Fuzzy Algorithms for Control

FUZZY ALGORITHMS FOR CONTROL

Edited by H.B. Verbruggen Delft University of Technology H.-J. Zimmermann RWTH Aachen R. BabuSka Delft University of Technology

Kluwer Academic Publishers Boston/Dordrecht/London

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Copyright 0 1999 by Kluwer Academic Publishers.

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Printed in the United States of America

INTERNATIONAL SERIES IN INTELLIGENT TECHNOLOGIES Prof. Dr. Dr. b.c. Hans-Jiirgen Zimmermann, Editor

European Laboratory for Intelligent Techniques Engineering

Aachen, Germany Other books in the series: Fuzzy Logic and Intelligent Systems edited by Hua Li and Madan Gupta Fuzzy Set Theory and Advanced Mathematical Applications edited by Da Ruan

Fuzzy Databases: Principles and Applications by Frederick E. Petry with Patrick Bose Distributed Fuzzy Control of Multivariable Systems by Alexander Gegov Fuzzy Modelling: Paradigms and Practices by Witold Pedrycz Fuzzy Logic Foundations and Industrial Applications by Da Ruan Fuzzy Sets in Engineering Design and Configuration by Hans-Juergen Sebastian and Erik K. Antonsson Consensus Under Fuzziness by Mario Fedrizzi, Janusz Kacprzyk, and Hannu Nurmi Uncertainty Analysis in Enginerring Sciences: Fuzzy Logic, Statistices, and Neural Network Approach by Bilal M . Ayyub and Madan M. Gupta Fuzzy Modeling for Control Robert BabuSka Trafic Control and Transport Planning: A Fuzzy Sets and Neural Networks Approach by DuSan Teodorovid and Katarina Vukadinovid

Contents

Preface Part I THE POSITION AND STATE OF T H E ART OF FUZZY SYSTEMS 1

Fuzzy Systems in Control Engineering

H. B. Verbruggen and P.M. Bruzjn 1.1 1.2 1.3 1.4 1.5 1.6

lntroduction Control engineering: solutions and limitations Advanced control Fuzzy control Misunderstandings and possibilities Conclusions

References 2 Fuzzy Logic, Control Engineering and Artificial Intelligence

D. Dubois, H. Prude and L. Ughetto 2.1 2.2 2.3 2.4 2.5 2.6

lntroduction Background on approximate reasoning Approximate reasoning vs. fuzzy logic controllers Validation of fuzzy rule bases Interpolation with fuzzy rules Conclusions

References

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3 Fuzzy Control Versus Conventional Control

K.-E. A r z ~ n M. , Johansson and R. BabuSka 3.1 3.2 3.3 3.4 3.5

lntroduction Fuzzy control systems Industrial fuzzy control Modern nonlinear fuzzy control Summary

References 4 Data-Driven Construction of Transparent Fuzzy Models

R. BabuSka and M. Setnes 4.1 4.2 4.3 4.4 4.5 4.6 4.7

lntroduction Fuzzy model structure Fuzzy clustering Extraction of an initial rule base Simplification and reduction of the rule base Example: Fuzzy modeling and control of an HVAC System Conclusions

References

Part II DESIGN AND ANALYSIS ISSUES 5 Fuzzy Logic Normal Forms for Control Law Representation

111

I. Perfilieva

111 lntroduction 113 Fuzzy logic control models and their normal forms 117 Functional realization of normal forms Approximation and representation of real-valued real continuous 119 functions 123 5.5 Conclusions 124 Appendix: Proof of Theorem 2

5.1 5.2 5.3 5.4

References 6 Stability Analysis of Fuzzy Control Loops

A. Ollero, J.P. Marin, A. Garcia-Cerezo and F. Cuesta 6.1 6.2 6.3 6.4 6.5

lntroduction Fuzzy control engineering practice and stability Input-output methods for stability analysis Lyapunov approaches Conclusions and perspectives

References

4

Contents

vii

7 Performance Criteria: Classical and Fuzzy Design

J.M. Sousa, U. Kaymak and H.B. Verbruggen 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Introduction Design specifications Classical performance specifications Classical performance criteria Fuzzy performance criteria Fuzzy performance criteria in model-based predictive control Summary and suggestions for further research

References

182

8 Complexity Reduction Methods for Fuzzy Systems

M. Setnes, V. Lacrose and A. Tztli 8.1 8.2 8.3 8.4 8.5 8.6

Introduction Elimination and selection Rule base simplification Dimensionality reduction Structured systems Concluding remarks

References Part Ill APPLICATION OF FUZZY SYSTEMS 9 Intelligent Data Analysis and Fuzzy Control H.-J. Zzmmermann, J. Angstenberger and R. Weber Basic principles of data analysis 9.1 Methods for fuzzy data analysis 9.2 9.3 Tools for data analysis Fuzzy data analysis and fuzzy control 9.4 9.5 Industrial applications 9.6 Conclusions References

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FUZZY ALGORITHMS FOR CONTROL

vlll

10 Fuzzy Control in Process Industry

E. K. JUUSO 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

lntroduction Fuzzy expert control Fuzzy modeling Fuzzy models in nonlinear control Adaptive fuzzy control Application: Solar power plant Application: Lime kiln Implementation Conclusions

References 11 Fuzzy Logic Applications in Mobile Robotics

A. Ollero, G. Ulivi and F. Cuesta 11.1 11.2 11.3 11.4

lntroduction Fuzzy logic in reactive navigation An assisted guidance wheelchair for disabled people Conclusions

References 12 Enhancing Flight Control using Fuzzy Logic

G. Schrarn, M.A. Ferncindez-Montesinos and H. B. Verbruggen 12.1 12.2 12.3 12.4 12.5

lntroduction Fuzzy control of a civil aircraft benchmark Control reconfiguration in case of failures Recovery from large disturbances Conclusions

References Index

Preface

This edited book gives an overview of the work performed by a number of European research groups, that worked together in a consortium called Falcon (Fuzzy Algorithms for Control) under the umbrella of the European Community program to stimulate activities in this rapidly developing field. New results of analysis and design and some new applications have been included as well to show the progress that has been made in the years after the formal termination of the working group. As a result of the Falcon activities, new research and information dissemination programs have been initiated by the European Community in which the participants of the Falcon consortium still play a leading role. After the seminal work of Zadeh in the sixties, it took a long time before fuzzy sets and fuzzy logic were accepted by the academia and the industry. Fuzzy sets and fuzzy logic caused a major paradigm shift that influenced many scientific, technical and non-technical areas. This is one of the reasons why this new theory had to face severe criticism from some scientific areas, which felt that their carefully built positions were attacked. Control engineering is one of the fields in which fuzzy logic was introduced early in the seventies. European researchers, such as Marndani in the UK, Holmblad and 0stergaard in Denmark and van Nauta Lemke in the Netherlands took the lead in this area. A general acceptance in the control engineering community could not be obtained before the nineties, however, after a breakthrough of fuzzy logic applications in consumer goods and in other areas, which was stimulated by a powerful research program in Japan. In 1992,prof. H.-J. Zimmermann took the initiative to form a European consortium to promote the research and the application of fuzzy sets and fuzzy logic. A working group on fuzzy algorithms for control (Falcon) was set up by research teams from Germany, France, the United Kingdom, Belgium, the Netherlands, Italy and Spain. This group was subsidized by the European Community. Later on, partners from Sweden and Finland became associated members and also researchers from Russia were involved in the working group. At the same time, a major European Congress on Intelligent Techniques and Soft Computing (EUFIT) was organized, and again Zimmermann was the initiator of this event. This conference is annually held in Aachen and attracts about 600 participants, both from the academia and the industry.

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This book contains 12 chapters clustered into three parts, which each contains four chapters. Most of the chapters could have been included in at least two parts of the book, but by using fuzzy clustering techniques in an intuitive way, we came to the current division of the chapters. Chapters in Part I address the position of fuzzy systems in systems and control engineering. It took a long time before fuzzy modeling and control techniques were accepted by the control community. However, it has been shown that these techniques can provide solutions to problems that would remain unsolved or yield unsatisfactory results with conventional methods. Fuzzy control also faces some criticism from the A1 community. In many cases, fuzzy sets and fuzzy logic are used to approximate a nonlinear mapping between input and output signals, rather than that human experience and reasoning are used along the seminal ideas of fuzzy sets and systems. The first chapter by Verbmggen and Bruijn gives an overview of areas where fuzzy modeling and control can be beneficial for control engineering. It describes general issues in control engineering problems which should be solved and compares the fuzzy-system approach to conventional and advanced approaches based on analytical descriptions of the problem. It shortly comments on why fuzzy control is still to some extent a controversial subject and cautions that fuzzy control is not a panacea for all unsolved problems. The second chapter by h z t n , Johansson and BabuSka starts with a short overview of the basic concepts of fuzzy control systems, such as the structure of the controller (including prefiltering and postfiltering blocks). A comparison with the commonly used industrial controllers is presented and some nonlinear fuzzy control structures are described. The application of fuzzy models can play an important role in nonlinear control. Construction of transparent fuzzy models from data is the subject of Chapter 3 by BabuSka and Setnes. It is shown that fuzzy models extracted from data in an automatic manner are to some degree redundant and thus can be simplified. Similarity measures are used to achieve this simplification and to provide transparentmodels. This approach is demonstrated on the modeling and control of an air-conditioning system. The fourth chapter by Dubois, Prade and Ughetto analyses the coherence between fuzzy-control methodologies and the field of approximate reasoning. It is shown that discrepancies exist between the two fields and that the pragmatic engineering approach based on a nonlinear mapping of input and output signals, which is now mostly used in fuzzy control, could benefit from a more formal approach based on knowledge representation. It is also shown that the notion of gradual rules makes it possible to recover the concept of interpolation as a special case of fuzzy-logic inference. Part 11, which also contains four chapters, is concerned with several analysis and design issues in fuzzy control systems. The control theory community built up an elegant and nearly complete framework for the analysis and design of linear control systems. Moreover, many concepts have been developed for nonlinear systems, but a complete framework for these systems does not exist. It is clear that in order to compete with the approach presented hitherto in the control community, fuzzy control must incorporate some of the major concepts of control theory, such as stability and performance evaluation.

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In Chapter 5, Perfilieva introduces fuzzy logic normal forms for the representation of control laws. The chapter also addresses using fuzzy-logic control models as universal approximators. Fuzzy logic normal forms are introduced for the uniform algebraic representation of these models. It is shown that a class of real-valued functions can be characterized by a certain fuzzy logic form. In control engineering, the concept of stability of nonlinear systems is very important and it is clear that stability analysis should be an important tool for analyzing the nonlinear behavior of fuzzy control systems. In Chapter 6 , Ollero, Marin, GarciaCerezo and Cuesta present a thorough overview of the stability analysis of fuzzy control loops. The chapter first reviews the basic stability concepts. Then, two different approaches are considered in more detail: input-output stability and Lyapunov stability. The relation between the two approaches is also highlighted. Although stability is a basic condition for a control system, in order to achieve the desired performance, a number of design specifications should be fulfilled. This is the subject of Chapter 7 by Sousa, Kaymak and Verbruggen. The main objectives to be fulfilled are related to the performance specifications with regard to the speed and accuracy of the control system and the ability to handle disturbances and model-plant mismatch (robustness of the overall system). It is shown that fuzzy decision-making algorithms are able to translate the objectives and constraints derived from the controldesign goals in a transparent way. In many applications in modern process-industry and manufacturing systems, the control problem cannot be divided into a number of single-input, single-output (SISO) systems and the interaction between variables in these systems has to be considered. The number of rules in such a MIMO system is large, as it is an exponential function of the number of input variables and the number of linguistic terms. The complexity of multivariable fuzzy rule-based systems is the central topic of Chapter 8 by Setnes, Lacrose and Titli. This chapter gives a survey of methods to reduce the complexity in fuzzy systems by pruning insignificantrules, or by using decentralized and hierarchical structures. It is shown that the methods presented in this chapter can reduce the complexity of the system considerably and can provide the user or designer with more insight in the system. In the third part of the book, a number of applications of fuzzy control are presented. Only those application areas are covered in which the Falcon group members were most active or where an approach is presented which is different from those presented elsewhere. Chapter 9 by Zimmermann, Angstenbergen and Weber is to a certain extent related to the previous chapter. Advanced data-analysis techniques can be applied to reduce the complexity of large amounts of data. The overall goal is to find structure in these data by partitioning them into relatively few classes of similar objects described by some attributes. A number of methods for fuzzy data analysis is described. In data analysis, it is not a priori clear which method should be applied to get the desired solution. Therefore, it is very important to pre-process the data before they become input to the data-analysis method. A software tool for data analysis, called DataEngine, is described in more detail. The relation between fuzzy data analysis and fuzzy control

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is considered and, finally, an industrial application is presented of a blast-furnace process. Modern process industry presents many challenges for the design of control systems. Because of the customer-driven approach, the minimization of energy consumption and waste of material and because of the restrictions set by environmental regulations, a complex control setting must be considered. One has to cope with problems such as process nonlinearity, time-varying characteristics, incomplete process knowledge, etc. Chapter 10 by Juuso describes some of the problems in the process industry which can be solved by fuzzy control. To handle fuzzy control, a linguistic equation approach has been developed. Fuzzy modeling based on expert knowledge is used, resulting in a compact linguistic equations type of model and it is shown how these models can be used in a fuzzy-control configuration. The benefits of the presented method are demonstrated in two real applications: a solar-plant control system and a lime kiln plant. The chapter concludes with an overview of tools available for the implementation of fuzzy controllers in the process industry. A number of challenging problems in which fuzzy logic can play an important role can be found in mobile robotics. Fuzzy logic has been used to consider the inaccuracy of the sensors and, in general, the uncertainties due to the limitations of the robot's perception system. It is also useful to express control commands and navigation strategies in a qualitative form. However, the complexity of the system becomes very high and requires a large number of rules to express behaviors of the complete mobilerobot system. The reactive navigation problem is addressed by applying the concept of virtual sensors in order to reduce the complexity and to add some flexibility to the implementation of the behaviors. Next, obstacle avoidance, map building, localization and planning problems are treated. Applications of fuzzy control to a mobile robot and a wheelchair for disabled people are presented. The last chapter of the book, by Schram, Aznar and Verbruggen, is concerned with the enhancement of flight control by means of fuzzy logic. In aeronautical systems, fuzzy logic can reduce the extensive efforts in the design of control systems for new aircraft. Moreover, unexpected situations, such as changing weather conditions and system failures, can be handled by fuzzy systems to a certain extent. It is expected that in the near future, problems with the certification of fuzzy-control applications in aircraft will be solved. The chapter first describes the design of a fuzzy MIMO controller for the lateral and longitudinal flight path. A comparison is made with conventional controllers. Next, the problem of flight safety is considered and possible solutions using fuzzy logic are presented. Two examples are given. First, a control reconfiguration is described, for a situation where an engine failure occurs; the induced moment and the decreased force must be compensated by the remaining actuators and control surfaces. The second example concerns the microburst phenomenon, an unexpected vertical displacement of air that is radiated outward as it reaches the ground. A recovery strategy has been developed which can be readily implemented in a fuzzy flight-control framework. It was not an easy task to complete the final text of this edited book. The authors are involved in many projects concerning fuzzy and conventional control systems and little time was available for most of them to write chapters on research, which is, in

PREFACE

xiii

some cases, still going on. Fuzzy control has finally become an accepted and respected method in the control-engineering community and this is at least partly the result of the efforts of the Falcon-group members. We wish to thank the authors for their contributions and for reviewing chapters of their colleagues. Finally, we want to thank Magne Setnes and Stanimir Mollov, Ph.D. students of the Control Laboratory of the Delft University of Technology, for their excellent job to produce in a short period a revised version of the chapters complying with the requirements of the publisher.

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THE POSITION AND

STATE OF THE ART OF FUZZY SYSTEMS

1

FUZZY SYSTEMS IN CONTROL ENGINEERING H.B. Verbruggen and P.M. Bruijn

Delft University of Technology Faculty of Information Technology and Systems Control Laboratory Mekelweg 4, PO Box 5031 2600 GA Delft, The Netherlands

1.1

INTRODUCTION

Classical control methods have shown their applicability in many practical control problems in industry. It is shown, however, that still unanswered questions remain, which can probably be solved with the fuzzy system approach. Modern production methods and modern production units require increased flexibility, resulting in highly nonlinear system behavior of partly unknown systems. Advanced control methods developed by system and control theorists are only partly able to satisfy the demands. It is in this area that fuzzy modeling and control methods can play an important role, because available qualitative operator and design knowledge can easily be implemented. In this chapter, the possible role of fuzzy systems in low level control and in more advanced control is indicated. The introduction of fuzzy methods has been a controversial subject and has resulted in many misunderstandings. This chapter tries to clarify this situation and to emphasize the possible cooperation between the various players in the game: conventional control theory, fuzzy control, the A1 community, and last but not least the end users.

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1.2 CONTROL ENGINEERING: SOLUTIONS A N D LIMITATIONS Control Engineering has been for centuries an art in which craftsmen like the builders of windmills and steam engine designers used experiments, their common sense, and their experience to control the speed of the wings of a mill, to set the mill automatically in the right direction and to govern the speed of steam engines. It was not until this century that a mathematical description was given of the principles, analysis and design of feedback systems. In the first decades of our century, much equipment was built, based on mechanical, hydraulic and pneumatic solutions. Ingenious flap and nozzle constructions served as controllers, while measurement equipment provided the information which was processed by the controller and delivered as a manipulated variable to the actuator acting on the flow, pressure, speed and other quantities which influence the system. The introduction of computers provided new and challenging possibilities and opened many application areas for control engineering. However, control engineering in practice was and is still based in the majority of applications on simple controllers and the application of simple tuning rules to set the parameters of these controllers. It is mainly in the last four decades that systems and control theory has been developed and has become a sophisticatedand highly respected science. However, few of the theories were applied in common practice. Only in very sophisticated areas such as the aeronautical and space industry one can find applications of advanced system and control theories, and also in sophisticated mass-produced (consumer) products are these new theories sometimes applied despite their high development costs. What are the reasons for the gap between sophisticated theory and common practice? What are the possibilities for fuzzy systems? To find an answer to these questions, we first go back to the basic questions which should be posed when solving a control problem: what should be controlled, what are the requirements and what type of control algorithm should be used? 1.2.1

What should be controlled?

This seems to be an easy question, because it boils down in most cases to simple answers such as: the temperature of a batch reactor, the flow in a supply pipe line, the speed of a motor, the position of an antenna, etc. But behind the asking of this question there is a hidden world of experience, reasoning and knowledge, because the chosen quantity is perhaps a substitute for the real quantity we want to control: the quality of a product, the maximization of the throughput, the minimization of the waste of material, or the minimization of the consumption of energy. In most processes, a number of quantities are controlled; these interact, have constraints and are sometimes only partly manageable. The dynamic properties of the system significantly influence the speed and the magnitude of the actual and possible behavior. Our simple question expands into many questions such as:

What is the dynamic behavior of the system to be controlled? is the system linear or nonlinear? time-varying? exhibits delay time?

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5

What are the disturbances acting on the system? can they be measured and compensated for? Which quantities are measurable and which quantities can be reconstructed from measurements? Which quantities describe to a reasonable extent the unknown or non-measurable quantities we actually want to control? Which quantity should be controlled by which manipulated quantity? Therefore, the answer to our simple question can be very complicated, and is to a certain extend sustained by methods developed in control theory. However, still, many questions can only be solved by human experience using qualitative information (reasoning).

1.2.2 What are the requirements? This question is related to the previous one and describes the final behavior of the controlled system. It is related to such questions as: What is the required range of the controlled quantity? m

How fast should the system react to a required set point change (tracking behavior) or diminish the influence of sudden disturbances (disturbance rejection) or diminish the influence of random disturbances (minimum variance control)?

These questions can sometimes be related to such simple criteria as the amount of overshoot allowed, the rise-time of the system to a set point change, the settling time of the response, the relative and absolute damping ratio, the reduction of the variance of the noise acting on the system and the ability to follow a desired response (model reference). The requirements can also be translated to performance criteria or cost functions that penalize the difference between the desired and the actual behavior of process variables together with the magnitude of the manipulated variables. Usually, quadratic criteria are used which are easy to manipulate mathematically when the system to be controlled is linear, thus resulting in a linear control law. Once the system is known and the cost function has fixed parameters, the controller design is a straightforward procedure that is solved analytically; the structure and the parameters of the control algorithm follow directly from the system and the requirements. Not much seems to be left to the creativity of the designer. This "ideal" situation is, however, hardly met because it requires the exact knowledge of the process and the disturbances acting on it, and it requires the translation of the real requirements into a quadratic cost function. The real system should, however, act within possible constraints on inputs, outputs and state variables. Again, even in this simple case, some intuitive knowledge about the parameters to be chosen should be available. Often we choose some reasonable values, simulate the system and evaluate the results. Based on these results, we change some "optimal" parameter values to get a more desired behavior, which is also based on our "hidden" requirements and previous experience.

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FUZZY ALGORITHMS FOR CONTROL

1.2.3 Which methodologies are available? Three categories of questions can be recognized to choose between existing methodologies:

1. What is the structure of the solution? Is it possible to control the system by a feedforward solution or should we use a feedback solution? Are intermediate signals available to implement a cascade control structure? Are disturbances measurable and can they be diminished by feedforward control, or should the average influence of the unmeasurable disturbances be leveled out by feedback? Is there such a high interaction between the controlled variables that a multi-input multi-output control structure is needed? Is the state of the system available or reconstructible and should it be used in a state-feedback structure? Although nearly all control engineering textbooks treat many of the above mentioned control structures, few or nothing is said about the question of which method should be preferred under certain circumstances. It seems an interesting possibility to apply fuzzy decision making techniques in a decision support system to help the designer in making the right choice. Recently, interesting research has been going on in the field of heterogeneous control, in which in-line a choice is made between a number of controllers (for regulator control and for servo control, for instance). The configuration is switched from one controller to another depending on the situation (Kuipers and Astrom, 1994). 2. What are the structural parameters of the controller configuration? Once the structure of the controller has been chosen, this question often boils down to such questions as: what is the order of the controller to be designed or how many and which states has to be reconstructed? In many cases, especially in common practice, we choose a fixed structure for the controller. In industry, by far the most popular controller is a PID-controller with a fixed structure and structural parameters determining the use of proportional action, derivative action and integral action. This controller is so popular that there are many rules of thumb available to set the parameters. This holds also for the structural parameters. Depending on the expected noise or the delay time, the structural parameters are fixed, and this implies that the controller is a P-controller, PI-controller, PD-controller or PID-controller. 3. The last question to be posed is what are the actual values of the parameters of the controller? Commonly, the parameter values of the controller have to be set or adjusted by the control engineer. There are many rules of thumb to set the three parameters of a PID-controller. These rules are developed either from experience, depending on the type of the controlled variable (pressure, temperature, flow, level, etc.), or on a rough estimate of the main parameters that describe the process: gain, delay time and dominant time constant. The controller parameters and the sampling period are then a function of these process parameters. If the process parameters are not available, the controller settings are based on the closed loop behavior of the controlled process, by bringing the process into oscillation with a P-controller and using the frequency of oscillation

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and the proportional gain of the P-controller by which this oscillation happens, as the two parameters that define the three parameters of the controller. In some commercially available controllers, such a procedure is done by an A1 technique. As indicated above, control engineering practice is based on the combination of theoretical background, supplied by the control and system theory community, and of expertise, built up over many decades and based on experience, experiments, and sound engineering solutions. This combination offers an interesting basis for the application of A1 methods (especially fuzzy systems), together with analytical analysis and design methods, as long as the methods support each other and do not try to compete in those areas in which one of the two has the better testimonials. 1.3

ADVANCED CONTROL

Modern process operation and production methods are characterized by an increasing demand forpexibility: operating the plant with varying throughput, product mix and product grade (customer-defined production instead of producer-defined production). The process is required to operate at different operating points, to change over fast from one operating point to another, to take into account constraints that have to be met. Therefore the system will exhibit a strongly nonlinear behavior and can often insufficiently be analyzed in comparison with the situation where the process was mainly required to operate at a few well-defined operating points. Moreover, there is an increasing need for supervision of complicated processes, for extensive fault detection and fault diagnosis methods, and for dynamic planning and scheduling methods. a strong demand for new production methods and the development of new production plants to decrease the waste of material, to minimize the energy consumption, to minimize the effects on the environment, and to cope with the ever-increasing competition. New production methods require complicated equipment with many inner loops and utility feedbacks to decrease energy consumption and waste of material. This leads to highly nonlinear systems, much interaction between the control loops, increased danger of instability and various phenomena acting simultaneously on the system. Many of these newly developed production methods can only partly be described by first principles and conservation laws. However, experience gained by pilot plant operations and available as expert rules and experimental data should be used for the plant description. In summary: strong emphasis should be placed on the combination of knowledge about the system in the form of a mathematical description based on first principles and conservation laws, experience gained from operators and pilot plant operation, and the results of experiments with similar processes. an integrated information system with sophisticated human interfaces that is plant-wide in operation, and can handle the various levels of automation: control, monitoring, optimization, supervision, scheduling, planning, management. This requires the ability to handle qualitative and quantitative information in one system with different levels of precision and complexity. These tasks are mainly

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FUZZY ALGORITHMS FOR CONTROL

fulfilled by different people responsible for the various levels of automation. The demand for flexibility and fast reactions to market situations will require a dynamic and reactive response on all levels of automation. Control engineering methods that cope naturally with dynamic systems are well equipped to address these problems and fuzzy methods can play an important role to improve the human interfaces. The use of the state variables approach to describe dynamical systems introduced a strong theoretical framework to analyze and design more complex controllers. The main advantage of this method is its universality in describing linear, nonlinear, multiinput multi-output, single-input single-output,continuous and discrete systems, within the same mathematical framework. Not only the input-output behavior of the system is described but also its internal behavior. Based on this system description, we got a complete framework for linear systems which provides universal analysis and design procedures for closed and open loop systems. The combination with quadratic performance criteria especially became quite popular, which let to optimal control and optimal noise rejection using, e.g., Kalman filtering techniques. What is needed is an adequate description of the system (linear or linearized), is a translation of the requirements into the parameters of the performance criterion, and the availability of the state of the system. Many techniques were developed to reconstruct or estimate unknown system parameters and states. The designer's experience was mainly introduced in the choice of the performance criterion parameters. The main problems encountered are the availability of a good model of the process that can be sufficiently linearized around an operating point, the availability of all state variables, and the choice of the right parameters in the performance criterion. Control problems in the aeronautical and space industry initiated the research on adaptive controllers that adapt their parameters to changing conditions. These autonomous adapting systems must, however, be protected by extensive safeguarding and jacketing measures. These measures are mainly based on experience, and are related to rule base systems. Within the process industries Model Based Predictive Control (MBPC) is successfully applied in a number of applications (Richalet, 1993). This control strategy is based on the prediction of the future system behavior by using a process model. The basic concepts appearing in all predictive control approaches are the following : Use of an available (nonlinear) model to predict the process outputs at future discrete time instants over a prediction horizon. Computation of a sequence of future control actions (over a control horizon) using the model of the system by minimizing a certain objective function, which is such that the predicted outputs errors are as close as possible to a desired reference trajectory under given operation constraints. Receding horizon principle, so that at each sampling instant the optimization process is repeated given new measurements. Only the first control action of the sequence of obtained actions, is applied to the process.

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9

Because of the explicit use of a process model and the optimization approach MBPC can be applied to complex processes, e.g., multivariable, non-minimum phase, openloop unstable, nonlinear processes or processes with a long time delay. Moreover, the method can efficiently deal with constraints. Further, MBPC is attractive, because it is an intuitive concept and relatively easy to tune. For all these reasons, MBPC has been well received both by the academic world and by the industry. Robust control is becoming a very important research area in control. Realizing that exact knowledge of a system is almost never available, robust control methods take into account the uncertainty of the model description. Generally, a quite conservative control algorithm will be the result. Finally, the research on nonlinear control and modeling techniques has been developing quite fast over the last decade and practical approaches are becoming available. So, there are many possibilities to apply advanced control methods in industry. The control and system theory community is supplying most interesting methods, which, however, need extensive mathematical process models, the determination of the most suitable performance criteria and the application of complicated (non-convex) optimization techniques. This is often not in line with industrial practice, in which mathematical models are mostly incomplete or only roughly known. In the following will be indicated that the application of fuzzy techniques can be an alternative that sometimes fits better the problem at hand. 1.4

FUZZY CONTROL

The early applications of fuzzy control, see (Mamdani and Gaines, 1981), were based on the idea to mimic the control actions of the human operator. In this case, a priori knowledge is used and the final controller performs as well as the best operators. Fuzzy control in this sense fits well when the system to be controlled is only partly known, difficult to describe by a white box model, and few measurements are available, or the system is highly nonlinear. However, extensive experience in operating the process should be available to the system designers. Many applications of fuzzy control are related to simple control algorithms, such as PID controllers. In a natural way, nonlinearities and exceptions are included which are difficult to realize with conventional controllers. In conventional control, many additional measures have to be included for the proper functioning of the controller: anti-reset windup, proportional kick, retarded integral action, etc. These enhancements of the simple PID controller are based on long-lasting experience and the newly made marriage between continuous control and discrete control. This bag of tricks can be built in in a very natural way in a fuzzy PID-like controller. Moreover, other types of local nonlinearities can easily be built in, because a fuzzy controller can be described as a nonlinear mapping (Buckley, 1992). As indicated in the previous section, (nonlinear) models play an important role in many advanced controllers. There are several possibilities to model a system by applying fuzzy techniques such as models based on Mamdani fuzzy rules, models based on Takagi-Sugeno rules, fuzzy relational models (Pedrycz, 1993) and a combination of these models (BabuSka and Verbruggen, 1997). Some approaches to determine a fuzzy model are:

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FUZZY ALGORITHMS FOR CONTROL

A fuzzy model can be obtained by using a priori knowledge about the system provided as rules by system designers and system operators. Knowledge acquisition is, however, sometimes cumbersome, costly, and time-consuming. A fuzzy model can be obtained by using available measurements and using identification methods, e.g., clustering methods to find the parameters and fuzzy terms of the rules describing the system. This method provide good results and can easily be interpreted in a linguistic way, thus providing a means for evaluating and validating the final model with knowledge from operators and experts (Yager and Filev, 1994). The resulting fuzzy models can be used to develop fuzzy controllers (Terano et al., 1994). An interesting application is the use of these models in Model-Based Predictive Controllers (MBPC) to calculate the future output of a system for different control sequences, and to find the optimal control action while taking into account a desired behavior and constraints on system variables. The model of the process must be able to predict the future process output, perform fast simulations, and, preferably, based on a physical background, such that it can be understood by an operator. In case conventional modeling approaches based on physical modeling or linear system identification can not derive reliable models for complex or partly known systems, fuzzy modelling give promising results. Fuzzy methods can also be used to describe the control goals. As seen in conventional control, it is necessary to express the requirements and desires in crisp values or mathematical expressions which should be optimized. The original minimization problem in MBPC can also be formulated as a fuzzy decision problem: Using a process model, a fuzzy decision making algorithm selects the control actions that best meet the specifications, see Fig. 1.1. Hence, a control strategy can be obtained that is able to push the process operation closer to the constraints and to force the process to a better performance based on goals and constraints set by the operator and by known conditions provided by the system's designers. The formulation of the control

Controller

t; knowledge

I

.

algorithm

Process

I

L--------------------

Figure 1.1

MBPC, described as objective evaluation and fuzzy decision making.

problem leads to a generalization of the objective function used in model-based predictive control. The quadratic cost function used in classical MBPC is generalized to a confluence of fuzzy goals and fuzzy constraints. It is possible to aggregate fuzzy

FUZZY SYSTEMS IN CONTROL

11

goals and constraints by using fuzzy operators, choosing the operator that best fulfills the desired combination of goals and constraints. In reality, some constraints should not be violated and some requirements should be kept. However, other constraints and requirements can be less important and fulfilling them regardless of the effort required is not really useful. Little research has been done in the application of fuzzy methods in multi-input, multi-output systems (Raymond et al., 1995). This has mainly been the domain of classical linear control, in which either a direct multi-dimensional controller is designed, or a decoupling mechanism is used to diminish the interaction between the loops. These methods are, however, based on a rather precise mathematical description of the process model and the performance requirements. Fuzzy methods can handle systems which are less precisely described and of which the interaction between variables is only approximately known (e.g. strong interaction, weak interaction, no interaction). Adaptive fuzzy control is a possibility to cope with time-varying and even nonlinear behavior of a system (Driankov et al., 1993). However, the measures to keep the adaptive controller always functioning in the right way are complicated. In fuzzy controllers, exceptions can be easily implemented and their interpretation is straightforward to the user and designer. Generally, it can be said that exception handling and safety guarding is implemented in a fuzzy controller in a transparent way with easy linguistic interpretations, while in conventional (adaptive) control these measures result from a different system view that are not easily integrated with the conventional control algorithm. When the actual parameters of the controller are adapted according to the behavior of the overall system, an adaptive supervisory control algorithm is used. The adaptation should be related to some performance measure of the system. Several possibilities to apply fuzzy techniques can be distinguished: The performance criterion provides information as membership functions, such as that the overshoot is too high, too low, within the specs. Most criteria can be used, such as overshoot, rise time, accuracy in steady state. The supervision is done by rules relating these fuzzy performance measures (premisses) to the settings of the parameters of the controller to be adapted (consequents). a

A fuzzy model is used as a representation of the time-varying system. This model is adapted and used in a fuzzy control strategy. Depending on the situation, a choice is made between different control strategies (strategy switching). A fuzzy decision maker realizes this selection based on the requirements and actual state of the system and takes care of transient behavior.

Because a more or less autonomous system will result in the supervisory methods described above, special attention should be paid to exception handling and safety nets, which can be described quite easily by rules. The whole supervisory system can then be realized in a fuzzy expert system.

12

1.5

FUZZY ALGORITHMS FOR CONTROL

MISUNDERSTANDINGS A N D POSSIBILITIES

Fuzzy control has always been a controversial subject, especially in the control theory community. However, in practice, fuzzy control is becoming increasingly popular, partly because of commercially available programming tools. The need for control methods for nonlinear systems is becoming very important because of modern production methods and new innovative industrial installations. There is still a large gap between the control theory community and industrial practice. It is frustrating for the control theory community that the elegant and comprehensive framework for system analysis and design is hardly ever applied in the process industry, which is still applying the well-known PID controller in most applications, and trusts to manual control in more complex situations. Industrial practice is, however, demanding solutions to problems which are not always in line with the methods available in system and control theory. Fuzzy control and the application of Artificial Neural Networks seem to promise some solutions, although this cannot be expected to be the panacea for all problems still existing in practice. There is a lack of mutual understanding between the fuzzy control community and the conventional control community, partly due to exaggerated claims made by the fuzzy control community and partly due to traditional control community's presumptuous attitude, through which the empirical nature of fuzzy control was designated an "unscientific" approach. The fuzzy control community boasts sometimes about the ability of fuzzy methods to handle all nonlinear systems and claims, together with the proponents of the artificial neural network approach, that they provide the sole solution to nonlinear system design. However, many interesting results have been obtained recently by applying nonlinear systems theory, of which the approach of exact linearization for affine models is only one of the breakthroughs. A match can be made between control theory and fuzzy methods when the nonlinear model is provided by fuzzy techniques and the controller is based on classical or advanced control, e.g. using the inverse nonlinear model found by fuzzy modeling or neural networks in the exact-linearization approach to nonlinear affine systems. Some people attack the fuzzy control community by stating that the final control algorithm just boils down to a nonlinear gain schedule which could actually be obtained by other interpolation methods. This is true, because it can be proven (see (Kosko, 1994))that fuzzy controllers are universal approximators. This is, however, also true for other methods such as neural networks, splines or wavelets. Fuzzy control provides a man-machine interface which very much facilitates the acceptance, validation and transparency of the control algorithm. That the tool boils down to a simple algorithm is not something that is questionable, but is very convenient from the point of view of computational effort. It has also been stated that fuzzy control cannot predict the p e l f o m n c e of fuzzy controllers. In the case of fuzzy control that models the operators' strategy, the performance is clearly related to the performance of the best operators, and thus predictable. In the general case of fuzzy control (in fact a nonlinear mapping from process output to process input), the performance of the final controlled system is more difficult to predict. However, predictable control is only possible when a good mathematical description of the system is available, and the control aims can be stated in crisp numbers or as a criterion to be optimized. By introducing fuzzy constraints

FUZZY SYSTEMS IN CONTROL

13

and performance criteria, new possibilities are introduced (Yasunobu and Miyamoto, 1985). Stability is a principal characteristic of feedback systems, and the research on how to design a controller which guarantees stability has been a major topic in control theory. There are many stability definitions which are based on the internal (state) or external (input-output) character of the system. For nonlinear systems in particular, much research has been performed over many decades, of which Lyapunov's direct method and the Circle Criterion are the most important methods. The control theory community might have the impression that stability is not an issue in fuzzy control. This is not true, as is shown in a later chapter. Much research has been done in applying Lyapunov's methods for fuzzy systems. Also, methods based on the Circle Criterion have been developed for fuzzy systems. A qualitative study of the stability of fuzzy systems can be performed by using linguistic trajectories, see Driankov et a1 (Driankov et al., 1993). An interesting point is that stability and robustness measures are introduced, comparable to the margins used in conventional linear control systems. The real design problem is not only to assess stability, but to describe the influence of the design parameters (the controller) and the process parameters on the stability, and to use stability criteria not only as an analysis tool but also as a design tool. This means that in the design phase, one is not interested in a crisp concept which provides a yeslno answer about stability but one is interested in how far the system is from the instability limits. Thus, it is important also for nonlinear systems, to define (fuzzy) measures to indicate how far the system is from instability. Thus far the relationship between classical control and fuzzy control. What about the relationship between the A1 community and fuzzy control? The origin of fuzzy control was within the perspective of Artificial Intelligence. The fuzzy logic controller was built to represent the knowledge and expertise of the operator and designer, and there was no relation to the classical control engineering approach. Today, fuzzy control is, by the control community, increasingly seen as a universal approximator, (see (Buckley, 1992)) which is competing with other approximation methods, such as artificial neural networks, and which is strongly competing with nonlinear control strategies. The main motivation to use fuzzy logic as an application of approximate reasoning is disappearing on the lower levels of control, and it is sometimes seen only as an interesting by-product. It is, therefore, interesting to see that especially in the area of modeling and identification, there is a tendency to blend information of a different nature (expertise of operators and designers, measurements and mathematical equations). It provides solutions in between black box and white box models, and fuzzy modeling plays an important role in this process of information fusion. Classical control has not been very much involved in solving higher-level control problems such as supervision, optimization, monitoring, planning and scheduling of dynamic complex systems, although these issues are very important from an economic point of view. Compared to conventional control techniques, fuzzy techniques are well equipped to solve these kinds of problems. They allow, for instance, for more uncertainty, which tends to reduce complexity and to increase the credibility of the

4

14

FUZZY ALGORITHMS FOR CONTROL

resulting model. Besides, fuzzy logic allows the inclusion on these levels of the expertise of experienced operators and plant managers. This expertise is often based on qualitative, uncertain and incomplete information. As an example, developments in the field of Fault Detection and Fault Diagnosis show that fuzzy methods are playing an increasingly important role. Fault Tolerance is gaining more importance in control systems. Especially when large faults occur, it will be necessary to reconfigure the control system, otherwise the whole system would break down.

1.6

CONCLUSIONS

In this chapter, authors' personal view on the role of fuzzy control in control engineering is given. The authors realize that they have exaggerated some statements in order to emphasize the differences between the classical control approach and the fuzzy control approach. They hope that this will provide food for thought, and will be beneficial for people involved in practical applications of control, for "conventional"and "fuzzy" control people, and people from the A1 community. It is to be hoped that they would all realize the advantages and disadvantages of the various methods and not stick solely to their own approach. References

BabuSka, R. and Verbruggen, H. (1997). Fuzzy modeling and model-based control for nonlinear systems. In Jamshidi, M., Titli, A., Boverie, S., and Zadeh, L., editors, Applications of Fuuy Logic: Towards High Machine Intelligence Quotient Systems, pages 49-74. Prentice Hall, New York. Buckley, J. (1992). Universal fuzzy controllers. Automatica, 28:1245-1248. Driankov, D., Hellendoorn, H., and Reinfrank, M. (1993). An Introduction to Fuzzy Control. Springer, Berlin. Kosko, B. (1994). Fuzzy systems as universal approximators. IEEE Trans. Computers, 43: 1329-1333. Kuipers, B. and Astrom, K. (1994). The composition and validation of heterogeneous control laws. Automatica, 30(2):233-249. Mamdani, E. and Gaines, B. (1981). Fuuy Reasoning and its Applications. Academic Press, New York. Pedrycz, W. (1993). Fuuy Control and Fuuy Systems (second, extended,edition).John Willey and Sons, New York. Raymond, C., Boverie, S., and Titli, A. (1995). Fuzzy multivariable control design from the fuzzy system model. In Proceedings Sixth IFSA World Congress, Sao Paulo, Brazil. Richalet, J. (1993). Industrial applications of model based predictive control. Automati c ~29: , 1251-1274. Terano, T., Asai, K., and Sugeno, M. (1994). Applied Fuzzy Systems. Academic Press, Inc., Boston. Yager, R. and Filev, D. (1994). Essentials of Fuuy Modeling and Control. John Wiley, New York.

.

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Yasunobu, S. and Miyamoto, S. (1985).Automatic train operation system by predictive fuzzy control. In Sugeno, M., editor, Industrial Applications of F u u y Control, pages 1-1 8. North-Holland.

2

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL

INTELLIGENCE D. Dubois, H. Prade and L. Ughetto*

lnstitut de Recherche en lnformatique de Toulouse (IRIT) - CNRS Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France

2.1 INTRODUCTION The termfuzzy logic is rather ambiguous because it refers to problems and methods that belong to different fields of investigation. When scanning the literature, it is possible to find three meanings for the expressionfuzzy logic. In its most popular perception, it refers to numerical computations based on fuzzy rules, for the purpose of modeling a control law in systems engineering. However, in the mathematical literature, fuzzy logic means multiple-valued logics, with the purpose of modeling partial truth values and vagueness. Lastly, in Zadeh's papers, fuzzy logic is better understood as fuzzy set-based methods for approximate reasoning at large, and approximate reasoning is a subtopic of Artificial Intelligence. This state of facts creates communication problems between researchers in the fuzzy set area and, consequently, outside the fuzzy world as well. Indeed the fields concerned withfuzzy logic, and that use this terminology, are

*L. Ughetto is currently with ENSAT, BP447,22305, Lannion Cedex, France.

18

FUZZY ALGORITHMS FOR CONTROL

Control Engineering, Formal Logic and Artificial Intelligence. Some of those fields almost never communicate with one another. Fuzzy logic, as understood by control engineers, is no logic at all, from the points of view of logicians. Moreover, the research programs of Artificial Intelligence and Control Engineering are quite divergent: the latter is based on numerical methods and is black-box oriented. The former insists on symbolic processing and knowledge representation. Fuzzy logic is devoted to knowledge representation and symbolic/numeric interface, and its status is rather ambiguous in that respect. Fuzzy set theory has brought together researchers that had little background in common and the temptation exists for each community (Artificial Intelligence, Logic and Control), to emphasize a narrow view of fuzzy logic that fits its own tradition. Interestingly, the original motivation of fuzzy logic control was to represent expert knowledge in a rule-based style and to build a standard control law that faithfully reflects this knowledge (Mamdani and Assilian, 1975). Fuzzy logic control was thus put from the start in the perspective of Artificial Intelligence, because it did not use the classical control engineering paradigm of modeling a physical system and deriving the control law from the model. As such, fuzzy logic control is viewed as an application of the approximate reasoning methodology proposed in (Zadeh, 1973), that exploits formal models of common-sense reasoning. Following this path may sound promising, even for control engineers, since they do employ heuristic knowledge in practice, be it to specify objectives to attain. Supervision also involves a lot of know-how, despite the existing sophisticated control theory. However, in the last five years, a significant deviation from original motivations and practice of fuzzy logic has been observed in the control engineering community (Nguyen et al., 1995; Chand and Chiu, 1995; Efstathiou, 1995). Namely, fuzzy rule-based systems are more and more considered as standard, non-fuzzy universal approximators of functions (Buckley and Hayashi, 1993; Castro, 1995; Kosko, 1992; Wang, 1992), and less and less as a means of extracting control laws from heuristic knowledge. This trend raises several questions for fuzzy logic. First, if fuzzy logic is to compete alternative methods in approximation theory, it faces a big challenge because approximation theory is a well-established field in which many results exist. An approximate representation of functions should be general enough to capture a large class of functions, simple enough (especially the primitive objects, here the fuzzy rules) to achieve efficient computations and economical storage, and should be amenable to capabilities of learning from data. Are fuzzy rules capable of competing with standard approximation methods on such grounds? The answer is far from clear. On the one hand the universal approximation results for fuzzy rule-based systems presuppose a large number of rules. This is good neither for the economy of representation nor for the linguistic relevance. On the other hand, the identificationbetween fuzzy rule-based systems with neural nets or variants thereof (radial basis functions and the like) (Kosko, 1991; Berenji and Khedkar, 1992; Bersini and Gorrini, 1994; Jiang and Sun, 1995), has created a lot of confusion as to the actual contribution of fuzzy logic. To some extent it is not clear that fuzzy logic-based approximation methods for modeling and control need fuzzy set theory any longer. Moreover, the connection to knowledge representation, part of which relies on the readability of fuzzy rules as knowledge chunks, is lost. This

t

t

t

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

19

incompatibility leads systems engineers into cutting off the links between Fuzzy Logic and Symbolic Artificial Intelligence, hence with fuzzy set theory itself (which aims at providing an interface between numerical data and symbolic labels). This is very surprising a posteriori since the incompatibility between high precision and linguistic meaningfulness in the description of complex systems behavior is exactly what Zadeh prompted to introduce fuzzy sets as a tool for exploiting human knowledge (Zadeh, 1973). It is questionable whether the present tendency of systems engineering to immerse fuzzy logic inside the jungle of function approximation and optimization methods (a trend also called so$ computing) will produce path-breaking results that make fuzzy rule-based systems outperform already existing tools. It is not clear either that it will accelerate the recognition of fuzzy set theory proper, since there is a clear trend to keep the name fuzzy and drop most of the contents of the theory. Yet, it seems that control engineering practice can benefit from the readability of fuzzy rule-based systems (Dubois and Gentil, 1994). The latter are easier to modify, they can serve as tools for integrating heuristic, symbolic knowledge about systems, and numerical control laws issued from mathematical modeling. Some interesting works have also been done in fuzzy rule-based tuning of PID controllers (Galichet et al., 1992). More generally the ranges of applicability of fuzzy controllers and classical control theory are complementary. Whenever modeling is possible, control theory offers a safer approach, although a lot of work is sometimes necessary to bridge the gap with practical problems. Fuzzy logic sounds reasonable when modeling is difficult or costly, but knowledge is available in order to derive fuzzy rules. This philosophy, which has led to successful applications in Europe before fuzzy logic became worldwide popular (for instance the cement kiln controllers (Holmblad and astergaard, 1995)), tends to disappear from the literature of fuzzy control, when one looks at the recent literature on neuro-fuzzy control. It must be noticed that while in the beginning of fuzzy control, fuzzy rule-based systems were understood as relevant to Artificial Intelligence, Artificial Intelligence had rejected fuzzy control as a non-orthodox approach that was not purely symbolic processing. To-date, fuzzy logic advocates tend to reject symbolic Artificial Intelligence as not capable of dealing with real complex systems analysis tasks. Proposing the recently emerged buzz-words soft computing or computational intelligence (Zurada et al., 1994): a mixture of fuzzy rules, neural nets and genetic algorithms, as a new scientific paradigm that would make traditional Artificial Intelligence research obsolete, sounds hasty and somewhat dangerous. First, this new stream of numerical modeling, while rejecting the methodology of symbolic AI, keeps the A1 vocabulary, thus leading to terminological confusion. Second, numerical methods and symbolic approaches are once again presented as competing while they are complementary. There is no way of getting rid of the language level when communicating with humans. Overemphasizing numerical modeling may result in cutting fuzzy logic from its roots and making fuzzy set theory obsolete as well. Zadeh himself recently advocated the idea of computing with words as being the ultimate purpose of tools such as fuzzy logic (Zadeh, 1996).

2Cornputational Intelligence is also the name of the Canadian symbolic A1 journal.

20

FUZZY ALGORITHMS FOR CONTROL

In order to achieve this program, it seems that a part of fuzzy logic research should go back to Artificial Intelligence problems, and that fuzzy logic should again serve as a bridge between Systems Engineering and Artificial Intelligence. Needless to say that in that perspective, control engineers should receive some education in logic, and Artificial Intelligence researchers interested in systems engineering should be aware of control theory. Such a shift in education and concerns would open the road to addressing, in a less ad hoc way, issues in the supervision of complex systems, a problem whose solution requires a blending between knowledge and control engineering, and not only tools for the approximation of real functions, be they non-linear. This chapter tries to maintain the links between modern fuzzy control methodologies and the field of approximate reasoning from which fuzzy control once emerged. Section 2.2 presents the basic principles of approximate reasoning and of fuzzy inference in the setting of possibility theory. Section 2.3 proposes a formal analysis of fuzzy control from the point of view of fuzzy inference. Section 2.4 is an example of what logic can bring to fuzzy rule-based engineering: a tool for checking the consistency of a set of fuzzy rules. Lastly, Section 2.5 discusses the issue of fuzzy rule-based interpolation. It explains how interpolation can be viewed as a particular case of fuzzy inference. It also shows that the extension principle of fuzzy set theory suggests a natural method for interpolating between ill-known (fuzzy) values.

2.2

BACKGROUND O N APPROXIMATE REASONING

Fuzzy rule-based systems have been proposed in (Zadeh, 1973). They were designed in order to model the knowledge of human operators, with a view to control complex processes from this expert knowledge. However, Zadeh's ambition later on (between 1975 and 1985) was to lay bare the formal basis of a theory for reasoning with linguistic knowledge pertaining to numerical universes and containing lexical imprecision. This theory, which includes fuzzy rule-based systems (Zadeh, 1979), is a generalization of classical logic, from a semantic point of view. It provides a rather natural tool for numeric 1 symbolic interfaces, representing symbols with fuzzy sets of numbers. 2.2.1 Linguistic variables

A number of terms from the natural language clearly refer to numerical scales, as for instance tall, hot, old, medium-sized . . . Their use in computer programs should avoid the two following dangerous temptations:

m

Reasoning only with symbols, without any numerical interpretation of them. This approach raises two main problems. On the one hand, it is impossible to exploit numerical information (e.g., from sensors) in a symbol processing program and, on the other hand, since there is no interpretation of the symbols, their meaning can be misunderstood. For instance, the meaning of tall depends on the speaker (children or adults) andlor on the context (people, buildings or trees).

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

21

Fixing a set A of values such that "if u E A then u is tall" and "if u $ A then u is not tall". This approach, adopted by most of the old expert systems, is very sensitive to thresholds variations and to the lack of precision of numerical data. This semi-numerical approach is very common in engineering applications since the symbolic knowledge they manipulate generally refer to measurable (and often measured) numerical variables. However, the part of knowledge manipulating numerical data can be expressed with symbols. In order to properly deal with symbolic pieces of information, an appropriate representation of their semantics is necessary. Linguistic terms often refer to sets of numbers with ill-defined boundaries. This is due to the gradual property of these terms which itself comes from the continuity of the corresponding numerical scale. Then a fuzzy set, i.e., a set whose characteristic function is defined on [O,l] instead of { O , l ) , appears to be an easy and natural tool to represent sets of numbers with ill-defined boundaries (Zadeh, 1965; Kaufmann, 1973; Dubois and Prade, 1980; Dubois and Prade, 1985; Yager et al., 1987). A linguistic term A represented by a convex fuzzy set on a totally ordered set of values U is called a simple category. U is usually an interval on a continuous universe. And A is convex means that its membership function is unimodal, i.e.:

V ( Uu', , u") E

u3such that u < u' 5 ul', pA(u')> min(pA(u),pA(ul')).

There are two main kinds of membership functions of simple categories: H

Monotone membership functions which generally represent extreme values on a linguistic scale (as for instance very tall in Fig. 2.2). In case of an increasing function, there are two thresholds u- and uS such that Vu 5 u-,pA(u) = 0 and Vu > - u S , pA(u) = 1. For decreasing functions (as small) 0 and 1 have to be permuted in the constraints.

H

Bell-shaped membership functions which represent intermediate terms of a linguistic scale (see Fig. 2.2). For instance, trapezoidal fuzzy intervals (shown in Fig. 2.1) are represented by four numbers which define:

1. the support of A: S ( A ) = { u E U / pA(u) > 0) is the set of values which correspond more or less to A. It is the complement of the elements which are not at all in A.

2. the core of A: C ( A ) = { u E U values of A.

/

pA(u) = 1 ) is the set of prototype

Linear transitions between 0 and 1are generally sufficient to express the gradual nature of A. Thus, pA(u)evaluates the closeness between u and the prototypes of A. It can be understood as a similarity degree. A symbolic representation of a numerical scale is often achieved by a rough partition made of linguistic terms. Note that these terms not only depend on the speaker or on the context as already said, but also on the number of terms in the partition. Tall is not the same in T = {small, middle-size, tallland in 7 ' = {very small, small, tall, very tall}(Farreny and Prade, 1986). This remark emphasizes the need for a numerical

22

FUZZY ALGORITHMS FOR CONTROL

.A

Figure 2.1

Fuzzy set with trapezoidal membership function

representation of linguistic terms. A variable ranging on a set of terms 7 is called a linguistic variable (Zadeh, 1975). A partition 7 = {A1 , . . . ,An) should: D

cover the entire universe U , i.e., Vu E U ,

3Ai E 7 such that P A ; ( u ) > 0

or equivalently: inf

max

u E U i=l, ...,n D

( u ) > 0.

be composed of well-contrasted terms, i.e., terms with non overlapping cores:

This is a weak version of a fuzzy partition. The most widely found understanding of a fuzzy partition verifies the following constraint (Ruspini, 1969):

and is often called a strict fuzzy partition. The overlap between adjacent linguistic tenns in 7 expresses the progressive variation from one term to the other, and the difficulty to determine a precise threshold between them. The partial membership of an element to several linguistic terms will prove to be useful. Indeed, it is not always necessary to choose only one of the corresponding symbols for further processing. More generally, some linguistic categories involve several dimensions or numerical scales. For instance, large involves the length, width and depth of an object. They are called complex categories and are represented by a logical combination of simple categories as for instance:

where * and * are (sometimes the same) fuzzy conjunction operators. Not all complex categories can be decomposed into simple ones, nor do they always refer to numerical universes. In this chapter, we only deal with fuzzy sets on numerical universes.

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE Srndll

M~ddle-Sized

23

Vcw Tall

Figure 2.2 An example of strict fuzzy partition of the universe U

Links between numerical variables are often symbolically described by conditional expressions of the " i f . . . then. . . " form. In the condition and conclusion parts of these rules, the value of the variables is expressed with simple categories. For instance, "ifthe pressure is high and the volume is normal then the temperature is high" is a rule of the form "if X1 E A1 and X2 E Aa then Y E B", where Al, A2 and B are simple categories. This rule is modeled by: pR(ul, us, v) = ( p (ul) ~ ~ (211)) -+ pB(v), where * is a fuzzy conjunction and + a fuzzy implication. Several kinds of fuzzy conjunctions exist for expressing the and in the condition part of fuzzy rules. The most usually found choice is a triangular norm (Yager and Filev, 1994), in practice the minimum operator, the product operator or the linear triangular norm max(0, a + b - I), which is very drastic. From a control engineering point of view, the product is the most popular choice because of its differentiability. However other types of less drastic conjunctions sometimes better express the linguistic and, namely an averaging operation such as the arithmetic mean or the geometric mean, as advocated in (Zimmermann, 1987), for instance. There is no theoretical constraint restricting the choice of an aggregation operation for combining the conditions appearing in a fuzzy rule. Several kinds of implications can be chosen as well for the purpose of modeling the link between the condition part and the conclusion part of a fuzzy rule, depending on the meaning of this " i f . . . then . . . " rule (Dubois and Prade, 1991), as will be seen later.

2.2.2 Principles of approximate reasoning In this section, a set IC = { P I , . . . ,Pk)of I; linguistic pieces of knowledge is considered. Moreover, the Pi's are either in the affirmative form "Xi is A" or in the conditional form "if Xi is A then Xj is B", and involve n (numerical) variables X I , . . . , X , defined respectively on the universes U1, . . . , U,. Zadeh suggests that the meaning of such a set K should be represented by a fuzzy relation R on the Cartesian product Ul x . . . x U, (Zadeh, 1979). This fuzzy relation R describes an elastic restriction on the possible tuples of values (uI , . . . ,u,) for these variables X I , . . . , X,, according to the knowledge expressed in K. Then, its membership function p~ should be understood as a possibility distribution on U1 x . . . x U,. Indeed, pR(ul,. . . ,u,), called possibility degree, expresses to what extent (ul, . . . ,u,) is a possible tuple of values for X1, . . . ,X, with respect to

24

FUZZY ALGORITHMS FOR CONTROL

the available knowledge described by the set K. For instance, p R ( u l , . . . , u,) = 0 means that u l , . . . ,u , are impossible values while p R ( u l , . . . , u,) = 1 means that these values are totally compatible with K. To each piece of knowledge P, in K corresponds a fuzzy relation denoted Ri. Relation R is constructed from these relations R1,. . . ,Rg since, in accordance with classical logic, K is considered as the conjunction of the formulas it contains. Each fuzzy relation Riis viewed as a constraint. Thus, the tuple of values ( u l ,. . . ,u,) is compatible with K if and only if it is compatible with each piece of knowledge. ~ h e s e values are then in the intersection of the fuzzy relations R1,. . . ,Rk:

where the intersection is performed on the membership functions, via the min operator, The choice of the conjunction operator min can be justified by its idempotency property. Indeed, this property is necessary to avoid problems due to the potential existence of redundant pieces of knowledge in K. However, it can also be justified from the minimal specijicity principle. A possibility distribution .rr is the membership function of a fuzzy set of (mutually exclusive) possible values for a variable X (here X = ( X I ,. . . , X,)). It expresses the available (generally incomplete) knowledge on X. To compare two sets of knowledge IC and IC', it is possible to compare their possibility distributions n and x',x is said to be more specific than x' if and only if n < n'. And n is said to be compatible with K (described by the fuzzy relation R) if and only if K 5 p ~ It. means that a possible value for n is also (at least as) possible for R. Then p~ is the least specific possibility distribution compatible with IC. And since min is the greatest fuzzy conjunction operator, r n i n ( p ~, ,. . . ,p ~ ) is , the least specific (and then the least arbitrary) distribution compatible with each piece of knowledge P I ,. . . ,Pk. In case of an empty set K, i.e., there is no piece of knowledge at all, the corresponding possibility distribution is uniformly equal to 1. It expresses total ignorance since then every value is considered as being entirely possible. Using an operation different from the minimum operation when combining the fuzzy relations representing fuzzy rules (for instance the product) presupposes that rules are independent, because the combination then produces a reinforcement effect. For instance, rule redundancy is prohibited, while the use of minimum allows for redundant rules since the latter do not affect the resulting relation. This remark sheds some light on the meaning of the minimum rule in possibility theory: its use avoids making any assumption on the dependency between the pieces of information that are combined. This is the meaning of non-interactivity, as introduced in (Zadeh, 1975). In order to compute the values of a variable X j induced by IC, the corresponding fuzzy relation R has to be projected on the universe of X j , namely U j . For each u; E U j , the most possible values ( ~ 1 ,. . ,u;,. . . , u,) of ( X I , .. . , X,) have to be computed. It comes down to computing the fuzzy set K j as follows:

t3

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

25

I

With p~ = mini=l ,...,k ,u~,it leads to:

where each Ri is defined on Ul x . . . x U, (by means of a cylindrical extension in case Rionly involves a subset of variables, say X 1 ,. . . ,X p ;then

I

. . , u p ,u p + l , .. . ,u,) = P R (~2 1 1 , . - . , u p ) ,Vup+l,.. . ,u,.

CLR;( ~ 1 , .

A particular case of combination/projection similar to (2.3) is known as the sup-min composition and is denoted by 0. For binary relations, it writes R1 o Rz with:

This formulation given in (Zadeh, 1979) is considered as the fundamental principle of fuzzy logic, when fuzzy relations represent information from natural language using simple or complex categories (as defined in Section 2.2.1). It comes down to a nonlinear optimization process since inference is achieved by finding out the optimum of some combination of membership functions under prescribed constraints.

2.2.3 Fuzzy implications for fuzzy rules In logic, conditional expressions of the form "if p then q" can be understood in two ways:

I

p is false or q is true, q is true each time p is true.

If p="X is A" and q="X is B", with X ranging on U , the first interpretation leads to understand the rule as: Vu E U, u E XU B , where 3 is the complement of A in U . Indeed, each value of X lies inside B or outside A. The second interpretation comes down to the inclusion B A. Both representations are equivalent for ordinary sets. In terms of truth table, the formulap -+ q is false only if p is true and q false. When p and q refer to different universes: p="X is A" on U and q="Y is B" on V, the expression "if p then q" defines a relation R on U x V. Namely, the images of the elements in A via R are in B, i.e., B A o R, where A o R = {v E V / u E A and (u,u) E R) is the image of A by R, on V. The following equivalence holds for ordinary sets: B 2 A o R if and only if R g U B.3. When A and B are fuzzy sets, this equivalence generally no longer holds. And then, depending on the considered formulation, two kinds of fuzzy rules can be derived, which are based on genuine extensions of implications.

>

>

3 0 n U x V, 3 U B stands for (A) U ( U x B). We do not exhibit the cylindrical extensions A x V and U x B for short.

I

26

FUZZY ALGORITHMS FOR CONTROL

Gradual rules.

The fuzzy translation of B

> A o R is:

It means that the fuzzy image of A by R is included in B in the sense of the fuzzy inclusion of Zadeh. The conjunction *is often taken as min (as done in (2.3)), and the sup means that the inequality should hold for all u . The least specific, non-fuzzy relation which verifies this inequality corresponds to Rescher-Gaines implication: P R ( % 21)

= P A ( 2 1 ) -SRG/AB

(21)

=

i f ~ A ( ~ ) L P B ( v ) l

0

otherwise.

(2.5)

The rule A + , B clearly means that if X = u and a = ,YA(u), then Y E B, = { v E V / p g ( v ) 2 a ) , where B, is the a-cut of B. Such a modeling of the rule then means that "the closer X to the typical values of A, the closer Y to the typical values of B", as in the rule "At a constant temperature, the bigger the volume, the lower the pressure". This kind of rule leads to a widening of the core of the conclusion. This widening is all the more important as the activation level a is low. Such rules are called gradual rules (Prade, 1988; Dubois and Prade, 1992b). Now, the least specificfuzzy relation which verifies (2.4) is based on a residuated implication: P R ( ~V, )

= PA(^) -+ P B ( V ) = sup ( 6 / P A ( U ) *6

5 PB(V)}

(2.6)

When the conjunction * is min, it corresponds to Godel implication:

In a numerical context, this implication leads to a discontinuous membership function for B'. For this reason, it can be questioned. However, it makes sense in a symbolic context where membership degrees belong to a finite set of totally ordered discrete values. When * is the product, (2.6) is Goguen implication: P R ( U , 21)

= PA(U)

+Go,

PB(V)

=

{

i n (1

)

P

A

#0

otherwise.

(,.,)

which preserves continuous membership functions. With a precise input X = u , both implications lead to a fuzzy output with the same core B, and the same support S ( B ) (if a > 0). Then the choice between them only depends on the need for a continuous membership function for the conclusion or not. These two kinds of rules also express gradation. They are called fuzzy gradual rules to differentiate them from the pure gradual rules obtained with the non-fuzzy Rescher-Gaines implication.

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

27

Certainty rules. Another kind of rules is induced by the material implication A U B. The corresponding fuzzy relation is of the form:

where max is a disjunctive fuzzy operator. This relation is called Kleene-Dienes implication. The maximum operation max(a,b) in (2.9) can be changed into any strict triangular co-norm like a b - a.b. For this kind of rules, the activation level a of the antecedent part of the rule A is used as a certainty degree about the conclusion, as in the rule: "the higher the temperature, the more certain the engine will break down soon ". Then these rules are called certainty rules (Dubois and Prade, 1992a; Dubois and Prade, 1996b). Then, by contrast with gradual rules, the conclusion part B can be non-fuzzy. The matching degree between the antecedent part of the rule and the input fact is used to describe the certainty of an event (i.e., a crisp fact), as for instance in the rule: "the later one gets up, the more certain one is to miss the morning train". This interpretation in terms of a certainty degree can be justified as follows. Given a possibility distribution n and a fuzzy set of values B on V, the possibility degree of B is H(B) = supvEBn(v)and its necessity degree N(B) = i d v e B1 - n(v)= 1 - H(B) (Dubois and Prade, 1985). Then, the fact "Y is B is (at least) a-certain" writes:

+

which is equivalent to Vv E V , x(v) 5 max(pB(v),1 - a ) . The least specific possibility distribution satisfying this constraint is n(v) = max(pB(v), 1 - a ) , Vv E V . Replacing a with ,uA(u),Kleene-Dienes implication is found. The uncertainty level 1 - a observed in Fig. 2.4.f evaluates the possibility for the conclusion to be outside B.

Rules expressing both certainty and graduality. There is a pair of operA o R and ators (conjunction, disjunction) for which the equivalence between B R & 2 U B holds. Namely, this is when the conjunction * in (2.4) is defined by a* b = max(0,a b - 1)and the disjunction is defined pointwise by the bounded sum min(1,a + b). The corresponding implication, known as Lukasiewicz implication, is pR(u,v ) = min(1,l - pA(u) pB(v))which can be seen both as an extension of material implication and as satisfying (2.6). Rules modeled with this implication produce at the same time a widening of the core and an uncertainty degree of the conclusion. Since it preserves the equivalence between the two formulations of a rule, this implication may appear as the most suitable from a mathematical point of view. However, such rules are at the same time gradual rules and certainty rules, and they are not adapted to situations where only core widening makes sense, and no certainty level must appear, or conversely.

>

+

+

2.2.4

The Generalized Modus Ponens

A particular form of approximate reasoning is the Generalized Modus Ponens (Zadeh, 1979) (GMP for short). The knowledge base K contains two pieces of knowledge:

28

FUZZY ALGORITHMS FOR CONTROL

"PI = X is A'" defined on U , and a fuzzy rule "P2= X is A then Y is B" defined on U x V. The inference pattern is then: if

X is A' X is A

then

Y is B

The possibility distributions of PI and P2 are TI = PA, and 1r2 = p ~ -+ p ~ The . values of Y on V are given from the projection of R = A' n ( A + B ) on V , i.e., by computing:

or equivalently B' = A' o ( A -t B ) , where o is the sup-min composition. When A' = { u O ) ,B' is then simply given by: ~ B( vI) = p ~ ( u Ov). , More generally, iC can be composed of n parallel fuzzy rules, i.e., rules of the form "Ri = if X is Ai then Y is Bi", and of one input fact (a piece of knowledge about X) " X is A'". Following Zadeh's approach, the inferred knowledge on Y is computed by:

which can write:

The rule by rule inference mechanism used in classical expert systems corresponds to the inference pattern: B' = = , , , , , , ,A' o (Ai + B i ) . It is important to notice that in case of rules modeled with a fuzzy implication, the result B' obtained by this method differs from the GMP.Indeed, even with crisp sets, only the following inclusion generally holds:

ni

For instance, if A' = Ai U Aj (with i # j ) , processing rule by rule leads to B' = V because whatever the rule Rk, (Ai U A j ) o (Ak + Bk) = V , while the left hand side of (2.14) leads to the right result B' = Bi U B j (with crisp sets). However, the equality in (2.14) holds if only singleton input values A' = { u O )are considered. Then approximate reasoning comes down to rule by rule inference. 2.3

APPROXIMATE REASONING VS. FUZZY LOGIC CONTROLLERS

This section tries to determine to what extent the inference methods used in fuzzy control are in accordance with the Zadeh's approach to approximate reasoning. As a preliminary remark, note that they do not pursue the same goal: m

The aim of approximate reasoning is to represent the meaning of linguistic knowledge in order to build automatic deductive processes. Such a process has to calculate conclusions (which can be imprecise) from imprecise facts

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

29

and fuzzy rules. It is interesting for qualitative descriptions of the behavior of systems (Wenstop, 1976), but also for information systems at large. In fuzzy engineering systems, there are different goals. It can be the determination of a control law from a set of fuzzy rules (Mamdani and Assilian, 1975). It can be the construction of an interpolation mechanism between linear, numerical models (Sugeno, 1985). Lastly, many papers are concerned with deriving approximate non-linear models from numerical data (Kosko, 1991; Wang and Mendel, 1992). Moreover, the inference process in fuzzy engineering is similar to the weight propagation in classical expert systems (Buchanan and Shortliffe, 1984) since most methods compute weights which are propagated to the conclusion, while approximate reasoning uses a combination-projectionprocess. The problem is to find out if these two inference methods are coherent. 2.3.1 Inference machinery in fuzzy control The principle of fuzzy control has been proposed by Mamdani and Assilian (Mamdani and Assilian, 1975). It is based on a very simple technique which involves only a small part of fuzzy set theory. A fuzzy system for control is composed of a set of n rules of the form: if X1 is Ai,1 and . . . and X, is Ai,, then Y is Bi,i = 1,.. . , n where the Xj's are the observable variables of the system to control (or their derivatives) and Y is a control variable (supposed unique for the sake of simplicity). In other words, the Xj's are the input variables of the control system and Y is the output variable, i.e., the input of the system to control. In general, the input variables are signal errors and their derivatives. The sets Ai,j are elements of a partition 7j of U j (universe of variable Xj). For instance, 7 = (Negative Large (NL), Negative Medium (NM), Negative Small (NS), Zero (Z), Positive Small (PS), Positive Medium (PM), Positive Large (PL)], as shown in Fig. 2.3.

Figure 2.3

A linguistic partition

7j,

of

Uj as used in fuzzy control.

For a precise observation (uy, . . . ,u$),the value of the control variable is computed by the following three-step method: In a first step, known asfu&ication, these values are compared to the prototype situations described by the rules. A matching degree ai = m i n j = ~...,,, P A ; , (u?) is computed for each rule Ri.

30

FUZZY ALGORITHMS FOR CONTROL

Then, in the inference step, a fuzzy set B' is computed as follows:

where A is a conjunction operator, usually min. Then, Bi = ai A Bi is given by p ~( v: ) = min(ai, p ~( v;) ) .The union operator is the maximum, applied to the

membership functions. This method suggests to choose the control value in the union of the values (weighted by the similarity degrees a i ) recommended for each typical situation (i.e., the situations corresponding to the core of the condition part of the rules). Sometimes, A is the product (Larsen, 1980) and U is the bounded sum (Kosko, 1991;Wang and Mendel, 1992). Lastly, the defuuijication step consists of choosing one particular value v0 of Y in B f . The center of gravity method is the most widely used one, and it goes back to early times of fuzzy control (Braae and Rutherford, 1979). It is given by:

where SvEv p p ( v ).dv = IBfI is a generalization of the cardinality of B f . Other defuzzification methods have been proposed; the mean of maxima of p ~ was , originally proposed in (Marndani and Assilian, 1975); some also use the center of area, that is, the median value which cuts the fuzzy output into two parts of equal areas. More sophisticated methods have recently been proposed (Foulloy, 1994; Yager and Filev, 1994). In the particular case where the Bi's are singletons (Bi = {bi)), the center of gravity method leads to:

It was used in particular by Sugeno and many others in case of rules with non-fuzzy conclusion parts (Sugeno, 1985). In case of strict partitions of triangular membership functions on each U j , at most two values P A ; + ,( u j ) are non-zero, whatever the precise input u j . Then, in the case of one-dimensionalinputs, the choice of v0 relies on an interpolation process between couples of rules. With trapezoidal membership functions, the center-of-gravityinterpolation is linear with the Sugeno's method and non-linear with Mamdani's method. In fact, Takagi and Sugeno introduced more general rules (Takagi and Sugeno, 1985). The conclusion parts of these rules are no more singletons, but (often linear) functions of the input variables. They are used for identification of non-linear systems or control functions. A fuzzy control system can work as a PID controller (Galichet et al., 1992) since it defines a numerical (linear or non-linear) function from the input variables to the single output variable. The difference actually lies in the way this function is found. It is the

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

31

result of a mathematical model of the process in traditional control engineering, while fuzzy control uses the expert knowledge of a human operator, in accordance with the A1 methodology. However, fuzzy logic differs from standard expert systems since it provides an interpolation mechanism between rules. Then, even with linguistic knowledge modeled by production rules, the continuity of control functions is ensured. Moreover, with a good set of rules, a reasonable control law can be obtained. This approach can be (and has proved to be) useful in case of complex and hard-to-handle process. Then, it can be easier to collect expert knowledge than to compute an optimal control function. More recently, fuzzy rule-based systems and neural networks have been used to model non-linear control laws from numerical data. In this case, fuzzy theory is used as a tool in a black-box, crisp input / crisp output model, and no more as an interface between numerical values and symbolic knowledge. This problem is far away from fuzzy control understood as expert control and is not in the scope of this chapter.

2.3.2 Ftrzzification Fuzzy control systems use both numerical values (e.g., from sensors) and symbolic representations of knowledge. An interface from a number (or an interval of numbers) to its symbolic representation and vice versa is then necessary. The fuzzification process converts a numerical measurement X = uO into its symbolic representation using a linguistic partition 7 = { A I ,. . . ,A,) of U , by (uO)of u0 to each Ai. computing the membership degree The wordfuuiJication is rather ambiguous since it wrongly suggests that a precise number is turned into a fuzzy one. In our case, matching or Jiltering seems to be more accurate words since symbolic terms act as filters which are more or less filtering the value uO.However, the fuzzy filtering of a precise value produces a fuzzy set of symbols. This justifies the word fuuiJication. The fuzzification step gives membership degrees oi = p~~(u).Before using them, it is important to correctly understand their meaning. ( u )evaluates (on [0,11) to what extent Ai is the right The membership degree linguistic term for representing u. It is a compatibility degree between Ai and u, a truth degree of the fact " X is Ai", with respect to the piece of certain knowledge X = u. It is not at all an uncertainty degree. In the literature, the use of terms like certainty coefficient or likelihood level makes it ambiguous and sometimes leads to a confusion between fuzzy sets and uncertainty measures. To understand why ai is not an uncertainty degree, remark that the uncertainty would be due to a lack of information on the value of X (e.g., from imprecise sensors) or to contradictory pieces of information (from different sensors for instance). In the ( u ) reflects the similarity fuzzification context, X = u is known for sure, and ,u~, between u and the prototypes of Ai. Thus, a linguistic partition of the [O, 11-interval can be compatible for values near 1, incompatible for values near 0 and partially compatible for values near 0.5. And since the ai's are not uncertainty measures, it becomes possible to use operators from multiple-valued logics to aggregate them. A first, although debatable, way of using these degrees ai comes from the classical expert systems methodology. It consists of choosing the more accurate linguistic term

32

FUZZY ALGORITHMS FOR CONTROL

Ai, i.e., the term which maximizes PA; (u). In this case, it is easy to argue thatfuuiness is useless. Indeed, it would come down to set the thresholds between linguistic terms

Ai and Aj in points u i j such that p ~( ~~ i ,=~ f) i ( ~~ i ,~>~ 0. ) The right way, as done in fuzzy control, is to keep all the significant fuzzy terms > 0. The ai's are then used to interpolate between several control values. The decision (choice between values) is then made after the symbolic processing, after the propagation of all the ai, not before. If the sensor gives an imprecise value of X, for instance an interval X E [u-, u+], then the matching weight p~~ (u) becomes an interval with lower bound Nx (Ai) and upper bound IIx (Ai) defined as follows:

Ai, i.e., all the Ai's such that ai

and are respectively the necessity and possibility degrees of Ai, knowing (incompletely) X. These degrees quantify the amount to which it is certain and possible respectively that Ai is the right term to represent X . They are both equal to p ~(u) ; only if X is known precisely ( X = u). They are the basic degrees for fuzzy filtering (Cayrol et al., 1982; Dubois et al., 1988b). More generally, if the input is a fuzzy value X = A', these degrees can be generalized as follows (Dubois and Prade, 1985):

IIx (Ai) is a very weak matching degree since only one common value for A' and Ai (with membership 1) is sufficient to have IIx(Ai) = 1. It only evaluates the existence of an overlap between A' and Ai. And even with IIx (Ai) = 1, a lot of possible values of X may not be compatible with Ai. By contrast, Nx(Ai) = 1 if and only if all possible values of X in A' are totally compatible with Aj, that is, S(A1) C C(Ai). Nx (Ai) > 0 if and only if the totally possible values of X are somewhat compatible with Ai. In other words, if and only if C(A1) 5 S(Ai) (where A' and Ai are fuzzy intervals, i.e., upper semi-continuousfuzzy sets). For this reason, N x (Ai) seems to be a more accurate matching degree (than IIx (Ai)) to be used in fuzzy control, if control engineers have to deal with fuzzy inputs (Palm and Driankov, 1995). 2.3.3 Mamdani rules for control

Even if fuzzy rules have been widely used in fuzzy control, the choice of the implication operator for modeling the rules has often been a source of confusion for control engineers, and conjunctions have been preferred to implications from practical points of view (see, for instance, (Mendel, 1995)). Figure 2.4.g shows the conclusion obtained with the kind of inference proposed by Mamdani for control. It is a truncation of B which comes down to understanding the rule as a conjunction (X, Y) E A x B instead of an implication, i.e.,

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

33

This formulation sounds counterintuitive, especially for logicians. Notably, p ~ t= m i n ( a , p B ) can hardly be understood as a possibility distribution. Indeed, p ~ being l sub-normalized, it would mean that no value is entirely possible for Y, or that the conclusion B' is partially contradictory (and totally contradictory if a = 0 since then the conclusion is empty). This is in opposition with laws of logic since if a rule does not match a situation, it should lead to the conclusion unknown (i.e., n = 1 everywhere, meaning that every value is possible) instead of contradictory (n = 0, meaning that every value is impossible). In some papers devoted to the presentation of fuzzy control, the meaning of the membership degrees 0 and 1 is sometimes misinterpreted. 0 is often systematically taken for unknown, as in (Mendel, 1995), instead of contradictory. This confusion is also due to the following result: when a rule modeled by means of a genuine implication, does not match the input fact, it does not apply and leads to an unknown result. If X = { u O )and p A ( u O )= 0, then with an implication operator, V v E V , pR(uO,v ) = 1, which should be interpreted as "the output is unknown" (and not the conclusion v is the true one). On the contrary, pR(uO,V ) = 0 means that value v is impossible. However, with Mamdani rules, V v E V , pR(uO,v ) = 0 means that value v is unknown as a conclusion since the figure represents a lower bound on a possibility degree. Using the theory of approximate reasoning, each rule is viewed as a constraint, and . the contrary, in Mamdani's the induced possibility distribution is n 5 P A + p ~ On approach, each rule is viewed as a piece of data of the form "if X is A then it is possible to assume Y is B". It leads to postulate the opposite inequality n 2 p ~ * p ~ for a suitable connective *. In the fuzzy case, Mamdani rules should be understood as "the more X is A, the more it is possible that Y is B". And the second part "it is possible that Y is B" means that each value in B is possible for Y (Dubois and Prade, 1992a). If B is non-fuzzy, II(B) only evaluates to which amount one element in B is possible, while all the elements in B are possible (at least) at the degree A(B)= i n f , , ~n ( v ) (Dubois and Prade, 1992a). The piece of information "each Y in B is a-possible" then writes:

i n f U E ~ n (2v )a.

(2.23)

This is so weak a constraint that the least specific solution is n = 1 everywhere since this equation is equivalent to n 2 m i n ( p B ,a). So, in this case, p ~ = m i n ( p B ,a ) is used, as the greatest lower bound of n , for representing the statement "Y lies in B is (at least) a-possible". It comes down to modeling the rule with a conjunction. It also means that each relation R satisfying the rule verifies: PR(U,V)

>~ ~ ~ ( P A ( U ) , P B ( ~ ) ) .

(2.24)

v ) = 0 should be understood as an inequality: / . L ~ ( u O , v ) 2 0, which Now, pR (uO, actually means that the possibility degree is unknown. This inequality also applies to a fuzzy B, when the membership degrees belong to an ordinal scale (Dubois and Prade, 1992a). In a less qualitative context, niirz can be replaced by the product. This is the formulation of Larsen (Larsen, 1980): p ~ ( u ,v) = p ~ ( u ).pg ( v ). This kind of rules is not very informative. Indeed, the inferred result gives values in B which are at least a-possible. It does not forbid some values in B to be more

34

FUZZY ALGORITHMS FOR CONTROL

possible (for some other reason), nor values outside B to be somewhat possible as well. This kind of view is not in accordance with logic, since here the more rules, the more data is collected, the larger the range of possible values for (X, Y). In logic, increasing the number of rules viewed as constraints leads to decrease the set of possible values for (X,Y). In fuzzy control, where the best precision of the result is desired, the rules which produce uncertainty are not suitable. Clearly, rules modeled with Kleene-Dienes implication make more sense with symbolic conclusions, in order to propagate uncertainty. By contrast, the gradual rules correspond both to the rules used in logic and to the intuitive meaning of control rules. In particular, pure gradual rules (modeled with Rescher-Gaines implication) give more precise results than a non-fuzzy rule built on the support of A and B (S(A) + S(B)), as clear from Fig. 2.4.c. For instance, if X = u0 such that p ~ ( u O )= 1, then B' = C(B), the core of B .

f[~-\

--------

I

"-

a) Condition part of the rule

c) Gradual rule

-

- Input

b) Conclusion part of the rule

Rescher-Gaines

e) Fuzzy gradual rule

d) Fuzzy gradual rule

- Goguen

--------

1

*

10

g) Mamdani rule

- Minimum

Figure 2.4

v

v

:I-\ 0 Certainty rule

a = /\

- ~dtlel

- Kleene-Dienes

V

a

h) Lukasiewicz

Inference with one rule and a precise input.

V

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

35

Despite these suitable properties, this lund of rules has not been widely used, on the basis of practical considerations. Indeed, if these rules are used regardless of their semantics, they can turn out to be inconsistent, while Mamdani's approach produces a result even in the presence of logical inconsistency. This advantage is only apparent since the user is not protected against hislher own mistakes. Moreover, the inference mechanism, explained in the next section, can be more difficult to handle and more time consuming than with Mamdani rules, but only in case of fuzzy inputs. When inputs are precise, reasoning with implication-based rules is as simple as with conjunction-based rules. 2.3.4 Inference with a set of parallel rules

Rule bases used in fuzzy control are most often composed of parallel rules, i.e., rules with the same input space U and the same output space V. The set of rules is then of the form K: = {if X is Ai then Y is Bi / i = 1 , . . . ,n]. More generally, the rules may involve compound conditions, but this makes no difference for the point discussed in this section. So, " X is AiWmay succinctly denote a compound condition here. Modeling the rules with a fuzzy implication -+,and applying Zadeh's approach to approximate reasoning, the fuzzy relation R induced by K: on U x V is obtained by aggregating the representations of the rules conjunctively:

This relation R is in accordance with the meaning of each rule as a constraint since:

(cf. Section 2.2). If

+ is Godel implication, R verifies the inequality:

and if it is Goguen implication, the following inequality holds:

When the rules are modeled with a fuzzy conjunction *, they are aggregated disjunctively. If * = min:

in accordance with the inequality defining the semantics of the rule as a piece of data forming a building block of R:

It also sanctions the fact that if Bi and Bj are both whatever possible, so is it for BiU B j . This mar-min model is usual in fuzzy control. However, the disjunction (mar) of the pieces of knowledge is not in accordance with approximate reasoning theory. Indeed,

36

I

FUZZY ALGORITHMS FOR CONTROL

in the framework of approximate reasoning, a piece of knowledge is a restriction on the possible values of the variables involved, i.e., an upper bound of possibility degrees as stated by (2.26). By contrast, in fuzzy control, fuzzy rules are generally not understood as constraints on the values of Y since each rule gives a lower bound of the possibility distribution, as stated by (2.28). Equation (2.27) should not be confused with the max-min composition (2.3). In the latter, the minimum is used for aggregating the pieces of knowledge, while in (2.27) it is used for modeling the rules. In the max-min composition, the maximum is used for defining the projection of possibility distributions, while in (2.27), it is a tool for aggregating imprecise observed data. If the rules are fired with a precise input X = {u), the activation level for each rule is a, = p ~(21). , In order to compute the conclusion, these activation levels &e used differently, depending on the kind of rule. For rules considered as constraints (i.e., modeled with an implication), the intersection of the conclusions of each rule is computed as:

4

>

with B: = {v E V / p ~(v), ai)for pure gradual rules, and pB:(v) = I n a ~ ( p B(v) , ,1ai) for certainty rules. With conjunction-based rules, the union of the partial results is computed as: t

with p ~(v) : = m i n ( p ~(v), , ai). Figure 2.5 shows the inferential behavior of each kind of rules for a set K of two adjacent rules and a precise input. Some remarks can be derived from these results: Inference in the style of Mamdani (2.30) is not in accordance with classical approach of expert systems. In particular, Fig. 2.5.g shows that the more rules are activated, the more imprecise the conclusion will be, due to the disjunctive aggregation of the rules. This is an accumulation process rather than an interpolation process. For instance, the set of knowledge K= ( A ,if A then B1, if A then Bz}leads to B1 U B2 instead of BI f l Bz,even with non fuzzy rules. This phenomenon is generally hidden by the defuzzification process which chooses only one value in the inferred fuzzy set. It is the case in the previous example when the precise value chosen in B' = B1 U B2 is included in B1 n B2.It contrasts with classical logic, where the more rules are activated, the more precise the conclusion is. m

If an imprecise rule "if A then B1 U Bz" is added to a set K containing a more precise one "if A then B1", the imprecise rule hides the precise one with Mamdani method. With the approximate reasoning methodology, the former is considered redundant and is then ignored. Indeed, the more informative conjunction-basedrule is the one with the largest conclusion, giving the biggest set of possible values. While the more informative implication-based rule is

4

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

37

the one with the smallest conclusion, restricting as much as possible the set of not-impossible values. w

Certainty rules, based on Kleene-Dienes implication, still produce an uncertainty level (see Fig. 2.5.0. Hence they are not suitable for interpolation. The conclusion with a set of pure gradual rules can be precise (Fig. 2.5s). In this case, the defuzzification step becomes useless. By contrast, conjunction - based rules never give a precise output.

w

Inference with pure gradual rules represents an interpolation process, as further explained in Section 2.5. Gradual rules are the only kind of fuzzy rules to underlie an interpolation process inherently. With Mamdani rules, interpolation is artificially produced by the defuzzification step. Implicative rules can be incoherent with one another (as in Fig. 2.5.0, i.e., the output B' can be empty (total incoherence), or sub-normalized (partial incoherence) for a given precise input. This is due to the conjunctive aggregation of the partial results. This occurrence is sometimes viewed as a drawback of the approach in control engineering since, then, no value can be given to the process to be controlled. However, it can be turned into an advantage since such implication-based rules allow the design of coherence checking procedures (Dubois et al., 1997; KinkiClClt, 1994). Indeed, incoherence can be detected and removed from sets of implication-based rules while it can be hidden in sets of conjunction-based fuzzy rules which always seem coherent (Dubois et al., 1997). Mamdani's method on a set of logically inconsistent set of rules may produce strange results.

2.3.5 Inference with fuzzy inputs Until now, only precise inputs are considered in fuzzy control. However, some situations could justify the use of imprecise or fuzzy inputs, for instance in systems receiving values from low precision sensors (e.g., which only give an interval of values) or even from symbolic sensors (Benoit and Foulloy, 1993). Another important situation concerns the use of systems chaining several fuzzy controllers. The use of fuzzy inputs avoids the defuzzification of the conclusion after each level, and then suitably allows the propagation of the fuzziness through the whole chain of inference. For a fuzzy input A', Zadeh's approach to approximate reasoning gives (equation (2.12)):

With Mamdani rules, the output B' can be computed as follows:

where I'Ix(Ai), defined in (2.21) is the possibility degree of " X is Ai" (Zadeh, 1978) based on the fuzzy restriction on the value of X expressed by "X is A'".

38

FUZZY ALGORITHMS FOR CONTROL 4

vb

ub al

al

a2

a2

0

uo a) Condition part of the rules

c) Gradual rules

;li-m,

-

Input

- Rescher-Gaines

e) Fuzzy gradual rules

b) Conclusion part of the rules

d) Fuzzy gradual rules

- Goguen

;:i;

f'J Certainty rules

vb

- ~bhel

- Kleene-Dienes

a2 0

g)Mamdani rules

Figure 2.5

- Minimum

h) Lukasiewicz

Inference with a set of two rules and a precise input.

v

V

vb

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

39

This is equivalent, in the framework of approximate reasoning, to combine A and R = UiEl,, ,n Ai x Bi and project the result on V, since then: ,,

PBI (v)

= SUPzLEu min(p~1 (u), m a i = ~...,, n m i n ( ~(u), ~ ; PB; (v))),

which can be written equivalently, since o and U commute:

With a certainty rule, i.e., a rule modeled with Kleene-Dienes implication, the GMP leads to: PB/(V) = sup,,^ m i n ( ~ (u), ~ 1 m a 4 1 - PA(U),~ = ma(PI3 (v), 1 - NX (A))

~ ( 2 1 ) )

(2.34)

where Nx (A) is the necessity degree of A based on the known restriction "X is A'" (equation (2.20)). However, this inference does not match the usual modus ponens when A' = A since then Nx(A) 0.5 only. Indeed, if values of X are fuzzily restricted by PA, it is not certain that the value of X is among the ones which totally (i.e., to degree I ) satisfy the condition " X is A". With one gradual rule and a fuzzy input A', the membership function of the output p ~ is, not a direct function of the conclusion of the rule B and a matching degree between A and A' as in (2.32) and (2.34). Indeed, with Rescher-Gaines implication, the GMP writes:

>

However, it can be checked that: The core of B' is B,, the a-cut of B, with a = inf,EC(A~)~ A ( u ) . If the support of A' is not included in the support of A, then B' exhibits uncertainty. This uncertainty level is given by s u p , e s ( ~ )PA!(u). w

With one gradual rule, if A' = A then B' = B.

Inference with a set of parallel implication-based rules and a fuzzy input is not simple in general. Indeed, as explained in Section 2.2.2, reasoning rule by rule and then aggregating conjunctively the results leads to a superset of the expected conclusion B' (sometimes the entire output space). This is clear since only an inclusion (and not an equality) generally holds in (2.14). The right inference algorithm first combines the rules, and then applies the supmin composition, following (2.13). Some algorithms have been proposed, for Godel implication in (Dubois et al., 1988a) or for Rescher-Gaines implication in (Ughetto et al., 1997). Due to special properties, it has been shown that GMP inference with certainty rules can come down to the rule by rule inference if some redundant rules are added to the rule base K (Dubois et al., 1997).

40

FUZZY ALGORITHMS FOR CONTROL

However, the conclusion B' is always included in the conjunctive combination of the results obtained with each implication-basedrule. In general (when a set of rules is considered), the conclusion B' is different from Bi, for a given i, whatever the operator modeling the rules, even if the input is of the form A' = Ai. With conjunction-based rules, the rules Ai x Bi involved in the inference process are such that S(A1)n S(Ai) # 0.The conclusion B' can be very imprecise since it is the disjunctive combination of the truncated conclusion parts B. of all these rules. With implication-based rules, only the rules Ai 4 Bi such that C(A1)C S ( A i ) are triggered. Moreover, the conclusion contains (in the sense of fuzzy sets inclusion) the intersection of the conclusion parts of these rules. However, for gradual rules the conclusion is included in the support of the intersection of the conclusion parts of the triggered rules. Then implicative rules seem the best ones since fewer of them are triggered and they lead to a more precise result. Gradual rules produce particularly precise results since they do not produce uncertainty levels in their conclusions, if the input matches the rule. 2.3.6 Defuzzification As shown in the previous section, the output of a fuzzy rule-based system is generally imprecise and fuzzy. As a fuzzy set cannot directly be used as a command input of a process which generally requires a precise value, the fuzzy conclusions of rulebased control systems have to be converted into one precise value. This is called the defuzzification. After the inference step, the only available piece of knowledge on Y is that it lies in the support of the fuzzy set B'. Thus the problem is to choose one precise value for Y which is representative of B'. With implication-basedrules, ~ B( vI )is the possibility degree of the (non-fuzzy) fact "Y = v". Of course, pB1 ( v ) = 0 means that "Y = v" is impossible and p ~ (tv )= 1 means that "Y = v" is totally possible. Most often, there are several values v E V such that / A B ~ ( v ) = 1, even if they are mutually exclusive. If Vv E V , p ~ (lv ) = 1, then the conclusion B' is uninformative. This is due to an incomplete set of rules or an input corresponding to an unplanned situation. Now, if flu E V , ~ B( vI )= 1, then the set of rules is not logically coherent (see Section 2.4). With conjunction-based rules, the output B' is usually not normalized. However, it does not mean that no value is entirely possible since pgt ( v )is only a lower bound of the possibility distribution ny. It means that with the current knowledge, no totally possible value is known for sure. Defuzzification is similar to the computation of the mean value from a probability density. Actually, the classical center of gravity method identifies B' with the probability density p ~ /I l B'I and the result v0 to the corresponding mathematical expected value. The center of area method computes the median of the density p~lI(B'1. It is not always very clear how to a priori advocate a defuzzification method against another. Only a posteriori justifications are generally proposed on the basis of practical applications. In particular, the center of gravity method cannot be justified by the approximate reasoning methodology in a possibilistic framework since the lower bound

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

41

B' defines a very large family of possibility distributions. Moreover, since this method comes down to computing a mean value, it is particularly not suitable to defuzzify multi-modal fuzzy sets (see the example in Section 2.4). Another defuzzification method, the mean of maxima, has been often criticized since it produces jumps in the control law that do not lead to the expected interpolation between rules. A comparative study of four defuzzification methods is detailed in (Jager et al., 1994). In order to approximate functions with fuzzy rules, Sugeno method is claimed to be the most interesting one, from a numerical point of view, since it does not produce irrelevant non-linearities. However, it no longer corresponds to approximate reasoning in the sense of Section 2.2, strictly speaking, particularly for rules with several conjunctive inputs. As already shown, possibility theory gives a good theoretical framework for formalizing fuzzy rules, according to fuzzy logic and natural language. In this framework, defuzzifying a fuzzy set B' means choosing a representative value for the possibility distribution r y = p ~ l . If qualitative possibility degrees are considered (i.e., the degrees belong to a linearly ordered scale), defuzzification is done by choosing one of the most plausible values. It is not suitable in control, since it resembles the mean of maxima method. However, it is the natural method in the framework of optimization or default reasoning. can be understood as the envelope of a With numerical possibility degrees, family of probability measures P ( B 1 )such that: VP E P ( B 1 ) ,VJ' C V , P ( F ) 5 n ( F ) = s U p U E ~ p g( vr ) . It can be checked that n ( F ) = S U P ~ ~ ~ ( (Dubois ~ , ~ Pand ( FPrade, ) 1987). As it is a family of probability measures, the defuzzification leads to a family of mean values, synthesized by a mean interval representing B'. This interval is bounded by Choquet integrals (see, for instance, (Denneberg, 1994)):

E*(B1) = inf{SV v.dP / P E P ( B 1 ) } = J; ( i n f ~ b ) . d a E*(B1) = sup{Sv v.dP / P E P ( B 1 ) } = ~ ; ( s u p ~ b ) . d a with Bb, = {v E V / ( v )2 a } . The interval [E,(B1), E*(B1)] is obtained by averaging the uncertainty contained in p g , , but the imprecision is still there in the form of the interval. To get rid of it, one may choose the representative value of Y as the midpoint of this interval. Such a proposal was made a long time ago in (Yager, 1981), but is has to our knowledge not been used as a defuzzification method in control. Yet, this choice seems to be less arbitrary than other defuzzification techniques and has nicer properties. It can be justified on the basis of a possibility-probability transformation which extend the indifferenceprinciple of Laplace (Dubois et al., 1993). It comes down to computing the center of gravity of the family P ( B r )which yields a E , (B')+E* (8') 2 (Chanas probability distribution whose mean value is indeed: E ( B 1 )= and Nowakowski, 1988; Dubois and Prade, 1996a). When B' is trapezoidal, this value and E*(B1)= is easy to compute. In the example of Fig. 2.1, E,(B1)= Moreover,this defuzzificationis linear in the sense that E(BI@Ba)= E(Bl)+E(B2),

9.

42

FUZZY ALGORITHMS FOR CONTROL

and E(c.B1)= c.E(B1),where CB is the addition of fuzzy numbers and c is a real value. This is due to the linearity of E, and E* (Yager, 1981; Dubois and Prade, 1987). The center of area of B' usually differs from the midpoint of the mean interval, and the density ~ B I / I Bgenerally 'J does not belong to the family P ( B t )(Dubois and Prade, 1980). This approach cannot be used with Mamdani rules. It is sensible only with implication-based rules, but with these rules (and more particularly with Goguen implication and gradual rules), the choice of a defuzzification method is not crucial. And it is even less crucial as the result is more precise. 2.4

4

VALIDATION OF FUZZY RULE BASES

The validation of a knowledge base consists of checking whether it obeys some properties and that it does not contain anomalies. It is an important step in the development of a rule-based system to avoid some unexpected behaviors. The possible anomalies and required properties in a set of rules have been classified, for instance, in (Aye1 and Rousset, 1990) in the case of classical rule bases, and their definition have been extended to the fuzzy case (KinkiClClC, 1994; Dubois and Prade, 1994). From the definitions, it is rather easy to see that with a set of parallel rules, only coherence, redundancy and coverage of the input space remain actual problems (Dubois et al., 1997).

2.4.1

Coherence of fuzzy rule bases

A set of n parallel production rules K = {Aj =$ Bi / i = 1, . . . ,n ) , where denotes material implication is classically consistent with an input A' if and only if the set {A', A1 B I ,. . . ,AS + Bz) has a model, i.e., is logically consistent. The set of rules is said to be coherent if this consistency condition holds whatever the input fact A' in a class of allowed inputs. Similarly, a set of fuzzy rules is said to be coherent if whatever the normalized input fact A', the output B' inferred with GMP is also normalized. In other words, whatever the input, there is at least one totally possible value for the output variable. In the possibilistic framework, this condition writes:

+

where T X , is~ the possibility distribution on X and Y, representing the knowledge base K. This definition of coherence takes place in a clear representation setting, and also presents the advantage of being general in its definition and rule independent in its formulation if the rule is understood as a constraint (since nothing is presupposed on the way nx,y is constructed). However, it leads in practice to conditions closely dependent on the operator used for modeling the rules. Indeed, X X ,y writes differently, depending on the kind of rules in IC. By contrast, other definitions of coherence in fuzzy rule bases (see (Dubois et al., 1997)for a brief overview) only consider properties that the antecedent and conclusion parts of the rules should verify, whatever the kind of rule. Generally speaking, implication-basedrules with too precise conditions and too

6

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

43

imprecise conclusions are clearly likely to be redundant. On the contrary, rules with too imprecise conditions may be dangerous when associated with conclusions which are too much focussed (since they are then more likely to lead to inconsistency when overlapping condition parts of other rules). Only some results on coherence are now summarized, for each kind of rule. See (Dubois et al., 1997) for more details and proofs. Implication-based rules. Incoherence (if any) can be detected with implicative rules. Then, it can be removed from the rule base. Checking the coherence of a rule base can be a procedure of high complexity. The two following results, which hold for every implication operator, allow to decrease significantly the amount of computations (Dubois et al., 1997): Two rules {A1-+ B1,A2 -+B 2 )can be incoherent only if their condition parts overlap. They are called adjacent rules. Moreover, the incoherence can only occur with an input fact overlapping the intersection of these condition parts. Thus S ( A 1 )n S ( A 2 )is called the potential conjicting zone. Coherence has to be checked only in the potential conflicting zones. If the outputs Bi's of the set of rules X: = {Ai -+ Bi / i = 1 , . . . , n ) are convex fuzzy sets, then IC is coherent if and only if the rules are pairwise coherent. This convexity condition generally holds in practical applications. Some other results are specific to each kind of fuzzy rules: Certainty rules. It has been shown in (Dubois et al., 1997) that a set IC = {Ai -+

Bi / i = 1, . . . , n) of n parallel certainty rules is coherent if and only if:

which means that coherence holds if and only if the classical rule base K' = { S ( A i )j C ( B i )/ i = 1,. . . ,n ) , where + is the classical-logic implication, is coherent. With convex outputs, this condition writes:

Checking the coherence of a set of parallel certainty rules is then very easy. Gradual rules. The case of gradual rules is more tricky to deal with. First, note that when coherence is understood in a crisp way (coherent or not coherent), the coherence conditions on gradual rules hold whatever the considered R-implication (2.6), since their core always correspond to Rescher-Gaines implication. In (Dubois et al., 1997), some general results are given, and checking algorithms are proposed when the Ai's and Bi's have trapezoidal membership functions. In particular, it is shown that if Al and A2 are adjacent elements, with trapezoidal membership functions, of a strict fuzzy partition of the input space (as shown in Fig. 2.2), then the set of two rules {A1 -+ B1,A2 + B 2 ) is coherent if and only if B1 and B2 are

44

FUZZY ALGORITHMS FOR CONTROL

supersets of two adjacent elements of a strict partition on V. In this particular case, checking the coherence of a set of parallel gradual rules is also very easy. Roughly, the complexity of the checking algorithms increases linearly with the number of input variables, and with the square of the number of rules. t

Conjunction-based rules. As pointed out in (Dubois and Prade, 1994), the notion of coherence in the sense of (2.36) does not apply to these rules. Indeed, since they represent lower bounds of the possibility distribution a X , y (as stated in equation (2.24)), no constraint forbids this possibility distribution to be normalized. This is rather normal for rules that are not considered as constraints, but that express only that things are possible. However, the defuzzification step may lead to results that make no sense if the inferred fuzzy set B' is not convex. As in the now popular example of obstacle avoidance, if the conclusion of a fuzzy expert system using Mamdani rules is "to avoid the obstacle in front, you can bypass on the left or on the right", a centroid defuzzification of "left or right" leads to "go ahead", right in the obstacle! t

2.4.2 Redundancy Redundancy in a set of rules is a less importantproblem than incoherence. It is generally considered as a drawback since it may have bad consequences from a computational point of view. However, it can make some consequences of the knowledge base more explicit and then be useful for the explanation of a deductive process. Moreover, wellchosen redundant rules have proved to be also useful to derive an efficient inference mechanism for certainty rules (Ughetto et a]., 1997). A piece of knowledge is considered redundant with respect to the knowledge base if it brings nothing new to K. In the approximate reasoning framework, a rule A t B is redundant with respect to K if:

(niZl,

(niz1

since then ...,, Ai -+ Bi) n ( A B ) = ,...,, A, + B,). Some basic results have been obtained on gradual and certainty rules: Adding a redundant rule to a coherent knowledge base K obviously leaves this base K: coherent. If a rule R is redundant with respect to the rule R 1 ,it is redundant with respect to any rule base containing R1. If A is normalized and B contains no uncertainty (i.e., 3v E V, pB(v) = O), then the certainty rule A -+ B is redundant with respect to Al -+ B1if and only ifAcA1andB>B1. For gradual rules modeled with Rescher-Gaines implication, A -+ B is redundant with respect to A1 + B1 if and only if:

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE w

45

For gradual rules modeled with Godel implication, A + B is redundant B1 and Vu E U , p ~ ( u 5 ) with respect to A1 -+ B1 if and only if B inf { P B ( ~ /) P B I ( v ) PA1 (21))-

>

>

Redundancy with respect to more than one rule is more tricky to deal with. Preliminary results are given in (Dubois et al., 1997). For conjunction-based fuzzy rules, A x B is redundant with respect to K: if:

It means that the information in A x B (the pairs of possible values) is already present in IC. Then, as for implicative rules, if a rule R is redundant with respect to another rule R1, it is redundant with respect to any rule base containing R1. From equations (2.22) (min-based definition of x ) and (2.37), it can be deduced that A x B is redundant with respect to Al x B1 if and only if A C A1 and B C B1.

2.4.3 Coverage of the input space The input space (which can be multi-dimensional) is totally covered by the rule base K if: (u)> 0. Vu E U , 3i E (1, . . . ,TL), An incomplete coverage may occur when the rules come from a human expert but also when the rules are automatically tuned, with a neural net for instance, and no coverage constraint has been enforced in the optimization process. Checking the total coverage of the input space is a trivial problem when U is mono-dimensional. However, the complexity increases rapidly with the number of input dimensions. With conjunction-based control rules, an incomplete coverage of the input space can be a rather important problem. Indeed, if no rule is triggered for a given input value u, B' = $ and no control value can be sent to the controlled system. It occurs for input values u such that Vi E { I , . . . , n ) , p ~( u, ) = 0, which expresses ignorance for conjunction-based rules (trivial lower bound on the resulting possibility distribution). With implication-basedrules, an incomplete coverage leads to an uncertainty degree on the conclusion B' for a fuzzy input A' such that A' Ui S ( A i ) . This uncertainty nA i ) ) .A total uncertainty B' = V is obtained when degree is given by height(ui(A1 the input triggers no rule. To avoid this problem, rule bases are often constructed using clustering methods or regular partitions on the input spaces. Then, no coverage checking is needed. Another solution uses membership functions with infinite support (as Gaussian membership functions) for the antecedent parts of the rules. However, Gaussian membership functions (or others with infinite support) do not really make sense with implication-based rules. An implication-based rule with Gaussian membership functions restricts the possible output values for the whole range of input values. Consider gradual rules. For any input value X = uo, there is a proper crisp subset B1(uO)of V that should contain the output. If X = uo is far enough from the core of the rule one expects to be free of specifying any value for the output variable Y. However it is not permitted to add a rule A" -+ B" such that the core of

46

FUZZY ALGORITHMS FOR CONTROL

A" is {uo) and the core of B" is disjoint with B1(uo).So the risk of logical incoherence is increased with implication-likerules with unbounded support. Due to their aggregation mode, implication-based rules are more adapted to a local representation of knowledge than for a global one. This is why they should be made only of fuzzy sets with finite support. In case of an incomplete rule base which cannot be completed, if an output value is required even for input values triggering no rules, some interpolation mechanisms can be used, as the one detailed in Section 2.5.3. 2.5

t

INTERPOLATION WITH FUZZY RULES

The interpolation mechanisms used in fuzzy control systems based on Mamdani and Sugeno rules are now briefly investigated. They are then compared to the interpolation performed by means of a set a gradual rules. In particular, it is shown that in fuzzy control, interpolation is artificially created by the defuzzification step while it is inherent to the inference mechanism for gradual rules, due to their close connection with the extension principle (Dubois et al., 1994). In this section, the input and output universes are mono-dimensional for the sake of simplicity. Moreover, they are assumed to be closed intervals on the real line: U C R and V R

2.5.1 Mamdani's and Sugeno's methods It is rather easy to see that inference with Mamdani-likefuzzy rules does not constitute

an interpolation process. Indeed, the output B' is the union of truncated conclusion parts of some rules. In the example of Fig. 2.6, for instance, the output B' is the union of the truncations of B1 and B2. When the input u moves from a1 to a2, only the truncation levels of B1 and B2 are modified (as shown by the arrows). Then, the maximal values of B' globally move from bl to bz,but not in a smooth and regular way. Besides, the mean of maxima is a discontinuous piecewise linear function (see Fig. 2.6). With this kind of rules, an interpolative behavior is often obtained by the centroid defuzzification. Even if the centroid of B' moves non-linearly from bl to b2, the function between ( a l ,b l ) and (as,b2) is continuous and monotonic, in situations like the one of Fig. 2.6. However, non-monotonic behaviors can be observed when the conclusion parts of the rules overlap differently, as shown in (Schott and Whalen, 1996). This is a very counterintuitive behavior. Moreover simple linear functions cannot be exactly represented by this method. They can be approximated only by non linear functions! A more particular case of these rules originally used in (Sugeno, 1985) consists in considering only crisp outputs B, = {b,). Rules with linear output functions B, = {b,.u + b:) (Takagi and Sugeno, 1985), which generalize the previous case, are nowadays the most widely used control rules. A rule "A, x b," can still be considered as a rule expressing possibility, whose meaning is "the more X is A,, the more it is possible that Y is b,". Since b, is a precise value, it really means "if X is A,, then Y = bi is totally possible". With strict partitions on a mono-dimensional input space, the centroid defuzzification leads

t

4

47

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

maxima -mean

Figure 2.6

of maxima

L....................................... : '

-

cenler rlf gravity

Interpolation with a set of two Mamdani rules.

to a linear interpolation between points, as stated by equation (2.17). With a multidimensional input space or a different kind of partition, the interpolation is no longer linear. The meaning of Sugeno rules using linear functions as outputs ("if X is Ai then Y is bi.u+ b!,")is more difficult to understand. In the examples of Fig. 2.7, what corresponds to the defuzzification step (and is often called inference with these rules) acts as a sum of the responses of adjacent rules. When using these rules, people look for a black-box model which is easy to use, a non-linear interpolation method more than an interpretable representation of knowledge. However, the interpolative behavior of this approach is questionable. Particularly strange cases are depicted in Fig. 2.7. The functions to be approximated are drawn with dashed lines. To construct the rule base, the following intuitive method is used: three points are taken on the graph, as well as the derivatives in these points. The conclusion part of each rule is then the straight line passing through one of these points, and whose slope is the derivative of the function at this point. In the first case, the result is particularly bad since the minimum of the function becomes a local maximum, and two minima are added. In the second case, the function is no more monotonic; a local minimum is added and a sharp transition is created. This behavior can be easily explained. Indeed, the interpolation between two Sugeno rules "if X is Al,then Y is bl.u bi" and "if X is A2,then Y is b2.u b;" (in case of a strict partition of the mono-dimensional input space) is the parabolic line going through bl .a1 bi, b2 .a2 bb and the intersection of the two straight lines bl .u bi and b2 .u b;. In this case, the right way for constructing the rule base would be to choose the bi's and hi's such that these three points are on the function to represent. Some other undesirable interpolative behaviors have been already pointed out (BabuSka et al., 1994). However, it has been shown that Takagi-Sugeno fuzzy systems are universal approximators (Buckley and Hayashi, 1993; Castro, 1995; Kosko, 1992; Wang, 1992).

+ +

+

+

+

+

48

FUZZY ALGORITHMS FOR CONTROL

"t

vt

-Obtained -

Figure 2.7

- - Expected

Strange interpolation with Sugeno rules.

Accurate approximations require many rules, more than the number which would be necessary for a linguistic representation. The previous example then only shows that despite their good approximation properties, these rules are not intuitive in their interpretation. In order to obtain a good representation of the previous function, the set of rules has to be tuned with automatic optimization procedures, as neural nets or gradient learning methods. This is necessary in order to avoid the unsuitable loss of monotonicity or the occurrence of local extrema as in Fig. 2.7. Generally, tuning methods do not preserve strict partitions of the input space, and rules overlap each other more (Ci=l,,,,,n p~~(u) > 1). Then, even with u = ai (the core of Ai), we do not find v = bi.ai bl. Rules, taken separately, are then meaningless and no longer correspond to the original linguistic-like approach to fuzzy reasoning proposed by Zadeh.

+

2.5.2 Interpolation with gradual rules Consider a function f from U to V (which are intervals of the real line), assumed to be bijective on U for the sake of simplicity, and a set P = {(ai, bi) / ai = f (bi), i = 1, . . . ,n} of n points in the graph r o f f . Suppose moreover that the ai's are ordered (Vi E (1, . . . , n}, ai < ai+l). Knowing that (ai, bi) belong to I? means that if the value u E U of X is close to ai, then the output value f (u) E V is close to bi. This semantics corresponds to gradual rules, since a rule "The more X is Ai, the more Y is Bi", noted Ai + Bi can be understood as "the more the value of X is close to the core of Ai, the more the value of Y is close to the core of Bi". Then, in order to represent the function f with a set of gradual rules, Ai should model "close to ai" and Bi "close to bi". Thus, the Ai's are chosen such that C(Ai) = {ai) and S(Ai) = [ai-1, ai+l], and the same for the Bi's. First, suppose that the membership functions of both the antecedent and the conclusion parts of the rules have linear transitions between 0 and 1. In other words, triangular

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

49

membership functions are considered. For a given input uO,the generalized modus ponens leads to linear interpolation between the points in P , as shown in Fig. 2.8. Indeed, only two cases can occur:

+ Bi

w

u0 = ai for some i. Then only rule Ai B' = C ( B i )= {bi).

applies, and since PA; = 1,

m

ai < u0 < ai+l. Then, since the input is a strict partition, PA; (uO)= a = 1 - , U A , + ~ ( ~ Oand ) , B' = (Bi), n ( B i + ~ ) l -= , {a.bi (1 - a).bi+l).

+

This is clearly a linear interpolation. It is similar to Sugeno interpolation with precise outputs for the rules. However, interpolation is inherent to the inference process with gradual rules. Moreover, no defuzzification is even necessary since the result B' is already a singleton value.

Figure 2.8

Linear interpolation with a set of gradual rules.

Gradual rules fully agree with the extension principle. Indeed, let us still assume triangular membership functions for the AiYs,but now the Bi's are supposed to be constructed as Bi = f (Ai),the image of Ai by f , applying the extension principle (Zadeh, 1965). It is defined by:

which also writes

( u )= p ~ (f; ( u ) ) or , since f is invertible:

t

50

FUZZY ALGORITHMS FOR CONTROL

From equations (2.4) and (2.5), a rule Ai + Bi expressing "the more X is Ai, the more Y is Bi = f (Ai)" or "the closer X to ai, the closer Y to bi = f (ai)" obeys the constraint:

PA^ (u)

5 PBi ( f (u)),

which fully agrees with (2.38). Moreover, applying the generalized modus ponens (GMP) with the set of gradual rules K: = {Ai + B, / Bi = f (Ai), i = 1,. . . , n) and a precise input value u0 can lead to two different cases: u0 = ai for some i. Then obviously B' = C(Bi) = {bi).

I

ai < u0 < ai+l. Then PA, (uO)= a = 1 - PA,+^ (uO)since Ai and Ai+l are adjacent elements of a strict partition of U . The output B' is computed by the GMP as:

The same result could be obtained with triangular membership functions on the Bi's instead of the Ai's, as shown in (Dubois et al., 1994). This means that a set of parallel gradual rules of the form Ai + f (Ai) can exactly represent a real-valued function of one real variable, which is continuous and monotonic on an interval. This result is only due to the fact that the function f is directly encoded by the membership function of the Bi's. In other words, there is no simplification at all, from a representation point of view. However, representing f by a set of rules allows a linguistic representation of it, if labels are given to the At's (leading to a linguistic partition of U , as in Fig. 2.3) and if the Bi's are approximated by a well-chosen linguistic partition on V. Moreover, a rule-based representation of a function allows to easily perform local modifications by altering some rules. The same operation on a parametrized function is much more difficult since changes generally affect its global behavior. The exact representation cannot be generalized to multiple-input functions. It has been shown in (Dubois et al., 1994) that rule bases constructed with the extension principle for such functions allow to retrieve exactly the function only for the reference points u = ai, and give a bracketing of the function for the other points. This result is explained by the fact that useful multiple-input functions are not bijective in practice, and equation (2.38) (one way to express the extension principle) obviously only holds for bijective functions. Usually, f (v) is not a singleton. In conclusion, the sets of gradual rules constructed with the extension principle verify two main properties which make them suitable for function representation:

-'

A set of rules constructed with the extension principle is always coherent.

t

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE m

51

The GMP used with a set of rules constructed from the function f , with the extension principle always give (at least) a bracketing (and not only an approximation) of the function: f (uO)E B'.

2.5.3 Interpolation between fuzzy rules in case o f an incomplete rule base In the previous section it has been shown that the usual concept of interpolation corresponds exactly to the logic of gradual rules. Conversely, a set of fuzzy rules "if X is Ai, then Y is Bi"can be viewed as a set of ill-located points on the graph of an unknown function. These ill-located points lie somewhere in the fuzzy Cartesian product Ai x Bi, for i = 1,. . . ,n (with x ~( u i ,v ) = min(pAi( u ) , p~~ ( v ) ) .This view is particularly natural when the fuzzy spots (Ai,Bi) are disjoint or even sparse. This type of problem has been studied in fuzzy data analysis, using different approaches: fuzzy linear regression, fuzzy least squares methods, etc. (Kacprzyk and Fedrizzi, 1992). Lowen (Lowen, 1990) has established a Lagrange interpolation theorem for fuzzy data of the form (ai,Bi) where only the Bi's are fuzzy intervals. Here, the problem of linear interpolation between two fuzzy points ( A l ,B1),(A2,B2) such that A1 n A2 = 0 is considered. Namely, the considered fuzzy points (Ai,Bi)are no more overlapping as in the previous section. They correspond to sparse pieces of knowledge between which we may wish to interpolate (in a linearway in the following) for dealing with some uncovered situation precisely described by u0 E U. Such sparsely described mappings have been studied in (K6czy and Hirota, 1993). The authors explicitly refer to distances in order to implement interpolative reasoning between the sparse pieces of knowledge. In case of precisely known points ( a l ,b l ) and (a2,b2), for an input value u0 E [al,a2](assuming a1 < a2),the output v0 is computed as follows:

v0 = X.bl

+ ( 1 - A ) .b2,

a2 - uO where X = -----, a2 - a1

+

or equivalently, (a2- al).vO= (a2- uo).bl (uO- ai).ba. Since al, a2, bl and b2 are ill-known it is possible to compute a fuzzy relation R between u0 and vO,by application of the extension principle to the above expression, changing al, a2, bl and b2 into Al, A2,B1 and B2,as follows:

where ul,u2,v l , v2 are such that: (u2 - uO).vl+ (uO- ul).v2 = (u2 - ul).vO In order to propose a constructive calculation method, we can first look for the possible values of a parameter A such that

where the possible values of vl and u2 are restricted by A1 and A2 respectively (we assume that u0 is in-between the supports of Al and A2). Then A is restricted by the fuzzy set A, applying the extension principle to the function ( u l ,u2) t A =

e:

52

FUZZY ALGORITHMS FOR CONTROL

Let [al7a:] (resp. [a2,a ; ] )be an a-level cut of A1 (resp. Az). Propagating the constraints ul E [a1,a:] and u2 E [a2,a;] along with (2.39) gives an interval of the form

which represents the a-level cut of A. When Al and A2 are LR fuzzy numbers with peaks a1 and a2, spreads al and yl , a2 and 72 respectively (Dubois and Prade, 1985), i.e.: pni (u) = L

(-)ai

-u

ai

, if u < ai and R

(7) , if u 2 ai,

p~ can be analytically computed by using the fact that X is increasing with respect to u1 and u2:

Once A is computed we can conclude by interpolationthat the values of v0 plausibly associated with the situation u0 are described by the fuzzy set B0 defined by:

Such an interactive expression can be easily computed in practice (Dubois and Prade, 1985, Chap. 2 Sect. 4). Note that B0 cannot be written as A O Bi $ ( 1 9 A) O Bi+1, where @, 9and @ denote extended sum, differenceand product, due to the interactivity between A and 18A. In particular it can be carried out on level-cuts, as in the example below. Example. Suppose that A1 = [l,21, B1 = [3,5],A2 = [5,7],B2 = [ B , 91. Consider u0 = 4. Then,

FUZZY LOGIC, CONTROL ENGINEERING AND ARTIFICIAL INTELLIGENCE

53

This approach can be readily extended when the situation uOis no longer precisely known but described by a fuzzy set S . Then we have to incorporate ps(u) in the min-part of (2.40) while the sup is now taken on ui,ui+l and u. We can still work with the a-level cuts and what is obtained then is a fuzzy set A which is intuitively larger than in the case where u0 is precisely known. It all comes down to computing the image of S via the fuzzy relation R using sup-min composition. The extension of this approach to rules with two conditions (like if X I is Ai,l and X2 is Ai,z, then Y is B;) requires the use of three rules that corresponds to three fuzzy points (Ai,1,Ai,2, Bi), i = 1 , . . . , 3 . The problem is then to compute the fuzzy values of 2 parameters A1 and X z that characterize the location of ( u Ov, O )with respects to the three fuzzy points (Ai,1,Ai,2) in the plane, that is, two equations similar to (2.39) can be constructed. Their solution requires the use of fuzzy constraint propagation methods, and deserves a careful study. Once A1 and A2 are obtained the fuzzy set B0 interpolated from the three rules would again be computed via a fuzzy number computation involving three interactive parameters Al, A2 and A3 summing to 1.

2.6

CONCLUSIONS

A control law described by fuzzy rules is more understandable by a human operator who knows the process but who is not an expert control engineer. It facilitates the integration of heuristic knowledge about the system (which is easily encoded by rules) and theoretic control laws from mathematical models, which are always idealizations of the process. Thus, fuzzy logic does not challenge the results or approach of classical control. It is more a complementary tool when the know-how and the heuristic knowledge on a process is more easily available than a good modeling. This chapter has studied the relation between approximate reasoning theory and the fuzzy controller "logic". However, it has been emphasized that even under the same terminology fuuy logic,A1 researchers and control engineers do not pursue the same goals. For the former, fuzzy logic is a tool for imprecise and/or uncertain knowledge representation and approximate reasoning, while for control engineers, fuzzy rulebased systems are used as a family of non-linear function approximators useful for the design of efficient control laws. In connection with this clash of interests, it has been shown that the fuzzy approach to approximate reasoning, which is consistent with classical logic is quite incompatible with the way control engineers exploit fuzzy rule-based systems. The core of the misunderstanding is that control engineers do not consider fuzzy rules as constraints because they aggregate them disjunctively, while constraints aggregate conjunctively. Mamdani or Sugeno's fuzzy rules model pieces of data that accumulate and the interpolation process is added on top by the defuzzification process, with no reference to fuzzy set or possibility theory. On the contrary, gradual rules seem to be particularly suited for the formalization of interpolative reasoning. Indeed, interpolation is embedded in the inference process of gradual rules (which obeys the generalized modus ponens). The close link between gradual rules and the extension principle of Zadeh, which defines the image of a fuzzy set by a function, gives an easy way to define a set of rules bracketing or approximating a given function. However, the interpolative capabilities of gradual rules in the multi-dimensional case lead to interval-valued outputs. Sets of gradual

54

FUZZY ALGORITHMS FOR CONTROL

rules are trivially universal approximators, but there is for now no particular family of (simple) membership functions which makes interpolation by gradual rules simpler and more efficient than other methods. Actually, the practical potential of gradual rules for approximation of functions is largely unexplored. However the logic of gradual rules gives both a tool for checking the coherence of a rule base given by a human operator, and an interpolation method which is more theoretically founded than the Mamdani's one, and corresponds exactly to the Sugeno's one in the mono-dimensional case. As the inference with gradual rules is more difficult only when non-precise inputs are considered (a rather rare situation in fuzzy control applications), the preference for conjunction-based rules in practical applications seems to be mainly due to cultural traditions and pragmatic considerations.

References Ayel, M. and Rousset, M. (1990). La cohkrence duns les bases de connaissances. Ctpadubs Editions, Toulouse, France. BabuSka, R., Jager, R., and Verbruggen, H. (1994). Interpolation issues in SugenoTakagi reasoning. In Proceedings of the 3rd IEEE International Conference on Fuzzy Systems (FUZZ-IEEE'94),pages 859-863. Benoit, E. and Foulloy, L. (1993). Capteurs flous multicomposants : Application a la reconnaissance. In Actes des 3tmes Joumkes Nationales sur les Applications des Ensembles Flous, EC2, Nanterre. Berenji, H. and Khedkar, P. (1992). Learning and tuning fuzzy logic controllers through reinforcements. IEEE Transactions on Neural Networks, 3:724-740. Bersini, H. and Gorrini, V. (1994). Mlp, rbf, flc: What's the difference. In Proeedings of the 2nd European Conference on Intelligent Techniques and Soft Computing (EUFIT194),pages 19-26, Aachen, Germany. ELITE-Foundation. Braae, M. and Rutherford, D. (1979). Theoretical and linguistic aspects of the fuzzy logic controller. Automatica, 15:15-30. Buchanan, B. and Shortliffe, E. (1984). Rule-based expert systems - The MYCIN experiments of the Stanford Heuristic Programming Projects. Addison Wesley. Reading. Buckley, J. and Hayashi, Y. (1993). Fuzzy input-output controllers are universal approximators. Fuuy Sets and Systems, 58:273-278. Castro, J . (1995). Fuzzy logic controllers are universal approximators. IEEE Transactions on Systems, Man and Cybernetics, 25:629-635. Cayrol, M., Farreny, H., and Prade, H. (1982). Fuzzy pattern matching. Kybernetes, 11:103-116. Chanas, S. and Nowakowski, M. (1988). Single value simulation of a fuzzy variable. Fuzzy sets and Systems, 25:43-59. Chand, S. and Chiu, S., editors (1995). Special issue on fuzzy logic with engineering applications. Proceedings of the IEEE, 83(3). Denneberg, D. (1994). Non-Additive Measure and Integral. Kluwer Academic Publishers, Dordrecht. Dubois, D. and Gentil, S. (1994). Intelligence artificielle et automatique. Revue dJIntelligence Artijcielle, 8(1):7-27.

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Dubois, D., Grabisch, M., and Prade, H. (1994). Gradual rules and the approximation of control laws. In H.T. Nguyen et al., editor, Theoretical Aspects of Fuuy Control, pages 147-1 8 1. Wiley, New York. Dubois, D., Martin-Clouaire, R., and Prade, H. (1988a). Practical computing in fuzzy logic. In Gupta, M. and Yamakawa, T., editors, Fuuy Computing - Theory, Hardware and Applications, pages 1 1-34. North-Holland, Amsterdam. Dubois, D. and Prade, H. (1980). Fuuy Sets and Systems: Theory and Applications. Academic Press, New York. Dubois, D. and Prade, H. (1985). Thkorie des Possibilite's -Applicationsa la Repre'sentation des Connaissances en Informatique. Masson, Paris. with collaboration of H. Farreny, R. Martin-Clouaire and C. Testemale, 26me Cdition 1987. Dubois, D. and Prade, H. (1987). The mean value of a fuzzy number. Fuuy Sets and Systems, 24:279-30. Dubois, D. and Prade, H. (1991). Fuzzy sets in approximate reasoning - part 1: Inference with possibility distributions. Fuuy Sets and Systems, 40: 143-202. Dubois, D. and Prade, H. (1992a). Fuzzy rules in knowledge-based systems - modeling gradedness, uncertainty and preference. In Yager, R. and Zadeh, L., editors, An Introduction to Fuuy Logic Applications in Intelligent Systems, pages 45-68. Kluwer Academic Publishers, Boston. Dubois, D. and Prade, H. (1992b). Gradual inference rules in approximate reasoning. Information Sciences, 6 1:103-122. Dubois, D. and Prade, H. (1994). On the validation of fuzzy knowledge bases. In Fuuy Reasoning in Information, Decision and Control Systems, pages 31-49. Kluwer Academic Publishers, Dordrecht. Dubois, D. and Prade, H. (1996a). Logique floue, interpolation et commande. J. Europ. Syst. Autom. (JESA),30(5):607-644. Dubois, D. and Prade, H. (1996b). What are fuzzy rules and how to use them. Fuzzy sets and Systems, 84(2): 169-1 86. Special issue in memory of Prof A. Kaufmann. Dubois, D., Prade, H., and Sandri, S. (1993). On possibility / probability transformations. In Lowen, R. and Roubens, M., editors, Fuuy Logic - State of the Art, pages 103-1 12. Kluwer Academic Publishers, Dordrecht. Dubois, D., Prade, H., and Testemale, C. (1988b). Weighted fuzzy pattern-matching. Fuuy Sets and Systems, 28(3):3 13-33 1. Dubois, D., Prade, H., and Ughetto, L. (1997). Checking the coherence and redundancy of fuzzy knowledge bases. IEEE Transactins on Fuuy Systems, 5(3):398-417. Efstathiou, J . (1995). Special issue on modern fuzzy control. Fuuy Sets and Systems, 70(2-3). Farreny, H. and Prade, H. (1986). Dealing with the vagueness of natural languages in man-machine communication. In Karwowski, W. and Mital, A., editors, Applicatiotzs of Fuuy Set Theory in Human Factors, pages 71-85. Elsevier, Amsterdam. Foulloy, L. (1994). Typologie des contr6leursflous - La logiquefloue. Rapport ARAGO number 14, pages 80-107. OFTA, Masson, Paris. Galichet, S., Dussaud, M., and Foulloy, L. (1992). ContrBleurs flous : Equivalences et Ctudes comparatives. In Actes des 2tmes Journe'es Nationales sur les Applications des Ensembles Flous, Nimes. EC2, Nanterre.

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Holmblad, L. and gstergaard, J. (1995). The FLS application of fuzzy logic. Fuzzy Sets and Systems, 70:135-1 46. Jager, R., Verbruggen, H., and Bruijn, P. (1994). Demystification of fuzzy control. In Tzafestas, S. and Venetsanopoulos, A., editors, Fuuy Reasoning in Information, Decision and Control Systems, pages 165-197. Kluwer Academic Publishers, Dordrecht. Jiang, J. and Sun, S. (1995). Neuro fuzzy modeling and control. Proceedings of the IEEE, 83:378406. Kacprzyk, J. and Fedrizzi, M., editors (1992). Fuuy Regression Analysis. Omnitech Press, Warsaw. Kaufmann, A. (1973). Introduction d la Thkorie des Sous-Ensembles Flous. Masson, Paris. Kinkiklklk, D. (1994). Vkn'jication de la cohkrence des bases de connaissances$oues. PhD thesis, Universitk de Savoie, Chambkry, France. Kbczy, L, and Hirota, K. (1993). Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases. Information Sciences, 7 1 :169-20 1. Kosko, B. (1991). Neural Networks and Fuuy Systems. Prentice-Hall, Englewood Cliffs, NJ. Kosko, B. (1992). Fuzzy systems as universal approximators. In Proc. of the 1st IEEE International Conference on Fuuy Systems (FUZZ-IEEE'92), pages 1 1 53-1 1 62, San Diego, CA. Larsen, P. (1980). Industrial applications of fuzzy logic control. International Journal of Man-Machine Studies, 12:3-10. Lowen, R. (1990). A fuzzy Lagrange interpolation theorem. Fuzzy Sets and Systems, 34:33-38. Mamdani, E. and Assilian, S. (1975). An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies, 7:1-1 3. Mendel, J. (1995). Fuzzy logic systems for engineering: A tutorial. Proceedings of the IEEE, 83:345-377. Nguyen, H., Sugeno, M., Tong, R., and Yager, R. (1995). TheoreticalAspects of Fuzzy Control. Wiley, New York. Palm, R. and Driankov, D. (1995). Fuzzy inputs. Fuuy Sets and Systems, 70:3 15-336. Prade, H. (1988). Raisonner avec des rbgles d'infkrence graduelle. Revue d'lntelligence Artijicielle, 2(1):29-44. Ruspini, E. (1969). A new approach to clustering. Information and Control, 15:22-32. Schott, B. and Whalen, T. (1996). Nonmonotonicity and discretization error in fuzzy rule-based control using COA and MOM defuzzification. In Proceedings of the 5th IEEE Internatinal Conference on Fuzzy Systems (FUZZ-IEEE196),pages 450-456. Sugeno, M. (1985). An introductory survey of fuzzy control. Information Sciences, 36:59-83. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics, 15:116-132.

t

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Ughetto, L., Dubois, D., and Prade, H. (1997). Efficient inference procedures with fuzzy inputs. In Proceedings of the 6th IEEE International Conference on Fuzzy Systems (FUZZ-IEEE'97),pages 567-572, Barcelona, Spain. Wang, L. (1992). Fuzzy systems are universal approximators. In Proceedings of the 1st IEEE Internatinal Conference on Fuuy Systems (FUZZ-IEEE'92),pages 11631169, San Diego, CA. Wang, L. and Mendel, J. (1992). Fuzzy basis functions, universal approximations and orthogonal least square learning. IEEE Transactions on Neural Networks, 3:807813. Wenstflp, F. (1976). Fuzzy set simulation models in a system dynamics perspectives. Kybernetes, 6:209-2 18. Yager, R. (198 1). A procedure for ordering fuzzy subsets of the unit interval. Information sciences, 24: 143-161. Yager, R. and Filev, D. (1994). Essentials of Fuzzy Modeling and Control. John Wiley & Sons, New York. Yager, R., Ovchinnikov, S., Tong, R., and Nguyen, H. (1987). Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh. John Wiley & Sons, New York. Zadeh, L. (1965). Fuzzy sets. Information and Control, 8:338-353. Zadeh, L. (1973). Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man and Cybernetics, 3:28-44. Zadeh, L. (1975). The concept of a linguistic variable and its application to approximate reasoning. Information Sciences. Part 1: 8:199-249; Part 2: 8:301-357; Part 3: 9:4380. Zadeh, L. (1978). Fuzzy sets as a basis for a theory of possibility. Fuuy Sets and Systems, 1:3-28. Zadeh, L. (1979). A theory of approximate reasoning. Machine Intelligence, 9:149174. Elsevier, New York. Zadeh, L. (1996). Fuzzy logic = computing with words. IEEE Transactions on Fuuy Systems, 4(2): 103-1 11. Zimmermann, H. (1987). Fuuy Sets, Decision Making, and Expert Systems. Kluwer Academic Publishers, Boston. Zurada, J., 11, R. M., and Robinson, C. (1994). Computational Intelligence Imitating Life. IEEE Press, NY.

3

FUZZY CONTROL VERSUS CONVENTIONAL CONTROL K.-E. Arzknl, M. Johanssonl and R. BabuSka2

'Lund Institute of Technology Dept. of Automatic Control

P.O. Box 118, S-221 00 Lund, Sweden, 'Delft University of Technology Faculty of Information Technology and Systems Control Laboratory Mekelweg 4, PO Box 5031

2600 GA Delft, The Netherlands

3.1

INTRODUCTION

Fuzzy sets, the foundation of fuzzy control, were introduced thirty years ago, (Zadeh, 1965), as a way of expressing non-probabilistic uncertainties. Since then, fuzzy set theory has developed and found applications in database management, operations analysis, decision support systems, signal processing, data classifications, computer vision, etc. The application area that has attracted most attention is, however, control. In 1974, the first successful application of fuzzy logic to control was reported (Mamdani, 1974). Control of cement kilns was an early industrial application (Holmblad and Bstergaard, 1982). Since the first consumer product using fuzzy logic was marketed in 1987, the use of fuzzy control has increased substantially. A number of CAD environments for fuzzy control design have emerged together with VLSI hardware for fast execution. Fuzzy control is being applied industrially in an increasing number of

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FUZZY ALGORITHMS FOR CONTROL t

cases, e.g., (Froese, 1993; Hellendoorn, 1993; Bonissone, 1994; Hirota, 1993; Terano eta]., 1994). The early work in fuzzy control was motivated by a desire to directly express the control actions of an experienced human operator in the controller, i.e., to mimic his behavior, and to obtain smooth interpolation between discrete controller outputs. Since then the application range of fuzzy control has widened substantially. However, the two main motivations still persevere. The linguistic nature of fuzzy control makes it possible to express process knowledge concerning how the process should be controlled or how the process behaves. The interpolation aspect of fuzzy control has led to the viewpoint where fuzzy systems are seen as smooth function approximation schemes. In most cases a fuzzy controller is used for direct feedback control. However, it can also be used on the supervisory level as, e.g., a self-tuning device in a conventional PID controller. Also, fuzzy control is no longer only used to directly express a priori process knowledge. For example, a fuzzy controller can be derived from a fuzzy model obtained through system identification. Therefore, it is difficult to define what a fuzzy controller is. A very general definition is: A Fuuy Controller is a controller that contains an, often non-linear, mapping that has been defined using fuzzy logic-based rules. The key issues in this definition are the non-linear mapping and the fuzzy logic-based rules. Fuzzy control has always been a controversial subject (IEEE, 1993a; IEEE, 1993b). This owes partly to lack of mutual understanding between the fuzzy control community and the traditional control community and partly to exaggerated claims in certain papers on fuzzy control. For example, in (Cox, 1993) fuzzy control is presented as a panacea that can solve 'all' control problems. Many people active in fuzzy control have no classical control background. This typically leads to reinventions of the wheel. Results that are well-known in classical control are presented as novelties in fuzzy control. At the same time many classical control engineers have a very 'fuzzy' idea of what fuzzy control really is. The empirical nature of fuzzy control where the importance of mathematical models is deemphasized is sometimes regarded as 'non-scientific'. The aim of this chapter is to compare fuzzy control with conventional control schemes both on the industrial arena and in academia.

4

4

4

Chapter outline In Section 1.2 the structure of fuzzy control systems is discussed. Most industrial control systems use PID control. In Section 1.3 a comparison is made between industrial PID control and fuzzy control. Fuzzy control system can be viewed as a special form of nonlinear control. This view is further discussed in Section 1.4.

3.2

FUZZY CONTROL SYSTEMS

Fuzzy set theory uses fuzzy inferencing to reason about linguistic variables, i.e., variables described by fuzzy sets. A number of different inference systems have been

FUZZY CONTROL VS. CONVENTIONAL CONTROL

61

developed. In fuzzy control it is, however, two inference systems that dominate: Mamdani fuzzy systems, also known as linguistic fuzzy systems, and Takagi-Sugeno fuzzy systems.

3.2.1 Mamdani fuzzy systems A Mamdani Fuzzy System contains a knowledge base consisting of fuzzy IF-THEN rules and membership function definitions, together with an inference engine that applies the fuzzy rules to the fuzzy input variables, generating fuzzy output variables according to Figure 3.1. The rules in a Mamdani fuzzy systems are on the form

I I

Knowledge Base

lFs1sHlGHdYnLOWhenuaZEA0

Figure 3.1

IF s s H I M a M Y h H I W h e n u b NEGITIVE

I I

The knowledge base and inference engine of a Mamdani fuzzy system

R(') : IF 2 1 IS A?) AND . . . AND x, IS A:) THEN y IS B(')

(3.1)

where xi are the input variables and y is the output variable. Both the rule antecedent and consequent are defined using fuzzy sets. In the above example only the AND connective is used. Rule bases on this form are known as being written in conjunctive form. In the general case, also other logical connectives may be used, e.g., OR and NOT. In fuzzy control the inputs and outputs are crisp rather than fuzzy. Therefore fuzzification and defuzzification are needed. Fuzzification transforms the numerical inputs into matching fuzzy sets, and defuzzification approximates the output fuzzy set with a single crisp number. The structure is extended according to Figure 3.2. The structure in Figure 3.2 provides an internal view of the fuzzy system. However, from an external view the fuzzy system implements a nonlinear, static mapping from x to y, according to Figure 3.3. The fuzzy sets and rules can be seen as a way of parameterizing this nonlinearity. In Mamdani fuzzy systems the antecedent and consequent fuzzy sets are often chosen to be triangular or Gaussian. It is also common that the input membership functions overlap in such a way that the membership values of the rule antecedents always sum to one. In this case, and if the rule base is in the conjunctive form, one can interpret each rule as defining the output value for one point in the input space. The input space

62

FUZZY ALGORITHMS FOR CONTROL Knowledge Base

4 0.2

Fuzz~flcat~on , -

Figure 3.2

Inference

0.6

B.

Deuzzlflcat~on

A Marndani fuzzy system.

Internal View

Figure 3.3

The internal and external views o f a fuzzy system

point is the point obtained by taking the centers of the input fuzzy sets and the output value is the center of the output fuzzy set. The fuzzy logic is used to obtain smooth interpolation between the points in the input space. Depending on which inference method is used, different interpolations are obtained. By proper choices it is even possible to obtain linear or multilinear interpolation. Mamdani fuzzy systems are quite close in nature to manual control. The controller is defined by specifying what the output should be for a number of of different input signal combinations. Each input signal combination is represented as a rule.

3.2.2 Takagi-Sugeno fuzzy systems A Takagi-Sugeno (TS) fuzzy system (Takagi and Sugeno, 1985) uses rules on the form

R(": IF X I IS A?) AND . . . AND x, IS A!) THEN y = h("(x)

(3.2)

The output functions h(')(x)could, in principle, be arbitrary functions of the inputs. However, in most cases they are chosen to be a linear combination of the inputs, i.e.,

t

FUZZY CONTROL VS. CONVENTIONAL CONTROL

63

This is called a linear Takagi-Sugeno (TS) system or a homogeneous TS system. Sometimes also a constant term is added, i.e.,

This structure is known as an affine TS system. In a TS fuzzy system, fuzzy logic is only used in the antecedent part of the rules. The consequent part is described using ordinary, crisp, numbers. The total output from a TS system is obtained as a weighted average of the crisp outputs of the individual rules with the weights taken as the degree of belief in the rules. Hence, the defuzzification is built-in to the inference system. When TS fuzzy systems are used it is common that the input fuzzy sets are trapezoidal. Each fuzzy set determines a region in the input space where, in the linear case, the output is determined by a linear function of the inputs. Fuzzy logic is only used to interpolate in the cases where the regions in the input space overlap. TS fuzzy systems are close in nature to gain scheduling. For each input region a linear controller is defined. The total controller is obtained by selecting one of the controllers based on the value of the inputs, or by interpolating between several of the linear controllers.

3.2.3 f i z z y nonlinear mappings Fuzzy systems implement static nonlinear input-output mappings. This implies that fuzzy control is a subset of nonlinear control. A major reason for nonlinear control is that the process is nonlinear. Basically all real processes are nonlinear, either through nonlinear dynamics or through linear dynamics in combination with constraints on, e.g., states and inputs. Another reason for nonlinear control is that the performance specifications are "nonlinear". For example, it may be of interest to have different small signal and large signal behavior. Also, several optimal control problems, e.g., the minimal time problem, have nonlinear controllers as solutions. The nonlinear mappings generated by fuzzy systems can in general be written as basis function expansions, i.e., as a weighted sum of basis functions gi(x) according

The exact nature of the basis functions is determined by which inference system that is used and how the fuzzy operations are defined. The basis function expansion can be viewed as canonical form that is common for several nonlinear function approximation methods, e.g., sigmoidal neural networks, radial basis function, splines, and wavelets. Under certain conditions it is possible to get an exact correspondence between certain fuzzy systems and other function approximation schemes, e.g., radial basis functions.

64

FUZZY ALGORITHMS FOR CONTROL t

3.2.4 f i z z y controller structure A fuzzy system mapping is just one part of a fuzzy controller. Often, signal processing is required both before and after the fuzzy system evaluation. A general structure that captures most applications of fuzzy systems to control is illustrated in Figure 3.4.

I F u u y Controller Pre-

-

I PostFlHm -+

Figure 3.4 A general fuzzy controller structure, consisting of a prefiltering device, fuzzy system mappings and a postfiltering device.

The structure consists of three parts: 1. a Prefiltering Device - for computing the fuzzy system inputs; 2. one (or several) Fuzzy System Mapping(s);

3. a Postfiltering Device - for computing the actual control signal. The prefiltering device represents the signal processing performed on the controller inputs in order to obtain the inputs of the fuzzy system. The prefiltering device may, for instance, perform some of the following operations on the input signals

4

Sampling. This includes time-sampling,quantization,and general AID conversion. Signal Conditioning. It is sometimes convenient to work with signals on a normalized domain, e.g., [-I, 11. This is accomplished by the introduction of normalization gains. The normalization gain is a linear gain that scales the input into the normalized domain [-I, 11. Values that fall outside the normalized domain are mapped onto the appropriate endpoint. Dynamic Filtering. This includes both linear and nonlinear filters. In a fuzzy PID controller, for instance, linear filters are used to obtain the control error, the error derivative and the error integral. Nonlinear filters are found in nonlinear observers, and in adaptive fuzzy control where they are used to obtain the fuzzy system parameter estimates. Feature Extraction. Through extraction of different features numeric transformations of the controller inputs is performed. These transformations may be Fourieror Wavelet-based transformations,coordinate transformationsor other basic operations performed on the fuzzy controller inputs. One interesting example is the linear trans, x being the input vector and V a possibly rectangular matrix. formation W = V X with

:

FUZZY CONTROL VS. CONVENTIONAL CONTROL

65

Function approximation on linear transforms of input variables are found in so called "Ridge Approximation" schemes (Sjoberg et al., 1995). Another interesting example is to compute the pairwise products xij = xixj,which allows correlations to be used as inputs to the subsequent parts of the controller. The postfiltering device represents the signal processing performed on the fuzzy system output to obtain the actual control signal. Operations that the postfiltering device may perform include

Precomputed Part of Control. In some fuzzy controllers, the purpose of the fuzzy system is to model the process dynamics. The "precomputed part of control" is then typically a model-based control scheme that uses the fuzzy model to compute the appropriate control action. Another example is when the fuzzy system is a supervisory tuning-device for a conventional PID controller. The "precomputed part of control" is then the PID algorithm, and the purpose of the fuzzy system is to select the appropriate PID parameters. Signal Conditioning. This can be a denormalization gain that scales the output of the fuzzy system to the physical domain of the actuator signal. Dynamic Filtering. In some cases, the output of the fuzzy system is the control increments. The actual control signal is then obtained by integrating the control increments. Of course, other forms of smoothing devices and even nonlinear filters may be considered. Sampling. This is typically hold devices and more general D/A conversion. Fuzzy control can in principle be applied at all levels in the control system hierarchy from the sensor and actuator level up to the production planning level. However, in practice fuzzy control is mostly used at the direct control level or at the supervisory control level. At the direct control level the inputs to the controller are the measured values and reference signals and the output from the controller is the control signal(s). Depending on which signals that are used as the input to the fuzzy mapping, different fuzzy controllers are obtained that are structurally equivalent to various conventional linear control schemes. For example, a fuzzy controller with the controller error, e, and the error derivative as inputs and the control signal as the output is structurally equivalent to a conventional PD-controller. A controller with the process state vector, x, as the inputs and the control signal as output is structurally equivalent to a state feedback controller. If the nonlinear mapping is designed to be linear then exact equivalence is obtained between the fuzzy controIIer and the corresponding conventional control scheme. At the supervisory control level the outputs from the controller are typically reference signals to underlying controllers. An example is the cascade control scheme. It is also possible to use fuzzy control in combination with a conventional controller. One example is to use a fuzzy system as a gain scheduling device that adjusts the parameters in a conventional controller. Another example is to use a fuzzy controller

66

FUZZY ALGORITHMS FOR CONTROL t

in parallel with a conventional controller. The outputs of the two controllers are added to form the total control signal. The fuzzy controller is designed to only contributeto the control signal in operating regions where the linear control action of the conventional controller is insufficient.

3.3

INDUSTRIAL F U Z Z Y CONTROL

Industrial process control is an area where several successful application of fuzzy control have been reported. There are several reasons for this. Linear PID control is the most widely used control structure in industry. The PID controller is usually presented as t

An alternative parameterization of the PID controller is the velocity (or incremental) form

The latter form has several advantages, e.g., less problems with integrator windup and bumpless mode changes. The reason why PID control has been so successful is that it is intuitive and close to the way humans manually control a process. The control action depends on the current controller error (the P-term), the time history of the error (the I-term), and a prediction of the future value of the error (the D-term). Furthermore, a general empirical observation is that most industrial processes can be controlled reasonably well with PID control provided that the demands on performance are not too high. Heuristic tuning rules exists for PID controllers that work reasonably well for a large class of processes. Hence, it is possible to use a PID controller also in the case when a detailed process model is not available. However, there are many cases when it is an advantage to use a nonlinear PID controller. If the plant is linear, we may for instance improve transient performance by crafting a controller nonlinearity that works as a time-optimal controller when the control error is large and as a linear controller when the error is small. If the plant is nonlinear, the controller nonlinearity can be designed so as to compensate for plant nonlinearities. A dead-zone is another example of a nonlinearity that is useful in connection with PID control. This reduces the wear on the actuators. Using fuzzy PID control it is possible to combine the advantages of ordinary linear PID control with the possibility to introduce nonlinearities in the control law. Using fuzzy control is also possible expand the basic PID scheme by introducing extra input and output signals, thus creating multivariable controllers. The different forms of fuzzy PID control can be summarized as:

4

*

FUZZY CONTROL VS. CONVENTIONAL CONTROL

Fuuy P-control: Fuzzy PD-control: Fuuy PI-control (absoluteform): Fuzzy PI-control (incrementalform): Fuzzy PID-control (absoluteform): Fuzzy PID-control (incrementalform):

67

u= F(e(t)) U =z.F ( e ( t ) $ , e(t)) U = Y. ( e ( t ) J , 47)) % = F (-$e(t),e ( t ) ) u = F ( e ( t )J, e(-r),$ e ( t ) ) $ =F . (-$e(t), e ( t ) ,-$e(t))

where 3(.) is a fuzzy system mapping. In fuzzy PID control the incremental forms are the most common when the controller contains integral action. The reason for this is that it is more intuitive to describe the controller actions in terms of changes to the control signal then in terms of the sum of the controller error. The rule base in a fuzzy PID controller builds up a nonlinearity that, in its basic form, is very close to being linear. A fuzzy PD controller could be obtained with the following nine rules: IF e IS N L AND e IS N L IF e IS Z E AND e IS N L IF e IS P L AND e IS N L IFeISNLANDeISZE IF e IS Z E AND e IS Z E IF e IS P L AND e IS Z E IF e IS N L AND e IS P L IF e IS Z E AND e IS P L IF e IS P L A N D e IS P L

THEN u IS THEN u IS THEN u IS THEN u IS THEN u IS THEN u IS THEN u IS THEN u IS THEN u IS

NL NS ZE NS ZE PS ZE PS PL

where we have introduced the labels N L , N S , Z E , PS and P L to mean Negative Large, Negative Small, Zero, Positive Small and Positive Large respectively. The readability is enhanced by presenting the rules in the table

The rule table has the typical stripe-diagonalform. In fuzzy control, a simple difference Ae = e ( k ) - e(k - 1 ) is often used as an approximation for the derivative. Starting from this, essentially linear fuzzy PD controller, the shape of the nonlinearity can be easily adjusted by modifying the membership functions or by introducing new rules. There are also other reasons why fuzzy control has been successful in industry. Fuzzy control is a direct approach to nonlinear control design. The rule-based formalism is intuitive and easy to understand for noncontrol engineers, especially in comparison with modern nonlinear control theory that is all but intuitive and user-friendly. Each rule represents local process knowledge about how the control signal should be selected for certain input signals. The local nature of the rules makes it possible to build up a controller in a step-wise fashion. Another reason for the success is the way

68

FUZZY ALGORITHMS FOR CONTROL

the technique is packaged. Commercial CAD environments for fuzzy controller development have user-friendly, graphical environments. They are available on industrially accepted hardware and they can automatically generate C code. This makes it easy for industry to apply fuzzy control. To start experimenting with non-linear control using existing industrial control systems is considerably more difficult. They often only provide pre-packaged PID blocks and anything in addition to that can be very difficult for the end-users to implement. 3.4

MODERN NONLINEAR FUZZY CONTROL

Fuzzy PID control can be viewed as a heuristic, or model-free, control approach. The controller is designed and tuned without any explicit model knowledge about the process. However, fuzzy control can also be used when an explicit process model is available. In the academic world there is a large amount of work being done on model-based fuzzy control. The key issues here are analysis and synthesis methods for fuzzy control systems. The predominating standpoint is to view fuzzy control as special case of nonlinear control where the fuzzy logic is used for parameterizing the nonlinearities. Alternatively, fuzzy systems are viewed as function approximation schemes that can be used for nonlinear system identification.

3.4.1 Fuzzy inverse control The simplest approach to design a controller for a nonlinear process when a process model is available is to use inverse control. This approach can also be used in fuzzy control, in which case the controller is based on the inverse of the fuzzy process model. The inversion is here only explained for SISO fuzzy models without any delay from the input to the output. In this case the fuzzy model can be written as

where x ( k ) is the state vector consisting of past process outputs and process inputs, i.e., x(k) = [ y ( k ) ,. . . ,y ( k - n , + l ) , u ( k - I ) , . . , u ( k - n u $ I ) ] ,u ( k ) is thecurrent .) is a fuzzy mapping. This model predicts the system's output process input, and 3(., at the next sample time. The objective of the inverse control method is to compute the input u ( k ) , such that the system output at the next sampling instant is equal to the desired (reference) output r ( k + 1).This can be achieved if the process model can be inverted according to: u(k)= 3-'(x(k), r(k 1))

.

+

If it is possible to obtain the inverse model, an open-loop (feedforward)control scheme can be used according to Figure 3.5. As with any open-loop control scheme, fuzzy inverse control only works if there is no disturbances or modeling errors. Inversion based approaches can also only be applied to stable systems with a minimum phase behavior (systems whose inverted dynamics are stable). Fuzzy inverse control has been proposed by a number of authors, e.g., (Braae and Rutherford, 1979; Pedrycz, 1993; Driankov et a]., 1993; Harris et al., 1993; Raymond

FUZZY CONTROL VS. CONVENTIONAL CONTROL

r(k+"

U(k'

Inverted Fuzzy Model

Figure 3.5

,

Y(k+l)

Process

69

w

Fuzzy inverse control.

et al., 1995). In most cases only an approximate inverse is used. In (Babugka, 1998) an exact fuzzy inverse scheme is suggested for the case of fuzzy singleton models.

3.4.2 Fuzzy internal model control Disturbances and model errors cause problems for open-loop schemes. The internal model control (IMC) scheme is one way of compensating for this. A fuzzy IMC controller scheme consists of of three parts: the controller based on an inverse of the process model, the fuzzy model, and a feedback filter, according to Figure 3.6.

"1

..............................................

Inverse fuzzy model

\ ', L

Process

................................................

F M

UJ - i

Fuzzy

-

model

Feedback filter < _...........................................................................................,

Figure 3.6

Fuzzy internal model control scheme.

The purpose of the fuzzy model working in parallel with the process is to subtract the effect of the control action from the process output. If the predicted and the measured process outputs are equal, the error e is zero and the controller works in the feedforward configuration. If a disturbance d acts on the process output, the feedback signal e is equal to the influence of the disturbance and is not affected by the effects of the control action. This signal is simply subtracted from the reference and the controller works in the open-loop feedback structure. With a perfect process model, the IMC scheme is hence able to cancel the effect of unmeasured output-additive disturbances and does not suffer from the disadvantages of feedforward controllers. However, the same is not true for input-additive disturbances, e.g., load disturbances. Two basic properties of the ideal IMC are inherent stability and perfect control. Inherent stability means that if the controller and the process are input-output stable and a perfect model of the process is available, the closed loop system is input-output stable. If the system is not input-output stable, but it can be stabilized by feedback, IMC still can be applied. Perfect control means that if the controller is an exact inverse of the model, and the closed-loop system is stable, then the control is error-free, i.e.

.

-

-

. . .-.

.

. . .

-.

70

FUZZY ALGORITHMS FOR CONTROL

y ( k ) = r(k), Vk. Control without steady-state offsets is attained for asymptotically constant references. However, in practice, the model is never an exact representation of the process. The feedback signal then contains both the effect of unmeasured disturbances and the effects of modeling errors, and it becomes a true feedback. For large modeling errors it deteriorates the performance of the control system and may introduce stability problems. An important difference between linear and nonlinear systems is that for linear systems, disturbances can be assumed to act additively at the output, while for nonlinear systems, unmeasured disturbances d p acting on the process generally lead to differences between the model and the process. The feedback filter is introduced in order to filter out the measurement noise and to stabilize the loop by reducing the loop gain. With nonlinear systems and models, the filter must be designed empirically.

3.4.3 fizzy model predictive control Model predictive control (MPC) is a general methodology of solving control problems in the time domain. It is based on three main concepts: 1. Explicit use of a model to predict the process output at future discrete time instants, over a prediction horizon.

2. Computation of a sequence of future control actions over a control horizon by minimizing a given objective function, such that the predicted process output is as close as possible to a desired reference signal.

3. Receding horizon strategy, so that only the first control action in the sequence is applied, the horizons are moved towards the future and optimization is repeated. The future process outputs are predicted over the prediction horizon H, using a model of the process. The predicted output values, denoted $(k i) for i = 1,.. . ,H,, depend on the state of the process at the current time k (for input-output models, for instance, represented by a collection of past inputs and outputs) and on the future control signals u ( k i) for i = 0, . . . , Hc - 1, where Hc is the control horizon. If Hc is chosen such that Hc < H,, the control signal is manipulated only within the control horizon and remains constant afterwards, i.e., u ( k i) = u ( k + Hc - 1) for i = Hc, . . . , H, - 1, see Figure 3.7.

+

+

+

Objective function. The sequence of future control signals u ( k + i) for i = 0,. . . , Hc - 1 is computed by optimizing a given objective (cost) function, in order to bring and keep the process output as close as possible to the given reference trajectory r , which can be the set-point itself or, more often, some filtered version of it. Most often used objective functions are modifications of the following quadratic function (Clarke et al., 1987):

The first term accounts for minimizing the variance of the process output from the reference, while the second term represents a penalty on the control effort (related

FUZZY CONTROL VS. CONVENTIONAL CONTROL

71

t

I

Figure 3.7

:

control input u

T h e basic principle of model predictive control.

for instance to energy). The latter term can also be expressed by using u itself or other filtered forms of u , depending on the problem (Soeterboek, 1990). The vectors cw and ,fl define the weighting of the output error and the control effort with respect to each other and with respect to the prediction step. Constraints, e.g., level and rate constraints of the control input or other process variables can be specified as a part of the optimization problem. Generally, any other suitable cost function can be used, but for a quadratic cost function, a linear, time-invariant model, and in the absence of constraints, an explicit analytic solution of the above optimization problem can be obtained. Otherwise, numerical (usually iterative) optimization methods must be used. Receding horizon principle. Only the control signal u ( k ) is applied to the process. At the next sampling instant, the process output y (lc 1) is available and the optimization and prediction can be repeated with the updated values. This is called the receding horizon principle. The control action u(lc 1 ) computed at time step k 1 will be generally different from the one calculated at time step k , since more up-to-date information about the process is available. Predictive control can be regarded as a generalization of the inverse model control approach. Without constraints and without penalizing the control action, the one-stepahead predictive control strategy is equivalent to the inverse model control strategy, where the inverse is computed numerically, by means of function minimization. Extending the prediction and control horizons, adding the control signal to the objective function and including constraints can be regarded as generalization of model inverse control. Because of the optimization approach and the explicit use of the process model, MPC can realize multivariable optimal control, deal with nonlinear processes, and can

+

+

+

72

FUZZY ALGORITHMS FOR CONTROL

efficiently handle constraints. Since a model of the process is a part of the control scheme, this model can be adapted on-line in order to minimize the difference between the expected and real process outputs. Fuzzy sets can be applied in several ways in the context of MPC, e.g., at the modeling level (Sousa et a]., 1997; Nakamori, 1994; Pottmann and Seborg, 1997;Fischer et al., 1997; de Oliveira and Lemos, 1995; Roubos et al., 1998), in optimization (Lu et al., 1997), and in the specification of the control objectives (Kaymak et al., 1997).

Fuzzy models in MPC. A fuzzy model acting as a numerical predictor of the process' output can be directly integrated in the MPC scheme shown in Fig. 3.8. The IMC scheme is usually employed to compensate for the disturbances and modeling errors. u

w

----*

Feedback filter Figure 3.8

Plant

Y

w

Fuzzy model

I

Fuzzy model in the MPC scheme with an internal model and a feedback t o

compensate for disturbances and modeling errors.

Since fuzzy models are in general nonlinear models, the associated optimization problem in MPC is inherently nonlinear and generally non-convex. The most straightforward way of minimizing (3.6) is to use some numerical optimization technique, such as the Nelder-Mead method (Walsh, 1975) or sequential quadratic programming (Gill et al., 1981). These algorithms, however, require a significant computing power which may be a serious obstacle in a real-time implementation. Moreover, convergence to a global minimum is not guaranteed and the algorithm can be trapped in a local minimum, which may result in undesired control actions and poor performance. Also the number of iterations needed to reach a solution can drastically differ from sample to sample. Premature termination of the iteration process is often necessary to fulfill the real-time constraints, which results in inferior solutions and hence a decreased control performance. Alternative optimization schemes are sought which can solve the optimization problem satisfactorily, i.e., find the global minimum and fulfill the real-time requirements. One possibility is to look for nonlinear models of a specific structure, e.g., nonlinear affine models. For instance, an input-output TS model which does not contain the inputs in the antecedent part is an affine nonlinear form. For affine models, inputoutput linearization techniques can be applied to obtain a globally linearized process

FUZZY CONTROL VS. CONVENTIONAL CONTROL

73

model (Nijmeijer and van der Schaft, 1990). The main idea of this technique is that, if the process is minimum-phase, it is possible to linearize it by means of a nonlinear static feedback law. Additionally, if the nonlinear process model is an affine inputoutput model, and assuming no modeling errors, the linearization scheme is exact, i.e., the resulting linear system is valid over the complete operating range. This is a main difference from the classical linearization through Taylor expansions, where resulting linear model is valid only in the neighborhood of the nominal operating point. In the absence of constraints, a linear predictive control law can be applied to the linearized process model.

3.4.4

Fuzzy state feedback

Intuitively, a plant with significant nonlinear dynamics should be controlled using a nonlinear controller. However, control design for nonlinear systems is a hard problem, and at present date there are no general design methods available. Some promising progress has been made using techniques like feedback linearization and backstepping designs. Both methods can be applied to certain classes of systems. We choose to illustrate a special case of these design methods which is the nonlinear equivalent to pole placement. Example 3.4.1 Consider a nonlinear system on the form

where f ( x ) and g ( x ) are nonlinear function of the process state vector. If we know f (a)and g ( x ) # 0 perfectly, the feedback

+

+

1 u=(- f ( x ) LTX T ) g(x) cancels the nonlinearity and gives the closed loop system the desired linear dynamics (Isidori, 1989)

If the functions f ( x )and g ( x ) are not known, one approach is to obtain approximate models f ( x ) and i ( x ) from an identification experiment. The approximations f ( z ) and i ( x ) ,can for instance be in terms of fuzzy systems whose parameters are optimized to fit the identification data. Using a certainty equivalence approach, we may then try the control

74

FUZZY ALGORITHMS FOR CONTROL

In this case the cancellation of the nonlinearity is only approximate. Present approaches to model based fuzzy control in the above spirit simply assume that the approximation of f ( x ) and g ( x ) is "sufficiently good". A more satisfactory approach would be to develop design algorithms that are robust with respect to bounded approximation errors. 3.4.5 Fuzzy sliding mode control

A simple and highly robust control structure for uncertain nonlinear systems is the so called sliding mode controller (Utkin, 1977) (Slotine and Li, 1991; Hung et al., 1993). The main idea behind sliding mode control is to transform the problem of stabilizing an nth order system (which is hard) into the problem of stabilizing a 1st order system (which is easier). Sliding mode control can be demonstrated on the problem of globally stabilizing the system (3.7). Let x ( t ) be the n-dimensional system state vector and consider the scalar function s ( x ,t ) = cTx(t). This function defines a surface S of dimension n-1 in the system state space

Substituting the n- 1 first state equations of (3.7) into (3.1 I), we have

Consequently, S also defines an ordinary differential equation in X I . If we define the surface S so that the equation (3.12) is exponentially stable, the system state is guaranteed to converge to the origin if it remains on the surface for all future times. Thus, the n-dimensional problem of driving the state vector to origin has been transformed into the 1-dimensionalproblem to forcing the system state onto the surface S and keeping the state on the surface. Forcing the system state to the sliding surface is equivalent of forcing the scalar s ( x ;t ) to zero in finite time. This can be accomplished by finding a control u such that the Lyapunov-like "sliding condition"

with 17

> 0 is fulfilled. Such a control forces the system state to the sliding surface

S, and once on the surface the system state remains there, obeying the dynamics of (3.12). This is illustrated in Figure 3.9. Next, we give an example of how sliding mode control can be used to guarantee stability for a nonlinear system in the presence of approximation errors.

Example 3.4.2 Consider the problem of stabilizing the following nonlinear system

Assume that a mathematical model of f ( x ) is unknown, but that we have some heuristic knowledge that we can formulate into a fuzzy model f ( x ) . We cannot expect

----

-

-

4

*

F

I&),

r

..

t $ +

-

--

FUZZY CONTROL VS. CONVENTIONAL CONTROL

Figure 3.9

75

State trajectories for a system under sliding mode control

this approximate model to be perfect, but assume that we can bound the approximation error by a function F ( x )

If ( x ) - f^(x)l5 F ( x ) A sliding mode controller can be designed for this system in the following steps. Define the sliding surface to be

with X > 0. If the system state is on the sliding surface, it remains on the surface provided that s = 0, i.e.

s = ( X i l + i 2=)(Ax2+ f ( x ) + u ) = 0 Since we do not know f ( x ) ,we use the control

The first part, 6, of the control is a certainty equivalence control (compare with (3.10)). To ensure that the state vector is forced onto the surface, we have added a second term, u * , discontinuous over the sliding surface. If we set

we can verify that this control satisfies the sliding condition (3.13):

1d --s2 2 dt

= 2s = (f ( x ) - f^(x))s - ksgn(s)s 5 -qlsI

From (3.14), we can see that the system state is directed towards the sliding surface "from both sides", by switching the control whenever the state trajectory crosses the sliding surface. In presence of noise or unmodelled dynamics, this results in a "chattering" control, inhibiting the direct implementation of sliding mode control. In

76

FUZZY ALGORITHMS FOR CONTROL

practice, chattering can be eliminated by approximatingthe switching sgn(s) by some smooth function, resulting in a "boundary layer" around the sliding surface. It has been suggested to design fuzzy controllers by approximating the function (3.14) by a fuzzy system, e.g., (Palm, 1992) and (Palm and Driankov, 1997). It is not always clear, however, why one would like to approximate the compact formula (3.14) by a fuzzy system, increasing the number of system parameters and the computational requirements.

3.4.6 Compensation of static nonlinearities and static scheduling In some cases, the main nonlinearitiesof a process are static nonlinearities on the input and output of the system, whereas the system dynamics is linear. In these situations, we can use the approximation capabilities of fuzzy systems to compensate for these nonlinearities. Example 3.4.3 Consider a SISO plant with linear dynamics and an input nonlinearity

One approach to control this system is to momentarily disregard the input nonlinearity and design a linear controller v = L(x) for the linear plant. We can then approximate the inverse (which may, or may not exist) of the input nonlinearity using a fuzzy system

and apply the control = f (L(x))

In some cases, it can be motivated to introduce static nonlinearities at the input of a linear system, as described by the next example. Example 3.4.4 In climate control systems, the temperature dynamics of a room can be modeled as a linear system x=Ax+bq where q is the heat delivered by the air conditioner (AC). It is straightforward to design a linear controller for this system. Typically, this controller measures the room temperature and computes the heat that the AC must provide. However, we cannot control the heat of the delivered air stream directly. The heat contained in the air delivered by the AC is proportional to the product of the air mass flow, w, and the temperature, T, of the air stream, i.e.

FUZZY CONTROL VS. CONVENTIONAL CONTROL

77

We may then use fuzzy logic rules to design a schedule

that for a desired heat q and auxiliary operating conditions p suggests the most comfortable combination of air mass flow and air temperature. This control structure is illustrated in Figure 3.10.

Figure 3.10

Static scheduling using fuzzy systems

3.4.7 Lyapunov-based analysis and synthesis Using fuzzy systems it is possible to define very general nonlinearities, that, for instance, contain discontinuities and jumps. In order to be able to derive any useful analytical results, it is necessary to constrain the classes of nonlinearities that one considers. The class of systems that has achieved most attention is the linear and affine Takagi-Sugeno system. For these systems both stability and synthesis results are available. The system type under consideration is described by

R(') : IF sl IS A:) AND . . . AND

2,

IS A:) THEN 5 = Alz

+ a1

(3.15)

In the linear case, the a1 terms are zero. The system type (3.15) can be viewed as a piecewise affine system, a natural extension of linear systems. A piecewise affine system consists of a decomposition of the state space into a set of regions, or operating regimes. Associated with each operating regime is an affine system. The regions can be overlapping or non-overlapping. The overlapping regions correspond to regions where fuzzy interpolation is performed between several affine dynamics. Asymptotic stability of nonlinear systems can in many cases be verified by an appropriate Lyapunov function. The standard approach for piecewise linear systems has so far been to try to consider them as linear differential inclusions, and try to find a single, globally quadratic, Lyapunov function for the system. The search for a quadratic Lyapunov function can be stated as a convex optimization problem in terms of Linear Matrix Inequalities (LMIs) (Boyd et al., 1994). Efficient numerical routines for solving LMIs are publically available (Gahinet et a]., 1995), and LMI based methods have been successfully applied to the stability analysis of fuzzy systems (Zhao, 1995; Tanaka et al., 1996).

78

FUZZY ALGORITHMS FOR CONTROL

When applied to piecewise affinesystems, these stability conditions are often found to be conservative in the sense that they fail to prove stability for a large class of stable systems, see (Johansson and Rantzer, 1996) for examples and (Corless, 1994) for significant results on quadratic stability. The conservatism of the global approaches to quadratic stability is twofold; no information of the region partition is taken into account and the Lyapunov function is restricted to be globally quadratic. A less conservative result has recently been suggested in (Johansson and Rantzer, 1996). The idea is to use a piecewise quadratic Lyapunov function that is tailored to fit the cell partition of the system. The search for a piecewic- quadratic Lyapunov function can also be formulated as an LMI-problem. The ideas can be extended to more general uncertain and nonlinear systems (Johansson and Rantzer, 1997), and similar techniques can also be used for performance analysis, such as Cz gain analysis of nonlinear systems (Rantzer and Johansson, 1997). Formulation of the stability results for affine fuzzy Sugeno systems can be found in (Johansson et al., 1997) and (Johansson et al., 1998). A related result which also is based on piecewise Lyapunov functions is described in (Cao et al., 1997). For linear Takagi-Sugeno systems the use of LMI methods also allows the design of various stabilizing state-feedback controllers (Zhao, 1995). Both nonfuzzy and fuzzy state-feedback controllers can be designed. The design is, however, based on the globally quadratic Lyapunov approach. 3.5

e

I

e

SUMMARY

Fuzzy control is becoming an increasingly used method for implementing nonlinear control systems. Industrial control systems are dominated by linear PID controllers. In many cases it is an advantage to add various nonlinearities to the basic PID scheme. This is one reason why fuzzy PID control has been successful in industry. Another reason is the user-friendly interface and computer-based design environments available. Also in the academic world a large amount of work is being done on fuzzy control. Here, the focus is on analysis and synthesis methods. For certain classes of fuzzy systems, e.g., linear Takagi-Sugenosystems, results are already available and one may expect that the number of results in this area will increase substantially in the near future.

t

References

BabuSka, R. (1998). Fuuy Modeling for Control. Kluwer Academic Publishers, Boston. Bonissone, P. P. (1994). Fuzzy logic controllers: an industrial reality. In Zurada, J. M., 11, R. J. M., and Robinson, C. J., editors, Computational Intelligence: imitating life, pages 3 16-327. IEEE Press, Piscataway, NJ. Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory. Siam Studies in Applied Mathematics. Braae, M. and Rutherford, D. (1979). Theoretical and linguistic aspects of the fuzzy logic controller. Autornatica, 15553-577.

t

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Cao, S., Rees, N., and Feng, G. (1997). Analysis and design for a class of complex control systems part 11: Fuzzy controller design. Automatica, 33(6): 1017-1028. Clarke, D., Mohtadi, C., and Tuffs, P. (1987). Generalised predictive control. part 1: The basic algorithm. part 2: Extensions and interpretations. Automatica, 23(2): 137-160. Corless, M. (1994). Robust stability analysis and controller design with quadratic Lyapunov functions. In Zinober, A. S., editor, Variable Structure and Lyapunov Control, Lecture notes in Control and Information Sciences, chapter 9, pages 181203. Springer Verlag. Cox, E. (1993). Adaptive fuzzy systems. IEEE Spectrum. de Oliveira, J. V. and Lemos, J. (1995). Long-range predictive adaptive fuzzy relational control. Fuuy Sets and Systems, 70:337-357. Driankov, D., Hellendoorn, H., and Reinfrank, M. (1993). An Introduction to Fuuy Control. Springer, Berlin. Fischer, M., Schmidt, M., and Kavsel-Biasizzo, K. (1997). Nonlinear predictive control based on the extraction of step response models from Takagi-Sugeno fuzzy systems. In Proc. of the American Control Conference, pages 1210-1 2 16. Froese, T. (1993). Applying of fuzzy control and neuronal networks to modern process control systems. In Proceedings of the EUFIT '93, volume 11, pages 559-568, Aachen. Gahinet, P., Nemirovski, A., Laub, A. J., and Chilali, M. (1995). LMI Control Toolbox for use with Matlab. The Mathworks Inc. Gill, P., Murray, W., and Wright, M. (198 1). Practical Optimization. Academic Press, New York and London. Harris, C., Moore, C., and Brown, M. (1993). Intelligent Control, Aspects of Fuuy Logic and Neural Nets. World Scientific, Singapore. Hellendoorn, H. (1993). Design and development of fuzzy systems at siemens r&d. In Proc. of the IEEE International Conference on Fuuy Systems, pages 1365-1 370. Hirota, K., editor (1993). Industrial Applications of Fuuy Technology.Springer, Tokyo. Holmblad, L. and Ostergaard, J. (1982). Control of a cement kiln by fuzzy logic. In Gupta, M. and Sanchez, E., editors, Fuzzy Information and Decision Processes. North-Holland, Amsterdam. Hung, J. Y., Gao, W., and Hung, J. C. (1993). Variable structure control: A survey. IEEE Transactions on Industrial Electronics, pages 2-2 1. IEEE (1993a). Reader's forum. IEEE Control Systems Magazine. IEEE (1993b). Reader's forum. IEEE Control Systems Magazine. Isidori, A. (1989). Nonlinear Control Systems: an Introduction. Springer-Verlag. Johansson, M., Malmborg, J., Rantzer, A., Bernhardsson, B., and Arztn, K.-E. (1997). Modeling and control of fuzzy, heterogeneous and hybrid systems. In Proc. of SICICA 97, Annecy, France. Johansson, M. and Rantzer, A. (1996). Computation of piecewise quadratic Lyapunov functions for hybrid systems. Technical report, Department of Automatic Control. Also available at http: / /www. control. lth. se/-rantzer. Johansson, M.and Rantzer, A. (1997). Computation of piecewise quadratic Lyapunov functions for hybrid systems. In European Control Conference, ECC97.

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Johansson, M., Rantzer, A., and i b z ~ nK.-E. , (1998).Piecewise quadratic stability for affine sugeno systems. In Proc. of FUZZ-IEEE'98, Anchorage. Kaymak, U., Sousa, J., and Verbruggen, H. (1997).A comparative study of fuzzy and conventional criteria in model-based predictive control. In Proc. of IEEE International Conference on Fuzzy Systems, volume 2, pages 907-914. Lu, Y.-Z., He, M., and Xu, C.-W. (1997). Fuzzy modeling and expert optimization control for industrial processes. IEEE Trans. Control Systems Tech., 5:2-12. Mamdani, E. (1974). Application of fuzzy algorithm for control of simple dynamic plant. Proc. IEE, 121:1585-1588. Nakamori, Y. (1994). Fuzzy modeling for adaptive process control. In Kanel, A. and Langholz, G., editors, Fuzzy Control Systems. CRC Press. Nijmeijer, H. and van der Schaft, A. (1990). Nonlinear Dynamical Control Systems. Springer-Verlag,New York, USA. Palm, R. (1992). Sliding mode fuzzy control. In Proc. of the IEEE International Conference on Fuzzy Systems, pages 519-526. Palm, R. and Driankov, D. (1997). Stability of fuzzy gain-schedulers: Sliding-mode based analysis. In Proc. of the IEEE International Conference on Fuzzy Systems, pages 177-183. Pedrycz, W. (1993).Fuzzy Control and Fuzzy Systems (second, extended,edition).John Willey and Sons, New York. Pottmann, M. and Seborg, D. (1997).A nonlinear predictive control strategy based on radial basis function models. Comp. Chem. Engng., 21 :965-980. Rantzer, A. and Johansson, M. (1997).Piecewise linear quadratic control. In American Control Conference, ACC'97. Raymond, C., Boverie, S., and Titli, A. (1995). Fuzzy multivariable control design from the fuzzy system model. In Proceedings Sixth IFSA World Congress, Sao Paulo, Brazil. Roubos, J., Babuska, R., Bruijn, P., and Verbruggen, H. (1998).Predictive control by local linearization of a Takagi-Sugeno fuzzy model. In Proc. of the IEEE International Conference on Fuzzy Systems. Sjoberg, J., Zhang, Q., Ljung,L., Benveniste, A., Deylon, B., Glorennec, P., Hjalmarsson, H., and Juditsky, A. (1995).Nonlinear black-box modeling in system identification: A unified overview. Autornatica, 3 1. Slotine, J.-J. and Li, W. (1991).AppliedNonlinear Control. Prentice Hall International. Soeterboek, A. (1990). Predictive Control - A Unijied Approach. PhD dissertation, Delft University of Technology, Delft, The Netherlands. Sousa, J., Babuska, R., and Verbruggen, H. (1997).Fuzzy predictive control applied to an air-conditioning system. To appear in Control Engineering Practice. Takagi, T. and Sugeno, M. (1985).Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, 15:116-132. Tanaka, K., Ikeda, T., and Wang, H. 0. (1996). Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability,H , control theory and linear matrix inequalities. IEEE Transactions on Fuzzy Systems, 4(1): 113.

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Terano, T., Asai, K., and Sugeno, M. (1994). Applied Fuuy Systems. Academic Press, Inc., Boston. Utkin, V. I. (1977). Variable structure systems with sliding modes: a survey. IEEE Transactions on Automatic Control, 22:212-222. Walsh, G. (1975). Methods of optimization. John Wiley & Sons, New York, USA. Zadeh, L. (1965). Fuzzy sets. Information and Control, 8:338-353. Zhao, J . (1995). System Modeling, Identification and Control using Fuzzy Logic. PhD thesis, UCL, UniversitC Catholique de Louvain.

4

DATA-DRIVEN CONSTRUCTlON OF TRANSPARENT FUZZY MODELS R. BabuSka and M. Setnes

Delft University of Technology Faculty of Information Technology and Systems Control Laboratory Mekelweg 4, PO Box 5031

2600 GA Delft, The Netherlands

4.1

INTRODUCTION

Since its introduction in 1965, fuzzy set theory has found applications in a wide variety of disciplines. Automatic control is a field in which fuzzy set techniques have received considerable attention, not only from the scientific community but also from industry (Mamdani, 1974; Yasunobu and Miyamoto, 1985; astergaard, 1990; Kandel and Langholz, 1994). While most of the early design methods for fuzzy control were based on heuristic considerations, recent research has focused on the development of model-based fuzzy control techniques (Palm et a]., 1997; Driankov and Palm, 1998; BabuSka, 1998). In the model-based approach, a fuzzy model is first developed to approximate the behavior of a complex process to be controlled. Based on this model, a controller can be designed. Fuzzy set techniques have been recognized as a powerful tool for the development of models for systems that are not amenable to conventional modeling approaches due to the lack of precise, formal knowledge about the system, due to strongly nonlinear behavior or time varying characteristics.

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FUZZY ALGORITHMS FOR CONTROL

The rule-based nature of fuzzy models allows the use of information expressed in the form of natural language statements. This makes the models transparent to qualitative interpretation and analysis. At the computational level, fuzzy models can be regarded as flexible mathematical structures, similar to neural networks, that can approximate a large class of complex nonlinear systems to a desired degree of accuracy (Kosko, 1994; Wang, 1994; Zeng and Singh, 1995).

4.1.1 Acquisition of fuzzy models Conventionally, fuzzy models have been built primarily by encoding expert knowledge into linguistic rules. The advantage is a transparent system that can be maintained and expanded by human experts. However, knowledge acquisition is not a trivial task. Experts are not always available, and when they are, their knowledge is not always consistent, systematic and complete, but often incomplete, episodic and time-varying. Hence, there is an increasing interest in obtaining fuzzy models directly from measured data. Recently, a great deal of research activity has been devoted to the development of methods to build or update fuzzy models from numerical data. Most approaches are based on neuro-fuzzy systems, which exploit the functional similarity between fuzzy reasoning systems and neural networks (Brown and Harris, 1994; Jang et a]., 1997). This enables an effective use of optimization techniques for building fuzzy systems, especially with regard to their approximation accuracy. However, the aspects related to the transparency and interpretation tend to receive considerably less attention. Consequently, most neuro-fuzzy models can be regarded as black-box models which provide little insight to help understand the underlying process [examples can be found in (Wang, 1994; Lin, 1994)l. The lack of interpretability is a major drawback, since many other techniques can be used for black-box modeling, such as standard nonlinear regression (Seber and Wild, 1989), spline techniques (de Boor, 1978), or neural networks (Haykin, 1994). 4.1.2 Overviewoftheapproach

The approach described in this chapter aims at the development of rule-based fuzzy models which can accurately predict the quantities of interest, and at the same time provide insight into the system that generated the data. Such a model can be used not only for the given situation, but can also be more easily adapted to changing design parameters and operating conditions. From the system identification point of view, a fuzzy model is regarded as a composition of local submodels. Fuzzy sets naturally provide smooth transitions between the submodels, and enable the integration of various types of knowledge within a common framework. Attention is paid to the aspects of accuracy, transparency and complexity reduction of the obtained fuzzy models. The latter aspect plays an important role in the design of model-based controllers for time-critical applications. The methodology described in this chapter has two main steps: data exploration by means of fuzzy clustering, and fuzzy set aggregation with the help of similarity analysis. First, fuzzy relationships are identified in the product space of the system's

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85

variables and described by means of fuzzy production rules. This initial rule base is usually to a certain degree redundant. Similar fuzzy sets are identified and aggregated to produce generalized concepts, giving a comprehensible rule base with improved semantic properties. Data

3

Structure Selection parameters

2.

Fuzzy Clustering

!I Clurtering parumeters

I

3.

Generation of Initial Fuzzy Model

I 4.

Similarity Analysis Fuzzy Set Aggregation

I

threshold

Model accepted Figure 4.1

Overview of the model construction procedure.

The individual steps of this modeling approach are shown in Fig. 4.1. The purpose of the different steps and the related methods are outlined below, assuming that proper data collection has been performed. Step 1: Structure selection. The relevant input and output variables are determined with respect to the aim of the modeling exercise. For dynamic systems, structure selection allows us to translate dynamic identification into a regression problem that can be solved in a static manner. Often, a reasonable choice can be made by the user, based on the prior knowledge about the process. Step 2: Clustering of the data. Fuzzy clustering is used to discover substructures in the product space of the available observations. Each cluster defines a fuzzy region in which the system can be approximated locally by a submodel. The location and the parameters of the submodels are derived from the clusters found in the data. By applying techniques such as cluster validity measures or cluster merging, an appropriate number of clusters can be found. Step 3: Generation of an initial fuzzy model. Fuzzy clustering partition the space of the available data into regions in which relations exists between the inputs and the output. A rule-based fuzzy model is derived from the resulting fuzzy partition matrix and from the cluster prototypes. The rules, the membership functions and other

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FUZZY ALGORITHMS FOR CONTROL

parameters that constitute the fuzzy model are extracted in an automated way. The exact procedure depends on the type of fuzzy model required. In Section 4.4, we focus on fuzzy models of the Takagi-Sugeno type. Step 4: Fuzzy set aggregation. The initial rule base obtained from data is often qualitatively poor as it is based on numerical optimization. The qualitative aspects are enhanced in a process where similarity analysis is used to identify fuzzy sets representing compatible or redundant concepts. By aggregating compatible fuzzy sets and removing the redundant ones, a model with improved semantical properties is obtained. A tradeoff can be introduced between numerical accuracy and transparency, making it possible to generate models with varying degrees of complexity for different purposes. This procedure is discussed in Section 4.5. Step 5: Model validation. By means of validation, the final model is either accepted as appropriate for the given purpose, or it is rejected. In addition to numerical validation by means of simulation, the interpretation of fuzzy models plays an important role in the validation step. The coverage of the input space by the rules can be analyzed, and, for an incomplete rule base, additional rules can be provided based on prior knowledge, local linearization, or models based on physical laws. Such interpretation is made easier by the simplification of the rule base performed in the previous step. The focus of this chapter is on steps 2, 3 and 4 of the algorithm. Steps 1 and 5 are based on rather standard approaches known from linear and nonlinear system identification(Ljung, 1987; Sjoberget al., 1995). An application to an air-conditioning system is presented in order to illustrate the aspect of fuzzy modeling as well as the computational issues of a predictive controller based on the obtained fuzzy model. 4.2

FUZZY MODEL STRUCTURE

Different approaches to the modeling of systems with the help of fuzzy sets include rule-basedfuzzy systems (Zadeh, 1973),fuzzy linear regression methods (Tanaka et al., 1982) and fuzzy models based on cell structures (Smith et al., 1994). In the context of control, most popular are probably rule-based fuzzy models, in which the relationships between variables are represented by means of fuzzy if-then rules of the following general form: If antecedent proposition then consequent proposition.

The antecedent proposition is always a fuzzy proposition of the type "x is A" where x is a linguistic variable and A is a linguistic constant (term). The proposition's truth value (a real number between zero and one) depends on the degree of match (similarity) between x and A. Depending on the form of the consequent, three main types of rule-based fuzzy models are distinguished: Linguisticfuzzy model: both the antecedent and the consequent are fuzzy propositions. Fuuy relational model: generalizes the linguistic model, the relation between the antecedent and consequent terms is a fuzzy one. Takagi-Sugeno (TS) fuzzy model: the antecedent is a fuzzy proposition, the consequent is a crisp function.

I

I

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87

In this chapter, the TS fuzzy model is considered. However, the methods presented apply to the linguistic-type models as well (BabuSka and Verbmggen, 1995; Setnes et al., 199%). The TS model consist of a set of fuzzy rules, which each describe a local input-output relation, typically in an affine linear form:

Ri : wi(IfX I is Ail and . . . and xn is Ain then yi = aix + bi) .

(4.1)

Here Ri is the ith rule, x = [ X I ,. . ,xnIT E X C Rn is the input (antecedent) variable, Ail, . . . ,Ain are fuzzy sets defined in the antecedent space, yi is the rule output variable, and wi is the rule weight. Typically, wi = l,Vi, but these weights can be adjusted during the model reduction. The entire rule base consists of K rules: R = {Rili = 1 , 2 , . . . ,K). The aggregated output of the model, y, is calculated by taking the weighted average of the rule consequents:

where pi is the degree of activation of the ith rule:

and PA;, ( x j ) : R antecedent of Ri.

4.3

+ [O, 11 is the membership function of the fuzzy set Aij

in the

FUZZY CLUSTERING

Fuzzy clustering is applied to discover fuzzy regions in the data space in which the system can be approximated locally by a linear submodel (BabuSka, 1998). It is assumed that the data has been generated by a system y = f ( x ) 6 , where f is an unknown deterministic function f : Rn -+ R and E is a zero-mean random noise. The aim is to use the data ( x ,y ) to construct a deterministic function y = F ( x ) that can serve as a reasonable approximation of f ( x ) . The function F is represented as a collection of fuzzy if-then rules (4.1). Note that for dynamic systems, the regression vector x contains besides the inputs of the systems also their past values and the past values of the output(s). To identify this model from data means to find the antecedent membership functions Ail,. . . , Ai,, Vi, the consequent parameters ai, bi, Vi, and the number of rules K. Besides obtaining a good3t (\IF - f 11 is sufficiently small on the domain of interest X), it is desired that the model is compact ( K is reasonably small) and transparent (Aijare sufficiently distinct). With such a model, the consequent parameters can often be interpreted as reliable local models of the identified process. To identify the model (4.1), the regression matrix X and the output vector y are first constructed from the available data:

+

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FUZZY ALGORITHMS FOR CONTROL

Here N >> n is the number of samples used for identification. Then, the pattern matrix Z to be clustered is composed by appending y to X:

zT = [X, y ] .

(4.5) t

Given Z and an estimated number of clusters K, a fuzzy clustering algorithm (Gustafson and Kessel, 1979) is applied to compute the fuzzy partition matrix U. The vectors zk, k = 1,2,. . . , N , contained in the columns of the data matrix Z, are partitioned into K clusters, represented by their prototypical vectors v, = E R ( ~ + ' ) ,i = 1 , . . . ,K. Denote V E R(n+l)xK the [ v i , ~. . . ,~ z , nvi,(,+l)] , matrix having v, in its ith column. This matrix is called the prototype matrix. The fuzzy partitioning of the data among the K clusters is represented as the fuuy partition matrix U E RK x N , whose elements (denoted pik E [0, 11) are the membership degree of the data vector zk in the ith cluster. A class of clustering algorithms search for the partition matrix and the cluster prototypes, such that the following objective function is minimized: K

N

subject to the following constraints:

In eq. (4.6), m > 1 is a parameter that controls the fuzziness of the clusters. With higher values of m the clusters overlap more, and as m approaches one from above, the partition becomes crisp (pik E (0,l)). The usual setting with m = 2 is suitable for most applications. The function d(zk, vi) is the distance of the data vector zk from the cluster prototype vi. The constraint (4.7) avoids the trivial solution U = 0 and the constraint (4.8) guarantees that clusters are neither empty nor contain all the points to degree 1. The optimization problem defined by the functional (4.6) subject to the constraints (4.7) and (4.8) can be solved by different nonlinear optimization techniques. The most popular one is alternating optimization, which leads to an iterative scheme, known as the fuzzy c-means algorithm (Bezdek, 1981). The shape of the clusters is determined by the particular distance measure d(zk, vi) involved. Gustafson and Kessel proposed a clustering algorithm based on an adaptive inner-product distance measure (Gustafson and Kessel, 1979):

Here, Mi is a positive definite matrix which is adapted according to the actual shape of the ith cluster. This shape is described by the cluster covariance matrix Fi estimated from data:

DATA-DRIVEN CONSTRUCTION O F TRANSPARENT FUZZY MODELS

89

It can be shown that the distance inducing matrix Mi is equal to the normalized inverse of the cluster covariance matrix:

The normalization by the determinant of Fi is involved in order to constrain Mi. Without this constraint, the objective function (4.6), which is linear with respect to Mi, could be made as small as desired by making Mi less positive definite. In the iterative optimization scheme of the GK algorithm given in Algorithm 4.3.1, the superscript (1) denotes the value of a given variable at the lth iteration.

Algorithm 4.3.1 (Gustafson-Kessel (GK) algorithm.) Given 2,choose K , 1 < K < A', m > 1 and t > 0. Initialize u(')randomly, such that conditions (4.7) and (4.8) hold. Repeat for 1 = 1 , 2 , . . .

Step 1: Compute cluster prototypes (means):

Step 2: Compute the cluster covariance matrices:

Step 3: Compute the distances:

Step 4: Update the partition matrix:

i f d ( z k , v ; ) > O for l < z < K ,

1< k
otherwise K

= o for d ( z k ,v i )

> 0 , otherwise

0 E [0,1] with pik

fij:) i=l

until (

IU~

-

~ ( - ' ) ( l < E.

=1

90

FUZZY ALGORITHMS FOR CONTROL

The number of clusters K determines the number of rules in the fuzzy model obtained. Two main approaches to find an appropriate number of clusters can be distinguished: Cluster the data for different values of K and then use validity measures to assess the goodness of the obtained partitions. Different validity measures have been proposed in connection with the adaptive-distance clustering techniques (Gath and Geva, 1989). Start with a sufficiently large number of clusters and successively reduce this number by merging clusters that are compatible with respect to some predefined criteria. This approach is called compatible cluster merging (Krishnapuram and Freg, 1992; Kaymak and BabuSka, 1995).

4.4

EXTRACTION O F A N INITIAL RULE BASE

Given the triplet, (U,V, I?), obtained by clustering, the antecedent membership functions Aiand the consequent parameters ai and bi are computed as described below.

4.4.1 Antecedent membership functions The fuzzy sets Aij in the antecedent of the rules are obtained from the partition matrix U. The ikth element pik E [O,1] of this matrix is the membership degree of the data object z k in cluster i. One-dimensional fuzzy sets Aij are obtained from the multidimensional fuzzy sets defined point-wise in the ith row of the partition matrix by projections onto the input variables x j :

where 'proj' is the point-wise projection operator (Kruse et al., 1994). The obtained point-wise fuzzy sets Aij are approximatedby suitable parametric functions (Fig. 4.2).

P 1

parametric function

Figure 4.2

Approximation of the projected data by a parametric membership function

DATA-DRIVEN CONSTRUCTION OF TRANSPARENT FUZZY MODELS

91

Piece-wise exponential membership functions proved to be suitable for the accurate representation of the actual cluster shape: exp P ( X ; C L ~ ~ ~ , W ~=, W ~exp )

-

2

- ( ~ ) 2 )

,

ifx

)

,


i f s > c,, otherwise.

(4.13)

Here cl and c, are the left and right shoulder, respectively, and wl, w, are the left and right width, respectively. For cl = c, and wl = w,, the Gaussian membership function is obtained. This function is fitted to the envelope of the projected data by numerically optimizing its parameters.

4.4.2 Consequent parameters The consequent parameters for each rule can be obtained either by solving a weighted least-square problem for each rule separately or by solving a single "global" leastsquares problem following from (4.2).

A set of weighted least-square estimates. Let OiT = [a:, bi] ,let X, denote the matrix [X,11 and let Wi denote a diagonal matrix in R N x N having the degree of activation, ,Oi(xk),as its kth diagonal element. If the columns of X, are linearly independent and Pi(xk)> 0 for 1 5 k 5 N, then the weighted least-squares solution of y = X,8 6 becomes

+

This estimate provides an optimal local fit for the individual rules, while the global fit is not optimal.

Global least-square estimate. In order to obtain an optimal global predictor, the aggregation of the rules has to be taken into account. With the defuzzification formula (4.2), an optimal estimate of the consequent parameters can be obtained by solving a linear least-squares problem. Denote ri a diagonal matrix in R N x N having the normalized degree of activation yik = wi&(xk)/ wj&(xk)as its kth diagonal element. Denote X' the matrix in IEtN K N composed from matrices ri and X,

Denote 0' the vector in R ~ ( ~given + ~by)

The resulting least-squares problem y = X's'

+ E has the solution:

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FUZZY ALGORITHMS FOR CONTROL

From eq. (4.16) the parameters ai and bi are obtained by: ai

= [8;+17 eZ.+2,- .. ,'L+nIT,

bi = [8q+n+~],with q = (i -

+ 1) .

(4.18)

The global least-squares method gives a minimal prediction error, and thus it is suitable for deriving prediction models. At the same time, however, it biases the estimate of the local model parameters. For the purpose of local interpretation and analysis, however, the weighted least-squares approach is preferable. 4

4.5

SlMPLlFlCATlON A N D REDUCTION OF T H E RULE BASE

The transparency of fuzzy rule-based models obtained from data is often hampered by redundancy present in the form of many overlapping (compatible) membership function (Fig. 4.7a). In (Setnes, 1995) we proposed to use a similarity measure to asses the compatibility (pair-wise similarity) of fuzzy sets in the rule base, in order to identify sets that can be merged. Fuzzy sets estimated from data can also be similar to the universal set, thus adding no information to the model. Such sets can be removed from the antecedent of the rules. These operations reduce the number of fuzzy sets in the model. Reduction of the rule base follows when the antecedents of some rules become equal. Such rules are combined into one rule. In the following, we describe three approaches to model simplification and reduction. The compatibility between the fuzzy sets Al, and A, in the rules R1 and R,, respectively, is assessed by the the fuzzy analog of the Jaccard index:

4

where 1, m = 1,2, . . . , K, and c,l, E [0, 11. Then and U operators are the intersection and the union, respectively, and I . 1 denotes the cardinality of a fuzzy set (Dubois and Prade, 1980). The measure c,l, is computed for discretized domains.

4.5.1 Iterative compatibility analysis This approach is based on iterative merging of compatible fuzzy sets (Setnes, 1995; BabuSka et al., 1996). It requires two thresholds from the user, A, y E ( 0 , l ) for merging compatible fuzzy sets, and removing fuzzy sets compatible with the universal set, respectively. In each iteration, the compatibility between all fuzzy sets in each antecedent dimension j is analyzed. The pair of fuzzy sets having the highest compatibility c > X are merged. A new fuzzy set is created by merging, and the rule base is updated by substituting this fuzzy set for the ones merged. The algorithm again evaluates the updated rule base, until there are no more fuzzy sets for which c > A. Fuzzy sets compatible with the universal set (c > y) are removed from the rules in which they occur. This algorithm is given in Algorithm 4.5.1. 4.5.2 Similarity relations

Also this approach requires the two thresholds X and y. For each antecedent dimension, j = 1,. . . n, a similarity relation between the fuzzy sets is obtained in two steps: First,

4

4

DATA-DRIVEN CONSTRUCTION OF TRANSPARENT FUZZY MODELS

93

Algorithm 4.5.1 (Iterative compatibility analysis.)

Given a rule base R = {Ri li = 1,. . . , K), where Ri is given by (4. l ) , select the thresholds A, y E (0,l): Repeatfor j = 1 , 2 , .. . ,n: Step 1: Select most compatible fuzzy sets:

Step 2: Merge selected fuzzy sets:

If cjlm

> X : Acj = Merge(A~j),VAlj E ALj, set Alj = A,.

Until: cjl, < A. Step 3: Remove fuzzy sets similar to the universal set:

where p a j = 1,Vxj. If c,j > y, remove Aij from the antecedent of Ri.

I

a K x K binary fuzzy compatibility relation C j = [cjlm] is calculated (4.19). C j is reflexive and symmetric. Second, a similarity relation, S j , is calculated as the max-min transitive closure, C T j ,of C j (Klir and Yuan, 1995):

2. If Ci

# C j ,set C j = C; and go to 1.

3. Stop: C T j = Ci, set S j = C T j . Here o is the max-min composition. The lmth element of S j , [ s j l m gives ] , the similarity between Ajl and Ajm. For each antecedent dimension, the fuzzy sets having similarity sjl, > X are merged. Fuzzy sets compatible with the universal set are removed. This algorithm is given in Algorithm 4.5.2. The first approach merges only one pair of fuzzy sets per iteration and the rule base is updated between the iterations. The second approach merges all similar fuzzy sets per dimension simultaneously. Hence, the use of the transitive similarity relation may give different results than the iterative approach. Merging of fuzzy sets is accomplished by letting the support of the union of the sets in A L j be the support of the new fuzzy set ACj.This guarantees the preservation of the coverage of the antecedent space. The kernel of Acj is given by averaging the kernels of the sets in ALj (Fig. 4.3). If the antecedents of p 2 2 rules become equal, the p rules can be replaced by one common rule R,. The consequent parameters of the reduced rule base can be re-estimated from training data by using (4.14) or (4.17). Another solution is to calculate the parameters of R, from the parameters of the p removed rules. The latter

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FUZZY ALGORITHMS FOR CONTROL

Algorithm 4.5.2 (Similarity relations.) Given a rule base R = {Rili = 1 , . . . , K ) , where Ri is given by (4.1), select the thresholds A, y E (0,l): Repeatforj = 1,2, ..., n:

Step 1: Calculate similarity relation:

Step 2: Merge similar fuzzy sets:

VAlj E A L ~set , Alj = A c j . Step 3: Remove fuzzy sets similar to the universal set:

where puj = 1, V x j . If c;j

Figure 4.3

> y, remove Aij from the antecedent of Ri.

Merging of similar membership functions.

DATA-DRIVEN CONSTRUCTION O F TRANSPARENT FUZZY MODELS

95

method does not depend on the availability of data. This approach is now described: Let Q C { 1 , 2 , .. . , K ) be a subset of rule indices such that Alj = A m j , V j E { 1 , 2 ,. . . , n ) ,V l , m E Q. RQ then denotes the set of rules with equal antecedents. The rule R, replaces the rules in RQ, and its antecedent part equals that of the rules RQ, i.e., Acj = Ah, j = 1 , 2 , . . . , n, 1 E Q. The common rule R, is created such that it accounts for all the rules RQ by weighting it with the total weight of the rules RQ, w, = wi,and by letting its consequent y, be an average of the consequents of RQ. Thus, the set of rules RQ is represented by a single rule R, with weight w, and consequent parameters

CiEQ

Let

={I,

. . . , K ) - Q , the model output (4.2) now becomes

The substitution of RQ by Rc does not alter the input-output mapping of the TS-model (4.1).

4.5.3 Linguistic approximation A fuzzy model can be interpreted and also simplified by means of linguistic approximation (Esragh and Mamdani, 1979). Using (4.19), the fuzzy sets in the model are compared to some reference fuzzy sets and their modifications by selected linguistic hedges. The model is described in terms of the labels of the reference fuzzy sets. By substituting the reference fuzzy sets for the original fuzzy sets, the model can be directly interpreted linguistically. Figure 4.4 gives an example of three reference fuzzy sets 'Small', 'Medium' and 'Big', together with the linguistic hedges of Table 4.1. Table 4.1

Linguistic hedges.

- --

hedge

operation

hedge

very A

P:

More than A

more or less A

fi

Less than A

4.6

operation P A (x),

if x

< minix 1 P A (x) = 1 )

otherwise (x), if x > max{x otherwise

1 P A (x)= 1 )

EXAMPLE: FUZZY MODELING A N D CONTROL OF A N HVAC SYSTEM

In this section, a fuzzy model of an HVAC unit is first developed from input-output measurements. Then, the simplifiction method is applied with several different values

96

FUZZY ALGORITHMS FOR CONTROL

Figure 4.4

Reference fuzzy sets and fuzzy sets obtained by applying some linguistic hedges.

t

of the thresholds. The resulting models are validated by comparison with the process data and are also applied in a predictive control scheme. The system consists of a fan-coil unit inside a test cell (van Paassen and Lute, 1993). Hot water at 65 "C is supplied to the coil, which exchanges the heat between the hot water and the surrounding air. In the fan-coil unit, the air coming from the outside (primary air) is mixed with the return air from the room (secondary air). The flows of primary and secondary air are controlled by the outside and return dampers, and by the velocity of the fan, which forces the air to pass through the coil, heating or cooling the air. The HVAC system is depicted in Fig. 4.5. t

-

-

pr~mary~

outslde damper

Figure 4.5

Heating, ventilation and air-conditioning system.

The global control goal for this system is to keep the temperature of the test cell

Ti, at a certain reference value, ensuring that enough ventilation and renovated air is supplied to the room. For this purpose three different control actions can be used:

1. Velocity of the fan. The fan has three different velocities: low, medium and high.

DATA-DRIVEN CONSTRUCTION OF TRANSPARENT FUZZY MODELS

97

2. Position of the dampers (outside and return). The dampers can be open in different discrete positions, controlling the amounts of air coming from outdoors and returned from the test cell. 3. Position of the heating valve. The amount of water entering the heat exchanger is controlled by the heating valve. If this valve is completely open, the quantity of hot water supplied is maximal, and if it is closed, no hot water is supplied to the coil. 4.6.1 Initial fuzzy model of the system

The global control problem is partially reduced in this example. The fan is kept at low velocity in order to increase human comfort by minimizing the noise level. Both dampers are half-open, allowing ventilation from the outside, and the return of some air from the test cell to the fan-coil. Only the heating valve is used as a control input. As shown in Fig. 4.5, temperatures can be measured at different positions in the fan-coil. The supply temperature T,, measured just after the coil, is chosen as the most relevant temperature to control. First, an initial TS fuzzy model of the system was constructed from process measurements. The input variables were selected on the basis of correlation analysis and physical understanding of the process. The model predict the supply air temperature T, based on its present and previous value, the mixed air temperature T,, and the heating valve position u. Hence:

The model consist of ten rules, each with four antecedent terms, of the form:

Ri :

If T,(lc) is Ail and T,(k - 1) is Ai2 and u(lc - 1) is Ai3

and T, (lc) is Ai4 Then T, (lc

+ 1) = yi ,

(4.23)

)~, where yi = ~ F [ x ( l c1IT. The antecedent membership functions and the consequent parameters were estimated from a set of input-output measurements by using the method presented in Section 4.4. The identification data set is shown in Fig. 4.6. It contains N = 800 samples with a sampling period of 30s. The data was collected at two different times of day (morning and afternoon), using the excitation signal u that was designed to cover the entire range of the control valve positions and to contain the important frequencies in the expected range of process dynamics. An initial fuzzy model was created with the fuzziness parameter m = 2. The numbers of clusters was selected by means of of cluster validity measures. Figure 4.6d shows three validity measures; the average within cluster distance (Krishnapuram and Freg, 1992), the fuzzy hypervolume (Gath and Geva, 1989) and the cluster flatness (BabuSka, 1998) for different numbers of clusters. The initial model contains a total of 40 antecedent fuzzy sets, shown in Fig. 4.7a. A separate data set consisting of 400 observations, which was measured on another day, was used to validate the model. The supply temperatures measured and recursively predicted by the model in simulation are compared in Fig. 4.7b.

98

FUZZY ALGORITHMS FOR CONTROL

(a) Valveposition.

(b) Mixed-air temperature.

Aver. within cluster distvlcr

20 0

100

200 Time [min]

300

(c) Supply temperature.

400

I 2

6

8

10

12

(d) Cluster validity measures.

Figure 4.6 Identification data used in the modeling of the HVAC system (a), (b), (c), and three cluster validity measures applied t o various partitions of the identification data (d). All measures indicate a local minimum for K = 10.

99

DATA-DRIVEN CONSTRUCTION OF TRANSPARENT FUZZY MODELS

4.6.2 Simplifying the model When we inspect the initial fuzzy model, we notice that there are a lot of overlapping and similar fuzzy sets in the rule base (Fig. 4.7a). In order to reduce the complexity of the model, and thereby its computation time, we apply the rule base simplification method presented in Section 4.5. Since the rule base simplification does not require additional knowledge or data acquisition, and no computationally intensive algorithms like fuzzy clustering are used, it is advised to run the algorithm several times with different thresholds. In the results reported below, the threshold y for removing fuzzy sets similar to the corresponding universal set was kept at y = 0.8, while the threshold X for merging fuzzy sets was varied. The consequent parameters of the rules in the obtained simplified and reduced models where re-estimated by minimizing the least squares error (4.14) using the same training data used to identify the initial model.

1

250

50

HeaIing valve u(k-I)

(a) Membership functions. Figure 4.7

100

Time [minl

150

200

(b) Validation by recursive simulation.

Initial fuzzy model.

The various simplified models were validated on the validation data in a recursive simulation. The membership functions of two simplified models M 1 and M2 (obtained with X = 0.8 and X = 0.6, respectively) and their validations are shown in Figures 4.8 and 4.9, respectively, together with the membership functions and the validation of the initial model. From the figures one can see that the accuracy of models M1 and M2 is just slightly lower than that of the initial model. If we consider model M2, it is significantly reduced in complexity, compared to the initial model, as it only consists of 4 rules and 9 different fuzzy sets. The antecedents of model M2 are given in Table 4.2. The significant simplification of the rule base has made model M2 easy to inspect. When looking at the premise of the rules, we notice an interesting result for the antecedent variables T, (k) and T,(k - 1). Membership functions Al , Ag are almost Bz (Fig. 4.9a). This suggests that one of the two equal to membership functions B1, variables could be removed from the model. If we remove T,(lc - 1) from model M2 and re-estimate the consequent parameters, we end up with a strongly reduced model M3, constituted by four rules and only seven fuzzy sets.

100

FUZZY ALGORITHMS FOR CONTROL

(a) Membership functions. Figure 4.8

(b) Recursive simulation. Simplified fuzzy model M I .

i

so

(a) Membership functions. Figure 4.9

Table 4.2 If

Ts(k) is,

R1: Rz: A1

100 Time [min]

1so

200

(b) Recursive simulation. Simplified fuzzy model M2.

Antecedents of the simplified model M2.

Ts(k - 1) is,

u(k - 1) is,

T,(k) is,

fhen Ts(k + 1) =

-

(71

-

Yl

B1

Cz

-

YZ

R3:

A2

B2

c 3

Dl

R4:

-

Y3

-

c4

-

Y4

DATA-DRIVEN CONSTRUCTION O F TRANSPARENT FUZZY MODELS

101

The antecedent fuzzy sets of model M3 are the same as for model M2, except for that the variable T, (lc - 1) is no longer in the model. The RMS error of model M3 is lower than that of the initial model. 4.6.3

Control results

The three simplified models MI, M2 and M3, and the initial model, were all implemented in the IMC predictive control scheme depicted in Fig. 4.10. The controller's inputs are the setpoint (reference), the predicted supply temperature T,, and the filtered mixed-air temperature Tmf. The error signal e ( k ) = T,(lc) - T, (lc) is passed trough a first order low-pass digital Butterworth filter Fl. Another first-order low-pass Butterworth filter F2 is used for the Tm to filter out the measurement noise.

--

Predictive Controller

--

Fan-coil unit

Tm b

T,

Figure 4.10 Implementation of the IMC predictive controller based on the fuzzy model for the fan-coil system.

The predictive controller uses a branch-and-boundalgorithm to compute the optimal control action (Sousa et al., 1997). The prediction and control horizons were set to H, = 4 and H, = 2, respectively. Simulation experiments with a step-like reference were carried out. All models did well with respect to the numerical performance. The result obtained with the initial model is shown in Fig. 4.1 1. The result obtained with the simplest model, M3, is shown in Fig. 4.12. As expected from the model's prediction performance, the control performance is slightly better than that of obtained with the initial model. More significantly, the FLOPS (floating-point operations) used by model M3 in the control simulation were only 15% of the FLOPS used by the initial model. 4.6.4

Discussion

The results of the simplification exercise are summarized in Table 4.3. The relative performances and the computational costs of the models are also visualized in Fig. 4.3. For each model, Table 4.3 lists the X used to obtain the respective simplified model, its number of inputs, rules, membership functions, the FLOPS used to predict the

102

FUZZY ALGORITHMS FOR CONTROL

1

I

0

10

20

30

0

10

20

30

1

Figure 4.11

40 50 Time [min]

60

70

80

60

70

80

1

40 50 Time [min]

Simulation using the predictive controller based on the initial model.

I

0

Figure 4.12

10

20

30

40 50 Time [min]

60

70

1 80

Simulation using the predictive controller based on the simplified model M3.

DATA-DRIVEN CONSTRUCTION O F TRANSPARENT FUZZY MODELS

103

validation data, the validation error, the FLOPS used in the control simulation, and the control error. Table 4.3

Model

Performance obtained with fuzzy models of varying complexity.

X

inputs

rules

m. functions

Initial

-

0.8 0.6 0.6

4 4 4 3

10

M1 M2 M3

40 13 9 7

5

4 4

L

Or~glnal

A

MI

Validation FLOPS Error

Control KOPS Error

1.0000 0.3621 0.2526 0.1660

1.0000 0.3807 0.2664 0.1536

1.0000 1.0831 1.0367 0.8883

1.0000 1.0147 1.0618 0.9649

-... L

Model

M2

M3

Figure 4.13 Normalized error and FLOPS used by the four models in predicting the validation data (dash-dotted line) and in the control simulation (solid line).

From the results, we notice that the error in the validation of the model and in the control simulation is more or less the same for all the models. As expected, increased error in validation means increased error in control. The same holds for the used FLOPS. This relationship is clearly illustrated in Fig. 4.3. The relative FLOPS used by the model in prediction correspond to the relative FLOPS used to solve the control problem. For a given H,, the optimization algorithm will simulate the model the same number of times, independently of the model used (assuming their predictions are the same). If the model is sped up twice, so is also the optimization. However, due to the exponential complexity of the branch-and-bound method, the gain in FLOPS is high.

4.7

CONCLUSIONS

This chapter has given an overview of a data-driven approach to the construction of rule-based fuzzy models. First, fuzzy clustering techniques are used to search for relations between the system's variables. From the clusters found in the data, an initial rule-based model is built, focusing mainly on the model's approximation properties.

104

FUZZY ALGORITHMS FOR CONTROL

Then, similarity analysis is applied to identify similar fuzzy sets in the initial rule base. These fuzzy sets are aggregated to produce new fuzzy sets representing generalized and more comprehensive linguistic terms. In this way, the number of qualitative concepts used by the rule base is reduced. In the modeling procedure, the user can introduce a trade-off between qualitative and quantitative aspects by adjusting the similarity threshold used to aggregate similar fuzzy sets. The resulting rule-based model combines good quantitative properties, inherited from the initial rule base, with inspectable qualitative rules resulting from the aggregation of similar fuzzy sets. This approach has been applied to several real-world problems. One example is presented in this paper, others can be found in (Setnes et al., 1997; Setnes et al., 1998a).

I

References

BabuSka, R. (1998). Fuzzy Modeling for Control. Kluwer Academic Publishers, Boston. BabuSka, R., Setnes, M., Kaymak, U., and van Nauta Lemke, H. (1996). Rule base simplification with similarity measures. In Proceedings Fifh IEEE International Conference on Fuuy Systems, pages 1642-1647, New Orleans, USA. BabuSka, R. and Verbruggen, H. (1995). A new identification method for linguistic fuzzy models. In Proceedings FUZZ-IEEWIFES'9.5,pages 905-912, Yokohama, Japan. Bezdek, J. (198 1). Pattern Recognition with Fuzzy Objective Function. Plenum Press, New York. Brown, M. and Harris, C. (1994). Neurofuuy Adaptive Modelling and Control. Prentice Hall, New York. de Boor, C. (1978). A Practical Guide to Splines. Springer-Verlag,New York. Driankov, D. and Palm, R., editors (1998). Advances in Fuuy Control. Springer, Heidelberg, Germany. Dubois, D. and Prade, H. (1980). Fuuy sets and systems: theory and applications, volume 144 of Mathematics in science and engineering. Academic Press. Esragh, F. and Mamdani, E. (1979). A general approach to linguistic approximation. Int. J. Man-Machine Studies, 11501-5 19. Gath, I. and Geva, A. (1989). Unsupervised optimal fuzzy clustering. IEEE Trans. Pattern Analysis and Machine Intelligence, 7:773-781. Gustafson, D. and Kessel, W. (1979). Fuzzy clustering with a fuzzy covariance matrix. In Proc. IEEE CDC, pages 761-766, San Diego, CA, USA. Haykin, S. (1994). Neural Networks. Macmillan Maxwell International, New York. Jang, J.-S., Sun, C.-T., and Mizutani, E. (1997). Neuro-Fuzzy and Soft Computing; a Computational Approach to Learning and Machine Intelligence. Prentice-Hall, Upper Sadle River. Kandel and Langholz, editors (1994). Fuzzy ControESystems. CRC Press, Boca-Raton, F1. Kaymak, U. and BabuSka, R. (1995). Compatible cluster merging for fuzzy modeling. In Proceedings FUZZ-IEEWIFES'9.5,pages 897-904, Yokohama, Japan. Klir, G. and Yuan, B. (1995). Fuzzy sets andfuzzy logic; theory and applications. Prentice Hall.

I

I

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DATA-DRIVEN CONSTRUCTION O F TRANSPARENT FUZZY MODELS

105

Kosko, B. (1994). Fuzzy systems as universal approximators. IEEE Trans. Computers, 43: 1329-1333. Krishnapuram, R. and Freg, C.-P. (1992). Fitting an unknown number of lines and planes to image data through compatible cluster merging. Pattern Recognition, 25(4):385-400. Kruse, R., Gebhardt, J., and Klawonn, F. (1994). Foundations of Fuzzy Systems. John Wiley and Sons, Chichester. Lin, C. (1994). Neural Fuuy Control Systems with Structure and Parameter Learning. World Scientific, Singapore. Ljung, L. (1987). System Identijication, Theoryfor the User. Prentice-Hall, New Jersey. Mamdani, E. (1974). Applications of fuzzy algorithms for control of simple dynamic plant. In Proceedings IEE, number 121, pages 1585-1 588. Gstergaard, J. (1990). Fuzzy ii - the new generation of high level kiln control. ZementKalk-Gips, (1 1). Palm, R., Driankov, D., and Hellendoorn, H. (1997). Model Based Fuzzy Control. Springer, Berlin. Seber, G. and Wild, C. (1989). Nonlinear Regression. John Wiley & Sons, New York. Setnes, M. (1995). Fuuy Rule Base Simplijication Using Similarity Measures. M.Sc. thesis, Delft University of Technology, Delft, the Netherlands. (A.95.023). Setnes, M., BabuSka, R., Kaymak, U., and van Nauta Lemke, H. (1998a). Similarity measures in fuzzy rule base simplification.IEEE Transactions on Systems, Man and Cybernetics, 28(3):376-386. Setnes, M., BabuSka, R., and Verbruggen, H. (1998b). Rule-based modeling: Precision and transparency. IEEE Transactions on Systems, Man and Cybernetics, Part C: Applications and Reviews, 28(1):165-169. Setnes, M., BabuSka, R., Verbruggen, H., SBnchez, M., and van den Boogaard, H. (1997). Fuzzy modeling and similarity analysis applied to ecological data. In Proceedings FUZZ-IEEE'97, pages 4 15-420, Barcelona, Spain. Sjoberg, J., Zhang, Q., Ljung, L., Benveniste, A., Delyon, B., Glorennec, P.-Y., Hjalmarsson, H., and Juditsky, A. (1995). Nonlinear black-box modeling in system identification: a unified overview. Automatics, 3 l(12):1691-1 1724. Smith, S., Nokleby, B., and Comer, D. (1994). A computational approach to fuzzy logic controller design and analysis using cell state space methods. In Kandel and Langholz, editors, Fuzzy Control Systems, pages 398-427. CRC Press, Boca-Raton, F1. Sousa, J., BabuSka, R., and Verbruggen, H. (1997). Fuzzy predictive control applied to an air-conditioning system. Control Engineering Practice, 5(10):1395-1406. Tanaka, H., Uejima, S., and Asai, K. (1982). Linear regression analysis with fuzzy model. IEEE Trans. Systems, Man & Cybernetics, 12(6):903-907. van Paassen, A. and Lute, P. (1993). Energy saving through controlled ventilation windows. In 3rd European Conference on Architecture, Florence, Italy. Wang, L.-X. (1994). Adaptive Fuuy Systems and Control, Design and Stability Analysis. Prentice Hall, New Jersey.

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Yasunobu, S. and Miyamoto, S. (1985). Automatic train operation system by predictive fuzzy control. In Sugeno, M., editor, IndustrialApplications of Fuuy Control, pages 1-1 8. North-Holland. Zadeh, L. (1973). Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans. Systems, Man, and Cybernetics, 1:28-44. Zeng, X. and Singh, M. (1995). Approximation theory of fuzzy systems - MIMO case. IEEE Trans. Fuuy Systems, 3(2):2 19-235.

II

DESIGN AND ANALYSIS ISSUES

5

FUZZY LOGIC NORMAL FORMS FOR CONTROL LAW REPRESENTATION I. Perfilieva

Moscow State Academy of Instrument-Making and Informatics Department of Applied Mathematics Stromynka 20, 107846 Moscow, Russia

5.1

INTRODUCTION

The main paradigm of fuzzy control is that the control law is formulated as a knowledge based algorithm expressed in the language of logical formulas with vague predicates. It allows us to express qualitative characterization of variables using fuzzy predicates, and functional dependencies between variables using conditional sentences with fuzzy predicates (rules). Such an approach makes it possible to implement human reasoning in a form of a sequence of rules in the control algorithm. This fundamental idea introduced by L.A. Zadeh brought a new fruitful methodology to control theory. A lot of examples can be found where expert knowledge can be represented in the form of a rule based control algorithm. It has been demonstrated in practice that fuzzy control can be efficiently used to control complex systems both technical as well as those in which the human element plays a significant role. To clearly understand what the main advantages of the fuzzy approach to control are, let us compare it with the conventional control techniques. We will address the following three points: (a) description - linguistic or formal - of the problem in the language of the chosen theory,

112

FUZZY ALGORITHMS FOR CONTROL

(b) algorithmic procedure for computing the desirable parameters, (c) methods leading to explicit description of the control law. An essential feature of conventional control is that the control law can be analytically described using equations of different types. The most suitable algebraic form is

u ( k ) = f ( e ( k ) ,e ( k - 1 ) , . . . ,e ( k - n ) ,u ( k - I ) , . . . ,u ( k - n ) ) ,

(5.1)

where u represents the control output, e = w - y is the error between the setpoint value w and the output y of the controlled system. The main task of the conventional control theory is to find a function f in (5.1). However, this is not our interest in this chapter. Note that the computation procedure for u ( k ) is induced by the formula on the right hand side of (5.1). The same dependence is described in fuzzy logic using the collection of logical formulas in the form of the following rule-base:

IF e ( k ) is A: AND . . . AND e ( k - n) is A: AND u ( k - 1 ) is ~f AND.. .AND u ( k - n ) is Br THENu(lc)is~;, l < i < N ,

(5.2)

<

where N is a number of rules, A!, . . . , AT, BY,. . . ,By,1 i 5 N, are fuzzy predicate symbols and IF-THEN and AND are logical connectives, also known as implication and conjunction, respectively. It is implicitly assumed that logical formulas (rules) in this collection are joined either with the connective OR (disjunction) or with the connective AND (conjunction). Usually, fuzzy predicate symbols in (5.2) are expressed linguistically using terms such as "positive small", "near zero", "negative big" etc. The complete set of linguistic terms for each variable forms a fuzzy partition of the (normalized) universe of discourse. The set of formulas (5.2) describes the dependence of u ( k ) on e ( k ) ,e ( k - I ) , . . . , e ( k n), u ( k - I ) , . . . , u ( k - n) in the logical language using qualitative or set-theoretic characterization of variables instead of the numerical one. As (5.2) is a set of logical formulas, the algorithmic procedure for the computation of the value u ( k ) follows the usual way of interpretation in the logical theory plus a certain procedure known as defuzzification. The latter converts a qualitative characterization into a numerical one. All this means that the following items have to be successively determined. Membership functions of fuzzy sets corresponding to each of the predicate symbols A!, . . . , A:, B:, . . . ,Br, 1 5 i N . We use the same letters both for the predicate symbols as well as for membership functions of fuzzy sets.

<

Operations on the set of truth values corresponding to each of the explicit logical connectives AND and IF-THEN as well as to the implicit logical connective

OR. Defuzzification algorithm. Now, we can make the comparison. From the computational point of view, it is more or less evident that the control law represented by formula (5.2) which has been

FUZZY LOGIC FOR CONTROL

113

decomposed into N , ( N 2 I), simple uniform logical formulas, is more efficient than formula (5.1). However, the answer to the following two questions is not evident:

I . How can an arbitrary function (5.1) be represented in the form (5.2)? 2. Is this representation unique? The answers to these two questions are very important for control engineers. The constructive answer to the first question provides the reliability and mathematical support of fuzzy control. The answer to the second question leads to efficiency analysis of the fuzzy control algorithms. In the sequel, we will try to answer these questions. The chapter is structured in accordance with the following general approaches. The first one aims at a description of the classes of functions which can be represented by using (5.2). The second approach considers the problem in the opposite manner: how to find a description of the form (5.2) for a given function. In Section 5.2, we consider three different types of fuzzy logic control models, and show how they can be interpreted uniformly by two normal forms. In Section 5.3, the ways of correspondence between functions and normal forms are described. In Section 5.4, the procedures for the approximation and realization of the given function by normal forms are described. FUZZY LOGIC CONTROL MODELS A N D THEIR NORMAL FORMS

5.2

The collection of fuzzy logic formulas (5.2) will be called the Fuzzy Logic Control Model (FLCM). It is used instead of a functional description of the control law. Thus, (5.2) is the formal description of the dependence between variables in the specific logical language. In this section, we consider different types of FLCMs, and suggest their algebraic interpretation using two normal forms. It means that we will show the way of computation of the control value using algebraic operations, instead of logical ones. 5.2.1

Types of fuzzy logic control models

Three different types of FXCMs are most frequently used in the literature, namely, singleton, linguistic and Takagi-Sugeno ones. To show the difference among them as well as to stress their common origin, we rewrite (5.2) into the general form

IF Anti THEN Consi , where 1 5 i 5 N . Now, the specificity of each type of model is determined only by a certain form of the consequent Consi, while all of them have the same form of the antecedent. To show this explicitly, we will give the corresponding expressions of logical formulas for each of them. I . Singleton FLCM is characterized by the consequent which is given by numerical value (represented by fuzzy singleton). The form of IF-THEN rules in this case

114

FUZZY ALGORITHMS FOR CONTROL

is

IF Anti THEN ~ ( kis)bi, where bi, 1 5 i

(5.3)

5 N, are real numbers.

2. Linguistic FLCM is characterized by the consequent which is given linguistically using fuzzy sets. The form of IF-THENrules in this case coincides with (5.2), i.e. IF Anti THEN u(k)is Bi,

where Bi, 1 5 i

5 N, are fuzzy sets.

3. Takagi-Sugeno FLCM is characterized by the consequent which is given by linear combination of the input variables. The form of IF-THEN rules in this case is

+ +

IF Anti THEN u(k) = aioe(k) . . . aine(k - n ) b.z ~ " J( k - 1 ) - . . . - b i n u ( k - n ) .

(5.4)

When observing these three types of FLCMs, we see that antecedent in each rule describes a single fuzzy constituent of a fuzzy partition of the input space. The IFTHEN rule as a whole describes the correspondence between a fuzzy constituent in the input space and a certain representation of the output variable - a crisp value, a fuzzy subset, or a linear form. All three types of rules can be unified into one form represented by scheme (5.2), because fuzzy singleton in FLCM of the first type and the linear form over input variables in FLCM of the third type can be considered as characterizations of special fuzzy subsets. In what follows, we will deal with the general scheme (5.2) from the point of view of its ability to represent functions. In connection with this, the representation of a control law immediately follows. On the basis of such an approach, we can rewrite the logical formulas (5.2) by replacing the variables by those which are usually used to denote the independent and dependent variables, i.e.,

xl is A: AND . . . AND x, is A: THEN y is Bi,

(5.5)

where 1 5 i 5 N. Scheme (5.5) will be considered as a logical form of the representation of some functional dependence. To be able to provide it with an algorithmic procedure for the calculation of the dependent variable y, we need a certain translation from logical formulas into algebraic expressions. This step is known as the interpretation of logical formulas. To interpret the dependence expressed in (5.3, the following must be determined: membership functions of the fuzzy sets A:, . . . , A:, Bi, 1 5 i

5 N,

logical operations on the set of truth values which correspond to logical connectives. Two basic approaches to the determination of membership functions can be distinguished. The first one takes membership functions as an interpretation of linguistic

FUZZY LOGIC FOR CONTROL

115

expressions. Such membership functions are determined by experts and can be modified when tuning the controller,if necessary. The second approach extracts membership functions from the data using cluster analysis, no matter whether they are interpretations of some linguistic terms. In this chapter, we will not deal with this aspect. We rather focus on the second problem, namely the determination of the logical operations on the set of truth values. 5.2.2 Disjunctive and conjunctive normal forms

The formal logical expression ( 5 3 , which we considered as an alternative representation of any functional dependence, or, more specifically, of the control law, is in essence nothing else than a set of sentences. To be used in computation, they should be provided with a certain algorithm. We suggest two possible ways of translating (5.5) into algebraic formulas. The names disjunctive and conjunctive normal forms are used in analogy with the structure of well-known boolean expressions. y ) on a First, each logical formula of (5.5) is interpreted by a fuzzy relation Ri(x, universe X 1 x . . . x X, x Y where x = (xl,.. . ,x,). Second, we combine fuzzy y), 1 i 5 N, into one relation, by using the appropriate operations. relations Ri(x, As already mentioned, the interpretation of both explicit (conjunction, implication) and implicit (disjunction) logical connectives will be stressed. The most general approach to the interpretation of logical connectives is based on the concept of a t-norm. In accordance with this concept, a logical conjunction is interpreted by a t-norm and a disjunction by the corresponding t-conorm. The interpretation of the logical implication is connected with a chosen t-norm in accordance with one of the two ways below.

<

Homogeneous: the implication is interpreted directly by a t-norm (and coincides with the conjunction). Let us stress that this interpretation does not fully correlate with the logical meaning of the implication. Residuated: the interpretation is given by the formula

Examples of t-norms and their corresponding residuations are known from the literature. Here we consider only continuous t-norms. The most suitable are the following cases determined on the unit interval [O,l]:

1. Godel connectives atb = aAb

2. Product connectives atb = a.b

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FUZZY ALGORITHMS FOR CONTROL

3. Eukasiewicz connectives

atb = OV(a+b-1) a+tb = l A ( 1 - a + b ) . Recall that the ith formula in the collection (5.5) can be interpreted as a fuzzy relation R i ( x ,y). In order to distinguish two major cases of the interpretation of implications, we will write R t ( x ,y) if the relation is derived in accordance with the homogeneous way, and R f ( x ,y) if it is derived on the basis of residuation. A fuzzy relation RF is given by a combination of the antecedent and the consequent using a t-norm for the interpretation of both conjunction (AND) and implication (IFTHEN) R;(x,9 ) = A: ( X I ) t ... t AY(xn)t ~ i ( ~ ) . (5.9)

a

I

On the other hand, the fuzzy relation

fits precisely the way how the ith logical formula in (5.5) is interpreted. It means that the conjunction and the implication are interpreted in correspondence with some t-norm and its residuation. The interpretation of the whole collection (5.5) can be obtained in one of the two general forms of fuzzy relations, depending on the interpretationof the implicit logical connectives among single formulas. They are given by the sup and inf operations, respectively, in accordance with the two major interpretations of the implication (homogeneous and residuated):

where x = ( x l , .. . x n ) ,xj E X j , 1 5 j 5 n, y E Y . Now, by combining the two forms of fuzzy relations (5.1 1) and (5.12) composed with (5.9) and (5. lo), we are able to make an important generalization. It is motivated by the analogy with two normal forms for Boolean functions representation, namely the disjunctive and the conjunctive normal forms.

Definition 5.2.1 A Disjunctive Fuzzy N o m l Form (DFNF) is an algebraic formula of the form N

V ( A f ( x 1 t) . . . t A : ( x n )t B i ( y ) ).

i=l

(5.13)

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117

Definition 5.2.2 A Conjunctive Fuzzy Normal Form (CFNF) is an algebraic formula of the form N

Specifying a t-norm in the expression (5.13), we come to some popular interpretations of a FLCM. For example, if the t-norm in (5.13) coincides with the min operator, then we obtain the Mamdani-type interpretation of a FLCM (Mamdani and Assilian,

This type of model plays a noticeable role in the development of the methodology of fuzzy logic models. It serves as a prototype for different models in fuzzy control. In particular, the specific types of the FLCMs considered above, namely singleton and Takagi-Sugenoones, can be interpreted in the same way as in (5.15) when choosing the membership function of Bi appropriately. (a) For the singleton FLCM we have B ~ ( Y=)

{1 0

ify=bi, otherwise,

(b) For the Takagi-Sugeno FLCM we have B d x , y) =

1 ify=ailxl+...+ainxn, 0 otherwise.

Hence, we have shown in this section that each FLCM can be interpreted by one of the two algebraic formulas with fixed forms. In the sequel, we will speak about functional realization of the FLCM, starting from their normalized algebraic interpretations. 5.3

FUNCTIONAL REALIZATION O F N O R M A L FORMS

So far, we have presented an interpretation of the logical formulas in the language of algebra. Strictly speaking, we have demonstrated how one (logical) kind of description can be replaced by another (algebraic) one. But we still have no algorithmic procedure for calculating the dependent variable y from the independent variables (xl, . . . , x,), given the description of their dependence (5.13) or (5.14). Note, that in these expressions the (functional) dependence between (XI,. . . ,x,) and y is described implicitly and to extract it is not an easy task. In general, the procedure of transformation a formula into the function represented by it will be called the realization. More precisely, in this case, we will say that the function is realized by a formula and also, that a formula realizes the function. In this section, the realization of the two normal forms will be algorithmically described. The suggested procedure is split into two steps: first, the membership

118

FUZZY ALGORITHMS FOR CONTROL

function of a fuzzy set B ( y ) restricting the values of y given ( x l , .. . ,x,) is extracted from (5.13) (or (5.14)); second, the unique value of y corresponding to ( X I , . . . ,x,) is determined. Let us fix a continuous t-norm, its residuation, and a set of membership functions {Af ( x l ) ,. . . ,A r ( x n ) ,Bi(y),1 i 5 N } , defined on the set of real numbers R

<

Definition 5.3.1 A function f D F N F (f C p N p ) : Rn --+ E%, is realized by a formula of the form DFNF (5.13) (or CFNF (5.14))based on the set of membershipfunctions {At ( x l ) ,. . . ,A;(x,), Bi(y), 1 i N } if for each vector point xo = (xy,.. ., x z ) E Rn a value yo = ~ D F N F ( X O ) ( y o = ~ C F N F ( X ' ) ) corresponding to it is computed in accordance with the following algorithm.

< <

( a ) Substitute x0 into the DFNF (CFNF)formula and compute the function B ( y ) using the algebraic operations t and +t as follows:

The B ( y ) is the membershipfunction of thefuzzy set corresponding to the point x0 when substituted into formula (5.13) or (5.14),respectively. (6) Defuuify B ( y )ol; equivalently,find the value of the defuuiJication methods.

E R in accordance with one

1. Center of gravity =

JYB(Y)~Y . / - B ( Y ) ~ 'Y

2. Mean of maxima

where Av is one of the possible average operations. It is not difficult to show that contrary to the Boolean case, the functions f D p N F and f C F N F based on the same set of membership functions {At (xl), . . . ,A; (x,), Bi(y), 1 i 5 N) are generally not equal. Moreover, the following proposition directly follows from Definition 5.3.1.

<

!

1 I

Proposition 5.3.1 Each collection of logical formulas (5.5) based on the given interpretation of logical connectives as well as on afixed set of membership functions { A f ( x l ).,. . ,A l ( x n ) ,Bi(y),1 5 i 5 N ) on R realizes at least one function on R Note that, in general, we cannot say anything specific about the properties of functions realized by logical formulas or by their normal forms. To do that, we would have to know the properties (i.e., continuity, etc.) of operations interpreting the logical connectives and also the properties of membership functions.

t

FUZZY LOGIC FOR CONTROL

119

5.4 APPROXIMATION AND REPRESENTATION OF REAL-VALUED REAL CONTINUOUS FUNCTIONS As shown in the previous sections, logical language with fuzzy predicates is rich enough to describe real-valued real functions. But to be able to pursuit well the applications (particularly in control), the converse task must be solved namely, given a function, find its logical description in the form(5.5). In this section we will present two solutions of this problem for continuous functions defined on R The first solution is approximate. This means that the suggested logical description realizes a function which approximates the given one. The second solution is precise but works for a subclass of continuous functions. Let us fix an interpretation of the logical conjunction and the implication by some continuous t-norm defined on [O,1] and its residuation, respectively. In this case, the class of real-valued real functions that can be realized by a formula of the form DFNF (5.13) (or CFNF (5.14)) is fully characterized by the set of membership functions

and the defuzzification method. We denote by D F (CF) a class of real-valued real functions realized by the formula of the form DFNF (5.13) (or CFNF (5.14)) based on F. In the following, we will see that, as far as approximation is concerned, general dependence on F does not mean the dependence on concrete shapes of membership functions in F, but the dependence on non-zero domains of them only. 5.4.1

Approximation o f real-valued real continuous functions

In this section, we will show that each continuous real-valued real function on a compact U can be approximated by a certain function, realized by DFNF or CFNF when the set of membership functions F is appropriately chosen. Definition 5.4.1 A class of real-valued real functions H given on the set U C Rn is &-completewith respect to a class of real-valued real functions C given on the same set U if VgEC 3 h ~ H Ig(x)-h(x))<~forallx~U. We will demonstrate that F can be chosen in such a way that the class DF as well as CF is &-completewith respect to the class of all continuous real-valued real functions on a compact U for all positive E . In other words, it means that each continuous realvalued real function on a compact U can be approximated by an appropriate function from DF or CF with a prescribed accuracy. Throughout this section, F denotes the class of continuous on R membership functions. To distinguish its special subclasses, we will complete F by subscripts. The following notation will be employed. 1. Fc is a class of all the functions continuous on R so that each function from Fc is positive on a certain interval [c - 6, c 61 and is equal to zero outside it.

+

2. FM is a superclass of Fc, which consists of all partially continuous functions, which are positive on a certain interval [a, b] and are equal to zero outside it.

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FUZZY ALGORITHMS FOR CONTROL

3. FG is a subclass of Fc, which consists of all Gaussian membership functions defined by the formula e-wfor x E [c - 6, e outside.

+ 6) and are equal to zero

4 . FT is a subclass of F c , which consists of all the triangular membership functions defined by the formula max (0, 1-

9).

Theorem 1 Each class of real-valued real functions D F ~D, F ~DFT , dejned on a compact U c En using a continuous t-norm is &-completewith respect to the class of all continuous real-valued real functions on U for any E > 0. The proof of Theorem 1 is essentially contained in (Nguen et al., 1996; Castro and Delgado, 1996) where, however, this theorem was formulated using another terminology. Both groups of authors use the terminology inspired by applications of fuzzy logic to control. Therefore, they use the term universal approximation, while we prefer to use the standard topological term &-completeness. Similar results, which are related to classes DF, and DF, with fixed algebra of logical operations based on certain continuous t-norm and defuzzification procedure can be found in (Buckley, 1993; Kosko, 1992). The attention paid by researchers to this kind of approximation theorems shows their importance to fuzzy logic control theory because they state that each control law represented by a continuous function can be realized using a certain DFNF with a prescribed accuracy. At the same time, the approximate realization of the continuous control law by a CFWF formula was proved only in tukasiewicz algebra of logical operations (Klawonn and Noviik, 1996). This result was more or less expected because of the continuity of each logical operation. No similar results are known in general case. But it is known from the classical Boolean algebra that both normal forms realizing the same logical function are equivalent. In fuzzy logic, however, this does not hold. However, we can prove &-completeness. Theorem 2 The class CF, of real-valued realfunctions dejnedon a compact U c En using a continuous t-norm, its residuation is &-completewith respect to the class of all continuous real-valued real functions on U for any E > 0.

I

I

I

The proof of this theorem leads to a special finite covering of U realized by the supports of the membership functions defined on U. Interested readers may find it in Appendix. It must be stressed once more that for the proof it does not matter what concrete shapes of membership functions in Fc are chosen as only the dependence on their non-zero domains is important.

5.4.2 Representation of real-valued real continuous functions In this subsection we show, how is it possible to obtain a precise representation of a function by non-excessive collections of fuzzy logic formulas. The following theorems (Perfilieva, 1996;Noviik and Perfilieva, 1999)demonstratesthat CFMincludes the class of all continuous functions defined on a certain compact space U. On the contrary to the previous result about approximation, the dependence on concrete shapes of

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FUZZY LOGIC FOR CONTROL

121

membership functions in FM is essential. For simplicity, we consider the case of functions with one variable.

Theorem 3 For any continuous and strictly monotonous real-valued real function g ( x ) : [a,b] --+ R there are membership functions of fuzzy sets A l ( x ) ,A ~ ( x ) , Bl ( y ) ,B 2 ( y )belonging to FM such that the collection offuzzy logic formulas (5.5) IF x is Ai THEN y is B i , where 1

5 i 5 2, leads to a fuzzy

relation in the conjunctive normal form

which realizes a function fcFNF (x) : [a,b] + R in such a way that fcFNF(x) = g ( x )for every x E [a,b]. PROOF:The proof is constructive and can be used as a basis for an algorithm. Assume that acontinuous t-norm and its residuation are fixed. Since g ( x ) is continuous and monotonous on [a,b] it defines a one-to-one correspondence between [a,b] and [g(a),g(b)].Therefore, the inverse function g-'(y) exists. Assume that g ( x ) monotonously increases. The proof consists in constructing membership functions of the fuzzy sets A1,A2 and B1, B2 in such a way that the ) g ( x ) is fulfilled for all x E [a,b]. equality ~ C F N F ( X= Choose membership functions for the fuzzy sets A1 , A2 and B 1 ,B 2 as follows:

x-a Al(x) = 1 - b-a'

x-a A 2 ( x ) = - x E [a,bl, b-a'

Let xO E [a,b] be an arbitrary, but fixed element. We show that f C F N F ( x O = ) ( x O ) .Indeed,

A 2 ( x o )-tt B Z ( y )= 1

*

A~(xO)

5B2(~)

* x O - a 5 g - lb(-ya) - a

Thus,

B ( y ) = R c d ( x O , y= ) 1

(j g(xO)= y,

and then

sup B ( y ) = ~ ~ ~ ( x ~ , =g 1.( x ~ ) ) Y

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FUZZY ALGORITHMS FOR CONTROL

By Definition 5.3.1 where the defuzzification is specified as the mean of maxima we come to the final equality ~ C F N F ( X O ) = g(zO), which completes the proof. To extend the class of functions which are non-excessively realized by a collection of fuzzy logic formulas we introduce the following definition.

Definition 5.4.2 By a piecewise monotonousfunction g on a compact domain U we mean a function, for which there exists afinite partition of U such that the restriction of g to each set of the partition is strictly monotonous. We again consider only the case of functions with one variable.

Theorem 4 For any continuous and piecewise monotonous real-valued real function g ( x ) : [a,b] --+R there exist a number N and membership functions o f f u u y sets Ail ( x ) ,Ai2 ( x ) ,Bil ( y ) ,Biz ( y ) , 1 5 i 5 N belonging to FM,such that the collection offuuy logicformulas (5.5)

IF x is Ai THEN y is Bij , where 1 form

5 i 5 N,

1

5 j 5 2, describes afuzzy relation in the conjunctive normal N

R " ~ ( x , Y= )

2

A A(Aij(x) i=l j=1

+t

Bij(y))

and realizes a function f C F (~x )~: [a,b] -+ R such that every x E [a,b].

~CFNF (x)

= g ( x ) for

PROOF: Since g ( x ) is continuous and piecewise monotonous on [a,b],there exists a finite partition of [a,b] into a finite number of subintervals J1, . . . , JN such that the restriction gJJi, 1 5 i 5 N, to each set from the partition is strictly monotonous. In accordance with Theorem 3, for each such restriction there exist fuzzy sets Ail, Ai2 defined on Ji and B i l , Biz defined on g(Ji) such that the fuzzy relation

realizes a function ~ C F N (Fx~)such that g ( x ) = fcFNFi ( x )for all x E Ji9 1 5 i 5 N. We extend the definitions of the membership functions Aij ( x ) ,Bij ( y ) ,1 5 i N , 1 5 j 5 2, to [a,b] and g([a,b]) by setting them equal to 0 outside Ji and g ( J i ) , respectively. Consider the fuzzy relation

<

FUZZY LOGIC FOR CONTROL

123

where x E [ a , b], y E g ( [ a , b ] ) and Aij ( x ) , Bij ( y ) are the extensions. Since any arbitrary xO E [ a , b] belongs to exactly one subinterval Ji (except for the bounds of subintervals) then not more than two membership functions Ail (x),Ai2( x ) (Ai-l,2 ( x ) , Ail ( x ) or A i z ( x ) , Ai+l,l ( x ) , 2 5 i 5 N - 1, for internal bounds) can differ from 0 in xO. Thus, R C d ( x 0 , y )= R C ~ ( X ' , Y ) , Y E g ( J i ) , and then ~ C F N F ( X ' )= ~ C F N F ~ ( X = ' ) g(xO).

Corollary 5.4.1 Let conditions of Theorem 4 be satisfied and, in addition, let g ( x ) be monotonously increasing or decreasing on the whole interval [ a , b]. Then there exist membership functions of fuzzy sets (x),. . ., AN+^ ( x ) , B1 ( Y ) , . . ., B N +(Y) ~ such that N+l

R'~(x,Y) =

/\

(Ai(x)+tBi(y)) i=l realizes ~ C F N F ( Xsuch ) that ~ C F N F ( X=) g ( x ) forevery E [a,b]. Let us stress that the way of proving Theorem 4 opens an interesting combination of fuzzy technique with ordinary numerical methods. In the case where the dependence between inputs and outputs is given by a point-to-point correspondence, any approximation function obtained using numerical methods is easily transformed into a formula of the CFNF type. Concerning the problem of how many representations exist for the given continuous function, we see from the text that not only one way is possible. In this case, it is worth to search for the most efficient (in some sense) way of representation. 5.5

CONCLUSIONS

The ability of fuzzy logic to describe functional dependencies in a transparent way was demonstrated on the example of the control law representation. Three different types of fuzzy logic control models have been considered, namely linguistic, singleton and Takagi-Sugeno ones. Two fuzzy logic normal forms generalizing the well known Boolean disjunctive and conjunctive normal forms have been suggested for uniform algebraic representation of the fuzzy logic control models. The concept of E-completeness has been introduced. By using this concept, an approximation property of a class of real-valued real functions realized by a certain fuzzy logic normal form is characterized. It means that each continuous real-valued real function can be approximated by some function from that class. It has been proved that classes of functions realized by both considered normal forms are &-completewith respect to certain classes of membership functions (Gaussian, triangular, interval). Finally, a theorem stating that each continuous real-valued real function on a compact U can be realized by a special CFNF was proved. Its proof, besides others, demonstrates the useful combination of numerical analysis and fuzzy logic description.

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FUZZY ALGORITHMS FOR CONTROL

Appendix: Proof of Theorem 2

Let n = 1,E > 0 and g(x) be a continuous function on U . Since g(x) is continuous on a compact U c $ it is uniformly continuous on U . Thus, for the given E > 0 there exists a 6 > 0 such that

lg(x) - g(z)I L

5E

if

lx - z1 < 6 for any x , z E U.

Since U is a compact, there exists a finite set of points

such that the following holds:

<

Let us denote yi = g(zi), 1 5 i N. For the given g(x) and E > 0 we will now show how a fuzzy relation RCd(x, y) of the type CFNF can be formed, such that function f C ~ ~ ~realized ( x ) by it (cf. Definition 5.3.1) approximates g(x) up to the given E . We put N

R C d ( x , y= )

A ( A i ( x )+ t B i ( ~ ) ) , i=l

(5.A.2)

where N = (21,A ~ ( x Bi(y) ), E Fc, 1 5 i 5 N , andAi(x) = 1 iff lx-zil 5 6, Bi ( y ) > 0 iff ly - yil 5 5 , t is a continuous t-norm and +t is its residuation. Let us take an arbitrary xo E U and show that

First, we will prove the fact that R ~ ~ ( X Oy ), > 0 i f f I y - g(zio)1 . 1x0 - zioI 5 6, and zio E Z . Substitution xo into (5.A.2)gives

<

5, where

2 2' ~

Suppose that ~

>

~ ~ (y ) 2 00 for , the certain

y. Then,

From the choice of subscript io it follows that Aio(xo) = 1, which together with condition (5.A.4) give us that Bio( y ) > 0 and that

For the values of y : ly - g(zio)l>

5 , or equivalently Bi,(y) = 0, we have

t

FUZZY LOGIC FOR CONTROL

which means that R(xo,y ) = 0. For the values of y : ly - g(zio)l 5

125

5 we have Bio(y) > 0 and so

while for other subscripts i : Ai(x0) = 0 and thus,

Therefore,

R(xo,Y ) = (Aio( X O ) +t Bio ( y ) ) > 0. and the desired fact is proved. The final inequality (5.A.3) now follows from the formula

which in accordance with the proved fact is none other than

Indeed,

References

Buckley, J. (1993). Sugeno type controllers are universal controllers. Fuzzy Sets and Systems, 53:299-304. Castro, J. L. and Delgado, M. (1996). Fuzzy systems with defuzzification are universal approximators. IEEE Trans. Systems, Man, and Cybernetics, 26: 149-152. Klawonn, F. and Nov6k, V. (1996). The relation between inference and interpolation in the framework of fuzzy systems. Fuzzy Sets and Systems, 81:331-354. Kosko, B. (1992). Fuzzy systems as universal approximators. In Proc. IEEE Int. Con$ on Fuzzy Systems, pages 1153-1 162, San Diego. Mamdani, E. and Assilian, S. (1975). An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies, 7 :1-13. Nguen, H. T., Kreinovich, V., and Sirisaengtaksin, 0. (1996). Fuzzy control as a universal control tool. Fuzzy Sets and Systems, 80:71-86. Novik, V. and Perfilieva, I. (1999). Evaluating linguistic expressions and functional fuzzy theories in fuzzy logic. In Zadeh, L. A. and Kacpryk, J., editors, Computing with Words in Systems Analysis. Springer, Heidelberg. Perfilieva, I. (1996). Minimization of the number of rules in fuzzy relational models. In Proc. 2nd Int. FLINS Workshop, pages 29-33, Mol, Belgium.

0

STABILITY ANALYSIS OF FUZZY CONTROL LOOPS A. Ollerol, J.P. Marin2, A. Garcia-Cerezo3 and F. Cuestal

' ~ e de ~ .lngenieria de Sisternas y Autornitica Universidad de Sevilla, Carnino de Los Descubrirnientos s/n E-41092, Sevilla, Spain L.A.A.S. du C.N.R.S. 7, Avenue du Colonel Roche 31077 Toulouse Cedex, France Dep. de lngenieria de Sisternas y AutornStica Universidad de MAlaga, Plaza El Ejido s/n E-29013, MSlaga, Spain

6.1 INTRODUCTION Stability of feedback control systems is a main problem in control theory but also in control engineering. It is well known that the feedback loop is the basic structure to track reference signals and to regulate systems in spite of external perturbations. The feedback structure decreases the sensitivity to parameter variation and external disturbances, however the feedback could strongly affect the stability of the system. Thus, an unstable system can be stabilized by means of an appropriated feedback control loop. On the other hand, an open-loop stable system could be destabilized by means of feedback. The interest of the stability studies in fuzzy control has become

128

FUZZY ALGORITHMS FOR CONTROL

a controversial question in the fuzzy logic literature. However, in many fuzzy logic industrial control applications, the practical interest of stability analysis cannot be questioned due to safety and reliability requirements. In fact, external disturbances and unexpected parameter changes are usually present in practical control system. Then, before efforts are made to satisfy any conventional control system performance related to speed or accuracy, the ability of a system to come to an equilibrium after external or internal disturbances (stability) is needed. Applications in power plants, chemical plants, vehicles, robots and many others require that the feedback control systems will satisfy this property, i.e., will be stable in all the possible working conditions. Furthermore, in many cases, it is very difficult to assess stability only by experimentation in several working conditions. Then, some tools to generalize the analysis and to be able to guarantee the stability of fuzzy control systems are required. This chapter summarizes several existing approaches to solve this problem. The chapter emphasizes stability studies for loops with the structure shown in Fig. 6.1, particularly when ref (t) = 0. In this structure the fuzzy logic controller is a nonlinear system , and the process to be controlled could be represented by a linear or a nonlinear system. Thus, the control loops are nonlinear and then it is difficult to obtain general results useful for the analysis and design. However, the stability of different significant classes of fuzzy control systems can be analyzed and practical experiments have shown the interest of the analysis.

Figure 6.1

I

Fuzzy control loop.

The next section of the chapter introduces some concepts related to the qualitative stability analysis in control engineering practice. Section 6.3 and Section 6.4 present a survey of existing methods for stability analysis in the framework of control theory. Section 6.3 is related to the input-output methods while Section 6.4 concentrates on Lyapunov methods. Finally, conclusions are presented in Section 6.5. 6.2

t

F U Z Z Y CONTROL ENGINEERING PRACTICE A N D STABILITY

Traditionally, fuzzy control has been claimed to be useful in situations where: 1. there is no acceptable mathematical model of the plant, and

2. there are experienced human operators who can satisfactorily control the plant and provide qualitative rules in terms of vague and fuzzy sentences. In fact, fuzzy control has been applied in many cases under these conditions. Furthermore, the fuzzy control methods are also useful in other situations than in l ) and 2). In fact, it is well known that the type of controllers with the structure proposed in (Takagi and Sugeno, 1985) can be derived from experimental data. Thus, even if the

I

STABILITY ANALYSIS OF FUZZY CONTROL LOOPS

129

process human operator is not able to provide the rules, fuzzy identification techniques can be applied to obtain a Takagi-Sugeno fuzzy controller. Many applications of these controllers have been presented in the last 15 years. Even if a conventional differential equation mathematical model exists, the model could be too complex to apply general results from control theory. In these cases, it is possible to benefit from the ability of fuzzy control to provide a flexible tool to design complex nonlinear control laws. Furthermore, fuzzy control makes possible to integrate into the controller functional qualitative knowledge in the shape of understandable rules. Obviously, the disadvantage is the lack of general design methods to assure the control system performance and particularly the stability. Control engineering practice is frequently a combination of well known control theory concepts and judicious heuristic rules on control engineering. Furthermore, the heuristic knowledge associated to the particular process to be controlled and data obtained in experimental results are frequently involved. Stability is usually considered in control engineering as a relative measure associated with the dynamic response of a system. Figure 6.2 shows different responses y ( t ) of systems with the structure shown in Fig. 6.1 for a step reference ref ( t ) = 1. The corresponding degree of stability is different. High potential of instability exists if the response exhibits high overshoot, the decay ratio of the oscillations is high (oscillations not damped), and the rise time is fast. Thus, the system in Fig. 6.2a has greater degree of stability than the system in Fig. 6.2b.

130

FUZZY ALGORITHMS FOR CONTROL

trajectory introduced in (Braae and Rutherford, 1979). This linguistic trajectory is given by the sequence of rules fired in the dynamic response. In (Aracil et al., 1988) a geometric interpretation based on the study of vector fields associated to the plant and to the fuzzy logic controller is given. Different dynamic behaviors (limit cycles, isolated areas, and oscillations) can be identified by the analysis of the trajectories. Then, qualitative conclusions about the stability or instability of the system can be derived. Furthermore, these conclusions can be used to modify the rules in order to obtain a desired dynamic behavior. Example 6.2.1 To illustrate the above ideas consider a regulation fuzzy control loop with the reference signal ref (t) = 0 in Fig. 6.1. Assume a second order process with state variables that measure deviations with respect to the origin (nominal working conditions). It is intended to maintain as much as possible the system at the origin in spite of perturbations. Thus if a perturbation separates the system from the origin, the fuzzy controller should react to drive the system to this point. Consider that the process to be controlled is a second order system described by a transfer function:

G ( s )=

K w : ( l + ac) s2 2c$wns w;

+

+

The controller is a Proportional plus Integral (PI) conventional fuzzy controller having as inputs the error (E) and the integral of the error (IE), and as output the control action u. The control structure is shown in Fig. 6.3a. The rules of the fuzzy controller are given in Fig. 6.3b. Figure 6.4 presents the definition of the fuzzy linguistic terms. Figure 6.5 provides the time response of E and IE and the trajectory in the phase plane (E-IE) for the following parameter values: w , = 1, = 0.05, K = 0.1, a = 2. Notice that the rules are also represented in the phase plane. l.If(IEisIENB)and@isEN)Ulen(UisUPB) 2. If (IE is ENS) and (E is EN) then (U is UPB) 3. If (IE is IEZ) and (E is EN)then (U is UPM) 4.If (IE is IEPS) and (E is EN) thcn (U is UPS) 5. If@ ia IEPB) and (E is EN) then (U is UNM)

6.If(IEisIENB)and@isEZ)thcn(UisUPB) 7.If(IEisENS)and(EisEZ)thcn(UisUPB) S.If(IEisWand@isEZ)thca(UisUZ)

controller

9.If(IEisIEPS)md(EisEZ)then(UisUNB) lO.If(IEisIEPB)and@isEZ)th~11(UisUNB) ll.If(IEisIENB)and(EisEF')thca(UisUPM) 12. If(IE is ENS) and (E is EF') thcn (U is UNS) 13.K(IEisIEZ)and@isEF')then(UisUNB) 14.K(IEisIEPS)and@isEF')then(UisUNB)

15.K(IEisIEPB)and(EisEF')then(UisUNB)

(a) Fuzzy regulation control loop. Figure 6.3

(b) Rules o f the fuzzy controller.

Fuzzy regulation controller (a) and the used rules (b)

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

131

Figure 6.4 Definition of fuzzy sets for the controller in the fuzzy regulation control loop. T h e controller inputs are the error (E) and the integral of the error (IE). T h e controller output is the control action (U).

132

FUZZY ALGORITHMS FOR CONTROL

Time (seconds)

10

5

20

15

25

30

Time (seconds)

Phase Plane

2 15-

1

-

UPM

UPB

-1.5

Figure 6.5

/

I

UNS

I UNB

(

UNB

I

UNB

UPB

l CUPMJ l

UPS

II

UNM

1

Temporal response of E and IE, and the phase plane.

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

133

Time isecondsl

Time lsecondsl Phase Plane

2

I

I i ! UPM , UNE UNB I UNB UNB I \ / i I .......................... 1 ......... I I UNB UPB UPM \ I UNM .........................1. .......... .I....... ...i................. j I j UPB / UPB I UPM 1 UPS / UNM ....................

!. ...............

1 i

..

-2

l 4

-2

,

I

0

, 2

!

* 4

6

Integral of the Error

Figure 6.6 troller.

Temporal responses of E and I E and the phase plane with the modified con-

134

FUZZY ALGORITHMS FOR CONTROL

I

-1 0

20

10

30

40

50

40

50

Time Isecondsl

-

0

10

20 30 Time (seconds\ Phase Plane

UPM

I UNS /

UP0

UP0

-1.5

-2

-6

-4

1

UNB

I UNB I

UNB

UP.

1

UNM

UPS

-2 0 2 lnteoral of the Error

1

4

1 6

Figure 6.7 Temporal responses of E (a) and IE (b) and the phase plane (c) showing a limit cycle.

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

135

The rules define fuzzy partitions in the phase plane that can be identified graphically and associated to the points of the closed loop trajectories. Then, it is possible to modify the consequents according to the desired behavior. Thus, for example, the oscillations can be decreased and the rise time shortened by modifying the consequent of the rules 7 and 9 from Positive Big (UPB) to Positive Medium (UPM) and from Negative Big (UNB) to Negative Medium (UNM) respectively. The resulting phase plane and the temporal response of the error are depicted in Fig. 6.6. It could be also noticed that if the consequences of rules 7 and 9 are changed from "If (IE is IENS) and (E is EZ) then (U is UPB)", and "If (IE is IEPS) and (E is EZ) then (U is UNB)", respectively to "If (IE is IENS) and (E is EZ) then (U is UNB)" and "If (IE is IEPS) and (E is EZ) then (U is UPB)", the resulting fuzzy controller generates a limit cycle behavior shown in the phase plane and temporal response of Fig. 6.7. This simple technique can be used to analyze other nonlinear effects (Aracil et al., 1988). Obviously, the main difficulty of the approach is the constraint on the number of inputs and outputs of the system under study and the dimension (number of state variables) which is required to describe the dynamic behavior. Note that the graphical interpretation is difficult when there are more than two state variables. Stability could be related to local stability around an equilibrium point, such as the origin in the above example, and global stability involving all the space in which the variables associated to the process to be controlled can vary. Many stability analysis techniques are only local stability techniques. The analysis in the state-space mentioned in the previous section could be extended to all the space by considering other equilibria, either stable or unstable, in the state-space and studying the behavior in all the space. This analysis has been formalized using concepts from the qualitative theory of nonlinear systems. Stability and robustness indices were defined using these concepts (Aracil et al., 1989). These indices give measures of how far is the system from the stability loss. Two indices are related to the stability of the origin (local stability) and are computed by using an estimation of the Jacobian of the system around the origin. There is also an index that measures how far is the system from the appearance of new equilibria. The indices have been applied to the design of robust fuzzy systems when a nonlinear model of the process is known (Ollero et al., 1995). In this case the determination of the Jacobian (derivativesof the nonlinear function with respect to the state variables) is easy. However, if the model is not available, the computation of the Jacobian by means of real experiments with the process to be controlled is also possible. That requires intensive measurement of the process variables and eventually the estimation of their increments, which is not always easy. It is also possible to identify a Takagi-Sugeno fuzzy model from the experiments and then design a fuzzy controller for the identified model. The stability of Takagi-Sugeno systems by computing the Jacobian matrix of the model linearized around an equilibrium point is studied in (Kang et al., 1998). In the context of the general stability theory, two general approaches have been used: input-output stability theory and Lyapunov theory. The first one deals with existing relations between the outputs of the system and the external inputs. The second one refers to properties of the internal representation of the system by means of state variables and the effect of perturbations resulting in changes of initial conditions.

136

FUZZY ALGORITHMS FOR CONTROL

From the mathematical point of view, the input-output approach is based on functional analysis and operators rather than on differential equations as the original formulation of the Lyapunov approach is. The following two sections study both the input-output stability and Lyapunov stability. Quite informal mathematical style is used. For precise statements, the reader should refer to the bibliography. 6.3

I

INPUT-OUTPUT METHODS FOR STABILITY ANALYSIS

The basic assumption is that in well-behaved systems bounded inputs should results in bounded outputs and that small changes in inputs should result in small changes in outputs. In the input-output methods the systems can be represented by operators G which produce the output y(t) which corresponds to the input x ( t ) ,or by relations G formed by all the pairs ( x ( t ) y(t)) , where y(t) is a possible output produced by the input x ( t ). The operator G maps an input space U into an output space Y . The concept of stability is based on the properties of U and Y . If a property L of the input is invariant under the transformation G, the system is said to be L-stable. For any fixedp E [I,m ) , f :+ R + R is said to belong to L, if, and only if, f is locally integrable and

When p = GO, f E L,

I

if, and only if,

l l f llw = SUP If t>o

(t)l < m.

Another usual norm is the 2-norm defined as

Ilf

112

=

(1, l f 0

I

( t ) l ~ d t )-

The space Lz represents the set of all finite-energy signals, and the space L, the set of all bounded signals. With the definitions above, the system represented by the operator G is said to be L, stable if u E L, is mapped into y E L,. When p = m , L,-stability is also referred to as Bounded-Input, Bounded-Output (BIBO) stability. 6.3.1 Small gain theorem: circle criterion and conicity criterion A basic result in input-output stability is the small gain theorem. This theorem arises from the concept of the gain g(G) of a system G. This gain is defined as:

where the truncated value of a function f ( t )for a time T is given by fT(t) = f ( t )for 0 5 t T and f ~ ( t = ) OforT < t.

<

I

.

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

137

A system is finite-gain stable when g(G) < oo. The Small Gain Theorem states that a sufficient condition for the stability of the general closed-loop system in Fig. 6.8 is that g ( G ) g ( H )< 1. That is, the product of the gains should be lower than 1.

Figure 6.8

General feedback structure.

The gains can be obtained easily for two classes of systems: nonlinear static systems and linear time invariant systems. Thus, if G is a linear time-invariant representation in the working point of the process to be controlled, its gain can be easily computed by using its frequency response. Furthermore, if H is a nonlinear static function H ( e ) , the gain can be obtained as:

Thus, if the system in Fig. 6.8 consists of a linear time-invariant process G with a static fuzzy controller, the computation of the gains is very easy. If the process is nonlinear it could be also possible to obtain the structure in Fig. 6.8. In this case, H includes the fuzzy controller and the nonlinearities of the process. If H is static the same concepts apply. Consider the system in Fig. 6.9 obtained by adding and subtracting C in the equations describing the system in Fig. 6.8. Notice that in Fig. 6.9 the block F is the feedback configuration of the blocks G and C, and the block N is H-C. It has been demonstrated that the stability conditions of the systems in Fig. 6.8 andFig. 6.9 are equivalent (Desoer and Vidyasagar, 1975). The application of the small gain theorem to the system in Fig. 6.9 leads to the conicity criterion (Safonov, 1980). According to this criterion the system is stable if there is a linear operator C and a positive number r > 0 such that s ( F ) < r and g ( N ) 5 $. The number r is called radius and the operator C is called the center. If H is nonlinear static, G is linear time-invariant, and C is static and linear, the computation of the gains g ( F ) and g ( N ) is easy. Thus, C should be chosen in the Multi-Input Multi-Output (MIMO) case as a constant matrix of appropriate dimensions. In the Single-Input Single-Output (SISO) case C is a scalar. The circle criterion, as proposed in (Ray and Majunder, 1984) for the stability analysis of both SISO and MIMO fuzzy control systems, can be considered as a particular case as shown in (Ollero et al., 1993). Let the block H in Fig. 6.9 be the nonlinear static function H ( e ) . Then, the following sufficient conditions for the stability of the fuzzy control system can be obtained (Aracil et al., 1993; Garcia-Cerezo et al., 1994):

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

139

Example 6.3.1 Consider the system in the Example 6.2.1 with the fuzzy controller in Fig. 6.3b to Fig. 6.4. Notice that this controller has two inputs and one output. It can be shown how the equations (6.3) to (6.5) are satisfied with the center:

In this case and the system is stable. However, when using the fuzzy controller with the rules 7 and 9 changed, it is not possible to find any center C such that r h / r g < 1. The minimum value obtained was:

The method can be used with nonlinear plants but in this case the original system with a nonlinear plant and a nonlinear feedback fuzzy controller is transformed to a system with a linearized component G with a nonlinear feedback including the nonlinear components of the plant and the nonlinear controller. The method has been applied to the design of control systems of robots. Both manipulators and robot vehicles have been considered. Particularly, the stability of fuzzy path tracking strategies of autonomous vehicles has been studied in the framework of the FALCON working group (Ollero et al., 1996). Notice that, in the procedure described in the above paragraph, a conventional differential equation model of the plant is still required for the design of the control system. However, a new procedure to overcome this requirement has been formulated in the framework of FALCON. In this method the input-output data collected in experiments with the plant to be controlled are used to verify the conicity conditions, as shown in (Garcia-Cerezo and Ollero, 1994; Ollero et al., 1996). Consider a nonlinear plant to be controlled described by

where the control signal u = -q5(x) is provided by a fuzzy controller. The system can be written as

where A is the Jacobian o f f (x) and

Then, the closed loop fuzzy control system can be represented by the feedback structure in Fig. 6.8 where (6.10) G(s) = ( s I - A)-'

140

FUZZY ALGORITHMS FOR CONTROL

and ul = u2 = 0. If H in (6.3) is substituted by (6.9), the condition (6.4) can be written as: llA + b4(x) - f (Y)- Cyll < ~ l l ~ l l where $(y) represents the nonlinear fuzzy controller and C is the center of the cone. Consider the stability around the equilibrium x = y = 0 and the following center

where

Then, the condition (6.3) can be rewritten as

Furthermore, (6.4) can be written as

F = GII + b&G]-I

The above expressions provide the basis for the design of stable and robust fuzzy controllers. Thus, in (Garcia-Cerezoand Ollero, 1994), the fuzzy controller is approximated around the equilibriumby a linear state feedbackcontrol law which is computed by applying linear control theory methods as for example pole placement techniques or linear optimal quadratic regulators. These methods assure the stability of the linear closed loop system (6.1 1). Furthermore, the conic robustness can be computed for this linear closed loop system as

The conic deviation is given by

An experimental procedure to compute 4(y) by using the free system response with Y = u2 = 0 is presented in (Garcia-Cerezo and Ollero, 1994). The center of the cone is given by the linear feedback around the equilibrium point. This center approximates the control surface around this equilibrium. Then, the remaining control surface is numerically computed to minimize r h / r g where r h is given by (6.13) and rg by (6.12). Later, fuzzy identification is applied to compute a Takagi-Sugeno system approximating the control surface. This method has been applied to the design of fuzzy control system of robot vehicles (Garcia-Cerezo et al., 1996).

t

141

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

6.3.2 Passivity theorem Another theory that has been applied to the analysis of fuzzy control systems is the Passivity Theorem. The theorem can be formulated for the general feedback system in Fig. 6.8. Let

be the truncated inner product of the two functions f and g. Assume that for the system in Fig. 6.8 there exist real constants such that:

where the subscript T stands for the truncated value and 11x11$, is the La-norm of XT. Then, the system of Fig. 6.8 is La-stable if E + 6 > 0. The intuitive interpretation of these conditions can be found in (Vidyasagar, 1993). Consider, in Fig. 6.8, that ul, z and u are currents, and u2, e and y are voltages. Then, the conditions state that if G has finite gain and dissipates at least as much energy as a resistor of value e (which could be negative), then the system is stable if the overall dissipation constant (which can be thought of as the sum of the "effective resistances" of G and H) is positive. Passivity concepts have been also applied to study the stability of fuzzy control systems. In (Melin and Ruiz, 1997) the stability of linear time-invariant plants with PI (two-inputs, single-output) fuzzy controllers is studied. Using similar block transformations than in Section 6.3.1, a structure with a parallel system and a feedback system similar to Fig. 6.8 is obtained. In this case the parallel system is hl ( s ) = (11K ) HI ( s ) where Hl ( s )is a cascade connection of linear systems composed of a pure integrator, that comes from the PI like structure of the controller, the plant transfer function and a stable filter. The feedback system h2 consist of Hz with the positive feedback ( l / K ) , where H2 is the nonlinear static system that satisfies an input-output real positive relation. Assuming hl ( s ) is a positive real transfer function, then it can be shown (using the Kalman-Yakubovich-Popov lemma) that hl ( s ) is passive. Assume also that hl (s) is proved to be passive with a constant K such that h 2 ( s )is positive real (i.e., K 2 kc). Suppose that the former proposition holds for some K > 0. If the subsystem h2 is positive real with K 2 kc, then the equilibrium point of the feedback system is globally asymptotically stable. A graphical interpretation has been also derived (Melin and Ruiz, 1997).

+

6.3.3 Hyperstability

Consider again the particular case of a linear time invariant system G with a nonlinear feedback H = as shown in Fig. 6.8 with ul = u2 = 0. Notice that the external

142

FUZZY ALGORITHMS FOR CONTROL

disturbances can be interpreted as initial conditions for the state variables. The hyperstability theory deals with the determination of conditions that have to be met for the linear system G to guarantee asymptotic stability. Assuming that a state-space model of G is available, the stability conditions have been also formulated in the time domain. Consider the state-space linear model

where y is a m-vector of process outputs and u is a p-vector of process inputs. It can be seen that:

I = l T u l y d t 5 c2

(6.18)

where the superscript ' stands for the transpose, gives a stability condition. In (Opitz, 1993), the integrand is transformed into a positive definite form. That could be done if the equations of the Kalman-Yakubovich lemma hold. A modified output vector is defined to avoid the limitation of the same number of input and outputs (see expression of the index I).Thus, if the dimension of y (number of process outputs) is m and the dimensions of u is p (number of process inputs) the modified output is

where M is a p x m matrix and N is a p x p matrix. If the nonlinear relation: u = -@(e),where the error signals e are given by e = -y, defines the fuzzy controller, the inequality can be written as:

<

/ ( @ ' N @- GIMe)dt c2

(6.20)

In the SISO case, the integral inequality has been shown to be equivalent to the fuzzy controller inequality: @ ( e )- a M 1 e elRe (6.21)

<

where a > 0 and R is a positive definite matrix. This equation gives a measure for the nonlinearity which can be tolerated in the fuzzy controller feedback. The stability conditions are given by this inequality and the equations of the Kalman-Yakubovich Lemma defined using the matrices A, B, C and D of the internal description of the linear system (Opitz, 1993). An important aspect continues to be the determination of the linear and nonlinear blocks in Fig. 6.8 from the initial nonlinear plant and nonlinear fuzzy controller, particularly if there is not an available model of the process to be controlled.

6.4

LYAPUNOV APPROACHES

6.4.1 Introduction In this section, some techniques to analyze the internal stability of various classes of fuzzy dynamical systems based on Lyapunov methods are presented. It can be shown

I

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

143

that, under some technical assumptions, input-output stability and internal stability are equivalent. Moreover, in the work of (Willems, 1972) the relations between input-output approaches and state-space approach are studied. However, these two approaches do not address the same problem and do not require the same assumptions on the dynamical system. Consequently, input-output stability and internal stability should be considered as complementary tools to study some stability properties of a given dynamical system. Thus, the analogies and differences between the two methodologies will be emphasized in the following. The basic concepts of Lyapunov theory are first presented. Then, various classes of Lyapunov function commonly used in the literature will be presented. Each class is associated with some classes of fuzzy dynamical systems. Computing techniques to check the internal stability of each case are presented. Finally, some remarks and extensions are included. 6.4.2 Basic concepts of Lyapunov theory Although there are numerous definitions of Lyapunov stability, this chapter concentrates on the most popular one: the Global Asymptotic Stability (GAS) of the equilibrium point 0 of a given dynamical system described by the differential equation:

x = f (x),with x(0)= xo where x E Rn is the state of the system and f : Rn -+ Rn. It is assumed that the system (6.22) is time-invariant, autonomous and admits a finite-dimensional statespace representation. All these assumptions are not required in some of the inputoutput approaches. Necessarily, f (0)= 0.The function f describes the dynamic of the system and can be parameterized by a classical algebraic equation and/or a fuzzy knowledge based system (or a combination of both of them). Then, the following definition can be stated: Definition 6.4.1 The equilibrium 0 is Globally Asymptotically Stable (GAS) with respect to the system (6.22) if

Vxo E Rn, x(t) -+ O a s t + co along all the trajectories of the system (6.22). Thus, the global asymptotic stability of the equilibrium point 0 in the sense of Lyapunov is related to an initial internal perturbation on the state vector and requires the state of the system to converge to the (unique) equilibrium point 0. On the contrary, the input-output stability is related to an external perturbation and requires the output of the system to be small enough with respect to the perturbation. In the following, a Lyapunov-function candidate is defined: Definition 6.4.2 V : Rn -+ R is called a Lyapunov-function candidate if V(0)= 0, V is continuously differentiable, V x E Rn - (01,V(x) > 0,3 r > 0,infllzl12r V(x)> 0 and V(x)+ co as llxll + CO. The Global Asymptotic Stability around the equilibrium 0 can be proved by the following result:

144

FUZZY ALGORITHMS FOR CONTROL

Theorem 6.4.1 The equilibrium 0 is GAS with respect to the system (6.22) if exists V, a Lyapunov-function candidate, such that V V ( x ) ' f( x ) < 0, V x E Rn - (0) where V is called a Lyapunov function of the system (6.22). V stands for the gradient column vector. Consequently, proving the equilibrium point 0 is GAS using Lyapunov method consists of finding a Lyapunov function V. Some connections between input-output stability and internal stability can be found in (Vidyasagar, 1993). It seems the most important result can be stated as follows: If a dynamical system is input-output stable and zero-state detectable (i.e., u 0, y 0 implies x ( t ) + 0 as t + co (Hill and Moylan, 1977)), then input-output and internal stability are equivalent. For linear systems, zero-state detectability is equivalent to detectability.

6.4.3 Some classes of Lyapunov function and applications Theoretically, a Lyapunov function may be any Lyapunov-candidate function. Consequently, it belongs to an infinite-dimensional space. Thus, the search of a general Lyapunov function is numerically intractable. In practice, in order to reduce the complexity of the problem, a Lyapunov function belonging to a predefined family of functions is searched. Such families are parameterized by a finite-set of scalar variables. Consequently, the problem turns out to be a numerical optimization one. In the next, some families of Lyapunov functions are presented. For each of them, their properties and the corresponding classes of fuzzy dynamical systems, which can be analyzed, will be discussed.

Quadratic Lyapunov function. The family of quadratic Lyapunov functions has the following form:

Consequently, finding a quadraticLy apunov function is equivalentto finding a symmetric positive definite matrix P . Such Lyapunov functions have two main applications that will be considered in the following:

Stability of Takagi-Sugeno type fuzzy systems. Quadratic Lyapunov function can deal with homogeneous Takagi-Sugeno dynamic models in the following (continuous-time) form: Rule i : If x = Li then x = Aix,

i = 1 . . .N

(6.24)

where Liis a multidimensional fuzzy subset and Ai is a matrix of appropriate dimension. The main result is the following:

+

Theorem 6.4.2 If 3 P = PI > 0 such that A:P PA, < -2aP, a > 0, a E R, i = 1 . . .N , then the equilibrium 0 is GAS with respect the system (6.24). 0 Thus, the stability analysis of such systems reduces to a Linear Matrices Inequalities (LMI) problem (Boyd et al., 1994). It can be shown that, under suitable variable change,

STABILITY ANALYSIS OF FUZZY CONTROL LOOPS

145

the later theorem leads to systematic synthesis procedure for fuzzy state feedback. A similar result applies to discrete-time systems. The main advantage of such approach is the use of advanced linear control theory to solve the synthesis problem and allows to extend it to nonlinear systems. However, it may lead to conservative results since the matrix P must be the same for each rule and the premise part of the rules are ignored. Extensive application can be found in (Tanaka and Sugeno, 1992; Tanaka and Sano, 1994; Tanaka et al., 1996; Zhao et al., 1996; Chen et al., 1993).

Example 6.4.1 Consider an inverted pendulum on a cart. The system is described by the following motion equations:

where xl denotes the angle (in radians) of the pendulum from the vertical position, g = 9.81m/s2is the gravity constant, m = 2kg is the mass of the pendulum, M = 8kg is the mass of the cart, 1 = 0.5m is the length of the pendulum, u is the force applied to the cart and a = l / ( m M ) . The approximated fuzzy model of the pendulum is given by the following Takagi-Sugeno system:

+

+ Blu If xl is E2 then x = A2x + B2u

If x1 is L1 then 2 = Alx

The fuzzy sets L1 and L2 are described in Fig. 6.10. A fuzzy state feedback controller of the form: If xl is L1 then u = Flx If xl is L2 then u = F2x has been designed. It can be shown that the closed-loop system is:

+

If xl is L1 n Z1 then j: = (Al B1Fl)x = Allx If X I is LZ n L2 then 5 = (A2 + B2F2)x= Az2x If X I is L1 n L 2 then j: = (A1 B1F2+ A2 + B2F1)5= A,x

+

Applying Theorem 6.4.2,the stability conditions are given by:

AiiP

+ PAii 5 -2aP,

i = 1 , 2 and AAP

+ P A , 5 -2aP,

P > 0, a > 0

146

FUZZY ALGORITHMS FOR CONTROL

Figure 6.10

Membership functions of L1 and

L2. t

The stability conditions turn out to be a Generalized Eigenvalue Problem when a is maximized (Boyd et al., 1994). Numerical solutions can be computed efficiently using standard algorithms (Gahinet et al., 1995). Computing the matrices F,, i = 1 , 2 by standard pole placement method and choosing the poles of the matrices Aii,i = 1,2 as [-2, -21, the following results are obtained:

Fl = [120.6667 22.66671 and F2 = [2551.6 7641 The stability conditions give:

A, =

[

101.9772 30.7426 20.4726 10.9083

1

and a = 1.1236.

Consequently, the closed-loop system is stable.

Stability of fuzzy Lur'e system. Another class of fuzzy systems, which can be analyzed by using Quadratic Lyapunov functions, is described by:

where 9 is an uncertain nonlinear (possibly time-varying) feedback. CcLis called a Lur'e system. Typically, @ represents the fuzzy controller. Notice that this system corresponds to the general feedback structure in Fig. 6.8 when zq = uz = 0 and u = z. The only information needed on 9 is an input-output property (or sector) of the form: Vt,Vy,u=(P(y,t), u 1 Q u + 2 u 1 S y + g 1 R y ~ 0 . (6.27) The stability problem of the system CCLis called the Lur'e problem. According to the matrices Q, S and R, several absolute stability criteria, which are summarized in Table 6.1, could be obtained.

t

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

Table 6.1

Criteria Small gain Positivity Circle Conicity

147

Absolute stability criteria of the Lur'e system.

Q

Sector

IIuII I IIYII ufy 2 0 (u-Kiy)'(u-Kzy)50 IICY - uII I ~ I I Y I I

In,

0 In,

In,

S

R

O -In,=-&

-I n v

-i(Ki+Kz) -C

&(K:Kz+KkKi) C'C - r21n,

0

Note that the matrices Q, S and R depend on the fuzzy controller parameters (membership functions and approximate reasoning). In general, the computation of Q, S and R for a given fuzzy controller is a difficult task. Preliminary results are given in (Marin and Titli, 1996). The main result is the following:

Theorem 6.4.3 If 3 P = P'

> 0 and E > 0 such that

RC+cI [ A IB'PP + -P A(S'- C+IRD)'C

PB-C1(S'+RD) -Q - D'S' - S D - D'RD

then the equilibrium 0 is GAS with respect to the system

(6.28)

xcL.

The condition (6.28) leads to an explicit formulation of the Lyapunov function. However, in case of minimality (in the sense of linear system theory) of EL,(6.28) can be expressed as an equivalent frequency-domain condition: vw

2 0,

F+G(I~~-A)-'B+B'(-I~~-A)-~G'~B'(-I~W-A')-~H(I~W>0 (6.29) where F = Q + D'RD + SD + D'S', G = SC + D'RC, H = C'RC, j 2 = -1. The frequency-domain approach allows us to easily deal with time-delay systems, providing a graphical interpretation in the Nyquist plane of the condition (6.28) and does not need the solution of an LMI problem to conclude about the stability of the closed-loop system. The equivalence between the condition (6.28) and the latter frequency-domain condition establishes a strong connection between Lyapunov and input-output approaches. The main advantage of the Lur'e problem is the weak assumption (only the inputoutput property (6.27)) required on the nonlinear uncertain part @ (i.e., the fuzzy controller). Thus, this method can deal with neural network controllers, TakagiSugeno controllers (Sugeno, 1985), Mamdani controllers (Mamdani and Assilian, 1974)and accepts various classes of approximate reasoning (Marin and Titli, 1996). However, the poor information on @ can lead to very conservative results. As a matter of fact, if a given controller satisfying the condition (6.27) leads to a stable closedloop system, then any controller satisfying the same input-output property lead also to a stable closed-loop system. Moreover, @ is allowed to be time varying whereas a fuzzy knowledge based system is generally time invariant. This fact increases the

148

FUZZY ALGORITHMS FOR CONTROL

conservatism of the methodology. Applications of this approach can be found in (Kitamura and Jurozumi, 1991; Aracil et al., 1993; Opitz, 1993; Wang and Langari, 1994). Example 6.4.2 Consider a system in the state-space form:

with

(6.3 1) Note that the plant is the same as the plant in the Example 6.2.1 with an additional integral term (11s). The output vector is y = (e,e)'. The first component of the output vector ( e ) corresponds to the integral of the error (IE) in Example 6.2.1, and the second component (e)is the error (E) in Example 6.2.1. The fuzzy logic controller is now based on the Takagi-Sugeno inference and is written in the following form:

+ Kde is En then u = -Kpe - Kdd = -Kpd [e el1 Rule 2: If Kpe + Kde is Z+ then u = -Urn Rule 3: If Kpe + Kde is L- then u = Urn Rule 1: If Kpe

8

where Kpd = [ K p Kd]. The fuzzy sets describing linguistic terms En, L+ and L- are given in Fig. 6.11.

Figure 6.11

Membership functions of

in, L+

and

L-.

The next step is the characterization of the possible sets of matrices (Q, S and R) satisfying the condition (6.28). The results of (Marin and Titli, 1996) can be applied. Using the symmetric nature of the control surface, a set of two Linear Matrices Inequalities can be obtained. Adding the LMI resulting from Theorem 6.4.3, the feasibility of a set of LMIs can be checked. However, in this example, the matrices ( K p d erKpd)and are taken for any value of U,,, and cx : Q = 11, 1 , S = ?j

+

R=~,K~~K~~.If~,=1,~=0.05,K=0.1,a=2,K~=1,K~=4where~,is

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

149

a small positive number. A solution of the stability condition is given by:

5.0022 0.5001 5.0017 0.5001 49.6678 4.1992 5.0017 4.1992 51.3320

1

; with E = low7,6 , = lop5.

(6.32)

It can be concluded that the closed-loop system is globally asymptotically stable. Numerically, the stability is checked for 2 0.435.

<

6.4.4 Lur'e-Postnikov Lyapunov function Another well-known family of Lyapunov function has the following form:

where K is a constant matrix which has to be determined. The closed-loop system under consideration is the Lur'e system presented in Section 6.4.3. This form of Lyapunov function is called the Lur'e-Postnikov Lyapunov function. The Popov criterion can be proved using such a Lyapunov function. Applications to fuzzy control systems can be found in (Katoh et al., 1995; Yarnashitaand Hori, 1991). The matrix K is called the Popov multipliers. Lur'e-Postnikov Lyapunov functions are more general than the quadratic ones. Thus, the results are less conservative and more powerful. However, the use of such a Lyapunov function requires the following (restrictive) assumptions: A1) 9 is time-invariant, A2) dim(u)=dim(y)=n,, A3) 9 ' ( a ) K is the gradient of a real valued function, A4) D = 0 and A5) @'(a)K a 0, Va. In practice, in order to satisfy assumption A3), @ must be diagonal (i.e., the ith component of 9 depends exclusively on the ith component of y). Consequently, it can be assumed without loss of generality that the matrix K is also diagonal. After a possible loop transformation (Vidyasagar, 1993), it can be assumed, without loss of generality, that @(y)'(@(y)- y1) 5 0 or equivalently: Q = I, S = + I ,R = 0. Then, the following result follows:

>

Theorem6.4.4 If 3 P = P' 1 , . . . , np, such that

[

+

> 0, 6 > 0, K = diag(ki) > 0, T = diag(ti) 2 0, i =

+

+

P B - C'(S1 RD) A'P P A - C'RC €1 -Q - D'S' - S D - D'RD B ' P - (S' RD)'C

+

]5

0

(6.34)

then the equilibrium 0 is GAS with respect to CCL. In (6.34), the elements of K , ki, are called the multipliers and the elements of T, ti,are called the scaling parameters. In the SISO case, it is possible to obtain an equivalent frequency-domain condition and derive a graphical test to check condition of Theorem 6.4.4 without solving an LMI problem or computing the Lyapunov function. The philosophy of the Lur'ePostnikov function is to use the fact that is time-invariant. The main advantage of Lur'e-Postnikov Lyapunov function is the reduced conservatism with respect to the

150

F U Z Z Y ALGORITHMS FOR CONTROL

quadratic Lyapunov function. However, the assumptions A2) - A3) are very restrictive and rarely satisfied in practical situations. In order to circumvent this difficulty, a new stability criterion has been proposed, which bridges the gap between the circle and the Popov criteria (Furutani et al., 1992). Example 6.4.3 The same system as in Example 6.4.2 is considered. Using the particular structure of the fuzzy controller and defining a new variable y* = Kpe + Kde, the closed-loop system can be described by x = A x B u , y* = C * x + D u with u = +*(y*). The matrices A , B and D are unchanged. Furthermore C* = - K C . The fuzzy logic controller is described by:

I

+

Rule 1: If y* is Ln then u = y*

=

Rule 2: If y* is L+ then u = 1

The fuzzy sets i n , L+ and i- are unchanged. It can be observed that all assumptions required for the application of Theorem 6.4.4 are satisfied. Stability of the closed-loop can be proven via Theorem 6.4.4 for > 0. With unchanged parameters except for [ = 0.001, the following results are obtained:

<

P=

[

10.0000 0.0200 10.0000 0.0200 124.3005 0.1458 10.0000 0.1458 134.3082

1

I

; with tl = 1 and kl = 0.8851.

Multi-quadratic Lyapunov function. The family of multi-quadratic Lyapunov functions has the following form: If x E Li then V ( x ) = xlPix, i = 1 , . . . , N

I

where Li is a crisp set such that nzi = 0 , uLi = Rn and Pi are positive definite matrices. The family of multi-quadratic Lyapunov function is not continuously differentiable. Consequently, a rigorous mathematical treatment requires the notion of a generalized dynamic system (Roxin, 1969) and is irrelevant in this discussion. In the following, the principal ideas and no the precise mathematical results of such an approach will be presented. The model of the open-loop system is assumed to be written in a homogeneous Takagi-Sugenoform: Rule i : If x E

Li then x = Aix + Biu,

i = 1,. . . ,N ,

(6.35)

where Li is a fuzzy set. Then, system (6.35) is transformed into the following form:

Ei then x = ( A , + AA,)x

+ ( B , + A B i ) u , i = 1,. . . ,N (6.36) where L, is a crisp set defined by x E L, @ p1;. ( x ) > p1;, (x),j # i and pL, ( x )stands Rule i : If x E

for the membership function of the fuzzy set L,. A A , and AB, are uncertainty matrices

I

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

151

computed by taking into account the overlaps between the fuzzy sets. Consequently, the fuzzy system (6.35) is converted into a family of local linear uncertain linear systems. Each linear system is valid in the given crisp region. Note that the systems (6.35) and (6.36) are not equivalent. More precisely, all possible trajectories of (6.36) contain all possible trajectories of (6.35) but the converse is not true. The next step is the use of advanced linear control theory to compute independently a family of linear state feedback matrices Ki, ensuring the robust quadratic asymptotic stability of each local linear uncertain system (proven by the family of quadratic Lyapunov functions Vi(x) = xlPix). The global (discontinuous) control law is defined by: Rule i : If a: E J?; then u = Kix, i = 1,.. . , N

(6.37)

Then, the following result can be obtained:

Theorem 6.4.5 If A l ) the number of traversing time instants of the closed-loop system among the region Li is finite, and A2) the closed-loop system stay in the ith region for a period t > 0, then the point xo is GAS with respect the system (6.36) (and, consequently, the system (6.35)) under the control law (6.37). The use of multi-quadratic Lyapunov function seems to be attractive since it reduces the analysis of a global nonlinear system to the analysis of independent linear local model. However, the method suffers from the following drawback: the assumption A l ) implies that the non-existence of a stable-limit cycle of the closed loop system has to be a priori assumed and the assumption A2) implies the a priori non existence of sliding hypersulface. Moreover, the control law is discontinuous. Application of such methodology can be found in (Cao et al., 1996a; Cao et al., 1996b).

Piece-wise linear Lyapunov function. The piece-wise linear Lyapunov functions can be divided into the following groups:

where F(x)= 0 if x is negative and F(x)= 1 if x is positive. Hjand cj are matrices and vectors to be determined. In this approach, the fuzzy control law must be = Kjx dj for x E Pj approximated by a polytop-wise affine form: u(x)= u~(x) where Pjis a polytop. Then, a numerical optimization procedure allows to determine the free parameters of the Lyapunov function (Kiendl and Ruger, 1995). Such an approach is possibly less conservative since the class of Lyapunov functions which can be approximated by the form (6.35) seems less restricted than other families of Lyapunov functions. Note that, while with the multi-quadratic Lyapunov function, no assumption is needed on the existence of sliding mode andlor stable limit cycle.

+

where Ai > 0 are parameters to be determined. This parameterization of the Lyapunov function has been suggested by (Langari and Tomizuka, 1990), and (Myszkorowski

152

FUZZY ALGORITHMS FOR CONTROL

and Longchamp, 1993). The use of such Lyapunov function applies to discrete-time linear uncertain systems (possibly) associated with fuzzy controllers. It seems to lead to conservative results and does not present any advantage with respect the use of Quadratic Lyapunov functions.

6.4.5 Remarks and extensions

t

Fuzzy stability criterion. One of the main characteristic of the presented Lyapunov approaches is that they address the problem of global stability (i.e., stability in the large) of an equilibrium point. In practical situations, such a requirement could be neither necessary nor sufficient. As a matter of fact, an acceptable dynarnical behavior can be characterized by: After an acceptable (i.e., not too big) perturbation on the initial state, the fuuy system must converge, with an sufJiciently high degree of possibility and fast enough, to a suficiently small region close to the origin. From the Lyapunov theory point of view, this means that an invariant, attractive and sufficiently small domain M C IWn around the origin and a corresponding sufficiently large attraction domain A C IW" have to be determined. The characterization of the convergence rate can be achieved, at no computational cost, using the concept of exponential stability (i.e., replacing the inequality V V ( x ) ' f( x ) < 0 in Theorem 6.4.1 by V V ( x ) ' f( x ) 5 € V ( X )€, > 0 Since, it can be shown (under some mild assumptions) that a fuzzy system is equivalent to a family of crisp uncertain nonlinear systems, it is possible to compute the domains M and A and the convergence rate e for each crisp uncertain nonlinear system using classical Lyapunov theory. Consequently, a family of crisp sets M, A and the convergence rate E can be obtained. These families define a fuzzy invariant and attractive domain M , the corresponding attraction domain A and the fuzzy convergencerate E. In this sense, the latter approach provides a fuzzy stability criterion. A more detailed theoretical discussion can be found in (Deglas, 1993; Deglas, 1984). Note also that (Kiska et al., 1985) used a closely related notion to Lyapunov theory to study stability of fuzzy linguistic systems. Preliminary results on the computation of the fuzzy domain, and convergence rate are presented in (Marin and Titli, 1997a; Marin and Titli, 1997b).

!

'

Performance criterion. So far, Lyapunov theory has been used to address the problem of the stability of an autonomous system with fuzzy control feedback. In practical situations, such property may be insufficient since the system may be affected by external perturbations. Generally, the requirement for an acceptable dynamical behavior of the system is that the external perturbation p affects to a small extent a given output error measurement e. The model of the system becomes x = f ( x , ~ ) , e = g(x,p),x(0) = xo. This problem is called the disturbance rejection problem.

A is a subset defined by its membership function pA(.) 5 a 2 , where Aa stands for the a-cut of A.

lIn fact a1

-+ [ O , 11 and satisfying Aal c

Aaz,

t

4

STABILITY ANALYSIS O F FUZZY CONTROL LOOPS

153

If a given signal s(t) is measured by its L2 norm (i.e., llsll$ = ~ ( t ) ~ d tthe ), performance can be characterized by the L2 gain. As pointed out in Section 6.3, it is said that the system has an L:! gain a if llell 5 allpll for all admissible perturbations p and xo = 0. Such a problem can be addressed by using the Lyapunov approach. As a matter of fact, it can be shown that if there exists a Lyapunov-candidate function satisfying: VV(x)'f (x,p) lle1I2 - a211P112 5 0, then the system has an Lg gain a (Van der Schaft, 1996). Consequently, all techniques presented in this section to prove global asymptotic stability of an equilibrium point can be directly extended to characterize the L2 gain of a given fuzzy system (Marin and Titli, 1996; Marin, 1996; Cao eta]., 1996a).

+

The necessity of local stability for the global one. The methodologies presented so far, based on quadratic or multi-quadratic Lyapunov functions, can be applied to study the stability of homogeneous Takagi-Sugeno systems through the analysis of the (quadratic) stability of each local linear systems to conclude on the stability of the overall system. This is due to the fact that the premise part of the rules is not used in the stability analysis. Recently, it has been proven that the stability of the local system is not necessary for the stability of the global system (Kim et al., 1995). Moreover, the use of the premise part of the rules in the stability analysis allows us to deal with non-homogeneous Takagi-Sugeno models (i.e., the conclusions of the rules are: x = A i x + bi where bi is a vector) (Marin and Titli, 1995; Marin, 1995).

6.5

CONCLUSIONS AND PERSPECTIVES

The stability analysis of fuzzy control systems is a topic that has attracted the attention of an increasing number of researchers in the control theory and mathematical system theory domains. This topic has a great practical interest in many applications in which safety and reliability are critical issues and where stability testing using only experimentation will not suffice to guarantee the stability property in all possible working conditions. Some concepts related to fuzzy control engineering practice and stability have been reviewed in the second section of this chapter. The techniques considered in this section are based on studying temporal and phase-plane responses. Section 6.3 introduced several different methods for input-output stability. The small gain theorem leads to the circle criterion and the conicity criterion. Sufficient stability conditions by applying the conicity criterion have been presented. The passivity theorem and the hyperstability concepts have also been introduced. Several techniques to study stability properties of fuzzy dynamical systems based on Lyapunov methods have been considered in Section 4. Despite recent important advances, this topic is far from being complete. It seems that the (multi-) quadratic approach associated with Takagi-Sugeno systems is a very promising one, but it needs future investigations to reduce the conservatism, to allow a better treatment of the assumption A1-2) in Theorem 6.4.5, and to integrate a fuzzy stability criterion. If the final objective of the analysis methods is the synthesis procedure, the formulation of the problem by the way of Linear Matrix Inequalities is a convenient and flexible

154

FUZZY ALGORITHMS FOR CONTROL

formalism, as it allows the search of the fuzzy controller in a direct form. The synthesis of fuzzy state feedback is now well known when the premise part of the open-loop system depends only on the state. The cases of fuzzy output feedback, fuzzy observers and control of fuzzy systems with premise part of the rule depending on the control vector remain an open challenge. It is interesting to highlight the information required for the application of the stability analysis approaches. In fact several stability measures are based on the assumption that a precise mathematical model of the system under consideration is known. In practice, such a mathematical model is often unknown or its determination is too expensive. In fact, for many authors the assumption of the mathematical model of the plant is being known contradicts one of the arguments supporting the application of fuzzy control systems, i.e., to control processes that are poorly understood from the mathematical point of view. However, new methods combining modeling techniques from input-output data and expert rule information can provide the models required for the stability analysis. Fuzzy identification techniques can be applied to obtain Takagi-Sugeno systems. In recent years, several stability studies of Takagi-Sugeno systems have been presented. This chapter introduces the application of quadratic and multi-quadratic Lyapunov functions, as mentioned above. Recently, input-output stability studies and frequency response methods to detect limit cycles of TakagiSugeno systems have been also presented. Furthermore, more research efforts are need to extend stability results to MIMO systems and to produce new practical methods for stability analysis and design of these systems.

t

4

Acknowledgments

The authors would like to thank K.E. ArzCn and A. Titli for their valuable comments and corrections of this chapter. References

Aracil, J., Garcia-Cerezo, A., Barreiro, A., and Ollero, A. (1993). Stability analysis of fuzzy control systems based on the conicity criterion. In Lowen, R., editor, Fuuy Logic: State of the art, System Engineering. Kluwer Academic Publisher. Aracil, J., Garcia-Cerezo, A., and Ollero, A. (1988). Stability of fuzzy control systems: A geometrical approach. In Kulikowski C.A., R. and Ferrate, G. A., editors, Artzjicial Intelligence, Expert Systems and Languages in Modelling and Simulation. Elsevier Science Publisher B.V., North Holland. Aracil, J., Ollero, A., and Garcia-Cerezo, A. (1989). Stability indices for the global analysis of expert control systems. IEEE Transactions on System Man and Cybernetics, SMC 19(5):988-1007. Boyd, S., Gahoui, L. E., Feron, E., and Balakrishnan, V. (1994). Linear matrices inequalities in systems and control theory. SIAM Studies in Applied Mathematics, 15. Braae, M. and Rutherford, D. (1979). Theoretical and linguistic aspects of the fuzzy logic controller. Automatica, 15553-577.

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Cao, S., Rees, N., and Feng, G. (1996a). Analysis and design of uncertain fuzzy control systems, part 11: Fuzzy controller design. In FUZZ-IEEE'96, pages 647-653, NewOrleans. Cao, S., Rees, N., and Feng, G. (1996b). Stability analysis and design for a class of continuous-time fuzzy control systems. Int. J. Control, 64(6):1069-1087. Chen, C.-L., Chen, P.-C., and Chen, C.-K. (1993). Analysis and design of fuzzy control system. Fuuy Sets and Systems, 57: 125-1 40. Deglas, M. (1984). Invariance and stability of fuzzy systems. Journal of Mathematical Analysis and Application, 99:299-319. Deglas, M. (1993). A mathematical theory of fuzzy systems. In Gupta, M. and Sanchez, E., editors, Information and Decision Processes. North-Holand Publishing Company. Desoer, C. and Vidyasagar, M. (1975). Feedback Systems. Input-Output Properties. Academic Press. Furutani, E., Saeki, M., and Araki, M. (1992). Shifted popov criterion and stability analysis of fuzzy control systems. In ACCJ92,pages 2790-2795, Tuscon, Arizona. Gahinet, P., Nemirovski, A., Laub, A. J., and Chilali, M. (1995). LMI Control Toolbox, For Use with MATLAB. The MATHWORK Inc. Garcia-Cerezo, A. and Ollero, A. (1994). Design of fuzzy control systems from experimental data. In EUFITJ94,volume 3, pages 1175-1 182. Garcia-Cerezo, A., Ollero, A., and Aracil, J. (1994). Dynamic analysis of fuzzy logic control structures. In Kandel, A. and Langholz, G., editors, Fuzzy Control Systems, pages 141-160. CRC Press. Garcia-Cerezo, A., Ollero, A., and Martinez, J. (1996). Design of a robust highperformance fuzzy path tracker for autonomous vehicles. International Journal of Systems Science, 27(8):799-806. Hill, D. and Moylan, P. J. (1977). Stability results for nonlinear feedback systems. Automatics, 13:377-382. Kang, G., Lee, W., and Sugeno, M. (1998). Stability analysis of TSK fuzzy systems. In FUZZ-IEEE'98, pages 555-560. Katoh, R., Yamashita, T., and Singh, S. (1995). Stability analysis of control system having PD type of fuzzy controller. Fuzzy Sets and Systems, 74:321-334. Kiendl, H . and Ruger, J. (1995). Stability analysis of fuzzy control system using facet functions. Fuzzy Sets and Systems, 70:275-285. Kim, W., Ahn, C., and Kwon, W. (1995). Stability analysis and stabilization of fuzzy state space models. Fuzzy Sets and Systems, 7 1 :131-142. Kiska, J., Gupta, M., and Nikiforuk, P. (1985). Energistic stability of fuzzy dynamic systems. IEEE Trans. on Systems, Man and Cybernetics, 5(15):783-792. Kitamura, S. and Jurozumi, T. (1991). Extended circle criterion and stability analysis of fuzzy control system. In IFES'91, pages 634-643. Langari, G . and Tomizuka, M. (1990). Stability of fuzzy linguistic control systems. In 29th CDC'90, pages 21 85-2190, Honolulu. Mamdani, E. and Assilian, S. (1974). Application of fuzzy algorithms for control of simple dynamic plant. In IEE'74, volume 121, pages 1585-1588.

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Marin, J. P. (1995). Conditions necessaires et suffisantes de stabilite quadratique d'une classe de sys3mes flous. In LFA'95, pages 240-247, Paris. Marin, J. P. (1996). H, performance analysis of fuzzy system using quadratic storage function. In AADECA'96, pages 7-1 1, Buenos-Aires. Marin, J. P. and Titli, A. (1995). Necessary and sufficient conditions for quadratic stability of a class of Tagaki-Sugeno fuzzy systems. In EUFIT'95, pages 786-790, Aachen. Marin, J. P. and Titli, A. (1996). Robust performances of closed-loop fuzzy systems: A global Lyapuov approach. In FUZZ-IEEE'96, pages 732-737, New-Orleans. Marin, J . P. and Titli, A. (1997a). Fuzzy stability analysis of fuzzy systems, part I : Quadratic parametrization of fuzzy systems. In EUFIT'97, volume 2, pages 12941299, Aachen. Marin, J. P. and Titli, A. (1997b). Fuzzy stability analysis of fuzzy systems, part 11: Fuzzification of Lyapunov theory and application. In EUFIT'97, volume 2, pages 1300-1305, Aachen. Melin, C. and Ruiz, F. (1997). A passivity framework for fuzzy control system stability: case of two-input-single-output fuzzy controller. In 6th International Conference on Fuzzy Systems, pages 367-370, Barcelona (Spain). Myszkorowski,P. and Longchamp, R. (1993). On the stability of fuzzy control systems. In 32th CDC'93, pages 1751-1752, San Antonio. Ollero, A., Garcia-Cerezo,A,, and Aracil, J. (1995). Design of robust rule-based fuzzy controllers. Fuuy Sets and Systems, 70:249-273. Ollero, A., Garcia-Cerezo, A., Aracil, J., and Barreiro, A. (1993). Stability of fuzzy control systems. In Driankov, D., editor, An Introduction to F u q Control, pages 245-292. Springer-Verlag. Ollero, A., Garcia-Cerezo,A., and Martinez, J. (1996). Design of fuzzy logic controllers from heuristic knowledge and experimental data. In XIII IFAC World Congress, volume A, pages 433-438. Opitz, H. (1993). Fuzzy control and stability criteria. In EUFIT'93, pages 130-136, Aachen, Germany. Ray, S. and Majunder, D. (1984). Application of circle criteria for stability analysis of linear SISO and MIMO system associated with fuzzy logic controllers. IEEE Transaction on Systems, Man, and Cybernetics, SMC-142:345-349. Roxin, E. (1969). Stability in general control system. J. Di~erentialEq, 1:115-150. Safonov, M. (1980). Stability and Robustness of Multivariable Feedback Systems. MIT Press, Cambridge, MA. Sugeno, M. (1985). An introductory survey on fuzzy control. Information Sciences, 3659-83. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its application to modelling and control. IEEE Transactions on Systems, Man, and Cybernetics, 1 5 116-132. Tanaka, K., Ikeda, K., and Wang, H. (1996). Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: Quadratic stabilizability, h, control theory and linear matrix inequalities. IEEE Trans on Fuzzy Systems, 4(1): 1-13.

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Tanaka, K. and Sano, N. (1994). Robust stabilization problem of fuzzy control systems and its application to backing up truck trailer. IEEE Trans on Fuzzy Systems, 2(2): 119-134. Tanaka, K. and Sugeno, M. (1992). Stability analysis and design of fuzzy control system. Fuzzy Sets and Systems, 45: 135-156. Van der Schaft, A. (1996). L2-gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and information Sciences, Vol218. Springer. Vidyasagar, M. (1993). Nonlinear Systems Analysis. Second edition. Prentice Hall, New Jersey, Englewood Cliffs. Wang, L. and Langari, R. (1994). Fuzzy controller design via hyperstability theory. In FUZZ-IEEE'94, pages 178-182, Orlando. Willems, J. (1972). Dissipative dynamical systems, part I: General theory. 22 Arch. Rational Mech. Anal., 145:321-35 1 . Yamashita, Y. and Hori, T. (1991). Stability analysis of fuzzy control system. In IECON'91, pages 1579-1584, Kobe. Zhao, J., Wertz, V., and Gorez, R. (1996). Fuzzy gain scheduling based on fuzzy models. In FUZZ-IEEEJ96,pages 1670-1676, New-Orleans.

/

PERFORMANCE CRITERIA: CLASSICAL AND FUZZY DESIGN J.M. Sousa*, Uzay Kaymakt and Henk B. Verbruggen

Delft University of Technology Faculty of Information Technology and Systems Control Laboratory Mekelweg 4, PO Box 5031 2600 GA Delft, The Netherlands

7.1

INTRODUCTION

The design of a control system generally involves different steps. First, the system to be controlled is studied in order to decide about the types of sensors and actuators to be used and their proper insertion in the system. Second, a model of the resulting system is derived using first principles or an identification procedure. The identification of a system involves usually the simplification and validation of the obtained model. Once the model is derived, the design specifications must be established, and the controller meeting the desired specificationscan be designed. With the model and the controller, the resulting control system can be simulated and implemented (Doyle et al., 1992). Figure 7.1 shows a general feedback control scheme for multivariable systems. Various input signals (actions) influence the process P resulting in output variables y.

'J.M. Sousa was on leave from Technical University of Lisbon, Faculty of Mechanical Engineering, GCARIIDMEC, Portugal. t ~Kaymak . is currently with Shell International, Exploration and Production, Rijswijk, The Netherlands.

160

FUZZY ALGORITHMS FOR CONTROL

-+ d1 V goals

r

d

P

Y

Ym

S

dm

4

+

I Figure 7.1

Block diagram of a feedback control scheme.

The input variables are usually divided in control actions or manipulated variables u and system disturbances d, which can not be influenced before entering the process. The goals to be achieved are imposed on the controller (indicated by the double arrow in Fig. 7.1), such that the system under control achieves the desired specifications. Note that the reference r can also be seen as a goal to be achieved by the control system. The plant under control and the actuators manipulated by u are included in the process. The sensors are represented by the operator S, having as inputs, the output variables from the process y and the measurement disturbances dm,generating the measured outputs y,. The controller C generates the control actions u based on the received information: the measured outputs y,, the references r to be followed, the disturbances d and dmif available, and the goals to be obtained. The purpose of a control system is to keep the values of the output variables y as close as possible to the respective target values r by manipulating the control actions u, and to minimize the effects of the disturbances d and dmon the controlled system. Sometimes this goal is translated to a cost function, where a term related to the control effort can be added. The objective of a control system must be accomplished taking into account the dynamic behavior of the process P, and the static constraints such as the flow rate in a tube that has its maximum value when a valve before that tube is fully open. Control problems are usually divided in two main categories.

4

4

1 . Regulation - In this type of problems, the controller, usually called a regulator,

should keep the system on an operating point or setpoint. 2. Tracking - The controller is designed such that the system should follow a predefined and time-varying trajectory or reference.

For both problems design specifications are usually the translation of the main goal in control design, i.e., the outputs y should be as close as possible to respective prespecified references r, suppressing the influence of the disturbances. This goal must be accomplished despite the fact that u or its change are limited due to some constraints present in the system. Several examples of constraints in control actions can be given, e.g., the flow rate has its maximum value when a valve is fully open, or the opening of a valve should be kept small to save energy.

4

PERFORMANCE CRITERIA

161

Design specifications fulfilling the design goals and objectives for the controller design must be specified. Usually, three main objectives are required: stability for the overall system, performance regarding the accuracy and the speed of the system's response and robustness to disturbances and dynamics which are not modeled. The design specifications are often combined in an optimal control problem, where several design criteria can be aggregated using different approaches. Design specifications are discussed in Section 7.2, where systematic approaches for designing linear control systems are presented, and general procedures for deriving controllers in the presence of nonlinear systems are discussed. Particular attention is devoted to performance specifications, because they are directly related to the general control objectives. Classical performance specifications are presented in Section 7.3. Performance specifications are formalized in performance criteria. These performance criteria are expressed by the size of certain signals of interest. There are different ways of defining the size of a signal, given by different norms or semi-norms for signals. An overview of the classical performance criteria using norms and semi-norms of signals and/or systems for defining the performance criteria is given in Section 7.4. Performance criteria, which are based on norms or semi-norms, are used for specifying classical design specifications. Sometimes, however, it is preferable to define informal design goals such as "the step response from the reference signal to the output should not overshoot too much" or "the sensor noise should not cause u to be too large", which may better describe the control goals. These type of control goals can be formally translated to performance criteria using the theory of fuzzy sets. The main goal of this chapter is then to explain how the theory of fuzzy sets can be used for formulating design specifications in terms of fuzzy performance criteria. These criteria are presented in Section 7.5. Using fuzzy sets for defining the goals of a control system, the concept of performance criteria is generalized. Fuzzy performance criteria are presented in Section 7.5, and can be seen as fuzzy goals and fuzzy constraints in a fuzzy multicriteria decision making environment. This approach can translate the objectives and constraints derived from the control design goals of a given system in a transparent way. In the control design, the decision goals and the constraints are defined on relevant system variables. The fuzzy decision making algorithm uses a process model to select the control actions that best meet the specifications. As the controller derived using fuzzy goals and constraints is a nonlinear controller, it is more able to derive a controller that is closer to the constraints, resulting in a better performance of the system. Note that the confluence of fuzzy criteria (goals and constraints) can be translated as a generalization of the objective function used in model-based predictive control. Section 7.6 describes the use of fuzzy performance criteria in model-based predictive control (MBPC). The chapter concludes with a summary of the state-of-the-art of control performance criteria, in classical and fuzzy terms, and presents suggestions for future lines of research in the field.

162

7.2

FUZZY ALGORITHMS FOR CONTROL

DESIGN SPECIFICATIONS

In general, the design goals, also called design objectives for controller design, are expressed by design specijications. These can have different forms, and are usually related to the architectures or configurations of the respective control systems. As an example, consider, for instance, an air-conditioning system, where the global design goal can be stated as obtaining and maintaining 'human comfort'. This goal must be translated in terms of temperature, humidity, ventilation and noise. Stating, for instance, that "the temperature should be around 20 "C"is already a design specification, because the control goal is specified for a certain variable. The simplest example is to consider just one goal, such as the minimization of the error between a given reference and the output(s) of the system. In this case, a single cost or objective function is optimized. Often, however, several goals are simultaneously considered and a multicriteria optimization approach must be applied, where the controller must perform mutually well on all these goals. Several criteria can also be combined in a single cost function as in the optimal control paradigm. A clear distinction between design specijication and design criterion is usually not made in control design, and especially for linear time-invariant systems both terms are used interchangeably. In this chapter the term design specijications is reserved for the imprecise design goals and objectives required by the control designer for the variables under control, and the term design criterion is used for the formal or mathematical description of the design specifications. The design specification stated as "the overshoot must be small", for instance, is translated to the precise design criterion: "the overshoot q50s must be smaller than 5%" (see the definition of overshoot in Eq. (7.7), Section 7.3). The rest of this section presents design specifications and criteria. The possible combination of design criteria for the design of controllers is discussed. First, generally used approaches of control design specifications for linear systems are presented. A generalization to nonlinear systems is discussed afterwards.

t

4

7.2.1 Design specifications for linear systems t

For linear systems, the design specifications can be translated to design criteria in a systematic way. The design criteria can be specified either in time or in frequency domain. Quantitative specifications of the closed loop system are established, and a controller meeting these specifications can be designed. The first problem posed is the feasibility problem, i.e. determining whether all design specifications can be simultaneously satisfied. A design specification is translated to a design criterion Ji, which depends on different variables of the system, such as the control actions u, the outputs of the system y, the states x, the disturbances d, etc. When the design criteria are defined for different variables, all of them must be satisfied, i.e. the problem must be feasible. This approach is sometimes called multicnteria optimization (Boyd and Barret, 1991). In this approach, a tradeoff between the separate parts of the criteria is made in order to find the possible solutions. Note however, that there is no ordering or priority among the design criteria in this approach. Therefore, several solutions can be obtained.

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163

Another approach that is more often utilized is optimal control. In this approach all the design criteria Jimust be translated to functions of only one variable, usually the control actions u. Moreover, an ordering of the several criteria must be given, and a unique solution of the optimal problem is obtained. Let v be a general variable under optimization. Each design criterion is thus translated to a function of this variable represented by Ji(v).The combination of all the design criteria is given by the cost (objective) function J(v)

where L is the number of criteria defined, and f is a function that combines the criteria. The two most used methods to combine criteria, the weighted-sum and the weightedmax, follow the formal definition of optimal control, briefly discussed in the following paragraph.

Optimal control problem. The general form of an optimization problem, usually known as nonlinear constrained optimization problem is defined as min

J(v)

vEV

subject to

C i ( v ) L O , i = 1 , 2 ,..., Nc;

(7.2)

where the objective function J(v) is defined as before, the constraint functions Ci(v) are real-valued scalar functions, v E V, and Nc is the number of constraints. The constraints in a system can be present for the control actions u ,the state variables x, the outputs of the system y, or the changes in these variables. Note that all constraints in the optimal control formulation must be expressed by the constraint functions Ciof the chosen optimization variable v. As an example, let the variable under optimization be the control actions, i.e., A

v = u. Let the design criterion be given by the error between the desired reference and the predicted outputs using the model of the system: J(u) = r - y. Considering a regulation problem, the references are constant. Thus, in this case it is sufficient to have a function relating y to u: Y = f(u), (7.3) in order to solve the optimization problem. As this function is actually a part of the model of the system, this problem is quite trivial. Unfortunately, the formulation of an optimal control problem is not always so simple. The constraints considered in Ci(v)are usually known as 'hard' constraints, contrary to the 'soft' constraints that can also be represented as additional design criteria Ji(v).This terminology of 'hard' and 'soft' constraints is generalized for fuzzy constraints in Section 7.5. Each criterion Ji has an optimal value, if only that specific criterion is considered. Therefore, a trade-off between the several design criteria for a suitable design of a control system is "searched". Thus, the specification of J(v) determines the trade-off between the several criteria. This is generally done interactively, often by repeatedly adjusting the weights in a weighted-sum or weighted-mar objective and evaluating the resulting optimal design. These two methods to combine design criteria are presented in the next paragraphs.

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Weighted-Sum Objective. A common method to combine the individual goals translated in a design criterion Ji is to add all of them, after they have been multiplied by non-negative weights Xi

The weights assign relative values among the functionals Ji.The objective function of the form (7.4) is called a weighted-sum objective. A typical example of the application of the weighted-sum objective is in model-based predictive control, where the sumsquared error added to a term minimizing the control effort is usually used as the cost function (see Section 7.6.1).

Weighted-Max Objective. Another approach called minimax design, is to form the objective function as the maximum of the weighted functions

where Xi are non-negative weights as before. The weights are meant to express the designer's preference among the criteria, just as in the weighted-sum objective. The combination of the several criteria for both methods of constructingthe objective function is usually chosen in such a way that they lead to closed-loop convex constraints. If J(v) is a convex function and the constraints are convex, the optimization is a convex programming problem (Gill et al., 1981), which is known to have efficient numerical solutions. Therefore, only convex problems are usually considered in the classical approach, even if the system under optimization is nonlinear. As a final remark, note that even for linear systems it can be quite complex to define the required design specifications. Moreover, the translation of them to design criteria is usually difficult or sometimes even impossible. Even when this stage is achieved, i.e. the design criteria are all defined, it is still necessary to choose a method to combine them, and choose the respective weights for the different criteria.

a

t

7.2.2 Design specifications for nonlinear systems The procedure for designing linear systems described in the previous section can be applied to nonlinear systems only in the time domain. In general, a response of a nonlinear system to a specific input signal does not reflect its response to a different input signal. Therefore, a description in the frequency domain is not adequate for this type of systems. In general, it is possible to look for some qualitative design specifications in the operating region of interest. For any type of system (linear or nonlinear) the design specifications can be divided in three main groups (Slotine and Li, 1991). 1. Stability for closed loop system under control both in local and global sense.

2. Performance which is described by the accuracy and speed of the time responses for some typical references such as the step response. For this particular response, the three most used specificationsare

=

rise time,

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165

overshoot, and settling time.

3 . Robustness to disturbances, measurement noise, model-plant mismatches, etc., where the system must still be able to satisfy the desired specifications when these effects are present. Some remarks should be made at this point. Note that stability for nonlinear systems is usually defined in a way that does not cope with persistent disturbances (Slotine and Li, 1991). The reason for this is that the stability of nonlinear systems is defined with respect to initial conditions, and only temporary disturbances can be translated to initial conditions. Therefore, robustness is used to cope with persistent disturbances. The three most important design specifications, i.e., robustness, performance and stability, may conflict to some extend, and a trade-off between them is usually required to obtain a good control system. This chapter does not address explicitly stability and robustness specifications, i.e. no design specificationsare specified concerning these features by themselves. Hence, only pelformunce specifications are explicitly treated here. It should be stressed, however, that although stability and robustness are not considered, they can be implicitly present in some performance specifications. This is one of the reasons why rule-based FLC, in the Marndani's sense (Mamdani, 1974), are widely applied in industry and performing so well. Note finally that performance specifications defined for nonlinear systems can be translated to performance criteria and combined into an optimal control problem using the weighted-sum or the weighted-max objective, as it is usually done for linear systems. The next section presents classical, i.e., non-fuzzy, performance specifications usually defined for linear and nonlinear systems.

7.3

CLASSICAL PERFORMANCE SPECIFICATIONS

One of the most important steps in the design of a control system is the choice of the performance specifications, which influences the type of controller to be used. Performance specifications by themselves can also be contradictory as design specifications. Hence, when performance specifications are translated to performance criteria, a trade-off between the different criteria must also be made, in order to find a suitable controller. Usually, the performance specifications are divided in the following groups: 1 . input/output (YO) specifications, related to the effect of the control actions u into the system's outputs y,

2. regulation specifications, measuring the effect of the disturbances d and dminto Y,and 3. actuator effort of the control actions u. Sometimes the combined effect of disturbances and control actions on the output is also considered. The following sections describe each of these three most utilized groups of performance specifications in more detail.

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7.3.1 Input-output specifications It is usual to express specifications on the system outputs y in terms of a given input response. Some of the most used specifications for linear systems are made in terms of the step response. Step responses give a good indication of the performance of the controlled variable to command inputs that are constant for long periods of time and occasionally change quickly to a new value (new setpoint). Let h(k) denote the unit step response of the SISO mapping describing a system P. A SISO system is considered for the sake of simplicity, but the next definitions are also valid for MIMO systems. Note that for nonlinear systems, different steps of the system present different behaviors. Thus, several working points of the system must be considered and the specifications described in the following must be done for all these working points. In linear systems this procedure is simplified and only the unit step response must be considered. The performance specifications defined in the following must thus be applied for the several working points when a nonlinear system is considered. The following specificationsare defined for discrete or discretized systems. A common specification for step responses is to assure asymptotic tracking, i.e., a zero steady-state error for the system, that can be translated as A $at (P, A) = lim Ah(k) = A , k+a,

where A E R is the amplitude of the step. Both the overshoot and the undershoot are defined as functions of P. The overshoot is defined as A $os(P, A) =sup - 1) , (7.7) k>O

and the undershoot as,

A

$us (P, A) = sup -Ah(k) . k>O

The rise time and settling time can be defined in various ways. In general, the rise time is defined as A

~ ! J ~ ~ ,= ( Pinf{K ) I Ah(k)

> XA,

k

> K),

(7.9)

where a common value for the parameter is X = 0.8. The settling time is given by A

$set(P)=inf{K

I

(Ah(k)- A [

< 6,

k 2 K),

where the parameter 6 is usually set to 0.05 or 0.02. Figure 7.2 presents an example of a step with amplitude A = 1,where the overshoot, the undershoot, the rise time with X = 0.8, and the settling time with c = 0.05 are illustrated. Other specifications normally used for the step response of linear systems are the general step response envelope specijication, the general response-time functional or the step response interaction. The readers interested in these specifications are referred to, e.g., (Boyd and Barret, 1991). Step response specifications are suitable for systems where the references to be followed are constant for long periods and change abruptly to new values after those

PERFORMANCE CRITERIA

Figure 7.2

167

Example of several I / O specifications.

periods. However, typical command signals can be more diverse, changing frequently in a way that is not completely predictable. For these systems, the goal is to have some system variables that follow or track a (continuously) changing reference. Usually, the outputs y should track the respective references r with small errors, ideally zero. The errors are thus defined as the difference between the references to be followed and the outputs of the system under control as

e ( k ) = r ( k ) - y ( k ).

(7.11)

Some norms of these error signals such as their root-mean square values, the averageabsolute norm or the oo-norm (peak), are commonly used as performance criteria for control purposes. The definitions of these performance criteria are given in Section 7.4.

7.3.2 Regulation specifications This type of specifications considers the effect of the disturbances d and dm on the outputs of the system, assuming that the control signals u are equal to zero or constant. This formulation is useful for linear systems, where the effects of different inputs can be studied separately and summed afterwards, due to the superposition principle. Ideally, the effect of the disturbances on the output should be as small as possible. For linear systems, some typical performance specifications are usually considered. The simplest case is to consider the disturbances constant, and requiring that the disturbances should be asymptotically rejected, i.e., the effect of the disturbances should converge to zero. When the disturbances can be described by a stochastic process, it is usual to require that the root-mean square (see the definition in Section 7.4.1) of the obtained outputs must be smaller than a certain constant value. Another common regulation specification in the frequency domain is the classical minimum regulation bandwidth, which is defined as the largest frequency below which the disturbance is

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largely damped. A detailed description of regulation specifications for linear systems can be found in (Boyd and Barret, 1991). For nonlinear systems the effects of the disturbances can not be studied separately from the control inputs, because the superposition principle is not valid for these type of systems. Therefore, the specifications dealing with disturbances are in the group of robustness specifications, which falls out of the scope of this chapter.

4

7.3.3 Actuator effort The size of the actuator signals is usually limited. Performance specifications must define the proper limits in the control signals or in their variations. The limitations of the actuators can have different reasons like the following. 4

Actuator heating. Excessive heating of an actuator can be caused by large or fluctuating actuator signals, damaging or causing wear to the system. Such constraints can be expressed in terms of a root-mean square norm of u, possibly with weights. Saturation. The limits of actuator signals should not be exceeded, because the actuators may be damaged. These specifications can be expressed in terms of criteria defined as a scaled or weighted oo-norm of u. Power or resource use. Large and high frequent actuator signals are usually associated with excessive power consumption or resource use. A scaled averageabsolute semi-norm of u is often used to express the criteria fulfilling these specifications. Mechanical or other wear. Frequent rapid changes in the actuator signal may cause undesirable stresses or excessive wear. These constraints may be expressed in terms of slew rate or the second derivative norms of u. A brief survey of the different performance specifications defined for a given system has been presented in this section. Performance criteria are the translation of performance specifications to a formal description. This translation can be made in classical or fuzzy terms. The next section describes classical performance criteria, while fuzzy performance criteria are presented in Section 7.5. 7.4

CLASSICAL PERFORMANCE CRITERIA

Usually, the control goals can be expressed in terms of the size of certain signals of interest. For example, tracking error signals, given by the difference between the references r and the system's outputs y must be "small", while actuator signals u should, normally, not be "too large". The criterion describing the performance of the tracking system can be measured, e.g., by the size of the error signal. The size of a signal can be precisely defined using norms, presented in the next section, which generalize the concept of the Euclidean length of a vector (Boyd and Barret, 1991).

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169

7.4.1 Norms and semi-norms of signals

Different norms for signals are described in this section. First, the concept of a norm is defined as follows. Let v ( t ) denote a time signal in a vector space V. A norm of v , represented by llvll maps the space V to R and has the following four properties: 1. Ilvll

>0

2. llvll = 0

(Nonnegativity), v = 0, (Positive definiteness),

3. llavll = lalllvII, V a E R (Homogeneity),

4.

IIv + wll 5 11011 + llwll

(Triangle inequality).

for any v ,w E V . If all the properties except the positive definiteness hold, then a semi-noim is defined. Several norms of signals are presented in the next paragraphs in

both time and frequency domain, where the physical meaning of each one is described. Note that the signals of interest in a system are usually obtained in a discrete or discretized way. Hence, discrete-to-continuous transformation of these signals using, e.g., a zero-order-hold or a first-order-hold must be applied, so that a certain norm or semi-norm of the signals can be computed. The most common norms are the 1-norm, 2-norm and co-norm. These norms can be derived as special cases of a p-norm defined as

1-norm. This norm is the integral of the absolute value of a signal v ( t ) :

and can be seen as a measure of the total fuel or resource consumption.

2-norm. The 2-norm of a signal gives the square root of the total energy, and is given by

If the system under control is linear, the 2-norm can be computed in the frequency domain using Parseval's theorem, see e.g. (Zhou et al., 1996). Note that the 1-norm and the 2-norm are appropriate for transient signals, which decay to zero as time progresses. The same happens for the integral of time multiplied by the absolute error (ITAE) norm defined below. The rest of the norms defined in this section are used for measuring the size of perbistent signals.

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oo-norm (Peak). One simple interpretation of "the signal v is small" is that it is small at all times, or equivalently, its maximum or peak absolute value is small. The oo-norm of v is thus the least upper bound (supremum) of the absolute value of a signal, given by

The oo-norm of a signal depends entirely on the extreme or large values the signal takes on. As the oo-norm depends on occasionally large values of the signal, it is a worst case norm. I T A E n o r m . Sometimes it is useful to introduce a time dependent weight in the norm, given a certain function of time w(t). The most simple example is the integral of time multiplied by the absolute error (ITAE norm), where w(t) = t. The ITAE-norm is defined as m I

I

V

I

AI ~

~

tlv(t)~dt. ~

(7.16)

This norm is given by the 1-norm of v weighted by the time. This weight emphasizes the importance of the signal v as time evolves, and de-emphasize the signal at the beginning of the response. Thus, for this norm the steady-state behavior of the signal is more important than the transient behavior. Root-Mean-Square. For signals with finite steady-state power (non transient signals) it is useful to define a measure that reflects its average size, which is given by the root-mean-square (RMS) value, defined by

provided that the limit exists. This semi-norm is a classical notion of the size of a signal, and it is widely used in many areas of engineering. Signals with small RMS norms can still exhibit occasional large peaks, if they are not too frequent and do not contain too much energy. The llvllms is thus an average measure of a signal. Hence, a signal with small RMS value can still be very large for some time period. Average-Absolute Value. The average-absolute value is a measure that puts even less emphasis on large values of a signal than the RMS norm, and it is defined by

supposing that the limit in (7.18) exists. The ( ( ~ (semi-norm (~a is useful in measuring the average resource used (like fuel), when the resource consumption is proportional to Iv(t) 1. The comparison of the three (semi-)norms: oo-norm, llvllms and I(v(laa,shows that they simply put different emphasis on large and small signal values. The oo-norm

PERFORMANCE CRITERIA

Figure 7.3

171

Input-output mapping of a subsystem.

puts all its emphasis on large values, the RMS semi-norm puts less emphasis on signal amplitudes, and the average-absolute semi-norm puts uniform emphasis on all signal amplitudes. Other (semi)-norms of signals can be defined, but the seven presented are probably the most commonly utilized to measure different characteristics of a signal. The notion of norm of a signal can be extended to the norm of a system.

7.4.2 Norms of systems Let H be a mapping from a given input w to an output z as in Fig. 7.3. The input can be, e.g., a control action u,a disturbance d, etc. The output can be a system's output y, for instance. Note that H can be a subsystem of the total considered system P. The notion of norm can be used for the mapping H as an extension of the induced norms usually defined for linear time-invariant(LTI) systems (Zhou et al., 1996). Thus, the induced p-norm of a mapping H is defined as A

max

liH1lip = ,IwII.<m

Il~llP

-,

llwllp

using the definition of p-norm as in (7.12). A norm for a particular mapping is used when an input signal, e.g., a step, sinusoidal, impulse, etc. is applied to the system, and the output z is measured. Note that if the system is nonlinear, dividing the system in subsystems does not simplify the analysis because the superposition principle is not valid. Hence, in general it is not possible to measure the effect of a particular control action or disturbance, without taking into account the influence of the remaining inputs (control actions or disturbances). Moreover, different input signals of the same type, e.g. different steps or impulses have in general different responses. Thus, the norms of nonlinear systems have little practical value, except for systems described by first principles, or systems for which some prior knowledge about its behavior is readily available. A completely different situation occurs when the system under control is linear. The norms defined in Section 7.4.1 can be applied, and the discussion presented for each norm can be extended to systems. For linear systems, when many input signals are applied with a probability distribution, the average size of the response can be measured using the expectation with respect to the distribution of the input signals. Another possibility for measuring the size of a linear system when several inputs are considered, is to take the worst case or the largest norm of the response of H to the given input signals. Classical performance criteria are used to translate the performance specifications in formal terms such that the behavior of the system is as close as possible to the desired behavior. Performance specifications are sometimes contradictory, and a compromise

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between them is then necessary. This is also the case in decision making problems, in which a compromise between competing criteria and requ.irements should also be obtained. For this type of problems it is useful to fuzzify the criteria and requirements, leading usually to better decisions. Considering the difficulty in translating the performance specifications to performance criteria in order to fulfill the designer requirements, even in the presence of linear systems, it seems reasonable to describe the performance specifications using fuzzy performance criteria. The definition of fuzzy performance criteria can be easier than classical performance criteria due to the inherent fuzziness present in the performance specifications and the flexibility introduced in the definition of the performance criteria. Moreover, the combination of different criteria can be made using different and more general operators than the ones presented in Section 7.2.1, by using decision functions.

7.5

F U Z Z Y PERFORMANCE CRITERIA

Classical performance criteria can only be defined using norms or semi-norms. However, sometimes informal design goals translate better the designer's intentions than the precise classical design criteria. In a non-classical approach, the design specifications can be seen as (fuzzy) objectives and (fuzzy) constraints. Some examples are "the error between the reference and the output should be very small" or "the control signal should not change too much". The linguistic terms used in the design specificationssuch as "small" and "not change too m u c h can be defined by using fuzzy sets. With the introduction of fuzzy sets for defining the goals of a control system, it is possible to use criteria that do not constitute a norm or semi-norm, generalizing the concept of design goals as used in the classical control theory. This type of goals, however, can be easily combined with norms in a fuzzy decision making environment.

7.5.1 f i z z y decision making Fuzzy performance criteria must be aggregated in order to find the optimal control actions for a given control system. An approach that transparently translates the objectives and constraints derived from the control design goals of a given system (classically known as performance specifications) to performance criteria is fuzzy multicriteria decision making. The concept of multicriteria decision making in a fuzzy environment is originally defined as a confluence of decision goals and constraints (Bellman and Zadeh, 1970). For example, in a single-input single-output system a control goal G can be the minimization of the error between a desired reference r and the output y (output error). The satisfaction of this goal can be represented by a membership function that is defined on the universe of the output error, as shown in Fig. 7.4(a). Fuzzy constraints can be defined on the universe of discourse U of the control variable u. An example is the 'soft' constraint 'u not substantially larger than 0.8' whose degree of satisfaction can be represented by a membership function as shown in Fig. 7.4(b). Note that the 'hard' constraints 0 5 u 5 1 are implicit in the given membership function. Both the goals and the constraints are represented by membership functions, which facilitates their aggregation. As goals and constraints

p:i A I

~:n\ PERFORMANCE CRITERIA

I

173

I

11

-K

11

K

Output error

(I

ox

I

Control achon

Figure 7.4 Example of fuzzy goals and constraints: (a) a fuzzy goal "minimal output error", (b) a fuzzy constraint "control action not substantially larger than 0.8".

are both represented by membership functions in a similar manner, they are usually called fuzzy decision criteria in the fuzzy decision making environment. The use of fuzzy performance criteria in control is presented in Section 7.5.2. Each performance criterion is described by a membership function. As an example, let each error in (7.11) be defined by the triangular membership function, as depicted in Fig. 7.4(a). Note that this definition is for a certain time step k, and a confluence of the different errors for the all response should be made to obtain a measure that is a norm or semi-norm. This procedure is however time consuming, and normally not necessary, because it is enough to consider the number of steps necessary to guarantee a good measure of the defined error. For step responses, for instance, this number of steps is equal to the settling time, as defined in (7.10). In fact, the VO specifications defined in Section 7.3.1 can be almost straightforwardly used as fuzzy performance criteria. Regulation effort and actuator effort are usually seen as fuzzy constraints, and fuzzy sets can be defined for these specifications. Classical performance criteria can not be directly used as fuzzy performance criteria, but the physical meaning behind their definition is of great interest. In fact, instead of using the norms and semi-norms defined in Section 7.4 directly, they can be used as an indication on how to aggregate the different fuzzy sets defining the fuzzy criteria, and also to define the fuzzy sets translating the different criteria. A system that is required to avoid peak errors, for instance, should aggregate the errors over the time horizon penalizing each large error, similar to the peak or m-norm as in (7.15). On the other hand, if only the average of errors is important, and an eventual peak is allowed, the aggregation of criteria must make a sort of average, using a procedure similar to the I-norm or the 2-norm, see (7.13) and (7.14).

7.5.2 fizzy decision making in control Assume now that one considers q decision goals GI, . . . ,G,, and r decision constraints Cl , . . . ,Cr. Let p ~, .,. . ,p~~ denote the membership functions that represent the satisfaction of the fuzzy goals, and let pc, , . . . ,pc, be the membership functions that represent the satisfaction of the fuzzy constraints. Each fuzzy goal G, and each fuzzy constraint Ciconstitute a decision criterion (,, i = 1,.. . , T, where T = q T is the total number of goals and constraints.

+

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FUZZY ALGORITHMS FOR CONTROL

+

Let the policy .rr = u ( k ) ,. . . ,u(k N - 1)be a sequence of control actions for the N steps in time. In the most general case, all the criteria must be applied at each time step i, with i = 1 , . . . ,N. Thus a criterion
+

~x

4

= PC,,

O . . . O pclqO ~ ~ l ( ~O+- .l. O ) PCIT O PC21 O . . . O PCzqO PCzcq+l) O . . .OPCZTO . ..

PCNI

0.. '0 P h q@ PCN(~+I) @. . . @ PCNT.

(7.20) 4

Various types of aggregation operations can be used as decision functions for expressing different decision strategies using the well-known properties of these operators (Klir and Yuan, 1995). The aggregation operator can be used to translate a linguistic description of the control goals into a decision function. In this way, various forms of aggregation can be chosen giving greater flexibility for expressing the control goals. The selection of the decision function is an important issue as it reflects the goals of the optimization. Understanding the influence of the various decision functions on the optimization and learning the effects of the different parameters is important for designing a good controller (Kaymak et al., 1996). The decision criteria in (7.20) should be satisfied as much as possible, which corresponds to the maximal value of the overall decision. Thus, the optimal sequence of control actions .rr* is found by the maximization of p,: T*

=

arg max

p,.

u ( k ) ,...,u ( k + N - 1 )

The formulation of the control problem as a confluence of (fuzzy) goals and (fuzzy) constraints, can be seen as a generalization of the cost function usually used in modelbased predictive control. Next section describes the application of fuzzy performance criteria in MBPC, showing the advantages of generalizing the objective function, usually at the cost of increasing computational time, to derive the optimal control actions.

7.6

FUZZY PERFORMANCE CRITERIA I N MODEL-BASED PREDICTIVE CONTROL

The control goals and constraints can be specified using fuzzy sets in fuzzy performance criteria. In this approach, human knowledge is involved in specifying the control objectives and constraints, rather than the control protocol itself (Meiritz et al., 1995; Kaymak and Sousa, 1997). Using a process model, a fuzzy decision making algorithm selects the control actions that best meet the specifications, see Fig. 7.5. Hence, a

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PERFORMANCE CRITERIA

control strategy can be obtained that is able to push the process operation closer to the constraints and to force the process to a better performance based on goals and constraints set by the operator and by known conditions provided by the system's designers. This prescriptive approach is closely related to predictive control. The

Figure 7.5

Controller based on objective evaluation and fuzzy decision making.

formulation of the control problem as a confluence of fuzzy goals and fuzzy constraints leads to a generalization of the objective function used in model-based predictive control. In the MBPC environment the number of steps is equal to the prediction horizon: N = H,. The optimal policy is now given by: run*

=

max

u ( k ) ,...,u ( k + H p - 1 )

CLn

.

(7.22)

The objective function used in classical MBPC is usually given by a sum of an error measure of one or more output variables. Using fuzzy goals and constraints it is possible to aggregate them using fuzzy operators, and choose the operator that best fulfills the desired combination of goals and constraints. Section 7.6.1 gives an overview of conventional model-based predictive control. Because the membership functions for the fuzzy criteria can have an arbitrary shape and because of the nonlinearity of the decision function, the optimization problem in Eq. (7.22) is usually nonconvex. Optimization problems are briefly discussed in Section 7.6.2, where conditions under which the optimization problem remains convex and a solution for the nonconvex optimization are presented. Finally, a simple example illustrate the advantages of using fuzzy performance criteria in Section 7.6.3. 7.6.1 Model- based predictive control Model-based predictive control (MBPC) is a general control methodology that uses a model to predict the process output at future discrete time instants, over a specified prediction horizon. This model is used in the computation of a sequence of future control actions over a specified control horizon by minimizing a given objective function under given operation constraints. Only the first control action in the sequence is applied, the horizons are moved towards the future and the optimization is repeated (receding horizon principle). Because of the optimization approach and the explicit use of a process model, MBPC can integrate multivariable optimal control, handle processes with nonlinearities, nonminimum phase behavior or a significant dead-time, and can efficiently deal with constraints.

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FUZZY ALGORITHMS FOR CONTROL

The future process outputs are predicted over the prediction horizon Hp using a model of the process. The predicted output values, denoted as G(k + i) for i = 1, . . . ,H p , depend on the state of the process at the current time k, and on the future control signals u(k i) for i = 0,.. . ,Hc - 1, where Hc is the control horizon. If H, is chosen such that H, < H,, the control signal is manipulated only within the control horizon and remains constant afterwards, i.e., u(k i) = u(k Hc - 1 ) for i = H c , ..., H p - 1 . The sequence of future control signals u(k i) is computed by minimizing or maximizing a given objective (cost) function, in order to bring and keep the process output as close as possible to the given reference trajectory r , which can be the setpoint itself or some filtered version of it. Usually, a quadratic objective function like

+

+

+

+

is used (Clarke et al., 1987), since it leads to a convex optimization problem for linear systems. In (7.23), the term containing the predicted error d(k i) = w(k + i) G(k i) minimizes the difference between the desired reference w(k i) and the predicted output, while the second term represents a penalty on the control effort. The parameters ai and Pi determine the weighting of the output error and the control effort with respect to each other and with respect to the prediction step. Additional 'hard' constraints, e.g., level and rate constraints of the control input or other process variables can also be specified as a part of the optimization problem. Generally, any other suitable cost function can be used, but for a quadratic cost function, linear, timeinvariant model, and in the absence of constraints, an explicit analytic solution of the above optimization problem can be obtained. Otherwise, numerical (usually iterative) optimization methods must be used. The basic MBPC control scheme is depicted in Fig. 7.6.

+

+

+

i Process

Figure 7.6 Predictive control scheme: r denotes the setpoint, (reshaped setpoint) and u is the calculated control action.

w

the reference signal

The performance of MBPC depends directly on the predictive accuracy of the model. As the model accuracy decreases, so does the performance of the controller. Consequently, the major part of the MBPC design effort and cost is related to modeling and identification (Richalet, 1993). In the presence of nonlinearities and constraints,

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usually a nonconvex optimization problem must be solved at each sampling period. The control horizon determines directly the dimension of the optimization problem, which may thus become very complex. Optimization problems in classical and fuzzy predictive control are discussed in the next section. 7.6.2 Optimization problems Because the membership functions for the fuzzy criteria can have an arbitrary shape and because of the nonlinearity of the decision function, the maximization of the membership function defining the control policy p, (see Eq. (7.22)) is usually nonconvex. However, in some special conditions it is possible to guarantee that the problem is convex. In fact, when the system to be controlled is linear, the goal of the optimization is the minimization of the prediction error over the prediction horizon, the constraints on the optimization space u(k) x . . . x u(k + H, - 1) are convex, the membership functions for the fuzzy goals at each step are given by triangular membership functions and the Yager t-norm (Yager, 1980) is used for the aggregation of the criteria, the optimization problem at each step of fuzzy predictive control is convex. A more detailed explanation, the proof and a simulation example are given in (Sousa et al., 1996). For the general nonconvex optimization, different iterative optimization techniques such as Nelder-Mead or sequential quadratic programming (SQP) can be used to circumvent this problem. Such methods, however, require significant computing power, which is a serious obstacle for a real-time implementation. Moreover, in nonconvex optimizations, SQP converges usually to local minima, giving poor solutions. A possible solution is to formulate the decision problem as a discrete choice problem where a selection has to be made out of a set of possible alternatives. For control purposes, however, the control actions are usually not restricted to a finite number of possible values but they can have any value within certain bounds. For formulating the discrete choice problem, the control space is thus discretized and the problem is reduced to searching the best control action in the discretized control space. Due to this discretization an approximate solution is obtained. Various techniques such as fuzzy dynamic programming (Baldwin and Pilsworth, 1982; Bellman and Zadeh, 1970) or branch-and-bound methods (Sousa et al., 1995) can be used for performing the search efficiently. The application of branch-and-bound optimization to control problems with fuzzy cost functions, can reduce significantly the search time, allowing the application of fuzzy predictive control to a much broader class of systems. A branch-and-bound algorithm for predictive control using fuzzy decision functions is introduced in (Sousa et al., 1997b). Generally, the computational load of these methods is larger than the computational load of the convex optimization methods. For this reason, the design of the performance criteria using fuzzy goals and criteria can only be applied, in general, to discrete processes with relatively long sampling periods. 7.6.3 Application of fuzzy performance criteria In this section fuzzy performance criteria in predictive control are applied to a linear system. A system of this type has been selected for the experiments in order to be

178

FUZZY ALGORITHMS FOR CONTROL

able to compare the control results when classical and fuzzy criteria are applied. The selected system is given by the transfer function

This is a nonminimum phase system and it has two complex poles in the right-half plane (unstable in open loop). The system, proceeded by a zero order hold circuit, has been discretized with a sample time of 1s. Model-based predictive controllers are designed for the systems by using both a conventional objective function and a fuzzy objective function. In general, the overall control goals for the time domain, can be stated as achieving a fast system response while reducing the overshoot and the control effort. For the considered system, these goals are represented in the objective function by terms which correspond to the minimization of the output error, the output change and the change of the control action, which is a generalization of Eq.(7.23). The objective function looks like

where A$ denotes the change in the predicted output and AU denotes the change in the control action. The factors a,, P, and y,are weighting terms that are application dependent. The parameters m l , n l , mn, nz, m3 and n3 must be selected appropriately depending on the application and they must satisfy 1 5 m, 5 ni 5 H,, i E { 1 , 2 , 3 ) . Usually, ml and m;!are chosen equal to one, m3 equal to zero, nl and n2 equal to H,, and n3 equal to H , - 1. When fuzzy multicriteria decision making is applied to determine the objective function, additional flexibility is introduced. Figure. 7.7 shows examples of membership functions used for the error, change in the predicted output and the change in the control action. The minimization of the output error and the control effort are represented by the triangular membership functions p,(d(k i)) and p , ( A u ( k i ) ) around zero, which are defined on the respective universes of discourse. When there is a crisp rate constraint on the control actions, this can be represented by suitably modifying the membership function p,(Au(k i ) ) as shown in Fig. 7.7. The satisfaction of the change in the output is indicated by the trapezoidal membership function p y ( A c ( k i)). The system is allowed to change the output freely within a margin, while large changes are penalized. In principle, each criterion can be defined for each time instant k + i , i = 1, . . . ,H,. Using T decision criteria, the number of total criteria in a fuzzy MBPC problem is given by T = H, . T (3 - H, for the given example). The membership functions are combined by using the Yager triangular norm (t-norm) (Yager, 1980). This aggregation operator has an additional parameter which influences the optimization results in a way that cannot be expressed by the weight factors. In this way, the objective function can be tuned with a single parameter which results in

+

t

I

I

+

+

+

t

PERFORMANCE CRITERIA

p"l/ ()\,\.

179

/

0

K;

e^(k+i) Figure 7.7 (7.26).

H; 0,H,+

~;(k+i)

K,'

Au(k+i)

Membership functions that represent the satisfaction of decision criteria in

improved control performance. The aggregation of performance criteria is thus given by p

=m a

0 ,1-

(5

(1 - p

+i

w y

+

n3

( 1 - p.(Au(k

( 1 - p,(.jj(.

+ sirwy

> 0,

(7.26)

a=m2

z=ml

+

n2

+z)))'~

)

l'wy)

,

wy

2=m3

where the parameters m,and n,, i E { 1 , 2 , 3 ) are defined as in (7.25). Since a t-norm is used, the decision goal is formulated as the simultaneous satisfaction of all the decision criteria. The response of the controllers is studied using simulations of the system. The membership functions and the parameters of the objective functions have been chosen in a way that leads to fast response while avoiding excessive oscillations and overshoot within the working range of the controller. The prediction horizon is kept as small as possible, since in practice the model-plant mismatch hampers the use of long horizons. In this study, the control space is discretized and the optimal control sequence is determined by the branch-and-bound algorithm, as described in (Sousa et al., 1997a; Sousa et al., 1997b). The control horizon is limited to two time steps in order to keep the computational load low. The predictive control scheme is applied to the linear system given by (7.24) without any constraints on the system. Both the conventional criteria and the fuzzy criteria are then able to control the system with a fast step response and no overshoot. However, when a rate constraint of lAul 5 0.5 is imposed on the system, the influence of the fuzzy criteria on the control problem becomes more dominant. For these experiments, H, is chosen equal to two and H, equal to six. It is required that the controller can bring the system to any level in the interval [-3,3]. Using the output error and the change in the output with ml = m2 = 1 and nl = n2 = H, was found to be sufficient for controlling the system. The following parameters are used for the conventional objective function: a, = 1 and p, = 5, i = 1 , . . . ,H,. For the fuzzy criteria, the following membership function parameters are used: K, = 10, Ky = 1 , Sy = 0.5 with w y = 2 for the Yager t-norm.

180

FUZZY ALGORITHMS FOR CONTROL

-II 0

50

100

150

200

250

300

350

300

350

Time [s]

1

-8 0

50

1

I

I

I

100

150

200

250

Time [s] Figure 7.8

Step responses for the linear system using the conventional objective function.

-1 1 0

50

1

I

100

150

I 200

250

300

350

Time [s]

-8l 0

50

I

100

I

150

I

200

I

250

I

300

I

350

Time [s] Figure 7.9

Step responses for the linear system using the fuzzy objective function.

PERFORMANCE CRITERIA

181

Figures 7.8 and 7.9 shows the response of the system for several step references. It is clear that the predictive controller with fuzzy criteria can improve the speed of the response considerably, while avoiding the overshoot. The response of the controller with conventional criteria can be made faster by changing the values of Pi,but this occurs at the expense of amplifying the oscillations due to the nonminimum phase behavior. Another solution can be extending the prediction horizon. However, a considerable increase of the prediction horizon is required, and this is in general undesired. Hence, this system benefits clearly from the additional flexibility introduced by the fuzzy criteria.

7.7 S U M M A R Y A N D SUGGESTIONS FOR FURTHER RESEARCH In classical control theory, the performance criteria of a designed controller is usually built by using different norms or semi-norms of the signals of interest, such as the control action or the output of system. Depending on the control goals, performance specifications are established for the system under control. The design goals given by the performance specifications are usually contradictory, and a trade-off between them must be made in order to choose the desired performance criteria. A different approach is to use fuzzy sets to define the imprecise control design goals. Control objectives defined as fuzzy goals and fuzzy constraints can be combined in a fuzzy decision making environment, because it is an approach that translates the objectives and constraints derived from the control design goals of a given system in a transparent way. As in the classical approach, the decision goals and the constraints are defined on relevant system variables. The formulation of the control problem as a confluence of (fuzzy) goals and (fuzzy) constraints can be seen as a generalization of the cost function usually used in modelbased predictive control. Various types of classical and fuzzy criteria can be used in MBPC. If fuzzy performance criteria are utilized, the choice of the decision function for aggregating the different criteria influences the control performance greatly. The investigation of different aggregation operators and its direct influence on the results form future research directions. Due to the various shapes of the membership functions used to define the fuzzy criteria, and due to the nonlinearity of the decision function, the optimization problem for determining the best control policy is usually nonconvex. Under certain conditions it is possible to guarantee the convexity of the optimization problem. For nonconvex optimization problems, the branch-and-bound approach can be used to determine the best control policy in a discretized domain. In the future research, the application of other algorithms such as dynamic programming and genetic algorithms will be studied, and their computational performance will be compared to the ones obtained from branch-and-bound. Acknowledgments

This work was partially supported by the Training and Mobility of Researchers programme.

182

FUZZY ALGORITHMS FOR CONTROL

References

Baldwin, J. and Pilsworth, B. (1982). Dynamic programming for fuzzy systems with fuzzy environment. Journal of mathematical analysis and applications, 85: 1-23. Bellman, R. E. and Zadeh, L. A. (1970). Decision making in a fuzzy environment. Management Science, 17(4):141-1 64. Boyd, S. P. and Barret, C. H. (1991). Linear Controller Design - Limits of Pelformance. Prentice Hall, Englewood Cliffs, New Jersey. Clarke, D., Mohtadi, C., andTuffs, P. (1987). Generalised predictive control. part 1:The basic algorithm. part 2: Extensions and interpretations.Automatica, 23(2): 137-160. Doyle, J. C., Francis, B. A., and Tannenbaum,A. R. (1992). Feedback Control Theory. Macmillan Publishing Company, New York. Gill, P. E., Murray, W., and Wright, M. (1981). Practical Optimization. Academic Press, New York and London. Kaymak, U. and Sousa, J. M. (1997). Model based fuzzy predictive control applied to a simulated gantry crane. In Proceedings of Second Asian Control Conference, ASCC'97, volume 111, pages 455-458, Seoul, Korea. Kaymak, U., Sousa, J. M., and Verbruggen, H. B. (1996). Influence of decision functions in fuzzy predictive control. In Proceedings of 4th European Congress on Fuzzy and Intelligent Technologies, EUFIT'96, pages 990-994, Aachen, Germany. Klir, G. and Yuan, B. (1995). Fuzzy sets and fuzzy logic; theory and applications. Prentice Hall. Mamdani, E. (1974). Applications of fuzzy algorithms for control of simple dynamic plant. In Proceedings IEE, volume 121, pages 1585-1588. Meiritz, A., Zimmermann, H.-J., Felix, R., and Freund, R. (1995). Goal-orientedcontrol based on fuzzy decision making. In Proceedings 3rd European Congress on Fuuy and Intelligent Technologies, pages 82-85, Aachen, Germany. Richalet, J. (1993). Industrial applications of model based predictive control. Automatica, 29: 1251-1274. Slotine, J. and Li, W. (1991). Applied Nonlinear Control. Prentice Hall, New Jersey, USA. Sousa, J. M., BabuSka, R., and Verbruggen, H. B. (1995). Some computational issues in fuzzy predictive control. In Proceedings IFAC International Workshop on Artificial Inteligence in Real-Time Control - AIRTC'9.5, pages 66-70, Bled, Slovenia. Sousa, J. M., BabuSka, R., and Verbmggen, H. B. (1997a). Branch-and-bound optimization in fuzzy predictive control: An application to an air conditioning system. Control Engineering Practice, 5(10): 1395-1406. Sousa, J. M., Kaymak, U., Bruijn, P. M., and Verbruggen, H. B. (1997b). Branchand-bound optimization in predictive control with fuzzy decision functions. In Proceedings of IFAC International Workshop on Artificial Inteligence in Real-Time Control - AIRTC'97, pages 48 1-486, Kuala Lumpur, Malasia. Sousa, J. M., Kaymak, U., Verhaegen, M., and Verbruggen, H. B. (1996). Convex optimization in fuzzy predictive control. In Proceedings of CDC'96 - 35th IEEE Conference on Decision and Control, pages 2735-2740, Kobe, Japan. Yager, R. (1980). On a general class of fuzzy connectives. Fuzzy Sets and Systems, 4:235-242.

4

.

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Zhou, K., Doyle, J., and Glover, K. (1996).Robust and Optimal Control. Prentice Hall, New Jersey, USA.

8

COMPLEXITY REDUCTION METHODS FOR FUZZY SYSTEMS M. Setnesl, V. Lacrose2 and A. Titli2

Delft University of Technology Faculty of Information Technology and Systems Control Laboratory Mekelweg 4, PO Box 5031 2600 GA Delft, The Netherlands L.A.A.S. du C.N.R.S. 7, Avenue du Colonel Roche 31077 Toulouse Cedex, France

8.1

INTRODUCTION

Conventionally, fuzzy rule-based systems have been built mainly by encoding expert knowledge into linguistic rules. Following the rule base completeness criterion, each possible combination of propositions in the condition part must be considered and the number of rules is an exponential function of the number of input variables. If there are n variables in the premise of the rules, and each of them take on m linguistic values, the corresponding complete rule base contains mn different rules. Even though some of the combinations are not relevant, the formal completeness criterion requires that all of them are inspected. This combinatorial explosion in the number of fuzzy rules involves a high design effort, hampers the inspectability and maintenance of the rule base, and is expensive in terms of memory and computation.

186

FUZZY ALGORITHMS FOR CONTROL

To decrease the design effort, different algorithms have been developed for the construction or tuning of fuzzy systems from numerical data. Typically, a model structure is created from expert knowledge in the form of IF-THEN rules. Parameters in this structure can be fine-tuned using input-output data. When no prior knowledge about the system under study is available, fuzzy models can be constructed entirely on the basis of numerical data. Most approaches, however, utilize only the function approximation capabilities of fuzzy systems, and little attention is paid to the complexity of the resulting models. Moreover, automated modeling techniques may introduce unnecessary redundancy into the rule base as a result of their focus on numerical optimization (Setnes et al., 1998a). When considering multivariable complex systems it is of great interest to reduce the number of fuzzy rules. In this chapter, a survey is given of some methods proposed in the literature for the purpose of complexity reduction in fuzzy systems. Section 8.2 discuss the pruning of insignificant rules and assumes that a rule base is already available, or that some initial structure has been chosen. The method discussed in Section 8.3 is generally applicable to existing rule based systems with fuzzy antecedents, and it seeks to reduce of the number of linguistic values, or terms, used in the rule base. If the rule base still has to be constructed, one can make the rule base synthesis easier and the resulting rule base less complex by reducing the problem's dimension through a mathematical fusion of the input variables or through symbolic fusion of propositions by constructing multidimensional fuzzy sets. Such methods are discussed in Section 8.4. Another way to reduce the complexity of a fuzzy system is to introduce a decentralized or hierarchical structure. This approach is discussed in Section 8.5, and some concluding remarks are given in Section 8.6. 8.2

ELIMINATION A N D SELECTION

We assume that a rule base has already been constructed or that some initial rule base structure (premise) has been obtained and is to be adapted by some supervised learning algorithm. The objective is to reduce the complexity of the rule base. The methods discussed in this section addresses this problem by removing, or pruning, rules in the rule base. In many cases, especially if a complete rule base has been constructed from combination of linguistic terms defined for the input variables, the rule base will have rules defined for regions of the input space that will not be visited by the physical system. 8.2.1

Pruning of rules

The simplest way to reduce the number of rules is to remove from the rule base the rules that are never fired, or fired only to a low degree. Often, such rules are related to conditions that are physically not realizable. Such rules can be detected by manual inspection, or by a data-driven approach. Even though such an approach work on an already constructed rule base, it can be applied during data-driven construction. In this case, pruning of rules is done in an intermediaterule base, decreasing the complexity of the remaining system that has to be trained, and as such also decreases the computation time of the training algorithm.

COMPLEXITY REDUCTION METHODS

187

8.2.2 Iterative parameter adaptation techniques Many training methods for fuzzy systems makes use of the fact that a wide class of fuzzy inference systems can be interpreted as a feed-forward network. This enables the use of parameter adaptation techniques known form the artificial neural networks theory, like back-propagation, in order to learn the parameters of a fuzzy system from training data. However, most such techniques focus only on the quantitative approximation of the given training data, while paying little attention to the complexity of the resulting rule base. There have been some attempts to remedy this by reducing the number of generated rules, and maintaining a transparent rule structure (Song et al., 1993; Lin, 1994; Chao et a]., 1996; Setnes et al., 1998~).Two such methods are described below.

Adaptive membership function fusion and annihilation. (Song et al., 1993) proposed a method for reducing the number of rules in a fuzzy Min-Max inference system without violating the completeness of the rule base. The method is applicable also during data-driven construction, where a (high) number of equidistant symmetrical membership functions are initially defined for all variables. These membership functions are then adapted using supervised learning applied to training data. Under the assumption that the inference surface is smooth, the process of adaption can reveal overdetermination (i.e. to many rules) of the fuzzy system in two ways. First, if two membership functions comes sufficiently close to each other they can be fused into a single membership function. In (Song et al., 1993), this closeness is determined the consistency measure (Dubois and Prade, 1980):

which determine the maximum of the intersection of the two membership functions P A and P B . Second, if a membership function becomes to narrow with respect to an adjacent membership function, it can be annihilated. In both cases, the number of rules is reduced as shown in Fig. 8.1.

Structure learning. In (Setnes et al., 1998c) a similar method is proposed as a part of the structure learning in fuzzy neural networks. Like the approach in (Song et al., 1993), the structure learning method also applies a similarity measure to detect compatible fuzzy sets that are then fused. Besides applying a different similarity measure1, the main difference is that the structure learning method does not remove narrow membership functions, but rather prunes individual rules without paying attention to completeness of the resulting rule base. This allows for pruning of rules defined for conditions that can not occur in the physical system. The similarity measure used in (Setnes et al., 1998c) is the fuzzy Jaccard-index. It measures the degree of overlap of two membership functions as the cardinality of their

an^ methods have been proposed to assess the similarity, or compatibility, of fuzzy concepts. A comparative analysis of different measures using human subjects can be found in (Zwick et al., 1987). and a mathematical analysis can be found in (Cross, 1993). A comprehensive study from the point of view of fuzzy modeling is reported in (Setnes, 1995).

188

FUZZY ALGORITHMS FOR CONTROL

X

Y

Intersection(N Z) > Threshold N and Z are fu'sed into NZ.

Figure 8.1

The membership function Z for input y is an~hilated.

Original rule table (left), after fusion (center) and after annihilation (right).

intersection divided by the cardinality of their union:

where min and max model the intersection and the union, respectively, and I . I denotes the cardinality of a fuzzy set2. To detect irrelevant rules, the structure learning approach evaluate the cumulative relative contribution (CRC) of each rule. The CRC of a rule Rj is measured by taking the sum of its relative degree of firing for each training data sample k in the used training data set:

where yj is the CRC of the jth rule. If the CRC of a rule is lower than a given threshold, yj < 7,then this rule is considered irrelevant for the model output and can be deleted. The CRC threshold T reflects the users willingness to trade accuracy for less complexity. Higher values of r remove more rules.

Example 8.2.1 Results obtained by the two previous methods are shown in Fig. 8.2. The studied system concerns the pressure dynamics of a fermenter tank (Fig. 8.2a). Systems measurements are obtained by keeping the air inlet at the bottom of the tank constant while the outlet valve is controlled. A model of the pressure dynamics is u(k))where y(k) and u(k)are the pressure sought of the form y(k 1) = f (y(k),

+

2 ~ h cardinality e of a fuzzy set A(z) is given by IAl = summation is replaced by an integration.

xT

p ~ ( x ) .For continuous domains X, the

q

t

COMPLEXITY REDUCTION METHODS

189

and valve position, respectively, at time sample k, and f (.) is a fuzzy system of the Takagi-Sugeno type (Takagi and Sugeno, 1985). The figures 8.2b - 8.2d show the training data projected onto the initial (complete) and final fuzzy premises of two rulebased models obtained by the methods presented in (Song et al., 1993) and (Setnes et al., 1998c), respectively.

rValve (ul)

air flow out

-

Pressure 6)

$

3

a

1.6

e a 1.4 Water

1.2 air flow in

20 40 60 80 Valve position u(k) [% closed]

valve ("2)

(a) Fermenter tank

20 40 60 80 Valve position u(k) [% closed]

(c) Trained with fusion and annihilation

100

(b) Initial rule base premise

100

20 40 60 80 Valve position u(k)[% closed]

100

(d) Trained with structure learning

Figure 8.2 The fermenter tank (a) and the premise of the initial rule base (b), and the results obtained with training using fusion and annihilation (c) and the structure learning method (d). Note that the structure learning has adapted the premise and pruned irrelevant rules in regions not populated by the training data (circles).

190

FUZZY ALGORITHMS FOR CONTROL

8.2.3 Two pass OLS In (Wang and Mendel, 1992) fuzzy basis functions and orthogonal least-squares (OLS) learning of a fuzzy basis function (FBF) expansion from data was introduced. This learning method was extended in (Hohensohn and Mendel, 1994) in order to reduce the initial FBF expansion and the corresponding rule-based system by selecting only the membership functions of the most significant rules to be a part of the final FBF expansion. This method of reduction is applicable also to an existing fuzzy rule-based system, as it can be written as a FBF expansion.

4

Fuzzy logic system. Consider a fuzzy logic system (FLS) with rules in the following form: ~j

:

IF x1 is F! and . - . and x, is F: THEN y is 8j , j = 1,2, - . . , M ,

(8.4)

where F: are fuzzy sets defined by their respective membership functions pF: (xi),i = 1 , 2 , . . . ,n, and Oj E R are singleton (crisp) rule consequents. When a product and-operator and a product implication method are used together with the center of gravity defuzzificationmethod, this leads to a FLS with the following form:

which coincides with the "zero-order" Takagi-Sugeno(TS) model (Takagi and Sugeno, 1985). When all the parameters of the FLS in (8.5) are considered free, methods such as back-propagation learning can be applied. The idea introduced in (Wang and Mendel, 1992) is to fix the premise parameters of the FLS such that the resulting fuzzy system is equivalent to a linear combination of nonlinear functions calledfuzzy basisfunctions (FBFs) and then to apply the more efficient OLS learning algorithm to determine the most significant FBFs and the corresponding rule consequents 8.

t

t

Fuzzy basis function expansion. Consider the fuzzy logic system in (8.5). Define a fuzzy basis function (FBF) as

where x = [xl,. . . ,x,]. Now the FLS (8.5) is equivalent to a linear combination of FBFs M

f (x) = C pj (x)8j.

(8.7)

3=1

From (8.5) and (8.6) we see that each fuzzy IF-THEN rule has a corresponding FBF that gives the relative degree to which the rule fires. The numerator gives the degree to which a particular rule fires, while the denominator in the FBF gives the sum of the

4

COMPLEXITY REDUCTION METHODS

191

degrees of all rules. Due to the normalization of the denominator in the FBF, each FBF is a function of all the membership functions in the FLS (8.5). Figure 8.3 shows an example of FBFs in a one dimensional premise space.

Figure 8.3 One dimensional premise space of four rules (top) and the corresponding fuzzy basis functions (bottom).

OLS learning. Consider the FBF expansion in (8.7) as a special case of the linear regression model M

d(k) =

C P, ( k ) ~+, e ( k ) ,

(8.8)

j=1

where d ( k ) is the system output, O j are real parameters, pj ( k ) = pj ( x ( k ) )are known regressors, and e ( k ) is an error signal. Given N input-output pairs [xO( k ) ,dO ( k ) ] ,k = 1 , - . , N, our task is to find an FBF expansion f ( x ) such that C, ( f ( x O ( k )) d'(k))2 is minimized. The linear regression model (8.8) can be written in a matrix form as ,

where d = [ d ( l ) ,. . . ,d ( ~ ) P] = ~ [pl , , . . . ,p M ]withpj = [ p j( I ) , . . . ,p j ( N ) ] 0~ = , [ O 1 , . . . ,ONIT, and e = [ e ( l ) ., . . ,e ( N ) ] .TheGram-Schmidt OLS algorithm performs an orthogonal decomposition for P, that is, P = WA, where W is an orthogonal matrix (wTW= I), and A is an upper-triangular matrix with unity diagonal elements. Substituting P = WA into (8.9), we have d = WAB e = W g e, where g = A0. The solution 0 can then be found through back-substitution or directly by 0 = A-'g since the matrix A is invertible. The only difference between the original Gram-Schmidt OLS algorithm and the OLS algorithm presented below is that here the OLS algorithm does not decompose the complete matrix P, but selects some dominant columns of P.

+

+

192

FUZZY ALGORITHMS FOR CONTROL

Starting from an initial rule base (8.5), the parameters of the FBF expansion are derived following (8.6) and (8.7)3. By fixing the parameters in pj(x), the OLS algorithm adopt the rule consequents parameters Oj and pick the Ms < M most significant FBF's. Algorithm 8.2.1 (OLS learning algorithm)

Step 1: Select the first basis vector wl For 1 5 i M, calculate

<

w(i) - pi

gli) =

and

and the error-reduction ratio (gii))

[err]?)=

(Wii))

Wy)

(dO)TdO

where pi = [pi(xO(I)),. . . , p i (xO( N ) ) ]is~given by the initial FBF expansion. Find the FBF with the largest error reduction ratio [err]?) = max ( [ e r r ] ? ) ) l
(8.12)

and select the first basis vector wl and the first element gl of the OLS solution vector w1 = w?l) = Pi1 and 91 = gl( i l l (8.13)

w

Step k: Select the next basis vector w k Repeat for 2 k 5 Ms: For 15 i M , i # i l , . . . , i # ik-1, calculate

<

<

3 ~ h method e presented in (Wang and Mendel, 1992) shows how to learn also an initial rule base from data using OLS.

I

COMPLEXITY REDUCTION METHODS

193

Find the remaining FBF with the largest error reduction ratio [err]!" =

lLiLM

max

,i#il

,...,i # i k - 1

r))

([err]

and select the kth basis vector w k and the kth element g k of the OLS solution vector. (8.18) W k = W (ki k ) and gk = g k( i k )

Last step: Solve the triangular system

where

The final FBF expansion is

where pij (x) are the subset of the FBFs determined by the initial FBF determination method with ij determined by the preceding steps.

Remark: The [err]( i ) -

(g'))2 ( d (w O:)'T ) )d~ ~w')

represents the error-reduction ratio due to

w!). Hence, the most significant FBFs are retained (based on their error-reduction ratio) in the final FBF expansion.

194

FUZZY ALGORITHMS FOR CONTROL

Reduction by two-pass OLS. However, as pointed out in (Hohensohn and Mendel, 1994),the FLS has not been totally reduced. As stated earlier, the denominator (8.6) of each of the M s selected FBFs contains all membership functions of the initial FLS, including those belonging to the M - M s non-selected FBFs. Thus, the FLS cannot be reduced to fewer than M rules, even though it contains fewer than M FBFs. To obtain the reduction from M to M s rules, a two-pass OLS algorithm was presented in (Hohensohn and Mendel, 1994). Like in the application of the one pass OLS learning, the initial FBF expansion is derived from the initial rule base with M rules, and the parameters in pj (x) are kept fixed. Algorithm 8.2.2 (Wo-pass OLS learning algorithm)

=

Step 1: Select Ms significant rules Run the OLS learning algorithm (Algorithm 8.2.1) to select the Ms rules out of the M initial ones. At this point, we only store the antecedent parameters of the Ms rules, and we do not solve the triangular system in (8.19). Remark: It is not necessary to prespecify Ms. The first OLS algorithm can [errIj < p, where 0 < p < 1 is a be ended at the Msth step when 1 prespecified tolerance.

4

xPl

=

Step 2: Remove rules Keep only the antecedent parameters of the Ms selected rules. Set M = M s , to reduce the number of rules. The corresponding FBFs are now normalized by (xi). We need the pi vectors for only the Ms rules. and the number of pi vectors is thus reduced.

zFln:=l

Step 3: Calculate rule consequents Run OLS for a second time to obtain the I ! I ( ~ s )parameters. OLS now uses all the remaining Ms FBFs. The objective here is not to find the most significant FBFs but to orthogonalize the modified set of M s FBFs by using equations (8.10), (8.14) and (8.15), and then to determine the I!I(Ms) vector from (8.19-8.21).

Remark: The second run of the OLS algorithm is faster than the first run, because Ms >> M . The reduced rule base makes the use of a bigger training set feasible for the second run, possibly improving the results. We now have acompletely reduced FLS containing only Ms rules. The computation time being proportional to the number of fuzzy sets, this represents a computation reduction of

9.

8.2.4 Surface approximation In (Dreier, 1994) a non-iterative method is described for generating the fuzzy rules to approximate the (smooth) hypersurface of any multiple-input-single-output (MISO) system described by a data set of input-output data. The method produces a "zeroorder" TS fuzzy model of the type in (8.4) starting from an initial rule base that is

t

t

COMPLEXITY REDUCTION METHODS

195

constructed by partitioning the domain of each input variable xi, i = 1 , . . . ,n by mi membership functions p~~(xi). A complete initial rule base of M = mi rules is obtained by taking all possible combinations of membership functions in the premise. The method then proceeds by both estimating the appropriate rule consequents and removing rules that does not contribute to the output of the model in a non-iterative

ni

way.

The rule reduction and consequent estimation method presented in (Dreier, 1994) is applicable to already existing rule-based models of the TS type, by taking the existing rule base as the initial rule base. Given N input-output pairs [x(k),y(k)], k = 1, . . . , N , let X = [x(l), . . . , x ( N ) ]be ~ a matrix containing the input vectors x(k) = [xl (k), . . . , xn(k)], and y = [y(l), . . . ,y ( ~ ) ] T a vector of corresponding output values. For an input x(k), let the degree of activation of the j'th rule be denoted by

The output of the TS model was given previously by (8.5), and for a given observation k, we would like to have

where 8, is the consequent of rule j . Given the (initial) rule base premise, the rule consequents can be determined from the training data. The method now proceeds by defining a coefficient matrix P with elements

The kjth element of P corresponds to the relative activation of rule j for the kth sample of the training data. This is equivalent to the FBF expansion in (8.7), and P in (8.25) is equivalent to P in (8.9). From equations (8.23) and (8.25) we can write

where 0 = [el, . . . ,OMIT and P is an N x M matrix. The rule consequents 8 are unknown, and must be found. Since there is no guarantee that all the proposed rules are fired (activated) by the training data, some columns of P may be zero and thus P ~ P may be singular. Singular value decomposition may fail to produce a pseudo-inverse of P. Therefore, the modal reduction method is used. Let W = pTP. If W is non singular, then its eigenvalues, A, and eigenvectors, stored column-wise in the modal matrix Z, satisfy

and have this property:

Z ~ W Z= Z ~ Z A = diag(A)

196

FUZZY ALGORITHMS FOR CONTROL

The eigenvectors stored in the columns of Z correspond to the diagonal elements of the eigenvalue matrix. Now, if W is singular or ill-conditioned, then some of the eigenvalues are zero or nearly so. In (Dreier, 1994), a near-zero eigenvalue is defined as an eigenvalue with a magnitude less than times the magnitude of the largest eigenvalue. If the columns in Z correspondingto the zero or near-zero eigenvalues are removed, and the reduced modal matrix is called Z,, then a reduced diagonal matrix of eigenvalues A, is defined by

t

where diag(A,) is invertible. The matrix 2, relates the physical singletons (rule consequents)8to modal singletons v by 8 = Z,v. (8.30) By first finding a solution for v, the consequents 8 can be obtained. First, pre-multiply P W. Next, pre-multiply by and use expression (8.26) by PT and replace P ~ by (8.29) and (8.30) to obtain

zT,

I

Equation (8.31) can be solved for v since the diagonal matrix diag(A,) is invertible. Once v is known, 8 is recovered from (8.30). Some of the physical singletons in 8 may be zero or nearly so. The associated rules can be removed since they do not contribute to the output of the model, and thus the rule base is reduced. Remark: The method presented here is based on fuzzy rules with singleton consequents. However, once the premise of the rule base has been established, other consequent models can be trained from the available data. Example 8.2.2 Consider the fuzzy model of the fermenter pressure dynamics from Example 8.2.1. The premise space partition in the initial model is shown in Fig. 8.2b. When we apply the method in (Dreier, 1994)to remove non-influencial rules, a reduced rule base is obtained, containing 12 in stead of the initial 20 rules. The premise partition of the reduced rule base is shown in Fig. 8.4. Note that this partitioning is similar to the one obtained by the structure learning method in (Setnes et al., 1998c), shown in Fig. 8.2d. The main difference is that while the surface approximation method only removes non-influencial rules, the structure learning method also adopt the membership functions by means of back-propagation learning.

8.3

8

RULE BASE SIMPLIFICATION

Fuzzy rule-based models obtained from data are often unnecessary complex as their construction is driven by the minimization of some cost function like, e.g., squared error. The complexity reduction methods presented in the previous sections mainly focus on the reduction of the number of rules. Methods such as the two pass OLS in (Hohensohn and Mendel, 1994) and the surface approximation method in (Dreier, 1994) provide good techniques for obtaining compact, initial rule bases whose premise partition can be further adapted. However, independent of the method used, whether

I

COMPLEXITY REDUCTION METHODS

197

Valve position u(k) [% closed]

Figure 8.4

Rule base premise obtained with surface approximation.

it is based on parameter adaptation or techniques like fuzzy clustering, the resulting rule base can contain unnecessary redundancy in the form of highly overlapping or compatible membership functions. This is often due to the fact that the rules defined in the multidimensional premise are overlapping in one or more dimensions. The resulting membership functions will thus be overlapping as well, and more membership functions will describe approximately the same concept. This is illustrated in Fig. 8.5. In (Setnes et al., 1998a) an iterative rule base simplification approach based on similarity between pairs of membership functions was proposed for reducing rule base complexity, applicable to all types of fuzzy rule bases in general. Based on this method, a one pass approach was presented in (Setnes et a]., 1997) and (Setnes et al., 1998b), and it is described in this section. 8.3.1 Similarity driven simplification A similarity relation is used to quantify the similarity between the fuzzy sets in the rule base. Similar fuzzy sets, representing compatible concepts, are aggregated (fused) in order to obtain a generalized concept represented by a common fuzzy set that can replace the similar ones in the rule base, thereby reducing the number of concepts (linguistic terms) used in the model. Fuzzy sets estimated from data can also possess high similarity to a universal set, that is, a set in which all elements of a domain have a membership close to 1. Such sets add no information to the model, and can be removed from the premise of the rules in which they occur. The process of aggregation is driven by the similarity among the fuzzy sets defined on the domain of the same antecedent variable, not in their product space. As such, a reduction of the model's term set is made possible without necessarily any rules being merged. Reduction of the number of rules some times follows as a result of the aggregation of similar fuzzy sets when the antecedents of two or more rules become equal. Such rules are aggregated into a new, general rule, which replaces the rules with the equal premise.

198

FUZZY ALGORITHMS FOR CONTROL

Algorithm

start"

Finish

f

(a) Parameter adaptation

I

(b) Fuzzy clustering

Figure 8.5 Example of redundancy in the fuzzy term set introduced during identification of a rule base by means of parameter adaptation (a) and fuzzy clustering (b).

COMPLEXITY REDUCTION METHODS

199

Similarity analysis. For each antecedent variable (input), the similarity relation is obtained in two steps. First a compatibility relation is constructed from pair-wise compatibility assessment of the fuzzy sets by means of a similarity measure. To assess the compatibility of the fuzzy sets F!, i = 1 , 2 , .. . , n , j = 1 , 2 , .. . ,M, the fuzzy analog to the Jaccard-index is applied (Dubois and Prade, 1980): Cilm

=

J(F; n F , " ) ~ z ~ J(F! U Fim)dxi '

where 1, m = 1 , 2 , . . . ,M. When the fuzzy sets F! and Fim are defined on a discrete domain by the membership function pF!( x i ) and p ~ ( xr i ) ,respectively, the fuzzy Jaccard index can be written as Cilm

=

I m i n ( ~( x~i:) ,P F (~x i ~) )I I m a ( ~ F( x/ i ) ,PF? ( x i ) )I '

(8.33)

where the min and max operators model the intersection and the union, respectively, and I . I denotes the cardinality of a fuzzy set (Klir and Youan, 1995). The compatibility measure takes on values cil, E [0,11, where c;lm = 1 reflects equality, and c;lm = 0 refers to non-overlapping fuzzy sets. This is the same measure that is used in the structure learning approach discussed in Section 8.2.2. For each input i = 1 , 2 , .. . , n, an M x M binary fuzzy relation C i = [cilm]is calculated, whose elements cilm are obtained by (8.33). This relation is reflexive and symmetric, but not transitive. In order to obtain a transitive similarity relation, Si, the max-min transitive closure, C T i ,of C i is calculated (Klir and Youan, 1995):

2. If Ci

# C i , set C i = Ci and go to 1.

3. Stop: C T i = C : , set Si = CTG Here o is the m a - m i n composition. The lmth element of the M x M binary fuzzy similarity relation Si = Isilm]gives the similarity between the concepts represented by the fuzzy sets Fi and Fim. Aggregation. For each antecedent variable, the aggregation of similar fuzzy sets takes place by applying a threshold X E ( 0 , l )to the similarity relation. The threshold represents the degree to which the user allows the concepts used in the model to be equivalent. The lower the value of A, the more concepts are combined, decreasing the term set of the model. Since the approach does not require additional data acquisition or computationally expensive optimization, the user can easy investigate the effect of different thresholds. The set of fuzzy sets having a similarity greater than the threshold X are aggregated to produce a fuzzy set representing a generalizationof the individual concepts represented by the similar fuzzy sets. After aggregation, the updated rule base is checked for fuzzy sets similar to the universal set. Such sets are removed from the premise of the rules in which they occur. The approach is summarized in Algorithm 8.3.1.

200

FUZZY ALGORITHMS FOR CONTROL

Algorithm 8.3.1 (Aggregation algorithm) Given a rule base R = {Rjlj = 1 , . . . ,M } , Rj: IF X I is F[ and . . . and z, is F i THEN . . . , where F:, i = 1 , . . . ,n, are fuzzy sets with membership functions p y , select A E ( 0 , l ) . Repeatfori = l , 2 , . . . , n Step 1: Calculate similarity relation:

Step 2: Aggregate similarfizzy sets: for1 = l , 2 , ..., M

~f

= {Fim I silm > A } , m E { I , 2 , . . . ,M ) Fhi = aggregate{Ff) ,

end

Step 3: Remove fuuy sets similar to universal sel

where pui = 1, Vxi. If > A, remove Fi from the antecedent of Rj.

4

Having identified a set Ff: of fuzzy sets Fim with a similarity higher than the given threshold, F!,= {Fim I silm > A ) , the aggregation of these sets is accomplished by letting the support of the general fuzzy set FAi be the support of UF r . This guarantees preservation of the premise space. The kernel of FAi is obtained by averaging of the kernels of F:. Example 8.3.1 illustrates the above method. Example 8.3.1 Consider the fuzzy sets Fl, Fz, ..., F:, in the fuzzy partition shown in Fig. 8.6a. From (8.33), a compatibility relation is obtained:

COMPLEXITY REDUCTION METHODS

(a) Initial partition o f five fuzzy sets

Figure 8.6

201

(b) Simplified fuzzy partition

Initial partition (a), and the corresponding simplified partition (b).

The corresponding similarity relation S is the max-min transitive closure of C :

Applying a threshold X = 0.5 to the similarity relation, we identify a set of similar fuzzy sets F = {Fmlslm > 0.5) = { F 2 ,F 3 ,F 4 ) . The concepts represented by the fuzzy sets { F 2 ,F 3 , F 4 ) are aggregated, creating a generalized concept represented by the fuzzy set FG. The simplified partition is shown in Fig. 8.6b. Recomputing rule consequents. If two or more rules end up with a common antecedent part as a result of the aggregation of similar fuzzy sets, they can be replaced by one general rule RG. The consequent parameters of the reduced rule base must be recomputed. One way to do this is to use training data and least-squares approximation. Another possibility is to let the parameters of RG be an aggregation of the parameters of all the rules it replaces. This method requires less computations, and it also does not depend on training data. Let Q C { 1 , 2 , . . . ,M ) such that for all I , m E Q, we have that Fi = F r , i = 1,2, . . . ,n, and let RQdenote a set of rules with equal antecedents. The antecedent of the general rule RG taken to represent the rules RQ is equal to the antecedent of the rules RQ. The consequent of RG must account for the consequents . a TS fuzzy system, this is done by weighting RG with the total of all the rules R ~For Q and let the parameters OG of of the weights of all the rules RQ,i.e. WG = C j E wj, the rules consequent function y ~ be a weighted average of the consequent parameters of RQ.Thus, the set of rules RQ is represented by a single rule RG with weight W G and a consequent part y ~ with the parameters

202 Let

FUZZY ALGORITHMS FOR CONTROL

= { j ( j = 1,2, . . . ,M ,j

6 Q), the output (8.24) now becomes

where ,Bj is the degree of activation of Rj as given in (8.23). For the TS structure, such a substitution of several rules RQ with common antecedent parts by one general rule as described above yields the same input-output mapping. Example 8.3.2 Consider the pressure system from Example 8.2.1. A fuzzy rule-based model has been obtained from systems measurements using the Gustafson-Kessel fuzzy clustering algorithm (Gustafson and Kessel, 1979) in the product-space of the systems in- and outputs. The initial rule base consist of nine rules and 18 membership functions obtained from nine clusters 4 . The membership functions of this rule base are shown in Fig. 8.7a, and the premise partition is depicted in Fig. 8.7b. After application of the similarity driven simplification,with X = 114, a reduced model with four rules and six membership functions is obtained. The membership functions are shown in Fig. 8 . 7 ~ and the premis partition of the reduced rule base is depicted in Fig. 8.7d. Even though the rule base has been significantly reduced, it is still capable of modeling the pressure dynamics of the fermenter tank. Figure 8.8 shows the free-run predictions (recursive simulation) of the training data for both the initial and the reduced rule base.

8.4

DIMENSIONALITY REDUCTION

In the following, we assume that no rule base is available for the problem under study. Two methods are addressed here that can help make the initial rule base synthesis less complex through reduction of the problems dimension. The first method seeks to reduce the number of input variables through mathematicalfusion, or combination of the sensory input variables, while the second method addresses symbolicfusion which involves the use of multidimensional fuzzy sets.

8.4.1 Mathematical fusion The mathematical fusion approach, also called sensoryfusion, is based on combining sensor signals (input variables) before presenting them as inputs to a fuzzy logic controller (KC). The variables are often fused linearly by some weighted sum. Various combinations of input variables can be possible, depending on the number of inputs. Some cases are shown in Fig. 8.9 to illustrate the method. It is assumed in Fig. 8.9 that the input signals of the FLC are represented by m = 5 linguistic labels each. The parameters of the fusion functions are positive parameters derived by physical considerations, designers knowledge or experience. Using sensory fusion, maximum reduction is obtained if all variables can be fused. However, it is clear that not all variables can be trivially combined. In practice, when

4The book by (BabuHka, 1998) gives a good introduction to product-space clustering for identification.

COMPLEXITY REDUCTION METHODS

4

Ba

203

2.2

1

.-

2

20.5

8

-

B

s1.8

2 0

X

5 1.6

rn

e

$ 1

5a

a 1.4

$05

1.2

8

B

0"

49 60 80 Valve pos~tionu(k) [% closed]

20

100

1

60

80

Valve position u(k) [% closed]

(a) Membership functions, initial RB

100

(b) Premise o f initial rule base

g0.s

8 1.2

1.4

1.6

Pressure y(k)

1.8

2

2.2

e

B

0"

20

49

60

80

Valve posltion u(k) 1% closed]

(c) Membership functions, reduced RB

100

20

40

60

80

Valve posltion u(k) [% closed]

100

(d) Premise o f reduce rule base

Figure 8.7 Membership functions (a) and the premise partition (b) of the rule base identified from clustering, and the membership functions (c) and premise partition (d) of the reduced rule base.

204

FUZZY ALGORITHMS FOR CONTROL

I 5m

laa

ism

xa

2000

Ti"" 1.

k"CI

lxa

2000

(b) Reduced model

(a) Initial model Figure 8.8 model (b).

I lmo

Predictions of the training data in free-run of the initial (a) and the reduced

x2

x2

Number of rules = 25

Number of rules = 5

(a) 2 variables

f

x2

"

Number of rules = 5

x3

Number of rules = 125 x3

Number of rules = 25

(b) 3 variables

x4 x4

Number of rules = 625

Number of rules = 25

(c) 4 variables

Figure 8.9

Mathematical fusion for n = 2.3 and 4.

COMPLEXITY REDUCTION METHODS

205

FLC problems are considered, only two variables are fused: generally these are the error and the change of the error. In order to explain the validity of the proposed error and change of error fusion, we will first introduce some basic concepts of variable structure systems (sliding mode control) (Slotine and Li, 1990; Driankov et al., 1993) before considering a FLC operating as a sliding mode controller.

Some basic concepts related to sliding mode control. Let us consider an nthorder nonlinear system described by

where x = [x, x, . . . ,x(*-l)IT is the state vector and u is a control variable. We denote the error between the current and the desired state vector xd by e = [e,ic, . . . ,e("-l)lT, where e = xd - X. The two basic principles of the variable structure system (VSS) is to: 1. define the so-called generalized error S as a linear function of the coordinates of the tracking error vector e:

The surface defined by S ( x ) = 0 is called the sliding sulface in the state space. 2. generate a control law that will bring the system back to the sliding surface and keep it there. The chosen control law is often:

Given this control law, we get the following three sliding mode conditions:

3. S > 0 if we are on one side of the sliding surface and S opposite side.

< 0 if we are on the

For a second order system (x = [x,xIT), the sliding surface S = 0 becomes a switching line defined by: S = cle c2e (8.39)

+

that can be written as

c1 S=e+Ae, A = - > O . c2

(8.40)

206

FUZZY ALGORITHMS FOR CONTROL

(a) 'Hard' sliding control law

Figure 8.10

(b) 'Soii' sliding control law

The 'hard' version of the sliding control law (a) and the 'soft' version (b).

Thus, the behavior of the controlled system on the sliding line is equivalent to that of a first order system of time constant A. The control law (8.38) has, however, one major disadvantage: it causes undesirable abrupt changes in control action (chattering) for each change of sign(S) (see Fig. 8.10a). In order to avoid this "bang-bang" control, the ideal sliding mode is smoothed by introducing a boundary layer that contains the sliding surface as illustrated in Fig. 8.1 1. Inside the boundary layer the control is proportional to S and outside it the control is

Figure 8.11

Commutation law in a sliding mode control.

saturated. The resulting control law is shown in Fig. 8. lob, and can be expressed by:

COMPLEXITY REDUCTION METHODS

207

We shall now interpret a FLC as a VSS realizing sliding mode control, and show the validity of the proposed variable fusion in the FLC.

FLC as a sliding mode controller. As in the VSS, for a 2ndorder system we define a generalized error S = cle cge that results from the fusion of an error signal and its derivative. As a result, instead of a 2ndorder control problem, we are faced with the simpler first order stabilization problem in S. Consider now the following simple FLC rule base:

+

IF S is: where the notations used are: N for Negative, P for Positive, ZE for Zero, S for Small, and M for Medium. Possible membership functions for S and u (singletons), together with the FLC output are presented in Fig. 8.12. -----.-........----------.......---

,"I

SNM

'PM

NS NM

:

"NS "NM:

"NM

.--_--_---_..-........----------..

S~~

Figure 8.12

j

'NS

'ZE

S~~

S~~

FLC as a sliding mode controller.

Thus, through variable fusion, a simple FLC can realize sliding mode control. The difference with the 'soft' sliding control law (Fig. 8.10b) is that, inside the boundary layer, the control action u is a general nonlinear function of S determined by the rule base of the FLC.

8.4.2 Symbolic fusion Another way to reduce the problem's dimension is to use multidimensional fuzzy sets (Foulloy et al., 1994; Passaquay, 1996). In such an approach, input variables are combined symbolically. Suppose the inputs X I , xz, . . ., x, are fused symbolically. If X = ( x l ,x2,. . . ,x,) takes on m linguistic values, then we have to defined m n-dimensional fuzzy sets, each of them describing one of the linguistic terms. An easy way to construct multidimensional fuzzy sets is based on the Delaunay triangulation (George and Hermeline, 1989).

208

FUZZY ALGORITHMS FOR CONTROL

Delaunay triangulation. Let Xn be an n-dimensional euclidean space where any point X E Xn is defined by its coordinate vector [ X I , x 2 , . . . ,x,IT. Denote P c Xn a set of m points P I ,P 2 , .. . ,Pm, where each point Pj is given by Pj = [xj l , xj2, . . .,x j n I T . The Voronoi' diagram associated with P is a sequence Vl ,V 2 , . . ,Vm of convex polyhedra covering X, where 6 consists of all the points X E Xd that have Pl as a nearest point in the set P: VX E Xd, X E

I4

iff VPjjZ1E P , 6 ( X , P l )5 6 ( X , P j ) , 1= 1 , 2 ,..., m ,

,/c;=,

where 6 denotes the euclidean distance: 6 ( X ,P j ) = (xi - x j i ) 2 . The geometrical dual of the Voronoi' diagram, obtained by linking the points Pj whose Voronoi' polyhedra are adjacent, is called the Delaunay triangulation of P. The joints are taken to be straight line segments and we use them as a framework for the subdivision of the space. The Delaunay triangulation is a collection of n-simplices covering the convex hull of P. Each n-simplice is defined by a set of (n + 1 ) vertices. Thus, for, e.g., a 2-dimensional euclidean space, triangles are used to partition the whole space. An illustration for a 2-dimensional case with ten points Pj is given in Fig. 8.13.

Figure 8.13

The Delaunay triangulation is the geometrical dual of the Voronoi'diagram.

Construction of multidimensional fuzzy sets. Let P be a set of m points in a n-dimensional euclidean space (Pj = [xjl,~ ~ 2. . ,,xjnIT . E X,). The points Pj can be selected such that they describe some aspect of a linguistic term A. The selection can be based on knowledge about the system at hand, or on some numerical optimization like, e.g., clustering (cluster centers). The membership function of the linguistic term A is defined from the Delaunay triangulation of P as follows: For j = 1,.. . ,m, if Pj is described by the linguistic term A then p ~( P j )= 1, else p ~ ( P i= ) 0. On each n-simplice of the Delaunay triangulation, the membership function p~ ( X )is defined according to a linear interpolation between the (n + 1 ) vertices

COMPLEXITY REDUCTION METHODS

209

of the n-simplice:

+

The coefficients a l , . . . ,a,+l are calculated from the ( n 1) vertices Pl = ( x l l ,2 1 2 , - . . ,x ~ , E ) ~P defining the n-simplice, by solving of the following equation: A = M-'a, (8.43) X1,l

"

where A = [al,. . . ,an+lIT,M =

'

x n

xn+1,1 " ' pi), . . . , P A ( ~ ~ ) , P A ( ~ ~ + I ) ~ .

Xn+l,n

1]

and /1 =

1

Example 8.4.1 Consider a 2-dimensional space ( X I x X2),with five characteristics points Pj = ( x j l ,xj2). The linguistic term A is completely described by the points P3,P4,and P5, while the points Pl and P2 do not describe this term:

The points Pj and the Delaunay triangulation associated with these points are shown in Fig. 8.14a. The membership function describing the linguistic term A is given by four linear functions, each applicable in one of the four regions defined by the n-simplices of the Delaunay triangulation. On the n-simplice defined by the vertices ( P I ,P3,P4),equation (8.43) becomes:

By inserting the obtained parameters in equation (8.42), we get the following membership function for this n-simplice:

Equivalent calculations for the three other n-simplices, ( P I ,P2, P3), (P2,P3, P5) and ( P 3 ,P4,P5),respectively, give the following membership functions:

The resulting membership function for the term A is shown in Fig. 8.14b and is given by

pA ( X ) = max[O,min(p$'2'3) (x),

( x )p, 2 1 3 1 5 ) (p2'4,5) ~ ) , ( x ) ). ]

210

FUZZY ALGORITHMS FOR CONTROL

x,

(a) Delaunay triangulation

(b) Fuzzy set A

Figure 8.14 Delaunay triangulation (a) of the five points P j . and the membership function p A ( X ) defined from the triangulation (b).

COMPLEXITY REDUCTION METHODS

8.5

21 1

STRUCTURED SYSTEMS

To facilitate the design of a multivariable FLC with a high number of variables, one can define a particular structure (decentralized, hierarchical) according to some knowledge about the process to be controlled. This knowledge can often be obtained from experienced process operators. By defining an appropriate structure, the design of a Multiple-Input-Multiple-Output (MIMO) FLC can be decomposed into the design of several Single-Input-Single-Output (SISO) decoupled FLCs (decentralized structure) and 1 or the design of several Multiple-Input-Single-Output(MISO) or smaller dimensional MIMO FLCs (hierarchical structure). An added advantage of such structured systems, is that their maintenance, e.g., tuning and extension, becomes much easier.

8.5.1 Decentralized In MIMO processes coupling effects can occur between the inputs and the outputs: one input can control more than one output, and one output can be controlled by more than one input. If the coupling effects are small, the design of the MIMO controller comes down to the design of a set of SISO controllers, each of them being designed for one pair of input-output variables. The MIMO control problem is then decomposed into a set of simpler SISO control problems. Many approaches based on the approximation or decomposition of multidimensional fuzzy relations in two-dimensional ones has been studied (Boverie et al., 1993;Jia and Zhang, 1993). In (Gegov and Frank, 1995) conditions for reducing multidimensional relations in fuzzy control systems to two-dimensional ones are studied for systems using the max-min composition operator. Besides providing a much simpler presentation, an approximation of the multidimensional relations by two-dimensional ones, also reduces the computational load. However, such approximation may lead to unsatisfactory results as some peculiarities of the process may be neglected. For this reason, in (Gegov and Frank, 1995; Gegov, 1996), conditions are investigated under which both presentations give equal results.

8.5.2 Hierarchical structure The hierarchical fuzzy controller was first introduced by (Raju and Zhou, 1990). In this hierarchical structure, the number of rules will increase linearly (not exponentially) with the number n of system inputs. The total number of rules reaches its minimum value if every level contains only two input variables as illustrated in Fig. 8.15. Deciding where the variables are put into the hierarchy is an important and sometimes difficult process. Often the hierarchy can be based on knowledge of the system or on some kind of sensitivity analysis. In practice, the variables are classified according to their importance. In that way, the process followed in the hierarchical structure is very similar to that used by an operator: a first decision is made based on the most important variables and then the operator modifies it according to the other parameters. A big advantage of the hierarchical FLC structure is that it is easy to add or subtract a variable without having to alter other rules in the rule base.

212

FUZZY ALGORITHMS FOR CONTROL

-

Figure 8.15

-

T h e hierarchical fuzzy controller.

8.5.3 Multi-level rules The structured hierarchical approach was introduced in (Sugeno et al., 1991; Sugeno et al., 1993) and was generalized in (K6czy and Hirota, 1992; K6czy and Hirota, 1993). It is based on the following observation: "a very complex muftivariable system might have a behavior that depends locally on only a subset of the system's variables". This subset varies according to the domain of the input space under consideration. Thus, the input space is divided into many domains and for each region "a local rule base" can be written. The local rule bases, called sub rule bases, consider only a subset of the system's variables. Thus, the structured hierarchical approach reduces the number of rules. In the higher levels (one or more), called meta levels, some metarules decide which sub rule bases to activate. All levels, excepted the bottom one are meta levels. The total input space is divided into domains and the domains form a partition of X,

X = u;='=,D,,

(8.44)

where Di are the domains. Every domain can be partitioned itself as, e.g.,

On the bottom level the rules have the usual form. However, the premise part of the rules here, instead of considering all the system's variables X = { x l ,x 2 , . . . ,x , ) considers only a subset of X. Figure 8.16 illustrates the hierarchical structure, and show the following rules: IF X I is All AND x2 is A22 THEN consider domain Dl IF xl is A12AND x2 is A22 THEN consider domain D2 Sub rule base of domain D2 : Ri: IF 25 is AS1THEN Y is . . .. Ri: IF xs is AE2THEN Y is . . .. IF xl is All AND 22 is A21 THEN consider domain D3

4

4

COMPLEXITY REDUCTION METHODS

213

IF zl is A12 AND 2 2 is Azl THEN consider domain D4 Sub rule base of domain D4: IF 2 3 is A31AND 2 4 is A42 THEN consider domain D41 Sub rule base of domain D4i : THEN Y is . . .. R;,: IF X G is Asl AND 27 is

... Ril : IF

XG

is As2 AND 2 7 is AT1THEN Y is . . ..

... IF 2 3 is A32 AND 2 4 is A41THEN consider domain Dq4.

Figure 8.16 Illustration of a structural hierarchy with four domains on the highest meta level. T w o rules are shown in domain D2 and four rules are shown in sub domain D4i.

In the simpler form of the structured hierarchical approach, domains form a boolean partition of the input space (Sugeno et al., 1993), such that one given situation corresponds to only one domain. Consequently, only one sub rule base is invoked. In the general approach (K6czy and Hirota, 1993), the input space partition is fuzzy. This means that more rule bases can be applicable at the same time. This fuzzy input space partition leads to a great flexibility for the choice of invoked sub rule bases but in the meantime raises two essential problems. One is how to combine sub rule bases referring to (partially) different input variables. In (K6czy and Hirota, 1993) it is proposes to find the least set containing all the variables involved in the different activated sub rule bases to define a subspace where all the rules can be represented by their cylindric extension. The union of all extended rules will form a temporary rule base to be used for reasoning for the input at hand. The other problem is how to treat contradictionsin the resulting temporay rule base? How to evaluate nonconvex conclusions? A defuzzification method like the center of gravity method may lead to an unacceptable conclusion, depending on the application, and consequently, the maximum method is suggested in (K6czy and Hirota, 1993). Another way to treat the problem is to consider separately each of the activated sub rule bases. Conclusions are calculated for each invoked sub rule base and are then

214

FUZZY ALGORITHMS FOR CONTROL

ordered according to preference. The best solution is then selected. In the application of hierarchical control reported in (Setnes and Emck, 1998), a combination is used: Each rule base is considered individually, and their individual outputs are combined in a weighted aggregation where the weights correspond to the degree of activation of each rule base.

8.5.4 Combination of techniques Combining both methods discussed in Section 8.4.1 and Section 8.5.2 leads to the so-called 'Hierarchical and sensory fusion approach' (Jarnshidi, 1996). An illustration of this is given in Fig. 8.17. Here, the variables are simply combined first, as in Fig. 8.9, and are then organized into a hierarchical structure similar to that of Fig. 8.15. The performances of some different approaches for rule reduction are illustrated graphically in Fig. 8.18 and summarized in Table 8.1. Table 8.1

Number of rules

n=l n > 1, n even n > 1,nodd

Number of rules for some rule-base reduction methods.

Sensory fusion

Hierarchical structure

m m(4)

m (n - 1) * m2

m(W)

Hierarchical and Sensory fusion m

(5- 1) * m2

(9 - 1) * m 2

Recall that ideal sensory fusion (fusing all variables), is usually impossible in practice. The simple fusion of an error signal e and its derivative e is usually exceedingly successful. It should also be pointed out that after the synthesis of a FLC, the number of fuzzy rules might be further reduced using some of the complexity reduction methods discussed in Sections 8.2 and 8.3.

8.6

CONCLUDING REMARKS

In this chapter a number of rule base reduction methods have been addressed. When facing multivariable complex systems it is, first of all, important to reduce the control problem's dimension (e.g. mathematical fusion) and then, to structure the control law. The structured approach (decentralized or hierarchical) splits the design of a multivariable FLC into the design of several lower dimensional FLCs. These rule bases are easier to design both manually and in a semi-automated manner. When considering data-driven construction,one should take care that the rule base is properly initialized, and avoid grid-like initialization that might produce irrelevant rules. After the construction of a rule base, it can be sought further simplified by the similarity driven simplification method studied in this chapter.

Acknowledgments This work was supported in part by the Research Council of Norway.

COMPLEXITY REDUCTION METHODS

215

F.L.C.

x3 Number of rules =I25 Number of rules = 25

(a) 3 variables

1

d

F.L.C.

-

x4

F.L.C. 2 u

x5 x5

-

Number of rules =3,125 Number of rules = 50

(b) 5 variables

1

F.L.C. 2 u2 -

,

x6

-

F.L.C. u3

x7

3

x8 U

-

x9

F.L.C.

4

x10 D

Number of rules = 9,765,625 Number of rules = 100 (c)

Figure 8.17

10 variables

Hierarchical and sensory fusion approach

,

216

FUZZY ALGORITHMS FOR CONTROL Rule-base reduction 120

I

Sensory fusion

-Hierarchical reduction

-1

Figure 8.18

2

3 Number of variables

4

5

Comparison of some rule-base reduction methods.

References

BabuSka, R. (1998). Fuzzy Modeling for Control. Kluwer Academic Publishers, Boston. Boverie, N., Narishkin, D., Lequellec, J., and Titli, A. (1993). Fuzzy control of high order systems using a parallel structure of second order blocks. In Preprints from IFAC World Congress, pages 573-576, Sidney, Australia. Chao, C. T., Chen, Y. J., and Teng, T. T. (1996). Simplification of fuzzy-neural systems using similarity analysis. IEEE Transactions on Systems, Man and Cybemetics Part B: Cybemetics, 26:344-354. Cross, V. (1993). An Analysis of Fuuy Set Aggregators and Compatibility Measures. Ph.D. thesis, Wright State University, Ohio. Dreier, M. (1994). A fast, non-iterative method to generate fuzzy inference rules from observed data. Technical report, Bell Helicopter Textron Inc., Texas, USA. Driankov, D., Hellendoorn, H., and Reinfrank, M. (1993). An Introduction to Fuuy Control. Springer-VerlagBerlin Heidelberg. Dubois, D. and Prade, H. (1980). Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York. Foulloy, L., Benoit, E., and Mauris, G. (1994). Observatoire Francais des Techniques Avanckes :Logique Floue, chapter Annexe IV. ARAGO 14, masson edition. Gegov, A. (1996). Distributed Fuzzy Control of Multivariable Systems. Kluwer, Dordrecht, The Netherlands. Gegov, A. and Frank, P. (1995). Reduction of multidimensional relations in fuzzy control systems. Systems and Control Letters, 25:307-3 13.

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George, P. L. and Hermeline, F. (1989). Maillage de delaunay d'un polybdre convexe en dimension d. extension i un polykdre quelconque. Rapport de recherche 969, INRIA. Gustafson, D. E. and Kessel, W. C . (1979). Fuzzy clustering with a fuzzy covariance matrix. In Proceedings IEEE CDC, pages 761-766, San Diego, USA. Hohensohn, J. and Mendel, J. M. (1994). Two-pass orthogonal least-squares algorithm to train and reduce fuzzy logic systems. In Proceedings FUZZ-IEEE, pages 696700, Orlando, USA. Jamshidi, M. (1996). Large-scale systems modelling, Control and Fuuy Logic. PrenticeHall, Englewood Cliffs, U.S.A. Jia, L. and Zhang, X. (1993). Identification of multivariable fuzzy systems through fuzzy cell mapping. In Preprintsfrom IFAC World Congress, pages 389-393, Sidney, Australia. Klir, G. J. and Youan, B. (1995). Fuuy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, New Jersey. Kbczy, L. T. and Hirota, K. (1992). Interpolation in structured fuzzy rule bases. In Proceedings FUZZ-IEEE'98, pages 402-405, San Francisco. Kbczy, L. T. and Hirota, K. (1993). Modular rule bases in fuzzy control. In EUFIT'93First European Congress on Fuuy and Intelligent Technologies, pages 606-610, Aachen, Germany. Lin, C. T. (1994). Neueal Fuuy Control Systems with Structure and Parameter Leaming. World Scientific, Singapore. Passaquay, D. (1996). Modtlisation et commande de proctdts i base de logique floue. Rapport de stage de fin d'Ctudes D960000, Institut National des Sciences Appliqutes de Toulouse. Raju, G. and Zhou, J. (1990). Fuzzy logic process controller. In IEEE International Conference on Systems Engineering, pages 145-147, Pittsburgh, USA. Setnes, M. (1995). Fuzzy rule-base simplification using similarity measures. M.Sc. Thesis, Delft University of Technology, Dep. of El. Eng., Control Laboratory, Delft, the Netherlands. (A.95.023). Setnes, M., BabuSka, R., Kaymak, U., and van Nauta Lemke, H. R. (1998a). Similarity measures in fuzzy rule base simplification.IEEE Transactions on Systems, Man and Cybernetics - Part B: Cybernetics, 28(3):376-386. Setnes, M., BabuSka, R., and Verbruggen, H. B. (1998b). Complexity reduction in fuzzy modeling. Mathematics And Computers In Simulation, 46(5-6):507-516. Setnes, M., BabuSka, R., Verbruggen, H. B., SBnchez, M. D., and van den Boogaard, H. F. P. (1997). Fuzzy modeling and similarity analysis applied to ecological data. In Proceedings FUZZ-IEEE'97, pages 415-420, Barcelona, Spain. Setnes, M. and Emck, F. (1998). Fuzzy control for the spry drying of washing powders. To appear in Journal-A, special issue on intelligent control. Setnes, M., Koene, A., BabuSka, R., and Bruijn, P. (1998~).Data-driven initialization and structure learning in fuzzy neural networks. In Proceedings FUZZ-IEEE'98, pages 1147-1 152, Anchorage, Alaska. Slotine, J.-J. and Li, W. (1990). Applied Nonlinear Control. Prentice Hall, Englewoods Cliffs. NJ.

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Song, B. G., Marks 11, R. J., Oh, S., Arabshahi, P., Caudell, T. P., and Choi, J. J. (1993). Adaptive membership function fusion and annihilation in fuzzy if-then rules. In Proceedings FUZZ-IEEUIFES'93, pages 961-967. Sugeno, M., Griffin, M., and Bastian, A. (1993). Fuzzy hierarchical control of an unmanned helicopter. In Fifh IFSA World Congress, pages 179-182, Seoul. Sugeno, M., T, M., Nisho, J., and Miwa, H. (1991). Helicopter control based on fuzzy logic. In Second Fuuy Symposium onfuzzy systems and their applications to human and natural systems, Tokyo. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modelling and control. IEEE Transactions on Systems, Man, and Cybernetics, 15:116-132. Wang, L. X. and Mendel, J. M. (1992). Fuzzy basis functions, universal approximation, and orthogonal least-squares learning. IEEE Transactions on Neural Networks, 3(5):807-813. Zwick, R., Carlstein, E., and Budescu, D. V. (1987). Measures of similarity among fuzzy concepts: A comparative analysis. International Journal ofApproximate Reasoning, 1:221-242.

III

APPLICATION OF FUZZY SYSTEMS

9

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL H.-J. Zimmermannl, J. Angstenberger2 and R. Weber2

R W T H Aachen Templergraben 64 52062 Aachen, Germany Management lntelligenter Technologien GmbH Promenade9 52076 Aachen, Germany

9.1 BASIC PRINCIPLES OF DATA ANALYSIS Data Analysis can be considered either as "the search for structure in data (J.C. Bezdek and Pal, 1992) or as a way to reduce the complexity of large masses of data. We shall focus in this paper on the second point of view. In order to clarify the terminology of data analysis used throughout this paper a brief description of its general process is given in what follows. In data analysis objects are considered which are described by some attributes. Objects can, for example, be persons, things (machines, products, ...), time series, sensor signals, process states, and so on. The specific values of the attributes are the data to be analyzed. The overall goal is to find structure (information) about these data. This can be achieved by classifying the huge amount of data into relatively few classes of similar objects. This leads to a complexity reduction in the considered application which allows for improved decisions based on the gained

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

225

decisions can be made. Here one could think of decision support for diagnosis problems (medical or technical), evaluation tasks (e.g. creditworthiness (Zimmermann, 1996)), forecast (sales, stock prices), and quality control as well as direct process optimization (alarm management, maintenance management, connection to process control systems, and development of improved sensor systems). In particular, the classes obtained in data analysis can be the input information to a fuzzy controller, which would become too big if the input information was not condensed before. Of course, this list of applications is by no means exhaustive; for more applications see e.g. (J.C. Bezdek and Pal, 1992). The process of data analysis described so far is not necessarily connected with fuzzy concepts. If, however, either features or classes are fuzzy the use of fuzzy approaches is desirable. In Fig. 9.1, for example, objects, features, and classes are considered. Both, features and classes can be represented in crisp or fuzzy terms. An object is said to be fuzzy if at least one of its features is fuzzy. This leads to the following four cases (Zimmermann, 1994): crisp objects and crisp classes; crisp objects and fuzzy classes; w

fuzzy objects and crisp classes; fuzzy objects and fuzzy classes.

Only the first case is considered by traditional (crisp) data analysis. The three last cases are approximated by the first. The last three cases, in all of which the memberships of objects in classes is gradual and not full or nil, are the focus of Fuzzy Data Analysis.

9.2

M E T H O D S FOR F U Z Z Y DATA ANALYSIS

Figure 9.2 indicates that some boxes - particularly those of feature analysis and classifier design - contain quite a number of classical dichotomous methods, such as clustering, regression analysis, etc., which for fuzzy data analysis have been fuzzified, i.e., modified to suit problem structures with fuzzy elements. The box "classification", in contrast, lists some approaches that originate in fuzzy set theory and that did not exist before. In modern fuzzy data analysis, three types of approaches can be distinguished. The first class contains algorithmic approaches, which in general are fuzzified versions of classical methods, such as fuzzy clustering, fuzzy regression, etc. The second class contains knowledge-based approaches, which are similar to fuzzy control or fuzzy expert systems. The third class, (fuzzy) neural net approaches, is growing rapidly in number and power. Increasingly combined with these approaches, but not discussed in this article, are evolutionary algorithms and genetic algorithms (Zimmermann, 1994). The major three classes mentioned above will be discussed in the following. Even though there also exist graph-theoretic and hierarchical clustering methods, we will in this chapter restrict ourselves to objective-function clustering methods. Objective-function methods allow the most precise formulation of the clustering criterion. The "desirability" of clustering candidates is measured for each c, the

226

FUZZY ALGORITHMS FOR CONTROL

number of clusters, by an objective function. Typically, local extrema of the objective function are defined as optimal clusterings. Many different objective functions have been suggested for clustering (crisp clustering as well as fuzzy clustering). The interested reader is referred in particular to the excellent book (Bezdek, 1981) for more details and many references. We shall limit our considerations to one frequently used type of (fuzzy) clustering method, the so-called c-means algorithm. Classical (crisp) clustering algorithms generate partitions such that each object is assigned to exactly one cluster. Often, however, objects cannot adequately be assigned to strictly one cluster (because they are located "between" clusters). In these cases, fuzzy clustering methods provide a much more adequate tool for representing real-data structures. To illustrate the difference between the results of crisp and fuzzy clustering methods let us look at one example usecr in the clustering literature very extensively: the butterfly.

Example 9.2.1 The data set X consists of 15 points in the plane, as depicted in Fig. 9.3. Clustering these points by a crisp objective-function algorithm might yield the picture shown in Fig. 9.4, in which "1" indicates membership of the point in the left-hand

1

Process Description

1

determination of membership function feature nomination scale levels

r '

Feature Analysis clustering structured modelling neural nets knowledge based

factor analysis discriminant analysis

rl I '

Classification

diagnosis ling. appmximaUon fuuitication defuzziication ranking

I Figure 9.2

neural nets

I

Scope of data analysis.

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

Figure 9.3

227

T h e butterfly.

cluster and "0" membership in the right-hand cluster. The x's indicate the centers of the clusters. Figure 9.5 and Fig. 9.6, respectively, show the degrees of membership the points might have to the two clusters when using a fuzzy clustering algorithm. We observe that, even though the butterfly is symmetric, the clusters in Fig. 9.4 are not symmetric because point xs,the point "between" the clusters, has to be (fully) assigned to either cluster 1 or cluster 2. In Fig. 9.5 and Fig. 9.6, this point has the degree of membership .5 to both clusters, which seems to be more appropriate. Details of the methods used to arrive at Fig. 9.5 - Fig. 9.6 can be found in (Bezdek, 1981).

><3 1

.'

e1

(I

.'

1

1

1

clustercenters :X i

Figure 9.4

Crisp clusters of the butterfly.

"

228

FUZZY ALGORITHMS FOR CONTROL

.. -94

,

.97

.

4

,.I

86

.

.06

..99

,

.SO

..86

.

01

rI4

.03

ok

-94

.06

.86

..I4

Clustercenters

? X

A

Figure 9.5

Cluster 1 of the butterfly.

,

-14

.86

..

,.06

..Ox

.

Xi0'

.14

.06

0.

SO

0'

86

94

.99

OX

.94

.14

.97

l

.

.86

Clustercenters

'X

Figure 9.6

Cluster 2 of the butterfly.

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

229

One of the best known fuzzy clustering algorithms is the fuzzy c-means algorithm (Bezdek, 1981). Even though the fuzzy c-means algorithm (FCM) performs better in practice than crisp clustering methods, problems may still have features that cannot be accommodated by the FCM. Exemplarily,'two of them shall be looked at briefly. Most crisp and fuzzy clustering algorithms seek in a set of data one or the other type of cluster shape (prototype). The type of prototype used determines the distance measurement criteria used in the objective function. Windham (Windham, 1983) presented a general procedure that unifies and allows the construction of different algorithms using points, lines, planes, etc. as prototypes. These algorithms, however, normally fail, if the pattern looked for is not in a sense compact. Dave (Dave, 1990) suggested an algorithm that can find rings or, in general, spherical shells in higher dimensions. His fuzzy shell clustering (FSC) algorithm modifies the variance criterion by introducing the radius of the "ring" searched for, arriving at

where Dik

=1

1 1 Xk

-Vi

11 - r i l .

r i is the radius of the cluster prototype shell, and all other symbols are as defined for the FCM algorithm. The algorithm itself has to be adjusted accordingly by including Ti.

Details are given in (Dave, 1990). This algorithm also finds circles if the data are incomplete. Figure 9.7 shows examples of it from Dave. The FCM as well as the FSC satisfy the constraint

Considering data sets shown in Fig. 9.8, this constraint would enforce that, for instance, two cluster points A and B would get the same degree of membership, p = .5, to clusters 1 and 2. The p i k would then express a kind of "relative membership" to the clusters, i.e., the membership of point B in cluster 1 compared to the membership of point B in cluster 2 (see also Fig. 9.8). From an observer's point of view it might, however, be inappropriate to assign the same degrees of membership to points A and B because he interprets those as (absolute) degrees of membership, e.g., degrees to which points A or B belong to clusters 1 or 2, respectively. Krishnapuram and Keller (Krishnapuram and Keller, 1993) suggest their possibilistic c-means algorithm clusters by modifying the definition of a fuzzy c-partition and, as a consequence, the objective function of the clustering algorithm. An interesting approach is also the semi-supervised approach by Bensaid, et.al. (Bensaid et al., 1996) that combines labeled and unlabeled data in clustering and thereby increases the stability of existing clusters. Knowledge based approaches resemble very much the classical and well-known Fuzzy Control systems. The real numbers obtained after defuzzification are here

230

FUZZY ALGORITHMS FOR CONTROL

Figure 9.7 Clusters by the FSC. (a) Data set; (b) circles found by FSC; ( c) data set, (d) circles found by FSC.

Figure 9.8

Data sets (Krishnapuram and Keller. 1993).

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

231

interpreted as degrees of membership of the elements to the classes to which they have to be assigned. Recently a lot of research efforts have been directed towards the combination of different intelligent techniques. Here the elaboration of neuro-fuzzy systems is one cornerstone for the future development of intelligent machines, see e.g. (Kosko, 1992). One of these methods is a fuzzy version of the Kohonen's network (Kohonen, 1989). It is expected that in the near future the areas of fuzzy technology, neural networks, and genetic algorithms will be combined to a higher degree. Especially for data analysis the combination of these methods could give promising results. Often a fuzzy clustering algorithm is not sufficient to solve real classification problems: the data to be clustered are generally not in a suitable form to be fed into the algorithm but they have to be pre-processed before.

Table 9.1

Neural and neuro-fuzzy methods for data analysis

Supervised learning: Neuro: Neuro-fuzzy:

9.3

Unsupervised learning:

Back-propagation

the Kohonen network

Fuzzy Back-propagation

Fuzzy the Kohonen network

T O O L S FOR DATA ANALYSIS

In many application areas it is known or can be predicted which methods can best be applied to determine the desired solution. In data analysis this is, however, generally not the case. The method to be applied in most cases depends very much on the type of information available and on the specific kind of patterns that are being searched. General recommendations can often be given e.g. that clustering methods should be used if one knows the number and shapes of the structures before, knowledge based approaches if expert knowledge is available and neural nets when enough learning data are available to train the respective neural nets. The specific method or pre-processing technique is, however, generally unknown in advance. Since in addition data analysis can only be performed computer supported, it is important that a variety of methods are at hand in the form of user-friendly software. Information about the data is not always provided in the form in which it is processed most effectively and efficiently. Therefore, it has turned out to be very important to "pre-process" the data before they become input to the data analysis methods. If, for example, in quality control some acoustic signals have to be investigated, it becomes necessary to filter these data in order to overcome the problems of noisy input. In addition to these filter methods some transformations of the measured data as, for example, Fast Fourier Transformation (FFT)could improve the respective results. Both, filter methods as well as FFT (Fast Fourier Transformation), belong to the class of signal processing techniques. Data Pre-processing includes signal processing and also conventional statistical methods.

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FUZZY ALGORITHMS FOR CONTROL

Statistical approaches could be used to detect relationships within a data set describing a special kind of application. Here correlation analysis, regression analysis, and discrimination analysis can be applied adequately. These methods could be used for example to facilitate the process of feature extraction. If, for example, two features from the set of available features are highly correlated, it could be sufficient for a classification to consider just one of these two. Software that contains the appropriate approaches in the form of CASE-tools is readily available for the area of fuzzy control. For fuzzy data analysis it is still very rare. In the following one of the few, maybe so far the only, existing CASE-tool is briefly described.

4

I

9.3.1 DataEngine - a software-tool for data analysis

-

The Software-Tool for Intelligent Data Analysis

p,l.

Preprocessing

Data

I -'\ File Serial Ports \\ Data Acquisition Boards Data Editor Data Generator

I

Algorithmic Data Analysis

\\

, /" -

Knowledge Based Data Analysis

I

File Serial Ports Data Acquisition Boards Printer

Neural Data Analysis

c++ Precompiler for HardwareplaUorms: - IBMCompatible(MS Wtndows) -Sun SPARC II (MOTIF) -other PlaUorms

- Algorithmic Classifiers - Rulebased Systems - Neural Nets

User Interface: -graphical Programming - interactive and automatic modes

Figure 9.9

Structure of DataEngine. 4

DataEngine is a software tool that contains methods for data analysis which are described above. Especially the combination of signal processing, statistical analysis, and intelligent systems for classifier design and classification leads to a powerful software tool which can be used in a very broad range of applications. DataEngine is written in an object oriented concept in C++ and runs on PCs. Interactive and automatic operation supportedby an efficient and comfortable graphical user

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

233

interface facilitates the application of data analysis methods. In general, applications of that kind are performed in the following three steps:

Modeling of a specific application with DataEngine. Each sub-task in an overall data analysis application is represented by a so called function block in DataEngine. Such function blocks represent software modules which are specified by their input interfaces, output interfaces, and their function. Examples are a certain filter method or a specific clustering algorithm. Function blocks could also be hardware modules like neural network accelerator boards. This leads to a very high performance in time-critical applications.

Figure 9.10

Screenshot of DataEngine.

Classifier design (off-line data analysis). After having modeled the application in DataEngine off-line analysis has to be performed with given data sets to design the classifier. This task is done without process integration. Classification. Once the classifier design is finished, the classification of new objects can be executed. Depending on the specific requirements this step can be performed in an on-line or offline mode. If data analysis is used for decision support (e.g. in diagnosis or evaluation tasks) objects are classified off-line. Data analysis could also be applied to process

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FUZZY ALGORITHMS FOR CONTROL

monitoring and other problems where on-line classification is crucial. In such cases, direct process integration is possible by configuration of function blocks for hardware interfaces. 9.4

FUZZY DATA ANALYSIS A N D FUZZY CONTROL

There are essentially two relationships between (fuzzy) data analysis and (fuzzy) control:

9.4.1 Input compactification and dynarnization Data Analysis can reduce the number of input variables or values of input variables. This happens if (Intelligent) Data Analysis is used in the design of a fuzzy controller and as well, if a data analyzer and a fuzzy controller are used in series, embedded in a system that has to be controlled. In the latter case it is primarily the "classification"-part of Data Analysis (see Fig. 9.2) which is used. In the former case it is predominantly the "Classifier Design" part. In both cases this can lead to smaller (and faster) fuzzy controllers or make systems controllable, which - due to their complexity - so far were not controllable by fuzzy controllers of whatever type. Whether a fuzzification module in the fuzzy controller is necessary or not will depend on the context. In any case, however, a very good tuning of data analyses and fuzzy controller is a prerequisite. The reduction of the dimensionality of the input by a prior use of Data Analysis might not only improve fuzzy controllers. It might also be beneficial for classical controllers. Since they are not the concern of this book, however, this questions shall not be discussed in more detail at this point. Fuzzy Controllers are supposed to be in principle static or one might even say "Markovian" in the sense that they are considering the (present) state of the input variables but not their history. It is our experience that often, however, as in traditional controllers it would be preferable or even necessary to consider as inputs not the present state of the input variables but rather their trajectories during a certain time interval. Integrating this into the knowledge base of a fuzzy controller is rather cumbersome and the presently available methods for Intelligent Data Analysis do not consider trajectories as inputs either. In this case methods of Dynamic Fuzzy Data Analysis (Joentgen et al., 1998) may be of help. The figure below illustrates the difference between classical (static) and dynamic data analysis. Consider a two-dimensional feature space with one additional time dimension and suppose that a set of objects is observed over time. States of objects at a point of time can be seen in the cut of this three-dimensional space at the current moment (Fig. 9.1 la). Two classes of objects can easily be distinguished in this plane. However, if the trace of each object from the initial to its current state (i.e. its trajectory) is considered and projected into the feature space (Fig. 9.1 lb), other classes may seem more reasonable.

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

feature 1

1

40

feature 2

(a) States o f objects at a point o f time

Figure 9.11

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feature 1

feature 2

(b) Projections o f trajectories over time into the feature space

States a t the current time (a) and trace from initial t o current time (b).

Methods or programs for Dynamic Fuzzy Data Analysis are not yet available commercially. Their more detailed description would exceed the scope of this chapter. The interested reader is referred to (Joentgen et al., 1998).

9.4.2 Common fuzzy inference Fuzzy Controllers of whatever type are generally considered to be "knowledge-based" or rule-based, independent of the formal representation of the rules. Data analysis obviously is using various approaches, algorithmic, rule based, and neural. If one only considers the rule-based part, however, there is a strong similarity between data analysis and fuzzy control. Input and output - and therefore also fuzzification and defuzzification - might be different, but the rule base andlor the inference engines might be similar or even identical. This raises the question of whether a (or several) data analyzer(s) and a fuzzy controller could not use the same inference engine, which may serve various rule bases. At least when both systems use the classical "min-max" inference there should not be any major problems for such a common use. If, however, different ways of approximate reasoning are used in data analysis and in fuzzy control, a joint inference engine does not make sense. Even if there is no common inference engine, knowledgebased data analysis and fuzzy control share some problems: Those problems do not concern the inference engine but the rule base. The number of rules in the rule base determines to a large extent the effort and the time needed for the inference. In fuzzy control it was already shown by simulation in the 80s that the performance of a fuzzy controller does not necessarily increase or decrease with the number of rules (see (Larkin, 1985)). Rather there seems to be an optimal number of rules, depending on other measures of performance chosen. In knowledge-based data analysis we have the same problem: while one tries to find the minimum number of rules one also tries to maximize the ratio of correctly

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classifying the objects. Different approaches have been used to solve this problem. Similar attempts in fuzzy control are not yet known to the authors but investigationsin this direction might be very desirable.

9.5

4

INDUSTRIAL APPLICATIONS

In the following we shall describe one industrial application with the focus on the data analysis part: Nowadays blast furnace operation is supervised by extensive measurements and controlled accordingly. Characteristicindications concerning process quality are given by the analysis of the radial temperature profile in the upper part of the furnace. Optimizing this temperature distribution would lead to considerable savings of input material. To achieve an optimization, quantitative relations between furnace parameters are needed. As those relationships are unknown, a process model can be provided using neural networks and fuzzy methods. In this section we show the application of fuzzy clustering and neural networks to classify temperature profiles and to build a model of the interdependence between process operation parameters and the resulting temperature profiles. These investigations have been carried out in a plant of a German steel producer (Angstenberger, 1996).

t

9.5.1 Problem formulation During the blast furnace process, ferric oxides contained in the input materials ore and sinter are reduced and melted. The output material is liquid pig-iron. Figure 9.12 shows a schematic overview of this process. The furnace is charged from its top. The input materials are sinking downwards while gas is flowing upwards through the furnace. The melting heat is supplied by this gas stream and by coke (which is also charged from above). Although studies have been made about the internal operating conditions of the furnace (Omori, 1987), the blast furnace operating personnel relies mainly upon measurements at the inputs and outputs of the process. A wide range of measuring data is acquired at the blast furnace. Blast furnace engineers check these data in order to find out if the change of a controllable furnace parameter is necessary. This inspection is done by expert knowledge, i.e. control of the blast furnace is based on the experience of the operating personnel. Quantitative knowledge of functional dependencies between furnace parameters is not absolutely required for that purpose. However, experience has shown that missing knowledge of quantitative functional interdependencies makes an optimized furnace operation almost impossible. The operation is considered to be optimal, if the fuel consumption (coke and coal) is minimized (cost savings) and the furnace shows a steady cycle without greater fluctuations of some parameters. One of these essential parameters is the radial temperature distribution in the upper part of the blast furnace. According to the experts' knowledge, this magnitude is the most important parameter to judge the condition of the furnace. The temperature profile should show a determined characteristic shape: Maximal value in the center of the furnace, fall off reaching the wall with a minimum briefly after the half radius and again rising reaching the wall. Past has shown that at the considered blast furnace

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INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

Figure

9.12

Schematic view of the blast furnace.

237

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FUZZY ALGORITHMS FOR CONTROL

among all of these profiles one specific shape is always connected with the best furnace operation (in the above-mentioned sense). The question indeed is how to receive such a shape and how to maintain it in the long run? For that purpose the shape influencing quantities have to be known as well as the order of magnitude of their quantitative influence on the furnace process. Due to the large number of furnace parameters and their highly non-linear coupling the identification of these influencing factors is difficult, which thus also holds for the modeling of the process itself. As there is no possibility to generate an adequatephysical model of the blast furnace, the relationship between these influencing parameters on one hand and the resulting temperature profile on the other hand were modeled using a neural network. By interviewing blast furnace experts the characteristics of the input materials sinter (iron supply) and coke (energy supply) as well as the charging program were identified as the main influence factors on the temperature profile. The charging program determines the order and the amount of the input materials brought into the furnace, thereby producing a series of sinter and coke layers. The thickness and the structure of these layers (shown at the upper left side of Fig. 9.12 as different hatchings) have great influence on the rising gas streams in the furnace and therefore determine indirectly the measured temperature profile. The two objectives of this analysis were: 1. A classification of the temperatureprofiles with the objective to assess how much the current state of the furnace corresponds to determined quality criteria.

2. The determination of the quantitativedependency of the cross-sectional temperature profile on the decisive magnitudes that constitute this profile (i.e. modeling the blast furnace). These two parts of the analysis can later on be combined into a complete blast furnace control system. In a first step the classification of the temperature profile gives information about the current state of the blast furnace. A suitable influencing of the profile can then be carried out using the furnace model. All examinations have been carried out with the help of the data analysis software DataEngine, which includes methods for fuzzy clustering as well as neural networks. Together with its data pre-processing, statistics and visualization facilities this tool enables a complete solution of the whole task.

9.5.2 Classification of temperature profiles in the blast furnace by fuzzy clustering For splitting the temperature profiles into classes 143 data records, each containing a profile of 8 temperature values recorded previously have been used. At the beginning, the number of different classes of profiles included in the data material was unknown, so the optimum number of classes had to be determined first. For the purpose of classification the fuzzy c-means algorithm was used, a fuzzy clustering procedure that allows gradual association of the considered objects (here: temperature profiles) to the respective classes (Zimmermann, 1996). The essential information this algorithm produces are the centers of the clusters found during clus-

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

239

tering. Each cluster center represents a (fictitious) sample that characterizes the respective class best. Here, fuzzy clustering results in typical temperature profiles that can be interpreted as prototypes of the respective classes. To determine the optimal number of classes, cluster validity measures can be used (Windham, 1981). The analysis of the given data records leads to an optimal partition of five clusters, whose centers are displayed in the following diagrams (Fig. 9.13). In each diagram the temperature is plotted against the measuring position (seen from the center of the furnace). Class I

Class 2

Tempiiip-y 200

: g $ $ -N :0 r 9 : P o r ltlon

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"--Class 4

Class 3

:"

Temp 400 200 0

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Figure 9.13

W z

N z

m O N

m O O N m m m N 0 9 9

Posltion

Types of temperature profiles (classif~cationinto 5 classes).

Using the criteria mentioned in the introduction above, it can be recognized that the profiles of classes 1, 2 and, with certain reservations, class 4 represent good furnace states, whereas classes 3 and 5 characterize undesired states of the furnace. The received class partition was validated by blast furnace experts. It turned out to represent a good characterization of typical states of the furnace (cf. also (Bulsari and Saxen, 1995)).

9.5.3 Modeling of the dependency between temperature profiles and input parameters by neural networks The available data records for modeling the blast furnace comprised the charging program, the distributions of the grain sizes of coke and sinter as well as several parameters

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FUZZY ALGORITHMS FOR CONTROL

of the material strength. As the grain spectrum of coke covers a relatively narrow area, the evaluation considered only the medium grain diameter. The distribution of the grain size of sinter plays an essential part for the furnace process, as it bears strongly upon the gas flow in the furnace. For that reason the distribution of the sinter grain size was used completely as an input parameter. Altogether 10 input parameters were used. Because of material transportation time and other effects the material parameters affect the temperature profile with a delay of approximately 24 hours. On the other hand, the charging program bears on the temperature profile with a delay of 8 hours due to process characteristics. An appropriate synchronization of the data had to be carried out. After further pre-processing (scaling, etc.) these data were used as inputs of a multilayer perceptron. Some data records were extracted from the total amount of data in order to be used for validation of the network performance. The output of the network consists of 8 values modeling the temperature at the respective measurement positions of the measuring system. Several network configurations (number of hidden layers, number of hidden neurons, learning parameters) have been trained and tested with regard to their efficiency. The neural network is able to approximate the temperature profiles with good precision. This is proven by the comparisons shown in Fig. 9.14. The diagrams show the temperature profiles calculated by the neural network (indicated as "Network" in Fig. 9.14) as opposed to the actually measured profiles ("Measurement" in Fig. 9.14) for four data records. The cases shown in Fig. 9.14 have not been used for training the neural network which proved to be able to recognize relatively unusual temperature profiles as well (like the one shown in Fig. 9.14 on the lower right).

Figure 9.14

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Comparison of measured and calculated temperature profiles.

However, the test results made clear as well that there are certain situations where the neural network is not able to achieve a reasonable result. Therefore, additional examinations have to be performed to quantify the influence of several additional magnitudes that have not been considered yet. In particular, these are magnitudes influencing the gas flow in the furnace and more detailed information about the input material. With inclusion of this information modeling and optimization of the blast furnace process can be largely improved. I

INTELLIGENT DATA ANALYSIS AND FUZZY CONTROL

9.6

241

CONCLUSIONS

The main goals of Data Analysis (DA) are to "detect structures in data" or to "reduce complexity of masses of data". Methods used to this end are numerous and heterogeneous. In the past it was generally assumed that objects are described in a dichotomous way (features exist or do not). If the objects or the classes to which they are assigned in DA are continuous rather than crisp, Intelligent Data Analysis (IDA) is a more appropriate tool than traditional DA. In IDA generally three classes of approaches are being used: algorithmic, knowledge based, and neural network approaches. Which of these methods is most appropriate depends on the context and the available knowledge of the expected structures. As manifold as the methods are the application areas of IDA. Fuzzy Control (FC) has so far not been considered as one of the main areas of application of IDA. Nevertheless, links between these areas exist and ought to be exploited. FC was originally conceived as the application of knowledge based (rule based) approaches to control engineering problems. The original Mamdami-controller was modified or extended (Buckley, Sugeno), but all existing FC-models are limited in applications by the manageable number of inputs. Since the number of inputs can make the rule base of FC grow exponentially, it seems to be more advisable to reduce the number of inputs in a kind of pre-processing before feeding them into an FC. This is exactly where one important link between IDA and FC is: too numerous inputs can be reduced to a small number of classes by IDA and the classes can then be used as inputs to the FCs. There is, of course, another link between FC and IDA: since IDA also uses knowledge based approaches, ideas of FC can be used in the framework of IDA. In this case the defuzzification modules of FC will generally be used to determine the grades of similarity between objects and classes, which are used to assign objects to classes. It is primarily the first link between IDA and FC which has not yet been considered and exploited sufficiently. It is to be hoped that FC can and will become even more applicable also in more complex systems by combining it appropriately with IDA. References

Angstenberger, J. (1996). Blast furnace analysis with neural networks. In C. von der Malsburg, W. v. S. J. S., editor, Art$cial Neural Networks - ICANN 96, pages 203-208, Berlin Heidelberg. Springer Verlag. Bensaid, A., Hall, L., Bezdek, J., and Clarke, L. (1996). Partially supervised clustering for image segmentation. Pattern Recognition, 29:859-871. Bezdek, J . (1981). Pattern Recognition with Fuuy Objective Function Algo-rithms. New York: Plenum Press. Bulsari, A. and Saxen, B. (1995). Classification of blast furnace probe temperatures using neural networks. Steel Research, 66(6):231-236. Dave, R. (1990). Fuzzy shell-clustering and applications to circle detection in digital images. Znt. J. Gen. Syst, 16:343 - 355.

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J.C. Bezdek, E. C.-K. T. and Pal, N. (1992). Fuzzy kohonen clustering networks. In IEEE International Conference on Fuzzy Systems, pages 1035-1043, San Diego USA. Joentgen, A., Mikenina, L., Weber, R., and Zimmermann, H.-J. (1998). Fuzzy dynamic data analysis based on similarity between functions. Submitted to Fuzzy Sets and Systems. Kohonen, T. (1989). Self-organization and associative memory. Springer Series in Information Sciences 8 (3rd ed.). Kosko, B. (1992). Neural Networks and Fuzzy Systems. Englewood Cliffs: Pren-ticeHall. Krishnapurarn, R. and Keller, J. (1993). A possibilistic approach to clustering. IEEE Trans. Fuzzy Syst., 1:98-110. Larkin, L. (1985). A fuzzy logic controller for aircraft flight control. In Sugeno, M., editor, Industrial Applications of Fuzzy Control, pages 87 - 103. Amsterdam. Omori, Y., editor (1987). Blast Furnace Phenomena and Modelling. The Iron and Steel Institute of Japan. London: Elsevier. Windham, M. (1981). Cluster validity for fuzzy clustering algorithms. Fuzzy Sets and Systems, 5: 177-185. Windham, M. (1983). Geometrical fuzzy clustering algorithms. Fuzzy Sets and Systems, 10:27 1-279. Zimmermann, H.-J. (1994). Hybrid approaches for fuzzy data analysis and configuration using genetic algorithms and evolutionary methods. In 11, Z. M. and Robinson, editors, Computational Intelligence - Imitating Life, pages 364 - 370. New York. Zimmermann, H.-J. (1996). Fuzzy Set Theory - And Its Applications (3rd rev. ed.). Boston: Kluwer.

10

FUZZY CONTROL IN PROCESS INDUSTRY: THE LINGUISTIC EQUATION APPROACH E.K. Juuso

Control Engineering Laboratory, lnfotech Oulu and Department of Process Engineering, University of Oulu FIN-90570 Oulu, FINLAND

10.1 INTRODUCTION The process industries face considerablecontrol challenges, especially in the consistent production of high quality products, more efficient use of energy and raw materials, and stable operation on different conditions. The processes are nonlinear, complex, multivariable and highly interactive. Usually, the important quality variables can be estimated only from other measured variables. Constraints, e.g. physical limitations of actuators must be taken into account. Significant interactionsbetween process variables cause interactions between the controllers. Various time-delays depend strongly on operating conditions and can dramatically limit the performance and even destabilize the closed loop system. Uncertainty is an unavoidable part of the process control in real world applications. These demands cannot be met by traditional control techniques only, and therefore various methodologies have been developed in order to extend the applicability of the control systems. Since PID algorithms cannot adequately compensate numerous interactive disturbances, these problems are sometimes solved by the introduction of

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various nonlinear PID modifications where the controller parameters depend on the error or on the parameter scheduling. Gain scheduling is an intuitively appealing approach which was successful in designing controllers for nonlinear systems. For linear time-varying systems, uncertainties are modelled by time-varying parameters, and several different time-delays can be taken into account explicitly. Gain scheduling provides a gradual adaptation of the gains, but is limited to a fixed control structure. vpically, a slow variation of system dynamics is assumed, which may lead to difficulties in compensating fast disturbances. In sliding mode control (SMC), a feedback controller is designed to provide convergence of a system's trajectory to a sliding surface which is based on the plant dynamics (Utkin, 1977). Actually, an infinite high switching frequency would be needed on the sliding surface for variable structure systems. Drastic control changes leading to chattering are the main obstacles for using SMC controllers in process industry. Model-based control is widely applied to industrial applications (Camacho and Bordons, 1995). Various modeling approaches try to combine the advantages of the physical and data-driven modeling techniques, e.g. parameters for mechanistic models are approximated by black-box techniques. Since the identification is on a practical level only for linear systems, a lot of work with local linear models is needed. Multiple model adaptive control (MMAC) allows different control structures, i.e. each mode corresponds to one model and one controller. Switching control strategies are based on selecting a controller from a finite set of fixed controllers. In Model-based Predictive Control (MPC), prediction of future plant behavior is used to compute the appropriate control action at each sampling instant over a prediction horizon. Various kinds of models can be used in MPC. Classical adaptive schemes do not cope easily with strong and fast changes unless the the adaptation rate is made very high. This is not always possible, some a priori knowledge about the plant dynamic behavior should be used. One alternative for these cases is a switching control scheme which selects a controller from a finite set of predefined fixed controllers. An important difference to the gain scheduling scheme is that each controller can have a completely different structure. The switching strategy can be based on heuristic rules or predictions with models. In the MUSMAR algorithm (Mosca et al., 1984), a supervisorcontroller selects the controller to be active by choosing the one, which best fits the plant model. Human operators can handle many processes which are very difficult to handle for conventional control methods. Rule-based programming is commonly used in the development of expert systems but this paradigm leads to serious maintenance and testing problems in practical applications where rule-based systems become really massive. Therefore, linking the rule-based systems to more efficient methods is essential for operability of the practical systems. Methodologies used in the development of expert systems are still useful in extracting expert knowledge from domain experts. Fuzzy logic control (FLC) has widely used in nonlinear and multivariable control. The first successful application (Mamdani, 1974) came almost 10 years after the introduction of fuzzy sets (Zadeh, 1965). Control of cement kilns was an early industrial application (Holmblad and astergaard, 1982). Fuzzy sets and possibility theory provide a unified framework for taking into accountthe gradual or flexible nature

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of variables, and the representation of incomplete information. The available prior knowledge can be combined with information obtained from numerical measurements. Various model-based methodologies can utilize also a fuzzy model. For small systems, e.g. a fuzzy logic controller,modeling the operators' actions is a quite useful procedure. For complex systems, acquiring the knowledge from the human operator is a tedious and time-consuming task at least if the rule-based procedure is used. Conflicting rules or a lack of knowledge or redundancy of rules have to be investigated. The intelligent methods have extended the toolbox of these hybrid, semi-mechanistic or gray-box modeling. Fuzzy clustering is an extension of fuzzy knowledge-based systems to data-driven techniques (BabuSka, 1998). Neuro-fuzzy modeling and identification techniques include fuzzy-logic-basedmethods to neural computing. Model-based fuzzy control is a feasible approach since nonlinear systems can be approximated by reasonably accurate fuzzy models. For small systems, models can be inverted into controllers. For large systems, a fuzzy model can predict the process behavior, and can be used in model-based predictive control (MPC). Linguistic equation approach introduced in 1991 (Juuso and Leiviska, 1992) has been used first as a development and tuning methodology for fuzzy set systems. The first direct industrial application was implemented in 1996 (Juuso et al., 1997b), and recently, the number of applications is increasing rapidly in modeling, intelligent analyzers, control and fault diagnosis. Chapter outline. In Section 10.2 the fuzzy expert control is discussed: most industrial applications are based on using expert knowledge. Fuzzy modeling can increase the performance of controllers extensively: the fuzzy modeling is viewed in Section 10.3, and Section 10.4 describes shortly how these models can be used in nonlinear control. The adaptive fuzzy control (Section 10.5) is also mainly related with model-based techniques. In all these sections, emphasis is given to the linguistic equation approach which extends the performance of the fuzzy control in many ways. The number of the fuzzy control applications has increased very fast, especially during the 90's, therefore, only two applications are treated in this chapter. A solar power plant described in Section 10.6 has been used as a test bed for direct feedback control. Various model-based techniques have been used in a lime kiln viewed in Section 10.7. The multilevel linguistic equation controller provides excellent results in both applications. Implementation is discussed in Section 10.8.

10.2

FUZZY EXPERT CONTROL

A large number of highly successful fuzzy control applications are implemented in process industry. A fuzzy logic control system can be interpreted as a real-time expert system combining symbolic if-then rules with qualitative, fuzzy variables which are linked to the real world by membership functions. Conventional expert systems can be represented by fuzzy expert systems with much less rules if nonlinearities are taken into account in membership functions. Therefore, complicated expert control strategies can be included to practical control. Traditionally, the rule base of the fuzzy controller is constructed from expert knowledge by trial-and-error, and most applications are still based on this approach. Various

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methodologies have been developed for automatic generation of fuzzy systems and inverting fuzzy models, i.e. fuzzy systems can be regarded as flexible mathematical approximation tools. However, the fuzzy controllers should be kept transparent to interpretation and analysis on the basis of understanding the process behavior since this insight is more important than close correlation to experimental data.

Derivative Ae (a) A symmetrical rule base.

Figure 10.1

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Changed TempnkmEM

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JO Cm%ITemprahlre

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(b)A control surfacebased on trapezoidal membership functions.

A PI type fuzzy controller.

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10.2.1 fizzy knowledge-based control Many fuzzy controllers are implemented as two dimensional fuzzy logic decision tables where control actions and input conditions are expressed in terms of membership functions. The rule base shown Figure 10.la is essentially linear, and the nonlinearity can be introduced by adjusting membership functions or by modifying rules. The nonlinear control surface shown in Figure 10.1b is based on trapezoidal membership functions. Modified rule bases are used in multivariable controllers, especially for feedforward purposes (Figure 10.2a). Some experiments with automatic control based on deterministic decision tables were carried out already before fuzzy control. For a cement plant, these decision tables were inspired by instructions found in a textbook for kiln operators, which contained the control rules for manual operation. First experiments with real cement plants were done since 1972 in Denmark (astergaard, 1996). After experiments in laboratory-scale processes, e.g. a small heat exchanger, FLS Automation introduced fuzzy logic to the control of a real cement kiln in 1978. The rule base was modified and the first two industrial applications, the Danish cement plant and a rotary kiln for reburning of lime in a Swedish paper mill, were implemented by the end of 1979. Many FLC applications are developed for rotary kilns (Sheridan and

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Burning end temperature (a) A symmetrical rule base.

Figure 10.2

(b) A control surface based on triangular membership functions.

A fuzzy feedforward controller.

Skjoth, 1983; Gadeberg and Holmblad, 1987; Ducay, 1988; Kemmerer, 1991; Erens, 1993; Dstergaard, 1993). A FLC implemented in 1992 for a cement kiln at Partek cement plant in Finland combined operator experience with process knowledge (Hantikainen, 1992). The rule base consists of several rule blocks corresponding different control objectives. The active rule blocks are selected by conventional meta-rules. The system contains also rules for quality prediction. The FLC was well accepted by the operators, and it has been used continuously without any changes after the on-line tuning. A knowledge-based FLC implemented in 1994 for a lime kiln at Wisaforest Pietarsaari mills in Finland proved to be very successful (see Section 10.7). Construction and tuning of the FLC was originally done in cooperation with the personnel (Haataja and Ruotsalainen, 1994), and later linguistic equations were used in further development (Juuso et al., 1996). The original system has been modified in order to take into account high level control and adaptation to changing operating conditions. Various models have been used in the tuning of the controllers (Juuso, 1998a). A need for continuous adaptation to changing operating conditions has been clearly seen. In 1994 a PI type FLC was implemented on the basis of the experience of the previous experiments for a solar power plant in Spain (Rubio et al., 1995). Recently this manually designed FLC has been replaced by an automatic genetic design technique (Gordillo et al., 1997). Although an additional feedforward controller was used together with the direct fuzzy control, adaptation to changing operating conditions would improve the performance considerably. Fuzzy control of the solar power plant is described in Section 10.6. In 1995 a FLC was installed for a bleaching plant at Kymmene Oy Wisaforest Pietarsaari mill together with Kajaani Elektroniikka Ltd and Valmet Automation

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(Larnpela et al., 1996). The plant is a nonlinear multivariable process with heterogeneous raw materials, complicated variable interactions and long delays. Fuzzy logic also facilitates the use of continuous measurements, different kind of analyzers and laboratory analyses together with operator and engineering knowhow. Using several control strategies the FLC provides the setpoint for the chlorine dioxide dosage controller. The standard deviations for crucial quality variables were decreased by 40-60 percent. Also chemical, yield and energy savings were resulted from the more stabilized control. A FLC implemented for a recovery boiler at Enso-Gutzeit Imatra mills in Finland (Lampela et al., 1996) consists of bed control and symmetry control. The target for the bed control is to optimize the air distribution and the liquor droplet size in order to improve the reduction degree, to decrease the sulphur emissions and to the improve heat exchange. For the symmetry control, the target is to reach the maximum burning capacity, keep the heat exchange surfaces clean and in effective use. Targets are achieved very well, and need for sootblowing has decreased considerably. As the FLC has stabilized the boiler run, the operators need to make only the essential changes. In a effluent treatment plant at Metsa Serla, Kyro board mill, in Finland, a FLC is controlling the return and excess sludge amounts and the nutrients dosage. The capacity has increased, and the plant indicates stable operation, elimination disturbances during load peaks, and diminishing environmental load. A clear improvement has been obtained in the suspended solids and phosphorus reduction, as well as a remarkable 35 percent decrease in the phosphorus acid consumption. A FLC for the heating control of a coke oven battery was developed in 1996 at Rautaruukki Raahe Steel in Finland in order to improve the adaptation of the heating control to changing coal blends (Palmu, 1998). The energy flow to the battery should be limited in order to avoid too high coke end temperatures. On the other hand, the energy demand of the next pushing sequence must be taken into account, otherwise the low temperatures will cause emission problems, or even pushing problems in the worst case. The control system consists of a dynamic scheduling model, a feedforward controller based on the energy model and a feedback controller based on coking index model. The FLC corrects the energy balance and removes contradictory data from the models. Since the rule-based FLC provides good insight to the control, it has been well accepted by the operators. The heating control is now used without any support from engineers. A multivariable fuzzy quality controller has been implemented for a thermomechanical pulping (TMP) plant at UPM Kymmene's Kajaani refiner line (Myllyneva et al., 1997). The quality control has to cope with two types of disturbances: slowly wearing of the refiner plates and variations in raw material quality. The control strategy based on freeness and mean fibre length control with specific energy uses on-line analyzers. Two controllers based on operator experience and process tests are used in a cascaded form: the first begins to raise freeness setpoint if the fibre length begins to go too short, and the second keeps the freeness on a desired level by changing the setpoint of the total specific energy. The first version was an advisory controller, and the closed-loop version is under testing.

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Fuzzy knowledge-based systems very similar to FLCs can be used in intelligent analyzers and software sensors to combine various sources of information. Digital image processing included to intelligent process analysis by a fuzzy logic detection algorithm introduced to the Burning Image Analyser (BIZER) of ABB for finding out the contour of the bed in recovery boilers (Murtovaara et al., 1996). Lack of appropriate on-line measurements, which is a serious problem in many processes, can be compensated by software sensors, e.g. A model-free trainable fuzzy conversion estimator was developed in 1995 for a set-up of lactic acid fermentation (Kivikunnas et al., 1996a). Fuzzy reasoning has also been used for analyzing temporal shapes in the same process (Kivikunnas et al., 1996b). The results of various applications have shown that the FLCs increase overall controllability of the processes by using process knowledge together with continuous measurements, different analyzers and laboratory analysis. The information can also be contradictory. A stable plant operation is achieved by early reactions to disturbances and the smooth control with considerably smaller steps than operators, and thus the operators can concentrate on higher level tasks or on drastic changes in operating conditions. The fuzzy expert control is easy to comprehend and adapt because of its inherent pragmatic and human modeling approach. The first versions of the controller are built quite easily but it is not a trivial task to improve the performance since the procedure is not very systematic. In principle, increasingly advanced and complicated control strategies can be introduced, but in the same time maintenance becomes more and more difficult. This was already observed in the early applications (Plstergaard, 1996). The trial-and-error approach is very time-consuming and forms a bottleneck in controller design. The situation becomes worse when the number of variables increases. For extensive rule bases, the trial-and-error procedure becomes impossible because of a huge number of rules. The number of rules included can be reduced by handling the nonlinearities properly by membership functions. However, the resulting fuzzy controller is fairly coarse.

10.2.2 High level fuzzy control Industrial processes have usually several control objectives which should be fulfilled as much as possible. Many FLC applications show that various control strategies can be balanced fairly successfully if the proposed control actions are not too contradictory, e.g. a combination of five strategies included in the FLC of the lime kiln (Haataja and Ruotsalainen, 1994; Juuso et al., 1996) are well balanced in normal operation but different tuning is needed in other operating points. If the objectives are frequently in conflict, priorities have to be assigned to the control objectives, i.e. the importance of each objective depends on the operating conditions. Naturally, a high level FLC can also be combined with the direct conventional control. FLS Automation launched in early 1990 a high level control strategy for cement kilns. The system under the name FUZZY I1 has the following elements (astergaard, 1996):

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State index values for the actual process conditions, e.g. stability, product quality and production level, combine various measurements into a single figure.

a

Arrangement of the overall control strategy into groups of control objectives. Priority management makes transitions between objectives smooth and gradual by determining the extent to which the control actions should be executed. Control objectives specifying the individual goals that the FLC has to fulfil are implemented as objective modules consisting of deviation calculation, rule block, output program and timing calculations. The structure facilitates a stepwise implementation of control objectives and index calculations when gradually more process control knowledge becomes available. The rule blocks are normally formulated as fuzzy rules, but other techniques, like PID, neural networks, mathematical models, etc. can be used as well. The output program selects the control actions on the basis of process constraints and indices. The time interval between the control actions is reduced when the deviation from the target value increases. Multi-objectivecontrollers can be interpreted as switching controllers which can be improved with fuzzy logic in various ways, e.g. fuzzy controllers are very powerful as individual controllers. In the FLC at Partek cement plant, switching is based on conventional meta-rules. A fuzzy switching strategy can make the operation smooth as the switching is gradual. High level fuzzy control acts as a human operator by taking over decisions on the coordination of setpoints and on the production level. This may work very well most of the time, but the system will never be complete since process is usually changing all the time. Therefore, it is necessary that the operator can switch off the fuzzy controller and control the process manually if he does not agree with the controller actions. However, the operator does not always judge the situation correctly. He can even consider the fuzzy controller a competitor which will be switched off at the first opportunity. Fuzzy controllers, especially high level controllers, should be tools used by operators, i.e. the development work should focus on "how to get the operator back into the driver's seat" (Plstergaard, 1996). The continuous operation of the fuzzy control is guaranteed only if there is a real co-operation between the control system and the operator, i.e. the operator can make both manual control actions and small adjustments of the fuzzy controller without switching off the FLC, and the FLC can continue operation after these interventions. These actions will improve the overall performance if the operator is familiar and agrees with the FLC actions most of the time.

10.2.3 Linguistic equation controller Linguistic equations (Juuso and Leiviska, 1995) have been used in replacing and developing control rules for FLCs. This approach can easily be combined with other approaches for FLC design, e.g. to a conventional way of acquiring knowledge from experts by interviews (Leiviska and Juuso, 1996). The traditional fuzzy systems described by if-then rules are represented by matrix equations if nonlinearities are handled

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by membership functions (Juuso and Leiviska, 1992). The original approach has been extended to fuzzy and real-valued linguistic equations with variable specific partition levels (Juuso, 1996). The first direct linguistic equation controller was implemented in 1996 for a solar power plant in Spain (Juuso et al., 1997b). The properties of the controller have been extended (Juuso et al., 1998c; Juuso et al., 1998d), and recently the multilevel LE controller has been installed for a lime kiln at UPM Kymmene Pietarsaari mills (Juuso, 1998a).

Derivative Ae (a) A rule base represented by integer numbers.

Figure 10.3

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Change of Temperature Enor

(b) A control surface based on real valued linguistic equations.

A fuzzy P I controller represented by linguistic equations

A f u u y PI type controller can be represented by control rules, as shown in Figure 10.la. All these rules can be obtained from a single linguistic equation

which is a special case of the matrix equation AX = 0 with the interaction matrix 1, and variables X = [ e Ae Au IT. In the rule base shown in Figure 10.3a, the linguistic values NB, NS, ZO, PS, PB are replaced by integer numbers -2, -1, 0, 1 and 2. The control surface of the fuzzy PI controller shown in Figure 10.lb can be obtained by using the integer valued linguistic equation together with trapezoidal membership functions. Equation 10.1 is also applicable on any fuzzy partition, and the procedure produces always a rule set which is complete, consistent, and continuous. If an incomplete set is satisfactory, a part of the rules can be deleted already before tuning. The best similarity with the LE controller is achieved if the positions of all the fuzzy sets are coordinated with the real-valued linguistic equations, i.e. linguistic values are replaced by real numbers. The real-valued LE controller provides a very smoothly changing control surface (Figure 10.3b).

A = [ 1 1 -1

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to slower the speed of error compensation already before the error goes to zero. The braking action is automatically activated when the error becomes large enough, and the system returns to the normal operation when the fast trend is removed. The steps activated during a one day operation of a solar power plant (Figure 10.5) modified membership definitions considerably (Figure 10.6). In changing operating conditions, some trends may push the controlled variable causing a slight offset which is removed very slowly if the control surface is symmetric. Since different asymmetry is needed in different operating conditions, an asymmetrical correction defined by the trend variable is included. If several trends effect to the offset, an asymmetry equation can be introduced. This action is not used simultaneously with the braking action. Actually, the modified membership definitions shown in Figure 10.6 include the asymmetrical effects as well.

Figure 10.5

An example of linguistic values of original error steps.

This implementation is very compact if compared with model-based predictive controllers which are alternative solutions for these situations. The overall operation corresponds to a three level cascaded controller: Basic PI type LE controller handles the normal operation with symmetrical membership definitions; Operation condition controller changes the control surface of the basic LE controller by modifying the membership definitions for the change of control variable Au. Predictive LE controller changes the membership definitions for the derivative of the error Ae. This level contains both the braking and asymmetrical actions. The strength of the braking and asymmetrical actions can be modified if the operating conditions are clearly changing. Effective delays of the process should be taken into account in selecting the strength of braking. The asymmetrical action is restricted to fairly stable conditions.

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.

.

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Figure 10.6 An example of membership definitions modified by the braking and asymmetrical actions.

The multilevel linguistic equation controller has been applied to a solar power plant (section 10.6) and a lime kiln (Section 10.7). Both implementations are very compact and easier to tune than corresponding fuzzy controllers. Membership definitions are required for error, change of error, change of control, original error, trend, correction coefficient and all the working point variables. Modularity is beneficial for the tuning of the controller to various operating conditions, and most important is that the same controller can operate on the whole working area. For the high level control, calculation of state indices can be represented by linguistic equations. Actually, the linguistic value of the control change is at the same time a measure of the state deviation. These linguistic values can be used in selecting priorities for the control actions obtained from different control objectives. This extension is used in the lime kiln control (see Section 10.7). 10.2.4 Adaptation

Adaptation to changing operating conditions is necessary in process industry, e.g. testing results obtained in the solar power plant demonstrate this very clearly (Section 10.6). Retuning is also needed for the FLCs of the lime kiln when the operating conditions change considerably (Section 10.7). A simple fuzzy controller outperforms easily PID controllers, and the operating area can be extended considerably by higher level structures: the high level control provides tools for selecting control objectives,

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and the switching control expands the selection of the control strategies. In principle, the predefined adaptation could be implemented by cascaded FLCs, but the solution is very time-consuming to tune and also to maintain. A basic linguistic equation controller corresponds to a normal FLC, and the working point control extends the operation to adaptive fuzzy control. The braking and asymmetrical actions correspond to switching control. All these properties are included in the same controller which operates smoothly since the adaptation of membership definitions, the switching between special modes and the objective selection are done simultaneously and gradually. Complicated self-tuning algorithms are not included in the controller implementation. Understanding the meaning of different operating conditions is essential for successful tuning.

10.3 FUZZY MODELING Mathematical modeling and simulation can improve dealing with materials, energy and information through better understanding of the underlying mechanisms in the system. The tradeoff between the necessary accuracy and resulting complexity becomes increasingly important when the nonlinear and multivariable behavior must be taken into account. Adaptation to various operating conditions would also be very useful for industrial practice (Juuso, 1998b). Physical (mechanistic, first-principle, "white-box") modeling is based on a thorough understanding of the system's nature and behavior which is represented by a suitable mathematical treatment. Real systems are usually too complex and poorly understood for a complete mechanistic modeling on an acceptable level of complexity. The obvious risk of unrealistic simulations is very dangerous in the subsequent steps of analysis, e.g. prediction and controller synthesis cannot be successful. Data-driven modeling approaches are based on general function approximators ("black-box" structures) which should capture correctly the dynamics and nonlinearity of the system. The identification procedure, which consists of estimating the parameters of the model, is quite straightforward and easy if appropriate process data is available. The structure and parameters of these models do not necessarily have any physical or chemical significance, and therefore, these models cannot be adapted to different operating conditions. Many nonlinear multivariable systems can be transformed into mechanistic models described by differential and algebraic equations. Also data-driven techniques are very useful in modeling, e.g. the linear regression is an efficient method in developing multivariable models. There are more difficulties with nonlinearities: interpreting, generalizing and adapting of nonlinear regression models is not an easy task. In addition to this, there are usually large amount of process knowledge in a qualitative form which cannot be transformed to mechanistic models. Computational intelligence can provide additional tools since humans can handle complex tasks including significant uncertainty on the basis of imprecise and qualitative knowledge. Intelligent methods are based on techniques motivated by biological systems and human intelligence, e.g. natural language, rules, semantic networks and qualitative models. Most of these techniques were already introduced by conventional expert systems. Practical techniques for handling uncertainties and qualitative information were achieved in the fuzzy modeling. Very complex nonlinear models can be

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constructed by fuzzy set systems. Data-driven modeling can also use ideas originating from neural networks, data analysis and conventional system identification. Building fuzzy models from prior knowledge involves various knowledge acquisition techniques originating from conventional expert system.

F u u y modeling is an extension of the expert system techniques to uncertain and vague systems. Fuzzy set systems continue the traditions of physical modeling on the basis of understanding the system behavior. Fuzzy rules and membership functions can represent gradually changing nonlinear mappings together with abrupt changes. Fuzzy models can help in extracting expert knowledge on an appropriate level (Juuso, 1996). Fuzzy models can also be constructed from data, which alleviates the knowledge acquisition problem. Various techniques have been used to fit the data with the best possible accuracy, but in most cases the interpretation of results is not addressed sufficiently. Fuzzy models can also be considered as a class of local modeling approaches, which attempts to solve a complex modeling problem by decomposing into number of simpler subproblems (BabuSka, 1998). Fuzzy modeling is usually based on the following rule-based models: m

Linguistic fuzzy model (Driankov et al., 1993), where both the antecedent and consequent are fuzzy propositions, suits very well to qualitative description of the process as it can be interpreted by using natural language, heuristics and common sense knowledge. Membership functions are defined by the expert or by experimentation. The input-output mapping is realized by the fuzzy inference mechanism equipped with conversion interfaces, fuzzification and defuzzification.

l

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Takagi-Sugeno (TS) fuzzy model (Takagi and Sugeno, 1985), where the consequent is a crisp function of the antecedent variables, can interpreted in terms of local models. Usually, the consequent is a parameterized function, whose structure remains constant and only the parameters vary. For widely used linear functions, the standard weighted mean inference must be extended with a smoothing technique. TS fuzzy models are suitable for identification of nonlinear systems. Fuzzy relational model (Pedrycz, 1984), which allows one particular antecedent proposition to be associated with several different consequent propositions, can be regarded as a generalization of the linguistic fuzzy model. Each element of the relation represents the degree of association between the individual reference fuzzy sets defined in the input and output domains, i.e. all the antecedents are tied to all the consequents with different weights. Singleton model, where the consequent is a crisp value, can be regarded as a special case of both the linguistic fuzzy model and the TS fuzzy model. Defuzzification reduces to the fuzzy-mean method. All these models can approximate static and dynamic nonlinear systems. There are several alternatives for representation of the system's dynamics.

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Prior knowledge can used in constructing rule-based fuzzy models: qualitative knowledge can be incorporated in linguistic fuzzy models, or in fuzzy relational models if there are several alternative rules; locally valid linear models can be collected by TS fuzzy models. Data-driven fuzzy modeling can be based on following methodologies: Fuzzy clustering (Bezdek, 1981) can be used as a tool to obtain a partitioning of data. Clustering techniques belong to the unsupervised learning methods. A large number of algorithms have been proposed, and applied to a variety of real-world problems. Methodologies, which decompose the problem into a set of locally linear models, are very suitable for constructing TS fuzzy models. Self-organizing maps (Kohonen, 1995) can be interpreted as a clustering technique suitable for preprocessing before fuzzy rule generation. Resulting neuron model can be generalized by linguistic equations (Juuso, 1996). Antecedent membership functions can be generated from the results of fuzzy clustering. The consequent part of a fuzzy TS model is developed as a linearization around the cluster center (BabuSka, 1998). Rule generation is usually based on membership functions defined in the procedure. Table-lookup scheme (Wang and Mendel, 1992; Wang, 1994) is a one-pass procedure for generating fuzzy rules from numerical 110-data with capability to combine linguistic information into a common rule base. This methodology does not offer any means to identify the structure of the system. Fuzzy rule generation can also take into account contradictory data (Krone and Kiendl, 1994; Krone and Schwane, 1996). Neurofuzzy methods provide various techniques for generating fuzzy set systems, e.g. ANFIS method (Adaptive-Network-basedFuzzy Inference Systems) is a well known neurofuzzy method which is suitable for tuning of membership functions (Jang, 1993). Fuzzy models on any fuzzy partition can be generated from linguistic equation models (Juuso et al., 1996). Different approaches can combined in the tuning phase, especially the linguistic equation approach is designed for combining different sources of information.

10.3.2 Linguistic equations Linguistic equations (LE) provide a compact method for extending the application areas of computational intelligence. According to the original framework (Juuso and Leiviskk 1992), a set of linguistic rules or relations can be changed into a compact equation m

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where Xj is a linguistic level for the variable j, j = l...m, i.e. the linguistic values very low, low, normal, high, and very high are replaced by numbers -2, -1,0, 1 and 2. The direction of the interaction is represented by coefficients A,, E {-1,0,1) if same level of fuzzy partition is used for all variables. Several sets of linguistic relations can be combined by matrix presentation AX = 0. Each equation represents a multivariable interaction: the directions and strengths of interactions are defined by coefficients of the interaction matrix. Only the variables with a nonzero coefficient belong to the interaction. Nonlinearities are taken into account by membership definitions which consists of two polynomial functions, one for positive and one for negative side labels. An example of membership definitions and membership functions is shown in Figure 10.7. With these definitions the values of input variables are mapped to the range -2...2 which correspond to the labels very low .... very high (or negative big ... positive big); normal (or zero) is always zero. After the matrix calculations, the outputs are mapped from the linguistic level to the real scale. Since only five parameters is needed for each variable, the LE systems can be adapted to various operating conditions. Usually, fuzziness is taken into account by membership definitions - linguistic equations approach does not necessarily need any uncertainty or fuzziness. However, also the linguistic equations can be used in fuzzy form (Juuso, 1996),e.g.

4

4

i.e. all the variables are not included to the model. The fact that experts do not always agree with interactions can also be taken into account by using several interaction matrices with different coefficient values. On the other hand, the directions of interactions can depend on the working area in nonlinear systems. In these cases, different interaction matrices have different degrees of membership. Naturally, a combination of these effects can be taken into account as well. The meaning of the linguistic values depends on the working point. Membership functions for finer partitions are more easily generated by a gradually refining set of levels. Fuzzy partition can be chosen independently for each variable, and some adaptive features can be achieved by this representation (Juuso, 1996), e.g. in a pressure modeling case four fuzzy models corresponding to different valve positions can be combined by a single LE model

if the partition level is one for both the valve position u and the current pressure pl; valve positions open, half open, almost closed, and closed are replaced by numbers 1 , 0, - 1 and -2, respectively, and pressure levels low, medium, high, and very high by numbers -1, 0, 1 and 2, respectively; the resulting new pressure pz is obtained on the partition level three. In the LE systems, the partition level is needed only for development of fuzzy set systems. Different strengths of the interactions are handled by using other positive and negative integers. Corresponding TS and singleton fuzzy models for the nonlinear pressure dynamics of this laboratory fermenter are presented in (BabuSka, 1998).

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real-valued linguistic equations (RLE) provide a basis for a sophisticated nonlinear systems where fuzzy set systems are used as a diagnostic tools. Fuzzification and defuzzification are integrated to the flexible scaling generated for membership definitions (Juuso, 1996). These systems can be easily tuned by neural networks, e.g. self-organizing maps and linear networks have been tested. Linguistic equations provide a method for generalizing results of neural networks. A real-valued linguistic equation system can be even considered as a new neural network type. Extension to real numbers was introduced because of difficulties to handle variables with dissimilar fuzzy partitions (Juuso, 1996). Actually, the example described by Equation 10.5 can be represented by even better by RLEs by using interaction matrix [4 - 1- 41 together with same membership definitions for both pressures. The RLEs are also used in a tuning algorithm for reducing the error between model and training data. Linguistic equation approach combines various intelligent modeling techniques on a unified framework: it was originally developed for handling large knowledge bases in process design (Juuso and Leiviska, 1992) ; a close connection to fuzzy set systems was important already in the early applications (Juuso, 1994); data-driven modeling properties have brought the approach close neural network techniques. Fuzzy modeling and control was the main application area. Properties of the LE approach are continuously improved and extended on the basis of industrial experience with various application areas, and recently, emphasis is moving from development to direct applications in control (Juuso et al., 1997b;Juuso and Balsa, 1997), in intelligent analyzers (Murtovaara et al., 1998), in fault diagnosis (Juuso et al., 1996; Juuso et al., 1998a), and in model-based control design (Juuso, 1998a; Juuso et al., 1998b). Fault diagnosis and intelligent analyzers are combined in model-based diagnostic process analysis (MDPA) (Juuso, 1997): the resulting systems can be used in various ways suitable for software sensors, risk analysis and detection of sensor failures. Sophisticated trend information can be utilized by temporal reasoning on the recent process history. The MDPA methodology has been tested with simulations, expert knowledge and real data. Fuzzy models can changed into linguistic equation models by replacing linguistic labels with real numbers. The FuzzEqu toolbox (Juuso, 1996) includes routines for building a single LE system from a large fuzzy systems including various rule blocks implemented in FuzzyCon or Matlab Fuzzy Logic Toolbox. Other fuzzy modeling approaches can be used as channels for combining different sources of information: local linear models are considered as Takagi-Sugeno fuzzy models, and fuzzy clustering and self-organizing maps are used for preprocessing of data. Fuzzy models on any fuzzy partition can be generated from LE models: rules or relations are developed either sequentially or simultaneously (Juuso et al., 1996), and membership functions are generated from the membership definitions on any location. In each equation, the locations of membership functions are defined by the model for one selected variable. Singleton models represent the LE model quite accurately if the locations of the membership functions are based on the shapes of the membership definitions in such a way that the linear surfaces are on appropriate areas. Takagi-Sugeno (TS) fuzzy models have the same requirement, but the locations of membership functions are different. Linguistic fuzzy models are developed from

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singleton models. Fuzzy relational models are useful for fuzzy LE models. There will be fairly few nonzero elements since nonlinearities are included to the membership functions. 10.3.3 Steady state modeling

In the lime kiln modeling, fuel and lime rates were used as inputs and the output was the burning end temperature (Juuso et al., 1997a). The data set was difficult for modeling since the lime feed changes rapidly from 35 t/h to 41 t/h, and otherwise there are no changes. The membership functions created by ANFIS were symmetrical and hardly changed by training. For the lime rate there are no mentionable changes in membership functions after training. The ANFIS model cannot deal with lime feed values between 35 and 41 if the lime feed has more than two membership functions, i.e. the model cannot be generalized. The interaction of the lime feed and the burning end temperature is incorrect when the lime is on lower area since the data does not include enough material on this area. The membership functions from linguistic equations approach are much more nonlinear (Figure 10.7). The model can be represented by a single linguistic equation with interaction matrix

In this case expert knowledge was used to compensate the weaknesses in input data. The feasible ranges generated from the data were modified to cover a wider operating range. The effect of lime feed was also changed negative and the rates for fuel feed and burning end temperature were increased to reduce the effect of lime feed to the final model. Again the initial system provides similar results as the ANFIS system for the training data. The error is reduced considerably by tuning: the result is very precise already after 2 epochs. The results shown in Figures 10.7 ... 10.1 1 were obtained after 10 epochs. The linguistic equation models can be used in many ways. Inverting of the model is easy as each of the variables can calculated on the basis of other two variables. Already the initial models worked very well both for temperature estimation and fuel rate control, and the results were further improved by tuning (Figures 10.8 and 10.9). The data did not include many alternatives for the lime rate, and therefore the results are not as good for that. Fuzziness of the equation system shows how well the equation represents the data. During the tuning process, the fuzziness is decreased, i.e. all the values of this curve will go towards zero. The system can be compared with interactive testing (Figure 10.10)and with surface plots (Figure 10.11). All this can be done either on the basis of membership definitions or on the basis of membership functions. In Figure 10.10, the actual modeling is based on membership definitions, and membership functions are used for diagnostic purposes. Model surfaces are very smoothly changing if membership definitions and real-valued linguistic equations are used. Model surfaces are important for assessing the directions of interactions, e.g. the incorrect effect of the lime feed was clearly seen from the surface plot.

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Figure 10.8 Comparison of linguistic equation model for burning end temperature after training 10 epochs.

10.3.4 Dynamic modeling Dynamic fuzzy models can be constructed on the basis of state-space models, inputoutput models or semi-mechanistic models (BabuSka et al., 1997). In the state-space

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models, fuzzy antecedent propositions are combined with a deterministic mathematical presentation of the consequent. The most common structure for the input-output models is the NARX /Nonlinear AutoRegressive with exogenous input) model which establishes a relation between the collection of past input-output data and the predicted

FUZZY CONTROL IN PROCESS INDUSTRY

Lime feed

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Fuel feed

10.11 Model surfaces of the linguistic equation for two input- one output model.

output:

where k denotes discrete time samples, n and m are integers related to the systems' order. MIMO systems can be built as a set of coupled MIS0 models. Delays can be taken into account by moving the values of input variables correspondingly. In the semi-mechanistic models, the modeling task is divided into two subtasks: wellunderstood mechanisms, e.g. mass and energy balances, are modelled with mechanistic methods, and approximations of partially known relationships and parameters with fuzzy methods. Dynamic neural networks and linguistic equations can be based on similar structures as the dynamic fuzzy models (Juuso, 1998b; Juuso, 1998a). The basic form of the linguistic equation (LE) model is a static mapping in the same way as fuzzy set systems and neural networks, and therefore dynamic models will include several inputs and outputs originating from a single variable. The dynamic behavior is provided by external dynamic models: Rather simple input-output models, e.g. the old value of the simulated variable and the current value of the control variable as inputs and the new value of the simulated variable as an output, can be used since nonlinearities are taken into account by membership definitions. For state-space modeling, each rule correspond to a different working point, and the deterministic equations on the consequent part is transformed into a linguistic equation model. Since the LE model can handle nonlinearities, at least some

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which are then used in generation of interaction alternatives. The equations with best fit to the data are selected to the interaction matrix: each new equation introduces at least one variable to the model. Any equation can be rejected or modified on the basis of expert knowledge. For each model, initial estimates of the delays are developed by correlation analysis. The method is suitable for handling data from process experiments where the changes are defined by the experimental program. For normal process data, levels of changes are not so clear, and therefore, similarities detected by correlation analysis can be accidental in some cases. For the lime kiln, the resulting delays were too large for some variables. The delays should be assessed against process knowledge, especially if normal on-line process data is used (Juuso, 1998b). The delays can also be tuned with linguistic equation models. In the present system, delays are handled as triangular fuzzy numbers already on each single model. For model development, the training data consist of 6 data sets corresponding 6 working point areas defined by production and fuel trend (Juuso, 1998a). Some overlap of the working point areas is automatically introduced since the fuel rate is not continuously increasing or decreasing as can be seen from simulation results shown in Figure 10.12. The delays were removed before tuning. The interaction vector is the same for all working areas, which is quite reasonable since the directions of interactions do not change considerably between different working points. The differences between the models are handled with membership definitions.

Time (min)

Figure 10.13

Simulation test with testing material.

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Multimodel approach A multimodel approach based on fuzzy LE models has been developed for combining specialized submodels (Juuso, 1998b; Juuso, 1998a). The approach is aimed for systems which cannot be sufficiently described with a single set of membership definitions because of very strong nonlinearities. Additional properties can be achieved since also equations and delays can be different in different submodels. In the multimodel approach, the working area defined by a separate working point model. In the FuzzEqu toolbox, the model can include n x m submodels if there are n working areas and m subareas. Each submodel is a single model whose output is a vector of singleton values. Singleton values of each output variable are aggregated by taking a weighted average. The weight of each submodel is equal to its degree of membership defined by the working point model. The calculation of the degree of membership can be a simple fuzzification procedure or a very complex LE system, e.g. in the web break indicator, the working point model consists of 25 cases, each modelled with about 20 equations (Juuso et a]., 1998a). The working point model is usually a steady state model based on direct measurements and some trends. For some variables, changes in signal level or variances and peak values could be useful. Each submodel has its own delay vector for the variables. In this way the delay is changing with time as the multimodel system switches gradually from one submodel to another. For slow processes, also trends are analyzed for some variables. The results are used as additional working point variables. Usually several process cases have a non-zero degree of membership, and these degrees are changing in time, i.e. the delays can be considered as smoothly changing fuzzy numbers. The multimodel approach is necessary for the modeling of the lime kiln, especially because of varying delays, but there are also considerable differences in membership definitions. Since the lime kiln process is fairly slow, and the working point is continuously changing (Figure 10.12), the individual single models should not be too different from each other in order to achieve a smoothly changing operation.

4

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Tuning Tuning is based on real-valued linguistic equations and membership definitions (Juuso, 1998b; Juuso, 1998a). Final membership definitions become consistent with the interactions presented by linguistic equations. Each equation model can be tuned and used to any direction, i.e. any variable can be calculated if sufficient number of other variables is known. For more complex systems, the limitations for solving a matrix equation are taken into account in the development environment. In the tuning of membership definitions, a polynomial function is fitted for the data points and the corresponding linguistic levels obtained by the current membership definitions and interactions. The basic form of the LE model is a static mapping, and therefore dynamic models will include several inputs and outputs originating from a single variable. For the tuning, all these inputs and outputs should have the same membership definitions. In the multimodel approach, submodels are tuned separately.

Testing Test results of the multimodel simulator with the training data are shown in (Juuso, 1998a). The one-step-ahead simulation is very accurate although the mul-

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timodel system does not always use the same model which was used in the tuning. Testing with independent data is essential in order to use the dynamic models in control design. One-step-ahead testing described in (Juuso, 1998a) produces also very accurate results, which is quite natural since the changes are fairly small on each simulation step. For controller tuning, more tests are needed since the accuracy requirement is very high. A more demanding test is again to use the calculated values always as an input on the following simulation step. The results are very good (Figure 10.13): there are only few considerable differences. According to an analysis with additional simulation tests, these differences can be caused by fairly short disturbances in fuel quality. Another reason for this kind of behavior could be an oscillation of kiln temperature caused by earlier control actions.

10.4

FUZZY MODELS I N NONLINEAR CONTROL

Nonlinear models can be developed for dynamic systems by various fuzzy methodologies. If a reasonably accurate model is available, it can be used to design a nonlinear controller for that process. Two particular techniques are considered in this Section: model inversion and model-based predictive control.

10.4.1 Control by model inversion For stable systems whose inverted dynamics are stable, a nonlinear controller can be designed by inverting a fuzzy model. Theoretically, a perfectly stable control exists in the ideal case over the entire operating range since the approach takes into account the process nonlinearities, including the inherent saturation levels of the control inputs and other process variables. Generally, it is difficult to find the inverse for a given fuzzy system. Fuzzy models with singleton consequents can efficiently be inverted by analytical manipulations, which is generally not possible with Takagi-Sugeno models, as both their antecedents and consequents depend on the inputs (BabuSka, 1998). An exact inversion is guaranteed if the membership functions and the fuzzy operators are chosen such that a multi-linear interpolation among the rules is obtained. The core of each membership function must be a single point, and the defuzzification method must be invertible (monotonic). Noninvertible rule bases are splitted into two or more invertible parts, and separate membership functions are built for each rule base. A model based on a single linguistic equation is always invertible, i.e. the control action can be calculated if all the other variables are known. Figures 10.8 and 10.9 shows an example of model inversion. The corresponding control surface is shown in Figure 10.4b. For a set of linguistic equations, inversion can be done as well. The control action can calculated if a sufficient number of variables is known. There are usually several feasible sets of input variables: the principles of the variable selection are the same as for ordinary sets of equations. For dynamic models, rather simple input-output models, e.g. the old value of the simulated variable and the current value of the control variable as inputs and the new value of the simulated variable as an output, can be used since nonlinearities are taken into account by membership

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definitions (Juuso, 1998a). For nonminimum phase systems, the models consist of two or more submodels, and a separate LE controller is developed for each submodel. For the lime kiln control, inverted steady-state models are used for getting the control variables to approximately correct values when the fuel type is changed. The normally used PI type LE controller cannot work immediately after the change because the feed rates of the different fuels have completely different ranges. In these cases, considerable changes have to be made to the speed of the flue gas fan. The inverted dynamic model is not useful because the range of the temperature values will also depend on the fuel type. After the transient period, the normal LE controller takes care of the feedback control. In the ideal situation with no modeling errors and disturbances, a controller obtained by model inversion yields perfect control with zero steady-state errors. In practice, there are both modeling errors and disturbances, and some type of feedback control is needed. The Internal Model Control (IMC) scheme is one way of handling the model-plant mismatch (BabuSka, 1998).

10.4.2 Model-basedpredictive control Model-based predictive control (MPC) has been widely applied in industry and has been accepted both by control engineers and plant operators. MPC is a general methodology for solving control problems in the time domain. Process models are used explicitly to predict the future outputs. An optimal control strategy is obtained by minimizing a cost criterion which usually includes a difference between the desired and predicted outputs, and a penalty on the control effort. Various constraints on the control inputs and other process variables or their rates can be taken into account as a part of the optimization problem. Prediction and control horizons, and the weights of the controlled and manipulated variables are appropriate tuning parameters of the optimization scheme. MPC is applicable to a wide range of processes including multivariablenonlinear systems with dead times, nonminimum phase, or unstable systems, and can deal with constraints in a systematic way. It can also make efficient use of known future references for path following, setpoint tracking, batch process control etc. Finally, it is a methodology open to extensions and new approaches in process modeling and optimization (Camacho and Bordons, 1995). Reasonably accurate models are needed since the performance of MPC depends on the accuracy of the model. In industrial applications, prediction of the process behavior is needed for large disturbances, i.e. nonlinear models are necessary. The complex, nonlinear multivariable models with uncertain parameters may require extensive computation time. The models should be simple to implement, cost-effective to simulate, and easy to understand. In fuzzy modeling both numerical and qualitative models can be combined, and the resulting models provide a certain transparency allowing interpretation and analysis of the process behavior. For instance, an input-output Takagi-Sugeno model is a very suitable model structure for this purpose (BabuSka, 1998). Understanding of the underlying phenomena is an essential feature in linguistic equation models which can represent input-output models in a very compact form (Juuso, 1998a). Even the

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multiple LE model simulations with changing effective delays are fast enough for real-time optimization with long prediction horizons. The predictive control scheme with linguistic equation models has been used only in the tuning of the braking action for large disturbances.

10.5

ADAPTIVE FUZZY CONTROL

In industrial applications, operating conditions are often changing so strongly that the changes in nonlinearities must be taken into account. Adaptive controllers generally contain two extra components compared to the standard controller (Driankov et al., 1993). The first is a process monitor which detects changes in the process characteristics either by a performance measure or by a parameter estimator. The second is the adaptation mechanism which updates the controller parameters. 10.5.1 Process monitor Changes in process characteristics can be detected through on-line identification of the process model, or by assessment of the controlled response. The model can be a transfer function, a discrete time linear model, a fuzzy model or a linguistic equation model. Adaptation mechanisms rely on parameter estimates of the process model, e.g. gain, dead time and time constant. The choice of perfarmance measures depends on the type of response the control system designer wishes to achieve. Alternative measures include overshoot, rise time, settling time, decay ratio, frequency of oscillations, gain and phase margins and various error integrals (Driankov et al., 1993). For complex problems, the performance is measured by the level of satisfaction on any number of goals and constraints. Importance of each performance measure depends on operating conditions task since industrial processes have usually several control objectives. The active measures should be selected on the basis of high level control strategies. Fuzzy controllers include diagnostic features (Juuso and Leiviskk 1994): on the coarse fuzzy partition level, the adaptive expert system is applicable to diagnostic purposes. If some variables become high or low, the fault diagnosis system activates the f u u y control system. If some variables become very high or very low, the working point has changed beyond the area where the fuzzy control system can take care of the process. The fuzzy control system is used on the finer fuzzy partition level. Diagnostical information can provide useful material for performance evaluation, but in real-world systems faults appear in various forms. Therefore, it may be difficult to find all the necessary information for the adaptation procedure. Model-based architectures of modern fault diagnosis systems as such are not very flexible, the flexibility must be taken into account in the adaptive controller (Rauma, 1997).

10.5.2 Adaptation mechanism Fuzzy controllers can already be designed to cope with a fairly large amount of nonlinearity in the process behavior along an operating condition, but the operating area can extended by the following mechanisms:

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The changes in the amount of the nonlinearity can be taken into account by changing scaling factors for appropriate variables. Altering the scaling factor is similar to gain tuning in standard PID controllers as it changes the sensitivity of the controller to the input.

4

Changing the membership functions will alter the shapes of the fuzzy sets, and thus the gain can be modified within a specific region. Altering rules can change drastically the controller operation, and thus it should be the last alternative in the adaptation procedure. This type of controllers are called self-organizing controllers. Scaling is suitable for on-line adaptation, but the other methods should be used with some constraints, and preferably off-line. The disadvantage of many self-organizing controllers is that the nonlinearity is captured by a very complex rule base because the membership functions are usually equally spaced symmetrical functions. The multilevel LE controllers already correspond to adaptive fuzzy controllers, but adaptation can be improved by the following mechanisms:

a

Altering the strengths of braking and asymmetrical action will fine-tune the controller operation for large disturbances and remove the steady-state error. The changes in strengths of the nonlinearity can be taken into account by scaling the membership definitions for appropriate variables. This mechanism, which is similar to the scaling in fuzzy controllers, is the basic technique in the operating point adaptation. m

Asymmetrical scaling is the main technique in braking and asymmetrical actions.

m

Changing membership definitions will alter the gain within a specific region. This mechanism is used if the adaptation of the multilevel LE controller is not sufficient.

4

Changing equations, which means that the control strategy is changed, is very seldomly used. The above also is the preferable sequence of use: changing the strengths of braking and asymmetrical actions and the scaling mechanisms are suitable for on-line adaptation, but the other techniques should be used with some constraints, and preferably off-line. In a meta-rule approach, parameters of a low-level controller are changed by a metarule supervisory system whose decisions are based on the performance of the low-level controller. Meta-rule modules consist typically a fuzzy rule base that describes the actions needed to improve the low-level FLC (Maeda and Murakami, 1992; Daugherity et a]., 1991; Isomursu and Rauma, 1994).

10.5.3 Controller tuning with adaptive fuzzy control An adaptive fuzzy controller can be run from a variety of initial conditions so that the trajectories of the controlled system cover the whole operating area, then the rules

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are constructed from the stored parameters of the adaptive fuzzy controller along the trajectories. Human decisions are needed only for deciding about the fuzzy sets for each variable. Wang has proposed a design method for linear sets of membership functions (Wang, 1998). This scheme can be modified to any parameterized controller by using the techniques described above. The adaptive tuning is a feasible solution if the process conditions can be varied in this way. Normally, this is not possible for industrial processes. However, this scheme can be used off-line if a reasonably accurate models are available.

10.6 APPLICATION: SOLAR POWER PLANT Solar power varies independently and cannot be adjusted to suit the desired demand. The aim of solar thermal power plants is to provide thermal energy for use in an industrial process such as sea water desalination or electricity generation. If such plants are to provide a viable, cost effective alternative to more polluting forms of power production, they must achieve this task despite fluctuations in their primary energy source, the sunlight. In addition to predictable seasonal and daily cyclic variations, the radiation intensity depends also on atmospheric conditions such as cloud cover, humidity, and air transparency. Solar thermal power plants can be based on distributed collectors. A distributed collector field is an ideal test-bed plant for control implementations as it represents complex dynamics and strong disturbances acting on the plant during the daily operation. This Section presents some results obtained by fuzzy logic and linguistic equation controllers in the ACUREX distributed solar collector field of the Plataforma Solar de Almeria (PSA) located in the desert of Tabernas (Almeria), in Southern Spain. There has been various research efforts to improve the efficiency from control and optimization viewpoints, recently within the European Union Training and Mobility of Researchers Program promoted by CIEMAT-Plataforma Solar de Almeria.

10.6.1 Plant description The ACUREX solar collectors field (Figure 10.14) supply thermal energy in the form of hot oil to an electricity generation system or a multi-effect desalination plant (Kalt, 1982; Zarza, 1995). The solar field consists of twenty rows of east-west oriented, one-axis elevation-tracking, parabolic collectors in ten parallel loops. There are a total of 480 modules in the field. The total reflective aperture area of the ACUREX collector field is 2,674 m2. The parabolic collector (Figure 10.15) uses a curved mirror surface to concentrate the solar radiation onto the receiver tube, which is located at the focal zone (line) of the parabola. The heat transfer fluid, synthetic oil, is pumped through the receiver tube, thus collecting the thermal energy gained through the receiver tube walls. The variations with the temperature of density, viscosity, and specific heat affect strongly the gain of the system. The field is provided with a sun tracking system which causes the mirrors to revolve around an axis parallel to that of the pipe to enable the varying inclination of the sun to followed.

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Figure 10.14

Acurex distributed solar collector field

Figure 10.15

A parabolic collector

The facility has a 4 MWh storage system which consists of a single 140 m3 thermocline oil storage tank which holds 110 m3 of Santotherm 55 synthetic oil and 30 m3 of inertizing N2. The cold inlet oil is extracted from the bottom of the storage tank and passed through the field using a pump located in the field inlet. The system is provided with a three way valve located at the field outlet that allows the oil to be recycled within the field until its outlet temperature is adequate for entering into the top of the storage tank. The aim of the tank is to provide the secondary process with a degree of independence from the fluctuating solar input to the field. This means that the secondary system can operate during short term interruptions of the solar input. Currently the secondary system is a desalination plant. It is a 14-cell multi-effect desalination plant capable of

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producing 72 rn3/day of distilled water. The hot oil from the top of the storage tank is sent to the low pressure boiler, thus producing 70 "C, 35 b a r steam which is sent to the first effect evaporator of the desalination plant. Though the plant could also be fed with the 16-to-26-bar steam produced in the PSA Electricity Generation System, this system is not working at present. 10.6.2

Control problem

The purpose is not to maintain a constant supply of solar produced thermal energy in spite of the disturbances because this is not a cost effective strategy, i.e. in theory at least an oversized collector should be used because of periods of low solar intensity. Rather the aim of the control scheme should be to regulate the outlet temperature of the collector field in order to supply steam to the turbine in a range as constant as possible despite the disturbances and uncertainties, changes of the solar radiation, ambient temperature, inlet oil temperature etc. (Juuso et al., 1997b; Meaburn, 1995; Zarza, 1995). This is beneficial in a number of ways. Firstly, it collects any available thermal energy in an useable form, i.e. at the desired temperature, which improves the overall system efficiency and reduces the demands placed on auxiliary equipment as the storage tank. Secondly, the solar field is maintained in a state of readiness for the resumption of full scale operation when the intensity of the sunlight rises once again; the alternative is unnecessary shutdowns and startups of the collector field which are both wasteful and time consuming. Finally if the control is fast and well damped, the plant can be operated close to the design limits thereby improving the productivity of the plant. The prime control requirement of solar power plant is to maintain the outlet oil temperature of the field as a constant value. Since the solar radiation can not be manipulated, this can only be achieved by means of varying the flow pumped through the pipes during the plant operation. The dynamics of the process depends on the general field operating conditions and characterized is by the following aspects: A time varying transport delay which depends on the manipulated variable (oil flow rate). The dynamics, in particular high frequency peaks in the frequency response of the plant, are difficult to model; the field presents a clear antiresonance frequency response. The plant has a nonlinear behavior, and therefore linearized models depend on the operating point. The solar radiation acts as a fast disturbance with respect to the dominant time constant of the process. A feedforward approach based directly on the steady state energy balance relationships reduces complexity since both solar radiation and inlet temperature are measured (Camacho et al., 1992). The control system implemented on a PC has been developed in-house at the PSA. This software written in C-programming language serves many purposes. Firstly, it

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logs measurements taken from the field via the data acquisition cards and displays them as a mimic diagram on the screen. Secondly, it monitors the collector field status and prevents potentially hazard situations, which might occur, e.g. oil temperatures above 300 OC. When a dangerous condition is detected the software automatically intervenes, warning the operator and defocusing the collector field. The controller presented in this Section is operating under this software. The dynamic characteristics of the plant vary significantly over the operating range making it difficult to obtain an adequate control performance by a fixed parameter PI controller. A controller tuned for medium flow conditions tends to oscillate for the low level conditions when a high setpoint coincides with low irradiation (Camacho et al., 1992). Several advanced approaches for controlling solar collector fields have been reported in the literature. Relative merits of one controller with respect to another are difficult to demonstrate since the exact solar radiation and inlet conditions can never be reproduced. Therefore, algorithms have been tested by simulation. Plant results with adaptive schemes, e.g. a self-tuning PI controller based on a pole-placement approach (Camacho et al., 1992) and a prescheduled adaptive control by resonance cancellation (Meaburn and Hughes, 1994), have clearly demonstrated the importance of adaptive tuning. Generalized predictive control (GPC) based on a gain schedulingalgorithm was able to handle different operating conditions and sudden perturbations caused by clouds (Camacho and Berenguel, 1994). An adaptive GPC approach provided fast response with high overshoots and some oscillations (Camacho et al., 1994). A nonlinear neural model-based predictive control was presented in (Arahal et al., 1997): the models are not accurate enough for eliminating the offset, but under extreme solar radiation conditions the controller worked fairly well since the neural network models introduced a kind of feedforward effect. The multi-predictive adaptive MUSMAR algorithm has shown to have robust characteristics with respect to unmodelled dynamics (Coito et al., 1997). In a cascade structure, the temperature of the storage tank is controlled to deal with different time constants. A dual version of the MUSMAR controller provides a smooth start-up transient. A MUSMAR based switching controller deals with fast dynamic changes by selecting the active controller on the basis of the predictive models (Rato et al., 1997). Each candidate controller has been tuned in order to match a region in the plant operating conditions. The feedforward controller has been an essential part of the control loop in these approaches. Although simulation tests have shown that the responses can be further improved by tuning, these improvements are not easy to realize online in changing operating conditions.

10.6.3 Fuzzy logic controller A PI type conventional fuzzy logic controller presented in (Rubio et al., 1995) was designed manually on the basis of the experience of the previous experiments: the error and the change of error were the input variables, and the output, the change of control, was obtained by the weighted average defuzzification. The FLC operated fairly smoothly on normal operating conditions.

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An automatic genetic design technique was later developed (Gordillo et a]., 1997) since the trial and error methodology make the improving task hard and difficult. The controller structure was also modified: the oil flow was included to the input variables, the number of membership functions was reduced, the membership functions of the input variables were kept unchanged, and the GA was used only to find the midpoints of the output membership functions (Gordillo et al., 1997). Simulation was used in the genetic tuning. The resulting FLC outperformed the experience based FLC also in the field tests as the additional input variable improves adaptation to the changing operating conditions. Automatic design techniques are necessary when additional variables are included to the rule base since the tuning becomes more complicated. A PI type conventional FLCs can also be improved by multiplying the inputs and outputs by coefficients depending on the operating conditions. According to the tests in 1998 at the ACUREX field, this technique can improve performance considerably. The linguistic equation controllers provide a very sophisticated techniques for further adaptation of fuzzy controllers.

10.6.4 Linguistic equation controller The linguistic equation controller first tested in 1996 has been improved in many ways, e.g, the offset was removed by the asymmetrical effect and the start-up has been smother due to the braking action. The experiments were carried out in June and September 1997. The controller is based on the PI-type fuzzy controller represented by (10. I), where the interaction matrix is A = [ 1 1 -1 1, and variables are defined as The error variable is the deviation of the outlet temperature X = [ e Ae Au from the setpoint. The control variable is the oil flow. Effective solar radiation and temperature difference between inlet and outlet temperatures are used as working point variables. The correction is obtained by an additional linguistic equation

IT.

correction = irradiation - temperature diflerence

(10.7)

which increases the absolute value of the oil flow change when the irradiation is high and the temperature difference low, i.e. a usual situation around the solar noon period. Before noon or after load disturbances, this equation decreases the absolute value of the oil flow changes in order to avoid oscillations. Operation condition correction is implemented as a linguistic equation controller, which changes the properties of the lower level controller by making the control surface either flatter or steeper. Linguistic equation controllers can be pre-tuned to various operating conditions. When difficult cloudy conditions occur, the working point is changing all the time (Figure 10.16) because of very abrupt changes in irradiation. Longer cloudy periods have also an effect on the temperature difference. The correction level changes between negative big and positive big (Figure 10.17): negative values are necessary to remove oscillations and positive values for speeding up operation. Values for the working point variables are obtained by a moving average on a selected time range. For large disturbances, this controller implementation can lead to a large overshoot, which is problematic if small control actions are required, e.g. in oscillatory situations. This problem is solved by taking the original temperature deviation into account. The

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0

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Linguistic levels of the working point variables on June 17,1997.

Local Time

Figure 10.17

Correction level based on the working point variables on June 17, 1997.

braking action is implemented by a correction, which increases the importance of the

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change of the temperature error when the temperature approaches the setpoint. The asymmetrical action is based on the time difference to the solar noon. All these properties are implemented into a very compact file by C-programming language. Membership definitions are required for eight variables: temperature error, change of temperature error, change of oil flow, irradiation, temperature difference, original temperature error and working point correction coefficient and trend variable. Tuning of the system was carried out with Fu&qu toolbox in Matlab TMenvironment. Although the controller does not need the membership functions, they are used in offline tuning and testing.

10.6.5 Controller testing The Acurex Solar Collectors Field of the Plataforma Solar de Almeria has been used as a test bed for various control methods. The Acurex field is the first direct application of linguistic equation controllers. Testing in the real plant was started in 1996 first with a single equation controller based on equation (10.1). The final linguistic equation controller corresponded to a three level cascaded fuzzy controller. Robustness of the controller was tested by trying difficult operating conditions, e.g. using high setpoint temperatures when the solar radiation was relatively low, or going to lower setpoints when the solar radiation was high. Robustness was clearly already obtained in the case of the single equation controller. The feedforward controller described in (Camacho et al., 1992) was not used in order to test a robust implementation of a feedback controller which could be applied to other plants as well. Start-up. The collector field is started every morning from the ambient temperature: with minimum flow it is easy to get the temperature to rise, but the oil flow must be increased in time to stop temperature increase. Actually, the speed of start-up is restricted by efficiency of stopping the temperature rise. The inlet oil temperature changes strongly during the recirculation period, and these irregular conditions are very difficult for many controllers. With the braking action, the start-up is very fast and accurate, and almost without any overshoot (Figures 10.18 and 10.20). In first tests, the response after the start-up was slightly oscillatory (Figure 10.18) but the oscillatory behavior was removed by further tuning of the controller (Figure 10.20). The combined braking and asymmetrical action resulted in a very smooth start-up even when the radiation has a lot of disturbances (Figure 10.23). The field can reach the full operation in 40 minutes with one setpoint change since the overshoot is minimal. The operation is very careful, e.g. the temperature rise was stopped for a while when an strong braking period was followed by a slower increase of the radiation intensity (Figure 10.20). Changing operating conditions requires an adaptive system, which is beyond the capabilities of a PID controller, and even a fuzzy controller without adaptation of membership functions has difficulties during the start-up period. The linguistic equation controller adapts its operation smoothly to working point changes. The operation corresponds to switching control approaches, and the tuning is done separately on several operating conditions. However, the controllers are combined at the higher level, and the final operation is very different, i.e. the switching is gradual.

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Figure 10.18 Test results of the linguistic equation controller (June 12, 1997): temperatures, oil flow and solar radiation.

Setpoint tracking. The performance of the collector field depends on how close to the upper limit the oil outlet temperature can be kept without overheating risk. The control surface must be different for different operating conditions, e.g. in 1996 the controller action was slightly too slow or too fast depending on the operating conditions. According to the tests in 1997 (Juuso et al., 1998c), the controller is very fast in setpoint changes as well, e.g. after a step of 20 degrees in the reference the controller was able to reach the new temperature with a rising time of 7.5 minutes, and the response was well damped with no overshoot (Figure 10.19). The irradiation level was high and fairly constant during the test. The slight offset, present in the results of the tests in 1996, was removed by adding the asymmetrical action, which has been implemented in the existing system on the basis of the previous tests. Even in the variable irradiation conditions, the setpoint is achieved very fast when the perturbations became smaller for a short while. In variable irradiation conditions, the controller is always ready for any changes, and therefore, it acts very carefully. In normal conditions, even faster responses could be achieved in some cases. The oscillatory behavior (Figures 10.18 and 10.19) was caused by the underestimation of the change of error, which resulted in too large control actions in some cases. In later tests, this unpleasant effect was removed by using narrow feasible ranges for the change of error (Figure 10.20). Also the large overshoots seen in Figure 10.19 were removed without making the response slower.

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The braking action is activated always when the error becomes large enough, and it operates until the error goes close to zero and the changing trend is stopped, e.g. the step levels shown in Figure 10.5 correspond to the test results of Figure 10.20. Since the membership definitions are modified by both the braking action and the unsymmetrical action, which activates when error is close to zero, the operation of the basic controller are most of the time modified (Figure 10.6). For PID controllers, the tuning should be different for each operating condition, which is far from the practical solution. Conventional fuzzy controllers can be tuned on wider working areas, e.g. normal operation throughout the day or a special type of disturbance. However, the performance will deteriorate outside the training area. For the linguistic equation controllers, the adaptation method is flexible. Even the strength of the braking and asymmetrical actions can be modified if the operating conditions are clearly changing, e.g. sky is clearing up or heavy clouds are coming.

Variable irradiation conditions. The single equation controller handled difficult cloudy conditions fairly well already during the test campaign in 1996 (Juuso et al., 1997b). The present multilevel controller works even better (Figure 10.21). The controller is able to maintain the outlet oil temperature in an acceptable range (within maximum 5 percent deviation) even when perturbations of almost 50 percent in the primary energy source, the solar radiation, were present. The start-up has not been affected by the unstable radiation (Figure 10.23).

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---

Outlet Temperature Set Point Temperature Inlet Temperature

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Figure 10.20 Test results of the linguistic equation controller (July 23, 1998): ternperatures, oil flow and solar radiation.

Very difficult operating conditions were tested in 1997 (Juuso et al., 1998c; Juuso et al., 1998d). The compensation of fast and big changes in the irradiation level was very efficiently solved. The setpoint is achieved very fast when the irradiation is stabilized (Figure 10.21). The operating point correction varies here between negative big and positive big (Figure 10.17). The overshoot could have been compensated if the stabilization had been taken into account by changing the strengths of the actions and the register size for averaging. During the test, the controller was all the time prepared to a new irradiation drop. The irradiation drop shown in Figure 10.21 was so long that the controller changed already to the minimum flow. Additional examples of the compensation of fast and big irradiation changes are seen from 14 to 15 hrs on June 17 (Figure 10.22). The operating point correction is now between negative small and positive medium (Figure 10.17), i.e. some speeding up is allowed, and there is no risk for oscillatory behavior. The robustness of the approach was demonstrated by the rejection of a strong disturbance caused by the combined action of a variation on the irradiation level and a big decrease in the inlet oil temperature caused by the start of the desalination plant (Figure 10.22). The solar field is maintained in a state of readiness for the resumption of full scale operation when the intensity of the sunlight rises once again; the alternative is unnecessary shutdowns and start-ups of the collector field which are both wasteful and time consuming. The temperature is maintained on a quite acceptable range even in the case of very low and varying irradiation (Juuso et al., 1998~).This can be seen in the results obtained on September 16, 1997 (Figure 10.23) when the controller was working the whole day without any parameter changes. The start-up was not disturbed

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Figure 10.21 Test results of the linguistic equation controller (June 17, 1997): temperatures, oil flow and solar radiation.

by the irradiation changes when the setpoint is on an appropriate temperature level. During heavy clouds, the temperature is unavoidably dropping even on the minimum flow level and the overshoot after clearing the sky is again caused by the careful operation requirements.

Variable inlet oil conditions. The linguistic equation controller handles all the disturbances in the same way, i.e. it reacts to the error and the change of error. The operation depends on working point conditions, which only change the membership definitions. The decreasing inlet temperature is compensated efficiently: the temperature error is very small (Figure 10.22), and the simultaneous irradiation disturbance is more difficult for the controller (Juuso et al., 1998c; Juuso et al., 1998d). The multilevel linguistic equation approach can handle easily multiple disturbances as shown in Figure 10.22. The results are very promising for the use of the controller with the secondary system, the desalination plant, operating even in conditions where there are some perturbations in irradiation as well. 10.6.6 Discussion

The linguistic equation controller reacts to variable irradiation conditions very efficiently since the control surface is changed only gradually. PID controllers cannot be tuned in this way. Fuzzy controllers can be tuned on variable conditions: the operation can be very smooth even for the whole day (Rubio et al., 1995), and the perform-

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Figure 10.22 Test results of the linguistic equation controller (June 17, 1997): temperatures, oil flow and solar radiation.

ance can be improved by genetic design (Gordillo et al., 1997). However, the same performance cannot be achieved if the operating conditions are different, i.e. each specialized controller should be different from the controller tuned for normal operation. Therefore, both PID and fuzzy controllers would require a switching strategy. A switching control scheme is applicable if there is some a priori knowledge about the plant dynamic behavior (Rato et al., 1997). However, high performance is difficult to achieve with methods based on a normal switching strategy: too frequent switching yields stability problems, and on the other hand, if the switching is too heavily constrained, the controller may operate most of the time far from optimal. Anti-swapping algorithms may become too complicated for fast disturbances. The multilevel linguistic equation controller is based on fuzzy switching which is represented by linguistic equations. Fuzzy switching could be added to both PID and fuzzy controllers, i.e. parameters of the controllers could be defined by fuzzy rules. A large number of pretuned controllers is needed in order to achieve a smooth operation. Cascaded structures available in some programming tools are also fairly complex. Handling multiple disturbances with PID and fuzzy controllers would mean additional requirements for specialized controllers and switching between them. A large number of controllers would be needed since each combination requires a slightly different tuning. The multilevel linguistic equation controller does not need any specific algorithms for the multiple disturbances since the high level control is embedded to the working point adaptation.

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Figure 10.23 Test results of the linguistic equation controller (September 16, 1997): temperatures, oil flow and solar radiation.

10.6.7 Conclusions Excellent results have been obtained with fuzzy logic controllers and their extensions, multilevel linguistic equation controllers. The multilevel linguistic equation controller has been tested in very adverse conditions, such as: fast start-up in the morning with high solar radiation, high fluctuations in the solar radiation, and changes in the inlet oil temperature. The controller is very robust in various difficult operating conditions. The results obtained in terms of disturbance rejection, speed of response, and setpoint tracking were fairly good. A smooth start-up of the plant without oscillations has been obtained by means of the braking action. Accurate set-point tracking is based on the asymmetrical action. Fast and very compact implementation was achieved, even for the three level controller structure. Different requirements put on the controlled system can be combined with the linguistic equation controller.

10.7

APPLICATION: LIME KILN

Lime kilns are used in paper industry to convert lime mud to lime in the recovery cycle. The lime mud reburning, i.e. lime mud filtering and lime mud calcination in a lime kiln, is an essential part of the chemical recovery system. Ideally, the kiln should be able to produce a sufficient amount of high quality lime to meet the needs of the recausticizing operation, while minimizing its energy consumption and flue gas emissions. In addition, certain process variables must remain within well-defined physical limits in order to assure safe operation of the kiln.

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In practice, those requirements set for the lime kiln operation are at least partially contradictory. Thus, it is in many respects a difficult and complicated control and optimization problem. On the other hand, for any given production rate, there is an optimal set of operating conditions that would provide the best achievableperformance regarding lime quality, energy consumption and environmental aspects. This Section deals with experiences of fuzzy control in an industrial scale lime kiln at UPM Kymmene Pietarsaari mills.

10.7.1 Plant description The causticizing process, which is part of the pulp mill chemical recovery produces white liquor from green liquor. This process consumes lime, CaO, and produces lime mud, principally CaC03, as a by-product. The purpose of the lime mud reburning plant is to convert the lime mud (CaC03) back into the lime (CaO) for reuse in the causticizing process. The quality of the reburned lime is a very important parameter in the production of low cost, high quality white liquor. For instance, the lime quality influences the rate of the slaking and the causticizing reactions, and the separation properties of the produced lime mud (Theliander, 1988). The rotary lime kiln (Figure 10.24), which is a large cylindrical oven with a length between 50 and 120m and a diameter between 2 and 4 m, has been the prime equipment for calcination in the pulp industry. Lime kilns are slightly inclined from the horizontal position and are slowly rotated on a set of bearings. The lime kiln at Pietarsaari mills is very long (diameter 3.6 m and length 106 m) and has a capacity of 500 tons limelday.

Figure 10.24

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Conceptually, the lime kiln operation may be divided into four process zones. These four zones are drying of the lime mud, heating of the lime mud solids up to the reaction temperature, calcification of the calcium carbonate into the calcium oxide and completing the calcification reaction (burning) (Jarvensivu, 1998). In the drying zone, residual unbound and bound moisture is removed thermally to yield a solid lime mud. Two processes occur simultaneously: transfer of heat from the hot flue gases to evaporate and vaporize the unbound surface moisture and transfer of internal bound moisture to the surface of the lime mud particles and its subsequent evaporation due to the first process. During the first stage of lime mud drying, the drying rate is nearly constant, i.e unbound moisture on the surface is evaporated and the rate of drying is determined by the rate of evaporation. For further drying, the moisture has to be transported from inside of the lime mud particles to the surface and when the critical moisture content has reached, i.e. when dry spots appear upon the surface, the drying rate falls. After the drying zone, i.e. when all moisture is removed from the lime mud, the lime mud solids temperature starts to rise. The temperature rise settles down, when the main reaction, i.e. calcium carbonate dissociation to calcium oxide and carbon dioxide, inside the lime kiln starts to consume heat energy. The calcification reaction starts already at about 800 "C, but typically in practice a temperature up to 1000 - 1100 "C is required in the hot end of the kiln to increase reaction rate and complete the reaction. After calcification, the reburned lime becomes a very fine powder. Therefore in order to produce a usable product, the lime kiln must not only dry the lime mud and heat it in order that calcification occurs, but it must heat calcined powder further up to about 1000 "C to achieve agglomeration of the powder into nodules.

10.7.2 Control problem The reaction time in a rotary kiln system is long and varying and there is a large quantity of lime mud and lime in the system at any given time. The control of the lime kiln is multivariable by nature. Classical model based supervisory control of process is nearly impossible, and the knowledge of experienced operator is often the best solution. Control changes should be made in small increments, allowing sufficient time to observe the effect after each change has been made. Overreaction leads to overcompensation, which generally causes oscillations and upsets which may take hours to bring back the process to normal operation. Open loop control of the burning end temperature, feed end temperature and excess oxygen has inefficiencies in fuel consumption, lime quality and dusting. An effective control scheme for the economical production of good quality lime can be divided into two levels. The high level control should define the optimal operating conditions to achieve optimal temperature profile and lime quality with minimum energy consumption and emissions. The purpose of the low level control is to maintain these conditions despite of fluctuations in the process. The control system is based on the following principles: The kiln production rate should be changed gradually by taking into account the levels of the mud storage tanks and the requirements of the other processes.

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Measurement or prediction is needed for the varying dry solids content of the incoming lime mud in order to maintain it as constant as possible. Varying heat energy content of the fuels is important when odor gases or wood dust is used as a fuel, especially when the fuel is changed. Changing of the fuel is a very challenging problem for automatic control. The bed depth of the kiln is controlled by adjusting the rotational speed of the kiln at changing production rates. The product quality is defined by the residual carbonate and the agglomeration of the powder, and both these depend on the temperature profile of the kiln. Measurements, laboratory analysis and visual inspection are needed in the development of intelligent quality analyzers.

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The temperature profile of the kiln is handled with three temperatures: the burning end temperature is primarily controlled by the fuel feed, the middle kiln temperature by the fuel feed and the flue gas fan, and the feed end temperature by the flue gas fan. Efficient combustion and excess oxygen control is based on the flue gas analysis, e.g. carbon monoxide, excess oxygen and TRS emissions. Smooth operation with gradually changing operating conditions is the best solution. The setpoints for temperatures and excess oxygen should be considered as goal values. Intelligent analyzers, e.g. product quality and flue gas emission analyzers, are useful especially in balancing the control objectives.

10.7.3 fizzy expert control The fuzzy logic controller (FLC) developed in 1994 for controlling the flue gas fan was implemented in Alcont-I1 system and tested in industry scale at Wisaforest Pietarsaari mills (Haataja and Ruotsalainen, 1994). The rule base of 22 rules was originally formulated on the basis of operator interview and tuned manually on-line in cooperation with the personnel. Compared to the earlier control approaches, the development and tuning was very fast (Juuso et al., 1996). Fuzzy logic in lime kiln control has proved to be successful. The software was designed to stabilize the production and furthermoreto increase the production capacity and to minimize the cost of production. The aims to stabilize the lime kiln, improve the quality of reburned lime, reduce the fuel consumption and make the kiln easier to handle have been realized. Fuzzy logic control has mitigated the process and made process operators work more easier and less time is consumed in routine duties. Process became more stabilized and the variation of process variables has been decreased. Process operators were very satisfied with the results of the control. Annual savings were estimated to be about 300 000 dollars. During the years the operating conditions of the lime kiln were changed, and also the tuning had to be changed. The tuning of the FLC was performed with linguistic equations (Juuso et al., 1996). The rule base for the FLC was replaced by linguistic

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equations shown in Figure 10.25. Each equation corresponds to a group of rules describing the effect of two input variables to the control variable (change in the fan speed). The variables are: (1) feed end temperature, (2) change in feed end temperature, (3) burning end temperature, (4) change in burning end temperature, (5) excess oxygen, (6) change in excess oxygen, (7) temperature after cyclone and (8) change in the fan speed. After including the high level control for setpoints, the FLC of flue gas fan has been represented as a PI type fuzzy controller. Fuzzy controllers have been tested also for the control of burning end temperature. Both applications have shown that a FLC can be tuned successfully to different operating conditions, and the tuning is needed every time when the operating conditions are changed considerably. Since tuning is time consuming, the FLC should be adaptive but adaptation algorithms are too risky for processes with various complicated interactions. For these reasons, the FLCs are now represented by multilevel linguistic equation controllers which can include very complex predefined adaptation procedures. 10.7.4 High level control

Various control objectives must be balanced in the control of lime reburning processes, e.g. the FLC of the flue gas fan consists of five rule blocks. Depending on the operating conditions, the importance of the different deviations may change. The ECS/FuzzyExpert system of FLS Automation focuses on these high level control strategies (Ostergaard, 1996). High level control is a natural area for the expert fuzzy control, also extensions to the switching control scheme can be efficient.

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The process variables have complex interactions,and therefore modeling of the lime kiln is necessary for successful balancing of control strategies (Juuso et al., 1997a; Juuso, 1998b; Juuso, 1998a). Intelligent modeling can combine expertise with process data. In a wide field survey performed at the pulp mill, a large amount of history data was collected from both normal operation and process experiments (Jiirvensivu, 1998). After preprocessing and comprehensive statistical analysis, the material including routine laboratory analyses and correspondingprocess measurements was examined by time series and multiple regression analysis, neural networks and linguistic equations. In the pulp mill, feedforward control of the kiln rotation and fan speed can be based on the nonlinear models, e.g. both neural network models and linguistic equation models are used in (Juuso and Jgirvensivu, 1998). The control action is adjusted by FLCs or expert system type of rules (Jarvensivu and Juuso, 1998) or linguistic equation controllers, also operators can change the control action. All these feedforward controllers are based either on experience or on inverted models. The high level control strategies are balanced by setting various target values (Jiirvensivu and Juuso, 1998). The target value for the production rate is based production maximization by taking into account the state of the lime mud storage and the kiln load. Product quality is maintained at the optimum level by adjusting the burning end and middle kiln temperature targets on basis of predicted residual carbonate and the operating conditions. The minimization of the energy consumption and environmental protection are taken into account by adjusting the target value for excess oxygen and wash water of the lime mud filters on the basis of the TRS emission level.

10.7.5 Linguistic equation controller In the lime kiln, linguistic equation controllers are used in both feedback and feedforward control. Although the feedforward controllers define the operating ranges for the feedback controllers, they are practically important only on drastic transient conditions. The normal operation is handled by feedback controllers since feedforward controllers cannot take into account all the disturbances (Section 10.4). The feedback controllers are based on the PI-type fuzzy controller represented by (10.1), where the interaction matrix is A = [ 1 1 1 1, and variables are defined as X = [ e Ae Au IT. For fuel feed control, the error variable is the deviation of the temperature from the setpoint, and the control variable is fuel feed. The controller of the flue gas fan consists of several PI type LE controllers with temperatures and excess oxygen as error variables. For all these controllers, the production rate was used as a working point variable, and the fuel trend as a trend variable. The linguistic value of the trend is useful even when the fuel is changed. The velocity of the flue gas fan and the quality of the dust have been included to the working point variables in order to take into account the effective fuel feed. The braking action depends on the original error in the same way as in the controller of the solar power plant. The asymmetrical action is based on the fuel trend analyzed from fuel feed history. The feedforward controllers based on inverted models are the only controllers when the fuel is changed. The adjustable range is also limited when the production rate is changed drastically, i.e. in the beginning the controller is as a feedforward controller and continues as a feedback controller when the temperature measurements start to be

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representative for the new fuel. The feedforward controller shown in Figure 10.4 has been extended to the whole operation area. The controller operates very well for smooth production rate changes as well. Compensation of disturbances caused by fast production rate changes take more time since the lime kiln process is slow. For safety reasons, the maximum cumulative change of control on predefined time intervals is limited, and this saturation can be seen in the cases of fast production rate changes. Model-based predictive control could be a useful method in tuning these limits for different operating areas. The linguistic equation controllers have developed and tested in FuuEqu Toolbox n t . the LE controllers have been implemented running in ~ a t l a b ~ ~ e n v i r o n m e All as functional blocks in ~ 2 ~ ~ r u n n on i nWindows g pla platform. The system has a modular structure where each module performs a specific part of the overall functionality (Jarvensivu and Juuso, 1998; Juuso and Jiirvensivu, 1998). The system also contains intelligent analyzers for the dust quality and the middle kiln temperature. The dust quality analyzer seems to be a very important addition to the controller. The burning end temperature is also estimated by sensor fusion, i.e. measurements obtained by a pyrometer are later verified with measurements obtained by a thermocouple.

10.7.6 Models in fuzzy control Attempts have been made to mathematically model the dynamics of the solids phase, the heat transfer phenomenon and mechanisms, and the kinetics of drying, heating and calcination reaction occurring inside a rotary kiln. However, an accurate and exact mathematical model of the lime kiln process is difficult to obtain. Furthermore, if successful it will lead to highly complicated nonlinear equations in the form of partial differential or algebraic equations. As a result the evolution of the kiln process continues to depend more on rule-of-thumb empiricism and information gained from the laboratory and full scale experiments. The modeling of the burning end described in Section 10.3.3 is a part of a larger study, which is aimed to further improve the fuzzy control of the lime kiln. In the burning end control, the modeling of the burning end temperature was found out to be necessary. In intelligent controI design, hybrid techniques combining different modeling methods in a smooth and consistent way are essential for successful comparison of alternative control methods (Juuso, 199%; Juuso, 1998a). The results of Section 10.3.4 have been used in controller tuning. Switching between different submodels in multiple model approaches should be as smooth as possible. For slow processes, predictive model-based technique is necessary at least on the tuning phase. Adaptation to various nonlinear multivariable phenomena requires an highly robust technique for the modeling and simulation. A multilevel linguistic equation controller has been tested with the simulation model of the lime kiln (Figure 10.26). For constant production rate, setpoint tracking is fast with very smooth fuel feed changes. Tuning with the robust dynamic simulator is needed because the process is slow, e.g. the simulation shown in Figure 10.26 corresponds to almost three days operation. The results of the steady state modeling (Section 10.3.3) have been used for the feedforward control. The feedforward controller can reduce the large temperature

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changes caused by the step changes of the production rate (Figure 10.27). Fast changes of the production rate are compensated very efficiently if the temperature setpoint is not changed. Late reactions will cause large temperature disturbances if the feedforward controller is not used (Figure 10.26). Large changes of the production rate should be done gradually by taking into account the requirements of other subprocesses. Lowering the temperature during the decreasing production is very slow, even keeping the setpoint is difficult if the change of the production rate is not taken into account(Figure 10.27). The trend of the production rate is now included to the asymmetrical action. 10.7.7 Conclusions

A proper combination of intelligent techniques on high and low level control can extend the functionality of the control system. The rotational speed of the kiln can be successfully controlled by the feedforward control based on the production rate. Fuzzy control of the flue gas fan resulted considerable savings, and later modifications have made the controller more flexible in changing operating conditions. Various intelligent models have extended the process insight and thus helped in balancing different high level control objectives. For low level control, models are necessary in fine-tuning since the lime kiln is a very slow process. Inverted models are also used in feedforward control in drastically changing operating conditions. According to the plant tests, adaptation to changing operating conditions is necessary for wide use of feedback fuzzy controllers.

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The most parts of the present intelligent control systems have been used at Pietarsaari pulp mill since April 1998. Considerable reduction of the excess oxygen makes the kiln more energy efficient and decreases draft in the kiln, and therefore reduces the dust circulation in the feed end, which in turn increases the attainable maximum production capacity. The variation of the specific energy consumption is also reduced over 30 percent, which reduces the lime quality variations by making the kiln process more stable. The present system can handle a wider range of operating conditions, and the difference between the automatic control and the manual control is significant. The latest results show even further improvement. The overall control system provides various tools for handling different control strategies in changing operating conditions. Multilevel linguistic equation controllers are feasible solutions for robust adaptation both by combining different control objectives and by changing the control surfaces. Linguistic equation models also provide tools for intelligent analyzers and fault diagnosis. The testing with new properties extending the operating area of the intelligent control system is continued.

10.8

IMPLEMENTATION

The first implementations were based on a block-oriented approach, but later FLS Automation developed a fuzzy control language including if-then rules and a compiler (astergaard, 1996). Many applications have been integrated to automation systems by

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Juuso, E. K. (1997). Intelligent methods in diagnostical process analysis. In Halttunen, J., editor, Proceedings of XIV IMEKO World Congress, New Measurements - Challenges and Visions, Tampere 1-6 June 1997, volume VII, pages 1-6. Juuso, E. K. (1998a). Intelligent dynamic simulation of a lime kiln with linguistic equations. In Zobel, R. and Moeller, D., editors, Proceedings of the ESM'98, Munchester; UK, June 16-19 1998, volume 2, pages 23-28, Delft, The Netherlands. SCS. Juuso, E. K. (1998b). Robust dynamic simulation with linguistic equations in intelligent control design. In Juslin, K., editor, Proceedings of the Eurosim'98 Simulation Congress, Espoo, Finland, April 14-15 1998, volume 2, pages 324-331. Juuso, E. K., Ahola, T., and Leiviska, K. (1996). Fuzzy logic in lime kiln control. In Yliniemi, L. and Juuso, E., editors, Proceedings of the TOOLMET'96 Workshop, Oulu, Finland, April 1-2, 1996, pages 1 1 1-1 19, Oulu. UOulu, Control Eng. Lab. Juuso, E. K., Ahola, T., and Leiviska, K. (1997a). Fuzzy modelling of a rotary lime kiln using linguistic equations and neuro-fuzzy methods. In Foulloy, L., editor, Proceedings of the 3rd IFACSymposium on Intelligent Components and Instruments for Control Applications - SICICA'97, Annecy, France, June 9-11,1997,pages 579584. Juuso, E. K., Balsa, P., and Leiviska, K. (1997b). Linguistic Equation Controller Applied to a Solar Collectors Field. In Proceedings of the European Control Conference -ECC'97, Brussels, July I - 4, 1997, volume 5, TH-E 14, paper 267 (CD-ROM), 6 PP. Juuso, E. K., Balsa, P., and Valenzuela, L. (1998~).Multilevel linguistic equation controller applied to a 1 mw solar power plant. In, Proceedings of the the 1998American Control Conference -ACC198,Philadelphia, PA, June 24-26, 1998., volume 6, pages 3891-3895. ACC. Juuso, E. K., Balsa, P., Valenzuela, L., and Leiviska, K. (1998d). Multilevel linguistic equation controller applied to a 1 mw solar power plant. In Dourado, A., Leite, F. S., de Lima, T. P., Cardoso, F., Dias, J., Oliveira, N., Ribeiro, B., and Afonso, J., editors, Proceedings of Controlo'98, the 3rd Portuguese Conference on Automatic Control, Coimbra, Portugal, September 9-11, 1998, volume 2, pages 62 1-626. APCA. Juuso, E. K. and Jarvensivu, M. (1998). Lime kiln process modelling with neural networks and linguistic equations. In Zimmermann, H.-J., editor, Proceedings of the 6th European Congress on Intelligent Techniques & Soft Computing -EUFIT198, Aachen, September 7 - 10, 1998, volume 3, pages 1601-1 605, Aachen. Mainz. Juuso, E. K. and Leiviska, K. (1992). Adaptive expert systems for metallurgical processes. In Jamsa-Jounela, s.-L. and Niemi, A. J., editors, Expert Systenzs in Mineral and Metal Processing, Proceedings ofthe IFAC Workshop, Espoo, Finland, August 26-28, 1991, IFAC Workshop Series, 1992, Number2, pages 119-124, Oxford, UK. Pergamon. Juuso, E. K. and Leiviska, K. (1994). Adaptive fuzzy controller with diagnostical features. In Zimmermann, H.-J., editor, Proceedings of the Second European Congress on Intelligent Technologies and Soft Computing -EUFIT'94, Aachen, September 21 - 23, 1994, volume 2, pages 994-998, Aachen. Augustinus Buchhandlung. Juuso, E. K. and Leiviska, K. (1995). A development environment for industrial applications of fuzzy control. In Zimmermann, H.-J., editor, Proceedings of the Third

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blocks. The fuzzy logic tools have a remarkable resemblance: fuuyTEcHTM (~nform) and T ~ ~ S h e l l ~ ~ ( O r tprovide e c h ) an extensive tool for prototype creation of conventional FLCs; D a t a E n g i n e T M ( ~ lhas ~ ) efficient tools for data-preprocessing and for combining neural networks and fuzzy logic in the system development. Properties of the tools are extended with PlugIns and Add-On modules: DataEngine has various PlugIns, f ~ z z ~ ~ ~ ~ ~ ~ ~ ~ a t a NeuroFuzzy ~ n a l ~ TMmodules. z e r ~ ~ aWINROSA n d (University of Dortmund) developed for generating relevant if-then rules from data can provide fuzzy systems for Matlab TM FLT, ~ U Z Z ~ T Eand CD H a~t a~~ n ~ iTM. ne Fuzzy controllers can be built in the development environment ~ 2 ~ ~ ( ~ e n s ~ m ) which provides a full freedom to make any modifications in real-time systems. The lime kiln control at UPM Kymmene Pietarsaaripulp mill has been tested in G2 TMenvironment running on windows N T ~The ~ overall . system combines various technologies: neural networks, expert rules, fuzzy set systems and linguistic equations. Fast prototyping can be used efficiently in this development environment. The development system is connected to the lime kiln process through a basic automation system and a plant information system. The basic automation system provides connections to the instrumentation and control outputs used for base-level regulatory control, pump and motor start and stop logic functions as well as the safety interlocking. The information management system XIS (Valmet) is used to collect data from the basic automation system and results from laboratory analysis. The information system also calculates short- and long-term average values, and archives the data in the real-time data base. The iterative prototype-based development is based on fast incremental prototyping and testing. Various tools support 3D surfaces, as well as interactive debugging and analysis in controller evaluation. Dynamic testing of the controller can be based on process simulation if a process model is available. The atl lab-SimulinkTMis a widely used environment: systems created with Fuzzy Logic Toolbox can naturally be integrated, but also fuuyTE~HTMprovides a fuzzy control block for atl lab-SimulinkTM. Dynamic Data Exchange (DDE) links are widely used in connecting the controller to the simulation system. Since DDE is available in various software tools, e.g. MatlabSirnulinkT M , ~ l c o n t ~ ~ e n ~ i n e eenvironment, ring Damatic TM, 1nTouchTMandFuzzyCon, it can be used in the final implementation, especially if the real-time requirements are not very demanding. Embedding to the target system can be done by generating portable C code, e.g. atl lab^^, f u ~ y T E ~ H ~ ~T la~nSdh e l l ~ ~that d oas a standard feature. Systems developed with DataEngineTMcan be included in other application programs by a runtime ADL modules available as a Dynamic Link Library (DLL). Another technique used in development tools is to download fuzzy logic function blocks to standard PLCs and process control systems. In ~ 2 ~ ~development t h e work is done directly in the target system since the environment is also the final platform for the control system. The modified system can be immediately included to the real-time operation. Linguistic equations (LE) were first used in the development and tuning of fuzzy systems: the FuzzEqu toolbox combines expertise, data and models. The LE controllers can be tested with dynamic LE simulators in atl lab-simulinkTM. Since LE controllers are very compact, they can integrated by programming languages. The first LE controller was implemented by C-language and embedded to the in-house control

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system developed at the Plataforma Solar de Almeria. For the lime kiln control, the present LE controller is implemented as functional blocks in the development environment G2TMconnectedto the process through the Alcont automation system and a plant wide information management system XIS (Valmet). Hybrid methods combining different intelligent techniques are increasingly important in the development of control systems. Free combination of fuzzy logic and neural networks can be done in various environments, e.g. DamaticTM,AlcontTM, m at lab TM,Data~ngineTMandG2 TM.Linguistic equations are used in Matlab TMand G2 T M , and implementation of a LE PlugIn to Data~ngine TMisunder progress. Also a fuzzy controller InFuuy with some LE-based tuning facilities is under development. The long term maintenance can be guaranteed if the control system can be modified on the final platform, i.e. fuzzy control can get wide acceptance in process industries only if it is an integrated and unified part of the DCS engineering tools. The integration is essential in large scale applications where strong support and built-in reliability is needed especially for maintenance. Special purpose software should be linked to various DCS engineering tools since the specification of a FLC can be presented in a quite compact form. The most critical issue is the control system configuration, i.e. the choice of the manipulated and measured/controlled variables and their interactions. According to the idea of partial control, all the specified variables are kept on a acceptable range by directly controlling only a small subset of variables at their exact setpoints.

10.9 CONCLUSIONS Robust industrial control can be achieved on the basis of fuzzy expert control and its extension multilevel linguistic equation control. Insight to the process dynamic operation is the most important issue. Automatic generation of controllers, modelbased techniques and adaptation techniques are very valuable in developing and tuning the controllers. Adaptation techniques and model inversion are suitable developing low-level control. Finally, the resulting controller should be transformed into an expert type controller. The model-based predictive control may be useful as an on-line technique for large disturbances and transients. The continuous operation of the fuzzy control is guaranteed only if there is a real co-operation between the control system and the operator, and if the controller is an integrated and unified part of the DCS engineering tools. The linguistic equation controllers have been tested extensively in two applications: the emphasis has been in the adaptive expert control for the solar power plant, and in various model based approaches for the lime kiln. Design principles of the expert control were transferred with very few differences from the solar application to the lime kiln application. The linguistic equation approach increases the performance of the fuzzy control considerably by combining various control objectives and adaptation to the changing operating condition into a very compact implementation. The LE approach is also very efficient as a modeling technique: models can be generated from data, various types of fuzzy rule-based models can be represented by LE models, and any LE model can be transformed to fuzzy rule-based models. Compact dynamic LE models suit very well to tuning of controllers, also with adaptive and predictive

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schemes. The model inversion is working well for LE models, and these results are used for the feedforward control. References

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Meaburn, A. (1995). Modelling and Control of a Distributed Solar Collector Field. Ph. D Thesis. UMIST. Meaburn, A. and Hughes, F. M. (1994). Prescheduled adaptive control scheme for resonance cancellation of a distributed solar collector field. Solar Energy, 52(2): 155166. Mosca, E., Zappa, G., and Lemos, J. M. (1984). Robustness of multipredictive adaptive regulators: Musmar. Automatica, 20:68 1-699. Murtovaara, S., Juuso, E., and Sutinen, R. (1996). Fuzzy logic detection algorithm. In Mertzios, B. G. and Liatsis, P., editors, Proceedings of lWISP'96 - Third International Workshop on Image and Signal Processing on the Theme of Advances in Computational Intelligence, 4-7 November 1996, Munchester; UK, pages 423426. Murtovaara, S., Juuso, E., Sutinen, R., and Leiviska, K. (1998). Modelling of pulp cooking characteristics by using fuzzy logic and linguistic equations. In Yliniemi, L. and Juuso, E., editors, Preprints of TOOLMET'98 Symposium - Tool Environments and Development Methods for Intelligent Systems, Oulu, April 16-17, 1998, pages 90-96, Oulu. Oulun yliopistopaino. Myllyneva, J., Leiviska, K., Kortelainen, J., and Nystedt, H. (1997). Fuzzy control of thermo-mechanical pulping. In Proceedings of the 20th International Mechanical Pulping Conference, Stockholm, June 9-13,1997. Nieminen, H. (1997). A block building approach to intelligent process control. In Yliniemi, L. and Juuso, E., editors, Proceedings of TOOLMETI97 - Tool Environments and Development Methods for Intelligent Systems, Oulu, April 17-18, 1997, pages 69-80, Oulu. Oulun yliopistopaino. astergaard, J.-J. (1993). Fuzzy control of cement kilns, a retrospective summary. In Zimmermann, H.-J., editor, Proceedings of the First European Congress on Fuuy and Intelligent Technologies -EUFlT'93,Aachen, September 7 - 10, 1993, Aachen. Augustinus Buchhandlung. Ostergaard, J.-J. (1996). High level control of industrial processes. In Yliniemi, L. and Juuso, E., editors, Proceedings of TOOLMETJ96 - Tool Environments and Development Methods for Intelligent Systems, Oulu, April 1-2, 1996, pages 1-27, Oulu. UOulu, Control Eng. Lab. Palmu, P. (1998). Experience of fuzzy logic control system at rautaruukki raahe steel. In Yliniemi, L. and Juuso, E., editors, Preprints of TOOLMET'98 Symposium - Tool Environments and Development Methodsfor Intelligent Systems, Oulu, April 16-17, 1998, pages 32-39, Oulu. Oulun yliopistopaino. Pedrycz, W. (1984). An identification algorithm in fuzzy relational systems. Fuuy Sets and Systems, 13: 153-167. Rato, L., Borrelli, D., Mosca, E., Lemos, J., and P.Balsa (1997). Musmar based switching con-trol of a solar collector field. In Proceedings of the European Control Conference -ECC197,Brussels, July I - 4, 1997, volume 5, TH-E 13, paper 266 (CD-ROM), 6 pp. Rauma, T. (1997). An adaptive control using diagnosis information. In Yliniemi, L. and Juuso, E., editors, Proceedings of TOOLMETJ97 - Tool Environments and Development Methods for Intelligent Systems, Oulu, April 17-18, 1997, pages 11612 1, Oulu. Oulun yliopistopaino.

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11

FUZZY LOGIC APPLICATIONS IN MOBILE ROBOTICS A. Ollerol,

G. Ulivi2 and F. Cuestal

Dep. de Ingenieria de Sistemas y Automitica Universidad de Sevilla, Spain Dip. di lnformatica e Automazione Universit di Roma Tre, Italy

11.1 INTRODUCTION

In this chapter several approaches for the application of fuzzy logic to autonomous mobile robots are considered. Many researchers have pointed out the interest of fuzzy logic for mobile robot control. Fuzzy logic has been used to consider the inaccuracy of the sensor and, in general, the uncertainties due to the limitations in the perception system. Moreover, it is useful to express control commands and navigation strategies in a qualitative way. The ability to interpolate between sensor readings and discrete control actions is also a very interesting property. A typical fuzzy logic application to control mobile robots is the definition of a set of rules relating the sensor inputs with the desired robot motion commands (Song and Tai, 1992). This simple approach will be considered in the first part of the Section 2 of this chapter. The main drawback is the very high number of rules that are required to express the behaviors. In general, fuzzy logic leads to strategies very complex to develop and debug when many sensors are considered individually. Solutions are either to split the knowledge base expressing the robot behaviors in several fuzzy rule banks or to introduce new concepts to formulate the rules at a higher

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level. A fuzzy system can also be used to encapsulate the set of rules in modules and to perform the inference in parts, with intermediate fuzzy variables (Mauris et al., 1994; von Altrok et al., 1992). A related technique is the application of the concept of virtual sensor, presented also in Section 2, as a mechanism to avoid rules with complex antecedents providing a technique for sensor fusion. Thus, a set of rules would provide an interpretation of the sensed data in the robot scenario, and a second set of rules would generate a reference value for the robot motion according to the fuzzy sensor interpretation of the environment. Fuzzy logic can also be applied in the context of the robotic reactive control approaches and particularly in the layered control system of the subsumption architecture that tightly couples sensing and action (Brooks, 1986; Arkin, 1990). The emergent behavior of the robot is the result of the cooperation of independent reactive modules, each one specialized in a particular basic behavior. Different techniques to manage, in a fuzzy frame, competing and cooperative agents could be considered. Fuzzy logic is useful to reduce the complexity and to add flexibility to the implementation of the behaviors. This approach will also be summarized in Section 2. On the other hand, fuzzy logic has a sufficient flexibility to allow formulating important behaviors of mobile robots simply using theoretical operators. This approach will be presented in Section 3 of the chapter. Obstacle avoidance, map building, localization and planning problems will be considered in this section. In the chapter, real experiments with the Romeo-3R mobile robot designed and built in the University of Seville are included in Section 2. Furthermore, experiments with an assisted guidance wheelchair for disabled people, which is developed in the University of "Roma Tre", illustrate Section 3.

11.2 FUZZY LOGIC IN REACTIVE NAVIGATION 1 1.2.1

Reactive navigation

The classical mobile robot navigation approach is based on the representation of the environment and the generation of plans to reach the goals taking into account the models. The problem can be decomposed in global route planning to determine route-passing points for the mobile robot and then path generation to determine the continuous path passing trough the route points. Once the path has been generated, path tracking strategies (Ollero et al., 1994) can be implemented to control the vehicle. This approach is appropriated when the mobile robot perception system is able to obtain an accurate representation of the environment and this environment is static. In these cases it is possible to determine optimal plans minimizing an appropriated cost index. However, in many applications, obtaining a suitable model of the environment is difficult or very expensive in terms both of the sensors required and of the computational resources. The assumption of static environment is also questionable in many cases. In theses circumstances it is preferred to use a reactive approach based on the continuous sensorial information and the ability to react fast enough even if these reactions are not optimal.

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303

This section deals with reactive navigation. The basic reactive approach is a matching between the stimulus and the appropriated reactions or commands. In the pure reactive approach, no environmentmodel is used and there is no planning system with the formulation of the classical approach. On the other hand, the ability to perceive fast enough is very important in the reactive approach. Therefore, appropriated sensors should be selected and located on the robot to satisfy this requirement. It should be noted that several approaches could be adopted to define the matching between stimulus and reactions. Thus, the problem can be formulated at several levels. At the lower level, the reactive approach consists of a direct coupling between the sensorial inputs and the low-level motion commands. This approach will be described in subsection 2.2. Later, reaction from a higher level description provided by the so-called virtual sensors will be described. Finally, the approach based on the behavior concept will be presented. 11.2.2 Direct fuzzy reaction from sensors The mobile robot's reactive control problem consist in a direct map between the input space (i.e. sensor data) into the output space (i.e. control commands). Fuzzy logic is used to implement this matching taking into account the imprecise information provided by the sensors and qualitative relations between inputs and outputs. The fuzzy controller has the same number of inputs as signals provided by the sensors (multi-input). Typical outputs are the command signals to the velocity and direction of the robotic vehicle (multi-output). The main problem usually consists in the design of such a multi-input multi-output (MIMO) controller. For example, consider a fuzzy controller allowing reaching a goal point, from a starting one, avoiding obstacles. The robot has a set of n proximity sensors providing the distance to obstacles. In particular ultrasonic proximity sensors will be considered. The controller inputs are the following ones: orientation error (the angle between the current direction of the robot and the direction to the goal), and the distances measured by each sensor. Controller outputs are the steering angle and the linear vehicle velocity. Consider the following parameters: Orientation error: Current Position - Goal Position. Domain: [-180, 1801 degrees. Linguistic values: (NegativeLarge, Negativesmall, Z, PositiveSmall, PositiveLarge] Distance to obstacles: Sonar readings Domain: [O,maxdistance]. Linguistic values: {Short, Medium, Large} The parameters of the controller outputs are: Steering angle Domain: [-180, 1801 degrees. Linguistic values:

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( NegativeLarge, Negativesmall, Z, Positivesmall, PositiveLarge]

m

Linear vehicle velocity Domain: [O, 10001 mmlsec. Linguistic values: {Slow,Medium, Fast) The rule base consists of rules such as: IF (Distl IS Short) AND ... AND (Distn IS Medium) AND (OrientationErrorIS NegativeSmall) THEN (Velocity IS Slow) AND (SteeringAngle IS NegativeLarge)

The aim of this rule, for example, is to decrease the mobile robot velocity and turn right in presence of obstacles close at left. The whole rule base was hand made taking into account the sensors location and using the orientation error to select the best motion direction. Figure 11.1 shows a LABMATE mobile robot navigating in an office by means of this controller: the dotted line is the path followed by the mobile robot in simulation while the solid line is the path followed in the real experiment.

Figure 11.1

Direct fuzzy control of a LABMATE mobile robot in an office.

Although this controller works properly, some questions arise: 1. What if position and/or orientation of a sensor is changed?

t

FUZZY LOGIC APPLICATIONS IN MOBILE ROBOTICS

305

2. How can another sensor be added to the system?

3. How can the controller be upgraded considering new goals?

4. What about different types of sensors'? The first question implies that all the rules involving such ultrasonic sensor should be changed as the meaning of the information provided by the sensor has changed. Furthermore, more rules could be needed to take into account all the information in order to compute velocity and steering control commands. In practice, a large number of rules could be affected. To solve the second problem, the new sensor should be included in the antecedent of every rule and new rules should be added to the system to properly evaluate the ~nfluenceof the new sensor. Nevertheless, this solution presents new problems: as the complexity of the antecedents grows the meaning and inf uence of each rule become lost. Even more, as it is well known, the growing of the rule base size is exponential with the number of inputs (Raju et a]., 1991). This implies that the time needed to evaluate the rule base grows too, which could make impossible a real time response. The third question refers how easy is to upgrade this monolithic controller. Assume that the mobile robot should go to a goal point avoiding obstacles, but now following a left wall when were possible. To do this, the old controller rules (allowing to go to a goal point avoiding obstacles) have to be changed in order to take into account this mc~dificationand, even more, the design of new rules is needed. Therefore, in practice, a new monolithic controller must be developed, tested and optimized again. Finally, the last question presents the problem of having heterogeneous sensors. When all of the sensors in the system have identical characteristics, the meaning or influence of each one in a rule is more or less obvious. However, when there are sensors with different characteristics (for example analog ultrasonic sensors having different ranges, digital ultrasonic sensors...) or even sensors of different nature (infrared sensors, laser range finder, video cameras, etc), the information provided by the sensors are very different. Therefore, the antecedents of each rule have to deal with this heterogeneous information to recognize the situation. Obviously, the complexity of the antecedent grows significantly and the difficulty to obtain appropriated rules could make impossible the design of the controller. Summarizing, the main drawbacks of this approach are: m

Number of rules is high (with exponential growing in the number of inputs).

a

Low flexibility.

These problems can be solved with the methods proposed in the following subsections.

11.2.3 Virtual sensors The main idea is to separate the environment perception problem from the control problem, in order to avoid rules with complex antecedents. The fuzzy controller behaviors are developed based on a unified representation of the environment. Virtual sensors, which are behavior oriented, provide this representation. The virtual sensors

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consist of another set of fuzzy rules taking into account the available sensor data (Ollero et a]., 1997; Mandow et al., 1996). This approach allows a linear growing of the number of rules with the number of inputs. The approach has been implemented and tested on the mobile robot ROMEO-3R (Robot Movil para ExteriOres, see Fig. 11.2). This robot has been developed by a Spanish research group at the University of Seville, as a part of a project on outdoor mobile robot navigation. This robot has non-holonomic tricycle locomotion and has been adapted from a conventional three-wheel car to cany out research in control, planning, perception and telerobotic issues.

Figure 11.2

ROMEO-3R mobile robot developed at the University of Seville.

ROMEO-3R performs autonomous navigation based on laser range finder, ultrasonic sensors, camera position estimation and teleoperation techniques. The mobile vehicle carries on board a heterogeneous configuration of ultrasonic sensors that was chosen considering the characteristics of the robot environment. Instead of a typical ring of the same sensors, its sensor configuration is a combination of three different types of ultrasonic sensors placed at different locations, see Fig. 11.3. In the experiments presented in this chapter ten ultrasonic sensors have been used. Four of them are Large Range Analogue (LRA) sensors (from 0.6 m to 3.0 m) and the other six are digital. Among the digital ones, four of them are of Mid-Range Digital (MRD, from 0.2 m to 1.0 m), and the other two are of Short-Range Digital (SRD, from 0.06 m to 0.3 m). For technical specifications of the ultrasonic sensors, see Table 1. These sensors are arranged in a way that six of them cover the front part of the vehicle and the other four cover its lateral sides (two in each side). Due to the low number of sensors, ROMEO-3R has relatively large blind sectors in between them. Consider a virtual sensor called LeftWallSensor that provides the distance to a wall at the left side of the robot. Then, comparing the data from this sensor with a reference distance (the distance at which the mobile robot must follow the wall) a fuzzy controller performing a left wall following behavior can easily be developed. The error and its first difference are used in a PI like fuzzy logic controller to compute

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ShcwtRangeW@al

nMddkRangeWs'tal 0

AniJogue

Figure

11.3 ROMEO-3R

Table 11.1

Sensing range Switching frequency Ultrasonic frequency Resolution Angle

with the ultrasonic sensors.

Technical specifications of sonars.

SRD

MRD

LRA

0.06 to 0.3 m 8 Hz

0.2 to 1.0 m 4 Hz

0.6 to 2.7 m 1 Hz

400 kHz 1 mm

200 kHz 4 mm

80 kHz

5 O

5 O

5 O

2 mm

the robot commands by means of very few rules as: IF (Error IS Negative) AND (AError IS Zero) THEN (Asteering IS Negative) AND (Velocity IS Medium) The reference distance could be a fuzzy value making possible, for example, to command the mobile robot to follow a left wall at a Medium distance. It is interesting to note that the number, type, and position of the available sensors on the mobile robot is completely transparent to the left wall following behavior fuzzy controller. So this controller can be used in spite of changes in the sensorial system configuration. These changes fall under the responsibility of the rules that build the virtual sensor. In fact, this implies a more robust control of the vehicle in the face of errors or malfunctions in a sensor as they can be corrected or compensated with other sensors in the virtual sensor building rules. The rule base that implements the virtual sensors performs the so-called sensor fusion. These rules take all sensed data into account to build the virtual sensors, which brings a fuzzy perception of the environment. This approach allows to increase the reliability of the sensorial system because several sensors are used to compensate the intrinsic sensor errors (for example, ultrasonic sensors brings accurate distance measurement but poor directional information) by means of complementary information. Even more, some kind of map (for example, by the use of occupancy grid (Elfes,

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1987)) could be built as the robot moves on, and use this recorded sensor data to help in the computation of the virtual sensor. This rule base is composed of rules such as: IF (Dist 1 IS High) AND (Dist2 IS Low) THEN (LeftWallSensor IS Close) This approach can be easily upgraded using several virtual sensors to create new ones, and/or combining previously designed behaviors to produce another ones. For example, assuming that there are available a LeftWallSensor and a RightWallSensor (developed to perform a left wall following and a right wall following respectively), they can be used by a corridor following module to perform its mission without developing new rules to compute a Corridorsensor. Figure 1 1.4 shows the application of this approach to command ROMEO-3R in a left wall following. The wall is followed using the LeftWallSensor that is implemented taking into account the measurements of the two left side sensors of the robot as shown in Fig. 1 1.4. Therefore, this approach offers some advantages over the direct fuzzy controller: Virtual behavior oriented sensors. Linear growing of the rule base size with the number of inputs. Easy to upgrade. Reusability. Independence of the controller from changes in the sensors. More robust environment perception. Before conclude this subsection, another interesting approach to the environment perception and reactive fuzzy control of a mobile robot is introduced. The general perception concept (Braunstingl et al., 1995), aims at constructing a so-called general perception of the surroundings from the measuring data provided by all the sensors and representing it as a vector called general perception vector, which provides a fuzzy description of the environment. Thus, the general perception characterizes situations in which the robot may find itself in rather an imprecise, but qualitatively appropriate way. However, if the complete perception is not available, the method fails due to fact that the robot does not recognize the situation in which it is. Therefore, an improvement of the method is needed which can be applied to mobile robots with few sensors. This method is called virtual perception memory as shown in (Braunstingl, 1996), where an ultrasonic sensor passes its perception data to the virtual perception system. In case the real sensor loses perception, the memory based virtual perception can be used as the robot moves on. Furthermore, data from different types of ultrasonic sensors can easily be combined into one perception. At the same time the virtual perception will be updated taking into account the robot's motion. This approach has been successfully implemented and validated over ROMEO-3R (Arrue et al., 1997).

FUZZY LOGIC APPLICATIONS IN MOBILE ROBOTICS

Figure 11.4

309

Sensors measurements and Leftwallsensor.

11.2.4 Fuzzy behavior based-approach In subsections 2.2 and 2.3 two approaches to the design of a mobile robot controller have been presented. Even when the later was much more modular and flexible, both of them was oriented to design a monolithic controller which performs a more or less specific task (for example, to follow a left wall or reach a goal point avoiding collision with obstacles). Those controllers were implemented in the traditional style SMPA

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(Sense, Map, Plan, Act) in order to carry out their mission. When a more complex and reliable reactive controller is needed, a new approach should be addressed. An interesting alternative approach is based on the original non-fuzzy layered control system of the subsumption architecture (Brooks, 1986). The basic idea is to have several simple, independent, and well tested basic reactive modules to perform different tasks (that are able to work by themselves), and then put all them together to build a more complex and unique controller. At work, every reactive module uses the sensed data to propose its control action, i.e. the control command needed to perform the behavior it was designed for. After that, a process is needed to compute a single action control to be applied to the mobile robot. Here, a competitive scheme, or a cooperative one, can be used. In the first case, just only the action proposed by one behavior is selected among the available ones, to take the control of the vehicle. In the second one, all the proposed commands are taking into account, in a higher or lower degree, to compute the final one (Saffiotti et al., 1993; Goodridge and Luo, 1994; Tunstel, 1996). In the implementation of this fuzzy behavior based approach, several reactive modules has been developed: left wall following, right wall following, corridor following, obstacle avoidance and go-to-goal point. All of these basic behaviors have been designed according to the modularity principles introduced in the previous subsection, i.e. virtual sensors have been developed for each behavior and inference in parts applied. The behavior combination has been implemented by means of a cooperative scheme, i.e. the outputs of all the behaviors are considered. Obviously, while navigating not all the behaviors have the same degree of relevance at the same time. For example, the action that is proposed by the left wall following behavior is more important in presence of a wall to be followed, but is less significant when an obstacle is detected. When the command fusion is performed the degree of relevance of each behavior is taken into account. This degree is translated into a weight that is applied to the behavior output. So the control action c is computed by:

In order to evaluate the degree of relevance of each behavior, a fuzzy rule base is designed. This is composed of rules that determine if the context is the best one for the application of the behavior, for example: IF LeftWallSensor IS Reference THEN Leftwallcontext IS High

... IF ObstacleSensor IS Far THEN ObstacleContext IS Low Then in this approach (see Fig. 11.5), a behavior is defined by three fuzzy systems: virtual sensors, behavior context evaluation and the behavior controller itself. Figure I 1.6 demonstrates what happens during a control behavior experiment avoiding two obstacles doing a left turn and later keep following a corridor, with ROMEO-3R finally being able to fulfil its mission going from point (rl) to point (r3). In the starting point (rl), the robot approach the left wall since the only available perception is the

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: HIGH LEVEL PLANNER: .........................

Figure 11.5

311

,.

Fuzzy behavior based architecture.

one provided by the left sensors. The robot keeps going in that direction until the front sensors detected an obstacle (01) and start turning right to avoid it. At this time the second obstacle (02) is detected and the robot start turning left avoiding this obstacle. While turning left a right wall is detected keeping that direction until reach point (r2) where the robot detects a left wall and the corridor behavior increases its activity keeping the vehicle straight ahead until point (r3). Figure 1 1.7 shows the levels of activation of each behavior context. Obstacle avoidance context has two peeks associated with the two obstacles that were found. The approach can be easily translated into a competitive scheme by means of selecting the behavior with maximum context value. Moreover, a hierarchical behavior structure can be implemented taking into account the context value of the others behaviors, giving in this way more priority to a behavior than other. Hybrid solutions combining reactivity with a high level plan to coordinate and supervise the overall mission execution is also a promising strategy. In this case global behaviors are defined by combining individual reactive behaviors. Instead of the definition of a sequence of individual behavior, fuzzy logic can be used to express this combination. Thus, the plan can also be expressed by means of a fuzzy description: Start to follow a short left wall at a medium distance, turn right, follow the corridor, follow the right wall and reach the point A. It is important to see that this planner is not a sequencer. All the behaviors work cooperatively, as before, but now the planner set a preferred behavior in such a way that the context value of that behavior is increased (see Fig. 11.5). That means the mobile robot tends to show this behavior among the others.

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L Figure 11.6

ROMEO-3R behavior based navigation.

Figure 11.7

Contexts activation levels.

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313

11.3 A N ASSISTED GUIDANCE WHEELCHAIR FOR DISABLED PEOPLE

Fuzzy logic techniques play an important role in the project of an assisted guidance wheelchair for disabled people which is developed in the University "Roma Tre". Its control system gathers the results of several researches in this field and is, therefore, a good example to show the common underlying philosophy. Some of the described components have already been implemented on the chair, others have been tested using a Nomad 200 mobile robot. The project, which is somewhat similar to NAVCHAIR (Bell et al., 1994), is aimed to disabled people experiencing difficulties in driving their chairs and, in particular, in using the standard joysticks. For example, traumatic events can produce damages at a high spinal level, causing total disability of all the limbs (quadriplegia); moreover elder people show sometimes a poor coordination of movements, or trembling, or sight problems.

F1

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Optional part

direction Desired

recognition

1 I

m\

Legend

I

item item

Ultrasonic readings

- --

11.8

already realized (maybe in other projects) standard realization

lectroni

Figure

to be realized

Block diagram

of

choose one input

t h e overall project.

A suitable assisted-driving system can help these persons. The most important function of this system is to guarantee the safety of the passenger, keeping the wheelchair

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at a proper distance from the obstacles. Moreover, higher level functions may provide help in difficult manoeuvers, as corridor following or narrow passages crossing. Note that a joystick is a good interface device, better than many alternative ones (e.g. a vocal input). So its substitution makes driving more difficult because other devices are generally slower and less accurate than the original one. Hence the assisted driving system is called into play again. Many ingredients of the system (obstacle avoidance, sensing, path planning) are ordinary issues in mobile robotic research and successful fuzzy implementations have been presented (see, e.g. (Ulivi, 1995), and the references therein). However there are also remarkable differences with the considered case, which make the fuzzy approach even more appealing. Indeed mobile robotics research addresses mainly the on-board intelligence, namely the autonomy of the vehicle. In the considered case, the friendliness of the system is of paramount importance. First of all, the wheelchair motion has to be as smooth as possible with the minimum amount of direction reversals. Moreover, the passenger is assumed to have the faculty of planning hislher activities and to choose the proper wheelchair paths to accomplish them. The assisted guidance system should, therefore, smoothly support the human plan, taking the control only when and if necessary, possibly giving explanations about its activities. Consider for example the case in which the user wants to push an unlocked door with the wheelchair. The assisting system has to recognize this desire and therefore to relax obstacle avoidance for the duration of the manoeuvre. The structure of the whole project is represented in Fig. 11.8 and clearly resembles the typical navigation system of a mobile robot. The part at the right of the heavy line is an optional one, which add more functionalities to the basic ones, as explained in the following. Note that the styles of the blocks have special meanings given in the legend. In particular rounded blocks single out parts where fuzzy logic is involved. Following a bottom-up description, one finds the sensors (ultrasonic devices that are cheap and rugged) and the two driving motors (one for each main wheel) equipped with incremental encoders. The motors are supplied by two PWM amplifiers, the reference of which is provided by the motor controllers. These implement a feedback law to control the speed of each wheel so to give the desired velocity (modulus and direction) to the wheelchair. These first parts are rather conventional. The upper block is in charge of the obstacle avoidance. It accepts as input the velocity desired by the user and, on the basis of the positions of the obstacles as reported by sensors, determines an admissible direction and modulates the velocity modulus according to the available free space around the chair. The user can communicate the desired velocity by several means, according to the kind and the entity of hisher disability. A simple joystick can still be used and in this case the obstacle avoidance module acts mainly as a safety function to avoid impacts. The interest in this configurationis not limited to laboratory testing as rather often accidents are reported due to driving mistakes or distractions, mainly in crowded environments or in complicated manoeuvers.

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The block "Map builder" has two functions. A map is indeed useful in the interpretation of the high level commands issued by the user (e.g. "follow the corridor"). Moreover short term, local maps can be used to recognize particular environments, if some 'a priori' knowledge is available. As a matter of fact, the wheelchair will be mainly used in the user's home. Typical flats show a few difficult passages (very narrow doors, small spaces between a table and a wall, ...) which can be easily recognized. In these cases, purposely-built motion plans can make the motion faster, safer and smoother, with a great improvement of the user's satisfaction.

11.3.1 Obstacleavoidance The importance of obstacle avoidance in the realization of any autonomous vehicle is witnessed by the number of related works and proposed algorithms. The speed required for this behavior rules out algorithms based on planning on some kind of map and call for very efficient solutions. Many of them are based on a simple and effective concept "if you perceive an obstacle, go in the opposite direction" which can be implemented in different ways. A rule-based system can readily implement the above simple strategy. Fuzzy logic, in particular, is very well suited to this aim, as it naturally deals with complex, real world cases, where more obstacles clutter the space near the wheelchair and it is necessary to trade among different directions they originate. Examples of this philosophy are given in (Maeda et al., 1993; Reignier, 1994; Pin and Watanabe, 1994; Demel et al., 1995).

Figure 11.9 Different attractive horizons: (a) leaves room for small adjustments of the desired trajectory, (b) gives t o obstacle avoidance an almost total freedom.

As shown in section 2, the main drawback of this approach is the combinatorial explosion of the number of rules as the number of sensors increases (see also, e.g., (von Altrok et al., 1992)). A different implementation strategy relies on modelling the obstacles as generators of repulsive force fields acting on the vehicle. According with the nature of the environment description, analytical functions can be used to describe the force field near each obstacle (De Medio and Oriolo, 1991) or more heuristic approaches can be adopted as in (Borenstein and Koren, 1991), which uses an occupancy grid to store the readings. Some of the ideas of the last quoted work are retained in the obstacle avoidance system of the wheelchair (adapted from (Balzarotti and Ulivi, 1996)), which however

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does not use the two-dimensional occupancy grid to achieve an higher efficiency. Fuzzy logic is used to develop a "functional model" of the problem, that embeds all the relevant aspects in one algorithm relying just on elementary operators. The resulting code therefore is not affected by combinatorial complexity explosion even if many sensors are used. According to the proposed approach, a one-dimensional fuzzy set represents the cost of moving in 128 directions evenly distributed over half a circle surrounding the wheelchair front. The cost is determined in term of (a) compatibility with the desired direction, (b) risk of collision and (c) smoothness of the resulting path. The actual motion command to the motors has the direction of the minimum cost and a magnitude dependent on the value of that cost. Following the standard fuzzy terminology the 128 directions are the elements of the universal set, over which the fuzzy sets portraying the necessary concepts are defined. After their typical shape and position (they circumscribe the chair) they all have been called horizons. For sake of clarity we assume now an omnidirectional zero dimension robot, equipped with a system of S ultrasonic sensors. They are all placed on the robot, and their axes are uniformly distributed over the range [-90" - 90°]. Some overlapping of radiation lobes is assumed. The following description makes use of triangular fuzzy sets -the one actually usedbut the use of other shapes is immediate.

4

4

The attractive horizon. In order to avoid obstacles, the desired direction has in general to be modified and the more obstacle avoidance modifies it, the higher is the "cost". Therefore, this direction is fuzzified transforming it in a triangular fuzzy set called the attractive horizon A. The support of this set represents the maximum deviation that the OA algorithm can apply and can be tuned according to the confidence in the desired direction and the angle spanned by the sensors. Fig. 9 shows different choices for this set. 4

The repulsive horizon. The repulsive horizon R is the most important data structure of the algorithm as this set is a one-dimension, egocentric representation of the perceived obstacles around the robot. It is obtained by a data reduction procedure. An overall risk of collision is determined for each direction merging the most recent readings. Its angular distribution is updated at each sampling instant taking into account the new measures and the robot movement. There are basically three sources of uncertainty when using an ultrasonic range finder for detecting the position of surrounding objects: the sensor's accuracy, the width of sonar beam, and the multiple path phenomena. The full characterization can be obtained by using the fuzzy set theory (Gambino et al., 1996) and their effects reduced making a large number of measurements. Here, to obtain a fast algorithm, a simpler model is used.

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Consider a single ultrasonic sensor placed in a point P, pointing to a direction Oo and reporting a measure d. The obstacle can belong to any point of an arc of circumference, typically of about 30°, centered in Oo at a distance d from P1. The angular possibility of occupancy is thus described by a triangular shape, defined over [-90°, 90'1. Its maximum, normalized to one, is in the direction Oo, and its support 821, in this case coinciding with [Oo - 15O, Oo 15"]. The set of the spans the arc ultrasonic sensor readings u j , j E [ I ,S ] performed from the same location is called a "perception" C. Perceptions are supposed to be performed at regular time intervals. The risk of collision with the perceived obstacles is related to the measured distance. A meaningful, simple risk estimation p(O) can be obtained from the possibility of occupancy, scaling it by:

+

[el,

Coefficient d S a f ,represents the minimum admissible distance from an obstacle. In the actual wheelchair it is different for each sensor. The membership function of the overall repulsive horizon (Fig. 1 1.10) can be set equal to:

R(0) =

U

j=l

~j

(0)mdiSt( d j )

...S

where Max has been used to implement the union operation. Moreover, the product can be easily interpreted as a fuzzy intersection. The choice of these two operators has been mainly dictated by computational considerations. The same considerations led to dismiss the FIFO buffer used in the previous work to memorize the prior measures. It provides some more robustness with respect to erroneous readings, at the cost of a severe increase of the execution time due to the trigonometric computation involved. The same goal has been at least partially achieved, giving more importance to the kinematic horizon, discussed below.

The kinematic horizon. Even in the case of an omnidirectional robot, it is convenient to avoid frequent changes of orientations and this become of paramount importance in the case of a manned wheelchair. This is the task of the kinematic horizon K. It is a triangular fuzzy set having its maximum coincident with the robot direction at the previous execution of the algorithm, and it is used to penalize large changes of direction. The direction horizon. The direction horizon 'D is a fuzzy set representing the "cost" to move the robot in a direction in terms of risk of collision, compatibility with the assigned motion commands and path smoothness. The set D,computed as

'The error on the measured distance d is considered negligible.

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I \

/ I

0

Figure 11.10 Right: the wheelchair approaches some obstacles. Dashed lines show sensors' lobes. Left: The resulting Repulsive horizon. Horizontal ticks represents sensors' axes.

contains all the information needed to compute the new velocity vector. Its direction is simply the one with the lower degree of membership to the set D, while its magnitude is dependent on this minimum value, with the law:

4

where D ( 4 ) is the degree of membership of 4 to the set V and X is a threshold. Figure 1 1.11 depicts a simple situation, where the kinematic horizon is neglected for ease of representation. The situation is the same than in Fig. 11.10, and the user wants to go slightly to the right. The presence of the obstacles, however, corrects his command and reduces a little the speed. When D(+) is higher than the threshold A, the velocity magnitude will go to zero. In fact a high value of V ( 4 )signifies that there is no "safe" direction sufficiently near to the desired and to the present motion directions. While in a completely autonomous robot this situation has to be used to generate an exception and to invoke higher control levels, in the case of the wheelchair it is supposed that the driver will issue a new direction command to escape from the stall. Moreover, as the driver must have the freedom of overriding the obstacle avoidance algorithm, he can modify the threshold simply repeating one or more times the previous command. In this way, when no obstacle is present, the wheelchair speeds up, and when the way is obstructed the driver can force the chair to push the obstacle, e.g. to move it. The chair, equipped with the described system, is able to move safely in typical cluttered environments, driven by the user with very simple vocal commands as "forward", "left", or "stop". In particular, it easily enters the door of a lift (60 cm wide, being the chair about 50 cm) and moves smoothly in the faculty corridors while people walk around.

FUZZY LOGIC APPLICATIONS IN MOBILE ROBOTICS

319

...............,

,,,,........,,,,

Figure 11.11 Upper: attractive horizon. Middle: repulsive and complemented attractive horizons. Lower: direction and repulsive horizons. The threshold is represented by the dashed line.

11.3.2 Implementation details The electric wheelchair is a standard unit with small modifications. It is 50 cm wide and 1 10 cm long and is powered by two independent 400W d.c. motors connected by irreversible gears (with ratio 96: 1) to the rear wheels. The motors are driven by two PWM amplifiers, whose duty ratio was originally controlled by the joystick through a simple analog circuit which maps joystick commands (forelaft, left/ right) in the two motor velocities. Seven ultrasonic sensors are installed on the front and the sides of the chair, giving a sufficient coverage for forward movements. Their locations had to be chosen taking also into account the use of blankets that may cover some sensor unsuitably placed. To ensure an easy development of the system, a laptop computer has been used to control the wheelchair, based on a 50 MHz 486. An external interface (DAQPad 1200, from National Instruments) allows the connections to sensors and to the motor amplifiers. Graphical language LabView (from the same firm)has been used as a frame to connect time critical routines (written in C) with user interface and to write other parts of the software. With this structure, a sampling time of 0.1 s can be easily achieved, if full screen plots are avoided. The cheap realization of the wheelchair had a huge impact on the overall implementation. Among the several faced problems, the most relevant are EM1 from PWM power amplifiers, non-backdriveability of the servos, dry frictions and wheel elasticity.

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Two simple encoders (100 pulseslrotation) have been purposely built and assembled on the motor shafts to measure the wheelchair speeds and displacements. Even if the theoretical resolution is quite satisfactory (0.17 mm), odometry is rather unreliable because of wheels' slippage. Using these measures two velocity servos have been implemented on the on-board computer, which moreover gathers the displacement information to monitor the wheelchair motion. 11.3.3 Fuzzy map building, localization and planning Knowing the environment (map building), finding its own position with respect to a known environment (localization), and searching the way to reach a goal (path planning) are complex functions that require a much richer description of the world. These three functionalities belong to the upper control levels of the wheelchair denoted as "optional", and up to now have not yet been implemented on board. They however have been studied on a mobile robot and, if needed, could also be adjusted for this project, provided a sufficient computing power is available. As all these subsystems have been already subjects of publications, here only a short outline will be given and their use on the wheelchair discussed. The original version of map building (Poloni et al., 1995) is based on a twofold description of the sensors and uses only simple theoretical operators. Consider a range finder (in particular an ultrasonic one) that provides a reading r. There is an evidence that some points located in the proximity of the arc of radius r and width equal to the sensor's radiation lobe are 'occupied'. At the same time, all the cells inside the circular sector with the same radius and width are likely to be 'empty'. The truth of these assertions are described by suitable functions, depending on the sensor physics. These two pieces of information can be accumulated while the robot moves around and explore the environment, giving origin to the fuzzy maps2 of occupancy 0 and of emptiness E . Accumulation can be performed by several kind of operators, and, for static maps, Union is an appropriate choice. Each point of the space belongs more or less to both E and 0, as "terzium non datur" does not apply. These sets can be combined generating other fuzzy maps which provide useful information on the safe parts S of the environment: which parts have not sufficiently explored, which parts have given contradictory measures, and so forth. In particular, for navigation purposes, the following simple definition for the dangerous areas can be adopted:

D=S

with S = E ~ D

even if, in the experiments, more complex expressions have been used to get rid of contradictory readings. The implementation of this method is straightforward, being the only complications due to the discretization of the maps and to the need of sweeping the circular sectors. It can be applied to different kinds of sensors, so representing a viable opportunity for sensor fusion (Fabrizi and Ulivi, 1997).

2 A fuzzy map is defined as a dimension 2 fuzzy set over a portion of cartesian space.

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APPLICATIONS IN MOHI1,E

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321

Conventional path planning algorithms cannot be used on grey-level maps so obtained, and generalized versions have been devised for fuzzy maps (Oriolo et al., 1994; Oriolo et al., 1997a). In particular, in the first paper the use of A* algorithm is proposed to single out the path of minimum risk over the navigation map V .The risk function can be fexibly defined, using the concept of fuzzy energy functions, in terms of the danger along the path. In the case of the wheelchair, some remarks are in order. First, the pri!ject goal is not full autonomy and moreover the chair moves very often in the same environment (e.g. the user's home). So an 'a priori' knowledge can be provided to the robot. avoiding the need for a complete, memory-consuming map building algorithm. Second, on long, complex paths, odometry of any robot becomes unreliable and this is -at a greater extent- the case of the wheelchair, where cost limitations do not allow expensive mechanical solutions. Finally, the typical environment is dynamic: tor example, a door can now be open and afterward closed. Building dynamic maps is a more complex task (Oriolo et al., 1997b) and, at the state of the art, it is advisable to rely on the user's planning capabilities. The main application of map building in this project is therefore limited to local1;lation, where small fuzzy maps are compared with an 'a priori' provided (crisp) map of all the admissible environment to recognize the vehicle position (Fortat-cz1.a ct ill.. 1995). This can be useful to ease traversing the narrow passages that arc present in rhc user's home. Their recognition can trigger specific preloaded plans which efficiently drives the chair after the passage in a way that recalls "Bchavior context detect~on" and activation described in the previous part. Searching for a common framework, we observe that the different ~nodulcsconstituting the control of the wheelchair can be arranged in a hierarchy with respect to complexity and sampling timc. At each level correspond a suitable implementation. Motor control requires a very fhst sampling time. Even if the sevel-c mechanical nonlinearities and the parameter uncertainties of the wheelchair make a f'uzzy logic approach appealing, it would require a dedicated processor, whilst a more conventionk~l controller can be implemented on the same computer used by the remaining softwarc components. Obstacle avoidance and map building are medium complexity modules and it is n o Inore possible to use direct analytical formulations. However they have a c.lei11definition and can be easily cast in term of, and etlicicntly implemented by, theoretical operators. Fast execution times are here important, in particular f'or the ohstaclc avoidance. The planner requires a hybrid approach, where the goal functions to hc rninimi~ctl arc defined by fuzzy concepts but standard techniques of graph sca~.cliinghrrvc to he used. Similarly, the environment recognition is based on the distance rncasurc h e t w ~ ~ c n "a priori" maps and the fuzzy one derived from sensors. It has to hc ~ ~ ~ i n l m i /hy ctl standard algorithms.

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11.4

CONCLUSIONS

In this chapter, some applications of fuzzy methods to mobile robotics have been shown. These systems mostly stress autonomy, a characteristic that call for "intelligent" controllers and, in this area, the shown methods result efficient and appealing. Interesting enough, searching a bibliographic database of the last three years, one can find hundreds of papers on the use of fuzzy techniques in mobile robots, while much few are those concerning applications to industrial manipulators. The latter are indeed much more oriented to precision and repeatability, and this is not the realm of fuzziness. According with the principle of increasing complexity-decreasingprecision, "intelligent" controllers find more space and motivations where the complexity of the controlled system and of its tasks is higher. As mobile robots heavily rely on data from many sensors, a direct implementation of fuzzy rule bases is generally unfeasible, as they would result in huge banks, and conclusions would be very slow to infer. Two approaches to this problem are presented in the chapter, one decompose the problem in smaller ones by introducing the concepts of virtual sensors and fuzzy behaviors. The other avoid the use of rules to express the required behaviors, and uses simple theoretical operators that result very often sufficient to describe the necessary operations. Both result in highly efficient systems, which perform quite satisfactorily. The chapter presents some results of experiments with the mobile robots ROMEO3R, designed and built at the University of Seville, and Labmate. Experiments have been also done with the Nomad 200 mobile robot and a wheelchair for disabled people developed in the University "Roma Tre".

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Acknowledgments

The authors would like to acknowledge to all the researchers that worked in fuzzy logic applications to robotics in the framework of the Falcon Working Group. Particularly we would like to mention the contribution of A. Garcia-Cerezo, J. Martinez, J. Gomez de Gabriel and A. Mandow (Universidad de Malaga, Spain),B. Anue (Universidad de Sevilla, Spain), and R. Braunstingl (Technical University of Graz, Austria), who are co-authors of the papers presenting the original work on virtual sensors described in section 2 of this chapter. Similarly, we wish to thank the researchers that have worked in the wheelchair project: G. Fortarezza (now at TIM, Roma, It), G. Oriolo (Universit di Roma "La Sapienza"),M. Vendittelli (now at LAAS, Toulouse, France), and the many students who developed their theses on this subject. References

Arkin, R. C. (1990). Integrating behavioral, perceptual and world knowledge in reactive navigation. Robotics and Autonomous Systems, 6:105-122. Arme, B. C., Cuesta, F., Braunsting, R., , and Ollero, A. (1997). Application of virtual perception memory to control a non-holonomic mobile robot. In 3rd IFAC SICICA 97, Annecy, France.

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Balzarotti, M. and Ulivi, G. (1996). The fuzzy horizons obstacle avoidance method for mobile robots. In Proc. of World Automation Con$ - ISRAMJ96,Montpellier, France. Bell, D., Borenstein, J., Levine, S., Koren, Y., and Jaros, A. (1994). An assistive navigation system for wheelchairs based upon mobile robot obstacle avoidance. In Proc. of 1994 IEEE Con$ on Robotics and Automation, S. Diego, CA. Borenstein, J. and Koren, Y. (1991). The vector field histogram-fast obstacle avoidance for mobile robots. IEEE Trans. on Robotics and Automation, 7:278-288. Braunstingl, R. (1996). Improving reactive navigation using perception tracking. In Proc. of 2nd World Automation Congress, Montpellier, France. Braunstingl, R., Sanz, P., and Ezkerra, J. M. (1995). Fuzzy logic wall following of a mobile robot based on the concept of general perception. In Proc. of the Seventh International Conference on Advanced Robotics, pages 367-376, Spain. Brooks, R. A. (1986). A robust layered control system for a mobile robot. IEEE Journal of Robotics and Automation, R.A-2: 14-23. De Medio, C. and Oriolo, G. (1991). Robot obstacle avoidance using vortex fields. In Stifter, S. and Lenar, J., editors, Advances in Robot Kinematics. Springer-Verlag, Wien. Demel, P., McCormac, S. E., and Uhl, A. (1995). A straightforward fuzzy collision avoidance strategy for autonomous vehicles. In Proc. of EUFIT'95, Aachen, Germany. Elfes, A. (1987). Sonar-based real-world mapping and navigation. IEEE Journal of Robotics and Automation, RA-3(3). Fabrizi, E. and Ulivi, G. (1997). Sensor fusion for a mobile robot with ultrasonic and laser range finders. In Proc. of 3rd IFAC SICICA'97, Annecy, France. Fortarezza, G., Oriolo, G., Ulivi, G., and Vendittelli, M. (1995). A mobile robot localization method for incremental map building and navigation. In Proc. 3rd lnt. Symp. on Intelligent Robotic Systems'95, Pisa, Italy. Gambino, F., Oriolo, G., and Ulivi, G. (1996). A comparison of three uncertainty calculus techniques for ultrasonic map building. In SPIE 1996, Int. Symp. on AerospaceDefence Sensing and Control, Orlando, FL. Goodridge, S. G. and Luo, R. C. (1994). Fuzzy Behavior Fusion for Reactive Control of an Autonomous Mobile robot: MARGE. IEEE. Maeda, Y., Tanabe, M., and Takagi, M. (1993). Behavior-decision fuzzy algorithm for autonomous mobile robots. Information Sciences, 71: 145-168. Mandow, A., Gomez-de-Gabriel, J., Martinez, J. L., Muiioz, V. F., Ollero, A., and Garcia-Cerezo, A. (1996). Greenhouse operation mobile robot aurora. IEEE Robotics and Automation Magazine. Mauris, G., Benoit, E., Foullow, L., and Josserand, J. F. (1994). Fuzzy range sensor fusion for the navigation of mobile robots. In Proc. of EUFIT'94, Aachen. Ollero, A., Garcia-Cerezo, A., and Martinez, J. L. (1994). Fuzzy supervisory path tracking of autonomous vehicles. Control Engineering Practice, 2:3 13-3 19. Ollero, A., Garcia-Cerezo, A., Martinez, J. L., and Mandow, A. (1997). Fuzzy tracking methods for mobile robots. In Towards high machine intelligence quotient, Enviromental and Intelligence Manufacturing Systems. Prentice Hall.

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Oriolo, G., Vendittelli, M., and Ulivi, G. (1994). Potential based motion planning on fuzzy maps. In Proc. of EUFIT'94, Aachen, Germany. Oriolo, G., Vendittelli, M., and Ulivi, G. (1997a). Fuzzy maps: A new tool for mobile robot perception and planning. Journal of Robotics Systems, 14:179-197. Oriolo, G., Vendittelli, M., and Ulivi, G. (1997b). Real-time map building and navigation for autonomous robots in unknown environments. IEEE Trans. on System, Man and Cybernetics, 28:301-3 15. Pin, F. and Watanabe, Y. (1994). Navigation of mobile robots using a fuzzy behaviorist approach and custom designed boards. Robotica, 12:491-503. Poloni, M . , Ulivi, G., and Vendittelli, M. (1995). Fuzzy logic and autonomous vehicles: experiments in ultrasonic vision. Fuuy Sets and Systems, 69: 15-27. Raju, G. V. S., Zhou, J., and Kisner, P. A. (1991). Hierarchical fuzzy control. International Journal of Control, 54:1201-1216. Reignier, P. (1994). Fuzzy logic techniques for mobile robot obstacle avoidance. Robotics and Autonomous Systems, 12:143-153. Saffiotti, A., Ruspini, E. H., and Konolige, K. (1993). Blending reactivity and goaldirectedness in a fuzzy controller. In IEEE International Conference on Fuuy Systems, pages 134-139. Song, K. and Tai, J. (1992). Fuzzy navigation of a mobile robot. In Proc. of 1992 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 141147. Tunstel, E. (1996). Mobile robot autonomy via hierarchical fuzzy behaviour control. In Proc. of 2nd World Automation Congress, Montpellier, France. Ulivi, G. (1995). Emergent techniques for mobile robots and autonomous vehicles. In Proc. of IFAC workshop ICASAV'95, Toulouse, France. von Aluok, C., Krause, B., and Zimmerman, H. J. (1992). Advanced fuzzy logic control of a model car in extreme situations. Fuuy Sets and Systems, 48:41-52.

12

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

G. Schram, M.A. Fernindez-Montesinos and H.B. Verbruggen

Delft University of Technology Faculty of Information Technology and Systems Control Laboratory Mekelweg 4, PO Box 5031 2600 GA Delft. The Netherlands

12.1

INTRODUCTION

An increasing amount of attention has been paid in recent years to the application of intelligent control methodologies in Aeronautical systems (Stengel, 1993; Steinberg, 1992). The integration of qualitative knowledge (heuristics) and quantitative knowledge (analytical models) through fuzzy modeling and control offers in particular many new possibilities. For example, the aggregation of human pilot heuristics and analytical aircraft models has yielded simple and transparent flight control designs (Chui and Chand, 1992; Chand and Chui, 1991; Larkin, 1985; Steinberg, 1993). Besides aircraft, fuzzy controllers have been developed for helicopters (Jiang et al., 1996; Phillips et al., 1996; Sugeno et al., 1993) and spacecraft (Berenji et al., 1993; Schram et al., 1996b; Woodard, 1996). Fuzzy modeling on the other hand has been applied to the identification of control system failures (Laukonen and Passino, 1994; Kwong et al., 1994; Youssef et al., 1996) and to the modeling of the human pilot (KrishnaKumar et al., 1995). However, it may be wondered why fuzzy logic could play a role in the control of aircraft. The dynamics of the aircraft are well described by first principles modeling and, although of nonlinear nature, the adaptation of controllers

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FUZZY ALGORITHMS FOR CONTROL

to varying dynamics has been successfully applied in practice. Therefore (classical) control structures are found in all present-day, flight-by-wire control systems (McRuer et al., 1973; Tischler, 1996). This is however only possible when extensive design efforts are executed for the various circumstances to be expected. Moreover, unexpected situations such as changing weather conditions and system failures are difficult to model and thus difficult to translate into appropriate classical control designs. For these reasons, the application of fuzzy logic can be justified as will be explored in this Chapter. The first part is devoted to the question whether fuzzy logic reduces the design effort that is necessary for the development of flight control systems. The design of current flight control system is mainly focussed on linear single-input-single-output (SISO) structures and only valid for limited regions of the flight envelope. Removing the interaction between the SISO loops and including extensive gain-scheduling procedures requires much additional design effort. In order to investigate whether advanced control techniques in general could increase the efficiency of the flight control design, the Group for Aeronautical Research and Technology in Europe (GARTEUR) recently formulated the so-called Research Civil Aircraft Problem (RCAM) (GARTEUR, 1997). The benchmark problem involves the control of a two-engined civil aircraft during the landing phase. Both horizontal and vertical flight paths must be followed accurately, under various conditions such as mass variations, passenger comfort and safety indexes, and control action limitations. With respect to the RCAM control problem, fuzzy logic shows beneficial properties because the control structure as well as the tuning process can be simplified through the additional expertise of the human pilot. The controller consists of linear SISO inner loops and fuzzy control law as outer loops. Both the outer loops are multi-input-multi-output (MIMO) of nature and gain scheduling is inherently achieved. Within the GARTEUR group, the fuzzy control design is extensively evaluated and compared to alternative control methodologies. In Section 12.2, an overview of the fuzzy control design including simulation results is provided. The second part of the Chapter is directed to the flight safety issue. In order to ensure future safety for the increasing number of passengers, aircraft, and flights, stringent safety requirements are demanded for future control systems (Shifrin, 1996). Robustness against (un)expected system failures and large atmospheric disturbances is required in the form of control reconfiguration and proper recovery, respectively. An example of a system failure is an engine failure; the induced moment and decreased force must be compensated by the remaining actuators. An example of a large atmospheric disturbance is the microburst, an unexpected vertical displacement of air that radiates outward as it reaches the ground. The microburst poses the greatest danger during the take off and landing phase of flight (Fujita, 1980). Because of the rule-based nature, reconfiguration and recovery strategies can be readily implemented in the fuzzy flight control framework. Moreover, the use of fuzzy sets makes the transition between nominal and exceptional situations smooth. In the Sections 12.3 and 12.4, control reconfiguration and microburst recovery are explored, respectively. The Chapter ends with concluding remarks in Section 12.5.

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

327

12.2 F U Z Z Y CONTROL OF A CIVIL AIRCRAFT BENCHMARK The Group for Aeronautical Research and Technology in Europe (GARTEUR) recently formulated the so-called Research Civil Aircraft Model (RCAM) design challenge (GARTEUR, 1997). The goal of the project is to investigate the efficiency of advanced control techniques for the design of flight control systems, in order to reduce the development cost. The control problem involves the control of a two-engined, large civil aircraft during the landing phase. The dynamic behavior of the aircraft is nonlinear, high-order, and strongly coupled. Both horizontal and vertical flight paths must be followed accurately, under various conditions. The performance specifications include rise time, settling time, overshoot, and steady state deviation requirements. Robustness is tested for mass, center-of-gravity, and airspeed variations. Passenger comfort and safety indexes include acceleration limits, damping specifications, and bounds on certain variables. The actuators have level and rate constraints. Twelve controller designs have been extensively evaluated and compared within the GARTEUR group (GARTEUR, 1997). One of the controller designs is based on fuzzy control and will be reviewed here (Schram and Verbruggen, 1997). Since our objective is to show how the human pilot's heuristics on flying aircraft can be incorporated, the Mamdani type of fuzzy control is selected (Mamdani, 1974). Experienced human pilots know how to handle an aircraft satisfactorily, i.e. they know how to combine competing criteria such as performance, robustness, safety, and passenger comfort without mathematical formulations. The Mamdani approach shows the particular benefits of applying fuzzy control to aircraft. The fuzzy controller consists of a longitudinal and a lateral part which will be described in the next Sections. Finally, several conclusions based on the GARTEUR evaluation results are discussed in Section 12.2.3.

12.2.1 Longitudinal controller design A simplified scheme of the aircraft model in the vertical plane is shown in Figure 12.1. The aircraft moves with an airspeed V, at an altitude h. The angle between the horizontal plane and the aircraft body is the pitch angle 0. The actuators are two engines and the elevator. The throttle setting Sth determines the thrust of the engines which initiates a deceleration or an acceleration. The deflection of the elevator 6, introduces a pitch angular velocity, called the pitch rate q. The objective of the longitudinal controller is to regulate the altitude and airspeed of the aircraft by throttle setting and elevator deflection. Pilot heuristics of flying an aircraft have been implemented in the design of the longitudinal fuzzy logic controller (FLC) (Schram et a]., 1996a). The design consists of a fuzzy logic outer loop tracking controller combined with a classical inner loop pitch attitude controller. In the high bandwidth inner loop controller, the pitch angle 0 is controlled by elevator deflections 6,. This increases the damping of the aircraft (phugoid) motion in the vertical plane. The (SISO) inner loop is tuned using a linearized model of the aircraft (McRuer et a]., 1973). The pitch angle commands or setpoints are then generated by the low bandwidth, outer loop tracking controller.

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FUZZY ALGORITHMS FOR CONTROL

elevator

&

engines

6th altitude

w

Figure 12.1

Simplified scheme of the aircraft model in vertical plane

The pitch attitude controller serves as the inner loop for the longitudinal outer loop controller. Suppose that the task of the pilot is to control the altitude h and airspeed VA of the aircraft. The control strategy of the pilot can be directly related to the total energy E of the aircraft (Larnbregts, 1983):

1 E = - m ~ :+ mgh 2 with m being the mass of the aircraft. For example, if the pilot realizes that the aircraft flies too low and too slow, he or she will increase thrust in order to increase the energy of the aircraft. On the other hand, if the aircraft flies too low but the velocity is too high, than the pilot will increase the pitch angle. In this way, the available energy is exchanged from airspeed to altitude. Based on the pilot heuristics, fuzzy rules of the Marndani type can directly be formulated, for example in the following form (Mamdani, 1974; Lee, 1990): If altitude low and airspeed slow then increase thrust and do not change pitch angle If altitude low and airspeed high then do not change thrust and increase pitch angle

The thrust in fact increases the total energy, while an exchange between kinetic and potential energy can be achieved by pitch angle changes (via elevator). With respect to reference values of (desired) altitude and airspeed, two rule bases are formed for throttle changes and pitch angle changes, respectively (Tables 12.1 and 12.2). The labels are defined as negative (N), positive (P), and extreme (E), very big (VB), big (B), medium (M), small (S), very small (VS), and zero (ZE). The number of rules is chosen relatively large in order to make local modifications of the fuzzy controller possible. The high number of rules provide enough "freedom" to tune the controller since only the rule consequents have been modified in order to keep the design procedure as simple as possible. This tuning is performed iteratively

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

329

Table 12.1 Rule base for throttle setting, e.g. IF velocity error is P M (aircraft too slow), and altitude error is ZE, THEN throttle change is PS (more thrust).

Table 12.2 Rule base for pitch angle commands, e.g. IF velocity error is PM, and altitude error is ZE, THEN pitch angle change is NVS (nose little down).

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FUZZY ALGORITHMS FOR CONTROL

by a low number of simulation runs while checking the RCAM design criteria (Schram and Verbruggen, 1997). The fuzzy propositions are related to a series of membership functions between [-I, 11. The membership functions are shown in Figure 12.2. To translate big and small errors into the [-I, 11 domain, scaling factors are defined using the physical relation between potential and kinetic energy. The control commands {-1,l) are scaled towards minimum and maximum control deflections for throttle and pitch angle changes. input: altitude error

input: airspeed error

outpum: pitch mgle and throme

E NVB

NB N M

NS

NVS

ZE

PVS

PS

PM

PB

PVB

P

0.5

-01

-0.8

-0.8

Figure 12.2

-0.4

-0.2

0

0.2

0.4

0.8

0.8

1

Membership functions for longitudinal FLC.

In Figure 12.3, the longitudinal controller is shown. The airspeed and altitude errors are denoted by V,,, and he, respectively. The aircraft block represents the aircraft dynamics including the inner loop pitch angle controller. The control actions are throttle setting $h and pitch angle commands 8,. For extra damping of the responses, the derivative signals of altitude and velocity are included. These signals could also be used directly in the FLC, but would increase the number of rules considerably. Small integral signal paths are required as well to guarantee zero steady state altitude and velocity errors. The additional input "priority" is used for a recovery procedure in Section 12.4. To show that the controller indeed mimics the human pilot's heuristics of flying an aircraft, an airspeed command of -3mIs and an altitude command of +25m are simultaneously given under nominal conditions (V, = 80m/s, h = 1000m). Moreover, the RCAM design challenge criteria are a settling time smaller than 45s and an overshoot smaller than 5%. The responses are plotted in Figure 12.4. Both the RCAM design challenge requirements are fulfilled. The actuator responses show that throttle setting remains practically constant, and that by an elevator deflection the energy is exchanged from airspeed to altitude. This corresponds directly to the underlying total energy principle of the human pilot. The closed-loop stability of the longitudinal fuzzy logic controller can be partly analyzed in one operating point using the so-called conicity-criterion (Schram et al., 1996a). The conicity-criterion is derived from the Small Gain Theorem (Garcia-

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

aircraft t inner loop

priorit

Figure 12.3

331

Block-schematic representation of longitudinal controller. Altitude response (--)

Airspeed response (--)

1

Figure 12.4

10301

elevator (-) and throttle (--)

I

Simultaneous airspeed and altitude command.

Cerezo and Ollero, 1992; Driankov et al., 1993). However, the stability analysis is very restricted because only a linearized model of the aircraft dynamics is considered and actuator constraints are not included. In fact, no methodologies are currently present to guarantee stability in an analytical way for such a complex plant with nonlinear dynamics, actuator constraints, etc. Therefore, flight controllers in general must always be analyzed by extensive (randomized) simulation studies. This also holds for the fuzzy control design.

12.2.2 Lateral controller design The top view of the aircraft is shown in Figure 12.5. The aircraft moves in the direction of the airspeed Va.With respect to the desired flight path, the lateral deviation yl,t should be zero, and the heading angle 1C, should compensate the sideslip angle P. The actuators are the ailerons and the rudder. In general, the (opposite) ailerons deflection 6, introduce a roll angular velocity, called the roll rate p, while a rudder deflection 6, causes a yaw angular velocity r. Similar to the longitudinal design, a classical inner loop controller is tuned first using an analytical aircraft model. In the high bandwidth inner loop controller, the roll angle is regulated by aileron deflections. The roll angle setpoints are generated by the outer loop controller. For the lateral fuzzy logic outer loop controller, pilot heuristics are again considered (Schram and Verbruggen, 1997). The selected variables to be controlled are heading angle $, lateral deviation ylat,and the sideslip angle P. A human

332

FUZZY ALGORITHMS FOR CONTROL desired flight path

t

lateral deviation

I 1 heading

roll angle

a'

.

$ .

.'

I' 1

I

Figure 12.5

aileron

..';I

1 rudder 1 6'

6a

I

I

Top view of the aircraft model.

pilot will generally handle the tracking angle and lateral deviation by rolVaileron commands, and sideslip by rudder deflection. For example, by rolling the aircraft clock-wise, a right turn is made that changes the course of the aircraft. On the other hand, small rudder deflections (pedals) can compensate the sideslip which results from e.g. a crosswind. Based on the pilot heuristics, fuzzy rules of the Marndani type can again be formulated, for example in the following form:

4

If desired tracking angle change positive (to the right) then roll aircraft clock-wise (positive) If sideslip angle positive then rudder deflection negative

The control strategy is implemented in two separate fuzzy controllers, a lateral tracking controller and a sideslip controller. In the lateral tracking controller, lateral deviation and heading angle errors are the inputs, and roll angle commands 4, the output. In the Table below, the rule base with heading error $, = $ T e - $ and lateral deviation yl,t as antecedents, and roll angle command as consequent is shown. In the second fuzzy controller, rules are defined to suppress the sideslip error /3, = P,., - P. These rules are shown in Table 12.4. The membership functions for heading, lateral deviation, and roll angle are shown in Figure 12.6. With the fuzzy sets which are used in the rule base, an initial linear function is obtained in the interval [-I, 11. Using physical relations between lateral deviation, heading angle, and airspeed, the scaling factors are defined. Airspeed is part of the scaling such that gain-scheduling as a function of airspeed is inherently achieved.

I

I

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

Table 12.3 Rule base for roll angle commands, e.g. IF heading error is PB (more t o the right), and lateral deviation is ZE, THEN roll angle is PB (clock-wise).

Table 12.4 Rule base for rudder commands, e.g. IF sideslip error NS (positive sideslip angle), THEN rudder command is NS (negative).

input: lat, dsviation

~nput:heading errol

I

I

output: roll angle

I

Figure

333

I

12.6 Membership functions for lateral tracking controller.

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FUZZY ALGORITHMS FOR CONTROL

For the sideslip error and rudder deflection, the membership functions are shown in Figure 12.7. The fuzzy sets (NM,NS,ZE,PS,PM)areconcentrated between [-0.25,0.25]. In case of an engine failure, the membership functions (NVB,NB,PB,PVB) are used to represent the more aggressive control behavior of the pilot (see Section 12.3). input: sideslip

output: rudder deflection

I

I

I

I

NM

NB

oiw -1 0

-05

0

4

0 5

-1

1

Figure 12.7

-08

-06

-04

-02

NS ZE

0

PB

PS PM

02

04

PVB

06

08

1

4

Membership functions for sideslip controller.

In Figure 12.8, the lateral controller is shown. The aircraft block represents the aircraft dynamics including the inner loop roll angle controller. The control actions are rudder deflection 6, and roll angle commands 4,. Notice that an extra integrator (Ki = 0.2) is added for zero steady state sideslip.

-,

ylat

fuzzy logic heading controller

W

$c

b

P

inner loop

Figure 12.8

8,

Block-schematic representation of lateral controller.

The lateral controller is tested under nominal conditions (V, = 80m/s, h = 1000m). An example of a lateral manoeuvre is to follow a lateral displacement of 50m with a settling time of t , < 30s. Furthermore, as part of the RCAM design criteria, there should be "little" overshoot, i.e. than 5%. The step response is plotted in Figure 12.9. Both RCAM specifications are met. Notice that the sideslip angle approximates 0 degrees in a short time period. 12.2.3 Evaluation results

Within the GARTEUR group, the fuzzy controller has been tested on all performance, robustness, ride quality, safety, and control activity criteria (Schram and Verbruggen, 1997). Extensive simulation results show that a compromise is found in which accuracy and robustness properties, for example with respect to mass, are met at the cost of a slightly too high vertical and lateral acceleration indicating passenger comfort (Schram and Verbruggen, 1997). In Figure 12.10, a plot is shown of a test scenario used in the RCAM controller evaluation. The flight mission is to follow a horizontal flight path

4

:pI-"

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

lateral dev~at~on (--)

335

head~ng(-),beta,(--),yaw rate(..)

20

0

-20

'

0

0

40

20

Figure 12.9

-1

60

0

',-.-20

40

60

Step response: desired lateral displacement 50m.

with strong lateral wind and engine failure (0-l), followed by a sharp right turn (1-2) and a descending (landing) flight path (2-4). The desired flight path and the simulated trajectory of the aircraft with fuzzy controller can be hardly distinguished, even in case of an engine failure. The routines for the engine failure will be further discussed in Section 12.3.

1500

2

0

y-position [krn] Figure 12.10

-20

-25

x-position [krn]

RCAM test scenario: following a three-dimensional flight path.

Additionally, the hybrid controller has been extensively evaluated and compared with respect to the other control methodologies (GARTEUR, 1997). Most of the controllers fulfilled the criteria and showed comparable results. This was exactly the objective of the design challenge. The aim was not to give the answer to the question "which control method is best", but rather to show, step by step, how modern control can be applied. The ultimate goal is to reduce the necessary design effort. All methods demonstrated improvements in specific areas compared to the classical approach. The main conclusions concerning the fuzzy logic controller are:

336 m

FUZZY ALGORITHMS FOR CONTROL

The translation of human heuristics into the fuzzy controller is a systematic procedure. The heuristics can directly be used to define the MIMO control structure. Furthermore, the use of linguistic terms makes the interpretation of the controller very straightforward. However, tuning the fuzzy controller must be done in an iterative way (in fact for all control methods). Linear control tools, e.g., in the frequency domain, can hardly be used to synthesize and analyze the fuzzy controller. The final verification of the controller has therefore to be done by extensive (randomized) simulations. On the other hand, starting with linear fuzzy controllers and only changing the rule consequents simplifies the tuning considerably. The application of a Takagi-Sugeno type of fuzzy controller in combination with (local-linear) fuzzy modeling could be a possibility to perform the design in an automatic way (BabuSka and Verbruggen, 1996; Takagi and Sugeno, 1985). Fuzzy control yields intuitive, simple rule-based controllers of low-order. This is very important, since black-box controllers like high-order state feedback or neural networks are not acceptable in the Aeronautical community. The visibility is of main importance in the validation of flight control systems. Therefore, no problems are expected concerning certification of fuzzy control systems.

I

I

Robustness is achieved indirectly. In first instance, the (nonlinear) controllers are of low gain. Secondly, through the implicit gain scheduling as a function of e.g. airspeed. This is exactly the way human pilots achieve their robustness, by low gains and adaptation. Beneficial future applications are expected in that area of flight control and guidance where pilot techniques can be copied and where precision requirements are not too high. In these areas, fuzzy controllers can directly be synthesized and possibly certificated. Additionally, in case of uncertainties which can not be modeled anymore, fuzzy control provides a very interesting solution through the easy exception handling capability. Therefore, the benefits of fuzzy control will be explored in the context of large atmospheric disturbances and control system failures in the remaining part of this Chapter.

I

12.3 CONTROL RECONFIGURATION IN CASE OF FAILURES Timely reconfiguration of flight control systems is critical to the safety of aircraft in case of failures (NTSB, 1979; RLD, 1994). Among many possible failures, sensor and actuator failures have a direct impact on the performance of the control system. Actuator and sensor failures can be detected and identified by the inherent (analytical) redundancy contained in the static and dynamic relations among the actuator commands and measured outputs, see e.g. (Frank, 1990; stengel, 1991). The identification results can directly be used to reconfigure the flight control system accordingly: failure tolerant control. The conventional approach is multiple model adaptive control (MMAC) (Athans, 1977; Maybeck and Stevens, 1991). In MMAC, each failure hypothesis corresponds to one model and one control mode. The model and control mode that has the highest likelihood (most "close" to the actual situation) is selected. However,

I

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

337

the MMAC approach is computationally very intensive since not only each component must be hypothesized but the type and the magnitude must be modeled as well. As a generalization of MMAC, a fuzzy approach for control configuration is presented (Schram et al., 1997; Schram et al., 1998a). Similar to MMAC, for the different failure types a control mode is derived beforehand. However, through the use of fuzzy measures that indicate the failure states, interpolation between the failure modes is achieved. In this way, much less control modes are required than in the MMAC approach and a more accurate reconfiguration of the control system could be obtained. Furthermore, a smooth transition between the control modes is achieved in case of a gradual degradation of system components. The concept is demonstrated on the RCAM model. Using the fuzzy controller of Section 12.2, additional supervisory rules are defined in the case of an engine failure.

12.3.1 Reconfiguration for engine failure The lateral controller in Section 12.2 shows bad responses in case of an engine failure and needs further modifications. For this failure situation, the lateral deviation from the flight path and the roll angle are too large. Moreover, large roll oscillations are present. In Figures 12.11 and 12.12 respectively, the lateral deviation and roll angle are shown under five test conditions. The test conditions are defined in Table 12.5. In all cases, one of the engines breaks down at t = Os, while at t = 100s, the engine is restarted. In order to decrease the lateral deviation and the roll angle, reconfiguration of the fuzzy controller is required.

Table 12.5

Flight conditions: variations of aircraft mass and airspeed.

No. 1

11 Mass (kg) ( 11 120,000 1

Airspeed (m/s) 80

120,000 120,000

90

I 1

line type thick solid dashed dotted dashdot

In general, in case of an engine failure, the human pilot will increase the power of the remaining engine in order to keep altitude and airspeed (total energy concept), and he or she will add extra rudder deflection in order to keep the desired heading angle with minimal sideslip. Rudder deflection in case of motor failure is a general pilot procedure. The power increase is achieved by the longitudinal fuzzy controller. The extra rudder deflection is in principle also achieved with the lateral controller. However, the human pilot will react more alertlaggressive than in case of no engine failure with respect to heading and sideslip errors. The pilot behavior is modelled by two extra rule bases, depicted in the Tables below. The consequents are more aggressive than in the rule bases for no engine failure. The effect of the modifications is that the gains of the lateral outer loops are implicitly increased.

338

FUZZY ALGORITHMS FOR CONTROL

Figure 12.11

Lateral deviation responses responses. mll awl.

20,

Figure 12.12

Table 12.6 Rule base for roll angle commands in case of engine failure, e.g. IF engine failure, and heading error is P B (more t o the right), and lateral deviation is ZE, THEN roll angle is PVB (clock-wise).

Roll angle responses.

I

Table 12.7 Rule base for rudder commands in case of engine failure, e.g. IF engine failure, and sideslip error NS (positive sideslip angle), THEN rudder command is N B (negative)

PB

PVB

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

339

Smooth switching between the rule bases for engine ON and OFF is automatically achieved by defining two membership functions for the extra input "engine failure". The membership functions are shown in Figure 12.13. In case of an engine failure, the rules for engine OFF are directly initiated by switching from 0 to 1. When the engine is restarted, the pilot switches gradually its control behavior from engine OFF to engine ON since helshe has to recover first. In Figure 12.13, the extra input signal for the controller is shown as well: the engine breaks down at t = 0s and is restarted at t = 100s. Input: engine failure

I

engine failure signal

I

1.

5

r

-

l

Figure 12.13 Membership functions for engine failure (left), and additional engine failure "input" during simulations (right).

The effect of the extra rules for engine failure is shown in the Figure 12.14and 12.15. Both the lateral deviation and roll angle responses are decreased by the more aggressive rudder commands. The roll angle does not exceed the limit of 10 degrees for all conditions, and has smaller oscillations than in the responses without reconfiguration. 10,

Figure 12.14

.

lateral deviation

Lateral deviation responses in case of reconfiguration.

12.3.2 Discussion The concept of control law reconfiguration within the fuzzy control framework has been demonstrated for an engine failure. In contrast to the conventional multiple model adaptive control (MMAC) approach, smooth transitions and a gradual interpolation

4

340

FUZZY ALGORITHMS FOR CONTROL

Figure 12.15

Roll angle responses in case of reconfiguration.

between the failure control modes is achieved by using fuzzy sets. Reconfiguration in case of other failure cases such as a disfunctioning of the ailerons or rudder can also be included (Schram et al., 1998a). The integration with failure detection and identification techniques is currently investigated as well (Schrarn et al., 1998b).

12.4

4

RECOVERY FROM LARGE DISTURBANCES

Adverse weather conditions are major hazards to aviation. The generic term windshear refers to a rapid change in the wind speed and/or direction (Fujita, 1980). The microburst, a dangerous form of windshear, is produced by a strong sudden downdraft of cool air which strikes the ground, producing winds that outflow in all directions. The microburst occurs below storms, sometimes even in clear air, with a very short, unpredictable lifetime. For arriving and departing aircraft, the microburst poses a serious risk (DGAC Portugal et al., 1994;NTSB, 1995). The hazard of the low-altitude microburst arises from the rapid change from a performance increasing headwind to strong performance decreasing tailwind and downdraft, see Figure 12.16. To cope with the headwind, the (auto)pilot intuitively takes actions to prevent the aircraft from climbing. These actions are then compounded by performance loss caused by the downdraft and tailwind, so that it may be impossible to avoid ground impact. From many accidents, it can be learned that a successful escape from such an inadvertent encounter requires a timely detectionlrecognitionand adequate control actions. Numerous methods and techniques in terms of avoidance, escape and recovery from these phenomena have been proposed by the airlines, aerospace industry, and research institutes (Hinton, 1992). As windshear may exceed the performance capability of aircraft, avoidance has been emphasized. Recent advances in windshear detection systems, has helped to reduce the hazard significantly by timely detection and avoidance (Bowles, 1990; Stratton and Stengel, 1995). However, due to the short-lived transient nature of this phenomenon, timely detection is not always possible. Hence, certain procedures and techniques must be applied to prevent a dangerous flight path situ-

4

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

'

341

l]k:K

burst

headwind

Figure 12.16

Effect of microburst during landing.

ation from occurring. Therefore, new guidance determination and guidance tracking techniques to enhance aircraft safety are still key factors of study.

12.4.1 A protection mechanism In (Aznar FernBndez-Montesinos et al., 1997), fuzzy logic has been applied to the guidance and control problem during the windshear phenomenon. It is hereby assumed that a reactive windshear detection system is present, which only alerts of the presence of windshear when the windshear already has been encountered. Since windshear is basically an energy management problem, the fuzzy controller of Section 12.2is used. Extra energy is automatically supplied during the microburst by increasing the thrust. Then, a recovery module is integrated with the fuzzy controller to enhance performance and hence safety. The recovery module performs a dual role: flight path determination and on-line protection of the flight envelope. Depending on the margins with respect to a minimum airspeed and a minimum altitude, the protection mechanism divides the available energy into airspeed and altitude. Once the flight bounds are guaranteed, the controller tracks the set-points indicated by the determined flight path. This flight path consists generally of an escape manoeuvre (ascending to a go-around altitude). In order to find an indicator for the flight envelope protection, the specific energy of the aircraft is considered (energy per unit weight), derived from equation (12.1):

where Va is the airspeed, mg is aircraft weight, and h is the altitude. With respect to altitude and airspeed changes Ah and AV,, the energy changes by:

342

FUZZY ALGORITHMS FOR CONTROL I

Considering a minimum airspeed Va,minand a minimum altitude hmin,the energy margins with respect to these values can be calculated as:

In the definition of a priority function f (h,V,), the following conditions are introduced:

f(h,Va) = -1 f (h,Va) = 0 f(h,Va) = 1

if =O (no margin altitude) if (Ah,)h = (Ahp)va (same energy margins) if(Ahp)vo=O (no margin airspeed)

(12.7)

,

I

Moreover, for the values in between, the energy distribution should be proportional to the ratio ( A h p ) h / ( A h p ) vSimple a. algebraic calculations then yield the following priority function that satisfies the above conditions: 4

f (h,v.) =

- aretan 7r

(-)(Ah, va

(Ah~)h

- I.

The new variable f (h,Va)in fact indicates the need for altitude either airspeed priority. The variable is used as input in the fuzzy controller, see Figure 12.3. To integrate the priority variable, three membership functions are defined. The membership functions are shown in Figure 12.17. Each membership function belongs to a rule base for pitch angle commands. The membership functions are defined such that a linear interpolation is achieved between one of the special rule bases and the normal rule base. Therefore, the altitude and airspeed functions are not overlapping. The rule bases are selected by fuzzy rules making the smooth interpolation between the rule bases possible. I I I I I

I I I I -1

0

1

I I I - - - - - - - - - - - - - - - - - - - - - - - - -

Figure 12.17

priority

I

I

one integrated I

fuzzy system

I

Fuzzy logic scheme of priority scheduling.

The rule base for nominal conditions was defined in Table 12.2. Two additional rule bases are defined for altitude and airspeed priority, respectively. For the altitude

I

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

343

priority case, the pitch angle commands are a function of altitude, see Table 12.8. In the rule base for airspeed priority, the pitch angle command is defined as function of airspeed errors, see Table 12.9.

Table 12.8 Rule base for pitch angle commands in case of altitude priority, e.g. IF velocity error is PM, and altitude error is PM (too low), THEN pitch angle change is PB (nose up).

Table 12.9 Rule base for pitch angle commands in case of airspeed priority, e.g. IF velocity error is P M (too slow), and altitude error is P M , THEN pitch angle change is NB (nose down).

12.4.2 Simulation results for recovery during landing In the simulations, the RCAM model is again used but only the vertical plane is considered. An aircraft model with mass = 150,000kg is trimmed for a glide slope of -3 degrees with an airspeed of 65m/s. For the microburst simulation, a realistic model is used (Oseguera and Bowles, 1988). In the definition of a safe flight envelope, the minimum airspeed Va,mi, is defined as 1.05 times the stall speed (58mls). Below the stall speed, the aircraft will fall out of the sky. For the determination of the minimum altitude hmi,, no straightforward guidelines are available. As an example, we use a glide slope deviation of 0.5 degrees. If the aircraft approaches the altitude bound during the recovery, altitude will automatically be preferred. The microburst strength is indicated by the so-called T-factor. In the simulations, the T-factor is varied between

344

FUZZY ALGORITHMS FOR CONTROL

0.25 (moderate intensity) and 0.35 (high intensity) corresponding to empirical values (Fujita, 1980). The results for the different microburst intensities are shown in Figure 12.18. Both the airspeed and altitude responses are shown. The nominal and minimum values for airspeed and altitude are depicted as dotted lines. In all cases, both the airspeed and altitude margins are respected because of the flight envelope protection mechanism. For the increasing microburst strength, more energy is automatically built up through the fuzzy controller. This energy is then used to compensate for the energy loss caused by the microburst.

Altitude ' /

,.'

-E

200 ' . . . . ....

100 -

'

0 0

20

40

60

80

' / _

100

'. .. .. .

120

Time (s) Airspeed

Frnax = 0.25 Frnax = 0.30 . - - Frnax = 0.35

I 0

20

40

60

80

100

120

Time (s)

Figure 12.18

Aircraft responses for varying microburst in abort landing scenario.

12.4.3 Discussion Complete avoidanceof a microburst is of course the best way to ensure safety. However, in case of a microburstencounter, guidance and control should be provided to cope with the energy loss by enhancing aircraft performance while maintaining flight envelope protection. The recovery concept introduced in this Section guarantees the flight envelope bounds while the controller tracks the set-points indicated by the go-around procedure. The flight envelope protection mechanisms has been readily integrated in the fuzzy logic control framework. Current research is directed to the application of this generic recovery concept to the take-off phase as well.

ENHANCING FLIGHT CONTROL USING FUZZY LOGIC

12.5

345

CONCLUSIONS

Fuzzy logic control has established as a useful tool for several aeronautical problems. In this Chapter, reduced design effort and safety have been emphasized as main factors in civil aircraft control design. The expertise of human pilot has been used to design longitudinal as well as lateral controllers for a recently formulated aircraft benchmark problem within a European research group. This yielded intuitive controllers of low complexity and good performance and robustness results satisfying the design challenge criteria. Furthermore, the design effort is reduced because of the systematic definition of the control structure and easy (linguistic) interpretation of the flight control laws. The second challenging area is the flight-safetyissue. This includes rare exceptional situations where conventional, linear control does not apply. Such situations include system failures and large atmospheric disturbances. Special rules for an engine failure have been readily included in the fuzzy control framework. Safe recovery from a microburst has been achieved by incorporating a special flight envelope protection mechanism in the form of additional rule-bases as well. The exception handing enhances the current flight control systems and could hereby increase the flight-safety in future applications. References

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NTSB (1995). National Transportation Safety Board Final report AAR-95/03 on McDonne11 Douglas DC-9-3 1 N954VJ, USAIR accident at Charlotte Airport, USA, July 2, 1994. Technical report. Oseguera, R. M. and Bowles, R. L. (1988). A simple, analytic 3- dimensional downburst model based on boundary layer stagnation flow. Technical report, NASA Technical Memorandum. Phillips, C., Karr, C. L., and Walker, G. (1996). Helicopter flight control with fuzzy logic and genetic algorithms. Engineering Applications of Artificial Intelligence, 9(2): 175-184. RLD (1994). Netherlands Aviation Safety Board aircraft Accident Report 92-1 1, El A1 Flight 1862, Boeing 747-258F 4X-AXG, Bijlmermeer, Amsterdam, October 4, 1992. Technical report. Schram, G., Copinga, G. J. C., Bruijn, P. M., and Verbruggen, H. B. (1998a). Failuretolerant control of aircraft; a fuzzy logic approach. In American Control Conference, pages 2281 - 2285, Philadelphia, USA. Schram, G., Gopisetty, S. M., and Stengel, R. F. (1998b). A fuzzy logic - parity space approach to actuator failure detection and identification. In AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA. AIAA 98-1014. Schram, G., IJff, J. M., Krijgsman, A. J., and Verbruggen, H. B. (1996a). Fuzzy logic control of aircraft; a straightforward mimo design. In AIAA Guidance Navigation and Control Conference, San Diego, CA, USA. AIAA 96-3847. Schram, G., van Buijtenen, W. M., BabuSka, R., and Verbruggen, H. B. (1 996b). Neurofuzzy control of satellite by reinforcement learning. In Computational Engineering in Systems Applications MultiConference (CESA-96), pages 1056 - 1060, Lille, France. Schram, G. and Verbruggen, H. B. (1997). RCAM: A fuzzy control approach. In Magni, J. F., Bennani, S., and Terlouw, J., editors, Robust Flight Control, A Design Challenge, Lecture Notes in Control and Information Sciences Series 224, pages 397 - 416. Springer Verlag. Schram, G., Verschoor, B., Sousa, J. M., and Verbruggen, H. B. (1997). Flight control reconfiguration using multiple fuzzy controllers. In Sixth IEEE International Conference on Fuuy Systems, pages 111 - 117, Barcelona, Spain. Shifrin, C. A. (1996). Aviation safety takes center stage worldwide. Aviation Week & Space Technology, pages 46 -48. Steinberg, M. (1992). Potential role of neural networks and fuzzy logic in flight control design and development. In AIAA Aerospace Design Conference, AIAA 92-0999, Irvine, CA, USA. Steinberg, M. (1993). Development and simulation of an f/a-18 fuzzy logic automatic carrier landing system. In 2nd IEEE Conference on Fuuy Systems, pages 797-802, San Fransisco, CA, USA. Stengel, R. F. (1991). Intelligent failure-tolerantcontrol. IEEE Control Systems, 1l(4): 1423. Stengel, R. F. (1993). Towards intelligent flight control. IEEE transactions on Systems, Man, and Cybernetics, 23(6): 1699-17 17.

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t

4

t

4

Index

Adaptive control, 11,254,269,275 Aggregation operator, 174 Algorithm aggregation algorithm, 200 branch-and-bound, 177, 179 Gustafson-Kessel algorithm, 89 iterative compatibility analysis, 93 maw-min transitive closure, 93 OLS learning, 192 similarity relations, 94 two-pass OLS learning, 194 Antecedent, 86 Application bleaching, 248 cement kiln, 247 coke oven, 248 effluent treatment, 248 fermentation, 249, 258 lime kiln, 247, 264, 284 paper machine, 266 recovery boiler, 248 rotary dryer, 264 solar power plant, 264,271,247 thermomechanical pulping, 248 air-conditioning, 95 civil aircraft benchmark, 327 fermenter pressure dynamics, 188, 196, 202, 200 microburst recovery, 340, 326 wheelchair, 3 13 Approximate reasoning, 20, 147 Asymmetrical action, 253,277,279, 288 Autonomous mobile robots, 301 navigation, 306 robot, 318 vehicle, 3 15 Basis function expansion, 63 Behavior, 301, 310 Braking action, 252,269, 275,279,288 Certainty equivalence approach, 73

Certainty rules, 27 Classification, 238 Classifier design, 233 Cluster covariance matrix, 88 merging, 90 validity measures, 90,98 Coherence of fuzzy rule bases, 42 Compatibility measure, 199 pair-wise, 199 relation, 199 Completeness, 119, 185 Complexity reduction, 103, 185, 224 Consequent, 86.91 Consistency measure, 187 Control reconfiguration, 326, 337 Control systems advanced, 3 , 7 classical, 3 dynamic behavior, 130 feedback, 127 frequency-domain, 147 heterogenous, 6 internal model, 101 linear control theory, 145 low level, 3 model-based predictive, 8, 10, 101 PID, 6.9 pole placement, 140 robust, 9 supervisory, 11 time invariant, 137 Control theory, 127 Control horizon, 176 policy, 174 problems, 160 Data analysis, 223, 241 attributes, 223

350

FUZZY ALGORITHMS FOR CONTROL

CASE-tool, 232 classes, 223 fuzzy expert system, 225 fuzzy regression, 225 neural net, 225,23 1 objects, 223 software, 23 1 Data analyzer, 234 Data pre-processing, 23 1 DataEngine, 232 classification, 233 classifier design, 233 function block, 233 Defuzzification, 40, 87, 118 Delaunay triangulation, 208-209 Design goals, 162 Design specifications, 160, 162 linear systems, 162 nonlinear systems, 164 Dimensionality, 234 reduction, 202 Dynamic fuzzy data analysis, 235 Dynamic planning and scheduling metho'ds, 7 Environment perception, 308 recognition, 32 1 Expert control fuzzy, 246,274,286 linguistic equations, 287,275,251 Failure tolerant control, 326,336 Fault detection, 7 Fault diagnosis, 7, 259,269 Feature extraction, 64,232 Feedforward control conventional, 274 linguistic equations, 290, 252 Filtering, 64-65 Flight control, 325,327, 345 Flight safety, 326, 336, 345 Function piecewise monotonous, 122 realized by formula, 117 Fusion and annihilation, 187 Fuzzification, 3 1, 234 Fuzzy back-propagation, 23 1 Fuzzy basis function, 190 Fuzzy behavior, 309-310 Fuzzy clustering, 87,225,236,257 c-means algorithm, 226,229 clustering methods, 231 fuzzy shell clustering, 229 Gustafson-Kessel algorithm, 89 possibilistic fuzzy c-means, 229 relative membership, 229 Fuzzy constraint, 172 Fuzzy control, 9.59, 127 state feedback, 73

fuzzy dynamical systems, 144 fuzzy identification, 140 hierarchical fuzzy controller, 21 1 industrial applications, 60 industrial fuzzy control, 66 modem approaches, 68 nonlinear feedback fuzzy controller, 139 PID-like, 66 robust fuzzy systems, 135 sliding mode, 74 static fuzzy controller, 137 Takagi-Sugeno fuzzy model, 135 Takagi-Sugeno inference, 148 Fuzzy controller, 306 definition, 60 structure, 64 Fuzzy decision making, 6, 172 fuzzy goal, 172 in control, 173 Fuzzy implications, 25 Fuzzy inputs, 37 Fuzzy map building, 320 Fuzzy model inversion, 267 linguistic equations, 257 linguistic, 256 neural, 257 relational, 256 rule generation, 257 singleton, 256 Takagi-Sugeno, 256,264 data-driven construction, 83, 85 linguistic, 86 relational, 86 Takagi-Sugeno, 86,97 validation, 86 Fuzzy model-based control internal model, 69 inverse model, 68 inverse model control, 68 predictive, 70 Fuzzy perception, 307 Fuzzy set, I48 multidimensional, 208 aggregation, 86 compatibility measure, 92 fuzzy partition, 135 linguistic terms, 148 projection, 90 Fuzzy system Mamdani, 61 Takagi-Sugeno, 62 Fuzzy the Kohonen network, 23 1 Gain scheduling, 76 Generalized rnodus ponens, 27 Gradual rules, 26 High level control, 249, 282, 287

INDEX High level plan, 3 11 Identification, 83, 98 Implementation, 29 1 Input nonlinearity, 76 Integrated information system, 7 Intelligent analyzers, 249, 259, 289 Intelligent data analysis, 241 Interaction, 7 Internal model control, 69 Interpolation, 46 Inverse model control, 68 Iterative compatibility analysis, 92 Least-squares estimation global, 91 weighted, 9 1 Linguistic equations dynamic model, 263 fuzzy, 258 interactions, 258 inversion, 268,290 membership definitions, 258 multimodel approach, 265 real-valued, 259 single model approach, 265 steady state model, 260 Linguistic approximation, 95 hedges, 95 model, 86 LMI, 148 Localization, 302, 320 Locomotion, 306 non-holonomic, 306 Map building, 302, 315, 32&321 range finder, 320 Mathematical fusion, 202 Mathematical model, 128 differential equation model, 139 Membership function merging, 94 acquisition from data, 90 adaptation, 187 aggregation, 199 annihilation, 187 closeness, 187 compatibility, 187 exponential, 91 fusion, 187 similarity, 187 Mobile robot, 320 Mobile robotics, 301, 314, 322 emergent behavior, 302 layered control, 302 motion command, 301 reactive control, 302 subsumption architecture, 302 Model-based predictive control, 175, 268,289

Motor control, 321 Navigation, 301, 3 14, 320 Neural networks, 147,236 Neuro-fuzzy system, 84 Nonlinear behavior, 7 Nonlinear systems, 128 nonlinear static systems, 137 pole placement, 73 qualitative theory, 135 Normal form conjunctive, 115 disjunctive, 115 Norms, 169 average-absolute value, 170 induced, 171 ITAE, 170 of systems, 171 root-mean-square, 170 semi-norm, 169 Obstacle avoidance, 302, 3 11.3 15,321 attractive horizon, 3 16 direction horizon, 3 17 kinematic horizon, 3 17 occupancy grid, 3 16 omnidirectional robot, 3 17 possibility of occupancy, 3 17 repulsive force fields, 3 15 repulsive horizon, 3 1 6 317 risk of collision, 316 ultrasonic sensor, 3 17 wheelchair, 3 15 Optimal control, 163 Optimization branch-and-bound, 101 Nelder-Mead, 177 sequential quadratic programming, 177 Orthogonal least-squares, 190 learning, 191 reduction by two-pass OLS, 194 Partition matrix, 88 Path planning, 320-321 dynamic maps, 321 fuzzy energy functions, 321 odometry, 32 1 Perception, 301, 306, 317 Performance criteria, 159 classical, 168 fuzzy, 172 in MBPC, 174 Performance specifications, 165 actuator effort, 168 input-output, 166 regulation, 167 Planning, 302,306 Prediction horizon, 176 Pruning, 186 Reactivity, 3 11

351

352

FUZZY ALGORITHMS FOR CONTROL

Receding horizon principle, 7 1, 175 Redundancy, 44 Representation of function, 120 Requirements disturbance rejection, 5 minimum variance, 5 tracking behavior, 5 Residuation, 115 Robot navigation goal point, 305 path generation, 302 path tracking, 302 reactive control, 303 reactive navigation, 303 route planning, 302 steering angle, 303 vehicle velocity, 303 Robustness, 135 Rule base completeness, 119, 185 complexity, 197 dimensionality, 202 redundancy, 197 simplification, 100, 92, 196 sub rule bases, 212 synthesis, 202 Rule table, 67 Rule cumulative relative contribution, 188 elimination and selection, 186 inspection, 186 multi-level, 212 pruning, 186 reduction, 190, 195 Sampling, 64-65 Sensors, 303, 305 general perception, 308 infrared, 305 laser range finder, 305-306 occupancy grid, 307 robot perception, 302 sensor fusion, 307, 320 ultrasonic, 303, 305-306 virtual perception memory, 308 virtual sensor, 302,305-307

Sensory fusion, 202 Signal conditioning, 64-65 Similarity driven simplification, 197 Similarity simplification, 197 analysis, 197 measure, 187 relation, 92, 197 Sliding mode control, 205 Stability, 13, 127 Bounded-Input Bounded-Output (BIBO), 136 circle criterion, 137 conicity criterion, 137 hyperstability, 141 input-output stability, 135 local stability, 135 Lur'e problem, 146 Lyapunov approach, 77 Lyapunov theory, 135, 143 stability analysis, 77, 128 Takagi-Sugeno type fuzzy systems, 144 State feedback control, 73 State variables approach, 8 State-space, 135 Structure learning, 187 Structure selection, 85 Structured systems decentralized, 21 1 hierarchical, 2 11 multi-level rules, 212 multidimensional relations, 21 1 multivariable E C ,21 1 Surface approximation, 194 Symbolic fusion, 207 Telerobotic, 306 Trajectories, 234 Transitive closure, 199 Tuning, 260,266,270,289 Uncertain systems, 152 Universal approximation, 13, 121 Validation of fuzzy rule bases, 42 Variable structure systems, 205 Video cameras, 305 Voronol diagram, 208 Weighted max, 164 Weighted sum, 164