Polynomial fuzzy model based control systems

Studies in Systems, Decision and Control 64Hak-Keung Lam Polynomial Fuzzy Model-Based Control Systems Stability Analysis and Control Synthesis Using Membership Function-Dependent Techniq

Studies in Systems, Decision and Control 64

Hak-Keung Lam

Polynomial Fuzzy Model-Based

Control Systems Stability Analysis and Control Synthesis Using Membership Function-Dependent Techniques

Studies in Systems, Decision and Control Volume 64

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Department of Informatics

King’s College London

London

UK

Studies in Systems, Decision and Control

ISBN 978-3-319-34092-0 ISBN 978-3-319-34094-4 (eBook)

DOI 10.1007/978-3-319-34094-4

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To my family, my beloved wife,

Esther Wing See Chan, and

my lovely daughter, Katrina Faye Lam, for standing by me

Hak-Keung Lam

I have researched on fuzzy model-based control systems since 1995 as a Ph.D.student During the past two decades, I have gained a rich experience andknowledge in the field, and have summarized the research achievements by anumber of publications.

I have witnessed the development of the research in the field of fuzzymodel-based control systems in the past 20 years In general, I divide it into fourstages In thisfirst stage, from early 1970s to late 1980s, fuzzy control has becomewell known as an intelligent control strategy for ill-defined and complex systemsdue to successful applications from household appliances to chemical plants Usingfuzzy logic concept, human spirit can be captured by linguistic rules which can berealized by machines Fuzzy logic controller is thus able to incorporate humanknowledge to control complex systems In the early applications, the fuzzy logiccontroller was designed heuristically without the need of the mathematical model ofnonlinear systems Although good performance can be demonstrated by someapplication examples, essential issues such as system stability and robustness arenot guaranteed which put the users and applications at risk

In the second stage, from early 1990s to mid-2000s, thanks to the T–S fuzzymodel fuzzy model-based control has become very popular and offered a systematicway for system analysis and control design Stability analysis has become a verypromising research topic since then Fruitful analysis results have been reported inmany articles Relaxation of stability conditions has drawn a great deal of attentionfrom the researchers in the fuzzy control community As the stability analysis hasnot considered the membership functions in most of the work during this period, thestability analysis/conditions are named as membership function-independent sta-bility analysis/conditions in my publications

The third stage was from mid-2000s to late 2000s I have proposed the bership function-dependent stability analysis which is able to bring the informationand characteristic of the membership functions into the stability conditions.Consequently, it is named as membership function-dependent stability conditions in

mem-my publications As the membership functions are the nonlinearity of the nonlinear

vii

system, it plays an important role to achieve more relaxed stability analysis resultscompared with the membership function-dependent stability analysis Furthermore,opposite to the concept of parallel distributed compensation, I have promoted theconcept of partially/imperfectly matched premises that the number of rules and/orpremise membership functions used in the fuzzy controller are different from those

of the fuzzy model to achieve greater control design flexibility and lower controlimplementation complexity (to reduce the implementation costs)

The fourth stage started from late 2000s The introduction of the polynomialfuzzy model takes the stability analysis and fuzzy control to another level usingsum-of-squares approach instead of linear matrix inequality

This book focuses on the work on the fourth stage which is the research on thestability analysis of polynomial fuzzy model-based control systems where the con-cept of partially/imperfectly matched premises and membership function-dependentanalysis are considered I would like to summarize my recent achievements on thistopic which present the latest research outcomes including findings, observations,concepts, ideas, research directions, stability analysis techniques, and controlmethodologies The membership function-dependent analysis offers a new researchdirection for the fuzzy model-based control systems by taking into account thecharacteristic and information of the membership functions (related to the nonlin-earity of the plant) in the stability analysis Membership function-dependent stabilityconditions are far more relaxed compared with some state-of-the-art membershipfunction-independent stability conditions It is more effective to deal with nonlinearcontrol problems as membership function-dependent approach considers the dedi-cated nonlinear system on hand rather than a family of nonlinear systems tackled inthe membership function-independent approach Through this book, I would like topromote the membership function-dependent analysis to be a new research directionand hope to see that it becomes a popular technique to deal with the stability analysisproblem for fuzzy model-based control systems

The content of this book is mainly at the research level presenting the mostrecent and advanced research results, which aims to promote the research ofpolynomial fuzzy model-based control systems, provide theoretical support, andpoint a research direction to postgraduate students and fellow researchers Theintroduction and preliminary parts of the book provides an overview of the topicsand technical materials are presented in a very detailed manner Numerical exam-ples are provided in each chapter to verify the analysis results, demonstrate theeffectiveness of the proposed polynomial fuzzy control schemes, and explain thedesign procedure This book is comprehensively written with detailed derivationsteps and mathematical details to enhance the reading experience, in particular, forreaders without extensive knowledge on the topics It is thus also recommended toundergraduate students with control background who are interested in polynomialfuzzy model-based control systems

This book has four parts consisting of ten chapters Thefirst part Introductionand Preliminaries provides the overview and technical background of the fuzzymodel-based control systems offering fundamental knowledge and mathematicalsupport for the subsequent parts The second part Stability Analysis Techniques

presents the latest techniques based on the membership function-dependent stabilityanalysis for polynomial fuzzy model-based control systems The third partAdvanced Control Methodologies extends the stability analysis techniques to morechallenging control problems The fourth part Advanced Lyapunov Functionsintroduces more effective Lyapunov functions for stability analysis and polynomialfuzzy control strategy for the control of nonlinear plants The content of eachchapter is briefly introduced below.

Part I Introduction and Preliminaries

• Chapter 1 gives a general overview of the fuzzy model-based control whichcovers the background, literature review, development of the field, fuzzymodels, fuzzy control methodologies, stability analysis approaches, and controlproblems

• Chapter 2 provides the technical and mathematical background for the fuzzymodel-based control which offers the equations of the fuzzy model andclosed-loop systems, definition of variables, published stability conditions interms of linear matrix inequalities, and sum of squares (SOS) These materialsare essential for the work in the subsequent chapters

Part II Stability Analysis Techniques

• Chapter 3 investigates the stability of polynomial fuzzy model-based controlsystems by treating the membership functions and system states as symbolicvariables The information of membership functions is considered in the stabilityanalysis and brought to the SOS-based stability conditions Techniques areproposed to introduce slack matrix variables carrying the information ofmembership functions to the SOS-based stability conditions without increasingmuch the computational demand

• Chapter 4 investigates the stability of polynomial fuzzy model-based controlsystems by bringing the approximated membership functions into theSOS-based stability conditions Various approximation methods of membershipfunctions are reviewed and their characteristics are discussed Using the Taylorseries expansion, the original membership functions are represented byapproximated membership functions which are a weighted sum of local poly-nomials in a favorable form for stability analysis SOS-based stability conditionsare obtained which guarantee the system stability if the fuzzy model-basedcontrol system is stable at all chosen Taylor series expansion points

• Chapter 5 investigates the stability of general polynomial fuzzy model-basedcontrol systems In Chaps.3 and4, a constraint that the polynomial Lyapunovfunction matrix is allowed to be dependent on some state variables determined

by the structure of the input matrices is required to obtain convex stability

conditions In this chapter, this constraint is removed and a two-step procedure

is proposed to search for a feasible solution to the SOS-based stability tions Consequently, the stability analysis results can be applied to a wider range

condi-of polynomial fuzzy model-based control systems

Part III Advanced Control Methodologies

• Chapter 6 considers a regulation problem for polynomial fuzzy model-basedcontrol systems An output-feedback polynomial fuzzy controller is employed todrive the system outputs to reach a desired level SOS-based stability conditionsfor the three cases (perfectly, partially and imperfectly matched premises) areobtained, which are facilitated by considering different information of mem-bership functions, to determine the system stability and synthesize the con-troller With the support of Barbalat’s Lemma, it is guaranteed that a stableoutput-feedback polynomial fuzzy controller will produce no steady state error

• Chapter 7 considers a tracking problem for polynomial fuzzy model-basedcontrol systems An output-feedback polynomial fuzzy controller is employed todrive the system outputs to follow a reference trajectory SOS-based stabilityconditions are obtained to determine the system stability and synthesize thecontroller where the tracking performance satisfies an H1 performance indexgoverning the tracking error

• Chapter 8 considers a sampled data output-feedback polynomial fuzzymodel-based control system which is formed by a nonlinear plant represented bythe polynomial fuzzy model and a sampled data output-feedback polynomialfuzzy controller connected in a closed loop SOS-based stability analysis con-sidering the effect due to sampling and zero-order-hold activities is performedusing the input-delay method SOS-based stability conditions are obtained todetermine the system stability and synthesize the controller

Part IV Advanced Lyapunov Functions

• Chapter 9 proposes a switching polynomial Lyapunov function candidate,which consists of a number of local sub-Lyapunov function candidates, for thestability analysis of polynomial fuzzy model-based control systems whereswitching is dependent on the system states When the system state vector fallsinto the pre-defined local operating domain, the corresponding localsub-Lyapunov function candidate is employed to take care of the system sta-bility Corresponding to each local sub-Lyapunov function candidate, a localpolynomial fuzzy controller is employed for the control of the nonlinear plantresulting in a switching polynomial fuzzy control strategy A favorable form ofswitching polynomial Lyapunov function candidate is proposed to make sure

that smooth transition among the local sub-Lyapunov function candidates takesplace at the switching boundary for a valid Lyapunov function candidate.SOS-based stability conditions are obtained to determine the system stabilityand synthesize the controller.

• Chapter10 proposes a fuzzy polynomial Lyapunov function candidate, whichconsists of a number of local sub-Lyapunov function candidates, for the stabilityanalysis of polynomial fuzzy model-based control systems where the contri-bution of each local sub-Lyapunov function candidate to the overall fuzzypolynomial Lyapunov function candidate is governed by the membershipfunctions and fuzzy rules Piecewise linear membership functions are proposedfor the implementation of membership functions in the fuzzy polynomialLyapunov function candidate to alleviate the difficulty in the stability analysiscaused by the time derivative of the membership functions Furthermore, thepiecewise linear membership functions divide the overall operating domain intooperating sub-domains A local polynomial fuzzy controller is proposed for thecorresponding operating sub-domains During the control process, the corre-sponding local polynomial fuzzy controller is employed for the control of thenonlinear plant resulting in switching control strategy SOS-based stabilityconditions are obtained to determine the system stability and synthesize thecontroller

November 2014

Thanks are delivered to my previous and current Ph.D students, Dr HugoNovais Carvalho Araujo, Dr Udeme Ekong, Dr Mohammad Narimani, Mr.Yuandi Li, Mr Chuang Liu, Mr Ahmad Ataka Awwalur Rizqi, Ms Ge Song, Mr.

Bo Xiao, Mr Xiaozhan Yang, Mr Yanbin Zhao, who have provided different kind

of support during the writing

In particular, I am greatly indebted to my wife, Esther Wing See Chan, and mydaughter, Katrina Faye Lam, for their patience, understanding, support, andencouragement that make this work possible

The work described in this book was substantially supported by King’s CollegeLondon

xiii

Part I Introduction and Preliminaries

1 Introduction 3

1.1 Background 3

1.2 Fuzzy Model-Based Control 5

1.2.1 Fuzzy Models 5

1.2.2 Fuzzy Controllers 8

1.2.3 Other Fuzzy Controllers 12

1.3 Stability Analysis 12

1.3.1 Types of Lyapunov Functions 14

1.3.2 Types of Stability Analysis 17

1.3.3 Techniques of Stability Analysis 18

1.4 Control Problems 22

1.4.1 Stabilization Control Problem 22

1.4.2 Regulation Control Problem 22

1.4.3 Tracking Control Problem 23

1.5 Design of FMB Control Systems 24

1.6 Conclusion 25

References 25

2 Preliminaries 39

2.1 Introduction 39

2.2 Notation 40

2.3 Fuzzy Models 41

2.3.1 T-S Fuzzy Model 41

2.3.2 Polynomial Fuzzy Model 43

2.4 State-Feedback Fuzzy Controller 45

2.4.1 Fuzzy Controller 45

2.4.2 Polynomial Fuzzy Controller 46

2.5 Various Types of FMB Control Systems 46

2.5.1 FMB Control System 47

2.5.2 PFMB Control System 47

xv

2.6 LMI/SOS-Based Stability Conditions 47

2.6.1 LMI-Based Stability Conditions for FMB Control Systems 48

2.6.2 SOS-Based Stability Conditions for FMB Control Systems 55

2.7 Conclusion 56

References 56

Part II Stability Analysis Techniques 3 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems with Mismatched Premise Membership Functions Through Symbolic Variables 61

3.1 Introduction 61

3.2 Stability Analysis 64

3.2.1 Property of Membership Functions 69

3.2.2 Boundary Information of Membership Grades 70

3.2.3 Boundary Information of Premise Variables 71

3.3 Simulation Example 72

3.4 Conclusion 81

References 81

4 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems with Mismatched Premise Membership Functions Through Taylor Series Membership Functions 85

4.1 Introduction 85

4.2 Taylor Series Membership Functions 88

4.3 Stability Analysis 91

4.4 Simulation Example 95

4.5 Conclusion 101

References 102

5 Stability Analysis of General Polynomial Fuzzy Model-Based Control Systems 103

5.1 Introduction 103

5.2 Stability Analysis 105

5.2.1 Perfectly Matched Premises 106

5.2.2 Partially Matched Premises 114

5.2.3 Imperfectly Matched Premises 120

5.3 Simulation Example 124

5.3.1 Perfectly Matched Premises 125

5.3.2 Partially Matched Premises 127

5.3.3 Imperfectly Matched Premises 128

5.4 Conclusion 133

References 133

Part III Advanced Control Methodologies

6 Output Regulation of Polynomial Fuzzy Model-Based

Control Systems 137

6.1 Introduction 137

6.2 Preliminaries 138

6.2.1 Polynomial Fuzzy Model with Output 138

6.2.2 Polynomial Fuzzy Controller with Integral Action 139

6.2.3 PFMB Control System 140

6.3 Stability Analysis 140

6.3.1 Perfectly Matched Premises 142

6.3.2 Perfectly Matched Premise with Regional Membership Function Information 148

6.3.3 Partially Matched Premises 152

6.3.4 Imperfectly Matched Premises 156

6.4 Simulation Example 158

6.4.1 Perfectly Matched Premises 159

6.4.2 Partially Matched Premises 161

6.4.3 Imperfectly Matched Premises 164

6.5 Conclusion 171

References 171

7 Output-Feedback Tracking Control for Polynomial Fuzzy Model-Based Control Systems 175

7.1 Introduction 175

7.2 Preliminaries 176

7.2.1 Polynomial Fuzzy Model 176

7.2.2 Reference Model 177

7.2.3 Output-Feedback Polynomial Fuzzy Controller 178

7.2.4 PFMB Control System 179

7.2.5 Error System 179

7.3 Stability Analysis 180

7.4 Simulation Examples 186

7.4.1 Simulation Example 1 186

7.4.2 Simulation Example 2 191

7.5 Conclusion 194

References 195

8 Sampled-Data Output-Feedback Fuzzy Controller for Nonlinear Systems Based on Polynomial Fuzzy Model-Based Control Approach 197

8.1 Introduction 197

8.2 Preliminaries 199

8.2.1 Polynomial Fuzzy Model 199

8.2.2 Sampled-Data Output-Feedback Fuzzy Controller 200

8.2.3 Sampled-Data Output-Feedback Polynomial Fuzzy Model-Based Control System 201

8.3 Stability Analysis 201

8.3.1 Stability Analysis with c¼ p 203

8.3.2 Stability Analysis with c6¼ p 209

8.4 Simulation Examples 211

8.5 Conclusion 217

References 218

Part IV Advanced Lyapunov Functions 9 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems Using Switching Polynomial Lyapunov Function 223

9.1 Introduction 223

9.2 Switching Polynomial Lyapunov Function 227

9.3 Switching Polynomial Fuzzy Controller 230

9.4 Polynomial Fuzzy Model-Based Control System 230

9.5 Stability Analysis 231

9.5.1 Relaxed Conditions for VðxÞ 232

9.5.2 Relaxed Conditions for _VðxÞ 232

9.6 Simulation Examples 235

9.6.1 Simulation Example: c6¼ p and fm1; ; Mcg 6¼ fw1; ; wpg 235

9.6.2 Simulation Example: c¼ p and fm1; ; Mcg ¼ fw1; ; wpg 244

9.7 Conclusion 256

References 256

10 Stability Analysis of Polynomial Fuzzy Model-Based Control Systems Using Fuzzy Polynomial Lyapunov Function 259

10.1 Introduction 259

10.2 Fuzzy Polynomial Lyapunov Function 260

10.3 Stability Analysis 263

10.3.1 Case c6¼ q and fm1; ; Mcg 6¼ fn1; ; nqg 263

10.3.2 Case c¼ q and fm1; ; Mcg ¼ fn1; ; nqg 267

10.4 Simulation Examples 270

10.5 Conclusion 291

References 291

Index 295

FLS Fuzzy Logic System

FMB Fuzzy Model-Based

FPLF Fuzzy Polynomial Lyapunov Function

IT2 Interval Type-2

LMI Linear Matrix Inequality

MFD Membership Function-Dependent

MFI Membership Function-Independent

MRAC Model Reference Adaptive Control

PDC Parallel Distributed Compensation

PFMB Polynomial Fuzzy Model-Based

PLF Polynomial Lyapunov Function

PLMF Piecewise Linear Membership Function

PMF Piecewise Membership Function

SDOF Sampled Data Output-Feedback

SDP Semi-Definite Program

SMF Staircase Membership Function

SOS Sum of Squares

SPLF Switching Polynomial Lyapunov Function

Part I

Introduction and Preliminaries

The basic concept, knowledge, and overview of the recent development of FMBcontrol systems are presented in Chap.1to provide a fundamental background toreaders Technical background and published stability analysis results for FMB con-trol systems are given in Chap.2, which offers preliminary details to support thestability analysis and control design presented in the subsequent chapters

Abstract This chapter gives a general overview of the fuzzy model-based control

which covers the background, literature review, development of the field, fuzzy els, fuzzy control methodologies, stability analysis approaches and control problems.The motivation of moving from basic fuzzy logic system for automatic control tofuzzy model-based control is first discussed It follows by discussing various types

mod-of fuzzy models such as T–S and polynomial fuzzy models Various types mod-of fuzzycontrollers are reviewed and their characteristics are discussed Combining variousfuzzy models and fuzzy controllers, a wide range of fuzzy model-based control sys-tems are formed Stability analysis of the fuzzy model-based control systems subject

to various types of Lyapunov functions, types of stability analysis and types of ity analysis techniques are gone through The characteristics of the three main types

stabil-of control problems including stabilization, regulation and tracking are discussed

1.1 Background

The theory of fuzzy sets was proposed by Prof Lotfi A Zadeh in 1965 Fuzzylogic generalizes the concept of traditional Boolean/multiple-valued logic A vague

or an ill-defined term such as linguistic terms, say, short, tall, small, big, etc., can

be measured by a fuzzy set characterized by a membership function sitting in the

Universe of Discourse with the membership grade (or membership value) in the

range of 0–1

With the concept of fuzzy sets, expert knowledge can be represented by a set

of fuzzy rules in an IF-THEN format In general, the IF part is referred to as the

antecedent or premise and the THEN part the consequent Taking a simple rule as an

example “IF distance x (t) is Long THEN force u(t) is Big”, the distance and force

are described by the fuzzy sets Long and Big, respectively, characterized by their responding membership functions The firing strength of each rule, which indicates

cor-how good the rule describing the situation, will be determined by the membershipgrades processed by the fuzzy operators

A general fuzzy logic system (FLS) [1] is shown in Fig.1.1 An FLS consists

of 4 components namely fuzzifier, rule base, inference engine and defuzzifier The

© Springer International Publishing Switzerland 2016

H Lam, Polynomial Fuzzy Model-Based Control Systems, Studies in Systems,

Decision and Control 64, DOI 10.1007/978-3-319-34094-4_1

3

Fuzzy Outputs

Fig 1.1 A block diagram of fuzzy logic system

fuzzifier maps the crisp inputs into fuzzy inputs using membership functions Therule base is a collection of rules in IF-THEN format, which describes the expertknowledge using linguistic rules The inference engine will perform reasoning based

on the fuzzy inputs and rules to generate the fuzzy outputs The defuzzifier thenconverts the fuzzy outputs into crisp outputs

The FLS in Fig.1.1was employed as a fuzzy controller proposed by Prof EbrahimMamdani [2 4] By incorporating the knowledge of control experts into the rule base,fuzzy controllers demonstrating human spirits were successfully applied to variousengineering applications such as sludge wastewater treatment [5] and control ofcement kiln [6]

It is recognized that the Mamdani-type fuzzy controller demonstrates the ing advantages:

follow-• The control action described by linguistic rules in IF-THEN format is understand

easy-to-• The design of the fuzzy controller is easy and simple that can be realized bycollecting expert knowledge without the need of complicated mathematical model

of the nonlinear plant

However, the main drawback of the Mamdani-type fuzzy controller is that thecontrol design is heuristic and the fuzzy controller is not favorable to theoretical sta-bility analysis [4] With the designed Mamdani-type fuzzy controller for the controlprocess, the stability of the overall control system is not guaranteed and the perfor-mance is not consistent (because different experts will have different knowledge andcontrol strategies for the same control process, which leads to different sets of rules).Fuzzy model-based (FMB) control approach [7] overcomes the disadvantages

of the Mamdani-type fuzzy control approach with the introduction of the Takagi–Sugeno (T–S)/Takagi–Sugeno–Kang (T–S–K) fuzzy model [8,9] The T–S/T–S–Kfuzzy model (hereafter T–S fuzzy model) offers a favorable theoretical analysis plat-form for the stability analysis and control design In the past two decades, extensiveresults on stability analysis, control synthesis, control problems and engineeringapplications have been reported More details of the FMB control are presented inthe following sections

1.2 Fuzzy Model-Based Control

An FMB control system is shown in Fig.1.2, which consists of a nonlinear plantrepresented by the T–S fuzzy model and a fuzzy controller connected in a closed loop

A nonlinear plant is a dynamical system with its dynamics described by differential

equations The system state vector x(t) or the system output vector y(t) (for

output-feedback case), combined with the input vector r(t), will be processed by the fuzzy

controller to generate the control signal u(t) The control signal is then input to the

nonlinear plant for the control purposes

1.2.1 Fuzzy Models

A fuzzy model is a mathematical tool to represent the nonlinear plant in a favorableform to facilitate the system analysis and control design Variations of fuzzy modelswere proposed with the combinations of 1) Type 1/2 fuzzy sets [10] in the antecedentand 2) Linear or Polynomial sub-systems in the consequent As shown in Fig.1.3,

+ Fuzzy Controller u(t) Nonlinear Plant representedby Fuzzy Model

Type-1 T-S

Fuzzy Model

Type-2 T-S Fuzzy Model

Type-2 Polynomial Fuzzy Model

Fig 1.3 Variations of fuzzy models

6 1 Introductionthere are 4 fuzzy models namely T–S fuzzy model, T–S polynomial fuzzy model,type-2 T–S fuzzy model and type-2 T–S polynomial fuzzy model.

1.2.1.1 Type-1 T-S/Polynomial Fuzzy Models

The (type-1) T–S fuzzy model [8,9] offers a systematic and effective way to model thedynamics of the nonlinear plant with some fuzzy rules The antecedent of each rule iswith the system states as the premise variables The membership functions are of type-

1 and divide the operating domain (Universe of Discourse) into a number of operatingsub-domains The consequent is a linear sub-system characterized by a linear state-space equation The contribution of each linear sub-system to the system modeling

is governed by the firing strength of the corresponding rule Mathematically, the T–Sfuzzy model describes the dynamics of the nonlinear plant as an average weightedsum of some local linear sub-systems The favorable form of the T–S fuzzy modelthat the consequents are linear sub-systems allows linear control theories and designtechniques to be used for stability analysis and control synthesis

With the sector nonlinearity concept [11,12], a mathematical model can be formed exactly to the form of T–S fuzzy model by considering a compact operat-ing domain (unless for some particular cases) Consequently, the T–S fuzzy model

trans-is considered as a local nonlinear model The analystrans-is results associated with theT–S fuzzy model are valid when the system is operating in the considered operatingdomain This drawback can be alleviated by replacing the linear sub-systems in theconsequent of the rules by polynomial sub-systems The resultant fuzzy model isreferred to as (type-1) polynomial fuzzy model [13–15] Because polynomials areallowed in the sub-systems, the T–S polynomial fuzzy model is able to represent awider class of nonlinear plants

T–S Switching Fuzzy Models

A T–S switching fuzzy model is a collection of local T–S fuzzy models which switchamong each other to describe the dynamics of the nonlinear plant If a nonlinearplant has switching elements [16], the local T–S fuzzy models can be constructed byconsidering each combination of the (on–off) status of the switching elements Forexample, considering 2 switching elements, the T–S switching fuzzy model will have

4 local T–S fuzzy models If a nonlinear plant does not have any switching elements[17], the operating domain of the nonlinear plant is first divided into a number oflocal operating sub-domains Corresponding to each local operating sub-domain, alocal T–S fuzzy model can be obtained, say, using the sector nonlinearity concept Inoperation, it will switch to the local T–S fuzzy model according to the working localoperating sub-domain to model the nonlinear plant As the local T–S fuzzy model is

less complicated in terms of nonlinearity compared with the nonlinear plant, it is infavor of the stability analysis and potentially leads to more relaxed stability analysisresults.

T–S Time-Delay Fuzzy Models

A time-delay nonlinear system is a dynamical system that depends on both thecurrent and time-delay system states of which the time delay can be constant or timevarying The T–S fuzzy model was extended to the time-delay T–S fuzzy model withthe inclusion of the time-delay system states [18–41] to model the nonlinear plantwith time-delay system states When the time delay vanishes, the time-delay T–Sfuzzy model is reduced to the original T–S fuzzy model

1.2.1.2 Type-2 T–S/Polynomial Fuzzy Models

The above mentioned T–S fuzzy model and polynomial fuzzy model are withtype-1 fuzzy sets in the antecedent of the rules The type-1 membership functionsare not good in capturing uncertainties, say the uncertainties of the input variables orparameter uncertainties, which limits its system modeling capability compared withtype-2 fuzzy sets So the type-1 fuzzy model will normally incorporate the uncertain-ties into the sub-systems, otherwise, the membership grades will become uncertain

in value which will lead to conservative analysis result

Theoretically, a type-2 FLS can be regarded as a collection of an infinite ber of type-1 FLSs [10] When the type-1 fuzzy model has membership functionswith uncertainties, by considering every single fixed value of the uncertainties (in

num-a bounded domnum-ain), it is equivnum-alent thnum-at there exist infinite number of membershipfunctions leading to infinite number of type-1 fuzzy models Thus, it is ideal thatthe type-2 fuzzy model is employed to represent a nonlinear plant with uncertaintiesusing a finite number of rules

1.2.1.3 Interval Type-2 T–S/Polynomial Fuzzy Models

By employing the type-2 fuzzy sets in the antecedent of the T–S fuzzy model andpolynomial fuzzy model, type-2 T–S fuzzy model and type-2 T–S polynomial fuzzymodel can be obtained One of the concerns of the type-2 fuzzy sets is the compu-tational demand for type reduction Interval type-2 (IT2) fuzzy sets, which are thereduced version of the type-2 fuzzy sets, are proposed to reduce the computationaldemand and complexity An IT2 T–S fuzzy model was proposed in [42–47] The ideacan be simply extended to IT2 T–S polynomial fuzzy model More general type-2 T–

S fuzzy models and type-2 polynomial fuzzy models can be obtained by employingthe type-2 fuzzy sets in both the antecedent and consequent of the rules

8 1 Introduction

1.2.2 Fuzzy Controllers

A fuzzy controller [11,12] is used to close the feedback loop performing the controlprocess The behaviour of the fuzzy controller is governed by a set of rules that theantecedent consists of membership functions with system states as the input variablesand the consequent is a sub-controller of any types Similar to the structure of theT–S fuzzy model, the fuzzy controller can be expressed mathematically as an aver-age weighted sum of sub-controllers of which the contribution of each to the overallcontrol process is governed by the firing strength of the corresponding rule Because

of the favorable structure of the T–S fuzzy model and the fuzzy controller, the overallclosed-loop system, i.e., FMB control system, can be expressed mathematically in

a systematic form, which facilitates the system analysis and control synthesis ious types of fuzzy controllers as shown in Fig.1.4such as state-feedback, output-feedback, observer-based, switching/sliding-mode, sampled-data and adaptive fuzzycontrollers have been proposed to deal with different control problems

Var-1.2.2.1 State-Feedback Fuzzy Controllers

A state-feedback fuzzy controller is a fuzzy controller with linear state-feedbackcontrollers using full state information as the sub-controllers in the consequent of

Fuzzy Controller

Adaptive Controller

Indirect Direct

Sampled-Data Controller

Variable Sampling Rate Constant Sampling Rate Switching/Sliding-mode Controller

Observer-based Feedback Controller

Partial State Full State Output-Feedback Controller

State-Feedback Controller

Dynamic Static

Fig 1.4 Various types of fuzzy controllers

the rule When full system states are not available, output-feedback or observer-basedfuzzy controller, which will be discussed later, can be employed.

In general, there are two types of state-feedback fuzzy controllers, namely staticand dynamic state-feedback fuzzy controllers For the static state-feedback fuzzycontrollers [11,12,48–60], the controller does not have dynamics For example, thefeedback gains of each linear state-feedback sub-controllers are constant during thecontrol process As the name implies, the dynamic state-feedback fuzzy controllers[61–64] demonstrate dynamics given by a dynamic compensator, which is governed

by a set of first-order differential equations, to its output This type of fuzzy troller is good in dealing with the reference tracking control and disturbance rejectionproblems

con-Using linear state-feedback controllers as the sub-controllers leads to an FMBcontrol system in the form of an average weighted sum of linear state-space sub-systems (hereafter linear sub-systems) This particular form extracts the linear andnonlinear parts of the nonlinear plant and fuzzy controller Thus, some linear controltheories can be employed to investigate the linear part (linear sub-systems) to studythe system stability and design the linear state-feedback sub-controllers

To facilitate the stability analysis and control design, the concept of parallel

distributed compensation (PDC) was proposed in [11,12], which suggests that (1)each linear sub-system is handled correspondingly by a linear sub-controller and (2)the fuzzy controller shares the same premise rules as those of the T–S fuzzy model.For the fuzzy controllers not satisfying conditions (1) and/or (2), analysis results andcontrol design were reported in [65–68] for T–S FMB control systems, in [13–15,

44, 57, 69–72] for polynomial fuzzy model-based (PFMB) control systems and in[43,44,73,74] for type-2 FMB control systems

1.2.2.2 Output-Feedback Fuzzy Controllers

An output-feedback fuzzy controller [25, 54, 61–64,75–94] is a kind of feedbackcontrollers Instead of using system states for feedback compensation, system outputsare used in the output-feedback fuzzy controllers This type of fuzzy controllers isparticularly useful when only the system outputs are available However, as full stateinformation is not available, it usually makes the system analysis and control designmore restrictive

1.2.2.3 Observer-Based Fuzzy Controllers

When the system states are not available for feedback compensation, a fuzzy observercan be used to estimate the system states The estimated system states are thenemployed by the fuzzy controller for the control process This kind of fuzzy con-trollers is referred to as observer-based fuzzy controllers [12,53,76,91,95–122]

As the fuzzy observer will add dynamics to the closed-loop system, it will complicatethe control scheme and make the system analysis and control design more difficult

10 1 Introduction

In general, there are two classes of observer-based fuzzy controllers The firstclass is that the fuzzy observer shares the same premise membership functions asthose of the T–S fuzzy model It is in favor of the system analysis that the separationprinciple [12] can be used for the design of the fuzzy observer and fuzzy controller.However, the constraint of sharing the same premise membership functions limitsthe range of nonlinear plants to be considered The second class of fuzzy observersdoes not require sharing the same premise membership functions It can be applied

to a wider range of nonlinear plants However, because of the mismatched number

of rules and/or premise membership functions, it makes the analysis more difficultand potentially leads to conservative result

1.2.2.4 Switching/Sliding-Mode Fuzzy Controllers

Switching/sliding-mode fuzzy controllers [16, 123–140] demonstrate an excellentproperty to deal with unknown but bounded parameter uncertainties The basic con-cept of the switching/sliding-mode control techniques [141] is employed to estimatethe values of the uncertain parameters using switching elements The estimated valuewill be used by the controller for the control process

This idea can also be applied to construct a switching T–S fuzzy model for a linear plant subject to parameter uncertainties In general, a switching fuzzy modelconsists of a number of switched local T–S fuzzy models [124], which switch amongeach other for modeling purposes As the switched local T–S fuzzy models are lesscomplicated compared with the nonlinear plant, it usually leads to less conservativestability analysis results and makes the design of stable fuzzy controllers easier.One of the drawbacks of the switching/sliding-mode fuzzy controllers is the highfrequency switching components, which leads to the undesired chattering effect[141] It can be alleviated by employing a saturation function to replace the hard-switching components [141] When the system is working in the saturation region,the switching behaviour will vanish and the chattering effect will disappear However,

non-a stenon-ady stnon-ate error mnon-ay be resulted

1.2.2.5 Sampled-Data Fuzzy Controllers

A sampled-data fuzzy controller consists of a sampler, a discrete-time fuzzy troller and a zero-order-hold (ZOH) A block diagram of sampled-data fuzzy con-troller is shown in Fig.1.5 Consider a state-feedback sampled-data fuzzy controller

which takes the system state vector x(t) of the nonlinear plant as input A sample

of x(K T ), K = 1, , ∞, is obtained after the sampler There are two types of

samplers considered in the literature, namely samplers with constant [72,91,142–

155] and variable [156–160] sampling rate The sampled system state vector x(K T )

is then processed by a discrete-time fuzzy controller to generate the control vector

u(K T ) After the ZOH, the control vector u(t) which value is kept constant during

the sampling period is employed for the control process

The sampled-data fuzzy controller can be implemented by a microcontroller ordigital computer, which is available at low cost, to reduce the implementation costand time However, because of the sampling activity, it introduces discontinuity tothe closed-loop system, which makes the analysis more difficult compared with thecontinuous-time FMB control systems

1.2.2.6 Adaptive Fuzzy Controllers

An adaptive fuzzy controller is a fuzzy controller with adaptive feature which willcorrect its parameter values (for example, feedback gains) in an online manner toadopt the change of the working environment This kind of fuzzy controllers aregood in dealing with nonlinear plants subject to parameter uncertainties and/or dis-turbances However, the update rule complicates the structure of the fuzzy controller,which increases the implementation cost

In general, there are two classes of adaptive fuzzy controllers, namely direct [161–

178] and indirect [162,163,168,169,179–184] adaptive fuzzy controllers In bothdirect and indirect adaptive controllers, the universal approximation capability of theFLS is used Generally speaking, in the direct adaptive fuzzy control approach, it

is assumed that there exists a perfect fuzzy controller which is able to stabilize theplant As the values of the parameters (for example, feedback gains) of the perfectfuzzy controller are unknown, an FLS is employed to estimate their values With theconsideration of the overall closed-loop system, an update rule is obtained to correctthe values of the parameters of FLS (for example, parameters of the membershipfunctions in the antecedent and consequent) for the estimation of the system parame-ters In the indirect adaptive fuzzy controller approach, the adaptive fuzzy controller

is characterized by the system parameters of the nonlinear plant An FLS is employed

to estimate the values of the plant parameters based on an adaptive update rule Theestimated values are then employed by a fuzzy controller for the control process.The direct/indirect adaptive fuzzy control schemes were extended to the direct/indirect fuzzy model reference adaptive control (MRAC) scheme [180,181] Based

on the update rule, the adaptive fuzzy controller will correct its parameter valuessuch that the system/output states follow those of a stable reference model

12 1 Introduction

1.2.3 Other Fuzzy Controllers

In the literature, more types of fuzzy controllers than the aforementioned ones can befound, for example, fault-tolerant controllers [27,185,186], networked controllers[187–190], stochastic controllers [191–194], positivity controllers [195–200], pre-dictive controllers [201–208], impulsive controllers [33, 209–215], etc A tree dia-gram shown in Fig.1.6gives a partial picture of FMB control systems found in theliterature

1.3 Stability Analysis

Stability analysis and control synthesis are the essential issues in the FMB controlproblems The most popular approach investigating the stability of FMB controlsystems is based on the Lyapunov method [141,216, 217] The stability analysiscan be done in general using the following steps:

1 Construct a fuzzy model representing the nonlinear plant

2 Choose the type of fuzzy controller for the control process

3 Form an FMB control system by connecting the fuzzy model and fuzzy controller

in a closed loop as shown in Fig.1.2

4 Define a Lyapunov function candidate, which is a scalar positive function

5 Obtain the stability conditions based on the Lyapunov stability theory [141,216,

217]

In this book, we shall study only the continuous-time FMB control systems.However, the concepts, analysis approaches and design techniques can be adoptedand applied to the discrete-time FMB control systems

Under the FMB control paradigm with the consideration of state-feedbackfuzzy controllers, there are 4 general types of Lyapunov function candidates (instep 4), namely quadratic, polynomial, piecewise linear/switching and fuzzy Lya-punov function candidates, used in the stability analysis After the Lyapunovfunction candidate is defined, two classes of stability analysis, namely member-ship function-independent (MFI)/dependent (MFD) stability analysis, are employed

in the literature to carry out the stability analysis It is then followed by different niques of stability analysis to obtain the stability conditions in terms of linear matrixinequalities (LMIs) or sum of squares (SOS) An overall picture of the Lyapunovfunction candidates and techniques of stability analysis is shown in Fig.1.7

tech-In the stability analysis of FMB control systems, conservativeness is related tothe following factors:

1 Types of Lyapunov Function: Lyapunov function is a mathematical tool to

investi-gate the stability of FMB control systems By employing different types or forms

of Lyapunov function candidates, which is used to approximate the domain of the

Control

Impulsive Control Predictive Control Model Reference Control Positivity Control Stochastic Control Netwoked Control Fault Tolerant Control

Time-Delay Control

Distributed Time Delay Control Variable Time Delay Control Constant Time Delay Control

Sampled Data Control

Variable Sampling Rate Control Constant Sampling Rate Control

Switching/Switched Control

Fuzzy Combined Model-Based Control Switching/Switched Model-Based Control Sliding-Mode Control

State-Feedback Control

Static/Dynamic State Feeback

Control

Full State Feedback Control

Observer-Based

Observer

Full-State Observer

Membership function -dependent Stability Analysis

P` olya’s Theorem

Membership Function Boundary

Membership Function

Membership Function Regional Information

2 Types of Stability Analysis: The types of stability analysis determine the

informa-tion of membership funcinforma-tions to be considered, which affects the conservativeness

of the stability conditions In the MFI stability analysis, the information of bership functions will not be used in the stability analysis On the contrary, inthe MFD stability analysis, the information of the membership functions will bebrought to the stability analysis/stability conditions Consequently, the stabilityanalysis results are dedicated to the nonlinear plant on hand and will usually lead

mem-to more relaxed stability conditions

3 Techniques of Stability Analysis: The techniques of stability analysis are the way

to carry out the stability analysis, for example, handling the membership tions; taking the information of membership functions into account Differenttechniques will determine the conservativeness of the stability conditions

func-1.3.1 Types of Lyapunov Functions

A Lyapunov function [141,217] is a non-negative function, which is used to gate the system stability of dynamic systems Lyapunov function can be regarded as

investi-a scinvesti-alinvesti-ar energy-like function, which is rinvesti-adiinvesti-ally unbounded An FMB control system

is guaranteed to be asymptotically stable [141, 216, 217] if the first time tive of the Lyapunov function is negative definite (zero is allowed at the equilib-rium points) The conditions leading to a stable FMB control systems are termed as

deriva-stability conditions Technical details of the Lyapunov functions will be discussed

in Theorem2.1

1.3.1.1 Quadratic Lyapunov Function

Denoting the system state vector x(t) =x1(t), , x n (t)T

, a quadratic Lyapunov

function candidate is defined as V (x(t)) = x(t) T

Px(t) > 0 for all x(t) = 0, where

P˙x(t) < 0 for all x(t) = 0, the FMB control system is guaranteed

to be asymptotically stable, i.e., x(t) → 0 as time t → ∞.

The advantages of using quadratic Lyapunov function for the stability analysisare as follows:

1 It is a simple form of Lyapunov function

2 It makes the stability analysis easier

3 It leads to simple stability conditions such that solving numerically a feasiblesolution to the stability conditions is less computationally expensive comparedwith other types of complicated Lyapunov functions

However, the main disadvantage is that it will lead to potentially conservativestability analysis results

When state-feedback fuzzy controller is considered, FMB control system isexpressed as an average weighted sum of linear control sub-systems Basic stabilityconditions in terms of LMIs were obtained in [11, 12,48] based on the quadraticLyapunov function With convex programming techniques, for example, Matlab LMItoolbox, a feasible solution can be found numerically (if there exists one) The FMBcontrol system is guaranteed to be asymptotically stable if there exists a common

matrix P such that all linear control sub-systems satisfy a set of Lyapunov inequities.

Based on the PDC design concept that the fuzzy controller shares the same premisemembership functions of the T–S fuzzy model, the LMI-based stability conditions[50–54] can be relaxed with the introduction of some slack matrices The stabilityconditions in [50–54,218] can be further relaxed and generalized by employing thePólya permutation theorem [55–57,219]

1.3.1.2 Polynomial Lyapunov Function

A polynomial Lyapunov function is an extension of the quadratic Lyapunov tion which can be viewed as a polynomial Lyapunov function of degree 2 Denoting

func-ˆx(x(t)) as a vector of monomials in x(t) with the largest degree of d, a

polyno-mial Lyapunov function is defined as V (x(t)) which is a polynomial of x(t) of

degree higher than or equal to 2d A polynomial Lyapunov function introduces some

16 1 Introductionhigher-order terms compared with the quadratic one for completing the squares It

is thus easier to construct a Lyapunov function that ˙V (x(t)) < 0 leading to relaxed

stability conditions

In the literature, the polynomial Lyapunov function is extended to investigatethe stability of PFMB control systems, which consists of a polynomial fuzzy modeland a polynomial fuzzy controller connected in a closed loop [13–15], using SOSapproach [220,221] As polynomials are allowed in the consequent of both the T–Spolynomial fuzzy model and the polynomial fuzzy controller, the system modelingand the feedback compensation capabilities are enhanced Details of the polynomialfuzzy model and polynomial fuzzy controller are given in Sects.2.3.1and2.4.2.Basic stability conditions in terms of SOS were obtained in [13, 14] androbust version of stability conditions can be found in [222] A feasible solution

of the SOS-based stability conditions can be found numerically by SOSTOOLS[223,224] Relaxed SOS-based stability conditions were obtained by transformation

of variables [15,67], consideration of operating domain [15,67] and approximation

of membership functions [71,72,225–227]

1.3.1.3 Piecewise/Switching Lyapunov Function

Piecewise or switching Lyapunov function [92, 136, 208, 228, 228–235] is aLyapunov function which consists of a number of local Lyapunov functions ofquadratic form switching among themselves According to some predefined workingconditions, for example, current working operating domain, the corresponding localLyapunov function will be adopted for local stability and control synthesis It should

be noted that only one local Lyapunov function will be activated at any instant Asdifferent local Lyapunov functions are used to adopt the working conditions, morerelaxed stability analysis results can be obtained compared with the quadratic one(which is considered as a particular case of the piecewise or switching Lyapunovfunction) However, the analysis will be comparatively more difficult because of theissues led by the switching activity

Stability analysis was carried out in [228–232] based on the piecewise Lyapunovfunction and in [136, 228, 233] based on the switching Lyapunov function LMI-based stability conditions were achieved to guarantee the system stability It is due tothe switching activity of the Lyapunov function that, for easy of control synthesis, aswitching fuzzy controller, which consists of some local fuzzy controllers associatedwith the corresponding local Lyapunov functions, was employed for the relaxation ofstability conditions Based on the active local Lyapunov function, the correspondinglocal fuzzy controller is employed for the control process Because of the switchingactivity, the fuzzy controller introduces discontinuity and jumps to the control signalwhich may not be suitable for some nonlinear plants Particularly, when the system

is working in the boundary, a high-frequency switching behaviour will happen

1.3.1.4 Fuzzy Lyapunov Function

A fuzzy Lyapunov function [67, 140, 236–258] is a more general form of thepiecewise/switching Lyapunov function Instead of using a hard switch, the fuzzyLyapunov function in quadratic form uses membership functions, which define theoperating sub-domains, to combine the local Lyapunov functions Mathematically,

a fuzzy Lyapunov function can be expressed as an average weighted sum of somelocal Lyapunov functions such that a smooth transition is ensured to overcome theproblems caused by the undesired hard-switching activity in the stability analysisapproach using piecewise/switching Lyapunov function However, the time deriva-tive of the membership functions will be generated which complicates the stabilityanalysis

The fuzzy Lyapunov function was first introduced in [236] to investigate the bility of FMB control systems LMI-based stability conditions were developed toguarantee the system stability To deal with the time derivative of the membershipfunctions, their boundary limits were used in the stability analysis When a partic-ular form of membership functions is used, the time derivative of the membershipfunctions will vanish to facilitate the stability analysis [239,242] Various enhance-ment techniques can be found in the literature to alleviate the limitations of the fuzzyLyapunov function and to further relax the analysis results [67, 240–244, 259].Discrete-time case was investigated in [237,238]

sta-1.3.2 Types of Stability Analysis

The approach of stability analysis based on the Lyapunov stability theory can

be classified generally into two types, namely, membership function-independent(MFI)/membership function-dependent (MFD) stability analysis The type of sta-bility analysis determines if the information of the membership functions is takeninto account in the stability analysis As the MFI stability analysis does not considerthe membership functions, it potentially leads to more conservative stability analysisresults compared with the MFD stability analysis which considers the membershipfunctions in the stability analysis

1.3.2.1 Membership Function-Independent Stability Analysis

The MFI stability analysis does not consider the information of the membershipfunctions but only the local control sub-systems of the FMB control systems Thus,the stability conditions does not involve any membership functions Once there exists

a feasible solution to the stability conditions, the FMB control system is guaranteed

to be stable for any shape of membership functions By ignoring the membershipfunctions in the stability analysis, some information of the nonlinearity is ignored.Therefore, the MFI stability results are potentially conservative

18 1 Introduction

1.3.2.2 Membership Function-Dependent Stability Analysis

The MFD stability conditions take the membership functions of both fuzzy modeland fuzzy controller into account in the stability conditions The stability conditionsobtained in this approach will contain the information of the membership functions

In a general case, more relaxed stability conditions can be obtained compared withthose obtained by the MFI stability analysis as more information of the FMB controlsystem is considered As the information of the membership functions is carried bysome slack matrices to the stability analysis and the number of stability conditionsare generally higher, computational demand on finding a feasible solution to the sta-bility conditions will be higher However, in the MFD stability analysis, the stabilityconditions obtained are not for any shape of membership functions but dedicated tothe FMB control system to be controlled

1.3.3 Techniques of Stability Analysis

The techniques of stability analysis determine the way of handling the membershipfunctions and introducing the slack matrices, which will lead to stability conditions

at different levels of conservativeness and computational demand The techniquesusing Pólya permutation theorem, membership function boundary and membershipfunction approximation will be discussed below The Pólya permutation theorem

is used for grouping terms sharing the same membership functions It is useful forFMB control system with the PDC design The techniques of stability analysis usingmembership function boundary/approximation is more effective even for non-PDCcase

1.3.3.1 Pólya Theorem

Basic MFI stability conditions were reported in [11,12,48] for mismatched premisemembership functions between the T–S fuzzy model and fuzzy controller As themembership functions are ignored in the stability analysis, the stability conditionsare comparatively conservative The PDC design concept was proposed in [11,12]which suggests that the T–S fuzzy model and fuzzy controller share the same set ofpremise membership functions The terms of the fuzzy summations corresponding tothe same membership functions can be collected to produce less conservative stabilityconditions By expanding the degree of the summation terms, the Pólya permutationtheorem was applied to collect the common terms to obtain further relaxed stabilityconditions [55–57]

Fig 1.8 Membership-function boundary information

1.3.3.2 Membership Function Boundary

The information of membership functions, which carries the nonlinearity information

of the FMB control system, plays an important role in the stability analysis for therelaxation of stability conditions For the stability analysis using Pólya permutationtheorem, it concerns the way of collecting the cross terms of membership functions inthe fuzzy summation but not the information of membership functions Consequently,the stability conditions are for a family of FMB control systems but not the one withthe specific membership functions to be controlled

To include the information of membership functions, their lower and/or upperbounds of the global operating domain [65–67, 70, 260–264] can be used in the

stability analysis Using the S-procedure [265,266], the information of the ship functions can be brought to the stability conditions An example of membershipfunction is shown in Fig.1.8 The smallest and largest grades of membership are con-sidered as the lower and upper bounds, which will be used in the stability analysisand appear in the stability conditions

member-1.3.3.3 Membership Function Regional Information

Regional information of membership-functions provides further information to thestability analysis compared with the global membership-function boundary Thebasic idea is to divide the membership functions into sub-regions Corresponding toeach sub-region, the local boundary information is used An example membershipfunction is shown in Fig.1.9where the operation domain (universe of discourse)

is divided into 3 sub-domains, namely, domain 1, domain 2 and domain 3 sponding to each sub-domain, it has its own local lower and upper bounds of thegrade of membership Referring to Fig.1.8, shifting the membership function to theleft or right will give the same lower and upper bounds but different local lower andupper bounds As a result, the local lower and upper bounds contain more specific

Fig 1.9 Membership-function regional information

information of the membership functions resulting in more relaxed stability analysisresults

By using the local membership-function boundary information, more relaxedstability conditions can be achieved [134,267,268] However, as more slack matricesare required to carry the regional information to the stability analysis, the number

of stability conditions is comparatively more than that of the stability conditionsusing global information When the number of sub-domains for the membershipfunctions tends to infinity, the whole membership function is taken into the stabilityconditions Consequently, the stability conditions are dedicated to the specific FMBcontrol system under consideration However, the number of stability conditions willtend to infinity that the solution cannot be found because the limitation of computer

is exceeded

1.3.3.4 Membership Function Approximation

The stability conditions led by stability analysis using Pólya permutation theoremonly contain the closed-loop control sub-systems but ignore the membership func-tions As each FMB control system to be investigated has its own specific membershipfunctions, it is expected that the stability conditions independent of the membershipfunctions will lead to conservative results However, when the membership functionsare brought to the stability conditions, the number of stability conditions becomesinfinity that it is impractical to employ convex programming techniques to searchfor a feasible solution numerically In order to overcome the difficulty, the technique

of membership function approximation was proposed in [44,68,71,225,269] Afavorable form of membership functions such as staircase [44,68,255], piecewiselinear [71, 270] and polynomial [225–227, 264, 269] membership functions areused to approximate the original membership functions, which are brought to thestability conditions These approximated membership functions demonstrate a niceproperty that their grades of membership can be computed based on sample points

Fig 1.10 Staircase membership function Solid line original membership function Dotted line

staircase membership function

Fig 1.11 Piecewise linear membership function Solid line original membership function Dotted

line piecewise linear membership function

Consequently, the infinite number of stability conditions can be approximated byfinite ones and a feasible solution can be found by using convex programming tech-niques

An example of staircase membership function is shown in Fig.1.10 which isemployed to approximate the original continuous membership function It approxi-mates the original membership function using a finite number of discrete grades ofmembership As a result, taking the approximation error into account, the overallsystem stability is implied by the system stability at all combinations of discretegrades of membership The approximation error is a source of conservativeness ofstability analysis Reducing the approximation error will reduce the conservative-ness, which can be achieved by reducing the step size However, the number ofstability conditions will increase Instead of reducing the step size, piecewise mem-bership function is a work around to reduce the approximation error An example

of piecewise membership function is shown in Fig.1.11 It can be seen from thisfigure that the original membership function is approximated by linear interpolation

using the sample points c1–c5 By taking into the account of the approximation error,

22 1 Introductionthe overall system stability is implied by the system stability at all sample points.Staircase and piecewise linear membership functions are polynomial functions ofzero and first degree, respectively The approximation error can be further reduced

by using polynomials of higher degrees

1.4 Control Problems

In general, there are 3 types of control problems, namely stabilization, regulation andtracking control problems, considered in most of engineering applications [271]

1.4.1 Stabilization Control Problem

The objective of stabilization control is to design a fuzzy controller to drive thesystem/output states to the origin as shown in Fig.1.12 This is the easiest controlproblem among the three as there are no external inputs involved and the targetvalues of the system states are zero For example, the engineering applications ofactive suspension systems [39,272–274], electromagnetic suspension systems [275],flexible-joint manipulators [180,276] and ship positioning systems [277] are a kind

of stabilization control problem

1.4.2 Regulation Control Problem

A regulation control problem [278] is similar to the stabilization control problem Thecontrol objective is to drive the system/output states to a constant level as shown inFig.1.13instead of the origin Referring to the closed-loop control system in Fig.1.2,

Fig 1.13 Regulation control Solid line Response of x (t) or y(t) Dotted line Reference r

the input vector r(t) is a constant control command, which indicates the expected

level of system/output level Because of the existence of the external input vector r(t),

it makes the stability analysis more difficult The engineering application of DC–DCpower converters [279–286] that the output voltage is regulated to a certain constantlevel despite the change of the input and load variation is one of the examples of theregulation control problems Another example is permanent-magnet synchronousmotors [287] for torque control

1.4.3 Tracking Control Problem

A tracking control problem [99, 288] is to design a controller to drive the tem/output states to follow those of a time-varying trajectory instead of a con-stant level in the regulation control problem as shown in Fig.1.14 Referring tothe closed-loop control system in Fig.1.2, the input vector r(t) is a time-varying

24 1 Introductioncontrol command The engineering applications of chaotic synchronization [289],induction motors [85], mobile robots [290] and robot arms [291,292] are examples

of the tracking control problem

1.5 Design of FMB Control Systems

In the above sections, various types of fuzzy models, fuzzy controllers, Lyapunovfunctions, stability analysis techniques and control problems are introduced to pro-vide mathematical tools and problem formulation for stability analysis and controlsynthesis for the achievement of control objectives The procedure of achieving astable FMB control system using these tools is summarized in Fig.1.15

In the design procedure, system identification or sector nonlinearity techniquesare employed to construct a (T–S, polynomial or type-2) fuzzy model representingthe nonlinear system Based on the fuzzy model, a fuzzy controller, for example, inFig.1.4is considered to realize closed-loop feedback control When a closed-loop isformed, stability analysis and control synthesis can be conducted through a Lyapunovfunction candidate and the stability analysis techniques Depending on the types offuzzy models, LMI- or SOS-based stability analysis can be performed which willoffer a set of stability conditions Solving the solution to the stability conditions willoffer the parameters of the fuzzy controller, which can guarantee the closed-loop

•Polynomial fuzzy model

•Type-2 fuzzy model

Fig 1.15 Design procedure of fuzzy model-based control system