Foundations of Fuzzy Control E-Book

Jan Jantzen

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 556

Veröffentlichungsjahr: 2013

Beschreibung

Foundations of Fuzzy Control: A Practical Approach, 2nd Edition has been significantly revised and updated, with two new chapters on Gain Scheduling Control and Neurofuzzy Modelling. It focuses on the PID (Proportional, Integral, Derivative) type controller which is the most widely used in industry and systematically analyses several fuzzy PID control systems and adaptive control mechanisms.

This new edition covers the basics of fuzzy control and builds a solid foundation for the design of fuzzy controllers, by creating links to established linear and nonlinear control theory. Advanced topics are also introduced and in particular, common sense geometry is emphasised.

Key features

• Sets out practical worked through problems, examples and case studies to illustrate each type of control system
• Accompanied by a website hosting downloadable MATLAB programs
• Accompanied by an online course on Fuzzy Control which is taught by the author. Students can  access further material and enrol at the companion website

Foundations of Fuzzy Control: A Practical Approach, 2nd Edition is an invaluable resource for researchers, practitioners, and students in engineering. It is especially relevant for engineers working with automatic control of mechanical, electrical, or chemical systems. 

Leseprobe

Contents

Cover

Title Page

Copyright Page

Dedication

Foreword

Preface to the Second Edition

Preface to the First Edition

Chapter 1: Introduction

1.1 What Is Fuzzy Control?

1.2 Why Fuzzy Control?

1.3 Controller Design

1.4 Introductory Example: Stopping a Car

1.5 Nonlinear Control Systems

1.6 Summary

1.7 The Autopilot Simulator*

1.8 Notes and References*

Chapter 2: Fuzzy Reasoning

2.1 Fuzzy Sets

2.2 Fuzzy Set Operations

2.3 Fuzzy If–Then Rules

2.4 Fuzzy Logic

2.5 Summary

2.6 Theoretical Fuzzy Logic*

2.7 Notes and References*

Chapter 3: Fuzzy Control

3.1 The Rule Based Controller

3.2 The Sugeno Controller

3.3 Autopilot Example: Four Rules

3.4 Table Based Controller

3.5 Linear Fuzzy Controller

3.6 Summary

3.7 Other Controller Components*

3.8 Other Rule Based Controllers*

3.9 Analytical Simplification of the Inference*

3.10 Notes and References*

Chapter 4: Linear Fuzzy PID Control

4.1 Fuzzy P Controller

4.2 Fuzzy PD Controller

4.3 Fuzzy PD+I Controller

4.4 Fuzzy Incremental Controller

4.5 Tuning

4.6 Simulation Example: Third-Order Process

4.7 Autopilot Example: Stable Equilibrium

4.8 Summary

4.9 Derivative Spikes and Integrator Windup*

4.10 PID Loop Shaping*

4.11 Notes and References*

Chapter 5: Nonlinear Fuzzy PID Control

5.1 Nonlinear Components

5.2 Phase Plot

5.3 Four Standard Control Surfaces

5.4 Fine-Tuning

5.5 Example: Unstable Frictionless Vehicle

5.6 Example: Nonlinear Valve Compensator

5.7 Example: Motor Actuator with Limits

5.8 Autopilot Example: Regulating a Mass Load

5.9 Summary

5.10 Phase Plane Analysis*

5.11 Geometric Interpretation of the PD Controller*

5.12 Notes and References*

Chapter 6: The Self-Organizing Controller

6.1 Model Reference Adaptive Systems

6.2 The Original SOC

6.3 A Modified SOC

6.4 Example with a Long Deadtime

6.5 Tuning and Time Lock

6.6 Summary

6.7 Example: Adaptive Control of a First-Order Process*

6.8 Analytical Derivation of the SOC Adaptation Law*

6.9 Notes and References*

Chapter 7: Performance and Relative Stability

7.1 Reference Model

7.2 Performance Measures

7.3 PID Tuning from Performance Specifications

7.4 Gain Margin and Delay Margin

7.5 Test of Four Difficult Processes

7.6 The Nyquist Criterion for Stability

7.7 Relative Stability of the Standard Control Surfaces

7.8 Summary

7.9 Describing Functions*

7.10 Frequency Responses of the FPD and FPD+I Controllers*

7.11 Analytical Derivation of Describing Functions for the Standard Surfaces*

7.12 Notes and References*

Chapter 8: Fuzzy Gain Scheduling Control

8.1 Point Designs and Interpolation

8.2 Fuzzy Gain Scheduling

8.3 Fuzzy Compensator Design

8.4 Autopilot Example: Stopping on a Hilltop

8.5 Summary

8.6 Case Study: the FLS Controller*

8.7 Notes and References*

Chapter 9: Fuzzy Models

9.1 Basis Function Architecture

9.2 Handmade Models

9.3 Machine-Made Models

9.4 Cluster Analysis

9.5 Training and Testing

9.6 Summary

9.7 Neuro-Fuzzy Models*

9.8 Notes and References*

Chapter 10: Demonstration Examples

10.1 Hot Water Heater

10.2 Temperature Control of a Tank Reactor

10.3 Idle Speed Control of a Car Engine

10.4 Balancing a Ball on a Cart

10.5 Dynamic Model of a First-Order Process with a Nonlinearity

10.6 Summary

10.7 Further State-Space Analysis of the Cart-Ball System*

10.8 Notes and References*

References

Index

This edition published in 2013 © 2013 John Wiley & Sons, Ltd

First Edition published in 2007 © 2007 John Wiley & Sons, Ltd

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Library of Congress Cataloging-in-Publication Data

Jantzen, Jan.  Foundations of fuzzy control: a practical approach / Jan Jantzen.    1 online resource.  Includes bibliographical references and index.  Description based on print version record and CIP data provided by publisher; resource not viewed.  ISBN 978-1-118-53557-8 (MobiPocket) – ISBN 978-1-118-53558-5 (Adobe PDF) – ISBN 978-1-118-53559-2 (ePub) – ISBN 978-1-118-50622-6 (hardback) 1. Automatic control. 2. Fuzzy systems. 3. Fuzzy automata. I. Title.  TJ213  629.8′312–dc23

2013023628

A catalogue record for this book is available from the British Library.

ISBN: 978-1-118-50622-6

In memory of Ebrahim (Abe) Mamdani (1 Jun 1942–22 Jan 2010) and Lauritz Peter Holmblad (23 Aug 1944–30 Mar 2005)

Figure 1 EH Mamdani (1942–2010)

Foreword

Since the objective of Foundations of Fuzzy Control is to explain why fuzzy controllers behave the way they do, I would like to contribute a historical perspective.

Before the 1960s, a cement kiln operator controlled a cement kiln by looking into its hot end, the burning zone, and watching the smoke leaving the chimney. The operator used a blue glass to protect his eyes. He controlled the fuel/air ratio in order to achieve steady operation of the kiln.

Central control was introduced in the cement industry in the 1960s. PID controllers were installed, mainly for uniform feeding of the raw materials and the fuel. Computers for process supervision and control were introduced in the cement industry in the late 1960s.

During experimental work in the 1970s, the fuel control strategy was programmed as a two-dimensional decision table with an error signal and the change in error as inputs.

The first time we heard about fuzzy logic was at the fourth IFAC/IFIP International Conference on Digital Computer Applications to Process Control, held in Zürich, Switzerland, in 1974. As a postscript to a paper on learning controllers, Seto Assilian and Abe Mamdani proposed fuzzy logic as an alternative approach to human-like controllers.

Experimental work was carried out at the Technical University of Denmark. The theoretical understanding and inspiration in relation to process control was gained mainly from papers written by Lotfi Zadeh and Abe Mamdani, and control experiments were performed in laboratory-scale processes such as, for example, a small heat exchanger. The rule based approach that underlies the decision tables was also inspired by the instructions that we found in a textbook for cement kiln operators, which contained 27 basic rules for manual operation of a cement kiln.

The first experiments using a real cement kiln were carried out at the beginning of 1978 at an FL Smidth cement plant in Denmark. At this stage of the development work, the attitude of the management was sceptical, partly because of the strange name, ‘fuzzy’. Other names were suggested, but eventually, with an increasing understanding by the management of the concept, it was decided to stay with the word fuzzy, a decision that has never been regretted since.

In 1980, FL Smidth launched the first commercial computer system for automatic kiln control based on fuzzy logic. To date, hundreds of kilns, mills and other processes have been equipped with high-level fuzzy control by FL Smidth and other suppliers of similar systems.

Jens-Jørgen Østergaard

FL-Soft, Copenhagen

Preface to the Second Edition

This second edition of Foundations of Fuzzy Control includes new chapters on gain scheduling, fuzzy modelling and demonstration examples. Fuzzy gain scheduling is a straightforward extension of the usual PID type fuzzy controllers in the sense that fuzzy rules can interpolate naturally between PID controllers. Broadly speaking, the concept of local fuzzy models is dual to fuzzy gain scheduling. The demonstration chapter includes five larger examples that can be used as teaching modules. Furthermore, the chapter on stability has been extended to include performance. The intent has been to reach farther than mere analysis, that is, to devise a design method that starts from specifications of performance. The book adopts a practical approach, which is reflected in the new subtitle, A Practical Approach.

The guiding principle has been to try to reach the bottom of the matter by means of geometry. Thus, the PID controller can be seen as an inner product. Together with viewpoints from adaptive control and the self-organizing controller, this has led to a set of tuning recommendations, where the starting point is a performance specification, namely, the desired settling time (Chapter 7). The tuning recommendations are applied to an unstable chemical reactor tank and for the control of the idle speed in a car engine, in order to test and demonstrate how it works (Chapter 10). Hopefully, the reader will find the second edition of the book even more fundamental and coherent than the first edition owing to the geometric approach.

My students requested more examples and illustrations, and this second edition tries to fulfil that wish. A simulator (Autopilot) was developed to illustrate concepts in nonlinear control, such as equilibria, and the tool can be used as a stand-alone teaching tool. The book contents have been reorganized, and each chapter consists now of two parts, clearly separated by a summary: the first part is intended for an introductory course, and the part after the summary is for an advanced course. The advanced part is also a research guideline for students who wish to write their thesis within fuzzy control.

I teach an introductory course on the Internet using one of the demonstration examples. Access to the course is through the companion website www.wiley.com/go/jantzen, which is devoted to this book. The website also contains downloadable material, such as the MATLAB® programs that produced the figures, lecture slides and error corrections.

Finally, I wish to acknowledge the inspiration and help I have received from Abe Mamdani, especially in connection with the idle speed project (Chapter 10). He died, much too early, in 2010, and he is sadly missed. This second edition is dedicated to him, as well as to Peter Holmblad – two giants in the history of fuzzy control.

Jan Jantzen

University of the Aegean at Chios, Greece

Preface to the First Edition

In summary, this textbook aims to explain the behaviour of fuzzy logic controllers. Under certain conditions a fuzzy controller is equivalent to a proportional-integral-derivative (PID) controller. The equivalence enables the use of analysis methods from linear and nonlinear control theory. In the linear domain, PID tuning methods and stability criteria can be transferred to linear fuzzy controllers. The Nyquist plot shows the robustness of different settings of the fuzzy gain parameters. As a result, a fuzzy controller can be guaranteed to perform as well as any PID controller. In the nonlinear domain, the stability of four standard control surfaces can be analysed by means of describing functions and Nyquist plots. The self-organizing controller (SOC) is shown to be a model reference adaptive controller. There is the possibility that a nonlinear fuzzy PID controller performs better than a linear PID controller, but there is no guarantee. Even though a fuzzy controller is nonlinear in general, and commonly built in a trial and error fashion, we can conclude that control theory does provide tools for explaining the behaviour of fuzzy control systems. Further studies are required, however, to find a design method such that a fuzzy control system exhibits a particular behaviour in accordance with a set of performance specifications.

Fuzzy control is an attempt to make computers understand natural language and behave like a human operator. The first laboratory application (mid-1970s) was a two-input-two-output steam engine controller by Ebrahim (Abe) Mamdani and Seto Assilian, UK, and the first industrial application was a controller for a cement kiln by Holmblad and Østergaard, FL Smidth, Denmark. Today there is a tendency to combine the technology with other techniques. Fuzzy control together with artificial neural networks provide both the natural language interface from fuzzy logic and the learning capabilities of neural networks. Lately hybrid systems, including machine learning and artificial intelligence methods, have increased the potential for intelligent systems.

As a follow-up to the pioneering work by Holmblad and Østergaard, which started at the Technical University of Denmark in the 1970s, I have taught fuzzy control over the Internet to students in more than 20 different countries since 1996. The course is primarily for graduate students, but senior undergraduates and PhD students also take the course. The material, a collection of downloadable lecture notes at 10–30 pages each, formed the basis for this textbook.

A fuzzy controller is in general nonlinear, therefore the design approach is commonly trial and error. The objective of this book is to explain the behaviour of fuzzy logic controllers, in order to reduce the amount of trial and error at the design phase.

Much material has been developed by applied mathematicians, especially with regard to stability analysis. Sophisticated mathematics is often required which unfortunately makes the material inaccessible to most of the students on the Internet course. On the other hand, application-oriented textbooks exist, easily accessible, and with a wide coverage of the area. The design approach is nevertheless still trial and error. The present book is positioned between mathematics and heuristics; it is a blend of control theory and trial and error methods. The key features of the book are summarized in the following four items.

Fundamental. The chapter on fuzzy reasoning presents not only fuzzy logic, but also classical set theory, two-valued logic and two-valued rules of inference. The chapters concerning nonlinear fuzzy control rely on phase plane analysis, describing functions and model reference adaptive control. Thus, the book presents the parts of control theory that are the most likely candidates for a theoretical foundation for fuzzy control, it links fuzzy control concepts back to the established control theory and it presents new views of fuzzy control as a result.Coherent. The analogy with PID control is the starting point for the analytical treatment of fuzzy control, and it pervades the whole book. Fuzzy controllers can be designed, equivalent to a P controller, a PD controller, a PID controller or a PI controller. The PD control table is equivalent to a phase plane, and the stability of the nonlinear fuzzy controllers can be compared mutually, with their linear approximation acting as a reference. The self-organizing controller is an adaptive PD controller or PI controller. In fact, the title of the book could also have been Fuzzy PID Control.Companion web site.1 Many figures in the book are programmed in MATLAB® (trademark of The MathWorks, Inc.), and the programs are available on the companion web site. For each such figure, the name of the program that produced the figure is appended in parentheses to the caption of the figure. They can be recognized by the syntax *.m, where the asterisk stands for the name of the program. The list of figures provides a key and an overview of the programs.Companion Internet course. The course concerns the control of an inverted pendulum problem or, more specifically, rule based control by means of fuzzy logic. The inverted pendulum is rich in content, and is therefore a good didactic vehicle for use in courses around the world. In this course, students design and tune a controller that balances a ball on top of a moving cart. The course is based on a simulator, which runs in the MATLAB® environment, and the case is used throughout the whole course. The course objectives are: to teach the basics of fuzzy control, to show how fuzzy logic is applied and to teach fuzzy controller design. The core means of communication is email, and the didactic method is email tutoring. An introductory course in automatic control is a prerequisite.

The introductory chapter of the book shows the design approach by means of an example. The book then presents set theory and logic as a basis for fuzzy logic and fuzzy reasoning, especially the so-called generalized modus ponens. A block diagram of controller components and a list of design choices lead to the conditions for obtaining a linear fuzzy controller, the prerequisite for the fuzzy PID controller.

The following step is into the nonlinear domain, where everything gets more difficult, but also more interesting. The methods of phase plane analysis, model reference adaptive control and describing functions provide a foundation for the design and fine-tuning of a nonlinear fuzzy PID controller.

The methods are demonstrated in a simulation of the inverted pendulum problem, the case study in the above-mentioned course on the Internet. Finally, a short chapter presents ideas for supervisory control based on experience in the process industry.

The book aims at an audience of senior undergraduates, first-year graduate students and practising control engineers. The book and the course assume that the student has an elementary background in linear differential equations and control theory, corresponding to an introductory course in automatic control. Chapters 1, 2, 3 and 9 can be read with few prerequisites, however. Chapter 4 requires knowledge of PID control and Laplace transforms and Chapters 5, 6 and 7 require more and more background knowledge. Even the simulation study in chapter 8 requires some knowledge of state-space modelling to be fully appreciated. Mathematical shortcuts have been taken to preserve simplicity and avoid formalism.

Sections marked by an asterisk (*) may be skipped on a first reading; they are either very mathematical or very practically oriented, and thus off the main track of the book.

It is of course impossible to cover in one volume the entire spectrum of topic areas. I have drawn the line between fuzzy control and neuro-fuzzy control. The latter encompasses topics such as neural networks, learning and model identification that could be included in a future edition.

Acknowledgements. I am pleased to acknowledge the many helpful suggestions I received from the late Lauritz Peter Holmblad, who acted as external supervisor on Masters projects at the Technical University of Denmark, and Jens-Jørgen Østergaard. They have contributed process knowledge, sound engineering solutions and a historical continuity. Thanks to Peer Martin Larsen, I inherited all the reports from the early days of fuzzy control at the university. I also had the opportunity to browse the archives of Abe Mamdani, then at Queen Mary College, London. I am also pleased to acknowledge the many helpful suggestions from Derek Atherton and Frank Evans, both in the UK, concerning nonlinear control, and in particular state-space analysis and describing functions. Last but not least, former and present students at the university and on the Internet have contributed collectively with ideas and suggestions.

Jan Jantzen

University of the Aegean at Chios

1www.wiley.com/go/jantzen