Foreword In the last 50 years, Automatic Control Theory has developed into a well-established engineering discipline that has found application in space technology, industry, household appliances and other technologi-cal implementations. It was designed to monitor and correct the perform-ance of systems without the intervention of a human operator. Lately, with the growth of digital computers and the universal acceptance of sys-tems theory, it was discovered and used in softer fields of human interest such as ecology, economics, biology, etc. In the meanwhile, being a dy-namic discipline, Automatic Control with the aid of the digital computer has evolved from simple servomechanisms to an autonomous self-organizing decision-making methodology that was given the name of In-telligent Control. Several manifestations of Intelligent Control have been proposed by various scientists in the literature. Fuzzy, Neural, Hierarchi-cal Intelligent, Cerebellar and Linguistic control systems are typical ex-amples of such theoretically developed Intelligent Controls. However, the application of such sophisticated methodologies to real life problems is far behind the theory. The areas with the highest need and the smallest tolerance for adopting the techniques resulting from such theoretical research are the industrial complexes. The main reason is the lack of suitable intelligent computational algorithms and in-terfaces designed especially for their needs. This book attempts to cor-rect this by first presenting the theory and then developing various com-putational algorithms to be adapted for the various industrial applications that require Intelligent Control for efficient production. vii

Foreword viii

The author, who was one of the first to actually implement Intel-ligent Control in industry, accomplishes this goal by developing step by step some of the most important Intelligent Computational Algorithms. His industrial experience, coupled with a strong academic background, has been channeled into creating a book that is suitable for graduate aca-demic education and a manual for the practicing industrial engineer. Such a book fills a major gap in the global literature on Computational Intelligence and could serve as a text for the developing areas of biologi-cal, societal and ecological systems. I am very proud to introduce such an important work.

George N. Saridis Professor Emeritus

Rensselaer Polytechnic Institute Troy, New York, 1999

Preface

Conventional control techniques based on industrial three-term control-lers are almost universally used in industry and manufacturing today, de-spite their limitations. Modern control techniques have not proved possi-ble to apply because of the difficulties in establishing faithful micro-scopic models of the processes under control. It is not surprising, there-fore, that manual control constitutes the norm in industry. In the early 1970s Intelligent Control techniques, which emulate the processing of human knowledge about controlling a process by machine, appeared and a new era of control was born. Intelligent Control has come a long way since then, breaking down the barriers of industrial conservatism with impressive results.

Intelligent Control, which includes Fuzzy, Neural, Neuro-fuzzy and Evolutionary Control, is the result of applying Computational Intel-ligence to the control of complex systems. This class of unconventional control systems differs radically from conventional (or hard control) sys-tems that are based on classical and modern control theory. The tech-niques of Intelligent Control are being applied increasingly to industrial control problems and are leading to solutions where conventional control methods have proved unsuccessful. The outcome of their application to industry and manufacturing has been a significant improvement in pro-ductivity, reduced energy consumption and improved product quality, factors that are of paramount importance in today’s global market.

The first Chapter presents an introduction to Computational In-telligence, the branch of Soft Computing which includes Expert Systems,

ix

Preface x

Fuzzy Logic, Artificial Neural Networks and Evolutionary Computation (Genetic Algorithms and Simulated Annealing) with special emphasis on its application to Control Engineering.

The theoretical background required to allow the reader to com-prehend the underlying principles has been kept to a minimum. The reader is expected to possess a basic familiarity with the fundamentals of conventional control principles since it is inconceivable that unconven-tional control techniques can be applied without an understanding of conventional control techniques. The book is written at a level suitable for both undergraduate and graduate students as well as for practicing engineers who are interested in learning about unconventional control systems that they are likely to see in increasing numbers in the next mil-lennium. The primary objective of the book is to show the reader how the fusion of the techniques of Computational Intelligence techniques can be applied to the design of Intelligent Systems that, unlike conven-tional control systems, can learn, remember and make decisions.

After many years of teaching in higher education, the author took leave to work in industry only to face the technology gap between control theory and practice firsthand. He is one of that rare breed of aca-demics who had a free hand to experiment on-line on large-scale chemi-cal processes. He spent considerable time trying to apply conventional modern control techniques but frustrated with the outcome, sought un-conventional techniques that could and did yield solutions to the difficult practical control problems that he faced. His search led him first to fuzzy control and later to neural control, which he applied to the process in-dustry with considerable success. Those pioneering years in industry proved critical to his thinking about control practice and the use of Com-putational Intelligence, which is proving to be a powerful tool with which to bridge the technology gap.

After some ten years in industry, the author returned to academe, applying reverse technology transfer instructing his students on Intelli-gent Control techniques that have proved effective in industry. This book is the result of the experience he gained during those years in industry and of teaching this material to his graduate class on Intelligent Control while many of the examples presented in this book are the result of this experience.

Chapter 1 is an introduction to the techniques of Computational Intelligence, their origins and application to Control Engineering. Con-ventional and Intelligent Control are compared, with a view to focusing

Preface xi

on the differences which led to the need for Intelligent Control in indus-try and manufacturing. Chapter 2 discusses Expert Systems with refer-ence to their engineering applications and presents some common appli-cations in industry and manufacturing. Chapter 3 discusses Intelligent Control Systems, their goals and objectives while Chapter 4 discusses its principal components.

The elements of Fuzzy Logic on which Fuzzy Controllers are based are presented in Chapter 5 while Chapter 6 discusses the mecha-nisms of Fuzzy Reasoning, i.e., the inference engine that is the kernel of every fuzzy controller. Chapter 7 defines the fuzzy algorithm, methods of fuzzification and de-fuzzification and outlines the principal fuzzy con-troller design considerations. The requirements for real-time fuzzy con-trollers, both supervisory as well as embedded, are discussed in Chapter 8, which also includes examples of industrial applications. Chapter 9 presents fuzzy three-term industrial controllers that are replacing many conventional three-term controllers in the industrial environment. Chap-ter 10 outlines the Takagi-Sugeno Model-Based Fuzzy Controller design technique and fuzzy gain-scheduling that fuse conventional and fuzzy control.

Neural Control, the second important technique of Intelligent Control, is presented in Chapter 11. The elemental artificial neuron and multi-layer artificial neural networks that form the kernel of neural con-trollers are introduced in this Chapter. The delta and back-propagation algorithms, two of the most common algorithms for training neural net-work, are described in Chapter 12. Chapter 13 discusses how neural con-trollers can be trained from linguistic control rules identical to those used in fuzzy control. Finally, the result of fusing fuzzy and neural techniques of Computational Intelligence in the design of hybrid neuro-fuzzy con-trollers is discussed in Chapter 14.

Evolutionary Computation, the latest entrant in the field of Com-putational Intelligence, and Genetic Algorithms, the best known example of stochastic numerical optimization techniques, are presented in Chapter 15. Chapter 16 introduces Simulated Annealing, a stochastic technique that has found considerable application in engineering optimization. Fi-nally, Chapter 17 demonstrates how these two techniques can be used to advantage in the design of conventional and intelligent controllers.

An extensive Bibliography on Computational Intelligence and its applications is presented in Chapter 18.

Preface xii

Appendix A offers a step-by-step study for the design of a fuzzy controller of a realistic non-linear dynamic plant using MATLAB and its Fuzzy Toolbox. Appendices B and C offer listings of the MATLAB m-files of Genetic and Simulated Annealing Algorithms. Finally, Appendix D presents a listing of a MATLAB m-file for training industrial neural controllers using the Neural Toolbox. Acknowledgments

This book would not have been written had it not been for two people: an anonymous kidney donor and. Mark Hardy M.D., Auchinloss Profes-sor of Surgery in the Department of Surgery at the College of Physicians & Surgeons of Columbia University in New York, who performed the transplant. Together, they gave him that most precious gift: Life. He is forever indebted to them.

The author gratefully also acknowledges the contributions of his colleagues and former students at the University of Patras in Greece, N. Antonopoulos to Chapter 2, K. Kouramas to Chapters 2 and 10, P. Skantzakis to Chapter 11, G. Tsitouras and G. Nikolopoulos to Chapter 13 and V. Goggos to Chapters 15, 16 and 17.

Robert E. King

reking@ieee.org October 2004

Series Introduction

Many textbooks have been written on control engineering, describing new techniques for controlling systems, or new and better ways of mathemati-cally formulating existing methods to solve the ever-increasing complex problems faced by practicing engineers. However, few of these books fully address the applications aspects of control engineering. It is the intention of this new series to redress this situation.

The series will stress applications issues, and not just the mathematics of control engineering. It will provide texts that not only contain an exposé of both new and well-established techniques, but also present detailed ex-amples of the application of these methods to the solution of real-world problems. The authors will be drawn from both the academic world and the relevant applications sectors.

There are already many exciting examples of the application of con-trol techniques in the established fields of electrical, mechanical (including aerospace), and chemical engineering. We have only to look around in to-day’s highly automated society to see the use of advanced robotics tech-niques in the manufacturing industries; the use of automated control and navigation systems in air and surface transport systems; the increasing use of intelligent control systems in the many artifacts available to the domestic consumer market; and the reliable supply of water, gas, and electrical power to the domestic consumer and to industry. However, there are currently many challenging problems that could benefit from wider exposure to the applicability of control methodologies, and the systematic systems-oriented basis inherent in the application of control techniques.

v

Series Introduction

vi

This new series will present books that draw on expertise from both the academic world and the applications domains, and will be useful not only as academically recommended course texts but also as handbooks for practitioners in many applications domains.

Neil Munro

Contents Cover Page

Series Introduction by Neil Munro v Foreword by George N. Saridis vii Preface ix

1. Introduction 1

1.1 Conventional Control 2 1.2 Intelligent Control 6 1.3 Computational Intelligence in Control 8

2. Expert Systems in Industry 13

2.1 Elements of an Expert System 15 2.2 The Need for Expert Systems 17 2.3 Stages in the Development of an Expert System 18 2.4 The Representation of Knowledge 20 2.5 Expert System Paradigms 20

2.5.1 Expert systems for product design 21 2.5.2 Expert systems for plant simulation and operator training 22 2.5.3 Expert supervisory control systems 23 2.5.4 Expert systems for the design of

industrial controllers 24 2.5.5 Expert systems for fault prediction

and diagnosis 24 2.5.6 Expert systems for the prediction

of emergency plant conditions 26 2.5.7 Expert systems for energy management 26 2.5.8 Expert systems for production scheduling 27 2.5.9 Expert systems for the diagnosis of malfunctions 28

3. Intelligent Control 31

3.1 Conditions for the Use of Intelligent Control 33 3.2 Objectives of Intelligent Control 34

4. Techniques of Intelligent Control 39

4.1 Unconventional Control 40 4.2 Autonomy and Intelligent Control 45 4.3 Knowledge-Based Systems 48

4.3.1 Expert systems 49 4.3.2 Fuzzy control 50 4.3.3 Neural control 51 4.3.4 Neuro-fuzzy control 51

5. Elements of Fuzzy Logic 53

5.1 Basic Concepts 54 5.2 Fuzzy Algorithms 59 5.3 Fuzzy Operators 60 5.4 Operations on Fuzzy Sets 63 5.5 Algebraic Properties of Fuzzy Sets 64 5.6 Linguistic Variables 64 5.7 Connectives 69

6. Fuzzy Reasoning 71 6.1 The Fuzzy Algorithm 74 6.2 Fuzzy Reasoning 76

6.2.1 Generalized Modus Ponens (GMP) 77 6.2.2 Generalized Modus Tollens (GMT) 77 6.2.3 Boolean implication 78 6.2.4 Lukasiewicz implication 78 6.2.5 Zadeh implication 79 6.2.6 Mamdani implication 79 6.2.7 Larsen implication 80 6.2.8 GMP implication 80

6.3 The Compositional Rules of Inference 81

7. The Fuzzy Control Algorithm 89 7.1 Controller Decomposition 90 7.2 Fuzzification 91

7.2.1 Steps in the fuzzification algorithm 96 7.3 De-fuzzification of the Composite

Controller Output Membership Function 98 7.3.1 Center of area (COA) de-fuzzification 98 7.3.2 Center of gravity (COG) de-fuzzification 99

7.4 Design Considerations 100 7.4.1 Shape of the fuzzy sets 100 7.4.2 Coarseness of the fuzzy sets 100 7.4.3 Completeness of the fuzzy sets 101 7.4.4 Rule conflict 102

8. Fuzzy Industrial Controllers 105

8.1 Controller Tuning 106 8.2 Fuzzy Three-Term Controllers 107

8.2.1 Generalized three-term controllers 108 8.2.2 Partitioned controller architecture 109 8.2.3 Hybrid architectures 112 8.2.4 Generic two-term fuzzy controllers 113

8.3 Coarse-Fine Fuzzy Control 117 9. Real-time Fuzzy Control 119

9.1 Supervisory Fuzzy Controllers 120 9.2 Embedded Fuzzy Controllers 123 9.3 The Real-time Execution Scheduler 124

10. Model-Based Fuzzy Control 135

10.1 The Takagi-Sugeno Model-Based Approach to Fuzzy Control 136 10.2 Fuzzy Variables and Fuzzy Spaces 137 10.3 The Fuzzy Process Model 139 10.4 The Fuzzy Control Law 141 10.5 The Locally Linearized Process Model 142

10.5.1 Conditions for closed system stability 144

10.6 The Second Takagi-Sugeno Approach 144 10.7 Fuzzy Gain-Scheduling 146

11. Neural Control 153 11.1 The Elemental Artificial Neuron 156 11.2 Topologies of Multi-layer Neural Networks 158 11.3 Neural Control 160 11.4 Properties of Neural Controllers 161 11.5 Neural Controller Architectures 162

11.5.1 Inverse model architecture 164

11.5.2 Specialized training architecture 165 11.5.3 Indirect learning architecture 166

12. Neural Network Training 169

12.1 The Widrow-Hoff Training Algorithm 170 12.2 The Delta Training Algorithm 173 12.3 Multi-layer ANN Training Algorithms 175 12.4 The Back-propagation (BP) Algorithm 176

13. Rule-Based Neural Control 181

13.1 Encoding Linguistic Rules 182 13.2 Training Rule-Based Neural Controllers 183

14. Neuro-Fuzzy Control 193

14.1 Neuro-Fuzzy Controller Architectures 194 14.2 Neuro-Fuzzy Isomorphism 195

15. Evolutionary Computation 203 15.1 Evolutionary Algorithms 205 15.2 The Optimization Problem 207 15.3 Evolutionary Optimization 208 15.4 Genetic Algorithms 211

15.4.1 Initialization 212 15.4.2 Decoding 212 15.4.3 Evaluation of the fitness 213 15.4.4 Recombination and mutation 214 15.4.5 Selection 215 15.4.6 Choice of parameters of a GA 217

15.5 Design of Intelligent Controllers Using GAs 221

15.5.1 Fuzzy controllers 221 15.5.2 Neural controllers 222

16. Simulated Annealing 225

16.1 The Metropolis Algorithm 226 16.2 Application Examples 228

17. Evolutionary Design of Controllers 235

17.1 Qualitative Fitness Function 236 17.2 Controller Suitability 237

18. Bibliography 247

A. Computational Intelligence 247 B. Intelligent Systems 247 C. Fuzzy Logic and Fuzzy Control 248 D. Fuzzy Logic and Neural Networks 251 E. Artificial Neural Networks 252 F. Neural and Neuro-Fuzzy Control 253 G. Computer and Advanced Control 254 H. Evolutionary Algorithms 254 I. MATLAB and its Toolboxes 257

Appendix A 259

Case Study: Design of a Fuzzy Controller Using MATLAB A.1 The Controlled Process 259 A.2 Basic Linguistic Control Rules 261 A.3 A Simple Linguistic Controller 261 A.4 The MATLAB fuzzy Design Tool 264 A.5 System Stabilization Rules 266 A.6 On the Universe of Discourse of the Fuzzy Sets 267 A.7 On the Choice of Fuzzy Sets 268

A.8 Compensation of Response Asymmetry 269 A.9 Conclusions 270

Appendix B 279

Simple Genetic Algorithm

Appendix C 285 Simulated Annealing Algorithm

Appendix D 289

Network Training Algorithm

Index 291 Back Cover

Chapter 1 Introduction Modern control theory, which has contributed so significantly to the ex-ploration and conquest of space, has not had similar success in solving the control problems of industry and manufacturing. Despite the progress in the field since the 1950s, the chasm between theory and practice has been widening and many of the needs of industry remain unsolved. In-dustry has had little choice, therefore, but to rely heavily on conventional (sometimes termed hard) control techniques that are based on industrial three-term controllers. Unfortunately, these simple and ubiquitous de-vices cannot always cope with the demands and complexity of modern manufacturing systems.

The chasm between theory and practice has led to a search for new and unconventional techniques that are not subject to the constraints and limitations of modern control theory to solve the control problems faced by industry and manufacturing,. The breakthrough came in the mid-1960s with the introduction of Fuzzy Logic by Zadeh. The applica-tion of Zadeh´s theory to control was to come almost ten years later and it was to take even more years before it received the respect and accep-tance that it rightly deserved. At about the same time, Widrow demon-strated the use of ADALINEs (Adaptive Linear Networks), which are a primitive form of Artificial Neural Networks (ANNs), in control. This was a radical departure from conventional control since a generic con-troller was trained to perform a specific task instead of being designed.

1

Chapter 1 2

The two approaches were developed independently and it was to take many years before these concepts were applied to any degree.

The application of Fuzzy Logic to Control Engineering was first demonstrated in Europe and Japan in the mid-1970s. Mamdani presented the first demonstration of Fuzzy Logic in 1974 on an experimental proc-ess. This demonstration of Fuzzy Logic Control (FLC) gave the impetus for a seemingly endless series of applications, which continues unabated to this day. With a few notable exceptions, Zadeh´s theory of Fuzzy Logic went unnoticed in the West for many years while, in the mean-time, there was a frenzy of activity in Japan applying the theory to such varied fields as home appliances, cameras and transportation systems. Not until the early 1980s did industries in the West seriously consider applying fuzzy control. At the forefront of this thrust was the process in-dustry and in particular the cement industry, which was the first to apply the new technique to control large-scale processes. The developments in the field since then have been impressive and today there are hundreds of plants worldwide being successfully controlled by such techniques. The field of Artificial Neural Networks, which evolved quite separately, has had a difficult evolution. Appearing in the 1970s as a field that offered much promise and potential, it was thwarted by inade-quate computational facilities and a lack of effective network training al-gorithms. Re-emerging in the 1980s, by which time significant progress had been made in both training algorithms and computer hardware, re-search and development in the field has evolved rapidly. Artificial Neu-ral Networks can be found today in a host of applications ranging from communications, speech analysis and synthesis, control and more. 1.1 Conventional Control Despite the advances in the theory of automatic control, most industrial plants, even to this day, are under the exclusive supervision and control of human operators. Their observations on the state of the plant from a host measurements taken from sensors in the plant coupled with their knowledge and experience of the plant lead them to decide on what con-trol strategy to take in order to achieve the desired product quality and production specifications.

In the past, industry has had little option but to use Classical Control theory that is based on macroscopic models of the plant in de-

Introduction 3

signing appropriate conventional controllers. These methods depend on empirical knowledge of the dynamic behavior of the controlled plant, de-rived from measurements of the control and manipulated variables of that plant. Traditionally industry has relied heavily on three-term (PID) controllers, that are incorporated today in most Remote Terminal Units (RTUs) and Programmable Logic Controllers (PLCs). The ubiquitous three-term controller is used to control all kinds of devices, industrial processes and manufacturing plants. Their tuning is based on simple ap-proximants of the controlled plant dynamics and on design methods such as the classical ones by Nichols and Ziegler or more modern techniques such as those of Persson and Astrom. Most often in practice turning is performed heuristically by expert tuners in situ.

Without doubt, these simple industrial controllers have offered sterling service for many decades and will continue to do so for many more, wherever simplicity and robustness are essential and control speci-fications permit. However, three-term controllers cannot always satisfy the increasing complexity of modern industrial plants and the demands for high flexibility, productivity and product quality, which are essential in today’s very competitive global market. The problem is further aggra-vated by the increasing environmental restrictions being placed on indus-try and manufacturing.

Modern Control was introduced in the early 1960s and is a rig-orous methodology that has proved invaluable for finding solutions to well-structured control problems. With a few notable exceptions, how-ever, its application to industry has been disappointing and few industrial controllers are designed with this methodology. The reasons for this dis-crepancy are the complexity, uncertainty and vagueness with which in-dustrial processes are characterized - conditions that do not allow for ready modeling of the controlled plant, essential to the application of modern control methodologies.

Despite more than five decades of research and development in the theory and practice of Control Engineering, most industrial processes are by and large still controlled manually. Today, Supervisory Control And Data Acquisition (SCADA) Systems and Distributed Control Sys-tems (DCS) make the operators’ task considerably easier. A partial sche-matic of such an information system using a distributed architecture, is shown in Figure 1.1.

Chapter 1 4

Figure 1.1 Distributed Control System architecture The operator console possesses one or more screens that display

the essential variables of the plant through a graphical user interface by which the operator interacts with the plant. A typical example of such a display is shown in Figure 1.2.

In plants where the various sub-processes interact, it is clear that the control problem can be severe, requiring operator skills that can only be acquired after years of experience. Today, Multimedia and Virtual Reality are finding their way into the control room, improving the man-machine interface and making decision-making considerably easier and the work environment more tolerable.

RTUs

LAN

Raw Materials

Clinker

Kiln

Operator Consoles

Fuel

Air

Introduction 5

One or more human operators normally supervise a cluster of sub-processes, receiving data on the state of the plant and sending cor-rections to the set points of the local controllers which are distributed throughout the plant so that the plant remains at its nominal state despite external disturbances. These local controllers are often embedded in RTUs that are also capable of performing sequential switching control, data acquisition and communications with the Supervisory Control Sys-tem and the operators’ consoles via a local area network.

Figure 1.2 A typical graphical user interface

In most industrial applications, human operators close the loop

between the controlled and the control variables of the controlled plant. Operators respond to observations of the principal variables of the plant and continuously stride to satisfy often-conflicting objectives, e.g., maximizing productivity and profit while minimizing energy demand. Proper operation of a process is thus very much dependent on the experi-ence of the operator, his knowledge about the process and its dynamics and the speed with which he responds to plant disturbances, malfunc-tions and disruptions. The yield of a process can vary quite significantly from operator to operator and less experienced operators are often unable

Chapter 1 6

to control a plant effectively, particularly under abnormal situations which they have never met before.

The control actions of a human operator are subjective, fre-quently incomprehensible and often prone to errors particularly when they are under stress. Indeed in the case of abnormal operating (i.e., alarm) conditions, their actions may be potentially dangerous and there is little margin for errors. Delays in making decisions can lead to disastrous results as was amply demonstrated in the Chernobyl nuclear reactor dis-aster.

Thus in modern complex plants there exists a very real need to assist operators in their decision-making, particularly in abnormal situa-tions in which they often are bombarded with conflicting signals. The advent of Computational Intelligence and unconventional control free operators of many of the tedious and complex chores of monitoring and controlling a plant, assuring them fast and consistent support in their de-cision-making. 1.2 Intelligent Control During the past twenty years or so, a major effort has been under way to develop new and unconventional control techniques that can often aug-ment or replace conventional control techniques. A number of unconven-tional control techniques have evolved, offering solutions to many diffi-cult control problems in industry and manufacturing. This is the essence of what has been termed Practical Control, which is a collection of tech-niques which practicing engineers have found effective and easy to use in the field. It is true to say that virtually all the techniques of unconven-tional control could not have been possible but for the availability of computationally powerful and high-speed computers.

Significant research has been carried out in understanding and emulating human intelligence while, in parallel, developing inference engines for processing human knowledge. The resultant techniques in-corporate notions gathered from a wide range of specialization such as neurology, psychology, operations research, conventional control theory, computer science and communications theory. Many of the results of this effort have migrated to the field of Control Engineering and their fusion has led to a rapid growth of new techniques such as inductive reasoning,

Introduction 7

connectionism and parallel distributed processing for dealing with vagueness and uncertainty.

This is the domain of Soft Computing, which focuses on stochas-tic, vague, empirical and associative situations, typical of the industrial and manufacturing environment. Intelligent Controllers (sometimes termed soft controllers) are derivatives of Soft Computing, being charac-terized by their ability to establish the functional relationship between their inputs and outputs from empirical data, without recourse to explicit models of the controlled process. This is a radical departure from con-ventional controllers, which are based on explicit functional relations. Unlike their conventional counterparts, intelligent controllers can learn, remember and make decisions. The functional relationship between the inputs and outputs of an intelligent controller can be specified either:

• indirectly by means of a relational algorithm, relational

matrix or a knowledge base, or • directly from a specified training set. The first category belongs to the domain of Fuzzy Systems while

Artificial Neural Networks belong to the second. Generality, in which similar inputs to a plant produce similar outputs so that sensitivity to per-turbations in the plant inputs is minimized, is an inherent feature of such systems. Generality implies that the controller is capable of operating correctly on information beyond the training set.

Intelligent controllers, whatever form they may take, share the following properties: they

• use the same process states, • use parallel distributed associative processors, • assure generality, and • are capable of codifying and processing vague data.

The principal medium of intelligent control is Computational In-

telligence, the branch of Soft Computing which includes Expert Systems, Fuzzy Logic, Artificial Neural Networks and their derivatives. Evolution-ary Computation (Genetic Algorithms and Simulated Annealing) is a very recent addition to this rapidly evolving field.

Chapter 1 8

1.3 Computational Intelligence in Control The field of Expert Systems, the first class of systems that this book dis-cusses, is the precursor to Computational Intelligence and is the most successful outgrowth of Artificial Intelligence. Expert systems use lin-guistic rules to specify domain knowledge and are used extensively to-day in industry in such diverse applications as fault prediction, fault di-agnosis, energy management, production management and supervisory control, among others.

Chronologically, fuzzy logic was the first technique of intelli-gent control. Neural, neuro-fuzzy and evolutionary control and their de-rivatives followed later, each technique offering new possibilities and making intelligent control even more versatile and applicable in an ever-increasing range of industial applications.

The third technique of intelligent control considered in this book appeared towards the end of the 1980s and is based on Artificial Neural Networks. Neural networks have had a varied history, progress having remained stagnant until the mid-1980s when efficient training algorithms were developed and fast computational platforms became readily avail-able. Since then, neural networks have had a remarkable resurgence, be-ing successfully used in a wide range of applications such as communi-cations, speech analysis and synthesis pattern recognition, system identi-fication and control.

Finally, in the mid-1990s Evolutionary Control, an outgrowth of Evolutionary Computing, emerged as a viable method for optimum con-trol. This technique, which is possible only because of the rapid devel-opments in computer hardware and software, uses stochastic methods.

Since the early 1990s a major effort has been underway to de-velop derivatives of these techniques in order to exploit the best features of each in the design of intelligent controllers. These new techniques have revolutionized the field of Control Engineering, offering new hope in solving many of the difficult control problems of industry and manu-facturing.

Computational Intelligence is based on concepts that practicing control engineers use on a daily basis and has played a major role in re-ducing the chasm between advanced control and engineering practice. The new control techniques based on Computational Intelligence no longer face the barrier of disbelief that they faced when they first ap-

Introduction 9

peared. Numerous successful applications in a variety of fields attest to the usefulness and power of these techniques.

Computational Intelligence uses numerical representation of knowledge in contrast to Artificial Intelligence, which uses symbolic representation. This feature is exploited in Control Engineering, which deals with numerical data since control and controlled variables are both defined numerically. Computational Intelligence adapts naturally to the engineering world, requiring no further data conversion. The techniques of Computational Intelligence share the following properties: they

• use a numerical representation of knowledge, • demonstrate adaptability, • have an inherent tolerance to errors, and • possess speeds comparable to those of humans.

Intelligent controllers infer the control strategy that must be ap-

plied to a plant in order to satisfy specific design requirements. This ac-tion can be the result of operations on a set of pre-specified linguistic control rules, as in the case of Fuzzy Controllers, or of training an artifi-cial neural network with numerically coded rules as in the case of Neural Controllers. In either case, the primary objective is to generate control actions which closely match those of an expert human operator. In this manner, the controller can assist the human operator to maintain the plant under his supervision at its nominal operating state while simulta-neously compensating for his inconsistency and unreliability brought about by fatigue, boredom and difficult working conditions.

Intelligent controllers can be trained to operate effectively in conditions of vagueness and uncertainty of both the plant state and plant environment and can respond to unforeseen situations autonomously, i.e., without intervention from the plant operator. They differ, however, from their human counterpart in their ability to learn new control rules or to adapt to new situations for which they have not been trained. Self-organizing controllers that have the ability to learn new rules on-line have been variously proposed in the literature and tried out in the labora-tory, but none has been commissioned so far in a manufacturing plant. The main reason is that this class of controllers assumes extended testing and experimentation on the controlled plant under normal operating con-ditions, a situation that few plant managers are likely to entertain.

Chapter 1 10

A variety of architectures have been proposed for the design and implementation of high level intelligent controllers for large-scale sys-tems. One of the most useful is the hierarchical architecture proposed by Saridis in the mid-1970s. In this, information from the controlled plant flows with decreasing frequency from the lowest to the highest layer of the hierarchy. In contrast, management directives (on such matters as production quotas, product qualities, etc.) flow in the reverse direction with increasing frequency as they descend the hierarchy, leading ulti-mately to selection of the best control strategy that must be imposed on the plant.

Saridis´ principle, on which a number of successful intelligent hierarchical process management and control systems have been devel-oped, can be paraphrased as:

“Increasing/decreasing precision is accompanied by decreasing/increasing intelligence”.

It is useful, finally, to note the features that every Intelligent Sys-

tem involving clusters of intelligent controllers must support: • Correctness - i.e., the ability to operate correctly for specific

sets of commands and plant safety constraints. • Robustness - i.e., the ability to operate acceptably despite

wide variations in plant parameters. The higher layers of the hierarchy must possess an inherent ability to deal with unforeseen variations.

• Extendibility - i.e., the ability to accept extensions to both hardware and software without the necessity for major modifications to either. Extendibility implies modularity, which is the partitioning of the system into easily modifiable software and hardware modules.

• Reusability - i.e., the ability to use the same software in different applications. To possess this feature, the system must be general or possess an open architecture.

The field of intelligent control is one of the most exciting and

promising new directions of automatic control that is opening up new frontiers for research and development in radical solutions to the control of industrial systems in the new millenium.

Introduction 11

Chapter 2 Expert Systems in Industry Expert Systems, which are the most commercially successful result of research in Artificial Intelligence, are software entities that emulate the cognitive abilities of human experts in complex decision making situa-tions. As one of the primary activities of Computer Science and depend-ent heavily on the rapid developments in computer technology, Expert Systems have been eagerly adopted by industry and applied to a wide range of applications. Expert Systems belong to the field of Intelligent Knowledge-Based Systems that constitute one of the principal fields of activity of Computational Intelligence, a field which has been referred to as the science that attempts to reproduce human intelligence using com-putational means. Computational Intelligence has also been referred to as the science that attempts to make computers perform tasks at which humans, for now at least, are better!

Computational Intelligence has many branches, one of the earli-est and most important of which belongs to Expert Systems. The other branches of Computational Intelligence are shown in Figure 2.1 and are introduced in subsequent chapters. Expert systems use a variety of methods to represent knowledge and derive decisions while they have the ability to manage knowledge from different sources of human thought and activity. The manner in which this knowledge is represented in the computational environment depends on the nature of the knowl-edge and the field of expertise. In an industrial environment knowledge

13

Chapter 2 14

is typically represented in the form of linguistic rules that describe the actions that must be taken in response to specified excitations.

Expert Systems

Fuzzy Systems

Evolutionary Computing

Neural Systems

Neuro-fuzzy Systems

Computational Intelligence

Figure 2.1 Branches of Computational Intelligence

There are many techniques for representing knowledge and each one has its advantages and disadvantages. The principal theoretical research issue is how to give Expert Systems the ability to search through the domain knowledge systematically and arrive at decisions rapidly. The following are techniques commonly used for representing knowledge:

• predicate logic, • semantic networks, • procedural representation, • production systems, and • frames.

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2.1 Elements of an Expert System Expert systems are the outcome of a major effort in computer science to emulate the cognitive faculty of humans. Artificial intelligence is the basis for this field of endeavor, which includes such areas as pattern recognition, artificial speech and artificial vision, among others. Conven-tional computer software can be viewed as the synergy of:

Software = Data + Algorithm

Here, the algorithm processes data in a top-down sequential manner until the result is arrived at. In contrast, computer software used in Expert Systems can be described as the synergy of:

System = Knowledge + Inference

In this case the system structure differs radically and the principal ele-ments are the knowledge base, which is a depository of all the available domain specific knowledge and the inference engine, the software whose function is to infer decisions.

An Expert System can be characterized as an intelligent knowl-edge-based system provided it reproduces knowledge in the form of rules. The most significant characteristic of this class of systems is that it draws on human knowledge and emulates human experts in the manner with which they arrive at decisions. One definition of an Expert System is thus:

“An Expert System is the embodiment of knowledge elicited from human experts, suitably encoded so that the computa-tional system can offer intelligent advice and derive intelli-gent conclusions on the operation of a system”. Production rules are a convenient form by which to represent the

knowledge of domain experts. Before describing this method, we note some alternative methods for representing knowledge that have been found useful in industrial applications. In general, knowledge that is useful in solving real industrial problems has two components:

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• facts, which constitute ephemeral information subject to

changes with time (e.g., plant variables) and • procedural knowledge, which refers to the manner in which

experts in the specific field of application arrive at their deci-sions.

Procedural knowledge (e.g., information flows, control se-

quences and actions, etc.) and the step-by-step procedure which must be followed in the specific manufacturing plant, is evidently known by production engineers and is the result of years of experience with work-ing with the plant or process. This is one of the principal reasons why Expert Systems have attracted so much attention in the industrial world. The use of rules is the simplest way to describe a manufacturing proce-dure, while linguistic rules of the classical if …then … else form are most commonly used by humans.

Knowledge Base

Explanation Sub-system

Man-Machine Interface

Knowledge Acquisition

System

Domain Experts

Inference Engine

Figure 2.2 Basic elements of an Expert System

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The basic elements of an Expert System are shown in Figure 2.2. An Expert System includes the following elements:

• the knowledge base, which comprises facts and rules with which to control a plant,

• the inference engine, which processes the data in the knowledge base in order to arrive at logical conclusions,

• the explanation sub-system, which is capable of a giving a rational explanation on how the decision was arrived at,

• the knowledge acquisition system, which is used by the knowledge engineers to help them analyze and test the knowledge elicited from human domain experts and

• the man-machine or user interface system through which the human operator interacts with the system.

2.2 The Need for Expert Systems The developments in the field of Expert Systems rapidly found propo-nents in industry despite the inherent reluctance to adopt new technol-ogy. The application of Expert Systems in industry and manufacturing was left to innovative manufacturers who were sufficiently broad-minded to take the risk in the expectation that the outcome would in-crease their competitive position and their market share. Despite some early failures of Expert Systems, which were touted for what they were supposed to do but didn’t, the positive results which were reported motivated more manufacturers to invest in knowledge-based technology. This in turn led to further research and development in the field of Ex-pert Systems in universities and research establishments. The reasons that has motivated industry to adopt knowledge-based techniques are the following:

• the lack of an explicit quantitative description of the physical plant,

• the existence of the knowledge and experience to control the plant, and

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• the ability of a class of knowledge-based systems to deal with vagueness and uncertainty that is characteristic of many indus-trial plants.

A common feature in industrial and manufacturing systems is that their quantitative models that are supposed to predict their dynamic behavior are either unknown or do not possess sufficient fidelity. This is particularly true in the case of large-scale industrial plants whose quanti-tative description is a difficult, tedious and occasionally impossible task for lack of sufficient deep knowledge. Deep knowledge is the result of microscopic knowledge of the physical laws that govern the behavior of a plant. In contrast, shallow knowledge is the result of holistic or macro-scopic knowledge and is readily available from human domain experts. This knowledge is acquired after years of experience in operating the plant and observing its peculiarities and nuances. 2.3 Stages in the Development of an Expert

System In developing a knowledge-based system using an Expert System, it is essential to concentrate first on the objectives of the Expert System and not how these objectives can be met. Great effort must therefore be made to specify these objectives and constrain the domain of the Expert Sys-tem. Inadequate specifications of the constraints of the Expert System over which it is expected to function and unwarranted expectations were the basic reasons for failure of many early Expert Systems to meet user requirements.

Once the domain of the Expert System has been specified, we are in a position to select the tools and methods with which to design the Expert System. During this phase of development, the knowledge engi-neer elicits the rules by which the plant is to be controlled from domain experts. Following interviews that invariably include questionnaires on what variables are observed and what controlling actions the domain experts would take in every conceivable situation, the knowledge so acquired is stored in a suitably coded form in the knowledge base of an Expert System shell. An Expert System shell is nothing more than a collection of software elements that perform all the tasks of an Expert System. While in the past Expert Systems were developed using object-

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oriented languages, notably LISP, it is inconceivable today to develop such a system without a shell. It should be noted that knowledge elicita-tion is one of the most painstaking tasks in the design procedure. Human domain experts are often reluctant to part with their knowledge, fearful that divulging knowledge gained after years of experience may lead to their redundancy and termination.

In the first stage of development of any Expert System, it is very useful to implement a rapid prototype. The objective here is not devel-opment of a complete Expert System, but a prototype that will form the basis of the final system under development. Once the knowledge engi-neer has a thorough understanding of the rules elicited from the domain experts and the manner in which decisions are arrived at and justified, he must then encode the knowledge in a form suitable for processing by the Expert System. It is noted that the rapid prototype need not possess all the features of the end product, but should incorporate the basic features that can be evaluated by both the domain experts and the end users. Should the prototype system demonstrate deficiencies and difficulties in inferring decisions, it is clearly preferable to make corrections and im-provements at this stage rather than in the end product when it may be very difficult and costly.

In the implementation stage of the Expert System, the knowledge elicited from the domain expert is transferred to the Expert System that runs on a suitable platform. Early Expert Systems invariably ran on powerful workstations or special purpose computers (such as the short-lived LISP machines) which were subsequently superceded by common microcomputers. Today, most Expert Systems can run on high end PCs or workstations. Once completed, the Expert System is tested off-line until the end users are convinced of its ability to infer correct results and support its decisions. It is noted that it is often difficult and uneconomic to test the Expert System exhaustively, i.e., for all possible conditions, in practice. For these reasons, end users must develop a close liaison with the Expert System designer, assisting him whenever some discrepancy is observed between their decision and that of the Expert System. Such discrepancies arise from rule conflict, misunderstandings or errors in the knowledge base.

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2.4 The Representation of Knowledge Simplicity in representing knowledge in an Expert System is essential and a variety of techniques have been proposed to this end. One of the most common representations of domain knowledge is the decision tree, each branch of which represents some action. Every branch of the tree emanates from a node where a condition is examined. Depending on the outcome of this condition, a specific branch of the tree is traversed until the next node. The tree may have many branches and nodes. Some early Expert Systems proposed for diagnosis in medicine had thousands of branches and nodes, making them cumbersome, slow and difficult to use.

Implementation of an Expert System can be either direct, using an object oriented programming language such as LISP, Prolog, C++, Visual Basic, Visual C++, Visual J++, Visual Fortran, etc. or, more conveniently using an Expert System shell such as G2, NEXPERT, etc.

As noted earlier, the rule base of an Expert System contains lin-guistic rules of the classical if ... then ... else form with which plant operators are trained and subsequently use to justify their decisions. These rules may appear as strings in their original form or encoded into numerical form. Quantitative descriptions of a plant are not always straightforward, particularly when only incomplete and vague data on the plant are available. To make descriptions possible in such cases, special techniques such as Fuzzy Logic (which is introduced in chapter 5) or probabilistic methods are used. 2.5 Expert System Paradigms Expert systems have been diffusing into industry rapidly since their introduction in the mid-1970s. Today, Expert Systems can be found in a variety of industrial applications, the most successful examples of which are described briefly below. The typical stages of a manufacturing plant are shown in Figure 2.3, which shows where Expert Systems can benefit production.

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2.5.1 Expert systems for product design Modern Flexible Manufacturing Systems (FMS) produce specialized products of high quality, limited production runs and short life cycles, i.e., lean production. These products undergo changes often and their design must be completed in very short times, imposing considerable stress on product designers. Expert computer-aided-design systems are now available to assist the designer, permitting him to exploit his crea-tive abilities to the utmost while advising him on design and materials constraints following extensive background computations.

Interpretation

Diagnosis

Control

Future behavior

Prediction

Supervision

Interpretation

Observed properties

Production Process

Desired properties

Programming

Knowledge

Product properties

Product specifications

Design

Figure 2.3 The manufacturing environment

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While conventional Computer Aided Design (CAD) software can process geometric shapes rapidly, the designer needs to know rapidly certain characteristics of the product being designed, such as strengths, thermal distributions, costs, etc. Expert CAD systems provide all this information while in addition advising the designer of alternative shapes from a priori experience with similar designs. The trend in product design today does not yet permit total design with expert CAD systems since design normally depends on the designer’s intuition and aesthetic knowledge, the prehistory of the product and economic factors that are difficult to incorporate in a knowledge base. The final product is a set of diagrams or plans, design specifications and various documents on which manufacturing will then proceed, as shown in Figure 2.3. 2.5.2 Expert systems for plant simulation and

operator training The training of operators to control modern industrial plants is important, time-consuming and very expensive when performed on the actual physical plants. Apart from the dangers involved should some wrong control action be taken with on-line training, the unevenness of produc-tion and uncertain quality of the product produced during operator training makes this procedure undesirable and very costly. Plant simula-tors, which simulate the plant and can be programmed to take into ac-count faults and malfunctions in the plant (quite similar, in fact, to flight simulators) are today being used extensively to train new plant operators. Usually an instructor, unseen to the trainee operator, enters malfunctions and observes the trainees’ reactions and performance.

The role of the instructor can be taken by Expert Systems, which can tirelessly repeat plant malfunctions and, like their human counter-parts, examine and instruct the trainee operators. The knowledge with which to operate a plant is embedded in a set of if … then … else rules that are used to operate the plant. Multimedia and Virtual Reality can be used in the man-machine interface in training plant operators, even before the plant has been commissioned. The same system can also be used to refresh old operators’ knowledge, much as pilots must undergo periodic training and certification using flight simulators.

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2.5.3 Expert supervisory control systems

Reference was made in Section 2.3 to the use of Expert Systems for the supervision and control of Computer Integrated Manufacturing (CIM) systems. The primary objective of any Supervisory Control And Data Acquisition (SCADA) system, which constitutes the kernel of any CIM system, is data acquisition, the overall supervision of the health of the plant, prompt alarming of out-of-range variables and control of the principal variables of the plant under control. Supervisory control sys-tems have revolutionized production plants, increasing productivity while significantly reducing production costs.

The next stage in their evolution was the introduction of Compu-tational Intelligence techniques that broadened their abilities signifi-cantly. The new generation of supervisory control systems exploits the knowledge and experience of domain experts in automatically correcting for plant malfunctions and discrepancies. New and advanced intelligent control techniques that were inconceivable until recently, are now com-monly incorporated into most commercially available SCADA systems, further improving product quality and productivity while simultaneously reducing production costs.

Expert systems are being used in industrial control, which is an integral part of any SCADA system, in the following fields:

• the design of industrial controllers, and • the supervision and control of manufacturing plants.

One of the major difficulties in the design of plant controllers, par-

ticularly in the case of large-scale multivariable plants, using conven-tional control techniques, is the unavailability of explicit models of the plants. For this reason industrial automation leans towards the use of three term (PID) controllers and various empirical and semi-empirical design techniques have been proposed to determine the parameters of these controllers. Examples of these design techniques are the well-known methods of Ziegler and Nichols and modern variants due to Persson and Astrom. In contrast, expert controller techniques, which can exploit the knowledge of expert controller tuners, can often offer supe-rior results. A number of vendors currently offer such software products.

The use of Expert Systems in the design of industrial controllers has two aspects. The first involves the rules on the most appropriate

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design technique to use in order to achieve the desired result. These rules are dependent on the specific plant to be controlled and criteria by which the control quality, i.e., the performance of the closed plant, is judged. The second aspect involves rules that specify the best control strategy to follow in any situation, given as advice to the operator. 2.5.4 Expert systems for the design of

industrial controllers Human operators are trained to use linguistic rules that involve the principal measured plant variables in order to maintain the plant at the desired state following some exogenous disturbance. The operator’s speed of reaction is critical in achieving a high quality of control and satisfactory product quality. Inaction, delays and inconsistencies in the actions of the operator due to fatigue or when under pressure, invariably leads to uneconomic operation and in the worst case, to disastrous re-sults, a prime example being the Chernobyl nuclear power plant.

The necessity to assist the operator in his routine tasks and ad-vise him on the best strategy to follow in extreme cases or in rare situa-tions which he may not have met earlier, was the motivation for the development of a new class of expert supervisory control systems. Cou-pled with the rapid developments in computer technology, this new generation of control systems is a reality that is finally diffusing into manufacturing.

Expert supervisory control systems do not require deep knowl-edge of the plant to be controlled, but are based on shallow knowledge of the form normally used by human operators. The fundamental require-ment is the existence of a conventional supervisory control system to which the Expert System is appended. 2.5.5 Expert systems for fault prediction and

diagnosis A very significant field of application of Expert Systems has been in equipment fault prediction and diagnosis, sometimes termed equipment health condition. Many such systems have augmented existing data acquisition systems and have proved invaluable for the prediction of faults in equipment. Examples of faults that are important to predict are

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increased wear of bearings of rotating machinery due to vibrations or excess friction due to overheating. This class of on-line, real-time Expert Systems is giving new meaning to the field of predictive maintenance. Productivity is benefiting through improved estimates of the time-to-go before catastrophic failure to the equipment is likely to occur. This is particularly important in the case of large equipment for which expensive spare parts have to be in stock, to be used in case of a breakdown. Expert systems can minimize and even eliminate stocks through timely pro-curement. The use of Expert Systems for fault prediction results leads to a drastic reduction in the mean time to repair equipment and a corre-sponding increase in the availability of the equipment and, most impor-tantly, an increase in plant productivity.

In preventive maintenance, historical data is gathered from suit-able sensors attached to the equipment (e.g., temperatures, pressures, vibrations, etc.). Real-time measurements of critical variables are com-pared with expected or desired values and any discrepancy is used to diagnose the possible cause of the discrepancy from rules embedded in the Expert System. Following spectral analysis of such measurements of bearing sounds by standard signal processing techniques, the Expert System suggests what maintenance will be required and when best to perform it.

Expert systems for fault diagnosis can be either off-line or on-line. In the former case, maintenance personnel enter into a dialog with the Expert System, supplying answers to questions posed by the Expert System on the health of the equipment. The Expert System then gives instructions on what further measurements and what actions should be followed that will focus on the source of the problem and then give advice on how to repair it. It is obvious that rapid fault diagnosis is of paramount importance in a manufacturing environment where every minute of lost production results in a loss of profit. It should be evident why expert fault prediction and diagnosis systems have been the subject of considerable commercial interest and have found such extensive application.

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2.5.6 Expert systems for the prediction of emergency plant conditions

Effective control of large complex industrial systems, such as nuclear reactors, power distribution networks and aircraft, is critically important since breakdowns can lead to unforeseen and potentially disastrous results. In recent history, the Chernobyl disaster stands out as a leading example of human error. Likewise, power blackouts over large areas of the power distribution network, are often the result of human error. But the most visible example is pilot error, when hundreds of lives are lost because of wrong pilot decisions that have been made in situations of immense pressure.

A human operator has great difficulty in making decisions when facing conflicting or excessive information, particularly if under stress. Real-time Expert Systems, using data from the plant and rules derived from logical reasoning and prior experience, advise the plant operator on the best course of action to take in order to avert a catastrophe and return the plant to its nominal operating state as quickly as possible, with minimal disruption of production and damage to the equipment. 2.5.7 Expert systems for energy management With the ever-increasing costs of energy, the management of energy in large industrial plants is of major concern and means to contain these costs are actively sought. Energy-intensive industries, such as the metal-lurgical, cement and petrochemical industries, have a very real need to contain their energy demand and most nowadays use some form of energy management system.

In large manufacturing plants the electric energy pricing policy is dependent on the power absorbed over, for instance, each 15-minute period. The power provider and the manufacturer agree on a pricing policy for every minute period during the day, the cost of energy being significantly lower in off-peak periods and prohibitively high in peak periods. In turn, the consumer agrees to restrict his energy intake to these limits. A significant penalty must be paid if the contractual limits are exceeded in any period. Such additional costs can make production non-competitive.

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It is therefore necessary to accurately predict what the power ab-sorbed over each period will be and to monitor the energy demand by shedding loads in time to avoid exceeding the contractual energy limit. The decision on which loads to shed and when to do so without disrupt-ing production, is a very difficult and tiring task for a human who would have to make this decision every 15 minutes throughout the day and night. The operator has to know which equipment can be shut down and which must, at all costs, be left running in order to avoid major disrup-tion of the production line or manufacturing plant and how long before each piece of equipment can be restarted without causing excess wear to it. In a large plant this is normally performed by shedding auxiliary equipment that is not considered absolutely essential to the manufactur-ing plant (e.g., circulation pumps, conveyor belts) and in the worst case by a total stoppage of production in periods of high energy cost. Many electric energy intensive plants today are forced to shut down production during peak hours in order to conserve energy.

Real-time expert energy management systems have been devel-oped and have been very successful in containing energy costs, replacing the human operator in this arduous task. Indeed, avoiding just one or two overload penalties often pays for the cost of the Expert System! The rules by which equipment can be operated, the order in which they may be shed, when and how many times per day they can be restarted are elicited from human operators and are embedded in the Expert System rule base. The real-time expert energy management system is then exe-cuted every few seconds following prediction of the energy absorbed at the end of the timing period. Naturally, the magnitude of the load that must be shed is critically dependent on the time-to-go before the end of the period: the shorter the time left, the larger must be the load that must be shed and the greater the malfunction that is incurred. Accurate predic-tion and effective and fast decisions from the Expert System are essential to proper operation. 2.5.8 Expert systems for production scheduling Production scheduling in manufacturing plants with multiple parallel production lines is essential in order to maintain high product throughput despite changes in production priorities, equipment malfunctions and variations in the raw materials. In deciding which production line can be

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used to manufacture a specific product, the production manager must know the production capacity and limitations of each production line, the overall production schedule, equipment and storage capabilities, etc. When a production line is disrupted for whatever reason, it is often necessary to switch production lines and change the priorities with which the product is produced, permitting high priority items to be completed first while lower priority items are placed in a queue.

The long-term production schedule is normally produced on a weekly or monthly basis but changes to it may be necessary due to equipment failures. When these failures are serious enough to cause extended production disruption it is necessary to re-compute the produc-tion schedule. Operational research techniques based on linear integer or mixed-integer programming is the conventional approach to this prob-lem, but these techniques are time-consuming.

An alternative way to reschedule production is through the use of empirical rules that are followed by production management. Expert scheduling systems using this knowledge and experience are considera-bly simpler to use and lead to equally feasible results much faster and have been used with excellent results. 2.5.9 Expert systems for the diagnosis of

malfunctions This application involves Expert Systems for the diagnosis of malfunc-tions in the sub-systems of a manufacturing system. The method requires decomposition of the manufacturing system into a set of interacting sub-systems and leads to the development of a knowledge base from which the cause of the malfunction can be inferred. The method is particularly useful in Flexible Manufacturing Systems (FMS) - discrete production lines, typical examples of which are beverage bottling, cigarette packing and food canning lines.

A malfunction in any sub-system can lead to a total shutdown of the production line and it is therefore critically important to diagnose the source of the malfunction as quickly as possible in order that the mal-functioning equipment be repaired rapidly and be put on-line once more. Thus sensors (photocells, inductive detectors, etc.) are placed at critical points along the production line, from which flow rates can be estimated continuously. It is obvious that rapid reinstatement of the production line

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is of paramount importance in order to maintain high equipment avail-ability and meet production schedules. In large bottling or canning plants, for instance, the sensors are linked to the Factory Data Acquisi-tion (FDA) system and measurements are continuously compared with the desired values.

Should some unit along the line malfunction, then clearly both the proceeding and succeeding units will suffer the consequences. Due to the interactive nature of most production systems and work cells, it is obvious that when any sub-system malfunctions, the sub-systems up-stream and down-stream will be affected sooner or later. Up-stream units must thus be stopped in time to avoid strangulation as a consequence of the accumulation of partially finished products which may exceed the capacity of the silos or queues if the malfunction persists for some time, while down-stream, units must be stopped because of starvation. Expert systems for the diagnosis of equipment malfunctions contain the rules by which a malfunction can be transmitted to adjacent units embedded in their knowledge base. The Expert System continu-ously monitors the materials flows and should the mass balance for each unit be essentially constant, then no alarm is issued. However, when some malfunction occurs, the Expert System is executed with the object of determining the source of the fault. Timing is clearly of the essence.

Chapter 3 Intelligent Control Intelligent control takes a radically different approach to the control of industrial processes and plants from conventional control. The knowl-edge and experience of human operators constitutes the basis for this new approach to Control Engineering for which Computational Intelli-gence provides the theoretical foundation. In this chapter we summarize the potential and some limitations of intelligent control and we attempt to address the questions on how, where, when and under what conditions can intelligent control be applied in practice.

Intelligent control seeks solutions to the problem of controlling plants from the viewpoint of the human-operator. In other words, the technique seeks to establish some kind of cognitive model of the human operator and not the plant under his control. This is the point at which in-telligent control departs from conventional control and it is undoubtedly true that the technique could not have been possible but for the rapid progress in computer technology. Computational Intelligence provides the tools with which to make intelligent control a reality. The reproduc-tion of human intelligence and the mechanisms for inferring decisions on the appropriate control actions, strategy or policy that must be followed are embedded in these tools.

Figure 3.1 shows how Computational Intelligence can be classi-fied according to the form of the knowledge (i.e., structured or unstruc-tured) and the manner in which this knowledge is processed (i.e., sym-bolic or numerical). For control applications, knowledge can be struc-

31

Chapter 3 32

tured or not, but processing is invariably numerical. Fuzzy and neural control form the core of intelligent control and are the principal compo-nents of computational intelligence.

PROCESSING

Symbolic Numerical

Neural Systems

Fuzzy Systems

Expert Systems

Structured

KNOWLEDGE

Unstructured

Figure 3.1 Classification of Computational Intelligence

In contrast to conventional control, intelligent control is based on advanced computational techniques for reproducing human knowl-edge and experience. Thus in intelligent control the focus of interest moves away from the tedious task of establishing an explicit, micro-scopic model of the controlled plant and the subsequent design of a cor-responding hard controller, to the emulation of the cognitive mechanisms used by humans to infer and support control decisions.

Intelligent control has been applied with considerable success in the process industry. Examples can be found in the petrochemical, ce-ment, paper, fertilizer and metals industries. With time, it is predicted that intelligent control will diffuse into most branches of industry by manufacturing and be adopted by progressive organizations that are seeking to improve their strategic position in the global market through improved productivity and product quality.

Intelligent Control 33

3.1 Conditions for the Use of Intelligent Control

Intelligent controllers use empirical models that form the framework on how and not why the controlled plant behaves in a particular manner, in-stead of relying on explicit mathematical models of the plant. The fun-damental problem in developing an intelligent controller is elicitation and representation of the knowledge and experience of human operators in a manner that is amenable to computational processing.

Intelligent systems invariably use a collection of heuristic and non-heuristic facts of common logic as well as other forms of knowledge in combination with inference mechanisms in order to arrive at and sup-port their decisions. A basic characteristic of this class of systems is that these systems are able to infer decisions from incomplete, inaccurate and uncertain information, typical of many industrial and manufacturing en-vironments. As noted in chapter 2, intelligent systems can be applied

• off-line, e.g., for controller design, for fault diagnosis and production management or

• on-line, e.g., for fault prediction and supervisory control. In conventional control, the knowledge about the controlled

plant (i.e., its model) is used implicitly in designing the controller, whereas in intelligent control, the knowledge about the plant is distinct from the mechanism that infer the desired control actions. This is shown schematically in Figure 3.2. It is obvious that an intelligent controller can therefore be easily configured for any plant by simply modifying its knowledge base.

The most common means to reproduce knowledge are linguistic rules, sometimes termed control protocols, which relate the state of the plant to the corresponding desired control actions. These rules represent shallow empirical knowledge, deep knowledge or model-based knowl-edge about the plant.

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Real-time Data Base

Inference Engine

Knowledge Base

Figure 3.2 Basic elements of an intelligent controller 3.2 Objectives of Intelligent Control The principal objectives of intelligent control are maximization of:

• the strategic success (i.e., profit) of the business and • productivity.

In order to succeed in these objectives, it is presumed that an intelli-

gent controller is technically correct. The success of any application is based on both technical and social factors that must co-exist and cooper-ate in order to bring about the desired results. The distinct characteristics of each must therefore be respected, otherwise conflicts will arise which are bound to minimize its impact and effectiveness. Thus continuing co-operation among plant management, production management and plant operators is essential to the acceptance of any such system. Doubts and differences of opinion, for whatever reason, by any one of those in-volved, may easily doom any system unless it has been studied in depth.

The optimum solutions using social criteria are by no means easy to determine or measure. For an intelligent system to be technically correct it is necessary that its technical specifications meet the demands of the particular application. There are a number of vendors that offer intelli-gent controllers for the process industry. All claim to meet the demands of industry and the choice of which one is the most suitable is not always

Intelligent Control 35

easy. Cost is one factor but prior experience with a similar plant is con-sidered the most important factor in making a decision on which system to purchase. The socially optimum solution is given by the support that plant management, production management and plant operators are pre-pared to give to a particular system to make it successful.

In considering intelligent control seriously for solving production control problems, which conventional control is unable to solve, answers must be sought to the following questions: will the proposed system

• repay its cost in finite time? • decrease the cost of production? • increase productivity? • lead to savings in energy? • improve equipment availability? • be simple to use or require specialized knowledge that is not

available in-house? • decrease the workload of plant operators?

It is implicitly assumed that the knowledge to control the plant is

available by the plant operators who have spent years operating the plant. It should be obvious that intelligent control is not a candidate when this knowledge is absent or incomplete, as in the case of an en-tirely new plant for which there is no prior experience. In practice this situation is not very likely to occur since most new plants are based on earlier designs for which some prior knowledge exists.

It is difficult to state all the technical properties of a successful intelligent system since they vary according to the application. The suc-cess of an intelligent system is very much dependent on the support of the users of the system. Intelligent controllers are underutilized and even ignored when user support is undermined.

Assuming that plant management has been convinced that intel-ligent control could lead to an improvement in the strategic position of the business, improve productivity and lead to a reduction in production costs, it is still necessary to convince plant operators to use the system. This is not always an easy task, as by tradition, operators are fearful of any new technology that may undermine their post, future and usefulness in the business. These are natural feelings that have to be taken into ac-count when any new system is introduced. This inbred fear can be

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greatly reduced by including the plant operators in the system develop-ment process and providing adequate training to alleviate his fears.

The older generation of plant operators spent years controlling plants from central control areas with classical instrumentation, adjusting the setpoints of conventional three term controllers and tediously logging plant activity manually. Today, the new generation of plant operators have been brought up in the era of computers, consoles with graphical user interfaces and all the benefits of Computer Integrated Manufactur-ing systems. Even the older plant operators have adapted, even though sometimes reluctantly, to the new environment. New plant operators no longer view the introduction of advanced technology as a threat but on the contrary, show great interest and an enviable ability to assimilate and use it effectively to improve their working conditions. This is especially true where management has had the foresight to provide the necessary training in advance. The days of pulling down control switches and turn-ing control knobs are gone, replaced by the touch of a light pen or a fin-ger on a screen or the click of a keyboard or a mouse. Report generation is a matter of seconds instead of hours, days or even weeks. Information is power and this can undoubtedly be enhanced through the use of intel-ligent techniques.

From the viewpoint of management, the success of an intelligent control system is judged solely on how rapidly the system will repay its investment. This is measured from the observed (and not assumed) in-crease in productivity, the energy reduction and the improvement in the mean-time-between-failures of the plant. Improvements of the order of 5-10% are not uncommon in the process industry. History has shown that since their introduction, manufacturers that have taken advantage of the new control techniques have benefited significantly on all counts. The specialization required to develop intelligent systems is Knowledge Engineering. It would be very wrong to conclude, however, that no knowledge of conventional control and system theory is neces-sary to design such systems. On the contrary, a very thorough knowledge of the abilities and limitations of classical and modern control techniques must constitute the background of the knowledge engineer. The most successful intelligent control systems that have been commissioned have been designed by control engineers with a very thorough background in conventional control techniques.

Knowledge Engineering requires the cooperation of knowledge engineers, domain experts and plant operators in the design phase, com-

Intelligent Control 37

missioning and operation of an intelligent system. Characteristics such as the quality, depth of knowledge, effectiveness of the inference engine and the suitability of the man-machine interface, are important to the effi-ciency and acceptance of an intelligent system.

An intelligent system based on computational intelligence uses linguistic rules with which to describe the knowledge about controlling the plant. Before eliciting the rules from human operators, it is very im-portant to stipulate the bounds of this knowledge, otherwise the system is likely to be unwieldy. It should be obvious, furthermore, that the intelli-gent system software can be written in any high-level language or be de-veloped on some expert system shell that simplifies the design process significantly.

Chapter 4 Techniques of Intelligent Control Conventional control systems design relies on the existence of an ade-quate macroscopic model of the physical plant to be controlled. The first stage in the analysis of such a system is therefore the development of an explicit mathematical model of the controlled plant that adequately re-produces the characteristics of the plant with fidelity. The model can be determined either from first physical laws or using some technique of identification from operating data mined from the plant. There exist a va-riety of design techniques that can then be used to design an appropriate hard controller. Today, this task is made simpler through the use of com-puter aided control systems design software with which the performance of the closed system can be rapidly evaluated and optimized. The ulti-mate objective is clearly the development of a hard controller that satis-fies specific performance specifications.

On completion of the design, the hard controller is then imple-mented in hardware or software that will execute in real-time on a proc-ess computer. The design of a conventional controller, particularly in the case of multivariable plants, is a tedious and painstaking process that re-quires repeated cycles of analysis, synthesis and testing. The design hopefully converges to an acceptable solution and ultimately to commis-sioning.

39

Chapter 4 40

In order to use conventional design techniques, it is essential that the model of the plant be simplified yet be sufficiently comprehensive so that it reproduces the essential dynamic features of the physical plant. Modern manufacturing plants have to meet increasing demands for more flexible production and improved quality while striving to meet stringent environmental constraints. Though there were high expectations that modern control theory would meet these demands, it has failed by and large to do so to any significant degree in industry and manufacturing, which thus far have had to be content with conventional industrial three-term controllers. The design of simple, practical and robust controllers for indus-try is usually based on low order holistic models of the physical plant. These approximants form the basis for the design of industrial controllers that satisfy relaxed performance criteria. Three-term controllers are the backbone of industrial control and these ubiquitous, simple and robust controllers have offered sterling service. However, these controllers can only perform at their best at the nominal operating point of the plant about which the approximant holds. When the operating point moves away from the nominal point, their performance is invariably degraded due to the inherent non-linearity of the physical plant. A number of tech-niques have been proposed to anticipate this problem, the most common example of which is gain-scheduling, a variant of which is considered in a later chapter. The objective here is one of extending the domain over which satisfactory controller performance is maintained. Adaptive controllers are another class of controllers whose parameters can be varied to track the changes in the operating point. Here, periodic identification is re-quired in order to follow the changes in the plant dynamics. The degree of autonomy of a controller is closely related to the range of operation of the controller and consequently to its robustness. The degree of auton-omy of a gain-scheduled controller is higher than that of a fixed control-ler but lower than that of an adaptive controller whose range of operation is correspondingly greater.

Techniques of Intelligent Control 41

4.1 Unconventional Control There are many situations in practice where, due to unforeseen changes in the controlled plant and its operational environment, a greater degree of autonomy is required. Conventional control techniques often fail to provide the necessary autonomy which these situations demand and this has led to an extensive search for new advanced control techniques which offer high autonomy and robustness despite the unfavorable oper-ating conditions, uncertainty and vagueness that characterize the plant and its environment. This is typically the domain of industry and manu-facturing. Coincidentally, this is also the domain of Intelligent Control. This recognition was not long in coming and it was not long before the beneficial results of applying Intelligent Control to industry became evi-dent.

Many industrial processes are so complex that any attempt at de-scribing them analytically is often futile. Even if much effort is expended in determining some form of explicit model, it is usually so complex as to make it of little use for the design of a suitable controller. To apply modern control design techniques it is necessary to simplify the model through linearization and then model reduction before proceeding to de-termine an appropriate linear controller. This design procedure used of-ten in design exercises leaves much to be desired in practice, as the original control problem is no longer attacked. Instead, some idealized controller for an idealized plant is determined and the probability that this controller is applicable to the real problem is small indeed. Invaria-bly resort has to be taken to parameter tuning on-line in order to obtain acceptable performance. Techniques for designing and analyzing non-linear plants are virtually non-existent and what techniques are available apply to very restricted situations. Thus the designer invariably falls on simulation to design an acceptable controller which must then be tested exhaustively in the field, an iterative, time-consuming procedure.

The field of Industrial Automation, which relies largely on Pro-grammable Logic Controllers (PLCs) and industrial three-term control-lers, is now being confronted with new control techniques which find their origins in Soft Computing and Computational Intelligence. The need to maintain tight production and quality control in large-scale in-dustrial and manufacturing plants producing products with high specifi-cations, and the inability of conventional control techniques to satisfy these requirements, has led to emergence of a new class of automatic

Chapter 4 42

control techniques. What sets these new unconventional techniques apart is their ability to arrive at control decisions and control strategies in ways that are radically different from those of conventional control. It is not unreasonable, therefore, that the new class of unconventional control techniques has caused considerable interest in industrial and manufactur-ing circles and has led to innovative controllers which have been applied to many difficult problems in industry. In this new class of controllers, the primary objective is minimization of the uncertainty and vagueness with which industrial processes are shrouded, leading to controllers with high autonomy and robustness.

The reproduction of the cognitive and decision making processes of a human operator of an industrial plant executing his control task has been the subject of intense research since the 1950s, reaching fruition in the 1970s with the implementation of the first experimental rule-based control system. The first practical unconventional industrial controllers were commissioned in the early 1980s in the cement industry, an indus-try with many difficult problems particularly in the critical kilning proc-ess. The development of unconventional controllers since then has been very rapid and they are to be found today not only in most process indus-tries but in all kinds of household appliances as well.

The new field of unconventional control, which is based on the knowledge and experience of human operators, is better known as Intel-ligent Control and is supported by fuzzy, neural, neuro-fuzzy and evolu-tionary control techniques. This book is devoted exclusively to these techniques and their application to practical industrial problems. In line with the use of the term Soft Computing in Control this field is some-times known as Soft Control.

Modern control theory, which is based on a microscopic descrip-tion of the controlled plant using differential or difference equations, can, in contrast, be described as Hard Control since it uses inflexible al-gorithmic computing techniques. There are fundamental differences be-tween conventional and intelligent control techniques. These differences are highlighted in this chapter. It is important to note that control in no way supercedes conventional control, but rather augments it in an effort to resolve some of the difficult and unsolved control problems which in-dustry and manufacturing face.

A fundamental difference between conventional and Intelligent Control is the manner in which the plant and controller are viewed. This is seen schematically in Figure 4.1. In conventional control, the plant and

Techniques of Intelligent Control 43

controller are viewed as distinct entities. The plant is assumed invariant and designed to perform a specific task without the controller in mind. The controller is subsequently designed independently and after the plant has been completed. In contrast, in intelligent control, the plant and the controller are viewed as a single entity to be designed and commissioned simultaneously. Thus plants which are unstable by nature are stable when a unified approach is taken. A typical example is a helicopter that is by nature an unstable plant, but so long as its controller is operational is perfectly stable.

Controller

Plant

Controller

Plant

(a) (b)

Figure 4.1 (a) Conventional and (b) Intelligent control configurations

The generalized closed control system is shown in Figure 4.2. Here block P depicts the controlled plant, block C depicts the controller and block S specifies the desired closed system performance specifica-tions. In conventional control blocks P and C are assumed linear (or lin-earized) and the block S defines the cost function, or criterion of per-formance, e.g., stability margin, rise time, settling time, overshoot, steady state error, integral squared error, etc. following some exogenous disturbance.

The following characteristics apply to industrial processes:

• the physical process P is so complex that it is either not known explicitly or is very difficult to describe in analytical terms, and

• the specifications S ideally demand a high degree of system autonomy so that the closed system can operate satisfactorily and without intervention despite faults in the system.

Chapter 4 44

Disturbance

Output

Specifications S Control Action

Controller C Input

Plant P

Figure 4.2 Closed control system variables

The intelligent control problem can be stated in precisely the

same way: given the plant P, find a controller C that will satisfy specifi-cations S that may be either qualitative or quantitative. The basic differ-ence here is that in intelligent control it is not necessary to have an ex-plicit description of the plant P. Furthermore, it may not always be pos-sible to separate the plant and controller. Since intelligent control is often rule-based, the control rules are embedded in the system and form an in-tegral element of the system. This structure presents new possibilities for improved integrated manufacturing systems.

As illustrated in Figure 4.3, Intelligent Control is the fusion of Systems Theory, Computer Science, Operations Research and Computa-tional Intelligence that bonds them. These techniques are now being called upon to solve problems of control engineering, which were hith-erto unsolvable. Intelligent control has been claimed to be able to repro-duce human-like properties such as adaptation and learning under unfa-vorable and uncertain conditions. There is considerable debate on this matter and many have refuted these claims. No one has refuted the fact, however, that intelligent control is capable of controlling large-scale in-dustrial processes that have hitherto been controlled only manually very successfully. Industry has only to take advantage of this fact to benefit significantly.

Techniques of Intelligent Control 45

Computational Intelligence

Operations Research

Systems Theory

Computer Science

Figure 4.3 Intelligent Control 4.2 Autonomy and Intelligent Control An intelligent system is designed to function in an uncertain and vague industrial or manufacturing environment with a view to increasing the success of the business. Success is defined as satisfaction of the system objectives, primarily the profit margin. Intelligent control systems are required to sense the environment, infer and justify the appropriate con-trol action on the system so as to improve productivity, reduce energy consumption and labor costs.

In advanced forms of autonomous systems, intelligence implies the faculties of perception and thought, the ability to make wise deci-sions and act correctly under a large range of unforeseeable conditions in order to survive and thrive in a complex and often hostile environment. In order to relieve the human operator of his often boring and tiresome control task, a high degree of autonomy is required in intelligent supervi-sory control systems. In the case of humans, these faculties are credited to his natural intelligence. In order to approach this level of intelligence

Chapter 4 46

under conditions of uncertainty and vagueness by mechanistic means, it is necessary to develop advanced inference and decision support tech-niques. Autonomy of operation is the objective and intelligent control is the means to this objective.

The theory of intelligent systems, which was developed by Saridis, fuses the powerful techniques for decision support of Soft Com-puting and advanced techniques of analysis and synthesis of conven-tional Systems Theory. The fusion of Computational Intelligence, Op-erations Research, Computer Science and Systems Theory offers a uni-fied approach to the design of intelligent control systems. The result of this fusion is Soft Control, which today is one of the most interesting ar-eas of research and development in Control Engineering.

Precision Intelligence

Organization

Coordination

Execution

Figure 4.4 Hierarchical structure of an Intelligent System

Intelligence is distributed hierarchically and in accordance with Saridis’ principle of “increasing precision with decreasing intelligence” as is depicted in Figure 4.4. Most practical hierarchical intelligent con-trol systems have three layers:

Techniques of Intelligent Control 47

• the Organization layer in which high level management deci-sions, e.g., production scheduling, are made,

• the Coordination layer for the tasks that have been decided on at the Organization layer. As in the case of the uppermost layer of the hierarchy, this layer normally possesses intelligence, and

• the Execution layer, which has little or no intelligence, is the layer in which the commands of the higher layers are executed. This layer involves low-level controllers embedded in the plant Remote Terminal Units (RTU). Recently, some vendors have added some degree of intelligence into this layer.

Coordinator Coordinator

Organizer

LAN

Executor Executor Executor

Figure 4.5 Distributed architecture of an intelligent system The uppermost layer of the hierarchy is activated rarely and at

random as need requires and uses qualitative reasoning to arrive at its decisions. The intermediate Organization level is activated by production management and has low repetition frequency, perhaps once or twice

Chapter 4 48

daily. In this layer, management arrives at a long-term production policy that must be followed to achieve production goals. The production pol-icy that has been decided in the highest layers is relayed to the Coordina-tion layer, which is responsible for carrying out this policy. In this inter-mediate layer, decisions on changes in the policy can be made should, for instance, a serious malfunction or breakdown occur in a production line, if raw material shortages are ascertained, or if changes are made in customer priorities. The Coordination layer is also responsible for prod-uct quality control, maximization of productivity of each unit in the fac-tory and for coordination between the various manufacturing units.

Even though the structure of an intelligent system is normally represented vertically in Figure 4.4 to indicate its hierarchical architec-ture, in practice such systems use a distributed architecture based on a client/server structure, as shown in Figure 4.5. Here, the Organizer acts as the server while the Coordinators and Executors are clients of the sys-tem.

4.3 Knowledge-Based Systems

Intelligent controllers use a qualitative description on how a process op-erates instead of an explicit quantitative description of the physical prin-ciples that relate the causes to the effects of the process. An intelligent controller is therefore based on knowledge, stated linguistically in the form of production rules, which are elicited from human experts. Appro-priate inference mechanisms must then be used to process this knowl-edge in order to arrive at suitable control decisions. One or more of the following techniques of Computational Intelligence may be used to this end:

• Expert Systems • Fuzzy Logic • Artificial Neural Networks • Neuro-fuzzy Systems • Evolutionary Computation

Techniques of Intelligent Control 49

4.3.1 Expert systems The objective of an expert system is to permit a non-expert user to ex-ploit the accumulated knowledge and experience of an expert in a spe-cific field of expertise. Knowledge-based expert systems use rules, data, events, simple logic and any other form of knowledge in conjunction with search techniques to arrive at decisions. Where an element of knowledge is missing then the expert system cannot but return a “don’t know” response, implying an inability to arrive at a decision. An expert system cannot extrapolate data and infer from similar or adjacent knowl-edge.

Expert systems operate either on-line when decisions are re-quired in real-time, as in the cases of fault prediction, energy manage-ment and supervisory control, or off-line, as in the cases of interactive, dialog-based systems for fault diagnosis and production management. In dialog-based expert systems using decision trees, forward and backtrack-ing mechanisms for searching the tree have proved successful in indus-trial applications. In the category of on-line expert systems, a number of vendors supply real-time expert supervisory control systems for indus-trial applications.

In the case where a complete decision tree that can account for every possible situation that may arise in practice is available, then such knowledge-based systems offer a simple and effective solution to the un-conventional control problem. Expert systems can be developed using object-oriented languages such as LISP, Prolog and C++ or any of the available expert system shells, such as G2, NEXPERT, etc. It should be obvious that situations involving uncertainty and vagueness cannot be treated effectively using conventional decision-tree based expert sys-tems.

In contrast, when insufficient or incomplete knowledge about a process is available and when uncertainty and vagueness characterize the plant and its measurements, then such knowledge-based expert systems are not always able to arrive at decisions and are consequently unaccept-able for real-time control purposes. It would be extremely frustrating to receive a “don’t know” or a conflicting decision in the case of an unfore-seen emergency when immediate action is required! In situations of un-certainty and vagueness, more effective mechanisms, capable of infer-ring decisions from incomplete data, are necessary. Fuzzy logic and arti-ficial neural networks and their hybrids are the primary examples of

Chapter 4 50

techniques that possess appropriate mechanisms to deal with uncertainty and vagueness. 4.3.2 Fuzzy control The assumptions that industrial processes can be modeled by sets of al-gebraic or differential equations and that the measurements from sensors are noise-free and exact rarely hold in practice. Likewise, the degree of uncertainty in the controlled process, which is characterized by the com-pleteness and vagueness of the information that is mined from the proc-ess, plays a major part in determining the behavior of a closed industrial system.

Many industrial processes can be controlled adequately by hu-man operators whose knowledge of mathematical models, algorithms and the underlying physical principles on which the process operates is limited to non-existent. Human operators have an inherent ability for de-ciding on acceptable or satisfactory actions to follow from qualitative in-formation received from a variety of apparently disparate sources. Natu-rally, where control of a process or plant is in the hands of human opera-tors, it is inappropriate to talk about optimum control decisions, but rather acceptable control decisions. It is unlikely that two operators will make identical decisions in any given situation and the only way to evaluate the quality of their decisions is to evaluate the productivity of the process under their control. Experience has shown that there is al-ways a best operator who is consistently capable of increased productiv-ity. This is the operator whose knowledge should be elicited since he clearly knows how to get the most out of the process!

Fuzzy control requires some qualitative description of the rules with which a human operator can control a process. Zadeh has noted that for many complex processes a high level of precision is not possible or even necessary in order to provide acceptable control. This is a pivotal concept in Computational Intelligence and forms the basis of unconven-tional control. Fuzzy logic, which in no way replaces probability theory, is the underlying theory for dealing with approximate reasoning in un-certain situations where truth is a matter of degree.

Techniques of Intelligent Control 51

4.3.3 Neural control Artificial Neural Networks (ANNs) were originally proposed in the 1940s but limitations in the hardware and in training methods at the time did not permit their practical application. Extensive and often dishearten-ing research followed over the next decades and it was not until the 1980s that ANNs became established as a viable technique of Computa-tional Intelligence. ANNs are made up of densely parallel layers of sim-ple non-linear neurons (or nodes) which are interconnected with varying weights (the synaptic weights) that completely specify the behavior of the network once it has been trained. Various network architectures have been proposed and their application to unconventional control was sim-ply a matter of time.

ANNs may be constructed with both analog and digital hardware and exhibit the properties of massive parallelism, robustness and fault tolerance, properties which make them ideally suited to control applica-tions. Furthermore ANNs possess:

• the ability to learn from experience instead of from models, • the ability to generalize and relate similar inputs to similar out-

puts, • the ability to generate any arbitrary non-linear functional rela-

tionship between their inputs and outputs, • a distributed architecture which is eminently suited to parallel

computation, • inherent stability, • the ability of absorbing new knowledge without destroying ex-

isting knowledge, and • the ability to learn on-line, despite disturbances in the process.

4.3.4 Neuro-fuzzy control Traditionally, fuzzy and neural systems were considered as distinct, since their origins are very different. This restrictive viewpoint has changed radically since the late 1980s when it was realized that both fields belonged to Computational Intelligence and had much in common.

Fuzzy systems are very powerful for representing linguistic and structured knowledge by means of fuzzy sets, but it is up to the experts

Chapter 4 52

to establish the knowledge base that is used in the system. Execution of the fuzzy algorithm is performed as a sequence of logical steps at the end of which a full explanation of the steps that were taken and the rules that were used in arriving at the conclusion is made available. In contrast, ANNs are ideal for the representation of an arbitrary nonlinear functional relationship with parallel processing methods, can be trained from train-ing sets but do not offer any mechanism for giving explanations on the decisions at which they arrive.

It is natural, therefore, to consider the advantages and benefits that a fusion of the two methods may present. This possibility is dis-cussed at length in a later chapter and it is shown how fuzzy control and neural control can be combined with interesting results.

Chapter 5 Elements of Fuzzy Logic Uncertainty can be traced to Heisenberg´s principle, which established multi-valued logic in the 1920s. In the late 1930s the mathematician Max Black applied continuous logic to sets of elements and symbols and named this uncertainty. The developments that followed allowed for de-grees of uncertainty, with truth and falsity being at the extremes of a continuous spectrum of uncertainty.

In his seminal publication entitled “Fuzzy Sets”, Zadeh pre-sented in 1965 the theory of multi-valued logic, which he termed Fuzzy Set Theory. Zadeh used the term Fuzzy Logic and established the founda-tions of a new area of scientific endeavor that continues to this day. Ini-tially, many have been critical of Zadeh´s theory, claiming that Fuzzy Logic is nothing but probability theory in disguise. Zadeh was to develop his theory into Possibility Theory, which differs significantly from prob-ability theory. In contrast, in Japan the theory of Fuzzy Logic was rap-idly assimilated into a host of applications that have yielded enormous profits! Kosko conjectures that the principles of Fuzzy Logic are much closer to the Far Eastern concept of logic than Aristotelian logic which the West espouses and this was the reason why the Japanese were more appreciative of Fuzzy Logic.

The theory of Fuzzy Logic establishes the basis for representing knowledge and developing the mechanisms essential to infer decisions on the appropriate actions that must be taken to control a plant. Since the late 1970s, Fuzzy Logic has found increasing application in the process

53

Chapter 5 54

industry, in traffic and train control systems and most notably in house-hold appliances.

The fundamental elements of Fuzzy Logic necessary to under-stand the techniques of Fuzzy Control are presented in this chapter. For further in depth study of the theory of Fuzzy Sets, the reader is referred to the numerous books and papers on the subject given in the Bibliogra-phy in chapter 18. 5.1 Basic Concepts

In classical set theory, a set consists of a finite or infinite number of ele-ments belonging to some specified set termed the universe of discourse. The elements of the universe of discourse may or may not belong to the set A, as shown in Figure 5.1.

Α x

f A

1

Figure 5.1 Characteristic function of a classical Boolean set

The crisp or Boolean characteristic function fA(x) in Figure 5.1 is

expressed as the discontinuous function fA (x) = 1 if x∈A = 0 if x∉A

Elements of Fuzzy Logic 55

Vagueness can be introduced in the theory of sets if the characteristic function is generalized to permit an infinite number of values between 0 and 1, as shown in Figure 5.2.

µΑ

x

1

Support set

Universe of Discourse

µΑ

x

1

Universe of Discourse

Support set

Figure 5.2 Examples of triangular and trapezoidal fuzzy sets

If Χ is the universe of discourse with elements x (i.e., the region to whic

A= µA(x)/x} or µA(x)/x} for x∈X

r the continuous and discrete cases respectively. Here µA(x) is termed

µA (x) : X→[0,1]

It is noted that the symbols ∫ and imply a fuzzy set and bear no relation to the integration and summation.

h the physical variable is confined), then we may state X={x}. A fuzzy set A on the universe of discourse X can be expressed symbolically as the set of ordered pairs

∫{ Σ{

fothe membership function of x on the set Α and is a mapping of the uni-verse of discourse Χ on the closed interval [0,1]. The membership func-tion is simply a measure of the degree to which x belongs to the set Α, i.e.,

Σ

Chapter 5 56

The support set Α of a fuzzy set is a subset of the universe of dis-course Χ for which µA(x)>0. Thus a fuzzy set is a mapping of the support set on the closed interval [0,1]. As an example, consider the temperature of water at some point in a plant. Consider for example the fuzzy vari-able Low. This can be described in terms of a set of positive integers in the range [0,100] and defined as Α={Low}. This set expresses the degree to which the temperature is considered Low over the range of all possible temperatures. Here, the membership function µA(x) has discrete values specified in degrees Centigrade by the set:

µA(0)= µΑ (5)= µΑ (10)= µΑ(15)= µ (20)=1.0, µ

ΑµΑ(40)=0.3, µΑ(45)=0.1,

Mor

1/20 + 0.9/25 + 0.8/30 + 50 + 0/55 +....0/100}

The symbe confused wi the

Α(25)=0.9, µΑ(30)=0.8, µΑ(35)=0.6, µΑ(50)= µΑ(55)= ..... µΑ(100)=0

e compactly, this e set can b expressed as:

µΑ(x)= {1/0 + 1/5 + 1/10 + 1/15 + 0.6/35 + 0.3/40 + 0.1/45 + 0/ bol ‘+’ represents the union operator in set theory and must not

th arithmetic addition. A graphical representation ofcorresponding fuzzy membership function µA(x) is shown in Figure 5.3.

µ (x) Α

1

10 20 30 40 . . . . . .

A={Low}

x

Figure 5.3 Discrete membership ion of the fuzzy set A={Low}

funct

Elements of Fuzzy Logic 57

A fuzzy variable is one whose values can be considered labels of zzy sets. Thus TEMPERATURE can be considered as a fuzzy variable

which c

dges. Thus th

Figure 5.4 The linguistic variable TEMPERATURE and some of its values

fuan take on linguistic values such as Low, Medium, Normal, High

and Very_High. This is precisely the way that human operators refer to plant variables in relation to their nominal values. It is shown in the fol-lowing that fuzzy variables can be readily described by fuzzy sets.

In general, any fuzzy variable can be expressed in terms of phrases that combine fuzzy variables, linguistic descriptors and he

e values of the fuzzy variable TEMPERATURE in the foregoing example can be described as High, NOT High, rather_High, NOT Very_High, extremely_High, quite_High etc., labels such as High, nega-tion NOT, connectives AND and hedges such as extremely, rather, quite etc. Figure 5.4 shows the variable TEMPERATURE with a few of its val-ues.

0 20 40 60 80 100

HighLow Very_Low

Linguistic Variable

Linguistic Values

Membership values

Universe of Discourse (degrees C)

TEMPERATURE

1

0.

0.6

8 0.6 0.8 0.9 1

Chapter 5 58

The dependence of a linguistic variable on another can be de-scribed by means of a fuzzy conditional statement of the form:

R : IF S1 THEN S2

or symbolically as: S1 →S2

where S1 and S2 are fuzzy conditional statements which have the general form:

S : Χ is Α

and Α∈Χ. A linguistic meaning can be given to the fuzzy subset Α to specify the value of Χ, for example:

IF the LOAD is Small THEN TORQUE is Very_High OR IF the ERROR is Negative_Large THEN OUTPUT is Negative_Large.

Two or more fuzzy conditional statements can be combined (or

included in another) so as to form a composite conditional statement such as:

R : IF S1 THEN (IF S2 THEN S3).

It should be obvious that the composite statement can be decomposed into the two simpler conditional statements:

R1 : IF S1 THEN R 2 AND R 2 : IF S2 THEN S3

The composite statement (or rule):

IF the ERROR is Negative_Large THEN (IF CHANGE_IN_ERROR is Positive_Large THEN OUTPUT is Positive_Large)

can be written more simply as a pair of rules:

Elements of Fuzzy Logic 59

R : IF CHANGE_IN_ERROR is Positive_Large THEN OUTPUT is Positive

ost useful rule structure in practice. Human opera-tors are invariably taught to control plants with linguistic rules of this

pe, rather than composite rules that appear to be far too complicated.

and depends on the complexity of the plant. Human plant perators rarely use more than approximately 30 rules for routine control

tasks, since rules that are rarel d to be quickly forgotten. To control a complex plant like a rotary iln as many as 60-80 rules may be

5.2 Fuzzy Algorithms

or ional statements can be combined with the OR r ELS N

For example a subset of linguistic rules used to control a steam engine can be written as:

EN CFUEL is Posi-tive_Large) OR IF SPE is Negati is (Positive_Large

CFUEL is Positive_Small)

is

R1 : IF ERROR is Negative_Large THEN R 2 2

_Large

This is the m

tyThe number of rules that are required to control a plant varies

enormouslyo

y used ten k

required but as few as 5 are necessary to control simple appliances like washing machines or cameras.

Two (o

more fuzzy conditE) connective to form a fuzzy algorithm R of the form:

RN : R1 OR R2 OR R3 OR .... OR Rn

IF SPE is Negative_Large THEN (IF CSPE is NOT (Nega-tive_Large OR Negative_Medium) TH

ve_Small THEN (IF CSPEOR Positive_Small) THEN

R OIF SPE is Negative_Small THEN (if CSPEPositive_Medium THEN CFUEL is Positive_Small)

where

Chapter 5 60

CFUEL

It is not

a∧b = min (a,b) = a if a≤ b = b if a>b

∨

ell known appli ound in program-mable controllers (AND (∧

in the sets C and D

D = A∨B = {max(a,b)} ∀ a∈A, b∈B

SPE = SPeed Error CSPE = Change in Speed Error

= Change in Fuel Intake

ed that linguistic control rules are of the familiar if… then … else form, where else is replaced by the connective OR.

5.3 Fuzzy Operators

The operators min (for minimum) and max (for maximum) can be used for either one or two elements. Thus the operations min and max on two elements a and b are defined respectively as:

a b = max (a,b) = a if a≥b = b if a<b

W cations of these operators are to be f

PLCs) for logical switching operations. These are the ) and OR (∨) functions whose operations are shown symboli-

cally below. A black tile indicates a 1 and a blank tile a 0.

OR (max ∨) AND (min ∧) operators min and max of two sets A and B result The

as follows:

C = A∧B = {min(a,b)} ∀ a∈A, b∈B

0 1

0 1

Elements of Fuzzy Logic 61

and are shown in Figure 5.5. The AND operator is therefore synonymous with the tor with the max operation. It is worth re hapters.

perators are used on one element only they imply the

a = ∧A = inf(A) a∈A and

sup(A) a∈A

a = ∧(a ,a ,......a m)

m

When the elements et are functions of a variable, then the operators are expressed as:

a = ∧x(a(x)) x∈X

in and max

min operation and the OR operamembering this in the following c

hen oWminimum (inf or infinum) or maximum (sup or supremum) of all the elements of the set, thus:

a = ∨A =

Operators can also be used as functions on elements of discrete sets, e.g.,

1 2

= a1 ∧a2 ∧ ..... a = ∧k( ak)

A = ∧(A1,A2,......A m)

= A1∧A2 ∧ ..... Am

= ∧k( Ak)

of the s

It is noted, finally, that expressions that involve the moperators use identical rules to those of arithmetic multiplication and ad-dition respectively.

Chapter 5 62

µΑ(x)

Figure

1

1

0

µΒ(x)

µC(x) - min

µD(x) - max

ical representation of the min and max operations

5.5 Graph

1

1

µ(x)

x 0

0

Elements of Fuzzy Logic 63

5.4 Operations on Fuzzy Sets

A fuzzy set n X is said to be a null set if its membership function is zero everywhere, i.e.,

A = ∅ if µA (x) = 0 ∀ x ∈ X The omplement⎯Α of a fuzzy set is simply:

µ⎯⎯A = 1 – µA(x) ∀ x ∈ X

Two fuzzy sets are considered identical if their membership functions are identical everywhere on the universe of discourse, i.e.,

A = B )≡ µB(x) ∀ x ∈ X A fuzzy set B is a subset of A if the membership function of B is smaller or equal to that of the fuzzy set A everywhere on X, i.e.,

A ⊂ B if µA (x)≤ µB(x) ∀ x ∈ X The nion of two fuzzy sets A and B on X is defined as:

µA ∪B(x) = µA(x)∨ µB(x) ∀

while the intersection of two fuzzy sets A and B on X is defined as:

µA∩B(x) = µA (x)∧ µB(x) ∀ x ∈ X

Finally, the product of two fuzzy sets A and B on X is defined as:

µA•B(x) = µA(x) B(x) ∀ x ∈ X

A o

c

if µA(x

u

x ∈ X

•µ

Chapter 5 64

5.5 Algebraic Properties of Fuzzy Sets

ly be extended to fuzzy sets. The classical prop-ties of sets can be expressed in terms of membership functions. For in-

stance, the distribution ts becomes:

∧ µC )

The standard definitions for the union, intersection and complement of classical logic can readier

property in terms of fuzzy se

(µA ∧ µA) ∧ µC = µA ∧ (µA while DeMorgan´s theorem becomes:

( )µ µ µ µΑ Β Α Β∨ = ∧

Th

e following properties apply only to fuzzy sets:

Χ Χ

Χ Χ EA or µ 0 0Α

A A or µ 0 µΑ Α

A E A or µ 1 µΑ Α

A E E o µ 1 1Α

∩

∩

≠ ∅

= ∧ =

where Ε is the unit se d ∅ is the null set.

.6 Linguistic Variables

As was noted earlier, a linguistic variable can take on values that are e of a

linguistic variable is specified in terms of the following terms:

primary terms which are labels of fuzzy sets, such as High, Low, Small, Medium, Zero,

• negation NOT and connectives AND and OR, • hedges such as very, nearly, almost and • markers such as parentheses ( ).

∪∩

≠∅ = ∅ ∧ =

∪∅ = ∨ =

∪ = ∨ =r

t specified by µA(x) = 1 ∀x∈X an

5

statements of a natural or artificial language. In general the valu

•

Elements of Fuzzy Logic 65

The primary terms may have either continuous or discrership functions. Continuous membership functions are norm

ete mem-ally de-b

fined by analytic functions. The Danish company F. L. Smidth in its fuzzy controllers designed for the cement industry uses Gaussian-like membership functions of the type shown in Figure 5.7 given by the ex-pression:

β

xγα

e1xµ)

||(

)( −−

−= A

µΑ(x)

1 0

Figure 5.7 Examples of membership functions used in the F. L. Smidth fuzzy controller

1

x

e triplet (α,β,γ) defining the

Th shape of the F. L.sets shown in Figure 5.7 is given in the following table:

1

-

Smidth fuzzy

Chapter 5 66

Linguistic Variable Acronym α β γ Positive_Large LP 0.25 2.5 1 Positive_Medium MP 0.25 2.5 0.7 Positive_Small SP 0.25 2.5 0.4 Positive_Zero ZP 0.1 6 0.1 Zero ZE 0.25 6 0 Negative_Zero ZN 0.1 6 -0.1 Negative_Small SN 0.25 2.5 -0.4 Negative_Medium MN 0.25 2.5 -0.7 Negative_Large LN 0.25 2.5 -1 Large HIGH 0.5 6 1 Normal OK 0.6 8 0 Low LOW 0.5 6 -1

An alternative way of definin tinuous membership functions

is through the generic S and Π functions shown in Figures 5.8(a) and 5.8(b). The first is monotonic and is specified by:

S(x,α,β,γ) = 0 for x ≤α = 2[(x -γ)/(γ-α)]2 for α≤x≤β = 1 - 2[(x -γ)/(γ-α)]2 for β≤x≤γ = 1 for x ≥γ

g con

µΑ(x)

1

Figure 5.8(a) The generic membership functions S(x,α,β,γ)

x

Elements of Fuzzy Logic 67

The second generic membership function is the Π function that changes o only his fu tion be defined in terms of S functions. In this case the pa ter res ts the width of the func e m n points re t em ship function has a va is fun n is given

,β,γ) x,γ-β,γ-β x γ - S(x,γ,γ+ β+γ x >

zzy sets can also be constructed from standardized

trapezoi ples of trapezoidal func-tions that represent the p sets all, M dium d Large) are shown in Figure 5.9. These fuzzy sets can be uniquely defined using four paramet

, discrete fuzzy sets are example, if the un

monotonicity at ne point . T nc canrame β rep en

tion between th edia whe he m berlue of 0.5. Th ctio by:

Π(x,α = S( /2,γ) for ≤ = 1 β/2, ) for γ

Continuous fudal or triangular functions. Three exam

rimary (Sm e an

ers, the inflexion points b and c and the left and right extinction points a and d that define the support set.

Figure 5.10 shows examples of some of the membership func-tions available in the MATLAB/Fuzzy Toolbox. Finally

sets of singletons on a finite universe of discourse. For erse oiv f discourse is given by the finite set:

X = {0 + 1 + 2 + 3 + 4 + 5 + 6}

Figure 5.8(b) The generic membership functions Π(x,α,β,γ)

x

µA(x)

Chapter 5 68

µ(x)

x x

1

a b c d xdcba

Neg_Large

µ(x)

Neg_Small Zero Pos_Small Pos_

x-x 0

Large

s the sets:

1}

Figure 5.9 Standardized trapezoidal membership functions then the fuzzy sets for the linguistic variables small, medium and large as shown in Figure 5.11 could be defined a

µsmall (x) = {0.3 + 0.7 + 1 + 0.7 + 0.3 + 0 + 0} µmedium(x) = {0 + 0 + 0.3 + 0.7 + 1 + 0.7 + 0.3} µlarge (x) = {0 + 0 + 0 + 0 + 0.3 + 0.7 +

Figure 5.10 Examples of membership functions provided in the MATLAB/Fuzzy_Toolbox

Elements of Fuzzy Logic 69

5.7 Connectives

Negation (NOT) and the connectives AND and OR can be defined in terms of the complement, union and intersection operations respectively. Usually the connective AND is used for fuzzy variables which have dif-ferent universes of discourse. If

Α = {µA(x)/x} for x∈X B = {µB(y)/y} for y∈Y

it follows that

The connect riable. It is obvious tha rse of dis-course.

A AND B = µA(x) ∧ µB(y)/(x,y) for x∈X, y∈Y = µA∩B (x,y)/(x,y)

ive OR connects linguistic values of the same vat both variables must belong to the same unive

x

µ(x)small

x

µ(x)medium

x

µ(x)large

Figure 5.11 Examples of discrete membership functions made up of singletons Thus if

Α = {µ (x) } for x∈X /xA

B = {µA(x)/x} for x∈X

Chapter 5 70

then

A OR B = µA(x) ∨ µB(x)/(x)

= µA∪B (x)/(x) for x∈X The connective OR can be used only when the fuzzy variables

have different universes of discourse except in the case where the vari-ables appear on the sam .... then statement as in

e rule:

if the PRESSURE is High or the SPEED is Low

The NOT operator negation in a natural lan-

uage. Thus if:

NOT Α =⎯Α= {1 – µA(x)/x}

The statement “PRESSURE NOT High” is clearly identical to the statement “PRESSURE is Not_High”.

Linguistic hedges are useful in generating a larger set of linguis-tic values from a smaller set of primary terms. Thus using the hedges very, the connectives NOT, AND and the primary term Large, we may generate the new fuzzy sets very_Large, NOT_very_Large, Large_ AND_NOT_very_Large, etc. In this manner it is possible to compute the membership function of complex terms such as:

Α = NOT_Small AND_NOT_Large

whose membership function is

µΑ(x) = [1 - µSmall(x)] ∧ [1 - µLarge(x)].

e side of a conditional if th

then FUEL_FEED must be Zero.

is synonymous withg

Α = {µA(x)/x} for x∈X

is

Chapter 6 Fuzzy Reasoning At the core of every fuzzy controller is the inference engine, the compu-tational mechanism with which decisions can be inferred even though the knowledge may be incomplete. It is this very mechanism that can give linguistic controllers the power to reason by being able to extrapolate knowledge and search for rules which only partially fit any given situa-tion for which a rule does not exist. Unlike expert systems that depend on a variety of techniques to search decision trees, fuzzy inference en-gines perform an exhaustive search of the rules in the knowledge base to determine the degree of fit for each rule for a given set of causes. The contribution to the final decision of rules that exhibit a small degree of fit is clearly small and may even be ignored while rules with a high degree of fit are dominant. It is clear that a number of rules may contribute to the final result to varying degrees. A degree of fit of unity means that only one rule has fired and only one unique rule contributes to the final decision, while a degree of fit of zero implies that the rule does not con-tribute to the final decision.

Inference engines can take different forms depending on the manner in which inference is defined. It is therefore prudent to review the fundamentals of fuzzy logic that will allow us to understand how the inference works and how it may be implemented. A fuzzy propositional implication defines the relationship between the linguistic variables of a fuzzy controller. Given two fuzzy sets Α and Β that belong to the uni-verses of discourse Χ and Υ respectively, then we define the fuzzy pro-positional implication as:

71

Chapter 6 72

R : IF Α THEN Β = Α → Β ≡ Α × Β

where Α × Β is the Cartesian product of the two fuzzy sets A and B. The Cartesian product is an essential operation of all fuzzy inference engines.

Using the conjunctive operator (min), the Cartesian product is defined as:

Α × Β = ∫ (µΑ(x) ∧ µB(y)) / (x,y) = ∫ m Χ×Υ Χ×Υ in(µΑ(x), µB(y)) / (x,y)

while for the case of an algebraic product the Cartesian product is:

Α × Β = ∫ (µΑ(x) • µB(y)) / (x y) , Χ×Υ

Thus, for example, given the discrete fuzzy sets:

Α = {1 + 2 + 3} and Β = {1 + 2 + 3 + 4} whose corresponding discrete membership functions (sometimes termed the grades of membership) are:

{µΑ(x)/x} = (1/1 + 0.7/2 + 0.2/3) and

{µΒ(y)/y} = (0.8/1 + 0.6/2 + 0.4/3 + 0.2/4), using the union (or more generally disjunction) operator, the Cartesian product using the conjunctive (min) operator is: RΛ = Α × Β = {min(1, 0.8)/(1,1), min(1, 0.6)/(1,2), min(1, 0.4)/(1,3), min(1, 0.2)/(1,4), min(0.7, 0.8)/(2,1), min(0.7, 0.6)/(2,2), min(0.7, 0.4)/(2,3), min(0.7, 0.2)/(2,4), ............ = {0.8/(1,1) + 0.6/(1,2) + 0.4/(1,3) + 0.2/(1,4) + 0.7/(2,1) + 0.6/(2,2) + 0.4/(2,3) + 0.2/(2,4) +

Fuzzy Reasoning 73

0.2/(3,1) + 0.2/(3,2) + 0.2/(3,3) + 0.2/(3,4)} This Cartesian product can be conveniently represented by means of the relational matrix:

x\y 1 2 3 4 1 0.8 0.6 0.4 0.2 2 0.7 0.6 0.4 0.2 3 0.2 0.2 0.2 0.2

which is shown in graphical form in Figure 6.1.

x

Figure Likewise, the R* = Α × Β whose relatio

RΛ

y

6.1 Graphical representation of the relational matrix RΛ

Cartesian algebraic product is computed as follows:

= {0.80/(1,1)+ 0.60/(1,2) + 0.40/(1,3) + 0.20/(1,4) + 0.56/(2,1) + 0.42/(2,2) + 0.28/(2,3) + 0.14/(2,4) + 0.16/(3,1) + 0.12/(3,2) + 0.08/(3,3) + 0.04/(3,4)}

nal matrix is:

Chapter 6 74

x\y 1 2 3 4 1 0.8 0.6 0.4 0.2 2 0.56 0.42 0.28 0.14 3 0.16 0.12 0.08 0.04

which is shown graphically in Figure 6.2. It is observed that the graphi-cal representations of the relational matrices of the two Cartesian prod-ucts have similarities. The Cartesian product based on the conjunctive operator min is much simpler and more efficient to implement computa-tionally and is therefore generally preferred in fuzzy controller inference engines. Most commercially available fuzzy controllers in fact use this method.

y

Figure 6.2 Grap 6.1 The Fuzzy The membership funterms of the individuaΑ and Β in different w

µR(x,y)

x

R*

hical representation of the relational matrix R*

Algorithm

ction that specifies fuzzy implication is given in l membership functions µΑ(x) and µΒ(y) of the sets ays, as described below. Assume that

= ψ(µΑ(x), µB(y)) for x∈ X and y∈Y

Fuzzy Reasoning 75

where ψ is some implication operator and

R = {µR(x,y)/(x,y)}.

In general, if Α1, Α2, ... ΑΝ are fuzzy sub-sets of Χ and Β1, Β2, ... ΒΝ are sub-sets of Υ (corresponding to the antecedents or causes and the conse-quents or effects respectively), then the fuzzy algorithm is defined as the set of rules:

RΝ : IF Α1 THEN Β1

OR IF Α2 THEN Β2

OR ...............................

IF ΑΝ THEN ΒΝ

This form is typically used in fuzzy control and is identical to the manner and terms in which human operators think.

The connective OR, abbreviated as φ, depends on the fuzzy im-plication operator ψ. Thus the membership function for N rules in a fuzzy algorithm is given by:

µRΝ(x,y) = φ(µR

1(x,y), µR2(x,y) .... )

= φ(ψ(µΑ1(x) µB

1(y)), ψ(µΑ2(x) µB2(y))....)

The foregoing relations apply to simple variables Α and Β. In

general, the part of the conditional statement of the form “IF … THEN … ELSE” involves more than one variable and can be expressed as a se-ries of nested statements of the form:

IF Α1 THEN (IF Α2 THEN .... (IF ΑN THEN Β))

or as a statement where the antecedents are related through the connec-tive AND, i.e.,

IF (Α1 AND Α2 AND .... ΑN) THEN Β

Chapter 6 76

whereupon:

µR(x1,x2,.....xN) = φ{µΑ1(x1),φ(µΑ2(x2)),...φ(φ(µΑN(xN),φ(µB(y))))} for x1,x2....xn ∈ X1, X2 ... XN, y∈Y or

µR(x1,x2,.....xn) = φ{µΑ1(x1)∧ µΑ2(x2) .... ∧( µΑN(xN), µB(y))}

= φ{∧k (µΑk(xk) µB(y))} for k=1,2..N

6.2 Fuzzy Reasoning As noted earlier, the knowledge necessary to control a plant is usually expressed as a set of linguistic rules of the form ‘IF (cause) THEΝ (ef-fect)’. These are the rules with which new operators are trained to control a plant and they constitute the knowledge base of the system. In practice, it is possible that not all the rules necessary to control a plant have been elicited, or are known. It is therefore essential to use some technique ca-pable of inferring the control action in this case, from available rules.

In classical propositional calculus the conditional statement (or rule):

IF Α THEN Β that can be expressed symbolically as:

Α → Β

is equivalent to the operation A∨ B, where Α and Β are sub-sets of the universes of discourse Χ and Υ respectively and A = NOT A. In fuzzy logic and approximate reasoning there are two fuzzy implication infer-ence rules:

Fuzzy Reasoning 77

6.2.1 Generalized Modus Ponens (GMP) In fuzzy logic there exist two principal categories of inferences. The first is Generalized Modus Ponens (or GMP) defined as follows: GMP: Premise 1: x is Α’ Premise 2: IF x is Α THEN y is Β Consequence: y is Β’

GMP is related to forward data-driven inference, which finds use in all fuzzy controllers. The objective here is to infer the effect given the cause. For the special case where Α’=Α and Β’=Β then GMP reduces to Modus Ponens. 6.2.2 Generalized Modus Tollens (GMT) The second category is Generalized Modus Tollens (or GMT) for which the following holds: GMT: Premise 1: y is B’ Premise 2: IF x is Α THEN y is Β Consequence: x is A’

GMT is directly related to backwards goal-driven inference mechanisms, which find application in expert systems. In contrast to GMP, the objective here is to infer the cause that leads to a particular ef-fect. For the special case where Α’ = A and Β’ = B then GMT reduces to Modus Tollens.

The most common inference relations, with which fuzzy infer-ence engines can be constructed, are given in the following. It will ob-served that they have differing degrees of complexity and simplicity of use and only the last two have found extensive application in practice.

Chapter 6 78

6.2.3 Boolean implication The classical binary implication due to Boole uses the conjunctive union operator in addition to negation and is defined as:

RΒoole = A × B = ( A × Y) ∪ (X × B) and

µR(x,y) = (1 - µΑ(x)) ∨ µΒ(y)

For the case of Ν rules, use is made of the connective AND in which case:

RN = ∧k Rk where k=1,2....Ν and

µRN(x,y) = ∧k (((1- µΑk(x))∨ µΒk(y)))

6.2.4 Lukasiewicz implication The Lukasiewicz implication (also known as Zadeh´s arithmetic rule of fuzzy implication) is chronologically one of the first and is based on multi-valued logic. It is similar to Boole´s implication intersection is re-placed by simple arithmetic addition but in this case, i.e.,

RL = ( A × Y) ⊕ (X × B)

and

µR(x,y) = 1 ∧ (1 - µΑ(x) + µΒ(y)) Likewise, the implication for the case of Ν rules use is made of the con-nective AND is given by:

RN = ∧k Rk where k=1,2....Ν and

µRΝ(x,y) = ∧k (1 ∧ (1 - µΑk(x)+ µΒk(y)))

Fuzzy Reasoning 79

6.2.5 Zadeh implication The Zadeh max-min implication involves the max and min operators and is defined as:

RZadeh2 = (A × B) ∪ ( A × X) and

µR(x,y) = (µΑ(x)∧ µΒ(y)) ∨ (1 - µΑ(x))

Zadeh´s fuzzy implication rule is difficult to apply in practice and it took several years before Mamdani proposed a simplification that made it especially useful for control applications. 6.2.6 Mamdani implication The Mamdani implication rule is a simplification of the implication pro-posed by Zadeh and uses only the min operator and is defined as:

RΜamdani = A × B and

µR(x,y) = µΑ(x) ∧ µΒ(y) = min(µΑ(x), µΒ(y))

For a fuzzy algorithm comprising N rules use is made of the con-nective OR, in which case the implication is:

RN = ∨k Rk where k=1,2....Ν and

µRN(x,y) = ∨k (µΑk(x)∧ µΒk(y))

Chapter 6 80

6.2.7 Larsen implication Finally, the Larsen implication rule uses arithmetic multiplication in the computation of the Cartesian product and is defined as:

RLarsen = A × B

and

µR(x,y) = µΑ(x) • µΒ(y)

For a fuzzy algorithm comprising N rules, use is made of the connective OR, i.e.,

RN = ∨k Rk where k=1,2....Ν and

µRΝ(x,y) = ∨k(µΑk(x) • µΒk(y)).

Both Mamdani´s and Larsen´s implications have found

extensive application in practical control engineering due to their compu-tational simplicity. Nearly all industrial fuzzy controllers use one or the other of these two fuzzy implications in their inference engines. Mam-dani´s implication, being computationally faster, is found most often. 6.2.8 GMP implication For completeness, it is useful to revisit Generalized Modus Ponens with a view to comparing it with the implication rules previously defined. Here

RGMP = A × Y → X × B

whereupon using the bounded product

Fuzzy Reasoning 81

1 if µΑ(x)≤ µΒ(y)

µR(x,y) = µΑ(x) > µΒ(y) = 0 if µΑ(x) > µΒ(y) Alternatively, using the algebraic product 1 if µΑ(x) ≤ µΒ(y)

µR(x,y) = µΑ(x) > µΒ(y) = 6.3 The Compositional Rules

Fuzzy controllers invariably involve numeroutherefore, to establish suitable mechanisms wirules in order to arrive at some consequent. Giquential fuzzy conditional rules:

R1 : IF Α THEN Β R2 : IF Β THEN C

it is possible to combine the two rules into a siintermediate result B and find the relationshipthe ultimate consequent directly, i.e.,

R 12 : IF A THEN C The composition of these two rules into one can

R12 = R1 o R2

µΑ(x) µΒ(y)

of In

s rules th whicven, fo

ngle rul betwe

be exp

if µA(y) > µΒ(y)

ference

and it is essential, h to process these r instance, two se-

e by absorbing the en antecedent and

ressed as:

Chapter 6 82

where o implies rule composition. In terms of the max-min operators used by Mamdani, the membership function of the resultant composi-tional rule of inference is:

µR12(x,z) = ∨y(µR

1(x,y) ∧ µR2(y,z))

= max (min (µR

1(x,y), µR2(y,z))

and in the case of the Larsen implication rule using the max-product op-erators:

µR12(x,z) = ∨y(µR

1(x,y) • µR2(y,z))

= max (µR

1(x,y) • µR2(y,z))

When discrete membership functions are used, the compositional

rule of inference in the case of Mamdani implication is analogous to the inner product of two matrices in which multiplication and addition are replaced by the min and max operators, respectively. For Larsen implica-tion, addition is replaced by the max operator while multiplication is arithmetic.

In the following, we discuss the procedure for determining the consequent (or effect), given the antecedent (or cause). Given

Α = {µΑ(x)/x} for x ∈ X Β = {µΒ(y)/y} for y ∈ Y

and the compositional rule of inference:

R = {µR(x,y)/(x,y)} for x ∈ Χ and y ∈ Y

We wish to infer the consequent if the antecedent is modified slightly to Α’, i.e.,

Α’ = {µΑ’(x)/x} for x ∈ X

Fuzzy Reasoning 83

Making use of the fuzzy compositional rule of inference using the max-min operators for instance: Β’ = Α’ o R

= ∨x(µA’(x)/x ∧ µR(x,y)/(x,y)) for x ∈ X and y ∈ Y or using the max-product operators: Β’ = Α’ o R

= ∨x(µA’(x)/x • µR(x,y)/(x,y)) for x ∈ X and y ∈ Y

By way of example consider the rule:

‘IF Α is Slow THEN Β is Fast’

given the fuzzy sets for Slow and Fast are given by the discrete member-ship functions:

µΑ(x) = {1 + 0.7 + 0.3 + 0 + 0 + 0} µΒ(y) = {0 + 0 + 0.3 + 0.7 + 1 + 1}

µΑ

x

µΒ

y

Figure 6.3 The fuzzy sets ‘Slow’ and ‘Fast’ in the example on the universes of discourse X,Υ = {0,1,2,3,4,5,6}. The discrete mem-bership functions are shown in Figure 6.3. We wish to determine the out-come if A = ‘slightly Slow’ for which there no rule exists.

Chapter 6 84

The procedure is straightforward, though tedious. The first step is to compute the Cartesian product and using the min operator this is simply:

RΛ = Α × Β = {min[µΑ(xi ), µB(yj)]} =

min[1, 0] min[1, 0] min[1, 0.3] min[1, 0.7] min[1, 1] min[1, 1] min[0.7, 0] min[0.7, 0] min[0.7, 0.3] min[0.7, 0.7] min[0.7, 1] min[0.7, 1] min[0.3, 0] min[0.3, 0] min[0.3, 0.3] min[0.3, 0.7] min[0.3, 1] min[0.3, 1] min[0, 0] min[0, 0] min[0, 0.3] min[0, 0.7] min[0, 1] min[0, 1] min[0, 0] min[0, 0] min[0, 0.3] min[0, 0.7] min[0, 1] min[0, 1] min[0, 0] min[0, 0] min[0, 0.3] min[0, 0.7] min[0, 1] min[0, 1]

0 0 0.3 0.7 1 1 0 0 0.3 0.7 0.7 0.7 = 0 0 0.3 0.3 0.3 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Thus if the antecedent A is modified somewhat, by displacing

the elements of the discrete fuzzy set one place to the left to represent the fuzzy set A = ‘slightly Slow’ then the original discrete membership func-tion:

µΑ(x) = {1 + 0.7 + 0.3 + 0 + 0 + 0} becomes

µ Α’ = {0.3 + 0.7 + 1 + 0.7 + 0.3 + 0} which is shown in Figure 6.4.

Using the fuzzy compositional inference rule:

Β’ = Α’ o R and the max-min operators (i.e., the Mamdani compositional rule):

µΒ’(y) = max (min (µA’(x), µRΛ(y)))

Fuzzy Reasoning 85

µΑ'

x

µΒ'

y

Figure 6.4 The compositional inference rule using the max-min operators then the discrete membership function of the new consequent Β’ can be readily computed. The relational matrix {min(µΑ’(x),µRΛ(y))} contains elements of the matrix RΛ and the discrete membership function µΑ’(x) and is given by:

µΒ’(y) =

min(0, 0.3) min( 0, 0.3) min(0.3, 0.3) min(0.7, 0.3) min(1, 0.3) min(1, 0.3) min(0, 0.7) min(0, 0.7) min(0.3, 0.7) min(0.7, 0.7) min(0.7, 0.7) min(0.7, 0.7) min(0, 1) min(0, 1) min(0.3, 1) min(0.3, 1) min(0.3, 1) min(0.3, 1) min(0, 0.7) min(0, 0.7) min(0, 0.7) min(0, 0.7) min(0, 0.7) min(0, 0.7) min(0, 0.3) min(0, 0.3) min(0, 0.3) min(0, 0.3) min(0, 0.3) min(0, 0.3) min(0, 0) min(0, 0) min(0, 0) min(0, 0) min(0, 0) min(0, 0)

0 0 0.3 0.3 0.3 0.3 0 0 0.3 0.7 0.7 0.7 = 0 0 0.3 0.3 0.3 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The final operation in determining the membership function

µΒ’(y) of the new consequence is selection of the largest element in each column, which is equivalent to applying the max operator in each col-umn. These are shown in bold. The result of the procedure is shown in Figure 6.4 and is:

µΒ’(y) = {0 + 0 + 0.3 + 0.7 + 0.7 + 0.7}

Chapter 6 86

It is noted that the displacement of the membership function for the new condition must be small, otherwise all the elements of the rela-tional matrix will be zero and no conclusion can be drawn. By way of comparison, the corresponding result using the max-product rule of com-positional inference:

µΒ ’(y) = max (µΒ ’(x) • µRΛ(y))

has a relational matrix:

R* = Α × Β = {µΑ(xi) • µB(yj)} =

[1* 0] [1* 0] [1* 0.3] [1* 0.7] [1* 1] [1* 1] [0.7* 0] [0.7* 0] [0.7*0.3] [0.7*0.7] [0.7* 1] [0.7* 1] [0.3* 0] [0.3* 0] [0.3*0.3] [0.3* 0.7] [0.3* 1] [0.3* 1] [0* 0] [0* 0] [0* 0.3] [0* 0.7] [0* 1] [0* 1] [0* 0] [0* 0] [0* 0.3] [0* 0.7] [0* 1] [0* 1] [0* 0] [0* 0] [0* 0.3] [0* 0.7] [0* 1] [0* 1]

0 0 0.3 0.7 1 1 0 0 0.21 0.49 0.7 0.7 = 0 0 0.09 0.21 0.3 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

whereupon

{µΑ’(xi ) • µB(yj)} =

(0* 0.3) (0* 0.3) (0.3*0.3) (0.7*0.3) (1* 0.3) (1*0.3) (0* 0.7) (0* 0.7) (0.21*0.7) (0.49*0.7) (0.7*0.7) (0.7*0.7) (0* 1) (0* 1) (0.09*1) (0.21* 1) (0.3* 1) (0.3* 1) (0* 0.7) (0* 0.7) (0* 0.7) (0* 0.7) (0* 0.7) (0* 0.7) (0* 0.3) (0* 0.3) (0* 0.3) (0* 0.3) (0* 0.3) (0* 0.3) (0* 0) (0* 0) (0* 0) (0* 0) (0* 0) (0* 0)

Fuzzy Reasoning 87

0 0 0.09 0.21 0.3 0.3 0 0 0.15 0.35 0.49 0.49 = 0 0 0.09 0.21 0.3 0.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

The maximum elements of each column are therefore:

µΒ’(y) = {0 + 0 + 0.15 + 0.35 + 0.49 + 0.49}

which are shown graphically in Figure 6.5.

µΑ'

x

µΒ'

y

Figure 6.5 Compositional inference using the max-product

In general, industrial fuzzy controllers are multivariable, involv-ing m inputs and p outputs. It is, however, simpler to think in terms of p parallel fuzzy controllers each with one output only. In practice this is achieved by multiplexing a multi-input single output controller since the precedents are the same and only the antecedents change in each case. For the case of a fuzzy controller with m inputs and one output only:

RN = {µRN(x1,x2....xm,y)/(x1,x2....xm,y)} for xk ∈ Xk and y∈Y. Now, given

Α’k = {µΑ’k(xk)/xk} for xk ∈ Xk, k=1,2...m

Chapter 6 88

the consequent Β’ is given by the relationship:

Β’ = (∧k A’k) o RN for k=1,2...m = {µΒ’(y)/y} for y∈Y where

µΒ’(y) = ∨ ∨ . . . ∨ [∧k (µΑ’k(xk) ∧ µRN(x1,x2....xm))] x1 x2 xm for k=1,2...m.

Finally, for completeness, if the max-product is used in the com-positional inference rule, the corresponding expressions for the resultant consequence is given by:

µΒ’(y) = ∨ ∨ . . . ∨ [Πk µΑ’k(xk) • µRN(x1,x2....xm)] x1 x2 xm for k=1,2...m where now

µRN(x1,x2....xm) = ∨j Πk [( µAkj(xk) • µB

j(y)] for k=1,2...m and j=1,2...n.

Chapter 7 The Fuzzy Control Algorithm Industrial processes are invariably multi-variable and considerable re-search has gone into developing conventional analytical techniques for multi-variable plants both in the time and frequency domains. All of these techniques assume the plant is linear, but with rare exceptions, these techniques have not found their way into industry because of their complexity and restrictions. There is no comparable theory for non-linear processes and thus industry has had to be content with various configurations of conventional industrial three-term controllers. It is ob-vious that this arrangement leaves much to be desired, leaving a gap be-tween control theory and practice which has been waiting to be bridged. Soft control is proving to be this very bridge.

Fuzzy control is by nature eminently suited to multi-variable and non-linear processes. Fuzzy linguistic rules required to control a multi-variable plant have already been considered in earlier chapters. It was observed there that whereas the antecedents of each rule differ for each output variable, the precedents are identical. This suggests that a multi-variable fuzzy controller can conveniently be decomposed into a set of single-output controllers equal to the number of outputs.

In practice, the measurements of plant variables even if contami-nated by noise and the control actions to the plant actuators are crisp. In order to apply the fuzzy algorithm it is therefore necessary first to fuzzify

89

Chapter 7 90

the measured plant variables and following completion of the fuzzy algo-rithm to de-fuzzify the result and thereby return to the engineering world. This chapter discusses the procedure of fuzzification and de-fuzzification as they apply to practical control. 7.1 Controller Decomposition The decomposition of a fuzzy multi-input multi-output (MIMO) control-ler into a set of multi-input single-output (MISO) fuzzy controllers (FC1, FC2 … FCn) as shown in Figure 7.1. The outputs of a multi-variable fuzzy controller can be computed in parallel in a multi-processor or se-quentially by multiplexing the MISO controllers when computational times are not critical.

FC1

FC2

FCn

FC3

x y1

y2

y3

yn

Figure 7.1 Decomposition of a multi-input multi-output incremental fuzzy controller into a set of multi-input single-output incremental controllers

The Fuzzy Control Algorithm 91

7.2 Fuzzification The algorithm for computing the crisp output of a fuzzy controller in-volves the following three steps:

(1) fuzzification (2) inference (3) de-fuzzification

To make these steps easier to understand, consider a fuzzy con-

troller with three inputs and a single output only. It should be obvious that the procedure that follows can be generalized for any number of in-puts. Given a MISO controller with inputs x1, x2, x3 and output y and as-suming that the linguistic control rules are of the form:

IF (Α1 AND Α2 AND Α3) THEN Β

then the membership function of the output of the controller is given by:

µR(x1,x2,x3) = φ(µΑ1(x1) ∧ µΑ2(x2) ∧ (µΑ3(xN), µB(y)))

= φ(∧k (µΑj(xk), µB(y))) for j=1,2,3 where the operator φ implies max-min or max-product. Using the inter-section operator, the degree of fulfillment of the j-th rule σj(k)∈[0,1] is defined by:

σj(k) = µΑ1(x1) ∧ µΑ2(x2) .... ∧ µΑN(xN)

and is computed every time the algorithm is executed. The degrees of fulfillment are thus a measure of how closely the inputs to the controller match the control rules. They can be viewed conveniently as weights that are assigned to every rule. In practice only a small number of the rules in the rule-base (typically less than 6) will exhibit non-zero degrees of fulfillment at any instant.

The expression for the degree of fulfillment given above applies when the Mamdani max-min implication rule is used. For the case of

Chapter 7 92

max-product implication, the corresponding expression for the degree of fulfillment is:

σj(k) = µΑ1(x1) • µΑ2(x2) .... • µΑN(xN)

The fuzzy implication rules yield the membership function of the output of the controller from knowledge of the current instantaneous measurements of the inputs to the controller x1(k), x2(k), x3(k). Thus at any instant k the membership function of the output of the controller us-ing the max-min fuzzy implication rule is:

Y(k) = (Χ1(k) ∧ Χ2(k) ∧ Χ3(k)) o RN

or, the max-product implication rule:

Y(k) = (Χ1(k) • Χ2(k) • Χ3(k)) o RN

where X1, X2 and X3 are the corresponding fuzzy sets of the controller in-puts. These computations are simplified significantly if the fuzzy sets of the inputs to the controller are taken as singletons defined as:

µΧi(k) = 1 if xi =xi(k)

= 0 otherwise whereupon:

µY(y) = ∨x1∨x2∨x3(1∧1∧1∧µRN(x1,x2,x3,y))

= µRN(x1,x2,x3,y)

Example 7.1 illustrates the procedure.

The Fuzzy Control Algorithm 93

Example 7.1 Graphical interpretation of

fuzzification The various operations required to establish the fuzzy set of the output of a fuzzy controller are illustrated graphically in this example. For simplic-ity assume that the controller has two inputs and a single output. Assume that the first input to the controller Input_1 (x1) is specified by 5 fuzzy sets, while Input_2 (x2) is specified by 3 fuzzy sets. The linguistic variables are assumed to be VL=Very_Low, LO=LOw, ZO=ZerO, LH=Little_High, MH=Medium_High, and VH=Very_High.

Assume that the following 15 control rules constitute the rule base:

R1: If Input_1 is LO and Input_2 is VL then Output is LO R2: If Input_1 is ZO and Input_2 is VL then Output is ZO R3: If Input_1 is LH and Input_2 is VL then Output is LH R4: If Input_1 is MH and Input_2 is VL then Output is LH R5: If Input_1 is VH and Input_2 is VL then Output is LH R6: If Input_1 is LO and Input_2 is ZO then Output is LO R7: If Input_1 is ZO and Input_2 is ZO then Output is LH R8: If Input_1 is LH and Input_2 is ZO then Output is MH R9: If Input_1 is MH and Input_2 is ZO then Output is MH R10: If Input_1 is VH and Input_2 is ZO then Output is VH R11: If Input_1 is LO and Input_2 is VH then Output is LH R12: If Input_1 is ZO and Input_2 is VH then Output is MH R13: If Input_1 is LH and Input_2 is VH then Output is MH R14: If Input_1 is MH and Input_2 is VH then Output is VH R15: If Input_1 is VH and Input_2 is VH then Output is VH

that can de depicted more compactly by means of the rule matrix:

x2\x1 LO ZO LH MH VH VL LO ZO LH LH LH ZO ZO LH MH MH VH VH LH MH MH VH VH

For simplicity, assume, furthermore that the fuzzy sets of the inputs and outputs are triangular and are as shown in Figure 7.2.

Chapter 7 94

The universes of discourse of Input_1 and Input_2 are as-sumed symmetric and are expressed as percentages of their maximum permissible values. Output y involves 5 fuzzy sets and is assumed asymmetric. Thus the inputs to the controller can take any value between ±100% of their maximum permissible values while the output can take any value between 0 and 100% of its maximum permissible value. For example, the Output y could represent the opening of a servo-valve, while Input_1 could be a pressure deviation and Input_2 could be the temperature deviation about their nominal values.

-100% 100%

LO ZO LH MH VH1

Input_1

-100% 100%

VL ZO VH1

Input_2

LO ZO LH MH VH 1

Output

100 %0

Figure 7.2 Fuzzy sets of inputs and outputs

The first five rules in the rule base are depicted graphically in Figure 7.3. Assume, furthermore, that at the instant of execution of the algorithm, the instantaneous inputs to the controller are -20% and -50% respec-tively.

The Fuzzy Control Algorithm 95

Every rule is now examined with a view to determine the degree to which it contributes to the final decision. This measure is termed the de-gree of fulfillment σj. The computational time required for this determina-tion is clearly dependent on the number of rules in the rule base. Fortu-nately, rarely are more than 20 to 50 rules required in practice and con-sequently computational time is minimal.

100%

LO ZO LH MH VH1

Input_1

-100% 100%

VL ZO VH1

Input_2

LO ZO LH MH VH 1

Output

0 100 %

R 1

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R 2

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R 3

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R4

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R 5

-100%

-100%

-100%

-100%

-100%

x (k) x (k)1 2

Figure 7.3 Graphical representation of first five rules in the rule base

For the given values of the inputs, it is clear that rules R1, R4 and R5 have no part in the final decision (and consequently the output) since these rules have not fired. The degrees of fulfillment of the non-fired rules are consequently zero. The intercepts of the vertical lines corre-sponding to the instantaneous value of Input_1 and Input_2 and the corresponding fuzzy sets specifying the membership value are:

Chapter 7 96

The Fuzzy Control Algorithm 97

µ1

1=0, µ21=0.66, µ3

1=0.33, µ41=0, µ5

1=0

and

µ12=0.5, µ2

2=0.5, µ32=0.5, µ4

2=0.5, µ52=0.5

respectively. The degree of fulfillment σj for every rule is computed from the membership values and the operation min(µj

1,µj2). Here

R1 : σ1= min(µ1

1,µ12)=min(0, 0.5)=0

R2 : σ2= min(µ21,µ2

2)=min(0.66, 0.5)=0.5 R3 : σ3= min(µ3

1,µ32)=min(0.33, 0.5)=0.33

R4 : σ4= min(µ41,µ4

2)=min(0, 0.5)=0 R5 : σ5= min(µ5

1,µ52)=min(0, 0.5)=0

etc.

7.2.1 Steps in the fuzzification algorithm The first step in the fuzzy controller algorithm is determination of the minimum intercepts for each input, i.e., their membership. If any rule has zero intercept then it is discarded in subsequent computations, as it does not contribute to the final decision.

The second step involves determination of the degrees of fulfill-ment of every rule. Using the min operator, this is equivalent to scanning the intercepts of every rule horizontally. Should any intercept be zero then this rule clearly does not contribute to the final conclusion.

The third involves determination of the composite membership function of the output of the controller. This operation is dependent on the choice of fuzzy implication rule. Thus in the case of Larsen implica-tion, the membership function of the output of the controller is the union of the individual membership functions for each rule that has fired weighted with the corresponding degree of fulfillment, i.e.,

µ(y) = σ1µ1(y) ∨ σ2µ2(y) ∨ σ3µ3(y) ∨ σ4µ4(y) ∨ σ5µ5(y)

= σ2µ2(y) ∨ σ3µ3(y)

Chapter 7 98

This is shown in graphical form in Figure 7.4. The resultant

composite membership function of the output of the controller is shown in the lower right hand side diagram of Figure 7.5. Using Mamdani im-plication, the composite membership function of the controller output is the union of the maximum values of the weighted membership functions of each rule that has fired, i.e.,

µ(y)= max (σ1,µ1(y))∨max(σ2,µ2(y))∨max(σ3,µ3(y)) ∨max(σ4 ,µ4(y))∨max(σ5, µ5(y)))

= max (σ2,µ2(y))∨max(σ3,µ3(y))

This procedure is shown in Figure 7.5 for comparison.

Input_1 Input_2 Output

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R 2

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R 3

-100%

-100%

x (k) x (k)1 2 1

0 100 %

µ(y)

Figure 7.4 Determination of the composite membership function of the controller output using the Larsen implication

The Fuzzy Control Algorithm 99

7.3 De-fuzzification of the Composite Controller Output Fuzzy Set

The final step in the fuzzy controller design procedure is de-fuzzification of the fuzzy set of the output to yield a single crisp value that uniquely specifies the desired control action. It must be noted that there is no theo-retical basis for deciding which is the best manner to perform de-fuzzification and there exist a number of schemes, each of which pre-sents a different degree of computational complexity. Simplicity and speed of computation are invariably the primary requirements in indus-trial controllers. Below, we note those de-fuzzification techniques that have found application in practical control applications:

Input_1 Input_2 Output

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R 2

100%

LO ZO LH MH VH1

-100% 100%

VL ZO VH1

LO ZO LH MH VH 1

0 100 %

R 3

-100%

-100%

x (k) x (k)1 2 1

0 100 %

µ(y)

Figure 7.5 Determination of the fuzzy set of the controller output using the Mamdani implication

7.3.1 Center of area (COA) de-fuzzification In this method the center of the area under the composite membership function of the output of the controller µY(y) is taken as the final output of the controller:

Chapter 7 100

∫

∫=

SΥ

SΥ

COA dyyµ

dyyµyy

)(

)(ˆ

where S is the support set of µY(y). In the case where the composite mem-bership function is discrete with I elements, this becomes:

∑

∑

=

== I

1iiΥ

I

1iiΥi

COA

yµ

yµyy

)(

)(ˆ

This method is sensitive to changes in the shape of the membership func-tions of the output. Because it yields intuitive results, this method has found extensive use in practical fuzzy control. 7.3.2 Center of gravity (COG) de-fuzzification In this method, the centers of gravity of the individual components of the composite fuzzy set for the rules that have fired, are combined by weighting them in accordance with their degrees of fulfillment. The out-come is thus:

∑

∑

=

== ƒ

1ii

iΥ

I

1ii

iΥi

COG

yµ

yµσy

)(

)(ˆ

On concluding de-fuzzification, the crisp output of the controller is ap-plied to the plant actuators.

The Fuzzy Control Algorithm 101

7.4 Design Considerations The principal factors that must be considered prior to implementation of the fuzzy control algorithm are the following:

7.4.1 Shape of the fuzzy sets

In the continuous case, the fuzzy sets are uniquely defined by some ana-lytic function while in the discrete case the fuzzy sets can be defined by arrays whose size depends on the number of quantization levels. The shape of the fuzzy sets used in the design of a fuzzy controller has been the subject of considerable research and it is fair to state that at this time there is no theory which can guide the designer on the best shape to use for a specific application. Experience in similar controllers and computa-tional ease appear to be the basic criteria in selecting the shapes of mem-bership functions. In practice it appears triangular and trapezoidal func-tions are generally used, though Gaussian-like functions are used by a number of vendors of fuzzy controllers for the process industry.

It has been claimed that the output of the controller is rather in-sensitive to the shape of the membership function. Comparison of the performance of a simple control system to changes in the membership functions of both the inputs and outputs of the fuzzy controller is made in Appendix A. 7.4.2 Coarseness of the fuzzy sets The number of fuzzy sets that are required to specify a variable is termed the coarseness of the controller and determines the accuracy of the con-troller. High accuracy requires a large number of fuzzy sets and corre-sponding memory requirements. In some cases it is desirable to use two or more levels of coarseness to control a process. Thus when the process variables are at some distance from the desired operating point, coarse control is applied through the use of very few fuzzy sets. As the trajec-tory of the process approaches the operating point, the number of fuzzy sets is increased. This results in finer control with increased accuracy. This technique is not unlike coarse-fine control used in classical control. An example of a pair of fuzzy sets for coarse-fine control is shown in Figure 7.6. Here, three fuzzy sets (NE – Negative, ZE - Zero, and PO -

Chapter 7 102

POsitive) are used for coarse control and five for fine control. This tech-nique has been applied with success in a number of processes requiring high terminal accuracy.

(a)

(b)

NE ZE PO

NB NSNM ZE PS PM PB

Figure 7.6 Coarse-fine fuzzy sets 7.4.3 Completeness of the fuzzy sets

The fuzzy control algorithm must lead to a unique control action for any set of inputs. This property is termed completeness and depends on the contents of the knowledge-base as well as the number and shape of the fuzzy sets used to describe the inputs and outputs of the controller.

The manner in which the fuzzy sets are defined on the universe of discourse, as well as the degree with which they overlap specifies the integrity of the controller, i.e. its ability to infer a plausible outcome un-der all circumstances. In the example of Figure 7.6, the overlap, defined as the intersection of the fuzzy sets, is at least 50%. In this case there will always be a dominant rule which has membership in excess of 0.5 so that an outcome will always be forthcoming. In the worst case, two rules at most will fire with an equal membership of 0.5 but still there will be no ambiguity as to the final result.

In contrast, the fuzzy sets in Figure 7.7 possess points on the universe of discourse where the intersection is less than 0.5. This leads to highly uneven control surfaces and irregular control actions. Indeed there

The Fuzzy Control Algorithm 103

are regions on the universe of discourse where membership is zero whereupon if the instantaneous value of the input falls in these regions no rule can be fired with the result that the controller is unable to infer any control action. This is clearly undesirable and indicates that fuzzy sets must overlap in order to obtain a continuous output.

ZE PS PM PB

Figure 7.7 Fuzzy set overlap 7.4.4 Rule conflict

The set of linguistic rules used to control a plant or process is normally elicited from expert human operators or domain experts. It is an undis-puted fact that the knowledge elicited from two human operators is rarely the same. Though they may have been trained with the same rules, with experience and time they have learned to modify them, believing that in this manner they can control the process better. Of course “better” is clearly a subjective criterion, since it may imply increased productiv-ity, reduced energy costs or even less trouble to the operator. Many of these criteria are conflicting.

Thus in eliciting the knowledge for a controller it is advisable to restrict interviews to one human operator, e.g., the supervisor whose knowledge on how to control the plant efficiently and effectively is un-disputed. If this is not possible, then there is little recourse but for the plant engineer to state the rules that he wishes to be followed. Even so, rule conflict is a common phenomenon and some means must be found for resolving this conflict. By simply writing the rules in sequential order it is virtually impossible to do so, however. Graphical means are very ef-fective when the number of inputs to the controller are three or less

Chapter 7 104

whereupon rules can be represented in the form of tiles whose colors specify the control action required. This is none other than the Fuzzy As-sociative Memory or FAM and Figure 7.8 shows an example of this sim-ple technique that has proved very useful in practice. The human eye is an excellent detector of abnormal color changes and thus by simply look-ing at the manner in which the colors of the tiles vary in control space it is possible to identify possible conflict.

Figure7.8 Representation of the knowledge base in tile form or Fuzzy Associative Memory (FAM)

Chapter 8 Fuzzy Industrial Controllers The industrial three-term controller without doubt constitutes the back-bone of industrial control, having been in use for over a century. Three- term controllers can take a number of forms, from the early mechanical to later hydraulic, pneumatic, analog and digital versions. The modern form takes the form of multi-tasking discrete-time three-term algorithms embedded in almost all industrial PLCs and RTUs. Undisputedly, indus-trial progress would have been greatly limited but for these ubiquitous controllers. To this very day, the majority of industrial plants rely almost exclusively on this inexpensive, robust and easy to use conventional con-troller.

The output of a conventional industrial controller normally in-volves three-terms: the first is proportional to its input (the P term), the second is proportional to the integral of the input (the I term) and the third is proportional to the derivative of the input (the D term). In most practical applications, the first two terms are sufficient and only a small fraction of industrial controllers make use of the derivative term. Three-term controllers can be configured in a variety of ways, from the sim-plest autonomous single-loop controller to cascade control when a single controller is insufficient to provide the necessary control due to the inter-actions in the controlled variables of the plant. Three-term controllers can also be configured so as to provide ratio or blending control when

105

Chapter 8 106

quantities must be maintained as percentages, typically found in materi-als blending and mixing processes.

Productivity is closely related to the quality of control. Low quality of control implies poor product quality with products that cannot meet standards, reduced productivity, loss of competitiveness and ulti-mately the collapse of the manufacturer. Effective control is thus of vital importance where high product quality and productivity are essential, high standards are to be maintained and market share assured. Effective operation of a plant implies correct tuning of the controllers to meet the product specifications while the efficiency of a plant is critically depend-ent on specifying the correct parameters of these controllers.

Traditionally, a three-term controllers are tuned on-line by hu-man experts who excite the plant by injecting appropriate disturbances to the set-points and then systematically adjust the parameters (i.e., gain constants) of the controller until the plant meets its design specifications. Re-tuning is normally necessary whenever the operating conditions of the plant are changed. Controller tuning requires considerable expertise and patience and takes time to master. The way in which a human tunes a control loop or plant is based on heuristic rules which involve such fac-tors as the rise time, settling time and steady state error of the closed sys-tem. Indeed, as will be seen later in this chapter, these rules are put to good use in the design of expert controller tuners, which a number of vendors offer today. 8.1 Controller Tuning Manual tuning can be supported by simple analytical techniques, which systematically determine the best (in some sense) controller parameters based on a simple macroscopic model (or approximant) of the controlled plant. Ziegler and Nichols offered design procedures to tune industrial controllers as far back as the 1940s that are still used to this day. By ob-serving the initial slope of the response of the plant when subjected to a step disturbance and the dead time of the plant before it responds, a com-plex plant can be approximated by the classical first order lag plus dead time approximant. This simple but very approximate approach has formed the basis for improved variants on the Ziegler-Nichols approach. Modern tuning techniques such as those by Persson and Astrom require additional information from the response of the controlled plant and re-

Fuzzy Industrial Controllers 107

quire more computational effort but yield vastly improved plant re-sponses. All these techniques assume that the controlled plant is scalar, i.e., has a single input and a single output and are not applicable to multi-variable plants. Multivariable plants, for which three-term controllers do not find ready application, require an entirely different approach to con-troller design.

For best performance three-term controllers must be tuned for all operating conditions. Unfortunately, the dynamic characteristics of most industrial plants depend on their operating state and production rates and these are far from linear or stationary. A three-term controller is nor-mally tuned for best (note that use of the word optimum is tactfully avoided) performance at a specific operating state. When the operating conditions of the plant change, so does the operating state, whereupon the parameters of the controller may no longer be the best and as a con-sequence performance is degraded. The degree to which such degrada-tion is acceptable clearly depends on the nature of the controlled plant, plants that are highly nonlinear being the most difficult to control effec-tively. The robustness of a controller is a measure of its ability to operate acceptably despite changes in the operating state of the plant.

Where the variations in the plant are severe, a three-term con-troller with fixed parameters is no longer effective and alternate tech-niques, which are capable of tracking the changes in the plant must be employed. Such techniques as gain-scheduling, auto-tuning and adaptive control are commonly used to extend the domain of effectiveness and thereby the robustness of the controller. Fuzzy logic can likewise be used to extend the domain of effectiveness of a three-term controller. 8.2 Fuzzy Three-Term Controllers The advent of fuzzy control motivated many researchers to reconsider the familiar robust three-term controller in the hope that “fuzzifying” it would improve its domain of effectiveness. This is a case of a technol-ogy retrofit, in which Computational Intelligence is applied to improve a well-known device. The result has been a new generation of intelligent three-term controllers with increased robustness that a number of ven-dors currently are offering.

Fuzzy logic can be applied to three-term controllers in a number of ways. One obvious approach is to fuzzify the gains of the three-term

Chapter 8 108

controller by establishing rules whereby these gains are varied in accor-dance with the operating state of the closed system. In this case the con-troller output has the generalized form

u = = fP(e,∫edt,De) + fI(e,∫edt,De)) + fD(e,∫edt,De) which when fuzzified can be expressed as the weighted sum

u = fuzzy(kP) e + fuzzy(kI) ∫edt + fuzzy(kD) De

Hybrid fuzzy three-term controllers in which only the propor-tional and derivative terms are fuzzified while the integral term remains conventional, have also been used. In this case the controller output is

u = uPD + kI ∫edt

An alternative class of fuzzy controllers, which possess the char-acteristics of a two-term PI controller is the generic fuzzy controller which has been used extensively in practice. Generic fuzzy controllers are very simple and require few rules to operate effectively. Using the closed system error and its derivative only, this class of fuzzy controllers is normally incremental with output

Du = f(e,De)

which must subsequently be integrated (or accumulated) to generate the final controller output. 8.2.1 Generalized three-term controllers The output of a conventional three-term controller contains a term pro-portional to the error e between the desired and the actual output of the controlled plant, a term proportional to the derivative of the error De=de/dt and a term proportional to the integral of the error ∫edt. In practice it is often preferable to use the derivative and the integral of the output of the process instead of the error in order to avoid sudden changes in the output when changing the setpoint, i.e., bump-less opera-tion. If the fuzzy sets of the error, derivative and integral terms are Ε, ∆Ε

Fuzzy Industrial Controllers 109

and ΙΕ respectively, then the control rules of a generalized fuzzy three-term controller can be expressed as:

Rr: IF e is Er AND De is ∆Εr AND ∫edt is ΙΕr THEN u is Ur Further, if the union operator relates the control rules, then the fuzzy al-gorithm reduces to the fuzzy implication rule

R = R1∪R2∪.....∪Rn=∪(Er×∆Er×IEr×Ur)

The fuzzy set of the output of the generalized fuzzy three-term controller is thus given by

U = (E×∆E×IE) ο R

whose membership function is consequently

µU(u)=∨(µΕ(e)∧ µ∆Ε(De)∧ µΙΕ(∫edt)∧ µR(e,De,∫edt,u))

for e∈E, DE∈∆Ε and ∫edt∈IE. A graphical display of the parameter sur-face of such a three-term controller would have to be three-dimensional and it would be difficult to comprehend the effect of each parameter on the controller output. 8.2.2 Partitioned controller architecture It is often more convenient to partition the three-term controller into two independent fuzzy sub-controllers that separately generate the signals uPD and uI that correspond to the proportional plus derivative term and the integral term respectively. The result is a fuzzy proportional plus de-rivative sub-controller FPD in parallel with a fuzzy integral controller FPI as shown in Figure 8.1. Both sub-controllers are fed with the error and its derivative. The second sub-controller requires an integrator (or accumulator) to generate the integral term. Both this controller and the generic two-term fuzzy controller that is discussed in a later section in this chapter require the error derivative De. In situations where the signal

Chapter 8 110

contains high frequency extraneous noise, there may be certain misgiv-ings in generating the derivative or difference term from the error since noise aggravates the situation. In this case it is clear that some form of low pass filtering or signal processing is necessary to reduce the effect of high frequency noise.

Figure 8.1 Decomposition of a three-term controller into two sub-controllers The rule matrix or Fuzzy Associative Matrix for the FPD fuzzy sub-controllers is:

e\De NB NM NS ZO PS PM PB NB NB NB NB NM NM NS ZO NM NB NB NB NM NS ZO PS NS NB NM NM NS ZO PS PM ZO NB NS NS ZO PS PM PB PS NM NS ZO PS PM PB PB PM NS ZO PS PM PB PB PB PB ZO PS PM PM PB PB PB

This FAM contains 7×7=49 rules, a number that must be compared to the case of a three-term controller which would require 7×7×7=343 rules and corresponding memory storage locations in its knowledge base. It is noted that the FAM shown above is symmetric about the principal di-agonal. It is possible to take advantage of this fact and store only half the rules if memory must be conserved. Rule pruning, in which adjacent rules are systematically eliminated, can further reduce the number of

e

De

u

FPD

FPI

Integrator

uPD

uI

Fuzzy Industrial Controllers 111

rules that have to be stored to about a quarter of the original number or about a dozen.

Applying the Mamdani compositional rule and confining the controller inputs e and De to the closed interval [-3,3] and quantizing the two inputs into 7 levels, results in the relational matrix shown below. The entries in the matrix are the numerical values of uPD that are stored in the controller memory: this form of look-up table control is very simple to implement and has been applied extensively in practice. By altering some of the entries in the matrix it is possible to trim the performance of the controller further, intentionally warping the control surface in order to compensate for inherent non-linearities in the plant characteristics. In the design study presented in Appendix A, it is shown how step-response asymmetry may be compensated for using this approach.

e\De -3 -2 -1 0 1 2 3 -3 -3 -3 -3 -2 -2 -1 0 -2 -3 -3 -3 -2 -1 0 1 -1 -3 -2 -2 -1 0 1 2 0 -3 -1 -1 0 1 2 3 1 -2 -1 0 1 2 3 3 2 -1 0 1 2 3 3 3 3 0 1 2 2 3 3 3

The relational matrix is shown graphically in Figure 8.2. This is

none other than the control surface of the FPD sub-controller. Due to quantization of the controller inputs, the controller output is not continu-ous and smooth and is sometimes referred to as a multilevel relay con-troller. Clearly the control surface can be smoothed and the controller performance improved by using extrapolation techniques in which case the control actions are defined everywhere in the bounded control space.

Chapter 8 112

1 2 3 4 5 6 7

-4

-2

0

2

4

Figure 8.2 Control surface of the FPD sub-controller 8.2.3 Hybrid architectures Various hybrid architectures that combine fuzzy and deterministic ele-ments have been proposed for the three-term controller. One such archi-tecture is shown in Figure 8.3. In this architecture the P+D sub-controller FPD is identical to the one considered earlier but the integral term involves a gain coefficient which is the result of using the simple rules given below. The proportional and derivative terms are determined from the simple set of rules

e NB NM NS ZO PS PM PB uPD NB NM NS ZO PS PM PB

while the integral gain is varied according to the following rules

e NB NM NS ZO PS PM PB kI PS PM PM PB PM PM PS

uPD

De

e

Fuzzy Industrial Controllers 113

The final output of this hybrid fuzzy controller is the sum

u = uPD + uI = uPD + fuzzy(kI)∫edt

Figure 8.3 Hybrid three-term controller

Extending the method further, it is possible to design a three-term controller using the configuration shown in Figure 8.4 with each of the controller parameters (kP, kI and kD) specified by independent rules, i.e.,:

U = fuzzy(kP)e + fuzzy(kI)∫edt + fuzzy(kD)De 8.2.4 Generic two-term fuzzy controllers The phase plane, which portrays the trajectory of the error between de-sired and actual output of a process, is a useful domain in which to spec-ify the linguistic control rules of the process. Figure 8.5(a) shows a typi-cal trajectory in phase-space, analogous to state space in which one of the states is related to its derivative. Figure 8.5(b) shows the error e(t) and the corresponding rate of change of error De(t) in response to a step excitation of the plant.

e

De u FPD

uPD

uI fuzzy(kI)

Integrator Multiplier

Chapter 8 114

Figure 8.4 Fuzzy controller with independent fuzzy parameters The temporal error response to a step excitation and its derivative can be broken up into regions defined by their zero crossover points. Thus the error may be coarsely described as positive, zero at the crossover or negative. Assume, therefore, that the three fuzzy sets (POsitive, ZerO and NEgative) suffice to describe each control variable. In Region 1 we can thus write the first generic control rule:

R1: IF e is PO AND De is NE THEN u is PO The objective of this rule is to apply maximum positive control action (e.g., torque in the case of a servomotor) to the controlled process in or-der to force it to accelerate to its final value with a minimum rise time. In Region 2 the corresponding generic control rule is:

R2: IF e is ΝΕ AND De is NE THEN u is ΝΕ

Here the objective is to apply maximum negative control action to decel-erate the process in order to minimize overshoot. Using similar reason-ing, the 11 rule Fuzzy Associative Memory that follows is derived.

Integrator

e

De

u fuzzy(kI)

fuzzy(kP)

fuzzy(kD)

Σ

Fuzzy Industrial Controllers 115

Figure 8.5 (a) Phase-space trajectory and (b) time-domain response

(b)

(a)

e

De

a b c d e f g h i j k l

i ii iii iv v vi vii viii ix

Chapter 8 116

Rule e De u Points

1 PO ZO PO a,e,i 2 ZO NE NE b,f,j 3 NE ZO NE c,g,k 4 ZO PO PO d,h,l 5 ZO ZO ZO origin 6 PO NE PO i,v 7 NE NE NE ii,vi 8 NE PO NE iii,vii 9 PO PO PO iv,viii

10 PO NE ZO ix 11 NE PO ZO ix

It is obvious that finer control can be achieved if the number of

control rules is increased. The number of rules is consequently a measure of the granularity of the controller.

Rule e De u Points

1 PL ZO PL a 2 PM ZO PM e 3 PS ZO PS i 4 ZO NL NL b 5 ZO NM NM f 6 ZO NS NS j 7 NL ZO NL c 8 NM ZO NM g 9 NS ZO NS k

10 ZO PL PL d 11 ZO PM PM h 12 ZO PS PS l 13 ZO ZO ZO origin 14 PL NS PM i 15 PS NL NM i 16 NL PS NM iii 17 NS PL PM iii 18 PS NS ZO ix 19 NS PS ZO xi

Increasing the number of fuzzy sets assigned to each control

variable to 19 and using 5 fuzzy variables PL for Positive_Large, PM for Positive_Medium, PS for Positive_Small, ZO for ZerO, NS for Nega-

Fuzzy Industrial Controllers 117

tive_Small, NM for Negative_Medium and. NL for Negative_Large leads to the FAM given above.

Figure 8.7 Generic two-term incremental fuzzy controller Figure 8.7 shows a schematic of the generic two-term architecture. This incremental fuzzy controller is generic since it can be applied to any dy-namic process that exhibits under-damped behavior. Being incremental, it is necessary to supply the nominal controller output u0 to the controller in order to generate the total control variable. 8.3 Coarse-Fine Fuzzy Control When a controller is required to operate under conditions of both large and small excursions of its inputs from their nominal values, it is con-venient to use two or more sets of fuzzy rules to effect improved control. For large excursions of the controller input variables, coarse control is applied with the objective of forcing the plant to return to its nominal operating point as rapidly as possible. Accuracy of control is of secon-dary importance under these circumstances and only a few rules are re-quired. When the plant variables reach some small region about the nominal operating point then fine control is applied. Here a new set of control rules necessary to effect the desired fine control actions are used and these involve a larger number of rules and fuzzy sets. Under normal operating conditions the controller uses fine control for small excursions about the nominal operating point.

An alternative way of achieving coarse-fine control is through zooming of the universe of discourse of each controller input variable. In

Generic Fuzzy Controller

Integrator

e

De

u

u0 Du

Chapter 8 118

this case the universe of discourse is varied, either in discrete regions in control space or smoothly as the plant approaches the desired operating point (see fuzzy gain-scheduling in chapter 10). This approach has been used to great effect for the control of high precision mechatronic devices.

Chapter 9 Real-time Fuzzy Control Chronologically, fuzzy control was the first form of intelligent control. It was to radically alter the course of Control Engineering with seemingly infinite applications. Fuzzy control appears today in real-time applica-tions ranging from domestic appliances (such as air-conditioners and re-frigerators where it has become a major selling point) to high-speed trains, automobiles and process control.

Zadeh´s theory of fuzzy sets laid the foundations of Fuzzy Con-trol and the revolution in Control Engineering that continues unabated to this day. Whereas the foundations of fuzzy set theory are on solid ground, fuzzy control is still very much an art, requiring knowledge and experience to implement correctly. Soft control is not a panacea for all control problems and it would be wrong to state that it will ultimately re-place conventional control. There are numerous cases where conven-tional control results in excellent quality of control and there is no need to replace it. There are however many cases, especially in industry, that have defied solution using any of the conventional control techniques and here fuzzy control has proved invaluable, offering solutions where none existed previously.

The control strategy of an industrial plant may often be de-scribed in linguistic terms by way of control rules or control protocols that relate the control actions to the controlled variables in linguistic terms. The resultant fuzzy controller must faithfully replicate the actions of the human operator from which the control rules were elicited.

119

Chapter 9 120

As noted earlier, the rules (sometimes referred to as production rules) by which a plant can be controlled are normally expressed as a set of relational statements of the form:

IF (Current_State_of_Plant) THEN (Control_Action) These rules, which are suitably encoded, are stored in the knowledge base of the fuzzy controller and are processed by a fuzzy inference en-gine based on one of the implication functions presented in chapter 6. 9.1 Supervisory Fuzzy Controllers The elements (i.e., building blocks) of a fuzzy controller are shown schematically in Figure 9.1 and are:

• a real-time data base RTDB where the current and the previous values of the control variables of the physical plant obtained from the local RTUs are deposited (usually following some form of signal processing to remove extraneous signals). The real-time database also contains the values of the controlled variables (i.e., control actions) and the time when the previous action was taken on the plant. The real-time database is the in-terface between the physical plant and the controller,

• the knowledge base KB in which the control rules, suitably coded, are stored,

• a database FS in which the fuzzy sets of the inputs and outputs of the controller are stored,

• a fuzzifier FZ where the inputs to the controller are transformed into the domain of fuzzy sets,

• an inference engine IE, the kernel of the fuzzy controller, is software with which the fuzzy sets of the controller outputs are computed,

• a de-fuzzifier DF where the fuzzy sets of the outputs are trans-formed back into the engineering domain and the numerical control actions are deposited in the real-time data base RTDB for transmittal to the RTUs which subsequently enforce these control actions on the physical plant and, finally

Real-time Fuzzy Control 121

• a development system DS through which the engineer interacts with the fuzzy controller during the development or whenever modifications have to be made. This module is removed once the controller is finalized.

Real-time Database RTDB

Effects Causes

Inference Engine IE

Fuzzifier FZ

Knowledge Base KB

S

Development System DS

Fuzzy Controller Kernel

Figure 9.1 Elements of a fuzzy con

De-fuzzifier

Fuzzy ets FS

troller

Chapter 9 122

In modern industrial plants, the link between the supervisory control system (whether SCADA or DCS) and the physical plant being controlled is through a Local Area Network (LAN) to which remote ter-minal units (RTUs) are attached. The RTUs include analog to digital converters for the conversion of the physical variables at rates that can reach thousands of samples per seconds. These units invariably include software with which three-term controllers (PID) can be implemented, as well as software with which the RTUs can communicate with the host computer (i.e., server) via the LAN. The client/server architecture shown in Figure 9.2 and is one of the most common architectures in use today. It is noted that in the new generation of industrial control systems, smart sensors and actuators with built-in micro-controllers that can be directly connected to the LAN are gradually replacing RTUs.

Figure 9.2 Architecture of a distributed supervisory control system

Distributed supervisory control systems involve a cluster of in-dustrial grade microcomputers, RTUs, micro-controllers and peripherals connected to a LAN. Each component of the cluster performs real-time control of a specific sub-process of the plant. The host computer usually

PC PC PC

LAN

RTU RTU

Clients

Real-time Fuzzy Control 123

contains the real-time database where all current data on the state of the plant is stored and can serve this data to any of the clients (sometimes re-ferred to as agents) in the cluster on demand.

An alternative architecture involves a distributed real-time data-base in which each client retains the data pertinent to the tasks it is as-signed. The client can transmit this data to any other client on demand. This leads to considerably more data having to be transmitted over the LAN between clients, resulting in data transmission delays. Finally, a hybrid architecture in which the local real-time databases are mirrored in the server can be used. In this case, each client must continuously update the data that has changed since the last transmission in the master data-base in the host computer. The advantage of this last architecture is that in the unlikely case that the host computer fails and its data is lost or cor-rupted, then the master database can be restored from the local databases in the clients when the host becomes operational again. 9.2 Embedded Fuzzy Controllers Fuzzy controllers are increasingly being embedded in micro-controllers for operation at the lowest level of the control hierarchy. Chip-size fuzzy micro-controllers have found their way into a variety of products such as disk drives, printers, video cameras, laser printers, cars, household appli-ances, and so forth. The use of real-time embedded micro-controllers is spreading rapidly for two reasons: convenience and cost. Except for memory restrictions, embedded controllers can be programmed to do just about anything their larger counterparts can do. Unlike their larger coun-terparts, which operate under such real time operating systems as Unix or Windows NT, there is no standardized real-time operating system for micro-controllers. Windows CE, which features a deterministic sched-uler, may change this situation, however. The programming language of choice for micro-controllers in the foreseeable future is likely to be Java, which is processor independent. Micro-controller hardware has generally been restricted to 16 and 32 bit processors but 4-bit and even 64-bit processors are in use, the former in domestic appliances.

The architecture of embedded intelligent micro-controllers fol-lows that presented in the previous section closely. Only the develop-ment system is independent and resides in a host PC where all develop-ment is performed off-line. Once the design has been completed, the ex-

Chapter 9 124

ecutable software, which includes the knowledge base, fuzzy sets, infer-ence engine, fuzzifier and de-fuzzifier, is downloaded to the micro-controller via a local link or is burned into an erasable programmable read-only memory (EPROM) which is then plugged into the micro-controller.

A new class of intelligent industrial three-term controllers is gradually replacing conventional industrial controllers in a number of critical applications that require increased autonomy. These controllers find use in situations where the operational demands of the application do not permit fixed or programmed gains. Intelligent industrial control-lers are implemented in software in micro-controllers (MC), programma-ble logical controllers (PLCs) or remote terminal controllers (RTUs) and a number of vendors today offer appropriate software for their develop-ment. In new plants it would be wise to consider incorporating intelligent controllers that have been shown to enhance control over conventional industrial controllers.

In order to minimize memory requirements as well as accelerate computation in embedded fuzzy controllers, many vendors restrict both the number and the shape of the permissible fuzzy sets for both inputs and outputs. Triangular fuzzy sets are almost universally used for the in-puts and outputs of the controller. Rules are coded numerically and the number of fuzzy sets is restricted. Singletons are often used to define the fuzzy sets of the outputs of the controllers as they simplify de-fuzzification considerably. 9.3 The Real-time Execution Scheduler Intelligent controllers reside in a devoted intelligent agent or are distrib-uted throughout the clients in the cluster. They can be executed either at regular intervals (analogous to discrete-time control) or on demand. In the case of a devoted intelligent agent, the executable program of the fuzzy controllers is scheduled to run at fixed time intervals. The execu-tion interval is clearly dependent on the dynamics of the plant under con-trol. The execution interval is typically set equal to one-tenth to one- twentieth of the value of the smallest time constant of the plant. In some cases, particularly in process control, where static control is acceptable, the execution interval is set equal to the settling time of the plant as de-termined from experimental observations.

Real-time Fuzzy Control 125

An alternative technique for scheduling the intelligent controller which is useful particularly when dealing with fast dynamic plants, is to have a control scheduler that continuously tracks the inputs to the fuzzy controller and their changes. Based on some measure of the rates of change (or differences) of these inputs, the control scheduler is pro-grammed to decide whether or not the controller should execute. Clearly if the inputs to the controller (i.e., the plant variables) do not change sig-nificantly with time, implying that the plant is stationary and operating at its nominal state, then there is no reason to effect control and the control-ler algorithm is not executed. Conversely, if the inputs are changing sig-nificantly, implying that the plant is in transition then immediate action must be taken in order to restore it to its nominal state. By continuously following the changes of the controller inputs, the control scheduler is therefore in a position to issue commands on when to execute the con-troller. If, following execution and after some minimum time interval, the control scheduler continues to observe that the rates of the controller inputs exceed pre-specified limits, then it sends another command to execute the controller and continues to do so until the plant returns to its nominal state. This scheme leads to a controller that executes at irregular intervals and on demand. The control scheduler is shown schematically in Figure 9.3.

Differencer

Execution Scheduler

Input

Figure 9.3 The real-time execution sch On executing the fuzzy algorithm

fuzzification, the new control variables are deposit

Controller

Output

eduler

and following de-ed in the real-time da-

Chapter 9 126

tabase. At regular intervals (measured in milliseconds) and provided there has been a significant change in the control variables since the last time the algorithm was executed, this information is transmitted via the LAN to the appropriate RTUs for immediate action on the plant. This fi-nal action closes the control loop.

The structure of a fuzzy controller can be seen to be very general and is applicable to a wide variety of plants and processes. The only thing that is required to change the controller is to modify the numbers of the inputs and outputs of the controller, the shape and number of the fuzzy sets and, most importantly, the rule base.

Example 9.1 Waste-water treatment control An interesting application of real-time fuzzy control to an environmental problem is outlined below. The system has been used in a municipal wa-ter-treatment plant, achieving significantly improved water treatment over existing methods. The system was implemented on a commercial PLC.

Through suitable modification of the biological treatment process, stabilization of the nitrogen compounds (nitrates, ammonia) can also be achieved. Biological treatment is classified by the type of micro-organisms that are used in the removal of the organic load and is either aerobic, anaerobic or both. Because of the anoxic zone in aerobic treat-ment, the simultaneous removal of both nitrates and ammonia is possi-ble. Interest is normally focused on secondary biological treatment and on aerobic treatment in particular. The wastewater treatment plant con-sidered in this paper involves extended aeration and is a variation of the active sludge method. A schematic of the plant is shown in Figure 9.6. Here the biological reactor possesses a tank with two zones as well as a secondary settling tank. Wet sludge is fed into the anoxic zone in which low oxygen conditions prevail and de-nitrification initially takes place. To operate satisfactorily, this stage requires low oxygen concentrations in the presence of nitrates.

Real-time Fuzzy Control 127

Sludge is transferred to a second aerated zone where organic load is

removed and nitrification take place. A fraction of this sludge is then re-turned to the anoxic zone while the remainder is fed to the secondary set-tling tank. The quantity of sludge returned for further treatment depends on factors that affect nitrification and de-nitrification and is therefore one of the process variables that must be controlled.

Finally, a fraction of the sludge in the secondary settling tank is re-turned to the biological reactor, while the rest is removed and fed to the fusion stage.

Solids storage

Biological treatment

Sludge feed Grit chamber

Bar racks

Skimming tank

Sludge Fusion

Sludge returns

Sludge Drying

Secondary settling tank Primary settling tank

Remaining sludge Suspended liquids

Figure 9.5 Schematic of a typical wastewater treatment plant The fraction of sludge fed back to the biological reactor is another

variable that must be controlled. In all wastewater treatment plants it is necessary to control the oxygen content in the aerated zone of the reac-tor. The oxygen content depends on the removal of the organic load and nitrification. The removal of organic load, nitrification and de-nitrification are the three principal quantities in a wastewater treatment plant that must be controlled. This is achieved by a suitable control strat-egy for the following three manipulated variables:

Chapter 9 128

1. the oxygen supply to the aerated zone (O2Feed), 2. the mixed liquid returns rate from the aerated zone to the anoxic

zone of the biological reactor (R_ml) and 3. the sludge returns rate from the secondary settling tank to the

biological reactor (R_sludge).

Settling tank

Remaining sludge for fusion

Measurement of BOD

+

Measurement of BOD

Measurement of: Temperature, DO and MLSS

ANOXICZONE

AERATEDZONE

Oxygen Feed

Sludge returns

Measurement of N=NO3 and N-NH3

Sludge feed

Reactor

Liquid Returns Pump for return

of liquids

Figure 9.6. The treatment process and its principal variables

The controlled variables of the plant and inputs to the controller are:

1. the ammonia concentration in the reactor (N-NH3), 2. the nitrate concentration in the reactor (N-NO3), 3. the dissolved oxygen in the reactor (DO), 4. the temperature in the reactor (TEMP), 5. the mixed liquid suspended solids concentration in the reactor

(MLSS), 6. the difference in biochemical oxygen demand D(BOD)between

the entrance and exit of the secondary settling tank.

Real-time Fuzzy Control 129

The integrity of the controller is directly related to the number of

fuzzy variables used. Increasing the number of fuzzy variables, however, increases the memory requirements of the controller logarithmically.

For wastewater plant control, the three fuzzy variables, HI (HIgh), OK and LO (LOw) normally suffice to characterize the controller inputs. Trapezoidal fuzzy sets (membership functions) are computationally sim-ple and effective in practice. Similarly, the manipulated variables or con-troller outputs must be allocated an appropriate number of linguistic de-scriptors. Five fuzzy variables, i.e., VH (VeryHigh), HI (HIgh), OK, LO (LOw) and VL (VeryLow) provide sufficiently fine control.

Finally, for computational simplicity, singletons are used to de-scribe the fuzzy sets of the controller outputs, leading to a particularly simple and fast procedure for de-fuzzification. The knowledge with which a given wastewater plant is controlled must first be elicited from plant operators. This is a painstaking task and one that is critical for proper operation.

If the 6 manipulated variables have 3 descriptors each, then the theoretical maximum number of rules is 36 or 729, a number which is clearly unmanageable and practically unnecessary. In practice 50 rules suffice to provide control of a wastewater treatment plant. Of these some 27 are required to stabilize the organic load (BOD), 11 to stabilize the ni-trification process while 12 rules similar to those for nitrification are necessary to stabilize de-nitrification.

The 50 rules, which form the knowledge base of the controller, are considered to be the minimum necessary to achieve acceptable control of a wastewater treatment plant under most operating conditions. A subset of these rules is shown in the Table on p. 130.

The control rules have the standard form:

R : IF (D(BOD) is Y1) AND (MLSS is Y2) AND (TEMP is Y3) AND (DO is Y4) AND (N-NH3 is Y5) AND (N-NO3 is Y6) THEN (O2Feed is U1) AND (R_Sludge is U2) AND (R_ml is U3) The controller outputs are all normalized to the range [0,100]%.

Under normal operating conditions the plant outputs have nominal values of 50% and the corresponding levels of reactor stabilization, nitrification and de-nitrification are 90%, 70% and 60% respectively.

Chapter 9 130

L=Low, OK= normal, H=High, VH=VeryHigh, B=Big, VB=VeryBig, S=Small, VS=VerySmall

D(BOD) MLSS O2Feed

Figure 9.7. Variations in oxygen feed in response

to a triangular perturbations in the D(BOD) and O2Feed Finally, Figure 9.7 shows the effect on the oxygen feed to the reactor of a triangular perturbation of 50% on D(BOD) and MLSS about their nominal values.

Real-time Fuzzy Control 131

Example 9.2 Rotary kiln control

The rotary kiln is the basic process in cement production. Raw feed is processed in a cyclone pre-heater comprising the rotary kiln and precal-ciner strings, both of which supply pre-heated feed to the rotary kiln, a long tubular structure with a refractory lining. Fuel, either coal dust or fuel oil is burnt at the exit of the kiln in a long flame that extends along most of the length of the kiln. Throughout the length of the kiln there is a hot airflow induced by an induction fan at the top of the precalciner. As a result of this hot air mass, there is a transfer of heat between the hot air and the raw feed. The raw feed undergoes crystallization as it reaches the end of the kiln where the temperatures are of the order of 1400 degrees. The kiln is ro-tated at a few revolutions per minute to allow for the lava-like fluid to exit the kiln and flow to the grate cooler where it solidifies into clinker. Clinker, when added to gypsum and ground to a fine powder in a finish-grinding mill, results in the final product, cement.

The cement production process is very energy-intensive, requiring large quantities of fuel in the kilning stage and electric power in the finish grinding process. As the cost of energy constitutes the largest component of the total cost of production, it is clear that every effort must be made to keep this cost to a minimum.

The kilning process is one of the most difficult processes to con-trol. It is significant that this process is one of the few remaining in the process industry, which generally relies on human operators. Many at-tempts have been made to model the process and then apply modern theory to control it. All these attempts have failed dismally due to the complexity, non-linearity and distributed nature of the process. Until the advent of fuzzy kiln control, first introduced by F. L. Smidth of Denmark in the early 1980s, all cement kilns were manually controlled. A kiln operator’s objective is to produce clinker of the desired specifi-cations and production rate while minimizing fuel feed, and at all times maintaining the process in a stable state.

Under-burning (i.e., using less fuel, which results in lower kiln temperatures) the raw material leads to clinker which is less reactive and thus undergoes an incomplete change of phase of the raw material. The lava-like material, which slowly slides down the kiln may then so-lidify, a catastrophic situation that results in kiln shut-down and repairs to the kiln lining. Over-burning, on the other hand, leads to excessive fuel consumption, less reactive clinker and high refractory lining wear.

Chapter 9 132

Most kiln operators prefer to operate the kiln in the region of over-

burning however, as it is more stable. In between the two extremes, however, there is a narrow region of stable operation in which high pro-ductivity and product quality can be achieved. Kiln operators find this state very difficult to maintain for long periods or time as the controls must be continuously manipulated to account for small changes in the operating state of the process. Kiln operators learn to control their plant from linguistic rules of the type:

IF the Kiln_KW demand is HIgh AND the outlet oxygen content O2 is Low AND the Previous_Fuel_Feed is OK THEN make a SMall reduction to the Fuel_Feed AND make NO change to the induced draft fan speed VIDF Here, Kiln_KW, outlet oxygen content O2 and Previ-

ous_Fuel_Feed constitute the input fuzzy variables, while current Fuel_Feed and induced draft fan speed VIDF are the output fuzzy variables. Approximately 50-60 linguistic rule are used to provide very good control of the kiln under normal operating conditions. For start-up and shut-down conditions, as well as abnormal situations, different sets of rules may be used.

The fuzzy controller can control this most difficult of processes as well as, and certainly more consistently than, a human operator, by ob-serving the same control variables and adjusting the same manipulated variables. Fuzzy kiln controllers can easily maintain control of the proc-ess in this stable intermediate state, consistently achieving marked fuel economy and high productivity. Fuzzy kiln controllers normally reside in a client on a client-server network, receiving information on the cur-rent values of the control variables and returning the manipulated vari-ables to the real-time data base on the file server for transmittal to the local RTUs. The fuzzy controller can be executed either at regular inter-vals or following an interrupt from the control scheduler, which moni-tors the temporal changes in the control variables.

Using fuzzy kiln control, fuel costs have been reduced by up to 5% while productivity has been increased by an equal amount. Today there is a very large number of kilns worldwide under fuzzy control. Similar fuzzy controllers have been used to control all of the processes associ-ated with cement production.

Real-time Fuzzy Control 133

Hierarchical intelligent control has also been used to control a

cluster of fuzzy controllers, each controlling a different sub-process of the kilning process.

The cement industry was the first process industry to adopt fuzzy control and today a number of vendors supply such controllers. There is certainly no doubt that fuzzy control has resulted in a major change in the cement industry and many process industries are now following suit, encouraged by the progress in the field.

VIDF

r

Rotary KiKiln motor

Raw Feed

Kiln KW

2

Prev_Fuel_Feed

Figure 9.10 The rotary kiln intelliand the principal control and co

Controller

Precalcine

ln

d

gent control s

C

ntrolled varia

Fuel Fee

l

ystem

linker flo

bles

Fue

O

Air flow

w

Chapter 10 Model-Based Fuzzy Control Two distinct trends in the design of fuzzy controllers have evolved in the last decade or so. The first, based on heuristic knowledge of the control actions necessary to control a process, has been dealt with at length in the preceding chapters. Heuristic fuzzy control does not require deep knowledge of the controlled process in order to be applied successfully. This feature has made heuristic fuzzy control very popular in the indus-trial and Fmanufacturing environments where such knowledge is often lacking. In general, soft control is possible only if heuristic knowledge of the control policy is known a priori. Thus for new processes, where there is no such prior knowledge, heuristic soft control is not a candidate. It is also clear that the heuristic approach cannot resolve such issues as overall system stability and system performance. This implies that this approach is case-dependent and it is therefore not possible to generalize the performance of fuzzy controllers from knowledge of their behavior in other applications.

The limitations of the heuristic approach led to a search for more rigorous methods that combine fuzzy logic and the theory of modern control. The result was Model-based Fuzzy Control which has attracted considerable attention and has been applied successfully in a number of applications, particularly in Japan, for the control of high-speed trains, helicopters, robotic arms and more. The technique assumes the existence of an explicit microscopic model of the controlled process of sufficient fidelity from which a series of linearized models can be derived for each

135

Chapter 10 136

nominal operating state. The model-based fuzzy control methodology is thus a fusion of soft and hard control techniques and offers advantages in situations where stability and transient behavior must be guaranteed.

In conventional gain-scheduling control, the selection of the scheduling procedure, i.e., the control law to use, is dependent on some exogenous variable. In contrast, fuzzy gain scheduling, which is a special form of model-based fuzzy control, uses linguistic rules and fuzzy rea-soning to determine the corresponding control law. Issues of stability, pole placement and closed loop dynamic behavior are resolved using conventional modern control techniques. 10.1 The Takagi-Sugeno Model-Based

Approach to Fuzzy Control In the mid-1980s Takagi and Sugeno proposed the use of fuzzy reason-ing to specify the control law of a conventional state feedback controller so that the overall system would have guaranteed properties. The original controller that Takagi and Sugeno proposed is characterized by a set of fuzzy rules that relate the current state of the process to its process model and the corresponding control law. These composite rules have the form:

R: IF (state) THEN (fuzzy process model) AND (fuzzy control law).

Consider a physical process described by its non-homogeneous state equations:

0xuxfx );,(= where x∈ℜn and u∈ℜm are the crisp n-dimensional process state vector and m-dimensional control vector respectively and x0 is the initial state. This explicit description of the process may be the result of deep knowl-edge about the process or the result of identification of the process from experimental data using any of the well-known system identification techniques.

One of the interesting features of the first Takagi and Sugeno technique for designing model-based fuzzy controllers is that under cer-

Model-Based Fuzzy Control

137

tain conditions that are, unfortunately, not always easy to satisfy, this technique guarantees stability of the closed system while specifying the transient behavior of the closed system through pole-placement. These are properties that are inconceivable with the heuristic fuzzy controller design technique. The difficulties in meeting the conditions for stability of the first method proposed by Takagi and Sugeno were eliminated in the second version that uses state differences. Both techniques are out-lined in this chapter. 10.2 Fuzzy Variables and Fuzzy Spaces Consider the crisp process state vector x define on a closed, real set Χ. The fuzzy state variables xi are fuzzy sets defined on Χ. The values of these variables are termed the fuzzy values of the fuzzy state variable and are written as ΦXij. For every fuzzy value of xi there exists a correspond-ing membership function µΦXij(x), which specifies the membership value of the crisp value xi

* of this variable. In general we define:

∫ Φ=ΦX Xij xxµX

ij/)(

for the continuous case and

∑ Φ=Φ xxµXijXij /)(

for the discrete case. The universe of discourse is given as the set:

TXi = {iik2i1i XXX ΦΦΦ ..., }

where ki is the number of fuzzy values xi. In order to simplify the analy-sis that follows it will be assumed that:

• the shapes of the fuzzy sets of ΦXi are identical for all i, • the number of fuzzy numbers k1 = k2 =...=kn and that • .... njiji2i1 XXXX Φ=Φ=Φ=Φ

Examples of such fuzzy sets are shown in Figure 10.1.

Chapter 10 138

……

ΦΧinΦΧi2ΦΧι1

µ

1

xi

Figure 10.1 Fuzzy sets of the state xi The state vector x of a process is defined over some space.

Every crisp value x* of the state vector corresponds to a specific state in state space. In the case of fuzzy controllers based on the Takagi-Sugeno model, the states take on fuzzy values and consequently the concept of state space must be modified to account for the fuzzy values of the state vector. Knowing that every fuzzy variable has a finite number of fuzzy values, we can then generate a finite number of fuzzy vectors that result from the combinations of the fuzzy values.

Each element of the crisp state variable x is fuzzified as in the heuristic fuzzy control case. In each fuzzy region of fuzzy state space, a rule uniquely defines the local process model in that region, e.g.,

RS

i : IF x=Φxi THEN =fix i(xi,ui) (10.1)

The symbolism x=ΦXi implies that the state of the process x be-

longs to the fuzzy region ΦXi. The consequent of each rule describes an explicit local model of the process in the corresponding fuzzy region

Model-Based Fuzzy Control

139

ΦΧi. Simple examples of fuzzy process rules for both continuous and discrete-time processes are:

IF u is Low AND x is High THEN the process model is = -0.7x - 0.2xx 3 - 3.1u

ELSE IF u is High AND x is Low

THEN the process model is = -0.6x - 0.3xx 3

- 2.8u and Given the process parametric model

Pressure(k+1)=a0 Pressure(k) + b1 Valve(k)+b2 Input_Flow_Rate(k)

IF Pressure(k+1) is High AND Valve(k) is Closed AND Input_Flow_Rate(k) is Very_High

THEN the process parameters are a0=-3, b1=2 and b2=-1 ELSE IF Pressure(k+1) is Low AND Valve(k) is OK AND Input_Flow_Rate(k) is High THEN the process parameters are a0=-4, b1=1 and b2=-2 ELSE etc.

10.3 The Fuzzy Process Model It is observed that whereas the antecedent of the fuzzy process rules are similar to those used in the heuristic fuzzy control case, the consequents are analytical expressions that describe a process model. Process rules can be expressed in terms of the elements of the crisp process state as:

RSi: IF x1=Φxi

1 AND x2=Φxi2 AND … xn=Φxi

n THEN =fix i(x,u)

Chapter 10 140

In any fuzzy region ΦXi the process can thus be specified by the state equation:

),()( uxfxµx ii

Si ⋅= (10.2)

where

)).(),...,(),(min(

)(...)()()(

nX2X1X

nX2X1Xi

S

xµxµxµ

xµxµxµxµ

in

i2

i1

in

i2

i1

ΦΦΦ

ΦΦΦ

=

∧∧∧=

(10.3)

are the degrees of fulfillment of the local models of the process using Mamdani’s fuzzy compositional rule.

For each nominal state of the process, a state equation of the form of (10.2) is determined. Using the set of fuzzy process rules (10.1) we establish the fuzzy open-loop model of the process, which is the weighted sum of the local models fi(x,u) , i.e.

∑ ⋅=i

ii

S uxfxwx ),()( (10.4)

where

∑=

i

iS

iSi

S xµxµ

xw)(

)()( ∈[0, 1]

(10.5)

are the normalized degrees of fulfillment or process function weights. Clearly the sum of the weights is unity, i.e.

∑ =i

iS 1xw )(

Model-Based Fuzzy Control

141

10.4 The Fuzzy Control Law Similarly, for every antecedent there exist two consequents, the second one of which specifies the state feedback control law that must be ap-plied in order to stabilize the process. This, of course, assumes that all states are measurable, a restrictive limitation of the technique in practice. Where it is not possible to measure all the state variables then observers or other forms of state estimators may be used to provide the missing states provided an explicit microscopic model of the process is available.

The second consequent of every rule in the fuzzy region ΦXj has the form:

RSj : IF x=Φxj THEN u =gi(x)

while the control law that must be applied to the process so that stability is assured at any nominal states is derived in an analogous manner and is:

)()( xgxµu jj

Cj ⋅= (10.6)

where

)).(),...,(),(min(

))(()(

nX2X1X

kXkj

C

xµxµxµ

xµxµ

jn

j2

j1

jk

ΦΦΦ

Φ

=

∧=

(10.7)

The overall control law is thus the weighted sum:

∑ ⋅=j

jj

C xgxwu )()( (10.8)

where wC

j are the control weights which are computed in Equation

(10.5).

Chapter 10 142

10.5 The Locally Linearized Process Model When the nonlinear model of the process is linearized (also termed Lyapunov linearization) about each nominal state of the process, the re-sult is a set of locally linearized models of the controlled process de-scribed by state equations:

(10.9) ii

ii

iii

i uBxAuxfx +== ),( where

iiii u,xii

iu,x

ii

i uf

B,xf

A∂

∂=

∂

∂=

are the usual partial derivatives of the nonlinear functions fi(•) evaluated at the nominal conditions (xi,ui). The set of locally linearized models at these nominal states defines the state equations of the controlled process at those states and is termed the overall fuzzy open-loop model of the process.

For each linearized local model there must be a corresponding linear control law that guarantees closed system stability while satisfy-ing time-domain criteria. The complete set of control laws constitutes the global control law of the closed system. In the Takagi-Sugeno approach, the decision on the process model and the control law to use clearly de-pends on the nominal state of the process and is based on fuzzy linguistic rules.

The linearized process model rule (10.2) now becomes:

RSP

i : IF x=Φxi THEN uBxAx ii += (10.10)

Modern control theory assumes that all the states of the process

are measurable or can be estimated, in which case a constant state feed-back control law of the form:

(10.11)

Model-Based Fuzzy Control

143

u Kii= x

can be applied, where Ki is a suitably gain matrix. Using any pole-placement technique, the elements of the gain matrix are chosen so that the poles of the closed system yield the desired transient behavior. The fuzzy control rule is therefore:

(10.12) :R jC RS

i : IF x=Φxi THEN ui =Kixi

Now, from the definitions of and it is seen that

these two are in fact identical, whereupon to simplify notation, let and . Substituting Equation (10.5) in

(10.4), we obtain the Overall Closed System (linearized) Fuzzy Model:

)(xw iS )(xw j

C

)()( xwxw iiS = )()( xwxw jj

C =

∑ +=i

iii uBxAxwx ))(( (10.13)

and likewise the overall control law:

∑=j

jj xKxwu )( (10.14)

Finally, substituting Equation (10.14) in (10.13) yields the over-

all homogeneous closed system state equations:

∑∑ +=i j

jiiji xKBAxwxwx ))(()(

(10.15)

or

∑∑=i j

ijji xAxwxwx )()( (10.16)

where

A A B Kij i i j= +

Chapter 10 144

is a Hurwitz matrix. It should be noted that with this technique, even though linearized models of the process have been used in the develop-ment, both the overall system state equations and the overall control law are not linear because of the nonlinearity of the normalized membership functions wi(x) and wj(x). This leads us to the conclusion that asymptotic stability of the closed system can be guaranteed only locally, i.e., around each nominal state, and cannot be inferred elsewhere. 10.5.1 Conditions for closed system stability The conditions for stability of the Takagi-Sugeno model-based fuzzy controller can be stated as follows:

Τhe closed system (10.15) is asymptotically stable and sta-bilizable at the origin x=0, if and only if

1. the matrix Aij is Hurwitz, i.e., all its eigenvalues possess

negative real parts and 2. there exists a positive definite matrix P such that

A P PAij

Tij+ < 0 ∀i,j (10.17)

. Even though these stability conditions are necessary and suffi-

cient, it is very difficult to determine a suitable positive definite matrix P. Furthermore, if the matrix Αij is not Hurwitz then a matrix P which satisfies Equation (10.17) does not exist and this technique is not appli-cable. 10.6 The Second Takagi-Sugeno Approach To overcome these limitations, Takagi and Sugeno proposed a second approach that follows the one already outlined in principle, but differs in the formulation of the fuzzy rules. Thus the rule which defines the fuzzy open loop model is modified to:

Model-Based Fuzzy Control

145

Rsi: IF xd=Φxi THEN ),( ii

ii uxfx = (10.18)

where xd is the desired state of the process. The modified model-based fuzzy control approach determines which fuzzy region the desired state of the process belongs to before deciding what control law to follow. It is noted that the desired state need not be stationary but may change con-tinuously without the need to re-compute the control law.

The desired state xd and the control law ud are the result of solv-ing the steady state equations:

0uxf dd

i =),(

Assuming that the desired state xd is constant or varies very slowly with time so that ≈0, the nonlinear system can then be lin-earized about (x

dxd,ud) to yield:

)()( dddd uuBxxAx −+−= (10.19)

which are linear state equations involving the state and control input de-viations from their nominal states xd and ud respectively, while

dd uuxx

iddd

xuxf

uxAA==∂

∂==

,

),(),(

and

dd uuxx

iddd

uuxf

uxBB==∂

∂==

,

),(),(

The modified technique has the following limitations that may

make application difficult:

• it is valid only for small deviations in the state and input vec-tors from their nominal values and

• for every change in the nominal state, the rules by which de-cisions are made must be changed and the linearization pro-cedure repeated.

Chapter 10 146

10.7 Fuzzy Gain-Scheduling Gain-scheduling is a well-known technique of conventional industrial control that is used when the plant is subject to changes in its operating state. Here the gains of the controller are varied in accordance with some exogenous variable. A typical example is aircraft control: as the altitude of the aircraft increases the influence of the control surfaces decreases due to the thinner air. This in turn requires a greater control action to achieve a particular result. If altitude (as measured by an altimeter) is therefore used as the exogenous variable to adjust the controller gains at given altitudes, then the controlling effect can scheduled to be essentially the same whatever the altitude. The adjustments to the controller gains are step-wise, resulting in bumpy control when the adjustments are ef-fected. Automatic transmission in vehicles is another well-known exam-ple. The distinct changes in the gear ratios while accelerating are a cause of jerky motion is undesirable. To avoid these sudden changes some manufacturers offer infinitely variable ratio transmissions that lead to smooth motion. Refering to the problem described in the previous section, if it were possible to generate a set D of known nominal states xi, for which a corresponding set S of linearized process models is computed, then fuzzy gain-scheduling can offer distinct advantages. In this scheme, the transi-tion from one nominal state to another is smooth since the system pa-rameters can be made to vary smoothly along the trajectory. These two sets are clearly invariant with changes in the nominal states. For any nominal state that does not belong to the set D, an approximating model can be derived from models belonging to the set without recourse to lin-earization.

For each nominal state xd the locally linearized model is stored in the model base S while the corresponding control law is stored in the control law base U. The nominal states xi in the nominal states base D can be conveniently chosen to be at the centers of the fuzzy regions ΦX,

i.e., the states xi∈ΦXi at which:

Model-Based Fuzzy Control

147

.),...,,min()( 1111xµ iLX i ==

Thus at the centers of each fuzzy region, the linearized system in

the consequents of the fuzzy rules are given by Equation (10.19) with xi in place of xd. The set of rules which describe the fuzzy open-loop model reduce to:

RSj: IF xd=ΦXi THEN =Aix i(x-xd)+Bi(u- ud) (10.20)

and the control law rules are defined by:

RCj: IF xd=ΦXj THEN u=Kj(x-xd)+ud (10.21)

Where the gain matrix K(xj,uj) is computed on the basis of the

linearized closed system defined in the corresponding fuzzy region LΧj. The fuzzy open loop model is now specified in terms of the deviations of the state and control actions from the nominal values, i.e.:

[ ]∑ −+−⋅=i

dididi uuBxxAxwx )()()( (10.22)

The overall control law is now given by:

∑ +−⋅=j

dddj uxxKxwu )()( * (10.23)

It is observed that both the closed system model and the control law are linear since the normalized membership functions wi(xd) and wj(xd) are constants less than unity. As in the previous case, the closed system is now given by:

[ ]∑∑ −+=i j

djiidjdi xxKBAxwxwx )()()( (10.24)

It follows that

∑∑∑∑ ===i j

djdi

j

dj

i

di 1xwxw xw xw )()()()(

Here

Chapter 10 148

∑=

i

idid AxwA )( ; ∑=i

idid BxwB )(

and ∑=

j

jdjd KxwK )(

In an analogous way

)()( dddd uuBxxAx −+−= and

ddd uxxKu +−= )(

The overall closed system is stable at the nominal state xd if and only if:

{ } n21m 0Aλm ,...,,*)(Re =< where λm are the eigenvalues of the matrix

A*=Αd+BdKd.

If these conditions hold, then we can state the following:

The fuzzy gain-scheduled closed system (10.24) is asymp-totically stable if and only if positive definite matrices Ρ and Q exist such that

QPAPA T −=+ ** (10.25)

In summary, the advantages of the second Takagi-Sugeno ap-proach are:

• the approach leads to linear descriptions of the open and closed system descriptions and the desired control law,

Model-Based Fuzzy Control

149

• computation of the control law and the conditions for stabil-ity are well established and

• the determination of the matrix Ρ is almost always possible. The second Takagi-Sugeno technique combines the simplicity of

heuristic fuzzy logic with rigorous hard control techniques and offers the possibility of specifying the desired closed system response as well as the conditions for stability. It suffers, however, from the fact that an ex-plicit analytical model of the process must be known.

Example 10.1 Fuzzy gain-scheduling of a

simple process

The second Takagi-Sugeno approach is illustrated by way of an exam-ple for fuzzy gain-scheduling. Aircraft exhibit dynamic characteristics which vary significantly with altitude: if a controller is tuned at low al-titudes then control performance is grossly degraded with altitude due to the thinning air. Thus gain-scheduling is resorted to in order to adapt the parameters of the controller with altitude. The example presented here assumes a linearized scalar model of the dynamics of the aircraft or ‘plant’. The explicit model of the plant is assumed known at ground level xd=0 and at some other specified altitude xd=1. The process rules are assumed, for simplicity, to be:

R0: IF xd=0 THEN (a) u50x50uxfx 1 ..),( +−== R1: IF xd=1 THEN (b) u2xuxfx 2 +−== ),(

At low altitude, case (a), the dynamics of the plant are slow with a nor-malized time constant of T=2 whereas at high altitude in case (b), the

Chapter 10 150

response of the plant is faster, with a time constant T=0.5 and is con-siderably more sensitive to control actions. The corresponding step re-sponses are shown in Figure 10.2.

Model-Based Fuzzy Control

151

Figure 10.2 Step respo

)

The transition of the dynamvary smoothly. Consider thewhose control rules are as fo

R0: IF xd=0 TH R1: IF xd=1 TH

In order to obtain the dtude extremes, the state feedthat the eigenvalue of the clo To simplify the analyshown in Figure 10.3 indicatplant follows the altitude. mathematical model of the ptitude increases the type (a) x=xd at which the model is e

(b

nses of the plant at the two altitudes

(a)

ics of the plant with altitude is assumed to use of a fuzzy gain-scheduling controller llows:

EN u1 = g1(x,xd)= k1(x-xd)

EN u2 = g2(x,xd)= k2(x-xd)

esired closed system response at both alti-back gains k1=0.6, k2=0.4 are selected so sed system is at (-0.2,0) in both cases. sis, let the fuzzy membership functions e how the transition of the dynamics of the It is clear that at very low altitudes the lant is predominantly of type (a) but as al-model increasingly fades into type (b) until ntirely type (b).

Chapter 10 152

Figure 10.3 Fuzzy sets of gain-scheduling

1 x 0

µ2

1

µ1

The gain-scheduling fuzzy sets can be described by the expres-

sions µ1(x)=1-x and µ2(x)=x. It is noted that µ1+µ2=1∀x. The overall fuzzy process model is therefore given by the weighted sum:

))(().)(.( u2xxwu50xx50wx d

2d

1 +−−++−−=

where w1 and w2 are the normalized membership functions:

i21

ii µ

µµµ

w =+

=

The overall fuzzy control law is therefore the weighted sum:

)(.)(. d2

d1 xxw40xxw60u −+−=

The fuzzy gain-scheduling controller provides the control actions to force the closed system to follow the desired response for all values of altitude. The response of the closed system to a very large step de-mand in altitude from x=0 to x=xd=1 is shown in Figure 10.4(b). Fig-ure 10.4(a) shows for comparison, the response of an invariant system governed by:

u2.0x2.0x +−=

Model-Based Fuzzy Control

153

(a)

)

Figure 10.4 Responses of (a) an invdynamic characteristics and (b) th

Since the closed system with the fuzzyear, it would be unreasonable to expFigure 10.3 to be the same. It is notedx=xd the response of the closed systeminvariant system, as desired.

(b

ariant system with the desired e fuzzy gain-scheduled plant

gain scheduler is clearly nonlin-ect the two responses shown in that in the proximity of x=0 and approaches the response of the

Chapter 11 Neural Control

The emulation of human cognitive ability is one of the primary fields of research interest in Computational Intelligence. The human is nowhere near as fast or as accurate in calculating as a modern computer, yet even at a very early age one can easily recognize objects and relate them in their natural environment even if they are distorted or partially hidden. Exactly the opposite is true with computers − they are far more capable than humans in performing tedious tasks that involve extensive numeri-cal computations yet they have difficulty performing cognitive tasks de-spite the impressive progress made in Artificial Intelligence.

The ability to learn through experience is one of the principal characteristics of humans. Humans have the capacity to store huge quan-tities and types of information, recall this data and process it in a very short time with little difficulty. One possible explanation for this superi-ority is the lack of suitable computer software to emulate the human’s ability to process information. A second explanation is the fact that the human brain and computers work in radically different ways, the brain being much more efficient at processing information. The human brain is extremely powerful, comprising a huge number (of the order of 109) of simple processing elements or neurons, each one of which is capable of communicating with the others. This is massive parallelism and it is in-teresting that modern computer designs are increasingly following this architecture.

Chapter 11 154

The neurons of an Artificial Neural Network (ANN) are ar-ranged in such a manner as to process information in parallel and simul-taneously. Each neuron sends activation or de-activation signals to other neurons while its activation depends on signals received from other neu-rons to which it is connected. The term synapses is commonly used for these connections. Suitable interconnection of these simple elements can yield powerful networks with the ability to learn, adapt and infer. The use of artificial neural networks is sometimes known as connectionism and ANNs can therefore be viewed as generalized connectionist ma-chines or generalized function mappers.

153

Since their re-emergence in the early 1980s, there has been an explosion of interest in the application of ANNs for qualitative reason-ing, which is at the core of the fields of Soft Computing and Intelligent Systems. This interest has been encouraged by the increasing availability of powerful parallel computing platforms capable of very high computa-tional speeds and parallel programming techniques. Multi-layer ANNs are finding use in an ever-increasing range of applications, from image recognition, voice analysis and synthesis to system identification and in-dustrial control. This chapter, and those that follow, present the most commonly used ANN architectures that have found application in Con-trol Engineering, the basic network-learning algorithms and examples of industrial applications.

The origins of ANNs are to be found some fifty years ago in the work of McCulloch and Pitts. The first contribution in the area of net-work learning was due to Hebb in 1949 who showed that learning in complex networks can be achieved by adapting the strengths of the syn-apses. Rosenblatt introduced the Perceptron, an early form of neuron, in the late 1950s.

The operation of multi-layer ANNs was not fully understood in those early days and research was restricted to structured perceptrons. Nilsson followed in the mid-1960s with learning machines and φ-machines that were made up of clusters of threshold logic units. In 1969 Minsky and Papert published their seminal work on Perceptrons, in which they proved that Perceptrons are limited in their ability to learn, pointing to their inability to represent a simple ΧΟR logic element. This work was to dampen the enthusiasm in ANNs and research in the field was virtually paralyzed for almost a decade.

Fortunately, ANNs re-emerged in the early 1980s due mainly to the work of Hopfield who continued his research on network training

Neural Control 155

methods. New ANN architectures and powerful learning algorithms were introduced in the field in the mid-1980s rekindling interest in ANNs and their application.

The rapid progress in very large-scale integrated circuitry (VLSI) and parallel computers aided the developments in ANNs and today the field of neural networks constitutes a thriving area of research and de-velopment. In a very comprehensive paper, Hunt et al. in 1992 (see the Bibliography in chapter 18) presented a host of applications of neural networks in Control Engineering and the reader is well advised to refer to this work. The properties that make ANNs particularly applicable to control applications are the following:

• being non-linear by nature, they are eminently suited to the control of non-linear plants,

• they are directly applicable to multi-variable control, • they are inherently fault tolerant due to their parallel structure, • faced with new situations, they have the ability to generalize and

extrapolate. These properties satisfy the fundamental requirements for their use in In-telligent Control. Neural controllers and fuzzy controllers, thus constitute the core of intelligent control.

An ANN is essentially a cluster of suitably interconnected non-linear elements of very simple form that possess the ability of learning and adaptation. These networks are characterized by their topology, the way in which they communicate with their environment, the manner in which they are trained and their ability to process information. ANNs are classified as:

• static when they do not contain any memory elements and their input-output relationship is some non-linear instantaneous func-tion, or

• dynamic when they involve memory elements and whose behav-ior is the solution of some differential equation

Chapter 11 156

11.1 The Elemental Artificial Neuron ANNs are constructed from elemental artificial neurons (which vaguely approximate physical neurons) that are suitably interconnected via branches. The synaptic weights are the gains or multipliers of these branches and uniquely specify the input-output transfer function (func-tional relationship or mapping) of the neural network. The synaptic weights of an untrained network are initially unknown and are deter-mined following training using some network training law which aims at minimizing some measure of the error between the output of the network and the desired output of the network.

A model of an elemental artificial neuron (also referred to as a node) is shown in Figure 11.1. A static neuron has a summer or linear combiner, whose output σ is the weighted sum of its inputs, i.e.:

bxw....xwxw nn2211 +++=σ where w and x are the synaptic weight and input vectors of the neuron respectively, while b is the bias or offset. A positive synaptic weight im-plies activation, whereas a negative weight implies de-activation of the input. The absolute value of the synaptic weight defines the strength of the connection. The weighted sum σ activates a distorting (or compres-sion) element f(i). One form that this element can take is the threshold logic unit, in which the output of the neuron is triggered when the inner product <w,x> exceeds the bias b. There are many variations of the non-linear distorting element, the most common of which are

1) Linear (ADALINE):

f(σ)=σ 2) Binary (Threshold Logic Unit or TLU):

f(σ) = 1 if σ>0 = 0 if σ≤0

Neural Control 157

3) Sigmoid:

f σe σ( ) =

+ −1

1 ∈[0,1]

4) Hyperbolic tangent:

σ

σ

e1e1σf −

−

+−

=)( =½(1+tanhσ) ∈[-1,1]

5) Perceptron:

f(σ) = σ if σ>0

=0 if σ≤0

f(σ) σ

xn wn

w2

w1

x2

x1

b

Figure 11.1 The elemental artificial neuron or no The input to the compression element σ may take on eithering forms, depending on whether the neuron is static or dy

• weighted sum (for the case of static memoryless n

y

de

of the follow-namic:

eurons):

Chapter 11 158

σ w x b w x ; x 1,b wi ii 1

n

i ii 1

n 1

n 1 n 1= + = = == =

+

+ +∑ ∑ • accumulated output (for the case of dynamic neurons with

memory):

∑+

=

+−=1

1

)()1()(n

iii kxwkk σσ

Here k is the time index and it is necessary to store the previous value of the weighted sum σ(k-1).

11.2 Topologies of Multi-layer Neural

Networks As noted earlier, ANNs are clusters of neurons structured hierarchically in a multiplicity of layers. The neurons of a network, depicted as circles in Figures 11.2 and 11.3, are normally structured in layers, resulting in multi-layered ANNs. The input layer of the network is at the lowest layer of the hierarchy while the highest layer corresponds to the output layer and yields the output of the network.

Feed-forward ANNs are networks where information flows suc-cessively from the lowest to the highest layers of the network and no feedback is involved. Figure 11.2 shows an example of a multi-layered ANN belonging to this class. It is observed that the internal or hidden layers of the network communicate with the environment only through the input and output layers. Though an ANN may, in principle, have any number of hidden layers, it has been proved that one hidden layer suf-fices to generate any arbitrary mapping between inputs and outputs.

Depending on the manner in which the various neurons in the net-work are connected, i.e., the network topology or network architecture, the following constitute the principal classes of ANNs:

• Hopfield recurrent network where the nodes of one layer inter-act with nodes of the same, lower and higher layers,

Neural Control 159

• feed-forward networks in which information flows from the lowest to the highest layers,

• feedback networks in which information from any node can re-turn to this node through some closed path, including that from the output layer to the input layer and

• symmetric auto-associative networks whose connections and synaptic weights are symmetric.

Inputs

Input Layer

Hidden Layer

Output Layer

Outputs

Figure 11.2 Feed-forward network

Figure 11.2 shows an example of a multi-layer feed-forward ANN involving an input layer, a single hidden layer and an output layer. This is a very common feed-forward network topology. Figure 11.3 shows a single-layered Hopfield network, which involves feedback. In contrast to feed-forward networks, every node of a Hopfield network is connected to all others. These ANNs can be useful in Soft Control because they possess the following three properties: they

Chapter 11 160

• can learn from experience rather than programming, • have the ability to generalize, • can generate arbitrary non-linear input-output mappings and

are distributed and inherently parallel.

Buses

Outputs

Inputs

Figure 11.3 Hopfield network with feedback

11.3 Neural Control In order to compare any unconventional control method with conven-tional methods it is necessary to enumerate the basic characteristics of each method and to specify a measure of comparison. Thus neural con-trol:

• is directly applicable to non-linear systems because of their ability to map any arbitrary transfer function,

• has a parallel structure thereby permitting high computational speeds. The parallel structure implies that neural controllers have a much higher reliability and fault tolerance than con-ventional controllers,

Neural Control 161

• can be trained from prior operational data and can generalize when subjected to causes that they were not trained with, and

• have the inherent ability to process multiple inputs and gen-erate multiple outputs simultaneously, making them ideal for multivariable intelligent control.

From the control viewpoint, the ability of neural networks to

cope with non-linear phenomena is particularly significant. As is well known, there is no unified theory of non-linear control and there exists a host of scattered techniques capable of giving solutions only to specific cases. ANNs can therefore be used to advantage in the design of nonlin-ear controllers for nonlinear plants, particularly since the design is the result of learning. 11.4 Properties of Neural Controllers Modern intelligent control systems are capable of some degree of auton-omy and the fusion of modern control techniques and Computational In-telligence assures them increased efficiency in changing, vague and un-certain environments. The ultimate objective is: autonomous intelligent systems capable of understanding and adapting to the changes in both the operating conditions and the environment in order to consistently maxi-mize performance. There are many situations where autonomous intelli-gent systems are considered essential, typically in cases of high danger such as in nuclear reactor and armament inspection, space and underwa-ter exploration where intelligent robots are already playing their part. This objective is feasible using the techniques of Computational Intelli-gence.

Both man and machine can control an industrial plant. There is, however, a fundamental difference in the manner in which they do so. Man processes large and seemingly disparate quantities of stimuli com-pared to machines whose stimuli are usually highly limited. The reason for this is not so much the absence of sensors but the manner in which these stimuli are processed. Humans have an inherent ability to process massive quantities of data quickly and efficiently, sorting what is rele-vant and ignoring what is not while fusing information from a variety of sources before arriving at a conclusion. Machines do not have this ability and it is uncertain whether they will in the foreseeable future.

Chapter 11 162

Finally, neural networks have the following properties that make them particularly useful for control: they

• possess a collective processing ability, • are inherently adaptable, • are easily implemented, • achieve their behavior following training, • can be used for plants that are non-linear and multivariable, • can process large numbers of inputs and outputs making them

suitable for multi-variable control, • are relatively immune to noise, • are very fast in computing the desired control action due to

their parallel nature and do not require an explicit model of the controlled process.

11.5 Neural Controller Architectures Widrow and Smith demonstrated the first application of a neural network in Control in the mid-1960s. They used a single ADALINE (ADaptive LInear NEuron) to control an inverted pendulum and showed that this elementary trainable controller was just as effective as a conventional controller after being trained.

The increasing demands for improved productivity, product quality and plant efficiency coupled with the increasing complexity of industrial plants have led inevitably to a search for more advanced con-trol techniques. Since the late 1980s, significant activity in the use of ANNs for identification and control of systems has been observed. Their ease of use, their inherent reliability and fault tolerance, has made ANNs a viable medium for control. Many architectures for the control of plants with ANNs have been proposed since the mid-1980s and the subject still presents considerable interest, not only in research but also practice. An alternative to fuzzy controllers in many cases, neural controllers share the need to replace hard controllers with intelligent controllers in order to increase control quality.

The problem of macroscopic identification of physical plants from normal operational data using multi-layer ANNs reduces to one of finding a dynamic functional relationship between the plant inputs and

Neural Control 163

outputs. The method is well established and is not limited to linear ap-proximants. By way of example, consider the case of a SISO discrete-time system that is to be identified by the ANN shown in Figure 11.4. The input to the ANN is fed from the input to the physical plant. If, fol-lowing training, the output of the ANN is identical to that of the plant then we say that the plant has been exactly identified. In practice, perfect identification is unlikely and the plant can be identified only approxi-mately. The fidelity of the approximation depends on the complexity of the network and is directly related to the order of the neural network and the number of past samples of the input that are used. ANN

Delayor

D2x

Dx

x

y

Figure 11.4 M

By placisignal one samptapped delay lineof it x(k-1), x(k-delay line are thenal y. This signaphysical plant. Tof the difference

Dnx

ulti-layer dynamic neural network model of a SISO plant

ng a series of n delayors D (i.e., elements that delay the le period) in series as shown in Figure 11.4, in effect a , we obtain the input signal x(k) and n delayed versions

2) … x(k-n). The n delayed signals from the transversal n fed to a multi-layer ANN that generates the scalar sig-l is then compared with the desired signal d from the he object of identification is to minimize some measure (d–y), by suitably adjusting (i.e., network training) the

Chapter 11 164

synaptic weights of the ANN. As will be seen in the following, a similar technique is used for the control of plants using ANNs. 11.5.1 Inverse model architecture The success of the back-propagation training algorithm for multi-layer neural networks, which is presented in some detail in the next chapter, was instrumental in opening the gates to a flood of applications of ANNs for control.

d u y

plicoANtivthethepininvinptra

e

Neural Controller

Plant

Figure 11.5 Inverse model structure

A neural controller that has found use in practice due to its sim-

city uses an inverse model of the plant. The method has much in mmon with conventional auto-tuning. During the training phase of the N, which is performed off-line with known training sets, the objec-

e is to establish the inverse relationship P-1 between the output(s) and input(s) of the physical plant, in effect the inverse transfer function if plant is linear. Thus if the physical plant is characterized by the map-g y=P(u), then following training the ANN will ideally generate the erse mapping u=P-1(y) so that the overall relationship between the ut and the output of the closed controlled system is unity, i.e., perfect cking! In keeping with accepted symbolism the output of the neural

Neural Control 165

controller, now the control signal to the plant has been renamed u whereas y refers to the plant output. Network training is based on some measure of the open system error between the desired and the actual out-puts eo=d-y of the closed system. A flow diagram of the method is shown in Figure 11.5.

It should be obvious that the very simplicity of the method is questionable. In theory the method should work but in practice it may not, because identification of an inverse mapping can never be exact, in which case the overall transfer relationship of the system is not unity. In order to work, even partially, the method requires repeated identification at regular intervals, a fact that makes the method impractical for most in-dustrial applications.

u eC

Controller Plant

d y

Figure 11.6 Specialized training architecture 11.5.2 Specialized training architecture The lack of robustness of the previous architecture can be compensated for by using what has been referred to as the “specialized training“ archi-tecture. The flow diagram of this architecture is shown in Figure 11.6 in which the closed system error becomes the driving force. In this architec-ture, the ANN is placed in series with the plant, as in the previous case, but within the closed loop. The result is increased robustness coupled with the advantages of conventional feedback, since training is now based on some measure of the closed system error ec=d-y. Training is

Chapter 11 166

considerably more difficult, however, with this structure, due to the feed-back action. 11.5.3 Indirect learning architecture

The indirect training architecture is more complicated than either of the preceding methods since it involves not one but two dynamic ANNs and its training is considerably more difficult. Here, one ANN is trained to model the physical plant following identification while the second ANN performs the controlling task using a feed-forward network. Both ANNs are trained on-line from normal operating records. A flow diagram of the architecture is shown in Figure 11.7.

Plant Controller

u

e*

e

y*

Plant Model

y d

Figure 11.7 Indirect learning architecture

During the training phase, the simulator ANN learns the func-tional relationship between input(s) and output(s) (i.e., the transfer func-tion) of the physical plant. This is the identification phase, which is based on some measure of the error between the output of the plant and that of the plant model simulator ANN, i.e., e*=y-y*.

Neural Control 167

Training can be either off-line or on-line with random or pseudo-random signals. Simultaneously, the overall error e=d-y is used to train the controller ANN. The advantage of this architecture is that it presents easier training of the controller ANN on-line since the error can be propagated backwards through the simulator ANN at every sampling in-stant.

Chapter 12 Neural Network Training Heuristic knowledge on how to control a plant can be used to train an ar-tificial neural network provided this knowledge is suitably encoded. The resultant neural controller is thus simply an ANN whose synaptic weights are trained with this encoded knowledge using an appropriate training algorithm. Network training is basic to establishing the func-tional relationship between the inputs and the outputs of any neural net-work and considerable effort has been spent on finding faster and more efficient training algorithms which will reduce the time required to train a network. There are two basic classes of network training:

• supervised learning that involves an external source of knowledge about the system or

• unsupervised learning that involves no external source of knowledge and learning relies on local information and internal data.

In the case of supervised learning, the desired outputs of the net-

work for every given input condition are specified and the network learns the appropriate functional relationship between them following repeated application of training sets of input-output pairs. The popular back-propagation algorithm, which is used in many applications, be-longs to this class. This algorithm gets its name from the fact that the synaptic weights of a multi-layer network are adapted iteratively by

169

Chapter 12 170

propagating some measure of the error between the desired and actual output of the network from its output back to its input.

In the case of unsupervised learning, no information on the de-sired output of the network that corresponds to a particular input is avail-able. Here, the network is auto-associative, learning to respond to the different inputs in different ways. Typical applications of this class are feature detection and data clustering. Hebb´s algorithm and competitive learning are two examples of unsupervised learning algorithms. A wide range of network topologies such as those due to Hopfield, Hamming and Boltzmann, also use the same method of learning. In general, these networks, with their ability to generate arbitrary mappings between their inputs and outputs, are used as associative memories and classifiers.

12.1 The Widrow-Hoff Training Algorithm A basic adaptive element is the ADAptive LInear NEuron (ADALINE) that was introduced by Widrow and Hoff in the 1960s. This element has a linear input-output relationship. The element is useful in introducing the fundamental concepts of network training. Clusters of ADALINES can be assembled into complex networks termed MADALINES (for Multiple ADALINEs) which were used in early classifiers and Learning Machines. Network training can be conveniently explained by consider-ing first how the synaptic weights of a single ADALINE may be adapted to generate a given functional relationship between its inputs and its out-puts. It will be seen that the same principle can be used for adapting nonlinear neurons and finally, multi-layer ANNs.

A single neuron with inputs:

x = {x1, x2 … xn, xn+1} ; xn+1=b which, weighted by an array of synaptic weights:

w = {w1, w2 … wn, wn+1} ; wn+1=1 results in the weighted sum:

Neural Network Training 171

σ = {w1 x1 + w2 x2 … + wnxn + wn+1xn+1} = <w,x>

An ADALINE has a linear functional relationship given by f(σ)=σ and can be trained by adapting its synaptic weights provided the desired output d is known. Training is based on some measure of the dis-crepancy or error e=d-y between the desired and actual output of the element.

Widrow and Hoff presented the first successful training algo-rithm for ADALINES in the 1960s. They named their training procedure the LMS algorithm (for Least Mean Squares), using the squared error for each training set as the objective function, i.e.

∑=

=m

1i

2ieJ

which is computed for all the elements of the training set. We seek the values of the synaptic weights that minimize the objective function. The inputs to the ADALINE may be either binary or real, whereas the values of the synaptic weights are always real and may be positive or negative, signifying activation and de-activation respectively. In terms of the in-puts and the synaptic weights of the element, the error is:

e = d - <w,x>

where <w,x> is the inner product of the synaptic weight vector and the input vector and thus the squared error is:

e2 = d2 – 2d<w,x> + <w,<x,x>w>

whose expected value is:

E(e2) = E(d2) - 2E(d<w,x>) + E(<w,<x,x>w>) The partial derivative of the expected value with respect to the synaptic weights is:

Chapter 12 172

=)( 2

weE

∂∂

-2p + 2Rw (12.1)

where

p = E(dx) and R = E(<x,x>)

R is the power of the input signal while p is the expectation of the prod-uct of the desired output and the input vector. Setting the partial deriva-tive in Equation 12.1 to zero yields the optimum synaptic weight vector:

w* = R-1p

which is known as the Wiener solution. If the matrix R and the vector p are known explicitly then the optimum synaptic weight vector w* can be computed directly. In practice, however, R and p are not known and the Wiener solution unfortunately cannot be used.

Widrow and Hoff observed that using the partial derivative of the squared instantaneous error

2k

kew

∂∇ =

∂

instead of its expectation, made little difference to the final result. This observation simplified network training significantly since the squared error sensitivity could now be expressed in terms of the synaptic weights and the error directly, i.e.,

2 2 ( , ) 2kk k k k k k k

k k

ee e d w xw w∂

ke w∂∇ = =− −< > =−

∂ ∂

The Widrow-Hoff algorithm can be expressed as the iteration:

kkkkkkk1k xe2 w~ w w ww λλ∆ +=∇−=+=+ (12.2) where λ is some positive constant that defines the rate of convergence of the algorithm. This expression shows how the synaptic weights at the

Neural Network Training 173

following iteration of the training procedure must be adapted. The search direction that must be followed is dependent on the sign of the gradient: thus if the gradient is negative then clearly the search trajectory is tend-ing towards some minimum. The algorithm is initialized with arbitrary initial synaptic weigh vector w0 and a new training set is applied to the element. The iteration is terminated once the squared error in that itera-tion falls below some acceptable value. At each iteration the change in the error is

∆ek = ∆(dk - <wk ,xk>) = -<xk ,∆wk>

and using the fact that the change in the synaptic weight is

∆wk = 2λek xk it follows that

∆ek = -2λ<xk ,xk>ek = -α ek

This means that the error is reduced by a factor of α at each it-eration. This factor thus plays a major role in the stability of convergence of the algorithm and must lie in the range 0<α<2. In practice values of α in the range of 0.1<α<1 are preferred. Too small a value of α leads to very slow convergence while a large value leads to oscillations and ulti-mately instability.

The Widrow-Hoff training algorithm has the advantage of sim-plicity but is plagued by slow convergence. Training algorithms are the subject of considerable research and algorithms with vastly increased training rates have been developed. The back-propagation algorithm, which was the major breakthrough that opened up the field of ANN, is one of the most popular, though by no means the fastest or most reliable. This algorithm is presented later in this chapter. 12.2 The Delta Training Algorithm The original Widrow-Hoff algorithm must be modified when the neural element contains a nonlinear element. However, in order to develop a systematic training procedure for the general case, it is necessary to place

Chapter 12 174

certain restrictions on the non-linear element. As will be seen below, the fundamental condition on the non-linear element is that it must be uni-formly differentiable, i.e., the non-linearity must not exhibit discontinui-ties.

Given a neuron with σ=<w,x>, output y=f(σ) and error e=d-y, the error sensitivity is given by the partial derivative:

( ) ( ) ( )e y f f g xw w w w

σ σ σ σσ

∂ ∂ ∂ ∂ ∂=− =− = =−

∂ ∂ ∂ ∂ ∂

It is obvious that for the error sensitivity to exist everywhere, it

is necessary that the gradient of the nonlinear function g(σ) = f’(σ) exist for all values of σ. This implies that the nonlinear function must be dif-ferentiable or smooth. In the special case of a linear neuron clearly g(σ)=1.

The development of the Delta algorithm follows that of the Wi-drow-Hoff algorithm closely. Here, the squared error sensitivity is

kkkkkk

kk

kk

k

2k

k xge2wσ

σfe2

wfe2

we

e2we

−=∂∂

∂∂

−=∂∂

−=∂∂

=∂∂

=∇~

which shows how the gradient g of the nonlinear function f enters the training algorithm. This expression is very similar to that of the Widrow-Hoff algorithm except that an additional term has been added.

The Delta training algorithm is given by the iteration

kkkk

kkkk1k xge2w~w∆www λλ +=∇−=+=+ (12.3)

In order to accelerate convergence, it is possible to vary the value of λ according to the progress achieved in training. Thus, at each iteration, for example, the value of λ may be made a function of the norm of the change in the synaptic weights. If the rate of change of the norm of the synaptic weights drops below some pre-specified lower limit, then the value of λ is doubled whereas if the error norm rises above some up-per pre-specified limit then the value of λ is halved. This is perhaps the simplest possible approach to adapting the value of the multiplier λ, and much more sophisticated ways are available but are beyond the scope of this book.

Neural Network Training 175

12.3 Multi-layer ANN Training Algorithms The training algorithm for a single nonlinear neuron is straightforward and is a variation of the original Widrow-Hoff algorithm. A multi-layer ANN comprises an input layer, a hidden layer and an output layer, each of which involves many nonlinear neurons. Indeed some ANNs used in pattern recognition, speech analysis and synthesis involve thousands of neurons. Often one or more of the layers of an ANN involves linear neu-rons. Unfortunately, however, there is no way to know a priori how many neurons are required in each layer with which to generate a speci-fied input-output mapping. At this time, the choice is arbitrary and a matter of experimentation. Training is performed first for a given num-ber of neurons, the number is then reduced and the ANN is re-trained. The number of neurons is further reduced at each successive training session until the measure of the error starts to increase.

For control applications, it is essential that the size of the ANN in the neural controller be limited both in the number of layers and the number of neurons per layer. Fortunately, minimal neural networks in-volving an input layer, a single hidden layer and an output layer, with a total of less than 10 nodes have proved quite successful in practical neu-ral controllers. This small size implies fast training and re-training should the circumstances require this.

The problem of supervised learning of a complex multi-layer ANN must be viewed as an iterative algorithm for systematically mini-mizing some measure of the discrepancy between the inputs and outputs of the network by adapting the synaptic weights and biases in each layer of the network. The structure of the iterative training algorithm for adapting the synaptic weights of the network is shown in Figure 12.1. The basic training algorithm for a multi-layer ANN is a variation of the Delta algorithm for a single neuron. In this case vectors are replaced by matrices whose dimensions depend on the layer number and the number of neurons in each layer. In the next section back-propagation, the most popular network-training algorithm is presented.

Chapter 12 176

12.4 The Back-propagation (BP) Algorithm A multi-layer neural network is characterized by the number of neurons it possesses in each layer. Thus, for instance, a 30-50-10 ANN has 30 neurons in the input layer, 50 in the second or hidden layer and 10 neu-rons in the output layer. An analysis of the back-propagation algorithm is presented be-low in simplified form so that it can be easily understood. Generalization of the algorithm to multi-layered ANNs with many neurons is relatively straightforward but requires multiple indices on the various synaptic weight vectors for each layer being considered. In order to avoid unnec-essary complications we will consider a simple 2-2-1 ANN involving two neurons in the input layer, two in the hidden layer and one in the output layer. This minimal ANN has, incidentally, been used success-fully in a neural controller for a large mill used for pulverizing coal for a cement kiln as well as a controller for a finish cutting lathe.

x y

e d

w

Training Algorithm

Figure 12.1 The structure of the network training procedure

Neural Network Training 177

The signals at the inputs and outputs of the neurons of the first layer are respectively:

)(;)(;

22232221122

11132211111

σfywxwxwσσfywxwxwσ

=++==++=

while the weighted signal at the output of the second (output) layer is:

σ3=v1y1+v2y2+v3 ; y3=f(σ3)

The synaptic weights of the first layer are given by the vector w and those of the second layer by the vector v. The error is

e = d - y3 = d - f(σ3)

v3

w23

w13

w12

w21

w11

w22

v1

v2

1

N

N 1

N

1

y3

x1

x2

Figure 12.2 An examp

l

e of a simple 2-2-1 ANN

Chapter 12 178

The partial derivatives of the squared error with respect to the synaptic weights in each layer therefore are

31j21iwee2

we

ijij

2

,;,; ==∂∂

=∂∂

31ivee2

ve

ii

2

,; =∂∂

=∂∂

Following the steps taken in developing the Delta training algo-

rithm presented in the previous section, the synaptic weights of the first layer must be adapted according to

kkkk1k λe2W∆WWW +=+=+kW

e∂∂

where W={wij} and the weights of the output layer as

kkkkk1+k e2+v=∆v+v=v λkv

e∂∂

The training algorithm thus requires evaluation of the partial de-

rivatives of the errors with respect to the synaptic weights of both layers, i.e. the error sensitivities

We

∂∂

and ve∂∂

Dropping the iteration index k to simplify notation, the chain rule is used to derive the first three partial derivatives for the first layer, which are:

Neural Network Training 179

3 3 3 3 3 1 13 1 1

11 11 3 11 3 1 1 11

3 3 3 3 3 1 13 1 1

12 12 3 12 3 1 1 12

3 3 3

13 13 3

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )

( )

y f fe f1

2

g g v xw w w f w

y f w fe f g g v xw w w f w

y few w

σ σ σ σ σ σ σ σσ σ σ σσ σ σ σ σ σ σσ σ σ σσ σσ

∂ ∂ ∂ ∂ ∂∂ ∂ ∂= − = − = − = −

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂∂ ∂ ∂

= − = − = − = −∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂∂= − = −

∂ ∂ ∂ ∂3 3 1 1

3 1 113 3 1 1 13

( ) ( ) ( ) ( )( )

f f g g vw f w

σ σ σ σ σ σσ σ σ

∂ ∂ ∂ ∂= − = −

∂ ∂ ∂ ∂

The reader is encouraged to derive the remaining three partial

derivatives following the same procedure. It is noted that given the in-put-output relationship of the compression function analytically, the de-rivative g can be readily derived. Thus for the tanh non-linearity, used very commonly as the distorting or compression function of the node:

σ

σ

e1e1σf −

−

+−

=)( and 2σe1

σ2σfσg)(

)(')( −+==

In the same manner, the partial derivatives for the neuron in the

second layer are

3 3 33 1

1 1 3 1

3 3 33 2

2 2 3 2

3 3 33

3 3 3 3

( ) ( )

( ) ( )

( ) ( )

y fe g yv v v

y fe g yv v v

y fe gv v v

σ σ σσσ σ σσσ σ σσ

∂ ∂ ∂∂= − = − = −

∂ ∂ ∂ ∂∂ ∂ ∂∂

= − = − = −∂ ∂ ∂ ∂

∂ ∂ ∂∂= − = − = −

∂ ∂ ∂ ∂

These expressions are especially simple if the propagation is

through one layer only. Thus given some random values for the synaptic weights in all layers of the ANN and the known input vector, the first pass is made during which the signals σ1, σ2, σ3 are computed. Computa-tion of the neuron outputs y1, y2, y3 then follows from which the various nonlinear functions and their derivatives (f,g) are computed.

At every iteration of the algorithm, the corrections to the synap-tic weight matrix ∆Wk for the first layer and the weight vector ∆vk for the second layer are computed. These corrections are then added to the pre-vious synaptic weights and the algorithm is repeated until convergence,

Chapter 12 180

defined as satisfaction of some error measure, is achieved, whereupon the algorithm is terminated. It is obvious that for complex ANNs con-taining many layers and neurons in each layer, it may be necessary to perform many thousands of iterations before convergence is achieved and clearly computers with high computational speeds are essential.

Finally, it must be noted that the back-propagation algorithm, though convenient and used extensively for training ANNs, is by no means the fastest training algorithm. Many variations of this algorithm have been developed by additional terms to the synaptic weight adapta-tion vector which take into account such factors as the momentum of learning. This additional term results in a significant increase in the rate of convergence of the training algorithm.

W0 v0 x

Training Algorithm

d

λ σ1 σ2 y1 y2

σ3 y3

∆W ∆v

First layer

Figure 12.3 Flow chart for the back-propagation algorithm for the 2

Sec-ond

-2-1 ANN

Chapter 13

Rule-Based Neural Control

Νeural controllers that can be trained from linguistic rules are a rela-tively new concept in Soft Control. As in the case of heuristic fuzzy con-trollers, it is expected that rule-based neural controllers will play their part in furthering the diffusion of Computational Intelligence to Control Engineering. As noted earlier, Soft Control is based on the knowledge and experience of human operators who are trained to control a plant us-ing linguistic rules of the classical IF (cause) THEN (effect) ELSE form. In the previous chapters, methods were presented for processing heuristic linguistic control rules of this form using Fuzzy Logic. In this chapter we discuss a simple, yet effective, technique with which an artificial neural network embedded in a neural controller can be trained from linguistic rules. This technique can be considered as a special case of neuro-fuzzy control that was examined in the previous chapter.

The neural networks used in control are either static, i.e., they contain no elements with memory, or dynamic, containing delayors that take account of the plant’s past history. Furthermore, these neural net-works are characterized by the fact that they are small involving only a small number of neurons and at most one hidden layer. Simplicity being of essence in industrial controllers where high reliability and robustness are of primary importance, it is not uncommon to find neural controllers with as few as 10 neurons.

181

Chapter 13 182

13.1 Encoding Linguistic Rules The training of a neural network is normally performed with a numerical training set. There are numerous commercially available software pack-ages from a number of vendors, notably MATLAB and its Neural Tool-box, Neural, Neuralware, etc., which make this task a relatively simple matter. As control rules involving linguistic variables cannot be used di-rectly for network training, it is therefore necessary to encode the lin-guistic rules into numerical form prior to using them in existing network training packages. The problem reduces to one of finding a suitable map-ping

T: R(L) → R(Q)

that maps the linguistic (i.e., qualitative) rule set R(L) into a correspond-ing numerical (i.e., quantitative) training set R(Q). This procedure has been proved very effective in training neural controllers for a number of applications, from large-scale industrial plants to mechatronic systems using even simple transformations.

The following simple example shows the procedure for trans-forming linguistic rules into numerical data. Defining the linguistic vari-ables NL = Negative_Large, NSM = Negative_Small, OK =Normal, PSM = Positive_Small, PL = Positive_Large then a one-to-one transfor-mation of linguistic variables into their numerical equivalents defined in the normalized range [-1,1] could be as follows:

[NL|NSM|OK|PSM|PL] → [-1.0|-0.5|0|0.5|1.0]

This is an example of a linear mapping that follows human intui-tion. Non-linear mapping may be used when necessary to compensate for any non-linear behavior of the plant being controlled. Thus a rule with three input variables, INPUT_1, INPUT_2 and INPUT_3 and output variable CONTROL_VARIABLE:

IF INPUT_1 is Negative_Small AND INPUT_2 is Positive_Small AND INPUT_3 is OK

Rule-Based Neural Control 183

THEN CONTROL_VARIABLE must be Positive_Large

will appear as the training string: [-0.5, 0.5, 0] → [1]. This training string can be used directly to form the training set of a neural network of specified topology. The corresponding MATLAB Neural Toolbox state-ment is simply:

P=[-0.5 0.5 0.0]; inputs

while the corresponding output is specified by the statement: T=[1.0] ; output

The training set is the collection of all the (P,T) pairs corresponding to every linguistic rule in the rule base. 13.2 Training Rule-Based Neural

Controllers The training of rule-based neural controllers of specified topology is per-formed off-line. The results of this training are the optimum values of the synaptic weights and the biases of the neural network that yield the re-quired control surface. These weights are subsequently downloaded to the real-time version of the controller. Should it prove necessary to mod-ify or add any rule then the network must be re-trained off line. Fortu-nately, small changes require very short training times since the network parameters will be initialized with its previous known values rather than entirely random values as in the case of a new controller.

In order to train the network it is necessary to map all the N lin-guistic rules contained in the rule base in accordance with the chosen mapping. For the earlier example, the numerical strings for each rule (i.e., the (P,T) pairs) are concatenated to form the training string:

P=[-1.0 -1.0 -1.0; -0.5 -0.5 0.5; 0 0.5 -0.5; ...] ; string of N triplets

Chapter 13 184

In a like manner, the corresponding output training set of a multi input - single output neural controller can be stated as follows:

T=[0 0.5 -0.5 ...... -1.0] ; string of N corresponding

outputs

This training set is used to train the neural network, hundreds or even thousands of times in random order until the synaptic weights con-verge. The initial estimates of the unknown synaptic weights are taken randomly. At every epoch (i.e., iteration of the training algorithm), the synaptic weights are updated in accordance with the training algorithm used. The back-propagation algorithm (BP) is by far the most popular training algorithm, though it often converges slowly, particularly when the network contains many neurons and layers. Fortunately, in the major-ity of control applications neural controllers normally contain very few neurons and training is fast. Learning is considered completed when some measure of the error between the desired and actual outputs of the network reaches some acceptable limit, or the number of epochs reach some upper limit.

Example 13.1 Design of a rule-based neural controller for a mechatronic system

A schematic diagram of a mechatronic device involving a finish lathe is shown in Figure 13.1. The cutting tool, on which a force sensor is attached, is forced to follow a desired force on the object being turned on the lathe. The quality of finish of the object is critically dependent on the rate with which the metal is cut. It is important where a high quality finish is required that the depth of cut be controlled with great accuracy. In most existing cutting lathes this is achieved with conventional two-term (PI) controllers. Here we will present how an unconventional controller may be used with equal or better results.

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185

The minimal neural network shown in Figure 13.2 will be used in

the neural controller. The network will be trained using linguistic rules. The controller has two inputs: the error between desired and actual force ek and the change in error Dek=ek-ek-1.

The incremental output of the network is Duk= F(gpek,gdDek) while the output of the neural controller is simply:

uk= uk-1+Duk=uk-1+F(giek,gpDek)

r

r

Controller

Lathe

Figure 13.1 A force-cutti

Finally, the output of the controller iplied to the plant as:

ck=gc uk

The parameters gi, gp and gc are thcontroller, necessary to convert the inputs to[-1,1]. The value for gc=(ck/uk)max while the(gi,gp) are tuned on-line or obtained in an ideNichols method. The linguistic control rules required process are shown in the form of the clinguistic rule matrix in Figure 13.3.

Force Senso

Actuato

ng lathe

s weighted before being ap-

e normalizing gains of the the controller into the range

pair of controller parameters ntical manner to the Ziegler-

to control the lathe cutting heckerboard pattern in the

Chapter 13

186

Figure 13.2 Minimal neural controller for the mechatronic system

Both the ERROR (e) and the CHANGE_OF_ERROR (De) are quan-tized into five elements (equal to the number of linguistic variables as-signed to each controller input) over the normalized ranges [-1,1]. Here, each controller input has been assigned seven linguistic variables and consequently the Linguistic Rule Matrix (LRM) or Fuzzy Associative Memory (FAM), contains 49 possible rules not every one of which is specified. Typically some 20-30% of the possible elements of the FAM are specified. In the example shown in Figure 13.3, the FAM contains 13 of the 49 possible rules.

De\e NL NM NMS OK PMS PM PL

PL OK PM PL

PM

PMS OK PMS

OK NM OK PM

NMS NMS OK

NM

NL NL NM OK

Figure 13.3 Linguistic Rule Matrix (LRM) or Fuzzy Associative Memory (FAM)

{v}

{w} e

De

Du

1

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187

Here PL = Positive_large, PM = Positive_Medium, OK = Normal, NS = Negative_Small, PS = Positive_Small, NM = Negative_Medium, PMS = Positive_Medium_Small and NMS = Negative_Medium_Small.

If the linguistic variables for the inputs to the controller are as-signed the following numerical values:

NM=-0.8. PM=0.8. OK=0. NS=-0.3. PS=0.3. NM=-0.8. PM=0.8

and the corresponding controller output:

NM=-1. PM=1. OK=0. NS=-0.35. PS=0.35. NM=-0.55. PM=0.55

then the numerical fuzzy associative memory (NFAM) matrix is shown in Figure 13.4. The magnitude of the control variable is specified in the darkened elements of the matrix. All other elements are left blank. The neural network will perform the necessary interpolation.

0

0

0

0

0

-1

+10.55

0.35

-0.35

0 +1 -1 e

0

+1

De

-1

0.55

-0.55

-0.55

Figure 13.4 Numerical FAM

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188

The training set comprising 13 rules is given below:

Rule e De Du

1 0.8 0.8 1 2 0.8 0 0.55 3 0.8 0.8 0 4 0.3 0.3 0.35 5 0.3 -0.3 0 6 0 0.8 0.55 7 0 0 0 8 0 -0.8 -0.55 9 -0.3 0.3 0 10 -0.3 -0.3 -0.35 11 -0.8 0.8 0 12 -0.8 0 -0.55 13 -0.8 -0.8 -1

The complete network training program in MATLAB is given in

Appendix D. On executing this program the following synaptic weights are obtained:

w11= 0.7233 w21= 1.0764 v1= 0.798 v2= 0.2237 w12= 0.2674 w22= -0.9706

where wij are the synaptic weights of the first layer and vi the synaptic weights of the second layer. The corresponding biases are:

b1=0.6312 b2=0.166 b3=-0.5403

Figure 13.5 shows the Network Learning Rate and the Network Er-

ror (i.e., the sum of squared errors) as functions of the number of ep-ochs. It is noted that less than 100 epochs are required to bring about conversion. This translates into a few seconds of computational time de-pending on the platform used. The performance of the closed system with the neural controller compares favorably with that of a conventional two-term controller in Figure 13.6, proving that unconventional control can be as good as con-ventional control and often superior.

Rule-Based Neural Control

189

0 20 40 60 80 1000

2

4

6

8

Epoch

Network Error

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

Epoch

Network Learning Rate

Figure 13.5 Network Errors and Network Learning Rates as functions of epochs

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

Time (second)

Figure 13.6 Comparison of step responses of the mechatronic system with the conventional controller (continuous line) and the neural controller (crosses)

Chapter 13 190

Example 13.2 Rule-based neural control

of a coal mill A schematic of a large vertical mill which is used to pulverize coal which is fed to a rotary kiln producing clinker at one of Europe’s largest cement manufacturers. This coal is injected into a rotary kiln where it self-ignites. The kiln produces clinker which is the principal constituent of cement (see also Example 9.2). Every cluster of cement kilns has one or more adjoined coal mills that feed them with pulverized coal. The coal mills normally operate discontinuously and on demand. When in operation, a coal mill normally produces more pulverized coal than a kiln can consume and thus an intermediate storage silo is used to store excess production. When the level of coal in the silo drops below some predetermined level, the coal mill is automatically started. Due to the danger of fire because of the combustibility of the coal, special care must be taken to maintain the coal mill environment inert. Conventional multivariable controllers have not been successful in controlling this process, a task normally left to human operators.

The process and the principal control and controlled variables are shown in Figure 13.7. The operator manipulates three principal vari-ables: the raw feed to the coal mill (FEED), the air feed to the mill (HADR) and the pulverized coal returns (RDPR) and his decision is based on measurements of the Differential Pressure in the mill (DP), the EXit Temperature of the pulverized coal (EXT) and the Under-Pressure in the mill (UP). The operator sets the nominal values for both the inputs and the outputs of the controller and the objective of the controller is to maintain the process in the desired state despite disturbances to the op-eration of the process from external sources.

Human operators are trained to control this process using a set of linguistic rules. It is interesting to note, however, that despite this com-mon training set, human operators develop their own control strategy over time which often vary from operator to operator. It is not surpris-ing, therefore, that plant performance and productivity vary accordingly. Operational consistency with human operators is desirable but rarely ob-tained in practice. Intelligent controllers, in contrast, assure such consis-tency on a 24-hour basis.

In the specific example a set of 125 rules (5×5×5) were elicited from human operators, examined for conflict and consistency and formed the knowledge from which the controllers were trained.

Rule-Based Neural Control 191

Five linguistic variables are used for each controller input, sufficient to yield the desired accuracy, whereupon the overall FAM has 3×5×5×5=375 rules. In practice this number is unwieldy and it is sim-pler to design three independent three-input single-output sub-controllers which are executed sequentially. Here the sub-controllers have identical causes but different effects. The three sub-controllers can consequently be trained independently. Fewer than 125 rules are neces-sary in practice to achieve satisfactory control. Rule pruning was sys-tematically performed with a view to reducing the number of control rules without loss of controller performance. This led to a reduced train-ing set of 65 rules, an example of which is given below: R: IF DP is Very_High AND EXT is HIgh AND UP is OK THEN FEED is Very_High AND HADR is OK AND RDPR is OK

RDPR

UP HADPR

EXT

DP

FEED

HADPR(0)FEED(0)RDPR(0)

UP(0)EXT(0)DP(0)

CONTROLLER

Figure 13.7 Rule-based neural control of a coal mill

Chapter 13 192

The controller algorithm is resident in the supervisory control sys-tem and is executed every 10 seconds. If the resultant incremental con-trol actions are small, indicative of stable operation, they are ignored and the process is maintained at its previous state.

Two such neural coal mill controllers have been in continuous op-eration since 1992 and have consistently led to energy demand reduction of approximately 5% and a comparable increase in productivity, food for thought for anyone still doubting the economics of intelligent control!

Chapter 14 Neuro-Fuzzy Control Neuro-fuzzy controllers constitute a class of hybrid Soft Controllers that fuse fuzzy logic and artificial neural networks. Though the principles of fuzzy logic and artificial neural networks are very different, the two techniques have important common features: fuzzy logic aims at repro-ducing the mechanism of human cognitive faculty while neural networks attempt to emulate the human brain at the physiological level.

In fuzzy controllers linguistic rules embody the knowledge on how to control a physical plant. In a neural controller this knowledge is embedded in the structure and the synaptic weights of the network. Feed-forward processing in ANNs is analogous to the inference engine in fuzzy logic. Fuzzy controllers use fuzzy compositional rules to arrive at their decisions, require fuzzification of the input variables and de-fuzzification of the composite fuzzy set of the output in order to obtain a crisp output from the controller. In contrast, neural controllers use simple arithmetic techniques, operating directly in the physical world. In both cases, current data from the physical plant being controlled is stored in a real-time database and then processed by an appropriate algorithm. Only in the manner with which the two techniques arrive at this control action do they differ radically. The ability to generalize, i.e., to extrapolate when faced with a new situation, is a feature common to both.

Evolving from very different origins, fuzzy and neural control-lers developed independently by researchers with very different back-grounds and very different objectives. Scant thought was given to the

3

19

Chapter 14 194

possibility of combining the two at the time. It did not take much time, however, for control engineers to realize that the operations of a fuzzy controller could be implemented to advantage with artificial neural net-works, which, because of their inherent parallelism and their superior computational speed, could lead to controllers capable of significantly higher bandwidth as required in a number of critical situations.

Neuro-fuzzy control has been the subject of numerous books and it is beyond the scope of this book to delve in depth into the various ar-chitectures that have been proposed. This chapter presents a brief intro-duction on how the two techniques can be combined and how the fuzzy controller algorithm can be implemented with artificial neural networks.

14.1 Neuro-Fuzzy Controller Architectures It is logical to examine the fusion of fuzzy logic and ANNs with a view to developing hybrid neuro-fuzzy controllers that possesses the best at-tributes of both techniques in the hope that this will lead to a superior class of intelligent controller. In principle it should be possible to “neu-ralize” a fuzzy controller or “fuzzify” a neural controller. It is useful, therefore, to state the principal characteristics of hybrid neuro-fuzzy con-trollers:

• they possess an architecture derived from both techniques, • they have elements of both fuzzy and neural controllers, each of

which performs a separate task and • their design methodology is a combination of the two

techniques.

A number of neuro-fuzzy controller architectures have been pro-posed, each with features that make them suitable for specific applica-tions. Considering fuzzy and neural elements as distinct entities, it is possible to construct a controller structured in layers, some of which are implemented with neural elements and others with fuzzy elements. A fuzzy element can, for instance, act as supervisor to a neural element that controls some conventional industrial three-term controller.

The following characteristics of ANNs are useful in implement-ing fuzzy controllers: they

Neuro-Fuzzy Control 195

• use a distributed representation of knowledge, • are macroscopic estimators, • are fault-tolerant and • can deal with uncertainty and vagueness.

14.2 Neuro-Fuzzy Isomorphism

The inference mechanism in a fuzzy controller follows a series of sys-tematic steps (the fuzzy algorithm) and in the process justifies its deci-sions by identifying which rules have been fired and what contribution each rule had on the final decision. There is no similar mechanism in a neural controller, which is consequently unable to justify its decisions. In this sense a neural controller is little more than a black box capable of performing an arbitrary functional mapping of its inputs to its outputs.

With the term “neuralization” we imply the use of ANNs in im-plementing fuzzy controllers. It is possible, for instance, to represent each linguistic rule with a separate ANN, maintaining thereby an iso-morphism (i.e., the same form) between the two techniques. Alterna-tively, it is possible to represent each linguistic rule with many neurons and synaptic weights using only one ANN. This destroys the isomor-phism, but leads to a much simpler implementation. In the first case, the resultant controller is unduly complicated, whereas in the second the re-sultant controller has a compact structure with few neurons and very few layers. With the term “fuzzification”, concepts of fuzzy logic are intro-duced in ANNs in which case the resultant controller will have neural equivalents of the basic fuzzy operators, e.g., min, max, max product and others.

The hybrid neuro-fuzzy controller that is described in the follow-ing comprises a multi-layered feed-forward network that implements the elements of a fuzzy controller using connectionist techniques. This ex-ample of neuralized design shows how it is possible, in principle, to sub-stitute the elements (or building blocks) of a fuzzy controller with neural equivalents. The resultant hybrid neuro-fuzzy controller uses a variety of dissimilar neurons and is certainly not the most practical or economical solution to neuro-fuzzy control. The structure of this neuro-fuzzy con-troller is interesting though impractical, however, because it retains the

Chapter 14 196

flow of operations of a fuzzy controller while emphasizing the concept of isomorphism.

d1 y1 dm ym

Figure 14.1 Multi-layered hybrid neuro-fuzzy controller (for simplicity not all branches are shown)

Figure 14.1 shows a multi-layer neural network of the hybrid

neuro-fuzzy controller. It is observed that each layer of the neural net-work has a fuzzy equivalent. The causes and effects (i.e., inputs and out-puts) of the network correspond to the input and output nodes of the net-work, while the hidden layers perform the intermediate operations on the fuzzy sets while embedding the knowledge base.

Every node in the second layer of the network performs a non-linear mapping of the membership functions of the input variables. The second layer involves a cluster of neurons that have been trained a priori

Layer 1

Layer 2

Layer 3

Layer 4

u1 u2 un

Neuro-Fuzzy Control 197

to map the desired membership functions. The nodes in the third layer perform the same function as the knowledge base in a fuzzy controller, while the connections between the second and third layers correspond to the inference engine in a fuzzy controller. The nodes in the third layer map the membership functions of the output variables. In the fourth and final layer, there is only one node for the output of the network and a node from which the training sets are introduced to the network.

The various neurons of each ANN are shown as nodes in Figure 14.1 and have properties which depend on the layer to which they be-long. The relation between the inputs and output of an elemental neuron is, as before, simply:

σ= wii=1

pui∑ where y = f(σ)

where y is the output, ui are the inputs, σ is the weighted sum, wi the syn-aptic weights, and f(•) the non-linear (compression) function of the ele-mental neuron. There are p inputs and p+1 synaptic weights for each neuron to account for the bias term.

In the first layer, each neuron distorts the weighted sum of the inputs so that it corresponds to the membership function. For example, if the fuzzy set is Gaussian, then

2ij

2iji

s

ξ(x -=σ

)− and f(σ)= eσ

where ξij are the centers and sij the standard deviations of the member-ship functions. Here, the synaptic weights of the first layer of the net-work must be equal to the centers of the membership functions, i.e., wij= ξij. Using Mamdani’s fuzzy compositional rule for instance, the nodes in the second layer of the controller perform the AND operator (i.e., min) in which

σ=min(u1, u2….up) and f(σ)=σ

The neurons in the third layer perform the OR operator required in the final stage of the fuzzy compositional rule. Here the synaptic weights in this layer are unity and

Chapter 14 198

σ==∑uii

p

1

and f(σ)=min(1,σ).

The fourth layer of the network possesses two sets of nodes. The

first set is required for de-fuzzification and in the case where the fuzzy sets of the output are also Gaussian then the synaptic weights of these nodes and the non-linear compression function are defined by

wij = ξijsij and ∑

=iijus

σσ)f (

The second set of nodes in this layer transfer the elements of the

training sets to the network in a reverse direction, i.e., from the output to the input of the network and here σi = di.

The training of the network is performed in two phases, at the end of which the parameters (ξij,sij) of the neurons in the first and third layer are determined. During this phase, the network also learns the con-trol rules, storing this knowledge in the synaptic weights of the connec-tions between the second and third layer.

Example 14.1 Design of a simple 2-input 1-output neuro-fuzzy controller

A simple example of a hybrid neuro-fuzzy controller is considered here. The controller has two inputs and one output. It will be assumed that each input is assigned three fuzzy variables LOw, OK and HIgh. As-sume, furthermore, that the fuzzy sets of the input variables are triangu-lar, as shown in Figure 14.2.

Each input to the network is fed to three ANNs, each of which has an input-output relationship (i.e., mapping) approximating the corre-sponding fuzzy set. The output of each ANN thus represents the corre-sponding membership function µ1

i, µ2i and µ3

i.

Neuro-Fuzzy Control 199

µ LO

OKHI

ui

i

Figure 14.2 Fuzzy sets of the input variables

Furthermore, assume that the knowledge base comprises five rules dis-tributed in control space (i.e., FAM) as shown in the tile diagram of Fig-ure 14.3.

u2\u1 LO OK HI

HI R1: OK R2: HI

OK R3: OK

LO R4: LO R5: OK

Figure 14.3 Rules in control space

Chapter 14 200

R1

2u

y

R2

R3

R4

R5

1u

σ

σ

σ

σ

σ

1

2

3

4

5

Figure 14.4 The inference mechanism

Given the membership values for each input, the nodes in the sec-ond layer compute the degree of fulfillment of each rule, as shown in Figure 14.4. For the given input values, only rules R3 and R5 are fired and the corresponding degrees of fulfillment are 0.2 and 0.5. Figure 14.5 shows the neuro-fuzzy controller with two inputs, five rules and one out-put. The five rules in the knowledge base are embodied in the second layer. The connections between the nodes in this layer and the next are determined by the rules. Thus, for example, the node corresponding to rule R3 has connections from the nodes representing the fuzzy sets OK and OK respectively while the node for rule R5 is linked to the nodes HI and LO of the first layer.

Neuro-Fuzzy Control 201

Output

3 5

De-fuzzification

Output Fuzzy Set

5 Rules

LO:OK:HI

R1:R2:R3:R4:R

LO:OK:HI

u1

Input Fuzzy Sets

u2

Figure 14.5 The simple neuro-fuzzy controller

with 2-inputs, 1-output and 5 rules

The outputs of the nodes of the second layer are the degrees of ful-

fillment of each rule. The fuzzy sets of the contributions from each rule that has fired are combined (using the union operator) in the third layer and finally the fourth and final layer performs the task of de-fuzzifying the output fuzzy set to yield a crisp output at the output of the neuro-fuzzy controller.

Chapter 15 Evolutionary Computation The design of intelligent controllers based on unconventional control techniques will undoubtedly become increasingly common in the near future and these developments will rely heavily on the use of the sto-chastic methods of Soft Computing in seeking optimum results. These hybrid methods offer a new and very exciting prospect for Control Engi-neering, leading to solutions to problems that cannot be solved by con-ventional analytical or numerical optimization methods.

Although stochastic methods of optimization are computer-intensive, the impressive progress that has been observed in computer hardware over the past decades has led to the ready availability of ex-tremely fast and powerful computers that make stochastic techniques very attractive. One of the ascending techniques of Intelligent Control is the fusion of Fuzzy and Neural Control with Evolutionary Computation.

Evolutionary Computation is a generic term for computational methods that use models of biological evolutionary processes for the so-lution of complex engineering problems. The techniques of Evolutionary Computation have in common the emulation of the natural evolution of individual structures through processes inspired from natural selection and reproduction. These processes depend on the fitness of the individu-als to survive and reproduce in a hostile environment. Evolution can be viewed as an optimization process that can be emulated by a computer. Evolutionary Computation is essentially a stochastic search technique with remarkable abilities for searching for global solutions.

203

Chapter 15 204

There has been a dramatic increase in interest in the techniques of Evolutionary Computation since their introduction in the mid-1970s. Many applications of the technique have been reported, including solv-ing problems of numerical and combinatorial optimization, the optimum placing of components in VLSI devices, the design of optimum control systems, economics, modeling ecological systems, the study of evolu-tionary phenomena in social systems, and machine learning, among oth-ers.

The idea behind Evolutionary Computation is best explained by the example quoted in Michalewicz (1992):

Do what nature does. Let us take rabbits as an example: at any given time there is a population of rabbits. These faster, smarter rabbits are less likely to be eaten by foxes, and therefore more of them survive to do what rabbits do best: make more rabbits. Of course, some of the slower, dumber rabbits will survive just be-cause they are lucky. This surviving population of rabbits starts breeding. The breeding results in a good mixture of rabbit genetic material: some slow rabbits breed with fast rabbits, some fast with fast, some smart rabbits with dumb rabbits, and so on. And on the top of that, nature throws in a ‘wild hare’ every once in a while by mutating some of the rabbit genetic material. The resulting baby rabbits will (on average) be faster and smarter than those in the original population because more faster, smarter parents survived the foxes ...

By analogy, in Evolutionary Computation, solutions that maxi-

mize some measure of fitness (the criterion or cost function) will have a higher probability of participating in the reproduction process for new solutions and it is likely that these solutions are better than the previous ones. This is a fundamental premise in Evolutionary Computation. Solu-tions of an optimization problem evolve by following the well-known Darwinian principles of “survival of the fittest”. The basic principles, the principal techniques and operators of Evolutionary Computation are in-troduced in this chapter and an example illustrates how an Evolutionary Algorithm can be used to determine the global optimum parameters of a complex problem. In Chapter 17 the technique is applied to the design of optimized control systems.

Evolutionary Computation 205

15.1 Evolutionary Algorithms

Historically, Evolutionary Algorithms were introduced at the end of the 1950s, but due to the lack of readily available fast computers, the whole field did not become known for many years. Initially, Evolutionary Pro-gramming was proposed as an alternative method of Artificial Intelli-gence, while Evolutionary Strategies were designed and used for the so-lution of difficult optimization problems. The seminal book by Holland entitled “Adaptation in Natural and Artificial Systems” published in 1975, laid the foundations of the technique and introduced the Genetic Algorithm, which is the most popular Evolutionary Algorithm.

Genetic Algorithms (GAs) derive their name from the genetic processes of natural evolution. They were developed from Holland in the mid-1960s and have been implemented successfully in a broad range of control applications, e.g., the design of neural and fuzzy controllers, for tuning of industrial controllers and for the creation of hybrid fuzzy/ evo-lutionary and neural/evolutionary controllers, etc. During the 1980s, the rapid progress in computer technology permitted the use of the Evolu-tionary Algorithms in difficult large-scale optimization problems and the method rapidly diffused into the scientific community. Today, new ap-plications of Evolutionary Algorithms are being reported in large num-bers and the field has finally achieved general acceptance.

The terminology in the field of Evolutionary Computation is de-rived from Biology and Genetics. This terminology may lead to some confusion to the engineer who must associate the new terminology with known engineering terms. Here, the terms used will be adapted to make them easier to comprehend. Although Evolutionary Algorithms appear to be extremely simple compared with their biological counterparts, they are, however, sufficiently complicated so as to yield solutions where conventional numerical methods have been known to fail.

Evolutionary Algorithms are a subset of Evolutionary Computa-tion and belong to the generic fields of the Simulated Evolution and Arti-ficial Life. The search for an optimum solution is based on the natural processes of biological evolution and is accomplished in a parallel man-ner in the parameter search space. The terminology used in Evolutionary Computation is familiar. Thus, candidate solutions of an optimization problem are termed individuals. The population of solutions evolves in accordance with the laws of natural evolution. After initialization, the population undergoes selection, recombination and mutation repeatedly

Chapter 15 206

until some termination condition is satisfied. Each iteration is termed a generation, while the individuals that undergo recombination and muta-tion are named parents that yield offsprings.

Selection aims at improving the average quality of the popula-tion, giving the individuals with higher quality increased chances for rep-lication in the next generation of solutions. Selection has the feature of focusing the search in promising areas of the parameter search space. The quality of every individual is evaluated by means of a fitness func-tion, which is analogous to an objective function. The assumption that better individuals have increased chances to reproduce even better off-spring is based on the fact that there is a strong correlation between the fitness of the parents and that of their offspring. In Genetics this correla-tion is termed heredity. Through selection, exploitation of the numeri-cal/genetic information is thereby achieved.

Through recombination, two parents exchange their characteris-tics through random partial exchange of their numerical/genetic informa-tion. The recombination of the characteristics of two parents of high fit-ness assumes that if a portion of the numerical/genetic information re-sponsible for high values of fitness recombines with an equivalent par-ent, then the chances that their offspring will have as high or even higher fitness values are correspondingly increased. Recombination is also re-ferred to as Crossover. Likewise, through mutation, an individual under-goes a random change in one of its characteristics, i.e., in a specific sec-tion of its structure. Mutation aims at introducing new characteristics to the population that does not necessarily exist in the parents, leading thereby to an increase in the variance of the population. Exploration of the search space is achieved through the operators of recombination and mutation

The cornerstone of Evolutionary Algorithms is the iterative pro-cedure in exploring the search space while simultaneously exploiting the information that is being accumulated during the search. This is in fact, where their functionality lies. Through exploration, a systematic sam-pling of the search space is achieved, while through exploitation the in-formation that has been accumulated during exploration is used to search for new areas of interest in which exploration can be continued. Unlike exploitation, exploration includes random steps. It should be emphasized that “random exploration” does not mean “exploration without direction” since the technique focuses on the most promising directions.

Evolutionary Computation 207

The most common types of Evolutionary Algorithms are:

• Genetic Algorithms • Evolutionary Strategies • Evolutionary Programming • Classifier Systems • Genetic Programming

The first three are used extensively in optimization problems,

while Classifier Systems are used in machine learning. Finally, Genetic Programming is used in the automatic production of computer programs. It is noted that Genetic Programming and Classifier Systems are often considered as special cases of Genetic Algorithms and not as special cases of Evolutionary Algorithms. 15.2 The Optimization Problem In the generalized optimization problem, the objective is a search of the values of a vector x ∈ M, whose cost or objective function is to be mini-mized, i.e.:

φ(x) → min The solution of the problem requires determination of the optimum solu-tion vector x*, for which:

∀ x ∈ Μ : φ(x*) ≤ φ(x).

In practice, the space of feasible solutions is often bounded, i.e., G ⊆ M by functions of the form gj : M→ℜ and an analytical solution to the problem is unlikely. Only when φ and gj are particularly simple func-tions can conventional numerical optimization methods such as linear and non-linear programming be used. Also, optimization problems met in practice often require simplification, resulting thereby in solutions that do not correspond to the original problem. This is one of the principal reasons for the adoption of unconventional stochastic optimization

Chapter 15 208

methods, such as Genetic Algorithms that are not bound by such con-straints.

15.3 Evolutionary Optimization The field of Evolutionary Optimization has generated considerable inter-est and excitement in the engineering community and is one of the as-cending fields of Control Engineering. The technique can yield solutions to optimization problems that cannot be solved otherwise. In Soft Con-trol, in particular, after the initial period of enthusiasm, it became neces-sary to search for optimum solutions where heuristics and prior knowl-edge could not be applied or were not always useful. For example, issues concerning the best form of the fuzzy sets to use in a Fuzzy Controller or the best topology and the number of neurons and layers of the ANN be-ing used in Neural Control are now being sought using Evolutionary Op-timization.

The well-known numerical optimization methods of nonlinear programming do not always lead to acceptable solutions in practical problems, often becoming entrapped in local minima instead of yielding global solutions. Evolutionary Algorithms, in combination with local hill-climbing techniques, in most cases are able to locate global optimum solutions and are rapidly superceding classical techniques.

What then are the benefits of Evolutionary Computation and what are the reasons for the intense interest in this field? The answer is simple: Evolutionary Algorithms are robust, flexible and adaptable and they can yield global solutions to any problem, whatever the form of the objective function. Conventional numerical optimization methods, in contrast, can yield excellent results in a specific class of problems and fail in all others. The main differences between Evolutionary Algorithms and conventional numerical optimization methods are the following: they

• seek the optimum solution by searching a population of points of the search (solution) space in parallel and not in an isolated space,

• do not require derivative information or any other informa-tion. The direction of search is influenced by the evaluations

Evolutionary Computation 209

of the objective function and of the respective fitness func-tion only,

• use stochastic (probabilistic) transition and not deterministic rules in the optimization procedure,

• are simple to implement and apply, • can yield a population of optimum feasible solutions in a

problem and not a unique one. The choice of the best solu-tion is then left to the user. This is very useful in practical problems where multiple solutions exist as well as in multi- objective optimization problems.

Figure 15.1 Structure of a simple Evolutionary Algorithm

The search process, which is followed by a simple Evolutionary

Algorithm for the solution of an optimization problem is shown in the flow chart of Fig. 15.1 and is summarized as follows:

Chapter 15 210

1. a population of initial solutions is created heuristically or randomly,

2. the fitness of every individual-solution is evaluated, using the fitness function, which depends strongly on the corresponding value of the objective function of every candidate solution,

3. the selection operator gives improved chances to the better solutions for survival in the next generation,

4. using the recombination operator, two parents, which have been chosen randomly using the selection operator, exchange numerical information, according to a pre-defined probability of recombination,

5. using the mutation operator, partial numerical information is perturbed according to a pre-defined probability of mutation,

6. the fitness values of the new population are re-evaluated and 7. if the termination criterion (statistical or temporal) is not

satisfied, then a return is made to the 3rd step, otherwise the algorithm is terminated and

8. the best solution from the set of optimum solutions is selected.

If the optimization problem has constraints, then there are two

alternative methods of approaching an optimization problem:

• the first uses penalties (solutions which violate the con-straints are penalized and their respective fitness function values are reduced) and

• the second uses a mapping of the candidate solutions with simultaneous use of the exploration operators (recombina-tion and mutation).

The pseudo-code of a simple Evolutionary Algorithm is shown

below:

k=0 ; % Initialization of the iteration index Init_population P(k) ; % Initialization of the random population Evaluate P(k) ; % Evaluation of the Fitness of the initial population until(done) % Iteration till the termination criterion be satisfied k:= k + 1; % Increment iteration index or epoch P':= select_parents P(k) ; % Selection of the sub-population for the

Evolutionary Computation 211

reproduction of the offspring recombine P'(k) ; % Recombination mutate P'(k) ; % Mutation evaluate P'(k) ; % Evaluation of the fitness of the new population P := survive P, P'(k) ; % Selection of survivors

15.4 Genetic Algorithms Here, we will limit our interest to the most popular Evolutionary Algo-rithm, the Genetic Algorithms. As was noted above, Holland proposed Genetic Algorithms in the 1970s, having studied the adaptability of or-ganisms as a natural phenomenon. Since the early 1980s, Genetic Algo-rithms have been used extensively in optimization problems. Meanwhile, the research efforts during the 1980s and 1990s resulted in the develop-ment of many new forms of Genetic Algorithms, which differ signifi-cantly from the initial version proposed by Holland. It is beyond the scope of this book to describe these variants and the reader is referred to the Bibliography in chapter 18 for further study.

The differences between Genetic Algorithms, Evolutionary Strategies and Evolutionary Programming lie in the operations that can-didate solutions are subject to during the evolutionary procedure. These differences may be critical in a successful implementation of an Evolu-tionary Algorithm but in practice, they are of lesser importance com-pared with the properties that they share. It is worthy of note that since the early 1980s each technique has borrowed principles from the others and their differences have tended to disappear.

In the design and use of a Genetic Algorithm, the following is-sues must be considered: the

• creation of the initial population – initialization, • representation-mapping of the candidate solutions, • evaluation of the fitness of every candidate solution, • implementation of the exploration operators –

recombination and mutation, • selection of the parents for reproduction of the offspring –

selection and • choice of the parameters of the Genetic Algorithm

Chapter 15 212

The stages of a simple Genetic Algorithm are discussed in some detail below. 15.4.1 Initialization

An initial population of N candidate solutions (one for each of the N un-knowns) is created and for every solution/individual/chromosome xi the corresponding objective function value φi is evaluated. Alternative meth-ods for the creation of the initial population or part of it, is through statis-tical analysis of the search space or through heuristic reasoning. 15.4.2 Decoding The N candidate solutions of the optimization problem are converted into binary strings of length L that are used to represent real numbers as fol-lows:

000….0 = minimum value of the parameter 000….1 = minimum value of the parameter + q×20

000…10 = minimum value of the parameter + q×21

……………………………………………………… 111…11 = maximum value of the parameter

where q = (max. value – min value)/(2L-1). Clearly the discretization step q specifies the precision of the representation while the length L of each representation need not be equal for all candidate solutions. When the optimization problem is multidimensional, then the partial strings are concatenated as shown in Figure 15.2 in order to create a single binary string. Other mappings that are commonly used are representations with real numbers and the Gray code.

By way of example, consider a simple two-dimensional optimi-zation problem whose objective function is quadratic:

φ(x)=x1

2+ x22

where 0≤xi≤7 are integers. Here we select the length of the binary string to be L=3 since 23=8, sufficient to represent the integer values of the so-

Evolutionary Computation 213

lutions. Thus the string 000 000 represents the solution (0, 0), the string 000 001 the solution (0,1), the string 111 111 the solution (7,7), etc.

Following accepted terminology, the binary string is named a genotype, the decoded information the phenotype, while every individual solution is a chromosome.

Fig. 15.2 Creation of the string in a multidimensional problem of optimization

15.4.3 Evaluation of the fitness Evaluation of the fitness of every candidate solution is fundamental in Genetic Algorithms. When a Genetic Algorithm is used for the optimiza-tion of an objective function, the fitness is analogous with the objective function. If the objective function takes positive values only, then the fit-ness values and the objective function values of an individual are syn-onymous. If not, then a transformation is necessary to reflect the qualita-tive difference of the candidate solutions while ensuring that the fitness values are always positive. Usually, we use the relative fitness fi of the individual-solutions defined as:

Chapter 15 214

∑=

N

1=jj

ii

φ

φf

15.4.4 Recombination and mutation The recombination and mutation operators perform the necessary explo-ration of the search space. Through recombination (or crossover), nu-merical information (i.e., the genetic material) is exchanged between two random individuals, while through mutation a bit-digit is perturbed and its value is changed.

For example, consider the following two 3-digits genotypes:

Q1= 0 1 1 0 0 1 and

Q2= 1 0 0 1 1 1

The crossover point is chosen randomly and in the example the result of recombination in the third gene (or digits) of the two binary strings are the new strings:

0 1 1 1 1 1 and

1 0 0 0 0 1

Crossover Point

Q1*=

Q2*=

In string Q1 the digits to the right of the crossover point are ex-

changed with these of the second string Q2, while the opposite is done with the digits of the second string. Crossover can be performed at one

Evolutionary Computation 215

or more points that are selected randomly. An alternative form of cross-over is uniform crossover where every digit of every offspring has equal probabilities of being taken from either parent. The frequency with which the crossover operator acts on the candidate solutions depends on some pre-defined crossover probability pcross∈[0, 1]. Practical values of the crossover probability are in the range [0.6-0.95], while techniques have been proposed where the same search process adapts the crossover probability.

During mutation, a random bit that is selected with some pre-defined mutation probability pmut, takes on values of 0 or 1 at random. This operation results in an increase in the variability of the population. Mutation is necessary since potentially useful genetic material may be lost at specific locations of the generation during previous operations. For instance, after mutation of the second digit/gene which is chosen randomly, the string,:

Q1= 1 0 0 0 0 1 is transformed into:

1 1 0 0 0 1 Q1*= Figure 15.3 depicts the operations of crossover and mutation in genotype space, in phenotype space in the objective function and correlated to the fitness function.

15.4.5 Selection During selection, N individuals are chosen for survival in the next gen-eration according to their fitness values from a population of Ν candidate solutions-individuals. Individuals with high fitness values have an in-creased probability of survival in the next generation/iteration, compared to those with low fitness values. Many methods of selection have been proposed but here only the popular roulette-wheel method is considered.

Consider the surface of a roulette wheel that is divided according to the fitness values, i.e., the angles of every sector of the roulette wheel are set proportional to the fitness. If the relative fitness values are equal (an unlikely situation in practice), then the roulette sectors will have

Chapter 15 216

equal angles, implying that there is an equal probability that the roulette ball will stop in any of the sectors.

Figure 15.3 Representation of the Crossover and Mutation Operators

The hypothetical case of the ball stopping on a dividing line is discounted! In reality, the fitness values are unequal in which case the sectors of the roulette will also be unequal, implying that the probability of the ball stopping in any given sector increases with the angle of the sector. It is not improbable, however, that the ball will stop in a sector with a small angle. Imagine now that the ball is rolled and finally stops in one of the N sectors of the roulette wheel. The sector where the ball stops defines the chromosome that will undergo evolution. It is obvious that the ball may fall two or more times in a sector which corresponds to a high fitness value and the corresponding solution/chromosome is se-lected an equal number of times for survival in the next generation. A so-

Evolutionary Computation 217

lution with low fitness may not be selected for survival in the next gen-eration.

25.25%

9.09%

13.13%

52.53%

A/A Chromosome Fitness % Totally 1 011100 25 25.25% 2 000011 9 9.09% 3 110100 52 52.53% 4 010011 13 13.13%

TOTAL 99 100%

Figure 15.4 Roulette-wheel selection for the objective function φ(x1,x2) =x1

2+x22 and a population of N=4.

15.4.6 Choice of the parameters of a GA

The main parameters that must be considered in the design of a Genetic Algorithm are the population size of the N solutions-individuals and the values of the probabilities of recombination/crossover and mutation. There are no general rules for selecting the appropriate probabilities but some general guidelines on values that give acceptable results have been established from experience. These guidelines suggest the following:

Ν∈[60, 100], pcross∈[0.6, 0.9] and pm∈[0.001, 0.01]

It is noted that the probability of recombination is referred to the population N, while the probability of mutation is referred to all the dig-its of the population. Also, it is worthy of note also that Genetic Algo-rithms have been used for the optimization of the parameters of the Ge-netic Algorithms (i.e., meta-GAs).

Chapter 15

218

Example 15.1 Constrained optimization

of a complex function

A difficult objective function is chosen for the demonstration of the po-tential of the Genetic Algorithms. Assume that the following objective function must be minimized:

φ(x1,x2)=x12+x2

2-0.3 cos(3πx1)-0.4 cos(4πx2)+0.7 where x1,x2 ∈ [-1,1]. As shown in Figure 15.5, the function φ(x1,x2) has multiple minima, which would be difficult to locate using conventional numerical methods of optimization. Even if the initial starting state in the search space were good, it would be difficult to avoid entrapment in some local optimum.

Figure 15.6 shows the two-dimensional contour of φ(x1,x2). Note that the minimum value of the function is at (0,0), while the search space is a square with axes (-1,1) and (-1,1). In Appendix C a program for the solution of the optimization problem of the function φ(x1,x2) is presented. The program is written in MATLAB and is based on routines that can be found at the Mathworks web site www.mathworks.com.

Executing the Genetic Algorithm to optimize the objective function φ(x1,x2) with random initial conditions leads to the results depicted in Figures 15.7 and 15.8. In Figure 15.7 the evolution of the optimization trajectory of the optimum solution in every generation is shown while in Figure 15.8 the evolution of the objective function in every generation (i.e., iteration) is shown. The upper, middle and lower curves correspond to the worst, average and minimum or optimum solution respectively.

It is noted that the Genetic Algorithm that has been implemented in this example follows an elitist strategy, i.e., it preserves the optimum so-lution, found so far, in every generation. Finally, the other parameters of the Genetic Algorithm are population_size=10, probability of mutation pm=0.01, probability of crossover pcross=0.5 and maximum number of generations maxgen= 40. The length of the binary string for every pa-rameter is 12, in which case the concatenated string has 24 digits.

Evolutionary Computation

219

Figure 15.5 The objective function φ(x1,x2)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 15.6 Contour graph of the objective function φ(x1,x2)

Chapter 15

220

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 15.7 Evolution of the trajectory of the optimum solution

in every generation

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

Figure 15.8 Evolution of the best/average/worst solution for every epoch

Evolutionary Computation 221

15.5 Design of Intelligent Controllers Using GAs

15.5.1 Fuzzy controllers The traditional design of fuzzy controllers is based primarily on heuristic techniques. The main problem in fuzzy control design is the difficulty in defining a host of parameters, such as the number and shape of the fuzzy sets of the inputs and outputs, the form of the inference engine and the de-fuzzification mechanism. Their choice has considerable influence on the overall behavior of the controller. Unfortunately, at present the theo-retical foundations for determining the optimum solutions does not exist and consequently only experience with similar problems can be used in their design. Often, the result is acceptable, but there is no guarantee that there does not exist a better solution.

The flexibility of Evolutionary Algorithms is the principal moti-vating factor for their use in determining the optimum values of the pa-rameters of a fuzzy controller. As was mentioned in previous chapters, the linguistic values of a fuzzy variable are defined by their membership functions. When the membership functions are triangular, then the three parameters a, b and c shown in Figure 15.9 uniquely specify the fuzzy set. The parameters of every fuzzy set are encoded into binary strings of sufficient length to give the desired precision. For n fuzzy sets and m variables and encoding with p digits, the total length of the chromosome is clearly nmp digits.

α γβ

µ(x)

x

Figure 15.9 Parameters of a triangular fuzzy set

Chapter 15 222

If the dynamic behavior of the process is known and there exists a macroscopic model of the process, then we may use the single objec-tive function (or criterion) to evaluate the performance of the closed sys-tem to a step disturbance:

dtetT

∫0 or ∫T

0

2dte

where T→∞. These are the familiar ITAE and ISE criteria respectively. Alternatively some criterion which involves overshoot, steady state error and rise time could be used.

15.5.2 Optimum neural controllers

The performance of a neural controller depends critically on the architec-ture of the ANN used, i.e., the number of neurons in every layer, the number of the layers and the topology of the network, the form of the compression function and the algorithm used to train the network.

The determination of these parameters is based on the knowl-edge and experience of the designer and any discussion on optimum de-sign is of no consequence. The determination of the parameters of a neu-ral controller can, however, be transformed into an optimization problem for which Evolutionary Algorithms appear very attractive.

Genetic Algorithms are considered very efficient in rapidly find-ing the approximate optimum solution of an optimization problem, but they are generally slow in finding the precise solution. For better con-vergence, Genetic Algorithms can be combined with local-search tech-niques, such as hill climbing, which are ideally suited to finding opti-mum solutions in the small. It is noted that for an Artificial Neural Net-work with m layers and ni neurons in the ith-layer, the total number of parameters in the optimization problem is:

∑−

=++=

1m

0i1ii n1nN )(

If every weight is decoded into a binary string with L digits, the

total length of the chromosome is clearly NL. It is obvious that the use of Genetic Algorithms in problems involving ANNs with thousands of neu-

Evolutionary Computation 223

rons becomes difficult and extremely time-consuming. However ANNs, which are used in Neural Controllers, usually have a very simple archi-tecture with rarely more than 30 neurons or more than one hidden layer, in which case GAs are very efficient.

The second application of Evolutionary Algorithms in the field of Neural Control has to do with the evolution of the topology of the network (i.e., the manner in which the neurons of the network are inter-connected) with or without parallel evolution of the weights of the net-work. This problem is practically unsolvable with conventional optimi-zation methods.

The main characteristic that makes Evolutionary Algorithms at-tractive for a broad class of optimization problems is their robustness be-cause:

• they do not require specific knowledge or derivative in-

formation of the objective function • discontinuities, noise or other unpredictable phenomena

have little impact on the performance of the method • they perform in parallel in the solution space, exploring

the search space with simultaneous exploitation of the information derived and they do not become entrapped in local optima

• they have good performance in multidimensional large-scale optimization problems and

• they can implement in many different optimization prob-lems without big changes in their algorithmic structure.

The main disadvantages of GAs are that they:

• face some difficulties in locating the precise global op-

timum, although it is easy for them to locate the vicinity where the global optimum exists and

• require a great number of evaluations of the objective function and therefore require considerable computa-tional power.

With regard to the first disadvantage, many hybrid evolutionary

algorithms, which embed local search techniques, have been proposed. Methods like hill-climbing or Simulated Annealing (presented in the

Chapter 15 224

next chapter) in combination with Evolutionary Algorithms have been developed in order to determine the exact global optimum and to im-prove the overall performance of the search algorithm. A notable genetic algorithm is that due to Moed and Saridis for global optimization. Con-cerning the second disadvantage, the dramatic evolution in computer technology in combination with the progress in parallel computer ma-chines is tending to minimize this disadvantage.

Chapter 16

Simulated Annealing

Simulated Annealing, which has much in common with Evolutionary Computation, is a derivative-free stochastic search method for determin-ing the optimum solution in an optimization problem. The method was proposed by Kirkpatrick et al. in 1983 and has since been used exten-sively to solve large-scale problems of combinatorial optimization, such as the well-known traveling salesman problem (TSP), the design of VLSI circuitry and in the design of optimum controllers. The main dif-ference between Evolutionary Computation and Simulated Annealing is that the latter is inspired by the annealing process for metals during cool-ing, while the former is based on evolutionary processes.

The principle of annealing is simple: at high temperatures the molecules in a metal move freely but as the metal is cooled gradually this movement is reduced and atoms align to form crystals. This crystal-line form actually constitutes a state of minimum energy. Metals that are cooled gradually reach a state of minimum energy naturally, while if they are forcibly cooled they reach a polycrystalline or amorphous state whose energy level is significantly higher. Metals that are annealed are pliable while the latter are brittle.

However, even at low temperatures there exists a small, but fi-nite probability that the metal will enter a state of higher energy. This implies that it is possible that the metal will leave the state of minimum energy for a new state where the energy is increased. During the cooling process, the intrinsic energy may rise or drop but as the temperature is lowered the probability that the energy level will increase suddenly is re-

225

Chapter 16 226

duced. The probability that a change in the state of the metal at some temperature T and initial energy level E1 to some other state with energy level Ε2 is given by:

TκEE 12

ep)( −

−= if Ε2>Ε1

=1 otherwise where κ is Boltzmann´s constant.

This thermodynamic principle was adapted to numerical analysis by Metropolis et al. in 1953 giving rise to the terms Simulated Anneal-ing. Simulated annealing attempts to minimize energy. This is similar to minimizing a Lyapunov function in modern control theory. In imple-menting the Metropolis algorithm the following must be known:

• the objective function φ (by analogy with the energy E of the metal) whose minimum is sought and,

• a control parameter T (the simulated temperature) whose temporal strategy defines the changes in the simulated tem-perature at every iteration of the algorithm.

16.1 The Metropolis Algorithm Although the analogy between the physical annealing process and Simu-lated Annealing is far from perfect, there is clearly much in common. In all the stochastic algorithms of optimization, an attempt is made to guide the solution from an initial random point to an optimum solution as rap-idly as possible. This can lead to entrapment in some local optimum from which it may be difficult if not impossible to extricate. Simulated Annealing does not suffer this problem, since the technique is stochastic and searches the solution space randomly. For the solution of an optimi-zation problem with Simulated Annealing, the following steps are re-quired:

1. an initial random solution vector x1 in the bounded parameter space is selected and its objective function φ(x1) is computed,

2. an initial temperature T(0)= Tinit is specified,

Simulated Annealing 227

3. using some stochastic or heuristic strategy, a new solution vector x2 is selected and the corresponding objective function value is evaluated φ(x2),

4. the difference of the objective function ∆φ = φ(x2)-φ(x1) is computed,

5. if ∆φ<0, then the solution vector x2 is accepted, otherwise if ∆φ>0 accept the solution vector according to the probability of

acceptance:

)()( kT∆φ

ekp−

= otherwise go to step 7,

6. set x1=x2 and φ(x1)=φ(x2) and weight the current simulated tem-perature with the coefficient λ, where 0<λ<1, decreasing the simulated temperature successively at every iteration, so that at the (k+1)st iteration:

T(k+1) = λT(k),

where k is the iteration index,

7. if the current simulated temperature is lower or equal to the final temperature, i.e., T(k) ≤ Tfinal, then accept the current solution vector as being optimum, otherwise return to Step 3 and repeat the process.

If the Simulated Annealing algorithm is to succeed, it is important

that the temporal annealing strategy that is followed, i.e., the simulated temperature profile, be suitable. The rate at which the simulated tem-perature is decreased depends on the weighting coefficient λ. Too high a simulated cooling rate leads to non-minimum energy solutions, while too low a cooling rate leads to excessively long computation times. The closer the value of λ is to unity, the slower simulated temperature de-creases. Figure 16.1 shows the probability of acceptance of the solution p(λ), as a function of the iteration index for different values of λ. In order to achieve effective exploration of the search space, it is advisable to use 0.95<λ<0.98. Finally, as in Evolutionary Computation, the trajectory of an optimization problem is critically dependent on the initial estimates of the optimum solutions that are heuristic or the result of statistical analy-sis.

Chapter 16 228

p(λ)

λ=0.5

λ=0.8

1

λ=0.95

1 10 19 28 37 46 55 64 73 82 91 100 109 118 k

Figure 16.1 Variation of the acceptance probability p(λ) as a function of the iteration index k.

16.2 Application Examples Two examples of the application of simulated annealing are presented below. In the first, the two-dimensional parameter optimization problem that was presented in the previous chapter on Evolutionary Computation is used. The second application example shows how Simulated Anneal-ing can be used to advantage to obtain the optimum coefficients of a two-term (PI) industrial controller.

Simulated Annealing 229

Example 16.1 Constrained optimization

of a complex function The problem statement in this example is identical to that of Example 15.1 in the previous chapter and refers to the problem of finding the pa-rameters x1 and x2 which minimize the multi-peaked objective function:

φ(x1,x2)=x12+x2

2-0.3 cos(3πx1)-0.4 cos(4πx2)+0.7 The parameter search space is bounded by the square x1,x2∈[-1,1]. Figure 15.5 showed that the objective function has multiple minima in the per-missible parameter search space. An example of the trajectory of the so-lutions in the bounded search space using simulated annealing algorithm is shown in Figure 16.2.

The evolution of the objective function φ(x1, x2) is shown in Figure 16.3. During the initial iterations of the algorithm and particularly when the simulated temperature is high, the trajectory appears random with large fluctuations. With successive decreases in the simulated tempera-ture, however, the algorithm is seen to converge. Without systematically decreasing the simulated temperature, the solution to the problem at every iteration would be totally random, something analogous to the Monte Carlo method and convergence is unlikely. Finally, Figures 16.4 and 16.5 show the trajectories of the parameters x1 and x2 as they con-verge to the null state (0,0) at which the objective function has a mini-mum. This is the state of minimum “energy”.

Convergence is achieved in approximately 150 iterations. It is noted that convergence is achieved without knowledge of the derivative of the objective function. This is particularly useful when the derivative of the objective function, required by most non-linear programming methods, is difficult or impossible to obtain analytically.

Chapter 16 230

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 16.2. Trajectory of the solution in the search space (x1,x2)

0 50 100 150 200 250 300 3500

0.5

1

1.5

2

2.5

3

φ(x1,x2)

Figure 16.3 Evolution of the objective function φ(x1,x2) as a function of the iteration index

Simulated Annealing 231

0 50 10 15 20 25 30 35-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 16.4 Evolution of x1 as a function of the iteration index

0 50 10 15 20 25 30 35-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Figure 16.5 Evolution of x2 as a function of the iteration index

Chapter 16 232

Example 16.2 Optimization of a two-term industrial controller

Given a simple SISO (Single Input - Single Output) plant whose step re-sponse, derived from experimental data, is shown in Figure 16.6. The plant is to be controlled by a classic industrial two-term controller (PI), whose input-output relationship is simply:

∫+=t

0ip τdτeKeKu )(

Here e is the error between the desired and the real output and u is

the control variable. The unknown parameters of the industrial controller are the gains Kp and Ki whose optimum values are sought. One way of determining the “best” gain pair is to use classical tuning methods such as those of Ziegler and Nichols or modern tuning techniques such as those of Persson and Astrom (see Bibliography in chapter 18). These methods (i) assume a simplified dynamic model of the plant and (ii) use heuristics to arrive at the “best” parameters instead of an analytical error criterion. Here in contrast, an analytical criterion is used directly and the optimum pair is determined using stochastic techniques

Figure 16.6 Discrete-time step response of the controlled process

Simulated Annealing 233

Using the ITAE criterion it is desired to obtain the values of the pa-rameter pair (Kp,Ki), which minimize the objective function (i.e., error criterion):

φ(•)= ( )∫=T

dttetITAE0

where T ∞. An example of the evolution of the criterion dependent from the iterations is depicted in Figure 16.7. Convergence is achieved in about 120 epochs (iterations).

Figure 16.7 Evolution of the objective function

Figure 16.8 Evolution of the parameters of the industrial controller

Chapter 16 234

Figure 16.8 shows the evolution of the parameter pair (Kp,Ki) of the industrial controller. It is noted that convergence is achieved, whatever the initial values of the unknown parameters. Finally, the step response of the closed-loop system is shown in Figure 16.9. This response must be compared with that of Figure 14.6 for the non-optimized neural control-ler. A cursory glance will confirm that the response of this system is su-perior.

Figure 16.9 Step response of the closed-loop system with minimum ITAE

Chapter 17 Evolutionary Design of Controllers The criteria used in the design of control systems, better known as objec-tive or cost functions, are normally formulated in terms of analytical functions. In practice, multiple engineering objectives are often difficult to express in closed analytical form and great effort is spent trying to find suitable expressions that result in acceptable system performance. In industrial three-term controller design, for instance, the optimum pa-rameters of the controller must satisfy not only specific closed system specifications but may also involve economical, ecological, production and other engineering factors. These multiple objectives are difficult to formulate analytically, conflicting and impossible to satisfy simultane-ously.

Controller design with multiple objectives can be attempted us-ing a composite objective function in which the various objectives are assigned weights or penalties and the design problem is transformed into a nonlinear optimization problem. Nonlinear programming is the tradi-tional method for obtaining numerical solutions to this problem, but there is no guarantee that the procedure will reach a global optimum. The problem is compounded if the parameter search space is bounded. Sto-chastic optimization using Evolutionary Programming or Simulated An-nealing is an attractive alternative when numerical solutions are required and significant inroads in its use have been made in recent years.

Chapter 17

236

Stochastic optimization with a qualitative measure of perform-ance of a closed system, instead of quantitative terms, offers distinct ad-vantages in controller design to the Control Engineer as he is able to re-late to the design problem directly in linguistic terms rather than through some abstract analytical formulation. 17.1 Qualitative Fitness Function Evolutionary Algorithms are used mainly in optimization problems and especially in constrained problems in which the objective function is complicated. This class of problems is usually difficult to solve with conventional numerical methods. In the stochastic approach using evolu-tionary programming, the multiple objective function is transformed into a composite fitness function, which then constitutes the driving force of the evolutionary search procedure.

Human knowledge and engineering objectives can normally be expressed in qualitative terms when a quantitative formulation is not possible. Considering the flexibility of Evolutionary Computation, it is evident that comparison of the candidate solutions can be performed us-ing fuzzy linguistic rules. The formulation of the optimization problem in such a way as to use linguistic objectives constitutes the difference be-tween the stochastic technique using Evolutionary Algorithms and fuzzy fitness criteria and conventional forms of Evolutionary Algorithms. In the qualitative design technique, fitness is expressed in terms of fuzzy linguistic rules that are processed using an inference mechanism whose outcome is de-fuzzified to yield crisp values for the fitness. It is recalled that in the classical Evolutionary Algorithms described in Chapter 15, the fitness values are computed from an analytical objective function. In the generalized optimization problem, the objective is a search for the values of a vector x ∈ M, whose objective function is minimized i.e.,

φ(x) → min

The function φ(x) expresses the engineering objectives in functional form and is some measure of the behavior of the system being studied. In the stochastic design technique, a Genetic Algorithm is used to search for the optimum values of the unknown vector x∈M, in the same manner

235

Evolutionary Design of Controllers

237

that was described in Chapter 15. The main difference here is that the evaluation of fitness does not follow the classical approaches, but is de-rived from a set of linguistic rules that express the multiple engineering objectives of the problem. The consequent of each fuzzy rule is taken as the objective function φ that can take m fuzzy linguistic values from the set:

w = {w1, w2 … wm} Assume also that the array

a = {a1, a2 … am} is the set of parameters, which consist of the antecedents, and

vi = {vi1 , vi

2 … viq}

is the set of q fuzzy values of the variable ai. Then, the linguistic rules that specify the fitness function in the Evolutionary Algorithm take the form:

R: IF a1 is v1i AND a2 is v2

i AND a3 is v3i …THEN φ is wi

The total number of design rules is thus equal to a1*n1+a2*n2+…+ak*nk. 17.2 Controller Suitability The knowledge about controlling a system exists in the form of linguistic rules whose antecedents (i.e., controller attributes) and consequents (i.e., controller suitability) uniquely specify a particular design. What is required therefore are rules that describe, in linguistic terms, how the controller attributes affect the performance of the closed system. Once this is done the stochastic design technique reduces to one of using fuzzy reasoning to infer the suitability of any design. De-fuzzification of the membership function of the suitability yields a crisp value for the fitness that is subsequently used as the fitness measure in the stochastic optimization procedure. A Genetic Algorithm is then used to determine the global optima of the parameters of the controller.

Chapter 17

238

The controller attributes are described by fuzzy sets, three to five being sufficient for most practical purposes. These fuzzy sets describe the desired attributes of the design. The specifications normally used in designing industrial controllers are overshoot, rise time (i.e., the time for the closed-loop response to reach some specified percentage of its final value) and settling time (i.e., the time required for the closed-loop response to reach some specified percentage of its final value). More design specifications can be added as necessary, e.g. steady-state error and the maximum permissible control actions that can be used, in which case the complexity of the computational problem is increased proportionally. Consider for example, the fuzzy sets for the Rise_Time, Overshoot and Settling_Time to be Small, Medium and Large while the fuzzy sets of the resultant Fitness to be Very_Small, Small, Negative_ Medium, Medium, Positive_Medium, Large and Very_Large. A sample of suitability rules can therefore be stated as follows:

R1: IF (Rise_Time is Small) AND (Overshoot is Small) AND (Settling_Time is Small) THEN (Fitness is Very_Large)

R2: IF (Rise_Time is Medium) AND (Overshoot is Small) AND (Settling_Time is Small) THEN (Fitness is Large)

R3: IF (Rise_Time is Medium) AND (Overshoot is Medium) AND (Settling_Time is Large) THEN (Fitness is Negative_Medium)

R4: IF (Rise_Time is Large) AND (Overshoot is Medium) AND (Settling_Time is Large) THEN (Fitness is Small))

R5: IF (Rise_Time is Large) AND (Overshoot is Large) AND (Settling_Time is Large) THEN (Fitness is Very_Small)

Examples of membership functions that can be used in the qualitative design technique are shown in Figure 17.1. Assuming that the settling time of the closed system is to remain constant, the variation of the fitness with rise time and overshoot is shown in Figure 17.2. The complete set of linguistic rules, containing 33=27 rules, constitutes the rule-base of the design procedure and is given in the Table that follows. These rules may be modified to satisfy any control objective and MATLAB and its Fuzzy Toolbox can be used to implement the design technique.

Evolutionary Design of Controllers

239

(a) Rise_Time

(b) Overshoot

(c) Settling_Time

Figure 17.1 Fuzzy sets used in the qualitative design technique

Chapter 17

240

(d) Fitness

Figure 17.1 (continued) Fuzzy sets used in the qualitative design technique

Figure 17.2. The fitness surface

Evolutionary Design of Controllers

241

1. IF (Rise_Time is Small) AND (Overshoot is Small) AND (Settling_Time is Small) THEN (Fitness is Very_Large) 2. IF (Rise_Time is Small) AND (Overshoot is Small) AND (Settling_Time is Medium) THEN (Fitness is Large) 3. IF (Rise_Time is Small) AND (Overshoot is Small) AND (Settling_Time is Large) THEN (Fitness is Positive_Medium) 4. IF (Rise_Time is Small) AND (Overshoot is Medium) AND (Settling_Time is Small) THEN (Fitness is Large) 5. IF (Rise_Time is Small) AND (Overshoot is Medium) AND (Settling_Time is Medium) THEN (Fitness is Positive_Medium) 6. IF (Rise_Time is Small) AND (Overshoot is Medium) AND (Settling_Time is Large) THEN (Fitness is Medium) 7. IF (Rise_Time is Small) AND (Overshoot is Large) AND (Settling_Time is Small) THEN (Fitness is Positive_Medium) 8. IF (Rise_Time is Small) AND (Overshoot is Large) AND (Settling_Time is Medium) THEN (Fitness is Medium) 9. IF (Rise_Time is Small) AND (Overshoot is Large) AND (Settling_Time is Large) THEN (Fitness is Negative_Medium) 10. IF (Rise_Time is Medium) AND (Overshoot is Small) AND (Set-tling_Time is Small) THEN (Fitness is Large) 11. IF (Rise_Time is Medium) AND (Overshoot is Small) AND (Set-tling_Time is Medium) THEN (Fitness is Positive_Medium) 12. IF (Rise_Time is Medium) AND (Overshoot is Small) AND (Set-tling_Time is Large) THEN (Fitness is Medium) 13. IF (Rise_Time is Medium) AND (Overshoot is Medium) AND (Set-tling_Time is Small) THEN (Fitness is Positive_Medium) 14. IF (Rise_Time is Medium) AND (Overshoot is Medium) AND (Set-tling_Time is Medium) THEN (Fitness is Medium) 15. IF (Rise_Time is Medium) AND (Overshoot is MEDIUM) AND (Set-tling_Time is Large) THEN (Fitness is Negative_Medium) 16. IF (Rise_Time is Medium) AND (Overshoot is Large) AND (Set-tling_Time is Small) THEN (Fitness is Medium) 17. IF (Rise_Time is Medium) AND (Overshoot is Large) AND (Set-tling_Time is Medium) THEN (Fitness is Negative_Medium) 18. IF (Rise_Time is Medium) AND (Overshoot is Large) AND (Set-tling_Time is Large) THEN (Fitness is Small) 19. IF (Rise_Time is Large) AND (Overshoot is Small) AND (Settling_Time is Small) THEN (Fitness is Positive_Medium)

Chapter 17

242

20. IF (Rise_Time is Large) AND (Overshoot is Small) AND (Settling_Time is Medium) THEN (Fitness is Medium) 21. IF (Rise_Time is Large) AND (Overshoot is Small) AND (Settling_Time is Large) THEN (Fitness is Negative_Medium) 22. IF (Rise_Time is Large) AND (Overshoot is Medium) AND (Set-tling_Time is SMALL) THEN (Fitness is Medium) 23. IF (Rise_Time is Large) AND (Overshoot is Medium) AND (Set-tling_Time is Medium) THEN (Fitness is NM) 24. IF (Rise_Time is Large) AND (Overshoot is Medium) AND (Set-tling_Time is large) THEN (Fitness is Small) 25. IF (Rise_Time is Large) AND (Overshoot is Large) AND (Settling_Time is Small) THEN (Fitness is Negative_Medium) 26. IF (Rise_Time is Large) AND (Overshoot is Large) AND (Settling_Time is Medium) THEN (Fitness is Small) 27. IF (Rise_Time is Large) AND (Overshoot is Large) AND (Settling_Time is Large) THEN (Fitness is Very_Small)

Rule-Base for the qualitative evaluation of controller fitness

using Rise_Time, Overshoot and Settling_Time.

Evolutionary Design of Controllers

243

Example 17.1 Design of an optimum

two-term controller for temperature control of a greenhouse

The qualitative controller design technique is applied to the design of a two-term (PI) controller to control the environment inside a greenhouse. Following a step demand in the reference temperature Tref, the temperature T at some point in the greenhouse shown in Figure 17.3, has the characteristic dead time followed by an exponential rise shown in Figure 17.4.

Figure 17.3 Schematic of the controlled greenhouse

The proposed technique is model-free and no attempt is made to obtain a low order approximant of the plant in order to tune the closed system. Here it is sufficient to know only the continuous or discrete step response. The design objective is to find the optimum parameters (Kp, Ki)* of a two-term controller that will lead to a closed system step response with a nominal rise time Trise=26 units, a nominal overshoot of p=10% and a nominal settling of Ts=20 units. Instead of defining some quantitative criterion with penalty functions, qualitative measures are used to describe the desired attributes of the closed system. Each attribute is therefore assigned a fuzzy variable, which define the suitability of the overall system.

Greenhouse C

Tref

T

+

-

Chapter 17

244

The fuzzy sets of the three attributes that form the inputs to the inference engine are assumed to be triangular while the fuzzy sets of the suitability function are Gaussian. The fuzzy sets are shown in Figure 17.1. In deriving the control surface, use is made of the Mamdani min-max compositional inference rule. De-fuzzification uses the Center of Gravity (COG). A simple Genetic Algorithm, which follows an elitist strategy, is finally used to obtain the global optima of the controller parameters. It is observed that fitness decreases with increasing overshoot and increasing rise time.

Figure 17.4 Normalized step response of the greenhouse

The step response of the optimum system designed with the proposed hybrid Evolutionary - Fuzzy (E-F) technique is compared in Figure 17.5 with that of the same controller designed for minimum ITAE. It is evident that the response of the E-F design is superior, being better damped, and reaches the steady state faster than the ITAE design. It is interesting to note that the E-F design has a higher ITAE index than that of the ITAE design. That the E-F design is superior to the ITAE design is no surprise since a multiple criterion was used in the former.

Evolutionary Design of Controllers

245

5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

Figure 17.5 Comparison of closed system step responses

for the optimum E-F and minimum ITAE controllers

Example 17.2 Design of an optimum neural controller for a lathe cutting process A rule-based neural controller for a cutting lathe was described earlier in chapter 16 and a schematic of the lathe cutting process was shown in Figure 16.7. The response of the process to a small step demand in feed rate was shown in Figure 16.8. The objective in this case study is a controller with a rise time of less than some specified value Trise, an overshoot that does not exceed p% of the steady state value and a settling time Tset less than some specified value. These design objectives can be achieved by (i) the proper choice of the control rules, (ii) the inference mechanism and (iii) optimization of the free parameters of the controller. Assume that the first part of the controller design procedure that involves rule elicitation and rule-encoding, has been completed and that a suitable neural network of specified topology has been designed and trained to generate the desired control surface, as in chapter 16. Our concern here is the second part of the design, i.e., the optimization of the free controller parameters.

ITAE E-F

Chapter 17

246

Twenty-seven rules relating the design attributes were necessary to specify the desired properties of the closed system completely and are displayed in Figure 17.3. The fuzzy sets of the three controller design attributes constitute the inputs to the inference engine and are assumed to be triangular while the fuzzy set for the fitness function is taken as Gaussian as shown in Figure 17.1.

The step response of the optimum controller designed with the qualitative Evolutionary - Fuzzy (E-F) design technique is compared with that of the same controller designed for minimum ITAE in Figure 17.6. The response of the qualitative design is superior, being better damped and faster in reaching the steady state than the ITAE design, presumably because a multi-objective criterion has been used.

Figure 17.6 Step responses of optimum controllers for the optimum E-F and minimum ITAE controllers

ITAE E-F

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Palm R., Driankov D. and Hellendoorn H. (1996) : Model Based Fuzzy Control, Springer, Berlin.

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Wu Q. H., Hogg. B. W. and Irwin G. W. (1992) : “A neural network regulator for turbo-generators”, IEEE Trans. on Neural Networks, Vol. 3, No.1, pp. 95-100.

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and Tuning, Instrument Society of America, Research Triangle Park, NC.

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Industrial Automation, Marcel Dekker, NY. Tzafestas S. G. (Ed) (1993) : Applied Control, Marcel Dekker, NY. H. Evolutionary Algorithms Angeline P. J., Saunders G. M. and Pollack J. B. (1994) : “An evolution

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Back T., Hammel U. and Schwefel H-P. (1997) : “Evolutionary Computation: Comments on the History and Current State”, IEEE Trans on Evolutionary Computation, Vol. 1, No. 1, pp 3 - 17. (contains over 220 references on Evolutionary and Genetic Algo-rithms).

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I. MATLAB and its Toolboxes

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Appendix A Case Study: Design of a Fuzzy Controller using MATLAB This Appendix considers a case study for the design of a generic fuzzy controller for the control of a simple, yet representative process. The dy-namic process exhibits nonlinear behavior and response asymmetry, for which it is desired to compensate. The objective is to show the reader the step-by-step procedure that must be followed and the ease with which it is possible to examine the performance of the resultant fuzzy controller using MATLAB and its Simulink and Fuzzy Toolboxes.

The steps in the design procedure are explained in sufficient de-tail so that the advantages and disadvantages of fuzzy control over con-ventional three-term control can be evaluated. The case study includes a comprehensive study of the effects of the choice of fuzzy sets on the per-formance of the closed system. The case study is based on the example given in the book by Gulley and Roger Jang (1995) on the use of the MATLAB Fuzzy Toolbox. 259

Appendix A 260

A.1 The Controlled Process The controlled process comprises a single storage tank shown in Figure A.1. Inflow is controlled by a flow control valve while outflow is through a fixed opening at the bottom of the tank. Outflow depends on the pressure in the tank and consequently on the level of the fluid in the storage tank.

The control objective is to maintain the level of the fluid in the storage tank h(t) constant so that the outflow rate remains constant de-spite perturbations in the inflow rate. In order to maintain the storage tank level constant, it is necessary to know the level of the fluid in the tank at all times. This can be achieved by using an inductive, capacitive or ultrasonic sensor, for instance. Even a floating ball connected to a po-tentiometer will do if the variations in the fluid level are relatively small. The controller has the error e(t)= hd - h(t) where hd and h(t) are the de-sired and actual levels of the fluid in the tank respectively as input and the rate u(t) with which the valve is opened or closed as output.

Desired fluid level hd

Actual level h(t)

Inflow

u(t)

Level sensor

Outflow rate y(t)

Figure A.1 The controlled process

Design of a Fuzzy Controller Using MATLAB 261

A.2 Basic Linguistic Control Rules Without knowing anything about the controlled process characteristics and using only basic qualitative knowledge, it is possible to control the level of fluid in the storage tank by following the three intuitive rules:

R1: IF the error_in_the_fluid_level is ZEro (i.e., the ac-

tual level is equal to the desired level) THEN the con-trol_valve_rate must be ZEro (i.e., do nothing),

R2: IF the error_in_the_fluid_level is LOw (i.e., the actual level is less than the desired level) THEN the con-trol_valve_rate must be POsitive_large (so that the tank will fill as quickly as possible),

R3: IF the error_in_the_fluid_level is HIgh (i.e., the actual level is higher than the desired level) THEN the con-trol_valve_rate must be NEgative_large (so that the tank will empty as fast as possible),

Since the controller must send a quantitative action to the control

valve, then the following additional information is required:

• the maximum permissible change σ (also termed the dead-zone) in the nominal value of the fluid level and

• the maximum rate of change of the fluid level ρ (which can be computed from the dimensions of the storage tank and the maximum inflow rate or by experimental measurements).

A.3 A Simple Linguistic Controller A very simple linguistic controller can be implemented very easily using the three rules presented above. Thus, for example, let the linguistic vari-able NORmal be assigned a tolerable discrepancy from the desired level of ±σ meters. If the level is greater than σ meters below the desired level then the valve must open at its highest rate POsitive_Large. By contrast if the level of fluid in the tank exceeds the desired level by σ then the rate of valve closure must be NEgative_Large. It is noted that this simple

Appendix A 262

principle is used in simple on-off controllers using relays in building thermostats for which no knowledge of the room dynamics or character-istics is required!

Unfortunately, this simple controller leads to undamped oscilla-tions in the level of fluid in the tank with a peak to peak amplitude of 2σ meters, as seen in Figures A.2(a) and A.2(b). Furthermore the step re-sponse, i.e., the fluid level as a function of time in response to a sudden change in the desired level, is asymmetric. As seen, an increase in the de-sired fluid level leads to a different response than a decrease in the de-sired level. This is characteristic of fluid systems since fluid pressure and level are nonlinearly related. There is no way of estimating the frequency of oscillation since there it is assumed that there is no knowledge of the dynamics of the controlled process. It is observed, however, that reduc-ing the dead-zone 2σ reduces the amplitude and increases the frequency of the oscillation. This is observed in Figures A.2(a) and A.2(b). Reduc-ing the dead-zone to zero will, in theory, lead to an infinite frequency of oscillation. In practice this is unlikely because of the inherent delays in the relay. In any case this continuous chattering is undesirable since it shortens relay life.

Figure A.2(a) Step response of linguistic controller with σ=0,1

Design of a Fuzzy Controller Using MATLAB 263

Figure A.2(b) Step response of linguistic controller with σ=0,025

The system can be modeled using MATLAB/Simulink/Fuzzy

Toolbox simply typing the instruction

sltank at which the flow diagram shown in Figure A.3 appears. In this file we are given the opportunity to compare the step responses of the fuzzy and fuzzy controllers for the same conditions.

The fuzzy sets in this simple case have the Boolean form shown in Figure 5.1. As a consequence the output of the controller can only take one of three possible values, i.e., POL, ZER or NEL. The qualitative controller is equivalent to a threshold logic unit with dead-zone, as shown in Figure A.4. If the fluid level error is within the dead-zone then the controller gives no output and the process coasts freely. However, when the absolute error exceeds the dead-zone then the controller gives a command of sgn(e).

Appendix A 264

Figure A.3 Simulink flow diagram using sltank

e

u

σ

-σ

1

-1

Figure A.4 Input-output relationship of qualitative controller A.4 The Design Procedure With the simple linguistic controller, the oscillations (or more precisely, limit cycles) of the fluid in the storage tank about its desired value are not acceptable except only where coarse control suffices. The constant chattering of the controller relay inevitably leads to wear of the control valve, which is in a state of continuous motion. We must find a suitable control strategy that will dampen these oscillations and ultimately elimi-

Design of a Fuzzy Controller Using MATLAB 265

nate them. It is also desirable to find some technique to compensate for the response asymmetry.

We could, of course, follow the road of conventional control, identifying the process dynamics by applying step disturbances to it, de-termining a simple approximant and using the cookbook techniques of Ziegler and Nichols, or more sophisticated methods in order to establish the parameters of the required three-term controller. This procedure is quite tedious and time-consuming and it is doubtful that all objectives can be satisfied simultaneously. It is certain, however, that a conven-tional three-term controller cannot compensate for the observed asym-metric response.

If we could change the control strategy by adding some addi-tional stabilizing rules and some advanced inference mechanism so that the control action changes smoothly, instead of abruptly as in the case of the simple controller, then we could achieve stabilization and significant improvement in the closed system response. The solution is to turn to unconventional control techniques and design a fuzzy controller with one of the available computer aided design packages. This case study uses MATLAB Simulink and the Fuzzy Toolbox.

In the following, we present the procedure that must be followed in the design of a simple generic fuzzy controller. Our objective is a ro-bust fuzzy controller which can be described by simple linguistic rules but which will not exhibit the problems that were encountered with the simple linguistic controller described earlier.

The next step in the design procedure following encoding of the control rules, is to decide on the number of fuzzy sets for every input and output variable of the controller. We must also specify the shape of the corresponding fuzzy sets, some of the available choices being shown in Figure 11.5. In this study we experiment with a number of shapes in an attempt to find acceptable overall performance. The FIS Editor in the Fuzzy_Toolbox shown in Figure A.5. is a simple and effective tool that makes designing fuzzy controllers simple.

As was noted in earlier chapters of this book, it is not necessary to specify many fuzzy sets for the inputs to the controller in order to ob-tain acceptable accuracy. Most industrial applications use 3 to 5 fuzzy sets. Where fine control is essential then many more fuzzy sets are re-quired. In the F. L. Smidth kiln controller, for instance, more than a dozen fuzzy sets are used. In the fuzzy controller proposed here, only

Appendix A 266

three fuzzy sets LOw, OK and HIgh are used for the input variables for simplicity.

A.5 System Stabilization Rules The oscillations in the fluid level in the storage tank which were ob-served with the simple qualitative controller resulting from the use of only three rules, can be damped by adding two more rules which involve the derivative of the level error de(t)/dt. This is a classical procedure even in conventional control. However, in order to avoid any problem through using derivatives of the set point, we prefer to use the derivative of the measured fluid level in the tank dh(t)/dt as the second input to the controller in addition to the level error e(t)=hd-h(t). In practice the de-rivative is replaced by the first backwards difference of the level ∆hk=hk-hk-1.

Figure A.5. The FIS Editor in the fuzzy design tool Adding two rules can, indeed, stabilize the closed system:

Design of a Fuzzy Controller Using MATLAB 267

R4: IF error_in_the_fluid_level is ZEro AND the

rate_of_change_of_level is NEgative_Small THEN the control valve must be closed slowly, i.e., the con-trol_valve_rate must be NEgative_Small

R5: IF error_in_the_fluid_level is ZEro AND the

rate_of_change_of_ level is POsitive_Small THEN the control valve must be opened slowly, i.e., the con-trol_valve_rate must be POsitive_Small

These 5 control rules may be specified in many ways, the most con-

venient of which is linguistic, as shown in Figure A.6.

1. If (level is OK) then (valve is no_change) 2. If (level is low) then (valve is

open_fast) 3. If (level is high) then (valve is

close_fast) 4. If (level is OK) and (rate_of_change_of

level is positive) then (valve is close_slow)

5. If (level is OK) and (rate_of_change_of level is negative) then (valve is open_slow)

Figure A.6. The control rules in linguistic form

A.6 On the Universe of Discourse of the

Fuzzy Sets The universe of discourse of every input to the controller depends on the permissible range of that variable. Errors are confined to e∈[-σ, σ] and the change of errors to dh(t)/dt|max∈[-ρ, ρ]. The two parameters σ and ρ thus uniquely specify the universes of discourse of the two controller in-puts.

The output of the controller is the rate with which the flow con-trol valve is opened or closed . Here we specify 5 fuzzy variables, i.e.,

Appendix A 268

NEg_Large, NEgative_Small, ZEro, POsitive_Small and POsitive_Large for finer control. The universe of discourse in this Case is normalized to [-1,1] where +1 indicates that the valve is entirely open (i.e., 100% open) and –1 that it is closed (i.e., 0% open). The median 0.5 implies that the valve is at the center of its range. A.7 On the Choice of Fuzzy Sets The fuzzy sets of the controller inputs and outputs can be given any of the most common shapes offered in the Fuzzy Toolbox, such as triangu-lar, trapezoidal, Gaussian, etc. (see Figure 5.12). Various combinations are tried and the resultant control surface and corresponding dynamic behavior of the closed system are compared.

The control surface is a graphical interpretation of the control ac-tions in input space, i.e., how the outputs of the controller vary as func-tions of the controller inputs. For a given set of control rules the control surface depends on the shape of the fuzzy sets of both the inputs and the outputs of the controller but this does not greatly affect the dynamic per-formance of the closed system.

The control surface of a generic controller with two inputs gives us an immediate indication of the magnitude of the control action in con-trol space but clearly when the controller has more than two inputs, the control surface becomes a manifold which is impossible to visualize. Should the performance of the closed system prove unsatisfactory, the control surface can be examined to see how the rules must be modified or what new rules must be added to bring about the desired behavior.

In the design example, the control surfaces were computed using an inference engine based on Mamdani’s compositional inference rule while COG (Center Of Gravity) was used to de-fuzzify the fuzzy set of the controller output. The Fuzzy Toolbox provides alternative forms for both the inference engine and de-fuzzification..

The following five cases were considered in the design study:

1. Case Α - Figure A.7: Inputs with 3 triangular fuzzy sets and outputs with 5 symmetric triangular fuzzy sets with large sup-port (50%) and no overlap.

Design of a Fuzzy Controller Using MATLAB 269

2. Case Β - Figure A.8: Inputs with 3 triangular fuzzy sets and outputs with 5 symmetric triangular fuzzy sets with small support and no overlap.

3. Case C - Figure A.9: Inputs with 3 Gaussian fuzzy sets and outputs with 5 symmetric triangular fuzzy sets with small support and no overlap.

4. Case D - Figure A.10: Inputs with 3 Gaussian fuzzy sets and outputs with 5 asymmetric triangular fuzzy sets with small support and no overlap.

5. Case E - Figure A.11 : Inputs with 3 Gaussian fuzzy sets and outputs with 5 asymmetric triangular fuzzy sets with small support and some overlap.

It is noted that triangular fuzzy sets with small support, i.e., σ,

ρ→0 approximate singletons (see chapter 5) that have been used in in-dustrial fuzzy controllers. The advantage of using singletons is the sim-plicity and speed of the computations for de-fuzzification. A number of vendors use singletons in their products, e.g., S5-Profuzzy, S7-Fuzzy_Control by Siemens and FuzzyTech by Allen Bradley.

For every one of the five cases we present the control surface and the corresponding step response. Figure A.12 shows the computer screen from the Fuzzy Toolbox which shows which rules are fired for a given controller input values and the corresponding controller output for the Case E. A.8 Compensation of Asymmetry Asymmetry in the step response of a closed system is not an uncommon phenomenon in practice due to the inherent nonlinearities in the con-trolled plants. In these cases, the step response for a positive demand dif-fers significantly from the response for a negative demand. A typical ex-ample is in the control of pressure, which is related by the square root of the observed height, as shown in the example analyzed in this Appendix. A conventional industrial three-term controller cannot easily compensate for this asymmetry. In contrast, asymmetry can be compensated for rela-tively easily in soft controllers (both fuzzy and neural) by warping the control surface appropriately. This can be achieved easily by displacing one or more of the fuzzy sets laterally on the universe of discourse. In

Appendix A 270

Figure A.10(c) for instance, the fuzzy set Positive_Small has been shifted to the left with the consequence that the control action for small positive errors in the liquid level is greater than that for small negative errors. Thus for small discrepancies in the liquid level about the nominal level, the rate with which the valve is changed is increased when the tank is filling and decreased when it empties. This leads to compensation of response asymmetry. A.9 Conclusions

It should have become clear that altering the shape of the fuzzy sets of the controller does not lead to major changes in the control surface. This is evident in Figures A.7(c) to A.11(c). In cases Α and Β, the control sur-faces shown in Figures A.7(c) and A.8(c) are almost flat in the dead-zone due the triangular nature of the fuzzy sets of both the inputs and the out-put. The corresponding step responses are shown in Figures A.11(a) and A.11(b) are unsatisfactory because their steady-state error is non-zero and response asymmetry is severe. It is noted, also, that in these two cases, both the triangular fuzzy sets with a large support set and the sin-gletons have comparable step responses. In contrast, in cases C, D and E the control surfaces are smooth because of the smooth (Gaussian) shape of the fuzzy sets of the controller inputs. The corresponding step re-sponses are seen to be superior. In cases D and E (see Figures A.11(d) and B.11(e)) the step responses are almost symmetric. Finally, in the last three cases the steady-state errors are essentially zero and overshoot is negligible. Case E appears to be the best as it demonstrates symmetric response, zero steady-state error and no overshoot.

As was seen in all five cases, the closed system step response is influenced significantly by the shape of the fuzzy sets of the inputs and to a lesser extent by the fuzzy sets of the controller output. In general, Gaussian fuzzy sets have smoother control surfaces, implying smoother control actions and improved responses. The best shapes of the fuzzy sets are not generally known and are the subject of ongoing research.

Design of a Fuzzy Controller Using MATLAB 271

CASE A

Figure A.7(a) Input fuzzy sets

Figure A.7(b) Output fuzzy sets

Figure A.7(c) Control surface

Appendix A 272

CASE B

Figure A.8(a) Input fuzzy sets

Figure A.8(b) Output fuzzy sets

Figure A.8(c) Control surface

Design of a Fuzzy Controller Using MATLAB 273

CASE C

Figure A.9(a) Input fuzzy sets

Figure A.9(b) Output fuzzy sets

Figure A.9(c) Control surface

Appendix A 274

CASE D

Figure A.10(a) Input fuzzy sets

Figure A.10(b) Output fuzzy sets

Figure A.10(c) Control surface

Design of a Fuzzy Controller Using MATLAB 275

CASE E

Figure A.11(a) Input fuzzy sets

Figure A.11(b) Output fuzzy sets

Figure A.11(c) Control surface

Appendix A 276

Figure A.12(a) Step response for Case A

Figure A.12(b) Step response for Case B

Design of a Fuzzy Controller Using MATLAB 277

Figure A.12(c) Step response for Case C

Figure A.12(d) Step response for Case D

Appendix A 278

Figure A.12(e) Step response for Case E

Figure A.13 Computations for case E

Appendix B A Simple Genetic Algorithm

The m-files given in the Appendices that follow can be downloaded from the author’s web site www.lar.ee.upatras.gr/reking

% Main_Program - ga.m popsize=10; % Population Size maxgen=50; % Number of Generations (iterations) length=12; % Length of genotype (i.e. number of bits

in the binary array) pcross=0.8; % Probability of Crossover pm=0.01; % Probability of Mutation bits=[length length]; vlb=[-1 -1]; vub=[1 1]; phen=init(vlb,vub,popsize,2); % Initialization of the population of phenotypes [gen, lchrom, coarse, nround] = encode(phen, vlb, vub, bits); % Conversion of phenotypes to binary string [fitness, object]=score(phen,popsize); % Evaluation of Fitness & Objective Function x=-1:0.1:1; % Display of the contour graph of the objective function y=-1:0.1:1; [x1,y1]=meshgrid(x,y); z=x1.^2+y1.^2-0.3*cos(3*pi*x1)-0.4*cos(4*pi*y1)+0.7; figure(1); contour(x,y,z); 279

Appendix B 280

[best_obj(1), index] = min(object); % Store the best candidate of the initial population best_gen=gen(index,:); best_phen=phen(index, :); [worst_obj(1), index1] = max(object); % Store the worst candidate of the initial population worst_cur_gen=gen(index1); worst_cur_phen=phen(index1); avg_obj(1)=0; % Calculate the average performance of the population for k=1:popsize avg_obj(1)=avg_obj(1)+object(k); end; avg_obj(1)=avg_obj(1)/popsize; best_x(1)=best_phen(1); best_y(1)=best_phen(2); for i1=1:2 fprintf(1,'%f ',best_phen(1)); end; fprintf('\n'); fprintf(1,'BEST : %f WORST : %f AVG : %f

\n',best_obj(1),worst_obj(1),avg_obj(1)); for i=1:maxgen % Start of the Genetic-Loop newgen=reproduc(gen,fitness); % Reproduction gen=mate(gen); % Mate two members of the population gen=xover(gen, pcross); % Crossover Operation gen=mutate(gen, pm); % Mutation Operation [phen, coa] = decode(gen, vlb, vub, bits); % Decode the genotype of the new population to phenotype [fitness, object]=score(phen,popsize); % Evaluation of the Fitness & Objective Functions [best_cur_obj, index] = min(object); % Store the best candidate of the current population best_cur_gen=gen(index, :); best_cur_phen=phen(index, :);

[worst_obj(i+1), index1] = max(object); % Store the worst candidate of the current population

worst_cur_gen=gen(index1); worst_cur_phen=phen(index1); avg_obj(i+1)=0; % Average performance of the current population for k=1:popsize avg_obj(i+1)=avg_obj(i+1)+object(k);

Simple Genetic Algorithm 281

end; avg_obj(i+1)=avg_obj(i+1)/popsize; if(best_cur_obj > best_obj(i)) % Apply Elitist Strategy phen(index1,:) = best_phen; gen(index1,:) = best_gen; object(index1) = best_obj(i); best_obj(i+1) = best_obj(i); elseif(best_cur_obj <= best_obj(i)) best_phen = best_cur_phen; best_gen = best_cur_gen; best_obj(i+1) = best_cur_obj; end; best_x(i+1)=best_phen(1); % Display evolution of the best solution on the contour graph best_y(i+1)=best_phen(2); hold; line(best_x,best_y); for i1=1:2 fprintf(1,'%f ',best_phen(i1)); end; fprintf(1,'---> %f\n',best_obj(i+1)); fprintf('\n'); fprintf(1,'BEST : %f WORST : %f AVG : %f \n',best_obj(i+1),worst_obj(i+1),avg_obj(i+1)); end xx=1:maxgen+1; % Display evolution of objective functions for the worst, average and best solutions figure(2); plot(xx,best_obj,xx,worst_obj,xx,avg_obj); grid; % File init.m - This function creates a random population function phen=init(vlb,vub, siz, sea) for i=1:siz phen(i,:)=(vub-vlb).*rand(1, sea) + vlb; end

% File score.m - This function computes the fitness and the objective function values of a population function [fitness, object]=score(phen, popsize) for i=1:popsize

Appendix B 282

object(i)=phen(i,1)^2+phen(i,2)^2-0.3*cos(3*pi*phen(i,1))-0.4*cos(4*pi*phen(i,2))+0.7;

fitness(i)=1/(object(i)+1); end

The following m-files called by the main

program can be downloaded directly from the MATHWORKS web site www.mathworks.com and bear the indication:

% Copyright (c) 1993 by the MathWorks, Inc. % Andrew Potvin 1-10-93.

% File encode.m - This function converts a variable from real to binary function [gen,lchrom,coarse,nround] = encode(x,vlb,vub,bits) lchrom = sum(bits); coarse = (vub-vlb)./((2.^bits)-1); [x_row,x_col] = size(x); gen = []; if ~isempty(x), temp = (x-ones(x_row,1)*vlb)./ ... (ones(x_row,1)*coarse); b10 = round(temp); nround = find(b10-temp>1e-4); gen = b10to2(b10,bits); end % File reproduc.m - This function selects individuals in accordance to their fitness function [new_gen,selected] = repro-duc(old_gen,fitness) norm_fit = fitness/sum(fitness); selected = rand(size(fitness)); sum_fit = 0; for i=1:length(fitness), sum_fit = sum_fit + norm_fit(i); index = find(selected<sum_fit); selected(index) = i*ones(size(index)); end new_gen = old_gen(selected,:); % File mate.m - This function mates two members of the population

Simple Genetic Algorithm 283

function [new_gen,mating] = mate(old_gen) [junk,mating] = sort(rand(size(old_gen,1),1)); new_gen = old_gen(mating,:);

% File xover.m - This function performs the Crossover operation function [new_gen,sites] = xover(old_gen,Pc) lchrom = size(old_gen,2); sites = ceil(rand(size(old_gen,1)/2,1)*(lchrom-1)); sites = sites.*(rand(size(sites))<Pc); for i = 1:length(sites); new_gen([2*i-1 2*i],:) = old_gen([2*i-12*i],1:sites(i)) ... old_gen([2*i 2*i-1],sites(i)+1:lchrom)]; end % File mutate.m - This function performs the Mutation operation function [new_gen,mutated] = mutate(old_gen,Pm) mutated = find(rand(size(old_gen))<Pm); new_gen = old_gen; new_gen(mutated) = 1-old_gen(mutated); % File decode.m - This function coverts a variable from binary to real function [x,coarse] = decode(gen,vlb,vub,bits) bit_count = 0; two_pow = 2.^(0:max(bits))'; for i=1:length(bits), pow_mat((1:bits(i))+bit_count,i) = two_pow(bits(i):-1:1); bit_count = bit_count + bits(i); end gen_row = size(gen,1); coarse = (vub-vlb)./((2.^bits)-1); inc = ones(gen_row,1)*coarse; x = ones(gen_row,1)*vlb + (gen*pow_mat).*inc;

% File b10tob2 - This function converts a variable from base 10 to base 2 function b2 = b10to2(b10,bits) bit_count = 0; b2_index = [];

Appendix B 284

bits_index = 1:length(bits); for i=bits_index, bit_count = bit_count + bits(i); b2_index = [b2_index bit_count]; end for i=1:max(bits), r = rem(b10,2); b2(:,b2_index) = r; b10 = fix(b10/2); tbe = find( all(b10==0) | (bits(bits_index)==i) ); if ~isempty(tbe), b10(:,tbe) = []; b2_index(tbe) = []; bits_index(tbe) = []; end if isempty(bits_index), return end b2_index = b2_index-1; end

Appendix C Simulated Annealing Algorithm % Main_Program sa.m Tinit=120; % Initial Simulated Temperature l=0.98; % Temperature Decrement Parameter Tfinal=0.00001; % Final Temperature % Computation and Display of objective Function Contour x=-1:0.1:1; y=-1:0.1:1; [x1,y1]=meshgrid(x,y); z1=x1.^2+y1.^2-0.3*cos(3*pi*x1)-0.4*cos(4*pi*y1)+0.7; figure(1); contour(x,y,z1); Tcur=Tinit; % Initialize Simulated Temperature x1=-1 + 2*rand; % Select First Solution (x1, x2) x2=-1+2*rand; z(1)=x1^2+x2^2-0.3*cos(3.*pi*x1)-0.4*cos(4*pi*x2)+0.7; r1(1)=x1; r2(1)=x2; i=1; while Tcur>Tfinal % Start of Simulated Annealing Loop

285

Appendix C 286

x_1=-1 + 2*rand; % Select New Solution (x_1, x_2) x_2=-1 + 2*rand; % Evaluate New Objective Function Value z_1=x_1^2+x_2^2 - 0.3*cos(3.*pi*x_1)-0.4*cos(4*pi*x_2)+0.7; g=exp(-((z_1-z(i))/Tcur)); % Acceptance Probability if ((z_1 < z(i))| (rand < g)) x1=x_1; x2=x_2; z(i+1)=z_1; else z(i+1) = z(i); end Tcur=Tcur*l; % New Simulated Temperature r1(i+1)=x1; r2(i+1)=x2; title('Search for the global optimum point'); xlabel('x axis'); ylabel('y axis'); i=i+1; end % End of Loop f=find(z==min(z)); fprintf(1,'The Minimum value has of the Obj. func. been observed so far is : %f in the %d iteration\n', min(z),f(1)); fprintf(1,'x=%f,y=%f\n',r1(f(1)),r2(f(1))); hold; line(r1,r2); title('Movement of x-y parameters in the search space'); xlabel('x parameter'); ylabel('y parameter'); figure(2); plot(z); title('Obective Function values versus Iterations'); xlabel('Iterations'); ylabel('Objective Function'); figure(3); plot(r1); title('Movement of the x parameter');

Simulated Annealing Algorithm 287

ylabel('x parameter'); xlabel('Iterations'); figure(4); plot(r2); title('Movement of the y parameter'); ylabel('y parameter'); xlabel('Iterations');

Appendix D Network Training Algorithm

% Main_Program net.m

P=[0.8 0.8; 0.8 0; 0.8 -0.8; 0.3 0.3; 0.3 -0.3; 0 0.8; 0 0; 0 -0.8; -0.3 0.3; -0.3 -0.3; -0.8 0.8; -0.8 0; -0.8 -0.8]; T=[1 0.55 0 0.35 0 0.55 0 -0.55 0 -0.35 0 -0.55 -1]; [R,Q] = size(P); S1 =2; [S2,Q] = size(T); [W10,B10] = rands(S1,R); Randomize network pa-rameters W20 = rands(S2,S1)*0.5; B20 = rands(S2,1)*0.5; disp_freq = 20; max_epoch = 9999; epoch limit err_goal = 0.01; error measure limit lr = 0.01; learning rate lr_inc = 1.05; learning increment lr_dec = 0.7; learning decrement err_ratio = 1.04; error ratio mom_const=0.95; TP = [disp_freq max_epoch err_goal lr lr_inc lr_dec mom_const err_ratio]; [W10,B10,W20,B20,epochs,TR]=trainbpx(W10,B10, 'purelin',W20,B20,'purelin',P,T,TP) ; training

algorithm with linear neurons in 289

Appendix D 290

both layers plottr(TR); plot results W10 ; print synaptic weights of first layer W20 ; print synaptic weights of second layer B10 ; print bias of first layer B20 ; print bias of second layer

Index

Adaptive linear networks (ADALINEs), 156, 162, 170 Artificial intelligence, 13 Artificial neural networks, 1, 51, 153 autoassociative, 159 feedback, 159 feed-forward networks, 159 generalized function mapping, 154 Hopfield recurrent networks, 158 multi-layer network topologies, 158 Artificial neurons, 153, 156 dynamic, 158 static, 156 Cartesian product, 72 Classical control, 2 Compositional rules of inference, 81 Computational intelligence, 6, 9, 13, 23, 31, 41, 153

Computer integrated manufacturing (CIM), 23, 36 Control: protocols, 119 rules, 119 Conventional control, 39 Deep knowledge, 18, 32 De-fuzzification, 98 center of area (COA), 98 center of gravity (COG), 98 Elemental artificial neuron, 156 Embedded fuzzy controllers, 123 Evolutionary: algorithms, 20 computation, 203 control, 8 controller suitability, 237 decoding, 212 design of conventional controllers, 235 design of intelligent controllers, 221

291

Index 292

[Evolutionary] operations, 205 crossover, 206, 214 mutation, 206, 214 recombination (see crossover) selection, 206, 215 simulated evolution, 205 optimization, 205, 208 programming, 211 strategies, 211 Expert systems, 13, 49 classification, 32 development, 18 diagnosis of malfunctions, 28 elements, 15 energy management, 26 fault diagnosis, 20, 24 fault prediction, 24 implementation, 19

industrial controller design, 24

LISP machines, 19 need, 17 operator training, 22 paradigms, 20 plant simulation, 22 prediction of emergency plant conditions, 26 predictive maintenance, 25 product design, 21 production scheduling, 27 representation of knowledge, 20 shells, 18, 20 supervisory control systems, 23 tools and methods, 18 Flexible manufacturing systems (FMS), 21, 28 Fuzzification, 91, 96

degree of fulfillment, 91, 96 graphical interpretation of, 93 Fuzzy: algorithm, 59 associative memory (FAM), 103, 115 conditional statements, 58 control, 54, 89 algorithm, 89 controllers, 105 coarse-fine, 117 decomposition, 90 embedded, 123 gain-scheduled, 40, 136 generalized three-term, 108 generic two-term, 113 hybrid architectures, 112 integrity, 101 optimization using genetic algorithms, 221 partitioned architecture, 109 real-time, 119 robustness, 107 Tagaki-Sugeno, 136, 144 three-term, 107 fitness criteria, 236 gain-scheduling, 136, 146 implications, 78 Boolean, 78 GMP, 80 Larsen, 80 Lukasiewicz, 78 Mamdani, 79 Zadeh, 79 inference engine, 71 degree of fit, 71 linguistic variables, 64 logic, 1, 7, 8 algorithm, 59, 74 basic concepts, 54 logic control (FLC) (see also Fuzzy controllers), 2

Index 293

operators, 60 conjunctive, 72 propositional implication, 71 reasoning, 71, 76 relational matrix, 73 sets, 55, 100 algebraic properties of, 64 choice of, 268 coarseness of, 100 completeness of, 101 linguistic descriptors of, 57 membership function of, 55 operations on, 63 complement, 63, 64 connectives, 69 DeMorgan's theorem, 64 intersection, 63, 64 product, 63 union, 63, 64 shape of, 100 support set of, 56 singletons, 67, 92 systems, 32 variable, 57 universe of discourse of, 55 Gain-scheduled controllers, 40 Generalized: Modus Ponens (GMP), 77 Modus Tollens (GMT), 77 Genetic algorithms (GAs), 8, 205, 211 fitness functions, 203, 206, 213 initialization, 212 parameters, 217 Hard control, 42 Human: intelligence, 6 operators, 59

Industrial: control, 23 controller optimization, 232 Inference engine, 6, 15, 17 effectiveness, 37 quality, 37 Intelligent: agents, 124 control, 6, 11, 31, 41, 43 autonomy, 45 basic elements, 34 conditions for use, 33 objectives, 34 techniques, 39 controllers, 7, 35 correctness, 10 extendibility, 10 precision, 10 reusability, 10 robustness, 10, 40 systems, 10 acceptance, 37 architecture, 46 design tools and methods, 18 distributed architecture, 47 efficiency, 37 hierarchical structure, 46 Knowledge: base, 76 based systems, 13, 48 embedded, 193 empirical, 33 engineering, 37 and experience, 31 heuristic, 135 Learning machines, 154 Linguistic: controller, 261 descriptors, 57

Index 294

[Linguistic] rule matrix, 186 rules, 8, 14, 16, 20, 32, 59 values, 57 variables, 57 Mamdani, 1, 79, 84, 111 Membership function, 55 generic S, 66 generic Π, 67 Model-based fuzzy control (see also Tagaki-Sugeno controllers), 135, 136 Modern control, 3 Multi-level relay controller, 111 Neural: control, 51, 153, 160 learning and adaptation, 161 parallel processing, 161 rule-based, 181 controllers, 160 architectures, 162 indirect learning, 166 inverse model, 164 specialized training, 165 design using genetic algorithms, 222 fidelity, 163 indirect training of, 166 inverse model of, 164 multi-variable, 161 properties of, 161 rule-based, 181 network training algorithms, 169 back-propagation (BP), 169, 176 flow chart, 180 Delta, 173 least mean squares (LMS), 171

multi-layer, 175 supervised learning, 169 unsupervised learning, 169 Widrow-Hoff, 170 Neuro-fuzzy control, 8, 51, 193 architectures, 194 isomorphism, 195 fuzzification of neural controllers, 195 neuralization of fuzzy controllers, 195 Numerical Fuzzy Associative Memory (NFAM), 187 Perceptron, 154 Procedural knowledge, 16 Real-time: expert systems, 26 execution scheduler, 124 fuzzy control, 119 Relational algorithm, 7 Representation of knowledge, 20 Response asymmetry compensation, 269 Rule: composition, 82 conflict, 102 encoding, 182 granularity, 116 Rule-based: neural control, 181 network training, 183 Saridis' principle, 10, 46 Shallow knowledge, 18, 24, 32 Simulated annealing, 8, 225 Metropolis algorithm, 226 optimization: constrained, 229 industrial controller, 232

Index 295

Soft: computing, 7, 41, 42, 154, 203 control, 42 Supervisory fuzzy controllers, 120 de-fuzzifier, 120 fuzzifier, 120 inference engine, 120 knowledge-base, 120 real-time data base, 120 Symbolic representation, 9 Synaptic weights, 156 Tagaki-Sugeno controllers (see also Model-based controllers), 136, 144 first approach, 136 fuzzy control law, 141

fuzzy process models, 139 fuzzy variables and fuzzy spaces, 137 locally linearized process model, 142 second approach, 144 stability conditions, 144 Uncertainty and vagueness, 7, 53 Unconventional control, 6, 40 Universe of discourse, 55 Waste-water treatment control, 126 Widrow-Hoff training algorithm,

170, 172, 173 Zadeh, 1, 50, 53, 119

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