Optimal LQG cost = 0.0922. Actual nominal H2/Popov controller cost = 0.0962. high together, or 2% low together, etc. The exact performance of perturbed systems with controllers from the D-K iteration is plotted in Figure 8.4. Notice that the range of stiffnesses for which the LQG controller can stabilize the system is very narrow. As expected, the H2/Popov controller achieves a significantly wider range of robustness. The controller was designed to be robust to ±2% variations in stiffness. In fact the H2/Popov controller maintains a level of near-optimal performance over more than twice the required range. Furthermore based on the nominal performance, it is clear that the controller has not sacrificed performance significantly for the sake of robustness. A similar plot was shown for the design in [204]. For perturbations in the ±2% range, the controller from [204] achieves inferior performance compared with the new controller. Furthermore, the controller from [204] maintains a flat level of performance for perturbations ranging in size from 0.1 down to below -0.25. It is more conservative than the new controller. Because the new SISO controller is relatively nonconservative, it is possible to design controllers with even greater robustness margins. By starting from the new controller's multipliers, the solution can be expanded to handle 110% and then 125% of the original uncertainty size. Also, [436] shows that the Bode plot of the new SISO controller is virtually indistinguishable from the LQG controller. This indicates that subtle changes in a controller can have significant effects on its robustness characteristics. For the MIMO MACE design, it was found that the initial conditions of H = 7, N = 0 could not start the initial synthesis routine. However, the initial condition did

173

8.7. Summary Table 8.2: MIMO MACE problem: Solution time (hrmin). Percentage of required uncertainty 65 82.5 100 Overall

Total solution time 5:11 2:35 4:11 11:57

#of synthesis steps 7 4 6 17

Avg. synthesis time 0:07 0:07 0:07 0:07

#of analysis steps 6 3 5 14

Avg. analysis time 0:44 0:44 0:43 0:44

start a design at 65% of the required uncertainty level, using the CG iteration. Therefore, a continuation method was used to extend this solution. The 65% solution was used to start a new design at 82.5%. A final design was then run at the required level. The final two designs were performed using the CGR iteration for the synthesis step. The time required for each of the designs to be completed is shown in Table 8.2. If the MIMO system could have been solved in one step, as the SISO problem was, then the design time would essentially be cut by two-thirds, down to perhaps 4 hours. Four hours is 20 times longer than was required for the SISO system. However, if design time for a system with n states scales with n3, then the MIMO case should take just 15 times longer than the SISO problem. In contrast, if the design time scales with n 4 , then the MIMO case should take 37 times longer than the SISO problem. Therefore, for this system, bringing a single D-K iteration to convergence is a problem that seems to scale at a rate better than n4. Due to a lack of data, detailed comparisons between the new MIMO controller and the MIMO controller of [204] are difficult to make. However, it is noted that the new design should theoretically achieve a 13.45 dB improvement in the cost over the nominal open-loop cost. As detailed in [204], the H^/Popov controller designed using the augmented cost function was implemented on the MACE system in the laboratory. It was experimentally found to achieve a 12.4 dB improvement over the open-loop system.

8.7

Summary

For an uncertain system, a multiplier analysis allows one to derive H2-performance bounds that the system is guaranteed to not exceed, even for the worst-case system pertubation (A). Although such results are necessarily conservative, there is evidence that, for practical systems, the conservatism is not overly restrictive. In particular, because the multiplier framework operates in the time domain, it is able to restrict the causality of disturbances acting upon the system, thereby reducing conservatism. The multiplier framework has been derived in the context of a generic optimization problem, rather than one specifically derived from a control analysis framework. Of course, the choice of the type of multipliers does depend heavily upon knowledge of the system and its uncertainties. The multiplier framework was shown to lead directly to an optimization of a cost function subject to LMI constraints. These constraints could also be written as Riccati equations. The analysis problem was solved using two different methods. LMI optimization routines were found to be limited to the analysis of small systems because of memory limitations. Therefore, a constrained gradient optimization routine was developed in which gradients were evaluated by solving Lyapunov equations.

174

Yang, Hall, and Feron

The problem of synthesizing a robust controller for a fixed set of multipliers was also studied. It was shown that the inherently bilinear problem could be reduced to an optimization subject to LMI constraints or to the solution of a set of coupled Riccati equations. The coupled Riccati equation solution can be derived by separating the problem into a robust full information control problem and a robust output estimation problem. Iterative methods to solve the coupled Riccati equations were investigated. Assuming that the iterative routine converges, the speed advantage of the iterative routine is expected to be even more significant as system size increases. A D-K iteration design algorithm was tested on two systems, including a large structural control problem. The iteration that employed coupled Riccati equation solvers for synthesis and a gradient optimization for analysis proved efficient and accurate. In a development setting, it is expected that the reduced computation time experienced with this routine would allow a control designer to implement and test a larger number of compensators, thereby improving the finished product.

Chapter 9

A Linear Matrix Inequality Approach to the Design of Robust H2 Filters Carlos E. de Souza and Alexandre Trofino 9.1 Introduction One of the fundamental problems in control systems is the estimation of the state variables of a dynamic system using available noisy measurements. Over the past three decades considerable interest has been devoted to estimation methods that are based on the minimization of the variance of the estimation error, i.e., the celebrated Kalman filtering approach [10]. These methods rely on the knowledge of a perfect dynamic model for the signal generating system in order to provide an optimal performance. In many cases, however, only an approximate model of the system is available. In such situations, it has been known [79], [91] that the standard Kalman filtering method fails to provide a guaranteed error variance. In order to cope with this problem, over the past few years interest has been devoted to the design of linear estimators that achieve a guaranteed upper bound on the error variance for any allowed modeling uncertainty; see, e.g., [182], [218], [331], [333], [369], [390], [429], [430], [431]. These filters are referred to as robust JJ.2 filters and can be regarded as extensions of the standard Kalman filter to the case of uncertain linear systems. The design of robust H2 filters on finite horizon for linear systems with an ellipsoidaltype parameter uncertainty in the state and input noise matrices has been studied in [218]. Robust H2 filters for linear systems with norm-bounded parameter uncertainty that provide an optimal bound for the error variance, in the stationary regime, have been developed in [182], [333], and [429]. The filter design in [333] allows for uncertainty in the state matrix only and can be viewed as a particular case of [182], which treated the design of robust, reduced-order filters in the presence of parameter uncertainty in both the state and the output matrices. On the other hand, [429] considered the general case of linear systems with norm-bounded parameter uncertainty in all the matrices of the system state-space model, including the coefficient matrices of the noise signals. An alternative technique of designing robust H2 filters for systems with norm-bounded uncertainty has been developed in [431]. However, the proposed filter is suboptimal 175

176

de Souza and Trofino

in the sense of the error variance upper bound. Recently, a general treatment of the robust minimum variance estimation problem for linear systems subject to norm-bounded parameter uncertainty in the state and output matrices has been presented in [369] for time-varying systems on finite horizon as well as the stationary infinite-horizon case. In this chapter we consider the design of robust H2 filters for linear continuoustime systems with parameter uncertainty in all the matrices of the system state-space model, including the coefficient matrices of the noise signals. The parameter uncertainty belongs to a given convex-bounded polyhedral domain and is allowed to be structured. The process noise and the measurement noise are assumed to be white signals with known statistics. The problem we address is the design of a linear stationary, asymptotically stable filter with an optimized upper bound for the estimation error variance, irrespective of the uncertainty. Both the design of full-order and reduced-order robust filters are analyzed. Linear matrix inequality (LMI)-based methodologies for designing such robust filters will be developed. The proposed design methodologies have the advantage that they can easily handle additional design constraints that keep the problem convex, for instance, when the filter is required to satisfy certain structure constraints. This is in contrast with the existing robust filter designs of [333], [369], [429], and [431], which are aimed at linear systems with norm-bounded parameter uncertainty and are given in terms of Riccati equations. We would like to stress that an alternative LMI-based solution to the robust filtering problem addressed in this paper has also been independently proposed in [156] and [157], which treated the continuous- and discrete-time cases, respectively. The chapter is organized as follows. A formal statement of the robust filtering problem is given in section 9.2. In section 9.3, an LMI-based methodology for designing full-order robust filters is developed. First, in order to gain insights for designing robust filters, the case of linear systems with a known state-space model is treated. Connections of the proposed LMI design and the celebrated Kalman filter will also be analyzed. This will be followed by the robust filter design. Section 9.4 parallels section 9.3 and deals with reduced-order robust filters.

9.2

The filtering problem

We consider the following linear time-invariant (LTI) system (E):

where x(t) G Rn is the state, XQ is a zero-mean random vector, w(t) € Rnt" is a zeromean, white noise signal with identity power spectral density matrix that is uncorrelated with XQ for all t > 0, y(t) € Rn" is the measurement, z(t) e Rn* is a linear combination of the state variables to be estimated, and A, B, C, D, and L are constant real matrices of appropriate dimensions, where L is known and A, 5, C, and D are unknown matrices such that the system matrix

belongs to a given polytope V described by

9.3. Pull-order robust filter design

177

i.e., any admissible system matrix S can be written as an unknown convex combination of na vertices 5,, i = 1, . . . , n s , given by

where Ai, Bi, C^, and Di, i = 1, . . . ,n s , are given matrices. Clearly, na = 1 corresponds to the case where the system (E) is perfectly known. We observe that the case where the input and measurement noise signals are different, say, vi(t) and V2(t), respectively, is a particular case of (9.1)-(9.3), where w(t) = [vf(t) V2(t)]T and the matrices B and D are replaced by [B 0] and [0 Z)], respectively. In this chapter we shall address the problem of designing a robust linear filter for estimating z with a guaranteed performance in the mean square error sense, irrespective of the uncertainty. More specifically, we look for a linear estimate z = T • y of the signal z, where F is a causal linear, asymptotically stable operator to be determined in order to minimize a suitable upper bound ^(F) for the worst-case asymptotic error variance, namely,

where $ is the set of the admissible filters. It is assumed that $ is the set of all LTI, asymptotically stable operators with state-space realization of the form

where the matrices Af € R n ' xn ', Kf € R n / x n v, and Lf e R n * xn / are to be determined and n/ is a given positive scalar. In the case where n/ = n, the filter will be referred to as a full-order robust filter and as a reduced-order robust filter when n/ < n. Observe that in terms of the state variables of (S) and (9.9), the estimation error e := z — z can be described by the following state-space model:

where

9.3

and

Full-order robust filter design

In this section we shall deal with the design of full-order robust filters. In order to pave the way for deriving the robust filter, initially we consider the case where the system matrix S is perfectly known, i.e., ns = 1. Considering the error system (9.10), it follows from well-known results that, subject to the asymptotic stability of this system, the variance of the estimation error as t —> oo is given by

178

de Souza and Trofino

where X is the solution to the Lyapunov equation

In light of the above, consider the following optimization problem:

Note that if such solution P exists, then the error system of (9.10) is asymptotically stable and P > X. Moreover, the objective function as above is strictly larger than the asymptotic error variance of (9.12). However, since the optimal solution to problem (9.14) is arbitrarily close to the solution to the Lyapunov equation (9.13), the gap between the objective function of (9.14) and the asymptotic error variance can be made arbitrarily small. In view of the above, we obtain using Schur's complements that the filtering problem of (9.7)-(9.8) can be reformulated as follows:

Note that the inequalities constraints of (9.16) and (9.17) are not convex on the decision variables P, A / , K f , and Lj. However, as it will be shown in the next theorem, by performing appropriate transformations and introducing new decision variables these inequalities can be transformed into LMIs. It should be observed that since we are dealing with the case where the system matrix S is perfectly known, it follows that the solution to our filtering problem is given by the Kalman filter. Thus, it is clear that the inequalities constraints of (9.16) and (9.17) should be feasible if the Kalman filter exists. Theorem 9.1. Consider the system (S) where A, B, C, and D axe known matrices. The full-order filter of the form of (9.9) that minimizes the asymptotic variance of the estimation error can be obtained in terms of the following LMIproblem in MA, PO, PI € R nxn , MK e R nxn «, ML e R n * xn , and N e R n - xn - ;

subject to

179

9.3. Full-order robust filter design With the optimal solution N, PQ, PI, MA, optimal filter are given by

, and MI found, suitable matrices of the

where P2 and P3 are any n x n matrices with P2 symmetric and such that P3P2~1P3r = PO . Moreover, the asymptotic estimation error variance satisfies E[eTe] = Tr[N] - e, where e > 0 is arbitrarily small. Proof 9.1. It will be shown that the inequalities of (9.16) and (9.17) are equivalent to the LMIs of (9.19)-(9.21). First, we partition P accordingly to A a , namely,

where P\ and P2 are n x n symmetric positive definite matrices. Note that, without loss of generality, the matrix P3 can be assumed nonsingular. To see this, suppose that the optimal P for the problem of (9.15)-(9.17) is such that P3 is singular. Let the matrix where a is a positive scalar and

Observe that since P > 0, we have that Q > 0 for a > 0 in a neighborhood of the origin. Thus, it can be easily verified that there exists an arbitrarily small a > 0 such that Q3 is nonsingular and the inequalities of (9.16) and (9.17) are feasible with P replaced by Q and such that the objective function of (9.15) will be increased only by an arbitrarily small quantity. Since Q3 is nonsingular, we thus conclude that there is no loss of generality to assume the matrix P3 to be nonsingular. In light of the definitions of the matrices Aa and Ca in (9.11), the inequality of (9.17) can be rewritten as

Premultiplying and postmultiplying (9.23) by J^ and Ji, respectively, where

and introducing the new variables

we have that the inequality of (9.23) becomes the LMI of (9.21). On the other hand, by Schur's complements, the condition P > 0 of (9.16) is equivalent to the inequalities of (9.19). We now prove the equivalence of the first inequality of (9.16) and (9.20). First, note that in view of the definition of Ba in (9.11), the inequality of (9.16) becomes

180

de Souza and Trofino

Premultiplying and postmultiplying (9.25) by Jf and Ji, respectively, where

and considering the definitions of PO and MK, we readily obtain that the inequality of (9.25) is equivalent to the LMI of (9.20). Finally, the filter matrices of (9.22) are obtained immediately from (9.24), which concludes the proof. D Remark 9.1. Note that the filter matrices of (9.22) can be rewritten as This implies that the matrix Pa can be viewed as a similarity transformation on the state-space realization of the filter and, as such, has no effect on the filter mapping from y to z. Thus, without loss of generality, we can set Pa = /„, and suitable state-space matrices of the optimal filter are as follows: Remark 9.2. Theorem 9.1 provides an LMI method for designing a full-order linear, stationary minimum variance filter for system (E) in the case where the system matrix S is known. It should be noted that in this situation, and subject to the detectability of (A,C) and stabilizability of (A,B), the Kalman filter also provides a state-space realization for the optimal filter. In the sequel we shall discuss the connections between the filter of Theorem 9.1 and the Kalman filter. We start by presenting an LMI formulation for the Kalman filter. First, note that in Kalman filtering, we set Af = A — KfC and Lf = L, which implies that the estimation error can be described by the following nth-order state-space model:

where x := x — x. The filter gain Kf is then determined in order to minimize the asymptotic variance of the estimation error, namely, to solve the following problem:

is asymptotically stable. The above can be reformulated in terms of the following optimization problem:

Hence, introducing the new variable Y = -WKf, we are led to the following LMI formulation for the Kalman filter:

9.3. Full-order robust filter design

181

Observe that since in Kalman filtering the estimation error can be described in terms of an n-dimensional state-space model, the state-space realization (9.10) for the estimation error is nonminimal. The implication of this is that the LMI of (9.21) can be relaxed to a nonstrict inequality as long as the (l,l)-block is positive definite. This will ensure the asymptotic stability of the estimation error dynamics as it will be shown below. Moreover, (9.21) can be reduced to an inequality of lower dimension. Premultiplying and postmultiplying the inequality of (9.21) by Jj and Ji, respectively, where

we get

where

If we introduce the constraint, P\ = PO -f £/, where £ is a positive scalar with e the inequality of (9.30) implies, in the limiting case, that

0,

Moreover, (9.30) reduces to

Hence, it follows from (9.31) that

Next, choosing PI — — PS, and considering (9.24), we get

Thus, we obtain

On the other hand, premultiplying and postmultiplying the inequality of (9.20) by Jj and Ja, respectively, where

it follows that (9.20) holds if and only if

182

de Souza and Trofino

Taking into account that PI —> PO,we obtain in the limiting case that (9.34) and (9.19) are equivalent to

Observe that the constraints of (9.32), (9.33), and (9.35) are exactly those for the LMI formulation of the Kalman filter of (9.27)-(9.29). Therefore, we conclude that by setting PQ = PI and choosing PI = —PS, the filter of Theorem 9.1 reduces to the Kalman filter. We now consider the design of full-order robust filters. To this end, we consider the system (E) and the uncertainty domain T> of (9.5). In light of the arguments developed in the case where the system matrix S of (9.4) is perfectly known, the robust filtering problem is formulated in terms of the optimization problem of (9.14) or, equivalently,

where Aa, Ba, and Ca are as in (9.11), and the system matrix 5 is any matrix belonging to the polytope T>. In this case, it follows that as t —» oo,

i.e., Tr[N] provides a suitable upper bound for the asymptotic error variance. Recall that when the system matrix 5" is known, i.e., ns = 1, this bound coincides with the variance of the estimation error. Our robust filter will thus be designed to minimize the upper bound of the asymptotic error variance of (9.39) or, equivalently, the robust filter design is formulated in terms of the optimization problem of (9.36)-(9.38). In light of the above arguments and considering Theorem 9.1, we have the following result. Theorem 9.2. The full-order filter of the form of (9.9) for the system (S) that minimizes the bound for the asymptotic error variance of (9.39) can be obtained in terms of the following LMI problem in MA,P0,Pi e R nxn , MK 6 R n x n «, ML € R n « xn , and N eR n <" xn ":

subject to

9.4. Reduced-order robust filter design

183

for i = 1, . . . , na, where the matrices [ £j ^ ] are the vertices of the polytope T>. With the optimal solution N, PQ, PI, MA, MK, and ML found, suitable matrices of the optimal robust filter are given by where P2 and P$ are any n x n matrices with P2 symmetric and such that P3P2~1P3r = PO- Moreover, the asymptotic estimation error variance satisfies E[eTe] < Tr[JV] for all admissible uncertainties. Proof 9.2. Using the same arguments as in the proof of Theorem 9.1, it follows that the optimization problem of (9.41)-(9.43) is equivalent to

where Aai and Bai are as in (9.11) with A, B, C, and D replaced by ^4i, J3i, Cj, and Z?i, respectively. Now, taking into account the convexity of the uncertainty domain and considering that the inequalities constraints of (9.46) and (9.47) are affine in the matrices Ai, Bi, Ci, and Di, we have the result that the optimization problem of (9.45)-(9.47) is equivalent to that of (9.36)-(9.38). D Remark 9.3. Theorem 9.2 provides an LMI-based design of a linear stationary filter for the system (£), which minimizes the upper bound of (9.39) for the asymptotic variance of the estimation error. This is in contrast with the existing robust H2 filter designs of [333, 369, 429, 431], which are aimed at linear systems with norm-bounded parameter uncertainty and are given in terms of Riccati equations. Remark 9.4. It should be noted that similarly to Remark 9.1, without loss of generality, suitable state-space matrices of the optimal robust filter of Theorem 9.2 are as follows:

9.4

Reduced-order robust filter design

In the previous section we addressed the problem of designing a robust filter with the same order as the system model. Now, we shall consider the design of reduced-order robust filters, i.e., the case where the order of the filter n/ is smaller than the order n of the system model. Similarly to section 3, we first consider the design for the case where the system matrix S is known. In order to be able to solve the optimization problem of (9.15)(9.17) via LMIs, we shall restrict the (1, 2)-block, P3, of P with dimensions n x n/ to be of the form P3 = [P4T 0]T. Note that now and the above inequality is not guaranteed to be tight. Thus, in general, Tr[7V] only provides an upper bound for the asymptotic error variance. In this situation, we have the following result.

184

de Souza and Trofino

Theorem 9.3. Consider the system (E) where A, B, C, and D are known matrices. A filter of order n/ < n of the form (9.9) that minimizes the bound for the asymptotic error variance of (9.48) can be obtained in terms of the following LMI problem in MA, PO €

subject to

where

and where P0 € R nxn ', M,i € R nxn ', and MK e R nxn «. With the optimal solution N, PO, PI, MA, MK, and M^ found, suitable matrices of the optimal filter are given by

where P2 and P4 are any n/ x n/ matrices with P2 symmetric and such that PjP2 1P^ = PO. Moreover, the asymptotic estimation error variance satisfies E[eTe] < Tr[AT]. Proof 9.3. The proof is along the same lines of that of Theorem 9.1. First, we partition P accordingly to Aa, namely,

where PI € R nxn and P2 e R n / x n / are positive definite matrices and P4 € R n ' xn '. Moreover, as in the proof of Theorem 9.1, without loss of generality, the matrix P4 is assumed to be nonsingular. Next introduce the matrix

Premultiplying and postmultiplying (9.52) by Jj and J^ respectively, it is easy to verify that this lead to the inequality of (9.17). Next, by Schur's complements and considering the definitions of PO and PS as above, it follows from the inequalities of (9.50) that P > 0. Finally, the remainder of the proof parallels that of Theorem 9.1.

9.5. Conclusions

185

In light of Theorem 9.3 and using similar arguments as in the case of the full-order robust filter design, we have the following result, which provides an LMI method for designing reduced-order, linear stationary filters for the system (E) that minimizes the upper bound of (9.39) for the asymptotic variance of the estimation error. Theorem 9.4. The filter of order n/ < n of the form (9.9) that minimizes the bound for the asymptotic error variance of (9.39) can be obtained in terms of the following LMI problem in MA,P0 e R n ' xn ', PI € R n x n , MK € R"/ x "«, ML € R n * xn ', and N eR n - x n -:

subject to

for i = 1,..., n s , and where PO, MA, and MK are as in (9.53). With the optimal solution N, PO, PI, MA, MK, and ML found, suitable matrices of the optimal robust filter are given by where P2 and Pj are any n/ x n/ matrices with P2 symmetric and such that PO. Moreover, the asymptotic estimation error variance satisfies E[eTe] < Tr[JV] for all admissible uncertainties. Remark 9.5. It can be easily verified that the matrix P± of Theorems 9.3 and 9.4 can be viewed as a similarity transformation on the state-space realization of the corresponding filters. Thus, without loss of generality, suitable state-space matrices of the reduced-order filters of Theorems 9.3 and 9.4 are as follows:

9.5

Conclusions

This chapter addressed the design of robust H2 filters for linear continuous-time systems with a state-space model subject to parameter uncertainty that belongs to a given convex bounded polyhedral domain. We developed LMI-based methodologies for designing linear stationary, asymptotically stable filters with an optimized upper bound for the asymptotic estimation error variance, irrespective of the uncertainty. These filters can be regarded as an extension of the celebrated Kalman filter to the case of systems with polytopic uncertainty in the matrices of the system state-space model. Both the design of full-order and reduced-order robust filters have been considered. The proposed design methodologies have the advantage that they can easily handle additional design constraints that keep the problem convex, for instance, when the filter is required to satisfy certain structure constraints.

This page intentionally left blank

Chapter 10

Robust Mixed Control and Linear Parameter-Varying Control with Full Block Scalings Carsten W. Scherer 10.1

Introduction

In the past years mixed control problems have attracted considerable interest, and the linear matrix inequality (LMI) approach to controller design has shown to be beneficial to tackle design specifications that have deemed untractable. For the design of output feedback controllers, it has been revealed quite recently how a suitable nonlinear change of controller parameters allows us to proceed in a straightforward fashion from an analysis specification for a controlled system formulated in terms of matrix inequalities to the corresponding synthesis inequalities for controller design [78, 264, 365]. One purpose of this chapter is to extend this general paradigm to robust performance specifications if the underlying system is affected by time-varying parametric uncertainties [265, 396]. We introduce a general technique that allows us to equivalently translate robust performance objectives formulated in terms of a common Lyapunov function into the corresponding analysis test with multipliers. Similar approaches, as those in [265, 396], are usually based on the so-called <S-procedure that introduces conservatism, since the resulting multipliers have, in general, a block-diagonal structure. Instead, we propose to use full block multipliers that are only indirectly described by LMIs such that the reformulation will not introduce conservatism. Hence we call the underlying abstract result for quadratic forms a full block 5-procedure. For robust stability specifications with affine dependence on the uncertainties, such an equivalent reformulation has been provided before in [341]. The main goal of this chapter is to generalize these ideas to robust performance specifications if the matrices describing the system can be rational functions of the parameters and to base the transformation on one abstract result that can be applied with 187

Scherer

188

ease to several performance specifications. Moreover, we will provide a discussion of the (straightforward) consequence for robust controller design. Related results for the robust stabilization problem have appeared in the inspiring paper [212]. Finally, we will turn to the design of linear parametrically varying controllers that has been fully tackled for block-diagonal multipliers [312, 191, 367]. Our purpose is to show how to overcome this structural restriction to arrive at a solution even if the multipliers are full block matrices that are only indirectly described by LMIs. Contrary to previous approaches one has to modify the structure of the scheduling function: One cannot just employ a copy of the parameters to schedule the controller, but one has to use a quadratic function thereof.

10.2

System description

We concentrate on systems that are affected by time-varying parametric uncertainties. Since we deal with linear fractional system representations, we assume that all these parameters are collected in the matrix A. The admissible set of values that can be taken by the uncertainties is denoted as

and assumed to be compact. Note that A captures both the size of the uncertainties as well as their structure. Typically, for a rational dependence of the system description on the parameters, the matrix can be taken to be block-diagonal, and the blocks take a real repeated structure; for the results in this chapter, such a structure is not required. Given the value set A, the actual time-varying parametric uncertainties consist of all continuous curves Let us now look at

a linear time-invariant (LTI) system in which the uncertainties enter in a linear fractional fashion to define the time-varying uncertain system under investigation. Obviously, Wi —» z\ constitutes the uncertainty channel, whereas Wj —> Zj, j — 2, . . . , m, are the performance channels used to describe the desired performance specifications, and u —> y is the control channel. In fact, u is the control input and y is the measured output, and any LTI system that closes the loop as

is said to be a controller. The resulting controlled system admits the description

10.3. Robust performance analysis with constant Lyapunov matrices

189

with the realization

The multiobjective robust controller design problem can be sketched as follows: Find a controller that exponentially stabilizes the closed-loop system and that achieves a mixture of various performance specifications on the different performance channels of the controlled system for all possible uncertainties affecting the system. In the next section we extend well-known analysis tests for nominal performance as described in [365] to the corresponding robust performance tests against time-varying uncertainties in terms of a constant quadratic Lyapunov function. The essential new ingredient will be the ease in which one can derive the corresponding multiplier test by just referring to the abstract full block 5-procedure that is presented in Lemma 10.1 in the appendix.

10.3

Robust performance analysis with constant Lyapunov matrices

In this section, we provide analysis tests that are based on finding a constant quadratic Lyapunov function in order to guarantee the following properties: • Well-posedness of the linear fractional transformation (LFT) used to describe the uncertain system. • Uniform (in the uncertainty) exponential stability. • Robust performance, specified as quadratic performance (such as bounding the 1/2gain or dissipativity) or as bounding the H-2-norm, the generalized #2-norm, or the peak-to-peak gain.

10.3.1

Well-posedness and robust stability

We call the description (10.3) well posed if

Then the channel Wj —> Zi of the uncertain closed-loop system admits the alternative representation

with the function

that is, due to (10.5), continuous (and even smooth) on A.

190

Scherer

If the interconnection is well posed, (10.3) or (10.6) are uniformly exponentially stable if there exist constants K and a > 0 such that, for every uncertainty A(.) and for every unforced (w^ = 0, . . . , wm = 0) system trajectory £(.), Under the hypothesis (10.5), it is well known that uniform exponential stability is guaranteed by the existence of an X > 0 that satisfies the Lyapunov inequality A.( A)T X+ XA.(&) < 0 VA in the admissible value set A. In the following result the Lyapunov inequality is rewritten into a form that facilitates the application of the full block <Sprocedure. Theorem 10.1. Suppose the interconnection (10.3) is well posed, and suppose there exists an X > 0 satisfying

Then the uncertain system (10.3) is uniformly exponentially stable. Proof 10.1. The very standard proof is only included for the convenience of the reader. By compactness of A, there exists an e > 0 such that

VA € A. Now suppose A(.) is any admissible uncertainty curve, and let £(.) be an arbitrary unforced system trajectory. With v(t) := £(t)TX£(t), we obtain by left multiplying £(t)T and right multiplying £(t). With the integrating factor ec*, this implies Due to Amin(^)||e(<)||2 < v(t] < A max (*)||£WII 2 , we finally obtain

which finishes the proof with explicit formulas for the constants a and K.

D

The condition (10.7) is formulated in terms of the rational function .A(A) on the whole set A such that it is pretty hard to verify directly. The main purpose of the full block «S-procedure proposed in this chapter is to equivalently reformulate this test into a more explicit condition that makes use of multipliers. Here, the relevant set of multipliers is defined as

Whenever required we will tacitly assume that any such multiplier is partitioned as.

For robust stability, we will reveal in detail how to apply Lemma 10.1 such that we can easily extend the technique to robust performance tests.

10.3. Robust performance analysis with constant Lyapunov matrices

191

Theorem 10.2. The interconnection (10.3) is well posed and X satisfies (10.7) if and only if there exists a multiplier P € P with

Proof 10.2. To apply Lemma 10.1, we introduce

and

Hence 5t/ = ker(E/T) n 5 is described as

Therefore, we conclude that Su is complementary to SQ if and only if / - A£>n is nonsingular. Moreover, if I—APu is nonsingular, we arrive at the alternative description of <Sjy as

Then it is obvious that the following conditions are equivalent: inequality (10.7) and N < 0 on Su] inequality (10.9) and N + TTPT < 0 on 5; and, finally, the condition P € P and P > 0 on ker(t/). The latter just follows from the observation that the kernel of U is nothing but the image of ( 7 ). All this shows that Theorem 10.2 is a reformulation of Lemma 10.1, which finishes the proof.

10.3.2

Robust quadratic performance

Suppose Ppi is a symmetric matrix that defines a performance index. Then the robust quadratic performance (QP) specification on the channel i is formulated as follows: There exists an e > 0 such that

holds for any trajectory of the uncertain system (10.3) with £(0) = 0. The following technical hypothesis on the performance index is both indispensable to apply the full block «S-procedure and to arrive at nominal controller synthesis procedures that can be based on solving LMIs:

Scherer

192

Among others, this hypothesis holds true for the following very important special cases: Robust QP with guarantees that the Z/2-gain of the channel Wi —> Zi is robustly smaller than 7. guarantees strict robust dissipativity of the channel Wi —> Zi, generalizing positive realness. Based on Lyapunov arguments, one can easily obtain a sufficient LMI condition for robust QP. Theorem 10.3. Suppose the interconnection (10.3) is well posed, and suppose there exists an X satisfying

V A G A. Then the uncertain system (10.3) is uniformly exponentially stable and satisfies the robust QP specification for the channel Wi —> z». Proof 10.3. The upper-left block of (10.12) reads as

At this point we exploit (10.11) to infer (10.7) such that we can conclude uniform robust exponential stability. The proof of robust performance is, again, straightforward. By continuity and compactness of A, we can add ( Q ^ ) on the left-hand side of (10.12) for some small e > 0 without violating the inequality. For any Wi € LI, let £(.) and Zi(.) be the corresponding state and output trajectory of the uncertain system (10.3) with £(0) = 0. By right and left multiplication with ( Jf/A ) and its transpose, we infer

Integrating over [0, T] and letting T go to infinity implies the desired inequality (10.10) if we recall f (0) = 0 and lim^oo t(T) = 0. On the basis of Lemma 10.1, we can obtain with ease the equivalent multiplier test. Theorem 10.4. The interconnection (10.3) is well posed, and X satisfies (10.12) if and only if there exists a P € P such that

10.3. Robust performance analysis with constant Lyapunov matrices

193

Proof 10.4. As in the proof of Theorem 10.2, apply Lemma 10.1 to

and

10.3.3

Robust H2 performance

If the disturbance Wi that affects the system is stochastic in nature, we can also consider the H2 criterion as a performance measure for the perturbed system. Let us assume that £(0) = 0, that Wi is white noise, and that we are interested in bounding the variance of the output Zi by a given number 7. We need to assume that the uncertainty model and the controller are such that

Then it is easy to arrive at the following analysis result. Theorem 10.5. Suppose the interconnection (10.3) is well posed, and suppose there exist X > 0, Z > 0 satisfying, VA € A,

Then the uncertain system (10.3) is uniformly exponentially stable, and the variance of the output Zi on the time interval [0, oo) is smaller than 7. Due to the zero row and column, the inequality in (10.16) could be simplified. We kept it in this form to display the similarity to the robust QP test. Proof 10.5. Uniform exponential stability follows as in the proof of Theorem 10.3. For verifying robust performance, one can easily rewrite (10.16) with y = X~l to

194

Scherer

Using Lemma 10.2, (10.17) is seen to be equivalent to

Let us now recall that the state covariance matrix of the uncertain system is given as the solution of the initial value problem

Standard comparison results for linear matrix differential equations imply )C(t) < ^y for t > 0 and hence we infer

which leads to the desired bound on the output variance. We end up with two inequalities in the parameter A. Therefore, we have to apply Lemma 10.1 to each of these inequalities individually. This leads to two independent multipliers to equivalently reformulate the robust H2 analysis condition to the multiplier version. Theorem 10.6. The interconnection (10.3) is well posed and X, Z satisfy (10.16)(10.17) if and only if there exist Pi,P2 G P such that

Remark. Instead of the stochastic interpretation, the criterion derived here also admits a deterministic interpretation; it guarantees a bound on the sum of the energy of the output responses to initial conditions taken as the columns of #i(A(0)).

10.3.4

Robust generalized H2 performance

The generalized H2 norm is the gain of the system mapping Wi € L% into z» 6 £«,, i.e., the energy-to-peak gain [347]. The corresponding performance specification is to robustly guarantee the bound

Note that the gain can only be finite if (10.15) holds, which is assumed throughout this section.

10.3. Robust performance analysis with constant Lyapunov matrices

195

Theorem 10.7. Suppose that the interconnection (10.3) is we]] posed and that there exists an X > 0 such that, V A € A, the inequality (10.16) and

hold true. Then (10.3) is robustJy exponentially stable, and the gain of L^ 3 Wi —* Zi € LOO is smaller than 7. Proof 10.6. As for QP, the proof of stability is obvious, and one can infer from (10.16) that there exists an e > 0 such that

for any wt e L2. Integrating over [0,T] leads to £(T)T#£(T) < (7 - e) /QT iyi(t) T tWi(t) dt and hence The second inequality (10.21) implies Ci(A) T Ci(A) < ^X and therefore, due to £>ii(A) = 0, Both relations taken together lead to (10.20).

D

Similarly as for the H2-performance specification, we have to introduce two independent multipliers to equivalently reformulate these conditions to the corresponding multiplier versions. Theorem 10.8. The interconnection (10.3) is we]] posed and X satisfies (10.16) and (10.21) if and on]y if there exist multipliers Pi,P2eP with (10.18) and

10.3.5

Robust bound on peak-to-peak gain

Let us finally include in our list the important time-domain specification to keep the peak-to-peak gain L^ 3 Wi —> Zi e LOO smaller than 7. To be precise, we intend to robustly guarantee

Again, simple Lyapunov arguments provide sufficient conditions for this inequality to hold. Theorem 10.9. Suppose the interconnection (10.3) is we]] posed. If there exist X, A > 0, /x such that, VA € A,

196

Scherer

then (10.3) is robustly exponentially stable, and the gain of LOO 3 robustly smaller than 7.

is

Proof 10.7. As before we observe that (10.24) implies

We infer £(t)TX£(t) < pf* e-A(*-T)|K(r)||2dT and hence

The inequality (10.25) leads (due to 7 - n > 0) to

for some small e > 0. If we combine the inequalities we infer and hence equation (10.23) holds. The reformulation into the corresponding multiplier version is, again, straightforward. Theorem 10.10. The interconnection (10.3) is well posed, and (10.24)-(10.25) hold if and only if there exist multipliers PI , P2 £ P that satisfy

Note that X and A enter these inequalities nonlinearly. In an analysis problem, it is advisable to search for the best upper bound on the peak-to-peak gain by fixing A > 0 and minimizing the bound 7 over the LMIs (10.26)-(10.27) (which is a convex optimization problem) to get the optimal value 7*(A). Then one can perform an additional line search over A to further minimize 7* (A) and to get to the smallest achievable bound.

10.4

Dualization

On the basis of Lemma 10.2, one can dualize all robust performance tests we have provided in the previous section. Let us concentrate on QP only with an index matrix that is nonsingular and whose inverse is denoted as

10.5. How to verify the robust performance tests

197

Note that this property can be enforced by a slight perturbation that changes neither the problem formulation nor its solution. For an arbitrary matrix M, we observe that

Hence the set of dual scalings has to be denned as

and each P € P is partitioned in the same fashion as P. Corollary 10.1. There exists an X and a multiplier P £ P with (10.12) if and only if there is a dual Lyapunov matrix y > 0 and a dual multiplier P G P with

The Lyapunov matrices and multipliers are related as

Proof 10.8. Suppose X and P satisfy (10.12). Since A = 0 is in the value set A, we conclude that R > 0. Hence we infer

This implies that S as denned in (10.14) is, in fact, a negative subspace of N of maximal dimension. Moreover, since (10.13) implies ( ^ }TP( pn ) < 0, P is nonsingular. Hence the same is true of N. Due to (10.28) (after a row permutation), the proof is finished by applying Lemma 10.2.

10.5

How to verify the robust performance tests

The set of multipliers P is obviously convex. However, since this set is described by infinitely many inequalities, and since it is not obvious, in general, how to reduce to finitely many conditions, it is hard to even decide whether a matrix P is indeed contained in P or not. This precludes the verification of the robust performance tests in Section 10.3 by standard algorithms.

198

Scherer

This is the motivation to confine, at the expense of conservatism, the search to a smaller set of multipliers that admits a simple description, preferably in terms of finitely many LMIs. For that purpose we assume that the value set A is the convex hull of finitely many prespecified matrices:

Let us then introduce the multiplier set

Note that, in general, this is a strict inclusion such that the restriction of the search to PI introduces conservatism. However, a straightforward convexity argument (based on Q < 0) reveals that

Hence, the set PI can indeed be fully described by finitely many LMIs, and the search for P G PI renders our robust performance test verifiable by solving standard LMI problems. Remark. If T>u = 0 such that the parameters enter (10.3) affinely, we observe that all the robust performance inequalities already imply that any multiplier in P satisfies Q < 0 and is, hence, contained in P\. Hence, in this case, P and P\ conincide and no conservatism has been introduced. It is possible to reduce the conservatism for block-diagonal uncertainties. For that purpose let us assume that

Then we obtain (10.29) where the N = 2m generators A,, are defined by letting each 6i vary in {—1,1}. Let us now partition the upper-left block Q of the multiplier P as A, which defines ra diagonal blocks Q\,..., Qm. If we introduce

we infer

on the basis of a simple multiconvexity argument. Again, the set P2 is described in terms of finitely many LMIs. As expected, one can demonstrate by simple examples that the set P2 is in general larger than PI, and that P2 can lead to less conservative robust performance tests. All these techniques apply in the same fashion to the corresponding dual inequalities as provided in section 10.9.2 for the corresponding sets of dual multipliers P, PI, and ft-

10.6. Mixed robust controller design

10.6

199

Mixed robust controller design

We have listed several important analysis tests for robust performance. In a typical multiobjective robust controller design problem one tries to find a controller that meets a selection of all these specifications on various channels of the controlled system. In order to render the underlying analysis problems algorithmically tractable, we constrain ourselves to the multiplier set P\ or, for block-diagonal uncertainties, to P%, respectively. Even for the nominal performance specifications, the corresponding multiobjective control problems are still hard and mostly open. It is well known that the main difficulty arises due to the fact that each of the performance specifications requires, in general, a different Lyapunov matrix to render it satisfied. The currently known design techniques do not easily overcome this obstacle. This has been the motivation to consider, instead, the so-called mixed control problems in which the goal is to render the specifications satisfied with a common Lyapunov function X for all specifications under investigation. To be specific, we consider the robust mixed QP/H2 problem: Try to robustly achieve QP with index Ppi on channel Wi —> Zi and an H2 bound 7 on channel Wj —* Zj. Note that the generalization to any other combination of (possibly repeated) performance specifications on other channels is obvious. The robust mixed QP/H2 control problem hence aims at finding a controller, multipliers P, PI, PI € P, and a Lyapunov matrix X as well as an auxiliary variable Z > 0 that render the LMIs (10.13) and (10.18)-(10.19) with i replaced by j satisfied.

10.6.1

Synthesis inequalities for output feedback control

Quite recently it has been observed [264, 78, 365, 361] how to step in a formal manner from the analysis to the corresponding synthesis inequalities. This procedure is based on transforming the Lyapunov matrix X and the controller parameters as into the new symmetric blocks X, Y and the transformed controller parameters K, L, M, N that are collected in the variable v. Let us now introduce the functions

and

that are affine in v. In order to transform the synthesis inequalities, one needs to find a formal congruence transformation involving Z such that the blocks in the analysis inequalities transform as

Then it suffices to perform the substitution

200

Scherer

to arrive at the synthesis inequalities in the new variables v and all other variables that appear in the analysis test (such as multipliers and auxiliary parameters). Once having solved the synthesis inequalities, we are led by the inversion of (10.30) back to the Lyapunov matrix and to the desired controller parameters. This inverse is easily calculated by finding nonsingular matrices U, V with / — XY = UVT and solving the equations

for X and Ac, Bc, Cc, Dc. If one performs these two steps for the robust mixed QP/H2 problem, one arrives at the synthesis inequalities

where we suppress the variable v for reasons of space. Unfortunately, testing the feasibility of these inequalities, in general, does not amount to solving a convex optimization or LMI problem. Hence one has to resort to controller scalings iteration as they are known from /^-theory [265]. One might suggest the following procedure: Consider the scaled uncertainty set rA and try to maximize r such that the synthesis inequalities are satisfied for the value set rA. Start with a nominal design for r = 0. Then the iteration proceeds as follows: In the first step, fix K, L, M, N and find X, Y and multipliers P, PI, PI, which correspond to rA such that the synthesis inequalities hold and r is maximal. If one parametrizes the multipliers by finitely many LMIs, then testing the feasibility for fixed r corresponds to an analysis problem and hence reduces to a standard LMI feasibility test; the maximization of r can be performed by bisection. In the second step, one fixes P, PI, PI and maximizes r by varying the whole variable v. For fixed r, this corresponds to a nominal performance design problem (such that it reduces to an LMI feasibility test), and the maximization of r is done by bisection.

10.6. Mixed robust controller design

201

One can immediately advise many variations of this principal procedure. In particular, one might choose other combinations of fixed and varying parameters in the optimization steps to increase r [361].

10.6.2 Synthesis inequalities for state-feedback control The procedure described in the previous section applies literally to state-feedback design. One just needs to employ the functions

and the equations to reconstruct the Lyapunov matrix and the (static) controller read as As such, the synthesis inequalities are not affine in all variables. However, as demonstrated for the analysis inequalities, one can straightforwardly dualize the synthesis inequalities that we have derived above. Due to the mere fact that Bj(v) and Dij(v) actually do not depend on v, the dual inequalities define convex constraints even if letting both the multipliers, the auxiliary parameter Z, and the whole parameter v vary; in fact, they can be easily rearranged to become affine in all unknowns. Hence, what is known from single-objective control problems or for block-diagonal multipliers completely generalizes to robust mixed control problems with full block scalings.

10.6.3

Elimination of controller parameters

It is well known how to eliminate the controller parameters in single-objective nominal design problems. This is also possible in single-objective robust control problems. Let us consider, as an example, the robust QP problem with index Pp on the channel w2 —> z2. We will provide a variation of the standard procedure based on the projection lemma that leads, even for a general QP index, to particularly simple formulas. We just apply Lemma 10.3 and observe that we can work with basis matrices K\ and K2 of the kernels of respectively. Then we arrive at the following equivalent synthesis test: Find X, Y and multipliers P € P, P €. P that satisfy

Scherer

202

and the duality coupling condition

Nonconvexity enters via the generally nonconvex constraint (10.34). Although this result has not appeared in the literature, it is a straightforward (and notationally simple) extension of those in [341, 212] to robust QP. The main purpose for being so detailed is to point out the relation to linear parameter-varying (LPV) control in the next section.

10.7

LPV design with full scalings

In LPV control, it is assumed that the parameters A(£) that enter the system are not unknown but that they can be measured online. This allows, among other applications, to approach (robust) gain-scheduling synthesis problems for nonlinear control systems. So far, the LPV problem has been solved for block-diagonal parameter matrices with the corresponding block-diagonal scalings [19, 191, 312, 367], In [364] we have sketched how to extend these techniques to full block scalings, and in this chapter we give for the first time the full problem solution. Adjusted to the structure of (10.1), we assume that the measured parameter curve enters the controller also in a linear fractional fashion. Hence an LPV controller is defined by scheduling the LTI system

with the actual parameter curve as

The controller is hence parameterized through the matrices Ac, Bc, Cc, Dc, and through a possibly nonlinear scheduling function AC(A). Note that previous approaches were based on AC(A) = A such that the controller is scheduled with an identical copy of the parameters. However, full block scalings require the extension to a more general scheduling function that will turn out a posteriori to be quadratic functions. The goal is to construct an LPV controller that renders the QP specification with index Pp for the channel w% —» z% for all possible parameter curves satisfied. As standard, the solution of this problem is obtained with a simple trick. In fact, the controlled system can, alternatively, be obtained by scheduling the LTI system

with the parameters as

10.7. LPV design with full scalings

203

and then controlling this parameter-dependent system with the LTI controller (10.35). Hence, with the chosen controller structure, the LPV problem we started out with is equivalently reformulated to a robust performance design problem as discussed previously: Find an LTI controller (10.35) that renders the system (10.36)-(10.37) uniformly exponentially stable such that the robust QP specification for tu2 —>• ^2 with nonsingular index Pp is satisfied. Due to the fact that the parameters are measured online, the uncertainties enter the system in a very specific form, which is reflected in the particular structure of the describing matrices in (10.36) and of the parameters in (10.37). This particular structure implies that the synthesis inequalities related to this robust performance problem are standard LMI problems. Hence they can be solved (without conservatism) using existing algorithms. For guaranteeing robust stability and performance of the closed-loop system, we employ extended multipliers adjusted to the extended uncertainty structure that are given as

and that satisfy

The corresponding dual multipliers Pe = P

l

are partitioned similarly as

As indicated by this notation, it will turn out that the LPV synthesis inequalities will be influenced only by the multipliers blocks in Pe and Pe without indices, and they will be actually identical to those of the robust control problem apart from the coupling condition (10.34). Therefore, testing these synthesis conditions indeed amounts to solving a standard LMI problem. Theorem 10.11. There exists a controller (10.35) and a scheduling function such that the system (10.36)-(10.37) controlled with (10.35) satisfies the analysis conditions for robust QP with multipliers (10.38)-(10.39) if and only if there exist X, Y and scalings P € Pi, P € Pi that satisfy the LMIs (10.31)-(10.33). Proof 10.9. The proof of "only if" is straightforward: Eliminate the controller parameters in the analysis inequalities. Due to the specific structure of the describing matrices in (10.36), the resulting synthesis inequalities simplify to (10.31)-(10.33) such that only the multiplier parts

204

Scherer

appear. The constructive proof of "if" is more involved. Let us assume that we have found a solution to (10.31)-(10.33). First step: Extension of scalings. Let us define the matrices

with the row partition of P. Note that im(Z) is the orthogonal complement of im(Z) and recall For the given P and P, we try to find an extension Pe with (10.38) such that the dual multiplier Pe — P~l is related to the given P as in (10.40). After a suitable permutation, this amounts to finding an extension

where the specific parametrization of the new blocks in terms of a nonsingular matrix T and some symmetric N will turn out to be convenient. Such an extension is very simple to obtain. However, we also need to obey the positivity/negativity constraints in (10.38) that amount to

and

We can assume without loss of generality (perturb, if necessary) that P — P~l is nonsingular. Then we set and observe that (10.42) holds for any nonsingular T. The main goal is to adjust T to render (10.43)-( 10.44) satisfied. We will in fact construct the subblocks TI = TZ and T2 = TZ of T - (Ti T2). Due to (10.41), conditions (10.43)-(10.44) read in terms of these blocks as

If we denote by n+(S) and n_(S'), respectively, the number of positive and negative eigenvalues of the symmetric matrix S, we have to calculate n_(AT — Z(ZTPZ)~1ZT) and n+(N - Z(Z?PZ}~1ZT}. Simple Schur complement arguments reveal that

equals By Lemma 10.2 and ZTPZ > 0, we infer ZTP~1Z < 0, and hence n-(ZTPZ) n-(ZTP~lZ}. This leads to

=

10.7. LPV design with full scalings

205

This implies that there exist TI, T2 with n_(AT), n+(N) columns that satisfy (10.45). Since n+(N) + n_(./V) equals the number of rows of T, these two blocks indeed define a square T that can be assumed (after perturbation, if necessary) to be nonsingular. This finishes the construction of the extended multiplier (10.38), where we observe that the dimensions of Q22/R22 equal the number of columns of T\/T2 which are, in turn, identical to n-(N)/n+(N). Second step: Construction of the scheduling function. Let us recall that

on A. By Lemma 10.3, we can hence infer that, for each A € A, there indeed exists a A C (A) that satisfies (10.39). Due to the structural simplicity, we can even provide an explicit formula that shows that A C (A) can be selected to depend smoothly on A. Indeed, by a straightforward Schur-complement argument, (10.39) is equivalent to

for U = Re- S^Q~lSe > 0, V = -Q"1 > 0, W = Q^lSe. Since there exists a solution, the inequality can be rearranged to

and it is clear that one solution is obtained by rendering the (2, l)-block to vanish:

Note that A C (A) has dimension n_(N) x n+(N}. Third step: LTI controller construction. After having reconstructed the scalings, the design of the LTI part of the controller amounts to solving a nominal QP problem that can be done along standard lines, as shown, e.g., in [365], Remark. The proof reveals that the scheduling function A C (A) has as many rows/ columns as there are negative/positive eigenvalues of P — P~l (if assuming without loss of generality that the latter is nonsingular). If it happens that P — P~l is posititve or negative definite, there is no need to schedule the controller at all; then we obtain a controller that solves the robust QP problem. The search for X and Y and for the multipliers P € PI and P € PI to satisfy (10.31)(10.33) amounts to testing the feasibility of a standard LMI. Moreover, the controller construction in the proof of Theorem 10.11 is constructive. Hence we conclude that we have found a full solution to the QP LPV control problem (including L2-gain and dissipativity specifications) for full block scalings Pe that satisfy Qe < 0. The more interesting general case without this still restrictive negativity hypotheses is dealt with in a forthcoming paper.

206

10.8

Scherer

Conclusions

In this chapter we have discussed how to handle multiobjective robust performance analysis and synthesis problems for systems that are affected by time-varying parametric uncertainties. We introduced a general technique, the so-called full block «S-procedure, that allows us to formally introduce full block multipliers in order to reduce the conservatism that could result from the restriction to standard block-diagonal multipliers. Finally, we provided a solution to the single-objective LPV control problem if we employ a subclass of full block scalings; the fully general case is left to a forthcoming paper.

10.9

Appendix: Auxiliary results on quadratic forms

10.9.1 A full block ^-procedure Suppose «S is a subspace of R n , T G R/ xn is a full row rank matrix, and U C Rkxl is a compact set of matrices of full row rank. Define the family of subspaces

indexed by U G U. Suppose N G R n x n is a fixed symmetric matrix. The goal is to render the implicit negativity condition explicit. We want to relate this property, under certain technical hypotheses, to the existence of a symmetric multiplier P that satisfies

As a technical hypothesis, we require that all subspaces Sy are complementary to a fixed subspace «Sb C 5 that has the two properties

In the intended applications, «S is an unperturbed system, T picks the interconnection variables that are constrained by the uncertainties, the elements ofU^U define kernel representations of the uncertainties, and Su is the uncertain system. The complementarity condition amounts to a well-posedness property of the uncertain system description, and the nonnegativity of N is a condition on the performance index of interest. Lemma 10.1. The two conditions

hold if and only if there exists a matrix P that satisfies

10.9.2

Dualization

The dualization of robust performance tests is most easily achieved with the following well-known auxiliary result that is the abstract version of a dualization argument in [212], which is based on manipulating block matrices.

10.9. Appendix: Auxiliary results on quadratic forms

207

Lemma 10.2. Suppose that N is symmetric and nonsingular, and that S is a negative subspace of N of maximal dimension. (N is negative definite on S, and the number of negative eigenvalues of N coincides with the dimension ofS.) Then is positive definite on

10.9.3

Solvability test for a quadratic inequality

The following explicit solvability characterization for a quadratic matrix inequality serves to conveniently eliminate controller parameters in synthesis tests. Consider the quadratic inequality

in the unstructured unknown X. We assume that and that

is nonsingular.

By Lemma 10.2, we can dualize to equivalently reformulate the inequality as

The solvability test makes use of matrices A± and J5j_ whose columns form a basis of the kernels of A and B, respectively. Lemma 10.3. The quadratic inequality (10.48) has a solution X if and only if

and

The proof of "only if" is obvious. The proof of the converse can be based on the projection lemma and is constructive; if a solution is known to exist, one can explicitly calculate it.

This page intentionally left blank

Chapter 11

Advanced Gain-Scheduling Techniques for Uncertain Systems Pierre Apkarian and Richard J. Adams 11.1

Introduction

The gain-scheduling problem has been the subject of a great deal of research over recent years, from both theoretical and practical viewpoints. This renewed interest probably stems from the development of new techniques and software that allow for a more rigorous and systematic treatment of the gain-scheduling problem. The classical approach to this problem essentially consists of repeated design syntheses associated with some scheduling strategy connecting locally designed controllers. Such schemes, however, lack supporting theories that guarantee the behavior of the scheduled controller. A significant contribution toward the elimination of such weaknesses is the formulation of the gainscheduling problem in the context of convex semidefinite programming [315], an elegant and solidly based branch of optimization theory [296, 292, 403]. Expressed in terms o linear matrix inequalities (LMIs), the gain-scheduling problem is readily and globally solved using currently available efficient optimization software [153]. LMI techniques now appear as very natural mechanisms for the formulation of gain-scheduling problems as well as for a vast array of other problems in the control field. Reference [64] gives a overview of the scope of application of such techniques. As emphasized in HQQ control theory, a key stage in the characterization of gainscheduled controllers is the search for adequate Lyapunov functions that establish stability and a performance bound for the closed-loop system. The linear-fractional transformation (LFT) gain-scheduling techniques in [312, 19, 252, 253] or the so-called quadrat gain-scheduled techniques in [36, 21] make use of a fixed Lyapunov function, as opposed to one that depends on the scheduled variables, to characterize stability and performance. According to [424], such approaches are potentially very conservative because they allow for arbitrary rates of variation in the scheduled variables. More dramatically, it has been shown in [424] that some systems are not even quadratically stabilizable, that is, are no stabilizable on the basis of a single Lyapunov function. A significant improvement over such techniques can be obtained by exploiting the concept of parameter-dependent Lya209

210

Apkarian and Adams

punov functions. This is discussed in the context of robustness analysis and synthesis in [149, 135, 191] and for the gain-scheduling problem in [424, 191]. Parameter-dependent Lyapunov functions allow the incorporation of knowledge on the rate of variation in the analysis or synthesis technique, and therefore lead to much less conservative answers. The reader is referred to [33, 424] for earlier work related to the approaches considered here. The discretization of continuous-time gain-scheduled controllers is considered in [17]. In this chapter we investigate two different techniques: [362, 78, 265] and an extension of [148, 147] to the gain-scheduling problem. These techniques impose no restriction on the plant and provide a simple and streamlined treatment of the gain-scheduling problem. Moreover, the technique in [362, 78] allows the incorporation of multiple specifications into the design problem such as H2 — HQQ , pole clustering, or control effort constraints. The second technique is more restrictive but offers computational advantages. The focus of this work is on the computational effort for controller calculation and on the practical issues of controller implementation. A special emphasis is placed on the development of scaling techniques that take advantage of the problem's structural properties and thus reduce the conservatism of the gain-scheduling approach. It is further shown that combining the capabilities of both techniques provides a comprehensive and effective methodology, encompassing all components of the gain-scheduling task from theoretical constructions to real-time implementations. The chapter is structured as follows. Sections 11.2 to 11.4 give a thorough discussion of gain-scheduling synthesis techniques together with some refinements and improvements in section 11.5. Finally, the validity and applicability of concepts and techniques are demonstrated for a two-link flexible manipulator application in section 11.6. The work here is also related to the linear parameter-varying (LPV) control problem considered in Chapter 10. In Chapter 10, the rates of variation of the parameter are not taken into account, hence, potentially leading to conservative answers. However, as is common to most modern control techniques, the refinements in this chapter increase the computational cost, and this could be prohibitive for large systems. Some techniques to alleviate the computational cost due to the gridding phase are considered in [22, 399].

11.2

Output-feedback synthesis with guaranteed Ly gain performance

In this section we recap some known results on the gain-scheduling technique with bounded parameter variations rates and point out connections between different approaches. We first give a general characterization of gain-scheduled controllers, the solution to which involves both intermediate controller matrices and Lyapunov variables X and Y. This formulation will be referred to as the basic characterization, emphasizing the fact that it can be easily extended to multiple objective problems, pole clustering problems, etc. [362, 78]. Next, a second formulation of gain-scheduled controllers is presented. It will be referred as the projected characterization, as the intermediate controller matrices have been eliminated through projections [148]. Reconstructing the controller state-space data from the projected conditions has been addressed in [148, 147] for the customary HQQ control problem. The reconstruction procedure is again described here, in the case of the gain-scheduling problem, for completeness of the discussion. The reader is referred to [33, 34, 424] for details, insights, and applications of analogous gain-scheduling techniques. The problem addressed throughout the chapter is the following. Suppose we are given

11.2. Output-feedback synthesis with guaranteed Z/2-gain performance

211

an LPV plant G(0) with state-space realization

where define the problem dimension. The time- varying parameter 6 := (0i, . . . , OL)T &s well as its rates of variation 0 are assumed bounded as follows: (a) each parameter 0; ranges between known extremal values 9t and 0*:

(b) the rate of variation 0* is assumed well defined at all times and satisfies

where v^ < v^ are known lower and upper bounds on 0$. The first assumption means that the parameter vector 0 is valued in a hypercube G. Similarly, (11.3) defines a hypercube O
The gain-scheduled output-feedback control problem consists of finding a dynamic LPV controller, K(0), with state-space equations

which ensures internal stability and a guaranteed Z/2-gain bound 7 for the closed-loop operator (11.1)-(11.5) from the disturbance signal w to the error signal z, that is,

and all admissible trajectories (0,0) and zero-state initial conditions. Note that A and AK have the same dimensions, since we restrict the discussion to the full-order case. The formulation of such controllers can be handled via an extension of the bounded real lemma with quadratic parameter-dependent Lyapunov functions V(xd, 0) = x^P(0)xc£, where Xd stands for the state vector of the closed-loop system. See [424, 149, 135, 362] for details. Note that the controller state-space matrices are allowed to depend explicitly on the derivative of the time-varying parameter 0. Different techniques to remove the dependence on 0 will be extensively discussed in section 11.3; see also [34]. Except for the usual smoothness assumptions on the dependence on 0, the problem data and variables will be unrestricted in the subsequent derivations. The basic characterization of gain-scheduled controllers with guaranteed L2-gain performance is presented in the next theorem where the dependence of data and variables on 0 and 0 has been dropped for simplicity.

Apkarian and Adams

212

Theorem 11.1 (basic characterization). Consider the LPVplant governed by (11.1), with parameter trajectories constrained by (11.2), (11.3). There exists a gain-scheduled output-feedback controller (11.5) enforcing internal stability and a bound 7 on the L2 gain of the closed-loop system (11.1) and (11.5), whenever there exist parameterdependent symmetric matrices Y and X and a parameter-dependent quadruple of statespace data (AK,BK,CK,DK) such that for all pairs (0,0) in & x 6
In such a case, a gain-scheduled controller of the form (11.5) is readily obtained with the following two-step scheme: • solve for N, M, the factorization problem

• compute AK, BK, CK with

Proof 11.1. See [362, 78]. Note that since all variables are involved linearly, the constraints (11.6) and (11.7) constitute an LMI system. This system is, however, infinite due to its dependence on (0,0) ranging over Q x G^. Using the projection lemma, detailed in [148], the controller variables can be eliminated, leading to a characterization involving the variables X and Y only. This is presented in the next theorem. Theorem 11.2 (projected solvability conditions). Consider the LPV plant governed by (11.1), with parameter trajectories constrained by (11.2) and (11.3). There exists a gain-scheduled output-feedback controller (11.5) enforcing internal stability and a bound 7 on the L2 gain of the closed-loop system (11.1) and (11.5) whenever there exist parameter-dependent symmetric matrices Y(0) and X(0) such that for all pairs (0,0) in G x Gd the following infinite-dimensional LMI problem holds:

11.2. Output-feedback synthesis with guaranteed I/2-gain performance

213

where A/x and A/y designate any bases of the null spaces of [ C2 D2\ ] and [ B% respectively. Proof 11.2. This is a straightforward application of the projection lemma [148] to the LMI (11.6), with respect to the matrix variable

Theorem 11.2 only provides existence conditions for controllers of the form (11.5). These conditions become necessary and sufficient if we confine the involved Lyapunov functions to the set of quadratic forms

As an immediate extension of the results in [147], the next theorem provides for controller construction. Once again, the dependence on 9 has been dropped to facilitate manipulations. It is further assumed that • (HI) Di2 and D2\ are full-column and full-row rank, respectively. This assumption is without restriction and greatly simplifies the presentation. The construction is easily extended to the singular case along the lines of [147]. Theorem 11.3 (controller construction from projections). Assume the conditions of Theorem 11.2 hold for a pair (X, Y) and some performance level 7. Then a gainscheduled controller can be constructed for any pair (0, Q] in 9 x 0^ by the following sequential scheme: Compute DK solution to

and set

Compute BK and CK solutions to the linear matrix equations

Compute

214

Apkarian and Adams Solve for N, M the factorization problem

Finally, compute AK, BK, and CK with the help of (11.8)-(11.10). It should be noted that in spite of their different structures, the characterizations given in Theorems 11.1 and 11.2-11.3 are equivalent and can virtually be used interchangeably for controller synthesis. In contrast, when the focus is on computational complexity or practical implementation, these techniques exhibit significant differences. This is discussed in section 11.4. Finally, the case where only some parameters Bi are subject to constraints on their derivatives is easily handled by removing the unconstrained parameters from the matrix functions X ( . ) and Y(.).

11.2.1

Extensions to multiobjective problems

A useful practical advantage of the basic technique is that it easily extends to multiobjective problems. Various channels of the closed-loop system can be specified independently with a rich list of specifications. See [365, 265] for a thorough discussion. As an example, it is possible to specify an L2 gain bound with regional pole constraints on the closed-loop dynamics of the underlying LTI systems (9 frozen). Such constraints consist of vertical and horizontal strips, disks, conic sectors, parabolas, or intersections of such regions. The LMIs (11.6)-(11.7) must then be complemented with

where the data Xjk and p,kj define the geometry of the region.

11.3

Practical validity of gain-scheduled controllers

It must be stressed that an LPV controller derived from Theorem 11.1 or 11.2-11. is not gain scheduled in the usual sense of the term. Its implementation requires not only the real-time measurement of the parameter 0, but also of its time derivative B. This is generally prohibitive, since parameter derivatives either are not available or are difficult to estimate during system operation. Gain-scheduled controllers that do not require a measurement of B will be called practically valid hereafter. As discussed in [33], there is no systematic and tractable approach for removing the dependence on B while maintaining the generality of Theorem 11.1 or 11.2-11.3. As suggested by the controller formula (11.8), a simple but conservative approach has been proposed in [34]. It consists of restricting the variable Y(B] to Y = 0, that is, Y not depending on B. This operation amounts to using a fixed Lyapunov function for the parameter-dependent control problem described in (11.12). It thereby sacrifices some performance, resulting in a higher 7. Keeping in mind that the dependence of the controller data on B stems from the term XY + JVMT, (11.8), the general characterization of Theorem 11.3 offers additional freedom that is worth pointing out. The discussion is summarized in Table 11.1. Row 1 of the table simply says that if the scheduled variable is assumed constant in time, a practically valid gain-scheduled controller can theoretically be constructed

215

11.4. Reduction to finite-dimensional problems Table 11.1: Selection of variables in the gain-scheduled control problem. Variables X, Y

Variables N, M

Practical validity

NMT = 1- X(d}Y(0] NMT = 1- X(e}Y(0] N := I - X(8)Y0, M :=I

Yes No

fee, fee.

X := X(0) Y := Y(9) X := X(9) Y := Y(0] X := X(0) Y := YQ X := X0 Y := Y(0)

N := /, M := / - F(0)X0

Yes Yes

4J£ unbounded

X :— XQ Y := YQ

NMT = 1- X0Yo

Yes

f =o f eed

using Theorem 11.1, or alternatively Theorems 11.2-11.3, for any matrix functions X(.) and Y(.) of 0. Such an approach ignores possible time variations of 9 and provides neither performance nor stability guarantees for the closed-loop system in the face of time variations. With the same choice of matrix functions X(.) and Y(.), but the rate of variations of 9 being confined to a compact 0^, row 2 says that there is no known techniques to compute a practically valid gain-scheduled controller. In rows 3 and 4, we have assumed the conservative choices that X or Y are constant matrix variables. In both cases, the gain-scheduling problem with bounded rate of variations admits practically valid controller solutions, provided the variables N and M are adequately selected in Theorems 11.1 and 11.3. With further conservatism, that is, 9 is unbounded, row 5 says that the problem is again tractable and solvable using the same techniques. The case of time-varying parameters with bounds on the rate of variation can be constructively handled by the choices of rows 3 and 4. However, due to the loss of duality in the variables X and F, such choices are not equivalent. As a consequence, there are some problems for which it is better to take a parameter-dependent X and a constant Y, while others will require the converse. Hence, both alternatives must be tried to get a less conservative design. In the controller construction scheme, the variables N and M are subject to the algebraic constraint / — XY = NMT from which one easily infers the identity

In light of this identity, a practically valid gain-scheduled controller in the cases of rows 3 and 4 can be derived using the same formulas (11.9) and (11.10) but with AK suitably updated to

The same formulas are still valid for the case of frozen-in-time parameters, row 1, and for arbitrarily varying parameters, row 5, the variables X and Y being replaced by their constant values XQ and YO, in the latter case. Summing up, Table 11.1 displays all options to handle any situations from the frozen-in-time parameters to arbitrarily time-varying parameters. However, the case in which both X and Y depend on 9 with a bounded 9 still resists a convex formulation for a practically valid controller.

11.4

Reduction to finite-dimensional problems

Even with the simplifications of Table 11.1 in place, the characterizations of Theorem 11.1 or 11.2-11.3 involve the solution of a convex but infinite-dimensional and infinitely constrained problem. This is the price to pay for allowing a general parameter dependence in the plant (11.1). Generally speaking, there is no systematic rule for selecting the

216

Apkarian and Adams

functional dependence of the matrix functions X and Y on 9. We are therefore led to some simple heuristics in order to simplify the computation of solutions to the LMI problems (11.6)-(11.7) or (11.11)-(!!. 13). A simple but practical technique has been proposed in [424]. The key idea is to "mimic" the parameter dependence of the plant in the Lyapunov function variables X and Y. Interestingly, the same idea can be used in the more general context of the basic characterization of Theorem 11.1. In return, this offers new potential approaches for the synthesis of gain-scheduled controllers with multiple objective constraints (mixed H2 - HOC, pole clustering, and others still to find). To be more specific, consider the class of plants (11.1) having a linear-fractional transformation (LFT) dependence on nonlinear functions of the scheduled variable, that is, whose statespace data further satisfy

where Pi(.},i = 1, . . . , N, are differentiate functions of 9. Note that such a description encompasses many practical situations, since most systems in aeronautics and robotics can be represented as an LFT in nonlinear functions of the time-varying parameters. Copies of the plant's nonlinear functions, /9i(.), can be introduced into the quadruple and the pair (X(.),Y(.)) in an affine fashion,

and

The functional dependence of X and Y being fixed, the matrices 4fir,o»-A/f,t>- • • play the role of decision variables in the infinitely constrained LMI problems (11.6)-(11.7) or (11. 11 )-(!!. 13). A simple remedy for turning such problems into a finite set of LMIs is to grid the value set of 9 [424]. Since the derivative 9 appears linearly in the LMIs (11.6) and (11.11)-(11.12), there is only need to check the extreme points of the set 0^, denoted T, for all admissible values of 9. The overall procedure can be described as follows. Step 1. Define a grid Q for the value set of 9. Step 2. Minimize 7 subject to the LMI constraints associated with Q x T . Step 3. Check the constraints with a denser grid. Step 4. If Step 3 fails, increase the grid density and return to Step 2. Computing solutions (11.21) and (11.22) to the LMI system associated with Q x T is a convex optimization that can be solved by polynomial-time algorithms [296, 63] and the software [153]. Such problems generally require a large number of variables and

11.4. Reduction to finite-dimensional problems

217

constraints that today limit the scope of application of such techniques. With this considered, available solvers are still efficient for problems of reasonable size, say, for up to 15 states and 2 or 3 scheduled variables. LMI-based gain-scheduling techniques have proven very powerful in a number of delicate applications [424, 33, 34, 21]. When restricted to the parameterization (11.21) and (11.22), the basic and projected characterizations are no longer equivalent. In the first one, we have further restrictions on the structure of the quadruple (AK(-), BK(-),CK(-),DK(-))- As a result, the first approach is generally more conservative, although we have observed very little difference in practice. See the application section 11.6 for comparisons. From a complexity viewpoint, the first technique requires a larger number of scalar variables to be optimized; the number of additional variables being approximately n(n + 7712 +P2)L, where L is number of blocks in the LFT (11.20). Its scope of application is therefore more restricted. In contrast, the controller equations resulting from the basic characterization are significantly less complex than those resulting from the projected characterization. Note that such controller constructions are essentially dominated by matrix inversions and QR decompositions in (11.8)-(11.10) and (11.15)-(11.17). At each sampling time, the basic characterization essentially requires 1 matrix inversion, whereas the projected characterization will require 2 QR decompositions and 3 matrix inversions for problem (11.14), 2 matrix inversions for the computation of AK, BK, CK by exploiting partitioning. Thus, in the light of these comments and because the expressions (11.21) essentially reduce to scalar-by-matrix multiplications, controllers resulting from the first technique are more easily implemented for rapidly varying LPV systems. In addition, these controllers have an LFT representation in terms of the nonlinear functions, Pi(.), and hence are computationally comparable with those of the LFT gain-scheduling approaches in [312, 19]. Note also that for both techniques, the most computationally demanding step comes from the inversion of the term / — X(0)Y(0), typically a large matrix. It is sometimes possible to exploit rank deficiencies in the X^s and the Yi's to further reduce computational efforts, for instance, by using inversion with rank correction formulas. A simple case that does not require the inversion of I — X(0)Y(9) is when the LFT system (11.20) depends on a single nonlinear function pi(9). Noting that

the inverse of / — X(0)Y(6) can be computed from the lower-left block in the above expression. Moreover, if we assume without restriction that 0 is in the image set of /f>i(.), then

The positive condition in (11.23) ensures that the matrices

218

Apkarian and Adams

are simultaneously diagonalizable. Hence there exists a congruence transformation T and a diagonal matrix AI, computed offline, such that for any value of the map pi(.)>

Therefore, a cheap way of computing (/ — XY}~1 at each sample of time is simply to invert the diagonal matrix in (11.24) and perform multiplications of corresponding blocks. Since they offer complementary advantages, the techniques described above can be used together to yield a more effective methodology. Confirmed by practical experience, the following rules have proven useful. 1. All necessary tunings requiring repeated computations should be based on the less costly projected technique. 2. The procedure is completed by running the basic technique for controller implementation purposes. Although the last phase may be very slow, it is run only once in the whole design process.

11.4.1

Overpassing the gridding phase

As discussed earlier, there is no direct technique to overpass the gridding phase, hence making the design more direct. Under special circumstances, LFT, affine, or polynomial parameter dependence of data and variables or polytopic approximation of the original plant, techniques such as the 5-procedure or multiconvexity concepts can be used repeatedly to get a finite number of (sufficient) LMI conditions. See the conference version of the present work [18] and also [135, 149] and references therein.

11.5

Reducing conservatism by scaling

As is common in robust control theory, it is possible to further enhance the design procedure by exploiting structural information on the operator relating the signals w and z. Yet, the conditions of Theorems 11.1 or 11.2 also provide robust stability conditions in face of • multiplicative memoryless time-varying uncertainties

• nonexpansive dynamic uncertainty (for instance LTI) [64]

This description, however, ignores potential structures of the operator A. We therefore assume, hereafter, that the plant is governed by (11.1) with w and z subject to

11.5. Reducing conservatism by scaling

219

where A is a multiplicative memoryless time-varying operator with structure

and confined to the compact set

The set of scalings associated with the structure (11.26) is defined as

We have implicitly assumed, without restriction, that the problem has been squared so that mi = pi in the subsequent derivations. With this notation in mind, a scaled version of the bounded real lemma can be established. Lemma 11.1 (structured robust stability). The LPV system governed by

and (11.25)-(11.27), with parameter trajectories 6(t) constrained with (a) and (b), is internally stable whenever there exists a parameter-dependent symmetric matrix P(6) and a parameter-dependent scaling S(d] in S& such that P(0) > 0 and

holds for all admissible values of the parameter vector 9 and of its time derivative 0. Proof 11.3. See Appendix A. Note that Lemma 11.1 provides robust stability conditions for nonexpansive uncertain operators Aj. These conditions also guarantee a robust L2-gain performance bound 7 with respect to any input/output channel (wi,Zi), with the remaining channels corresponding to uncertainties of the form described earlier. See, for instance, [64] for details Other extensions to repeated-scalar uncertainties Aj = <5j/Si are straightforward. Also important is the fact that the scaling matrix S(0) is allowed to vary with 9. Hence, the conservatism of the gain-scheduling technique can potentially be reduced for all values of 6 independently. As an immediate consequence of the above lemma, the gain-scheduling techniques in Theorems 11.1 and 11.2-11.3 are readily extended to handle the structural constraint (11.25)-(11.26). Such extensions simply follow from the substitutions

in Theorems 11.1 and 11.3 and

in Theorem 11.2.

220

Apkarian and Adams

Owing to the dependence of the modified characterizations on both S and S 1, they are no longer standard LMI problems. Such problems are in the class of LMI problems with rank-reduction constraints hence difficult to solve. Here, we adopted a simple computational scheme in the spirit of n synthesis techniques. Recall that the LMI (11.6), with the scaling 8(9} in place, can take the form

Note^that this is an LMI in the variables S"1, y, AK, and CK provided that the variables X, BK, and DK are maintained fixed. Therefore, coupled with Theorem 11.1, this result suggests a first scheme to compute a best possible S"1. Now our goal is to obtain the counterpart of the projected characterization Theorem 11.2 for computing S~l. In view of the discussion of section 11.4, this will be at a reduced computational cost. Interestingly, such a scheme cannot be directly derived from the LMIs (11.11)-(11.12). We shall make use of a partially projected characterization as follows. Applying the projection lemma [148], with respect to the (2,1) free term in (11.31), equivalent LMI conditions are

Assuming now that (11.32) holds and projecting again with respect to the controller variable CK in (11.33), the following equivalent LMI condition is obtained:

Gathered together with (11.13), the matrix inequalities (11.32) and (11.34) form an LMI system in the variable Y and S'1 with fixed X, BK, and DK- It turns out that the conditions (11.31) and (11.7) on one side or the conditions (11.13), (11.32), and (11.34) on the other side form the basis of two possible schemes for iteratively reducing the conservatism of the gain-scheduling techniques of section 11.2. Such schemes proceed as follows. Step 0. Setting 5(0) = /, minimize 7 with the basic or the projected technique. Step 1. Compute a scaling S(9)~1 minimizing 7 with the help of (11.31) and (11.7) or alternatively (11.32), (11.34), and (11.13).

11.6. Control of a two-link flexible manipulator

221

Step 2. With S(0] being fixed, perform a gain-scheduling synthesis with the basic or projected technique. Step 3. Iterate over Steps 1 to 3 until convergence. As before, we are using a grid of the set 0 to perform the optimizations and a user-defined functional dependence of S~l on 0. For LFT plants of the form (11.20), a practical choice is the affine expansion

It is important to mention that the previously described iterative procedure, although not giving a global solution to the problem, has proven very efficient in practice. A demonstration is given in the following section. Unlike the standard D — K iteration procedure, it involves not only scalings and Lyapunov variables, but also some controller variables in the same optimization step. In our opinion, this is a central factor favoring convergence, both in speed and accuracy.

11.6

Control of a two-link flexible manipulator

11.6.1

Problem description

The gain-scheduled control of a two-link flexible manipulator is a nontrivial problem. The dynamics of such a system include both rigid body and lightly damped structural modes. The problem is complicated by uncertainty in the high frequency dynamics of the system and by the variation of dynamics with manipulator geometry. The first of these complications drives the requirement for closed-loop robust stability, while the second drives the requirement for gain scheduling. In addition, a rapid closed-loop response to position commands is desired. The ability of a control synthesis approach to handle the trade-offs between robustness, performance, and gain scheduling with the least possible conservatism is thus critical for such a system. For example, if the gain-scheduling parameters are allowed to vary infinitely quickly, closed-loop performance and robustness will suffer. If the uncertainty structure of the design model is not considered, it will be impossible to find a controller that meets design objectives. These trade-offs are studied and addressed in this section. SECAFLEX is a two-link flexible planar manipulator driven by geared DC motors, used as a laboratory platform for control-structure interaction experiments at CERTONERA in Toulouse, France. The two flexible members are homogeneous beams. There is a concentrated mass at the elbow due to the DC motor and a concentrated mass at the tip of the second beam, which is the payload. The modeling of the manipulator has been studied extensively [4]. A simple drawing of the two-link manipulator is shown in Figure 11.1. #1 and 6-2 are the shoulder and elbow joint angles, respectively. r\ and r^ are the corresponding control torques. The second-order form of the manipulator equations of motion are

where M is the inertia matrix, D is the damping matrix, and K is the stiffness matrix. u(t) is the input vector, u = (TI T2)T. Due to the variable geometry of the system, the inertia matrix is a function of the second joint angle, #2- This dependence causes

Apkarian and Adams

222

Figure 11.1: Two-link flexible manipulator.

Figure 11.2: cri(G(ju),02)) for different manipulator geometries. 62 = 0: solid; 0% = f: dashed; 02 = TT: dotted. significant changes in the response of the system to input torques over the range of possible configurations, 02 € [0, TT] (rad.). If we consider an output vector, y = (Oi 02)T, then we can define the transfer function from u to y as G(s,
where

11.6. Control of a two-link flexible manipulator

223

Figure 11.3: Synthesis model. and the damping, stiffness, and control effectiveness matrices, respectively, are

The manipulator equations of motion can be rewritten in first-order LFT form in 02 [1]>

To provide closed-loop command tracking, a simple weighted minimization of the sensitivity function is used. A frequency-dependent weight, Wp, penalizes the error, e, between angular position commands, iui, w2, and the system response, y. By forcing this weighted sensitivity function to be less than unity, the complementary sensitivity function approaches identity at low frequencies, thus providing good command tracking. In order to account for uncertainties in high frequency dynamics, an additive uncertainty model is incorporated into the synthesis model. The additive uncertainty weight is formulated by considering the difference between the full-order geometry-dependent model, G(s, 92), and some reduced-order design model, G>(s, 92), of lower order but still dependent on manipulator geometry. Consider W/, the additive uncertainty weighting function and A/, a complex uncertainty block, scaled such that ||A/||oo < 1. We can define the error between the full-order and reduced-order models as E(s,02):

The additive uncertainty weight must then provide a frequency domain bound on this error; that is,

The LPV design model has the form (11.1) and is built by combining the above performance and robustness formulations into a single multiobjective synthesis model; see Figure 11.3. The design weights used for the following examples are

224

Apkarian and Adams Table 11.2: Performance comparisons (7) with ignored structure. Basic technique with X(Qi) and ¥(62) Projected technique with X (#2) and ^(#2) LFT technique in [19]

3.82 3.82 3.82

It should be noted here that, in the manipulator example, 0 = 6-2 is not an independently evolving external parameter, it is a state of the system. By treating #2 as an external parameter, we are actually immersing the "quasi-linear" dynamics (11.39) into the larger class of LPV dynamics (11.29). Therefore, in guaranteeing stability and performance for a class of parameter trajectories (through bounds on 82 and ^)> we ignore the fact that the trajectories themselves are defined by the plant dynamics. The result is thus a degree of conservatism in the design. The resulting controller is in the class of nonlinear controllers since 6 := 62 is here a subvector of the measurement y := (#1, ^)TIt is usually referred to as a scheduled-on-the-plant-output controller [371]. It captures the nonlinearity of the plant (11.39) in 62- The controller matrices (AK,BK,CK,DK) are instantaneously updated with respect to the plant's state #2* as described in section 11.2.

11.6.2 Numerical examples Different cases have been investigated that highlight the merits and some of the concerns in using the gain-scheduling methodologies detailed in sections 11.2 to 11.5. Comparisons with existing gain-scheduling techniques are also given. Case 1. In this first case, the two-block structure associated with robustness and performance is ignored. The basic and projected techniques are applied with both X and Y depending on #2:

As discussed earlier, this amounts to assuming that the scheduled variable #2 is frozen in time. The corresponding 7 levels are compared with the LFT gain-scheduling technique in [19], a technique that puts no bound on the parameter variation rates. The results are presented in Table 11.2. Surprisingly, all techniques give the same result. The achieved value of 7 is actually a lower bound that can be checked by performing an HQQ synthesis on a nominal model (62 = f). Therefore, if the structure of the problem is ignored, the techniques here or [312, 19] may offer no advantages over existing LFT gain-scheduling techniques. Case 2. In this second case, the problem's uncertainty structure is explicitly taken into account. The performance and robustness objectives are relaxed by introducing a fixed scaling S := diag(0.5e-3/2X2,-^2x2), found by performing a standard n synthesis with constant scaling on the nominal plant (#2 = f )• The application of gain-scheduling techniques to this scaled problem leads to the bound 7 = 0.58. Results are presented in Table 11.3 for different selections of the Lyapunov variables X and Y and assuming 62 is fixed in time (#2 = 0). Both the basic and projected techniques guarantee the same performance bound for all admissible values of QI, provided that both X and Y depend on #2- A slight degradation occurs when the dependence on 62 is restricted to X, but the design is still acceptable.

11.6. Control of a two-link flexible manipulator

225

Table 11.3: Performance comparisons of basic and projected techniques (9 — 0). 7

Basic technique Projected technique

x(o2), y(02) 0.58 0.58

X(62], Yo 0.95 0.95

X0, Y(e2} 25.51 25.35

XQ, YQ

380.60 380.60

Conversely, when Y only depends on Q^, both techniques appear very conservative. As predicted in section 11.3, the selections of columns 2 and 3 are not equivalent and both must be tested to reduce conservatism. The designs in column 4 use fixed X and Y matrices and thus yield much more conservative answers. The corresponding value, 7 = 380.60, was also obtained using the LFT technique in [19]. The consequences of this discussion are threefold. • Ignoring the uncertainty structure introduces undesirable limitations on the power of advanced gain-scheduling techniques. One must compromise between robustness/performance requirements and gain-scheduling objectives in order to fully benefit from such techniques. • Trade-offs can be systematically handled through the use of scalings. Good initial guesses for such scalings can often be found on the basis of a nominal system issued from the LPV plant. • The basic and the projected techniques give about the same results, the first one being preferable in a real-time implementation perspective. Case 3. We investigate here the effects of a bound on the rate of variation of 62 on the achievable robustness/performance level with the projected synthesis technique. The scaling 5 is as defined previously, and we assume symmetric bounds

It is intuitively clear that increasing the bound on the variation of 62 degrades both robustness and performance. For our manipulator system a realistic bound is 100 deg./sec. As can be observed in Figure 11.4, which describes 7 as a function of a, the proposed techniques perform well for variations of up to an order of magnitude greater than those expected. Case 4. In this last case, the iterative schemes proposed in section 11.5 are used to further improve robustness and performance. We exploit the scheme based on the matrix inequalities (11.32), (11.34), and (11.13). Recall that such schemes take advantage of parameter-dependent scalings. According to the definition of SA in (11.28), the scaling assumes here the special form

The Lyapunov variables have been selected in the form X — XQ + cos(02}Xi and Y = YQ. Practical validity of the controller is thus ensured from the analysis in section 11.3. The evolution of 7 during the alternate iterations is depicted in Figure 11.5. Convergence required 12 elementary steps as described in section 11.5 and led to a best 7 value of 0.30. This is obviously the best design among those attempted. The result in Figure 11.5 was achieved with the projected technique and the characterizations given in (11.32) and (11.34) for the scaling computation. The resulting scaling matrix is

Apkarian and Adams

226

Figure 11.4: Performance level 7 vs. bound on the rate of variations. *: value of 7 for unbounded derivatives; o: value of 7 for frozen parameter.

Figure 11.5: Evolution of 7 vs. alternate iterations. Finally, using the above scaling, we recomputed the gain-scheduled controller with the basic technique. The same value, 7 = 0.30, was obtained. Since it is of a simpler form and provides satisfactory performance, this last gain-scheduled controller is used in the subsequent analysis and simulations.

11.6.3 Frequency and time-domain validations In this section, the best-7 controller from case 4 is analyzed. In Figure 11.2 we saw how the manipulator dynamics changed with 6-2- It is now interesting to examine how the resulting gain-scheduled controller varies with manipulator geometry. Figure 11.6 shows the singular values of the underlying LTI controllers K(s, #2) at three different values of0 2 . We see that the gain-scheduled controller evolves as physically expected, applying higher gains when the manipulator inertias are greater (62 = 0) and reduced gains when the inertias are smaller (#2 = TT). Notice that at frequencies above 10 rad/sec, both the manipulator system and the controller are relatively independent of the parameter. Extensive nonlinear simulations of the response of the closed-loop system to various commands have been performed. For the sake of brevity, we will present only one of these. The high frequency flexible modes that were removed from the synthesis model are

11.6. Control of a two-link flexible manipulator

227

Figure 11.6: cri(K(ju>,02}) for different manipulator geometries. 6-2 = 0: solid; #2 = f ' dashed; 62 — TT: dotted.

Figure 11.7: Nonlinear simulation results. Solid: 0\, T\\ dashed: #2?

reintroduced in the simulation model. With initial conditions of #i(0) = 0 and #2(0) = 0, a step command of 180 degrees in both angles is given. This maneuver was chosen to take the manipulator through the entire range of possible dynamics as quickly as possible. The angular responses and corresponding control inputs are depicted in Figure 11.7. The rise time (3.5 seconds), settling time (5 seconds), and overshoot (< %5) are markedly superior to those observed using either robust LTI controllers or heuristically motivated gain-scheduling approaches [1]. The maneuver is completed without violating the limits on control authority, \r\\ < lOOJV/m and \T?\ < 20JV/m.

228

Apkarian and Adams

11.7 Conclusions Advanced gain-scheduling design approaches for LPV systems have been presented with emphasis on the practical goals of reduced computational burden and ease of implementation. Two complementary techniques for the calculation of such controllers have been investigated that, when used together, achieve these two objectives. The methodology is completed with a new scaling technique that takes into account the uncertainty structure of multiobjective synthesis problems. The challenging problem of the control of a two-link flexible manipulator is introduced in this context and used to demonstrate the validity of the theoretical solutions.

Appendix A Proof of Lemma 11.1. The system governed by (11.29) and (11.25)-(11.27) is stable whenever there exists a quadratic Lyapunov function V(x, 0} = xTP(9}x such that P(ff) > 0 and

for all admissible trajectories x(t), w(t), z(t), 0(t), and A(£). The signals w and z related by (11.25)-(11.27) satisfy the quadratic constraints

where S is in «SA and may depend on 9. Therefore, stability is ensured whenever (11.43) holds for all trajectories x(t), 9(t) and any pair (w(t),z(t)) characterized by (11.44). By an ^-procedure argument [434, 433], this is guaranteed whenever

This expression can be rewritten in the form

which is nothing else than the quadratic form condition associated with the matrix inequality in (11.30). This completes the proof of the lemma.

Acknowledgment We are thankful to P. Gahinet and G. Becker for their helpful and stimulating support during this work.

Chapter 12

Control Synthesis for Well-Posedness of Feedback Systems Tetsuya Iwasaki 12.1

Introduction

Active use of linear matrix inequalities (LMIs) for analysis and synthesis of robust control systems dates back at least to the late 1980s [51, 69,158, 228]. A new trend for computeraided control design had also been started during the same period, motivated by [61, 59], which showed the nontrivial impact of convex programming on control design. Recent recognition of LMIs as a powerful tool for robust control [108] has put a tremendous impetus on the development of new control analysis and synthesis methods [64, 377] with the aid of efficient computational algorithms [63, 153, 122, 296]. A wide variety of control synthesis problems have been addressed using LMI approaches in the state-space setting, including the HQQ [148, 215], the scaled HQQ [316], the gain-scheduled HQQ [19, 312], the positive real control [398], and other problems related to the H2-performance [209, 363]. A remarkable fact is that all these different control problems reduce to the same type of computational problems involving LMIs: Standard problem (SP). Find positive definite (possibly structured) matrices X and Y satisfying LMIs and additional rank constraints. From this observation, it is tempting to state that "many" of the control synthesis problems posed in the state-space setting can be reduced to the same computational problem stated above when a certain LMI approach is employed. Our main objective is to show explicitly a general class of control problems that reduce to SP. The class is defined by the closed-loop specifications given in terms of the well-posedness of feedback systems [211, 212, 213] and includes the H^, the positive real, nominal/robust/gain-scheduled, continuous-time/discrete-time control problems as special cases. We show that the existence of a controller, yielding a closed-loop system whose well-posedness can be proved by a P-separator [212], can be checked by solving the SP. 229

230

Iwasaki

Figure 12.1: Feedback system for analysis.

A fundamental fact to be revealed here is that the troublesome rank constraints in the SP disappear and the resulting problem becomes convex if the controller is allowed to use sufficient information on the plant. This fact can also be anticipated from the Packard's result [312], which showed that the gain-scheduled H^ control problem can be reduced to an LMI optimization problem if the controller is allowed to be scheduled by the same (or greater) number of the repeated scalar parameters of the plant. Our results extend Packard's result for the more general class of control problems including the HQQ control problem considered in [312] as a special case. The generality of our problem formulation also allows us to interpret the rank constraint arising in the reduced-order control problem [148, 215] as the lack of sufficient plant information.

12.2

Well-posedness analysis

12.2.1 Exact condition Consider the feedback system shown in Figure 12.1, where M € Hrxq is a known constant matrix and V is an unknown matrix belonging to a known compact set V C C 9Xr . A large class of uncertain systems can be described by this feedback system [102]. For instance, linear time-invariant (LTI) continuous-time systems subject to structured uncertainties are given by the feedback system with M consisting of the state-space matrices of the nominal part and V containing the frequency variable s and the uncertain component. The feedback system in Figure 12.1 is said to be well posed if Det(7 —VM) ^ 0 for all V € V. The notion of well-posedness is in fact equivalent to robust stability/performance when M and V are chosen appropriately [108, 211]. The following result gives a necessary and sufficient condition for well-posedness. Theorem 12.1. Let M € Rrxq and V C C 9Xr be given. Suppose that V is a compact set. The following statements are equivalent. such that

Proof 12.1. The result is proved in [213]. Here, we give a proof for completeness. Suppose statement (i) holds. Let

12.2.

Well-posedness analysis

231

for sufficiently small e > 0. Then, using the compactness of V, it can readily be verified by substitution that (iia) and (iib) hold. Suppose (i) does not hold, but (ii) holds. We show a contradiction as follows. First note that there exist z ^ 0 and V G V such that

Let w := V*z ^ 0. Then using

we see that (iia) and (iib), respectively, imply C*©C < 0 and C*©C > 0, which is a contradiction. This result is closely related to recent results on the integral quadratic constraints (IQCs) [273] and the linear-fractional transformation (LFT) scaling [26]. One of it implications is that robust stability of a certain class of systems can be examined by searching for a matrix 6 that satisfies the conditions in statement (ii). This search is, however, nontrivial in general due to the infinitely many constraints in (iib).

12.2.2

P-separator

The computational difficulty caused by statement (iib) of Theorem 12.1 can be avoided by restricting our search for 0 within a certain class depending upon a specific V. To this end, consider the following special class of V:

where Vn repeats kn times in V£", and

with given real matrices Rn — R^, Qn = Qj[, and 5n. For this class of V, the following result shows a class of O for which condition (iib) is automatically satisfied. Lemma 12.1. Consider the set V given in (12.2) and define

Then, for any 0 € 0, statement (iib) of Theorem 12.1 holds. Proof 12.2. Let O € 0 and V € V be given. Note that

Iwasaki

232

Figure 12.2: Feedback configuration for control synthesis.

is a block diagonal matrix whose nth block matrix I5n is given by

Using Pn > 0 and Wn > 0, it can be verified that I3n > 0, and hence we conclude that U>0. This result provides a sufficient condition for well-posedness of the feedback system in Figure 12.1; if there exists G G 0 such that statement (iia) of Theorem 12.1 holds, then the feedback system is well posed for all V € V. This condition can be verified by searching for the parameters Pn appearing in the definition of 0, which is a standard LMI feasibility problem. We call the class of G defined in Lemma 12.1 the "P-separator," since it establishes the topological separation of the graphs of M and V [166, 211, 348]. The P-separator provides a unified description of the HQQ condition with the J9-scaling and the positive realness condition with multiplier. In general, the P-separator introduces conservatism; there may be no 9 € 0 such that (iia) of Theorem 12.1 holds even if statement (ii) of Theorem 12.1 holds true. However, this gap vanishes if 2s + / < 3, where s and / are, respectively, the numbers of the repeated scalar blocks and the (unrepeated) full blocks in V. This fact can be shown using the corresponding result for the .D-scaling [314]. See [212] for the detail.

12.3

Synthesis for well-posedness

12.3.1 Problem formulation Consider the feedback system in Figure 12.2, where M is a given real matrix and VM and VK belong to a known compact subset of complex matrices V:

It can readily be seen that the feedback system in Figure 12.2 can be recognized as the feedback connection of V and F(M,K). We then consider the following synthesis problem:

12.3. Synthesis for well-posedness

233

Well-posedness synthesis problem (WSP). Given a real matrix M and a compact subset of complex matrices V, find a matrix K such that the feedback system of V and F(M,K) is well posed, i.e.,

holds. This problem is a general mathematical problem that includes many control synthesis problems as special cases, as shown in section 12.4. In the context of control design, Fu(M, VM) is the plant, while TL(^ K, K) is the controller. In particular, M and K correspond to state-space matrices of the plant and the controller, respectively. Choosing the "perturbation" V appropriately, various classes of control problems can be generated. For instance, V may dictate the frequency variable, uncertain dynamics and/or parameters, scheduling parameters, nonlinear elements, and infinite-dimensional dynamics, leading to such control problems as reduced-order, robust, nonlinear, delay /preview, gain-scheduled controls. Unfortunately, it is difficult to solve the WSP exactly. We will restrict our attention to a subclass of the feasible controllers defined by the P-separator and give an exact solution to the auxiliary problem. To this end, the rest of this section is devoted to defining the auxiliary problem. From Theorem 12.1, condition (12.4) holds if and only if there exists a real symmetric matrix G such that

Hence, WSP reduces to the problem of finding matrices K and B satisfying the above inequalities. However, it is difficult to directly solve these inequalities numerically due to the following two main reasons: (i) condition (12.5) imposes a nonconvex constraint on K and 0, and (ii) condition (12.6) is essentially described by infinitely many inequality constraints unless V has only a finite number of elements. Below, we introduce a specific V and overcome the second difficulty by imposing the structure of the P-separator on 0 such that (12.6) is automatically satisfied. Thus, in the statement of the auxiliary problem given later, the latter difficulty will not show up while the former remains. Consider the following class of "perturbations" :

where t = mn or kn (n = 1,... ,a), Vn repeats i times in V£, and Vn is a subset of complex matrices defined by (12.3) with given real matrices Rn — R^, Qn — Qj, and Sn such that

The significance of condition (12.8) may be found in [212]. For this class of V, we present the following definition.

234

Iwasaki

Definition 12.1. Consider the set V given by (12.7). Define a subset of real symmetric matrices 0 as follows. A given matrix 0 belongs to 0 if there exist real symmetric positive definite matrices Pn e R( m «+ A: «) x ( m «+ fc «) (n = 1,... ,a) such that

where 72 is given by

and similarly for 5 and Q. Lemma 12.2. Let real matrices Rn = R%, Qn = Q^, and Sn (n = 1,... ,a) be given such that (12.8) holds, and define the set V by (12.7). Consider the set 0 in Definition 12.1. For any 060, condition (12.6) holds. Proof 12.3. Fix V := diag(VM, VAT) € V and 0 € 0. Let T be an orthogonal matrix that transforms V into the following block-diagonal matrix V:10

Noting the identities

and similar identities for S and Q, we have

where the last inequality can be verified in a similar manner to the proof of Lemma 12.1. Thus (12.6) holds. We have shown that any 0 € 0 satisfies (12.6). Hence, by restricting our search for 0 within the set 0, the troublesome constraint (12.6) is automatically satisfied and WSP reduces to the following. 10

For example, when a = 3, we have

12.3. Synthesis for well-posedness

235

Auxiliary well-posedness synthesis problem (AWSP). Given a real matrix M and a subset of complex matrices 0 in Definition 12.1, find 0 6 0 and K such that (12.5) holds. This problem is only a (conservative) approximation of the WSP, and there is a gap between the two in general. That is, any solution K of the AWSP is also a solution of the WSP, but the converse may not be true. Nevertheless, as shown later, there are certain classes of control problems (e.g., HOC control, positive real control, and robust stability/performance synthesis against linear time- varying (LTV) dynamic uncertainties [270, 370]) that can be formulated as the AWSP exactly. Finally, we impose an assumption on the problem data M. From Figure 12.2, we see that M has two inputs and two outputs. Accordingly, M can be partitioned as

The star product F(M,K] in (12.5) can be written by the following LFT:

where matrix L is defined by

where

x

is the size of

We assume that M22 = 0. In this case, TL(K, L) becomes affine in K as follows:

This assumption simplifies the proof of our main results and can be removed in a standard manner (e.g., [215]).

12.3.2

Main results

This section presents a solution to the AWSP. Proofs of the results will be given in the next section. The following theorem provides a necessary and sufficient condition for solvability of the AWSP. Theorem 12.2. Let integers a > 0, mn > 0, kn > 0, and real matrices Rn = R^, Qn = Qn, Sn (n = 1,..., a), and M be given. Define the set 0 by Definition 12.1. The following statements are equivalent. (i) There exist 9 e 0 and K satisfying (12.5).

Iwasaki

236

(ii) There exist real symmetric matrices

such that

where

Assuming that statement (ii) of Theorem 12.2 holds, a solution for the AWSP can be computed as follows. Procedure for computing 0 € © and K 0. AWSP data. M, ©, where the set © is defined by Definition 12.1 in terms of a, ?nn, kn, Rn, on, and CJn (71= 1,..., a). 1. Compute matrices X and Y satisfying condition (ii) of Theorem 12.2. Denote by Xn and Yn the nth block matrices on the diagonal of X and V, respectively. 2. For each n = 1,..., a, if kn ^ 0, let Fn € Rm"xkn be such that and define Pn by If fcn = 0, then just let Pn := Xn. 3. Compute 7^, 5, and Q that specify 9 as in Definition 12.1, using Pn. 4. Let

where

and L is given by (12.11).

12.3. Synthesis for well-posedness

237

In this procedure, steps 2, 3, and 4 are straightforward and require as much computation as several singular value decompositions. On the other hand, step 1 is not easy. Constraints (12.12)-(12.14) on X and Y are LMIs, while the rank conditions in (12.15) are not convex. It is these rank constraints that make step 1 computationally nontrivial. This type of nonconvex problem has been shown to be NP-hard [144], and no efficient algorithms are currently known to guarantee global convergence. Nevertheless, there are several algorithms that work well in practice [159, 128, 176, 217]. The troublesome rank condition disappears for certain cases. An important special case is that kn > mn. Note that the size of matrices Xn and Yn in (12.15) is ran. Hence, if kn > m n , the rank condition holds automatically and the computational problem in step 1 becomes an LMI feasibility problem for which efficient algorithms exist [153, 122]. Now, a natural question is, What is the "physical" significance of condition kn > m n ? By definition, kn and m n , respectively, denote the numbers of nth "perturbation" Vn in the "controller" FL^K, K} and the "plant" Fu(M, VM)- Hence, condition kn > mn means that the controller has sufficient information on the plant perturbation. As we see later, the rank condition remains for a robust control setup since the information on the plant perturbation is unknown to the controller, while for the gain-scheduling problem the rank condition disappears since the plant parameters are assumed to be measurable on line. The reason why the fixed (low) order control design is difficult can also be explained by a similar argument.

12.3.3

Proof of the main results

In this section, we prove Theorem 12.2. The procedure for computing a solution to the AWSP will also be justified constructively by the proof. Consider the matrix inequality in (12.5) where the variables are 6 and K. First, we ask the following question: For a fixed 6, under what condition does there exist K such that (12.5) holds? To answer the question, we use the following lemma.

Lemma 12.3. Let matrices A, B, and C be given. The following statements are equivalent:

When statement (ii) holds, a matrix K satisfying the condition in statement (i) is given by where Proof 12.4. See, e.g., [95, 446]. Using Lemma 12.3 and noting that F(M,K) in (12.5) can be written as Fi,(K,Ij) with L given by (12.11), we obtain the condition for the existence of K in (12.5) as follows. Lemma 12.4. Let a reaJ symmetric matrix G and a matrix L be given as

Suppose Q > 0. Then the following statements are equivalent.

238

Iwasaki

(i) There exists a matrix K such that

(ii) The following three conditions hold:

Proof 12.5. Let Then we have

where Since Q > 0, the left-hand side of inequality (12.22) is nonnegative definite, and hence $ > 0. Therefore, condition (12.22) can be rewritten

Then applying Lemma 12.3, we see that there exists K satisfying (12.24) if and only if

hold. It is easy to verify that these inequalities can be written as (12.20) and (12.21). D Lemma 12.4 provides a necessary and sufficient condition for the existence of K satisfying condition (12.18). When the condition is met, one such K can be found by applying Lemma 12.3 to (12.24), as described in step 4 of the procedure for computing 0 6 0 and K in the previous section. AWSP has now reduced to the problem of finding 9 6 0 such that (12.19)-(12.21) hold, where L is defined by (12.11) in terms of the problem data M. It should be noted here that the supposition Q > 0 in Lemma 12.4 holds for any 0 6 0 . Next we show that condition (12.19) is automatically satisfied by any 0 € 0. Let T be the transformation matrix introduced in the proof of Lemma 12.2. Noting the relationships in (12.10), we have

Using the identities [73]

12.3. Synthesis for well-posedness

239

we have

where the last inequality holds due to (12.8) and Pn > 0 [73]. Thus (12.19) holds. In the remaining part of the proof, we will show that conditions (12.20) and (12.21) reduce to (12.12) and (12.13), respectively. For this purpose, the following lemma is useful. Lemma 12.5. Consider the set 0 in Definition 12.1 and fix any G € 0. Then G"1 is given by

where it is defined by

and similarly for S and Q. Proof 12.6. Let an orthogonal matrix T be such that

holds for all 6 € 0 (see the proof of Lemma 12.2). Let 7£n := P~l ®Rn, and similarly for 5n and Qn. Noting the relation

we have

where we used TTT = 7. Note that, from (12.11), matrices L^2 and L^ can be chosen as

Using these choices and substituting 9 € 0 in Definition 12.1 and G"1 in Lemma 12.5 into (12.20) and (12.21) yields (12.12) and (12.13), respectively, where Xn and Yn are the mn x mn matrices defined by

(Here * denotes irrelevant entries.) Now we need the following lemma.

240

Iwasaki

Lemma 12.6. Given integers m > 0, k > 0, and mxm real symmetric matrices X and Y, the following statements are equivalent. (i) 3 P 6 R(™+ fc ) x ( m + fc ) such that and Rank

(ii)

hold.

Moreover, if (ii) hoJds, then one such P in (i) can be constructed as

where F e R mxfc is a matrix factor such that Proof 12.7. See, for example, [215, 315]. From Lemma 12.6, we see the necessity of conditions (12.12)-(12.15). Conversely, if there exist X and Y satisfying these conditions, one can find Fn e R m " xfc " satisfying (12.16). Then Pn can be constructed as in (12.17), which in turn specifies 0 as in Definition 12.1. Clearly, this © satisfies (12.20) and (12.21). Moreover, from (12.19)(12.21), the existence of K such that (12.24) holds is guaranteed. These K and 9 will satisfy (12.5).

12.4

Applications to control problems

In this section, we show how the general results obtained in the previous section can be specialized to solve various control problems.

12.4.1

Fixed-order stabilization

Consider the npth-order LTI plant

and the ncth-order LTI controller

where A is either the differentiation operator s for continuous-time systems or the delay operator z for discrete-time systems. Fixed-order stabilization problem. For the npth-order plant in (12.25), design an ncth-order controller in (12.26) such that the closed-loop system is stable, where nc < np. This problem is a special case of WSP defined and solved in the previous section. The matrices M and K in (12.4), respectively, correspond to the plant and the controller as follows:

12.4.

Applications to control problems

241

The set V is given by

where (continuous-time case), (discrete-time case).

This V is a special case of the general class defined by (12.7). In particular, if we specify the data in (12.7) as

(continuous time), (discrete time),

with sufficiently small scalar e > 0, then we have the above V (except for the difference caused by t ^ 0). Note that assumption (12.8) is satisfied due to the introduction of e > 0. This auxiliary parameter will be removed later. As shown above, the fixed-order stabilizationFixed-order stabilization problem is a special case of WSP. In fact, for this particular class of V, the use of the P-separator does not introduce conservatism, and therefore the problem reduces to an AWSP exactly. Hence, a solution to the problem can be obtained by applying Theorem 12.2 to this special case. Theorem 12.3. The fixed-order stabilization problem admits a solution if and only if there exist real symmetric matrices X and Y satisfying

and either for the continuous-time case, or

for the discrete-time case. This result is known [209]. It is straightforward to reduce the result from Theorem 12.2 except for the treatment of e > 0, which deserves some explanation. If we specialize (12.12) and (12.13) for the continuous-time case, we get

Since X > 0 and y > 0, there exists a sufficiently small e > 0 satisfying these inequalities if and only if the inequalities hold for e = 0. Thus we obtain the conditions in Theorem 12.3.

Iwasaki

242

Figure 12.3: Robust control system.

12.4.2

Robust control

Consider the uncertain system described by ^t/(P(A),A), where A is the uncertain component, P(A) is the nominal transfer function, and A is either the differentiation operator s or the delay operator z. Assume that A belongs to the set

where 7 > 0 is a given scalar. The nominal transfer function P(A) is of order np and has the following state space realization:

Note that ^c/(P(A),A) is the transfer function from u to y when the feedback loop w = A.Z is added to the above state-space system. Our objective is to design an ncth-order controller C(X) with state-space realization

such that the closed-loop system in Figure 12.3 is robustly stable against A € A. Robust control problem. Consider an uncertain system described by the LFT ^rc/(P(A), A) of the uncertainty set A in (12.30) and the nominal plant P(A) in (12.31). Design a controller C(A) in (12.32) such that the closed-loop system in Figure 12.3 is robustly stable against A € A. This is a robust stability problem. Note, however, that the robust performance specification can also be incorporated by adding a fictitious "performance block" in A [104]. This robust control problem can be reduced to a WSP. Let M and K be given by

and define the set V as

12.4.

Applications to control problems

243

where A is the "unstable pole region" defined by (12.28). Then we see that the specification of the robust control problem is exactly given by (12.4). Thus, with the problem data defined above, the robust control problem reduces to the WSP. To solve the problem, we "approximate" the original problem by the corresponding AWSP and apply Theorem 12.2. In general, this step introduces conservatism into the result. If the uncertainty is LTV dynamic systems, however, this step does not change the problem and, in this case, the WSP and AWSP are equivalent11 [270, 370]. Applying Theorem 12.2 to the AWSP with the problem data12

which defines the above V, we obtain the following result. Theorem 12.4. The robust control problem admits a solution if there exist real symmetric matrices X, Y, V, and W satisfying

and either

for the continuous-time case, or

for the discrete- time case, where "P is the set of scaling matrices

commuting with the structure of A. 11

In this case, V becomes an operator on La and condition (12.4) must be interpreted appropriately. Here, e > 0 is a sufficiently small scalar. Its role is the same as that discussed in section 12.4.1.

12

Iwasaki

244

This result is first shown in [124, 316]. A proof of this theorem can be obtained by specifying the relevant parameters such as Rn, Sn, and Qn appropriately in Theorem 12.2. Note, however, that such a procedure leads to the following conditions instead of (12.35): Rank Rank where

The equivalence of this condition and (12.35) is clear from Lemma 12.6.

12.4.3 Gain-scheduled control Consider the linear parameter-varying (LPV) system described by the following state equation: (12.40)

where A is the differentiation operator or the delay operator, and the time-varying parameter A(£) belongs to the set

for all t. Note that this LPV system is given by the LFT , where P(A) is the transfer function for system (12.40). Our objective is to design a controller such that the L2-induced operator norm of the closed-loop mapping from w\ to z\ is strictly less than a given level 7 for all possible parameter variation A(£) e A. This problem is basically the same as the robust control problem in the previous section if no information on A(£) is assumed to be given other than A(t) 6 A. Here, we assume that additional information for A(i) is available. In particular, we consider the situation where the value of A(£) can be measured in real time but is not known exactly when designing a controller. A possible structure for the controller that fits naturally into the situation is the following [312]:

Note that this controller has the same LFT structure as the plant and can be described as Tu(C, A), where C(A) is the transfer function of the state-space system given by the first equation in (12.43). The feedback system consisting of the plant in (12.40) and (12.41) and the controller in (12.43) is depicted in Figure 12.4.

12.4. Applications to control problems

245

Figure 12.4: Gain-scheduled control system.

Gain-scheduled control problem. Consider the LPV plant described by (12.40) and (12.41). Assume that the time-varying parameter A(£) belongs to the set A defined in (12.42) for all t. Design an LPV controller in (12.43) such that the closed-loop system in Figure 12.4 is robustly stable against A(£) € A and

holds, where 7 > 0 is a given scalar. This problem can be reduced to an AWSP with the problem data

where the partitions in M and K indicate how ^(M, K} in AWSP is defined, we have that Ap is the "performance block" inserted to accommodate the performance specification, J-Y := diag(7,7/7) is the scaling that normalizes the maximum norm of the performance block Ap, and A is the "unstable pole region" defined in (12.28). Note that the above V is specified by the following choice of parameters:

246

Iwasaki

where e > 0 is a sufficiently small scalar and its role is as before. Using these parameters, Theorem 12»2 «an be specialized to known results [19, 312] as follows. Theorem 12.5. The gain-scheduled control problem admits a solution if there exist real symmetric matrices X, Y, V e T>, and W e T> satisfying

for the continuous-time case, or

for the discrete-time case, where

and £> is the set of scaling matrices

commuting with the structure of A.

12.5

Concluding remarks

We have defined a general algebraic problem, WSP, of finding a matrix K that makes a certain feedback connection well posed. A (partial) solution to WSP is given in terms of the exact solution to the auxiliary problem, AWSP. The solvability condition is given by an SP indicated in section 12.1; that is, AWSP is solvable if and only if there exist real symmetric matrices satisfying certain LMIs and additional rank constraints (see Theorem 12.2). In general, SP is not a convex problem due to the rank constraints. No efficient algorithms are known to guarantee global convergence to a feasible solution of SP whenever the solution set is nonempty. Nevertheless, there are several heuristic algorithms that seem to perform well in practice [159, 128, 176, 217]. The result presented here indicates

12.5. Concluding remarks

247

the importance of SP and should motivate further improvement of existing algorithms and development of new algorithms. A fundamental fact implied by our main result (also anticipated from the result of [312]) is the following: AWSP can be solved exactly via convex programming if all the information on the plant "perturbation" is available for the controller; otherwise, AWSP reduces to a nontrivial SP with nonconvex rank constraints. We stress that the plant "perturbation" may include not only the actual parameter variation as in [312] but also more general dynamical elements such as the frequency variable and infinite-dimensional, nonlinear components. In section 12.4, we have shown that our result can be specialized to existing results that solve important control problems. The basic strategy is to describe the control problem in question as a special case of WSP or AWSP by appropriately specifying the "perturbation" V. Typically, the set V is defined in terms of the frequency variable A and the uncertainty A. In fact, the class of control problems reducible to WSP or AWSP can be enlarged by considering a wide variety of possible V. For instance, V may dictate infinite-dimensional system components such as e~ST and nonlinear components such as the limiter.

This page intentionally left blank

Part V

Nonconvex Problems

This page intentionally left blank

Chapter 13

Alternating Projection Algorithms for Linear Matrix Inequalities Problems with Rank Constraints Karolos M. Grigoriadis and Eric B. Beran 13.1

Introduction

A large number of controller synthesis problems, such as full-order stabilization, HQQ, ^-synthesis with constant scaling, robust gain-scheduling, and other full-order control design problems, have been formulated as convex feasibility or optimization problems involving linear matrix inequalities (LMIs) [315, 64, 216, 215, 148, 19, 43]. The book by Skelton, Iwasaki, and Grigoriadis [377] presents a collection and a unified formulation of many of these problems. Recently developed interior-point algorithms provide efficient computational tools for numerical solution [401, 404, 296, 403] and newly develop computer-aided control design software packages have been based on these algorithms; see, for example, the LMI Control Toolbox for Matlab [153, 150] and the Semidefin Programming Package [401, 402] with user friendly interfaces [68, 122]. These algorithms converge in polynomial time, and the problem structure is exploited to increase computational efficiency. The above LMI control design problems provide controllers of order equal to the order of the generalized plant. However, controller implementation constraints often dictate the use of low-order controllers because of simplicity, hardware limitations, or computational reasons; e.g., [448] discusses the controller-order limitations of the Hubble Space Telescope pointing control system. Many control design problems where the order of the controller is less than the order of the generalized plant can be formulated as LMI problems with an additional matrix coupling rank constraint that destroys the convexity of the optimization problem. Hence, interior-point algorithms with guaranteed convergence cannot be used to obtain a solution. Heuristic algorithms to address rankconstraint problems for low-order control synthesis have been proposed recently [94,124, 217, 128, 282], but convergence of these algorithms to a solution is not guaranteed. In this context, we present an approach based on alternating projection techniques 251

Grigoriadis and Beran

252

for low-order control design problems described by LMIs and coupling matrix rank constraints. Similar techniques have been used successfully in the past to solve image reconstructions and statistical estimation problems [84, 440]. In a general formulation, the idea behind these techniques is the following: Given a family of closed convex sets the sequence of alternating projections onto these sets converges to a point in the intersection of the family. Several variations of this standard approach have been proposed to solve minimum-distance problems with respect to a family of convex sets [70, 188] and to accelerate the rate of convergence using information about the direction toward the intersection [180]. We are interested to solve feasibility and optimization problems involving nonconvex matrix rank constraints. However, in such a case convergence of the alternating projection algorithms can be guaranteed only locally, i.e., when the initial starting point is in a neighborhood of a feasible solution [84]. Alternating projections for nonconvex constraint sets have been used in the past in image reconstruction problems with satisfactory performance [416]. In this work, the method of alternating projections combined with efficient semidefinite programming (SP) algorithms is presented and applied to the solution of fixed-order control synthesis problems. Extensive numerical experimentation is provided to assess the computational efficiency and reliability of the methods.

13.2

LMI-based control design with rank constraints

In this section we formulate the fixed-order stabilization with prescribed degree of stability and the HQQ control problems in terms of LMIs and a coupling matrix rank constraint. An algebraic approach is followed to eliminate the unknown matrix parameters from the design problem. When this LMI problem with the rank constraint is solved, then the desirable fixed-order controller is obtained via the solution of an additional LMI problem or via the use of explicit formulas. This approach follows the work of Iwasaki and Skelton [215] and Apkarian and Gahinet [19]. A similar mathematical formulation can be obtained for a large class of fixed-order control design problems that result in LMIs with additional rank constraints [377]. Hence, a unified approach can be followed for the solution of these problems.

13.2.1 Control problem formulation We consider linear time-invariant (LTI) systems of the form

where x € Rn is the state vector of the system, u € Rn" is the control input, w € Rni" is the external input (that includes plant disturbances and measurement noise), z € Rn* is the regulated output, and y G Rn" is the measured output. Without loss of generality we assume that Dyu = 0. We seek to design LTI controllers of fixed order nc

where xc € Rnc is the controller state. The matrix G contains all the unknown controller parameters.

13.2. LMI-based control design with rank constraints

253

If we augment the open-loop system (13.1) with the states corresponding to the controller (13.2), we obtain the augmented system

We have introduced the abbreviations

which allow us to write the control law (13.2) as u = Gy. Connecting the system (13.1) and the controller (13.2), and eliminating the variables u and y, we obtain the closed-loop system

where the closed-loop system matrix is an affine function of the unknown controller parameters as follows:

13.2.2 Stabilization with prescribed degree of stability To examine the stabilization problem consider the following representation of the system (13.1) that ignores the external input and the regulated output variables

The closed-loop system x = Ax — (A + BuGCy)x can be computed via (13.5) by deleting the appropriate columns and rows that correspond to the external input and the regulated output. We will say that the system x = Ax is stable with prescribed degree of stability a (or a-stable) if all the eigenvalues of A are to the left of the line -aj in the complex plane, where a is a given positive scalar. The following result provides necessary and sufficient conditions for a-stability. Theorem 13.1. Let A be a given square matrix and a be a given positive scalar. Then the following statements are equivalent: (i) The system x = Ax is a-stable. (ii) There exists a matrix Y > 0 such that (A + a/)T Y + Y (A + al) < 0. Notice that a-stability guarantees that the free response of the system decays to zero faster than e~at. We seek to design a controller of fixed-order nc to guarantee that the closed-loop system x = Ax is a-stable; that is, we are interested to solve the following problem:

254

Grigoriadis and Beran

Fixed-order a-stabilization problem. Consider the system (13.6) and let a be a given positive scalar. Find a controller (13.2) of order nc, if it exists, such that the closed system is a-stable. Using the matrix inequality condition (ii) of Theorem 13.1 we obtain the followin formulation of the fixed-order a-stabilization problem: Find a parameter matrix Y > 0 and a controller matrix G such that

Notice that condition (13.7) is a bilinear matrix inequality (BMI) in terms of the matrix variables Y and G. Techniques to solve these type of BMIs have been considered in the literature; e.g., see [45], [400], or [281]. However, only very low order problems can be addressed effectively. In this work, our objective is to eliminate the unknown control variables from the BMI (13.7). To this end, rewrite (13.7) as follows:

The following result provides necessary and sufficient conditions for the solvability of this matrix inequality (13.8) with resect to G. Lemma 13.1 (elimination lemma). Let matrices B € R nxm , C e R fcxn , and Q = QT G R n x n be given matrices. Consider the set of matrices

Then the following statements are equivalent: (ii) The following conditions hold:

Proof 13.1. This result is an extension of Finsler's theorem; see [209]. To use this elimination lemma notice that the orthogonal complements of the coefficient matrices Ba and C% can be computed as follows:

Hence, we obtain the following necessary and sufficient conditions for the existence of a controller that solves the a-stabilization problem

By introducing the notation

13.2. LMI-based control design with rank constraints

255

where X,Y e R n x n and X 22 ,y 22 6 R,«cxn Cj aft,ei simpie matrix manipulations and the choice of we obtain the conditions

Note that the above conditions are independent of the controller order. In addition, it is possible to show via the Schur's complement formula [64, page 7] and the matrix inversion lemma [446, section 2.3] that the relationship between X and Y in (13.11) provides the conditions

and Notice also that via the Schur's complement formula, condition (13.14) is equivalent to Rank(7 — XY) < nc. The rank condition (13.14) is obtained by observing that

and noting that

Conditions (13.12), (13.13), and (13.14), where X > 0 and Y > 0, are the necessary and sufficient conditions for the existence of an a-stabilizing controller of order nc. Notice that the rank constraint (13.14) is trivially satisfied when we seek for a full-order controller of order nc = n. We will denote the set of matrices X and Y that satisfy the constraints (13.12) and (13.13) by Fa. Since Fa is the feasibility set of a coupled set of LMIs, it is a convex set. If matrices X and Y that satisfy the conditions (13.12), (13.13), and (13.14) are obtained, then the unknown controller parameter matrix G that solves the a-stabilization problem can be obtained by solving the LMI (13.7) for G, where Y is obtained by (13.11). Alternatively, algebraic expressions that parameterize all controllers that correspond to a feasible solution X and Y are provided in [377].

13.2.3

HOC control

In this section, we consider the problem of designing a controller (13.2) of order nc to bound the L2 to L2 induced norm of the system (13.1) from w to z. It is known that this induced norm is the HQQ norm of the transfer function from w to z; see, for instance, [446, Chapter 4]. We seek to solve the following control problem. Fixed-order HQQ control problem. Consider the system (13.1) and let 7 be a given positive scalar. Find a controller of the form (13.2) of order nc, if it exists, such that the closed-loop system (13.4) has HQQ norm less than or equal to 7.

Grigoriadis and Beran

256

The following bounded real lemma provides necessary and sufficient conditions for a system to satisfy a given HQQ norm bound. Theorem 13.2 (bounded real lemma). Given a system of the form

then the following statements are equivalent: the system from w to z;

is the transfer function of

(ii) there exists a positive definite matrix Y such that

By inserting the expressions (13.5) for the closed-loop matrices in the bounded real lemma condition (ii) we obtain the following BMI formulation of the HQQ control problem: Find a parameter matrix Y > 0 and a controller matrix G such that

or, equivalently,

where the second term is repeated in (*) to provide a symmetric expression. Using the elimination Lemma 13.1 and following an algebraic procedure similar to the a-stabilization problem of section 13.2.2 we obtain the following necessary and sufficient conditions for the HOC control problem: There exists a controller that solves the fixedorder HOQ control problem if and only if there exist positive definite matrices X and Y such that

13.3. Alternating projecting schemes

257

and

Notice that the coupling constraints (13.19) and (13.20) are same as in the a stabilization problem. We will denote the set of matrices X and Y that satisfy the convex constraints (13.17), (13.18), and (13.19) by r Hoo . When the feasibility problem with constraints (13.17)-(13.20) is solved with respect to X and Y, then a fixed-order H^ controller that solves the design problem is obtained by solving the LMI (13.15) with respect to G or via the explicit formulas presented in [215, 377]. It is emphasized that a large class of many other fixed-order control design problems such as linear quadratic control, covariance control, positive-real control, /i-synthesis with constant scaling, and linear parameter-varying control can be formulated in a similar mathematical framework via linear matrix inequalities and a coupling matrix rank constraint; see [377].

13.3

Alternating projecting schemes

Based on the formulation provided in section 13.2, the fixed-order control design problem reduces to a feasibility problem of obtaining a matrix pair that satisfies a family of LMIs and a coupling matrix rank constraint. In this section, alternating projection methods are presented for the solution of this type of feasibility problem. These methods exploit the geometry of the design space to find feasible solutions, and they have been used successfully to address image restoration and statistical estimation problems in signal processing [84, 440, 416]. Both the standard alternating projection method and the directional alternating projection method are presented [70, 188, 180].

13.3.1

The standard alternating projection method

Consider a family C\, £2, • • • , Cm of closed, convex sets in the space of symmetric matrices. We suppose that the sets have a nonempty intersection, and we seek to solve the feasibility problem of finding a matrix in the intersection C\ {\Ci H • • • (\Cm. Let Pct denote the orthogonal projection operator onto the set C$, where i = l,...,ra. Hence, for any n x n symmetric matrix X, the matrix Pc^X) denotes the orthogonal projection of X onto Cj, that is, the matrix in Ci that has minimum distance from the matrix X. The orthogonal projection theorem [255] guarantees that this projection is unique. The question we would like to answer is the following: Is it possible to provide a solution to the feasibility problem by making use of the orthogonal projections onto each constraint set? The answer is yes and is provided by the following result, which we call the standard alternating projection theorem [76, 180]. Theorem 13.3. Let XQ be any n x n symmetric matrix and C\,C<2,... ,Cm be a family of closed, convex sets in the space of symmetric matrices. Then if there exists an intersection, the sequence of alternating projections

258

Grigoriadis and Beran

converges to a point in the intersection of the sets, i.e., Xj —»• X, where If no intersection exists, the sequence does not converge. Hence, starting from any symmetric matrix, the sequence of alternating projections onto the constraint sets converges to a solution of the feasibility problem, if one exists. The case of no intersection can be detected by examining the convergence of the even and odd subsequences of the above sequence of alternating projections. A schematic representation of the standard alternating projection method is shown in Figure 13.1. It can be easily verified that the limit X of the alternating projection sequence depends on the starting point XQ, as well as on the order of the projections. Hence, by rearranging the sequence of projections we can obtain a different feasible point.

Figure 13.1: Standard alternating convex projection algorithm. An important feature of the standard alternating projection algorithms (13.21) is that the algorithms can be implemented very easily, and usually the amount of calculations in one iteration is very small. However, in some cases the algorithms may suffer from slow convergence. For example, consider the case of two planes intersecting with a small angle. In this case the standard alternating projection algorithm (13.21) might oscillate for many iterations between the two sets before it converges to a point in the intersection. An effective remedy is often obtaioned by the directional alternating convex projection Algorithm, described below [180].

13.3.2 The directional alternating projection method The directional alternating projection method uses information about the geometry of the constraint sets to provide an algorithm with accelerated convergence to solve the matrix feasibility problem. The basic idea behind this approach is to utilize in each iteration the tangent plane of one of the constraint sets, so that the sequence of points

13.4. Mixed alternating projection/SP (AP/SP) design for low-order control

259

we obtain approaches the intersection of the sets more rapidly (see Figure 13.2). For simplicity, we will consider the case of two closed and convex constraint sets C\ and £2The directional alternating convex projection algorithm is described next.

Figure 13.2: Directional alternating projection algorithm. Theorem 13.4. Let XQ be any nxn symmetric matrix. Then the sequence of matrices {Xi}, i = 1, 2, . . . , oo, given by

converges to a point in the intersection of the sets Ci and €2Hence, starting from any symmetric matrix, the sequence of directional alternating projections (13.22) provides an accelerated numerical algorithm to obtain a feasible matrix in the intersection of the constraint sets C\ H^2- In fact, it can be easily verified that when the two sets C\ and €2 are hyperplanes in the space of symmetric matrices, then the alternating projection algorithm converges to a feasible point in one cycle, independently of the angle between the two hyperplanes.

13.4

Mixed alternating projection/SP (AP/SP) design for low-order control

In this section, we describe how to solve low-order control problems by exploiting the alternating projection technique described in the previous section and efficient SP algorithms. The presentation given here was originally presented in [44].

260

Grigoriadis and Beran

Recall that the fixed-order control design problem has a solution if and only if there exist a matrix pair (X, Y) in the intersection of a family of LMI constraint sets and a rank-constraint set. Let's denote by FConvex the LMI constraint that the matrices X and Y need to satisfy and define the constraint set

Hence, a necessary and sufficient condition for the existence of a controller for fixed order nc is that there exists (X, Y) € FCOnvex n ZUc. The set ZUc that restricts the controller order is a nonconvex constraint set. Also, notice that, depending on the specific control design problem, the set Fconvex corresponds to the set Fa or the set FH^ defined in section 13.2. The alternating projection scheme presented in the last section can now be used to find a matrix pair (X, Y) in the intersection of the convex constraint set FCOnvex and the nonconvex rank-constraint set Znc. Our first task is to compute the projections onto the convex constraint set Fa or FH^ • One approach is to decompose the LMI constraints as intersections of sets of simpler geometry and to find analytical expressions for the projection operators onto these simpler sets. This approach was followed in [176]. However, the iterative projections onto these multiple sets require a large number of iterations for convergence resulting in computationally expensive algorithms. Alternatively, the projection onto the set Fa or FH^ can be computed using SP for convex optimization. Hence, the multiple projections required in [176] onto the set FCOnvex are eliminated, and faster convergence can be achieved. The following result, see [64], provides the projection onto a general LMI constraint set F as the solution to an SP problem. Proposition 13.1. Let F be a convex set described by an LMI. Then the projection X* = PpX can be computed as the unique solution Y to the SP problem

In our problem, our objective is to compute the projection onto the joint set Fconvex C Sn x <Sn of matrix pairs (X, Y). This projection can be found by solving the following SP problem, where (XQ,YQ) are the given matrices that we seek to find the projection, and X, y, 6", T € <Sn are the free variables

We denote the minimizing solutions by (X*,y*); that is, the projection onto FCOnvex is written as

13.4. Mixed alternating projection/SP (AP/SP) design for low-order control

261

In addition to the above LMI constraints sets, we seek to compute the orthogonal projection onto the nonconvex constraint set ZUc. To this end, define the following sets in the space of symmetric matrices

and

Then the connection between

Notice that the sets T> and P are closed convex sets, where 72.^ is the only nonconvex set. The expressions for the orthogonal projections onto these sets are provided next.

Theorem 13.5. Let The orthogonal projection, Z* = PpZ, of Z onto the set X> is given by

The orthogonal projection onto the set P is provided by the following result, which follows from [193]. Theorem 13.6. Let Z G <Sn and ]et Z + J = LKLT be the eigenvalue-eigenvector decomposition of Z + J, where A is the diagonal matrix of the eigenvalues and L is the orthogonal matrix of the normalized eigenvectors. The orthogonal projection, Z* = PpZ, of Z onto the set P is given by

where A_ is the diagonal matrix obtained by replacing the negative eigenvalues in A by zero. Hence, this projection requires an eigenvalue-eigenvector decomposition of the 2n x 2n symmetric matrix Z + J. We note that the rank-constraint set 72.^, defined by (13.27), is a closed set, but it is not convex. Therefore, given a matrix Z in <S2n, there might be several matrices in 72* that minimize the distance from Z. We will call any such matrix a projection of Z on 7£fc. The following result provides a projection onto the set 72* [198]. Theorem 13.7. Let Z e Sn and let Z + J = UY,VT be a singular value decomposition of Z + J. The orthogonal projection, Z* = Pnk Z, of Z onto the set 72* is given by

where S& is the diagonal matrix obtained by replacing the smallest In — k = n — nc singular values in Z -f J by zero.

262

Grigoriadis and Beran

Notice that the sequence of the projections onto the two sets P and Tlk can be computed in one step via the eigenvalue-eigenvector decomposition of Z + J followed by zeroing the appropriate number of the smallest eigenvalues. If we denote this sequence of projections by Pp-R.k Z, then the directional alternating projection can be used to find the projection onto the set ZUc via the following sequence of iterations:

We will call this step of our alternating projection algorithm an inner iteration. Hence, the above inner iteration provides the projection Pznc (X, Y) of (X, Y) onto Zn<. . The alternating projection algorithm for the fixed order control problem can now be programmed utilizing SP for the projection onto rconvex and the above inner iteration scheme for the projection onto ZUc . The proposed procedure is the following: First find a solution that corresponds to a full-order controller. This is simply done by solving an LMI feasibility problem for the constraint set rc0nvex- Next, obtain a solution that corresponds to a controller of order at most nc — 1. This can be done via the SP problem

The obtained solution will be the starting point for our alternating projection algorithm. The steps of this alternating projection algorithm are as follows. Notice that in order to have fast convergence the directional alternating projection algorithm is used as mentioned in section 13.3.2. Step 1. Solve the SP problem (13.34) that corresponds to a controller of order at most nc = n — 1. The solution (Xo,Y0) will be our starting point. Step 2. Consider the problem where the controller order is reduced by one; i.e., set nc = nc — 1. Compute the following iterative sequence of projections:

Continue until (X^Yi) G rconvex n Znc, or until infeasibility has been detected. Step 3. Return to Step 2, until the controller order nc is the desired one.

263

13.5. Numerical experiments

We will call each sequence of iterations in Step 2 an outer iteration, as opposed to the inner iterations for the projection onto ZHc. It is emphasized that global convergence of the above alternating projection schemes is not guaranteed. Hence, it is possible that a specified low-order controller order might not be achieved even though such a controller exists.

13.5

Numerical experiments

In this section extensive numerical experimentation is provided to assess the efficiency and the complexity of the proposed alternating projection schemes for the solution of LMI problems with rank constraints. The fixed-order a-stabilization problem for a nth-order spring-mass system, a helicopter system, and the static output feedback stabilization problem for randomly generated systems are considered. The results were obtained on an HP735 workstation using Matlab version 4.2 and the SP package by Boyd and Vandenberghe [401].

13.5.1 Randomly generated stabilizable systems The first numerical experiment considers randomly generated static output-feedback stabilizable systems. A stabilizable system is obtained by reflecting the eigenvalues of randomly generated matrices via an eigenvalue-eigenvector decomposition, where the positive eigenvalues are replaced with their negative values. Then a product of arbitrary input, feedback gain, and output matrices is subtracted from this matrix to guarantee that the result is static-output-feedback stabilizable. By shifting all eigenvalues by an amount —a, where a is a positive scalar, desired a-degree of stability can be introduced in the static output feedback problem. Table 13.1: Sizes for the random generated experiments. Case a b c d e f

System order n 4 6 8 4 6 8

Number of inputs nu I 2 3 3 4 5

Number of outputs ny 1 2 3 2 3 4

We seek to obtain the lowest-order a-stabilizing controller obtained via the proposed alternating projection methods. We will compare this result with the following Kimura bound &{,, which provides an upper bound on the control order for the stabilizing control problem [229] This bound is for a = 0. Table 13.1 shows the cases we have considered. Notice that the Kimura bound kf, is equal to k\> = 3 for cases a, b, and c, and kb = 0 for cases d, e, and f. For each one of the above six cases 200 hundred random experiments were carried out. A degree of stability a = 0.1 has been introduced in the randomly generated

Grigoriadis and Beran

264

systems, and the objective is to obtain the lowest-order stabilizing controller that places the closed-loop poles to the left of —a. Tables 13.2 and 13.3 show the results for each one of the cases considered in Table 13.1. Table 13.2: The results for the random example cases a, b, and c. All controllers were of order 0 or 1, as expected with the Kimura bound kb = 3. The CPU time is the average time used on all examples. Controller order

0

Outer iterations 0 1 2 3 4 5 6 7-38

1 Average CPU time

Case a Rate # 176 88.00 0 0.00 3 1.50 3 1.50 4 2.00 3 1.50 0 0.00 6 3.00 5 2.50 18.71s

Case b Rate # 177 88.50 1.00 2 3.50 7 4 2.00 3 1.50 0.00 0 1.00 2 5 2.50 0 0.00 15.93s

Case c Rate # 184 92.00 3 1.50 7 3.50 2 1.00 1 0.50 0 0.00 1 0.50 1 0.50 1 0.50 26.15s

From these results it is observed that in the majority of the experiments, the lowestorder achievable controller is obtained in 0 iterations, that is, by solving the convex problem (13.34) as described in section 13.4. In all the experiments, the lowest-order achievable controller is of order lower than or equal to the Kimura bound &{,. In 5 experiments for case a, the lowest-order achievable controller was 1, instead of the zeroth order that is guaranteed by the construction of the experiments. Table 13.3: The results for the random example cases d, e, and f. All controllers were of order 0, as expected with the Kimura bound kb = 0. The CPU time is the average time used on all examples. Outer iterations 0 1 2 3 4 CPU time

Case d Rate # 200 100.00 0 0.00 0 0.00 0.00 0 0 0.00 1.17s

Case e Rate # 199 99.50 0 0.00 0 0.00 1 0.50 0 0.00 3.19s

Casef Rate # 196 98.00 0 0.00 3 1.50 0 0.00 1 0.50 11.18s

13.5.2 Helicopter example The following example is from [226]. The goal is to obtain a static state feedback controller for the following helicopter model such that the closed-loop poles are located to the left of -a = -0.1 at the complex plane. The system is of the form (13.6), and the

13.5. Numerical experiments

265

data are the following:

The algorithm converged in zero iterations; that is, the solution of the convex minimization problem (13.34) provides a static controller

that achieves the desired objective. The closed-loop poles are -20.8379, -0.1042, and -0.2572 ± 0.9738J. The required CPU time to obtain this controller is 1.27 sec.

13.5.3

Spring-mass systems

In this numerical experiment interconnected spring-mass system models were generated. The order of the system is equal to twice the number of the interconnected masses. We first consider the ^-stabilization problem for a 2-mass/spring system. We assume that the two bodies have equal mass mi = m-i = 1 and they are connected by a spring with stiffness k = 1. We assume that only the position of body 2 is measured and a control force acts on body 1; i.e., the problem is noncollocated. A state-space representation of this system, for the noise-free case (w = 0), in the form (13.6) is the following:

where the state variables x\ and Xs are the position and velocity, respectively, of body 1; and X2 and x\ are the position and velocity of body 2. We seek fixed-order a-stabilizing controllers using the mixed AP/SP method. The results are presented in Table 13.4. The convex approach (13.34) provides controllers of order 2 for 0 < a < 0.16. The same approach gives controllers of order 3 for a between 0.16 and 1.0. In fact a controller of order 3 is in theory possible for all possible a's, since the system is controllable and observable. Controllers of order 2 are obtained for 0.16 < a < 0.42 via our proposed AP/SP method. The number of flops required to obtain the given solution was estimated as follows: The SP program provides two types of CPU time, "user time" and "system time," but only the "user time" is included in this measure. Moreover, the number of flops per CPU second is estimated to be about 105. Computing the value in flops in Matlab gives the value of 6.6104. The total amount of flops is therefore estimated to be CPU time x!05-f Matlab counted mflops. We give the best controller computed with the mixed AP/SP method. For a = 0.42 and a desired controller order of 2, the mixed AP/SP method required 13.18 mflops and 12 iterations. The computed controller is

Grigoriadis and Beran

266

Table 13.4: Results of the combined alternating projection and semidefinite programming algorithm.

3

Effort required Mega flops Outer iter.

2

Mega flops Outer iter.

nc

Prescribed degree of stability a 0.16 0.2 0.3 0.42 0.01 Same as Same as 0.29 0.33 0.29 below 0 0 0 below

0.17 0

0.16 0

3.33 4

3.58 3

13.18 12

1.0 0.26 0

NC

and the closed-loop eigenvalues

We now proceed to examine higher-order spring-mass systems. Tables 13.5 and 13.6 provide the lowest-order achievable controller and the number of iterations needed for a = 0.001 and a = 0.1 for different numbers of the system masses. Hence, either zero or one iteration of the AP/SP algorithm is enough to provide a static output feedback controller depending on the desired degree of stability a.

Table 13.5: Results for degree of stability a = 0.001. Number of masses

Number of iterations

2 3 4 5

0 0 0 0

Lowest-order achievable controller 2 3 4 5

Table 13.6: Results for degree of stability a = 0.1. Number of masses

Number of iterations

2 3 4 5

0 0 1 1

Lowest-order achievable controller 2 3 5 6

13.6. Conclusions

13.6

267

Conclusions

Alternating projection algorithms are proposed to solve LMI control design problems with coupling rank constraints. These problems appear in many reduced-order control synthesis problems. The above problems are formulated as matrix feasibility problems of finding matrix parameters in the intersection of a family of LMI and matrix rank constraint sets. Alternating projection methods combined with SP algorithms utilize the geometry of the design space along with efficient LMI solvers to obtain matrix parameters that satisfy the design constraints. Directional alternating projections provide a more efficient implementation. However, only local convergence of the alternating projection algorithms to a matrix solution that satisfies the above constraints is guaranteed. Computational experiments have been used to demonstrate the applicability of the proposed algorithms for addressing fixed-order control problems. Fixed and random benchmark examples have been developed for this objective. These results indicate that the proposed algorithms combined with SP are effective in addressing low-order controllers for small and medium-size problems.

This page intentionally left blank

Chapter 14

Bilinearity and Complementarity in Robust Control Mehran Mesbahi, Michael G. Safonov, and George P. Papavassilopoulos 14.1

Introduction

In this chapter, we present an overview of the key developments in the methodological, structural, and computational aspects of the bilinear matrix inequality (BMI) feasibility problem. In this direction, we present the connections of the BMI with robust control theory and its geometric properties, including interpretations of the BMI as a rankconstrained linear matrix inequality (LMI), as an extreme form problem (EFP), and as a semidefinite complementarity problem (SDCP). Computational implications and algorithms are also discussed. The simultaneous appearance of the (unknown) variables x and y in the matrix inequality

for a given set of symmetric matrices Fij G R pxp (i = 1, . . . , n; j = 1, . . . , m) not only provides an unexpectedly powerful formulation for a wide range of robust control problems, but it also introduces new and elegant structural and computational questions. A possible initial attempt to rename the products X^T/J as Zij in (14.1) and rewriting it in terms of z^ 's as an LMI [64] ,

introduces yet another twist to this problem, since the unknown variables Zij's are now constrained to be related in a rather peculiar manner, for example,

269

270

Mesbahi, Safonov, and Papavassilopoulos

Since at the present time this approach seems to present neither aesthetic insights nor suggest a computational approach for finding the variables x and y in (14.1) (or prove the nonexistence of them), we abandon our initial temptation and decide to treat the problem with complete regard to its most distinguished property: bilinearity; we shall refer to this problem as the BMI. What makes the BMI an extremely important problem in robust control, however, is far beyond an initial mathematical curiosity to make the LMI "bilinear." The BMI is introduced in response to some of the most important issues facing the field of robust control and in particular those that cannot be addressed in the LMI framework [167], [354]. The chapter presents various facets of the research performed by the authors and their colleagues on the BMI problem. The topics to be covered are categorized as 1. methodological, 2. structural, and 3. computational. Under the first category, we present the very important results pertaining to the formulation of the robust synthesis problems as a BMI. In particular, we cover reformulating the n/km synthesis [103], [349], along with specifications on the controller order and structure, to a BMI. Our presentation in this part is based on the results reported in Safonov, Goh, and Ly [354] and is influenced by the dissertations of Goh [164] and Ly [260]. We then proceed to present some of the structural aspects of the BMI. It turns out that the investigations into the geometry of the solution set of the BMI lead to some very interesting nonconvex and convex programming problems over cones. In particular, we discuss the results pertaining to the equivalence of the BMI with examining the magnitude of the diameter of a certain convex set [356]. The results connecting the BMI to a class of cone optimization problems, namely, the SDCPs [208], [280], [281], [279], [283], will also be given particular attention. The computational methods for solving the BMI are briefly reviewed in the final section. Section 14.1.1 introduces the relevant preliminaries.

14.1.1

Preliminaries on convex analysis

Let H be a finite-dimensional Hilbert space equipped with the inner product {-, •) : H x H —» R (e.g., the n-dimensional Euclidean space or the space of n x n matrices with the appropriate notion of an inner product defined on them). A set 1C C H is a cone if for all a > 0, atC C 1C. fC is a convex cone if /C is a cone and it is convex, i.e., for all a G [0,1], a/C + (1 — a)/C C /C, or, equivalently, if 1C is a cone and /C -f /C C /C. A convex cone 1C is called pointed if 1C n (-1C) = {0} and solid if it has a nonempty interior. An extreme form (or an extreme ray) of a convex cone 1C is a subset E = {ax : a > 0} of /C, such that if x = ay + (1 — a)z for 0 < a < 1 and y, z e /C, one can conclude that y,z G E [185], [198]. The set of extreme forms (rays) of the cone 1C is denoted by /C. The dual cone of a set 5 C H, denoted by S*, is defined to be

If S is a pointed closed convex cone, then the interior of its dual cone, int S*, is given by

A closed convex cone in a finite-dimensional Hilbert space is pointed if and only if its dual is solid [46]. In particular, Sj. is a pointed, closed, and self-dual cone—the cone

14.1. Introduction

271

that induces the ordering used in the LMI and the BMI formulations. It can easily be shown that S* is always a convex set, and that if Si C 52, then 5£ C 5J". In addition, S = (5*)* if and only if 5 is a closed convex cone. Given a pointed closed convex cone 1C C n, a linear map M : U -» H is called /C-positive if VO ^ X e /C, Af (X) € int /C*. Furthermore, a linear map M : H —> H. is called /C-copositive if (X, M(X)) > 0 VX € /C [46], [173], [208]. We are now ready to formulate the cone problems that are considered in section 14.3.2. The cone-LP is formulated as follows: Given a cone /C C H, a linear map M : H —>• W, and the elements Q and C in 7i, find Z eH (if it exists) as a solution to

Similarly, the cone-LCP is formulated as follows: Given a cone /C C H, a linear map M : H -* H, and Q € ft, find Z e ft (if it exists) such that

The above instances of the cone-LP and the cone-LCP shall be referred to as the cone-LPjc(C, Q,M) and cone-LCP/c(Q, M). When 1C is the nonnegative orthant in the n-dimensional Euclidean space, the cone-LP/c(C, Q,M) (14.3)-(14.5) and the coneLCPjc(Q,M) (14.6)-(14.8) are equivalent to the familiar linear programming (LP) and the linear complementarity problems (LCPs) [85], [90]. A problem that serves as a bridge between the BMI and the cone-LP/LCPs is the EFP: Given a pointed closed convex cone /C C H, a linear map M : H —> 7i, find X 6 H (if it exists), such that

The above instance of the EFP is referred to as the EFPjc(M). When /C is the nonnegative orthant in the n-dimensional Euclidean space, the EFP is a trivial problem. It should be noted that the solution set of an EFP is generally nonconvex; given the two extreme forms of /C that solve the EFP/c(M), a strict convex combination of them is not an extreme form of /C. It is also important to note that the EFP requires M(X) to lie in the interior of the dual cone. This is in light of the fact that for certain important classes of linear maps M, including the map that is encountered in the context of the BMIs, M(X) is known to lie in, but possibly on the boundary of, the dual cone, for all the extreme forms X of the cone /C. As it is shown in section 14.3.2, the EFP formulation, in spite of its nice geometrical interpretation, includes as its special case the BMI problem. Since the extreme forms of the matrix classes that are considered in the chapter can be characterized by their rank, the EFP formulation of the BMI also translates directly to a rank minimization problem over a convex set of matrices [281] (refer to section 14.3.2).

14.1.2

Background on control theory

We use Ti and Si to denote the dimensions of ith input and the ith output of the finitedimensional linear time invariant (LTI) plant C(s), respectively; the order of G(s) is

272

Mesbahi, Safonov, and Papavassilopoulos

Figure 14.1: T(s) = Fl{G(s),K(s)}.

denoted by n; similarly q and qM designate the order of the controller and the multiplier to be synthesized. Consider the LTI plant G(s) partitioned as

where

and A e R nxn , Dn e R S l X r i , D22 € R 32Xr2 , and Gy(«) := A, + Ci(Is - A}-lBj. With respect to this particular partition of the transfer matrix G, the feedback connection shown in Figure 14.1 has the transfer function

where K(s) denotes a controller of order q given by

and

The inclusion of the uncertainty in this framework is now really an extra bonus; see Figure 14.2. The resulting transfer function from u\ to y\ is simply

provided that all the appropriate inverses exist. Motivated by the way that uncertainty manifests itself in the generalized plants (those that include the nominal plant, the sensors, and the actuators), one is led to consider uncertainties of the form

273

14.1. Introduction

Figure 14.2: T := Fu{ (s),fl{G(s),K(s)}}

for some prescribed positive integers L, F, and ni, . . . ,n^, such that ||A||oo < 1. The <5j's are real or complex valued and in general correspond to parametric uncertainties; Ai's on the other hand are time invariant operators used to account for such things as unmodeled dynamics. Given that A has arisen from our modeling technique and/or our inadequate knowledge about the nature of things, and that G is a physical reality, K is our only hope to make the system in Figure 14.2 operate in a way that is desirable to us: the controller K is chosen to provide internal stability and (external) performance for all possible perturbations A; performance in robust control setting is often taken as the ability of the system to reject the disturbances lumped in the term u\ , not only necessitating that HTHoo < 1 (as the result of the small gain condition, the so-called bounded realness of the operator T [360], [444]) but also requiring IIT^Hcx, to be as small as possible. Let us shift our attention back to the unperturbed configuration of Figure 14.1. It is well known that the bilinear sector transform T(s) = sect{T(s)} := (/ — T(s))(7 4T(s))"1 maps bounded real systems HT^S)!^ < 1 into positive real systems Herm(T(jo;)) > 0 and vice versa. A routine calculation reveals that the state-space matrices of f ( s ) = sect{T(s)} and T(s) are related by

where

the corresponding bilinearly transformed open-loop plant has the state-space system matrix

It is known that we may linearly parameterize the set of realizable S~ matrices as follows [315], [316]: (14.21)

274

Mesbahi, Safonov, and Papavassilopoulos

where SQ is the state-space system matrix of

and

Thus we have all the necessary machinery to translate between positive real and bounded real conditions. The principle tool in translating frequency domain results related to positive real conditions in terms of the corresponding matrix inequalities is the following important lemma. Lemma 14.1 (generalized strong positive real LMI: see [13], [354]). LetG(s) = C(Is — A)~1B + D be a minimal state-space realization. Then for some e > 0, Herm( if and only if there exists P = PT such that

Moreover, G(s) is stable if and only if P > 0.

14.2

Methodological aspects of the BMI

Mathematics often takes a precise problem formulation as its starting point; theoretical research in engineering more often than not takes it close to its ending. It is common to see statements like "since this problem is equivalent to solving a system of linear equations, we consider the problem solved" in the engineering literature. But when should we consider an engineering problem solved? Certainly, one could respond to this question by saying that an engineering problem is solved when the corresponding mathematical problem is solved; however, this correspondence is never one to one and, moreover, there are still many questions as to when a mathematical problem, especially in optimization, is declared to be solved. Let us then take the following as our starting point: An engineering problem is solved when a computationally reasonable method (on a reasonable model of computation) exists for at least one of its equivalent mathematical formulations. Given that the statement above is a reasonable approximation to our actual motivation in control research, we strive for a problem formulation that leads to an efficient computational method for the solution of a control problem. On the other hand, we strongly prefer a mathematical formulation and a solution method that are directly linked to our engineering intuition and judgment. The frequency domain techniques in control system design are prime examples of such preferences. To make matters even more interesting, we often search for a formulation that not only captures our engineering considerations,

14.2. Methodological aspects of the BMI

275

but that also has a nice mathematical representation. Alas often these considerations cannot be satisfied at the same time, although exceptions exist.13 The engineering problem considered in this chapter is the synthesis of controllers for systems whose models are not precisely known. We adopt the framework of the robust control theory and in particular that of fi/km analysis and synthesis. Roughly speaking, with reference to Figure 14.1, p,/km synthesis is the problem of finding the controller K such that Fu{A,T} is stable for all admissible A's. Since one would like to exploit the structure of the uncertainty A (of the form (14.17)) in order to establish stability, we are led beyond requiring \\T\\oo < 1 to such conditions as solving [446]

where D is a stable, minimum phase scaling matrix such that D(s)A(s) = A(s)D(s); the infimum is guaranteed to provide a lower bound (upper bound) for the multivariable stability margin /CM [351], [352] (respectively, n [102]), where for an asymptotically stable transfer matrix T(s),

and

The D-K iteration is a solution method for the n/km synthesis, which proceeds by alternating between solving an HQQ optimization problem by fixing D(s) in (14.25), followed by a convex optimization with K fixed [31], [77], [349]. An improvement in the conservativeness of the D-K iteration is known as the D, G-K iteration [131], [314]. Both design techniques can be enhanced by considering the n/km synthesis in the positive real framework, resulting in what is known as the M-K iteration [353]. The M-K iteration approach has a very nice property of bypassing the curve fitting step in the D-K iteration; this latter approach in fact forms the basis for the proof of Theorem 14.1 (below).

14.2.1

Limitations of the LMI approach

It remains true, however, that much difficulty remains in the robust synthesis of practical, nonconservative controllers. At least three very important classes of robust control design issues have not been found to be readily transformable into the LMI framework, which has offered a very promising direction to study many robust analysis problems. These are (1) /i/fcmsynthesis via dynamical scalings/multipliers, (2) fixed order control synthesis, and (3) decentralized controller design (i.e., synthesis of controllers with "block diagonal" or other specified structure). • Consider the robust control problem of p,/km synthesis. Current n/km techniques [31], [77], remain inherently conservative, i.e., suboptimal, because the D-K iteration approach of alternately synthesizing first a controller K(s) and then diagonal scalings D(s) is in no way guaranteed to achieve a globally or, it turns out, even a locally optimal solution. The D-K iteration can, in theory, get stuck at points that are local minima with respect to D alone and with respect to K alone, but are not a true local minima with respect to joint variations in D and K. 13

LP, the simplex algorithm, and the corresponding economic interpretation, are beautiful examples of such an interaction, even though finding a polynomial time algorithm for LP that corresponds to a certain pivoting strategy of the simplex method is still an open problem.

Mesbahi, Safonov, and Papavassilopoulos

276

• One of the main objections to the modern control synthesis theories, such as LQG, HOQ, and n/km, is that the resultant controllers are typically of relatively high order—at least as high as the original plant and often much higher in the usual case where the plant must be augmented by dynamical scalings, multipliers, or weights in order to achieve the desired performance. • Control system designs for very large or complex systems must often be implemented in a decentralized fashion; that is, local loops are decoupled and closed separately with little or no direct communication among local controllers. The synthesis of optimally robust decentralized systems has obvious benefits. However, even the synthesis of optimal decentralized HQQ and LQG control systems has remained beyond the scope of the existing theories. Neither the LMI framework nor other existing theories has yet proved to be sufficiently flexible to handle problems in the foregoing classes. The purpose of this section is to demonstrate that the BMI framework is sufficiently flexible to simultaneously accommodate all three of the foregoing types of control design specifications, in addition to handling all those that the LMI handles. In particular, we provide a proof of the following theorem. Theorem 14.1 (see [354]). The fj,/km synthesis problem can be formulated as the following matrix inequality: Given real matrices Raug, Uaug, Vaug, and R^t, U^ , V^ find matrices Z, SQ such that,

where

and P = PT and X = XT € R("+9)x ("+<*), and W is constrained to have the structure consistent with the uncertainty structure of the underlying p,/km problem (14.17). The rest of this section is devoted to the proof of Theorem 14.1 and one of its important consequences in the design of the decentralized controllers.

14.2.2

Proof of Theorem 14.1

We first make a note of the following: Any constraint of the form Xi = 1, yi = 1 for an instance of the BMI can be transformed to an instance with no such constraints [354]. Proof 14.1 (see [354]). In the n/km synthesis procedure outlined in [353], the p,/km problem is shown to be solved if there exists a suitably structured block-diagonal rational multiplier matrix M(s) such that

No constraint is placed on the stability of M(s), but it is required to be uniformly bounded on the ju-axis and to satisfy the generalized strong positive real condition; i.e.,

14.2. Methodological aspects of the BMI

277

for some e > 0,

It is supposed in [259] (without loss of generality if N = oo) that M(s) is a weighted sum of certain suitably structured block-diagonal transfer function matrices Mi(s),

where

and W := [Wi,..., WN] G R S l X < J S l . In most cases Wi G R, but more generally, as when there are repeated uncertainty blocks, Wi may be block diagonal matrices of a certain specified form. The specific details of the structure constraints on Wi and Mi(s) required for the various types of real and/or complex uncertainty block structures are described in [259] and [355]. One may form the "augmented" closed-loop system

then Taug(s) has a state-space realization of the form (14.21),

In view of the form of (14.34), -Aaug naturally assumes the form

where A^ denotes the A-matrix of M(s) and At G R( n+ 9) x ( n+ 9) is the A-matrix of the bilinear sector transformed closed-loop plant sect{T(s)}. Under (14.30)-(14.31), the stability of sect{T(s)} is equivalent to the stability of the original untransformed closed-loop plant T(s) [353]. In view of (14.36) and (14.35), one has

where

Theorem 14.1 is now proved by applying Lemma 14.1 for checking the positive realness °f -^aug-

We note that the matrix P in the above theorem is the solution to the LMI that results from the application of Lemma 14.1. Since we do not require A&ug to be stable, no definiteness condition is imposed on P. Instead, stability of the closed-loop At (and

278

Mesbahi, Safonov, and Papavassilopoulos

hence T(s)) is ensured by testing existence of X = XT > 0 such that Herm(—XA t ) > 0 (e.g., [289, page 63]); this is the role of the matrix X in the BMI (14.28). Clearly (14.28) is a BMI feasibility problem. It is jointly linear in the parameters of the matrices W, X, P. It is affine in the controller parameter matrix SQ. Some special cases of this problem have been found by Packard et al. [315], [316] to be reducible to LMIs via the Parrott theorem. These include full-state feedback H^ and full-order HQO with constant diagonal scalings (M(s) = "constant matrix"), as well as certain simultaneous stabilization and related gain-scheduling problems. But optimal solution to even the fixed order (q < n) HQQ problem (i.e., M(s) = I) has remained previously elusive despite some determined efforts. The foregoing BMI formulation (14.28) provides a simple formulation of these and the related synthesis problems. Interestingly, the BMI formulation is flexible enough to accommodate constraints on controller structure as well. To simplify matters, we make the following assumption (refer to (14.13)): Hence by (14.22), Note that no significant loss of generality results from this assumption since it can always be made to hold via a singular perturbation of the plant (given that our constraints do not require an infinite bandwidth controller). With D22 = 0, it is clear that the controller K(s) inherits the same block structure as Q(s). In particular, if the state-space matrices AQ,BQ,CQ,DQ are constrained to have a block-diagonal structure, then K(s) will be block diagonal, too; i.e., K(s) will be a "decentralized" controller.

14.2.3

Why is the BMI formulation important?

We conclude this section with a recapitulation of the reasons why the BMI formulation is important: • The main attraction of the BMI formulation is its simplicity and generality. It allows the controller and multiplier/scaling optimization in fj,/km synthesis to be formulated as a single finite-dimensional optimization over the controller parameters SQ and the multiplier/scaling and Lyapunov parameters W, X, P. Consequently, application of nonlinear programming techniques to the BMI at least assures convergence to a joint local optimum in D and K—the D-K iteration of ^/km synthesis cannot make this claim. The BMI formulation also eliminates the curve fitting step of the traditional D-K iteration approaches to n/km. • A broad spectrum of robust control synthesis problems can be formulated within the BMI framework. These include order-constrained controller (J,/km synthesis with specifications requiring such additional properties as decentralized control. Gain scheduling and simultaneous n/km synthesis for several plants also fall in this framework. We note, however, that the BMI formulation is not without its drawbacks. One major concern is that biconvex optimization problems are in general difficult to solve. Moreover, much available structure is hidden in the BMI formulation. For example, we know that the BMI for the full-order output feedback HQQ synthesis may be reduced to an LMI. It seems unlikely, however, that order-constrained or decentralized control problems will admit an LMI embedding. It will be interesting to see just how broad a class of BMIs will admit an embedding within the LMI framework.

14.3. Structural aspects of the BMI

14.3

279

Structural aspects of the BMI

Optimization often provides a very convenient framework for constructing computational procedures for a wide range of problems. Applying a standard trick that changes a feasibility problem to an optimization one results in rewriting the BMI as (14.40)

infa

subject to

the BMI has a feasible point if and only if the value of the infimum is negative. At this point we can theoretically apply some general-purpose global optimization technique to (14.40)-(14.41). Our intuition, however, suggests that the geometrical properties of the BMI should be useful in the construction of the computational procedures. Our goal is to gain as much insight into the geometrical and analytic properties of the BMI to the extent that our choice of the algorithm for its solution is judicious and transcends beyond a rather blind application of some global optimization technique. There are at least three issues that have to be addressed in connection with the BMI and the global optimization methods: 1. What are the geometric interpretations of the BMI? 2. What are the specific properties of the global optimization problem that arises from the BMI, and can these properties be used to devise more efficient algorithms for the BMI? 3. Which instances of the BMI can be solved efficiently? Moreover, are there instances for which certain "structural" properties can be established? All of the above issues can be addressed by studying the BMI on its own. We believe, however, that many important structural and computational issues of the BMI can be studied by establishing a connection between the BMI and problems that are more well understood in optimization theory. This section is devoted to such investigations. In the first part, we present the result showing that the BMI can be formulated as a convex maximization problem. The second part establishes the connection between the BMI and two optimization problems over cones, namely, the EFP and the SDCPs. The two approaches discussed in this section, besides providing very nice geometrical insights into the structure of the solution set of the BMI, also suggest computational procedures for its solution, a subject that we shall comment on in section 14.4.

14.3.1

Concave programming approach

It is shown that a BMI has a nonempty solution set if and only if the diameter of a certain convex set is greater than two. The convex set in question is simply the intersection of ellipsoids centered at the origin. In this avenue we end up addressing the first two issues raised above about the relationship between the BMI and the global optimization methods. Specifically, we prove the following theorem. Theorem 14.2 (see [356]). The BMI (14.1) is feasible if and only if the diameter of a suitably constructed convex set C is strictly greater than 2; i.e.,

280

Mesbahi, Safonov, and Papavassilopoulos

The implications of Theorem 14.2 are twofold. First, it opens up a wide range of possibilities for the application of the concave minimization algorithms for the solution of the BMI. At the same time, the concave minimization result indicates, in a rather transparent way, why it is not a good idea to spend research time looking for polynomial time BMI solvers.14 This is due to the fact that concave minimization belongs to a class of computational problems for which the existence of a polynomial time algorithm is highly unlikely (concave minimization is NP-hard [410]). The NP-hardness of the BMI was also explicitly proved in [394]. The rest of the section is devoted to the proof of Theorem 14.2 and some of its implications [356]. We begin by noting that the BMI problem, that of finding the vectors x and y such that 53-. XiyjFij > 0 (14.1), is equivalent to finding the corresponding vectors such that Z^ XiUjFij < 0> which can be written as

where Let p be a real positive number such that

and let C C R"+™ be the convex set

where (14.47) Notice that by (14.45) one has that the matrix Q(z) > 0 for all \\z\\ = 1. It follows that the set C is the intersection of ellipsoids in Rn+m. Suppose that C has a diameter strictly greater than 2. C is the intersection of an infinite number of ellipsoids and thus its maximum diameter is achieved at some wi — (x,y) on the boundary of C and, by symmetry, also at w% = —w\. It thus holds that

where

and

Inequality (14.48) is equivalent to

14 It is generally accepted in computer science that polynomial running time of an algorithm is equivalent to efficiency.

14.3. Structural aspects of the BMI

281

from which it follows that

Thus,

and consequently (—x,y) satisfy (14.1). On the other hand, suppose (—x, y) satisfy (14.1). Without loss of generality we may

uniformly in z, ||z|| = 1. Thus the radius of the set C is strictly greater than (||x||2 + ||y||2) = 1. Since C is Hermitian and centered about the origin, the diameter is precisely twice the radius. Hence, the diameter of C is strictly greater than 2. The p's used in (14.47) may all be the same or they may be chosen to depend on z. All we need is to guarantee that

and thus we can take p to be z-dependent, e.g.,

Finding a different p for each z is a laborious task. A single constant p that will satisfy (14.45) for all z can easily be computed via the matrix inequality

where the ijih entry of the n x m matrix Q is given by

In view of Theorem 14.2, the following statement is obvious: Consider the optimization problem

subject to

Then there exists a feasible pair (x, y) (14.1) if and only if the global optimum of (14.59)(14.60) is strictly greater than 1; note that since all points of the form (x,y) = (x,0) or (x,y) = (0,y) with ||x|| = ||y|| = 1 satisfy (14.60), the optimum of (14.59) is always greater than or equal to 1. Actually, instead of solving the problem (14.59) for its global optimum, all we need is a point (x,y) where J(x,y) > 1. Note that (14.59)-(14.60) is

282

Mesbahi, Safonov, and Papavassilopoulos

a nonlinear programming problem where a convex function is to be maximized subject to an infinite number of quadratic constraints (parameterized by z) all of which are ellipsoids centered at the origin. We note that the BMI can be formulated as follows: Find an n x m matrix N of rank 1 (i.e., N = xyT] such that Vz, with ||z|| = 1,

that is, in the Hilbert space of n x m real matrices find a hyperplane that strictly separates the origin from the set W = {G(z) \ \\z\\ = 1}, and in addition it should hold that the matrix N that defines the perpendicular to this hyperplane is of rank one. In the absence of the restriction rank(N) = I , the problem may be interpreted as the LMI

subject to

a solution exists if and only if the minimal cost e* < 0. If the rank of N is restricted to be less than min{ra,n}, the LMI formulation fails. We notice that the (14.62)-(14.64) formulation of the BMI comes remarkably close to an LMI. At the same time, however, this problem formulation signifies the role that rank-restricted LMIs play in the control synthesis problems in a rather transparent way. We shift our attention now to the cone programming formulations of the BMI.

14.3.2

Cone programming approach

We present an approach for solving the BMI based on its connections with certain problems defined over matrix cones. These problems are, among others, the cone generalization of the LP and the LCP (referred to as the cone-LP and the cone-LCP, respectively). Specifically, we show that solving a given BMI is equivalent to examining the solution set of a suitably constructed cone-LP or cone-LCP. In this direction we end up addressing all of the issues raised above pertaining to the BMIs and global optimization techniques. The main results proved in this section are the following. Theorem 14.3 (see [281]). The BMI is an instance of the EFP. Theorem 14.4 (see [281]). The BMI has a solution if and only if a suitably constructed SDCP with a copositive linear map has a rank-one solution. The emphasis of this section is on the matrix theoretic aspects of the cone problems that the BMI leads to. In this direction, we spent quite an effort to understand the geometry of a certain classes of matrix cones and subsequently to establish their relevant properties, helpful in our understanding of the BMI. The rest of this section is devoted to the proofs of the above two theorems. Few initial steps Let us rewrite the BMI (14.1) as

14.3. Structural aspects of the BMI

283

where

As it becomes apparent by the subsequent developments, it is convenient to assume that m = p, and when necessary, that y^s (1 < j < m) are nonnegative. The first assumption is made to avoid defining inner products between matrix classes of different dimensions. The second assumption is made in section 14.3.2 to facilitate the formulation of the problems in terms of dual cones. In section 14.3.2 we shall drop the nonnegativity assumption on the vector y for the cone-LCP formulation. These assumptions are warranted for the following reasons. First, note that if we define F* = Y^ixi^ij ^ $P (1 ^ J' ^ m )> then ^ZjXiF^ = ^jUjF*. But the last sum is a linear inequality in -FJ's. Thus, as is customary in LP, one can assume that m < p and that yj 's are positive (by an appropriate augmentation). Now we would only need to define F»ji = 0 (1 < i < n;m < j < p) for the assumption m = p to be justified. Recall that Gordan's theorem of alternative [85], [382] relates the solvability of the following two systems of linear inequalities: Given A € R m x n , the system Ax > 0 has a solution if and only if the system yTA = 0, y > 0,y ^ 0, has no solution. This theorem can be generalized for the linear inequalities over matrix cones as follows. Proposition 14.1. Given the symmetric matrices A'^s e Sp (1 < i < n), the system SiLi xiAi > 0 has a solution if and only if the system

has no solution. From Gordan's theorem of alternative over the cone of symmetric positive semidefmite matrices, one concludes that the BMI (14.1) does not have a solution if and only if

Therefore, the BMI (14.1) has a solution if and only if

Now let

and y=diag(y) € Sp. Since (Ff )T = F? = Y^j yjFij (recalling that F^s are symmetric matrices),

therefore

284

Mesbahi, Safonov, and Papavassilopoulos

Combining (14.67) and (14.70) we conclude that (14.1) has a solution if and only if there exists Y > 0, V=diag(y), for some y > 0, such that for all Z > 0, Z =£ 0,

We observe the following: 1. According to (14.68) for all p x p skew-symmetric matrices Z G o , Fi Vec Z = 0 (i = 1, . . . , n). Consequently, if there exists a matrix Y € R pxp such that (14.71) holds, then one can assume that Y is symmetric, since the skew-symmetric part of Y does not contribute to the left-hand side of inequality (14.71): Let Y = Y\ + YI, with Y\ and Y% being the symmetric and skew-symmetric part of Y, respectively. Then for 1 < i < n, Fi Vec Y = F^Vec YI + Vec Y2) = Fi Vec YI. 2. According to (14.69), for all matrices Y e R pxp , Fi Vec Y = Vec Wit for some Wi € 5P. Therefore, if X = (Vec F)(Vec F)T, then M(X) can be represented by

Remark 14.1. Suppose that the vector y is not required to be nonnegative in the above analysis. It is clear that the above steps are still valid with the obvious modifications and that the end result would read as follows: The BMI has a solution if and only if there exists a diagonal matrix Y such that VZ > 0, Z ^ 0, inequality (14.71) holds. We shall use this observation later in section 14.3.2. Inequality (14.71) can be interpreted as requiring M(X) to belong to a certain matrix class. The matrices in this class are symmetric (given that X is symmetric) and have quadratic forms that are positive over the vec form of the nonzero matrices in «Sph. This observation justifies the introduction of the following matrix classes. Denote by PST> the class of p2 x p2 symmetric positive semidefinite matrices, i.e., matrices for the which the quadratic form is nonnegative over the vec form of the pxp matrices (Rpxp),

Let PST>o denote the subset of p2 x p2 symmetric matrices with quadratic forms nonnegative over the vec form of the symmetric pxp matrices («SP), and with the vec form of the skew-symmetric matrices (o ) in their null space; i.e.,

Clearly both PST> (14.74) and PSP0 (14.75) are closed convex cones. Moreover, it can be shown that certain essential features of the PSD cone can be generalized for the class of PST>o matrices, including rank-one decomposition property, self-duality, and the unity rank of the extreme forms [198], [281],

14.3. Structural aspects of the BMI

285

Let C denote the class of symmetric PSD-copositive matrices,

A particular subset of C which be useful in our cone formulations is a subset of matrices in Cs which have the p x p skew-symmetric matrices in their null space, i.e.,

Finally, let B denote the class of symmetric PSD-completely positive matrices

One can establish the following results. Lemma 14.2 (see [281]). The matrix classes 13, C, and CQ are closed convex cones in 5p2. Moreover, B* = C,C* =B,C is solid, and B is pointed. Lemma 14.3 (see [281]). The extreme forms of B are matrices (Vec Z)(Vec Z)T, Z>0. Note that It is now observed that (14.71) states whether a nonlinear combination of matrices 2 2 F/s € Rp xp (1 < i < n) belongs to the interior of the cone of PSD-copositive matrices C. In fact, due the particular form of the linear map M (14.73), M(X) is required to be in Ci := C0 n int C. EFP formulation of the BMI We present the proof of Theorem 14.3 in this section, i.e., the reformulation of the BMI as the EFPs(M), where M is defined by (14.72), and B is the cone of PSD completely positive matrices. Proof 14.2 (see [281]). If the BMI has a solution X, then there exists Y = diag(y), y > 0, X = (Vec Y)(Vec Y)T such that M(X) G int C, and, hence, the EFPB(M) has a solution. Conversely, suppose that the EFPg(M) has a solution X. Then there exists V > 0 such that X = (VecV)(Vec V)T and M(X) € int C; i.e., VZ > 0, Z ^ 0,

Let F = TTYT be such that y is diagonal, T is nonsingular, and TT = T'1. Then VecF = (T
286

Mesbahi, Safonov, and Papavassilopoulos

The last inequality follows from the fact that if Vec W = (T T) Vec Z, Z > 0, and T is nonsingular, then W > 0, since W = TZTT. Therefore the diagonal matrix Y is a solution to the EFP#(M). Now define the vector Remark 14.2. Suppose that the nonnegativity assumption on the vector y is dropped. It then follows that the BMI is also equivalent to finding an extreme form X of the PST>Q cone (14.75) such that M(X) e hit £* = int C, and, in fact, M(X] € Ci := C0 n int C. This observation shall be used in section 14.3.2. The implication of Theorem 14.3 is that the BMI is equivalent to checking whether the image of an extreme form of the matrix cone B under the linear map M (which is constructed from the original data of the BMI) is in the interior of the dual cone B*. This equivalence thus provides a rather simple geometric interpretation of the BMI feasibility problem. An immediate consequence of the EFP formulation is the following characterization of the BMI instances for which a solution exists. Corollary 14.1 (see [281]). The BMI has a solution if the linear map M (14.72) is B-positive. Proof 14.3 (see [281]). If M is B-positive, then every (nonzero) extreme form of B is mapped to the interior of B* = C. Therefore, the EFPg(M), and consequently the BMI, have a solution. Cone-LCP formulation of the BMI In this section we explore the idea of establishing a connection between the BMI and the class of LCPs over matrix cones (cone-LCP). The EFP formulation of the BMI discussed in section 14.3.2 is our main tool in this direction. In particular we shall provide a proof of Theorem 14.4. Our motivation for the cone-LCP approach is twofold. First, the complementarity formulation enables one to address the two important structural issues (2) and (3) mentioned in the opening paragraph of section 14.3. The basic idea is to use the rich theory that has been developed for the LCPs over the last few decades to examine the properties of the BMI. Additionally, we shall show that the cone-LCP that arises from the BMI can be formulated on the PST> cone, rather than on the less (computationally) understood matrix cone B of section 14.3.2. The complementarity problem over the PST) cone also has the advantage of being readily amenable to an interior-point approach [234]. However, one has to note that a "general" cone-LCP is a difficult computational problem, as is the LCP itself. One of the main advantages of the cone-LCP formulation, however, is the ability to recognize efficiently solvable instances. In this section we shall assume for the moment that the cone-LCPs of the form coneLCPpsv(Q, M), with M being a PSX>-copositive (see section 14.1.1), can be solved with "reasonable" efficiency. This assumption is guided by the fact that a usual copositive LCP is a more tractable problem than a general LCP.

14.3. Structural aspects of the BMI

287

The starting point for the cone-LCP approach is the remark made after the proof of Theorem 14.3: The BMI has a solution if and only if the image of an extreme form of the matrix cone PST>Q under the linear map M is in the interior of C or in fact in C\ . As the next proposition states, the cone PST> can substitute the cone PSV^ in the above statement. Proposition 14.2 (see [281]). There exists an extreme form of the matrix cone PSD, X, such that M(X) 6 int C if and only if there exists an extreme form of the matrix cone PSV0, Y, such that M(Y) e int C. Let us denote by p = p(p + l)/2 the dimension of the space of symmetric p x p matrices. Before stating the main result of this section we make the following observation. Lemma 14.4 (see [281]). Let Y be an extreme form of the cone PST>Q. Then there exists a symmetric W e PST>Q such that Tr(YW) = 0 and rank(W) = p - 1. Consider the cone-LCP-p,s£>(Q,M) and let M be defined by equation (14.72): Find 2 X 6 Sp (if it exists) such that

Theorem 14.5 (see [281]). The BMI has a solution if and only if there exists a matrix Q € int (-C} (or Q e —C\), such that the cone-LCPp$T>(Q,M) has a rank one solution. Proof 14.4 (see [281]). Suppose that the BMI has a solution X*; that is, we have that X* = (Vec Y)(Vec Y}T, Y e <SP, M(X*) € int C, and, in fact, M(X*) € Ci. By Lemma 14.4, there exists W e PSVQ, ranW = p-l, such that Tr(WX*) = 0. Without loss of generality, assume that \\W\\ = 1. Let Qa = aW - M(X*}. Note that since M(X) is symmetric (for X € PST>0], Qa is also symmetric. Moreover, Qa(Vec Z) = 0 VZ € o , since both M(X) and W are in the PST>Q (see (14.73)). It suffices to show that there exists an a > 0 such that QQ G int(-C) (or Q € -Ci). Since M(X*) € intC and B is closed, there exists 0 > 0 such that

Therefore, choosing a < /?, we see that int (-C) (in fact Q& e -Ci). Moreover,

a)

< 0. Hence Q&

By construction, X* € ^51), rankX* = 1, and On the other hand suppose that there exists a rank-one matrix X* in the solution set of the cone-LCP (Q, M), with Q e int (-C). Then there exists Z* € R pxp such that

288

Mesbahi, Safonov, and Papavassilopoulos

Since Q & int (-C) and Q 4- M(X*) G PST>, using the inclusion B C PST>, one has the following:

The last inequality follows from the fact that, for all A G B (A ^ 0), Tr(AQ) < 0. Consequently, M(X*) G int C. In view of Proposition 14.2, there exists a rank-one matrix

such that M(y*) € int C. Therefore, the BMI has a solution. The above proof can be modified in an obvious way to conclude that it is only sufficient to take Q G -C\. We shall refer to the special case of the LCP over the PSD cone (14.79)-(14.81), as the semidefinite complementarity problem (SDCP). An immediate consequence of the above theorem is that if a matrix Q G int(— C) cannot be found for which the corresponding SDCP has a solution, then the BMI does not have a solution. Corollary 14.2 (see [281]). The BMI does not have a solution if the SDCP (14.79)(14.81) is not solvable for any Q G int (— C) (or in fact Q G — C\). It is noteworthy that the linear map M in the SDCP formulation, which arises in the context of the BMI, is itself copositive with respect to the matrix cone PST>. Proposition 14.3 (see [281]). The linear map M denned by (14.72) is PSV-copositive. Consequently if we define M*(X] = XlILi FfXFii an^ ^ne implication holds, then VQ G —C\, the cone-LCPpsx>(Q, M} is solvable if it is feasible. Another cone-LCP formulation of the BMI, in addition to the one mentioned above, is to incorporate the problem of finding the matrix Q in Theorem 14.5 in setting up the corresponding cone-LCP. For this purpose it is convenient to associate to the matrix cones PST>, B, and C (subsets of «Sp2), the cones PST>, B, and C (subsets of Rp4), which are obtained by applying the vec operator to these matrix cones, i.e.,

and

It is easy to verify that PST>, B, and C are closed convex cones in Recall that

Therefore, in view of the relation PSD = PSD*, the only matrices in PSV* that are the Vec form of a symmetric matrix are those in PST). Let F = 2?»i ft ®Fi € R p4xp4 . For the linear map M defined by (14.72) and using the property of the Kronecker products, Vec M(X) = FVec X. Combining the above ideas with the result of Theorem 14.5, one readily obtains the following corollary.

14.4. Computational aspects of the BMI

289

Corollary 14.3 (see [281]). Let

and

Then the BMI has a solution if and only if the homogeneous cone-LCP^ xpj^(Q, M) has a solution of the form

where — Q € int C, and X has rank one. We note that if Q € — C and X e PSV, then Q + M(X) is automatically symmetric and therefore if Vec (Q + Af (X*)) € PSP*, then Q + M(X*) € PSX>. The above corollary reduces the BMI feasibility problem to the problem of examining the solution set of a certain cone-LCP. This can be a "tractable" problem if the solution set is finite or if the linear map M enjoys certain "additional" properties. Since there are various results in the complementarity theory that pertain to the cardinality of the solution set of a cone-LCP [208] , classification of the efficiently solvable instances of the BMI can be based on these results as well.

14.4

Computational aspects of the BMI

This part of the chapter is devoted to a brief overview of the computational methods for solving the BMI, those that are directly related with the presentation of sections 14.2 and 14.3. We shall first provide a few words on the approaches that are not touched upon in this section. Motivated by observing that the function

is convex in x for a fixed y and vice versa, Goh, Safonov, and Papavassilopoulos [168] proposed a global optimization algorithm based on the branch and bound strategy for solving the BMI; in this approach, the bilinearity of the function (14.83) is successively employed in the bounding part of each iteration. The dissertation of Liu [249] discusses many interesting aspects of the parallel implementation of BMI solvers that are based on the branch and bound strategy (see also [250]). In [45] a global optimization technique based on the generalized Benders decomposition, which is used in bilinear and biconvex programming [138], [155], [412], is proposed. Finally we should mention the alternating LMI method [214], but this later class of algorithms is not guaranteed to find a feasible point of the BMI (14.1), even if one exists. Back to the methods that are linked directly to the aspects of the BMI investigated in this chapter, we present a brief overview of each. Starting with the optimization problem (14.40)-(14.41) obtained trivially from the BMI feasibility problem, one can proceed to devise a computational method based on the following strategy. Initially pick an arbitrary XQ and yo and then choose ao > 0 such

290

Mesbahi, Safonov, and Papavassilopoulos

that (14.41) is satisfied (thus we obtain a feasible point for the optimization problem); proceed by trying to reduce o^ at each step without leaving the feasible region. In the spirit of the interior-point methods for solving the LMIs and SDP problems, Goh et al. [169] proposed using the logarithmic barrier functional to ensure that the iterations stay inside the feasible region. A barrier method for the BMI

(2) Choose some (XQ, yo) and CXQ such that (3) Until

The local minimization step (3a) is initiated from (ajk,£fc>2/fc)- The convergence of the above algorithm is contingent upon a wise choice of the parameter 0. Note that as Hk —* oo, the first term of the objective functional in (3a) approaches that of minimizing the parameter a, whereas its second term ensures that this minimization is performed without leaving the feasible region of (14.40)-(14.41). Under some mild conditions, the barrier method described above is guaranteed to converge to a local minimum of (14.40)-(14.41). The choice of the parameter 0 is guided by the methods for solving LMIs and it is closely related to the self-concordant barrier parameter for the cone of PSD matrices [296] . We observe that our choice of the initial points XQ and yo can be influenced by the results of the more conservative approaches, such as the alternating LMI approach or the D-K iteration. In any event the algorithm above, which is based on the BMI formulation of the n/km synthesis problem along with a methodology borrowed directly from convex programming, is guaranteed to provide improvements over that of the existing methods [164]. Barrier methods are not the only variant of the interior-point methods that can be used for obtaining local solutions to BMIs, although they are probably among the most well understood. Other interior-point solution methods for locally solving the BMI is presented in the dissertation of Goh [164], and in particular one that corresponds to the method of centers in the spirit of [63] and [206]. Computational examples are also provided in [164]. Motivated by the reformulation of the BMI as a concave minimization problem presented in section 14.3.1, Safonov and Papavassilopoulos [356] proposed the following conceptual algorithm for finding a feasible point of (14.1). The approach is based on the optimization problem (14.59)-(14.60), which is obtained directly from Theorem 14.2. In order to solve (14.1), we can solve a sequence of problems of the type (14.59)-(14.60), each one having a finite number of inequalities. At the beginning of step fc, assume that the points z^\ . . . , z^k~1^ have been generated from an arbitrary initial guess z^ with \\zW\\ = 1. Then the kth step of the algorithm follows: Solve for a globally maximizing pair (or, y) in (14.84)

14.5. Conclusion

291

subject to

If the global minimum J* = 1, stop; the problem (14.59) is infeasible. If J* ' > 1, let the solution be ( z ^ y ) and solve

where

Note that the maximal value in (14.86) is the maximal eigenvalue Amax(J9r). Take z^ € Rfc to be a maximizing z in (14.86); i.e., z^> is any unit norm eigenvector of the matrix (14.87) associated with Amax(#). If Amax(#) < 0, we stop and the pair (z( fc) ,7/ fc) ) provides a solution of BMI (14.1); otherwise, we choose pW > A max (H) and go to step fc + 1. It can be shown that this process will stop in a finite number of steps if (14.1) has a solution; otherwise, (14.1) is infeasible. Note that solving (14.84) for the global maximum may be quite a time-consuming problem, although there exist several algorithms for solving nonconvex maximization problems of this type [200]. Finally we observe that it is not necessary to solve (14.84) for the global maximum, but we can stop as long as a pair (x^, y^) with \\x^ \\2+\\y^ \\2 > 1 has been generated. This may be detrimental to the speed of convergence of the algorithm but avoids spending a lot of time in finding the global maximum of (14.84). If one chooses to do this, it might be advisable now and then to solve (14.84) globally. Last, we mentioned the solution method based on the SDCP approach, which is based on examining the solution set of a given SDCP. We note that often an SDCP has only a finite number of solutions; in fact, there are many classes of SDCPs for which one can guarantee even the uniqueness of the solution. This is the main advantage of the SDCP approach, since examining the feasible region defined by an LMI for the existence of a rank-one matrix is itself a difficult computation problem (the rank-minimization problem under LMI constraints is a powerful framework for studying many robust control synthesis problems as well [124], [282]). The results of section 14.3.2 provide an explicit expression for the linear maps that appear in the SDCP formulation of the BMI based on the matrices F'^s (i = 1, . . . , n; j = 1, . . . , m). The complementarity theory can be used to classify instances of the BMI that reduce to a convex optimization problem over the PSD cone. For example, as a result of Corollary 14.1, 5-positivity guarantees the existence of a feasible point for the BMI, thus reducing the BMI to characterizing a single matrix. The existence of the interior-point methods for monotone SDCPs also provide local and in some cases globally convergent methods for certain classes of the BMIs.

14.5

Conclusion

Rooted in some of best traditions in control theory, the BMI provides a framework to address the most important issues facing the field of robust control. The BMI formulation

292

Mesbahi, Safonov, and Papavassilopoulos

offers direct interpretation of the robust control synthesis problem, exemplified by a search for the parameters of the controller on one hand, and the multiplier on the other— and at the same time, it has led to elegant mathematical investigations in optimization theory, some of which were presented in this chapter. Last, and most important, the BMI formulation promises efficient, reliable, and automated procedures for the synthesis of nonconservative robust controllers, improving upon those obtained by employing the existing methods, including the D-K and the M-K iterations.

Part VI

Applications

This page intentionally left blank

Chapter 15

Linear Controller Design for the NEC Laser Bonder via Linear Matrix Inequality Optimization Junichiro Oishi and Venkataramanan Balakrishnan 15.1

Introduction

Over the past few decades, rapid strides have been made in control theory, resulting in the solution of several important controller design problems: LQR, LQG, HQQ, to name a few [388, 107, 175]. Many of these controllers have been successfully implemented in industrial applications. However, a disadvantage with most of these methods is that they are optimal in only a narrow sense, and actual engineering specifications (which are usually stated as competing constraints) must be translated or reinterpreted so as to fit into the narrow framework of these methods. In parallel with the theoretical developments in systems and control, there have been significant advances in optimization theory and algorithms, as well as an almost exponential growth in computing power, so that numerical controller design methods, especially those based on convex optimization, have become increasingly relevant [59, 65, 62, 61, 284, 303, 30]. Such numerical methods enjoy the advantage that several commonly encountered design requirements can be specified directly in a natural manner and that the design interaction between various competing performance specifications can be readily studied. In this chapter, we describe the application of one such computer-aided design method for controlling the NEC Laser Bonder. This method combines the Youla parametrization of the set of achievable stable closed-loop maps with convex optimization based on linear matrix inequalities (LMIs) to numerically design optimal linear time-invariant (LTI) controllers under multiple design specifications. The significance of Youla parametrization for computer-aided control system design has long been recognized; see, for example, [59, 61]. In these references, the Youla parametrization is used to reformulate the problem of LTI controller design for LTI 295

296

Oishi and Balakrishnan

systems as a quadratic programming problem, which is then solved numerically. The approach that we present in this chapter is very close in spirit to the one in [59, 61]. The main difference is that here the LTI controller design problem is reformulated instead as an LMI optimization problem; we describe this reformulation in section 15.2. We apply this technique, in section 15.3, to design an LTI controller for the NEC Laser Bonder.

15.2

Controller design using the Youla parametrization and LMIs

A standard framework for controller design is shown in Figure 15.1. P is the model of the plant, i.e., the system to be controlled. K is the controller that implements the control strategy for improving the performance of the system, y is the signal that the controller has access to, and u is the output of the controller that drives the plant, w and z represent inputs and outputs of interest. Note that z can include components of the control input u, so that specifications on the control input such as bounds on the control effort can be handled. In other words, the map Hzw from w to z contains all the input-output maps of interest. The controller design problem is the design of an LTI controller K such that the closed-loop map Hzw, with the controller K in place, satisfies some design specifications. Often an "optimal" controller is sought, one that yields an optimal performance measure subject to the design specifications.

Figure 15.1: A standard controller design framework. The Youla parameterization (see [411] for details) yields a convex (in particular affine) parameterization, in terms of the Youla parameter Q, of the set of achievable stable closed-loop maps Hzw from w to z for the system in Figure 15.1. This fact is of great significance, as the numerical search for the "optimal" Youla parameter Q—one that minimizes an objective that is a convex function of Hzw subject to convex constraints on Hzw—is a convex optimization problem. In this section, we show how by restricting Q to lie in a finite-dimensional subspace, we can reformulate several important practical controller design specifications as LMI constraints on Q. Some of the design specifications that we will consider are constraints on the response z for some reference input w, asymptotic tracking constraints, bounds on the H^ and HOO norms of Hzw, etc. Thus several important LTI controller design problems can be solved numerically using optimization over LMIs.

15.2. Controller design using the Youla parametrization and LMIs

297

15.2.1 Youla parametrization We begin with a brief review of the Youla parametrization of the set of achievable stable closed-loop maps for LTI plants with LTI controllers. Let the state equations describing the plant P be

Let the open-loop plant transfer matrix in Figure 15.1, denoted P, be partitioned as

Then, with K(X) denoting the transfer function of the LTI controller, the transfer function from w to z is

The set of achievable, stable closed-loop maps from w to z is given by is stable and satisfies The set of the controllers K that stabilize the system is in general not a convex set. Thus optimizing over H using the description (15.2), with K as the (infinite-dimensional) optimization variable, is a difficult numerical problem. However, the theory of Youla parameterization enables us to give a convex parameterization of 7i. It turns out (see [61] and the references therein for details) that the set H can be also written as

where TI, T2, and T3 are fixed, stable transfer matrices that can be computed as follows. Let Knom and Lnom be real matrices of appropriate sizes such that A — BuKnom and A — LnomCy are stable (i.e., have all their eigenvalues in the open unit disk). Then TI, T2, and T3 are given by

where

The most important observation about this reparametrization of Ji is that it is affine in the infinite-dimensional parameter Q; it is therefore a convex parameterization of the set of achievable stable closed-loop maps from w to z. (The parameter Q is also referred to as the Youla parameter.) This fact has an important ramification—it is possible now

298

Oishi and Balakrishnan

to use convex optimization techniques to find an optimal parameter Qopt and therefore an optimal controller Kopt. The general procedure for designing controllers using the Youla parameterization proceeds as follows. Let 0o> i> • • • > 4>m be (not necessarily differentiate) convex functional on the closed-loop map that represent performance measures. These performance measures may be norms (typically H2 or Hoonorms), certain time-domain quantities (step response overshoot, steady-state errors), etc. Then the problem

is a convex optimization problem (with an infinite-dimensional optimization variable Q). This problem corresponds to minimizing a measure of performance of the closed-loop system subject to other performance constraints. In practice, problem (15.3) is solved by searching for Q over a finite-dimensional subspace. Typically, Q is restricted to lie in the set

where QI,--.,QN are fixed, stable transfer matrices. For convenience, we let G = [&i • • • 9N], and Q(&) = diQi + ---- h 0NQN. This enables us to solve problem (15.3) "approximately" by solving the following problem with a finite number of scalar optimization variables:

The transfer matrices Qi and their number, N, should be so chosen that the optimal parameter Q can be approximated with sufficient accuracy. With 0opt denoting the optimizer of problem (15.5), we can compute optimal controller Kopt as follows:

where

and (AQ, BQ, CQ, DQ) is a realization of <5(©opt)The book [61] describes perhaps the best-known work—the computer-aided control system package QDES —that combines Youla parametrization and convex optimization for LTI controller design (see also [59]). QDES consists of a front-end compiler that translates a control design problem into problem (15.5); this optimization problem is solved numerically by further reducing it to a quadratic program. In order to reformulate problem (15.5) into a quadratic program, some approximation and sampling of frequency domain constraints is necessary. However, the positive and bounded real lemmas and

15.2. Controller design using the Youla parametrization and LMIs

299

their variations (see, for example, [13, 432, 225]) can be used to exactly formulate a number of frequency domain constraints into LMI constraints. This motivates the use of LMI techniques to solve problem (15.5), which we develop in the rest of the chapter.

15.2.2

Controller design specifications as LMI constraints

We now show how some typical constraints on Hzw can be reformulated as LMI constraints. Of course, since quadratic programs can be directly translated into LMI optimization problems [64], every controller design constraint listed in [59] can be reformulated as an LMI constraint. In addition, we will show how certain frequency domain constraints can be naturally reformulated as LMI constraints. We will restrict the Youla parameter Q to lie in the finite-dimensional subspace Qp.n) given in (15.4). The corresponding set of achievable closed-loop maps is

Constraints on the response to specific inputs Suppose that a certain exogenous output (i.e., a component of z] is required to lie within specified upper and lower bounds for a given reference input signal (i.e., a component of w). Let HZWtTef be the corresponding transfer function. Then the response constraint is simply an affine constraint on the Hzw,Tef and therefore on G (see [59, 61]).

LMIs for asymptotic tracking constraints Suppose that a certain exogenous output is required to asymptotically track a specific input signal. Let -H^™, track be the corresponding transfer function. Then the tracking constraint is simply an equality constraint on Hzw track(l) and therefore on 6 (see [59, 61]).

LMIs for HZ norms objectives The H2 norm of a transfer function H of a stable linear system is defined as

Prom Parseval's theorem, the H2 norm can also be calculated as the square root of the sum of the squares of the impulse response coefficients. As the H2 norm measures the root mean square (RMS) value of the output of the system when it is driven by white noise with unit power spectral density, constraints on ||/fztt;||2 provide a way of incorporating noise sensitivity constraints on the design. Suppose the transfer function H has a state-space realization (A, B,C, D). Then \\H\\2 can be calculated as follows. Define the controllability Gramian W as the solution to the Lyapunov equation Then the H2 norm is Now consider the constraint that the H2 norm of the transfer function from some (or all) of the components of w to some (or all) of the components of z be less than some 7. Let this transfer function be denoted Hzw. Since Hzw € Wfi n , the transfer function Hzw

300

Oishi and Balakrishnan

has a state-space realization (AZW,BZW,CZW(&), DZW(Q)), where Azw and Bzw are real matrices, and Czw and Dzw are affine functions of 0. Then the constraint ||/rztl,||2 < 7 is equivalent to

This is easily rewritten as the LMI constraint

LMIs for HOQ norms objectives The HQO norm of a transfer function H of a. stable linear system is defined as

where crmax(M) denotes the maximum singular value (spectral norm) of a matrix M. The HQO norm measures the L2- or energy-gain of the system; therefore, constraints on ||#ziu||oo provide a way of incorporating noise amplification constraints on the design. In addition, HQQ constraints provide a way of incorporating robustness requirements into the design; see, for example, [98, 106, 175]. Suppose the transfer function H has a state-space realization (A, B,C,D). Then the bounded real lemma (see [64] and the references therein) enables the reformulation of the constraint ||#||(x> < 7 as the following LMI constraint on P:

Now, consider the constraint that the HQQ norm of the transfer function Hzw from some (or all) of the components of w to some (or all) of the components of z be less than some 7. Using a state-space realization (Azw, Bzw, CZW(Q), Dzw(&)} for Hzw, where Azw and Bzw are real matrices, and Czw and Dzw are affine functions of 0, the constraint ll-^Tztulloo < 7 is equivalent to the existence of P = PT such that

holds. This is easily rewritten as the LMI constraint

The implication of section 15.2.2 is that the problem of LTI controller design with several common design specifications can be posed as an optimization problem (15.5) with LMI constraints (and objective) and solved numerically using standard software tools such as [126, 153]. We demonstrate the application of this approach toward controller design for the NEC Laser Bonder.

15.3. Design of an LTI controller for the NEC Laser Bonder

15.3

301

Design of an LTI controller for the NEC Laser Bonder

A laser bonder is a soldering machine that connects the leads of an integrated circuit (1C) chip with pads on the 1C board. The bonding head consists of a small tool that presses a lead of the 1C on the corresponding pad. Lasers then melt the solder of the pad thereby completing the connection. Once a lead is successfully connected, the tool moves up to its initial position. Then the positioning stage that holds the 1C board moves above the next lead, and the process is repeated in order for the new connection to be made. A schematic model of the bonder head is shown in Figure 15.2. The bonder head consists of a tool, a linear voice coil motor (VCM), which drives the tool up and down, and a position sensor (with a linear scale), which measures the vertical position of the tool. Our objective is the design of a controller that achieves the up-down positioning with high speed and precision.

Figure 15.2: Schematic model of the NEC Laser Bonder.

15.3.1

Modeling of NEC Laser Bonder

The equation governing the motion of the bonder head is

where F is the thrust force of the linear motor, ks is the spring constant, and h is the tool position. The thrust force is given by F(i) = kfi(t), where kf is the thrust force constant and i is the current through the motor. This current, in turn, satisfies

where Vm is the voltage input to the motor, R is the resistance of the motor, L is its inductance, and ke is its velocity-current coefficient. The values of the various constants are shown in Table 15.1.

Oishi and Balakrishnan

302

Table 15.1: Model parameter values for the NEC Laser Bonder. Mass M Resistance R Inductance L Spring constant ks Thrust force constant kf Velocity-current coefficient ke

1.91 Kg 5.96ft 2.25 mH 422 N/m 12.8 N/A 12.8 V-s/m

Equations (15.7) and (15.8) can be combined to yield the following state equation:

The controller to be designed has access to the position reference input /iref and the output of the tool position sensor /ise, which is simply the actual tool position with some additive sensor noise nse. The control input, i.e, the output of the controller, is a motor voltage Vi n ; this signal, corrupted by an additive noise nact, drives the motor. In addition, there is a disturbance fd that acts on the motor. For convenience, we scale this disturbance by 1/M and define Td = fd/M. In our framework, the position reference input /iref, the scaled motor disturbance Td, the position sensor noise nse, and the actuator noise nact form the exogenous input w; the computed voltage Vjn is the control input u; the actual position h and the actual motor input voltage Vm form the regulated variables z; and the reference input href and the sensed tool position hse form the measured variable y (see Figure 15.3). From (15.9), we then have

Substituting the parameter values shown in Table 15.1, we have the following state equations that describe a linear model for the Laser Bonder:

15.3. Design of an LTI controller for the NEC Laser Bonder

303

Figure 15.3: Block diagram of the NEC Laser Bonder. Discretizing the system with a sampling frequency of 1KHz, we obtain

where

and

15.3.2

Design specifications

There are three main design specifications: high speed, high precision, and high reliability. More precisely, we have the following design constraints. SP-1 Tracking a given reference input. In the absence of disturbances and noises, for the reference input href, shown in solid lines in Figure 15.4, the position h is required to lie within the limits shown in dashed lines. This constraint ensures that the design satisfies a number of specifications: - Tracking delay. The delay in tracking the reference input must not exceed 3ms. — Settling interval. The response is required to lie within ±3/L/m from 20ms onward. - Overshoot. The overshoot must not exceed 3/Ltm. SP-2 Asymptotic tracking. In the absence of disturbances and noises, the step response from the reference input hret to the position h must eventually settle at one.

Oishi and Balakrishnan

304

Figure 15.4: Constraints on the response to a given reference input. The solid line shows the reference input and the dashed lines the limits within which the response is required to lie. SP-3 Bounds on input signal Vm. In the absence of disturbances and noises, the voltage input V to motor is limited to ±10V, reflecting the limitations of the transformer driving the motor. The objective follows: OBJ Minimize the HOC norm of transfer matrix from reference input (/iref) and disturbance (TO) to position (h) and motor voltage (Vm). This serves to mitigate the effect of disturbances at the signals of interest.

15.3.3

Setting up the LMI problem

The open-loop system is stable, and Knom and Lnom can be taken to be zero in order to generate TI, T%, and T3. However, since the open-loop system is very lightly damped (i.e., the poles of the open-loop system were very close to but less than one), we choose the feedback gain Knom and observer gain Lnom, using standard LQG control, to increase the damping on the system (i.e., the LQG controller served to decrease the magnitude of the poles of the closed-loop system). This step, strictly speaking, is unnecessary; however, our experience has shown that the optimization problems are much better conditioned with this additional step. The set Q was chosen as

This set can be easily rewritten in the form of (15.4), with 9 = [B\ • • • 0$]. From the argument in section 15.2.2, the specifications SP-1 and SP-2 each lead to two linear constraints on 6; in the notation of problem (15.5), we obtain four LMI

15.3. Design of an LTI controller for the NEC Laser Bonder

305

constraints 0 i , . . . , 4. The specification SP-3 is an equality constraint on 9, which can be represented in the form of two LMI constraints ^5 and 06The objective OBJ is incorporated in problem (15.5) by defining 4>7 as the LMI constraint (15.6) and setting 00 = 7 2 -

15.3.4

Numerical results

We used the Matlab LMI Control Toolbox [153] to solve problem (15.5) to obtain 0opt and thus Kopt. Figure 15.5 shows the nominal (i.e., noise- and disturbance-free) response of the closed-loop system with the optimal controller in place. The response lies between the specified upper bound and lower bounds. In particular, the tracking delay, settling interval, and overshoot constraints are satisfied. Moreover, the response converges to the steady-state reference input value of 0.001m.

Figure 15.5: The response to the reference input, shown in a solid line. The dashed lines show the constraints. Figure 15.6 shows the motor voltage Vm corresponding to the reference input. Note for all time the magnitude is under 10V, as required. The optimal value of the objective OBJ subject to the constraints SP-1 through SP-3, i.e., the smallest HOC norm from [/iref rj] to [h Vm] subject to the various design constraints, is 4.4 x 105. For purposes of comparison, we attempted to find the smallest HQO norm achievable with LQG controllers that satisfied the design constraints and obtained a value of 5.8 x 105. Thus the H^ norm with the best LQG controller that we could design is about 20% larger in this example; moreover, there is no systematic way of incorporating multiple design constraints in the LQG design procedure (we had to resort to trial and error). It is possible that a more experienced LQG control designer could match the LMI design; however, this makes the point that with the assistance of the computer-aided procedure presented herein, even inexperienced users can quickly create useful designs.

Oishi and Balakrishnan

306

Figure 15.6: The measured voltage of the motor to the reference input, shown in a solid line. The dashed lines show the motor voltage constraints. Next, we study numerically the interaction between two competing constraints, viz., the HOQ norm of transfer matrix from [hre{ Td] to [h Vm], and the tracking delay. Clearly, increasing the allowable tracking delay, i.e., relaxing the tracking delay constraint, offers the potential of reducing the H^ norm. We study the trade-off between these quantities for three values for the settling interval. The trade-off curves are shown in Figure 15.7. These results show that in the case of a settling interval of ±3/^m, the HQQ norm does not get smaller when the tracking delay constraint is relaxed beyond 4ms, suggesting that other design constraints are then binding. Similar comments can be made in the case of setting intervals of ±10//m and ±30^um. As expected, for every value of the tracking delay, the HQQ norm decreases when the settling interval is allowed to be larger. Besides quantifying the interaction between competing constraints, the trade-off curves serve to illustrate the limits of performance of LTI controllers; every point (d, 7) above a trade-off curve is an achievable design specification, meaning that there is an LTI controller that simultaneously satisfies the maximum actuator input and settling interval constraints and results in a closed-loop response that has a tracking delay that does not exceed cf, with an HQQ norm from [hre{ Td] to [h Vm] that does not exceed 7. Every point below the curve represents an (almost certainly)15 unachievable design specification.

15.4

Conclusions

We have presented a numerical control design method for designing optimal LTI controllers for LTI plants with multiple design constraints. The techniques combine convex optimization over LMIs with the Youla parametrization of the set of achievable stable closed-loop maps. The advantages of this approach are that engineering specifications 15 Recall that we are restricting Q to lie in the finite-dimensional subspace Qfin, hence the parenthetical qualification.

15.4. Conclusions

307

Figure 15.7: Trade-off between the tracking delay and HOC norm of the transfer function from [href rd] to [h Vm}. can be directly incorporated in the design, without calling on the experience or intuition of the designer to "tune" the design parameters. The particular advantage with the use of LMIs is that several important frequency domain constraints can be exactly reformulated as LMI constraints using the bounded real lemma. In addition to designing controllers, it is possible to study the limits of performance of linear controllers, that is, to numerically determine the best achievable performance using linear controllers. We can also study the interaction between competing design constraints by numerically determining trade-off curves. The task of implementing the controllers presented herein on the "real" NEC Laser Bonder system remains. Another issue that is of interest is the incorporation of robustness requirements on the design so as to account for parameter and operating condition variations.

This page intentionally left blank

Chapter 16

Multiobjective Robust Control Toolbox for Linear-MatrixInequality-Based Control Stephane Dussy 16.1

Introduction

When applying linear matrix inequality (LMI) methods to control problems, see, e.g., [64, 108, 315, 148], the LMI conditions that arise are often complicated, making a direct application of LMI solvers (LMI Control Toolbox [153], SDPsol [426], Imitool [126]) cumbersome. The user needs to understand a wide field of theoretical knowledge before starting to compute a control law. Moreover, this knowledge shall include both a strong background in control theory and a deep understanding of the recent advances in optimization, namely, the polynomial time interior-point algorithms. In view of this theoretical difficulty inherent to the use of LMI techniques, the main purpose of the Multiobjective Robust Control Toolbox (MRC Toolbox) [116, 113] is to provide a set of functions that allow us to synthesize a robust control law under various constraints without expert knowledge on robust control theory. These user friendly functions just require the practitioner to enter the model state-space representation and the physical specifications. The toolbox functions are based on recent advances in control theory, in particular, those pertaining to multiobjective robust gain scheduling [115, 114, 111]. We consider a wide class of nonlinear uncertain systems for which we can derive a linear fractional representation (LFR). From this standard representation, we describe a formulation of the control problem based on the gain-scheduled framework introduced by Packard [35, 312]. This formulation includes various specifications such as a norm bound on the command input and some saturations on specific outputs. The requirements in terms of performance will be specified via an a-stability condition, i.e., exponential stability with a decay rate a, or via an L2-gain condition. The public domain MRC Toolbox [113] is a set of Matlab functions built on top of Imitool [126] and SP [402]. For the output-feedback case, these M-functions use a cone complementarity algorithm [128] adapted to robust control. They may be complemented with recent LMI-based tools, see, e.g., the toolbox developed by Beran [43] for linear 309

310

Dussy

time-invariant (LTI) systems. The theoretical framework of the methodology is defined in section 16.2. The main functions of the MRC Toolbox are described in section 16.3. Finally, numerical examples are given in section 16.4 with the control of an inverted pendulum [114] and a nonlinear benchmark called RTAC [115].

Notation For a given integer vector r = [7*1, . . . , rn], r» € N, i = 1, . . . , n, we define the sets

16.2 16.2.1

Theoretical framework Specifications

To describe the methodology, let us consider a parameter-dependent nonlinear system of the form

where x is the state; u is the command input; y is the measured output; z is the controlled output; Q, i = 1,...,JV, are some outputs of interest; and w are the disturbances. The vector £ contains time- varying parameters, which are known to belong to some given bounded set V. Also, we assume that A(-,-), B u (-,-), C y (-,-), Bw(-,-), Dyw(-,-), and Dzu(-,-} are rational functions of their arguments. We seek a dynamic, possibly nonlinear, output-feedback controller, with input y and output u such that a number of specifications are satisfied. To define our specifications, we consider a given polytope P of initial conditions. We also consider the set of admissible disturbances, chosen to be of the form

where wmax is a given scalar. To P, W and the uncertainty set V, we associate the family X(XQ, Wmax) of trajectories of the above uncertain system in response to a given XQ € P. Our design constraints are as follows: 5.1 The system is well posed along every trajectory in A^zoj^max); that is, for every t > 0, the system matrices A(x,£(t)), Bu(x, £(£))» etc. are well defined. 5.2 Every trajectory in X(XQ,$) decays to zero at rate a, that is, limt-»oo eatx(t) = 0. 5.3 For every trajectory in X(x$, w ma x)> the command input u satisfies W > 0, ||w(<)ll2 < S.4 For every trajectory in X(xo,wmgLX), the outputs ?i = EiX, i = 1,...,./V, are bounded; that is, ||?t(0ll2 < Ci.max for every t > 0.

16.2. Theoretical framework

311

S.5 For every trajectory in X(Q,wma.x), the closed-loop system exhibits good disturbance rejection; i.e., a maximum L2-gain 7 from z to it; is guaranteed. This is equivalent to impose a bound 7 such that

We recall that the above specifications should hold robustly, that is, for every £(t) G T>.

16.2.2

LFR of the open-loop system

The first step is to derive an LFR for the system. In other words, we seek a set of equations

that is formally equivalent to the nonlinear equations (16.1). We will refer to [114, 129], where the framework for the construction and the normalization of an LFR for such a system is described. For this example, we considered the case of bounded uncertainties. The LFR, and therefore the associated LMI-based conditions, are different when the uncertainties are expressed through sector conditions. In this case, the complete methodology is described in [112]. The next sections describe the basic idea of this methodology for nonlinear systems with bounded uncertainties.

16.2.3

LFR of the controller

We take our measurement-scheduled controller to be in the LFR format

where p, q are fictitious measured inputs and outputs of the controller. Note the crucial assumption here: The controller is scheduled with respect to the measured element in the nonlinear block A, namely, A m (y) (rank i/m), and it is robust with respect to the uncertain part, namely, A u (y) (rank vu}. As described in the first chapter of the book, our theoretical framework refers to the mixed parameter-scheduled/robust problem, also called robust measurement scheduling. The closed-loop system is shown in Figure 16.1.

16.2.4

LMI conditions

We seek quadratic Lyapunov functions ensuring specifications S.I—5 for the closed-loop system. As seen in [111, 129], it is possible to formulate the synthesis conditions to ensure these specifications in the form of a set of LMI conditions associated with a nonconvex constraint. For the specific problem described in the previous sections, these conditions are as follows.

Dussy

312

Figure 16.1: LFR of the closed-loop system. Let N be a matrix whose columns span the nullspace of [Cy Dyp Dyw]T. The synthesis conditions are

16.2.

Theoretical framework

313

This quite complex formulation is easy to solve thanks to efficient LMI solvers. The above LMIs are interpreted as follows: An input norm bound is specified with (16.8), (16.9), and (16.10). Bounds on specific outputs are ensured with (16.8), (16.9), and (16.11). The remaining matrix conditions specify the stability and the guaranteed L2gain of the system. Every condition above is an LMI, except for the nonconvex equations (16.7) and (16.9). When (16.4) holds, enforcing this condition can be done by imposing TrSuTu + w tn i *^e additional LMI constraint

In fact, the problem belongs to the class of "cone-complementarity problems," which are based on LMI constraints of the form

where V, W, Z are matrix variables ( V, W being symmetric and of same size) and F( V, W, Z) is a matrix- valued, affine function, with F symmetric. The corresponding cone-complementarity problem is The heuristic proposed in [128], which is based on solving a sequence of LMI problems, can be used to solve the above problem. This heuristic is guaranteed to converge to a stationary point. Algorithm 7i. 1. Find VQ, WQ, ZQ that satisfy the LMI constraints (16.13). If the problem is infeasible, stop. Otherwise, set k = 1. 2. Find Vfc, Wk, Zk that solve the LMI problem

3. If the objective Tr(Vfc_i Wk + Wk-\Vk) has reached a stationary point, stop. Otherwise, set k = k + 1 and go to (2).

16.2.5

Main motivations for the toolbox

As described in the first chapter of the book, there are several key features in the proposed approach. • Efficient numerical solution. LMI optimization problems can be solved very efficiently using recent interior-point methods (the global optimum is found in modest computing time). This brings a numerical solution to problems when no analytical or closed-form solution is known. • A systematic method. The approach is very systematic and can be applied to a very wide variety of nonlinear control design problems. This includes (and is not reduced to) systems whose state-space representations are rational functions of the state vector.

314

Dussy

• Multicriteria problems. The approach is particularly well suited to problems where several (possibly conflicting) specifications are to be satisfied by the closed-loop system. This is made possible by use of a common Lyapunov function proving that every specification holds. The main problem remains the complexity of the LMI conditions. Therefore, it appears necessary that this part of control synthesis remains "transparent" for the user. In view of the above features, the approach opens the way to CAD tools and user friendly interfaces for the analysis and the control of nonlinear uncertain systems. All of these key features allow us to consider the LMI formulation as one of the most promising approaches in order to reduce the gap between the engineering and research communities. During the last decade, control engineers have been frustrated by the theoretical advances that could appear to them more and more complicated and cumbersome to apply to industrial problems. In fact, from an engineering point of view, every robust control problem can be summarized by the description of the system (including its variation) and the design goals (through performance specifications). The main purpose of the MRC Toolbox is to build some user friendly functions that only require the plant description and the physical specifications before synthesizing the appropriate control law. Then, these "engineerlike requirements" are translated by the toolbox into some advanced LMI-based conditions. To do so, the underlying theoretical background of these Matlab functions attempts to make the best trade-off between calculability and complexity of the synthesis conditions. Quadratic Lyapunov functions represent for the moment the best solution to get some LMI-based conditions (therefore, easy to solve numerically) for a wide class of specifications and systems (systems with bounded uncertainties [114], Lur'e system [112], discrete-time systems [117]).

16.3

Toolbox contents

16.3.1

General structure of the toolbox

The main driver, named mrct, can be called directly. The toolbox also contains the functions mrstsf, mrstof, mr!2sf, and mr!2of that solve specific problems, mrct invokes these M-functions and also some other functions that remain hidden. The latter are referenced with block letters MR, e.g., MRspec that asks interactively the specifications. The data of the system, i.e., the matrices A, Bu, Bp, Bw, Cq, Dqu, Dqp, Dqw, Cy, Dyu, Dyp, Dyw, Cz, Dzu, Dzp, and Dzw from LFR (16.2), can be packed and unpacked in a convenient way with the routines mrpck and mrunpck.

16.3.2

Description of the driver mrct

The use of mrct is straightforward and just requires following on-line instructions. This function invokes MRspec that asks the user to enter the specifications and store them in a data file. According to these specifications, the driver follows the theoretical steps of LMI-based robust synthesis described in section 16.2.4: 1. Build the LMI conditions, each one reflecting a specification. 2. Solve the LMI problem. In output feedback, it requires the resolution of the cone complementarity algorithm. 3. Reconstruct the controller thanks to the solutions of the LMI conditions. This can be achieved by solving another LMI problem (see [111] for more details).

315

16.3. Toolbox contents Table 16.1: The main specific functions. Problem description

State-feedback Output-feedback

Bounded uncertainties Real bounded uncertainties Sector conditions Bounded uncertainties Sector conditions

Function a-stab. L2-gain mrstsf

mr!2sf

mrstof

mr!2of

Type

'b' 'br' 's' 'b' 's'

16.3.3 Description of the main specific functions The computation of a controller can be performed directly through the functions mrstsf, mrstof, mr!2sf, and mr!2of, where the specifications are entered as arguments of corresponding M-functions. Each function can solve different kinds of problems, depending on what the user specifies in the argument type of the function. Table 16.1 summarizes the various problems that can be handled with the MRC Toolbox. The specifications of the control problem appear through the arguments of the functions as described in the following example »[K,opt,info] = mr!2of(type,G,alpha,gamma,normw,rho,cz,czb,v,nnm,r,lambda) where K contains the desired controller matrices packed with mrkpck,

opt is the optimized value of the minimization problem, and 1. type specifies the kind of problem to be handled: 'b' for bounded uncertainties (choice by default), 's' for sector conditions, 'br' for real bounded uncertainties. 2. G contains the matrix representation of the system A, Bu, Bp, Bw, Cq, Dqu, Dqp, Dqw, Cy, Dyu, Dyp, Dyw, Cz, Dzu, Dzp, Dzw. This single matrix can be packed/unpacked with the functions mrpck and mrunpck. 3. alpha specifies the required decay rate of the closed-loop responses. 4. gamma specifies the imposed maximum Li2-gain. gamma = ' opt' for a minimization problem. 5. normw specifies the norm bound on the disturbance energy. 6. rho represents the norm bound on the command input, rho = ' opt' for a minimization problem. 7. czb represents the bounds on outputs of interest stored rowwise in cz. 8. v contains the polytope of initial conditions stored columnwise. 9. nnm is the number of unmeasured uncertainties in A = diag(A m , A u ), so that nnm = rank(A u ).

Dussy

316

Figure 16.2: Inverted pendulum on a cart. 10. r contains the structure of A stacked in the row vector. It contains the way some of the uncertain parameters are repeated in A. 11. lambda represents the norm bound on the variables P, Q, 5", and T defined in section 16.2.4. Remark. The controller order for the output-feedback case is a priori equal to the order of the plant. The synthesis of a low-order controller can be achieved by using some well-known LMI-based algorithms [176, 414] that are not included in the toolbox.

16.4

Numerical examples

16.4.1 An inverted pendulum benchmark The inverted pendulum is built on a cart moving in translation without constraint (see Figure 16.2). Our goal is to stabilize the inverted pendulum in the vertical position, with an uncertain but constant mass m. ( is the position of the cart, 0 is the angle position of the pendulum with the vertical axis, and u is the control input acting on the cart translation. We seek to control the angle 0 without limitation on the cart's horizontal position. We assume that the mass m is constant and unknown but bounded: m € [mo — Am , mo + Am], where mo, Am are given. The parameter vector here is taken to be f = (m - m0)/Am, so that f € T> = [-1 , 1]. Our specifications are described by S.l-4, where a, umax and xmax = [Omax Omax]T are given. The polytope of admissible initial conditions is determined by two given scalars

A detailed description of the methodology for this example is described in [114]. The necessary steps for the synthesis of the desired controllers, and in particular the construction of the LFR, are described in the following Matlab file:

16.4. Numerical examples

317

7. LFR of the system [0 ; -0.87]; A = [0 1 ; 12.43 0]; Bu -0.03 0.05 0.04 -0.261.07 -0.07 0.11 1.13 -0.26] Bp= [ 0 0 0 0 0 0 0 0 0 C q = [ 0 0 0 0 0 0 0 0 1 ; 0 5 . 7 4 7.46 0 1 0 -5.52 0.92 0]' Dqu - [0 -0.40 -0.52 1 0 0 0.38 -0.00 0] 2.52 0 Dqp = [0 0 0 0 0 0 0 5 2 -0.12 0.05 0.. -0.01 0.02 0.25 -0.12 0.49 -0.03 5 2 -0.02 0.03 0.02 -0.16 0.64 -0.04 0.07 0.68 -0 16 0 0; 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 1.9 0 0 0 0 0 0.23 0 0.01 0.21 0 .16 0.11 -0.48 0.03 -0.05 -0.500.1

0 ; -0.00 -0.00 -0. 0.08 -0.00 0.00 0.00 0 0 0 0 0 0 0 0]; Cy = [ 1 0 ] ; Dyu = 0; Dyp = zeros(l,9); 7. fill matrices (z,w) with zeros [Bw,Dqw,Dyw,Cz,Dzu,Dzp,Dzw] - mrzerozw(A,Bu,Bp,Cy) ; G = mrpckCA.Bu.Bp.Bw.Cq.Dqu.Dqp.Dqw.Cy.Dyu.Dyp.Dyw.Cz.Dzu.Dzp.Dzw) ; 0 0

7. Specifications 7. type = 'b'; Alph =0.15 Rho = 50 Cz - [1 0 ; 0 1] ; Czb - [13]; v = [0.3 0.3 ; 0.3 -0.3]

nnm = 4 r = [4131]; Lamb - 2000;

7. bounded uncertainties 7. required decay-rate 7. input norm bound '/. bounded outputs 7. associated norm bounds 7. it means I [1 0]X| < 1 and I [0 1]X|<3 7. polytope of initial conditions 7. it means 1X1(0) l<0.3 and 1X2(0) I <0.3 7. last 4 uncertainties are unmeasured 7. structure of Delta 7. norm bound on the variables

7t Controller Synthesis y. [K,opt,info] = mrstof(type,G,Alph,Rho,Cz,Czb,v,nnm,r,Lamb); [Ab,Bbp,Bby,Cbq, Dbqp,Dbqy,Cbu]=mrkunpck(K)

The above code gives the solution

The plots of Figure 16.3 show the envelopes (solid line) of the closed-loop responses in 0(t) and u(t) of the inverted pendulum, with m varying within its two extremal values, i.e., from -50% to -1-50% of its nominal value mo. They also show (dotted line) the responses of the system for the nominal value m = mo-

318

Dussy

Figure 16.3: Envelopes of 9(i) in rad and u(i) in Newton for the mixed robust/gainscheduled controller.

Figure 16.4: Rotational actuator to control a translational oscillator.

16.4.2 The rotational/translational proof mass actuator The RTAC is a nonlinear benchmark proposed by Bupp, Bernstein, and Coppola [75]. Consider the unbalanced oscillator described in Figure 16.4. It is built with a cart (mass M) fixed to the wall by a linear spring (stiffness k) and constrained to have onedimensional travel along the Z-axis. An embedded proof mass actuator (mass m and moment of inertia /) is attached to the center of mass of the cart and can rotate in the horizontal plane. The radius of rotation is e. The cart is submitted to a disturbance F, and a control torque N is applied to the proof mass. We seek a state-feedback controller such that S.I and S.3—5 are ensured for the closed-loop system. We also want to minimize the upper bound on command input um&x. For more details, the complete methodology is described in [115]. The solution has been computed with the following Matlab file:

16.4.

Numerical examples

319

7. LFR of the system A = [0 1 0 0 ; -1.04 0 0 0 ; 0 0 0 1 ; 0.19 0 0 0 ] ; Bu = [0 ; -0.19 ; 0 ; 1.04]; Bp = [0 0 0 0 ; 0.01 -0.01 -0.01 0.05 ; 0 0 0 0 ; 0.01 0.01 0.00 -0.01]; Bw = [0 ; 1.04 ; 0 ; -0.19]; Cq = [0.84 0 0 0 ; 1 . 23 0 0 0 ; 0 0 0 0 ; 0 0 0 1.00]; Dqp = [-0.01 0 0.01 -0.04 ;0 0.02 0.02 -0.06 ; 0 0 0 0 ; 0 0 0 0 ] ; Dqu = [0.16 ; 0.23 ;1.00 ; 0]; Dqw » [-0.84 ; -1.23 ; 0 ; 0] ; Cy - [1 0 0 0; 0 0 1 0]; Dyu = [0 ; 0] ; Dyp = zeros (2, 4); Dyw = [0 ; 0] ; Cz = [0.32 0 0 0; 0 0 0.32 0; 0 0 0 0 ] ; Dzu = [0 ; 0 ; 1] ; Dzp = zeros(3,4); Dzw = [0 ; 0 ; 0] ; G = mrpck(A,Bu,Bp,Bw,Cq,Dqu,Dqp,Dqw,Cy,Dyu,Dyp,Dyw,Cz,Dzu,Dzp,Dzw) ;

'/. Specifications 7. -------------type = 'br' ; Alph = 0; Gam = 30; Rho = ' opt ' ; cz ™ [1 0 0 0 ; 0010; 0001]; czb = [1.28 0.5 0.5]; v = [0.05 ; 0 ; 0 ; 0] ; r = [31]; Lamb = 5000;

7. real bounded uncertaintie 7. required decay-rate 7. L2-gain 7. Minimize input norm bo 7. bounded output 7. and associated norm bound 7. it means |X1|<1.28, |X3|<0. 7. and |X4|<0. 7. polytope of initial condition 7. it means |X1(0)I<0.05 7. structure of Delt 7. norm bound on the variable

7. Controller Synthesis y. -------------------[K, opt, info] = mr!2sf (type, G, Alph, Gam, Rho, cz, czb, v,r,

The string "opt" for Rho means that this variable will be minimized; that is, we seek to minimize the command input bound. We obtained the state-feedback controller

with the guaranteed bound -/Vmax = 0.4586. The plots of Figure 16.5 show the closed-loop responses in Z(t) and N(t) of the RTAC for nonzero initial conditions. These plots point out one of the main drawbacks of this approach based on quadratic Lyapunov functions. Depending on the problem, the LMI conditions may be very conservative and lead to some constraints more stringent than we expected them to be (this is the case here for the bound on the command input). Therefore, to be optimal, this approach requires some guidelines to relax the design parameters [115]. Then it generally provides very helpful trade-off curves for numerous problems of control design. Note that the LMI approach is particularly well suited for designing trade-off curves, since it provides a solution in reasonable time, it guarantees that all the criteria are robustly respected, and it allows us to minimize a parameter and thus to provide an optimal solution. It does not require any longer to run numerous simulations (corresponding to the different possible variations of the uncertain plant) in order to check that the specifications hold. Figure 16.6 shows trade-off curves decay rate/control effort with 7 = oo (right-hand side) and performance/control effort with a = 0 (left-hand side). Performance is measured by 7, the upper bound on the closed-loop Li2-gain, and control effort is measured

Dussy

320

Figure 16.5: Closed-loop system responses with w = 0, zero initial conditions except for Z(0) = 0.01, Z in meter, N in Newton.

Figure 16.6: Trade-off plots command input/decay rate/L2-gain for the state- (solid line) and output-feedback (dashed line) controller. On the right side, 7 = oo, and on the left side, a = 0. Nm&x is in Newton. by the upper bound on peak command input ATmax. In each plot, two curves are shown: one for state feedback (solid line) and the other for output feedback (dashed line).

16.5

Concluding remarks

The MRC Toolbox provides a user friendly interface for the robust control of a wide class of uncertain nonlinear systems under constraints. It is based on recent results on LMI-based robust measurement scheduling and bounded multiobjective control. This toolbox provides an easy and fast controller synthesis under various specifications. It handles nonconvex problems via an efficient heuristic. The main idea that gave birth to this software was to make a compromise between complex theoretical foundations and a friendly use for an engineer without expert knowledge of the recent results in control theory. The future updates of this software shall include the implementation of an algorithm for the synthesis of low-order controllers [176, 414] and the construction of new functions for discrete-time systems [117]. These updates shall also include a graphical interface for the description of the system (via block diagrams and Simulink-like icons), its variation, and the design goals.

Chapter 17

Multiobjective Control for Robot Telemanipulators Jean Pierre Folcher and Claude Andriot 17.1

Introduction

Teleoperation is the branch of robotics dedicated to manipulation in inaccessible environments. Such environments can be distant or hostile (space or nuclear) or can be at a different power scale (microsurgery or electronic assembly); see [80, 146]. A standard teleoperator system consists of a slave robot tracking a master robot via a bilateral link (the controller and the actuators). The master robot is manipulated by a human operator and the slave robot interacts with the environment (the load). Ensuring good tracking performance and maintaining system stability are the control objectives usually expressed in terms of network-based properties. It has been shown in [82] that passiv ity theory is useful to analyze the teleoperator stability for a wide range of operators and environments. A stability criterion was derived in [81] by verifying constraints on the scattering or admittance matrices of the teleoperator. A passivity approach for the design of a bilateral controller has been presented in [82]. A related method, based on HOQ optimization, is examined in [16, 435]. The use of convex optimization for a passive design is shown in [205]. A key point in the design of a bilateral controller is to ensure good trade-offs between the conflicting objectives of stability and performance. We propose a multiobjective design approach expressed in terms of linear matrix inequalities (LMIs). This formulation appears well suited for many control problems and especially for multiobjective ones; see [123, 365]. The present approach is close to the one discussed in [123] and generalizes the results obtained for linear time-invariant (LTI) systems subject to "structured" dissipative perturbations. The central idea is that a number of design specifications, such as robust (H2 norm) performance, robust input peak bound, etc., can be translated into LMI constraints associated with a nonconvex constraint of the form ST = /, where 5, T are two (block-diagonal) matrix variables. Our problem can thus be solved using LMI optimization associated with an efficient cone complementarity linearization algorithm described in [128]. This makes our multiobjective robust design control problem amenable to an efficient numerical solution. Finally, the multiobjective approach is illustrated with a prototype telerobotic sys321

322

Folcher and Andriot

tern: the RD500 of the Commissariat a 1'Energie Atomique (CEA). The force control problem of a single-joint slave manipulator is considered. A proportional integer (PI) force controller derived in [299] is also designed in this chapter to compare performances. The organization of the chapter is the following. In section 17.2, main ideas of telemanipulation, background on network theory, and network-based telemanipulation control objectives are given. The multiobjective design approach for the control of linear systems subject to dissipative perturbation is developed in section 17.3. The proposed approach is applied in section 17.4 to design the force control law of a single-joint slave manipulator.

17.2

Control of robot telemanipulators

17.2.1 Teleoperation systems A general model of a teleoperation system is shown in Figure 17.1.

Figure 17.1: Teleoperation system. A teleoperation system includes the human operator, the environment, and the teleoperator. This last one is composed of the master manipulator, the slave manipulator, and the bilateral link. In the case of the mechanical bilateral link, the intrinsic reversibility property implies that the action of the environment on the slave manipulator is transmitted to the master manipulator. The operator feels interactions between the slave manipulator and the environment: this is kinesthetic feedback. To obtain good performance, manipulators need to be mechanically reversible (small inertia and friction terms), and the bilateral link has to be rigid. To increase distance between the master and the slave space, electrical teleoperators have been developed. They include two manipulators and a bilateral link, which consists of electrical actuators and a controller. Control laws implemented on the calculator achieve the kinesthetic feedback. In the sequel we study the design of control laws for electric teleoperators.

17.2.2

Background on network theory

Telemanipulation systems can be seen as subsystems interacting dynamically. These composite systems are intrinsically structured, and network theory appears to be a con-

17.2. Control of robot telemanipulators

323

venient tool to describe teleoperator properties [408]. Widely known as an analysis tool for electrical systems [13], its use for mechanical and robotics systems is restricted. In this section we review basic concepts of network theory extended to robotics systems analysis. Networks. This representation is based on a particular choice of physical variables: Force / and flow v are chosen such that quantity P = fv represents a power. Mechanical basic systems can be described by elementary LTI 2-port networks shown in Figure 17.2.

Figure 17.2: 2-port network. fit f0 are input and output force vectors, respectively; Vi, v0 are the input and output speed vectors. Examples of basic networks are given: - An inertial and viscous friction in a series; see Figure 17.3(a). - A stiffness connected in parallel with viscous friction; see Figure 17.3(b). - A geared drive with a gear ratio n; see Figure 17.3(c).

Figure 17.3: Basic networks. These networks, respectively, are governed by the following equations:

Folcher and Andriot

324

Y(s), Z(s), and H(s) correspond to a particular choice of input and output variables. They are called, respectively, the admittance matrix, the impedance matrix, and the hybrid matrix. Introducing the input wave vector a = [ai, a0]T = [fi — Vi, f0 — VO]T and the reflected wave vector 6 = [6j, b0]T = [fi + Vi, f0 + v0]T the network equation becomes

where 5(s) is called the scattering matrix. Matrices Y(s), Z(s), H(s), and S(s) are interrelated: For instance, Z(s) = ^(s)"1 (when Y(s) is invertible) and S(s) = (Y(s) —

Passivity. A multiport network is said to be passive if and only if for any set of the motion v and effort / satisfying the network function, the inequality

is ensured; see [339], E(T] is the energy provided to the network, and we note that a passive network can only dissipate energy. Passive LTI networks possess positive real admittance matrix Y(s) or bounded real scattering matrix S(s). That is, for Re s > 0, entries of Y(s), S(s) are analytical and Y(s)* + Y(s) > 0, / - S(s)*S(s) > 0; see [13].

17.2.3

Toward an ideal scaled teleoperator

We consider that geometric similarity is enforced for the telemanipulation system. The control problem is broken down into independent problems for each coupled joint: Their behavior is modeled as networks; see Figure 17.4.

Figure 17.4: Single-joint teleoperation network model. are the networks related to, respectively, operator, master manipulator, bilateral link, slave manipulator, and the environment. /m, vm are the effort and the velocity of the master joint; fa, va are the effort and the velocity of the slave joint, vim, vis are the velocities seen by the bilateral link and /j m , fia are auxiliary forces (when the manipulator has no local control law it corresponds to the effector force). Teleoperator transparency. The goal is to obtain constant velocities and effort ratio between the master joint and the slave joint for all the frequencies: This is transparency.

17.2. Control of robot telemanipulators

325

The associated ideal network will be described by the hybrid representation

where n/, nv are the force ratio and the velocity ratio. For micromanipulations n/, nv > 1, and for large scale n/, nv < 1 are fixed. When n/ = nv = 1 the operator is virtually connected to the network Na. Teleoperation system stability. The teleoperation system represented in the network framework in Figure 17.5 has to be stable. Modeling the environment (network Ne) and the operator (network N0) accurately in a simple way is difficult. In [81], Colgate proposed a stability test based on passivity that we recall.

Figure 17.5: Networks interconnection. We suppose that N0 modeling the operator and Ne modeling the environment are passive networks; that is, VT > 0,

The interconnected system will be stable if and only if there exists a scalar (3 > 0 such that

where Y(s) is the admittance matrix of the network Nt. Structured network synthesis.

Transparency of the telemanipulator is ensured when

- the bilateral link is rigid; - the master and the slave joints are backdrivable. To check the first specification, the network -/Vj, which represents the bilateral controller, is designed as a stiffness K\ connected in parallel with viscous friction BI\ see Figure 17.3(b). This network can be interpreted as a proportional integral speed control law. The second requirement is achieved by considering force control strategies (see [197, 417, 418]), which ensure transparent properties for networks JVm, Ne. In the next section we present a multiobjective robust control design approach to satisfy teleoperation control specifications and take into account additional time-domain properties useful for coping with noise measurement and actuators saturation.

Folcher and Andriot

326

17.3

Multiobjective robust control

17.3.1

Problem statement

We consider the system defined as follows:

where x G Rn is the state, u G R"u is the command input, w G Rnt" is the exogeneous input, z G Rn* is the regulated output, and y G Rn» is the measured output. We assume that Dqw, Dzp, Dzw are zero. This system can be seen as a linear time-invariant system connected to m operators A* from C <7m]TEach Ai belongs to different class of operators (defined in [368] where we refer the reader for more details). Ai may be a linear or nonlinear system, time varying or not, possibly memoryless. In any case, we assume that A* is {Ui, Vi, Wi}-dissipative; i.e., for every pi, ^ G C^e with pi = Aj(^) we have

where Ui < 0, Vi, Wi > 0 are given real matrices. For instance, {— /, 0, <72/}-dissipativity property expresses that operator A^ has an £-2 gain less than or equal to a. Note that the composite operator A defined as

is also {5f7, SV, 5W}-dissipative with

and

17.3.2

Control objectives

Our purpose is to find a strictly proper LTI controller of the form

17.3. Multiobjective robust control

327

where x € R™ is the controller state, and A, B, C are constant matrices of appropriate size, such that the closed-loop system (with x = \XT XT}T) defined by

with A {St/, 5V, 5W}-dissipative, achieves simultaneously a number of desired properties.

Figure 17.6: Closed-loop system. We partition vectors w and z as w = [w{ w%^r, z — [zf 2^]T, and we define our control objectives as follows. O.I Robust stability. For every A {U, V, W}-dissipative, the closed-loop system (17.7) with w = 0 is stable. O.2 Robust performance. For every A {£/, V, W}-dissipative, for every t > 0, the responses z\ of the closed-loop system (17.7) to zero initial conditions, and a unit impulse in the zth coordinate of w\, satisfy

where 7 > 0 is a prespecified number. (This control objective can be interpreted as a "robust H2 norm" condition on the closed-loop system.) O.3 Robust output peak bound. For every A {£/, V, W}-dissipative, the responses z\ of the closed-loop system (17.7) to zero initial conditions, and a unit impulse in the ith coordinate of 1^2, the other inputs being zero, must satisfy

where zmax is a given positive number.

328

Folcher and Andriot

O.4 Robust input peak bound. For every A {£/, V, W}-dissipative, for the trajectories defined in O.3, the resulting command input satisfies, for some prespecified number

17.3.3 Conditions for robust synthesis Analysis results are based on quadratic functions V(x) = xTPx, and they lead to matrix inequalities involving the closed-loop system (17.7) state-space description. This approach is similar to the one developed in [123], where the system included a timevarying norm bounded gain that we can interpret as pointwise quadratic constraints on fictitious signals p and q. Here the constraints on A involve integral quadratic constraints on signals p and q, and the matrix inequalities we obtain are closely related to the ones obtained in [123]. However, we cannot interpret V as a quadratic Lyapunov function in the conventional sense because V decreases to zero but not necessarily monotically. We assume that there exist symmetric, positive-definite matrices P € R 2nx2n and an 5 given by (17.5) such that with w = 0,

Integrating both sides of this relation from 0 to T, we obtain

As A is {SU, SV, 5W}-dissipative, the last right-hand-side term is negative and the above relation implies that the ellipsoid £p -^ = {x \ xTPx < x(0)TPx(0)} is invariant. Thus, the system (17.7) satisfies control objective O.I. Condition (17.9) holds if and only if

In addition, condition (17.11) ensures that the output energy in z\ in response to an impulse input is bounded above by x(0)TPx(0). To be precise, a trajectory such as considered in O.2 satisfies

Condition (17.11) together with

guarantees that control objectives O.1-O.2 hold. Condition (17.11) also implies that for every v > 0, the ellipsoid Sp v = {x \ xTPx < v} is invariant [64]. That is, for every trajectory with zero input w and af(0) € £pv,

17.3. Multiobjective robust control

329

we have x(i] G £p v for every t > 0. Applying this to a trajectory in response to initial condition x(0) = BW2ti, we see that the condition x(0) <E £p;l/,

implies that z(t) G £p v for every t > 0. If, in addition,

then the bound (17.8) holds. We conclude that the condition

together with (17.11) and (17.13), imply that control objectives O.I and O.3 hold. Likewise, the input peak bound can be enforced by considering u as an output u = Cux, where Cu = [0 C}. We obtain that the additional condition

with (17.11) and (17.13), enforces O.I and O.4. The above results are summarized in the following theorem. Theorem 17.1. If there exists a symmetric, positive-definite matrix P G R 2nx2n such that conditions (17.11), (17.12), (17.13), (17.14), and (17.15) hold, then control objectives O.1-O.4 hold for the closed-loop system (17.7). We present a useful lemma needed to establish inequality conditions depending on data matrices (A, Bp,...) of system (17.1). Lemma 17.1 (completion [312]). Let P, Q G R n x n be two positive-definite matrices. There exists a positive symmetric matrix P G R 2nx2n such that the upper-left n x n bloclc of P is P, and that of P~l is Q if and only if Q > 0, P > Q~l. For every Q > 0, P > Q"1, the set of matrices P and their inverse Q satisfying the conditions above are parameterized by

where M G R nxn is an arbitrary invertible matrix and N = (I - QP}M~T. Without loss of generality, we may assume that P > Q~l (this restricts our search to full-order controllers only; see [148]). We can define K, Z as follows:

and we obtain that inequality (17.12) is equivalent to the existence of a symmetric matrix X such that

330

Folcher and Andriot

Similarly, condition (17.13) is equivalent to the existence of a scalar v such that

Condition (17.14) is equivalent to the existence of scalars r, v such that

Finally, the control input peak condition (17.15) can be written as

or

17.3.4 Reconstruction of controller parameters We seek A,B,C such that inequality (17.11) holds using the approach in [359], based on the following lemma. Lemma 17.2. Let the matrices G € R 2nx2n , G E R2"*2", and f € R 2nx2n ,

where R G R nxn is inversiWe. When conditions

hold, then G < 0. Using the Schur complement, inequality (17.11) is ensured if

and

17.3. Multiobjective robust control

331

hold. Without loss of generality, we choose R = M with M2 = P - Q l (the choice of M has no effect on the controller's transfer function, since it corresponds to a change of coordinates in the controller's states, x —> MX). Condition Gn < 0 of Lemma 17.2 becomes

Note that inequality (17.22) is ensured by the (2,2) block of the left-hand-side term of inequality (17.24). Condition G\\ < 0 of Lemma 17.2 can be written as

Condition G\2 = -G\\R~T is verified for P, Q,y, Z, S given by fixing A as

We obtain the following result. Theorem 17.2. If there exist P, Q, y, Z,S, T,r, v that satisfy constraints (17.18), (17.19), (17.20), (17.21), (17.24), and (17.25), then_ there exists a controller of the form (17.6) with A given by (17.26), B = M~1Z and C = -YQ~lM~l with M2 = P - Q~l such that the closed-loop system (17.7) satisfies O.1-O.4.

17.3.5

Checking the synthesis conditions

The conditions of Theorem 17.2 are not convex, due to the coupling conditions ST = / and rv = 1. However, they are simple LMI conditions for fixed S and T. We may use the cone complementarity linearization algorithm proposed in [128]. Consider the LMIs

332

Folcher and Andriot

The following heuristic is guaranteed to converge to a stationary point of the objective (see [128]). 1. Find P, Q, y, Z, S0, TO, TO, ^0 that satisfy constraints (17.18), (17.19), (17.20), (17.21), (17.24), and (17.25), where the nonconvex equalities ST = I, TV = 1 are replaced by the LMIs (17.27). If the problem is infeasible, stop. Otherwise, set k = 1. 2. Find Sjt, Tk r/t, Vk that solve the LMI problem of minimizing Tr(7);_15 + Sk-iT) + i/k-iT + Tk-\v subject to the LMI constraints of step 1. 3. If the objective Tr(Tk-iS-\-Sk-iT) + i'k-iT+Tk-ii' has reached a stationary point, stop. Otherwise, set k = k + 1 and go to step 2.

17.4

Force control of manipulator RD500

Ensuring the backdrivability is the main objective of the control of the slave manipulator RD500 of CEA. This control problem is addressed by reducing the manipulator impedance (inertia and friction terms). For instance, the behavior of a rigid joint modeled by an inertia M is described by

where ve, /„/, /e, fu are the manipulator speed, the friction effort, the exogeneous effort, and the control effort, respectively. When this joint is controlled by the force feedback law where g > 0 is a constant gain, then the closed-loop system behavior is governed by

The resultant inertia and friction terms of the closed-loop system are divided by the factor g+1. This means that force feedback law is relevant to impose the backdrivability property for a rigid joint. This force control law feature also holds for a flexible joint: this is the case that we encounter in the sequel of this section.

17.4.1

Single-joint slave manipulator model

The single-joint slave teleoperator that we consider is shown in Figure 17.7. This manipulator is composed of a motor, a geared drive, a torque sensor, and a joint load. The motor is a current-controlled torque source associated with inertia Mm and viscous friction Bm. It delivers an effort /m, and its angular speed is vm. The geared drive with ratio n has a rotational compliance and friction, modeled by inertia Md, stiffness Kd, and viscous friction Bd- A torque sensor delivers the measured effort fa and introduces a vibrational mode, modeled by stiffness Ka. The joint load is defined by inertia Mj and viscous friction Bj. fe and ve are the effort applied by the environment and the angular speed of the joint, respectively.16 A convenient representation of this model based on network theory is shown in Figure 17.8. 16

The effort fe and the flow ve, respectively, equal the bilateral force fa and the bilateral speed — v9 defined in the single-joint teleoperation network model; see Figure 17.4.

17.4. Force control of manipulator RD500

333

Figure 17.7: Single-joint robot mechanical model.

Figure 17.8: Single-joint robot network model.

This network is built with basic subnetwork (an inertia and viscous friction in a series, a stiffness, a geared drive) governed by elementary relations given in section 17.2.2. By combining these relations, we obtain the state-space equation

where the states are the motor speed v m , the geared drive speed t^, the joint load speed i>e, the effort in the geared drive fd, and the effort in the torque sensor fs. Signals /e, da, fm, respectively, are the effort applied by the environment, a motor speed disturbance, and the torque delivered by the motor. The sensed signal fy = fa 4- ns represents the measure of the torque sensor effort fa corrupted by additive noise n a . The reference

Folcher and Andriot

334

effort17 fr and the sensed signal fy are processed by the force controller Kf to produce the motor torque signal fm such that

The ratio ^ allows us to scale the controller K/ output as an effort in the joint coordinate. The resultant closed-loop system forms a one-port network (with force variable fe and flow ve) that we called the closed-loop network in the sequel. Parameter values for the RD500 of CEA are given in Table 17.1. Table 17.1: RD500 parameter values. Symbols Motor inertia Mm Motor friction Bm Geared drive ratio n Geared drive stiffness Kd Geared drive inertia Md Geared drive friction Bd Sensor drive stiffness K3 Joint load inertia MI Joint load friction BI

17.4.2

Values 3.1 1(T3 9.4 1(T3 275.3947 64077 2.152 0

12146 10.98 25.798

Units Kgm2 Nms rad~ l Nmrad~l Kgm2 Nmsrad~l Nmrad~l Kgm2 Nms rad~ l

Controller design

The slave manipulator is in contact with a wide range of of environments represented in Figure 17.8 by the network Ne. The first approach is to model the environment as a stiffness. That is, we obtain the additional relation

Two extremal configurations are considered: - When the slave manipulator is free (no contact with the environment), then fe is negligible and the value of Ke is equal to zero. - When the slave manipulator is locked, then ve is negligible and the value of Ke is large. The free configuration is associated with tasks where the bilateral (position) control only is active. The locked configuration corresponds to a high impedance environment: The use of Thevenin's theorem shows that the system gain is high. For a given force controller, instability behavior may occur in this configuration. We "incorporate" the environment stiffness in the system by defining

and also a new interaction port with effort /* and flow ve for this "augmented" closed-loop network. The following control specifications are required for the blocked configuration: 17 The reference effort fr equals the bilateral force /ja (generated by the bilateral controller), which is defined in the single-joint teleoperation network model; see Figure 17.4.

17.4. Force control of manipulator RD500

335

(i) The "augmented" robot in closed loop with the force controller is stable when connected to any passive environment. (ii) The step response has no overshoot (no damage to the environment), and the settling time is minimized. (iii) The magnitude of the control input is reasonable (the maximum torque delivered by the motor is limited to 5 Nm). (iv) In the absence of tracking input, the control input is insensitive to the sensor noise (motor lifetime). Specification (i) is ensured when the "augmented" closed-loop network is strictly passive; see [82]. By using Tellegen's theorem given in [14], it is easy to prove that the closed-loop network is strictly passive (because the stiffness Ke is lossless). Therefore, the robot in closed loop with the force controller is stable when connected to any passive environment for the free configuration. Specification (i) is related to the stability problem for a linear time-invariant system (the "augmented" closed-loop network) connected to a dissipative operator /* = A(v e ) defined in section 17.3 with A a {0, — /, 0}-dissipative operator. This closed-loop system is represented in Figure 17.9. The control input signal u is the scaled motor torque in the joint coordinate; that is, u = nfm and the measured output is fr — fs- Note the similarity between Figure 17.6 in section 17.3 and Figure 17.9. Therefore, we can impose specifications (i)-(v) using the multiobjective design approach shown in section 17.3.

Figure 17.9: Closed-loop system. Specification (i) is directly control objective O.I. The tracking specification (ii) is expressed as a regulation problem (we consider fr = 0) with control objective O.3. The regulated output is z-z = fa, and the perturbation input is w% = da. Low values of the design parameter zmax ensure good tracking performance. Actuator saturation

Folcher and Andriot

336

specification (iii) is directly taken into account by control objective O.4 with u = fm. The parameter umax reflects for the design the bound of the control input peak. Specification (iv) is reflected in control objective O.2 with z\ = u and w\ = na. The design parameter 7 adjusts the level of the control noise power induced by the sensor noise na. The open-loop system G is described by (17.1) in section 17.3, where

stands for

with the state x = [vm, i^, ve, /
17.4.3

• In the sequel of this chapter the value of

Results

We choose design specifications: The LMIs were solved using the code SP [402] and its Matlab interface LMITOOL [126]. After 31 iterations using the heuristic presented in section 17.3.5, we obtain a stationary point. We compute state-space representation for the multiobjective controller K^ using formulas in Theorem 17.2. The associated controller has been reduced to obtain a 3states controller with negligible degradation of the open-loop frequency response in the bandwidth. That is, we obtain the following K¥(S) transfer function:

Proportional integral controller. For performance comparison purposes we consider a PI force controller A T 7 ( s ) derivedfrom[299]

17.4.

Force control of manipulator RD500

337

which guarantees specification (i) for one compliance joint with the assumption Bj = 0. This system is related to the single-joint robot represented in Figure 17.8, where the torque sensor stiffness, the joint load inertia MI, and the viscous friction BI are incorporated in the environment. According to Tellegen's theorem, it can be shown that this PI controller ensures specification (i) for the the slave manipulator RD500. The Bode diagram of the controller transfer functions is plotted in Figure 17.10 in the solid line for K1(s) and in the dashed line for

Figure 17.10: Kf(s)

and K^(s) Bode plots.

Figure 17.11 shows the Bode plots of GKf(s) in the solid line and GK^(s) in the dashed line. The multiobjective force controller K¥ enjoys a 13.83 dB gain margin, a 78.69 degree phase margin, and a 1.71 rad/s bandwidth, which are reasonable values. In Figure 17.12, we plot the Nyquist diagram of the closed-loop admittance matrices Ye(s) = v e ( s ) / f e ( s ) to check its strict positivity. That is, for the two controllers, closedloop networks are passive and specification (i) is ensured. Tracking performance ensured by force controllers K¥ and K?I are evaluated in Figures 17.13(a) and 17.13(b), respectively. Closed-loop systems responses to a 1000 Nm reference step fr are shown. The noise-free-sensed outputs fs are plotted in the solid line. The normalized controller outputs in the joint coordinate nfm are plotted in the dashed line. For the two controllers, no overshoot appears in the time response; that is, specification (ii) is ensured. Specification (iii) in the joint coordinate leads to a 1377 N m saturation level, and the normalized controller output peaks do not exceed this saturation level. We compare the transient response plotted in Figures 17.13(a) and 17.13(b). Note that damped oscillations appear in 17.13(b) and that the transient is slower. This fact is confirmed by the values of the 5% settling time, which are, respectively, 2.9 s and 1.4 s.

Folcher and Andriot

338

Figure 17.11: GKf(s} and GK^(s) Bode plots.

Figure 17.12: Closed-loop admittance matrice Ye(s) Nyquist plots.

17.5

Conclusion and perspectives

In this chapter, we present the design of a controller for robot telemanipulators based on a multiobjective approach involving LMI optimization problems. Telemanipulation control objectives (stability and tracking performance) are usually expressed in terms of network-based properties. The proposed approach, which considers the control of an LTI

17.5. Conclusion and perspectives

339

Figure 17.13: Time responses to a 1000 Nm reference step fr system subject to dissipative perturbation, can be related to the traditional networkbased approach founded on passivity theory. Furthermore, the approach incorporates many practical specifications as H2 norm bounds or time-domain bounds (output and command input peak) in the design. In our opinion, this approach directly uses teleoperation engineering expertise and allows us to analyze trade-offs between the different performances and stability objectives. Finally, we have applied the multiobjective approach to the force controller design of the slave teleoperator RD500 for a single joint. The achieved performances have been evaluated in view of performances ensured by a PI controller designed earlier. The multiobjective control law gives satisfactory results. Future work will include the master manipulator force controller and the bilateral controller designs to complete the structured network-based design. To keep a clear physical interpretation of the telemanipulation controller design, a multiobjective decentralized control approach is under study.

This page intentionally left blank

Bibliography [1] R. J. Adams, P. Apkarian, and J. P. Chretien, Robust control approaches for a two-link flexible manipulator, in 3rd International Conference on Dynamics and Control of Structures in Space, Cranfield, UK, May 1996, pp. 101-116. [2] I. Adler and F. Alizadeh, Primal-Dual Interior Point Algorithms for Convex Quadratically Constrained and Semidefinite Optimization Problems, Technical Report RRR 46-95, RUTCOR, Rutgers University, New Brunswick, NJ, December 1995. [3] M. Aitrami and L. El Ghaoui, LMI optimization for non standard Riccati equations arising in stochastic control, IEEE Trans. Automat. Control, 41:1666-1670, 1996. [4] D. Alazard and J. P. Chretien, The impact of local masses and inertias on the dynamic modelling of flexible manipulators, in International Symposium on Artificial Intelligence, Robotics and Automation in Space, Toulouse, Prance, 1992. [5] F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SI AM J. Optim., 5:13-51, 1995. [6] F. Alizadeh, J.-P. A. Haeberly, M. V. Nayakkankuppam, M. L. Overton, and S. Schmieta, SDPpack User Guide (Version 0.9 Beta), Technical Report 737, Courant Institute of Mathematical Sciences, New York, June 1997. Available at http://www.cs.nyu.edu/faculty/overton/sdppack. [7] F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton, Complementarity and nondegeneracy in semidefinite programming, Math. Programming (Ser. B), 77:111-128, 1997. [8] F. Alizadeh, J.-P. A. Haeberly, and M. L. Overton, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM J. Optim., 8:746-768, 1998. [9] K. D. Andersen, An efficient newton barrier method for minimizing a sum of euclidean norms, SIAM J. Optim., 6:74-95, 1996. [10] B. D. O. Anderson and J. B. Moore, Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, 1979. [11] B. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, Englewood Cliffs, NJ, 1990. [12] B. D. O. Anderson, S. Dasgupta, P. P. Khargonekar, F. J. Kraus, and M. Mansour, Robust strict positive realness: Characterization and construction, IEEE Trans. Circuits Systems, 37:869-876, 1990. 341

342

Bibliography

[13] B. D. O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis—A Modern Systems Theory Approach, Prentice-Hall, Englewood Cliffs, NJ, 1973. [14] R. J. Anderson, Dynamic damping control: Implementation issues and simulation results, in Proc. IEEE International Conference on Robotics and Automation, 1990, pp. 68-77. [15] R. J. Anderson and M. W. Spong, Bilateral control of teleoperators with time delay, in Proc. IEEE International Conference on Decision and Control, Austin, TX, 1989. [16] C. Andriot, J. Dudragne, R. Fournier, and J. Vuillemey, On the bilateral control of teleoperators with flexible joints by the HQQ approach, in Proc. SPIE Conference on Teleoperation and Control, Boston, 1992, pp. 80-91. [17] P. Apkarian, On the discretization of LMI-synthesized linear parameter-varying controllers, Automatica, 33:655-661, 1997. [18] P. Apkarian and R. Adams, Advanced gain-scheduling techniques for uncertain systems, in Proc. American Control Conference, June 1997. [19] P. Apkarian and P. Gahinet, A convex characterization of gain-scheduled HOO controllers, IEEE Trans. Automat. Control, AC-40:853-864, 1995. [20] P. Apkarian, P. Gahinet, and G. Becker, Self-scheduled HQQ control of linear parameter-varying systems, in Proc. American Control Conference, 1994, pp. 856860. [21] P. Apkarian, P. Gahinet, and G. Becker, Self-scheduled H^ control of linear parameter-varying systems: A design example, Automatica, 31:1251-1261, 1995. [22] P. Apkarian and H. D. Tuan, Parameterized LMIs in control theory, SIAM J. Control Optim., to appear. [23] K. J. Astrom and T. Hagglund, PID Control: Theory Design and Tuning, Instrument Society of America, Research Triangle Park, NC, 1995. [24] V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys, 26:2943, 1971. [25] W. F. Arnold and A. J. Laub, Generalized eigenproblem algorithms and software for algebraic Riccati equations, Proc. IEEE, 72:1746-1754, 1984. [26] T. Asai, S. Kara, and T. Iwasaki, Simultaneous modeling and synthesis for robust control by LFT scaling, IF AC World Congress, G:309-314, 1996. [27] B. D. Bainov and P. S. Simeonov, Systems with Impulse Effect: and Applications, Wiley, New York, 1989.

Stability, Theory

[28] V. Balakrishnan, Linear matrix inequalities in robustness analysis with multipliers, Systems Control Lett, 25:265-272, 1995. [29] V. Balakrishnan, Y. Huang, A. Packard, and J. C. Doyle, Linear matrix inequalities in analysis with multipliers, in Proc. American Control Conference, Baltimore, MD, 1994, pp. 1228-1232.

Bibliography

343

[30] V. Balakrishnan and A. Tits, Numerical optimization-based design, in The Control Handbook, W. Levine, ed., CRC Press, Boca Raton, FL, pp. 749-758, 1995. [31] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, /j,-Analysis and Synthesis Toolbox fa-Tools), MathWorks, Inc., Natick, MA, 1991. [32] B. A. Bamieh and J. B. Pearson Jr., The HZ problem for sampled-data systems, Systems Control Lett, 19:1-12, 1992. [33] G. Becker, Parameter-dependent control of an under-actuated mechanical system, in Proc. IEEE Conference on Decision and Control, Los Angeles, 1995. [34] G. Becker, Additional results on parameter-dependent controllers for LPV systems, in IFAC World Congress, San Francisco, 1996. [35] G. Becker and A. Packard, Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback, Systems Control Lett., 23:205-215, 1994. [36] G. Becker, A. Packard, D. Philbrick, and G. Balas, Control of parametricallydependent linear systems: A single quadratic Lyapunov approach, in Proc. American Control Conference, San Francisco, 1993, pp. 2795-2799. [37] R. Bellman and K. Fan, On systems of linear inequalities in Hermitian matrix variables, in Proc. Sympos. Pure Math., V. L. Klee, ed., 1963, pp. 1-11. [38] S. Benson, Y. Ye, and X. Zhang, Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization, Technical Report, Department of Management Sciences, The University of Iowa, Iowa City, IA. Revised October 13, 1997. [39] A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust semidefinite programming, in Semidefinite Programming and Applications, H. Wolkowicz, R. Saigal, and L. Vandenberghe, eds., 1998. [40] A. Ben-Tal and A. Nemirovski, Robust Convex Programming, Technical Report 1/95, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology, Technion City, Haifa, Israel, 1995. [41] A. Ben-Tal and A. Nemirovski, Robust truss topology design via semidefinite programming, SI AM J. Optim., 7:991-1016, 1997. [42] A. Ben-Tal and A. Nemirovski, Robust convex programming, IMA J. Numer. Anal, 1998. [43] E. Beran, Induced Norm Control Toolbox (INCT)—Users Manual, 1995. Available at http://www.iau.dtu.dk/Staff/ebb/INCT. [44] E. Beran and K. Grigoriadis, A combined alternating projection and semidefinite programming algorithm for low-order control design, in Proc. IFAC 96, San Francisco, Vol. C, July 1996, pp. 85-90. [45] E. Beran, L. Vandenberghe, and S. Boyd, A global BMI algorithm based on the generalized Benders decomposition, in European Control Conference, Brussels, Belgium, July 1997. [46] A. Berman, Cones, Matrices, and Mathematical Programming, Springer-Verlag, Berlin, New York, 1973.

344

Bibliography

[47] D. Bernstein and W. Haddad, LQG control with an HQQ performance constraint: A Riccati equation approach, in Proc. American Control Conference, 1988, pp. 796802. [48] D. S. Bernstein and W. H. Haddad, LQG control with an HOO performance bound: A Riccati equation approach, IEEE Trans. Automat. Control, 34:293-305, 1989. [49] D. Bernstein and W. Haddad, Robust stability and performance analysis for linear dynamic systems, IEEE Trans. Automat. Control, AC-34:751-758, 1989. [50] D. S. Bernstein and W. H. Haddad, Robust stability and performance analysis for state-space systems via quadratic Lyapunov bounds, SI AM J. Matrix Anal. Appi, 11:239-271, 1990. [51] J. Bernussou, P. L. D. Peres, and J. C. Geromel, A linear programming oriented procedure for quadratic stabilization of uncertain systems, Systems Control Lett, 13:65-72, 1989. [52] V. Blondel and J. N. Tsitsiklis, NP-hardness of some linear control design problems, European J. Control, 1, 1995. [53] V. Blondel and J. N. Tsitsiklis, A Survey of Computational Complexity Results in Systems and Control, Technical Report, LIDS, Massachusetts Institute of Technology, Cambridge, MA, October 1998. [54] P. Bolzern, P. Colaneri, and G. De Nicolao, Optimal robust state estimation for uncertain linear systems, in Proc. IF AC Symposium on Robust Control Design, Rio de Janeiro, Brazil, 1994, pp. 152-161. [55] J. F. Bonnans, R. Cominetti, and A. Shapiro, Pertubed Optimization under the Second Order Regularity Hypothesis, Application to Semidefinite Optimization, 1996, manuscript. [56] N. K. Bose, Applied Multidimensional System Theory, Van Nostrand Rheinhold, New York, 1982. [57] C. Boussios and E. Feron, Estimating the conservatism of the Popov criterion, in Proc. AIAA GNC Conference, 1996. [58] C. Boussios and E. Feron, Estimating the conservatism of Popov's criterion for real parametric uncertainties, Systems Control Lett, 31:173-183, 1997. [59] S. P. Boyd, V. Balakrishnan, C. H. Barratt, N. M. Khraishi, X. Li, D. G. Meyer, and S. A. Norman, A new CAD method and associated architectures for linear controllers, IEEE Trans. Automat. Control, 33:268-283, 1988. [60] S. Boyd, V. Balakrishnan, E. Feron, and L. El Ghaoui, Control system analysis and synthesis via linear matrix inequalities, in Proc. American Control Conference, Vol. 2, San Francisco, June 1993, pp. 2147-2154. [61] S. Boyd and C. Barratt, Linear Controller Design: Limits of Performance, Prentice-Hall, Englewood Cliffs, NJ, 1991. [62] S. Boyd, C. Barratt, and S. Norman, Linear controller design: Limits of performance via convex optimization, Proc. IEEE, 78:529-574, 1990.

Bibliography

345

[63] S. Boyd and L. El Ghaoui, Method of centers for minimizing generalized eigenvalues, Linear Algebra Appi, 188:63-111, 1992. [64] S. P. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SI AM Studies in Applied Mathematics 15, SI AM, Philadelphia, 1994. [65] S. Boyd, N. Khraishi, C. Barratt, S. Norman, V. Balakrishnan, and D. Meyer, QDES Version 1.5, Electrical Engineering Dept., Information Systems Laboratory, Stanford University, Stanford, CA, 1990. [66] S. P. Boyd and L. Vandenberghe, Introduction to Convex Optimization with Engineering Applications, Course Notes, Information Systems Library, Stanford University, Stanford, CA, 1997. Available at http://www.stanford.edu/class/ee364/. [67] S. Boyd, L. Vandenberghe, and M. Grant, Efficient convex optimization for engineering design, in Proc. IFAC Symposium on Robust Control Design, September 1994, pp. 14-23. [68] S. Boyd and S. Wu, SDPSOL: A Parser/Solver for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure, User's Guide, Version Beta, Stanford University, Stanford, CA, 1996. Available at http://www. stanford.edu/~boyd/SDPSOL.html. [69] S. Boyd and Q. Yang, Structured and simultaneous Lyapunov functions for system stability problems, Internat. J. Control, 49:2215-2240, 1989. [70] J. P. Boyle and R. L. Dykstra, A method for finding projections onto the intersection of convex sets in Hilbert space, Lecture Notes in Statist., 37:28-47, 1986. [71] R. D. Braatz, P. M. Young, J. C. Doyle, and M. Morari, Computational complexity of /i calculation, Proc. American Control Conference, June 1993, pp. 1682-1683. [72] R. Braatz, P. Young, J. Doyle, and M. Morari, Computational complexity of /i calculation, IEEE Trans. Automat. Control, 39:1000-1002, 1994. [73] J. W. Brewer, Kronecker products and matrix calculus in system theory, IEEE Trans. Circuits Systems, CAS-25:772-781, 1978. [74] N. Brixius, F. A. Potra, and R. Sheng, Solving Semidefinite Programs with Mathematica, Technical Report 97/1996, Department of Mathematics, University of Iowa, 1996. [75] R. Bupp, D. Bernstein, and V. Coppola, A benchmark problem for nonlinear control design: Problem state, experimental testbed, and passive nonlinear compensation, in Proc. IEEE American Control Conference, Seattle, WA, June 1995, pp. 43634367. [76] W. Cheney and A. A. Goldstein, Proximity maps for convex set, Proc. Amer. Math. Soc., 12:448-450, 1959. [77] R. Y. Chiang and M. G. Safonov, Robust Control Toolbox, Ver. 2.0, Math Works, Inc., Natick, MA, 1992. [78] M. Chilali and P. Gahinet, HOC design with pole placement constraints: An LMI approach, IEEE Trans. Automat. Control, 41:358-367, 1995.

346

Bibliography

[79] R. C. Chung and P. R. Belanger, Minimum-sensitivity filter for linear time-invariant stochastic systems with uncertain parameters, IEEE Trans. Automat. Control, AC21:98-100, 1976. [80] J. E. Colgate, Power and impedance scaling in bilateral teleoperation, in Proc. IEEE International Conference on Robotics and Automation, 1991, pp. 2292-2297. [81] J. E. Colgate, Robust impedance shaping telemanipulation, IEEE Trans. Robotics Automation, 9:374-384, 1993. [82] J. E. Colgate and N. Hogan, Robust control of dynamically interacting systems, Internat. J. Control, 48:65-88, 1988. [83] E. G. Collins Jr., L. D. Davis, and R. Stephen, Homotopy algorithm for maximum entropy design, AIAA J. Guidance Control Dynamics, 17:311-321, 1994. [84] P. L. Combettes and H. J. Trussell, Method of successive projections for finding a common point of sets in metric spaces, J. Optim. Theory Appl., 67:487-507, 1990. [85] R. W. Cottle, J. S. Pang, and R. E. Stone, The Linear Complementarity Problem, Academic Press, New York, 1992. [86] G. E. Coxson and C. L. DeMarco, Testing robust stability of general matrix polytopes is an NP-hard computation, in Proc. Annual Allerton Conference on Communication, Control and Computing, Allerton House, Monticello, IL, 1991, pp. 105106. [87] J. Cullum, W. E. Donath, and P. Wolfe, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Math. Programming Study, 3:35-55, 1975. [88] M. A. Dahleh and I. Diaz-Bobillo, Control of Uncertain Sytems: A Linear Programming Approach, Prentice-Hall, Englewood Cliffs, NJ, 1995. [89] R. D'Andrea, Generalized L2 synthesis: A new framework for control design, in Proc. 35th IEEE Conference on Decision and Control, 1996. [90] G. B. Dantzig, Linear Programming and Extensions, Princeton University, Princeton, NJ, 1963. [91] J. A. D'Appolito and C. E. Hutchinson, Low sensitivity filters for state estimation in the presence of large parameter uncertainties, IEEE Trans. Automat. Control, AC-14:310-312, 1969. [92] S. Dasgupta, G. Chockalingam, B. D. O. Anderson, and M. Fu, Lyapunov functions for uncertain systems with applications to the stability to time varying systems, IEEE Trans. Circuits Systems, 41:93-106, 1994. [93] J. David, Algorithms and Design for Robust Controllers, Ph.D. thesis, Catholic University of Leuven, 1994. [94] J. David and B. De Moor, The opposite of analytic centers for solving minimum rank problems in control and identification, in Proc. 32nd Conference on Decision and Control, 1993. [95] C. Davis, W. Kahan, and H. Weinberger, Norm preserving dilations and their applications to optimal error bounds, SIAM J. Numer. Anal., 19:445-469, 1982.

Bibliography

347

[96] B. de Moor, Total least squares for affinely structured matrices and the noisy realization problem, IEEE Trans. Acoust., Speech, Signal Processing, 42:3104-3113, 1994. [97] M. A. H. Dempster, Stochastic Programming, Academic Press, New York, 1980. [98] C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Academic Press, New York, 1975. [99] O. Didrit, L. Jaulin, and E. Walter, Guaranteed analysis and optimization of parametric systems, with application to their stability degree, European J. Control, 1997. [100] O. Diekmann, S. A. von Gils, S. M. Verduyn Lunel, and H. O. Walther, Delay Equations. Functional, Complex, and Nonlinear Analysis, Appl. Math. Sci., Springer-Verlag, New York, 1995. [101] J. Doyle, Guaranteed margins for LQG regulators, IEEE Trans. Automat. Control, AC-23:756-757, 1978. [102] J. Doyle, Analysis of feedback systems with structured uncertainties, IEEE Proc., 129-D:242-250, 1982. [103] J. C. Doyle, Synthesis of robust controllers and filters with structured plant uncertainty, in Proc. IEEE Conference on Decision and Control, Albuquerque, NM, December 1983. [104] J. C. Doyle, Structured uncertainty in control system design, Proc. IEEE Conference on Decision and Control, December 1985, pp. 260-265. [105] J. C. Doyle, Uncertainty and robustness out of control, Talk delivered at the Workshop on Robust Control, Siena, Italy, July 1998. [106] J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory, Macmillan, New York, 1992. [107] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, State-space solutions to standard H2 and H^ problems, IEEE Trans. Automat. Control, 34:831-847, 1989. [108] J. Doyle, A. Packard, and K. Zhou, Review of LFT's, LMI's and /i, in Proc. IEEE Conference on Decision and Control, Vol. 2, Brighton, December 1991, pp. 12271232. [109] J. C. Doyle and G. Stein, Multivariable feedback design: Concepts for a classical/modern synthesis, IEEE Trans. Automat. Control, AC-26:4-16, 1981. [110] J. Doyle, K. Zhou, K. Glover, and B. Bodenheimer, Mixed H^ and H^ performance objectives II: Optimal control, IEEE Trans. Automat. Control, 39:1575-1587, 1994. [Ill] S. Dussy, An LMI Approach for Multiobjective Robust Control, Ph.D. thesis, University of Paris IX-Dauphine, Paris, March 1998. [112] S. Dussy, LMI-based robust control of Lur'e system, IASTED Control Comput. J., 26:9-12, 1998. [113] S. Dussy and L. El Ghaoui, Multiobjective Robust Control Toolbox (MRCT): User's Guide, 1996. Available at http://www.ensta.fr/~gropco/stafF/dussy/ gocpage.html.

348

Bibliography

[114] S. Dussy and L. El Ghaoui, Multiobjective bounded control of uncertain nonlinear systems: An inverted pendulum example, in Control of Uncertain Systems with Bounded Inputs, Springer-Verlag, Berlin, New York, 1997, pp. 55-73. [115] S. Dussy and L. El Ghaoui, Measurement-scheduled control for the RTAC problem, Internat. J. Robust Nonlinear Control, 8:377-400, 1998. [116] S. Dussy and L. El Ghaoui, Multiobjective robust control toolbox for LMI-based control, in Proc. IFAC Symposium on Computer Aided Control Systems Design, Gent, Belgium, April 1997, pp. 353-358. [117] S. Dussy and L. El Ghaoui, Multiobjective robust measurement-scheduling for discrete-time systems: An LMI approach, in Proc. IFAC Conference on Systems Structure and Control, Bucharest, Romania, October 1997, pp. 343-348. [118] A. El Bouhtouri and A. J. Pritchard, Stability radii of linear systems with respect to stochastic perturbations, Systems Control Lett., 19:29-33, 1992. [119] L. El Ghaoui, LMITOOL: An Interface to Solve LMI Problems, ENSTA, France, 1995. [120] L. El Ghaoui, State-feedback control of systems with multiplicative noise via linear matrix inequalities, Systems Control Lett., 24:223-228, 1995. [121] L. El Ghaoui and M. Aitrami, Robust state-feedback stabilization of jump linear systems, Internat. J. Robust Nonlinear Control, 6:1015-1022, 1996. [122] L. El Ghaoui, F. Delebecque, and R. Nikoukhah, LMITOOL: A User-Friendly Interface for LMI Optimization. Available at ftp.ensta.fr in pub/elghaoui/lmitool, February 1995. [123] L. El Ghaoui and J. P. Folcher, Multiobjective robust control of LTI systems subject to unstructured perturbations, Systems Control Lett., 28:23-30, 1996. [124] L. El Ghaoui and P. Gahinet, Rank minimization under LMI constraints: A framework for output feedback problems, in Proc. European Control Conference, July 1993, pp. 1176-1179. [125] L. El Ghaoui and H. Lebret, Robust solutions to least-squares problems with uncertain data, SIAM J. Matrix Anal. Appi, 18:1035-1064, 1997. [126] L. El Ghaoui, R. Nikoukha, and F. Delebecque, LMITOOL: A Front-End for LMI Optimization, User's Guide, 1995. Available via anonymous ftp at ftp.ensta.fr, under /pub/elghaoui/lmitool. [127] L. El Ghaoui, F. Oustry, and H. Lebret, Robust solutions to uncertain semidefinite programs, SIAM J. Optim., 9:33-52, 1998. [128] L. El Ghaoui, F. Oustry, and M. Aft Rami, A cone complementary linearization algorithm for static output-feedback and related problems, IEEE Trans. Automat Control, 42:1171-1176, 1997. [129] L. El Ghaoui and G. Scorletti, Control of rational systems using linear-fractional representations and linear matrix inequalities, Automatica, 32:1273-1284, 1996. [130] M. K. H. Fan, A quadratically convergent local algorithm on minimizing the largest eigenvalue of a symmetric matrix, Linear Algebra Appl., 188/189:231-253, 1996.

Bibliography

349

[131] M. K. H. Fan, A. L. Tits, and J. C. Doyle, Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics, IEEE Trans. Automat. Control, 36:25-38, 1991. [132] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math., 86:149-175, 1997. [133] E. Feron, Linear Matrix Inequalities for the Problem of Absolute Stability of Control Systems, Ph.D. thesis, Stanford University, Stanford, CA, October 1993. [134] E. Feron, Analysis of robust H2 performance using multiplier theory, SIAM J. Control Optim., 35:160-177, 1997. [135] E. Feron, P. Apkarian, and P. Gahinet, Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions, IEEE Trans. Automat. Control, 41:1041-1046, 1996. [136] E. Feron, V. Balakrishnan, S. Boyd, and L. El Ghaoui, Numerical methods for U.2 related problems, in Proc. American Control Conference, Vol. 4, Chicago, June 1992, pp. 2921-2922. [137] R. Fletcher, Semidefinite matrix constraints in optimization, SIAM J. Control Optim., 23:493-513, 1985. [138] C. A. Floudas and V. Visweswaran, A primal relaxed dual global optimization approach, J. Optim. Theory Appl, 10:237-260, 1972. [139] B. K. R. Fourer and D. Gay, AMPL—A Modeling Language for Mathematical Programming, Scientific Press, San Francisco, 1993. [140] A. Fradkov and V. A. Yakubovich, The /-Procedure and Duality Theorems for Non Convex Problems of Quadratic Programming, Technical Report 1, Vestnik Leningrad University, 1973 (in Russian). [141] A. L. Fradkov and V. A. Yakubovich, The 5-procedure and duality relations in nonconvex problems of quadratic programming, Vestnik Leningrad Univ. Math., 6:101-109, 1979 (in Russian, 1973). [142] B. A. Francis, A Course in HQQ Control Theory, Lecture Notes in Control and Inform. Sci. 88, Springer-Verlag, New York, 1987. [143] M. Fu, H. Li, and S. I. Niculescu, Robust stability analysis and stabilization for time-delay systems using the integral quadratic constraint approach, in Stability and Control of Delay Systems, Lecture Notes in Control and Inform. Sci., SpringerVerlag, London, 1997. [144] M. Fu and Z. Q. Luo, Computational complexity of a problem arising in fixed order output feedback design, Systems Control Lett, 30:209-215, 1997. [145] K. Fujisawa and M. Kojima, SDPA (Semidefinite Programming Algorithm) User's Manual, Technical Report B-308, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 1995. [146] T. Fukuda, K. Tanie, and T. Mitsuoka, A new method of master-slave type of teleoperation for a micro-manipulator system, in Proc. IEEE Micro Robots and Teleoperation Workshop, Hyannis, MA, 1987.

350

Bibliography

[147] P. Gahinet, Explicit controller formulas for LMI-based HQQ synthesis, in Proc. American Control Conference, 1994, pp. 2396-2400. [148] P. Gahinet and P. Apkarian, A linear matrix inequality approach to HQQ control, Internal. J. Robust Nonlinear Control, 4:421-448, 1994. [149] P. Gahinet, P. Apkarian, and M. Chilali, Parameter-dependent Lyapunov functions for real parametric uncertainty, IEEE Trans. Automat. Control, 41:436-442, 1996. [150] P. Gahinet and A. Nemirovskii, General purpose LMI solver with benchmarks, in Proc. 32nd Conference Decision Control, San Antonio, TX, 1993. [151] P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, The LMI control toolbox, in Proc. IEEE Conference on Decision and Control, December 1994, pp. 2038-2041. [152] P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, The LMI control toolbox, in Proc. 33rd IEEE Conference on Decision and Control, 1994, pp. 2038-2041. [153] P. Gahinet, A. Nemirovskii, A. J. Laub, and M. Chilali, LMI Control Toolbox, The Math Works, Inc., Natick, MA, 1995. [154] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, San Francisco, CA, 1979. [155] A. M. Geoffrion, Generalized benders decomposition, J. Optim. Theory Algorithms, 10:237-260, 1972. [156] J. C. Geromel, Optimal linear filtering under parameter uncertainty, IEEE Trans. Signal Processing, 47:168-175, 1999. [157] J. C. Geromel, J. Bernussou, G. Garcia, and M. C. de Oliveira, H2 and HQQ robust filtering for discrete-time linear systems, in Proc. IEEE Conference on Decision and Control, 1998, pp. 632-637. [158] J. C. Geromel, P. L. D. Peres, and J. Bernussou, On a convex parameter space method for linear control design of uncertain systems, SI AM J. Control Optim., 29:381-402, 1991. [159] J. C. Geromel, P. L. D. Peres, and S. R. Souza, Output feedback stabilization of uncertain systems through a min/max problem, in Proc. IFAC World Congress, 1993. [160] M. X. Goemans, Semidefinite programming in combinatorial optimization, Math. Programming, 79:143-161, 1997. [161] M. X. Goemans and D. P. Williamson, .878-approximation for MAX CUT and MAX 2SAT, in Proc. 26th ACM Symposium on the Theory of Computing, 1994, pp. 422-431. [162] J. L. Coffin, A. Haurie, and J. Ph. Vial, Decomposition and nondifferentiable optimization with the projective algorithm, Management Sci., 38:284-302, 1992. [163] J. L. Coffin, Z. Q. Luo, and Y. Ye, Complexity analysis of an interior cutting plane method for convex feasibitlity problems, SI AM J. Optim., 6:638-652, 1996. [164] K. C. Goh, Robust Control Synthesis via Bilinear Matrix Inequalities, Ph.D. thesis, University of Southern California, Los Angeles, May 1995.

Bibliography

351

[165] K. C. Goh, J. H. Ly, L. Turan, and M. G. Safonov, n/Km-synthesis via bilinear matrix inequalities, in Proc. IEEE Conference on Decision and Control, December 1994, pp. 2032-2037. [166] K.-C. Goh and M. G. Safonov, Robust analysis, sectors, and quadratic functionals, in Proc. IEEE Conference on Decision Control, 1995, pp. 1988-1993. [167] K. C. Goh, M. G. Safonov, and J. H. Ly, Robust synthesis via bilinear matrix inequalities, Internat. J. Robust Nonlinear Control, 6:1079-1095, 1996. [168] K. C. Goh, M. G. Safonov, and G. P. Papavassilopoulos, A global optimization approach for the BMI problem, in Proc. IEEE Conference on Decision and Control, Lake Buena Vista, FL, December 1994. [169] K. C. Goh, L. Turan, M. G. Safonov, G. P. Papavassilopoulos, and J. H. Ly, Biaffine matrix inequality properties and computational methods, in Proc. American Control Conference, July 1994. [170] K.-C. Goh and F. Wu, Duality and basis functions for H2 performance analysis, Automatica, 33:1949-1959, 1997. [171] G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed., Johns Hopkins University Press, Baltimore, 1989. [172] J. Gondzio, Multiple centrality corrections in a primal-dual method for linear programming, Comput. Optim. Appl, 6:137-156, 1996. [173] M. S. Gowda, Complementary problems over locally compact cones, SIAM J. Control Optim., 27:836-841, 1989. [174] A. Grace, Optimization Toolbox User's Guide, MathWorks, Inc., Natick, MA, 1992. [175] M. J. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Englewood Cliffs, NJ, 1995. [176] K. M. Grigoriadis and R. E. Skelton, Low order control design for LMI problems using alternating projection methods, Automatica, 32:1117-1125, 1996. [177] S. C. O. Grocott, J. P. How, and D. W. Miller, A comparison of robust control techniques for uncertain structural systems, in Proc. AIAA Guidance Navigation and Control Conference, August 1994, pp. 261-271. [178] M. Grotschel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin, New York, 1983. [179] K. Gu, Discretized Imi set in the stability problem of linear uncertain time-delay systems, Internat. J. Control, 68:923-934, 1997. [180] L. G. Gubin, B. T. Polyak, and E. V. Raik, The method of projections for finding the common point of convex sets, USSR Comput. Math. Phys., 7:1-24, 1967. [181] S. Gupta, Robust stability analysis using LMIs: Beyond small gain and passivity, Internat. J. Robust Nonlinear Control, 6:953-968, 1996. [182] W. M. Haddad and D. S. Bernstein, Robust, reduced-order, nonstrictly proper state estimation via the optimal projection equations with Petersen-Hollot bounds, Systems Control Lett, 9:423-431, 1987.

352

Bibliography

[183] W. M. Haddad and D. S. Bernstein, Parameter-dependent Lyapunov functions, constant real parameter uncertainty, and the Popov criterion in robust analysis and synthesis: Parts 1 and 2, in Proc. Conference on Decision and Control, 1991, pp. 2262-2263, 2274-2279. [184] J.-P. A. Haeberly, M. V. Nayakkankuppam, and M. L. Overton, Extending Mehrotra and Gondzio higher order corrections to mixed semidefinite-quadratic-linear programming, Optim. Methods Softw., to appear. [185] M. Hall, Jr., Combinatorial Theory, Wiley, New York, 1987. [186] J. K. Hale and S. Verduyn Lunel, Intoduction to Functional Differential Equations, Springer-Verlag, New York, 1991. [187] S. Hall and J. How, Mixed T^/A* performance bounds using dissipation theory, in Proc. 1993 Conference on Decision and Control, San Antonio, pp. 1536-1541. [188] S. P. Han, A successive projection method, Math. Programming, 40:1-14, 1988. [189] C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SI AM J. Optim., to appear. [190] C. Helmberg, F. Rendl, R. Vanderbei, and H. Wolkowicz, An interior-point method for semidefinite programming, SI AM J. Optim., 6:342-361, 1996. [191] A. Helmersson, Methods for Robust Gain Scheduling, Ph.D. thesis, Linkoping University, Linkoping, Sweden, 1995. [192] U. Helmke and J. B. Moore, Optimization and Dynamical Systems, Springer-Verlag, London, 1994. [193] N. J. Higham, Computing the nearest symmetric positive semidefinite matrix, Linear Algebra AppL, 103:103-118, 1988. [194] J. B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin, 1993. [195] J.-B. Hiriart-Urruty and D. Ye, Sensitivity analysis of all eigenvalues of a symmetric matrix, Numer. Math., 70:45-72, 1995. [196] G. Hirzinger, Direct digital robot control using a force torque sensor, in Proc. IFAC Symposium on Real Time Digital Control Applications, Guadalajara, Mexico, 1983, pp. 243-255. [197] N. Hogan, Impedance control: An approach to manipulation, parts I, II, III, J. Dynamic Systems Measurement Control, 107:1-24, 1985. [198] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990. [199] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. [200] R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, 2nd ed., Springer-Verlag, Berlin, 1993.

Bibliography

353

[201] J. P. How, Robust Control Design with Real Parameter Uncertainty using Absolute Stability Theory, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, February 1993. [202] J. P. How, W. M. Haddad, and S. R. Hall, Robust control synthesis examples with real parameter uncertainty using the Popov criterion, in Proc. American Control Conference, June 1993, pp. 1090-1095. [203] J. P. How and S. R. Hall, Connections between the Popov stability criterion and bounds for real parameter uncertainty, in Proc. American Control Conference, 1995, pp. 1084-1089. [204] J. P. How, S. R. Hall, and W. M. Haddad, Robust controllers for the Middeck active control experiment using Popov controller synthesis, IEEE Trans, on Control Systems Technology, 2:73-87, 1994. [205] Z. Hu, S. E. Salcudean, and P. D. Loewen, Optimisation-based teleoperation controller design, in Proc. IEEE of the International Federation of Automatic Control Congress, San Francisco, CA, June 1996, pp. 405-410. [206] P. Huard, Resolution of mathematical programming with nonlinear constraint by the method of centers, in Nonlinear Programming, J. Abadie, ed., North-Holland, Amsterdam, 1967. [207] E. F. Infante and W. B. Castelan, A Lyapunov functional for a matrix differencedifferential equation, J. Differential Equations, 29:439-451, 1978. [208] G. Isac, Complementarity Problems, Springer- Verlag, New York, Berlin, 1992. [209] T. Iwasaki, A Unified Matrix Inequality Approach to Linear Control Design, Ph.D. thesis, Purdue University, West Lafayette, IN, December 1993. [210] T. Iwasaki, Robust performance analysis for systems with norm-bounded timevarying structured uncertainty, in Proc. 1994 American Control Conference, Baltimore, MD. [211] T. Iwasaki and S. Kara, Well-posedness of feedback systems: insights into exact robustness analysis, in Proc. IEEE Conference on Decision and Control, Kobe 1996, pp. 1863-1868. [212] T. Iwasaki and S. Kara, Well-posedness of feedback systems: insights into exact robustness analysis and approximate computations, IEEE Trans. Automat. Control, 43:619-630, 1998. [213] T. Iwasaki, S. Kara, and T. Asai, Well-posedness theorem: A classification of LMI/BMI-reducible control problems, in Proc. International Symposium on Intelligent Robotic Systems, Bangalore, India, 1995, pp. 145-157. [214] T. Iwasaki and R. E. Skelton, A complete solution to the general Hooproblem, LMI existence conditions and state-space formulas, in Proc. American Control Conference, San Francisco, CA, June 1993. [215] T. Iwasaki and R. E. Skelton, All controllers for the general HQQ control problem: LMI existence conditions and state space formulas, Automatica, 30:1307-1317, 1994.

354

Bibliography

[216] T. Iwasaki and R. E. Skelton, Parametrization of all stabilizing controllers via quadratic Lyapunov functions, J. Optim. Theory Appl, 85:291-308, 1995. [217] T. Iwasaki and R. E. Skelton, The XY-centering algorithm for the dual LMI problem: A new approach to fixed order control design, Internal. J. Control, 62:12571272, 1995. [218] B. N. Jain, Guaranteed error estimation in uncertain systems, IEEE Trans. Automatic Control, AC-33:230-232, 1975. [219] F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control Optim., 31:1360-1377, 1993. [220] U. Jonsson, Robustness Analysis of Uncertain and Nonlinear Systems, Ph.D. thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, 1996. [221] U. Jonsson, Stability analysis with Popov multipliers and integral quadratic constraints, Systems Control Lett, 31:85-92, 1997. [222] U. Jonsson and A. Rantzer, On duality in robustness analysis, in Proc. 34th IEEE Conference on Decision and Control, New Orleans, LA, 1995, pp. 1443-1448. [223] U. Jonsson and A. Rantzer, Duality bounds in robustness analysis, Automatica, 33:1835-1844, 1997. [224] U. Jonsson and A Rantzer, A Matlab Toolbox for System Analysis via Integral Quadratic Constraints, Technical Report TFRT-7556, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, January 1997. [225] R. E. Kalman, Lyapunov functions for the problem of Lur'e in automatic control, Proc. Nat. Acad. Sci. U.S.A., 49:201-205, 1963. [226] L. H. Keel, S. P. Bhattacharyya, and J. W. Howze, Robust control with structured perturbations, IEEE Trans. Automat. Control, AC-33:68-78, 1988. [227] M. Khammash and J. B. Pearson, Performance robustness of discrete-time systems with structured uncertainty, IEEE Trans. Automat. Control, AC-36:398-412, 1991. [228] P. Khargonekar and M. Rotea, Mixed Hz/Hoo control: A convex optimization approach, IEEE Trans. Automat. Control, 36:824-837, 1991. [229] H. Kimura, Pole assignment by gain output feedback, IEEE Trans. Automat. Control, AC-20:509-516, 1975. [230] K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Math. 1133, Springer-Verlag, Berlin, New York, 1985. [231] K. C. Kiwiel, A linearization algorithm for optimizing control systems subject to singular value inequalities, IEEE Trans. Automat. Control, AC-31:595-602, 1986. [232] K. C. Kiwiel, A direct method of linearizations for continuous minimax problems, J. Optim. Theory Appl., 55:271-287, 1987. [233] M. Kojima, M. Shida, and S. Shindoh, Local convergence of predictor-corrector infeasible-interior-point algorithms for sdps and sdlcps, Math. Programming, 80:129-161, 1998.

Bibliography

355

[234] M. Kojima, S. Shindoh, and S. Kara, Interior-Point Methods for the Monotone Linear Complementarity Problem in Symmetric Matrices, Technical Report, Department of Information Science, Tokyo Institute of Technology, Japan, 1994. [235] M. Kojima, S. Shindoh, and S. Kara, Interior point methods for the monotone linear complementarity problem in symmetric matrices, SIAM J. Optim., 7:86-125, 1997. [236] I. E. Rose, F. Jabbari, and W. E. Schmittendorf, A direct characterization of L^gain controllers for LPV systems, Proc. IEEE Conference on Decision and Control, 1996, pp. 3990-3995. [237] H. Kushner, Introduction to Stochastic Control, Holt, Rinehart and Winston, Inc., 1971. [238] V. Lakshmikantam and S. Leela, Differential II, Academic Press, New York, 1969.

and Integral Inequalities, Vols. I and

[239] A. J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control, AC-24:913-921, 1979. [240] H. Lebret, Antenna pattern synthesis through convex optimization, in Advanced Signal Processing Algorithms, SPIE-The International Society for Optical Engineering, 1995, pp. 182-192. [241] F. C. Lee, H. Flashner, and M. G. Safonov, Positivity-based control system synthesis using alternating LMIs, in Proc. American Control Conference, Seattle, WA, June 1995. [242] F. C. Lee, H. Flashner, and M. G. Safonov, Positivity embedding for noncolocated and nonsquare flexible systems, Appl. Math. Comput., 70:233-246, 1995. [243] F. C. Lee, H. Flashner, and M. G. Safonov, An LMI approach to positivity embedding, in Proc. American Control Conference, Seattle, WA, June 1995. [244] C. Lemarechal, A. Nemirovskii, and Yu. Nesterov, New variants of bundle methods, Math. Programming, 69:111-147, 1995. [245] C. Lemarechal, F. Oustry, and C. Sagastizabal, The W-Lagrangian of a convex function, Trans. Amer. Math. Soc., to appear. [246] C. Lemarechal and C. Sagastizabal, Variable metric bundle methods: From conceptual to implementable forms, Math. Programming, 76:394-410, 1997. [247] A. S. Lewis and M. L. Overton, Eigenvalue optimization, Acta Numer., 5:149-190, 1996. [248] H. Li, S. I. Niculescu, L. Dugard, and J. M. Dion, Robust HQQ control of uncertain linear time-delay systems: A linear matrix inequality approach with guaranteed a-stability, Part II, in Proc. 35th IEEE Conference on Decision and Control, December 1996. [249] S-M. Liu, Parallel Algorithms for Global Nonconvex Minimization with Applications to Robust Control Design via BMIs, Ph.D. thesis, University of Southern California, Los Angeles, August 1995. [250] S-M. Liu and G. P. Papavassilopoulos, Numerical experience with parallel algorithms for solving the BMI problem, in Proc. IF AC World Congress, July 1996.

356

Bibliography

[251] C. Livadas, Optimal JJ.2/Popov Controller Design Using Linear Matrix Inequalities, Master's thesis, Massachusetts Institute of Technology, Cambridge, February 1996. [252] W. M. Lu and J. C. Doyle, HOQ control of LFT systems: An LMI approach, in Proc. IEEE Conference on Decision and Control, Tucson, AZ, 1992, pp. 1997-2001. [253] W. Lu and J. C. Doyle, HQQ control of nonlinear systems: A convex characterization, IEEE Trans. Automat. Control, 40:1668-1674, 1995. [254] W. M. Lu, K. Zhou, and J. C. Doyle, Stabilization of uncertain linear systems: An LFT approach. IEEE Trans. Automat. Control, 1:50-65, 1996. [255] D. G. Luenberger, Optimization by Vector Space Methods, John Wiley, New York, 1968. [256] D. G. Luenberger, Investment Science, Prentice-Hall, Englewood Cliffs, NJ, 1997. [257] Z.-Q Luo, J. F. Sturm, and S. Zhang, Duality and Self-Duality for Conic Convex Programming, Technical Report El 9620/A, Econometric Institute, Erasmus University Rotterdam, The Netherlands, 1996. [258] A. I. Lur'e and V. N. Postnikov, On the theory of stability of control systems, Appl. Math. Mech., 8:464-465, 1944 (in Russian). [259] J. H. Ly, M. G. Safonov, and R. Y. Chiang, On computation of multivariable stability margin using generalized Popov multipliers—LMI approach, in Proc. American Control Conference, July 1994. [260] J. H. Ly, A Multiplier Approach to p/km Synthesis, Ph.D. thesis, University of Southern California, Los Angeles, February 1995. [261] J. H. Ly, K. C. Goh, and M. G. Safonov, LMI approach to multiplier km/^isynthesis, in Proc. American Control Conference, Seattle, WA, June 1995. [262] J. H. Ly, M. G. Safonov, and R. Y. Chiang, Real/complex multivariable stability margin computation via generalized Popov multiplier-LMI approach, in Proc. American Control Conference, Baltimore, MD, 1994, pp. 425-429. [263] H. M. Markowitz, Mean-Variance Analysis in Portfolio Choice and Capital Markets, Blackwell, Oxford, 1987. [264] I. Masubuchi, A. Ohara, and N. Suda, LMI-based controller synthesis: A unified formulation and solution, in Proc. American Control Conference, 1995, pp. 34733477. [265] I. Masubuchi, A. Ohara, and N. Suda, Robust multi-objective controller design via convex optimization, in Proc. IEEE Conference on Decision and Control, 1996, pp. 263-264. [266] MATLAB: High-performance Numeric Computation and Visualization Software, Version 4.1, MathWorks, Inc., Natick, MA, 1993. [267] A. Megretsky, Power Distribution Approach in Robust Control, Technical Report TRITA/MAT-92-0027, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden, 1992.

Bibliography

357

[268] A. Megretski, Power distribution approach in robust control, in Proc. IF AC Congress, Sydney, Australia, 1993, pp. 399-402. [269] A. Megretsky, S-Procedure in Optimal Non-Stochastic Filtering, Technical Report TRITA/MAT-92-0015, Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden, March 1992. [270] A. Megretsky, Necessary and sufficient conditions of stability: A multiloop generalization of the circle criterion, IEEE Trans. Automat. Control, AC-38:753-756, 1993. [271] A Megretski, Integral Quadratic Constraints for Systems with Rate Limiters, Technical Report LIDS-P 2407, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, 1997. [272] A Megretski, C. Kao, U. Jonsson, and A. Rantzer, A Guide to IQC-Beta: Software for Robustness Analysis, Laboratory for Information and Decision Systems, Masschusetts Institute of Technology, Cambridge, MA, 1997. Available at http://www.mit.edu/people/ameg/home.html. [273] A. Megretski and A. Rantzer, System analysis via integral quadratic constraints, IEEE Trans. Automat. Control, 42:819-830, 1997. [274] A. Megretsky and S. Treil, Power distribution inequalities in optimization and robustness of uncertain systems, J. Math. Systems Estim. Control, 3:301-319, 1993. [275] S. Mehrotra, Higher Order Methods and Their Performance, Technical Report 90-16R1, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1991. [276] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM J. Optim., 2:575-601, 1992. [277] G. Meinsma, Y. Shrivastava, and M. Fu, Some properties of an upper bound of /z, IEEE Trans. Automat. Control, 41:1326-1330, 1996. [278] D. Meizel, Stabilisation robuste des systemes a dynamique perturbee, in Systemes Non-Lineaires 2: Stabilite—Stabilisation, vol. 2, A. J. Fossard and D. NormandCyrot, eds., Masson, Paris, 1993. [279] M. Mesbahi, Matrix Inequalities, Stability Theory, Interior Point Methods, and Distributed Computation, Ph.D. thesis, University of Southern California, Los Angeles, June 1996. [280] M. Mesbahi, Solving a class of rank minimization problems as semi-definite programs with applications to fixed order output feedback synthesis, Systems Control Lett, 33:31-36, 1998. [281] M. Mesbahi and G. P. Papavassilopoulos, A cone programming approach to the bilinear matrix inequality and its geometry, Math. Programming, 77:247-272, 1997. [282] M. Mesbahi and G. P. Papavassilopoulos, On the rank minimization problem over a positive semi-definite linear matrix inequality, IEEE Trans. Automat. Control, 42:239-243, 1997.

358

Bibliography

[283] M. Mesbahi, G. P. Papavassilopoulos, and M. G. Safonov, Matrix cones, complementarity problems, and the bilinear matrix inequality, in Proc. IEEE Conference on Decision and Control, New Orleans, LA, December 1995. [284] D. G. Meyer, V. Balakrishnan, C. Barratt, S. Boyd, N. Khraishi, X. Li, and S. Norman, Intelligent specification language compilers, the Q-parametrization, and convex programming: Concepts for an advanced computer-aided control system design method, in Advanced Computing Concepts and Techniques in Control Engineering, M. J. Denham and A. J. Laub, eds., NATO Adv. Sci. Inst. Ser. F Comput. Sys. Sci. 47, Springer-Verlag, Berlin, 1987, pp. 487-496. [285] D. Miller, J. de Luis, G. Stover, and E. Crawley, MACE: Anatomy of a modern control experiment, in Proc. IFAC 13th World Congress, July 1996. [286] R. D. C. Monteiro, Polynomial convergence of primal-dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions, SIAM J. Optim., 8:797-812, 1998. [287] R. E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979. [288] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, New York, 1993. [289] K. J. Narendra and J. H. Taylor, Frequency Domain Criteria for Absolute Stability, Academic Press, New York, 1973. [290] A. Nemirovski, Subject: Quality of approximations, private communication, 19xx. [291] A. Nemirovskii, Several NP-hard problems arising in robust stability analysis, Math. Control Signal Systems, 6:99-105, 1993. [292] A. Nemirovski and P. Gahinet, The projective method for solving linear matrix inequalities, in Proc. American Control Conference, 1994, pp. 840-844. [293] A. Nemirovski and P. Gahinet, The projective method for solving linear matrix inequalities, Math. Programming, 77:163-190, 1997. [294] Yu. Nesterov, Complexity estimates of some cutting plane methods based on the analytic barrier, Math. Programming, 69:149-176, 1995. [295] Yu. Nesterov and A. Nemirovskii, An interior-point method for generalized linearfractional problems, Math. Programming, Ser. B, 1993. [296] Yu. Nesterov and A. Nemirovskii, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM, Philadelphia, 1994. [297] Yu. Nesterov and M. J. Todd, Primal-dual interior-point methods for self-scaled cones, SIAM J. Optim., 8:324-364, 1998. [298] Yu. E. Nesterov, M. J. Todd, and Y. Ye. Infeasible-Start Primal-Dual Methods and Infeasibility Detectors for Nonlinear Programming Problems, Technical Report, Department of Management Sciences, University of Iowa, Iowa City, IA, February 1998; Math. Programming, to appear. [299] W. S. Newman and Y. Zhang, Stable interaction control and coulomb friction— Compensation using natural admittance control, J. Robotic Systems, 11:3-11, 1994.

Bibliography

359

[300] S.-I. Niculescu, On Some Frequency Sweeping Tests for Delay-Dependent Stability: A Model Transformation Case Study, manuscript. [301] S. I. Niculescu, Systemes a retard. Aspects qualitatifs sur la stabilite et la stabilisation, Nouveaux Essais, Diderot, Paris, 1997. [302] G. Niemeyer and J. J. E. Slotine, Stable adaptative teleoperation, in Proc. IEEE American Control Conference, 1991. [303] S. Norman and S. Boyd, Numerical solution of a two-disk problem, in Recent Advances in Robust Control, P. Dorato and R. K. Yedavalli, eds., IEEE Press, Piscataway, NJ, pp. 285-287. [304] B. Oksendal, Stochastic Differential

Equations, Springer-Verlag, New York, 1985.

[305] F. Oustry. The W-Lagrangian of the maximum eigenvalue function, SIAM J. Optim., 9:526-549, 1999. [306] F. Oustry, Vertical developments of a convex function, J. Convex Anal., 5:153-170, 1998. [307] F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function, Math. Programming, 1998, submitted. [308] M. L. Overton, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appi, 9:256-268, 1988. [309] M. L. Overton, Large-scale optimization of eigenvalues, SIAM J. Optim., 2:88-120, 1992. [310] M. L. Overton and R. S. Womersley, Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM J. Matrix Anal. Appl., 16:667-718, 1995. [311] M. L. Overton and X. Ye, Toward second-order methods for structured nonsmooth optimization, in Advances in Optimization and Numerical Analysis, S. Gomez and J.-P. Hennart, eds., Kluwer Academic Publishers, Norwell, MA, 1994, pp. 97-109. [312] A. Packard, Gain scheduling via linear fractional transformations, Systems Control Lett., 22:79-92, 1994. [313] A. Packard and J. C. Doyle, Robust control with an H2 performance objective, in Proc. American Control Conference, 1987, pp. 2141-2146. [314] A. Packard and J. Doyle, The complex structured singular value, Automatica, 29:71-109, 1993. [315] A. Packard, K. Zhou, P. Pandey, and G. Becker, A collection of robust control problems leading to LMI's, in Proc. IEEE Conference on Decision and Control, Brighton, UK, 1991, pp. 1245-1250. [316] A. Packard, K. Zhou, P. Pandey, J. Leonhardson, and G. Balas, Optimal, constant I/O similarity scaling for full-information and state-feedback control problems, Systems Control Lett, 19:271-280, 1992. [317] F. Paganini, Sets and Constraints in the Analysis of Uncertain Systems, Ph.D. thesis, California Institute of Technology, Pasadena, December 1995.

360

Bibliography

[318] F. Paganini, A set-based approach for white noise modeling, IEEE Trans. Automat. Control, 41:1453-1465, 1996. [319] F. Paganini, Frequency domain conditions for robust H2 performance, IEEE Trans. Automat. Control, 44:38-49, 1999. [320] F. Paganini, Convex methods for robust E^ analysis of continuous time systems, IEEE Trans. Automat. Control, 44:239-252, 1999. [321] F. Paganini, R. D'Andrea, and J. Doyle, Behavioral approach to robustness analysis, in Proc. American Control Conference, Baltimore, MD, June 1994, pp. 27822786. [322] F. Paganini and E. Feron, Analysis of robust H2 performance: Comparisons and examples, in Proc. IEEE Conference on Decision and Control, December 1997. [323] C. H. Papadimitriou, Computational Complexity, Addison-Wesley, Reading, MA, 1994. [324] P. Park and T. Kailath, Dual ^-spectral factorization, Internat. J. Control, 1997. [325] P. C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Lyapunov, Proc. Cambridge Philos. Soc., 58:669-672, 1962. [326] G. Pataki, On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues, Technical Report MSRR-604(R), Management Science Research Group, GSIA, Carnegie Mellon University, Pittsburgh, PA, August 1995. [327] R. P. Paul. Problems and research issues associated with the hybrid control of force and displacement, in Proc. IEEE International Conference on Robotics and Automation, 1987, pp. 1966-1971. [328] P. L. D. Peres and J. C. Geromel, H2 control for discrete-time systems, optimality and robustness, Automatica, 29:225-228, 1993. [329] P. L. D. Peres, S. R. Souza, and J. C. Geromel, Optimal H2 control for uncertain systems, in Proc. American Control Conference, Chicago, June 1992. [330] M. A. Peters and A. A. Stoorvogel, Mixed E^/Hoo control in a stochastic framework, Linear Algebra Appl, 205-206:971-996, 1994. [331] I. R. Petersen and D. C. McFarlane, Optimal guaranteed cost filtering for uncertain discrete-time linear systems, Internat. J. Robust Nonlinear Control, 6:267-280, 1991. [332] I. R. Petersen and D. C. McFarlane, Optimal guaranteed cost control of uncertain linear systems, in Proc. American Control Conference, Chicago, June 1992. [333] I. R. Petersen and D. C. McFarlane, Robust state estimation for uncertain systems, in Proc. 30th IEEE Conference Decision and Control, Brighton, UK, 1991, pp. 2630-2631. [334] I. R. Petersen, D. C. McFarlane, and M. A. Rotea, Optimal guaranteed cost control of discrete-time uncertain systems, in Proc. 12th IFAC World Congress, Sydney, Australia, July 1993, pp. 407-410.

Bibliography

361

[335] B. T. Polyak, Subgradient methods: A survey of soviet research, in Nonsmooth Optimization, C. Lemarechal and R. Mifflin, eds., Pergamon Press, Elmsford, NY, 1977. [336] K. Poolla and A. Tikku, Robust performance against time-varying structured perturbations, IEEE Trans. Automat. Control, 40:1589-1602, 1995. [337] V. M. Popov, Absolute stability of nonlinear systems of automatic control, Automat. Remote Control, 22:857-875, 1962. [338] F. A. Potra and R. Sheng, Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming, Technical Report 86, Department of Mathematics, University of Iowa, Iowa City, November 1996. [339] G. Raisbeck, A definition of passive linear networks in terms of time and energy, J. Appl. Physics, 25:1510-1514, 1954. [340] A. Rantzer, On the Kalman-Yakubovich-Popov lemma, Systems Control Lett, 28:7-10, 1996. [341] A. Rantzer and A. Megretski, System analysis via integral quadratic constraints, in Proc. 34#i IEEE Conference on Decision and Control, December 1994, pp. 30623067. [342] A. Rantzer and A. Megretski, Stability criteria based on integral quadratic constraints, in Proc. IEEE Conference on Decision and Control, Kobe, Japan, 1996, pp. 215-220. [343] F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover, New York, 1990. [344] R. T. Rockafellar, Convex Analysis, Princeton, Princeton, NJ, 1970. [345] J. Rohn, Systems of linear interval equations, Linear Algebra Appi, 126:39-78, 1989. [346] J. Rohn, Overestimations in bounding solutions of perturbed linear equations, Linear Algebra Appl., 262:55-66, 1997. [347] M. A. Rotea, Generalized HI control, Automatica, 29:373-385, 1993. [348] M. G. Safonov, Stability and Robustness of Multivariable Feedback Systems, Massachusetts Institute of Technology Press, Cambridge, MA, 1980. [349] M. G. Safonov, L°° optimal sensitivity vs. stability margin, in Proc. IEEE Conference on Decision and Control, Albuquerque, NM, December 1983. [350] M. G. Safonov, Stability margins of diagonally perturbed multivariable feedback systems, IEEE Proc., 129:251-256, 1982. [351] M. G. Safonov and M. Athans, Gain and phase margin for multiloop LQG regulators, in Proc. IEEE Conference on Decision and Control, Clearwater Beach, FL, December 1976. [352] M. G. Safonov and M. Athans, A multiloop generalization of the circle criterion for stability margin analysis, IEEE Trans. Automat. Control, 26:415-422, 1981.

362

Bibliography

[353] M. G. Safonov and R. Y. Chiang, Real/complex A"m-synthesis without curve fitting, in Control and Dynamic Systems, vol. 56, C. T. Leondes, ed., Academic Press, New York, 1993, pp. 303-324. [354] M. G. Safonov, K. C. Goh, and J. H. Ly, Control system synthesis via bilinear matrix inequalities, in Proc. American Control Conference, Baltimore, MD, June 1994. [355] M. G. Safonov and P. H. Lee, A multiplier method for computing real multivariable stability margin, in IFAC World Congress, Sydney, Australia, July 1993. [356] M. G. Safonov and G. P. Papavassilopoulos, The diameter of an intersection of ellipsoids and BMI robust synthesis, in Proc. IFAC Symposium on Robust Control Design, Rio de Janeiro, Brazil, September 1994. [357] M. G. Safonov and G. Wyetzner, Computer-aided stability criterion renders Popov criterion obsolete, IEEE Trans. Automat. Control, AC-32:1128-1131, 1987. [358] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [359] M. Sampei, T. Mita, and M. Nakamichi, An algebraic approach to HQQ output feedback control problems, Systems Control Lett., 14:13-24, 1990. [360] I. W. Sandberg, On the Z/2-boundedness of solutions of nonlinear functional equations, Bell System Tech. J., 43:1581-1599, 1964. [361] C. W. Scherer, Prom LMI analysis to multichannel mixed Imi synthesis: A general procedure, Selected Topics Identification Model. Control, 8:1-8, 1995. [362] C. W. Scherer, Mixed HijH^ control, in Trends in Control, A European Perspective, A. Isidori, ed., Springer-Verlag, Berlin, 1995, pp. 173-216. [363] C. Scherer, Mixed H^jB.^ control for linear parametrically varying systems, in IEEE Conference on Decision and Control, 1995, pp. 3182-3187. [364] C. W. Scherer, Robust generalized HI control for uncertain and LPV systems with general scalings, in Proc. IEEE Conference on Decision and Control, 1996, pp. 3970-3975. [365] C. W. Scherer, P. Gahinet, and M. Chilali, Multiobjective output-feedback control via LMI optimisation, IEEE Trans. Automat. Control, 42:896-911, 1997. [366] H. Schramm and J. Zowe, A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results, SIAM J. Optim., 2:121-152, 1992. [367] G. Scorletti and L. El Ghaoui, Improved linear matrix inequality conditions for gain scheduling, in Proc. IEEE Conference on Decision and Control, 1995, pp. 36263631. [368] G. Scorletti and L. El Ghaoui, Improved LMI conditions for gain scheduling and related control problems, Internat. J. Robust Nonlinear Control, 8:845-877, 1998. [369] U. Shaked and C. E. de Souza, Robust minimum variance filtering, IEEE Trans. Signal Process., 40:2474-2483, 1995.

Bibliography

363

[370] J. S. Shamma, Robust stability with time-varying structured uncertainty, IEEE Trans. Automat. Control, 39:714-724, 1994. [371] J. F. Shamma and M. Athans, Analysis of gain scheduled control for nonlinear plants, IEEE Trans. Automat. Control, 35:898-907, 1990. [372] A. Shapiro, Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton, Appl. Math. Optim., 18:215-229, 1988. [373] A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Math. Programming, Ser. B, 77:301-320, 1997. [374] A. Shapiro and M. K. H Fan, On eigenvalue optimization, SIAM J. Optim., 5:552568, 1995. [375] N. Z. Shor, Minimization Methods for Non-Differentiable Verlag, Berlin, 1985.

Functions, Springer-

[376] N. J. Shor, Quadratic optimization problems, Soviet J. Circuits Systems Sci., 25:111, 1987. [377] R. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, Taylor & Francis, London, 1998. [378] R. S. Smith and M. Dahleh, The Modeling of Uncertainty in Control Systems, Lecture Notes in Control and Inform. Sci., Springer-Verlag, Berlin, 1993. [379] G. Sonnevend, New algorithms in convex programming based on a notion of "center" (for systems of analytic inequalities) and on rational extrapolation, in Trends in Mathematical Optimization, K. H. Hoffmann, J. B. Hiriart-Urruty, C. Lemarechal, and J. Zowe, eds., Internat. Ser. Numer. Math. 84, Birkhauser-Verlag, Basel, 1988, pp. 311-327. [380] A. G. Sparks and D. S. Bernstein, The scaled Popov criterion and bounds for the real structured singular value, in Proc. IEEE Conference on Decision and Control, December 1994, pp. 2998-3002. [381] R. J. Stern and H. Wolkowicz, Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations, SIAM J. Optim., 5:286-313, 1995. [382] J. Stoer and C. Witzgall, Convexity and Optimization in Finite Dimension, Springer-Verlag, Berlin, New York, 1970. [383] A. A. Stoorvogel, The robust ££2 problem: A worst case design, in Con/. Decision and Control, Brighton, England, December 1991, pp. 194-199. [384] A. A. Stoorvogel, The robust H^ control problem: A worst-case design, IEEE Trans. Automat. Control, 38:1358-1370, 1993. [385] M. K. Sundareshan and M. A. L. Thathachar, Instability of linear time-varying systems—Conditions involving noncasual multipliers, IEEE Trans. Automat. Control, 17:504-510, 1972. [386] M. Sznaier, An exact solution to general SISO mixed H2/H infinity problems via convex optimization, IEEE Trans. Automat. Control 39:2511-2517, 1994.

364

Bibliography

[387] M. Sznaier and J. Tierno, Is set modeling of white noise a good tool for robust H2 analysis?, in Proc. 37th IEEE Conference on Decision and Control, Tampa, FL, 1998, pp. 1183-1188. [388] G. Stein and M. Athans, The LQG/LTR procedure for multivariable feedback control design, IEEE Trans. Automat. Control, AC-32:105-114, 1987. [389] K. Takaba and T. Katayama, Robust H2 control of descriptor system with timevarying uncertainty, in Proc. American Control Conference, Philadelphia, June 1998, pp. 2421-2426. [390] Y. Theodor and U. Shaked, Robust discrete-time minimum variance filtering, IEEE Trans. Signal Process., 44:181-189, 1996. [391] M. J. Todd, K. C. Toh, and R. H. Tiitiincu, On the Nesterov-Todd direction in semidefinite programming, SIAM J. Optim., 8:769-796, 1998. [392] K. C. Toh, M. J. Todd, and R. H. Tiitiincu, SDPT3 : A MATLAB Software Package for Semidefinite Programming, 1996. Available at http://www.math.nus.sg/ "mattohkc/index.html. [393] O. Toker, On the complexity of the robust stability problem for linear parameter varying systems, in 13th Triennial IFAC World Congress, Vol. H, 1996, pp. 53-57. [394] O. Toker and H. Ozbay, On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback, in Proc. American Control Conference, 1995. [395] O. Toker and H. Ozbay, Complexity issues in robust stability of linear delaydifferential systems, Math. Control Signals Systems, 9:386-400, 1996. [396] H. Tokunaga, T. Iwasaki, and S. Hara, Multi-objective robust control with transient specifications, in Proc. IEEE Conference on Decision and Control, 1996, pp. 34823483. [397] M. Torki, Epi-differentiabilite du second ordre de certaines fonctions spectrales et conditions d'optimalite en optimisation de valeurs propres, Technical Report, LAO, Universite Paul Sabatier, Toulouse, Prance, December 1997. [398] L. Turan, C. H. Huang, and M. G. Safonov, Two-Riccati positive real synthesis: LMI approach, in Proc. American Control Conference, 1995, pp. 2432-2436. [399] H. D. Tuan and P. Apkarian, Relaxations of parameterized LMIs with control applications, Internat. J. Robust Nonlinear Control, to appear. [400] H. D. Tuan, S. Hosoe, and H. Tuy, D.C. Optimization Approach to Robust Control: Feasibility Problems, Technical Report 9601, Nagoya University, Nagoya, Japan, October 1996. [401] L. Vandenberghe and S. Boyd, SP: Software for Semidefinite Programming. Users Guide, Stanford University, Stanford, CA, 1994. Available at http://www.stanford. edu/~boyd/SP.html. [402] L. Vandenberghe and S. Boyd, SP, Software for Semidefinite Programming, User's Guide, December 1994. Available via anonymous ftp at isl.stanford.edu, under /pub/boyd/semidef_prog.

Bibliography

365

[403] L. Vandenberghe and S. Boyd, A primal-dual potential reduction method for problems involving matrix inequalities, Math. Programming Ser. B, 69:205-236, 1995. [404] L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38:49-95, 1996. [405] L. Vandenberghe, S. Boyd, and A. El Gamal, Optimal wire and transistor sizing for circuits with non-tree topology, in Proc. IEEE/A CM International Conference on Computer-Aided Design, San Jose, CA, November 1997, pp. 252-259. [406] L. Vandenberghe, S. Boyd, and A. El Gamal, Optimizing dominant time constant in RC circuits, IEEE Trans. Computer-Aided Design, 2:110-125, 1998. [407] L. Vandenberghe, S. Boyd, and S.-P. Wu, Determinant maximization with linear matrix inequality constraints, SIAM J. Matrix Anal. Appl., 19:499-533, 1998. [408] J. J. van Dixhoorn and F. J. Evans, Physical Structure in Systems Theory, Academic Press, New York, 1974. [409] P. Van Dooren, A generalized eigenvalue approach for solving Riccati equations, SIAM J. Sci. Statist. Comput, 2:121-135, 1981. [410] S. A. Vavasis, Nonlinear Optimization: Complexity Issues, Oxford University Press, New York, 1991. [411] M. Vidyasagar, Control System Synthesis: A Factorization Approach, MIT Press, Cambrigde, 1985. [412] V. Visweswaran, C. A. Floudas, M. G. lerapetritou, and E. N. Pistikipoulos, A Decomposition-Based Optimization Approach for Solving Bilevel Linear and Quadratic Programs, Kluwer Academic Publishers, Norwell, MA, 1996. [413] P. G. Voulgaris, Optimal h2/h control via duality theory, IEEE Trans. Automat. Control, 11:1881-1888, 1995. [414] W. Wang, J. Doyle, C. Beck, and K. Glover, Model reduction of LFT systems, in Proc. IEEE Conference on Decision and Control, Brighton, England, December 1991, pp. 1233-1238. [415] Z. Q. Wang and S. Skogestad, Robust controller design for uncertain time delay systems, in Analysis and Optimization of Systems: State and Frequency Domain Approaches for Infinite-Dimensional Systems, Lecture Notes in Control and Inform. Sci., Springer-Verlag, Berlin, 1993, pp. 610-623. [416] C. Weber and J. Allebach, Reconstruction of frequency offeset Fourier data by alternating projections, in Proc. 24th Allerton Conference, Monticello, IL, 1986, pp. 194-201. [417] D. E. Whitney, Force feedback of manipulator fine motion, J. Dynam. Systems and Measurement Control, 9:91-97, 1977. [418] D. E. Whitney, Historical perspective and state of the art in robot force control, Internat. J. Robotics Res., 6:3-14, 1987. [419] J. C. Willems, The Analysis of Feedback Systems, Research Monographs 62, MIT Press, Cambridge, 1969.

366

Bibliography

[420] J.C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control, AC-16:621-634, 1971. [421] J. C. Willems, On the existence of a nonpositive solution to the Riccati equation, IEEE Trans. Automat. Control, AC-19:592-593, 1974. [422] W. M. Wonham, Random differential equations in control theory, in Probabilistic Methods in Applied Mathematics, Vol. 2, A. T. Bharucha-Reid, ed., Academic Press, New York, 1970, pp. 31-212. [423] S. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, 1997. [424] F. Wu, X. Yang, A. Packard, and G. Becker, Induced L2-norm control for LPV system with bounded parameter variations rates, in Proc. American Control Conference, Seattle, WA, 1995, pp. 2379-2383. [425] S.-P. Wu and S. Boyd, sdpsol: A Parser/Solver for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure. User's Guide, Beta Version, Stanford University, Stanford, CA, June 1996. [426] S.-P. Wu and S. Boyd, Design and implementation of a parser/solver for SDPs with matrix structure, in Proc. Conference on Control and Applications, Dearborn, MI, September 1996, pp. 240-245. [427] S.-P. Wu, L. Vandenberghe, and S. Boyd, MAXDET: Software for Determinant Maximization Problems. User's Guide, Alpha Version, Stanford University, Stanford, CA, April 1996. [428] L. Xie and C. E. de Souza, LMI approach to delay-dependent robust stability and stabilization of uncertain linear delay systems, in Proc. 35th IEEE Conference on Decision and Control, December 1995. [429] L. Xie and C. E. de Souza, On robust filtering for linear systems with parameter uncertainty, in Proc. 34th IEEE Conference on Decision and Control, New Orleans, LA, 1995, pp. 2087-2092. [430] L. Xie, C. E. de Souza, and Y. C. Soh, Robust Kalman filter for uncertain discretetime systems, IEEE Trans. Automat. Control, 39:1310-1314, 1994. [431] L. Xie and Y. C. Soh, Robust Kalman filtering for uncertain systems, Systems Control Lett, 22:123-129, 1994. [432] V. A. Yakubovich, The solution of certain matrix inequalities in automatic control theory, Soviet Math. DokL, 3:620-623, 1962 (in Russian, 1961). [433] V. A. Yakubovich, Nonconvex optimization problem: The infinite-horizon linearquadratic control problem with quadratic constraints, Systems Control Lett., 19:1322, 1992. [434] V. A. Yakubovich, The 5-procedure in non-linear control theory, Vestnik Leningrad Univ. Math., 4:73-93, 1977 (in Russian, 1971). [435] J. Yan and S. E. Salcudean, Teleoperation controller design using optimization with application to motion-scaling, IEEE Trans. Control Systems Tech., 4:244258, 1996.

Bibliography

367

[436] K. Y. Yang, Efficient Design of Robust Controllers for JJ.2 Performance, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, February 1997. [437] K. Yang, S. Hall, and E. Feron, A new design method for robust H2 controllers using Popov multipliers, in Proc. American Control Conference, Albuquerque, NM, June 1997, pp. 1235-1240. [438] K. Y. Yang, C. Livadas, and S. R. Hall, Using linear matrix inequalities to design controllers for robust H2 performance, in Proc. AIAA Guidance, Navigation, and Control Conference, July 1996. [439] T. Yang and L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication, IEEE Trans. Circuits Systems, 44:976-988, 1997. [440] D. C. Youla and H. Webb, Image restoration by the method of convex projections, IEEE Trans. Medical Imaging, 1:81-94, 1982. [441] P. Young, Robustness with Parametric and Dynamic Uncertainty, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1993. [442] P. M. Young and J. C. Doyle, Properties of the mixed /i problem and its bounds, IEEE Trans. Automat. Control, 155-159, 1996. [443] P. M. Young, M. P. Newlin, and J. C. Doyle, Practical computation of the mixed p, problem, in Proc. American Control Conference, Chicago, IL, 1992. [444] G. Zames, On the input-output stability of nonlinear time-varying feedback systems—Parts I and II, IEEE Trans. Automat. Control, 11:228-238, 465-467, 1966. [445] G. Zames and P. L. Falb, Stability conditions for systems with monotone and slope-restricted nonlinearities, SIAM J. Control, 6:89-108, 1968. [446] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ, 1995. [447] K. Zhou, K. Glover, B. Bodenheimer, and J. Doyle, Mixed H.2 and H.^ performance objectives I: Robust performance analysis, IEEE Trans. Automat. Control, 39:1564-1574, 1994. [448] G. Zhu, K. M. Grigoriadis, and R. E. Skelton, Covariance control design for the Hubble space telescope, J. Guidance Control Dynam., 18:230-236, 1996.

This page intentionally left blank

Index Control channel, 188 Control gain iteration, 169, 170 Control gain with relaxation iteration, 169, 173 Controller order, 270 Convex analysis, 58 Convex cone, 270 Convex optimization, 3 Convexity, 57 Coupled Riccati equations, 156, 167 Cumulative spectrum, 141 Cutting-plane method, 60

a-stability, 253 Affine multiplier, 97 Affine quadratic stability, 97 Affine quadratic stability test, 105 Alternating projection methods, 257 Analytic centers, 61 Asymptotic properties, 57 Asymptotic quadratic convergence, 69 Barrier, 61 Barrier methods, 290 Bilinear matrix inequality, 34, 156, 269 computational aspects, 289 methodological aspects, 274 structural aspects, 279 Bilinear sector transform, 273 Bounded real, 274 £-positivity, 286, 291 Bundle, 57 Bundle methods, 24, 61

Decentralized, 276 Decentralized controller design, 275 Decision variables, 5 Delay differential systems, 27 Determinant maximization problems, 22 D, G-K iteration, 275 Differential geometry, 70 Discrete-time parametric Kalman-YakubovichPopov lemma, 106 D-K iteration, 156, 167, 171, 275 Dominant delay, 51 Dominant harmonics, 115 D-scaling, 232 Dual map, 15 Dual problem, 83 Duality, 21 Dynamical systems over cones, 35

Central path, 23, 45 Certainty channel, 188 Clock mesh design, 51, 52 Combinatorial optimization, 50 Combinatorial problems, 29 Combinatorial problems with uncertain data, 29 Complementarity, 70 Complementarity conditions, 23, 43 Concave minimization, 290 Concave programming, 279 Cone optimization problems, 270 Cone programming, 282 Cone-LCP, 271, 287, 288 Cone-LCP formulation of the bilinear matrix inequality, 286 Cone-LP, 271 Conic complementarity problems, 34 Conservatism, 173 Conservative, 172 Constraints, 5

Eigenvalue optimization, 57 Elimination lemma, 165 Ellipsoid algorithm, 60 Embedding, 5 Equality constraints, 85 Estimation, 175 Experiment design, 88 Extreme form, 270, 287 Extreme form problem, 269, 271, 282, 285 Extreme ray, 270 369

Index

370

Finite gain, 157 Fixed order (q < n] HOQ problem, 278 Fixed order control synthesis, 275 Force control, 332 Full block ^-procedure, 187 Full-order robust filter, 177 Gain-scheduled control, 245 Gain-scheduled output-feedback control problem, 211 Gain-scheduling techniques, 209 Generalized eigenvalue problem, 22, 90 Generalized Kalman-Yakubovich-Popov lemma, 98 Global convergence, 69 Global maximum, 291 Gordan's theorem of alternative, 283 Graph problem, 59 H2, 145 H2 norm, 130 impulse response interpretation, 133 stochastic white noise interpretation, 135 H2-performance, 129, 155 H2/Popov controllers, 156,164,165,169, 170 Hoc, 129 Homogeneous cone, 289

Impulse differential systems, 26 Integral quadratic constraints, 13, 110, 231 Interior-point methods, 22, 290 Intermediate oracle, 59 Interval calculus, 30 Kalman filter, 176, 180 Kalman-Yakubovich-Popov lemma, 28, 98 Li, 129 L2-gain performance, 210 L2-induced operator norm, 244 Lagrange multipliers, 133, 159 Lagrange relaxation, 7, 15, 19 Large-scale, 57 Limitations of the LMI approach, 275 Linear complementarity problem, 271 Linear Lyapunov function, 21 Linear matrix inequality, 3,129, 229, 274 Linear parameter-varying system, 244

Linear program, 16 Linear program with implementation constraints, 16 Linear programming, 271 Linear time-invariant, 95 Linear-fractional model, 9 Linear-fractional transformation, 10, 231, 235 Lovasz ^-function, 50 Lovasz problem, 74 Lyapunov equation, 168 Lyapunov exponent, 90 Lyapunov functions, 18, 19, 103 Lyapunov inequality, 86 Lyapunov-Krasovskii functional, 28 H/km analysis and synthesis, 275 H/km synthesis, 270, 275, 276 Matlab, 168 MAX-CUT, 29, 50 Max-det problem, 79 Maximum eigenvalue function, 59 Maximum eigenvalue minimization, 59 Mehrotra's formula, 46 Middeck Active Control Experiment (MACE), 156, 170 Mixed parameter-scheduled/robust problems, 6 Mixed semidefimte-quadratic-linear programs, 41 M-K iteration, 275 Moreau-Yosida, 62 Multiaffine Lyapunov function, 97 Multiaffine Lyapunov matrix, 98 Multiobjective robust control, 326 Multiobjective Robust Control Toolbox, 309 Multiplier formulation, 156 Multipliers, 98, 109, 156, 160 Network theory, 322 Nonconservative, 172 Nondegeneracy, 70 Nondifferentiable optimization methods, 24 NP-hardness of the bilinear matrix inequality, 280 Numerical experiments, 61 Numerical illustrations, 57 Optimal control, 25

371

Index Optimality conditions, 23 Parallel implementation of bilinear matrix inequality, 289 Parameter-dependent Lyapunov functions, 210, 211 Parameter-scheduled synthesis, 6 Parametric Kalman-Yakubovich-Popov lemma, 98 Parametric Lyapunov functions, 96 Parametric multiplier approach, 95 Passive, 157 Passivity, 157, 325 Path following, 23 Path-following algorithm, 44 Peak-to-peak gain, 195 Performance block, 242, 245 Performance channels, 188 Pointed closed convex cone, 270 Polynomial interpolation, 31 Polytopic linear differential inclusions quadratic stability, 75 Polytopic model, 14 Poor oracle, 59, 76 Poor-oracle methods, 58 Popov analysis, 86 Popov multiplier, 111, 156 Popov multipliers, 160, 162 Positive, 157 Positive real, 274 Positivity, 157 Potential function, 77 Predictive robust control, 28 Predictor-corrector scheme, 44 Primal-dual interior-point method, 22, 44 Projection operator, 257 Proximal point, 62 PSD-completely positive matrices, 285 •pSD-copositive, 288 PSD-copositive matrices, 285 P-separator, 232, 241 Quadratic stability, 20 Quadratic-sdp problem, 68 Quality of relaxation, 18, 36 Rank relaxation, 16 Rank-constrained linear matrix inequalities, 269 Rank-constrained problems, 34

Rank-minimization problem, 291 Rank-restricted linear matrix inequalities, 282 Rayleigh variational formulation, 58 Reduced-order robust filter, 177 Riccati equality, 163 Riccati equations, 156 Rich oracle, 59, 64, 77 Robot telemanipulators, 322 Robust control, 242 Robust controller design, 199 Robust decision problems, 6 Robust feasibility problem, 6 Robust full-information problem, 166 Robust H 2 , 129 Robust H2 control, 155 Robust H2 niters, 175 Robust H2-performance, 131, 155, 193 Robust H2 synthesis, 164 Robust least squares, 31, 51, 53 Robust linear programming, 16 Robust measurement scheduling, 311 Robust optimality problem, 6 Robust output estimation problem, 166 Robust performance, 116 Robust performance analysis, 113 Robust portfolio optimization, 33 Robust quadratic performance, 191 Robust semidefinite program, 14 Robust stability, 158, 189 Robustness, 129 Robustness analysis, 6, 111 SDPpack, 47 Second-order analysis, 57, 70 Second-order bundle methods, 24 Second-order schemes, 70 Self-concordant barrier, 290 Semidefinite complementarity problem, 269, 282

Semidefinite program, 3 Semidefinite program duality, 25 Semidefinite-quadratic-linear program, 41 Separation principle, 166, 167 Software for solving semidefinite programs, 24

Solid, 270 5-procedure, 8, 129, 130, 159 Star product, 235 Stochastic Lyapunov functions, 20 Stochastic optimization, 7

372 Structured linear algebra, 31 Structured total least squares, 32 Subdifferential, 58 Subgradient methods, 24, 60 Support function, 58 Symmetrized Kronecker product, 45 Toolbox for IQC analysis, 125 Topological separation, 232 Truncation operator, 110 Truss topology design, 53 W-Hessian, 77 ZY-Lagrangian theory, 70 Uncertain systems, 5 Uncertainty, 272 Uncertainty model, 9 Uncertainty set assumption, 9 Unsymmetric eigenvalue problem, 17 Upper bound, 159, 160 Upper bounds on robust ^-performance, 156 Well-posedness, 9, 189, 230 Worst-case H/2 norm, 159 Worst-case H2-performance, 156, 158 Worst-case asymptotic error variance, 177 Youla parameterization, 166

Index