Modern Adaptive Fuzzy Control Systems 3031173929, 9783031173929

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Studies in Fuzziness and Soft Computing

Ardashir Mohammadzadeh · Mohammad Hosein Sabzalian · Chunwei Zhang · Oscar Castillo · Rathinasamy Sakthivel · Fayez F. M. El-Sousy

Modern Adaptive Fuzzy Control Systems

Studies in Fuzziness and Soft Computing Volume 421

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related fields. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover significant recent developments in the field, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Ardashir Mohammadzadeh · Mohammad Hosein Sabzalian · Chunwei Zhang · Oscar Castillo · Rathinasamy Sakthivel · Fayez F. M. El-Sousy

Modern Adaptive Fuzzy Control Systems

Ardashir Mohammadzadeh Multidisciplinary Center for Infrastructure Engineering (MCIE) Shenyang University of Technology Shenyang, Liaoning, China Chunwei Zhang Multidisciplinary Center for Infrastructure Engineering (MCIE) Shenyang University of Technology Shenyang, Liaoning, China Rathinasamy Sakthivel Department of Applied Mathematics Bharathiar University Coimbatore, Tamil Nadu, India

Mohammad Hosein Sabzalian LabREI—Smart Grid Laboratory Department of Systems and Energy FEEC—School of Electrical and Computer Engineering University of Campinas (UNICAMP) Campinas, Brazil Oscar Castillo Division of Graduate Studies Tijuana Institute of Technology Tijuana, Baja California, Mexico Fayez F. M. El-Sousy Department of Electrical Engineering College of Engineering Prince Sattam Bin Abdulaziz University Al Kharj, Saudi Arabia

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-031-17392-9 ISBN 978-3-031-17393-6 (eBook) https://doi.org/10.1007/978-3-031-17393-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Fuzzy systems, especially type-2 neuro-fuzzy systems, are now used extensively in various engineering fields for different purposes. In plain language, this book aims to practically explain fuzzy systems and different methods of training and optimizing these systems. For this purpose, type-2 neuro-fuzzy systems are first analyzed along with various methods of training and optimizing these systems through implementation in MATLAB. These systems are then employed to design adaptive fuzzy controllers. The author tries to present all well-known optimization methods clearly and code them in MATLAB. Furthermore, all materials are available at http://www. simref.org, which can be visited by aficionados for faster in-depth learning. All dear readers of this book are kindly asked to share their views about writing flaws and scientific problems with us via the above website so that we can refine the book in the next editions. In the end, we would like to thank Dr. Sahraneh Ghaemi, the esteemed associate professor at University of Tabriz, and Dr. Ali Ahmadian, the esteemed assistant professor of University of Bonab, who have helped us scientifically edit this book. Shenyang, China Campinas, Brazil Shenyang, China Tijuana, Mexico Coimbatore, India Al Kharj, Saudi Arabia

Ardashir Mohammadzadeh Mohammad Hosein Sabzalian Chunwei Zhang Oscar Castillo Rathinasamy Sakthivel Fayez F. M. El-Sousy

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Contents

1

An Introduction to Fuzzy and Fuzzy Control Systems . . . . . . . . . . . . 1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What is Adaptive Fuzzy Control? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Why Adaptive Fuzzy Control? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Problems in Adaptive Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 3 3

2

Classification of Adaptive Fuzzy Controllers . . . . . . . . . . . . . . . . . . . . . 2.1 Direct Adaptive Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Indirect Adaptive Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Integrating Adaptive Fuzzy Controller with Other Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Integrating Direct and Indirect Adaptive Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Integrating Hybrid Fuzzy Controller with Other Controllers to Compensate for Estimation Error . . . . . . . 2.3.3 Integrating Hybrid Fuzzy Controller with Output Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Integrating Adaptive Fuzzy Controller with H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Integrating Adaptive Fuzzy Controller with Supervised Controller . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Integrating Adaptive Fuzzy Controller with Other Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Different Classes of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Affine Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Non-affine Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Nonlinear Feedback Systems . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Nonlinear Pure-Feedback Systems . . . . . . . . . . . . . . . . . . 2.4.5 Nonlinear Single-Input–Single-Output and Multi-Input–Multi-Output Systems . . . . . . . . . . . . . .

5 5 5 6 6 6 6 7 7 7 8 8 9 9 10 11

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2.4.6 Nonlinear Output and State Feedback Systems . . . . . . . . 2.4.7 Discrete and Continuous Systems . . . . . . . . . . . . . . . . . . . 2.5 Adaptation Mechanism in Fuzzy Systems . . . . . . . . . . . . . . . . . . . . 2.5.1 Setting Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Setting Structure and Parameter . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 12 12 12 13 13 14

3

Type-2 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Singleton Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Non-singleton Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Features of Type-2 Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Basic Operations in Type-2 Fuzzy . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Fuzzification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Type Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Implementation in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Designing a General Type-2 Fuzzy System with an Example . . . 3.12 Interval Type-2 Fuzzy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17 19 20 21 22 22 23 24 28 36 44 46 46

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Training Interval Type-2 Fuzzy Systems Based on Error Backpropagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Training Fuzzy Systems with Nie-Tan Type-Reduction . . . . . . . . 4.2.1 Implementation in MATLAB . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fuzzy System with KM-EKM Type-Reduction . . . . . . . . . . . . . . . 4.4 Training Type-2 Fuzzy System with Extended Kalman Filter . . . 4.5 Training Type-2 Fuzzy System Based on Genetic Algorithm . . . . 4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Calling Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Jargons of GA Toolkit in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 GA-Based Optimization of Neuro-Fuzzy System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Training Neural Networks Based on PSO . . . . . . . . . . . . . . . . . . . . 4.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Formulation of Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Implementation in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Training Type-2 Fuzzy System Through Second-Order Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 49 51 51 52 56 56 59 62 68 71 71 72 74 79 79 79

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4.11.3 Levenberg–Marquardt Algorithm . . . . . . . . . . . . . . . . . . . 4.11.4 Conjugate Gradient Method . . . . . . . . . . . . . . . . . . . . . . . . 4.11.5 Implementation in MATLAB . . . . . . . . . . . . . . . . . . . . . . . 4.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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80 80 81 81 93

5

Baseline Indirect Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.1 Problem Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2 Designing Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.3 Designing Moderation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.4 Application in Moderation of Inverted Pendulum . . . . . . . . . . . . . 99 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6

Type-2 Indirect Adaptive Control with Estimation Error Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Resistant Adaptive Fuzzy Control with Estimation Error Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Problem Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Estimating Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Designing Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Designing Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Analysis of Stability and Inference of Adaptive Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Switching Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 103 103 104 104 104 106 110 112 115 116 117 117

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Direct Adaptive Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Adaptive Fuzzy Control with Fewer Limitations . . . . . . . 7.2.2 Type-2 Fuzzy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 119 120 121 129 130 132

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Direct Adaptive Fuzzy Control with a Self-regulated Structure . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Description of the Self-regulated Structure Algorithm . . . . . . . . . 8.4 Adaptation Rules in Self-regulated Adaptive Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Application in Inverted Pendulum Control . . . . . . . . . . . . . . . . . . .

135 135 135 136 140 142

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8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 9

State Limitation Through Supervised Control . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Supervised Control for Indirect Adaptive Fuzzy Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Supervised Control for Fuzzy Control Systems in General . . . . . . 9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145

10 Adaptive Sliding Fuzzy Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Designing a Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 151 154 157

145 147 149 149

Chapter 1

An Introduction to Fuzzy and Fuzzy Control Systems

1.1 Historical Background In the early 1990s, successful applications of the fuzzy logic increased in the automated control. For instance, developments were observed in washing machines, advanced cameras, automated transmission, and underground trains [1]. In fact, fuzzy logic controllers (FLCs) are used as a method of controlling complicated nonlinear systems that cannot be controlled easily through conventional methods. They need no system models but can use a competent person’s information regarding the system. Nevertheless, it is difficult to analyze the stability and resistance of this controller because there are no systematic design methods, a flaw which has limited the applications of FLCs. Moreover, there is a profound background to adaptive control with respect to the proof of stability, resistant design, and performance analysis [2]. In the 1960s, major breakthroughs were achieved in the theory of stability and control in adaptive controllers. In the middle 1980s, studies of adaptive control were mainly focused on the problem of resistance to the non-modeled dynamics and bounded distortions. With early achievements in the adaptive control of linear systems, developments to nonlinear systems gained in popularity from the late 1980s to the 199s. Hence, adaptive control presented powerful mathematical methods for analyzing stability and resistance of nonlinear control systems. Therefore, it appears logical to integrate fuzzy logic control and adaptive control in order to obtain a better control method, which is called the adaptive fuzzy control (AFC). This hybrid method can both employ human knowledge and analyze stability and resistance.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Mohammadzaheh et al., Modern Adaptive Fuzzy Control Systems, Studies in Fuzziness and Soft Computing 421, https://doi.org/10.1007/978-3-031-17393-6_1

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1 An Introduction to Fuzzy and Fuzzy Control Systems

1.2 What is Adaptive Fuzzy Control? Wang defined an adaptive fuzzy system through a training algorithm whose parameters (and structure) would be regulated through numerical information. Accordingly, the neuro-fuzzy systems, in which fuzzy systems are integrated with neural networks, are also adaptive fuzzy systems. An adaptive fuzzy controller can be defined as a controller used in adaptive fuzzy systems through the adaptive control theory to develop training algorithms that guarantee the stability and performance of a closed-loop system. Lyapunov stability methods play a major role in designing and analyzing the stability of adaptive systems [2]. The stability of adaptive fuzzy systems can be monitored by analyzing the behavior of the candidate Lyopunov function. In brief, a controller is called an adaptive fuzzy controller if: • An adaptive fuzzy system is employed; • The Lyopunov stability technique is adopted to develop training algorithms that guarantee the stability of the closed-loop system.

1.3 Why Adaptive Fuzzy Control? The advantages of adaptive fuzzy control include the advantages of both fuzzy controls and adaptive control: • Fuzzy control uses an expert operator’s information. In fact, an expert operator can describe the system behavior or presents the control knowledge regarding how to control the system. This information can easily be presented as if–then rules. • Fuzzy systems are good nonlinear general estimators. In the classic linear resistant adaptive control, linear estimators are employed to estimate some unknown functions that are assumed to be linear. The linearity assumption of functions is rejected when fuzzy systems are used in adaptive control. Therefore, the adaptive fuzzy control represents a nonlinear resistant control scheme, in which the system does not have to be assumed linear in response to unknown parameters [3]. • Fuzzy control is easily interpretable, for it follows a control strategy through human knowledge, the principles of which are perceivable to unskilled people. The classic control theory uses complicated mathematical methods that are not interpretable. Hence, engineers prefer to use simply understandable methods in practical applications [3]. • Fuzzy control is easy to implement. Developing the ICs of VLSI type facilitated and accelerated the implementation of fuzzy controllers. • The necessary software and hardware for the implementation of fuzzy controllers are inexpensive [3].

References

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• Adaptive fuzzy control does not depend on the system model. Adaptive algorithms are used for the online configuration of parameters emerging in the controller section. Therefore, the mathematical model of the system is not required [1]. • Adaptive control guarantees stability and resistance, which are considered very important in the control theory. Stability means that the output of each bounded input remains bounded at any time, whereas resistance refers to the ability of a control system to keep the system stable while encountering the non-modeled dynamics and external distortions. The conventional fuzzy control fails to guarantee the stability and resistance of the control system. In adaptive control, the Lyopunov stability technique provides a mathematical framework for developing adaptive algorithms that guarantee stability and resistance. • Adaptive control presents a systematic control method. There are no standard methods for systematic design in conventional fuzzy control, and parameters are often configured through trial and error.

1.4 Problems in Adaptive Fuzzy Controller According to the aforesaid advantages, an adaptive fuzzy controller is a suitable candidate for the control of nonlinear dynamic systems with uncertainty. Nevertheless, there are still some problems that limit the practical application of this controller. The most important problem is the fixed general structure of fuzzy controllers, which are usually determined through trial and error. Inadequate attempts have been made to develop adaptive fuzzy controllers with the self-regulated structure, and the main topics such as stability, computational efficiency, and implementation have not been analyzed completely. In particular, the stability of systems with variable structures has not been proven; hence, designing an adaptive fuzzy controller with a self-regulated structure can be really useful. The other problems include applicability to specific classes of nonlinear systems, exponential growth of rules with the increased number of membership functions, constraints imposed on design parameters that are hard to determine in practice, and complexity of controllers in nonlinear systems in the form of triangular membership functions.

References 1. L.-X. Wang, Adaptive fuzzy systems and control: design and stability anyalysis (Prentice-Hall: Englewood Cliffs, New Jersey, 1994) 2. S.S. Ge, C.C. Hang, T.H. Lee, T. Zhang, Stable adaptive neural network control (Kluwer Academic Publishers, London, 2002) 3. J.T. Spooner, M. Maggiore, R. Ordonez, K.M. Passino, Stable adaptive control and estimation for nonlinear systems: neural and fuzzy approximation techniques (Wiley, New York, 2002)

Chapter 2

Classification of Adaptive Fuzzy Controllers

Since the early 1990s, adaptive fuzzy control has actively been under research. In fact, many researchers have analyzed this field, and many policies, methods, schemes, and practical applications of control have been published in books, journals, and conferences. Therefore, it is impossible to completely describe adaptive control in one seminar, and a brief description of the wide range of adaptive fuzzy control will be provided in this chapter. In the simplest case, adaptive fuzzy controllers are developed only through adaptive fuzzy systems that are generally divided into two classes: direct adaptive fuzzy controllers and indirect adaptive fuzzy controllers.

2.1 Direct Adaptive Fuzzy Controller Direct adaptive fuzzy controllers use adaptive fuzzy systems as controllers [1]. The adaptive mechanism for regulating the parameters of an adaptive fuzzy system is designed to stabilize the controlled system and enhance the closed-loop system performance. Direct adaptive fuzzy controllers were analyzed in [1–3].

2.2 Indirect Adaptive Fuzzy Controller Unlike direct adaptive fuzzy controllers, indirect adaptive fuzzy controllers benefit from adaptive fuzzy systems for system modeling. In fact, a controller is based on the assumption that the fuzzy system describes the controlled system well [2–7].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Mohammadzaheh et al., Modern Adaptive Fuzzy Control Systems, Studies in Fuzziness and Soft Computing 421, https://doi.org/10.1007/978-3-031-17393-6_2

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2 Classification of Adaptive Fuzzy Controllers

2.3 Integrating Adaptive Fuzzy Controller with Other Controllers Direct and indirect adaptive fuzzy controllers are simple; however, they have some problems. Therefore, adaptive fuzzy controllers have been integrated with the other control methods in recent years.

2.3.1 Integrating Direct and Indirect Adaptive Controllers The hybrid adaptive fuzzy control methods have been analyzed in [8–11]. The control output is a weighted hybrid of direct and indirect adaptive fuzzy controllers. This hybrid controller provides a framework for combining the system knowledge and the control knowledge.

2.3.2 Integrating Hybrid Fuzzy Controller with Other Controllers to Compensate for Estimation Error In general, the estimation error emerges when nonlinear functions are estimated through fuzzy systems. The estimation error may adversely affect the stability and performance of adaptive fuzzy control systems. To solve this problem, previous controllers are integrated with the other controllers. In [12], a fuzzy sliding mode controller was designed by integrating the adaptive fuzzy controller with the sliding mode controller to compensate for the estimation error. In [13–18], an adaptive fuzzy controller was described along with a new control term designed through the estimation error boundary. This term is added to the control output to compensate for the estimation error; however, it is very difficult to determine the estimation error boundary in practice. Therefore, some of the adaptive mechanisms were introduced in the next step for the online estimation of these boundaries [19, 20].

2.3.3 Integrating Hybrid Fuzzy Controller with Output Feedback Controller In many applications, it is either very difficult or impossible to measure all states of the controlled system, a problem which can be solved by the output feedback control. The only variable that needs to be measured is the system output, and many adaptive fuzzy control methods have been designed on the basis of output feedback [14, 21].

2.3 Integrating Adaptive Fuzzy Controller with Other Controllers

7

2.3.4 Integrating Adaptive Fuzzy Controller with H∞ Control External distortions play a major role in the practical applications of control by not only weakening the control performance but also causing instability. The optimal H∞ control is a technique used in the classic control theory to minimize the effects of external distortions. In [22–28], an adaptive fuzzy controller was integrated with the H∞ control technique to mitigate the effects of external distortions.

2.3.5 Integrating Adaptive Fuzzy Controller with Supervised Controller Sometimes, an adaptive fuzzy controller may not be fast enough. In this case, the system state variables may exit the desirable range. This problem can be solved by adding the adaptive gain, which cannot be very large. At the same time, increasing the adaptive gain increases noise sensitivity and results in the output chattering of the controller. Therefore, some researchers integrated adaptive fuzzy control with a supervised controller to keep the states of the controlled system within a desirable range without needing a large adaptive gain [1, 11, 29, 30]. This supervised control also has another control term that is designed under the assumption that the upper and lower bounds of nonlinear functions are known. When the states move out of a desirable range, the supervised controller starts forcing the states to stay within the desirable range.

2.3.6 Integrating Adaptive Fuzzy Controller with Other Control Methods An adaptive fuzzy control method was proposed in [10] where the controller’s output was a combination of direct adaptive fuzzy control, indirect adaptive fuzzy control, and another control term to compensate for the estimation error. The boundaries used in the control term of the variable structure are estimated online; hence, there is no need for the prior knowledge about boundaries. The integration of direct adaptive fuzzy control, indirect adaptive fuzzy control, and supervised control was described in [11]. The integration of adaptive fuzzy control with the control term of a variable structure and the H∞ control was introduced in [15] to mitigate the effects of estimation error and external distortions as much as necessary. In general, adaptive fuzzy control is integrated with the other control methods to eliminate the flaws of direct adaptive fuzzy control and those of indirect adaptive fuzzy control. Nonetheless, they are more complicated in terms of stability analysis and implementation. Thus, for a particular application, a designer should decide what control technique is appropriate for integration with adaptive fuzzy control.

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2 Classification of Adaptive Fuzzy Controllers

2.4 Different Classes of Nonlinear Systems There are different classes of nonlinear systems in the nonlinear control theory. These classes have different characteristics; hence, they need various control methods. There are a few good techniques for a class of nonlinear systems. For instance, the feedback linearization techniques can be employed to control the nonlinear systems that can be linearized. The strict-feedback nonlinear systems can be controlled through back stepping design. In addition, feedback control can be adopted to control the nonlinear systems in which all state variables are not measurable. The results of nonlinear control compelled researchers to design the adaptive fuzzy control methods for this class of systems based on the existing techniques. The following subchapters analyze the nonlinear systems to which adaptive fuzzy control methods can be applied.

2.4.1 Affine Nonlinear Systems Under certain conditions, the input–output solution for a class of nonlinear single– input–single–output (SISO) systems can be formulated as below: 

x˙i = xi+1 , i = 1, · · · , n − 1 x˙n = f (x) + g(x)u + d(t)

(2.1)

where x = [x1 , x2 , . . . , xn ]T ∈ R n , u ∈ R, y ∈ R denote the state variables, the input, and the output, respectively. Moreover, f (x), g(x) refer to the unknown flat functions, whereas d(t) denotes the bounded external distortions when |d(t)| ≤ d0 . The nonlinear systems that can be represented as this form are called the affine nonlinear systems, the input of which emerges as a line in equations. If f (x), g(x) are known, the feedback linearization technique can be employed to design controllers. The general structure of this technique is as follows: u=

1 [− f (x) + v] g(x)

(2.2)

where v is a new control variable which will be explained further in the next chapters. If f (x), g(x) are unknown, it is possible to use adaptive fuzzy control. In [1–5], the indirect adaptive fuzzy control methods were employed to describe ˆ affine nonlinear systems, and adaptive fuzzy systems fˆ(x|θ f ), g(x|θ g ) were utilized to estimate f (x), g(x). The Lyopunov stability analysis was adopted to obtain adaptive rules. In this method, cautionary aspects should be taken be taken into account to ˆ prevent singularity, i.e. when the controller approaches infinity as g(x|θ g ) becomes zero. For instance, Wang [1] proposed an algorithm to regulate θ g and solve this problem.

2.4 Different Classes of Nonlinear Systems

9

In [22, 23, 30], direct adaptive fuzzy control methods were proposed for affine v|θ ) to estimate the ˆ indirect systems by using only one adaptive fuzzy system u(x, 1 control rule u = g(x) [− f (x) + v]. The direct adaptive fuzzy control methods can completely solve the singularity problem in the indirect state, which might occur as the denominator becomes zero in the control rule. However, further constraints must be imposed on the control gain g(x) than the indirect methods.

2.4.2 Non-affine Nonlinear Systems Another extensive class of nonlinear systems includes non-affine nonlinear systems in which the input does not emerge as the affine form. In fact, the SISO non-affine nonlinear system is defined as below: ⎧ ⎨ x˙i = xi+1 , i = 1, · · · , n − 1 x˙ = f (x, u) ⎩ n y = x1

(2.3)

where x = [x1 , x2 , . . . , xn ]T ∈ R n , u ∈ R, y ∈ R denote the state variables, the input, and the output, respectively. Moreover, f (x, u) is a flat unknown function; therefore, it is fair to state that affine nonlinear systems are classified as a specific category of non-affine nonlinear systems. In recent years, researchers have proposed different adaptive fuzzy control methods for controlling non-affine nonlinear systems [31, 32]. Since the input does not emerge as linear, the feedback linearization technique is not executable. It is very difficult and challenging to perform adaptive fuzzy control on non-affine nonlinear systems. In general, advanced mathematical techniques should be used.

2.4.3 Nonlinear Feedback Systems Many of the practical nonlinear systems can be defined as a state space called the strict-feedback form: ⎧ ⎨ x˙i = f i (x i ) + gi (x i )xi+1 , 1 ≤ i ≤ n − 1 (2.4) x˙ = f n (x n ) + gn (x n )u n≥2 ⎩ n y = x1 where x = [x1 , x2 , . . . , xn ]T ∈ R n , u ∈ R, y ∈ R denote the state variables, the input, and the output, respectively. Furthermore, f i (·), gi (·) i = 1 . . . n are the flat unknown functions. The purpose is to design the control input in a way that the output can approximate to the reference input as much as possible.

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2 Classification of Adaptive Fuzzy Controllers

In the recent decade, the back stepping method has been known as an important technique for designing feedback systems because it guarantees general stability, traceability, and responsible performance for an extensive class of feedback systems with unknown parameters [33]. The main idea of back stepping design is to select some of the state variables as the virtual input of control for the subsystems with lower dimensions than those of the general system. In fact, each step of the back stepping design is a virtual control design based on the control terms obtained from the previous steps. A heuristic feedback design of the actual control input is obtained from the final Lyopunov function when the design process ends, and the general system stability is proven [34]. In [34], the idea of back stepping was used with adaptive fuzzy control to control a multi–input–multi–output (MIMO) industrial system. Distortion was also taken into account, and stability was analyzed well through the Lyopunov method. The proposed controller is a resistant method. However, the classic back stepping control is mainly based on the assumption that the unknown functions f i (x i ), gi (x i ) should be linear in relation to the unknown parameters. This assumption can be eliminated through fuzzy systems and adaptive neural networks. Adaptive neural back stepping control was analyzed in [34, 35]. Neural networks were employed in each step to estimate unknown functions, although a main problem with the adaptive neural back stepping control method is the increasing complexity. In fact, the complexity of controllers increased dramatically as the number of states (n) increased. The drastic increase in complexity is due to the need to estimate the derivatives of nonlinear functions [36]. In each step, partial derivatives should be calculated and used as the neural network inputs for this purpose. After each step, the number of partial derivatives increases greatly, and so does the complexity of the controller. To solve this problem, a dynamic control technique was proposed in [36] by introducing a first-order filter to avoid estimating the derivatives of nonlinear functions. Recently, adaptive intelligence control was also developed for discrete systems. In [37], a feedback-state adaptive neural control scheme was proposed through the idea of back stepping, whereas MIMO systems were considered in [38, 39].

2.4.4 Nonlinear Pure-Feedback Systems These systems are defined as below: ⎧ ⎨ x˙i = f i (x i , xi+1 ), i = 1, . . . , n − 1 x˙ = f n (x n , u) ⎩ n y = x1

(2.5)

2.4 Different Classes of Nonlinear Systems

11

where x = [x1 , x2 , . . . , xn ]T ∈ R n , u ∈ R, y ∈ R denote the state variables, the input, and the output, respectively. Moreover, f i (x i , xi+1 ) i = 1 . . . n refers to the flat unknown functions. The literature on the control of such systems is scant [40]. In [40], the adaptive neural control of these systems was proposed by integrating back stepping with the input–state stability analysis and the small gain theory. However, this method is prone to the increased complexity. In [41], adaptive neural control was described through the Nussbaum-gain functions and the idea of back stepping. The disadvantage of this method is long responsiveness.

2.4.5 Nonlinear Single-Input–Single-Output and Multi-Input–Multi-Output Systems Using the results from nonlinear SISO systems, researchers developed adaptive intelligent control to nonlinear MIMO systems with uncertainty. In general, it is more difficult to control nonlinear MIMO systems with uncertainty because of dealing with the input matrix and transactions between subsystems. In [42], an adaptive fuzzy controller was proposed for a class of nonlinear systems including affine subsystems under the assumption that the inputs were not interlinked and that the internal connections of the system were known. In [43–47], adaptive neuro-fuzzy control was proposed for a class of nonlinear systems in which the constraints were lifted from their internal connections. Nevertheless, the number of inputs must be equal to the number of outputs, and the inputs must be in the affine format. In other words, they must emerge as linear. In [48, 49], adaptive neural controllers were proposed for a specific class of nonlinear robotic MIMO systems which proper features. In [50], adaptive fuzzy control was introduced for two classes of nonlinear MIMO block-triangular systems with uncertainty in which input constraints were interlinked to properly remove the internal connections. Most of the relevant studies assumed that the inputs would emerge as affine. The control of nonlinear MIMO systems with uncertainty and non-affine inputs is still an open-ended problem.

2.4.6 Nonlinear Output and State Feedback Systems The state feedback control is used for the systems in which all state variables are assumed to be measurable. In practice, it is sometimes very difficult or impossible to measure some states. The output feedback control is applied to the systems in which it is only necessary to measure the output. For nonlinear SISO systems, the adaptive fuzzy output feedback control was proposed in [37, 51] by estimating the necessary output derivatives with high-gain

12

2 Classification of Adaptive Fuzzy Controllers

observers. Due to the use of high-gain observers, the phenomenon of mutation might be detected in the transient solution. To solve this problem, the saturation methods were introduced in [52, 53]. Linear observers were used in [54, 55] to estimate error dynamics. In [32], a nonlinear observer was introduced by using the prior input–output as the neural network input instead of the system output derivatives. The adaptive intelligent output feedback control was also considered for more extensive classes of nonlinear systems. The MIMO systems were also used in [39, 56, 57].

2.4.7 Discrete and Continuous Systems Since most of the controllers are implemented on digital computers, control plays a key role in a discrete space. The adaptive intelligent control of discrete time systems has interested many researchers. Due to the problems with these systems such as noncausality in back stepping design, the discrete time methods have been less popular than the continuous time methods [39]. In the SISO discrete time systems, adaptive intelligent control describes a class of affine nonlinear systems for discrete time [58, 59]. In [60], output and state feedback controllers were analyzed for a class of discrete time nonlinear systems with a general relative degree and bounded distortion. Back stepping design was analyzed in [61] for discrete time systems of the appropriate format. Regarding MIMO discrete time systems, adaptive neural control was analyzed in [62]. Moreover, the state feedback neural control method was analyzed in [38] for a class of MIMO discrete time nonlinear systems. This paper analyzed output feedback control in a similar system with a similar controller only through inputs and outputs.

2.5 Adaptation Mechanism in Fuzzy Systems 2.5.1 Setting Parameters In adaptive intelligent control, smart systems (e.g., neural networks, fuzzy systems, and adaptive fuzzy systems) are employed to estimate some unknown functions. The parameters of smart systems should be adjusted online to guarantee stability. In a smart system, there are usually two types of parameters: linear and nonlinear. For instance, the consequent parameters of a fuzzy system are linear, whereas the parameters of input membership functions (centers and variance) are nonlinear. In a neural network, the middle layer weights are nonlinear, whereas the output layer parameters are linear.

2.6 Conclusion

13

Most of the studies on the applications of smart systems have focused on the settings of linear parameters, and there are few studies on the settings of nonlinear parameters. In [33], adaptive control was described through multilayer neural networks to train the middle layer weights. Adaptive fuzzy control was described in [63–65] in addition to setting the parameters of membership functions. The linear parameters of smart systems can easily be set and analyzed; however, dimension explosion is a potential problem. In other words, the number of adjustable parameters will increase exponentially as the input dimension enlarges. There are fewer nonlinear parameters, although they are harder to set. Not only are they set slowly, but they are also more difficult to analyze. Hence, given the specific application and the necessary accuracy, it is necessary to determine whether to train nonlinear parameters.

2.5.2 Setting Structure and Parameter Most of the systems used in adaptive control have fixed structures. There are also fixed quantities of membership functions in fuzzy systems and neurons in neuron networks. It is important select the right structure, which affects the estimation of a smart system. It is difficult to select the right structure for a specific application. In fact, a designer should test several structures to find the right one. A few attempts have been made to develop smart systems with self-regulating structures. Park described adaptive fuzzy control with a self-regulating structure in [66, 67] by adding rules based on the input. In [45], adaptive neuro-fuzzy control was introduced with a self-regulating structure to add or delete rules. Neural control with a self-regulated structure was analyzed in [38] under the conditions in which the middle layer neurons were divided into two sections. Nevertheless, these methods face certain limitations. For instance, the stability analysis was applied only to a fixed structure. No studies addressed the effects of structure change on stability. No algorithms were proposed in [66–68] to limit the dimensions of smart systems. Therefore, if the initial conditions are not appropriate, the dimensions of smart systems will exceed the implementation capability. Gao employed the error reduction ratio technique in [45] to decrease or increase the rules. Adaptive intelligent control with a self-regulating structure is an open-ended research avenue.

2.6 Conclusion This chapter classified nonlinear systems in terms of dynamic equation structure and described the applications of fuzzy systems in their control briefly to prove dear readers with a general insight into the research literature.

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Chapter 3

Type-2 Fuzzy Systems

3.1 Introduction The fuzzy logic is a successful method for modeling uncertainty, vagueness, and imprecision [1]. Since the introduction of the fuzzy logic, great breakthroughs have been made in the application of fuzzy systems over more than four decades. Although the type-1 fuzzy systems have been used widely in practical applications, many researchers have started studying type-2 fuzzy systems in recent years. Based on the type of input fuzzification, the fuzzy systems are divided into singleton and non-singleton classes.

3.2 Singleton Fuzzy Systems The type-1 singleton fuzzy system was the first type of fuzzy systems proposed. In these systems, type-1 membership functions are used in antecedent and consequent sections, and the inputs have crisp values. There are different sources of uncertainty which fuzzy systems handle in practical applications and real-world environments. Some of these sources of uncertainty are as follows [2]: • • • •

Uncertainty in inputs due to noise and conditions of observers and sensors Uncertainty caused by changes in the conditions of operation controllers Use of noise data for training parameters Uncertainty in modeling through verbal variables.

A flaw of the type-1 fuzzy set is that uncertainties are expressed with membership functions, whereas the outputs of these membership functions are crisp values [3]. Hence, type-1 fuzzy systems cannot model high levels of uncertainty. To solve this problem, type-2 fuzzy systems were introduced by Zadeh in 1975; however, fewer studies had been published on these systems by the end of the previous century [4]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Mohammadzaheh et al., Modern Adaptive Fuzzy Control Systems, Studies in Fuzziness and Soft Computing 421, https://doi.org/10.1007/978-3-031-17393-6_3

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Type-2 fuzzy systems are the developed version of type-1 fuzzy systems, in which the antecedent and consequent membership functions are of type 2 [5]. The output of a type-2 membership function is a value between 0 and 1. In fact, unlike the type-1 fuzzy system, the membership degree is a fuzzy number. This degree of freedom helps model further uncertainties. A type-2 fuzzy set A˜ is defined with the membership function 0 ≤ μ A˜ (x, u) ≤ 1 in which x ∈ X and u ∈ Jx ⊆ [0, 1]: } { A˜ = (x, u), μ A˜ (x, u)| ∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]

(3.1.3)

where Jx is the primary membership function of x when Jx ⊆ [0, 1] for ∀x ∈ X . Uncertainty in the membership degree of the primary type-2 membership function, which includes a range, is called the footprint of uncertainty. In fact, a type-2 membership function can be interpreted as a type-1 set. If the secondary membership degree of type-2 membership functions is continuous, it can be equal to a countless number of type-1 membership functions [5]. The interval type-2 fuzzy systems [6] are a special version of type-2 fuzzy systems in which the secondary membership functions are of the interval form. The footprint of uncertainty, which was defined earlier, can be introduced as the terms of the upper-bound and lower-bound membership functions in the interval form. The use of type-2 membership functions in antecedent and consequent sections resulted in the following superiorities to type-1 membership functions: • The outputs of type-2 membership functions are fuzzy; thus, these functions can model further and more complicated uncertainties [7]. • Since one type-2 membership function with the fuzzy secondary membership degree can be used instead of multiple type-1 membership functions, there will be fewer rules [8]. • As discussed earlier, a type-2 fuzzy set can be expressed as a large number of type-1 fuzzy sets. Hence, using typ-2 membership functions will improve the control performance and yield a better control signal [2]. In the recent decade, type-2 fuzzy systems have interested many researchers [9]. Different papers have reported many practical applications of type-2 fuzzy systems such as the induction motor control, biaxial motion control, temperature control in a part of a lathe, diesel engines, DC-to-DC convertors, and moving robots. In these applications, type-2 fuzzy systems outperformed type-1 fuzzy systems. Although many papers have been published on type-2 fuzzy systems, most of them have used singleton fuzzy systems, which disregard numerical uncertainty. In fact, numerical uncertainty indicates an input with noise as well as the imprecision of sensors and input tools. Therefore, non-singleton fuzzy systems should be employed while dealing with uncertain inputs.

3.3 Non-singleton Fuzzy Systems

19

3.3 Non-singleton Fuzzy Systems The major difference between non-singleton and singleton fuzzy systems lies in fuzzification. Since 1974, fuzzy systems with singleton fuzzification were used extensively in practical applications due to simplicity and small-scale computations when Mamdani employed the first fuzzy controller to control a steam engine. However, many of the practical applications were prone to noisy data and uncertainties in the outputs of sensors, which could not be ignored. Hence, non-singleton fuzzification was proposed to handle uncertain data. At first, the non-singleton type-1 fuzzy (NSF1F) system was utilized in some applications. For instance, it was used in [10] to analyze time series, in [11] to model nonlinear systems, and in [12] to control a DC motor. In [13], a feedback structure (output feedback) was proposed by using the gradient descent to train the parameters. In [14], the genetic algorithm was employed for optimization, and the proposed system was used in classification problems. In [15], non-singleton type-2 fuzzy systems were analyzed by Mendel for the first time ever. In this paper, the type-1 fuzzy system was first introduced, and type-2 operators and definitions were then presented. Most of the studies on type-2 nonsingleton fuzzy systems were conducted by Mendez. These studies mainly include training through different methods. The least error squares method was employed to train the parameters of a type-2 fuzzy system with type-1 non-singleton fuzzification in [12, 16–18] and with type-2 non-singleton fuzzification in [18]. In [19], the Kalman filter was utilized to train the parameters of a type-2 fuzzy system with type1 non-singleton fuzzification, in which the inputs were modeled on the type-1 fuzzy numbers. In [20], a type-2 non-singleton fuzzy system was used for image noise elimination through the modified PSO algorithm for parameter setting. In [15], the studies of fuzzifiers were first reviewed, and a type-2 fuzzy system with non-singleton fuzzification was then proposed without using conventional membership functions such as Gaussian and triangular functions. In the proposed method, the membership function for the input data is inferred from the stored data and a histogram. In [21], the genetic algorithm was employed to train the type-2 non-singleton parameters, and the proposed system was used for pattern classification. In [22], a comparison was drawn between a type-1 fuzzy system with non-singleton fuzzification and a type-2 fuzzy system with singleton fuzzification. According to the results, the nonsingleton type-2 fuzzy system outperformed the non-singleton type-1 fuzzy system in a noisy environment. The most recent study by Mendez in 2013 was conducted on the proposal of hybrid algorithms to train non-singleton type-2 fuzzy systems [23]. In this paper, the recursive square root algorithm and the gradient descent algorithm were adopted to set consequent parameters and antecedent parameters, respectively.

20

3 Type-2 Fuzzy Systems

3.4 Features of Type-2 Fuzzy Systems This subchapter analyzes the important features of type-2 fuzzy systems. At first, type-2 membership functions are introduced and compared with type-1 membership functions. Figure (3.1) demonstrates a Gaussian type-1 membership function. If this function is blurred in Fig. (3.2), a type-2 membership function is obtained. Assume that x has the value of x’, the membership degree (u ' ) has a specific value in the type-1 membership function, whereas u ' has different values in the type-2 membership function. Hence, the type-2 membership function is a 3D membership function. In other words, the membership degree of a type-2 membership function is a type-1 membership function itself. ˜ is defined as below: The type-2 membership function ( A) A˜ =

{(

} ) (x, u), μ A˜ (x, u) | ∀x ∈ X, ∀u ∈ Jx ⊆ [0, 1]

(3.1)

where 0 ≤ μ A˜ (x, u) ≤ 1. In fact, Jx ⊆ [0, 1] indicates the primary membership of x, whereas μ A˜ (x, u) is a type-2 membership function that represents the secondary membership. If the degree of the secondary membership function is 1, i.e., μ A˜ (x, u), ∀u ∈ Jx ⊆ [0, 1]„ there will then be an interval type-2 membership 1

Membership

0.8 0.6 0.4 0.2 0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

1

1.5

2

2.5

3

Input

Fig. 3.1 Gaussian type-1 membership function

Membership

1 0.8 0.6 0.4 0.2 0 -2

-1.5

-1

-0.5

0

0.5

Input

Fig. 3.2 Type-2 membership function

3.5 Basic Operations in Type-2 Fuzzy

21

Membership

1 0.8 0.6 0.4 0.2 0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Input

Fig. 3.3 An interval type-2 membership function

Fig. 3.4 The structure of a type-2 fuzzy system

function (refer to Fig. (3.3)). Figure (3.4) depicts the structure of a type-2 fuzzy system, each component of which is described here.

3.5 Basic Operations in Type-2 Fuzzy Before some of the basic operations are described, a few basic concepts of the interval type-2 fuzzy system are analyzed. Consider an interval type-2 membership function shown in Fig. (3.5). The upper-bound membership function (UMF) and the lower-bound membership function (LMF) indicate the highest and lowest degrees of primary membership, respectively. In general, a type-2 membership function can be considered the combination of type-1 membership functions, each of which is called an embedded membership function. If the type-2 membership function is sliced horizontally, the lowest level is called the footprint of uncertainty (FOU). Consider two interval type-2 membership functions of A˜ and B˜ in the form shown ˜ intersection ( A˜ ∩ B), ˜ and by Fig. (3.5). The operations of their union ( A˜ ∪ B), ˜ are defined as below: complement ( A)

3 Type-2 Fuzzy Systems Secondary membership

22 1 0.8 0.6 0.4 0.2 0 1 0.5

primary membership

0

-10

-8

-6

-4

-2

0

2

4

6

8

10

input

Fig. 3.5 An interval type-2 membership function

[ ] A˜ ∪ B˜ = 1/ μ A(x) ∨ μ B(x) , μ A(x) ∨ μ B(x) ∀x ∈ X ˜ ˜ ˜ ˜

(3.2)

[ ] A˜ ∩ B˜ = 1/ μ A(x) ∧ μ , μ ∧ μ ˜ ˜ A(x) B(x) ∀x ∈ X ˜ ˜ B(x)

(3.3)

[ ] A˜ = 1/ 1 − μ A(x) ∀x ∈ X ˜ ˜ , 1 − μ A(x)

(3.4)

where ∨ and ∧ represent t-conorm and t-norm, respectively.

3.6 Fuzzification )T ( A fuzzifier maps an input vector x = x1 , . . . , x p onto a membership function A˜ x . If A˜ x is a type-1 membership function, fuzzification is considered type-1 nonsingleton. If it is a type-2 membership function, fuzzification is considered type-2 non-singleton. Moreover, if it is a singleton membership function (i.e., μ A˜ x (x) = 1/1 for x = x ' and μ A˜ x (x) = 1/0 for x /= x ' ), then fuzzification is considered singleton.

3.7 Rules Type-1 and type-2 fuzzy systems share the same structure of rules; however, antecedent and consequent membership functions are of type-2. Consider a type2 fuzzy system with p inputs (x1 , . . . , x p ) and one output (y). The Lth rule is defined as below: R l : I F x1 is F˜1l and · · · and x p is F˜ pl , T H E N y is G˜ l , l = 1, . . . , M where F˜ and G˜ are type-2 membership functions.

(3.2)

3.8 Logics

23

3.8 Logics In type-2 fuzzy systems, the inference engine combines the rules and maps an input type-2 fuzzy set onto an output type-2 fuzzy set. If F˜11 × · · · × F˜ p1 = A˜ l , Eq. (3.2) can be rewritten as below: R l : F˜11 × · · · × F˜ p1 → G˜ l

A˜ l → G˜ l

=

l = 1, . . . , M

l where through the membership functions μ Rl (x, y) ( R is described ) μ Rl x1 , . . . , x p , y :

μ Rl (x, y) = μ A˜ l →G˜ l (x, y)

(3.3) =

(3.4)

Equation (3.4) can then be rewritten as below: μ Rl (x, y) = μ A˜ l →G˜ l (x, y) ⊓ ⊓ ( )⊓ ··· μ F˜ pl x p = μ F˜1l (x1 ) μG˜ l (y) [⊓ p ]⊓ μ F˜il (xi ) = μG˜ l (y) i=1

(3.5)

In general, the p-dimensional input to R l is written with a type-2 membership function A˜ x : A˜ x = μx˜1 (x1 )

⊓

···

⊓

( ) ⊓p μx˜ p x p = μx˜i (xi ) i=1

(3.6)

where x˜i (i = 1, . . . , p) denotes the fuzzy sets corresponding to the inputs. Each rule R l determines a type-2 fuzzy set B˜ l = A˜ x ◦ R l when: μ B˜ l (y) = μ A˜ x ◦Rl =

[

⊔ x∈X

μ A˜ x (X )

⊓

]

μ Rl (X, y)

(3.7)

Fig. (3.8) demonstrates this input–output dependency. In most of the applications, the interval type-2 fuzzy set is used with t-norm; therefore, the degrees of fire are as below: ] ( ) [ ( ) l ( )] [ l F˜ l X ' = f l X ' , f X ' = f l , f

(3.8)

( ) ( ) ( ) f l X ' = μ F˜ l x1' ∗ · · · ∗ μ F˜ l x 'p

(3.9)

where 1

and

p

24

3 Type-2 Fuzzy Systems

( ) ( ) l( ) f X ' = μ F˜1l x1' ∗ · · · ∗ μ F˜ pl x 'p where * denotes the product operator.

3.9 Type Reduction In this subchapter, a type-1 set is generated in the output. There are different methods for type reduction: 1. Centroid Type Reduction In a centroid defuzzifier, the type-1 output sets are combined through t-conorm to generate a centroid. If the resultant output fuzzy set is called B, then: ΣN yc (x) = Σi=1 N

yi μ B (yi )

i=1

μ B (yi )

(3.10)

where the number of partitions is B. In the centroid type reduction, type-2 output fuzzy sets are combined through their union. The membership degree (y ∈ Y ) is defined as below: μ B˜ (y) =

⊔M l=1

μ B˜ l (y)

(3.11)

where μ B˜ l is obtained from Eq. (3.5). ˜ In fact, the centroid type reduction obtains the centroid of B:  Yc (x) =

θ1

/Σ N [ ] i=1 yi θi μ D1 (θ1 ) ∗ · · · ∗ μ D N (θ N ) ΣN i=1 θi

 ···

θN

(3.12)

where Di = μ B˜ (yi ) and θi ∈ μ B˜ (yi ), i = 1, . . . , N . Consider the following definitions: ΣN a = Σi=1 N

yi θi

i=1 θi

, b = μ D1 (θ1 ) ∗ · · · ∗ μ D N (θ N )

(3.13)

According to Eq. (3.12), a large number of (a, b) calculations should be performed to determine Yc (x). Assume that (a, b) is calculated for α times: (a1 , b1 ), . . . , (aα , bα ). The necessary and important information for type reduction includes only the greatest and smallest values of (a, b), i.e., a L = min ai , a R = max ai , b L = min bi , and b R = max bi . The next chapters will propose simpler method with much lower volumes of calculations. The following operations should be implemented step by step to calculate Yc (x):

3.9 Type Reduction

25

(1) Obtain the combined output set through Eq. (3.11) (for this purpose, calculate μ B˜ l first through Eq. (3.5)). (2) Divide the output space into N sections (y1 , . . . , y N ). (3) Divide μ B˜ (yi ), i = 1, . . . , N into M i points. ⊓N (4) The number of membership set will be i=1 Mi , and calculate the pairs of ⊓N (ai , bi ), i = 1, . . . , i=1 Mi through Eq. (3.13). Finally, obtain Yc (x) from Eq. (3.12). Furthermore, the minimum t-norm should be used in this method. 2. Height Type Reduction In a height defuzzifier, the output membership set of each rule is replaced with a singleton in a point which as the largest membership degree in the set. The output of the defuzzifier is obtained from the following equation: ΣM yh (x) =

( )

l l l=1 y μ B l y ( l) ΣM l=1 μ B l y

(3.14)

where y l is a point that has the highest membership degree in the Lth output set (if there ( ) number of similar points is more than one, their mean is considered). For μ B l y l , the following equation is considered: ( ) ( ) p μ B l y l = μG l y l ∗ Ti=1 μ Fil (xi )

(3.15)

where T and * represent the t-norm (product or minimum t-norm). In the height type reduction method, the output set is replaced with a type-2 singleton, the range of which is a point. Its membership degree is a type-1 set within [0, 1]. The Lth output set has a singleton at y l with the highest membership degree in the principal membership function of the output set B l (the principal membership function has the highest secondary membership function). Hence, Eq. (3.12) is simplified as below:  Yh (x) =

θ1

 ···

θM

[

/Σ M l ] l=1 y θl μ D1 (θ1 ) ∗ · · · ∗ μ D M (θ M ) ΣM l=1 θl

(3.16)

where θl ∈ Dl , l = 1, . . . , M. Perform the following calculations step by step to determine Yh (x): ( ) (1) In each output set l = 1, . . . , M, select y l and determine μ B l y l . ( ) (2) Like the previous method, divide the range of μ B l y l into an appropriate number of points. However, the number of points on the horizontal axis is M in this ( )method. (3) If μ B l y l is divided into N points, the total number of possible combinations ⊓M in Eq. (3.16) will be i=1 Ni . (4) Determine Yh (x) through Eq. (3.16).

26

3 Type-2 Fuzzy Systems

In Eq. (3.16), the number of calculations is that of the previous method.

⊓M i=1

Ni , which is much smaller than

3. Center-of-sets (COS) Type Reduction In the COS defuzzification method, the consequent set of each rule is replaced with a singleton located in its centroid, and the centroid of the type-1 set constructed from these singletons will then be obtained. The output is as follows: ΣM l=1

ycos (x) = Σ M

p

cl Ti=1 μ Fil (xi )

l=1

(3.17)

p

Ti=1 μ Fil (xi )

where T and cl represent t-norm and the lth consequent set. If the consequent sets ( ) are symmetric, normal, and convex, there will be cl = y l and μG l y l = 1 for l = 1, . . . , M; therefore: ycos (x) = yh (x). Similar to defuzzification, each consequent set is replaced with its centroid (if the consequent set is type-2, its centroid will be a type-1 set) in the COS type reduction, and the weighted mean of these centroids will be determined.⊓ The weight of the lth p centroid of the firing strength corresponds to the lth rule, i.e., i=1 μ Fil (xi ). Finally, the output of this type reduction method is as follows:  Ycos (x) =

 ···

d1

··· dM

e1

/Σ





eM

M Tl=1 μcl (dl )

∗

M μ El (el ) Tl=1

M l=1

ΣM

dl el

l=1 el

(3.18)

where T and * denote t-norm. Moreover, dl = cl = cG˜ l is the lth centroid of the ⊓p consequent set, whereas el ∈ El = i=1 μ Fil (xi ) indicates the firing strength corresponding to the lth consequent set. The following calculations should be performed step by step to find the output: (1) Divide the output space Y into the right number of points, and determine the centroid of each consequent set (cG˜ l ). These centroids can be obtained once and then used along the process. ⊓p (2) Determine the firing strength El = i=1 μ Fil (x i ) corresponding to the lth consequent set. (3) Divide the range of each cG˜ l into the right number Ml . (4) Divide the range of each El into the right number Nl . (5) Determine the number of possible states {c1 , . . . , c M , e1⊓ , . . . , e M } for el ∈ El and dl ∈ cG˜ l . The total number of states will be equal to M j=1 M j N j . (6) Finally, obtain the output from Eq. (3.18) 4. Type Reduction for Interval Type-2 Systems The general type-2 fuzzy systems are very complicated due to the large amounts of type reduction calculations. In general, the most important method of type reduction is as follows:

3.9 Type Reduction

27

Y (Z 1 , . . . , Z M , W1 , . . . , W M ) /Σ     M l=1 wl z l M M ··· ··· Tl=1 μzl (zl ) ∗ Tl=1 μWl (wl ) = ΣM z1 z M w1 wM l=1 wl

(3.19)

where T and * denote t-norm with wl ∈ Wl , zl ∈ Z l . If the secondary degrees of membership functions are of the interval type, the amounts of calculations will noticeably decrease. For an interval type-2 fuzzy system, each of Wl , Z l , l = 1, . . . , M in Eq. (3.18) is an interval type-1 set. Hence, the following equation can be written due to the fact that μzl (zl ) = μWl (wl ) = 1: 

 Y (Z 1 , . . . , Z M , W1 , . . . , W M ) =

··· z1



 zM

w1

···

/Σ 1

wM

M l=1

ΣM

wl zl

l=1

wl

(3.20)

Since memberships are crisp numbers in an interval type-1 set, an interval set is determined with upper and lower bounds. In general interval type-2 fuzzy systems, each zl in Eq. (3.20) is an interval type-1 set, the centroid of which is cl . This set expands from both sides as much as sl . Moreover, each wl is an interval type-1 set with the centroid of h l and the expansion of Δl . Since Y is an interval type-1 set, the first and last points of this interval must only be determined (yl and yr ). An iterative method is now presented to find yl and yr . Consider the following definition: ΣM wl zl S(w1 , . . . , w M ) = Σl=1 M l=1 wl

(3.21)

where wl ∈ [h l − Δl , h l + Δl ], h l ≥ Δl , and zl ∈ [cl − sl , cl + sl ]. The maximum values of S and yl are determined through the following steps by considering zl = cl + sl and z 1 ≤ z 2 ≤ · · · ≤ z M : (1) Obtain S ' = S(h 1 , . . . , h M ) from Eq. (3.21) by considering wl = h l . (2) Find k(1 ≤ k ≤ M − 1) when z k ≤ S ' ≤ z k+1 . (3) Obtain S '' = S(h 1 − Δl , . . . , h k − Δk , h k+1 + Δk+1 , . . . , h M + Δ M ) from Eq. (3.21) by considering wl = h l − Δl for l ≤ k and wl = h l + Δl for l ≥ k + 1. (4) If S '' = S ' , the algorithm will stop, and S '' will be the maximum value of S; otherwise, go to the next step. (5) Go to the second step by considering S '' = S ' . Mendel et al. indicated that the algorithm would reach a solution in at most M steps.

28

3 Type-2 Fuzzy Systems

3.10 Implementation in MATLAB Different methods of type reduction are implemented through an example. Consider the consequent membership functions in Fig. (3.6). There are two Gaussian membership functions at each point within y ∈ [1, 5]. For instance, there are two Gaussian membership functions 0.9∗ N (y, 2, 0.4) and 0.8∗ N (y, 2, 0.4) at y = 2. The methods of rank reduction are applied to these membership functions, and the results are compared. (1) Centroid In this method, the union of membership functions should first be determined as below: (Fig. 3.7). If y ∈ [1, 5] is divided into 21 points, there will totally be 221 embedded type-2 membership functions, the centroids of which are shown in Fig. (3.8). Table (3.1) presents the right and left bounds of these membership functions. Table (3.2) indicates the script of this method in MATLAB. (2) Height In this method, the activated output membership functions are replaced with a singleton membership function with the highest primary membership degree. 1 1

1 0.5

0.5

0.8 1

0

0.4

1

1.5

2

2.5

3

3.5

4

4.5

5

3

3.5

4

4.5

5

Fig. 3.6 Consequent membership functions

1 0.8 0.6 0.4 0.2 0

1

1.5

2

2.5

Fig. 3.7 The union of consequent membership functions

3.10 Implementation in MATLAB

29

Fig. 3.8 Rank reduction with the centroid method Table 3.1 The results of rank reduction with the centroid method

Centroid method

Right bound point

Left bound point

2.4448

2.2828

Table 3.2 The script of the centroid method for rank reduction in MATLAB

MATLAB script for the centroid rank reduction clear all clc x=1:0.01:5; y1_u=@(x)0.9*exp(-0.5.*(x-2).^2./0.4.^2); y1_l=@(x) 0.8*exp(-0.5.*(x-2).^2./0.4.^2); y2_u=@(x) 0.8*exp(-0.5.*(x-3).^2./0.2.^2); y2_l=@(x) 0.6*exp(-0.5.*(x-3).^2./0.2.^2); y3_u=@(x) 0.2*exp(-0.5.*(x-5).^2./0.2.^2); y3_l=@(x) 0.1*exp(-0.5.*(x-5).^2./0.2.^2); %union_MF= subplot(2,1,1) plot(x,y3_u(x)); hold on plot(x,y3_l(x)); plot(x,y2_u(x)); plot(x,y2_l(x)); plot(x,y1_u(x)); plot(x,y1_l(x)); N=21; x1=x; %% Union y1(1,:)=y1_u(x1); y1(2,:)=y2_u(x1); y1(3,:)=y3_u(x1);

(continued)

30

3 Type-2 Fuzzy Systems

Table 3.2 (continued) a1=find(y1(1,:)100)=100; cl(cl>100)=100; wu(wuwr(1:M-1)); if ~isempty(R) R=R(end); else R=1; end yr_prim=(wr(1:R)'*fl(1:R)+wr(R+1:end)'*fu(R+1:end))/(sum(fl(1:R))+sum(fu(R+1:end))); if abs(yr_prim-yr)wl(1:M-1)); if ~isempty(L) L=L(end); else L=1; end yl_prim=(wl(1:L)'*fu(1:L)+wl(L+1:end)'*fl(L+1:end))/(sum(fu(1:L))+sum(fl(L+1:end))); if abs(yl_prim-yl)