LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems with Time Delay (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari 351 Chong Lin, Qing-Guo Wang, Tong He...

Author: Chong Lin | Guo Wang | Tong Heng Lee | Yong He

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Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari

351

Chong Lin, Qing-Guo Wang, Tong Heng Lee, Yong He

LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems with Time Delay

ABC

Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Authors Chong Lin

Yong He School of Information Science and Engineering Central South University Changsha 410083, China Email: [email protected]

Institute of Complexity Science Qingdao University Qingdao 266071, China Email: [email protected]

Qing-Guo Wang Tong Heng Lee Department of Electrical and Computer Engineering National University of Singapore 10 Kent Ridge Crescent Singapore 119260 Email: [email protected]

Library of Congress Control Number: 2006937682 ISSN print edition: 0170-8643 ISSN electronic edition: 1610-7411 ISBN-10 3-540-49552-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-49552-9 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, 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. Typesetting: by the authors and SPS using a Springer LATEX macro package Printed on acid-free paper

SPIN: 11822301

89/SPS

543210

Preface

Since initiated by Lotﬁ A. Zadeh in 1965, fuzzy set theory has triggered a considerably large body of areas to blossom. A fuzzy system is, in a very broad sense, any fuzzy logic-based system where fuzzy logic can be used either as the basis for the representation of diﬀerent forms of system knowledge or the model for the interactions and relationships among the system variables. Fuzzy systems have proven to be an important tool for modeling complex systems for which, due to complexity or imprecision, classical tools are unsuccessful. There have been diverse ﬁelds of applications of fuzzy technology from medicine to management, from engineering to behavioral science, from vehicle control to computational linguistics, and so on. Fuzzy modeling is a conjunction to understand the system’s behavior and build useful mathematical models. Diﬀerent types of fuzzy models have been proposed in the literature, among which the Takagi-Sugeno (T-S) fuzzy model is a rule-based one suitable for the accurate approximation and identiﬁcation of a wide class of nonlinear systems. There has been an increasing amount of work on analysis and synthesis of fuzzy systems based on T-S fuzzy models. Since 2000, T-S fuzzy model approach has been extended to tackle analysis and control problems of nonlinear systems with time delay. So far extensive results have been presented for investigating T-S fuzzy systems with time delay, many of which adopt an easy and popular scheme, say, linear matrix inequality (LMI) based method. However, there lacks of a monograph in this direction to provide the state-of-the-art of coverage of this new growing area. This book serves as a comprehensive monograph on T-S fuzzy systems with time delay. It is not intended as a collection of existing results in the literature but to cover as many as possible interesting topics and establish systematic structures towards analysis and control methods. The book is mainly based on the recent research work carried on by the authors. It includes the latest developments and advances for analysis techniques and synthesis methods, brings out the characteristic systematism in them, and points out further insight to solve relevant problems. Topics on T-S fuzzy systems with time delay cover a wide range including stability analysis, stabilization, tracking control, variable structure control, observer design and ﬁlter design.

VI

Preface

The book is a useful source of reference for all those, from graduate students to senior researchers, from mathematicians to human and social science scholars, interested in or working with fuzzy control methods. The prerequisites for the book are modest and they are fundamental knowledge of systems, control, matrix theory and basic fuzzy set theory. We are grateful to the Department of Electrical and Computer Engineering of the National University of Singapore for providing plenty of resources for our research work. We also appreciate the support from the “Taishan Scholar” Programs Foundation of Shandong Province, P.R.China.

Chong Lin, Qingdao Univ., P.R.China Qing-Guo Wang, National Univ. of Singapore Tong Heng Lee, National Univ. of Singapore Yong He, Central South Univ., P.R.China September 2006

Symbols

R

ﬁeld of real numbers

Rn

n-dimensional real Euclidean space

Rn×m

space of n × m real matrices

I

identity matrix

In

identity matrix of dimension n × n

0n×m

zero matrix of dimension n × m

AT

transpose of matrix A

A−1

inverse of matrix A

det(A)

determinant of matrix A

deg(·)

degree of a polynomial

rank (A)

rank of matrix A

A>0

symmetric positive deﬁnite

A≥0

symmetric positive semi-deﬁnite

A<0

symmetric negative deﬁnite

A≤0

symmetric negative semi-deﬁnite

A1/2

symmetric square root of A ≥ 0, i.e., A1/2 A1/2 = A

diag {A1 , · · · , An }

diagonal matrix with Ai as its ith diagonal element

λ(A)

eigenvalues of square matrix A

σmax (A)

largest singular value of matrix A

σmin (A)

smallest singular value of matrix A

ρ(A)

spectral radius of square matrix A

VIII

Symbols

∀

for all

∈

belong to

⊆

subset

|·|

absolute value (or modulus)

·

spectral norm

· ∞

induced l∞ -norm

:=

deﬁned as

→

tend to, or mapping to (case sensitive)

lim

limit

f (t− )

= lim→0+ f (t − )

f (t+ )

= lim→0+ f (t + )

sup

supremum

inf

inﬁmum

LMI

linear matrix inequality

BMI

bilinear matrix inequality

PDC

parallel distributed compensation

sum

Contents

1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 History of Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 T-S Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5

2.

Stability Analysis of T-S Fuzzy Systems . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Augmented Fuzzy Weighting-Dependent Lyapunov-Krasovskii Functional Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Further Augmented Lyapunov-Krasovskii Functional Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Conclusions and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 10 11

18 22 25

3.

Extension to Fuzzy Large-Scale Systems . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 29 36 37

4.

Stabilization Methods for T-S Fuzzy Systems . . . . . . . . . . . . . . . 4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Stabilization Method Using Razumikhin Theorem . . . . . . . . . . . . 4.3 Other Stabilization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Fuzzy Weighting-Dependent Approach . . . . . . . . . . . . . . . . 4.3.2 Using Updated Bounding Methods . . . . . . . . . . . . . . . . . . . 4.3.3 Choosing Appropriate Functionals . . . . . . . . . . . . . . . . . . . . 4.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 40 45 45 45 47 49 53

17

X

Contents

5.

Variable Structure Control for T-S Fuzzy Systems . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Stabilization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Matched Uncertainty Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Mismatched Uncertainty Case . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 57 57 62 67 71

6.

Observer-Based Fuzzy Control: Delay-Independent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Delay-Free Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Two-Step LMI Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Single-Step LMI Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Time-Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Bounded Real Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Conclusion and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 76 78 78 80 85 86 89 96 99

Observer-Based Fuzzy Control: Delay-Dependent Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation and Preliminaries . . . . . . . . . . . . . . . . . . . . . 7.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 106 108 119 122

7.

8.

9.

Output Tracking Control for T-S Fuzzy Systems . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Formulation and Preliminaries . . . . . . . . . . . . . . . . . . . . . 8.3 H∞ Output Tracking Control Design for Delay-Free Systems . . . 8.4 H∞ Output Tracking Control Design for Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusion and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 126 129 133 139 143

Fuzzy Filter Design for T-S Fuzzy Systems . . . . . . . . . . . . . . . . . 149 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.2 Fuzzy Weighting-Dependent Approach to Filter Design . . . . . . . . 150

Contents

XI

9.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 General Fuzzy Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150 152 158 158 160 167 171

10. Descriptor Method for T-S Fuzzy Control Systems . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Fuzzy Descriptor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Stability Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Fuzzy Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conclusion and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 174 176 180 183 188

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

1. Introduction

1.1 History of Fuzzy Systems Fuzzy theory is ﬁrmly grounded in the mathematics of fuzzy sets and fuzzy logic developed by Lofti A. Zadeh in 1965 [166]. Since its introduction, fuzzy set theory has grown to become a major scientiﬁc domain. A fuzzy system is referred to as any static or dynamic system which makes use of fuzzy logic and of the corresponding mathematical framework. Such a system allows a gradual and continuous transition, say, from 0 to 1, rather than a crisp and abrupt change between binary values of 0 and 1. It is known that in an ordinary set, an element of the universe either belongs to or does not belong to the set. This shows the membership of an element is crisp. A fuzzy set is a generalization of an ordinary set by allowing a degree of membership for each element, which is a real number on [0, 1]. The membership function of a set maps each element to its degree. Accordingly, fuzzy logic is an extension of ordinary logic, which can represent fuzzy implications. There are a lot of publications to address the details of fuzzy theory, see, e.g., [65][66][67] [71] [124][168][176]. Fuzzy theory also experienced a period of sparking controversy. In the 1960s, many scholars objected to the idea and viewed the new theory as against basic scientiﬁc principles. Some mathematicians claimed that probability theory could work equally well or better than fuzzy theory. Other criticisms claimed that fuzzy theory is something of an illusion because it merely hides the complexity of control system design in determining the rule set, or that fuzzy logic is little more than a restatement of traditional statistical methods in new and obscure nomenclature. In the late 1960s and early 1970s, fuzzy theory continued to growth as an independent ﬁeld, and there appeared some other fundamental fuzzy concepts, like fuzzy algorithms[167], fuzzy decision making[2], etc. In 1973, Lofti A. Zadeh published a paper to establish the foundation for fuzzy control[168]. In this work, he introduced the concept of linguistic variables and proposed to use fuzzy IF-THEN rules to formulate human knowledge. Meanwhile, initial applications of fuzzy control for real systems, like the steam engine controller[102] and the cement kiln controller in Denmark, showed that the fuzzy ﬁeld was promising. Although fuzzy theory was still ignored by some researchers because it was associated with artiﬁcial intelligence, a ﬁeld resulting in a lack of credibility with industrial ﬁrms, the later big events largely encouraged people working at fuzzy control. In 1980s, Sugeno applied fuzzy control to a Fuji Electric water puriﬁcation plant and pioneered the work on a fuzzy robot which is C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 1–7, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

2

1. Introduction

a self-parking car controlled by calling out commands [114][117]. Yasunobu and Miyamoto from Hitachi demonstrated the superiority of fuzzy control systems for the Sendai underground railway and used their idea to control accelerating, braking, and stopping when the line opened in 1987. During the Second Annual International Fuzzy Systems Association Conference held in Tokyo in 1987, Hirota displayed a fuzzy robot arm playing two-dimensional ping-pong game[60], and Yamakawa demonstrated the use of fuzzy control in balancing an inverted pendulum[155]. In 1990s, the fast development of fuzzy idea in Japan helped promote interest in fuzzy systems. Several scientiﬁc societies engaged in fuzzy ﬁeld have been established. Practical applications were not only in industry but also in the manufacture of consumer products, like video camera, microwave cooker, and washing machine, etc [176]. Work on fuzzy systems was also proceeding in the United States and Europe. The US Environmental Protection Agency has investigated fuzzy control for energy-eﬃcient motors, and NASA has studied fuzzy control for automated space docking. Large companies such as Boeing, General Motors, and Whirlpool have worked on fuzzy logic for use in low-power electrical appliances, energy-eﬃcient electrical motors, and so on. In February 1992, the ﬁrst IEEE International Conference on Fuzzy Systems was held in San Diego, and in 1993, the IEEE Transactions on Fuzzy Systems was inaugurated. These events symbolized the acceptance of fuzzy theory in a wide community of engineering society. Up to now, fuzzy systems have been successfully used in a number of realword applications, such as Penicillin-G conversion [1], prediction of river water ﬂow [118], and many other examples in ecological systems and biomedical ﬁeld [1][112]. Meanwhile, numerous publications have been reported in providing theoretical support. Various methodologies have been proposed for analysis, modeling, design, control and monitoring of fuzzy systems.

1.2 T-S Fuzzy Systems Fuzzy ideas are useful for modeling complex nonlinear systems in which, due to complexity or imprecision, classical tools are unsuccessful. Like neural networks, fuzzy systems have been recognized as powerful universal approximators due to their capability of approximating a given system with arbitrary accuracy [68][69] [135][136][137]. Various techniques such as fuzzy clustering [1][161], neuro-fuzzy learning methods [92], and orthogonal least squares [118][136] have been proposed for data-driven fuzzy modeling. There are numerous other signiﬁcant publications which discuss the fuzzy modeling [4][7][23] [25][91][111] [112][115][119][154]. Fuzzy models can be static or dynamic. Diﬀerent types of fuzzy models have been reported in the literature. The widely used are rule-based fuzzy systems, in which the relationship between variables are represented by means of fuzzy IF-THEN rules of the form:

1.2 T-S Fuzzy Systems

3

IF antecedent proposition THEN consequence proposition. Rule-based fuzzy systems include Mamdani models (or linguistic fuzzy model) [168] [101], fuzzy relation models [109] [157], and Takagi-Sugeno (T-S) fuzzy models [119]. This book focuses on T-S fuzzy models, where the consequences are crisp functions of the antecedent variables rather than fuzzy propositions. T-S fuzzy systems are popular and well used tools in recent years. A general T-S fuzzy model employs an aﬃne fuzzy model with a constant term in the consequence[119]. It is known that smooth nonlinear dynamic systems can be approximated by aﬃne T-S fuzzy models[5][159]. Most recent developments are based on T-S models with linear rule consequences (here and after, such models are generally called T-S fuzzy models). The main feature of T-S fuzzy models is to represent the nonlinear dynamics by simple (usually linear) models according to the so-called fuzzy rules and then to blend all the simple models into an overall single model through nonlinear fuzzy membership functions. Each simple model is called a local model or a sub-model. The output of the overall fuzzy model is calculated as a gradual activation of the local models by using proper defuzziﬁcation schemes [111][119][154]. It has been proved that T-S fuzzy models can approximate any smooth nonlinear dynamic systems[29][122]. A continuous-time T-S fuzzy model is of the following form: Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Bwi w(t) + Bui u(t), z(t) = Ci x(t) + Dwi w(t) + Dui u(t), y(t) = Ei x(t) + Dywi w(t) + Dyui u(t),

(1.1) i = 1, 2, . . . , r,

where x ∈ Rn , u ∈ Rm1 , w ∈ Rm2 , z ∈ Rn1 and y ∈ Rn2 are the state, the control input, the disturbance, the controlled output and the measured output, respectively; r is the number of IF-THEN rules; Ai , Bwi , Bui , Ci , Dwi , Dui , Dywi , Dyui and Ei are real constant matrices with appropriate dimensions; θj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables and the fuzzy sets. By fuzzy blending, the overall fuzzy model is inferred as follows:

x(t) ˙ = z(t) = y(t) =

r i=1 r i=1 r i=1

hi (θ)[Ai x(t) + Bwi w(t) + Bui u(t)], hi (θ)[Ci x(t) + Dwi w(t) + Dui u(t)], hi (θ)[Ei x(t) + Dywi w(t) + Dyui u(t)],

(1.2)

4

1. Introduction

where θ = [θ1 , . . . , θp ], r νi (θ), hi (θ) = νi (θ)/ i=1

νi (θ) =

p

µij (θj ),

(1.3)

j=1

and where µij (θj ) is the grade of membership of θj in µij . It is seen that νi maps Rp onto [0, 1], which is called the membership function corresponding to plant rule i, i = 1, . . . , r. Moreover, the fuzzy weighting functions hi (θ) satisfy r

hi (θ) = 1,

hi (θ) ≥ 0.

(1.4)

i=1

The design of controllers to achieve the synthesis purpose is performed through the parallel distributed compensation (PDC) [134]. Details will be provided in related chapters. Recent advances witness the extension of T-S fuzzy model approach to nonlinear time-delay systems. The analysis and synthesis method is originated in [9]. So far, Various methodologies have been proposed for investigation of TS fuzzy systems with time-delay in a wide range of research topics, including stability and state-feedback stabilization [10][142][158], static output-feedback control [12], dynamic output-feedback control [73][150], linear quadratic statefeedback control [139], variable structure control [81], H∞ control [73][150][160], and H∞ ﬁlter design [149]. In case of unavailability of the states, fuzzy observers play an important role in analysis and synthesis of fuzzy systems. Accordingly, observer-based fuzzy control have also attracted much attention for T-S fuzzy systems with time-delay, see [9][10][138] for observer-based stabilization, [13] for observer-based reliable control and [14] for observer-based guaranteed cost control. A continuous-time T-S fuzzy model with state delay is of the following form: Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bwi w(t) + Bui u(t),

(1.5)

z(t) = Ci x(t) + Cτ i x(t − τ (t)) + Dwi w(t) + Dui u(t), y(t) = Ei x(t) + Eτ i x(t − τ (t)) + Dywi w(t) + Dyui u(t), x(t) = φ(t), t ∈ [−τ0 , 0], i = 1, 2, . . . , r, where notations are as in (1.1) and Aτ i , Cτ i and Eτ i are real constant matrices corresponding to delay terms; φ is the initial condition; The time-delay τ (t) may be unknown but is assumed to be continuous function of time with an upper bound τ0 : τ (t) ≤ τ0 .

(1.6)

1.3 The Book

5

By fuzzy blending, the overall fuzzy model is inferred as follows: x(t) ˙ = z(t) = y(t) =

r i=1 r i=1 r

hi (θ)[Ai x(t) + Aτ i x(t − τ ) + Bwi w(t) + Bui u(t)],

(1.7)

hi (θ)[Ci x(t) + Cτ i x(t − τ ) + Dwi w(t) + Dui u(t)], hi (θ)[Ei x(t) + Eτ i x(t − τ ) + Dywi w(t) + Dyui u(t)],

i=1

x(t) = φ(t), t ∈ [−τ0 , 0], where the notations are as in (1.2). Recently, in the direction of investigating T-S fuzzy systems with time delay, we have done a lot of work for analysis and control problems. These work includes stability analysis, stabilization, fuzzy variable structure control, observer-based fuzzy control, fuzzy tracking control, fuzzy ﬁlter design, and so on. Most of the present results either provide new LMI-based solutions to certain fuzzy control problems or give up-to-date LMI-based methods that reduce conservatism entailed in previous methods. This monograph is thus formed according to our recent research work.

1.3 The Book Since 2000, T-S fuzzy model approach has been extended to tackle analysis and control problems of nonlinear systems with time delay. So far extensive results have been presented for investigating T-S fuzzy systems with time delay. However, there lacks of a monograph in this direction to make necessary insight. This book aims to provide up-to-date developments in the analysis and control area for nonlinear time-delay systems using T-S fuzzy model approach. It serves as a comprehensive monograph on T-S fuzzy systems with time delay. It intends to present, in a single volume, a fairly comprehensive, up-to-date and detailed treatment of stability analysis, control and estimation. The materials included in the book are mainly based on research results of the authors and their co-workers in the domain. See [49]-[59] and [76]-[90]. The content is to look at the latest developments and advances for analysis techniques and synthesis methods, to bring out the characteristic systematic in them, and to point out further insight to solve relevant problems. Topics on T-S fuzzy systems with time delay cover a wide range including stability analysis, stabilization, fuzzy variable structure control, fuzzy tracking control, fuzzy observer design and ﬁlter design. It covers many interesting topics and establishes systematic structures towards analysis and control methods. Indeed, most of the results presented in this book reduce to testing the solvability of LMI problems, a standard convex optimization problem [3], achievable in polynomial-time. Eﬃcient interior-point methods have been developed and are available as toolboxes in widely spread control-oriented scientiﬁc softwares, such as LMI toolbox in Matlab [36].

6

1. Introduction

For presentation, we have made technical development of the results as selfcontained as possible. Knowledge of the matrix theory and the linear system theory is assumed from readers. Illustrative examples of diﬀerent degrees of complexity are given to facilitate understanding. The book is a useful source of reference for all those, from graduate students to senior researchers, from mathematicians to human and social science scholars, interested in or working with fuzzy control methods. This book is organized into 10 relevant but independent chapters. Chapter 1 introduces the history of fuzzy systems with emphasis on T-S Fuzzy Systems. Chapter 2 studies the asymptotic stability of T-S fuzzy systems with statedelay. It focuses on the delay-dependent stability analysis based on the fuzzy weighting-dependent Lyapunov-Krasovskii functional method. It provides the state-of-the-art of stability results that reduce the conservatism entailed in previous results using quadratic Lyapunov-Krasovskii functionals. Also, we will present augmented Lyapunov-Krasovskii functional approaches to further reduce the conservatism. Delay-independent stability criteria are listed as easy corollaries of the delay-dependent results. Chapter 3 deals with fuzzy large-scale systems. It can be viewed as extensions of the contents in Chapter 2. It provides an improved delay-dependent method by choosing appropriate Lyapunov-Krasovskii functionals and adopting free-weighting matrix techniques. Chapter 4 focuses on the stabilization problem for T-S fuzzy systems with time-delay. In particular, it aims to present delay-dependent design of state feedback stabilizing fuzzy controllers for T-S fuzzy systems with state delay and input delay, using the Razumikhin Theorem. It also discusses other stabilization methods for time-delay T-S fuzzy systems. Chapter 5 investigates the stabilization problems of uncertain fuzzy timedelay systems via variable structure control approach. The considered systems are with parameter perturbations and external disturbances. The uncertainties are assumed to be matched or mismatched in the state variables and the timedelay is assumed to be smooth or non-smooth. Various design methods of variable structure controllers are presented to achieve the robust stabilization purpose. The suﬃcient conditions are given in terms of LMIs which are easy to solve. Chapter 6 deals with the observer-based fuzzy H∞ control through delayindependent method. In case of unavailability of the states, fuzzy observers play an important role in analysis and synthesis of fuzzy systems. This chapter presents a single-step LMI method for the fuzzy control design, which overcomes the drawback of the two-step LMI approach often encountered in the literature. The derivation relies mainly on a matrix decoupling technique using which a resultant matrix inequality can be equivalently converted to strict LMIs. Chapter 7 deals with the observer-based fuzzy H∞ control through delaydependent method. It provides consistent design methods for time-delay T-S fuzzy systems. It is a complement to the delay-independent method, and

1.3 The Book

7

overcomes the drawback that delay-dependent control designs are usually inconsistent with the analysis method. Chapter 8 investigates the H∞ output tracking control for T-S fuzzy systems. It addresses both cases of delay-free and state-delay systems. The proposed designs of fuzzy controllers depend on feasible sets of solutions to a group of matrix inequalities. Then, it establishes a two-step LMI method to solve these matrix inequalities. Chapter 9 considers the H∞ ﬁlter design for T-S fuzzy systems. Firstly, it presents a fuzzy weighting-dependent method and gives the design results in terms of LMIs. The main feature is the use of fuzzy weighting-dependent Lyapunov functionals which can reduce the conservatism arisen from the quadratic Lyapunov functional approach. Secondly, it considers the H∞ ﬁlter design in the general case, and establishes delay-dependent design results in terms of LMIs. The main technique is the use of matrix transformation and the free-weighting matrix method. The results for rate-independent case, delay-independent case and delay-free case are easy corollaries. Chapter 10 studies the descriptor method for T-S fuzzy control systems. The considered class of fuzzy systems is of the form of time-delay fuzzy descriptor systems that is in the extended Takagi-Sugeno fuzzy model. It provides suﬃcient conditions in terms of LMIs for the stability and stabilization problems by using appropriate analysis methods for descriptor systems.

2. Stability Analysis of T-S Fuzzy Systems

This chapter studies the asymptotic stability of Takagi-Sugeno (T-S) fuzzy systems with state-delay. It focuses on the delay-dependent stability analysis based on the fuzzy weighting-dependent Lyapunov function method. This method provides the state-of-the-art of stability results that reduce the conservatism entailed in previous results using quadratic Lyapunov-Krasovskii functionals. Also, we will present an augmented fuzzy weighting-dependent Lyapunov-Krasovskii functional to further reduce the conservatism. Delay-independent stability criteria are listed as easy corollaries of the delay-dependent results. We give several examples to illustrate the use of corresponding results.

2.1 Introduction The well-known Takagi-Sugeno (T-S) fuzzy model [119] has been recognized as a popular and powerful tool in approximating a complex nonlinear system. As a consequence, the study of T-S fuzzy systems has been attracted an increasing interest in the past decades. Lots of eﬀorts have been made to develop updated stability analysis methods and various techniques have been obtained [17][64][80][98][126][134] in terms of linear matrix inequalities (LMIs) [3] (which can be easily solved using existing solvers such as the LMI Toolbox in Matlab [36]). However, the use of a common Lyapunov matrix brings much conservativeness in the stability test. Due to the nonlinear fuzzy weighting functions, the parameter-dependent stability methods for dealing polytopic systems [113] are not applicable for T-S fuzzy systems. In order to reduce the conservatism entailed in the previous results using quadratic ideas, a fuzzy weighting-dependent approach is presented recently in [120] for fuzzy systems with fuzzy weighting functions being C 1 functions. As clariﬁed in [122][120], this requirement (i.e., fuzzy weighting functions are C 1 functions) is met for fuzzy models constructed from smooth nonlinear systems by using a sector nonlinearity approach. The sector nonlinearity approach is eﬀective in global or semiglobal fuzzy modeling which can exactly represent the dynamics of a nonlinear system. In fact, this assumption has been also used in [11] where the system is of polytopic settings and the connections are C 1 functions similar to [120]. Recently, the T-S fuzzy model approach has been extended to deal with nonlinear systems with time-delays [9][10][42][73][75][81][158]. Diﬀerent delayindependent methodologies have been proposed for analysis and synthesis of C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 9–25, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

10

2. Stability Analysis of T-S Fuzzy Systems

T-S fuzzy systems with time-delay [9][10][73][158]. It is generally recognized that delay-dependent results are usually less conservative than delay-independent ones, especially when the size of delay is small. Quite recently, delay-dependent techniques have been also reported in [42][75] by adopting the bounding method in [105]. Notice that all these analysis methods are based on quadratic LyapunovKrasovskii functionals which bring conservativeness in establishing stability test. Moreover, the bounding techniques used in these methods also bring conservativeness. In this chapter, we aim to extend the idea in [11][120] to deal with time-delay T-S fuzzy systems by using fuzzy weighting-dependent Lyapunov-Krasovskii functionals. We ﬁrst derive delay-dependent stability conditions using a recent technique called free-weighting matrix method [57], which does not impose any bounding techniques on some cross product terms. This method is simpler or sometimes even more eﬀective than the descriptor model transformation method [32][34] and gains fruitful for time-delay systems [58][145][151]. As a result by using fuzzy weighting-dependent Lyapunov-Krasovskii functionals combining the introduction of free-weighting matrices, our delay-dependent method improves existing ones in [42][75]. Delay-independent stability results are easy corollaries of the present delay-dependent results, which are more eﬀective than existing delayindependent results in [9][10][73][158]. Next, to further reduce the conservatism, we propose an augmented fuzzy weighting-dependent Lyapunov-Krasovskii functional. This new approach will produce better results due to some additionally introduced “slack” variables. Finally, we give several examples to illustrate the less conservativeness of the present methods.

2.2 Problem Formulation Consider a nonlinear system with time-delay which could be approximated by a time-delay T-S fuzzy model with r plant rules [9][10][42][73][75][158]: Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)), x(t) = φ(t), t ∈ [τ0 , 0],

(2.1)

where x ∈ Rn is the state; Ai and Aτ i are constant real matrices with appropriate dimensions; θj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables (which are the functions of state variables) and the fuzzy sets; φ(t) is a continuous vector-valued initial function on [−τ0 , 0]; The time-delay τ (t) may be unknown but is assumed to be smooth function of time as considered in [9][73][75]: 0 ≤ τ (t) ≤ τ0 ,

τ˙ (t) ≤ d < 1,

(2.2)

where τ0 > 0 and d are two scalars. By fuzzy blending, the overall fuzzy model is inferred as follows:

2.3 Stability Analysis

r i=1

x(t) ˙ = =

r

11

wi (θ(t))(Ai x(t) + Aτ i x(t − τ (t)) r i=1 wi (θ(t))

hi (θ(t))(Ai x(t) + Aτ i x(t − τ (t)))

i=1

:= A(t)x(t) + Aτ (t)x(t − τ (t)),

(2.3)

x(t) = φ(t), t ∈ [−τ0 , 0], where θ = [θ1 , . . . , θp ], wi : Rp → [0, 1], i = 1, . . . , r, is the membership function of the system with respect to plant rule i, and hi (θ(t)) = wi (θ(t))/ ri=1 wi (θ(t)). It is obvious that the fuzzy weighting functions hi (θ(t)) satisfy hi (θ(t)) ≥ 0,

r

hi (θ(t)) = 1.

(2.4)

i=1

For brevity, we will use hi (t) to denote hi (θ(t)) in the sequel. The purpose of this chapter is to establish updated delay-dependent stability criteria for fuzzy system (2.3). As in [11][120], we will assume that the fuzzy weighting functions hi (t) are C 1 functions. When this assumption is removed (i.e., reduced to quadratic method), we will show that our delay-dependent methods are still less conservative than existing one in [75] which is the same as that in [42].

2.3 Stability Analysis The delay-dependent stability result in this section is stated as follows. Theorem 2.3.1. Assume that h˙ k (t) ≤ βk , k = 1, 2, . . . , r,

(2.5)

where βk ≥ 0. System (2.3) is asymptotically stable if there exist matrices Pj > 0, Q > 0, Z > 0, Yj and Tj , j = 1, 2, . . . , r, such that the following LMIs hold: Pk ≥ Pr , k = 1, 2, . . . , r − 1, Ωij + Ωji < 0, i ≤ j, where

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ωij = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

r−1 k=1

βk (Pk − Pr )

+Pj Ai + ATi Pj +Yj + YjT + Q

(2.6) (2.7) ⎤

⎥ ⎥ τ0 ATi Z −τ0 Yj ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, −Tj − TjT − (1 − d)Q τ0 ATτi Z −τ0 Tj ⎥ ⎥ ⎥ ⎥ −τ0 Z 0 ⎥ ⎦ −τ0 Z Pj Aτ i − Yj + TjT

and where an ellipsis denotes a block induced easily by symmetry.

12

2. Stability Analysis of T-S Fuzzy Systems

Proof. Choose the fuzzy weighting-dependent Lyapunov-Krasovskii functional as

t

V (xt ) = x(t)T P (t)x(t) +

0

x(s)T Qx(s)ds t−τ (t)

t

x(s) ˙ T Z x(s)dsdθ, ˙

+ −τ0

(2.8)

t+θ

where xt = x(t + α), α ∈ [−τ, 0] and P (t) =

r

hj (t)Pj .

(2.9)

j=1

The Newton-Leibniz formula gives x(t−d(t)) = x(t)−

t t−d(t)

x(s)ds, ˙ which yields

T T 2 x(t) Y (t) + x(t − τ (t)) T (t) x(t) − x(t − τ (t)) −

t

x(s)ds ˙

t−τ (t)

≡ 0,

(2.10)

where Y (t) = rj=1 hj (t)Yj and T (t) = rj=1 hj (t)Tj and where Yj and Tj are arbitrary matrices with appropriate dimensions, j = 1, 2, . . . , r. With (2.10), the time derivative of V (xt ) along solutions to (2.3) is computed as ˙ V˙ (xt ) = x(t)T P˙ (t)x(t) + 2x(t)T P (t)x(t) +x(t)T Qx(t) − (1 − τ˙ (t))x(t − τ (t))T Qx(t − τ (t))

t T +τ0 x(t) ˙ Z x(t) ˙ − x(s) ˙ T Z x(s)ds ˙ ≤

r

t−τ0

˙ h˙ k (t)x(t)T Pk x(t) + 2x(t)T P (t)x(t)

k=1

+x(t)T Qx(t) − (1 − d)x(t − τ (t))T Qx(t − τ (t))

t +τ0 x(t) ˙ T Z x(t) ˙ − x(s) ˙ T Z x(s)ds ˙ t−τ (t)

+2 x(t)T Y (t) + x(t − τ (t))T T (t)

t

× x(t) − x(t − τ (t)) −

x(s)ds ˙ ,

(2.11)

t−τ (t)

where the inequality is caused only by τ˙ (t) ≤ d and τ (t) ≤ τ0 . Noticing that (2.4) implies h˙ r (t) = −

r−1 k=1

h˙ k (t),

(2.12)

2.3 Stability Analysis

13

which yields r

h˙ k (t)x(t)T Pk x(t) =

k=1

r−1

h˙ k (t)x(t)T Pk x(t) + h˙ r (t)x(t)T Pr x(t)

k=1

=

r−1

h˙ k (t)x(t)T (Pk − Pr )x(t).

(2.13)

k=1

Under (2.5)-(2.6), one has, from (2.11) with (2.13) and substitution of x(t) ˙ with state equation (2.3), that + τ0 M (t)Z −1 M (t)T η(t) V˙ (xt ) ≤ η(t)T Ω(t)

t

η(t)T M (t) + x(s) ˙ T Z Z −1 M (t)T η(t) + Z x(s) ˙ ds, (2.14) − t−τ (t)

where η(t)T = [x(t)T , x(t − τ (t))T ], M (t)T = [Y (t)T , T (t)T ] and ⎡ ⎤ r−1 β (P − P ) k r k=1 k ⎢ ⎥ ⎢ ⎥ T ⎢ +P (t)A(t) + A(t) P (t) P (t)Aτ (t) − Y (t) + T (t)T ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ +Y (t) + Y (t)T + Q ⎥ +τ0 A(t)T ZAτ (t) ⎥. =⎢ Ω(t) ⎢ ⎥ ⎢ ⎥ +τ0 A(t)T ZA(t) ⎢ ⎥ ⎢ ⎥ T ⎢ −T (t) − T (t) − (1 − d)Q ⎥ ⎣ ⎦ +τ0 Aτ (t)T ZAτ (t) Furthermore, (2.7) implies that ⎡ ⎤ r−1 β (P − P ) k k r k=1 ⎢ ⎥ P (t)Aτ (t) ⎢ ⎥ τ0 A(t)T Z −τ0 Y (t) ⎥ ⎢ +P (t)A(t) + A(t)T P (t) ⎢ ⎥ T −Y (t) + T (t) ⎢ ⎥ ⎢ +Y (t) + Y (t)T + Q ⎥ ⎢ ⎥ ⎢ ⎥ T ⎢ ⎥ −T (t) − T (t) Ω(t) := ⎢ T ⎥ τ A (t) Z −τ T (t) 0 τ 0 ⎢ ⎥ ⎢ ⎥ −(1 − d)Q ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 −τ0 Z ⎣ ⎦ −τ0 Z =

=

r r

hi (t)hj (t)Ωij i=1 j=1 r r 1 2

< 0,

hi (t)hj (t)(Ωij + Ωji )

i=1 j=1

(2.15)

14

2. Stability Analysis of T-S Fuzzy Systems

−1 which is equivalent to Ω(t)+τ M (t)T < 0 by Schur complement. There0 M (t)Z ˙ fore, from (2.14) we have V (xt ) < 0, which proves the theorem by Theorem A.1 or stability theory of [44] (see pp. 293, Theorem 8.1).

We remark here that, as pointed out in [120], it is in practice diﬃcult to obtain the bounds βk . This largely depends on the information of the membership functions, and needs to estimate their variations. So far it is lack of common methods and one has to determine the bounds according to each practical fuzzy model. A reference method has been provided in [120]. However, in case of sharp variations of fuzzy weighting functions (i.e., βk is large), the present fuzzy weighting-dependent Lyapunov-Krasovskii functional method in Theorem 2.3.1 is still less conservative than the quadratic Lyapunov-Krasovskii functional method. In details, if βk ’s are large, then Pk could be chosen such that Pk − P tend to zero. In the extreme case of nonexistence of the derivative of the fuzzy weighting functions, we have the following direct corollary using the quadratic Lyapunov-Krasovskii functionals. Corollary 2.3.1. System (2.3) is asymptotically stable if there exist matrices P > 0, Q > 0, Z > 0, Y and T , such that the following LMIs hold for i = 1, 2, . . . , r: ⎡ ⎤ P Ai + ATi P ⎢ P Aτ i − Y + T T τ0 ATi Z −τ0 Y ⎥ ⎢ ⎥ T ⎢ +Y + Y + Q ⎥ ⎢ ⎥ ⎢ ⎥ T T ⎢ −T − T − (1 − d)Q τ0 Aτ i Z −τ0 T ⎥ < 0. (2.16) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎥ −τ0 Z ⎣ ⎦ −τ0 Z Proof. It is obvious by setting Pi = P , Yi = Y and Ti = T in (2.6)-(2.7). Remark 2.3.1. Noting that (2.16) is equivalent to ⎤ ⎡ P Ai + ATi P + τ0 X P Aτ i + τ0 X1 ⎢ τ0 ATi Z ⎥ ⎥ ⎢ ⎥ ⎢ +Y + Y T + Q −Y + T T ⎥ ⎢ ⎥ ⎢ ⎥ < 0, ⎢ −T − T T + τ0 X2 T ⎢ τ0 Aτ i Z ⎥ ⎥ ⎢ ⎥ ⎢ −(1 − d)Q ⎥ ⎢ ⎦ ⎣ −τ0 Z ⎡ ⎤ X X1 Y ⎢ ⎥ ⎢ ⎥ ⎢ X2 T ⎥ ≥ 0, ⎣ ⎦ Z

(2.17)

(2.18)

2.3 Stability Analysis

15

for some matrices X ≥ 0, X1 and X2 ≥ 0, the result in [75] (see Corollary 2.5.3) is recovered by setting T = 0, X1 = 0 and X2 = 0 in (2.17)-(2.18). Moreover, it is easy to see that the requirement of d < 1 can be removed in Corollary 2.3.1 and Theorem 2.3.1 due to the introduction of variables Ti . Later we will use an example to further illustrate the less conservativeness of Corollary 2.3.1 and Theorem 2.3.1 compared with [75]. Theorem 2.3.1 can easily lead to a delay-independent (τ0 -independent) result or a rate-independent (d-independent) result. For instance, setting Z = εI with suﬃciently small ε > 0, Yj = 0 and Tj = 0 (j = 1, 2, . . . , r), Theorem 2.3.1 reduces to the following delay-independent result. Corollary 2.3.2. (τ0 -independent result). Assume that (2.5) holds. System (2.3) is asymptotically stable if there exist matrices Pj > 0 and Q > 0, j = 1, 2, . . . , r, such that the following LMIs hold: Pk ≥ Pr , k = 1, 2, . . . , r, ⎤ ⎡ r−1 2 k=1 βk (Pk − Pr ) + 2Q ⎢ Pj Aτ i + Pi Aτ j ⎥ ⎥ ⎢ ⎥ < 0, ⎢ +Pj Ai + ATi Pj + Pi Aj + ATj Pi ⎦ ⎣ −2(1 − d)Q

(2.19) i ≤ j.

(2.20)

On the other hand, a rate-independent (d-independent) result can be easily obtained as a direct corollary of Theorem 2.3.1 by removing the terms containing matrix Q. Corollary 2.3.3. (d-independent result). Assume that (2.5) holds. System (2.3) is asymptotically stable if there exist matrices Pj > 0, Z > 0, Yj and Tj , j = 1, 2, . . . , r, such that the following LMIs hold: Pk ≥ Pr , k = 1, 2, . . . , r − 1, ij + Ω ji < 0, i ≤ j, Ω where

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ij = ⎢ Ω ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

r−1 k=1

βk (Pk − Pr )

+Pj Ai + ATi Pj +Yj + YjT

(2.21) (2.22) ⎤

⎥ ⎥ Pj Aτ i − Yj + TjT τ0 ATi Z −τ0 Yj ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. −Tj − TjT τ0 ATτi Z −τ0 Tj ⎥ ⎥ ⎥ ⎥ −τ0 Z 0 ⎥ ⎦ −τ0 Z

Remark 2.3.2. We remark that the τ0 -independent result is applicable when the bound of time-delay is unknown while the d-independent result is for the case when the derivative of time-delay is unavailable. It is obvious that Corollary

16

2. Stability Analysis of T-S Fuzzy Systems

2.3.2 is less conservative than existing delay-independent stability results in [9][10][73][158]. This is because those LMI stability conditions are included in (2.19)-(2.20) with selecting Pj = P , j = 1, 2, . . . , r. Theorem 2.3.1 can be improved by adopting a relaxed LMI technique similar to that used in [98]. In a more detail, under condition (2.5), system (2.3) is asymptotically stable if (2.6) and the following LMIs hold: i ≤ j, ⎤

(2.23)

⎥ ⎥ · · · X2r ⎥ ⎥ .. ⎥ ≤ 0, .. . . ⎥ ⎦ · · · Xrr

(2.24)

Ωij + Ωji < Xij ⎡ X11 X12 ⎢ ⎢ ⎢ X22 := ⎢ X .. ⎢ .. ⎢ . . ⎣

T + Xij ,

· · · X1r

for additional matrices Xij with Xii = XiiT , i = 1, . . . , r, j = i, . . . , r, where the notations and variables are as in Theorem 2.3.1. This statement is due to the fact that, under (2.23)-(2.24), the last step (2.15) in the proof of Theorem 2.3.1 is reﬁned as r r 1 Ω(t) = hi (t)hj (t)(Ωij + Ωji ) 2 i=1 j=1 1 T hi (t)hj (t)(Xij + Xij ) 2 i=1 j=1 r

<

r

[h1 (t)I, · · · , hr (t)I]T = [h1 (t)I, · · · , hr (t)I] X ≤ 0,

(2.25)

When restricted to system (2.3) without delay terms (i.e., Aτ i = 0), the following stability result is straightforward. Corollary 2.3.4. Assume that (2.5) holds. System (2.3) with Aτ i = 0 is asymptotically stable if there exist matrices Pi > 0 and Xij with Xii = XiiT , i = 1, . . . , r, j = i, . . . , r, such that (2.6), (2.24) and the following LMIs hold: 2

r−1

βk (Pk − Pr ) + Pj Ai + ATi Pj + Pi Aj + ATj Pi

k=1 T < Xij + Xij ,

i ≤ j.

Proof. It is obvious from (2.23)-(2.24).

(2.26)

Remark 2.3.3. Corollary 2.3.4 employs a relaxed LMI technique and therefore is less conservative than the stability result in [120] for delay-free fuzzy systems. This is because the result in [120] is recovered by setting Xij = 0, i = 1, . . . , r, j = i, . . . , r, in conditions of Corollary 2.3.4. In case of unknown derivatives of fuzzy weighting functions, Corollary 2.3.4 reduces exactly the stability result in

2.4 Augmented Fuzzy Weighting-Dependent

17

[98] and thus is less conservative than other stability results such as [64]. Later we will use an example to further illustrate the less conservativeness of Corollary 2.3.4 compared with [64][98][120].

2.4 Augmented Fuzzy Weighting-Dependent Lyapunov-Krasovskii Functional Method In this section, we propose an augmented fuzzy weighting-dependent LyapunovKrasovskii functional and present an improved method for the stability test. For simplicity, we only consider constant delay case. Result for time-varying delay case can be obtained accordingly. The main development is stated as follows. Theorem 2.4.1. Assume that (2.5) holds. System (2.3) with d = 0 is asymptotically stable if there exist matrices Pj > 0, Wj , Sj ≥ 0, Q > 0, U , R > 0, Z > 0, Yj , Tj and Nj , j = 1, 2, . . . , r, such that the following LMIs hold: ⎤ ⎡ ⎤ ⎡ Pr Wr Pk Wk ⎦≥⎣ ⎦ , k = 1, 2, . . . , r, ⎣ (2.27) Sk Sr ⎡ ⎤ QU ⎣ ⎦ ≥ 0, (2.28) R i ≤ j,

Ψij + Ψji < 0, where

⎡

(11)

Ψij

⎢ ⎢ ⎢ ⎢ ⎢ Ψij = ⎢ ⎢ ⎢ ⎢ ⎣ (11)

Ψij

(12)

Ψij

(22)

Ψij

=

=

=

r−1 k=1 r−1 k=1 r−1

(12)

Ψij

(22)

Ψij

ATi (R + τ0 Z)

(2.29)

Wj + NjT

−τ0 Yj

⎤

⎥ ⎥ ATτi (R + τ0 Z) Sj − NjT − U −τ0 Tj ⎥ ⎥ ⎥ −R − τ0 Z 0 0 ⎥, ⎥ ⎥ −R −τ0 Nj ⎥ ⎦ −τ0 Z

βk (Pk − Pr ) + (Pj + U )Ai + ATi (Pj + U T ) + Yj + YjT + Q, βk (Wk − Wr ) + (Pj + U )Aτ i + ATi Wj − Yj + TjT , βk (Sk − Sr ) − Tj − TjT + WjT Aτ i + ATτi Wj − Q.

k=1

Proof. The proposed augmented fuzzy weighting-dependent Lyapunov-Krasovskii functional is of the following form

18

2. Stability Analysis of T-S Fuzzy Systems

⎡ V (xt ) = ⎣

⎤T ⎡ x(t)

⎤⎡ P (t) W (t)

⎤ x(t)

⎦ ⎣ ⎦⎣ ⎦ x(t − τ ) S(t) x(t − τ ) ⎡ ⎤⎡ ⎤T ⎡ ⎤

t x(s) Q U x(s) ⎣ ⎦⎣ ⎦ ⎣ ⎦ ds + t−τ x(s) ˙ UT R x(s) ˙

0 t x(s) ˙ T Z x(s)dsdθ, ˙ + −τ0

(2.30)

t+θ

where ⎡ ⎤ ⎡ ⎤ r Pj Wj P (t) W (t) ⎦. ⎣ ⎦= hj (t) ⎣ S S(t) j j=1

(2.31)

Using the Newton-Leibniz formula, we modify (2.10) as ⎡ ⎤ r r r hj (t)Yj + x(t − τ )T hj (t)Tj + x(t ˙ − τ )T hj (t)Nj ⎦ 2 ⎣x(t)T j=1

× x(t) − x(t − τ ) −

j=1

t

j=1

x(s)ds ˙

t−τ

≡ 0,

(2.32)

where Yj , Tj and Nj are arbitrary matrices with appropriate dimensions, j = 1, 2, . . . , r. Accordingly, η(t) is modiﬁed as ζ(t) = [x(t)T , x(t − τ )T , x(t ˙ − τ )T ]T . The rest of the proof follows a similar line to the proof of Theorem 2.3.1. We would like to point out that Theorem 2.4.1 improves Theorem 2.3.1 in case of d = 0 since the LMIs (2.27)-(2.29) reduces to (2.6)-(2.7) by setting Wj = 0, Sj = 0, U = 0, Nj = 0 and R = εI with suﬃciently small ε > 0, j = 1, 2, . . . , r. Remark 2.4.1. Similar to obtaining Corollary 2.3.1 form Theorem 2.3.1, an improved stability result based on augmented quadratic Lyapunov-Krasovskii functional method can be obtained as a corollary of Theorem 2.4.1. Also, an improved delay-independent result can be obtained as a corollary of Theorem 2.4.1. The details are omitted here and left as exercises for interested readers.

2.5 Further Augmented Lyapunov-Krasovskii Functional Methods It is veriﬁed in the preceding sections that the stability condition obtained by augmented Lyapunov functionals leads to better results. Recently, for linear time-delay systems, it is shown in [83] that better robust stability result could be produced by augmenting some terms in the Lyapunov functionals. It is also shown in [49] that the conservatism can be reduced for both nominal and uncertain cases by choosing the type of further augmented Lyapunov functionals.

2.5 Further Augmented Lyapunov-Krasovskii Functional Methods

19

The conservatism of delay-dependent stability conditions for T-S fuzzy systems can be further reduced by adopting this type of Lyapunov functionals. In this section, we brieﬂy address the further augmented Lyapunov-Krasovskii functional approach. For simplicity, we still consider constant delay case only. Result for time-varying delay case can be obtained accordingly. The main result is stated as follows. Theorem 2.5.1. System (2.3) with d = 0 is asymptotically stable if there exist matrices P > 0, Q > 0, R > 0, S > 0, Z > 0, U, W, X, Y , Nlj and Tk j, j = 1, . . . , r, l = 1, . . . , 4, k = 1, . . . , 6, such that the following LMIs hold: ⎡ ⎤ P Y ⎣ ⎦ ≥ 0, (2.33) T Y X ⎤ ⎡ Q U ⎦ ≥ 0, ⎣ (2.34) UT R Ωij + Ωji < 0, where

i ≤ j,

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ωij = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(2.35) ⎤

T Ω11 Ω12 Ω13 Ω14 τ ATi T5j Ω16

⎥ ⎥ T Ω22 Ω23 Ω24 τ ATτi T5j Ω26 ⎥ ⎥ ⎥ T T Ω33 T4j −τ T5j Ω36 ⎥ ⎥, ⎥ −R 0 τ N4j ⎥ ⎥ ⎥ −τ S −τ W ⎥ ⎦ −τ Z

where T T Ω11 = Q + N1j + N1j − T1j Ai − ATi T1j + τ S, T T Ω12 = N2j − N1j − ATi T2j − T1j Aτ i , T T Ω13 = P + N3j + T1j − ATi T3j + τ W + U, T T Ω14 = N4j − ATi T4j + Y, T Ω16 = τ N1j + τ ATi T6j , T T Ω22 = −Q − N2j − N2j − T2j Aτ i − ATτi T2j , T T Ω23 = −N3j + T2j − ATτi T3j + Y T, T T Ω24 = −N4j − ATτi T4j − U + X, T Ω26 = τ N2j + τ ATτi T6j , T Ω33 = τ Z + T3j + T3j + R, T Ω36 = τ N3j − τ T6j .

20

2. Stability Analysis of T-S Fuzzy Systems

Proof. The further augmented Lyapunov-Krasovskii functional is of the following form: ⎡ ⎤⎡ ⎤T ⎡ ⎤ x(t) P Y x(t) ⎦⎣ ⎦ ⎣ ⎦ V (xt ) = ⎣ x(t − τ ) YT X x(t − τ ) ⎡ ⎤⎡ ⎤T ⎡ ⎤

t x(s) Q U x(s) ⎣ ⎦⎣ ⎦ ⎣ ⎦ ds + t−τ x(s) ˙ UT R x(s) ˙ ⎡ ⎤⎡ ⎤T ⎡ ⎤

0 t x(s) S W x(s) ⎣ ⎦⎣ ⎦ ⎣ ⎦ dsdθ. (2.36) + −τ t+θ x(s) ˙ WT Z x(s) ˙ The rest of the proof can be made following a similar line to the previous theorems, and the details are omitted here. The following result is a direct corollary of Theorem 2.5.1. It is a simpler case by using the Lyapunov-Krasovskii functional of the form

t x(s)T Qx(s)ds V (t) = x(t)T P x(t) +

0

t−τ t

x(s) ˙ T Z x(s)dsdθ. ˙

+ −τ

(2.37)

t+θ

Corollary 2.5.1. System (2.3) with d = 0 is asymptotically stable if there exist matrices P > 0, Q > 0, Z > 0, Nk and Tk , k = 1, 2, 3, such that the following LMIs hold: ⎡ ⎤ Φ11 Φ12 Φ13 τ N1 ⎢ ⎥ ⎢ ⎥ ⎢ Φ22 Φ23 τ N2 ⎥ ⎢ ⎥ < 0, i = 1, 2, . . . , r, (2.38) ⎢ ⎥ ⎢ Φ33 τ N3 ⎥ ⎣ ⎦ −τ Z where Φ11 = Q + N1 + N1T − T1 Ai − ATi T1T , Φ12 = N2T − N1 − ATi T2T − T1 Aτ i , Φ13 = P + N3T − ATi T3T + T1 , Φ22 = −Q − N2 − N2T − T2 Aτ i − ATτi T2T , Φ23 = −N3T + T2 − ATτi T3T , Φ33 = τ Z + T3 + T3T . It can be shown as in [49] that Corollary 2.5.1 is equivalent to the following result.

2.5 Further Augmented Lyapunov-Krasovskii Functional Methods

21

Corollary 2.5.2. System (2.3) with d = 0 is asymptotically stable if there exist matrices P > 0, Q > 0, Z > 0, N1 and T2 , such that the following LMIs hold: ⎡ ⎤ Ψ11 Ψ12 τ ATi Z τ N1 ⎢ ⎥ ⎢ ⎥ ⎢ Ψ22 τ ATτi Z τ N2 ⎥ ⎢ ⎥ < 0, i = 1, 2, . . . , r, (2.39) ⎢ ⎥ ⎢ −τ Z 0 ⎥ ⎣ ⎦ −τ Z where Ψ11 = P Ai + ATi P + N1 + N1T + Q, Ψ12 = P Aτ i − N1 + N2T , Φ22 = −N2 − N2T − Q. Note that (2.39) is equivalent to ⎡ ⎤ P Ai + ATi P ⎢ ⎥ P Aτ i − N1 ⎢ ⎥ τ ATi Z ⎥ ⎢ +N1 + N1T ⎢ T ⎥ +N2 + τ X1 ⎢ ⎥ ⎢ +τ X + Q ⎥ ⎢ ⎥ < 0, ⎢ ⎥ ⎢ ⎥ −N2 − N2T T ⎢ τ Aτ i Z ⎥ ⎢ ⎥ ⎢ ⎥ −Q + τ X2 ⎣ ⎦ −τ Z ⎡ ⎤ X X1 N1 ⎢ ⎥ ⎢ ⎥ ⎢ X2 N2 ⎥ ≥ 0, ⎣ ⎦ Z

i = 1, 2, . . . , r,

(2.40)

(2.41)

for some matrices X ≥ 0, X1 and X2 ≥ 0. Therefore, the result in [75] is recovered by setting N2 = 0, X1 = 0 and X2 = 0 in (2.40)-(2.41), which is listed below. Corollary 2.5.3. System (2.3) with d = 0 is asymptotically stable if there exist matrices P > 0, Q > 0, Z > 0, Y and X, such that the following LMIs hold: ⎡ ⎤ P Ai + ATi P ⎢ ⎥ ⎢ ⎥ ⎢ +Y + Y T P Aτ i − Y τ ATi Z ⎥ ⎢ ⎥ ⎢ ⎥ (2.42) ⎢ +τ X + Q ⎥ < 0, i = 1, 2, . . . , r, ⎢ ⎥ ⎢ ⎥ ⎢ −Q τ ATτi Z ⎥ ⎣ ⎦ −τ Z ⎡ ⎤ XY ⎣ ⎦ ≥ 0. (2.43) Z

22

2. Stability Analysis of T-S Fuzzy Systems

Theoretically, Theorem 2.5.1 covers Corollaries 2.5.1 and 2.5.2, while Corollaries 2.5.1 and 2.5.2 cover Corollary 2.5.3.

2.6 Illustrative Examples We have shown theoretically that the method in this chapter covers some of existing ones for fuzzy systems with/without time-delay. In this section, we use several examples to further verify the less conservativeness. The solver used is the LMI Toolbox in Matlab [36]. Example 2.6.1. Consider a nonlinear system ¨ = −6θ(t) ˙ sin2 (θ(t)) − 8θ(t) + θ(t − τ ) − 2θ(t ˙ − τ ), θ(t) ˙ ˙ where the range of θ(t) is assumed to satisfy |θ(t)| < θ0 . The purpose is to test the stability for a given delay τ > 0. ˙ Introduce new variables x(t) = [x1 (t), x2 (t)]T with x1 (t) = θ(t) and x2 (t) = θ(t). Then the nonlinear system is described by ⎡ ⎤ ⎡ ⎤ 0 1 0 0 ⎦ x(t) + ⎣ ⎦ x(t − τ ). x(t) ˙ =⎣ −8 −6 sin2 (θ(t)) 1 −2 The above can be expressed exactly by the following fuzzy model x(t) ˙ =

2

hi (Ai x(t) + Aτ i x(t − τ )),

i=1

where

⎡

A1 = ⎣

Aτ 1

⎤ 0

1

⎡

⎤ 0 1

⎦ , A2 = ⎣ ⎦, −8 −6 −8 0 ⎡ ⎡ ⎤ ⎤ 0 0 0 0 ⎦ , Aτ 2 = ⎣ ⎦, =⎣ 1 −2 1 −2

h1 = sin2 (θ(t)),

h2 = cos2 (θ(t)).

Using the existing delay-dependent method in Corollary 2.5.3, the largest delay is computed to be τ = 0.2570 such that the system is stable. However, using Corollary 2.3.1, the largest delay is computed to be τ = 0.3078, which is larger than 0.2570. Next we use our fuzzy weighting-dependent methods in Theorems 2.3.1 and 2.4.1 to test if the delay can be further larger. It is seen that |h˙ i | = ˙ sin(2θ(t))| ≤ θ0 , i = 1, 2. Assume θ0 = 2. Then, by using Theorems 2.3.1 |θ(t) and 2.4.1, we compute the largest delay values as τ = 0.3195 and τ = 0.3222, respectively. This reveals that the fuzzy weighting-dependent methods could produce better results. It also reveals that Theorem 2.4.1 is less conservative than Theorem 2.3.1. The next example is borrowed from [75] in which a delay-dependent method is provided for the stability of fuzzy systems.

2.6 Illustrative Examples

23

Example 2.6.2. Consider a system of the form (2.3) with ⎡ ⎡ ⎤ ⎤ −3.2 0.6 −1 0 ⎦ , A2 = ⎣ ⎦, A1 = ⎣ 0 −2.1 1 −3 ⎡ ⎡ ⎤ ⎤ 1 0.9 0.9 0 ⎦ , Aτ 2 = ⎣ ⎦. Aτ 1 = ⎣ 0 2 1 1.6 The purpose is to compute the maximum value of τ0 such that the fuzzy system is asymptotically stable. Note that the existing delay-independent methods in [9][10][73][158] fail to verify the stability of this system. Table 2.1 lists the computation results using delay-dependent conditions in Corollary 2.3.1 and Theorems 2.3.1 and 2.4.1, compared with the existing delay-dependent method in Corollary 2.5.3 for diﬀerent values of d. It is seen from Table 2.1 that the largest values of τ0 are obtained by using our methods. Moreover, Theorem 2.4.1 produces better results than Theorem 2.3.1 (the computations are based on the same upper bound β for all βi , i.e., βi ≤ β). Note that when d ≥ 0.1, the method in Corollary 2.5.3 fails to test the stability of this system.

Table 2.1. Comparison results of maximum τ0 for Example 2.6.2 Methods

d = 0 d = 0.02 d = 0.1 d = 0.9

Corollary 2.5.3

0.5837 0.3637

—

—

Corollary 2.3.1

0.6148 0.5592 0.4809 0.4454

β = 1 Theorem 2.3.1 0.6185 0.5618 0.4809 0.4513 Theorem 2.4.1 0.6345 0.5645 0.4826 0.4524 β = 0.5 Theorem 2.3.1 0.6248 0.5630 0.4814 0.4537 Theorem 2.4.1 0.6399 0.5663 0.4831 0.4556

The third example shows that the stability cannot be tested by quadratic methods while can be veriﬁed by our fuzzy weighting-dependent methods in Theorems 2.3.1 and 2.4.1. Example 2.6.3. Consider a system of the form (2.3) with ⎡ ⎡ ⎤ ⎤ −5 −4 −2 −4 ⎦ , A2 = ⎣ ⎦, A1 = ⎣ −1 −2 20 −2 ⎡ ⎡ ⎤ ⎤ 0.5 0.6 0.3 −0.4 ⎦ , Aτ 2 = ⎣ ⎦. Aτ 1 = ⎣ 1 0.4 0.5 0

24

2. Stability Analysis of T-S Fuzzy Systems

The purpose is to compute the maximum value of τ0 such that the fuzzy system is asymptotically stable. The existing delay-dependent quadratic stability method in [75] and delay-independent methods in [9][10][73][158] fail to verify the stability of this system. The delay-dependent quadratic stability method in Corollary 2.3.1 fails as well. Now, we use our fuzzy weighting-dependent methods in Theorems 2.3.1 and 2.4.1. Table 2.2 lists the computation results for diﬀerent values of d under the same upper bound βi ≤ β = 0.8. It reveals from Table 2.2 that Theorem 2.4.1 produces better results than Theorem 2.3.1.

Table 2.2. Comparison results of maximum τ0 for Example 2.6.3 Methods

d = 0 d = 0.5 d = 0.9

Corollary 2.5.3

—

—

—

Corollary 2.3.1

—

—

—

β = 0.8 Theorem 2.3.1 0.3883 0.3225 0.2518 Theorem 2.4.1 +∞

0.5221 0.2844

In order to show the less conservativeness of our delay-independent result of Corollary 2.3.2 over existing ones in [9][10][73][158], let us examine the following numerical example. Example 2.6.4. Consider a system of the form (2.3) with ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 2 3 −7 1 0.5 0 ⎦ , A2 = ⎣ ⎦ , Aτ 1 = Aτ 2 = ⎣ ⎦. A1 = ⎣ −8 −4 −3 −1 0 0.3 For this system with d = 0, the existing delay-independent stability methods in [9][10][73][158] fail to test the asymptotic stability. Using our delay-independent result of Corollary 2.3.2, we verify that this system is asymptotically stable with an allowable common upper bound 1.5 for the variation rate of fuzzy weighting functions, i.e., βi ≤ 1.5. This convinces the statement in Remark 2.3.2 that Corollary 2.3.2 is less conservative than the above mentioned delay-independent stability methods. The following example is taken from [120]. We use this example to illustrate the less conservativeness of Corollary 2.3.4 (as stated in Remark 2.3.3) compared with existing stability results for delay-free fuzzy systems. Example 2.6.5. Consider a delay-free fuzzy system of the form (2.3) with Aτ 1 = Aτ 2 = 0 and ⎡ ⎡ ⎤ ⎤ −5 −4 −2 −4 ⎦ , A2 = ⎣ ⎦. A1 = ⎣ −1 −2 20 −2

2.7 Conclusions and Notes

25

For this fuzzy model, the maximum allowable common upper bound of βi obtained by the method of [120] is 2.57. Other quadratic stability methods such as those in [64][98] fail to test the stability of this fuzzy system. Using our result in Corollary 2.3.4, the maximum allowable common upper bound of βi is computed to be 2.8192. This shows the less conservativeness of Corollary 2.3.4. We now use a numerical example to further illustrate the use of Theorem 2.5.1 and Corollaries 2.5.1-2.5.3 and the comparisons among these results. The example is borrowed from [75]. Example 2.6.6. Consider a system of the form (2.3) with d = 0 and ⎡ ⎡ ⎤ ⎤ −3.2 0.6 −1 0 ⎦ , A2 = ⎣ ⎦, A1 = ⎣ 0 −2.1 1 −3 ⎡ ⎡ ⎤ ⎤ 1 0.9 0.9 0 ⎦ , Aτ 2 = ⎣ ⎦. Aτ 1 = ⎣ 0 2 1 1.6 The purpose is to compute the maximum value of τ such that the fuzzy system is asymptotically stable. Corollary 2.5.3 produces τmax = 0.5837 while Corollaries 2.5.1 and 2.5.2 result in τmax = 0.6148. Using Theorem 2.5.1 we have τmax = 0.6289, which is larger than above values.

2.7 Conclusions and Notes Delay-dependent stability problems for time-delay T-S fuzzy systems have been studied in this chapter. Various delay-dependent stability conditions have been established using the fuzzy weighting-dependent Lyapunov-Krasovskii functional method combining with the introduction of free-weighting matrices. Furthermore, augmented fuzzy weighting-dependent Lyapunov-Krasovskii functionals have been proposed to further reduce the conservatism. Also, further augmented Lyapunov-Krasovskii functional approach has been addressed. Delay-independent stability criteria are easy corollaries of the present delay-dependent results. Several examples have been given to illustrate the less conservativeness of the present methods. Note that only the case of single time-varying delay τ (t) is considered in this chapter. However, the method can be easily extended to handle multiple delay cases. Readers could refer to [48] for how to deal with the multiple time-delay cases. Also, researchers may seek other ways that can reduce the conservatism by introducing additional matrix variables (or say “slack variables”). However, the price under additional variables and large dimensions of LMIs is the growth of computational burden in the meantime. When using the LMI Control Toolbox, as discussed in [3] and [36], the number of ﬂoating point operations or the time required to test the feasibility of the set of LMIs is proportional to α3 β, where α and β are, respectively, the numbers of variables and rows. This reminds us that in practice we have to seek an acceptable trade-oﬀ between reducing conservatism and computational complexity.

3. Extension to Fuzzy Large-Scale Systems

This chapter deals with fuzzy large-scale systems. It extends the delay-dependent methods of Chapter 2 to large-scale fuzzy models. It provides a delay-dependent method for the stability analysis by choosing appropriate Lyapunov-Krasovskii functionals and adopting the free-weighting matrix techniques.

3.1 Introduction Practical systems, such as power systems, economic systems and process control systems, usually possess “large” dimensions. Large-scale systems show their importance in representing those systems with large dimensions. The characteristic of independent subsystems connected by a number of interconnections makes it possible to reduce computational burdens arisen from the analysis and control of large dimensional systems. So far, various techniques have been sought for the analysis and synthesis of large-scale systems, among which the well-known Takagi-Sugeno fuzzy model approach [119] is a popular one in dealing with nonlinear settings. See [96][140] and the references therein. Indeed, the T-S fuzzy model has already been recognized as a powerful tool in approximating and describing complex nonlinear systems. As a consequence, the study of T-S fuzzy systems has been gained much attention in the past decades [80][122]. In view of time delays being commonly residing in practical systems, the study of T-S fuzzy systems with time delays has been attracted an increasing interest these years. Lots of eﬀorts have been made to develop both delay-independent methods [9][10][73][81][84][152][158] and delay-dependent methods [42][75]. Most of the proposed methods are in terms of linear matrix inequalities LMIs which can be solved by eﬃcient solvers. Recently, eﬀective schemes have also been developed for T-S fuzzy large-scale systems with time delays [61][97][171]. An eﬃcient LMI-based delay-dependent stability analysis test is presented in [97] by adopting the generalized Park’s bounding technique [105]. In this chapter, we focus on the investigation of T-S fuzzy large-scale systems with time delays and aim to develop improved delay-dependent stability criteria over that in [97]. In order to reduce the conservatism entailed in [97], we will adopt the free-weighting matrix method [57], which, unlike the generalized Park’s bounding method [105], does not impose any bounding techniques on cross product terms. This method has been shown more eﬀective than previous ones (such as the descriptor model transformation method [34]) for time-delay C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 27–37, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

28

3. Extension to Fuzzy Large-Scale Systems

systems [58][144][145][151]. In order to serve systems in more general settings, we assume the time delays are smooth and time-varying other than merely constant. We ﬁrst present a delay-dependent stability criterion which is an obvious improvement over that in [97]. Then, when the time delays meet special requirements, we give a further improved delay-dependent criterion. All results in this Chapter are in terms of strict LMIs. Finally, we use an illustrative example to verify the less conservativeness of the present results.

3.2 Problem Formulation Consider a fuzzy large-scale system with time delays in interconnections [97], in which the ith (i = 1, 2, . . . , J) fuzzy subsystem is described by x˙ i (t) =

ri

hij (t)Aij xi (t) +

j=1

J

Bki xk (t − dki (t)),

k=1

xi (t) = φi (t), t ∈ [τ, 0],

(3.1)

where xi is the state; Aij and Bki are constant real matrices with appropriate dimensions; hij is the fuzzy weighting function related to the membership function by (3.5); φi is a continuous vector-valued initial function; dki (t) is the time-delay satisfying 0 ≤ dki (t) ≤ τki ,

d˙ki (t) ≤ µki ,

k, i = 1, 2, . . . , J,

(3.2)

where τki > 0 and µki are scalars. Scalar τ is deﬁned as τ = max1≤k,i≤J {τki }. In the above fuzzy large-scale system, each isolated subsystem is represented by a T-S fuzzy model with ri plant rules: Plant Rule j: IF σ1i is Mj1i and · · · and σpi is Mjpi , THEN x˙ i (t) = Aij xi (t),

(3.3)

where σli (x) and Mjli (l = 1, . . . , p) are respectively the premise variables (which are the functions of state variables) and the fuzzy sets. By fuzzy blending, the overall fuzzy model of the ith isolated subsystem is inferred as follows: ri ri j=1 wij (t)Aij xi (t) ri x˙ i (t) = = hij (t)Aij xi (t) := Ai (t)xi (t), (3.4) j=1 wij (t) j=1 where wij : Rp → [0, 1] is the membership function (dependent on σli ) of the ith isolated subsystem with respect to plant rule j, and hij (t) = wij (t)/

ri j=1

wij (t),

hij (t) ≥ 0,

ri

hij (t) = 1.

(3.5)

j=1

The purpose of this chapter is to establish updated delay-dependent stability criteria for fuzzy system (3.1).

3.3 Stability Results

29

3.3 Stability Results The following result provides a delay-dependent stability criterion for time-delay fuzzy large-scale system (3.1). Theorem 3.3.1. System (3.1) is asymptotically stable if there exist matrices Pi > 0, Qki ≥ 0, Zki ≥ 0, Cki ≥ 0, Nki ≥ 0, Ski ≥ 0, Yki , Rki and Tki , k, i = 1, 2, . . . , J, such that the following LMIs hold: ⎡ ⎤ Nki Rki Yki ⎢ ⎥ ⎢ T ⎥ (3.6) ⎢ Rki Ski Tki ⎥ ≥ 0, ⎣ ⎦ T YkiT Tki Zki Ωij < 0, where

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ωij = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

i = 1, . . . , J, j = 1, . . . , ri ,

T Y1i . . .

−C1i .. Y1i

.

···

(i, i) . . .

.. · · · YJi ..

···

Zi Aij

.

T −T1i − T1i

+τ1i S1i ..

. . .

.

−(1 − µii )Qii T −Tii − Tii

+τii Sii ..

..

.

. . .

. −(1 − µJi )QJi T −TJi − TJi

TJi

Zi B1i

(i, i) = Pi Aij + ATij Pi + Yii + YiiT − Cii +

···

J l=1

J l=1

τil Zil .

+τJi SJi

Zi Bii

···

Zi BJi

with

Zi =

−(1 − µ1i )Q1i

+τii RT ii + Tii

+τJi RT Ji

T T Pi − Yii Bii

T T Pi − YJi BJi

···

−CJi

. . .

. . .

.

+τ1i RT 1i .

. ..

T T B1i Pi − Y1i

..

⎤

T YJi

T1i

(3.7)

(τli Nli + Qil + Cil ),

−Zi

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

30

3. Extension to Fuzzy Large-Scale Systems

Here and after, a blank entry denotes a zero block and an ellipsis denotes a symmetric block as usual. Proof. Choose the Lyapunov-Krasovskii functional as V (xt ) =

J

Vi (xt ),

(3.8)

i=1

with J

Vi (xt ) = xi (t)T Pi xi (t) +

+

J

k=1

0

−τki

k=1

t

xk (s)T Qki xk (s)ds

t−dki (t)

t

x˙ k (s)T Zki x˙ k (s)dsdθ, t+θ

where xt = [x1 (t + α1 )T , · · · , xJ (t + αJ )T ]T , αi ∈ [−τ, 0]. It is easy to see that there exist two constants c1 and c2 such that c1 xi (t)2 ≤ Vi (xt ) ≤ c2 max

sup

1≤i≤J αi ∈[−τ,0]

xi (t + αi )2 .

We next compute the time derivative of V (xt ). For any k, i = 1, 2, . . . , J, the t Newton-Leibniz formula gives xk (t−dki (t)) = xk (t)− t−dki (t) x˙ k (s)ds, which yields

2 xi (t)T Yki + xk (t − dki (t))T Tki

t × xk (t) − xk (t − dki (t)) − x˙ k (s)ds t−dki (t)

≡ 0,

(3.9)

where Yki and Tki , k, i = 1, 2, . . . , J, are arbitrary matrices with appropriate dimensions. With (3.6) and (3.9), the time derivative of Vi (xt ) along solutions to (3.1) is given by V˙ i (xt ) = 2xi (t)T Pi x˙ i (t) J + xk (t)T Qki xk (t) − (1 − d˙ki (t))xk (t − dki (t))T Qki xk (t − dki (t)) k=1

+

J k=1

≤

J

τki x˙ k (t) Zki x˙ k (t) −

⎡

τki ⎣

⎤T ⎡ xi (t)

t

T

T

x˙ k (s) Zki x˙ k (s)ds t−τki

Nki Rki

⎤⎡

⎤ xi (t)

⎦⎣ ⎦ ⎣ ⎦ T (t − d (t)) R S (t − d (t)) x x k ki ki k ki k=1 ki ⎡ ⎤⎡ ⎤T ⎡ ⎤ J t xi (t) Nki Rki xi (t) ⎣ ⎦⎣ ⎦ ⎣ ⎦ ds − T Rki Ski xk (t − dki (t)) k=1 t−dki (t) xk (t − dki (t))

3.3 Stability Results

+

31

J

2 xi (t)T Yki + xk (t − dki (t))T Tki k=1

t

× xk (t) − xk (t − dki (t)) −

x˙ k (s)ds t−dki (t)

+2xi (t)T Pi x˙ i (t) +

J k=1

+

J

xk (t)T Qki xk (t) − (1 − µki )xk (t − dki (t))T Qki xk (t − dki (t))

t

τki x˙ k (t)T Zki x˙ k (t) −

x˙ k (s)T Zki x˙ k (s)ds ,

(3.10)

t−dki (t)

k=1

where the inequality is caused only by dki (t) ≤ τki and d˙ki (t) ≤ µki . Noticing that J J

xk (t)T Qki xk (t)

i=1 k=1

=

J J

xi (t)T Qik xi (t) + xi (t)T Cik xi (t) − xk (t)T Cki xk (t) ,

(3.11)

i=1 k=1

and J J

τki x˙ k (t)T Zki x˙ k (t)

i=1 k=1

=

J J

τik x˙ i (t)T Zik x˙ i (t) =

i=1 k=1

J

x˙ i (t)T Zi x˙ i (t),

(3.12)

i=1

substituting x˙ i (t) with state equation (3.1), one has from (3.10)-(3.12) that V˙ (xt ) ≤

J

ηi (t)T Ωi (t) + Mi (t)T Zi Mi (t) ηi (t)

i=1

−

J

J

⎡

⎤T xi (t)

⎢ ⎥ ⎢ ⎥ ⎢ xk (t − dki (t)) ⎥ ⎣ ⎦ i=1 k=1 t−dki (t) x˙ k (s) ⎡ ⎤⎡ ⎤ xi (t) Nki Rki Yki ⎢ ⎥⎢ ⎥ ⎢ T ⎥⎢ ⎥ × ⎢ Rki Ski Tki ⎥ ⎢ xk (t − dki (t)) ⎥ ds, ⎣ ⎦⎣ ⎦ T YkiT Tki Zki x˙ k (s) t

(3.13)

32

3. Extension to Fuzzy Large-Scale Systems

where ηi (t)T = [x1 (t)T , · · · , xJ (t)T , x1 (t − d1i (t))T , · · · , xJ (t − dJi (t))T ], Mi (t) = [0, · · · , 0, Ai (t), 0, · · · , 0, B1i , · · · , BJi ], ⎡ T ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ωi (t) = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−C1i

.. Y1i

.

···

Y1i . . .

(i, i) . . .

.. · · · YJi ..

.

T −T1i − T1i

+τ1i S1i ..

.

−(1 − µii )Qii

T T Bii Pi − Yii

+τJi RT Ji

−(1 − µ1i )Q1i

T −Tii − Tii

+τii RT ii + Tii

T T BJi Pi − YJi

···

. . .

. . .

−CJi

+τ1i RT 1i .

···

..

T T B1i Pi − Y1i

..

.

.

T YJi

T1i

⎤

+τii Sii ..

..

.

. −(1 − µJi )QJi T −TJi − TJi

TJi

(i, i) = Pi Ai (t) + Ai (t)T Pi + Yii + YiiT − Cii +

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

+τJi SJi J

(τli Nli + Qil + Cil ).

l=1

J Furthermore, (3.7) implies that j=1 hij (t)Ωij < 0 which is equivalent to Ωi (t)+ Mi (t)T Zi Mi (t) < 0 by Schur complement. Therefore, from (3.7) and (3.13) we have V˙ (xt ) < −εx(t) for suﬃciently small scalar ε > 0, where x(t) = [x1 (t)T , · · · , xJ (t)T ]T . This proves the theorem by stability theory of [44] (See Appendix). The above result provides a delay-dependent stability test in terms of LMIs, in which the condition can be fulﬁlled using existing LMI toolbox. It can be easily veriﬁed that Theorem 3.3.1 covers the result for nominal systems in [97] which is listed below. Corollary 3.3.1. System (3.1) is asymptotically stable if there exist matrices Pi > 0, Qki ≥ 0, Zki ≥ 0, Cki ≥ 0, Nki ≥ 0, and Yki , k, i = 1, 2, . . . , J, such that the following LMIs hold: ⎡ ⎤ Nki Yki ⎣ ⎦ ≥ 0, (3.14) YkiT Zki

3.3 Stability Results

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−C1i .. Y1i

.

···

⎤

T Y1i . . .

(i, i) . . .

· · · YJi ..

T YJi

···

−CJi −(1 − µ1i )Q1i ..

T T Bii Pi − Yii . . .

−

···

.

T T B1i Pi − Y1i . . .

T Pi BJi

33

. . .

.

−(1 − µii )Qii .

. . .

−(1 − µJi )QJi

..

T YJi

Zi Aij

Zi B1i

···

Zi Bii

···

Zi BJi

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−Zi

i = 1, . . . , J, j = 1, . . . , ri ,

(3.15)

where (i, i) = Pi Aij + ATij Pi + Yii + YiiT − Cii +

J

(τli Nli + Qil + Cil ),

l=1

Zi =

J

τil Zil .

l=1

In fact, Corollary 3.3.1 is recovered by setting Rki = 0, Ski = 0 and Tki = 0 in (3.6)-(3.7). Therefore, Theorem 3.3.1 is less conservative than Corollary 3.3.1. Later we will use an example to further illustrate the less conservativeness of Theorem 3.3.1 compared with Corollary 3.3.1. Theorem 3.3.1 is for the case of multiple delays in each subsystem state xk . For the special case of a single delay in each subsystem state xk , i.e., 0 ≤ dki (t) = dk (t) ≤ τk ,

d˙k (t) ≤ µk ,

k, i = 1, 2 . . . , J,

(3.16)

where τk > 0 and µk are scalars, the result in Theorem 3.3.1 can be improved as follows. Theorem 3.3.2. System (3.1) with the time delays satisfying (3.16) is asymptotically stable if there exist matrices Pi > 0, Qi ≥ 0, Zi > 0, Yki , and Tki , k, i = 1, 2, . . . , J, such that the following LMIs hold for jk = 1, . . . , rk , k = 1, . . . , J: ⎤ ⎡ (i) J Φ ⎥ ⎢ i=1 11 J ⎥ ⎢ J (i) ⎥ ⎢ i=1 Φ(i) Φ 21 22 i=1 ⎥ < 0, ⎢ (3.17) Φ(j1 , · · · , jJ ) := ⎢ ⎥ ⎢ Φ31 Φ32 Φ33 ⎥ ⎦ ⎣ Φ42 0 Φ44 Φ41

34

3. Extension to Fuzzy Large-Scale Systems

where

(i)

Φ11

⎡

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Pi Aiji + ATiji Pi ⎢ = ⎢ Y1i · · · · · · YJi ⎥ ⎥, T ⎢ ⎥ +Yii + Yii + Qi ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ T YJi ⎡

(i)

Φ21

⎤

Y1iT .. .

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T = ⎢ T1i · · · Bii Pi − YiiT + Tii · · · TJi ⎥ , ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ T T BJi Pi − YJi ⎡

(i)

Φ22

⎤

T B1i Pi − Y1iT .. .

⎤

T T1i .. .

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ −T1i · · · −(1 − µi )Qi − Tii − TiiT · · · −TJi ⎥ , ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ T −TJi ⎡

⎢ ⎢ Φ31 = ⎢ ⎣

⎡

⎤ τ1 Z1 A1j1 ..

⎥ ⎥ ⎥, ⎦

.

⎢ ⎢ Φ32 = ⎢ ⎣

τJ ZJ B1J · · · τJ ZJ BJJ

τJ ZJ AJjJ ⎡

Φ41

⎤ T T τ1 Y11 · · · τ1 Y1J ⎢ ⎥ .. ⎥ ⎢ . = ⎢ .. ⎥, . ⎣ ⎦ T T · · · τJ YJJ τJ YJ1 ⎡

⎢ ⎢ Φ33 = ⎢ ⎣

⎡

Φ42

⎤ T T τ1 T11 · · · τ1 T1J ⎢ ⎥ .. ⎥ ⎢ . = ⎢ .. ⎥, . ⎣ ⎦ T T · · · τJ TJJ τJ TJ1

⎤

−τ1 Z1 ..

⎥ ⎥ ⎥, ⎦

. −τJ ZJ

τ1 Z1 B11 · · · τ1 Z1 BJ1 .. .. . .

Φ44 = Φ33 .

⎤ ⎥ ⎥ ⎥, ⎦

3.3 Stability Results

35

Proof. Choose the Lyapunov-Krasovskii functional as

t J xi (t)T Pi xi (t) + xi (s)T Qi xi (s)ds V (xt ) = i=1

0

t−di (t)

x˙ i (s)T Zi x˙ i (s)dsdθ .

t

+ −τi

(3.18)

t+θ

For any k, i = 1, 2, . . . , J, we have from the Newton-Leibniz formula that J J

2 xi (t)T Yki + xi (t − di (t))T Tki k=1 i=1

t

× xk (t) − xk (t − dk (t)) −

x˙ k (s)ds t−dk (t)

≡ 0,

(3.19)

where Yki and Tki , k, i = 1, 2, . . . , J, are arbitrary matrices with appropriate dimensions. With (3.19), the time derivative of V (xt ) along solutions to (3.1) is given by V˙ (xt ) =

J

[2xi (t)T Pi x˙ i (t) + xi (t)T Qi xi (t)

i=1

−(1 − d˙i (t))xi (t − di (t))T Qi xi (t − di (t))

t +τi x˙ i (t)T Zi x˙ i (t) − x˙ i (s)T Zi x˙ i (s)ds] t−τi

≤

J J

2 xi (t)T Yki + xi (t − di (t))T Tki

k=1 i=1

t

× xk (t) − xk (t − dk (t)) −

x˙ k (s)ds t−dk (t)

+

J

[2xi (t)T Pi x˙ i (t) + xi (t)T Qi xi (t)

i=1

−(1 − µi )xi (t − di (t))T Qi xi (t − di (t))

t +τi x˙ i (t)T Zi x˙ i (t) − x˙ i (s)T Zi x˙ i (s)ds].

(3.20)

t−di (t)

Substituting x˙ i (t) with state equation (3.1), one has from (3.20) that J J τi Mi (t)T Zi Mi (t) + τi Ni Z −1 N T ξ(t) V˙ (xt ) ≤ ξ(t)T Φ(t) + i

−

J

i=1

i=1 t

i

i=1

ξ(t)T Ni + x˙ i (s)T Zi Zi−1

t−di (t)

× NiT ξ(t) + Zi x˙ i (s) ds,

(3.21)

36

3. Extension to Fuzzy Large-Scale Systems

where ξ(t)T = [x1 (t)T , · · · , xJ (t)T , x1 (t − d1 (t))T , · · · , xJ (t − dJ (t))T ], T T , · · · , TiJ ], NiT = [Yi1T , · · · , YiJT , Ti1 ⎡ ⎤ J (i) Φ 11 ⎦, Φ(t) = ⎣ i=1 (i) (i) J J Φ Φ i=1 21 i=1 22 ⎡ ⎤ Y1iT ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ (i) = ⎢ Y · · · P A (t) + A (t)T P + Y + Y T + Q · · · Y ⎥ Φ i i i i ii i Ji ⎥ , 11 ii ⎢ 1i ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎣ ⎦ T YJi (i)

(i)

and Mi (t) is as in (3.13), Φ21 and Φ22 are as in (3.17). If (3.17) holds, then J J Φ(t) + i=1 τi Mi (t)T Zi Mi (t) + i=1 τi Ni Zi−1 NiT < 0 by Schur complement. Therefore, from (3.17) and (3.21) we have V˙ (xt ) < −εx(t) for suﬃciently small scalar ε > 0.

3.4 Illustrative Example We have theoretically shown, as stated just after Corollary 3.3.1, that the method in this chapter covers that in [97]. In this section, we use a numerical example to further illustrate the less conservativeness of the present results. Consider a fuzzy large-scale system of the form (3.1) with J = 2, r1 = 2, r2 = 2, and ⎡ ⎡ ⎤ ⎤ −2.1 0.2 −1.3 0 ⎦ , A12 = ⎣ ⎦, A11 = ⎣ 0.1 −0.9 0.3 −1.1 ⎡ ⎡ ⎤ ⎤ −6.1 2 −1 1 ⎦ , A22 = ⎣ ⎦, A21 = ⎣ 3.7 −4 1 −2 ⎡ B11 = ⎣

B21

−1.1 0.3

⎡

⎤

⎤ 20

⎦ , B12 = ⎣ ⎦, −0.9 −1 03 ⎡ ⎡ ⎤ ⎤ 1.2 0 −0.8 −0.1 ⎦ , B22 = ⎣ ⎦. =⎣ 0 0.3 −0.9 −0.9

Suppose the time delays satisfy 0 ≤ dki (t) = dk (t) ≤ τ,

d˙k (t) ≤ µ,

k, i = 1, 2 . . . , J,

(3.22)

3.5 Conclusion

37

where τ > 0 and µ are scalars. Therefore, both Theorems 3.3.1 and 3.3.2 are applicable for the stability test of this fuzzy large-scale system. The purpose is to compute the maximum value of τ such that the system is asymptotically stable. For constant delay case (i.e., µ = 0), the maximum value of τ = 0.3100 is obtained using the method in Corollary 3.3.1. However, Theorems 3.3.1 and 3.3.2 produce τ = 0.4337 and τ = 0.4887, respectively, which are larger than that obtained using the method in Corollary 3.3.1. For time-varying delay case (i.e., µ = 0), the method in Corollary 3.3.1 is not applicable. Table 3.1 lists the maximum delay bounds (for diﬀerent values of µ) obtained using Theorems 3.3.1 and 3.3.2. It is seen from Table 3.1 that Theorem 3.3.2 produces better results than Theorem 3.3.1. Table 3.1. Comparisons of maximum delay bound τ Methods

µ = 0 µ = 0.5 µ = 0.9 µ ≥ 1

Corollary 3.3.1 0.3100

—

—

—

Theorem 3.3.1 0.4337 0.3688 0.3471 0.3463 Theorem 3.3.2 0.4887 0.3939 0.3516 0.3506

3.5 Conclusion The stability problem for fuzzy large-scale systems with time delays has been studied in this chapter. Updated delay-dependent stability criteria have been established by using appropriate Lyapunov-Krasovskii functionals combining with the introduction of free-weighting matrices. An illustrative example has been given to demonstrate the use and the less conservativeness of the present methods.

4. Stabilization Methods for T-S Fuzzy Systems

The preceding chapter addresses the stability analysis for T-S fuzzy systems with time-delay. This chapter focuses on the stabilization problem for T-S fuzzy systems with time-delay. In particular, it aims to present delay-dependent design of state feedback stabilizing fuzzy controllers for T-S fuzzy systems with state delay and input delay.

4.1 Problem Formulation Consider a nonlinear time-delay system that can be represented by the following fuzzy model with state and input delays: Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Di x(t − τi (t)) + Bi u(t) + Ei u(t − τi (t)),

(4.1)

x(t) = φ(t), t ∈ [−τ, 0], where x ∈ Rn and u ∈ Rm are the state and control input, respectively; Ai , Di and Bi are constant real matrices with appropriate dimensions; r is the number of plant rules; θj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables (which are the functions of state variables) and the fuzzy sets. It is assumed that the premise variables are independent of the input variables u. τi (t), i = 1, 2, . . . , r, are continuous functions of t and τ ≥ 0 is an upper bound of τi (t), i.e., τi (t) ≤ τ ; φ(t) is a continuous vector-valued initial function on [−τ, 0]. By fuzzy blending, the overall fuzzy model is inferred as follows: r wi (θ)(Ai x(t) + Di x(t − τi (t)) + Bi u(t) + Ei u(t − τi (t))) r x(t) ˙ = i=1 i=1 wi (θ) r hi (θ)(Ai x(t) + Di x(t − τi (t)) + Bi u(t) + Ei u(t − τi (t))), (4.2) = i=1

x(t) = φ(t), t ∈ [−τ, 0], where θ = [θ1 , . . . , θp ]T , wi : Rp → [0, 1], i = 1, . . . , r, is the membership function of the system with respect to the ith plant rule, and hi (θ) = rwi (θ) with i=1 wi (θ) r hi (θ) ≥ 0 and i=1 hi (θ) = 1. The design of state feedback stabilizing fuzzy controllers for the fuzzy system (4.2) is based on the parallel distributed compensation [134]. C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 39–53, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

40

4. Stabilization Methods for T-S Fuzzy Systems

Regulator Rule i: IF θ1 is µi1 and · · · and θp is µip THEN u(t) = −Fi x(t),

i = 1, 2, . . . , r.

(4.3)

The overall state feedback control law is inferred by r r w (θ)Fi x(t) r i =− hi (θ(x(t)))Fi x(t). u(t) = − i=1 i=1 wi (θ) i=1

(4.4)

The aim is to determine the local feedback gains Fi such that the closed-loop system x(t) ˙ =

r r

hi hj [(Ai i=1 j=1 r r

− Bi Fj )x(t) + Di x(t − τi (t))]

hi hj (t − τi (t))Ei Fj x(t − τi (t)),

−

(4.5)

i=1 j=1

x(t) = φ(t), t ∈ [−τ, 0], is asymptotically stable.

4.2 Stabilization Method Using Razumikhin Theorem In view of the possible non-smooth delay (i.e., nonexistence of the derivative of time-delay), the Razumikhin Theorem (see Theorem A.2 in Appendix) is a useful tool for the stabilizing controller design. The design result in this section is as follows. Theorem 4.2.1. There exists a fuzzy control law (4.4) such that the closed-loop fuzzy system (4.5) is asymptotically stable if there exist matrices Q > 0, Ui > 0, Vi > 0, Wi > 0, Si > 0 and Yi , i = 1, 2, . . . , r, such that the following LMI’s hold: Θij + Θji < 0, ⎤ ⎡ −Q QATk − YlT BkT ⎦ ≤ 0, ⎣ Ak Q − Bk Yl −Ui ⎤ ⎡ −Q QDkT ⎦ ≤ 0, ⎣ Dk Q −Vi ⎤ ⎡ −Q YlT EkT ⎦ ≤ 0, ⎣ Ek Yl −Wi ⎤ ⎡ −Q YlT ⎦ ≤ 0, ⎣ Yl −Si

(4.6) (4.7)

(4.8)

(4.9)

(4.10)

4.2 Stabilization Method Using Razumikhin Theorem

41

for i, k, l = 1, 2, . . . , r and i ≤ j, where Θij = (Ai + Di )Q + Q(Ai + Di )T − Bi Yj − YjT BiT + Ei Si EiT +τ Di (Uj + Vj + Wj )DiT + (3τ + 1)Q,

(4.11)

If this is the case, the local feedback gains Fi are given by Fi = Yi Q−1 ,

i = 1, 2, . . . , r.

(4.12)

Proof. The proof is based on the Razumikhin Theorem (Theorem A.2). Let the Lyapunov function be V (x) = xT (t)P x(t), where P = Q−1 . In the sequel, we simply use hi , hi (s) and hi (s − τj (s)) to denote hi (θ(x(t))), hi (θ(x(s))) and hi (θ(x(s − τj (s)))), respectively. Without loss of generality, deﬁne x(σ) := φ(−τ ) for σ ∈ [−2τ, −τ ) and hi (θ(x(σ))) := hi (θ(x(0))) for σ ∈ [−τ, 0). Rewrite (4.5) as

t r r hi hj [(Gij + Di )x(t) − Di x(s)ds] ˙ x(t) ˙ = i=1 j=1 r r

−

t−τi (t)

hi hj (t − τi (t))Ei Fj x(t − τi (t)),

i=1 j=1

where Gij := Ai − Bi Fj , i, j = 1, 2, . . . , r. The derivative of V along the solutions of (4.5) is given by V˙ = 2xT (t)P x(t) ˙

r r T hi hj [(Gij + Di )x(t) − Di = 2x (t)P i=1 j=1 r r

−2xT (t)P

t

x(s)ds] ˙

t−τi (t)

hi hj (t − τi (t))Ei Fj x(t − τi (t))

i=1 j=1

=

r r

hi hj xT (t)[P (Gij + Di ) + (Gij + Di )T P ]x(t)

i=1 j=1 r r

−

i=1 j=1

× −

t

2xT (t)P Di

hi hj t−τi (t)

r r

hk (s)hl (s)(Gkl x(s) + Dk x(s − τk (s)))

k=1 l=1 r r

hk (s)hl (s − τk (s))Ek Fl x(s − τk (s)) ds

k=1 l=1 r r

−2xT (t)P

i=1 j=1

hi hj (t − τi (t))Ei Fj x(t − τi (t)).

42

4. Stabilization Methods for T-S Fuzzy Systems

Using the bounding method in (4.13), − 2cT d ≤ cT Xc + dT X −1 d,

∀X > 0,

(4.13)

where c and d are two vectors with compatible dimensions, we have

t r r −2xT (t)P Di hk (s)hl (s)Gkl x(s)ds t−τi (t)

≤ τi (t)x

t +

T

k=1 l=1 T (t)P Di Ui Di P x(t) r r

hk (s)hl (s)xT (s)GTkl Ui−1 Gkl x(s)ds

t−τi (t) k=1 l=1

for any matrices Ui > 0, i = 1, 2, . . . , r. Similarly, it holds that

t r r −2xT (t)P Di hk (s)hl (s)Dk x(s − τk (s))ds t−τi (t)

k=1 l=1

≤ τi (t)xT (t)P Di Vi DiT P x(t)

t r hk (s)xT (s − τk (s))DkT Vi−1 Dk x(s − τk (s))ds, + t−τi (t) k=1

t

2xT (t)P Di

r r

hk (s)hl (s − τk (s))Ek Fl x(s − τk (s))ds

t−τi (t)

≤

k=1 l=1 T τi (t)x (t)P Di Wi DiT P x(t)

t r r

hk (s)hl (s − τk (s))xT (s − τk (s))

+

t−τi (t) k=1 l=1 ×FlT EkT Wi−1 Ek Fl x(s

− τk (s))ds,

and −2xT (t)P

r r

hi hj (t − τi (t))Ei Fj x(t − τi (t))

i=1 j=1

≤

r

hi xT (t)P Ei Si EiT P x(t)

i=1

+

r r

hi hj (t − τi (t))xT (t − τi (t))FjT Si−1 Fj x(t − τi (t))

i=1 j=1

for any matrices Vi > 0, Wi > 0 and Si > 0, i = 1, 2, . . . , r. Hence, we have from the above that

4.2 Stabilization Method Using Razumikhin Theorem

V˙ ≤

r r

hi hj xT (t)Lij x(t)

i=1 j=1 r

i=1

+

r r

t

hi

+

r

hi

i=1 r

43

hk (s)hl (s)xT (s)GTkl Ui−1 Gkl x(s)ds

t−τi (t) k=1 l=1

t r

hk (s)xT (s t−τi (t) k=1

t r r

− τk (s))DkT Vi−1 Dk x(s − τk (s))ds

hk (s)hl (s − τk (s))xT (s − τk (s)) t−τi (t) k=1 l=1 ×FlT EkT Wi−1 Ek Fl x(s − τk (s))ds r r hi hl (t − τi (t))xT (t − τi (t))FlT Si−1 Fl x(t − τi (t)), + i=1 l=1 +

hi

i=1

where Lij = P (Gij + Di ) + (Gij + Di )T P +τ P Di (Ui + Vi + Wi )DiT P + P Ei Si EiT P. Note that (4.6) implies Ωij (δ) + Ωji (δ) < 0 for a suﬃciently small scalar δ > 0, where Ωij (δ) = Lij + (3τ + 1 + 2τ δ)(1 + δ)P, for i = 1, 2, . . . , r and i ≤ j. By using the Razumikhin Theorem, suppose V (x(t+ σ)) < (1 + δ)V (x(t)) for σ ∈ [−τ, 0]. Then, if (4.7)-(4.10) also hold which are equivalent to GTkl Ui−1 Gkl ≤ P,

DkT Vi−1 Dk ≤ P, FlT EkT Wi−1 Ek Fl ≤ P, FlT Si−1 Fl ≤ P,

we have V˙ ≤

r r i=1 j=1 r

+2

i=1

+

r i=1

hi hj xT (t)Lij x(t) +

hi

r

i=1 t

r

t

xT (s)P x(s)ds

hi t−τi (t)

hk (s)xT (s − τk (s))P x(s − τk (s))ds

t−τi (t) k=1

hi xT (t − τi (t))P x(t − τi (t))

44

4. Stabilization Methods for T-S Fuzzy Systems

≤

r r i=1 j=1 r

+2

hi hj xT (t)Lij x(t) +

≤

hi τi (t)xT (t)(1 + δ)P x(t)

i=1 t

xT (s)(1 + δ)P x(s)ds + xT (t)(1 + δ)P x(t)

hi t−τi (t)

i=1 r r

r

hi hj xT (t)Lij x(t) + τ xT (t)(1 + δ)P x(t)

i=1 j=1

+2τ xT (t)(1 + δ)2 P x(t) + xT (t)(1 + δ)P x(t) r r hi hj xT (t)[Lij + (3τ + 1 + 2τ δ)(1 + δ)P ]x(t) = i=1 j=1

=

r

h2i xT (t)Ωii (δ)x(t) +

i=1

r

hi hj xT (t)(Ωij (δ) + Ωji (δ))x(t)

i<j

< 0.

This completes the proof.

If there is no input delay terms (i.e., Ei = 0), we have the following corollary. Corollary 4.2.1. There exists a fuzzy control law (4.4) such that the closed-loop fuzzy system (4.5) with Ei = 0 is asymptotically stable if there exist matrices Q > 0, Ui > 0, Vi > 0 and Yi , i = 1, 2, . . . , r, such that the following LMI’s hold: ij + Θ ji < 0, Θ (4.14) ⎤ ⎡ −Q QATk − YlT BkT ⎦ ≤ 0, ⎣ (4.15) Ak Q − Bk Yl −Ui ⎤ ⎡ −Q QDkT ⎦ ≤ 0, ⎣ (4.16) Dk Q −Vi for i, k, l = 1, 2, . . . , r and i ≤ j, where ij = (Ai + Di )Q + Q(Ai + Di )T − Bi Yj − YjT BiT Θ +τ Di (Uj +

Vj )DiT

+ 2τ Q.

(4.17) (4.18)

If this is the case, the local feedback gains Fi are given by Fi = Yi Q−1 ,

i = 1, 2, . . . , r.

Proof. It is obvious from the proof of Theorem 4.2.1.

(4.19)

Corollary 4.2.1 gives a delay-dependent LMI design result for T-S fuzzy models with state delay only. It can be easily shown that Corollary 4.2.1 is less conservative than the delay-independent fuzzy control design results in [10][158] for small delays. This point will be further illustrated by examples later. Remark 4.2.1. If τ = 0 (i.e., no delay), the LMI’s in (4.14)-(4.16) reduce to the following 12 r(r + 1) LMI’s:

4.3 Other Stabilization Methods

(Ai + Aj + Di + Dj )Q + Q(Ai + Aj + Di + Dj )T −Bi Zj − ZjT BiT − Bj Zi − ZiT BjT ≤ 0,

45

(4.20)

for i = 1, 2, . . . , r and i ≤ j. Remark 4.2.2. Theorem 4.2.1 and Corollary 4.2.1 contain, respectively, 12 r(r + 1)(4r + 1) and 12 r(r + 1)(2r + 1) LMI’s. If Bi = B and Ei = E, i = 1, 2, . . . , r (i.e., with the same input matrices and the same delayed-input matrices in all plant rules), the closed-loop fuzzy system (4.5) is simpliﬁed as r x(t) ˙ = hi [(Ai − BFi )x(t) + Di x(t − τi (t))] i=1

−

r r

hi hj (t − τi (t))EFj x(t − τi (t)),

(4.21)

i=1 j=1

x(t) = φ(t), t ∈ [−τ, 0]. In this case, the number of LMI’s in Theorem 4.2.1 reduces to r(4r+1), while that in Corollary 4.2.1 reduces to r(2r+1). Furthermore, if Ui , Vi , Wi and Si are chosen common matrices, respectively, i.e., Ui=U , Vi=V , Wi=W and Si = S, i = 1, 2, . . . , r, only 5r LMI’s in Theorem 4.2.1 and 3r LMI’s in Corollary 4.2.1 are left.

4.3 Other Stabilization Methods In this section, we discuss other stabilization methods for the design of state feedback stabilizing fuzzy controllers for the time-delay fuzzy systems. 4.3.1

Fuzzy Weighting-Dependent Approach

As shown in Chapter 2, the fuzzy weighting-dependent approach works well in establishing LMI criteria for stability analysis of time-delay fuzzy systems. However, when applied to the stabilization problem, this approach could not lead to consistent LMI conditions for the stabilizing controller design. (Here, by “consistent LMI conditions” we mean that the LMI control design is fully based on the LMI analysis method without any conservative steps involved.) Moreover, the control design is quite complicated and the well-used PDC method is not applicable anymore. Even for delay-free fuzzy systems, the complexity and multiplicity of the design are inevitable. See such attempts in [120] for continuous-time case and in [19] for discrete-time case. 4.3.2

Using Updated Bounding Methods

In analyzing retarded type systems, the well-used Lyapunov-Krasovskii functional to establish delay-dependent LMI conditions for stability is of the form

t V (xt ) = x(t)T P x(t) + x(s)T Qx(s)ds

0

t−τ (t) t

x(s) ˙ T Z x(s)dsdθ, ˙

+ −τ

t+θ

(4.22)

46

4. Stabilization Methods for T-S Fuzzy Systems

where xt = x(t + α), α ∈ [−τ, 0], and matrices P > 0, Q > 0 and Z > 0 are to be chosen. Based on (4.22), Park [108] and Moon et al. [105] have shown that better results over previous ones can be produced with the use of newly developed bounding methods. The following is a bounding method introduced in [105]. Let a ∈ Rna , b ∈ Rnb and N ∈ Rna ×nb . Then, for any matrices X, Y and Z with appropriate dimensions, the following holds: ⎡ ⎤T ⎡ ⎤⎡ ⎤ a 0 −N a ⎦⎣ ⎦ − aT N b = ⎣ ⎦ ⎣ T 0 b −N b ⎤⎡ ⎤ ⎡ ⎤T ⎡ X Y −N a a ⎦⎣ ⎦, (4.23) ≤⎣ ⎦ ⎣ Z Y T − NT b b where ⎤ ⎡ X Y ⎦ ≥ 0. ⎣ YT Z

(4.24)

The bounding method in (4.23) relaxes that in [108], and is also (especially when −2aT N b < 0) less conservative than the well-known one given by − 2cT d ≤ cT Xc + dT X −1 d,

∀X > 0,

(4.25)

where c and d are two vectors with compatible dimensions. Shortly, the bounding technique in [105] is further employed by Fridman (see, e.g., [32][34]) to establish less conservative stability test. The involved idea is to introduce a similar type of Lyapunov-Krasovskii functionals which is based on a descriptor form representation of the system. Fridman’s descriptor approach can reduce conservatism in dealing with time-delay systems with or without uncertainties [33][34][74][146]. Later, a new method was proposed by He et al., Wu et al. and Xu et al., respectively, in [57], [145] and [151], which adopt the type of Lyapunov-Krasovskii functionals in (4.22) but without imposing any conservative bounding techniques on cross-terms. The main idea in these work is the introduction of freeweighting matrices which bring much ﬂexibility in solving LMIs. These free-weighting matrices t are mainly introduced from the Newton-Leibniz formula x(t − τ (t)) = x(t) − t−τ (t) x(s)ds, ˙ which yields

˙ T N3 + x(t ˙ − τ (t))T N4 2 x(t)T N1 + x(t − τ (t))T N2 + x(t)

t × x(t) − x(t − τ (t)) − x(s)ds ˙ t−τ (t)

≡ 0,

(4.26)

for any matrices Ni (i.e., the free-weighting matrices) with appropriate dimensions, i = 1, . . . , 4. This method has been shown more eﬀective in reducing conservatism entailed in previous results, especially for uncertain systems, see [57]

4.3 Other Stabilization Methods

47

for polytopic uncertainty and [145][151] for norm-bounded uncertainty. In fact, the same idea has been used in [58][144][153] as well to provide better robust stability tests for uncertain neutral delay systems. Note that the methods (for both uncertain and nominal cases) in [145] and [151] are the same, and the results for nominal systems in [57] and [145] are equivalent (see, e.g., [48]). However, in dealing with polytopic uncertain systems, the method in [57] is less conservative than those in [145] and [151]. All the above mentioned techniques rely mainly on the bounding methods, and they can result in strict LMI stability criteria for time-delay systems. However, when applying to solving stabilization problems, the above techniques usually fail to lead to consistent LMI stabilizing control designs. Optimization algorithms or conservative matrix transformations have to be involved to achieve the LMIbased design purpose. See, e.g., [32][105][144] for details. Anyway, these methods can be adopted in the design of state feedback stabilizing fuzzy controllers for time-delay fuzzy systems. See, e.g., [42] that adopts the bounding method in (4.23). If the time-delay is not a smooth function of time, we just need to modify the functional of (4.22) to

0 t V (xt ) = x(t)T P x(t) + x(s) ˙ T Z x(s)dsdθ. ˙ (4.27) −τ

t+θ

Then, the stabilizing fuzzy control design procedure can be carried out using methods analogous to the above mentioned techniques. 4.3.3

Choosing Appropriate Functionals

As discussed in the last subsection, a better method or bounding technique can lead to less conservative LMI conditions to test stability. But, it usually produces non-strict LMI conditions to design controllers. In fact, the control design usually encounters solving bilinear matrix inequalities (BMI). See, e.g., [133] for discussions on BMIs. Although BMI methods are also popular in recent control applications, approaches relying on BMIs beyond the scope of this book and will be not discussed here. While the technique used is essential in reducing conservatism, the choice of Lyapunov-Krasovskii functionals also plays an important role in the analysis and synthesis of T-S fuzzy time-delay systems. We would like to stress that in general it is hard to theoretically compare the results obtained using diﬀerent methods and adopting diﬀerent functionals. What practitioners can do is partial comparisons or numerical comparisons. The frequently used functional is of the quadratic form (4.22) which enables us to establish delay-dependent LMI conditions. In the following, we demonstrate how to use an alternative functional to derive consistent LMI conditions for the stabilizing controller design. Let us consider fuzzy model (4.2) without delayed-input terms (i.e., Ei = 0). Assume the delay is constant (i.e., τ (t) = τ = τ0 ). The aim is to determine the local feedback gains Fi such that the state feedback control law

48

4. Stabilization Methods for T-S Fuzzy Systems

u(t) = −

r

hi (θ)Fi x(t),

(4.28)

i=1

stabilizes the closed-loop system x(t) ˙ =

r r

hi (θ)hj (θ)[(Ai − Bi Fj )x(t) + Di x(t − τ )],

(4.29)

i=1 j=1

x(t) = φ(t), t ∈ [−τ0 , 0]. Let us choose the Lyapunov-Krasovskii functional to be: V (xt ) = V1 + V2 + V3 + V4 , where V1 = xT (t)P x(t),

0 t r ¯j x(s)dsdα, hj (s)xT (s)U V2 = −τ

V3 =

0

−τ

t+α j=1 r t

hj (s)xT (s − τ )T¯j x(s − τ )dsdα,

t+α j=1

t

¯ xT (α)Sx(α)dα,

V4 = t−τ

and where P = Q−1 , S¯ = Q−1 SQ−1 , U¯i = Q−1 Ui Q−1 , T¯i = Q−1 Ti Q−1 , and Q > 0, S > 0, Ui ≥ 0, and Ti ≥ 0 are to be chosen, i = 1, 2, . . . , r. Since

t r r T ˙ hi hj [(Gij + Di )x(t) − Di x(s)ds] ˙ V1 = 2x (t)P t−τ

i=1 j=1

=

r r

hi hj xT (t)[P (Gij + Di ) + (Gij + Di )T P ]x(t)

i=1 j=1 r r

−

t

2xT (t)P Di

hi hj

i=1 j=1 r r

×

t−τ

hk (s)hl (s)(Gkl x(s) + Dk x(s − τ ))ds,

k=1 l=1

where Gij := Ai − Bi Fj , using the bounding method in (4.25), we have V˙ (xt ) ≤

r r

hi hj xT (t)[P (Gij + Di ) + (Gij + Di )T P

i=1 j=1

¯ ¯i + S]x(t) +τ0 P Di (Xi + Yi )DiT P + τ0 U r hi xT (t − τ )(−S¯ + τ0 T¯i )x(t − τ ) + i=1

4.4 Illustrative Examples

+

r

hi

i=1

+

r i=1

hi

t

r r

49

¯k + GTkl X −1 Gkl )x(s)ds hk (s)hl (s)xT (s)(−U i

t−τ k=1 l=1

t r

hk (s)xT (s − τ )(−T¯k + DkT Yi−1 Dk )x(s − τ )ds.

t−τ k=1

After some manipulations and matrix transformations, we can arrive at that the closed-loop fuzzy system (4.29) is asymptotically stable if there exist matrices Q > 0, S > 0, Xi > 0, Yi > 0, Ui ≥ 0, Ti ≥ 0 and Zi , i = 1, 2, . . . , r, such that the following LMI’s hold: (Ai + Di )Q + Q(Ai + Di )T − Bi Zi − ZiT BiT +S + τ0 Ui + τ0 Di (Xi + Yi )DiT < 0,

(4.30)

T

(Ak + Al + Dk + Dl )Q + Q(Ak + Al + Dk + Dl ) −Bk Zl − ZlT BkT − Bl Zk − ZkT BlT + 2S + τ0 (Uk + Ul ) +τ0 Dk (Xk + Yk )DkT + τ0 Dl (Xl + Yl )DlT ≤ 0, −S + τ0 Ti ≤ ⎤ ⎡ QATj − ZjT BjT −Uj ⎦≤ ⎣ Aj Q − Bj Zj −Xi ⎤ ⎡ −Uk − Ul QATk − ZlT BkT QATl − ZkT BlT ⎥ ⎢ ⎥ ⎢ ⎥≤ ⎢ Ak Q − Bk Zl −Xi 0 ⎦ ⎣ 0 −Xi Al Q − Bl Zk ⎡ ⎤ −Tj QDjT ⎣ ⎦≤ Dj Q −Yi

(4.31)

0,

(4.32)

0,

(4.33)

0,

(4.34)

0,

(4.35)

for i, j = 1, 2, . . . , r and k < l, k, l = 1, 2, . . . , r. If this is the case, the local feedback gains Fi are given by Fi = Zi Q−1 ,

i = 1, 2, . . . , r.

(4.36)

There are 12 r(r + 1)(r + 3) LMI’s in (4.30)-(4.35). If Bi = B, i = 1, 2, . . . , r, the number of LMIs reduces to 2r(r + 1), involving those of (4.30), (4.32), (4.33) and (4.35). Furthermore, if Xi and Yi are also chosen common matrices, respectively, i.e., Xi = X and Yi = Y , i = 1, 2, . . . , r, only 4r LMI’s are left to be solved.

4.4 Illustrative Examples In this section, we use examples to illustrate the use of the present results in Section 3.2. The ﬁrst example is taken from [10], which is used therein to illustrate the T-S fuzzy control method for the backing up control of a continuous-time trucktrailer model with time-delay. We directly adopt the fuzzy model for our use.

50

4. Stabilization Methods for T-S Fuzzy Systems

Example 4.4.1. Consider the following fuzzy model: x(t) ˙ =

2

hi (Ai x(t) + Di x(t − τi (t)) + Bi u(t)),

(4.37)

i=1

where

⎡

¯

vt −a Lt 0 0 0

⎤

⎢ ⎥ ⎢ vt¯ ⎥ A1 = ⎢ a Lt ⎥, 0 0 0 ⎣ ⎦ 2 ¯2 ¯ v t vt 0 a 2Lt t 0 0 ⎡

¯

vt −a Lt 0 0 0

⎤

⎢ ⎥ ⎢ vt¯ ⎥ A2 = ⎢ a Lt ⎥, 0 0 0 ⎣ ⎦ 2 ¯2 ¯ dv t dv t a 2Lt0 t0 0

⎡

¯

vt −(1 − a) Lt 00 0

⎢ ⎥ ⎢ ⎥ v t¯ D1 = ⎢ (1 − a) Lt ⎥, 0 0 0 ⎣ ⎦ 2 ¯2 v t 0 0 (1 − a) 2Lt 0 ⎡

¯

⎡

⎤

vt −(1 − a) Lt 00 0

⎤

⎢ ⎥ ⎢ ⎥ B1 = B = ⎢ 0 ⎥ , ⎣ ⎦ 0 ⎡

⎤

⎢ ⎥ ⎢ ⎥ v t¯ D2 = ⎢ (1 − a) Lt ⎥, 0 0 0 ⎣ ⎦ 2 ¯2 dv t (1 − a) 2Lt0 0 0

v t¯ lt0

v t¯ lt0

⎤

⎢ ⎥ ⎢ ⎥ B2 = B = ⎢ 0 ⎥ , ⎣ ⎦ 0

with l = 2.8, L = 5.5, v = −1.0, t¯ = 2.0, t0 = 0.5, d = 10t0 /π and a = 0.7. Now, suppose that there are perturbed delayed-inputs in system (4.37) and the overall model is given as x(t) ˙ =

2

hi [Ai x(t) + Di x(t − τi (t)) + Bi u(t) + Ei u(t − τi (t))],

(4.38)

i=1

where Ei = 10%Bi , i = 1, 2. Next, we show the eﬀectiveness of our result in 1 Theorem 4.2.1. For simulation purpose, we simply choose h1 (x) = 1+exp(x 1 +0.5) and h2 (x) = 1 − h1 (x) where x(t) = [x1 (t), x2 (t), x3 (t)]T . Also suppose the time-delay is a constant given by τi (t) = τ = 0.1. It is seen that the unforced open-loop system of (4.38) is not stable. Since Bi = B and Ei = 0.1B, by Remark 4.2.2, there are eighteen LMI’s to be solved. Using LMI Toolbox, it is computed that there is a set of feasible solutions with ⎡ ⎤ 6.0676 1.4459 −1.7207 ⎢ ⎥ ⎢ ⎥ Q = ⎢ 1.4459 0.9162 3.3196 ⎥ , ⎣ ⎦ −1.7207 3.3196 33.9657 Y1 = −7.6858 0.1648 7.3117 , Y2 = −6.8677 0.5504 7.9082 . By Theorem 4.2.1, we conclude that the fuzzy model (4.38) is stabilizable and the stabilizing fuzzy controller is given by u(t) = −

2

hi Fi x(t),

(4.39)

i=1

where the local state feedback gains Fi are computed according to (4.12) as F1 = −3.6857 9.1238 −0.8632 , F2 = −3.7138 9.7537 −0.9086 .

4.4 Illustrative Examples

51

The response of the closed-loop system (4.38) with control law (4.39) is shown in Figure 4.1(a) under initial condition x(σ) = φ(σ) = [4, − 1, 2]T , σ ∈ [−0.1, 0]. The response of controller is shown in Figure 4.1(b). It is seen from Figure 4.1(a) that the closed-loop fuzzy system is asymptotically stable. Next, we examine an example without delayed-input terms. We will use this example to illustrate the eﬀectiveness and merits of the controller design method in Corollary 4.2.1. Example 4.4.2. Consider the following fuzzy system: x(t) ˙ =

2

hi (Ai x(t) + Di x(t − τi (t)) + Bi u(t)),

(4.40)

i=1

with

⎡

⎡ ⎤ 0 ⎦ , A2 = ⎣ ⎦ , B1 = B2 = ⎣ ⎦ , A1 = ⎣ 0.4 1 0 −2 1 ⎡ ⎡ ⎤ ⎤ −0.8 −0.1 −0.6 0 ⎦ , D2 = ⎣ ⎦. D1 = ⎣ 0 −0.3 −0.2 −0.5 ⎤

0 0

⎡

−1 0.3

⎤

The time-delays are chosen to be τ1 (t) = τ2 (t) = 0.7 sin2 (x1 (t) + π/3) where x(t) = [x1 (t), x2 (t)]T . Then, τi (t) ≤ τ = 0.7. The membership functions are selected as h1 = sin2 (x2 ) and h2 = cos2 (x2 ). It is seen that this fuzzy model has the same form as studied in [10][158]. However, it has been veriﬁed that all stabilizing fuzzy control design methods in [10][158] fail to work. This is because, by applying to this example, the delayindependent LMI conditions in Theorem 2 of [10] and Theorem 3 of [158] cannot produce a feasible solution, and the algebraic inequality conditions in Theorem 4 of [158] are not satisﬁed. Now, we use our delay-dependent result in Corollary 4.2.1 (because of no delayed-input terms) to examine the stabilizability of the fuzzy model (4.40). Since B1 = B2 , by Remark 4.2.2, we have to solve 10 LMI’s in using Corollary 4.2.1. It is computed that there is a set of feasible solutions to these LMI’s with ⎡ ⎤ 0.9813 2.0596 ⎦, Q=⎣ 2.0596 5.4497 Y1 = 2.6632 8.9426 , Y2 = −4.1264 −8.5287 . Hence, we conclude that this fuzzy model is stabilizable and the stabilizing fuzzy controller is given by u(t) = −

2 i=1

hi Fi x(t),

(4.41)

52

4. Stabilization Methods for T-S Fuzzy Systems

where the local state feedback gains Fi are computed according to (4.19) as F1 = −3.5304 2.9752 , F2 = −4.4506 0.1170 . The simulation result is depicted in Figure 4.2. The response of the closed-loop system with control law (4.41) is shown in Figure 4.2(a) under initial condition x(σ) = φ(σ) = [−0.5, 1]T , σ ∈ [−0.7, 0]. The response of the stabilizing controller is shown in Figure 4.2(b). It is seen from Figure 4.2(a) that the closed-loop fuzzy system is asymptotically stable. 8

x (t) 1 x (t) 2 x (t) 3

6

x(t)

4

2

0

−2

−4

0

1

2

3 Time t

4

5

6

(a) 30

25

u(t)

20

u(t)

15

10

5

0

−5

−10

0

1

2

3 Time t

4

5

6

(b) Fig. 4.1. Response of closed-loop system (4.38) under fuzzy controller (4.39)

4.5 Conclusion

53

1

x (t) 1 x (t) 2

x(t)

0.5

0

−0.5

0

0.5

1

1.5

2

2.5 Time t

3

3.5

4

4.5

5

(a) 0

−0.5

−1

u(t)

u(t)

−1.5

−2

−2.5

−3

−3.5

−4

0

0.5

1

1.5

2

2.5 Time t

3

3.5

4

4.5

5

(b) Fig. 4.2. Response of the closed-loop system in Example 4.4.2 under fuzzy controller (4.41)

4.5 Conclusion In this chapter, we have studied the delay-dependent stabilization methods for T-S fuzzy systems with time-delay. We focus on the state feedback stabilizing controller design using Razumikhin Theorem. The bounded time-varying delays are assumed to be in both the state and the control input. We also discuss other stabilization methods for time-delay T-S fuzzy systems. The advantages and eﬀectiveness of the present results have been illustrated by several examples. In general, the disadvantage of the results lies in that a large number of LMI’s is involved when the number of IF-THEN rules, r, is big. It is a challenging work to seek more eﬀective methods to reduce such computational burdens.

5. Variable Structure Control for T-S Fuzzy Systems

This chapter aims to study the stabilization of uncertain fuzzy time-delay systems in Takagi-Sugeno fuzzy model via variable structure control (VSC) approach. Both cases of matched uncertainties and mismatched uncertainties in the state variables are considered. We will use appropriate Lyapunov functionals and Razumikhin Theorem [44] to solve the stabilization problem. We will present various control schemes in terms of LMIs for the VSC design.

5.1 Introduction In practice, the inevitable uncertainties may enter a nonlinear system in a much more complex way. The uncertainty may include modeling error, parameter perturbations, fuzzy approximation errors, and external disturbances. In such circumstances, especially in the existence of external disturbances, the above established methods to control fuzzy time-delay systems could not work well anymore. However, we point out that the variable structure control (VSC) [20] is a reasonable approach to take eﬀect if the lumped uncertainties are known to be bounded by smooth functions. This is owing to that the plant is intentionally changed under the VSC to achieve a desired behavior. In a more detail, the VSC system could drive the trajectories onto the so-called sliding surface (or switching surface) in a ﬁnite time and maintain on it thereafter, and on the sliding surface the system is insensitive to internal parameter perturbations and external disturbances. VSC approach has been successfully adopted in the control of time-delay systems these years [26][41][173]. Depending on the nature of the delayed system, the so-called switching manifold [26][173] other than the traditional switching surface [41] have to be designed. Quite recently, VSC approach has been also applied to solve the stabilization and tracking problems for fuzzy systems with matched uncertainties [174]. In this chapter, we study a T-S fuzzy time-delay model with parameter perturbations and external disturbances. We extend the stabilization results in [174] to time-delay fuzzy systems. The type of time-delay is assumed to be smooth or non-smooth. Moreover, compared with [174], not only matched uncertainties but also mismatched uncertainties in the state variables are considered in this chapter. Four VSC design methods (given in four Theorems separately) are presented to achieve the robust stabilization of the fuzzy time-delay system. These various controller design methods are presented in terms of LMIs. C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 55–73, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

56

5. Variable Structure Control for T-S Fuzzy Systems

5.2 Problem Formulation Consider a nonlinear time-delay system which could be described by the following T-S fuzzy time-delay model with r plant rules. Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Di x(t − τi (t)) + Bi u(t), x(t) = φ(t), t ∈ [−τ0 , 0],

(5.1)

where x ∈ Rn and u ∈ Rm are the state and control input, respectively; Ai , Di and Bi are constant real matrices with appropriate dimensions; θj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables (which are the functions of state variables) and the fuzzy sets. It is assumed that the premise variables are independent of the input variables u(t). τi (t) is the time delay and τ0 is the upper bound of τi (t), i = 1, 2, . . . , r; φ(t) is a continuous vector-valued initial function on [−τ0 , 0]. In this chapter we consider two diﬀerent cases for the time delay: Case I. τ˙i (t) ≤ βi < 1, i = 1, 2, . . . , r. Case II. τi (t) is continuous and bounded by τ0 , i.e., τi (t) ≤ τ0 , i = 1, 2, . . . , r. Case I means that the derivative of τi (t) is available and less than 1. If such information is not available, we have to consider Case II. By fuzzy blending, the overall fuzzy model is inferred as follows: r wi (θ)(Ai x(t) + Di x(t − τi (t)) + Bi u(t)) r x(t) ˙ = i=1 i=1 wi (θ) r hi (θ)(Ai x(t) + Di x(t − τi (t)) + Bi u(t)), (5.2) = i=1

x(t) = φ(t), t ∈ [−τ0 , 0], function where θ = [θ1 , . . . , θp ], wi : Rp → [0, 1], i = 1, . . . , r, is the membership of the system with respect to plant rule i, and hi (θ) = wi (θ)/ ri=1 wi (θ). It is obvious that r hi (θ) ≥ 0, hi (θ) = 1. i=1

The above model does not include any perturbations. If the original nonlinear system has uncertainties, it is reasonable to use a T-S fuzzy model with uncertainties to approximate the system. In this chapter, let us consider the following uncertain T-S fuzzy model which includes (5.2) as its nominal fuzzy system: x(t) ˙ =

r

hi (θ){Ai x(t) + Di x(t − τi (t)) + fi (x, t) + gi (x(t − τi ), t)

i=1

+Bi (I + ∆i (t))u(t) + di (x, x(t − τi ), t)}, x(t) = φ(t), t ∈ [−τ0 , 0],

(5.3)

5.3 Stabilization Results

57

where fi ∈ Rn and gi ∈ Rn stand for the lumped uncertainties corresponding to x and x(t − τi ), respectively, ∆i (t) ∈ Rm×m satisfying ∆i (t) ≤ δBi < 1 is the uncertainty in the input, di ∈ Rn is the external disturbance. Both cases for fi and gi satisfying matching conditions and mismatching conditions will be studied in this chapter. Let δB = max{δBi , i = 1, 2, . . . , r}. In the following, we make the following assumptions. Assumption 5.1. In the model (5.3), assume that Bi = B, i = 1, 2, . . . , r, and B has full column rank. Assumption 5.2. Assume that the disturbances di in the model (5.3) satisfy the following matching conditions: there exist di (x, x(t−τi ), t) ∈ Rm with di (x, x(t− τi ), t) ≤ ei (t) where ei (t) ≥ 0, i = 1, 2, . . . , r, are uniformly bounded continuous functions of t, such that di (x, x(t − τi ), t) = B di (x, x(t − τi ), t),

i = 1, 2, . . . , r.

(5.4)

Assumption 5.1 means that each nominal local system shares the same input channel. It is required to facilitate the design of variable structure controllers. Note that all the examples in [9][10][27][73][121][174] fall into the special class of nonlinear systems of the form x˙ = f (x(t), x(t − τ ), t) + Bu(t), which satisﬁes Assumption 5.1 after fuzzy modeling. Assumption 5.2 means that the disturbances di enter the corresponding local system through the same input channel.

5.3 Stabilization Results 5.3.1

Matched Uncertainty Case

In this subsection, we consider the case that fi and gi satisfy the following matching conditions: Assumption 5.3. There exist fi (x, t) ∈ Rm and gi (x(t − τi ), t) ∈ Rm with fi (x, t) ≤ δfi x and gi (x(t − τi ), t) ≤ δgi x(t − τi ), i = 1, 2, . . . , r, where δfi and δgi are nonnegative scalars, such that the following hold: fi (x, t) = B fi (x, t),

(5.5)

gi (x(t − τi ), t). gi (x(t − τi ), t) = B

(5.6)

Our ﬁrst result is with respect to smooth delays and matched uncertainties. Theorem 5.3.1. Consider uncertain system (5.3) with Case I and assume Assumptions 5.1, 5.2 and 5.3. If there exist matrices Q > 0, Si > 0, Xi and Yi , i = 1, 2, . . . , r, satisfying LMIs ⎡ ⎤ r Ai Q + QATi + BYi + YiT B T + i=1 Si Di Q + BXi ⎦ < 0, Li := ⎣ (5.7) QDiT + XiT B T −(1 − βi )Si i = 1, 2, . . . , r,

58

5. Variable Structure Control for T-S Fuzzy Systems

then the variable structure controller ⎧ ⎨ − r h H(A x(t) + D x(t − τ )) − i i i i=1 i u(t) = r ⎩− i=1 hi H(Ai x(t) + Di x(t − τi )),

1 1−δB (ϕ1

σ + ) σ , σ = 0,

(5.8)

σ = 0,

stabilizes system (5.3), where > 0 is a user deﬁned scalar, and H = (B T Q−1 B)−1 B T Q−1 ∈ Rm×n , σ(t) = Hx(t), r hi [(δB HAi + δfi )x(t) ϕ1 (x, x(t − τi ), t) = i=1

+(δB HDi + δgi )x(t − τi ) + ei (t)].

(5.9)

Proof. The proof is twofold: ﬁrstly, we show that the trajectory of system (5.3) under (5.8) will reach the sliding surface S: S := {x ∈ Rn : Hx = 0}

(5.10)

in a ﬁnite time and will stay on it thereafter. Then, we show that on the sliding surface S, the motion of (5.3) is asymptotically stable. Let us prove the ﬁrst part. The derivative of σ(t) along the trajectory of the closed-loop system given by (5.3) and (5.8) is σ(t) ˙ = H x(t) ˙ r = hi H(Ai x(t) + Di x(t − τi )) i=1

+

r

hi (fi + gi + di ) + u(t) +

i=1

r

hi ∆i u(t)

i=1

r r 1 σ + (I + hi ∆i )(ϕ1 + ) hi (fi + gi + di ) 1 − δB σ i=1 i=1 r r hi ∆i ) hi H(Ai x(t) + Di x(t − τi )) . −(

=−

i=1

i=1

Noticing that ϕ1 (x, x(t − τi ), t) ≥ 0 and > 0, the above gives ˙ σ T (t)σ(t)

r r hi ∆i (ϕ1 + ) ϕ1 + σ + i=1 σ + hi fi + gi + di σ ≤− 1 − δB 1 − δB i=1 r r hi ∆i ) hi (HAi x(t) + HDi x(t − τi )) σ +( i=1

i=1

δB (ϕ1 + ) ϕ1 + σ + σ + ϕ1 σ ≤− 1 − δB 1 − δB = −σ, which means that the reachability condition σ T (t)σ(t) ˙ < 0 for σ = 0 (see [20] and the references therein) is satisﬁed. This proves the ﬁrst part.

5.3 Stabilization Results

59

We now prove the second part. Let the Lyapunov-Krasovskii functional be r t T V (xt ) = x (t)P x(t) + xT (αi )S¯i x(αi )dαi , i=1

t−τi (t)

where P = Q−1 , S¯i = Q−1 Si Q−1 , i = 1, 2, . . . , r, and xt = x(t + α), α ∈ [−τ0 , 0]. Note that the above type of Lyapunov-Krasovskii functional is suitable only for Case I (since the derivative of τi (t) is available) and is not suitable for Case II (since the derivative of τi (t) may be invalid). In order to use the stability theory for time-delay systems [44], by choosing two positive constants cmin = λmin (P ) r and cmax = λmax (P ) + rτ0 i=1 λmax (S¯i ), it is seen that cmin φ2 ≤ V (φ) ≤ cmax

sup

φ(α)2 .

α∈[−τ0 ,0]

The derivative of V along the solution of (5.3) is given by V˙ (xt ) r r = 2xT (t)P x(t) ˙ + xT (t)S¯i x(t) − (1 − τ˙i (t))xT (t − τi )S¯i x(t − τi ) i=1

≤ 2x (t)P x(t) ˙ + T

=

r

i=1

x (t)S¯i x(t) − T

i=1

⎡ hi ⎣

r

⎤T ⎡

r

(1 − βi )xT (t − τi )S¯i x(t − τi )

i=1

P Ai + ATi P P Di

x(t)

⎤⎡

⎤ x(t)

⎦ ⎣ ⎦⎣ ⎦ DiT P 0 x(t − τi ) x(t − τi ) ⎤⎡ ⎤T ⎡ ⎤ ⎡ r ¯i x(t) x(t) S ⎥⎢ ⎥ ⎢ i=1 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ −(1 − β1 )S¯1 ⎥ ⎢ x(t − τ1 ) ⎥ ⎢ x(t − τ1 ) ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ +⎢ .. .. ⎥⎢ ⎥ ⎢ ⎥ .. ⎥⎢ ⎥ ⎢ ⎥ ⎢ . . . ⎦⎣ ⎦ ⎣ ⎦ ⎣ −(1 − βr )S¯r x(t − τr ) x(t − τr ) i=1

+2

r

hi xT (t)P B(fi + gi + (I + ∆i )u(t) + di )

i=1

=

r

⎡

hi ⎣

⎤T ⎡ x(t)

P (Ai + BYi P ) + (Ai + BYi P )T P ∗

⎤⎡

⎤ x(t)

⎦ ⎣ ⎦⎣ ⎦ 0 (Di + BXi P )T P x(t − τi ) x(t − τi ) ⎤⎡ ⎤T ⎡ ⎤ ⎡ r ¯i x(t) x(t) S ⎥⎢ ⎥ ⎢ i=1 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ −(1 − β1 )S¯1 ⎥ ⎢ x(t − τ1 ) ⎥ ⎢ x(t − τ1 ) ⎥ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ +⎢ .. .. ⎥⎢ ⎥ ⎢ ⎥ .. ⎥⎢ ⎥ ⎢ ⎥ ⎢ . . . ⎦⎣ ⎦ ⎣ ⎦ ⎣ ¯ −(1 − βr )Sr x(t − τr ) x(t − τr ) i=1

+2

r i=1

hi xT (t)P B(fi + gi + (I + ∆i )u(t) − Yi P x(t) − Xi P x(t − τi ) + di ),

60

5. Variable Structure Control for T-S Fuzzy Systems

where ‘∗’ denotes a symmetric block. The last equality holds for any matrices Xi and Yi with appropriate dimensions. On the switching surface (5.10), it holds 2

r

hi xT (t)P B(fi + gi + (I + ∆i )u(t) − Yi P x(t) − Xi P x(t − τi ) + di )

i=1

= 0. Therefore, if the conditions in (5.7) hold, we continue to have V˙ (xt ) ≤

r i=1

⎡ hi ⎣

⎤T ⎡ x(t) x(t − τi )

⎦ ⎣

⎤ P

⎡

⎦ Li ⎣ P

⎤

⎤⎡ P

⎦⎣ P

x(t) x(t − τi )

⎦

≤ λmax (diag {Li })λmin (P )x(t)2 . Since λmax (diag {Li })λmin (P ) < 0, the above shows the motion of (5.3) is asymptotically stable. Theorem 5.3.1 provides a stabilization result by using VSC approach. The key idea is to design an appropriate sliding surface and a stabilizing variable structure controller, and then apply the controller to drive the system trajectories onto the sliding surface on which the closed-loop system state tends to zero. In our design method, the sliding surface is of the form (5.10) which is related to matrix Q if (5.7) is feasible. And on the sliding surface the state x(t) satisﬁes B T Q−1 x = 0 which enables the controller in (5.8) to render the closed-loop system asymptotically stable. Remark 5.3.1. For the special case when the time delays in each local system are equal, i.e., τ (t) := τi (t) and thus β := βi , i = 1, 2, . . . , r, the LMIs in (5.7) can be simpliﬁed as ⎡ ⎤ Ai Q + QATi + BYi + YiT B T + S Di Q + BXi ⎣ ⎦ < 0, i = 1, 2, . . . , r, QDiT + XiT B T −(1 − β)S for matrices Q > 0, S > 0, Xi and Yi . If there is no perturbations in B, i.e., ∆i (t) = 0 and thus δB = 0, the controller in (5.8) can be simpliﬁed. If there is no time delay and there is no perturbations in B, the LMIs in (5.7) is reduced to Ai Q + QATi + BYi + YiT B T < 0,

i = 1, 2, . . . , r,

for matrices Q > 0 and Yi , and the controller in (5.8) is also reduced much. In this special case, Theorem 5.3.1 reduces to a stabilization result for standard fuzzy systems via variable structure control, which is similar to that provided in [174].

5.3 Stabilization Results

61

Next, we give a result below with respect to non-smooth delays and matched uncertainties. Theorem 5.3.2. Consider uncertain system (5.3) with Case II and assume Assumptions 5.1, 5.2 and 5.3. If there exist matrices Q > 0, Si > 0, Xi and Yi , i = 1, 2, . . . , r, satisfying LMIs ⎡ ⎤ Ai Q + QATi + BYi + YiT B T + Q Di Q + BXi ⎣ ⎦ < 0, (5.11) QDiT + XiT B T −Si Q ≥ Si , i = 1, 2, . . . , r,

(5.12)

then the variable structure controller of the same form as in (5.8) stabilizes system (5.3). Proof. The proof is also twofold: the ﬁrst part of the proof, which shows that the trajectory of system (5.3) under the variable structure controller as in (5.8) will reach the sliding surface deﬁned in (5.10) in a ﬁnite time and will stay on it thereafter, remains the same as that for Theorem 5.3.1. Next we show the second part, which proves that on the sliding surface, the motion of (5.3) is asymptotically stable. Let P = Q−1 and let the Lyapunov function be V (x(t)) = xT (t)P x(t). We will use a method similar to that adopted in [10] for the asymptotic stability analysis. It is based on the Razumikhin Theorem (see Appendix, or [10] and [44]). It is seen that λmin (P )x(t)2 ≤ V (x(t)) ≤ λmax (P )x(t)2 . The derivative of V along the solution of (5.3) is given by V˙ (x(t)) =

r

hi [xT (t)(P Ai + ATi P )x(t) + 2xT (t)P Di x(t − τi (t))]

i=1

+2

r

hi xT (t)P B(fi + gi + (I + ∆i )u(t) + di )

i=1

=

r

hi [xT (t)(P (Ai + BYi P ) + (Ai + BYi P )T P )x(t)

i=1

+2xT (t)(P (Di + BXi P )x(t − τi )] r hi xT (t)P B(fi + gi + (I + ∆i )u(t) +2 i=1

−Yi P x(t) − Xi P x(t − τi ) + di ).

62

5. Variable Structure Control for T-S Fuzzy Systems

On the switching surface in (5.10), we have V˙ (x(t)) =

r

hi [xT (t)(P (Ai + BYi P ) + (Ai + BYi P )T P )x(t)

i=1

+2xT (t)P (Di + BXi P )x(t − τi )] ≤

r

hi [xT (t)Ωi x(t) + xT (t − τi )P Si P x(t − τi )],

i=1

where Ωi = P (Ai + BYi P ) + (Ai + BYi P )T P +P (Di + BXi P )(P Si P )−1 (Di + BXi P )T P. Note that the conditions in (5.11) and (5.12) are equivalent to Ωi + P < 0, P Si P ≤ P, for i = 1, 2, . . . , r, which gives that Ωi + (1 + δ)P < 0, P Si P ≤ P, for a suﬃciently small scalar δ > 0. Hence, by using Razumikhin Theorem, if V (x(t − α)) < (1 + δ)V (x) for α ∈ [−τ0 , 0], then we have V˙ (x) ≤

r

hi [xT (t)Ωi x(t) + xT (t − τi )P x(t − τi )]

i=1

<

r

hi [xT (t)Ωi x(t) + xT (t)(1 + δ)P x(t)]

i=1

≤ max{λmax (Ωi + (1 + δ)P ), i = 1, 2, . . . , r}x(t). Since max{λmax (Ωi +(1+δ)P ), i = 1, 2, . . . , r} < 0, the above shows the motion of (5.3) is asymptotically stable. 5.3.2

Mismatched Uncertainty Case

In this subsection, we consider the case that fi and gi satisfy the following assumption: Assumption 5.4. There exist nonnegative scalars ρfi and ρgi , i = 1, 2, . . . , r, such that the following hold: fi (x, t) ≤ ρfi x(t), gi (x(t − τi ), t) ≤ ρgi x(t − τi ),

i = 1, 2, . . . , r.

(5.13)

Under the above mismatched uncertainties and smooth delays, we have the following result.

5.3 Stabilization Results

63

Theorem 5.3.3. Consider uncertain system (5.3) with Case I and assume Assumptions 5.1, 5.2 and 5.4. If there exist matrices Q > 0, Si > 0, Xi , Yi and scalars i > 0, i = 1, 2, . . . , r, satisfying LMIs ⎤ ⎡ 1/2 Ξi Di Q + BXi ρfi Q 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ QDiT + XiT B T −(1 − βi )Si 0 ρ1/2 Q gi ⎥ < 0, i = 1, 2, . . . , r, ⎢ (5.14) ⎥ ⎢ 1/2 ⎢ 0 −i I 0 ⎥ ρfi Q ⎦ ⎣ 1/2 0 −i I 0 ρgi Q where Ξi = Ai Q + QATi + BYi + YiT B T +

r

Si + i I,

i=1

then the variable structure controller ⎧ ⎨ − r h H(A x(t) + D x(t − τ )) − i i i i=1 i u(t) = ⎩ − r hi H(Ai x(t) + Di x(t − τi )),

1 1−δB (ϕ3

σ + ) σ , σ = 0,

i=1

(5.15)

σ = 0,

stabilizes system (5.3), where > 0 is a user deﬁned scalar, and H = (B T Q−1 B)−1 B T Q−1 ∈ Rm×n , σ(t) = Hx(t), r hi [(δB HAi + ρfi H)x(t) ϕ3 (x, x(t − τi ), t) = i=1

+(δB HDi + ρgi H)x(t − τi ) + ei (t)].

(5.16)

Proof. The proof is twofold: ﬁrstly, we show that the trajectory of system (5.3) under (5.15) will reach the sliding surface S: S := {x ∈ Rn : Hx = 0}

(5.17)

in a ﬁnite time and will stay on it thereafter. Then, we show that on the sliding surface S, the motion of (5.3) is asymptotically stable. Let us prove the ﬁrst part. The derivative of σ(t) along the trajectory of the closed-loop system given by (5.3) and (5.15) is σ(t) ˙ = H x(t) ˙ r = hi H(Ai x(t) + Di x(t − τi ) + fi + gi ) i=1

+

r i=1

hi di + u(t) +

r i=1

hi ∆i u(t)

64

5. Variable Structure Control for T-S Fuzzy Systems

1 σ (I + hi ∆i )(ϕ3 + ) 1 − δB σ i=1 r r −( hi ∆i ) hi H(Ai x(t) + Di x(t − τi )) r

=−

i=1

+

r

i=1

hi H(fi + gi ) +

i=1

r

hi di .

i=1

Noticing that ϕ3 (x, x(t − τi ), t) ≥ 0 and > 0, the above gives ϕ3 + δB (ϕ3 + ) σ + σ + ϕ3 σ 1 − δB 1 − δB = −σ,

σ T (t)σ(t) ˙ ≤−

which means that the reachability condition is satisﬁed. This proves the ﬁrst part. We now prove the second part. Let the Lyapunov-Krasovskii functional be r t V (xt ) = xT (t)P x(t) + xT (αi )Si x(αi )dαi , i=1

t−τi (t)

where P = Q−1 , S¯i = Q−1 Si Q−1 , i = 1, 2, . . . , r, and xt = x(t + α), α ∈ [−τ0 , 0]. It is seen that r λmin (P )φ2 ≤ V (φ) ≤ (λmax (P ) + rτ0 λmax (S¯i )) sup φ(α)2 . α∈[−h,0]

i=1

Similar to the proof of Theorem 5.3.1, the derivative of V along the solution of (5.3) satisﬁes ⎡ ⎤T ⎡ ⎤ r r x(t) x(t) ⎦ Ψi ⎣ ⎦+2 V˙ (xt ) ≤ hi ⎣ hi xT (t)P (fi + gi ) x(t − τi ) x(t − τi ) i=1 i=1 +2

r

hi xT (t)P B((I + ∆i )u(t) − Yi P x(t) − Xi P x(t − τi ) + di ),

i=1

where

⎡

Ψi = ⎣

P (Ai + BYi P ) + (Ai + BYi P )T P +

r

¯ i=1 Si P (Di + BXi P )

(Di + BXi P )T P

−(1 − βi )S¯i

⎤ ⎦.

By incorporating the fact that 2xT (t)P (fi + gi ) ≤ i xT (t)P P x(t) + −1 i fi + gi

−1 T ≤ xT (t)(i P 2 + ρfi −1 i I)x(t) + ρgi i x (t − τi )x(t − τi ) ⎡ ⎤T ⎡ ⎤⎡ ⎤ x(t) I 0 i P 2 + ρfi −1 x(t) i ⎦ ⎣ ⎦⎣ ⎦ =⎣ I ) x(t − τi ) 0 ρgi −1 x(t − τ i i

holds for any scalar i > 0, we continue to have that on the switching surface (5.17),

5.3 Stabilization Results

V˙ (xt ) ≤

r i=1

where

⎡

Φi = Ψi + ⎣

⎡ hi ⎣

⎤T x(t) x(t − τi )

⎡

⎦ Φi ⎣

65

⎤ x(t) x(t − τi )

⎦,

⎤

i P 2 + ρfi −1 i I

0

0

ρgi −1 i I

⎦.

Note that by Schur complement, the LMIs in (5.14) are equivalent to that ⎡ ⎤ ⎤ ⎡ Q 0 Q 0 ⎣ ⎦ < 0, i = 1, 2, . . . , r, ⎦ Φi ⎣ 0 Q 0 Q which yields V˙ (xt ) ≤ max{λmax (Φi ), i = 1, 2, . . . , r}x(t)2 . Since max{λmax (Φi ), i = 1, 2, . . . , r} < 0, the above shows the motion of (5.3) is asymptotically stable. With respect to Assumption 5.4 and non-smooth delays, we have the following result. Theorem 5.3.4. Consider uncertain system (5.3) with Case II and assume Assumptions 5.1, 5.2 and 5.4. If there exist matrices Q > 0, Si > 0, Xi , Yi and scalars i > 0, i = 1, 2, . . . , r, satisfying LMIs ⎡ ⎤ 1/2 Ai Q + QATi + BYi + YiT B T + Q + i I Di Q + BXi ρfi Q ⎢ ⎥ ⎢ ⎥ (5.18) ⎢ −Si 0 ⎥ < 0, QDiT + XiT B T ⎣ ⎦ 1/2 0 −i I ρfi Q ⎡ ⎤ 1/2 S − Q ρgi Q ⎣ i ⎦ ≤ 0, i = 1, 2, . . . , r, (5.19) 1/2 ρgi Q −i I then the variable structure controller of the same form as in (5.15) stabilizes system (5.3). Proof. The proof is also twofold: the ﬁrst part, which is to prove that the trajectory of system (5.3) under the variable structure controller as in (5.15) will reach the sliding surface deﬁned in (5.17) in a ﬁnite time and will stay on it thereafter, remains the same as that for Theorem 5.3.3. Next we show the second part, which proves that on the sliding surface, the motion of (5.3) is asymptotically stable. Let P = Q−1 and let the Lyapunov function be V (x(t)) = xT (t)P x(t).

66

5. Variable Structure Control for T-S Fuzzy Systems

We will use the Razumikhin Theorem again. It is seen that λmin (P )x(t)2 ≤ V (x(t)) ≤ λmax (P )x(t)2 . The derivative of V along the solution of (5.3) is given by V˙ (x(t)) r hi [xT (t)(P Ai + ATi P )x(t) + 2xT (t)P Di x(t − τi (t)) = i=1

+2xT (t)P (fi + gi )] r hi xT (t)P B(u(t) + ∆i u(t) + di ) +2 i=1

=

r

hi [xT (t)(P (Ai + BYi P ) + (Ai + BYi P )T P )x(t)

i=1

+2xT (t)(P (Di + BXi P )x(t − τi ) + 2xT (t)P (fi + gi )] r hi xT (t)P B(u(t) + ∆i u(t) − Yi P x(t) − Xi P x(t − τi ) + di ). +2 i=1

Taking into account 2xT (t)P (fi + gi ) −1 T ≤ xT (t)(i P 2 + ρfi −1 i I)x(t) + ρgi i x (t − τi )x(t − τi ), for i > 0, and 2xT (t)(P (Di + BXi P )x(t − τi ) ≤ xT (t)P (Di + BXi P )(P Si P )−1 (Di + BXi P )T P x(t) +xT (t − τi )P Si P x(t − τi ), on the switching surface in (5.17), we have V˙ (x(t)) ≤

r

hi [xT (t)Υi x(t) + xT (t − τi )(P Si P + ρgi −1 i I)x(t − τi )],

i=1

where Υi = P (Ai + BYi P ) + (Ai + BYi P )T P + i P 2 + ρfi −1 i I +P (Di + BXi P )(P Si P )−1 (Di + BXi P )T P. Note that, by Schur complement, the conditions in (5.18) and (5.19) are equivalent to Υi + P < 0, P Si P + ρgi −1 i I ≤ P, for i = 1, 2, . . . , r, which gives that Υi + (1 + δ)P < 0, P Si P + ρgi −1 i I ≤ P,

5.4 Illustrative Example

67

for a suﬃciently small scalar δ > 0. Hence, by using Razumikhin Theorem, if V (x(t − α)) < (1 + δ)V (x) for α ∈ [−τ0 , 0], then we have V˙ (x) ≤

r

hi [xT (t)Υi x(t) + xT (t − τi )P x(t − τi )]

i=1

<

r

hi [xT (t)Υi x(t) + xT (t)(1 + δ)P x(t)]

i=1

≤ max{λmax (Υi + (1 + δ)P ), i = 1, 2, . . . , r}x(t). Since max{λmax (Υi + (1 + δ)P ), i = 1, 2, . . . , r} < 0, the above shows the motion of (5.3) is asymptotically stable. Remark 5.3.2. We would like to point out that the fast switching (or say the chattering) behavior often occurs in using VSC approach, see [26][41][173][174] and the examples therein. Therefore, this behavior is also inevitable in our synthesis methods due to incorporation of VSC approach. However, the chattering phenomenon could be reduced by introducing a small boundary layer around the switching surface S as mentioned in Section 4.1 of [26]. For instance, the controller in (5.8) could be replaced by ⎧ ⎨ − r h H(A x(t) + D x(t − τ )) − 1 (ϕ + ) σ , σ ≥ ε, i i i 1 i=1 i 1−δB σ u ˜(t) = ⎩ − r h H(A x(t) + D x(t − τ )), σ < ε, i=1

i

i

i

i

(5.20) where ε is a small positive scalar.

5.4 Illustrative Example To illustrate the use of the results in this chapter, we consider an example of a continuous stirred tank reactor (CSTR). We depart directly from the following model, which is taken from [9]. x˙ 1 (t) = f1 (x) + 0.25x1 (t − τ ), x˙ 2 (t) = f2 (x) + 0.25x2 (t − τ ) + 0.3u(t), where x = [x1 , x2 ]T with x1 and x2 corresponding respectively to the conversion rate of the reaction and the dimensionless temperature, τ is related to the recycle delay time, and x2 f1 (x) = −1.25x1 (t) + 0.072(1 − x1 ) exp( ), 1 + x2 /20 x2 f2 (x) = −1.55x2 (t) + 0.576(1 − x1 ) exp( ), 1 + x2 /20 xi (t) = φi , t ∈ [−τ, 0]. For detailed explanations, see [9]. There are three steady states for u = 0. With respect to the expected operating point (xd , ud ), an approximation to the above

68

5. Variable Structure Control for T-S Fuzzy Systems

system is given in [9], using the Teixeira-Zak’s method [126], by the following three IF-THEN rules: Rule 1: IF the temperature is low (i.e., x2 (t) is about 0.8862), THEN z(t) ˙ = A1 z(t) + D1 z(t − τ ) + B1 v(t); Rule 2: IF the temperature is middle (i.e., x2 (t) is about 2.7520), THEN z(t) ˙ = A2 z(t) + D2 z(t − τ ) + B2 v(t); Rule 3: IF the temperature is high (i.e., x2 (t) is about 4.7052), THEN z(t) ˙ = A3 z(t) + D3 z(t − τ ) + B3 v(t), where z(·) := [z1 (·), z2 (·)]T = x(·) − xd , v(t) = u(t) − ud , and ⎡ ⎡ ⎤ ⎤ −1.4274 0.0757 −2.0508 0.3958 ⎦ , A2 = ⎣ ⎦, A1 = ⎣ −1.4189 −0.9442 −6.4066 1.6168 ⎡ ⎡ ⎤ ⎤ −4.5279 0.3167 0.25 0 ⎦ , D1 = D2 = D3 = ⎣ ⎦, A3 = ⎣ −26.2228 0.9837 0 0.25 ⎡ ⎤ 0 ⎦ := B. B1 = B2 = B3 = ⎣ 0.3 The membership functions are selected as ⎧ ⎪ ⎪ 1 if x2 ≤ 0.8862 ⎪ ⎨ x −0.8862 2 w1 = 1 − 2.7520−0.8862 if 0.8862 < x2 < 2.7520 , ⎪ ⎪ ⎪ ⎩0 if x2 ≥ 2.7520 ⎧ ⎨ 1 − w if x ≤ 2.7520 1 2 , w2 = ⎩ 1 − w if x ≥ 2.7520 3 2 ⎧ ⎪ ⎪ 0 if x2 ≤ 2.7520 ⎪ ⎨ x2 −2.7520 w3 = 4.7052−2.7520 if 2.7520 < x2 < 4.7052 , ⎪ ⎪ ⎪ ⎩1 if x ≥ 4.7052 2

which gives that hi = wi , i = 1, 2, 3. Now, we allow modeling and approximation errors, parameter perturbations and external disturbances in the above fuzzy model. Suppose the overall uncertain fuzzy model is given by z(t) ˙ =

3

hi {Ai z(t) + Di z(t − τi (t)) + fi (z, t) + gi (z(t − τi ), t)

i=1

+Bi (I + ∆i (t))v(t) + di (z, z(t − τi ), t)}, z(t) = φ(t), t ∈ [−τ0 , 0],

(5.21)

5.4 Illustrative Example

69

where fi and gi are lumped uncertainties corresponding to z(t) and z(t − τi ), respectively, ∆i (t) is the uncertainty caused from the input, satisfying ∆1 (t) = 0.1 sin(t),

∆2 (t) = 0.2 sin(t),

∆3 (t) = 0.1 cos(t),

di is the external disturbance satisfying di = B di with d1 = 0.1 sin(t),

d2 = 0.03 cos(t),

d3 = 0.05 sin(t).

Then, δB = 0.2,

e1 (t) = 0.1| sin(t)|,

e2 (t) = 0.03| cos(t)|,

e3 (t) = 0.05| sin(t)|.

Due to the incorporation of parameter perturbations and external disturbances, the control methods in [9][10][27][126][158] do not work anymore. The stabilization result in [174] is also invalid due to the time-delay and the possible mismatched uncertainties. In the following, we consider two diﬀerent cases, and show that our results work well to achieve the stabilization of system (5.21). Case 1: The recycle delay time is known and τi = τ is a constant. Suppose that τ = 2, fi and gi satisfy Assumption 5.3 with fi (x, t) = 0.1 sin(z2 (t)),

gi (x(t − τ ), t) = 0.06 sin(z2 (t − τ )),

i = 1, 2, 3.

Now, we use Theorem 5.3.1 to verify if the system is stabilizable via variable structure controllers. One set of feasible solutions to LMIs (5.7) is computed to be ⎡ ⎡ ⎤ ⎤T 0.6224 0.0546 −0.0455 ⎦ , X1 = X 2 = X 3 = ⎣ ⎦ , Q=⎣ 0.0546 0.9691 −0.8076 ⎡ ⎡ ⎤ ⎤ 0.5354 −0.0016 0.5346 −0.0012 ⎦ , S2 = ⎣ ⎦, S1 = ⎣ −0.0016 0.9846 −0.0012 0.9844 ⎡ ⎡ ⎤ ⎤T 0.5317 0.0005 3.1388 ⎦ , Y1 = ⎣ ⎦ , S3 = ⎣ 0.0005 0.9834 −3.2537 ⎡ ⎡ ⎤T ⎤T 12.1004 54.0365 ⎦ , Y3 = ⎣ ⎦ . Y2 = ⎣ −10.6189 −4.9665 Hence, from Theorem 5.3.1, the fuzzy system (5.21) is stabilizable under controller of the form (5.8) which is given by ⎧ ⎨ − 3 h M z(t) − M z(t − 2) − 1 (ϕ + ) σ , σ = 0, d 1 i=1 i i 1−δB σ v(t) = (5.22) ⎩ − 3 hi Mi z(t) − Md z(t − 2), σ = 0, i=1

70

5. Variable Structure Control for T-S Fuzzy Systems

where M1 = −4.3122 −3.1695 , M2 = −20.7556 5.2736 , M3 = −86.0851 3.1864 , Md = −0.0731 0.8333 , ϕ1 =

3

hi [(δB Mi + δfi )z(t) + (δB Md + δgi )z(t − 2) + ei (t)],

i=1

σ(t) = [−0.2925 3.3333]z(t), δfi = 0.1,

δgi = 0.06,

i = 1, 2, 3,

and is a positive scalar. Figure 5.1 shows the convergence process of z(t) → 0 under controller (5.22) for = 2 and initial condition φ(0) = [−2, 3]T . It reveals that the fuzzy controller works well to drive the modeled fuzzy system to approach zero state. When there is no uncertainties, the comparison of closed-loop responses of z(t) with x(t) − xd is shown in Figure 5.2. Here, xd = [0.1440, 0.8862]T is a steady state when u = 0. It is seen from Figure 5.2 that the trajectory of x(t)−xd goes apart from its fuzzy modeling variable z(t) in the very beginning and after a few seconds it goes towards z(t) and tends to zero. This indicates that the trajectory of the original closed-loop variable x(t) tends to its steady state xd . The large deviation of x(t) − xd and z(t) at the beginning is due to the modeling error provided in [9] by using the Teixeira-Zak’s method [126]. The modeling error contains the exponential part of the form exp( 1+xx22/20 ) which aﬀects system trajectories under circumstance of x2 = 0 and time-delay τ = 2. However, for this example, the modeling error does not aﬀect the control purpose, i.e., the closed-loop trajectory of x(t) − xd will follow z(t) and tend to zero, as shown in Figure 5.2. Case 2: The derivative of the recycle delay time is not available and only an upper bound of τ is known. Suppose that an upper bound of τi (t) is τ0 = 2, fi and gi satisfy Assumption 5.4 with gi (x(t − τ ), t) = 0.15 sin(z(t − τ )),

fi (x, t) = 0.1 sin(z(t)),

i = 1, 2, 3,

giving ρfi = 0.1, ρgi = 0.15, i = 1, 2, 3. Now, we use Theorem 5.3.4 to verify if the system is stabilizable via variable structure controllers. One set of feasible solutions to LMIs (5.18) is computed to be ⎡ Q=⎣

⎤ 9.1837 0.3888

⎡

⎤ 5.7694 0.1568

⎦ , S1 = ⎣ ⎦, 0.3888 17.2634 0.1568 8.5789 ⎡ ⎡ ⎤ ⎤ 5.7700 0.1555 5.7368 0.1563 ⎦ , S3 = ⎣ ⎦, S2 = ⎣ 0.1555 8.6717 0.1563 8.7295

5.5 Conclusion

⎡ X1 = ⎣

⎡

⎤T

−0.3230

−0.3221

⎤T

⎦ , X2 = ⎣ ⎦ , −14.3862 −14.3862 ⎡ ⎡ ⎤T ⎤T 40.7481 172.4858 ⎦ , Y2 = ⎣ ⎦ , Y1 = ⎣ −17.7607 −163.9241 1 = 11.4835,

2 = 14.9787,

⎡ X3 = ⎣ ⎡ Y3 = ⎣

−0.3049 −14.3870

71

⎤T ⎦ ,

787.3871 −105.0909

⎤T ⎦ ,

3 = 17.1598.

Hence, from Theorem 5.3.4, the fuzzy system (5.21) is stabilizable under controller of the form (5.15) which is given by ⎧ ⎨ − 3 h (F z(t) + F z(t − τ )) − 1 (ϕ + ) σ , σ = 0, i d i 3 i=1 i 1−δB σ v(t) = (5.23) 3 ⎩− h (F z(t) + F z(t − τ )), σ = 0, i=1

i

i

d

i

where

F1 = −4.5282 −3.1580 , F2 = −21.0659 5.3335 , F3 = −86.7704 3.2343 , Fd = −0.0353 0.8333 , σ(t) = F z(t), F = −0.1411 3.3333 , ϕ3 =

3

hi [(δB Fi + ρfi F )z(t)

i=1

+(δB Fd + ρgi F )z(t − τi ) + ei (t)], and is a positive scalar. Figure 5.3 shows the convergence process of z(t) → 0 under controller (5.23) for = 1 and initial condition φ(0) = [−2.5, 2.5]T . It reveals that the fuzzy controller drives the modeled fuzzy system to approach zero state. When there is no uncertainties, the comparison of closed-loop responses of z(t) with x(t) − xd is shown in Figure 5.4. Also, xd = [0.1440, 0.8862]T is a steady state when u = 0. It is seen from Figure 5.4 that the trajectory of x(t) − xd goes apart from its fuzzy modeling variable z(t) in the beginning (due to the reasons similar to case 1) and then goes towards z(t) and tends to zero. This shows that the trajectory of the original closed-loop variable x(t) tends to its steady state xd .

5.5 Conclusion In this chapter, we have studied the stabilization problem for uncertain T-S fuzzy time-delay systems in the presence of parameter perturbations and external disturbances. The uncertainties are assumed to be matched or mismatched in the state variables and the time-delay is assumed to be smooth or non-smooth. Various design methods of variable structure controllers are presented to achieve the robust stabilization purpose. The suﬃcient conditions are given in terms of LMIs which are easy to solve. An illustrative CSTR example is given to show the use and eﬀectiveness of the present results.

72

5. Variable Structure Control for T-S Fuzzy Systems

3

z1(t) z2(t)

2

1

0

−1

−2

4

3 Time t

2

1

0

5

6

5

6

(a) 40

v(t)

20

0

−20

−40

−60

−80 0

1

2

3 Time t

4

(b) Fig. 5.1. Response of system (5.21) with case 1 under controller (5.22)

3

Z2 2

Z1

1

0

−1

X1−0.1440 X2−0.8862

−2

−3

0

2

4

6

10

8

12

14

16

18

Time t

Fig. 5.2. Response of original closed-loop system corresponding to case 1

5.5 Conclusion 3

z1(t) z2(t)

2

1

0

−1

−2

−3

0

1

2

3

4

5 Time t

6

7

8

9

10

8

9

10

(a) 60

v(t)

40

20

0

−20

−40

−60

−80 4

3

2

1

0

5 Time t

6

7

(b) Fig. 5.3. Response of system (5.21) with case 2 under controller (5.23)

3

Z2 2

Z1 1

0

X1−0.1440

−1

X2−0.8862

−2

−3

0

2

4

6

8

10 Time t

12

14

16

18

20

Fig. 5.4. Response of original closed-loop system corresponding to case 2

73

6. Observer-Based Fuzzy Control: Delay-Independent Method

In case of unavailability of the states, fuzzy observers play an important role in analysis and synthesis of fuzzy systems. In this chapter, we will study the observer-based fuzzy control problem for T-S fuzzy systems with time delay. It presents a single-step LMI method for the fuzzy control design, which overcomes the drawback of the two-step LMI approach often encountered in the literature. The derivation relies mainly on a matrix decoupling technique using which a resultant matrix inequality can be equivalently converted to strict LMIs.

6.1 Introduction As well known, the Takagi-Sugeno (T-S) fuzzy model [119] has been recognized as a popular and powerful tool in approximating and describing complex nonlinear systems. Recently, the T-S fuzzy model approach has been extended to deal with nonlinear systems with time-delays. Various methodologies have been proposed for investigation of T-S fuzzy systems with time-delay in wide research topics, e.g., stability and state-feedback stabilization [9][10][142][158][175], static output-feedback control [12], dynamic output-feedback control [73][150], linear quadratic state-feedback control [139], variable structure control [81], H∞ control [73][150][160], and H∞ ﬁlter design [149]. Delay-dependent method has appeared as well [42]. In case of unavailability of the states, fuzzy observers play an important role in analysis and synthesis of fuzzy systems. Accordingly, observer-based fuzzy control has also attracted much attention for T-S fuzzy systems with time-delay, see [9][10][138] for observer-based stabilization, [13] for observer-based reliable control, [14] for observer-based guaranteed cost control, and [84] for observer-based tracking control. Most of the above analysis and control design methods developed for T-S fuzzy systems aim to seek conditions in terms of LMIs. However, there has been a drawback so far in the observer-based H∞ control design. That is, the derived criteria usually involve non-strict LMIs, see [98][99][106][131] for delayfree fuzzy systems and [73][84] for time-delay fuzzy systems. In this circumstance, the usual way is to set the Lyapunov matrix to be diagonal and use two-step LMI approaches under the price of bringing conservatism [98][106][131]. Quite recently, the observer-based H∞ control problem considered in [98] is improved by [80] in which a single-step LMI approach is presented. In this chapter, we study the observer-based H∞ control problem for T-S fuzzy systems with time-varying delay. We aim to extend the idea in [80] to time-delay C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 75–103, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

76

6. Observer-Based Fuzzy Control: Delay-Independent Method

case and use the proposed matrix decoupling technique to establish strict LMI conditions. The main technique to achieve this purpose is a proposed matrix decoupling approach using which the matrix inequality containing coupled variables is equivalently converted to strict LMIs. We believe that the inherent idea in the present method could beneﬁt the observer-based fuzzy control design for both certain and uncertain T-S fuzzy systems with/without time-delay.

6.2 Problem Formulation Consider a nonlinear system with state-delay which could be represented by a T-S fuzzy time-delay model described by Plant Rule i (i = 1, 2, . . . , r): IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bwi w(t) + Bui u(t),

(6.1)

z(t) = Ci x(t) + Cτ i x(t − τ (t)) + Di u(t), y(t) = Ei x(t) + Eτ i x(t − τ (t)), x(t) = φ(t), t ∈ [−τ0 , 0], where x ∈ Rn , u ∈ Rm1 , w ∈ Rm2 , z ∈ Rn1 and y ∈ Rn2 are the state, the control input, the disturbance, the controlled output and the measured output, respectively; r is the number of IF-THEN rules; Ai , Aτ i , Bwi , Bui , Ci , Cτ i , Di , Ei and Eτ i are real constant matrices with appropriate dimensions; θj (x(t)) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables and the fuzzy sets; φ is the initial condition; The time-delay τ (t) may be unknown but is assumed to be smooth function of time: τ˙ (t) ≤ β < 1,

τ (t) ≤ τ0 ,

(6.2)

where β and τ0 are two non-negative scalars. Let θ = [θ1 , . . . , θp ] and νi (θ) : Rp → [0, 1], i = 1, . . . , r, be the membership function corresponding to plant rule i. Denote hi (θ) = νi (θ)/ ri=1 νi (θ). It is obvious that ri=1 hi (θ) = 1 with hi (θ) ≥ 0. In the sequel, for simplicity we use hi (t) to denote hi (θ(x(t))). By fuzzy blending, the overall fuzzy model is inferred as follows: r νi (θ)[Ai x(t) + Aτ i x(t − τ ) + Bwi w(t) + Bui u(t)] r x(t) ˙ = i=1 i=1 νi (θ) r hi (t)[Ai x(t) + Aτ i x(t − τ ) + Bwi w(t) + Bui u(t)], (6.3) = i=1

z(t) = y(t) =

r i=1 r

hi (t)[Ci x(t) + Cτ i x(t − τ ) + Di u(t)], hi (t)[Ei x(t) + Eτ i x(t − τ )],

i=1

x(t) = φ(t), t ∈ [−τ0 , 0].

6.2 Problem Formulation

77

The design of observer-based H∞ controllers is performed through the parallel distributed compensation [134]: Controller Rule i (i = 1, 2, . . . , r): IF θ1 is µi1 and · · · and θp is µip THEN (t) + Aτ i x (t − τ (t)) + Bwi w(t) + Bui u(t) x ˙ (t) = Ai x +Li (y(t) − y(t)), (t) + Eτ i x (t − τ (t)), y(t) = Ei x

(6.4)

(t), u(t) = Ki x x (t) = φ(t), t ∈ [−τ0 , 0], where Li and Ki are, respectively, the observer gains and controller gains to be designed. The overall observer-based control law is thus inferred as x ˙ (t) =

r

hi (t)[Ai x (t) + Aτ i x (t − τ (t)) + Bwi w(t) + Bui u(t)

i=1

+Li (y(t) − y(t))], r hi (t)[Ei x (t) + Eτ i x (t − τ (t))], y(t) =

(6.5)

i=1

u(t) =

r

hi (t)Ki x (t).

i=1

Combining (6.3) with (6.5), the closed-loop fuzzy system is written as x ˙ (t) = z(t) =

r r i=1 j=1 r r

hi (t)hj (t)[Gij x (t) + Mij x (t − τ (t)) + Bi w(t)],

(6.6)

hi (t)hj (t)[Hij x (t) + Ni x (t − τ (t))],

i=1 j=1

x (t) = φ(t), t ∈ [−τ0 , 0], where

⎡

x (t) = ⎣ ⎡ Gij = ⎣

⎤ x(t) x(t) − x (t)

=⎣ φ(t)

Ai + Bui Kj −Bui Kj

⎡ Bi = ⎣

⎦,

⎡

⎤ Bwi 0

0

⎦,

Ai − Li Ej

⎤

⎤ φ(t) φ(t)

⎦,

⎦,

(6.7) ⎡

Mij = ⎣

⎤ Aτ i

Hij = Ci + Di Kj −Di Kj ,

0

0 Aτ i − Li Eτ j

⎦,

Ni = Cτ i 0 .

The purpose is to design controllers (6.5) such that the equilibrium solution x (t) = 0 of the closed-loop fuzzy system (6.6) with w = 0 is asymptotically stable and the following H∞ performance is satisﬁed:

78

6. Observer-Based Fuzzy Control: Delay-Independent Method

T

T

z(t)2 dt ≤ γ 2 0

w(t)2 dt

(6.8)

0

for all T > 0 and w ∈ L2 (0, ∞) under zero initial conditions (i.e., φ(t) ≡ 0, t ∈ [−τ0 , 0]). If this is the case, the fuzzy controller is said to render the closedloop fuzzy system (6.6) asymptotically stable with disturbance attenuation γ.

6.3 Delay-Free Case In this section, we consider a simple case, i.e., delay-free case. We rewrite the overall fuzzy model without delay terms as x(t) ˙ = z(t) = y(t) =

r i=1 r i=1 r

hi (t)[Ai x(t) + Bwi w(t) + Bui u(t)],

(6.9)

hi (t)[Ci x(t) + Di u(t)], hi (t)Ei x(t),

i=1

x(t) = φ(t), t ∈ [−τ0 , 0]. The observer-based fuzzy control law is of the following form: x ˙ (t) = u(t) =

r i=1 r

hi (t)[Ai x (t) + Bwi w(t) + Bui u(t) + Li (y(t) −

r

hi (t)Ei x (t))],

i=1

hi (t)Ki x (t).

(6.10)

i=1

6.3.1

Two-Step LMI Method

In this subsection, we list a previous result taken in [98] for the observer-based H∞ fuzzy control of system (6.9). The following basic lemma can be easily established using standard Lyaponuv technique, see [98]. Lemma 6.3.1. There exists an observer-based fuzzy control law (6.10) which renders the closed-loop fuzzy system (6.9) asymptotically stable with disturbance attenuation γ if there exist matrices Ki , Li , X > 0, Y > 0, and Xij with Xii symmetrical, i, j = 1, 2, . . . , r, j > i, such that the following matrix inequalities hold: ⎡ ⎤ T X XBui Ki ΛTii X + XΛii + γ −2 XBwi Bwi ⎢ ⎥ (6.11) ⎣ ⎦ < Xii , T KiT Bui X ΓiiT Y + Y Γii

6.3 Delay-Free Case

⎡

79

⎤ ΛTij X + XΛij

⎢ X(Bui Kj + Buj Ki ) ⎥ ⎢ ⎥ T T ⎢ +γ −2 X(Bwi Bwj ⎥ + Bwj Bwi )X ⎢ ⎥ ⎣ ⎦ T T + KjT Bui )X ΓijT Y + Y Γij (KiT Buj T ≤ Xij + Xij ,

⎡ X11 ⎢ ⎢ .. ⎢ . ⎢ ⎢ T ⎢ X1r ⎣ U1k

j > i,

⎤ T · · · X1r U1k ⎥ .. ⎥ . . .. . . . ⎥ ⎥ < 0, ⎥ T ⎥ · · · Xrr Urk ⎦ · · · Urk −I

(6.12)

k = 1, 2, . . . , r,

(6.13)

where Λii = Ai + Bui Ki ,

Λij = Ai + Aj + Bui Kj + Buj Ki

Γii = Ai − Li Ei , Γij = Ai + Aj − Li Ej − Lj Ei , Uik = [Ci + Di Kk Di Kk ]. In order to solve the nonlinear matrix inequalities in Lemma 6.3.1 for the design of Ki and Li , the next result provides a suﬃcient condition, which contains two steps in solving LMIs. The detailed proof is omitted here and it can be referred to [98]. Theorem 6.3.1. There exists an observer-based fuzzy control law (6.10) which renders the closed-loop fuzzy system (6.9) asymptotically stable with disturbance attenuation γ if (Step 1) there exist matrices Z > 0, Mi , Nij and Zij with Nii and Zii symmetrical, i, j = 1, 2, . . . , r, j > i, such that the following LMIs hold: ⎡ ⎤ ZATi + Ai Z + Bui Mi ⎢ Bui Mi ⎥ ⎢ ⎥ T T ⎢ +MiT Bui ⎥ < Zii , i = 1, 2, . . . , r, + γ −2 Bwi Bwi (6.14) ⎢ ⎥ ⎣ ⎦ T MiT Bui Nii ⎡

Z(Ai + Aj )T + (Ai + Aj )Z

⎤

⎢ ⎥ ⎢ T Bui Mj + Buj Mi ⎥ + Buj Mi +Bui Mj + MjT Bui ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T T T + γ −2 (Bwi Bwj + Bwj Bwi ) ⎢ +MiT Buj ⎥ ⎢ ⎥ ⎣ ⎦ T T MiT Buj + MjT Bui Nij + NijT T ≤ Zij + Zij ,

j > i,

(6.15)

80

6. Observer-Based Fuzzy Control: Delay-Independent Method

⎡

Z11 · · · Z1r ⎢ ⎢ .. . . .. ⎢ . . . ⎢ ⎢ T ⎢ Z1r · · · Zrr ⎣

⎤ T V1k ⎥ .. ⎥ . ⎥ ⎥ < 0, ⎥ T ⎥ Vrk ⎦

k = 1, 2, . . . , r,

(6.16)

V1k · · · Vrk −I where Vik = [Ci Z + Di Mk Di Mk ], and (Step 2) after solving the LMIs in Step 1, there exist matrices Y > 0 and Ji , i = 1, 2, . . . , r, such that the following LMIs hold: ATi Y + Y Ai − EiT JiT − Ji Ei < Z −1 Nii Z −1 ,

i = 1, 2, . . . , r,

(6.17)

(Ai + Aj )T Y + Y (Ai + Aj ) − EiT JjT − Jj Ei − EjT JiT − Ji Ej < Z −1 (Nij + NijT )Z −1 ,

j > i.

(6.18)

In this case, the controller gains Ki and the observer gains Li are given by Ki = Mi Z −1 , 6.3.2

Li = Y −1 Ji ,

i = 1, 2, . . . , r.

(6.19)

Single-Step LMI Method

In this subsection, we will present another design method which improves that in Theorem 6.3.1. We ﬁrst establish a useful lemma and then present the main result. The following lemma gives a new matrix inequality condition which is equivalent to that of Lemma 6.3.1. Lemma 6.3.2. There exist matrices Ki , Li , X > 0, Y > 0, and Xij with Xii symmetrical, i, j = 1, 2, . . . , r, j > i, such that the matrix inequalities (6.11)¯ > 0, Y¯ > 0, Pij and Qij (6.13) hold if and only if there exist matrices Ki , Li , X with Pii and Qii symmetrical, i, j = 1, 2, . . . , r, j > i, such that the following matrix inequalities hold: ¯ T + Λii X ¯ + γ −2 Bwi B T < Pii , XΛ (6.20) ii

wi

ΓiiT Y¯ + Y¯ Γii < Qii , i = 1, 2, . . . , r, ¯ + γ −2 (Bwi B T + Bwj B T ) ≤ Pij + P T , ¯ T + Λij X XΛ ij wj wi ij T ¯ T ¯ Γij Y + Y Γij ≤ Qij + Qij , j > i, ⎤ ⎡ ¯U ¯T P11 · · · P1r X 1k ⎥ ⎢ .. .. ⎥ ⎢ .. . . ⎢ . . . . ⎥ ⎥ < 0, k = 1, 2, . . . , r, ⎢ ⎥ ⎢ T ¯U ¯T ⎥ ⎢ P1r · · · Prr X rk ⎦ ⎣ ¯1k X ¯ ··· U ¯rk X ¯ −I U ⎡ ⎤ Q11 · · · Q1r ⎢ ⎥ ⎢ .. . . .. ⎥ ⎢ . . . ⎥ < 0, ⎣ ⎦ T Q1r · · · Qrr

(6.21) (6.22) (6.23)

(6.24)

(6.25)

6.3 Delay-Free Case

81

¯ik = Ci + where Λii , Γii , Λij and Γij remain the same as in Lemma 6.3.1, and U Di K k . Proof. Necessity. Suppose there exist matrices Ki , Li , X > 0, Y > 0, and Xij with Xii symmetrical, i, j = 1, 2, . . . , r, j > i, such that the matrix inequalities (6.11)-(6.13) hold. Partition Xij as ⎡ ⎤ ij ⊗ X ⎦, Xij = ⎣ (6.26) ⊗ Qij T , Qii = QT , i = 1, 2, . . . , r, and the notation ’⊗’ denotes a block ii = X where X ii ii which will not be used in the sequel. Substituting (6.26) into (6.11), we have ⎡ ⎤ T X XBui Ki ΛTii X + XΛii + γ −2 XBwi Bwi ⎢ ⎥ ⎣ ⎦ T KiT Bui X ΓiiT Y + Y Γii ⎡ ⎤ ii ⊗ X ⎦ , i = 1, 2, . . . , r. <⎣ (6.27) ⊗ Qii Multiplying both sides of (6.27) from the left and right, respectively, by [I 0] ¯X ii X ¯ leads to (6.20). ¯ = X −1 and Pii = X and its transpose, and then setting X Multiplying both sides of (6.27) from the left and right, respectively, by [0 I] and its transpose, and then setting Y¯ = Y leads to (6.21). Similarly, from (6.12), we can verify (6.22) and (6.23) by substituting (6.26) into (6.12) and setting ¯X ij X. ¯ Next we show (6.24) and (6.25). It is seen that (6.13) with Pij = X substitution of (6.26) implies (6.25) and ⎡ ⎤ 11 1r X ··· X (C1 + D1 Kk )T ⎢ ⎥ .. .. .. ⎢ ⎥ .. ⎢ ⎥ . . . . ⎢ ⎥ ⎢ ⎥ T T ⎥ ⎢ X1r ··· Xrr (Cr + Dr Kk ) ⎥ ⎢ ⎣ ⎦ C1 + D1 Kk · · · Cr + Dr Kk −I < 0,

k = 1, 2, . . . , r.

(6.28)

Multiplying the matrix of (6.28) from the left and right, respectively, by ¯ · · · , X, ¯ I} and its transpose, yields exactly (6.24). This shows the nediag {X, cessity. ¯ > 0, Y¯ > 0, Pij and Qij Suﬃciency. Suppose there exist matrices Ki , Li , X with Pii and Qii symmetrical, i, j = 1, 2, . . . , r, j > i, such that the matrix inequalities (6.20)-(6.25) hold. We ﬁrst show that (6.11) holds. To this end, denote for simplicity ¯ T + Λii X ¯ + γ −2 Bwi B T , Φi = XΛ ii wi i = 1, 2, . . . , r. Υi = ΓiiT Y¯ + Y¯ Γii ,

82

6. Observer-Based Fuzzy Control: Delay-Independent Method

Without loss of generality, assume that Υi (i = 1, 2, . . . , r) is invertible (Otherwise, set Υi as Υi + Y¯ with > 0 being a suﬃciently small scalar and the following deductions still work). Then, (6.20) and (6.21) imply that ⎡ ⎤ T 0 Φi − δ −1 Bui Ki Υi−1 KiT Bui ⎣ ⎦ 0 δΥi ⎤ ⎡ T 0 Pii − δ −1 Bui Ki Υi−1 Qii Υi−1 KiT Bui ⎦, (6.29) <⎣ 0 δQii holds for all i = 1, 2, . . . , r, and for all δ ≥ δ1 with δ1 > 0 being a suﬃciently large ⎡ scalar. Multiplying ⎤ both sides of (6.29) from the left and right, respectively, by ⎣

I δ −1 Bui Ki Υi−1 0 ⎡ ⎣ ⎡

<⎣

⎦ and its transpose, gives

I ⎤ Φi

Bui Ki

T KiT Bui

δΥi

⎦

Pii

Bui Ki Υi−1 Qii

T Qii Υi−1 KiT Bui

δQii

⎤ ⎦,

i = 1, 2, . . . , r.

(6.30)

Multiplying both sides of (6.30) from the left and right, respectively, by ¯ −1 , I} and its transpose, and setting diag {X ¯ −1 , X =X ⎡ Xii = ⎣

Y = Y (δ) = δ Y¯ ,

∀δ ≥ δ1 ,

¯ −1 Pii X ¯ −1 X

¯ −1 Bui Ki Υ −1 Qii X i

T ¯ −1 Qii Υi−1 KiT Bui X

δQii

⎤ ⎦,

i = 1, 2, . . . , r,

(6.31)

gives exactly (6.11). Similar to the process for obtaining (6.11) from (6.20) and (6.21), next we show (6.12) by using (6.22) and (6.23). Note that in this dealing process, we need to ¯ := P¯ij and Qij + 1 Y¯ := Q ¯ ij , respectively, such perturb Pij and Qij to Pij + 1 X that (6.22) and (6.23) are replaced by ¯ T + Λij X ¯ + γ −2 (Bwi B T + Bwj B T ) < P¯ij + P¯ T , XΛ ij wj wi ij ¯ ij + Q ¯ T , j > i. Γ T Y¯ + Y¯ Γij < Q ij

ij

(6.32) (6.33)

By using (6.32) and (6.33), and following a procedure analogous to that from (6.29) to (6.31), we can arrive at ⎡ ⎤ T T + Bwj Bwi )X X(Bui Kj + Buj Ki ) ΛTij X + XΛij + γ −2 X(Bwi Bwj ⎢ ⎥ ⎣ ⎦ T T T T δΥij (Ki Buj + Kj Bui )X T < Xij + Xij ,

j > i,

(6.34)

6.3 Delay-Free Case

83

for all δ ≥ δ2 with δ2 > 0 being a suﬃciently large scalar satisfying δ2 > δ1 , where ¯ −1 , X =X (6.35) Υij = ΓijT Y¯ + Y¯ Γij , j > i, ⎡ ⎤ ¯ −1 P¯ij X ¯ −1 ¯ −1 (Bui Kj + Buj Ki )Υ −1 Q ¯ ij X X ij ⎦. Xij = ⎣ ¯ ij ¯ T Υ −1 (Bui Kj + Buj Ki )T X ¯ −1 δ Q Q ij ij Thus, we see that (6.34) implies (6.12) by letting Y = Y (δ) = δ Y¯ for any δ ≥ δ2 . Finally, it remains to show that (6.13) holds with variables (6.31) and (6.35) by choosing some suﬃciently large δ. To its end, substituting (6.31) and (6.35) into the matrix of (6.13), and after some exchanges of rows and columns, the inequality (6.13) is equivalent to ⎡ ⎤ ¯ −1 P11 X ¯ −1 P¯1r X ¯ −1 · · · X ¯ −1 U ¯T ⊗ · · · ⊗ X 1k ⎢ . ⎥ .. .. .. . . .. .. ⎢ ⎥ . .. ⎥ . ⎢ . . . . ⎢ ⎥ ⎢ ¯ −1 ¯ T ¯ −1 ¯ −1 Prr X ¯ −1 U ¯T ⊗ · · · ⊗ ⎥ ⎢ X P1r X · · · X ⎥ rk ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ¯rk ¯1k ⎢ ··· U −I ⊗ · · · ⊗ ⎥ U ⎢ ⎥ ⎢ ¯ 1r ⎥ ⎢ ⎥ ⊗ ··· ⊗ ⊗ δQ11 · · · δ Q ⎢ ⎥ ⎢ .. . . . . .. .. .. .. . . . .. ⎥ ⎢ ⎥ . . ⎣ ⎦ T ¯ ⊗ ··· ⊗ ⊗ δ Q1r · · · δQrr < 0,

k = 1, 2, . . . , r,

(6.36)

which is further equivalent to ⎡ ¯U ¯T ⊗ P11 · · · P¯1r X 1k ⎢ . .. .. .. .. ⎢ . . ⎢ . . . . ⎢ ⎢ ¯T ¯U ¯T ⊗ ⎢ P1r · · · Prr X rk ⎢ ⎢ ⎢¯ ¯ ¯rk X ¯ −I ⎢ U1k X · · · U ⊗ ⎢ ⎢ ⎢ ⊗ ··· ⊗ ⊗ δQ11 ⎢ ⎢ . . .. .. . .. .. ⎢ .. . . ⎣ ⊗

···

⊗

⊗

··· .. . ··· ··· ··· .. .

¯T · · · δQ 1r

⎤ ⊗ .. ⎥ ⎥ . ⎥ ⎥ ⎥ ⊗ ⎥ ⎥ ⎥ ⎥ ⊗ ⎥ < 0, ⎥ ¯ 1r ⎥ ⎥ δQ ⎥ .. ⎥ . ⎥ ⎦ δQrr

k = 1, 2, . . . , r,

(6.37)

¯ik is as given in Lemma 6.3.2, ‘⊗’ still denotes a block which will not where U used later and such a block is a ﬁxed matrix independent of δ. So, to show (6.13), it is suﬃcient to show that (6.37) holds with some suﬃciently large δ. Now, from ¯ ij , we can easily make (6.24) and (6.25), when perturb Pij and Qij to P¯ij and Q the following hold:

84

6. Observer-Based Fuzzy Control: Delay-Independent Method

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ψk := ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

¯U ¯T P¯12 · · · P¯1r X 1k

P11

⎤

⎥ ⎥ T ⎥ ¯ ¯ ¯ P22 · · · P2r X U2k ⎥ ⎥ .. .. ⎥ .. .. < 0, . . . ⎥ . ⎥ ⎥ T T ¯U ¯T ⎥ P¯1r · · · Prr X P¯2r rk ⎥ ⎦ ¯ ¯ ¯ ¯ ¯ ¯ U1k X U2k X · · · Urk X −I T P¯12 .. .

¯ 1r ¯ 12 · · · Q Q11 Q

⎢ ⎢ ⎢Q T ⎢ ¯ 12 Q22 Ω := ⎢ . .. ⎢ . ⎢ . . ⎣ ¯T Q ¯T Q 1r

2r

··· .. . ···

k = 1, 2, . . . , r,

(6.38)

⎤

⎥ ⎥ ¯ Q2r ⎥ ⎥ < 0. .. ⎥ ⎥ . ⎥ ⎦ Qrr

(6.39)

From Schur complements, it is easy to see that there exists a scalar δ3 > 0 such that ⎤ ⎡ Ψ ∆ ⎥ ⎢ k (6.40) ⎦ < 0, k = 1, 2, . . . , r, ⎣ T ∆ δΩ hold for all δ ≥ δ3 , where ∆ denotes the compatible block matrix in (6.37) which is independent of δ. Hence, (6.37), and thus (6.13), holds. So far, we have arrived that (6.11)-(6.13) hold with variables given in (6.31) and (6.35) for any δ satisfying δ ≥ max{δ2 , δ3 }. This shows the suﬃciency and thus completes the proof. With Lemma 6.3.2 in hand, we are now in a position to present the main result of this subsection. Theorem 6.3.2. There exists an observer-based fuzzy control law (6.10) which renders the closed-loop fuzzy system (6.9) asymptotically stable with disturbance ¯ > 0, Y¯ > 0, M ¯ i , J¯i , Pij and Qij with Pii attenuation γ if there exist matrices X and Qii symmetrical, i, j = 1, 2, . . . , r, j > i, such that the following LMIs hold: ¯ T B T + γ −2 Bwi B T < Pii , ¯ + Bui M ¯i + M ¯ T + Ai X XA i i ui wi ATi Y¯ + Y¯ Ai − EiT J¯iT − J¯i Ei < Qii , i = 1, 2, . . . , r,

(6.41) (6.42)

T ¯ i + Aj )T + (Ai + Aj )X ¯ + Bui M ¯ jT Bui ¯j + M X(A T T T ¯ iT Buj ¯i + M +Buj M + γ −2 (Bwi Bwj + Bwj Bwi )

≤ Pij + PijT , (Ai + Aj )T Y¯ + Y¯ (Ai + Aj ) − EiT J¯jT − J¯j Ei − EjT J¯iT − J¯i Ej

(6.43)

≤ Qij + QTij ,

(6.44)

j > i,

6.4 Time-Delay Case

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

P11 .. .

··· .. .

P1r .. .

¯ T DT ¯ T +M XC 1 1 k .. .

85

⎤

⎥ ⎥ ⎥ ⎥ ⎥ T T T T ⎥ ¯ ¯ ··· Prr XCr + Mk Dr ⎥ P1r ⎦ ¯ ¯ ¯ ¯ C1 X + D1 Mk · · · Cr X + Dr Mk −I

< 0, ⎡ Q11 ⎢ ⎢ .. ⎢ . ⎣ QT1r

k = 1, 2, . . . , r, ⎤ · · · Q1r ⎥ . . .. ⎥ < 0. . . ⎥ ⎦ · · · Qrr

(6.45)

(6.46)

In this case, the controller gains Ki and the observer gains Li are given by ¯ iX ¯ −1 , Ki = M

Li = Y¯ −1 J¯i ,

i = 1, 2, . . . , r.

(6.47)

¯i = Proof. The result follows immediately from Lemmas 1 and 2 by setting M ¯ ¯ ¯ Ki X and Ji = Y Li , i = 1, 2, . . . , r, in (6.20)-(6.25). Compared with Theorem 6.3.1, Theorem 6.3.2 contains some new improvements as follows. (i) The result of Theorem 6.3.2 is less conservative than that of Theorem 6.3.1. This is because the conditions (6.41)-(6.46) of Theorem 6.3.2 are equivalent to the conditions (6.20)-(6.25) of Lemma 6.3.2 while (6.14)-(6.18) of Theorem 6.3.1 are only suﬃcient to ensure (6.11)-(6.13) of Lemma 6.3.1. (Note that the conditions of Lemmas 6.3.1 and 6.3.2 are equivalent, see Lemma 6.3.2.) This point will be further illustrated by an example that can be examined by Theorem 6.3.2 but cannot by Theorem 6.3.1. (ii) Theorem 6.3.2 proposes a single step to solve LMIs. This overcomes the drawback of the two-step approach of Theorem 6.3.1. (iii) The matrix dimensions are largely reduced. In details, the matrix dimensions in (6.41) and (6.43) are n × n while, correspondingly, those in (6.14) and (6.15) of Theorem 6.3.1 are 2n × 2n. The matrix dimension in (6.45) is (nr + q) × (nr + q) while, correspondingly, that in (6.16) of Theorem 6.3.1 is (2nr + q) × (2nr + q). Although we have one more LMI (r2 + 2r LMIs in Theorem 6.3.1 but r2 + 2r + 1 in Theorem 6.3.2) to be solved, due to the largely reduced matrix dimensions, the computational burden is hence reduced.

6.4 Time-Delay Case In this section, we design observer-based fuzzy controllers for the time-delay model (6.3) and present strict LMI conditions. To this end, we ﬁrst consider the problem of H∞ performance analysis for the unforced system described by (6.6).

86

6. Observer-Based Fuzzy Control: Delay-Independent Method

6.4.1

Bounded Real Lemma

The following result actually presents two delay-independent versions of bounded real lemma for T-S fuzzy systems with state-delay. Lemma 6.4.1. The fuzzy time-delay system described by (6.6) is asymptotically stable with disturbance attenuation γ if any of the following two conditions holds: Condition L1. There exist matrices P > 0, S > 0 and Xij with Xii symmetrical, i, j = 1, 2, . . . , r, j ≥ i, such that the following matrix inequalities hold: ⎡ ⎤ T Ωij + Ωji UijT + Uji T ⎣ ⎦ < Xij + Xij , j ≥ i, (6.48) Uij + Uji −2I ⎤ ⎡ X11 · · · X1r ⎥ ⎢ ⎢ .. . . .. ⎥ (6.49) ⎢ . . . ⎥ ≤ 0, ⎦ ⎣ T X1r · · · Xrr where

⎡

Ωij = ⎣

⎤

P Gij + GTij P + S + γ −2 P Bi BjT P

P Mij

MijT P

−(1 − β)S

Uij = Hij Ni .

⎦,

Condition L2. There exist matrices P > 0, S > 0 and Yij with Yii symmetrical, i, j = 1, 2, . . . , r, j ≥ i, such that the following matrix inequalities hold: Ωij + Ωji < Yij + YijT , j ≥ i, ⎤ ⎡ T Y11 · · · Y1r U1k ⎥ ⎢ .. ⎥ ⎢ .. . . .. ⎥ ⎢ . . . . ⎥ ≤ 0, k = 1, 2, . . . , r, ⎢ ⎥ ⎢ T T ⎥ ⎢ Y1r · · · Yrr Urk ⎦ ⎣ U1k · · · Urk −I

(6.50)

(6.51)

where Ωij and Uij are the same as in Condition L1. Proof. For Condition L1. We ﬁrst show that the H∞ performance is guaranteed and then choose a Lyapunov-Krasovskii functional to prove the asymptotic stability. Deﬁne a function J(t) as

t J(t) = x T (t)P x (t) + x T (α)S x(α)dα. (6.52) t−τ (t)

6.4 Time-Delay Case

87

Then we have ˙ + z T (t)z(t) − γ 2 wT (t)w(t) J(t) x(t) + M (t) x(t − τ (t)) + B(t)w(t)] =x T (t)P [G(t) T x(t) − (1 − τ˙ (t)) xT (t − τ (t))S x(t − τ (t)) + x (t)S +z T (t)z(t) − γ 2 wT (t)w(t) ⎡ ⎤T ⎡ ⎤ x (t) x (t) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T ≤ ⎢x [ Ω(t) + U (t) U (t)] ⎥ ⎢ (t − τ (t)) x (t − τ (t)) ⎥ , ⎣ ⎦ ⎣ ⎦ w(t) w(t)

(6.53)

where G(t) =

B(t) = U (t) =

r r

hi (t)hj (t)Gij ,

M (t) =

i=1 j=1 r

r r

hi (t)hj (t)Mij ,

i=1 j=1

hi (t)Bi , i=1 r r

hi (t)hj (t) Hij Ni ,

(t) = [U (t) 0], U

i=1 j=1

⎡ ⎢ =⎢ Ω(t) ⎢ ⎣

⎤ T

P G(t) + G(t) P + S

P M (t)

P B(t)

M (t) P

−(1 − β)S

0

B(t)T P

0

−γ 2 I

T

⎥ ⎥ ⎥. ⎦

(6.54)

Denote ⎡ Ω(t) = ⎣

⎤

P G(t) + G(t)T P + S + γ −2 P B(t)B(t)T P

P M (t)

M (t)T P

−(1 − β)S

⎦.

If (6.48) and (6.49) hold, we have that ⎤

⎡ ⎣

Ω(t) U (t)T U (t) −I

r r

⎦ ⎡

hi (t)hj (t) ⎣

⎤ Ωij UijT

⎦ Uij −I ⎡ ⎡ ⎤ ⎤ r r r T T T Ω Ω U + Ω U + U ii ii ij ji ij ji ⎦ ⎦+ h2i (t) ⎣ hi (t)hj (t) ⎣ = Uii −I Uij + Uji −2I i=1 i=1 j>i

=

i=1 j=1

(6.55)

88

6. Observer-Based Fuzzy Control: Delay-Independent Method

<

r

h2i (t)Xii +

i=1

r r

T hi (t)hj (t)[Xij + Xij ]

i=1 j>i

⎡

X11 ⎢ ⎢ .. = [h1 (t)I, · · · , hr (t)I] ⎢ . ⎣ T X1r

⎤⎡ ⎤ h1 (t)I · · · X1r ⎥⎢ ⎥ . ⎥⎢ . ⎥ .. . .. ⎥ ⎢ .. ⎥ ⎦⎣ ⎦ · · · Xrr hr (t)I

≤ 0,

(6.56) T which is, by Schur complement, equivalent to Ω(t)+ U(t) U (t) < 0. This implies from (6.53) that

T J(T ) − J(0) + (z(t)2 − γ 2 w(t)2 )dt ≤ 0. (6.57) 0

Due to J(T ) ≥ 0 and J(0) = 0 when subject to zero initial conditions for equation (6.6), the above inequality (6.57) veriﬁes (6.8). Next, we prove the asymptotic stability of the unforced system (6.6) with w = 0. Choose a Lyapunov-Krasovskii functional V ( xt ) = J(t) as in (6.52), where x t = x (t + α), α ∈ [−τ0 , 0]. It is seen that there exist two scalars c1 > 0 and c2 > 0 such that x(t)2 ≤ V ≤ c2 c1

sup

xt 2 .

α∈[−τ0 ,0]

If (6.48)-(6.49) hold, a deduction similar to the above leads to that the time derivative of V along the solutions of (6.6) with w = 0 satisﬁes V˙ < 0. This proves the motion of (6.6) with w = 0 is asymptotically stable according to the stability theory of [44] (See Theorem A.1 in the Appendix). This completes the proof of the lemma under Condition L1. For Condition L2. From (6.51), we have ⎡ ⎤ r T Y11 ··· Y1r h (t)U k 1k k=1 ⎢ ⎥ .. .. .. ⎢ ⎥ .. ⎢ ⎥ . . . . ⎢ ⎥ ≤ 0, ⎢ ⎥ r T T ⎥ ⎢ Y · · · Y h (t)U rr 1r rk ⎦ k=1 k ⎣ r r −I k=1 hk (t)U1k · · · k=1 hk (t)Urk which, by Schur complement, gives ⎡ ⎤ Y11 · · · Y1r ⎢ ⎥ ⎢ .. . . .. ⎥ ⎢ . . . ⎥ ⎣ ⎦ T Y1r · · · Yrr ⎡ ⎤ r T h (t)U k 1k ⎢ k=1 ⎥ r .. ⎢ ⎥ r +⎢ ≤ 0, ⎥ h (t)U , · · · , h (t)U . k 1k k rk k=1 k=1 ⎣ ⎦ r T k=1 hk (t)Urk

6.4 Time-Delay Case

89

and thus, r

h2i (t)Yii +

i=1

r r

hi (t)hj (t)[Yij + YijT ] + U (t)T U (t)

i=1 j>i

⎤⎡ ⎤ h1 (t)I Y11 · · · Y1r ⎢ ⎥⎢ ⎥ . ⎥⎢ . ⎥ ⎢ . = [h1 (t)I, · · · , hr (t)I] ⎢ .. . . . .. ⎥ ⎢ .. ⎥ + U (t)T U (t) ⎣ ⎦⎣ ⎦ T Y1r · · · Yrr hr (t)I ⎡

≤ 0,

(6.58)

where U (t) is as in (6.54). Now, choose the same function J(t) as in (6.52) and deﬁne notations as in (6.54) and (6.55). If (6.50)-(6.51) hold, one gets from (6.58) that Ω(t) + U (t)T U (t) r r r h2i (t)Gii + hi (t)hj (t)[Gij + GTij ] + U (t)T U (t) = <

i=1 r i=1

h2i (t)Yii +

i=1 j>i r r

hi (t)hj (t)[Yij + YijT ] + U (t)T U (t)

i=1 j>i

≤ 0,

(6.59)

(t) < 0. The rest of +U (t)T U which is, by Schur complement, equivalent to Ω(t) the proof for Condition L2 follows a similar line to that for Condition L1. This proves the lemma. We would like to point out that Conditions L1 and L2 can produce diﬀerent results: not only lead to diﬀerent control design criteria but also generate diﬀerent H∞ performances. This will be clearly seen later from illustrative examples. 6.4.2

Main Results

It is seen that Conditions L1 and L2 are matrix inequalities in which design variables Li and Ki are coupled with other variables P , S and Xij (or Yij ). We next seek methods to separate these coupled variables by equivalently converting the non-convex conditions to strict LMI conditions. In the literature of observer-based H∞ fuzzy control for fuzzy systems without time-delay, two-step LMI approach has been employed to solve matrix inequalities with coupled variables, see [98], [99], [106] and [131]. It is known that the two-step LMI approach brings much conservatism. The usual way is to partition P as diagonal P = diag {P1 , P2 } > 0. In the last section, we have proved that choosing P with full structures is equivalent to choosing P diagonal in solving the matrix inequalities therein. For fuzzy time-delay systems, by choosing variables P and S diagonal, a single-step LMI method is provided in [13] and [14] for reliable control and guaranteed cost control, respectively. In the following, we will solve the observer-based H∞ control problem based on Lemma 6.4.1, in the sense that necessary and suﬃcient LMI conditions are given to solve the matrix

90

6. Observer-Based Fuzzy Control: Delay-Independent Method

inequalities in Lemma 6.4.1. In another word, the matrix inequalities in Lemma 6.4.1 can be equivalently transformed into strict LMIs. The next lemma clearly addresses this point. Lemma 6.4.2. Condition L1 (respectively, Condition L2) holds if and only if Condition T1 (respectively, Condition T2) holds, where Conditions T1 and T2 are as follows: Condition T1. There exist matrices Ql > 0, Rl > 0, Wli and Xlij with Xlii symmetrical, l = 1, 2, i, j = 1, 2, . . . , r, j ≥ i, such that the following LMIs hold: ⎤ ⎡ T T + U Φ1ij + Φ1ji U ij ji ⎥ ⎢ T ⎥ < X1ij + X1ij ⎢ , (6.60) ⎦ ⎣ −2I Uij + Uji T , Φ2ij + Φ2ji < X2ij + X2ij ⎤ ⎡ Xl11 · · · Xl1r ⎥ ⎢ . ⎥ ⎢ .. . . ⎢ . . .. ⎥ ≤ 0, ⎦ ⎣ T Xl1r · · · Xlrr

where

(6.61)

(6.62)

⎡

⎢ ⎢ Φ1ij = ⎢ ⎢ ⎣ ⎡ Φ2ij = ⎣

⎤ T T Ai Q1 + Q1 ATi + Bui W1j + W1j Bui

∗

T +γ −2 Bwi Bwj + R1

−(1 − β)R1

Q1 ATτi

T Q2 Ai + ATi Q2 − W2i Ej − EjT W2i + R2

T ATτi Q2 − EτTj W2i ij = C Q + D W C Q . U i 1 i 1j τi 1

⎥ ⎥ ⎥, ⎥ ⎦

∗ −(1 − β)R2

⎤ ⎦,

If this is the case, the matrices Ki and Li in Condition L1 are given by Ki = W1i Q−1 1 ,

Li = Q−1 2 W2i ,

i = 1, 2, . . . , r.

(6.63)

Condition T2. There exist matrices Ql > 0, Rl > 0, Wli and Ylij with Ylii symmetrical, l = 1, 2, i, j = 1, 2, . . . , r, j ≥ i, such that the following LMIs hold: T , Φ1ij + Φ1ji < Y1ij + Y1ij

(6.64)

T Φ2ij + Φ2ji < Y2ij + Y2ij , ⎤ ⎡ T Y111 · · · Y11r U 1k ⎥ ⎢ .. .. ⎥ ⎢ .. . . ⎢ . . . . ⎥ ⎥ ≤ 0, ⎢ ⎥ ⎢ T T ⎥ ⎢ Y11r · · · Y1rr U rk ⎦ ⎣ rk −I 1k · · · U U

(6.65)

k = 1, 2, . . . , r,

(6.66)

6.4 Time-Delay Case

⎤ Y211 · · · Y21r ⎢ ⎥ . ⎥ ⎢ .. . . ⎢ . . .. ⎥ ≤ 0, ⎣ ⎦ T Y21r · · · Y2rr

91

⎡

(6.67)

ij are the same as in Condition T1. If this is the case, where Φ1ij , Φ2ij and U the matrices Ki and Li in Condition L2 are given by Ki = W1i Q−1 1 ,

Li = Q−1 2 W2i ,

i = 1, 2, . . . , r.

(6.68)

Proof. We only prove the equivalence of Condition L1 and Condition T1, since the proof for the equivalence of Condition L2 and Condition T2 follows similarly with slight modiﬁcations. Necessity. Suppose Condition L1 holds, i.e., (6.48)-(6.49) are satisﬁed for some matrices P > 0, S > 0, Ki , Li and Xij with Xii symmetrical, i, j = 1, 2, . . . , r, j ≥ i. Let P and S be ⎡ ⎤ ⎡ ⎤ P1 ∗ S1 ∗ ⎦, S = ⎣ ⎦, P =⎣ (6.69) ⊗ ⊗ ⊗ ⊗ where X1 > 0, S1 > 0 and ⊗ denotes a block which will not be used in the sequel. Substituting (6.69) into (6.48), we have 1ij + Ω 1ji < Xij + X T , Ω ij

(6.70)

where ⎡

1ij Ω

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

P1 (Ai + Bui Kj ) + (Ai + Bui Kj )T P1 T +S1 + γ −2 P1 Bwi Bwj P1

⊗ ATτi P1 ⊗ Ci + Di Kj

⎤ ∗ ∗ ⎥ ⎥ ⎥ ⎥ ⎥ ⊗ ∗ ∗ ∗ ⎥ ⎥ ⎥ ⎥. ⊗ −(1 − β)S1 ∗ ∗ ⎥ ⎥ ⎥ ⎥ ⊗ ⊗ ⊗ ∗ ⎥ ⎥ ⎦ ⊗ Cτ i ⊗ −I ∗

∗

Multiplying (6.70) from the left and right, respectively, by ⎡

P1−1 0 0 0 0

⎤

⎢ ⎥ ⎢ ⎥ P1 = ⎢ 0 0 P1−1 0 0 ⎥ ⎣ ⎦ 0 0 0 0I

92

6. Observer-Based Fuzzy Control: Delay-Independent Method

and its transpose, and then setting Q1 = P1−1 , R1 = P1−1 S1 P1−1 , W1i = Ki P1−1 and X1ij = P1 Xij P1T lead to (6.60). It also results in (6.62) for l = 1, i.e., ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ P1 X11 · · · X1r P1T X111 · · · X11r ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥⎢ ⎥ ⎢ .. . . ⎥ ⎢ .. . . .. .. ⎥⎢ . ⎥≤0 ⎢ . . .. ⎥ = ⎢ . .. ⎥ ⎢ . . ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ T T T X11r · · · X1rr X1r · · · Xrr P1 P1 due to (6.49). For obtaining the rest of LMIs, we have to rewrite (6.48) as ⎡ ⎤ (Gij + Gji )P −1 ⎢ ⎥ ⎢ ⎥ +P −1 (Gij + Gji )T ⎢ ⎥ −1 −1 T T ⎥ ⎢ (M + M )P P (H + H ) ij ji ij ji ⎥ ⎢ +2P −1 SP −1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −2 T T ⎢ +γ (Bwi Bwj + Bwj Bwi ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −1 T −1 −1 −1 T T ⎥ ⎢ P (Mij + Mji ) −2(1 − β)P SP P (Ni + Nj ) ⎥ ⎢ ⎣ ⎦ −1 −1 (Ni + Nj )P −2I (Hij + Hji )P T < P (Xij + Xij )P ,

(6.71)

where P = diag {P −1 , P −1 , I}. Let P −1 and P −1 SP −1 be ⎡ ⎡ ⎤ ⎤ ⊗ ∗ ⊗ ∗ ⎦ , P −1 SP −1 = ⎣ ⎦, P −1 = ⎣ ⊗ P2 ⊗ S2

(6.72)

where P2 > 0 and S2 > 0. Substituting (6.72) into (6.71), we have T 2ji < P (Xij + Xij 2ij + Ω )P , Ω

where

2ij Ω

⎡

⊗

(6.73) ∗

⎢ ⎢ ⎢ ⊗ (A − L E )P + P (A − L E )T + S i i j 2 2 i i j 2 ⎢ ⎢ ⎢ =⎢ ⊗ ⎢⊗ ⎢ ⎢ ⎢⊗ P2 (Aτ i − Li Eτ j )T ⎢ ⎣ ⊗ ⊗

∗ ∗ ⊗ ⊗ ⊗

∗

∗

⎤

⎥ ⎥ ∗ ∗⎥ ⎥ ⎥ ⎥ ∗ ∗⎥ ⎥. ⎥ ⎥ −(1 − β)S2 ∗ ⎥ ⎥ ⎦ ⊗ ⊗

Multiplying (6.73) from the left and right, respectively, by ⎡ ⎤ 0 P2−1 0 0 0 0 ⎦ P2 = ⎣ 0 0 0 P2−1 0 0

6.4 Time-Delay Case

93

and its transpose, and then setting Q2 = P2−1 , R2 = P2−1 S2 P2−1 , W2i = P2−1 Li and X2ij = P2 P Xij P P2T leads to (6.61). It also results in (6.62) for l = 2, i.e., ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ P2 P PP2T X11 · · · X1r X211 · · · X21r ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ . ⎥ ⎢ . ⎥⎢ ⎥ ⎢ .. . . ⎥ ⎢ .. . . .. .. ⎥≤0 ⎢ . ⎥⎢ . . .. ⎥ = ⎢ . .. ⎥ ⎢ . . ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ T T X21r X1r P P2T P2 P · · · X2rr · · · Xrr due to (6.49). This proves the necessity. Suﬃciency. Suppose Condition T1 holds, i.e., LMIs (6.60)-(6.62) are true for some matrices Ql > 0, Rl > 0, Wli and Xlij with Xlii symmetrical, l = 1, 2, i, j = 1, 2, . . . , r, j ≥ i. Let Ki and Li be as in (6.63). Setting P1 = Q−1 1 , S2 = R2 ,

−1 P2 = Q2 , S1 = Q−1 1 R1 Q1 , = diag {Q−1 , Q−1 , I}, Q 1

(6.74)

1

we see that (6.60)-(6.61) give T 1ij + Φ 1ji < Q(X 1ij + X1ij Φ )Q,

(6.75)

2ji < X2ij + 2ij + Φ Φ

(6.76)

where

1ij Φ

2ij Φ

T X2ij ,

⎤

⎡ P1 (Ai + Bui Kj ) + (Ai + Bui Kj )T P1

∗ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ −(1 − β)S1 ∗ ⎥ ATτi P1 ⎥ ⎦ Cτ i −I Ci + Di Kj ⎡ ⎤ ∗ P2 (Ai − Li Ej ) + (Ai − Li Ej )T P2 + S2 ⎢ ⎥ =⎣ ⎦. −(1 − β)S2 (Aτ i − Li Eτ j )T P2 ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

T +S1 + γ −2 P1 Bwi Bwj P1

∗

From Schur complements, it is easy to obtain from (6.75) and (6.76) that there exists a scalar λ0 > 0 such that ⎤ ⎡ 1ji Γ T + Γ T 1ij + Φ Φ ij ji ⎦ ⎣ Γij + Γji λ(Φ2ij + Φ2ji ) ⎡ ⎤ 1ij + X T )Q Q(X 0 1ij ⎦, <⎣ (6.77) T 0 λ(X2ij + X2ij ) hold for all λ ≥ λ0 , where ⎡ ⎤ T −KjT Bui P1 0 −KjT DiT ⎦. Γij = ⎣ 0 0 0

94

6. Observer-Based Fuzzy Control: Delay-Independent Method

We next show that after some exchanges of rows and columns, (6.77) yields exactly (6.48). In detail, denote ⎡ ⎤ I0 0 00 ⎢ ⎥ ⎢ ⎥ ⎢0 0 0 I 0⎥ ⎢ ⎥ ⎢ ⎥ I = ⎢ 0 I 0 0 0 ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢0 0 0 0 I ⎥ ⎣ ⎦ 0 0 In1 0 0 Multiplying (6.77) from the left and right, respectively, by I and IT , we have that (6.48) holds for the following matrices P > 0, S > 0 and Xij with any λ ≥ λ0 (i, j = 1, 2, . . . , r, j ≥ i): ⎡ ⎤ ⎡ ⎤ P1 0 S1 0 ⎦, S = ⎣ ⎦, P =⎣ 0 λP2 0 λS2 ⎡ ⎤⎡ ⎤⎡ ⎤ 0 0 Q Q X1ij 0 ⎦⎣ ⎦⎣ ⎦ IT . (6.78) Xij = I ⎣ 1/2 1/2 0 λ I 0 X2ij 0 λ I It can be easily veriﬁed that (6.49) also holds for the above Xij . This proves the suﬃciency and hence completes the proof of the lemma. Now, we are in a position to present the main result for the observer-based H∞ control design. Theorem 6.4.1. There exists a fuzzy control law (6.5) which renders the resulting closed-loop fuzzy system (6.6) asymptotically stable with disturbance attenuation γ if any of Conditions T1 and T2 holds. In this case, the local feedback gains Ki and the observer gains Li are given as in Conditions T1 and T2. Proof. The result follows immediately from Lemmas 6.4.1 and 6.4.2.

Theorem 6.4.1 provides a delay-independent method of observer-based H∞ control for time-delay fuzzy systems. The design scheme only needs a single-step to solve strict LMIs. We remark that the present results can be used to ﬁnd the smallest γ and the procedures are related to convex optimization problems which need no tuning of any parameter. When restricted to delay-free fuzzy systems, the following corollary is straightforward. Corollary 6.4.1. For delay-free case, there exists a fuzzy control law (6.5) (without delay terms) which renders the resulting closed-loop fuzzy system (6.6) (without delay terms) asymptotically stable with disturbance attenuation γ if any of the following Conditions C1 and C2 holds.

6.4 Time-Delay Case

95

Condition C1. There exist matrices Ql > 0, Wli and Xlij with Xlii symmetrical, l = 1, 2, i, j = 1, 2, . . . , r, j ≥ i, such that the following LMIs hold: ⎡ ⎤ ∗ ⎥ Π1ij + Π1ji ⎢ T (6.79) ⎣ ⎦ < X1ij + X1ij , (Ci + Cj )Q1 + Di W1j + Dj W1i −2I T , Π2ij + Π2ji < X2ij + X2ij ⎡ ⎤ Xl11 · · · Xl1r ⎢ ⎥ . ⎥ ⎢ .. . . ⎢ . . .. ⎥ ≤ 0, ⎣ ⎦ T Xl1r · · · Xlrr

(6.80)

(6.81)

where T T T Π1ij = Ai Q1 + Q1 ATi + Bui W1j + W1j Bui + γ −2 Bwi Bwj , T Π2ij = Q2 Ai + ATi Q2 − W2i Ej − EjT W2i .

Condition C2. There exist matrices Ql > 0, Wli and Ylij with Ylii symmetrical, l = 1, 2, i, j = 1, 2, . . . , r, j ≥ i, such that the following LMIs hold: T Π1ij + Π1ji < Y1ij + Y1ij , T Y2ij ,

Π2ij + Π2ji < Y2ij + ⎡ Y111 ··· Y11r (C1 Q1 + D1 W1k )T ⎢ ⎢ .. .. .. .. ⎢ . . . . ⎢ ⎢ T ⎢ ··· Y1rr (Cr Q1 + Dr W1k )T Y11r ⎢ ⎣ C1 Q1 + D1 W1k · · · Cr Q1 + Dr W1k −I ≤ 0, ⎡ Y211 ⎢ ⎢ .. ⎢ . ⎣ T Y21r

k = 1, 2, . . . , r, ⎤ · · · Y21r ⎥ .. ⎥ .. . . ⎥ ≤ 0, ⎦ · · · Y2rr

(6.82) ⎤

(6.83)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (6.84)

(6.85)

where Π1ij and Π2ij are the same as in Condition C1. In any case of Conditions C1 and C2, the local feedback gains Ki and the observer gains Li are given as in Conditions T1 and T2, i.e., of the form (6.68). Proof. It is easy from Theorem 6.4.1.

Note that Condition C2 is exactly that of Theorem 6.3.2. It is not easy to theoretically compare Condition C1 with Condition C2. But later we will use an example (see Example 6.5.2) to verify that Condition C1 can be less conservative than Condition C2 for certain systems. This point reveals that the delay-free results in Corollary 6.4.1 are better than or equivalent to the results in Theorems 6.3.1 and 6.3.2.

96

6. Observer-Based Fuzzy Control: Delay-Independent Method

6.5 Illustrative Examples In this section, we use several examples to illustrate the LMI-based results of the chapter. The ﬁrst example is to show the use and the merits of Theorem 6.3.2. Example 6.5.1. Consider the well-studied example of balancing an inverted pendulum on a cart, see [6] and [98]. The pendulum angle (x1 ) and the angular velocity (x2 ) satisﬁes x˙ 1 = x2 ,

x˙ 2 =

g sin(x1 ) − amlx22 sin(2x1 )/2 − a cos(x1 )u + w, 4l/3 − aml cos2 (x1 )

(6.86)

where g = 9.8 is the gravity constant, u is the force applied to the cart, w is the external disturbance which is assumed to be w = cos(2πt), 2l is the length of the pendulum, m is the mass of the pendulum, M is the mass of the cart, and a = 1/(m + M ). This nonlinear system can be described by a fuzzy model as in [98] with two IF-THEN rules: Plant Rule 1: IF x1 is about 0 THEN x(t) ˙ = A1 x(t) + Bw1 w(t) + Bu1 u(t), z(t) = C1 x(t) + D1 u(t), y(t) = E1 x(t), Plant Rule 2: IF x1 is about ±π/2 THEN x(t) ˙ = A2 x(t) + Bw2 w(t) + Bu2 u(t), z(t) = C2 x(t) + D2 u(t), y(t) = E2 x(t), where

⎡

A1 = ⎣

⎤ 0

1

⎡

⎤ 0

1

⎦ , A2 = ⎣ ⎦, 17.2941 0 12.6305 0 ⎡ ⎤ ⎡ ⎡ ⎤ ⎤ 0 0 0 ⎦ , Bu2 = ⎣ ⎦, Bw1 = Bw2 = ⎣ ⎦ , Bu1 = ⎣ 1 −0.1765 −0.0779 C1 = C2 = 1 0 , D1 = 0.008, D2 = 0.006, E1 = E2 = 1 0 .

The membership functions are h1 (t) = (1 − 1/(1 + exp(−7(x1 − π/4))))(1/(1 + exp(−7(x1 + π/4)))) and h2 (t) = 1 − h1 (t). For a performance index γ = 1, Theorem 6.3.1 works well using LMI toolbox in MATLAB (see [98] for details). However, for γ = 0.2, Theorem 6.3.1 does not lead to a conclusion since the two-step LMI conditions cannot produce a set of feasible solutions. Now, by applying Theorem 6.3.2 to solve the corresponding LMIs for γ = 0.2, we obtain a set of feasible solutions with

6.5 Illustrative Examples

97

K1 = 103 · 1.3904 0.2534 , K2 = 103 · 2.0021 0.3717 , ⎡ ⎡ ⎤ ⎤ 1.2918 1.2918 ⎦ , L2 = ⎣ ⎦. L1 = ⎣ 18.8996 14.2360 So, we conclude that the observer-based fuzzy control law (6.10) with above gains renders the resulting closed-loop fuzzy system asymptotically stable with disturbance attenuation γ = 0.2. In fact, using Theorem 6.3.2, the minimum value of γ can be tuned to be 0.08. This reveals the less conservativeness of Theorem 6.3.2 with respect to Theorem 6.3.1. We have theoretically shown that, for delay-free systems, Condition C2 in Corollary 6.4.1 is exactly that of [80]. The next example is used to illustrate that Condition C1 can produce better result than Condition C2, and is therefore better than those of [80] and [98]. Example 6.5.2. Consider a delay-free fuzzy model modiﬁed from Example 6.5.1 with two IF-THEN rules where ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 1 0 0.3 ⎦ , Bu1 = ⎣ ⎦ , Bw1 = ⎣ ⎦, A1 = ⎣ 17.2941 0 −0.1765 1 ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 0 1 0 0.5 ⎦ , Bu2 = ⎣ ⎦ , Bw2 = ⎣ ⎦, A2 = ⎣ 12.6305 0 −0.0779 0.2 C1 = 1 0 , C2 = 2 0 , D1 = 0.1, D2 = 0.03, E1 = 0.7 1 , E2 = 1 0.5 . Using Condition C2 (i.e., the result of [80]), the minimum γ is computed to be γ = 1.3332. However, using Condition C1, the minimum γ is computed to be γ = 1.3095. This shows that Condition C1 is less conservative than Condition C2 for Example 6.5.2. The next example is to show the merits and the use of Theorem 6.4.1. Example 6.5.3. Consider the above Example 6.5.1 of balancing an inverted pendulum on a cart. In order to illustrate the use of Theorem 6.4.1, we assume that the delay terms are perturbed along values of the scalar λ ∈ [0, 1], and the fuzzy time-delay model considered here is as follows. x(t) ˙ =

z(t) =

y(t) =

2 i=1 2 i=1 2 i=1

hi (t)[(1 − λ)Ai x(t) + λAi x(t − τ (t)) + Bwi w(t) + Bui u(t)], hi (t)[(1 − λ)Ci x(t) + λCi x(t − τ (t)) + Di u(t)], hi (t)[(1 − λ)Ei x(t) + λEi x(t − τ (t))].

(6.87)

98

6. Observer-Based Fuzzy Control: Delay-Independent Method

This fuzzy model is of the form of (6.3). Let λ = 0.1 and τ (t) = 1+0.2 sin(t), and thus τ˙ (t) ≤ 0.2 = β. Using the design method in Theorem 6.4.1, both Conditions T1 and T2 produce the H∞ performance γ as small as 0.0771. For instance, given γ = 0.13, the local feedback gains and observer gains are computed to be By Condition T1 : K1 = 103 · 1.0384 0.2165 , K2 = 103 · 1.6706 0.3465 , ⎡ ⎡ ⎤ ⎤ 1.3978 1.3997 ⎦ , L2 = ⎣ ⎦, L1 = ⎣ 18.6276 13.9651 By Condition T2 : K1 = 103 · 1.8932 0.3753 , K2 = 103 · 2.5505 0.5165 , ⎡ ⎡ ⎤ ⎤ 1.3973 1.3995 ⎦ , L2 = ⎣ ⎦. L1 = ⎣ 18.6282 13.9658 The simulation results are shown in Figures 6.1-6.5. Figure 6.1 shows the response of the state x(t) under initial condition φ(t) = [3, 0]T for t ≤ 0 and initial error e0 = [1, 0]T . Figure 6.2 shows the response of the error e(t) between the true state x(t) and the observed state x (t). Figure 6.3 shows the response of the corresponding control eﬀort u(t). Figure 6.4 shows the ratio of the energy of t t z T (t)z(t) to the energy of wT (t)w(t), i.e., 0 z T (α)z(α)dα/ 0 wT (α)w(α)dα. It √ is seen that the ratio is less than 0.006, which reveals that 0.006 ≈ 0.0775 is less than the prescribed γ = 0.13. When there is no external disturbance (i.e., w(t) = 0), the response of the true state x(t) of the closed-loop fuzzy system is depicted in Figure 6.5 under initial condition φ(t) = [2, 0]T for t ≤ 0 and initial error e0 = [0.5, 0]T . It is seen from Figure 6.5 that the closed-loop fuzzy system is asymptotically stable. Conditions T1 and T2, applied to Example 6.5.3, produce the same minimum γ = 0.0771. The next example is to show the diﬀerence between Conditions T1 and T2. Example 6.5.4. Consider the nominal fuzzy model given in [14] and [73] with two IF-THEN rules and ⎡ ⎡ ⎤ ⎤ −0.1125 −0.02 −0.0125 −0.005 ⎦ , Aτ 1 = ⎣ ⎦, A1 = ⎣ 1 0 0 0 ⎡ ⎡ ⎤ ⎤ −0.1125 −1.527 −0.0125 −0.23 ⎦ , Aτ 2 = ⎣ ⎦, A2 = ⎣ 1 0 0 0 ⎡ ⎤ 1 Bu1 = Bu1 = ⎣ ⎦ , C1 = C2 = 0 1 , Cτ 1 = Cτ 2 = 0 0 . 0

6.6 Conclusion and Notes

99

4 2 0 −2

x (t) 1

x (t) 2

−4 −6 −8 0

15

10

5 Time t

(a) 4 2 0 −2

x (t) 1

x (t) 2

−4 −6 −8 0

10

5

15

Time t

(b) Fig. 6.1. Response of the state x(t) for Example 6.5.3 (a) by Condition T1 Condition T2

(b) by

Let the disturbance matrices and other system matrices be ⎡ ⎡ ⎤ ⎤ 0.3 0.5 ⎦ , Bw2 = ⎣ ⎦ , D1 = 0.5, D2 = 0.3, Bw1 = ⎣ 1 0.2 E1 = 0.7 1 , E2 = 1 0.5 , Eτ 1 = Eτ 2 = 0 0 . For β = 0.2, the minimum γ’s computed from Conditions T1 and T2 in Theorem 6.4.1 are γ = 0.8121 and γ = 0.8304, respectively. This reveals that Condition T1 produces better result than Condition T2 for Example 6.5.4.

6.6 Conclusion and Notes The problem of observer-based H∞ control for T-S fuzzy systems with time-delay has been studied in this chapter. A strict LMI method has been presented for the

100

6. Observer-Based Fuzzy Control: Delay-Independent Method 1

e1(t) e (t) 2

0.5

0

−0.5

−1 0

15

10

5 Time t

(a) 1

e1(t) e (t) 2

0.5

0

−0.5

−1 0

10

5

15

Time t

(b) Fig. 6.2. Response of the error e(t) = x(t) − x (t) for Example 6.5.3 (a) by Condition T1 (b) by Condition T2

fuzzy control design. This method overcomes the drawback of the two-step LMI approach. Also, when conﬁned to delay-free fuzzy systems, the corresponding results are easy corollaries of the present results for delay systems. Illustrative examples have been given to demonstrate the use and merits of the present results. We should point out that the method of this chapter applies to fuzzy control designs when the premise variables in the controller rules are the same as those in the plant rules. For the general case when the premise variables in the controller rules are diﬀerent from those in the plant rules (as in [106]), it is very hard to obtain strict LMI conditions without bringing conservativeness in solving corresponding matrix inequalities. This has to be left as further research. Also notice that other methods, such as fuzzy weighting-dependent method (see Chapter 2 and [120]) and delay-dependent techniques (see, e.g., [42][57] or Chapters 2 and 3), may be useful for reducing conservatism. However, how to adopt the matrix decoupling technique presented in this chapter to establish delay-dependent strict LMI conditions remains unanswered.

6.6 Conclusion and Notes

101

2000 u(t)

1500

1000

500

0

−500 0

5

10

15

Time t

(a) 3000 2500 u(t)

2000 1500 1000

500 0 −500

−1000 0

5

10

15

Time t

(b) Fig. 6.3. Response of the control u(t) for Example 6.5.3 (a) by Condition T1 (b) by Condition T2

102

6. Observer-Based Fuzzy Control: Delay-Independent Method −3

6

x 10

5

4 3 2 1 0 0

5

10

15

10

15

Time t

(a) −3

5

x 10

4

3

2

1

0 0

5 Time t

(b) t t Fig. 6.4. Ratio of 0 z T (α)z(α)dα/ 0 wT (α)w(α)dα for Example 6.5.3 (a) by Condition T1 (b) by Condition T2

6.6 Conclusion and Notes

103

2 1 0 −1

x (t)

−2

1

x2(t)

−3 −4 −5 −6 0

2

4

8

6

Time t

(a) 2 1 0 −1

x1(t)

−2

x (t)

−3

2

−4 −5 −6 0

2

4

6

8

Time t

(b) Fig. 6.5. Response of closed-loop state x(t) for Example 6.5.3 when w(t) = 0 (a) by Condition T1 (b) by Condition T2

7. Observer-Based Fuzzy Control: Delay-Dependent Method

This chapter studies the problem of observer-based H∞ control for TakagiSugeno (T-S) fuzzy systems. It aims to provide a delay-dependent LMI-based method for the control design. It is known that the key important problem in the literature, even for delay-independent case, lies in the diﬃculty of decoupling matrix variables in corresponding matrix inequalities. This chapter suggests a decoupling technique for solving matrix inequalities with coupled variables, and provides an LMI-based algorithm by adopting the idea of the cone complementarity problem. The derivation relies on the appropriate choice of LyaponuvKrasovskii functionals which incorporate the intersections among fuzzy local systems. Illustrative examples are given to show the eﬀectiveness of the present delay-dependent result.

7.1 Introduction In this chapter, we will study observer-based H∞ control for T-S fuzzy systems with state-delay and aim to present delay-dependent design methods. The motivation mainly stems from two facts: (1) There is less work addressing delaydependent control design for T-S fuzzy systems [15][42], and, even less work for observer-based H∞ control for T-S fuzzy systems; (2) The existing design methods for observer-based H∞ control often encounter the diﬃculty of decoupling matrix variables in corresponding matrix inequalities [73][84]. This problem even occurs in delay-free T-S fuzzy systems [80][106][131]. So, in this chapter, we will develop delay-dependent techniques for observer-based H∞ analysis and H∞ synthesis of nonlinear time-delay systems which are represented by a very general T-S fuzzy model with state-delay. It is known that delay-dependent criteria are considered less conservative than delay-independent ones in the analysis and control of dynamical systems with time-delays, particularly when the delay is small. So far, various delay-dependent techniques have been reported for the analysis of time-delay systems, see [37][57][58][24][151]. However, when applying to control design, the current analysis methods usually lead to inconsistent designs or bring conservativeness. In this chapter, we use a diﬀerent type of Lyaponuv-Krasovskii functionals which incorporate the intersections among fuzzy local systems. As a result, a consistent delay-dependent method for the H∞ analysis and H∞ synthesis is presented. The delay-dependent result is in terms of matrix inequalities and an eﬀective LMI-based algorithm, adopting C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 105–124, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

106

7. Observer-Based Fuzzy Control: Delay-Dependent Method

the idea of a so-called cone complementarity problem [40], is developed to solve these matrix inequalities. The delay-dependent result overcomes the drawbacks entailed in existing methods in which non-strict LMI conditions or two-step LMI approaches have to be involved, see [106][131] for delay-free fuzzy systems and [73][84] for time-delay fuzzy systems. Two illustrative examples are given to show the eﬀectiveness and merits of the present results.

7.2 Problem Formulation and Preliminaries Consider a nonlinear system with time-delay which could be approximated by a T-S fuzzy time-delay model of the following form: Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bwi w(t) + Bui u(t), z(t) = Ci x(t) + Cτ i x(t − τ (t)) + Dwi w(t) + Dui u(t),

(7.1)

y(t) = Ei x(t) + Eτ i x(t − τ (t)) + Fwi w(t) + Fui u(t), x(t) = φ(t), t ∈ [−2τ0 , 0], i = 1, 2, . . . , r, where x ∈ Rn , u ∈ Rm1 , w ∈ Rm2 , z ∈ Rn1 and y ∈ Rn2 are the state, the control input, the disturbance, the controlled output and the measured output, respectively; r is the number of IF-THEN rules; Ai , Aτ i , Bwi , Bui , Ci , Cτ i , Dwi , Dui , Ei , Eτ i , Fwi , and Fui are real constant matrices with appropriate dimensions; θj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables and the fuzzy sets; φ is the initial condition; The time-delay τ (t) may be unknown but is assumed to be smooth function of time: τ˙ (t) ≤ β < 1,

τ (t) ≤ τ0 .

By fuzzy blending, the overall fuzzy model is inferred as follows: r νi (θ)[Ai x(t) + Aτ i x(t − τ ) + Bwi w(t) + Bui u(t)] r x(t) ˙ = i=1 i=1 νi (θ) r hi (θ)[Ai x(t) + Aτ i x(t − τ ) + Bwi w(t) + Bui u(t)], = z(t) = y(t) =

i=1 r i=1 r

(7.2)

(7.3)

hi (θ)[Ci x(t) + Cτ i x(t − τ ) + Dwi w(t) + Dui u(t)], hi (θ)[Ei x(t) + Eτ i x(t − τ ) + Fwi w(t) + Fui u(t)],

i=1

x(t) = φ(t), t ∈ [−2τ0 , 0], where θ = [θ1 , . . . , θp ], νi : Rp → [0, 1], i = 1, . . . , r, the membership function is r corresponding to plant rule i, and h (θ) = ν (θ)/ i i i=1 νi (θ). It is obvious that r h (θ) = 1 with h (θ) ≥ 0. i i i=1

7.2 Problem Formulation and Preliminaries

107

The design of observer-based H∞ controllers in this chapter is performed through the parallel distributed compensation [134]: Controller Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x ˙ (t) = Ai x (t) + Aτ i x (t − τ (t)) + Bwi w(t) + Bui u(t) + Li (y(t) − y(t)), (t) + Eτ i x (t − τ (t)) + Fwi w(t) + Fui u(t), y(t) = Ei x (t), u(t) = Ki x x (t) = φ(t), t ∈ [−2τ0 , 0], i = 1, 2, . . . , r,

(7.4)

where Li and Ki are, respectively, the observer gains and controller gains to be chosen. The overall observer-based control law is thus inferred by x ˙ (t) =

r

hi (θ)[Ai x (t) + Aτ i x (t − τ (t)) + Bwi w(t) + Bui u(t)

i=1

+Li (y(t) − y(t))], y(t) = u(t) =

r i=1 r

hi (θ)[Ei x (t) + Eτ i x (t − τ (t)) + Fwi w(t) + Fui u(t)], hi (θ)Ki x (t).

(7.5)

i=1

Combining (7.3) with (7.5), the closed-loop fuzzy system is written as x ˙ (t) = z(t) =

r r i=1 j=1 r r

hi (θ)hj (θ)[Gij x (t) + Mij x (t − τ (t)) + Bi w(t)],

(7.6)

hi (θ)hj (θ)[Hij x (t) + Ni x (t − τ (t)) + Di w(t)],

i=1 j=1

x (t) = φ(t), t ∈ [−2τ0 , 0], where

⎡

x (t) = ⎣

⎤ x(t)

⎦,

⎡ =⎣ φ(t)

⎤ φ(t)

⎦, φ(t) − φ(t) ⎡ ⎡ ⎤ ⎤ Ai + Bui Kj −Bui Kj Aτ i 0 ⎦ , Mij = ⎣ ⎦, Gij = ⎣ 0 Ai − Li Ej 0 Aτ i − Li Eτ j ⎡ ⎤ Bwi ⎦ , Hij = Ci + Dui Kj −Dui Kj , Bi = ⎣ 0 Ni = Cτ i 0 , Di = Dwi . x(t) − x (t)

(7.7)

Our aim in this chapter is to develop techniques for the design of controllers (7.5) such that the equilibrium solution x (t) = 0 of the closed-loop fuzzy

108

7. Observer-Based Fuzzy Control: Delay-Dependent Method

system (7.6) with w = 0 is asymptotically stable and the following H∞ control performance is satisﬁed:

T

T z(t)2 dt ≤ γ 2 w(t)2 dt (7.8) 0

0

for all T > 0 and w ∈ L2 (0, ∞) under zero initial conditions. If this is the case, the fuzzy controller is said to render the closed-loop fuzzy system (7.6) asymptotically stable with disturbance attenuation γ. In the sequel, we simply use hi to denote hi (θ(x(t))) and hi (s) to denote hi (θ(x(s))). Without loss of generality, deﬁne hi (θ(x(σ))) := hi (θ(x(0))) for σ ∈ [−τ0 , 0].

7.3 Main Result In this section, we establish delay-dependent conditions for the design of observer-based fuzzy controllers. As known in the analysis of dynamical systems with time-delays, the key important step is to choose an appropriate LyaponuvKrasovskii functional [24][57][105][151]. However, for T-S fuzzy systems with time-delay, analogous attempts may lead to quite conservative results due to the nonlinear connections between the local linear systems. In this section, we introduce a new type of Lyaponuv-Krasovskii functional which takes into account the nonlinear intersections among the local fuzzy models. We ﬁrst consider the problem of H∞ performance analysis for the unforced system described by (7.6). The following result presents a delay-dependent version of bounded real lemma for T-S fuzzy systems with state-delay. Lemma 7.3.1. The fuzzy time-delay system described by (7.6) is asymptotically stable with disturbance attenuation γ for any time-delay satisfying (7.2) if there exist matrices P > 0, R > 0, S > 0, T > 0 and U > 0 such that the following matrix inequalities hold: ⎡ ⎤ Ωii ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ ⎢√ ⎥ ⎢ τ G P −U ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ 0 ii ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 −S ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ √ ⎢ (7.9) τ0 Mii P −T ∗ ∗ ∗ ⎥ 0 0 ⎢ ⎥ < 0, i = 1, 2, . . . , r, ⎢ ⎥ ⎢ ⎥ ⎢ Hii P 0 Ni P 0 −I ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ T 2 ⎢ BT 0 0 0 Di −γ I ∗ ⎥ i ⎢ ⎥ ⎣ ⎦ √ τ0 Bi −R 0 0 0 0 0

7.3 Main Result

⎡

Ωkl + Ωlk ∗

∗

∗

⎢ ⎢√ ⎢ τ G P −U ∗ ∗ ⎢ 0 kl ⎢ ⎢√ ⎢ τ G P 0 −U ∗ ⎢ 0 lk ⎢ ⎢ ⎢ 0 0 0 −2S ⎢ ⎢ ⎢ √ ⎢ τ0 Mkl P 0 0 0 ⎢ ⎢ ⎢ √ ⎢ τ0 Mlk P 0 0 0 ⎢ ⎢ ⎢ ⎢ Hkl P 0 0 Nk P ⎢ ⎢ ⎢ ⎢ Hlk P 0 0 Nl P ⎢ ⎢ ⎢ ⎢ BT + BT 0 0 0 l ⎢ k ⎢ ⎢ ⎢ 0 0 0 0 ⎢ ⎣ 0 0 0 0

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

−T ∗

∗

∗

∗

∗

∗

0 −T ∗

∗

∗

∗

∗

0

0 −I ∗

∗

∗

∗

0

0

∗

∗

∗

0

0 DkT DlT −2γ 2 I ∗

∗

0

0

0

0

√ τ0 Bk −R ∗

0

0

0

0

√ τ0 Bl 0 −R

0 −I

k < l, k, l = 1, 2, . . . , r,

109

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (7.10)

where Ωij = (Gij + Mij )P + P (Gij + Mij )T +

τ0 Mij (U + T + R)MijT + S, 1−β

i, j = 1, 2, . . . , r. Proof. We ﬁrst show that the H∞ performance is guaranteed and then chose a Lyapunov-Krasovskii functional to prove the asymptotic stability. Set w(t) = 0 for t < 0. Introduce a functional J(t) as J(t) = J1 + J2 + J3 + J4 + J5 ,

(7.11)

where T (t)P −1 x (t), J1 = x

0 t r r hk (s)hl (s) xT (s)GTkl U −1 Gkl x (s)dsdα, J2 = −τ (t)

J3 =

−τ (t)

J4 =

0

0

T −1 hk (s)hl (s) xT (s − τ (s))Mkl T Mkl x (s − τ (s))dsdα,

t+α k=1 l=1

t r

−τ (t)

t+α k=1 l=1 r r t

t

J5 = t−τ (t)

hk (s)wT (s)BkT R−1 Bk w(s)dsdα,

t+α k=1

x T (α)P −1 SP −1 x (α)dα,

110

7. Observer-Based Fuzzy Control: Delay-Dependent Method

and matrices P > 0, R > 0, S > 0, T > 0 and U > 0 are to be chosen. The derivatives of Ji are given by ˙ (t), xT (t)P −1 x J˙1 = 2 J˙2 = τ (t)

r r i=1 j=1

hi hj x T (t)GTij U −1 Gij x (t)

r r

t

−(1 − τ˙ (t))

hk (s)hl (s) xT (s)GTkl U −1 Gkl x (s)ds,

t−τ (t) k=1 l=1

J˙3 = τ (t)

r r i=1 j=1

hi hj x T (t − τ (t))MijT T −1 Mij x (t − τ (t))

r r

t

−(1 − τ˙ (t))

hk (s)hl (s) xT (s − τ (s))

t−τ (t) k=1 l=1 T −1 T Mkl x (s ×Mkl

J˙4 = τ (t)

r

− τ (s))ds,

hi wT (t)BiT R−1 Bi w(t)

i=1

r

t

−(1 − τ˙ (t)) T (t)P −1 SP J˙5 = x

hk (s)wT (s)BkT R−1 Bk w(s)ds,

t−τ (t) k=1 −1

x (t) − (1 − τ˙ (t)) xT (t − τ (t))P −1 SP −1 x (t − τ (t)).

˙ as Next, to the end of our derivation, we need to bound J˙1 . Rewrite x

t r r x ˙ (t) = x ˙ (s)ds + Bi w(t)]. hi hj [(Gij + Mij ) x(t) − Mij t−τ (t)

i=1 j=1

Then, J˙1 =

r r i=1 j=1 r

+2

hi hj x T (t)[P −1 (Gij + Mij ) + (Gij + Mij )T P −1 ] x(t) hi x T (t)P −1 Bi w(t)

i=1

−

r r

t

2 xT (t)P −1 Mij

hi hj

i=1 j=1

t−τ (t)

r r

hk (s)hl (s)

k=1 l=1

(s) + Mkl x (s − τ (s)) + Bk w(s)]ds. ×[Gkl x Since ±2a b ≤ a M a + b M −1 b holds for compatible vectors a and b, and any compatible matrix M > 0, we have

t r r − 2 xT (t)P −1 Mij hk (s)hl (s)Gkl x (s)ds T

t−τ (t)

T

T

k=1 l=1

τ (t) T x (t)P −1 Mij U MijT P −1 x (t) ≤ 1−β

7.3 Main Result

t

+(1 − β)

r r

111

hk (s)hl (s) xT (s)GTkl U −1 Gkl x (s)ds,

t−τ (t) k=1 l=1

where we set aT = x T (t)P −1 Mij , b = Gkl x (s) and M = (1 − β)−1 U . Similarly, we have

t r r − 2 xT (t)P −1 Mij hk (s)hl (s)Mkl x (s − τ (s))ds t−τ (t)

k=1 l=1

τ (t) T x (t)P −1 Mij T MijT P −1 x (t) ≤ 1−β

t r r T −1 hk (s)hl (s) xT (s − τ (s))Mkl T Mkl x (s − τ (s))ds, +(1 − β) t−τ (t) k=1 l=1

and

t

−

2 xT (t)P −1 Mij

t−τ (t)

≤

r r

hk (s)hl (s)Bk w(s)ds

k=1 l=1

τ (t) T x (t)P −1 Mij RMijT P −1 x (t) 1−β

t r hk (s)wT (s)BkT R−1 Bk w(s)ds. +(1 − β) t−τ (t) k=1

Noticing the expression of z T (t)z(t) in 7.6, it follows that ˙ + z T (t)z(t) − γ 2 wT (t)w(t) J(t) ⎡ ⎤T ⎡ ⎤ x (t) x (t) r r ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ≤ hi hj ⎢ x (t − τ (t)) ⎥ Υij ⎢ x (t − τ (t)) ⎥ ⎣ ⎦ ⎣ ⎦ i=1 j=1 w(t) w(t) where

⎡ Υij11

T Hij Ni

(7.12)

−1

⎤ T Hij Di ⎥

P Bi + ⎢ ⎢ ⎢ ⎢ −P −1 SP −1 + NiT Ni ⎢ NiT Hij NiT Di ⎢ Υij = ⎢ +τ0 MijT T −1 Mij ⎢ ⎢ ⎢ ⎢ T −1 −γ 2 I + DiT Di ⎣ Bi P + DiT Hij DiT Ni +τ0 BiT R−1 Bi

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

with T Υij11 = P −1 (Gij + Mij ) + (Gij + Mij )T P −1 + τ0 GTij U −1 Gij + Hij Hij τ0 P −1 Mij (U + T + R)MijT P −1 + P −1 SP −1 . + 1−β

112

7. Observer-Based Fuzzy Control: Delay-Dependent Method

Therefore, if (7.9) and (7.10) hold, which by Schur complement are equivalent to Υii < 0 and Υkl + Υlk < 0 for i = 1, 2, . . . , r and k < l, k, l = 1, 2, . . . , r, then we obtain from (7.12) that the performance in (7.8) is satisﬁed (due to J(T ) ≥ 0 and J(0) = 0 when subject to zero initial conditions). Next, we prove the asymptotic stability of system (7.6) with w = 0. Choose 4 a Lyapunov-Krasovskii functional V ( xt ) = i=1 Vi , where Vi = Ji , i = 1, 2, 3, V4 = J5 , and Ji ’s remain the same as in (7.11). Here, x t = x (t+σ), σ ∈ [−2τ0 , 0]. It is seen that there exist two scalars d1 > 0 and d2 > 0 such that x(t)2 ≤ V ≤ d2 d1

sup

xt 2 .

σ∈[−2τ0 ,0]

For instance, d1 and d2 can be chosen as d1 = λmin (P −1 ), d2 = λmax (P −1 ) + τ0 λmax (P −1 SP −1 ) + τ02 max{λmax (GTkl U −1 Gkl )} k,l

+τ02

T −1 max{λmax (Mkl T Mkl )}. k,l

If the conditions in (7.9) and (7.10) hold, a deduction similar to the above procedure leads to that the time derivative of V along the solutions of (7.6) with w = 0 satisﬁes V˙ < 0. This proves the motion of (7.6) with w = 0 is asymptotically stable. This completes the proof. In case of constant Gij , Mij and Hij , the conditions (7.9) and (7.10) in Lemma 7.3.1 are strict LMIs with respect to variables P > 0, R > 0, S > 0, T > 0 and U > 0. Since Gij , Mij and Hij actually contain design variables Li and Ki , we next develop a decoupling technique to equivalently transform (7.9) and (7.10) to alternative forms of matrix inequalities and give an LMI-based algorithm for computation of Li and Ki . Lemma 7.3.2. There exist matrices P > 0, R > 0, S > 0, T > 0, U > 0, Ki and Li such that (7.9) and (7.10) hold if and only if there exist matrices X1 > 0, X2 > 0, R1 > 0, S1 > 0, S2 > 0, T1 > 0, T2 > 0, U1 > 0, U2 > 0, Yi and Wi such that the following matrix inequalities hold: ⎡ ⎤ Ψ1ii ∗ ∗ ∗ ∗ ∗ ∗ ⎢ ⎥ ⎢√ ⎥ ⎢ τ0 (Ai X1 + Bui Yi ) −U1 ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ ⎥ 0 0 −S1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ √ ⎢ ⎥ < 0, (7.13) 0 0 τ A X −T ∗ ∗ ∗ 0 τ i 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 Cτ i X1 0 −I ∗ ∗ ⎥ ⎢ Ci X1 + Dui Yi ⎢ ⎥ ⎢ ⎥ ⎢ T T 2 Bwi 0 0 0 Dwi −γ I ∗ ⎥ ⎢ ⎥ ⎣ ⎦ √ 0 0 0 0 0 τ0 Bwi −R1

7.3 Main Result

⎡ ⎣

−S2

⎤

∗

√ τ0 (X2 Aτ i − Wi Eτ i ) −T2

⎦ < 0,

⎡

(7.14)

∗

Ψ2ii

∗

⎤

∗

⎥ ⎢ ⎥ ⎢ √ ⎥ ⎢ τ0 (X2 Ai − Wi Ei ) −U2 ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ < 0, ⎢ ⎥ ⎢ −1 τ0 T T T ∗ ⎥ ⎢ 1−β (Aτ i X2 − Eτ i Wi ) 0 −X2 U2 X2 ⎥ ⎢ ⎦ ⎣ −1 τ0 T T T (A X − E W ) 0 0 −X T X 2 2 2 τi 2 τi i 1−β ⎡

Ψ1kl + Ψ1lk

⎢ √τ0 (Ak X1 + Buk Yl ) ⎢ ⎢√ ⎢ τ0 (Al X1 + Bul Yk ) ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ C X +D Y ⎢ k 1 uk l ⎢ ⎢ Cl X1 + Dul Yk ⎢ ⎢ T + BT Bwk ⎢ wl ⎢ ⎣ 0 0

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

−U1

∗

∗

∗

∗

∗

∗

∗

∗

∗

0

−U1

∗

∗

∗

∗

∗

∗

∗

∗

−2S1

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

−T1

∗

∗

∗

∗

∗

0

0

0

0

√ τ0 Aτ k X1 −T1

0

√ τ0 Aτ l X1

0

0

Cτ k X1

0

0

−I

∗

∗

∗

∗

0

0

Cτ l X1

0

0

0

−I

∗

∗

∗

−2γ 2 I

∗

∗

0

0

113

0

0

0

0

0

T T Dwk Dwl

0

0

0

0

0

0

0

√ τ0 Bwk −R1

0

0

0

0

0

0

0

√ τ0 Bwl

0

∗

(7.15)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−R1

(7.16) ⎡

−2S2

∗

∗

⎤

⎢ ⎥ ⎢√ ⎥ ⎢ τ (X A − W E ) −T ∗ ⎥ < 0, k τl 2 ⎢ 0 2 τk ⎥ ⎣ ⎦ √ τ0 (X2 Aτ l − Wl Eτ k ) 0 −T2 ⎡

Ψ2kl + Ψ2lk

∗

⎢ √τ0 (X2 Ak − Wk El ) −U2 ⎢ ⎢ √τ (X A − W E ) 0 ⎢ 0 2 l l k ⎢ ⎢ τ0 (ATτk X2 − EτTl WkT ) 0 ⎢ 1−β ⎢ τ0 ⎢ 1−β T T (AT 0 τ l X2 − Eτ k Wl ) ⎢ ⎢ τ0 ⎢ 1−β T T (AT 0 τ k X2 − Eτ l Wk ) ⎣

τ0 T T (AT τ l X2 − Eτ k Wl ) 1−β

0

(7.17)

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

−U2

∗

∗

∗

∗

0

−1 −X2 U2 X2

∗

∗

∗

0

0

−1 −X2 U2 X2

∗

∗

0

0

0

−1 −X2 T2 X2

∗

0

0

0

0

−1 −X2 T2 X2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (7.18)

114

7. Observer-Based Fuzzy Control: Delay-Dependent Method

for i = 1, 2, . . . , r and k < l, k, l = 1, 2, . . . , r, where T Ψ1ij = (Ai + Aτ i )X1 + X1 (Ai + Aτ i )T + Bui Yj + YjT Bui τ0 Aτ i (U1 + T1 + R1 )ATτi , +S1 + 1−β Ψ2ij = X2 (Ai + Aτ i ) + (Ai + Aτ i )T X2 + S2 −Wi (Ej + Eτ j ) − (Ej + Eτ j )T WiT ,

i, j = 1, 2, . . . , r. If this is the case, the matrices Ki and Li are given by Ki = Yi X1−1 ,

Li = X2−1 Wi ,

i = 1, 2, . . . , r.

(7.19)

Proof. The proof involves a block matrix partition method. Necessity. Suppose there exist matrices P > 0, R > 0, S > 0, T > 0, U > 0, Ki and Li such that (7.9) and (7.10) hold. Firstly, Partition P , R, S, T and U as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⊗ ∗ ⊗ ∗ ⊗ ∗ ⎦, R = ⎣ ⎦, S = ⎣ ⎦, P =⎣ 2 2 ⊗X ⊗R ⊗ S2 ⎤ ⎡ ⎤ ⎡ ⊗ ∗ ⊗ ∗ ⎦, U = ⎣ ⎦, (7.20) T =⎣ 2 ⊗U ⊗ T2 2 > 0, S2 > 0, T2 > 0, U 2 > 0 and ⊗ denotes a block that is not 2 > 0, R where X used in the proof. Substituting (7.20) into (7.9), and after some eﬀorts similar to the proof of Lemma 7.3.2, we have ⎡ ⎤ 2ii Φ ∗ ∗ ∗ ⎢ ⎥ ⎢ ⎥ ⎢√ ⎥ 2 −U 2 ⎢ τ0 (Ai − Li Ei )X ∗ ∗ ⎥ ⎢ ⎥ < 0, (7.21) ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ⎥ 0 0 −S2 ⎣ ⎦ √ 2 −T2 τ0 (Aτ i − Li Eτ i )X 0 0 where 2ii = Φ

τ0 2 + T2 + R 2 )(Aτ i − Li Eτ i )T + S2 (Aτ i − Li Eτ i )(U 1−β 2 + X 2 [Ai + Aτ i − Li (Ei + Eτ i )]T . +[Ai + Aτ i − Li (Ei + Eτ i )]X

−1 , S2 = It is easy to obtain that (7.21) gives (7.14) and (7.15) by setting X2 = X 2 −1 S2 X −1 T2 X −1 U −1 Li . Similarly, −1 , T2 = X −1 , U2 = X 2 X −1 and Wi = X X 2 2 2 2 2 2 2 (7.17) and (7.18) can be veriﬁed.

7.3 Main Result

115

The rest of the necessity is to verify (7.13) and (7.16). Rewrite (7.9) as ⎡

⎤

P −1 (Gii + Mii )

⎢ ⎢ +(Gii + Mii )T P −1 ⎢ ∗ ∗ ∗ ∗ ⎢ ⎢ +P −1 SP −1 ⎢ ⎢ ⎢ +τ0 GTii U −1 Gii ⎢ ⎢ ⎢ τ0 ⎢ MiiT P −1 −U −1 ∗ ∗ ∗ ⎢ 1−β ⎢ ⎢ τ0 ⎢ MiiT P −1 0 −T −1 ∗ ∗ ⎢ 1−β ⎢ ⎢ τ0 ⎢ MiiT P −1 0 0 −R−1 ∗ ⎢ 1−β ⎢ ⎢ ⎢ ⎢ −P −1 SP −1 ⎢ 0 0 0 0 ⎢ ⎢ +τ0 MiiT T −1 Mii ⎢ ⎢ ⎢ Hii 0 0 0 Ni ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 0 B T P −1 i

∗

∗ ∗ ∗

∗ −I DiT

< 0.

⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ τ0 BiT R−1 Bi ⎥ ⎦ −γ 2 I

(7.22)

We partition P −1 , P −1 SP −1 , U −1 , T −1 and R−1 as ⎡ P −1 = ⎣ ⎡ U −1 = ⎣

1 ∗ X ⊗ ⊗ 1 ∗ U ⊗ ⊗

⎡

⎤ ⎦, ⎤ ⎦,

P −1 SP −1 = ⎣ ⎡ T −1 = ⎣

T1 ∗ ⊗ ⊗

S1 ∗

⊗ ⊗ ⎤ ⎦,

⎤ ⎦, ⎡

R−1 = ⎣

1 ∗ R ⊗ ⊗

⎤ ⎦,

(7.23)

1 > 0, T1 > 0 and U 1 > 0. Substituting (7.23) 1 > 0, S1 > 0, R where X into (7.22), and after some matrix manipulations, we have (7.13) by setting −1 , S1 = X −1 S1 X −1 , R1 = R −1 and Yi = Ki X −1 , T1 = T−1 , U1 = U −1 . X1 = X 1 1 1 1 1 1 1 Similarly, (7.16) can be veriﬁed. This proves the necessity. Suﬃciency. Suppose there exist matrices X1 > 0, X2 > 0, R1 > 0, S1 > 0, S2 > 0, T1 > 0, T2 > 0, U1 > 0, U2 > 0, Yi and Wi such that (7.13)-(7.18) hold. Set Ki and Li as in (7.19). From (7.14) and (7.15), we have that

116

7. Observer-Based Fuzzy Control: Delay-Dependent Method

⎡

(1, 1)

∗

⎢ ⎢√ ⎢ τ0 (Ai − Li Ei )X2−1 −X2−1 U2 X2−1 ⎢ ⎢ ⎢ 0 0 ⎣ 0

0

∗

∗

∗

∗

−X2−1 S2 X2−1 √

τ0 (Aτ i −

Li Eτ i )X2−1

< 0,

∗

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−X2−1 T2 X2−1

(7.24)

where (1, 1) := [Ai + Aτ i − Li (Ei + Eτ i )]X2−1 + X2−1 [Ai + Aτ i − Li (Ei + Eτ i )]T τ0 (Aτ i − Li Eτ i )X2−1 (U2 + T2 )X2−1 (Aτ i − Li Eτ i )T + 1−β +X2−1 S2 X2−1 . Denote the left-hand matrices in (7.13) and (7.24) by Θi and Ξi , respectively. Letting ⎡ ⎤ ⎡ ⎤ X1 0 0 S1 ⎦, S = ⎣ ⎦, P =⎣ (7.25) 0 X2−1 0 X2−1 S2 X2−1 ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 0 T1 R1 0 U1 ⎦, T = ⎣ ⎦, ⎦, U = ⎣ R=⎣ 0 1 I 0 X2−1 U2 X2−1 0 X2−1 T2 X2−1 and submitting into the left-hand matrices in (7.9), and after exchanges of rows and columns, yields the following matrix: ⎡ ⎤ Θ ∗ ∗ ⎢ i ⎥ ⎢ ⎥ (7.26) ⎢ Γi Ξi + 1 Ξ i ∗ ⎥ , ⎣ ⎦ 0 0 −1 In where Γi is a ﬁxed matrix independent of and 1 , and ⎡ ⎤ τ0 (Aτ i − Li Eτ i )(Aτ i − Li Eτ i )T 0 1−β ⎦. Ξi = ⎣ 0 0 It is easy to verify that the matrix of (7.26) is negative deﬁnite for suﬃciently small scalars > 0 and 1 > 0. This implies that (7.9) holds for the above matrices P > 0, R > 0, S > 0, T > 0, U > 0, Ki and Li with suﬃciently small scalars > 0 and 1 > 0. Similarly, it can be shown that (7.10) also holds for these matrices with suﬃciently small scalars > 0 and 1 > 0. This completes the proof. Now, we are in a position to present the delay-dependent result for the H∞ control design.

7.3 Main Result

117

Theorem 7.3.1. There exists a fuzzy control law (7.5) which renders the resulting closed-loop fuzzy system (7.6) asymptotically stable with disturbance attenuation γ for any time-delay satisfying (7.2) if there exist matrices X1 > 0, X2 > 0, R1 > 0, S1 > 0, S2 > 0, T1 > 0, T2 > 0, U1 > 0, U2 > 0, Yi and Wi , i = 1, 2, . . . , r, such that the matrix inequalities in (7.13)-(7.18) hold. In this case, the local feedback gains Ki and the observer gains Li are given by (7.19). Proof. The result follows immediately from Lemmas 7.3.1 and 7.3.2.

Note that (7.13), (7.14), (7.16) and (7.17) are strict LMIs with respect to variables X1 > 0, X2 > 0, R1 > 0, S1 > 0, S2 > 0, T1 > 0, T2 > 0, U1 > 0, U2 > 0, Yi and Wi while (7.15) and (7.18) are not. We need to develop LMI-based algorithm to solve the matrix inequalities in Theorem 7.3.1. Remark 7.3.1. If we require U2 ≤ X2 and T2 ≤ X2 , then (7.15) and (7.18) could be represented as strict LMIs with all terms of X2 U2−1 X2 and X2 T2−1 X2 being replaced by X2 . This is due to the fact that U2 ≤ X2 and T2 ≤ X2 yield X2 U2−1 X2 ≥ X2 and X2 T2−1 X2 ≥ X2 . But this strict LMI method brings conservativeness arose from the constraints U2 ≤ X2 and T2 ≤ X2 . Next we give an eﬃcient LMI-based method to solve the matrix inequalities in Theorem 7.3.1. Let us introduce two matrices Qu > 0 and Qt > 0, and let X2 U2−1 X2 ≥ Qu and X2 T2−1 X2 ≥ Qt . Then, (7.15) and (7.18) are equivalent to ⎤ ⎡ Φ1ii ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ √ ⎢ τ0 (X2 Ai − Wi Ei ) −U2 ∗ ∗ ⎥ ⎥ ⎢ ⎥ ⎢ (7.27) ⎥ < 0, ⎢ ⎥ ⎢ τ0 T T T ⎢ 1−β (Aτ i X2 − Eτ i Wi ) 0 −Qu ∗ ⎥ ⎥ ⎢ ⎦ ⎣ τ0 T T T (A X − E W ) 0 0 −Q 2 t τ i τ i i 1−β ⎡ Φ1kl + Φ1lk

∗

∗

∗

∗

∗

∗

⎤

⎢ ⎥ ⎢ ⎥ ⎢ √ ⎥ ⎢ τ0 (X2 Ak − Wk El ) −U2 ∗ ∗ ∗ ∗ ∗ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ √ ⎥ τ0 (X2 Al − Wl Ek ) 0 −U2 ∗ ∗ ∗ ∗ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ τ0 T T T 0 −Qu ∗ ∗ ∗ ⎥ < 0, ⎢ 1−β (Aτ k X2 − Eτ l Wk ) 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ τ T T T 0 ⎢ 1−β ⎥ (A X − E W ) 0 0 0 −Q ∗ ∗ u τl 2 τk l ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ τ0 T T T ⎢ 0 0 0 −Qt ∗ ⎥ ⎢ 1−β (Aτ k X2 − Eτ l Wk ) 0 ⎥ ⎢ ⎥ ⎣ ⎦ τ0 T T T 0 0 0 0 −Qt 1−β (Aτ l X2 − Eτ k Wl ) 0

(7.28)

118

7. Observer-Based Fuzzy Control: Delay-Dependent Method

⎡ ⎣

¯2 ¯u X Q ¯2 U ¯2 X

⎤

⎡

⎦ ≥ 0,

⎣

⎤

¯2 ¯t X Q

⎦ ≥ 0,

¯ 2 T¯2 X

(7.29)

where ¯ 2 = X −1 , X 2

¯2 = U −1 , U 2

T¯2 = T2−1 ,

¯ t = Q−1 . Q t

¯ u = Q−1 , Q u

(7.30)

Now, we adopt the idea of solving a so-called cone complementarity problem which was initiated in [40] and has been further used in [105] to design delaydependent controllers for retarded type systems. The nonlinear minimization problem is proposed as follows: ¯ 2 + U2 U ¯2 + T2 T¯2 + Qu Q ¯ u + Qt Q ¯ t) Minimize Trace(X2 X subject to LMIs (7.13), (7.14), (7.16), (7.17), (7.27), (7.28), (7.29) and ⎤

⎡ ⎣ ⎡ ⎣

X2 I ¯2 I X Qu I ¯u I Q

⎡

⎦ ≥ 0,

⎣

⎤

⎡

⎦ ≥ 0,

⎣

⎤ U2 I ¯2 I U Qt I ¯t I Q

⎡

⎦ ≥ 0,

⎣

⎤

⎤ T2 I I T¯2

⎦ ≥ 0,

⎦ ≥ 0.

(7.31)

With respect to the above minimization problem, we develop the following algorithm which is analogue to that of [40] with some modiﬁcations suitable for solving our problem. In this algorithm, (7.15) and (7.18) are used as a stopping criterion. For simplicity, in each iteration k = 0, 1, . . ., we will use the notation Sk to denote a feasible set of solutions to LMIs (7.13), (7.14), (7.16), (7.17), (7.27), (7.28), (7.29) and (7.31), i.e., (k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

u

u

t

t

Sk := {X1 , X2 , R1 , S1 , S2 , T1 , T2 , U1 , U2 , (k) (k) ¯ (k) ¯ (k) ¯ (k) ¯ (k) , Q(k) , Q ¯ (k) }. Y ,W ,X , U , T , Q(k) , Q i

i

2

2

2

Algorithm 7.1. Step 1. Find a feasible set of solutions Sk to LMIs (7.13), (7.14), (7.16), (7.17), (7.27), (7.28), (7.29) and (7.31). If none, then stop with the conclusion that the matrix inequalities (7.13)-(7.18) are not solvable. Otherwise, set k = 0 and go to Step 2. If (7.15) and (7.18) are satisﬁed under Sk , then stop with the conclusion that the set of solutions Sk solves matrix inequalities (7.13)-(7.18). Otherwise, go to Step 3. Find a set of solutions Sk+1 that solves the following LMI problem: (k) ¯ (k) ¯ (k) ¯ ¯ (k) ¯ (k) ¯ (k) Minimize Tr(X2 X 2 + X2 X2 + U2 U2 + U 2 U2 + T 2 T 2 + T 2 T 2 + (k) ¯ (k) (k) (k) ¯ u Qu + Q Q ¯ Qt )) ¯t + Q Qu Qu + Q t t subject to LMIs (7.13), (7.14), (7.16), (7.17), (7.27), (7.28), (7.29) and (7.31). Step 4. If the trace in Step 3 is 10n, then stop with the conclusion that the set of solutions Sk+1 solves (7.13)-(7.18). Otherwise, go to

7.4 Illustrative Examples

119

Step 5. If (7.15) and (7.18) are satisﬁed under Sk+1 , then stop with the conclusion that the set of solutions Sk+1 solves (7.13)-(7.18). Otherwise, go to Step 6. If k is equal to a speciﬁed number (say kmax ), then stop. Otherwise, if k < kmax , then set k = k + 1 and go to Step 3. Regarding Algorithm 7.1, we give some remarks below. (a) Note that, by virtue of Theorem 2.1 of [40], the sequence of traces to be minimized in Step 3 is decreasing and bounded below by 10n. So, the sequence converges to some value Tropt ≥ 10n if the algorithm continues, where equality holds if and only if the condition in (7.30) is satisﬁed at the optimum. (b) We should point out that if the matrix inequalities (7.13)-(7.18) are solvable, then Algorithm 7.1 will ﬁnd a set of solutions by choosing a large kmax . Conversely, if Algorithm 7.1 converges after a large number of iterations and approaches the optimum Tropt > 10n, then (7.13)-(7.18) are not solvable. However, Tropt > 10n does not mean that (7.13)-(7.18) are not solvable. In case of Tropt > 10n, the algorithm may still stop at some iteration and ﬁnd a set of solutions to (7.13)-(7.18). This shows that the algorithm is powerful and guaranteed to generate a set of solutions to (7.13)-(7.18) if exists. Theorem 7.3.1 and Algorithm 7.1 provide a delay-dependent LMI-based method for observer-based H∞ control of the considered time-delay fuzzy systems. We remark that these results can be used to ﬁnd the smallest γ for a given τ0 , or the largest τ0 for a given γ. The ﬁnding procedures are related to convex optimization problems which need no tuning of any parameter. The potentials of the present delay-dependent results will be illustrated by examples in the next section.

7.4 Illustrative Examples In this section, we use two examples to illustrate the use and the eﬀectiveness of the results. The ﬁrst example is taken from [73] and only the nominal model is studied here. Example 7.4.1. Consider the nominal fuzzy model given in [73], which is of the form (7.1) with two IF-THEN rules and where ⎡ ⎡ ⎤ ⎤ −0.1125 −0.02 −0.0125 −0.005 ⎦ , Aτ 1 = ⎣ ⎦, A1 = ⎣ 1 0 0 0 ⎡ A2 = ⎣

−0.1125 −1.527 1

0

⎤ ⎦,

⎡ Aτ 2 = ⎣

−0.0125 −0.23 0

0

⎤ ⎦,

120

7. Observer-Based Fuzzy Control: Delay-Dependent Method

Bw1 = Bu1 = Bw2 = Bu2 ⎡ C1 = C2 = ⎣

⎡ ⎤ 1 = ⎣ ⎦, 0

⎤ 01

⎦,

Cτ 1 = Cτ 2 = 02×2 ,

00

Dw1 = Dw2 = 02×1 , E1 = E2 = 0 1 ,

Du1 = Du2

⎡ ⎤ 0 = ⎣ ⎦, 1

Eτ 1 = Eτ 2 = 01×2 ,

Fw1 = Fw2 = Fu1 = Fu2 = 0. The time-delay is τ (t) = 4 + 0.5 cos(0.9t), and thus τ˙ (t) ≤ 0.5 = β. Now, we use our delay-dependent methods to examine this example. Firstly, let us adopt the strict LMI method stated in Remark 7.3.1. For γ = 1.0001, it is computed that the maximum value of τ0 is 0.0028 which ensures the feasibility of corresponding LMIs. In this case, the local gains are computed to be K1 = 107 · −1.2068 −1.2389 , K2 = 107 · −1.2068 −1.2389 , ⎡ ⎡ ⎤ ⎤ 1.5808 0.0106 ⎦ , L2 = ⎣ ⎦. L1 = ⎣ 0.9996 1.0811 So, the closed-loop system of the form (7.6) with delay bound τ0 = 0.0028 is asymptotically stable with disturbance attenuation γ ≥ 1.0001. If γ is allowed to be 1.5, it is computed that the maximum value of τ0 is 0.89, in which case, the local gains are computed to be K1 = −15.1941 −12.5248 , K2 = −15.3505 −12.6134 , ⎡ ⎡ ⎤ ⎤ 1.6279 −0.3502 ⎦ , L2 = ⎣ ⎦. L1 = ⎣ 2.7529 2.2884 This shows the eﬀectiveness of the delay-dependent method in Remark 7.3.1. However, if γ is allowed to be further increased (i.e., γ > 1.5), the method in Remark 7.3.1 could not provide an increase of the delay bound τ0 = 0.89. We now adopt Algorithm 7.1, which is powerful in solving the matrix inequalities in Theorem 7.3.1, to further examine this example. For γ = 1.0001, it is computed that the maximum value of τ0 remains 0.0028. But, in this case, the local gains are computed small as given by K1 = −1.0000 0.0000 , K2 = −1.0000 0.0000 , ⎡ ⎡ ⎤ ⎤ 2.3260 0.8533 ⎦ , L2 = ⎣ ⎦. L1 = ⎣ 1.9527 2.0421

7.4 Illustrative Examples

121

However, for γ = 1.5, it is computed that the value of τ0 could be allowed much larger than 0.89. For instance, under τ0 = 9, the closed-loop system of the form (7.6) is still veriﬁed asymptotically stable with disturbance attenuation γ = 1.5, in which case the local gains are computed to be K1 = −59.4038 −42.3036 , K2 = −59.4013 −42.1938 , ⎡ ⎡ ⎤ ⎤ 2.7415 0.5926 ⎦ , L2 = ⎣ ⎦. L1 = ⎣ 3.7236 3.7497 The above is obtained after 60 iterations under the same initial feasible solutions as that for γ = 1.0001. Please note that, in this case, the corresponding minimum trace in Step 3 of Algorithm 7.1 is 20.0797 while the optimum 20 could be reached at the 66th iteration. If γ is allowed to be further large, Algorithm 7.1 could still ﬁnd feasible solutions for very large τ0 . This shows the method using Algorithm 7.1 is quite eﬃcient in solving the matrix inequalities in Theorem 7.3.1. In fact, it is not only less conservative than using the method in Remark 7.3.1 but also guaranteed to generate a feasible set of solutions to the matrix inequalities in Theorem 7.3.1 if the solutions do exist. The next example is an artiﬁcial one and is used to show that the existing delayindependent stabilization techniques cannot apply to the fuzzy system while the present delay-dependent design method works well. Example 7.4.2. Consider the fuzzy model of the form (7.1) with two IF-THEN rules and where ⎡ ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎤ 00 −1 −1 0 0 ⎦ , Aτ 1 = ⎣ ⎦ , Bw1 = ⎣ ⎦ , Bu1 = ⎣ ⎦ , A1 = ⎣ 01 0 −0.9 0.1 1 ⎡ ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎤ −5 −1 −0.1 −0.2 0.2 1 ⎦ , Aτ 2 = ⎣ ⎦ , Bw2 = ⎣ ⎦ , Bu2 = ⎣ ⎦ , A2 = ⎣ 1 0 0 0 0 1 ⎡ ⎡ ⎤ ⎤ 0 0.5 0.1 0.1 ⎦ , C2 = ⎣ ⎦ , Cτ 1 = Cτ 2 = 02×2 , C1 = ⎣ 0 1 0.5 0.2 Dw1 = Dw2 = Du1 = Du2 = 02×1 , , Fw1 = Fw2 = Fu1 = Fu2 = 0, E1 = E2 = 0.6 −0.1 , Eτ 1 = Eτ 2 = 0.3 0 . We assume that the time-delay τ (t) is unknown constant (thus β = 0) but bounded by τ0 > 0. Note that the ﬁrst local system (A1 , Aτ 1 , Bu1 ) is the well-known example studied in [24] and the references therein. It was shown in [24] that the existing delay-independent stabilization methods and H∞ control techniques could not be applied to this local system as (A1 , Bu1 ) is not stabilizable. Therefore, for this fuzzy model without disturbance terms, the delayindependent fuzzy control methods in [9][10][158] are not applicable due to the same reason that (A1 , Bu1 ) is not stabilizable.

122

7. Observer-Based Fuzzy Control: Delay-Dependent Method

Now, we use the delay-dependent H∞ control method in Theorem 7.3.1 to verify the closed-loop asymptotic stability with disturbance attenuation under appropriate τ0 and γ. For τ0 = 0.3 and γ = 0.9, we use Algorithm 7.1 to generate local gains as K1 = −0.5147 −66.1897 , K2 = −0.6054 −63.1429 , ⎡ ⎡ ⎤ ⎤ 2.4335 3.3474 ⎦ , L2 = ⎣ ⎦. L1 = ⎣ −3.7559 −4.1641 The above is obtained after 8 iterations in which the corresponding minimum trace in Step 3 of Algorithm 7.1 is 21.7556. It is seen that, for this example, although the optimum trace 20 is not reached at the 8th iteration, Algorithm 7.1 still generates a set of solutions to the matrix inequalities in Theorem 7.3.1. For simulation purpose, we simply choose w(t) = cos(t)/(t + 1)2 as the disturbance and select the membership functions as h1 = sin2 (t) and h2 = cos2 (t). The ratio of the energy of z T (t)z(t) to the energy of wT (t)w(t), i.e., t T t T 7.1(a). It is seen that the 0 z (α)z(α)dα/ 0 w (α)w(α)dα, is shown in Figure √ −4 ratio is less than 1.4235 × 10 , which reveals that 1.4235 × 10−4 = 0.0119 is less than the prescribed γ = 0.9. The corresponding control signal is depicted in Figure 7.1(b). When w(t) = 0, the responses of the state x(t) and the observed state x (t) of the closed-loop fuzzy system are depicted in Figure 7.2, under initial = [0.8, − 0.2]T . Figure 7.2 shows that the condition φ(t) = [1, − 0.4]T and φ(t) closed-loop fuzzy system is asymptotically stable.

7.5 Conclusion The problem of observer-based H∞ control for T-S fuzzy systems with timedelay has been studied in this chapter. Consistent delay-dependent methods have been developed for the H∞ performance analysis and H∞ control design. The derivation is based on a new decoupling technique which can equivalently transform related matrix inequalities into desired ones. An eﬃcient LMI-based algorithm has been given to solve corresponding matrix inequalities. Illustrative examples have been presented to demonstrate the potentials of the method.

7.5 Conclusion

123

−4

1.5

x 10

1

0.5

0

6

5

4

3

2

1

0

Time t

(a) 0.005

0

−0.005

−0.01

−0.015

−0.02

−0.025

0

1

2

3

5

4

6

7

8

9

10

Time t

(b) Fig. 7.1. (a) Ratio of energy of z T (t)z(t) to energy of wT (t)w(t) control u(t)

(b) Response of

124

7. Observer-Based Fuzzy Control: Delay-Dependent Method 1.2

x (t)

1

1

0.8

hat(x) (t) 1

0.6

0.4

hat(x)2(t) 0.2

0

x2(t)

−0.2

−0.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time t

Fig. 7.2. Responses of x(t) and observed state x (t) of closed-loop fuzzy system when w(t) = 0

8. Output Tracking Control for T-S Fuzzy Systems

This chapter studies the problem of H∞ output tracking control for nonlinear systems using Takagi-Sugeno fuzzy model approach. Firstly, an LMI-based design method is proposed for achieving the output tracking purpose. Then, the design method is extended to state-delay case.

8.1 Introduction The study of model following and tracking problems for ﬁnite-dimensional systems keeps attracting considerable attentions due to demands from practical dynamical processes in mechanics, economics and biology. Over the two decades, a variety of approaches have been proposed for output tracking problems for nonlinear systems. A notable approach is based on diﬀerential geometric methods (see [63] for a summary). The diﬀerential algebraic method is originated by Fliess [30]. Meanwhile, the output feedback linearization technique and adaptive scheme are incorporated to achieve the tracking purpose. In general, the exact feedback linearization approach takes into account only cases of the standard output-feedback canonical form and other forms like triangular and parameter strict feedback form [70][103][107]. These forms make use of the backstepping method in designing various tracking laws. The variable structure control approach is also a frequently used method in case of existence of parameter variation and disturbances [62][93]. This method in fact has a long history in model following control [162]. To attenuate the eﬀect of parameter variation and external disturbance, the H∞ tracking methods also pervade in the literature [16][93]. These years, fuzzy technique has been widely and successfully used in nonlinear system modeling and control [136]. Accordingly, the tracking problems have thereby attracted many researchers using adaptive fuzzy-neural idea to achieve the tracking goals [16][93][163]. Among various fuzzy modeling methods, the well-known T-S fuzzy model [119] is recognized as a popular and powerful tool in approximating a complex nonlinear system. In the T-S fuzzy modeling approach, some linear models are blended into an overall single model through nonlinear fuzzy membership functions to represent the nonlinear dynamics. So far, various techniques have been developed for stability analysis and stabilization of T-S fuzzy systems [64][120][126][132][134]. The issue of H∞ control for T-S fuzzy systems has also been studied extensively [6][17][72][98][100][106]. The nonlinear C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 125–147, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

126

8. Output Tracking Control for T-S Fuzzy Systems

tracking problems have been considered through T-S fuzzy model approach as well [47][131][174]. In general, the incorporation of linearization technique and adaptive scheme usually needs system’s perfect knowledge and produces complicated adaptation laws [16][47][163]. The variable structure control method also exhibits its drawback since the resulting chattering phenomenon is inevitable which may cause unstable factors of the system [93][174]. To avoid these drawbacks, a simple LMI-based method is used in [131] to study the model following control via T-S fuzzy model approach. Recently, the T-S fuzzy model approach has been also shown eﬀective and powerful in dealing with nonlinear systems with time-delays [9][10][73][158]. Diﬀerent methodologies have been proposed for analysis and synthesis of T-S fuzzy systems with time-delay [9][10][158]. The H∞ control for T-S fuzzy timedelay systems is studied in [73] and a design method for output feedback H∞ controllers is presented in terms of matrix inequalities. However, little work has been done to study the output tracking problem for T-S fuzzy time-delay models. In this chapter, we study the H∞ output tracking problem for nonlinear timedelay systems via T-S fuzzy model approach. The proposed design of fuzzy controllers depends on feasible sets of solutions to a group of strict LMI’s which are easy to solve using existing LMI Toolbox. We stress that a two-step approach has to be employed in designing appropriate tracking laws. Although the twostep method may be conservative, it has been shown eﬀective to solve complex matrix inequalities in the fuzzy control literature, see, e.g., [17][98][106][131].

8.2 Problem Formulation and Preliminaries Consider a nonlinear system with time-delay which could be approximated by a time-delay T-S fuzzy model of the following form: Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bwi w(t) + Bui u(t), y(t) = Ci x(t) + Cτ i x(t − τ (t)) + Dwi w(t) + Dvi v(t), x(t) = φ(t), t ∈ [−τ0 , 0],

(8.1)

i = 1, 2, . . . , r,

where x ∈ Rn , u ∈ Rnu , w ∈ Rnw , v ∈ Rnv and y ∈ Rny are the state, the control input, the bounded external disturbance, the bounded noise and the output, respectively; r is the number of IF-THEN rules; Ai , Aτ i , Bwi , Bui , Ci , Cτ i , Dwi and Dvi are real constant matrices with appropriate dimensions; θj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables and the fuzzy sets; φ is the initial condition; The time-delay τ (t) may be unknown but is assumed to be smooth function of time: τ˙ (t) ≤ β < 1,

τ (t) ≤ τ0 .

(8.2)

8.2 Problem Formulation and Preliminaries

By fuzzy blending, the overall fuzzy model is inferred as follows: r νi (θ)[Ai x(t) + Aτ i x(t − τ (t)) + Bwi w(t) + Bui u(t)] r x(t) ˙ = i=1 i=1 νi (θ) r hi (θ)[Ai x(t) + Aτ i x(t − τ (t)) + Bwi w(t) + Bui u(t)], = y(t) =

i=1 r

127

(8.3)

hi (θ)[Ci x(t) + Cτ i x(t − τ (t)) + Dwi w(t) + Dvi v(t)],

i=1

x(t) = φ(t), t ∈ [−τ0 , 0],

i = 1, 2, . . . , r,

the membership function where θ = [θ1 , . . . , θp ], νi : R → [0, 1], i = 1, . . . , r, is r corresponding to plant rule i, and h (θ) = ν (θ)/ i i i=1 νi (θ). It is obvious that r h (θ) = 1 with h (θ) ≥ 0. i i i=1 The purpose of this chapter is to control the above fuzzy time-delay system such that the output tracks a reference signal to meet the required tracking performance. Suppose the reference signal yd (t) is generated by p

yd (t) = Cxd (t) x˙ d (t) = Axd (t) + r(t),

(8.4)

where yd has the same dimension as y; xd , r ∈ Rnr are respectively the reference state and the bounded reference input; A and C are appropriately dimensional constant matrices with A Hurwitz. Tracking Requirements (i) The eﬀect of w(t), v(t) and r(t) on the tracking error y(t) − yd (t) is attenuated below a desired level in the sense of

Tf (y(t) − yd (t))T Q(y(t) − yd (t))dt 0

≤γ

Tf

2

[wT (t)w(t) + v T (t)v(t) + rT (t)r(t)]dt

(8.5)

0

for any ﬁnal time Tf > 0, where Q ≥ 0 is a weighting matrix and γ > 0 is a performance index. (ii) The state x(t) is uniformly globally bounded, in the sense that, given initial condition φ, there exists ξφ > 0 such that x(t, φ) ≤ ξφ for all t ≥ 0. In view of unmeasurable state variables x(t), we need to construct observers x (t) for the state. The fuzzy controller is assumed to incorporate information from x (t) and xd (t), i.e., the design of observer-based controllers is performed through the parallel distributed compensation as follows: Controller Rule i: IF θ1 is µi1 and · · · and θp is µip THEN x ˙ (t) = Ai x (t) + Aτ i x (t − τ (t)) + Bui u(t) + Li (y(t) − y(t)), (t) + Cτ i x (t − τ (t)), y(t) = Ci x

(8.6)

128

8. Output Tracking Control for T-S Fuzzy Systems

u(t) = Ki x (t) + Fi xd (t), i = 1, 2, . . . , r, x (t) = φ(t), t ∈ [−τ0 , 0], where the observer gains Li and controller gains Ki and Fi are to be chosen. The overall observer-based control law is thus inferred as r x ˙ (t) = hi (θ)[Ai x (t) + Aτ i x (t − τ (t)) + Bui u(t) + Li (y(t) − y(t))], i=1

y(t) = u(t) =

r i=1 r

hi (θ)[Ci x (t) + Cτ i x (t − τ (t))], hi (θ)[Ki x (t) + Fi xd (t)].

(8.7)

i=1

Combining (8.3) with (8.7), the closed-loop fuzzy time-delay system is written as x ˙ (t) =

r r

hi (θ)hj (θ)[Gij x (t) + Mij x (t − τ (t)) + Nij w(t)],

(8.8)

i=1 j=1

x (t) = φ(t), t ∈ [−τ0 , 0], where

⎡

⎡

⎤

⎤ w(t)

x(t)

⎡

⎤ φ(t)

⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ , φ(t) = , w(t) = x (t) = ⎢ x(t) − x ⎢ ⎢ ⎥ ⎥ φ(t) − φ(t) ⎥ , v(t) (t) ⎣ ⎣ ⎦ ⎣ ⎦ ⎦ xd (0) r(t) xd (t) ⎡ ⎡ ⎤ ⎤ Ai + Bui Kj −Bui Kj Bui Fj Aτ i 0 0 ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ Gij = ⎢ 0 Ai − Li Cj 0 ⎥ , Mij = ⎢ 0 Aτ i − Li Cτ j 0 ⎥ , ⎣ ⎣ ⎦ ⎦ 0 0 A 0 0 0 ⎡ ⎤ Bwi 0 0 ⎢ ⎥ ⎢ ⎥ (8.9) Nij = ⎢ Bwi − Li Dwj −Li Dvj 0 ⎥ . ⎣ ⎦ 0 0 I Thus, the tracking requirements are modiﬁed as (i) The eﬀect of w(t) on the tracking error y(t) − yd (t) is attenuated below a desired level,

Tf (y(t) − yd (t))T Q(y(t) − yd (t))dt 0

+ γ2 T (0)P φ(0) ≤φ

Tf

w T (t)w(t)dt

(8.10)

0

(0) is also for any ﬁnal time Tf > 0. Here, the eﬀect of initial condition x taken into account, and P > 0 is a certain weighting matrix.

8.3 H∞ Output Tracking Control Design for Delay-Free Systems

129

(ii) The augmented state x (t) is uniformly globally bounded (which implies that the state x(t) is uniformly globally bounded). We say the H∞ output tracking performance index γ is achieved if the above two requirements are met.

8.3 H∞ Output Tracking Control Design for Delay-Free Systems In this section, we consider the delay-free case, i.e., the form of (8.3) with Aτ i = 0 and Cτ i = 0. In this case, the overall fuzzy model is as follows: x(t) ˙ = y(t) =

r i=1 r

hi (θ)[Ai x(t) + Bwi w(t) + Bui u(t)],

(8.11)

hi (θ)[Ci x(t) + Dwi w(t) + Dvi v(t)].

i=1

The overall observer-based control law is r x ˙ (t) = hi (θ)[Ai x (t) + Bui u(t) + Li (y(t) − y(t))], i=1

y(t) = u(t) =

r i=1 r

hi (θ)Ci x (t), hi (θ)[Ki x (t) + Fi xd (t)].

(8.12)

i=1

Combining (8.11) with (8.12), the closed-loop fuzzy system is written as x ˙ (t) =

r r

hi (θ)hj (θ)[Gij x (t) + Nij w(t)],

(8.13)

i=1 j=1

where x (t), w(t), Gij and Nij are as in (8.9). The tracking requirements remain the same with replacement of φ(0) by x (0). In the following, we ﬁrstly establish matrix inequality conditions for the design of fuzzy controllers so that the H∞ output tracking performance index γ is achieved. Then, we give an LMI-based procedure to solve the matrix inequalities. Theorem 8.3.1. Consider the fuzzy system described by (8.13). The H∞ output tracking performance index γ is achieved if there exists a matrix P > 0 such that the following matrix inequalities hold: ⎡ ⎤ P Gii + GTii P P Nii HiT Q1/2 ⎢ ⎥ ⎢ ⎥ (8.14) ⎢ NiiT P −γ 2 I RiT Q1/2 ⎥ < 0, i = 1, 2, . . . , r, ⎣ ⎦ Q1/2 Hi Q1/2 Ri −I

130

8. Output Tracking Control for T-S Fuzzy Systems

⎡ ⎢ ⎢ ⎢ ⎣

⎤ P (Gij + Gji ) + (Gij + Gji )T P

⎥ ⎥ ⎥ ≤ 0, ⎦ Q1/2 (Ri + Rj ) −2I −2γ 2 I

(Nij + Nji )T P Q1/2 (Hi + Hj )

i < j,

(8.15)

where Hi = [Ci , 0, − C], Ri = [Dwi , Dvi , 0], Gij and Nij are as in (8.9). Proof. We ﬁrst show that the L2 gain relation in (8.10) is satisﬁed with replace ment of φ(0) by x (0) and then verify the augmented state x (t) is uniformly globally bounded. Deﬁne a function J(t) as J(t) = x T (t)P x (t).

(8.16)

The time derivative of J is given by J˙ =

r r

hi hj 2 xT (t)P [Gij x (t) + Nij w(t)],

i=1 j=1

where, for simplicity, we use hi to denote hi (θ(x(t))). Noting that (y(t) − yd (t))T Q(y(t) − yd (t)) ⎡ ⎤ ⎡ ⎤T ⎡ ⎤ r r x x (t) HiT (t) ⎦ Q Hj Rj ⎣ ⎦ ⎣ ⎦ hi hj ⎣ = T R w(t) w(t) i=1 j=1 i ⎡ ⎤T r r x (t) 1 ⎦ hi hj ⎣ = 2 i=1 j=1 w(t) ⎡ ⎛⎡ ⎤ ⎤ ⎞⎡ ⎤ HjT HiT x (t) ⎦ Q Hj Rj + ⎣ ⎦ Q Hi Ri ⎠ ⎣ ⎦ × ⎝⎣ RiT RjT w(t) ⎡ ⎤ ⎡ ⎤T ⎡ ⎤ r r x x (t) HiT + HjT (t) 1 ⎦ Q Hi + Hj Ri + Rj ⎣ ⎦ ⎣ ⎦, hi hj ⎣ ≤ 4 i=1 j=1 RT + RT w(t) w(t) i

j

it follows that ˙ + (y(t) − yd (t))T Q(y(t) − yd (t)) − γ 2 w T (t)w(t) J(t) ⎡ ⎡ ⎤T ⎤ r r x (t) x (t) ⎦ Ωij ⎣ ⎦ hi hj ⎣ ≤ w(t) w(t) i=1 j=1 ⎡ ⎡ ⎤T ⎤ r x (t) x (t) ⎦ Ωii ⎣ ⎦ h2i ⎣ = w(t) w(t) i=1 ⎡ ⎡ ⎤ ⎤T r r−1 x (t) x (t) ⎦, ⎦ (Ωij + Ωji ) ⎣ hi hj ⎣ + w(t) w(t) i=1 j=i+1

(8.17)

8.3 H∞ Output Tracking Control Design for Delay-Free Systems

where

⎤

⎡

⎢ Ωij = ⎣

131

⎤ ⎡ T T + H H 1 ⎥ i j ⎦ Q Hi + Hj Ri + Rj . ⎦+ ⎣ 4 RT + RT i j −γ 2 I

P Gij + GTij P P Nij NijT P

Therefore, if (8.14) and (8.15) hold, which by Schur complement are equivalent to Ωii < 0,

i = 1, 2, . . . , r,

Ωij + Ωji ≤ 0,

i < j, i, j = 1, 2, . . . , r,

we obtain from (8.17) that

Tf ((y(t) − yd (t))T Q(y(t) − yd (t)) − γ 2 w T (t)w)dt J(T ) − J(0) + 0

≤ 0.

(8.18)

(0), the above inequality veriﬁes (8.10) Due to J(T ) ≥ 0 and J(0) = x T (0)P x with replacement of φ(0) by x (0). Next, we prove that the augmented state x (t) is uniformly globally bounded. To this end, we show that the motion of (8.13) with w = 0 is asymptotically stable. Choose a Lyapunov function V ( x) = J(t) as in (8.16). If the conditions in (8.14) and (8.15) hold, a similar deduction leads to that the time derivative of V along the solutions of (8.13) with w = 0 satisﬁes V˙ < 0. This proves the motion of (8.13) with w = 0 is asymptotically stable. Since w(t), v(t) and r(t) (and thus w) are all bounded, one obtains immediately that x (t) is uniformly globally bounded. This completes the proof. Theorem 8.3.1 provides a matrix inequality condition for the considered H∞ output tracking problem. Since Gij and Nij contain design variables Li , Ki and Fi as given in (8.9), we need to seek methods to decouple them in (8.14) and (8.15). These years, many problems involving observer-based control design will ﬁnally encounter the decoupling of coupled variables of observer gains and controller gains. Some comparatively simple cases could be equivalently transformed into strict LMI conditions such as observer-based stabilization control design [3] (Chapter 7.6). However, more complicated cases have to be solved through two-step LMI methods, for instance, H∞ fuzzy control [98][106] and H∞ fuzzy model following control [131]. In the latter situations, to facilitate the design, the matrix variable P > 0 is chosen diagonal with respect to appropriate matrix blocks. In the sequel, we choose P > 0 diagonal and, as in [98][106][131], we give a two-step LMI approach to solve the matrix inequalities in (8.14) and (8.15). Letting P = diag {P1 , P2 , P3 } and substituting it into (8.14) and (8.15), after some exchanges of rows and columns we have respectively, ⎡ ⎤ Φii ΓiiT ⎣ ⎦ < 0, i = 1, 2, . . . , r, (8.19) T Γii P2 (Ai − Li Ci ) + (Ai − Li Ci ) P2

132

8. Output Tracking Control for T-S Fuzzy Systems

⎡

T ΓijT + Γji

Φij + Φji

⎢ ⎢ ⎢ ⎣ Γij + Γji P2 (Ai + Aj − Li Cj − Lj Ci ) +(Ai + Aj − Li Cj − Lj Ci )T P2 where

⎤ ⎥ ⎥ ⎥ ≤ 0, ⎦

i < j.

(8.20)

⎤

⎡ P1 (Ai + Bui Kj )

⎢ ∗ ∗ ∗ ∗ ⎢ ⎢ +(Ai + Bui Kj )T P1 ⎢ ⎢ ⎢ T P1 P3 A + AT P3 ∗ ∗ ∗ FjT Bui ⎢ ⎢ ⎢ Φij = ⎢ T P1 0 −γ 2 I ∗ ∗ Bwi ⎢ ⎢ ⎢ ∗ 0 0 0 −γ 2 I ⎢ ⎢ ⎢ 0 0 −γ 2 I 0 P3 ⎣ −Q1/2 C Q1/2 Dwj Q1/2 Dvj 0 Q1/2 Cj

∗ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥, ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎦ −I

T Γij = −KjT Bui P1 0 P2 (Bwi − Li Dwj ) − P2 Li Dvj 0 0 , i, j = 1, 2, . . . , r. The strict LMI design result is as follows. Theorem 8.3.2. There exists a fuzzy control law (8.12) such that the H∞ output tracking performance index γ is achieved for the resulting closed-loop fuzzy system (8.13), if the following two steps hold: Step 1. There exist matrices X1 > 0, P3 > 0, Yi and Fi , i = 1, 2, . . . , r, such that the following LMI’s hold: Πii < 0,

i = 1, 2, . . . , r,

Πij + Πji ≤ 0, where

⎡

i < j,

Ai X1 + X1 ATi

⎢ ∗ ∗ ∗ ∗ ⎢ T ⎢ +Bui Yj + YjT Bui ⎢ ⎢ ⎢ T P3 A + AT P3 ∗ ∗ ∗ FjT Bui ⎢ ⎢ ⎢ Πij = ⎢ T 0 −γ 2 I ∗ ∗ Bwi ⎢ ⎢ ⎢ ∗ 0 0 0 −γ 2 I ⎢ ⎢ ⎢ 0 0 −γ 2 I 0 P3 ⎣ −Q1/2 C T Q1/2 Dwj Q1/2 Dvj 0 Q1/2 Cj X1 i, j = 1, 2, . . . , r.

(8.21) (8.22) ⎤ ∗ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥, ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎦ −I

8.4 H∞ Output Tracking Control Design for Time-Delay Systems

133

Step 2. After solving Step 1 for X1 > 0, P3 > 0, Yi and Fi , let P1 = X1−1 and Ki = Yi X1−1 . Then, there exist matrices P2 > 0 and Zi , i = 1, 2, . . . , r, such that the following LMI’s hold: ⎡ ⎤ Φii ΓiiT ⎢ ⎥ ⎢ ⎥ T (8.23) ⎢ ⎥ < 0, i = 1, 2, . . . , r, ⎣ Γii P2 Ai + Ai P2 ⎦ −Zi Ci − CiT ZiT ⎡ Φij + Φji

T ΓijT + Γji

⎤

⎢ ⎢ ⎢ ⎢ P (A + Aj ) + (Ai + Aj )T P2 ⎣ Γij + Γji 2 i −Zi Cj − CjT ZiT − Zj Ci − CiT ZjT

⎥ ⎥ ⎥ ≤ 0, ⎥ ⎦

j > i,

(8.24)

where Φij remain the same as in (8.19) and (8.20), and Γij are given by T Γij = −KjT Bui P1 0 P2 Bwi − Zi Dwj − Zi Dvj 0 0 , i, j = 1, 2, . . . , r. If this is the case, the matrices Fi ,

Ki = Yi X1−1 ,

Li = P2−1 Zi ,

i = 1, 2, . . . , r,

(8.25)

are ready for constructing the fuzzy controller (8.12). Proof. If the LMI’s in (8.21) and (8.22) hold for some matrices X1 > 0, P3 > 0, Yi and Fi , then Φii < 0 and Φij + Φji ≤ 0 by setting P1 = X1−1 and Ki = Yi X1−1 , where Φii and Φij are the same as in (8.19) and (8.20). If further (8.23) and (8.24) hold for some matrices P2 > 0 and Zi , then (8.19) and (8.20) are true by setting Li = P2−1 Zi . This yields that the matrix inequalities in (8.14) and (8.15) hold for above Ki , Fi , Li and P = diag {P1 , P2 , P3 }. The result follows immediately from Theorem 8.3.1.

8.4 H∞ Output Tracking Control Design for Time-Delay Systems In this section, we extend the results for delay-free systems to time-delay systems. We ﬁrstly establish matrix inequality conditions for the design of fuzzy controllers so that the H∞ output tracking performance index γ is achieved. Then, we give an LMI-based procedure to solve the matrix inequalities.

134

8. Output Tracking Control for T-S Fuzzy Systems

Theorem 8.4.1. Consider the fuzzy system described by (8.8). The H∞ output tracking performance index γ is achieved if there exist matrices P > 0 and S > 0 such that the following matrix inequalities hold: ⎡ ⎤ 1 P Gii + GTii P + 1−β S P Mii P Nii HiT Q1/2 ⎢ ⎥ ⎢ ⎥ ⎢ T T 1/2 ⎥ ⎢ Mii P −S 0 Hτ i Q ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ NiiT P 0 −γ 2 I RiT Q1/2 ⎥ ⎣ ⎦ 1/2 1/2 1/2 Q Hi Q Hτ i Q Ri −I < 0, i = 1, 2, . . . , r, ⎡ 2 P (Gij + Gji ) + 1−β S ⎢ ⎢ T ⎢ +(Gij + Gji ) P ⎢ ⎢ ⎢ (Mij + Mji )T P ⎢ ⎢ ⎢ ⎢ (Nij + Nji )T P ⎣ Q1/2 (Hi + Hj ) ≤ 0,

⎤

(8.26)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −2S ⎥ ⎥ ⎥ 0 −2γ 2I ⎥ ⎦ Q1/2 (Hτ i + Hτ j ) Q1/2 (Ri + Rj ) −2I

i < j,

(8.27)

where Gij , Mij and Nij , i, j = 1, 2, . . . , r, are given in (8.9), and Hi = [Ci , 0, − C],

Hτ i = [Cτ i , 0, 0],

Ri = [Dwi , Dvi , 0].

(8.28)

Proof. We ﬁrst show that the L2 gain relation in (8.10) is satisﬁed and then verify that the augmented state x (t) is uniformly globally bounded. Deﬁne a function J(t) as

t 1 T J(t) = x (t)P x (t) + x T (α)S x(α)dα. (8.29) 1 − β t−τ (t) For simplicity we use hi to denote hi (θ(x(t))), and η to stand for [ x(t)T , x (t − T T T τ (t)) , w(t) ] . Without loss of generality, deﬁne xd (σ) := xd (0) for σ ∈ [−τ0 , 0]. The derivative of J(t) is given by J˙ =

r r

hi hj 2 xT (t)P [Gij x (t) + Mij x (t − τ (t)) + Nij w(t)]

i=1 j=1

1 1 − τ˙ (t) T x T (t)S x (t − τ (t))S x(t) − x(t − τ (t)). 1−β 1−β Noting that +

(y(t) − yd (t))T Q(y(t) − yd (t)) ⎡ ⎤ T H i r r ⎢ ⎥ ⎢ ⎥ hi hj η T ⎢ HτTi ⎥ Q Hj Hτ j Rj η = ⎣ ⎦ i=1 j=1 T Ri

8.4 H∞ Output Tracking Control Design for Time-Delay Systems

⎛⎡

⎤

⎡

HiT

⎤T

⎡

HjT

⎤

⎡

HjT

⎤T ⎞ HiT

⎢ ⎜⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎜⎢ T ⎥ ⎢ T ⎥ ⎥ ⎢ ⎥ ⎜⎢ Hτ i ⎥ Q ⎢ Hτ j ⎥ + ⎢ HτTj ⎥ Q ⎢ HτTi ⎥ ⎣ ⎝⎣ ⎦ ⎣ ⎦ ⎦ ⎣ ⎦ RiT RjT RjT RiT ⎡ ⎤ ⎡ ⎤T T T T T H H + H + H i j i j r r ⎢ ⎥ ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ≤ hi hj η T ⎢ HτTi + HτTj ⎥ Q ⎢ HτTi + HτTj ⎥ η, ⎣ ⎦ ⎣ ⎦ 4 i=1 j=1 RiT + RjT RiT + RjT 1 hi hj η T 2 i=1 j=1 r

r

=

135

⎟ ⎟ ⎟η ⎠

it follows that ˙ + (y(t) − yd (t))T Q(y(t) − yd (t)) − γ 2 w J(t) T (t)w(t) r r hi hj η T Θij η ≤ i=1 j=1

=

r

h2i η T Θii η +

i=1

where

Θij

r r−1

hi hj η T (Θij + Θji )η

(8.30)

i=1 j=i+1

⎡ ⎢ ⎢ ⎢ =⎢ ⎢ ⎣

⎤ P Gij + GTij P +

⎡

1 1−β S

P Mij P Nij

MijT P

−S

0

NijT P

0

−γ 2 I

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

HiT + HjT

⎥ 1⎢ ⎢ ⎥ + ⎢ HτTi + HτTj ⎥ Q Hi + Hj Hτ i + Hτ j Ri + Rj . ⎦ 4⎣ RiT + RjT Therefore, if (8.26) and (8.27) hold, which by Schur complement are equivalent to Θii < 0,

i = 1, 2, . . . , r,

Θij + Θji ≤ 0,

i < j, i, j = 1, 2, . . . , r,

(8.31)

one can obtain from (8.30) that

Tf J(T ) − J(0) + [(y(t) − yd (t))T Q(y(t) − yd (t)) − γ 2 w T (t)w(t)]dt 0

≤ 0.

(8.32)

the above inequality veriﬁes (8.10). Due to J(T ) ≥ 0 and J(0) = φT (0)P φ(0), Next, we prove that the augmented state x (t) is uniformly globally bounded. To this end, we show that the motion of (8.8) with w = 0 is asymptotically stable. Choose a Lyapunov-Krasovskii functional V ( xt ) = J(t) as in (8.29),

136

8. Output Tracking Control for T-S Fuzzy Systems

where x t = x (t + α), α ∈ [−τ0 , 0]. It is seen that there exist two scalars c1 > 0 and c2 > 0 such that c1 x(t)2 ≤ V ≤ c2

xt 2 .

sup α∈[−τ0 ,0]

If the conditions in (8.26) and (8.27) hold, a deduction similar to the above leads to that the time derivative of V along the solutions of (8.8) with w = 0 satisﬁes V˙ < 0. This proves the motion of (8.8) with w = 0 is asymptotically stable. Since w(t), v(t) and r(t) (and thus w) are all bounded, one obtains immediately that x (t) is uniformly globally bounded. This completes the proof. Theorem 8.4.1 provides a matrix inequality condition for the considered H∞ output tracking problem. Since Gij , Mij and Nij contain design variables Li , Ki and Fi , we need to seek methods to decouple them in (8.26) and (8.27). Similar to the delay-free case, we give a two-step LMI approach to solve the matrix inequalities in (8.26) and (8.27). Choose P > 0 diagonal and let S = [Skl ] > 0, i.e., ⎡ ⎤ T T S11 S21 S31 ⎢ ⎥ ⎢ T ⎥ (8.33) P = diag {P1 , P2 , P3 }, S = ⎢ S21 S22 S32 ⎥. ⎣ ⎦ S31 S32 S33 Substituting (8.33) into (8.26) and (8.27), after some exchanges of rows and columns, we have respectively, ⎡ ⎤ Ψii ΞiiT ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎡ ⎤⎥ ⎢ ⎥ 1 ⎢ ⎥ < 0, i = 1, 2, . . . , r, P (A − Li Ci ) + 1−β S22 (8.34) ⎢ ⎥ ⎢ 2 i ⎥ ∗ ⎢ ⎢ ⎥⎥ T ⎢ Ξii ⎢ +(Ai − Li Ci ) P2 ⎥⎥ ⎢ ⎢ ⎥⎥ ⎣ ⎣ ⎦⎦ −S22 (Aτ i − Li Cτ i )T P2 ⎤

⎡ T T Ξij + Ξji

Ψij + Ψji

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ξij + Ξji ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡

2 P (A − Li Cj ) + 1−β S22 ⎢ 2 i ⎢ ⎢ +P2 (Aj − Lj Ci ) ⎢ ⎢ ⎢ +(Ai − Li Cj )T P2 ⎢ ⎢ ⎢ +(Aj − Lj Ci )T P2 ⎢ ⎢ ⎢ ⎢ (Aτ i − Li Cτ j )T P2 ⎣ +(Aτ j − Lj Cτ i )T P2

⎥ ⎥ ⎥ ⎤⎥ ⎥ ⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥ ∗ ⎥ ⎥ ⎥ ≤ 0, ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ −2S22 ⎦ ⎦

i < j,

(8.35)

8.4 H∞ Output Tracking Control Design for Time-Delay Systems

137

where ⎡

(11)

Ψij

∗

∗

∗

∗

∗

∗

⎢ ⎢ (22) ⎢ Ψ (21) Ψij ∗ ∗ ∗ ∗ ∗ ⎢ ij ⎢ ⎢ ⎢ AT P 0 −S11 ∗ ∗ ∗ ∗ ⎢ τi 1 ⎢ ⎢ ⎢ ∗ ∗ ∗ 0 0 −S31 −S33 ⎢ Ψij = ⎢ ⎢ ⎢ B T P1 0 0 0 −γ 2 I ∗ ∗ ⎢ wi ⎢ ⎢ ⎢ ∗ 0 0 0 0 0 −γ 2 I ⎢ ⎢ ⎢ ⎢ 0 0 0 0 −γ 2 I 0 P3 ⎢ ⎣ Q1/2 Cj −Q1/2 C Q1/2 Cτ j 0 Q1/2 Dwj Q1/2 Dvj 0 ⎡

T −KjT Bui P1

⎢ ⎢ + 1 S Ξij = ⎢ ⎢ 1−β 21 ⎣ 0

∗ ∗ ∗ ∗ ∗ ∗ ∗

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−I ⎤

1 T 1−β S32

0

0

0

P2 Bwi −P2 Li Dwj

T −S21 −S32

0

−P2 Li Dvj 0 0 ⎥ ⎥ ⎥, ⎥ ⎦ 0 00

i, j = 1, 2, . . . , r, with (11)

= P1 (Ai + Bui Kj ) + (Ai + Bui Kj )T P1 +

(21)

T = FjT Bui P1 +

Ψij Ψij

(22)

Ψij

1 S11 , 1−β

1 S31 , 1−β 1 S33 . = P3 A + AT P3 + 1−β

The strict LMI design result is as follows. Theorem 8.4.2. There exists a fuzzy control law (8.7) such that the H∞ output tracking performance index γ is achieved for the closed-loop fuzzy time-delay system (8.8), if the following two steps hold: Step 1. There exist matrices X1 > 0, P3 > 0, S11 > 0, S33 > 0, S31 , Yi and Fi , i = 1, 2, . . . , r, such that the following LMI’s hold: Υii < 0,

i = 1, 2, . . . , r,

Υij + Υji ≤ 0,

i < j,

(8.36) (8.37)

138

8. Output Tracking Control for T-S Fuzzy Systems

where

⎡

(11)

Υij

∗

∗

∗

∗

∗

∗

⎢ ⎢ (22) ⎢ Υ (21) Υij ∗ ∗ ∗ ∗ ∗ ⎢ ij ⎢ ⎢ ⎢ X AT 0 −S11 ∗ ∗ ∗ ∗ 1 τi ⎢ ⎢ ⎢ ⎢ −S33 ∗ ∗ ∗ 0 0 −S31 ⎢ Υij =⎢ ⎢ T ⎢ 0 0 0 −γ 2 I ∗ ∗ Bwi ⎢ ⎢ ⎢ ⎢ ∗ 0 0 0 0 0 −γ 2 I ⎢ ⎢ ⎢ ⎢ 0 0 0 0 −γ 2 I 0 P3 ⎢ ⎣ Q1/2 Cj X1 −Q1/2 C T Q1/2 Cτ j X1 0 Q1/2 Dwj Q1/2 Dvj 0

∗ ∗ ∗ ∗ ∗ ∗ ∗

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−I

i, j = 1, 2, . . . , r, with (11)

T = Ai X1 + X1 ATi + Bui Yj + YjT Bui +

(21)

T = FjT Bui +

(22)

= P3 A + AT P3 +

Υij Υij Υij

1 S11 , 1−β

1 S31 , 1−β 1 S33 . 1−β

Step 2. After solving Step 1 for X1 > 0, P3 > 0, S11 > 0, S33 > 0, S31 , Yi and Fi , let P1 = X1−1 , Ki = Yi X1−1 , S11 = X1−1 S11 X1−1 and S31 = S31 X1−1 . Then, there exist matrices P2 > 0, S22 > 0, S21 , S32 and Zi , i = 1, 2, . . . , r, such that the following LMI’s hold: ⎡

T Ξ ii

Ψii

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢Ξ ⎢ ii ⎣

⎡

1 P A + ATi P2 + 1−β S22 ⎢ 2 i ∗ ⎢ T T ⎢ −Zi Ci − Ci Zi ⎢ ⎣ −S22 ATτi P2 − CτTi ZiT

⎤ ⎥ ⎥ ⎥ ⎤⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥⎥ ⎥⎥ ⎥⎥ ⎦⎦

i = 1, 2, . . . , r,

(8.38)

8.5 Illustrative Examples

⎡ Ψij + Ψji

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢Ξ ⎢ ij + Ξji ⎢ ⎢ ⎢ ⎣

139

⎤

T T + Ξ Ξ ij ji

⎥ ⎥ ⎥ ⎡ ⎤⎥ ⎥ ⎥ P2 (Ai + Aj ) + (Ai + Aj )T P2 ⎢ ⎥⎥ ⎥ ⎢ ⎥ 2 ∗ ⎥ ⎥ ≤ 0, ⎢ + 1−β S22 − Zi Cj − Zj Ci ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ −CjT ZiT − CiT ZjT ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥⎥ (Aτ i + Aτ j )T P2 ⎣ −2S22 ⎦ ⎦ −CτTj ZiT − CτTi ZjT

ij are where Ψij remain the same as in (8.34) and (8.35), and Ξ ⎡ T −KjT Bui P2 Bwi P1 1 T ⎢ 0 0 −Zi Dvj 0 ⎢ 1−β S32 1 ij = ⎢ + 1−β S21 −Zi Dwj Ξ ⎢ ⎣ T 0 0 0 0 0 −S21 −S32

i > j,

(8.39)

given by ⎤ 0⎥ ⎥ ⎥, ⎥ ⎦ 0

i, j = 1, 2, . . . , r. If this is the case, the matrices Fi ,

Ki = Yi X1−1 ,

Li = P2−1 Zi ,

i = 1, 2, . . . , r,

(8.40)

are ready for constructing the fuzzy controller (8.7). Proof. If the LMI’s in (8.36) and (8.37) hold for some matrices X1 > 0, P3 > 0, S11 > 0, S33 > 0, S31 , Yi and Fi , then we have Ψii < 0 and Ψij + Ψji ≤ 0 by setting P1 = X1−1 , Ki = Yi X1−1 , S11 = X1−1 S11 X1−1 and S31 = S31 X1−1 , where Ψij are the same as in (8.34) and (8.35). If further (8.38) and (8.39) hold for some matrices P2 > 0, S22 > 0, S21 , S32 and Zi , then (8.34) and (8.35) are true by setting Li = P2−1 Zi . This gives that the matrix inequalities in (8.26) and (8.27) hold for above Ki , Fi , Li , and P > 0 and S > 0 being constructed as in (8.33). The result follows immediately from Theorem 8.4.1. In case of Aτ i = 0 and Cτ i = 0, i.e., system (8.3) reduces to a delay-free system (8.11), it is easy to verify that Theorem 8.4.2 reduces exactly to Theorem 8.3.2.

8.5 Illustrative Examples To illustrate the eﬀectiveness and the use of the present results, let us consider two examples: the ﬁrst one is a delay-free example and the second one is with state-delay. Example 8.5.1. Consider a continuous stirred tank reactor (CSTR) which has been well-studied in the literature. We depart directly from the following nonlinear system, which is taken from [174].

140

8. Output Tracking Control for T-S Fuzzy Systems

¯ a (1 − z1 ) exp(− z˙1 (t) = −z1 (t) + D

1 ), z2 + γ¯

¯ 2 + HD ¯ a (1 − z1 ) exp(− z˙2 (t) = −(1 + β)z

1 ¯ u(t) + h), ¯ ) + β(¯ z2 + γ¯

where z = [z1 , z2 ]T with z1 and z2 corresponding respectively to the conversion rate of the reaction and the dimensionless temperature, u ¯ is associated with ¯ is related to external disturbance which may be caused by the the inputs, h uncontrollable change in the ambient temperature, and the parameters are taken ¯ a = 0.072, H ¯ = 8, γ¯ = 20 and β¯ = 0.3. For detailed explanations as in [9] as D about these parameters, please refer to [9][174]. By choosing the above group of parameters, the system possesses only one equilibrium, in case of zero inputs u ¯ = ue = 0, as ze = [ze1 , ze2 ]T = [0.0642, 0.3948]T . With respect to this steady state, an approximation to the nonlinear system is given in [174], using the Teixeira-Zak’s method [126], by the following three IF-THEN rules: Rule 1: IF the temperature is low (i.e., x2 (t) is about -1), THEN ¯ x(t) ˙ = A1 x(t) + Bu1 (u(t) + h); Rule 2: IF the temperature is middle (i.e., x2 (t) is about 0), THEN ¯ x(t) ˙ = A2 x(t) + Bu2 (u(t) + h); Rule 3: IF the temperature is high (i.e., x2 (t) is about 1), THEN ¯ x(t) ˙ = A3 x(t) + Bu3 (u(t) + h), ¯ − ue , and where x := [x1 , x2 ]T = z − ze , u = u ⎡ ⎡ ⎤ ⎤ −1.0684 0.0002 −1.0686 0.0002 ⎦ , A2 = ⎣ ⎦, A1 = ⎣ −0.5471 −1.2987 −0.5484 −1.2988 ⎡ ⎡ ⎤ ⎤ −1.0687 0.0002 0 ⎦ , Bu1 = Bu2 = Bu3 = ⎣ ⎦. A3 = ⎣ −0.5497 −1.2988 0.3 The membership functions are selected as Gaussian functions: exp(−(x2 + 1)2 /σ 2 ) , exp(−(x2 + + exp(−x22 /σ 2 ) + exp(−(x2 − 1)2 /σ 2 ) exp(−x22 /σ 2 ) , ν2 = 2 2 exp(−(x2 + 1) /σ ) + exp(−x22 /σ 2 ) + exp(−(x2 − 1)2 /σ 2 ) exp(−(x2 − 1)2 /σ 2 ) , ν3 = exp(−(x2 + 1)2 /σ 2 ) + exp(−x22 /σ 2 ) + exp(−(x2 − 1)2 /σ 2 )

ν1 =

1)2 /σ 2 )

where σ = 0.8. It is obvious that hi = νi , i = 1, 2, 3. It has been shown clearly [174] that the fuzzy model approach yields much smaller errors than the linearization approach in approximating the original nonlinear system.

8.5 Illustrative Examples

141

Here, our objective is to make the temperature follow a desired signal to meet the H∞ tracking performance and the state x(t) is uniformly globally bounded. Hence the system output is y(t) = [0, 1]x(t) + Dv v(t) where we allow bounded noise term in the output with Dv = 0.1. The desired signal is assumed to be generated by (8.4) with A = −1 and C = 0.6. We also allow mismatched disturbances in the above fuzzy model and let the overall fuzzy model be given by x(t) ˙ =

3

hi {Ai x(t) + Bwi w(t) + Bui u(t)},

(8.41)

i=1

where Bw1 = Bw2 = Bw3 = [1, 3]T . Next, we use the observer-based fuzzy control result in Theorem 8.3.2 to design appropriate controllers. Let γ = 0.9 and the weighting matrix be Q = I. It is computed by using LMI Toolbox that there is a set of feasible solutions to the two step LMI conditions (8.21)-(8.24) with F1 = 3.2468,

F2 = 3.2468, F3 = 3.2468, K1 = = 1.4035 −75.8430 , K2 = Y2 X1−1 = 1.4078 −75.8426 , K3 = Y3 X1−1 = 1.4122 −75.8426 , ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 156.6093 156.6039 156.5973 ⎦ , L2 = ⎣ ⎦ , L3 = ⎣ ⎦. L1 = ⎣ 470.9087 470.8924 470.8728 Y1 X1−1

Therefore, the fuzzy control law of the form (8.7) with above Fi , Ki and Li , i = 1, 2, 3, is such that the H∞ output tracking performance index γ = 0.9 (indeed γ can be tuned to be as small as 0.61 by using this method) is achieved for the resulting closed-loop fuzzy system. In order to show the simulations, we select w(t) = sin(t), v(t) = 0.05 cos(10t) and r(t) = 0.4 sin(t + 1). The initial conditions are x0 = [1, 0.3]T , e0 = [0, 0]T and xd (0) = −0.3. Figure 8.1 shows the responses of the system output y(t) and the reference signal yd (t). Figure 8.2 shows the ratio of the energy of ((y(t) − T yd (t))T Q(y(t)−yd (t)) to the energy of w T (t)w(t), i.e., 0 f ((y(t)−yd (t))T Q(y(t)− T T (t)w(t)dt. It is seen that the ratio tends to a scalar less than 0.1, yd (t))dt/ 0 f w √ which reveals that 0.1 ≈ 0.32 is less than the prescribed γ = 0.9. In fact, it can be shown that Tf T (0) x T (0)P x 0 ((y(t) − yd (t)) Q(y(t) − yd (t))dt < γ 2 + Tf Tf T (t)w(t)dt T (t)w(t)dt 0 w 0 w

142

8. Output Tracking Control for T-S Fuzzy Systems

holds for any Tf > 0. So, the requirement in (8.10) is met. Figure 8.3 shows the responses of the state x(t) and the error e(t) between the true state x(t) and the observed state x (t). It is seen that x(t) is uniformly globally bounded. Hence, our control purpose is achieved. The response of the corresponding controller is shown in Figure 8.4. It should be clariﬁed that the control input varies according to the periodic inputs and disturbances. It has nothing to do with chattering or fast switchings which may occur in variable structure control designs (see the examples in [174]).

0.3

y(t) 0.2

yd(t)

0.1

0

−0.1

−0.2

−0.3

−0.4

0

2

4

6

8

10 Time t

12

14

16

18

20

Fig. 8.1. Response of y(t) and yd (t) for Example 8.5.1

The next example is to illustrate the use of the present state-delay results. Example 8.5.2. Consider a fuzzy time-delay model which is obtained by adding state-delay terms to the fuzzy model in Example 8.5.1 with same parameters. Suppose the matrices of the delay terms are 20% perturbations of those of nonedelay terms, i.e., Aτ i = 0.2Ai for i = 1, 2, 3. Let the time-delay be τ = (3t + 4)/(t+2) which gives τ˙ (t) ≤ 0.5 < 1. To increase the tracking accuracy, we could set Q = 4I. After solving the LMIs in Theorem 8.4.2, a set of feasible solutions with respect to Ki , Fi and Li , i = 1, 2, 3, are given by K1 = 1.5272 −227.1815 , K2 = 1.5311 −227.1931 , K3 = 1.5351 −227.2044 , F1 = 2.4651, F2 = 2.4650, F3 = 2.4648, ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ 94.6507 94.6491 94.6477 ⎦ , L2 = ⎣ ⎦ , L3 = ⎣ ⎦. L1 = ⎣ 281.8913 281.8865 281.8823

8.6 Conclusion and Notes

143

4

3.5

3

2.5

2

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1 Time t

1.2

1.4

1.6

1.8

2

Fig. 8.2. Ratio of energy of ((y(t) − yd (t))T Q(y(t) − yd (t)) to energy of w T (t)w(t) for Example 8.5.1

Therefore, by Theorem 8.4.2 we conclude that the fuzzy control law of the form (8.7) with above Ki , Fi and Li , i = 1, 2, 3, is such that the H∞ output tracking performance index γ = 0.9 (indeed γ can be tuned to be as small as 0.64 for this example) is achieved for the resulting closed-loop fuzzy time-delay system. For simulations, with the same settings as in Example 8.5.1 (the initial condition φ(t) is set to be [1, 0.3]T for t ≤ 0), Figures 8.5-8.8 show, respectively, the responses of y(t) and yd (t), the ratio of the energy of ((y(t)−yd (t))T Q(y(t)−yd (t)) to the energy of w T (t)w(t), the response of the corresponding controller u(t) and the responses of x(t) and x(t) − x (t).

8.6 Conclusion and Notes The problem of H∞ output tracking control for T-S fuzzy systems with statedelay has been studied in this chapter. Suﬃcient conditions have been established in terms of matrix inequalities. A two-step LMI method has been given to solve the these matrix inequalities. Illustrative examples have been given to demonstrate the use and eﬀectiveness of the present results. It should be noted that the two-step LMI approach is based on a suﬃcient condition to solve the involved matrix inequalities. Although the two-step LMI design method brings conservativeness, it has been shown eﬀective in solving severely coupled complex matrix inequalities related to fuzzy control design (see [17][98][106][131]). We point out that the matrix decoupling technique presented in Chapters 6 and 7 is not applicable for solving the matrix inequalities in this chapter. This is because the system structure and controller type in this chapter are diﬀerent and therefore the involved matrix inequalities are more complicated. Up to now, it is still left open in the literature how to solve such type of complicated matrix inequalities without bringing conservativeness.

144

8. Output Tracking Control for T-S Fuzzy Systems 1

x1(t)

0.8

x2(t)

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

10 Time t

8

6

4

2

0

12

14

16

18

20

(a) 0.02

e1(t)

0.015

e2(t)

0.01

0.005

0

−0.005

−0.01

−0.015

−0.02

0

2

4

6

8

10 Time t

12

14

16

18

20

(b) Fig. 8.3. (a) Response of x(t) for Example 8.5.1 (b) Response of the error x(t) − x (t) for Example 8.5.1

8.6 Conclusion and Notes 1

u(t) 0.5

0

−0.5

−1

−1.5

−2

0

2

4

6

8

10 Time t

12

14

16

18

20

Fig. 8.4. Response of control u(t) for Example 8.5.1

0.6

0.5

y(t) 0.4

y (t) d

0.3

0.2

0.1

0

−0.1

−0.2

0

2

4

6

8

10 Time t

12

14

16

18

Fig. 8.5. Response of y(t) and yd (t) for Example 8.5.2

20

145

146

8. Output Tracking Control for T-S Fuzzy Systems 8

7

6

5

4

3

2

1

0

0

0.2

0.4

0.6

0.8

1 Time t

1.2

1.4

1.6

1.8

2

Fig. 8.6. Ratio of energy of ((y(t) − yd (t))T Q(y(t) − yd (t)) to energy of w T (t)w(t) for Example 8.5.2

2.5

u(t)

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

−2.5

0

2

4

6

8

10 Time t

12

14

16

18

Fig. 8.7. Response of control u(t) for Example 8.5.2

20

8.6 Conclusion and Notes

147

1

x1(t) 0.8

x2(t)

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

−1

10 Time t

8

6

4

2

0

12

14

16

18

20

(a) 0.02

e1(t) 0.015

e2(t) 0.01

0.005

0

−0.005

−0.01

−0.015

0

2

4

6

8

10 Time t

12

14

16

18

20

(b) Fig. 8.8. (a) Response of x(t) for Example 8.5.2 (b) Response of the error x(t) − x (t) for Example 8.5.2

9. Fuzzy Filter Design for T-S Fuzzy Systems

This chapter considers the H∞ ﬁlter design for T-S fuzzy systems with timedelay. Firstly, it presents a fuzzy weighting-dependent method and gives the design results in terms of LMIs. The main feature is the use of fuzzy weightingdependent Lyapunov functionals which can reduce the conservatism arisen from the quadratic Lyapunov functional approach. Secondly, it considers the H∞ ﬁlter design in the general case, and establishes delay-dependent design results in terms of LMIs. The main technique is the use of matrix transformation and the free-weighting matrix method. The results for rate-independent case, delayindependent case and delay-free case are easy corollaries.

9.1 Introduction Nonlinear ﬁltering keeps attracting researchers for decades as it is a crucial issue in signal processing. It has been an active research area and has wide practical applications such as communications and sensor networks. So far, various analysis and synthesis methodologies have been developed. The prevailing methods include the linear matrix inequality (LMI) approach [143][147] and the polynomial approach [170], which can overcome the drawback of numerical diﬃculty arisen from other methods such as the Riccati equation approach. Recently, the LMI approach has gained fruitful results in linear time-delay/delay-free system settings, see [35][39][156] and the references therein. The system performance (H2 and/or H∞ ) is usually taken into account in the designs. This actually provides much convenience in nonlinear ﬁltering developments for speciﬁc nonlinear systems. However, for complex nonlinear systems, there lacks of common techniques in ﬁlter designs. Fuzzy ideas are useful for modeling complex nonlinear systems in which, due to complexity or imprecision, classical tools are unsuccessful. It is known that fuzzy systems have been recognized as powerful universal approximators due to their capability of approximating a given system with arbitrary accuracy [137]. Among various rule-based fuzzy models, T-S fuzzy systems are popular and well used tools in recent years. The main feature of T-S fuzzy models is to represent the nonlinear dynamics by simple (usually linear) models according to the so-called fuzzy rules and then to blend all the simple models into an overall single model through nonlinear fuzzy membership functions. The output of the overall fuzzy model is calculated as a gradual activation of the local simple models by using proper defuzziﬁcation schemes [119]. It has been proved that T-S fuzzy models can approximate any smooth nonlinear dynamic systems with arbitrary accuracy (and indeed for many systems, the approximation is exact)[122]. Consequently, lots of eﬀorts C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 149–171, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

150

9. Fuzzy Filter Design for T-S Fuzzy Systems

have been made in the control literature to investigate T-S fuzzy systems and various techniques have been obtained during the past decades [17][122][126][134]. Recently, the T-S fuzzy model approach has been extended to deal with nonlinear systems with time-delays, see [9][10][73][81][158] for delay-independent methods and [15][42][75] for delay-dependent methods. However, results seldom appeared to cope with the nonlinear ﬁltering problem. In [130], a quadratic method is used to study H∞ fuzzy estimation for discrete-time delay-free T-S fuzzy systems. In [149], a delay-independent LMI approach is proposed for exponential H∞ ﬁlter design for T-S fuzzy systems with time delay. In [94], mixed H2 /H∞ ﬁltering design is presented in terms of delay-independent LMIs for discrete-time delay fuzzy systems. So far, quite less work has been reported to study the problem of delaydependent H∞ ﬁlter design for T-S fuzzy systems with time delay. This observation motivates the research in this chapter to present delay-dependent LMI schemes for the H∞ ﬁlter design via T-S fuzzy model approach.

9.2 Fuzzy Weighting-Dependent Approach to Filter Design In studying T-S fuzzy systems, most existing methods, for both control and ﬁltering problems, rely on a common Lyapunov matrix. This brings much conservativeness in achieving the analysis and synthesis purpose. In order to reduce the conservatism entailed in the use of quadratic ideas, in this section, we will employ a fuzzy weighting-dependent approach. This idea is essentially an extension of that in [120] for delay-free case. As clariﬁed in [122] and [120], the requirement that the fuzzy weighting functions are C 1 functions is met for fuzzy models constructed from smooth nonlinear systems by using a sector nonlinearity approach. The sector nonlinearity approach is eﬀective in global or semiglobal fuzzy modeling which can exactly represent the dynamics of a nonlinear system. Moreover, we will adopt the free-weighting matrix method [57][58][145] to avoid imposing bounding techniques on some cross product terms. As a result, we will present a delay-dependent LMI scheme for the H∞ ﬁlter design via T-S fuzzy model approach. The main technique is the use of fuzzy weighting-dependent Lyapunov functionals combined with the free-weighting matrix method. 9.2.1

Problem Formulation

Consider a time-delay T-S fuzzy model with r plant rules: Plant Rule i: IF σ1 is µi1 and · · · and σp is µip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bi ω(t),

(9.1)

y(t) = Ci x(t) + Cτ i x(t − τ (t)) + Di ω(t), z(t) = Ei x(t) + Eτ i x(t − τ (t)), x(t) = φ(t), t ∈ [−τ0 , 0], where x ∈ Rn , y ∈ Rn1 , z ∈ Rn2 and ω ∈ Rm are the state, the measurement, the output signal to be estimated, and the disturbance signal which belongs

9.2 Fuzzy Weighting-Dependent Approach to Filter Design

151

to L2 [0, ∞), respectively; Ai , Aτ i , Bi , Ci , Cτ i , Di , Ei and Eτ i are constant real matrices with appropriate dimensions; σj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables (which are the functions of state variables) and the fuzzy sets; φ(t) is a continuous vector-valued initial function on [−τ0 , 0]; It is assumed that the time-varying delay τ (t) is a smooth function of time: 0 ≤ τ (t) ≤ τ0 ,

τ˙ (t) ≤ d,

(9.2)

where τ0 ≥ 0 and d are two scalars. We remark here that this chapter also addresses the case when the bound for the derivative of the delay is unavailable. In this case, the corresponding ﬁlter design results are merely direct corollaries of the main results. By fuzzy blending, the overall fuzzy model is inferred as follows: x(t) ˙ = y(t) = z(t) =

r i=1 r i=1 r

hi (σ(t))[Ai x(t) + Aτ i x(t − τ (t)) + Bi ω(t)],

(9.3)

hi (σ(t))[Ci x(t) + Cτ i x(t − τ (t)) + Di ω(t)] hi (σ(t))[Ei x(t) + Eτ i x(t − τ (t))]

i=1

x(t) = φ(t), t ∈ [−τ0 , 0], where σ = [σ1 , . . . , σp ], hi (σ(t)) = wi (σ(t))/

r

wi (σ(t)),

i=1 p

wi (σ(t)) =

µij (σj ),

j=1

with µij (σj ) being the grade of membership of σj in µij and wi : Rp → [0, 1] denoting the membership function corresponding to plant rule i. It is obvious that the fuzzy weighting functions hi (σ(t)) satisfy hi (σ(t)) ≥ 0,

r

hi (σ(t)) = 1.

(9.4)

i=1

In this section, we consider the following ﬁlter form. Case I: The following ﬁlter form is sought for the estimate of (9.3): x ˙ (t) = y(t) =

r i=1 r i=1

hi (σ(t))[Ai x (t) + Aτ i x (t − τ (t)) + K(t)(y(t) − y(t))], hi (σ(t))[Ci x (t) + Cτ i x (t − τ (t))],

(9.5)

152

9. Fuzzy Filter Design for T-S Fuzzy Systems

z(t) =

r

hi (σ(t))[Ei x (t) + Eτ i x (t − τ (t))]

i=1

x (t) = φ(t), t ∈ [−τ0 , 0], where K(t) is the ﬁlter function to be designed. In a special case of K(t) = r i=1 hi (σ(t))Ki , we have the following ﬁlter form: Case II: The ﬁlter form is as in Case I by changing the ﬁrst ﬁlter equation to x ˙ (t) =

r

hi (σ(t))[Ai x (t) + Aτ i x (t − τ (t)) + Ki (y(t) − y(t))],

(9.6)

i=1

where Ki , i = 1, 2, . . . , r, is the ﬁlter gain to be designed. Note that the apparent diﬀerence between the two forms in Cases I and II is that Case I seeks a general ﬁlter function with a time-varying ﬁlter gain while Case II adopts the idea of parallel distributed compensation (PDC) [134] and seeks constant local ﬁlter gains. The inherent diﬀerence lies in that the general ﬁlter form in Case I can be found using the fuzzy weighting-dependent method proposed in this chapter while that in Case II can be easily design using the quadratic method. Combining (9.3) with (9.5), we have the ﬁlter error system ˙ = [A(t) − K(t)C(t)]ξ(t) + [Aτ (t) − K(t)Cτ (t)]ξ(t − τ (t)) ξ(t) +[B(t) − K(t)D(t)]ω(t),

(9.7)

e(t) = E(t)ξ(t) + Eτ (t)ξ(t − τ (t)) ξ(t) = φ(t) − φ(t), t ∈ [−τ0 , 0], where (t), e(t) = z(t) − z(t), and the notation M (t) stands r ξ(t) = x(t) − x for i=1 hi (σ(t))Mi for A(t), Aτ (t), B(t), C(t), Cτ (t), D(t), E(t) and Eτ (t). The purpose is to design ﬁlters in Case I and/or Case II such that the equilibrium solution ξ(t) = 0 of system (9.7) with w = 0 is asymptotically stable and the following H∞ performance is satisﬁed for a given index γ > 0:

L

L 2 2 e(t) dt ≤ γ ω(t)2 dt (9.8) 0

0

for all L > 0 and ω ∈ L2 [0, ∞) under zero initial conditions (i.e., ξ(t) ≡ 0 for t ∈ [−τ0 , 0]). If this is the case, we say that the H∞ ﬁlter design problem is solved for Case I and/or Case II.

9.2.2

Main Results

The main result of delay-dependent H∞ ﬁlter design is stated as follows. As in [120], assume that the fuzzy weighting functions hi (σ(t)) are C 1 functions.

9.2 Fuzzy Weighting-Dependent Approach to Filter Design

153

Theorem 9.2.1. Assume that |h˙ k (σ(t))| ≤ βk ,

k = 1, 2, . . . , r,

(9.9)

where βk ≥ 0. The H∞ ﬁlter design problem is solved for Case I if there exist matrices Pj > 0, Q > 0, Z > 0, Wj , Yj , Tj and Uj , j = 1, 2, . . . , r, such that the following LMIs hold for a given scalar δ > 0: Pk ≥ Pr ,

k = 1, 2, . . . , r − 1, i ≤ j,

Θij + Θji < 0,

(9.10) (9.11)

where ⎡

Θij

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(11)

Θij

=

Θij

(11)

Θij

Θij

r−1

(12)

(22)

⎤ √ τ0 (ATi Pj − CjT WiT ) ⎥ ⎥ √ T √ T T T ⎥ −Uj τ0 Tj τ0 (Aτ i Pj − Cτ j Wi ) ⎥ ⎥ ⎥ √ 2 √ T T T ⎥, −γ I τ0 Uj τ0 (Bi Pj − Dj Wi ) ⎥ ⎥ ⎥ ⎥ −Z 0 ⎥ ⎦ −2δPj + δ 2 Z (13)

Θij

√

τ0 Yj

βk (Pk − Pr ) + Pj Ai + ATi Pj + Q

k=1

−Wi Cj − CjT WiT + Yj + YjT + EiT Ej , (12)

= Pj Aτ i − Wi Cτ j − Yj + TjT + EiT Eτ j ,

(13)

= Pj Bi + UjT − Wi Dj ,

(22)

= −(1 − d)Q + EτTi Eτ j − Tj − TjT ,

Θij Θij Θij

and an ellipsis denotes a block induced by symmetry. In this case, the ﬁlter function is given by r −1 r K(t) = (9.12) hi (σ(t))Pi hi (σ(t))Wi . i=1

i=1

Proof. The proof is twofold: we ﬁrst show that (9.8) is satisﬁed under the given conditions, and then prove that system (9.7) with w = 0 is asymptotically stable. Deﬁne a functional J(t) as

t J(t) = ξ(t)T P (t)ξ(t) + ξ(s)T Qξ(s)ds

0

t−τ (t) t

˙ ˙ T Z ξ(s)dsdθ, ξ(s)

+ −τ0

t+θ

(9.13)

154

9. Fuzzy Filter Design for T-S Fuzzy Systems

where P (t) =

r

hj (σ(t))Pj .

(9.14)

j=1

The Newton-Leibniz formula gives ξ(t−τ (t)) = ξ(t)−

t t−τ (t)

˙ ξ(s)ds, which yields

2 ξ(t)T Y (t) + ξ(t − τ (t))T T (t) + ω(t)T U (t)

t ˙ × ξ(t) − ξ(t − τ (t)) − ξ(s)ds ≡ 0,

(9.15)

t−τ (t)

where Y (t) =

U (t) =

r j=1 r

hj (σ(t))Yj ,

T (t) =

r

hj (σ(t))Tj ,

j=1

hj (σ(t))Uj ,

j=1

and Yj , Tj and Uj (j = 1, 2, . . . , r) are arbitrary matrices with appropriate dimensions. By adding (9.15), the time derivative of J(t) along solutions to (9.7) is ˙ + ξ(t)T Qξ(t) ˙ = ξ(t)T P˙ (t)ξ(t) + 2ξ(t)T P (t)ξ(t) J(t) −(1 − τ˙ (t))ξ(t − τ (t))T Qξ(t − τ (t))

t ˙ − ˙ ˙ T Z ξ(t) ˙ T Z ξ(s)ds, +τ0 ξ(t) ξ(s) t−τ0

˙ + ξ(t)T Qξ(t) ≤ ξ(t)T P˙ (t)ξ(t) + 2ξ(t)T P (t)ξ(t) −(1 − d)ξ(t − τ (t))T Qξ(t − τ (t))

t ˙ − ˙ ˙ T Z ξ(t) ˙ T Z ξ(s)ds +τ0 ξ(t) ξ(s) t−τ (t)

+2 ξ(t)T Y (t) + ξ(t − τ (t))T T (t) + ω(t)T U (t)

t ˙ × ξ(t) − ξ(t − τ (t)) − ξ(s)ds ,

(9.16)

t−τ (t)

where the inequality is caused only by τ˙ (t) ≤ d and τ (t) ≤ τ0 . After substitution ˙ with state equation (9.7), one has from (9.16) that of ξ(t) ˙ + e(t)T e(t) − γ 2 ω(t)T ω(t) J(t) + τ0 M (t)Z −1 M (t)T η(t) ≤ η(t)T Θ(t)

t ˙ T Z Z −1 M (t)T η(t) + Z ξ(s) ˙ η(t)T M (t) + ξ(s) ds, − t−τ (t)

(9.17)

9.2 Fuzzy Weighting-Dependent Approach to Filter Design

155

where η(t)T = [ξ(t)T , ξ(t − τ (t))T , ω(t)T ], M (t)T = [Y (t)T , T (t)T , U (t)T ] and ⎡ ⎤ P (t)[B(t) − K(t)D(t)] P (t)[A (t) − K(t)C (t)] τ τ ⎢ Θ(t) ⎥ (11) ⎢ ⎥ T T ⎢ ⎥ +U (t)T −Y (t) + T (t) + E(t) Eτ (t) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ =⎢ Θ(t) ⎥ −(1 − d)Q − T (t) − T (t)T T ⎢ ⎥ −U (t) ⎢ ⎥ T ⎢ ⎥ +Eτ (t) Eτ (t) ⎢ ⎥ ⎣ ⎦ −γ 2 I ⎡ ⎤ ⎡ ⎤T (A(t) − K(t)C(t))T (A(t) − K(t)C(t))T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ +τ0 ⎢ (Aτ (t) − K(t)Cτ (t))T ⎥ Z ⎢ (Aτ (t) − K(t)Cτ (t))T ⎥ , ⎣ ⎦ ⎣ ⎦ (B(t) − K(t)D(t))T (B(t) − K(t)D(t))T (11) = P˙ (t) + P (t)[A(t) − K(t)C(t)] + [A(t) − K(t)C(t)]T P (t) Θ(t) +Q + Y (t) + Y (t)T + E(t)T E(t). r−1 Noticing that (9.4) implies h˙ r (σ(t)) = − k=1 h˙ k (σ(t)), we have P˙ (t) =

r

h˙ k (σ(t))Pk

k=1

=

=

r−1 k=1 r−1

h˙ k (σ(t))Pk + h˙ r (σ(t))Pr h˙ k (σ(t))(Pk − Pr ).

(9.18)

k=1

With (9.9), (9.10) and (9.18), (9.11) implies that ⎡ ⎢ Θ(t)(11) Θ(t)(12) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Θ(t)(22) ⎢ ⎢ ⎢ ⎢ Θ(t) := ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

P (t)B(t) + U (t)T √ −W (t)D(t) −U (t)T

−γ 2 I

√ τ0 A(t)T P (t) τ0 Y (t) √ − τ0 C(t)T W (t)T

⎤

⎥ ⎥ ⎥ ⎥ ⎥ √ T τ0 Aτ (t) P (t) ⎥ √ ⎥ τ0 T (t) √ ⎥ T T ⎥ − τ0 Cτ (t) W (t) ⎥ ⎥ ⎥ √ ⎥ T τ B(t) P (t) 0 √ ⎥ τ0 U (t) √ ⎥ T T ⎥ − τ0 D(t) W (t) ⎥ ⎥ ⎥ ⎥ −Z 0 ⎥ ⎦ 2 −2δP (t) + δ Z

156

9. Fuzzy Filter Design for T-S Fuzzy Systems

≤

r

h2i (σ(t))Θii +

i=1

r

hi (σ(t))hj (σ(t))(Θij + Θji )

i<j

< 0,

(9.19)

where Θ(t)(11) = P˙ (t) + P (t)A(t) + A(t)T P (t) − W (t)C(t) − C(t)T W (t)T +Y (t) + Y (t)T + Q + E(t)T E(t), Θ(t)(12) = P (t)Aτ (t) − W (t)Cτ (t) − Y (t) + T (t)T + E(t)T Eτ (t), Θ(t)(22) = −(1 − d)Q − T (t) − T (t)T + Eτ (t)T Eτ (t). Furthermore, noticing that −P (t)Z −1 P (t) ≤ −2δP (t) + δ 2 Z holds for any scalar δ > 0 due to (δZ − P (t))Z −1 (δZ − P (t)) ≥ 0, the above inequality (9.19) leads < 0 where Θ(t) is as Θ(t) by changing the (5,5) term −2δP (t) + δ 2 Z to to Θ(t) −1 <0 −P (t)Z P (t). Setting K(t) as in (9.12) gives W (t) = P (t)K(t). Thus, Θ(t) + τ0 M (t)Z −1 M (t)T < 0 by Schur complement. So, from is equivalent to Θ(t) (9.17) we have

L J(L) − J(0) + (e(t)2 − γ 2 Θ(t)2 )dt ≤ 0. (9.20) 0

Since J(L) ≥ 0 and J(0) = 0 when subject to zero initial conditions for equation (9.7), the above inequality veriﬁes (9.8). Next, we prove the asymptotic stability of system (9.7) when ω = 0. Choose a Lyapunov-Krasovskii functional V (ξt ) = J(t) as in (9.13), where ξt = ξ(t + α), α ∈ [−τ0 , 0]. It is seen that there exist two scalars c1 > 0 and c2 > 0 such that c1 ξ(t)2 ≤ V ≤ c2

sup

ξt 2 .

α∈[−τ0 ,0]

If (9.9)-(9.11) hold, a deduction similar to the above line leads to that the time derivative of V along the solutions of (9.7) with ω = 0 satisﬁes V˙ < 0. This proves the asymptotic stability of system (9.7) with ω = 0. This completes the proof. Theorem 9.2.1 gives a delay-dependent LMI design scheme using fuzzy weighting-dependent method. Its applicability relies on the existence of the derivative of the fuzzy weighting functions. When the derivatives of the fuzzy weighting functions are not available, we have to use the quadratic method to seek the ﬁlter form in Case II. In this case, we have the following result which is a direct corollary of Theorem 9.2.1. Corollary 9.2.1. The H∞ ﬁlter design problem is solved for Case II if there exist matrices P > 0, Q > 0, Z > 0, Wj , Yj , Tj and Uj , j = 1, 2, . . . , r, such that the following LMIs hold for a given scalar δ > 0:

9.2 Fuzzy Weighting-Dependent Approach to Filter Design

i ≤ j,

Ωij + Ωji < 0,

157

(9.21)

where ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ωij = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Ωij

(11)

Ωij

(12)

P Bi + UjT − Wi Dj

Ωij

(22)

−UjT

−γ 2 I

⎤ √ √ τ0 Yj τ0 (ATi P − CjT WiT ) ⎥ ⎥ √ √ T T T ⎥ τ0 Tj τ0 (Aτ i P − Cτ j Wi ) ⎥ ⎥ ⎥ √ √ T T T ⎥, τ0 Uj τ0 (Bi P − Dj Wi ) ⎥ ⎥ ⎥ ⎥ −Z 0 ⎥ ⎦ −2δP + δ 2 Z

(11)

= P Ai + ATi P + Q − Wi Cj − CjT WiT + Yj + YjT + EiT Ej ,

(12)

= P Aτ i − Wi Cτ j − Yj + TjT + EiT Eτ j ,

(22)

= −(1 − d)Q + EτTi Eτ j − Tj − TjT .

Ωij Ωij Ωij

In this case, the ﬁlter gains are given by Ki = P −1 Wi ,

i = 1, 2, . . . , r.

(9.22)

Proof. It is easy by choosing a functional J(t) in quadratic form of

t

J(t) = ξ(t)T P ξ(t) +

0

ξ(s)T Qξ(s)ds t−τ (t)

t

˙ ˙ T Z ξ(s)dsdθ. ξ(s)

+ −τ0

(9.23)

t+θ

It is easy to see that the LMI condition in Theorem 9.2.1 is theoretically less conservative than that in Corollary 9.2.1. This is because the LMIs of (9.10)(9.11) reduce exactly that of (9.21) by setting Pk = P for k = 1, 2, . . . , r. When the bound for the derivative of the delay is unavailable, we can easily establish rate-independent (i.e., d-independent) results for Cases I and II from Theorem 9.2.1 and Corollary 9.2.1. Here, we only list the following result which is straightforward from Corollary 9.2.1. Corollary 9.2.2. The H∞ ﬁlter design problem is solved for Case II if there exist matrices P > 0, Z > 0, Wj , Yj , Tj and Uj , j = 1, 2, . . . , r, such that the following LMIs hold for a given scalar δ > 0: Ω ij + Ω ji < 0,

i ≤ j,

(9.24)

158

9. Fuzzy Filter Design for T-S Fuzzy Systems

where

Ω ij

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(11)

(12)

P Bi + UjT − Wi Dj

Ω ij

(22)

−UjT

−γ 2 I

Ω ij

Ω ij

⎤ √ √ τ0 Yj τ0 (ATi P − CjT WiT ) ⎥ ⎥ √ √ τ0 Tj τ0 (ATτi P − CτTj WiT ) ⎥ ⎥ ⎥ ⎥ √ √ T T T ⎥, τ0 Uj τ0 (Bi P − Dj Wi ) ⎥ ⎥ ⎥ ⎥ −Z 0 ⎥ ⎦ −2δP + δ 2 Z

(11)

= P Ai + ATi P − Wi Cj − CjT WiT + Yj + YjT + EiT Ej ,

(12)

= P Aτ i − Wi Cτ j − Yj + TjT + EiT Eτ j ,

(22)

= EτTi Eτ j − Tj − TjT .

Ω ij Ω ij Ω ij

In this case, the ﬁlter gains are given by (9.22). t Proof. It is easy by removing the second item t−τ (t) ξ(s)T Qξ(s)ds in (9.23).

9.3 General Fuzzy Filter Design In this section, we consider the H∞ ﬁlter design in the general case. To achieve our purpose, we will adopt the free-weighting matrix method [57][58][145] and the matrix decoupling technique similar to that in [39]. As a result, delay-dependent LMI schemes are presented for the H∞ ﬁlter design for T-S fuzzy systems. Results for rate-independent case, delay-independent case and delay-free case are hence obtained as easy corollaries. 9.3.1

Problem Formulation

Consider a time-delay T-S fuzzy model with r plant rules: Plant Rule i: IF σ1 is µi1 and · · · and σp is µip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bi ω(t),

(9.25)

y(t) = Ci x(t) + Cτ i x(t − τ (t)) + Di ω(t), z(t) = Ei x(t) + Eτ i x(t − τ (t)), x(t) = φ(t), t ∈ [τ0 , 0], where x ∈ Rn , y ∈ Rn1 , z ∈ Rn2 and ω ∈ Rm are the state, the measurement, the output signal to be estimated, and the disturbance signal which belongs to L2 [0, ∞), respectively; Ai , Aτ i , Bi , Ci , Cτ i , Di , Ei and Eτ i are constant real matrices with appropriate dimensions; σj (x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables (which are the functions of state variables) and the fuzzy sets; φ(t) is a continuous vector-valued initial function on [−τ0 , 0];

9.3 General Fuzzy Filter Design

159

It is assumed that the time-varying delay τ (t) is a smooth function of time as follows: 0 ≤ τ (t) ≤ τ0 ,

τ˙ (t) ≤ d,

(9.26)

where τ0 ≥ 0 and d are two scalars. When the bound for the derivative of the delay is unavailable, the corresponding ﬁlter design results are merely direct corollaries of the main results. By fuzzy blending, the overall fuzzy model is inferred as follows: x(t) ˙ = y(t) = z(t) =

r i=1 r i=1 r

hi (σ(t))[Ai x(t) + Aτ i x(t − τ (t)) + Bi ω(t)],

(9.27)

hi (σ(t))[Ci x(t) + Cτ i x(t − τ (t)) + Di ω(t)] hi (σ(t))[Ei x(t) + Eτ i x(t − τ (t))]

i=1

x(t) = φ(t), t ∈ [−τ0 , 0], r where σ = [σ1 , . . . , σp ], hi (σ(t)) = wi (σ(t))/ i=1 wi (σ(t)) with wi : Rp → [0, 1], i = 1, . . . , r, denoting the membership function of the system with respect to r plant rule i. It is obvious that hi (σ(t)) ≥ 0 and i=1 hi (σ(t)) = 1. In this section, we consider the following ﬁlter form which is analogous to the fuzzy control form through parallel distributed compensation: x˙ f (t) = zf (t) =

r i=1 r

hi (σ(t))[Af i xf (t) + Bf i y(t)],

xf (0) = xf 0 ,

(9.28)

hi (σ(t))Cf i xf (t),

i=1

where Af i , Bf i and Cf i , i = 1, 2, . . . , r, are ﬁlter parameters to be designed. Combining (9.27) with (9.28) and deﬁning ξ(t) = [x(t)T , xf (t)T ]T , e(t) = z(t) − zf (t), we have the ﬁlter error system ˙ = A(t)ξ(t) τ (t)ξ(t − τ (t)) + B(t)ω(t), ξ(t) +A τ (t)ξ(t − τ (t)), e(t) = E(t)ξ(t) +E

(9.29)

where ξ(t) := [φ(t)T , xTf0 ]T for t ∈ [−τ0 , 0], and ⎡ ⎤ ⎡ ⎤ r r Aj 0 A(t) 0 = ⎦ := ⎣ ⎦, hi (σ(t))hj (σ(t)) ⎣ A(t) Bf i Cj Af i Bf (t)C(t) Af (t) i=1 j=1 ⎡ ⎤ ⎤ ⎡ r r Aτ j 0 Aτ (t) 0 τ (t) = ⎦ := ⎣ ⎦, hi (σ(t))hj (σ(t)) ⎣ A Bf i Cτ j 0 Bf (t)Cτ (t) 0 i=1 j=1

160

9. Fuzzy Filter Design for T-S Fuzzy Systems

= B(t)

r r

⎡ hi (σ(t))hj (σ(t)) ⎣

⎤ Bj

⎡

⎦ := ⎣

⎤ B(t)

⎦, Bf (t)D(t) r r = hi (σ(t))hj (σ(t)) Ej −Cf i := E(t) −Cf (t) , E(t) Bf i Dj

i=1 j=1

i=1 j=1 r

hj (σ(t)) Eτ j 0 := Eτ (t) 0 .

τ (t) = E

(9.30)

j=1

Our objective is to design ﬁlters of the form (9.28) such that the equilibrium ξ(t) = 0 of system (9.29) with w = 0 is asymptotically stable and the following H∞ performance is satisﬁed for a given index γ > 0:

L

L e(t)2 dt ≤ γ 2 ω(t)2 dt (9.31) 0

0

for all L > 0 and ω ∈ L2 [0, ∞) under zero initial conditions (i.e., ξ(t) ≡ 0 for t ∈ [−τ0 , 0]). If this is the case, we say that the H∞ ﬁlter design problem is solved. 9.3.2

Main Results

The following lemma is useful in the development. Lemma 9.3.1. The H∞ ﬁlter design problem is solved if there exist matrices > 0, Z > 0, and matrix functions Af (t), Bf (t), Cf (t), Y (t), T(t) and P > 0, Q (t), such that the following inequality holds for a given scalar δ > 0: U ⎡ ⎤ τ (t) PA √ √ T T T +U (t) ⎢ (1, 1) ⎥ τ0 Y (t) τ0 A(t) PB(t) P E(t) ⎢ ⎥ −Y (t) + T(t)T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −(1 − d)Q √ √ ⎢ T τ (t)T P E τ (t)T ⎥ ⎢ ⎥ −U(t) τ0 T(t) τ0 A ⎢ ⎥ −T(t) − T(t)T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ √ √ T ⎢ 2 −γ I τ0 U (t) τ0 B(t) P 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −Z 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 ⎢ 0 ⎥ −2δ P + δ Z ⎢ ⎥ ⎣ ⎦ −I < 0,

(9.32)

+ A(t) T P + Q + Y (t) + Y (t)T , and the other parameters where (1, 1) := P A(t) are as in (9.29).

9.3 General Fuzzy Filter Design

161

Proof. The proof is twofold: we ﬁrst show that (9.31) is satisﬁed under the given conditions, and then prove that system (9.29) with w = 0 is asymptotically stable. Deﬁne a functional J(t) as J(t) = ξ(t)T Pξ(t) +

0

t

ξ(s)T Qξ(s)ds

t−τ (t) t

+ −τ0

˙ TZ ˙ ξ(s)dsdθ. ξ(s)

(9.33)

t+θ

By adopting the free-weighting tmatrix method, we have from the Newton-Leibniz ˙ formula, ξ(t − τ (t)) = ξ(t) − t−τ (t) ξ(s)ds, that (t) 0 ≡ 2 ξ(t)T Y (t) + ξ(t − τ (t))T T(t) + ω(t)T U

t ˙ ξ(s)ds , × ξ(t) − ξ(t − τ (t)) −

(9.34)

t−τ (t)

(t) are arbitrary matrix functions with appropriate diwhere Y (t), T(t) and U mensions. With (9.34), the time derivative of J(t) along solutions to (9.29) is computed as ˙ + ξ(t)T Qξ(t) ˙ = 2ξ(t)T P ξ(t) − τ (t)) J(t) − (1 − τ˙ (t))ξ(t − τ (t))T Qξ(t

t ˙ T Zξ(t) ˙ TZ ˙ − ˙ ξ(s)ds +τ0 ξ(t) ξ(s) t−τ0

˙ + ξ(t)T Qξ(t) − τ (t)) − (1 − d)ξ(t − τ (t))T Qξ(t ≤ 2ξ(t) P ξ(t)

t ˙ T Zξ(t) ˙ TZ ˙ − ˙ ξ(s)ds +τ0 ξ(t) ξ(s) T

t−τ (t)

(t) +2 ξ(t)T Y (t) + ξ(t − τ (t))T T(t) + ω(t)T U

t ˙ ξ(s)ds , × ξ(t) − ξ(t − τ (t)) −

(9.35)

t−τ (t)

˙ where the inequality is due to τ˙ (t) ≤ d and τ (t) ≤ τ0 . After substitution of ξ(t) with state equation (9.29), one has from (9.35) that ˙ + e(t)T e(t) − γ 2 ω(t)T ω(t) J(t) −1 M (t)T η(t) ≤ η(t)T Ω(t) + τ0 M (t)Z

t ˙ T Z Z ˙ −1 M (t)T η(t) + Zξ(s) η(t)T M (t) + ξ(s) ds, − t−τ (t)

(9.36)

162

9. Fuzzy Filter Design for T-S Fuzzy Systems

(t)T ] and where η(t)T = [ξ(t)T , ξ(t − τ (t))T , ω(t)T ], M (t)T = [Y (t)T , T(t)T , U ⎤ ⎡ + A(t) T P P A(t) ⎥ ⎢ τ (t) − Y (t) PA ⎢ +U (t)T ⎥ PB(t) ⎥ ⎢ +Q + Y (t) + Y (t)T ⎥ ⎢ TE τ (t) +T(t)T + E(t) ⎥ ⎢ T ⎥ ⎢ +E(t) E(t) ⎥ ⎢ ⎥ ⎢ Ω(t) = ⎢ ⎥ T − T(t) − T(t) ⎥ ⎢ −(1 − d)Q ⎥ ⎢ (t)T − U ⎥ ⎢ τ (t)T E τ (t) ⎥ ⎢ +E ⎥ ⎢ ⎦ ⎣ −γ 2 I ⎡ ⎤ T A(t) ⎢ ⎥ ⎢ T ⎥ +τ0 ⎢ A τ (t) ⎥ Z A(t) Aτ (t) B(t) . ⎣ ⎦ T B(t) −1 P ≤ −2δ P + δ 2 Z holds for any scalar δ > 0 due to (δ Z − Noticing that −PZ −1 < 0 where Ω(t) is as in (9.32) P )Z (δ Z − P ) ≥ 0, inequality (9.32) leads to Ω(t) 2 −1 to −PZ P . It is easy to verify that by changing the (5,5) block −2δ P + δ Z −1 M (t)T < 0 by Schur complement. Ω(t) < 0 is equivalent to Ω(t) + τ0 M (t)Z Hence, we have from (9.36) that

L

J(L) − J(0) +

(e(t)2 − γ 2 ω(t)2 )dt ≤ 0.

(9.37)

0

This veriﬁes (9.31) since J(L) ≥ 0 and J(0) = 0 under zero initial conditions. Next, we prove the asymptotic stability of system (9.29) when ω = 0. Choose a Lyapunov-Krasovskii functional V (ξt ) = J(t) as in (9.33), where ξt = ξ(t + α), α ∈ [−τ0 , 0]. It is seen that there exist two scalars c1 > 0 and c2 > 0 such that c1 ξ(t)2 ≤ V ≤ c2

sup

ξt 2 .

α∈[−τ0 ,0]

Similar to the above deduction, we can show from (9.32) that the time derivative of V along the solutions of (9.29) with ω = 0 satisﬁes V˙ < 0. This proves the asymptotic stability of system (9.29) with ω = 0. This completes the proof. In order to decouple the variables in (9.32), we will use a decoupling technique similar to that in [39]. In such way, inequality (9.32) can be equivalently transformed into another form. > 0, Z > 0, and matrix functions Lemma 9.3.2. There exist matrices P > 0, Q Af (t), Bf (t), Cf (t), Y (t), T (t) and U (t), such that (9.32) holds if and only if there exist matrices P > 0, F > 0, Q > 0, Z > 0, and matrix functions Af (t),

9.3 General Fuzzy Filter Design

163

B f (t), C f (t), Y (t), T (t) and U (t), such that the following inequality holds: ⎡ ⎤ √ √ (1, 1) Φ12 − Y (t) + T (t)T Φ13 + U (t)T τ0 Y (t) τ0 ΦT11 Φ16 ⎢ ⎥ ⎢ ⎥ √ ⎢ −(1 − d)Q − T (t) − T (t)T −U (t)T √τ T (t) T τ0 Φ12 Φ26 ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ √ √ ⎢ 2 T −γ I τ0 U (t) τ0 Φ13 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −Z 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 ⎢ −2δΦ55 + δ Z 0 ⎥ ⎢ ⎥ ⎣ ⎦ −I < 0,

(9.38) ΦT11

T

where (1, 1) := Φ11 + + Q + Y (t) + Y (t) and ⎡ ⎡ ⎤ ⎤ P A(t) + B f (t)C(t) Af (t) P Aτ (t) + B f (t)Cτ (t) 0 ⎦ , Φ12 = ⎣ ⎦, Φ11 = ⎣ F A(t) + B f (t)C(t) Af (t) F Aτ (t) + B f (t)Cτ (t) 0 ⎡ ⎡ ⎤ ⎤ P B(t) + B f (t)D(t) E(t)T ⎦, ⎦ , Φ16 = ⎣ Φ13 = ⎣ −C f (t)T F B(t) + B f (t)D(t) ⎡ ⎡ ⎤ ⎤ Eτ (t)T P F ⎦ , Φ55 = ⎣ ⎦. Φ26 = ⎣ 0 F F > 0, Z > 0, and matrix Proof. Necessity. Suppose (9.32) holds for P > 0, Q functions Af (t), Bf (t), Cf (t), Y (t), T (t) and U (t). Partition P as ⎡ ⎤ P S ⎦, P = ⎣ (9.39) ST W where P > 0, W > 0 and S is invertible (Otherwise, S can be made invertible through slight perturbation). Let ⎡ ⎤ I 0 ⎦. H=⎣ (9.40) 0 SW −1 Multiplying (9.32) from the left and right, respectively, by diag {H, H, I, H, H, I} and its transpose, yields (9.38) with changes of variables as T , Z = H ZH T, F = SW −1 S T , Q = H QH T Y (t) = H Y (t)H T , T (t) = H T(t)H T , U (t) = U(t)H , Af (t) = SAf (t)W −1 S T , This proves the necessity.

B f (t) = SBf (t),

C f (t) = Cf (t)W −1 S T .

(9.41)

164

9. Fuzzy Filter Design for T-S Fuzzy Systems

Suﬃciency. Suppose (9.38) holds for P > 0, F > 0, Q > 0, Z > 0, and matrix functions Af (t), B f (t), C f (t), Y (t), T (t) and U (t). Choose two matrices W > 0 and S invertible such that F = SW −1 S T . Let P and H be deﬁned as in (9.39) and (9.40), respectively. Then, P > 0 due to Φ55 > 0 inferred by (9.38). Multiplying (9.38) from the left and right, respectively, by diag {H −1 , H −1 , I, H −1 , H −1 , I} and its transpose, yields (9.32) with changes of variables as = H −1 QH −T , Z = H −1 ZH −T , Q Y (t) = H −1 Y (t)H −T , T(t) = H −1 T (t)H −T , Af (t) = S −1 Af (t)S −T W,

Bf (t) = S −1 B f (t),

(9.42) −T U(t) = U (t)H , Cf (t) = C f (t)S −T W.

This completes the proof.

With the aid of the above lemmas, we are now in a position to present the main result. Theorem 9.3.1. The H∞ ﬁlter design problem is solved if there exist matrices P > 0, F > 0, Q > 0, Z > 0, Af i , B f i , C f i , Yi , Ti and Ui , i = 1, 2, . . . , r, such that the following LMIs hold for a given scalar δ > 0: Θij + Θji < 0, where

⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Θij = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(1, 1)

i ≤ j,

(9.43)

Ψ12 − Yi + TiT −(1 − d)Q − Ti −

Ψ13 + UiT TiT

−UiT

−γ 2 I

√ τ0 Yi

√ T τ0 Ψ11

√ τ0 Ti

√ T τ0 Ψ12

Ψ16

⎤

⎥ ⎥ Ψ26 ⎥ ⎥ ⎥ ⎥ √ √ T τ0 Ui τ0 Ψ13 0 ⎥ ⎥ ⎥, ⎥ −Z 0 0 ⎥ ⎥ ⎥ ⎥ −2δΨ55 + δ 2 Z 0 ⎥ ⎥ ⎦ −I

T with (1, 1) := Ψ11 + Ψ11 + Q + Yi + YiT and ⎡ ⎡ ⎤ ⎤ P Aj + B f i Cj Af i P Aτ j + B f i Cτ j 0 ⎦ , Ψ12 = ⎣ ⎦, Ψ11 = ⎣ F Aj + B f i Cj Af i F Aτ j + B f i Cτ j 0 ⎡ ⎡ ⎤ ⎤ P Bj + B f i Dj EjT ⎦ , Ψ16 = ⎣ ⎦, Ψ13 = ⎣ T F Bj + B f i Dj −C f i ⎡ ⎡ ⎤ ⎤ EτTj P F ⎦ , Ψ55 = ⎣ ⎦. Ψ26 = ⎣ 0 F F

9.3 General Fuzzy Filter Design

165

In this case, the ﬁlter parameters in (9.28) are given by Af i = F −1 Af i ,

Bf i = F −1 B f i ,

Cf i = C f i ,

i = 1, 2, . . . , r.

(9.44)

r Proof. Set ∆(t) = i=1 hi (σ(t))∆i where ∆ denotes matrix Af , B f , C f , Y , T or U . For simplicity, denote the matrix in the left side of inequality (9.38) by Θ(t). Then, we have from (9.43) that Θ(t) =

r

h2i (σ(t))Θii +

i=1

r

hi (σ(t))hj (σ(t))(Θij + Θji )

i<j

< 0.

(9.45)

By virtue of Lemmas 9.3.1 and 9.3.2, the H∞ ﬁlter design problem is solved, and the ﬁlter matrix functions are given by Af (t) = S −1 Af (t)S −T W, Cf (t) = C f (t)S −T W,

Bf (t) = S −1 B f (t), (9.46)

where matrices W > 0 and S are such that F = SW −1 S T . Or equivalently under transformation S −T W xf (t), the ﬁlter matrix functions can be of the following form: Af (t) = S −T W (S −1 Af (t)S −T W )W −1 S T = F −1 Af (t), Bf (t) = S −T W (S −1 (S −1 B f (t)) = F −1 B f (t), Cf (t) = (C f (t)S −T W )W −1 S T = C f (t).

(9.47)

Hence, the ﬁlter parameters in (9.28) are given by (9.44). This completes the proof. Theorem 9.3.1 provides a delay-dependent LMI condition for solving the H∞ ﬁlter design problem. Under diﬀerent situations, Theorem 9.3.1 implies various results. Firstly, when the derivative or the bound for the derivative of the delay is unavailable, the following rate-independent (i.e., d-independent) result can be easily obtained as a direct corollary of Theorem 9.3.1. Corollary 9.3.1. (rate-independent result). The H∞ ﬁlter design problem is solved for Case I if there exist matrices P > 0, Z > 0, Wj , Yj , Tj and Uj , j = 1, 2, . . . , r, such that the following LMIs hold for a given scalar δ > 0: c1 c1 Θij + Θji < 0,

i ≤ j,

(9.48)

c1 is deﬁned as Θij in (9.43) by setting Q = 0. In this case, the ﬁlter where Θij parameters in (9.28) are given by (9.44).

Proof. It is easy by removing the item tional (9.33).

t

T t−τ (t) ξ(s) Qξ(s)ds

in the chosen func

166

9. Fuzzy Filter Design for T-S Fuzzy Systems

Secondly, when the condition is required for arbitrary size of delay, the following delay-independent result is straightforward from Theorem 9.3.1. Corollary 9.3.2. (delay-independent result). The H∞ ﬁlter design problem is solved if there exist matrices P > 0, F > 0, Q > 0, Af i , B f i , C f i , i = 1, 2, . . . , r, such that the following LMIs hold for a given scalar δ > 0: c2 c2 Θij + Θji < 0,

where

c2 Θij

i ≤ j,

(9.49)

⎡

⎤ T Ψ + Ψ + Q Ψ Ψ Ψ 11 12 13 16 11 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −(1 − d)Q 0 Ψ26 ⎥ ⎢ ⎥ =⎢ ⎥, ⎢ ⎥ 2 ⎢ −γ I 0 ⎥ ⎢ ⎥ ⎣ ⎦ −I

and Ψ11 , Ψ12 , Ψ13 , Ψ16 and Ψ26 are the same as deﬁned in (9.43). In this case, the ﬁlter parameters in (9.28) are given by (9.44). 0 t ˙ TZ ˙ ξ(s)dsdθ Proof. It is easy by removing the item −τ0 t+θ ξ(s) in the chosen functional (9.33) and setting Yi = 0, Ti = 0 and Ui = 0. Finally, for delay-free case, the following result is straightforward from any of Theorem 9.3.1, Corollary 9.3.1 or Corollary 9.3.2. Corollary 9.3.3. (delay-free case). The H∞ ﬁlter design problem is solved in case of no delay if there exist matrices P > 0, F > 0, Af i , B f i , C f i , i = 1, 2, . . . , r, such that the following LMIs hold for a given scalar δ > 0: c3 c3 Θij + Θji < 0,

where

c3 Θij

i ≤ j,

⎡ ⎢ ⎢ ⎢ =⎢ ⎢ ⎣

(9.50) ⎤

T Ψ11 + Ψ11 Ψ13 Ψ16

⎥ ⎥ ⎥ −γ I 0 ⎥ , ⎥ ⎦ −I 2

and Ψ11 , Ψ13 and Ψ16 are the same as deﬁned in (9.43). In this case, the ﬁlter parameters in (9.28) are given by (9.44). Proof. Obvious.

9.4 Illustrative Examples

167

9.4 Illustrative Examples In order to illustrate the use of the ﬁlter design methods in this chapter, we now give two examples. Example 9.4.1. Consider the example of a continuous stirred tank reactor (CSTR). We depart directly from the following autonomous model, which is taken from [9][81]. η2 η˙1 (t) = −1.25η1(t) + 0.072(1 − η1 ) exp( ) + 0.25η1 (t − τ (t)), 1 + η2 /20 η2 ) + 0.25η2 (t − τ (t)) + ω(t), η˙2 (t) = −1.55η2(t) + 0.576(1 − η1 ) exp( 1 + η2 /20 ηi (t) = φi , t ∈ [−τ, 0], where η = [η1 , η2 ]T with η1 and η2 corresponding respectively to the conversion rate of the reaction and the dimensionless temperature, τ (t) is related to the recycle delay time, ω(t) = cos(2t) is the disturbance signal. With respect to the expected steady state ηd = [0.1440, 0.8862], an approximation to the above system is given in [9], using the Teixeira-Zak’s method [126], by the following three IF-THEN rules: Rule 1: IF the temperature is low (i.e., η2 (t) is about 0.8862), THEN x(t) ˙ = A1 x(t) + Aτ 1 x(t − τ (t)) + B1 ω(t); Rule 2: IF the temperature is middle (i.e., η2 (t) is about 2.7520), THEN x(t) ˙ = A2 x(t) + Aτ 2 x(t − τ (t)) + B2 ω(t); Rule 3: IF the temperature is high (i.e., η2 (t) is about 4.7052), THEN x(t) ˙ = A3 x(t) + Aτ 3 x(t − τ (t)) + B3 ω(t); where x(·) := [x1 (·), x2 (·)]T = η(·) − ηd , and ⎡ ⎡ ⎤ ⎤ −1.4274 0.0757 −2.0508 0.3958 ⎦ , A2 = ⎣ ⎦, A1 = ⎣ −1.4189 −0.9442 −6.4066 1.6168 ⎡ ⎡ ⎤ ⎤ −4.5279 0.3167 0.25 0 ⎦ , Aτ 1 = Aτ 2 = Aτ 3 = ⎣ ⎦, A3 = ⎣ −26.2228 0.9837 0 0.25 ⎡ ⎤ 0 B1 = B2 = B3 = ⎣ ⎦ . 1 The membership functions ⎧ ⎪ ⎪1 ⎪ ⎨ η2 −0.8862 w1 = 1 − 2.7520−0.8862 ⎪ ⎪ ⎪ ⎩0

are if η2 ≤ 0.8862 if 0.8862 < η2 < 2.7520 , if η2 ≥ 2.7520

168

9. Fuzzy Filter Design for T-S Fuzzy Systems

⎧ ⎨ 1 − w if η ≤ 2.7520 1 2 , w2 = ⎩ 1 − w if η ≥ 2.7520 3 2 ⎧ ⎪ ⎪ 0 ⎪ ⎨ w3 =

if η2 ≤ 2.7520

η2 −2.7520 , 4.7052−2.7520 if 2.7520 < η2 < 4.7052 ⎪ ⎪ ⎪ ⎩1 if η2 ≥ 4.7052

which gives that hi = wi , i = 1, 2, 3. Suppose the measurement and the signal to be estimated are y(t) = [8, 2]x(t) + [1, 2]x(t − τ (t)) + ω(t) and z(t) = [0, 1]x(t), respectively. Now, we use our results to design H∞ ﬁlters. Please note that the design procedure can be directly fulﬁlled through LMI Toolbox in Matlab [3][36]. Assume that the time-varying delay is τ (t) = 0.5 + 0.2 sin(t) and |η˙ 2 | ≤ 1, which gives 0 < τ (t) ≤ 0.7, τ˙ (t) ≤ 0.2 and β1 = β2 = β3 = 1. For δ = 2, we verify that there is a set of feasible solutions to LMIs (9.10)-(9.11) with a minimum performance level γ = 0.91 and ⎡ P1 = ⎣ ⎡ P3 = ⎣

98.3898 −7.7824 −7.7824 1.2150 98.1568 −7.7590 −7.7590 1.2105

⎡ W1 = ⎣

⎤ 0.9893 0.2189

⎦,

⎡

⎤ ⎦,

P2 = ⎣

115.3083 −10.9675 −10.9675 1.8120

⎤ ⎦,

⎤ ⎦, ⎡

W2 = ⎣

⎤ 1.9443 0.5222

⎦,

⎡ W3 = ⎣

⎤ 1.1427

⎦.

0.2217

Therefore, we conclude from Theorem 9.2.1 that the H∞ ﬁlter design problem is solved with H∞ performance level γ = 0.91, and the corresponding ﬁlter is of the form (9.5) where K(t) is computed from (9.12) with above Pi and Wi . Figure 9.1 shows the convergence process of ξ(t) = x(t) − x (t) → 0 under ω(t) = 0 and initial condition ξ(0) = [1, −1.5]T . Figure 9.2 shows the response t t of the ratio 0 e(s)2 ds/ 0 ω(s)2 ds with e(t) = z(t) − z(t). It is seen that the √ ratio is less than 0.045 which reveals that the H∞ performance level 0.045 = 0.2121 is less than the prescribed level 0.91. Note that the choice of diﬀerent δ may lead to diﬀerent minimum γ. For instance, if δ = 10, the minimum γ = 1.08 can be obtained. However, so far there is no eﬀective method to determine the optimum δ that produces the minimum γ. Anyway, the tuning of δ makes it more ﬂexible in solving corresponding LMIs than the ﬁxed scalar δ = 1. In some other papers (see, e.g., [42]), δ = 1 is always assumed and this certainly brings conservatism. In this example, if choosing

9.4 Illustrative Examples

169

1

ξ1(t) ξ2(t)

0.5

0

−0.5

−1

−1.5

−2

0

1

2

3

4

5

6

7

8

9

10

Time t

Fig. 9.1. Response of estimate error ξ(t) under ω(t) = 0

0.045

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0

0

2

4

6

8

12

10

14

16

18

20

Time t

Fig. 9.2. Response of the ratio

t 0

e(s)2 ds/

t 0

ω(s)2 ds

δ = 1, the minimum γ obtained by using Theorem 9.2.1 is 1.21 which is larger than those under choices of δ = 2 and δ = 10. Moreover, the minimum γ’s by Corollary 9.2.1 are 0.93 and 1.14 for δ = 2 and δ = 10, respectively. It reveals that Theorem 9.2.1 (based on fuzzy weightingdependent method) produces better results than Corollary 9.2.1 (based on quadratic method). Next, we give a numerical example to illustrate the use of Theorem 9.3.1.

170

9. Fuzzy Filter Design for T-S Fuzzy Systems

Example 9.4.2. Consider a system of the form (9.27) with two plant rules (r = 2) and ⎡ A1 = ⎣

−2.1 0.1

⎡ Aτ 1 = ⎣

1

−2

⎡

⎤

A2 = ⎣

⎦,

−1.1 0.1

−1.9

⎤ 0

⎦, −0.2 −1.1 ⎡ ⎤ −0.9 0 ⎦, =⎣ −1.1 −1.2 ⎤

⎤

⎦ , Aτ 2 −0.8 −0.9 ⎡ ⎡ ⎤ 1 0.3 ⎦ , B2 = ⎣ ⎦, B1 = ⎣ −0.2 0.1 C1 = 1 0 , C2 = 0.5 −0.6 , Cτ 1 = −0.8 0.6 , Cτ 2 = −0.2 1 , D1 = 0.3, D2 = −0.6, E1 = 1 −0.5 , E2 = −0.2 0.3 , Eτ 1 = 0.1 0 , Eτ 2 = 0 0.2 .

The time-varying delay is τ (t) = d0 + d sin(t) with d = 0.2 and d0 ≥ d, which gives 0 < τ (t) ≤ τ0 := d0 + d and τ˙ (t) ≤ d = 0.2. Let the fuzzy weighting functions be h1 (σ(t)) = sin2 (t) and h2 (σ(t)) = cos2 (t). The purpose is to test whether or not the H∞ ﬁlter design problem can be solved for this example. Suppose τ0 = 0.5 (i.e., d0 = 0.3). For a ﬁxed scalar δ = 1, we verify that there is a set of feasible solutions to LMI (9.43) with a minimum attenuation level γ = 0.38 and ⎡ F =⎣

0.5148 −0.3430

⎤

⎦, −0.3430 0.4626 ⎡ ⎡ ⎤ ⎤ −1.8398 0.9579 −1.2569 0.7952 ⎦ , Af 2 = ⎣ ⎦, Af 1 = ⎣ 1.4697 −1.2828 0.6763 −0.7643 ⎡ ⎡ ⎤ ⎤ −0.6606 −0.5558 ⎦ , Bf 2 = ⎣ ⎦, Bf 1 = ⎣ 0.4952 0.4513 C f 1 = −0.8921 0.5531 , C f 2 = 0.2275 −0.2757 .

Therefore, we conclude from Theorem 9.3.1 that the H∞ ﬁlter design problem can be solved and the corresponding ﬁlter is of the form (9.28) where the parameters are computed from (9.44) as

9.5 Conclusion

⎡ Af 1 = ⎣

−2.8792 0.0263

⎡

⎤

−2.8996 0.8771

171

⎤

⎦ , Af 2 = ⎣ ⎦, 1.0423 −2.7535 −0.6879 −1.0020 ⎡ ⎡ ⎤ ⎤ −1.1262 −0.8487 ⎦ , Bf 2 = ⎣ ⎦, Bf 1 = ⎣ 0.2355 0.3464 Cf 1 = −0.8921 0.5531 , Cf 2 = 0.2275 −0.2757 .

Note that diﬀerent δ may lead to diﬀerent minimum γ. For several values of τ0 , Table 9.1 lists the computation results of minimum γ under diﬀerent values of δ. Table 9.1. Minimum index γ for diﬀerent δ δ = 0.7 δ = 1 δ = 2 δ = 5 δ = 10 δ = 20 τ0 = 0.5

0.59

0.38

0.35

0.34

0.34

0.37

τ0 = 0.6

1.03

0.43

0.36

0.34

0.35

0.45

τ0 = 0.8

11.98

0.83

0.38

0.35

0.37

1.01

τ0 = 1

−−

2.22

0.41

0.37

0.45

−−

9.5 Conclusion Delay-dependent H∞ ﬁltering problem for T-S fuzzy systems with time-delay has been investigated in this chapter. LMI design schemes have been established using the fuzzy weighting-dependent Lyapunov-Krasovskii functional method and the matrix transformation method. Some useful results have also been given for cases with delay constraints or without delays.

10. Descriptor Method for T-S Fuzzy Control Systems

This chapter studies the descriptor method for T-S fuzzy control systems. The considered class of fuzzy systems is of the form of time-delay fuzzy descriptor systems that is in the extended Takagi-Sugeno fuzzy model. In this chapter, we provide suﬃcient conditions in terms of LMIs for the stability and stabilization problems by using appropriate analysis methods for descriptor systems.

10.1 Introduction During the last two decades, fuzzy technique has been widely used in nonlinear system modeling, especially for systems with incomplete plant information. Fuzzy logic systems serve well as universal approximators [136]. The well-known Takagi-Sugeno (T-S) fuzzy model [119] is a popular and convenient tool in functional approximations. Accordingly, the stabilization problem for systems in T-S fuzzy model has been studied extensively [121][132][134]. Recently, a wider class of fuzzy systems described by the descriptor form is considered in [123], where the model is in the extended T-S fuzzy model. It is known that a descriptor model describes a practical system better than a standard dynamic model [22]. Descriptor systems can be found in electrical circuits, moving robots and many other systems which have to be modeled by additional algebraic constraints. An important characteristic of descriptor systems is the possible impulse behavior, which is harmful to the physical system and is undesired in system control. In [123], a fuzzy model in the descriptor form is introduced, and stability and stabilization problems for the system are addressed. It is shown that the method therein could lead to simpler conditions than other methods for some systems. However, the results developed in [123] are not valid if the original system considered therein is in the pure descriptor form (i.e., the derivative matrix is not of full rank). This is because the conditions presented in [123] imply that all Ek ’s are invertible. We further notice that the starting point of [123] is the transformation of the considered system into an augmented system which stabilization problem is not equivalent to the original one. Such a newly introduced system may not be impulse-free for some rank deﬁcient matrix Ek . This problem causes unavailable stability analysis for the unforced system. In addition, if conﬁned to the augmented system, the invertibility of Ek ’s implies that all the local pairs (E ∗ , A∗ik ) are regular and impulse-free, which brings much conservativeness to the results for the augmented system. C. Lin et al.: LMI Approach to Analysis and Control, LNCIS 351, pp. 173–188, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

174

10. Descriptor Method for T-S Fuzzy Control Systems

In this chapter we study a true descriptor type fuzzy model with time-delay, and establish stability and stabilization results for the fuzzy time-delay descriptor system. The stabilization problem is solved without the constraint that all the local pairs (E, Ai ) are regular and impulse-free. The stabilization problem for standard fuzzy time-delay systems has already been solved [9] where the control design is based on the parallel distributed compensation (PDC) [134]. Based on the descriptor type Lyapunov-Krasovskii functionals and some particular developments for descriptor systems, this chapter presents suﬃcient conditions for the stability analysis and fuzzy controller design using PDC. The present results are in terms of LMI’s and are shown to be extensions of some existing developments such as in [9] and [148].

10.2 Fuzzy Descriptor Systems Let the r plant rules be described as follows. Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN ˜ ix ˜ i u(t), ˜(t) + D ˜(t − τ (t)) + B ˜˙ (t) = A˜i x Ei x ˜ x ˜(t) = φ(t), t ∈ [−τ0 , 0],

(10.1)

where x ˜(t) ∈ Rn and u(t) ∈ Rm are the state and control input, respectively; ˜ i and B ˜i are constant real matrices with Ei ∈ Rn×n with rank Ei ≤ n; A˜i , D appropriate dimensions; θj (˜ x) and µij (i = 1, . . . , r, j = 1, . . . , p) are respectively the premise variables (which are the functions of state variables) and the fuzzy sets. It is assumed that the premise variables are independent of the input ˜ is variables u(t). τ (t) is the time delay and τ0 is the upper bound of τ (t); φ(t) a compatible continuous vector-valued initial function on [−τ0 , 0] (Note that the system described by (E, A) may have initial jump for non-compatible initial condition [95]). In this chapter we assume that τ˙ (t) ≤ β < 1.

(10.2)

This type of delay is also considered in many other papers (such as [9]) for standard fuzzy time-delay systems. By fuzzy blending, it is seen that the derivative matrix may not be a constant matrix. If the derivative matrix in a descriptor system is perturbed, the system may exhibit much complicated behaviors such as discontinuous pole distribution and irregular impulse solutions. So far, in the analysis of descriptor systems, only a few works have been reported in case of perturbed derivative matrix (see [78][76] and the references therein) due to the complexities. As in [78][76], in this chapter we only consider a special case which meets the following assumption. Assumption 10.1. There exists an invertible matrix Q such that Ei Q = [Ei1 0] with Ei1 ∈ Rn×n1 , i = 1, 2 . . . , r, having full column rank.

10.2 Fuzzy Descriptor Systems

175

Under Assumption 10.1, it is easy to see that there are r invertible matrices Qi , i = 1, . . . , r, such that Qi Ei Q = diag {In1 , 0} := E.

(10.3)

Accordingly, with the transformation x = Q−1 x ˜, the r plant rules are equivalent to the following. Plant Rule i: IF θ1 is µi1 and · · · and θp is µip THEN E x(t) ˙ = Ai x(t) + Di x(t − τ (t)) + Bi u(t), x(t) = φ(t), t ∈ [−τ0 , 0], where E is as in (10.3) and ⎡ ⎤ Ai11 Ai12 ⎦, Ai = Qi A˜i Q := ⎣ Ai21 Ai22 ˜ φ(t) = Q−1 φ(t),

˜i , Bi = Q i B

(10.4)

⎡ ˜ i Q := ⎣ Di = Q i D

⎤ Di11 Di12

⎦,

Di21 Di22 i = 1, 2, . . . , r,

with Ai11 , Di11 ∈ Rn1 ×n1 , Ai22 , Di22 ∈ Rn2 ×n2 and n1 + n2 = n. The overall fuzzy model is inferred as follows: r wi (θ)(Ai x(t) + Di x(t − τ (t)) + Bi u(t)) r E x(t) ˙ = i=1 i=1 wi (θ) r hi (θ)(Ai x(t) + Di x(t − τ (t)) + Bi u(t)), =

(10.5)

i=1

x(t) = φ(t), t ∈ [−τ0 , 0], where θ = [θ1 , . . . , θp ], wi : Rp → [0, 1], i = 1, . . . , r, is the membership function of the system with respect to plant rule i, and wi (θ) hi (θ) = r . i=1 wi (θ) It is obvious that hi (θ) ≥ 0,

r

hi (θ) = 1.

i=1

We will call allowable hi (θ) to mean that hi (θ) satisﬁes the above requirements. The purpose of this chapter is to establish stability conditions for the unforced system: E x(t) ˙ =

r

hi (θ)(Ai x(t) + Di x(t − τ (t))),

i=1

x(t) = φ(t), t ∈ [−τ0 , 0], and design fuzzy controllers to stabilize system (10.5).

(10.6)

176

10. Descriptor Method for T-S Fuzzy Control Systems

We introduce below some preliminaries for later use. Consider a delay-free descriptor system described by E x(t) ˙ = Ax(t),

(10.7)

and a time-delay descriptor system described by E x(t) ˙ = Ax(t) + Dx(t − τ )

(10.8)

x(t) = φ(t), t ∈ [−τ 0]. Deﬁnition 10.2.1. The system (10.7) (or say, the pair (E, A)) is said to be regular if det(sE − A) = 0, and impulse-free if deg(det(sE − A)) = rank E. The regularity and impulse-free property of (E, A) ensure the existence and uniqueness of impulse-free solution to (10.7) on [0, ∞) [22], and also ensure the existence and uniqueness of impulse-free solution to (10.8) on [0, ∞) [148]. With respect to system (10.8), we give the following deﬁnition, which can be referred to [31][148]. Deﬁnition 10.2.2. The system described by (10.8) is said to be regular and impulse-free if the pair (E, A) is regular and impulse-free; The system (10.8) is said to be stable if for any > 0 there exists a scalar δ() > 0 such that, for any compatible initial condition φ(t) with sup−τ ≤t≤0 φ(t) < δ() the solution x(t) of (10.8) satisﬁes x(t) < for t ≥ 0, and furthermore, limt→∞ x(t) = 0.

10.3 Stability Result To arrive at the main stability result in this section, the following two lemmas are needed. The ﬁrst lemma provides a well-known stability result for descriptor systems, which can be found in [104]. Lemma 10.3.1. The descriptor system described by E x(t) ˙ = Ax(t) is regular, impulse-free and stable if and only if there exist a matrix P such that P T E = E T P ≥ 0, P T A + AT P < 0.

(10.9)

In fact, if (10.9) is true, then the Lyapunov function is chosen as V = x(t)T E T P x(t). Conversely, if E x(t) ˙ = Ax(t) is regular, impulse-free and stable, then a matrix P can be chosen such that (10.9) holds. See [104] for details. The next lemma is summarized from [31]. If a functional V : Cn [−τ, 0] → R is continuous and x(t, φ) is a solution to (10.8), we deﬁne 1 V˙ (φ) := lim sup (V (x(t + h, φ) − V (φ)). h + h→0

10.3 Stability Result

177

Lemma 10.3.2. Let ⎡ ⎤ ⎡ ⎤ ⎤ ⎡ In1 0 D11 D12 A11 A12 ⎦, ⎣ ⎦) ⎦, ⎣ (E, A, D) = (⎣ A21 A22 D21 D22 0 0 with A22 , D22 ∈ Rn2 ×n2 . Assume that A22 is invertible and ρ(A−1 22 D22 ) < 1. Then the system described by (10.8) is stable if there exist positive numbers α, µ, ν and a continuous functional V : Cn [−τ, 0] → R such that µφ1 (0)2 ≤ V (φ) ≤ νφ2 , V˙ (xt ) ≤ −αxt 2 , where xt = x(t + θ) with θ ∈ [−τ, 0] and φ = [φT1 , φT2 ]T with φ1 ∈ Rn1 . With the above lemmas in hand, we are now in a position to present the following stability result for the unforced system (10.6). Theorem 10.3.1. The unforced fuzzy descriptor system (10.6) is regular, impulse free and stable if there exist matrices P and S > 0 satisfying LMI’s P T E = EP ≥ 0, ⎤ ⎡ P T Ai + ATi P + S P T Di ⎦ < 0, ⎣ DiT P −(1 − β)S

(10.10) i = 1, 2, . . . , r.

(10.11)

Proof. If the LMI’s in (10.11) hold, then P T Ai + ATi P < 0, i = 1, 2, . . . , r. r Due to hi ≥ 0 and i=1 hi = 1, we have r r P ( hi Ai ) + ( hi ATi )P < 0, T

i=1

i = 1, 2, . . . , r,

i=1

which together with (10.10) implies that system (10.6) is regular, impulse-free by virtue of Lemma 10.3.1 and Deﬁnition 10.2.2. We now prove the asymptotic stability of system (10.6). Firstly, from (10.10) ⎡ ⎤ with E given as (10.3), we see that P has the form ⎣

P11 0

⎦ with 0 ≤ P11 ∈ P21 P22 Rn1 ×n1 . By (10.11), one obtains that P is invertible, and hence P22 is invertible and 0 < P11 . Substituting P into (10.11) and multiplying it on the left and the right, respectively, by [0, In2 , 0, In2 ] and its transpose, yields ⎡ ⎤ T T Ai22 + ATi22 P22 + S22 P22 Di22 P22 ⎣ ⎦ < 0, i = 1, 2, . . . , r, T Di22 P22 −(1 − β)S22

178

10. Descriptor Method for T-S Fuzzy Control Systems

and therefore ⎡ ⎤ T T P22 Ai22 + ATi22 P22 + S22 P22 Di22 ⎣ ⎦ < 0, T Di22 P22 −S22

i = 1, 2, . . . , r,

(10.12)

where S22 > 0 is the n2 × n2 block in S, i.e., S22 = [0, In2 ]S[0, In2 ]T . Thus, ⎡ ⎤ r r T P22 ( i=1 hi Ai22 ) + ( i=1 hi ATi22 )P22 + S22 ⎣ ⎦ < 0. (10.13) T )P22 −S22 ( ri=1 hi Di22 One sees from (10.13) that r r T ( hi Ai22 ) + ( hi ATi22 )P22 < 0 P22 i=1

i=1

r r T which gives that P22 ( i=1 hi Ai22 ), and thus i=1 hi Ai22 , is invertible. Multiplying (10.13) on the left and the right, respectively, by ⎡ ⎤ T r r ⎣− ( hi Ai22 )−1 ( hi Di22 ) , In2 ⎦ i=1

i=1

and its transpose, yields r r r r hi Ai22 )−1 ( hi Di22 ))T S22 (( hi Ai22 )−1 ( hi Di22 )) − S22 < 0, (( i=1

i=1

i=1

i=1

which shows that r r hi Ai22 )−1 ( hi Di22 )] < 1 ρ[( i=1

(10.14)

i=1

holds for all allowable hi . Now, let the descriptor type Lyapunov-Krasovskii functional be

t T xT (θ)Sx(θ)dθ, V (xt ) = x (t)EP x(t) + t−τ (t)

where xt = x(t + θ), θ ∈ [−τ0 , 0]. It is seen that λmin (P11 )φ1 2 ≤ V (φ) ≤ (2λmax (EP ) + τ0 λmax (S))

sup

φ(α)2 .

α∈[−τ0 ,0]

The derivative of V along the solution of (10.6) is given by V˙ (xt ) =

r

hi [xT (t)(ATi P + P T Ai )x(t) + 2xT (t)P T Di x(t − τ (t))]

i=1 T

+x (t)Sx(t) − (1 − τ˙ (t))xT (t − τ (t))Sx(t − τ (t)).

10.3 Stability Result

Noticing that V˙ (xt ) ≤

r

r

i=1

179

hi = 1 and τ˙ (t) ≤ β < 1, we have

hi [xT (t)(ATi P + P T Ai )x(t) + 2xT (t)P T Di x(t − τ (t))]

i=1 T

+x (t)Sx(t) − (1 − β)xT (t − τ (t))Sx(t − τ (t)) ⎡ ⎤ ⎤T ⎡ r x(t) P T Ai + ATi P + S P T Di ⎦ ⎦ ⎣ hi ⎣ = DiT P −(1 − β)S x(t − τ (t)) i=1 ⎡ ⎤ x(t) ⎦. ×⎣ x(t − τ (t)) So, if the LMI’s in (10.11) hold, there exists α > 0 such that V˙ (xt ) ≤ −αxt . By Lemma 10.3.2, the proof is completed. This theorem provides a suﬃcient delay-independent condition for the stability of the fuzzy time-delay descriptor system (10.6). It is noted that in [123], the conditions in all the theorems (Theorems 1-7) imply that all Ei ’s are invertible. In this case, the results given therein are not valid for true descriptor systems (i.e., systems with rank (E) < n), and indeed they can be dealt with using the method for standard systems. The invertibility of Ek ’s also brings conservativeness in establishing stabilization results which will be shown in the next section. Unlike [123], Theorem 10.3.1 gives a stability result for fuzzy time-delay systems of true descriptor type. Remark 10.3.1. When restricted to standard systems, i.e., E = I, Theorem 10.3.1 reduces to the stability result (Theorem 1) in [9]. Furthermore, when r = 1 and τ (t) ≡ τ , the condition of Theorem 10.3.1 reduces to that of Theorem 1 in [148] for linear descriptor systems with constant delay. Remark 10.3.2. Let EL = [0, In2 ]. Then the conditions in (10.10) and (10.11) can be combined and are equivalent to the following LMI’s: ⎡ ⎣

⎤ (XE + ELT Y )T Ai + ATi (XE + ELT Y ) + S (XE + ELT Y )T Di DiT (XE + ELT Y )

i = 1, 2, . . . , r,

−(1 − β)S

⎦ < 0, (10.15)

for some matrices X > 0, S > 0 and Y ∈ Rn2 ×n . This is because (10.15) implies (10.10) and (10.11) by letting P = XE + ELT Y ; and conversely, (10.10) and (10.11) ensure (10.15) by noting that P can be decomposed as XE + ELT Y with X = diag {P11 , In2 } and Y = [P21 , P22 ].

180

10. Descriptor Method for T-S Fuzzy Control Systems

10.4 Fuzzy Controller Design In this section we design fuzzy controllers for the fuzzy system (10.5) based on the parallel distributed compensation. Consider the following fuzzy control law: Regulator Rule i: IF θ1 is µi1 and · · · and θp is µip THEN u(t) = −Fi x(t),

i = 1, 2, . . . , r.

(10.16)

The overall state feedback control law is inferred by r r w (θ)Fi x(t) r i =− hi (θ)Fi x(t). u(t) = − i=1 i=1 wi (θ) i=1

(10.17)

The aim is to determine the local feedback gain Fi such that the closed-loop system E x(t) ˙ =

r r

hi (θ)hj (θ)[(Ai − Bi Fj )x(t) + Di x(t − τ (t))],

(10.18)

i=1 j=1

x(t) = φ(t), t ∈ [−τ0 , 0], is regular, impulse-free and stable. With the aid of the stability analysis method presented in the last section, the result for the stabilizing controller design is as follows. Theorem 10.4.1. The closed-loop fuzzy descriptor system (10.18) under fuzzy control law (10.17) is regular, impulse free and stable if there exist matrices X, Z > 0 and Yi with X invertible satisfying LMI’s EX = X T E ≥ 0, Ωii < 0, i = 1, 2, . . . , r,

(10.19) (10.20)

Ωij + Ωji ≤ 0,

(10.21)

where

⎡

Ωij = ⎣

i < j, hi hj ≡ 0, ⎤

Ai X + X T ATi − Bi Yj − YjT BiT + Z

Di X

X T DiT

−(1 − β)Z

⎦,

i, j = 1, 2, . . . , r.

(10.22)

In this case, the local feedback gain Fi is given by Fi = Yi X −1 ,

i = 1, 2, . . . , r.

(10.23)

Proof. Let P = X −1 and S = X −T ZX −1 . Denote Gij = Ai − Bi Fj , i, j = 1, 2, . . . , r. If the LMI’s in (10.19)-(10.21) hold, then with (10.23) we have P T E = EP ≥ 0, P T Gii + GTii P < 0,

i = 1, 2, . . . , r,

P (Gij + Gji ) + (Gij + Gji )T P ≤ 0, T

i < j, hi hj ≡ 0.

10.4 Fuzzy Controller Design

181

Hence, r r r r PT( hi hj Gij ) + ( hi hj Gij )T P i=1 j=1

i=1 j=1

r r h2i Gii ) + ( h2i Gii )T P = PT( i=1

i=1

j−1 j−1 r r +P T ( hi hj (Gij + Gji )) + ( hi hj (Gij + Gji ))T P i=1 j=2

i=1 j=2

< 0, which shows that system (10.18) is regular, impulse-free by virtue of Lemma 10.3.1 and Deﬁnition 10.2.2. We now prove the asymptotic stability of system (10.18). Similarly to the proof of Theorem 10.3.1, it can be shown from (10.19)-(10.21) that r r r ρ[( hi hj Gij22 )−1 ( hi Di22 )] < 1 i=1 j=1

(10.24)

i=1

where Gij22 = [0, In2 ]Gij [0, In2 ]T . Let

t

V (xt ) = xT (t)EP x(t) +

xT (θ)Sx(θ)dθ.

(10.25)

t−τ (t)

Then, λmin (P11 )φ1 2 ≤ V (φ) ≤ (2λmax (EP ) + τ0 λmax (S))

sup

φ(α)2 .

α∈[−τ0 ,0]

The derivative of V along the solution of (10.18) is given by V˙ (xt ) =

r r

hi hj [xT (t)(GTij P + P T Gij )x(t) + 2xT (t)P T Di x(t − τ (t))]

i=1 j=1

+xT (t)Sx(t) − (1 − τ˙ (t))xT (t − τ (t))Sx(t − τ (t)) r r hi hj [xT (t)(GTij P + P T Gij )x(t) + 2xT (t)P T Di x(t − τ (t))] ≤ i=1 j=1

+xT (t)Sx(t) − (1 − β)xT (t − τ (t))Sx(t − τ (t)) r h2i [xT (t)(GTii P + P T Gii )x(t) + 2xT (t)P T Di x(t − τ (t))] = i=1

182

10. Descriptor Method for T-S Fuzzy Control Systems

+

j−1 r

hi hj [xT (t)((Gij + Gji )T P + P T (Gij + Gji ))x(t)

i=1 j=2

+ 2xT (t)P T (Di + Dj )x(t − τ (t))] +xT (t)Sx(t) − (1 − β)xT (t − τ (t))Sx(t − τ (t)) ⎡ ⎡ ⎤ ⎤T r x(t) x(t) ⎦ ⎦ (I2 ⊗ X −T )Ωii (I2 ⊗ X −1 ) ⎣ h2i ⎣ = x(t − τ (t)) x(t − τ (t)) i=1 ⎡ ⎤T j−1 r x(t) ⎦ hi hj ⎣ + x(t − τ (t)) i=1 j=2 ⎡ ⎤ x(t) ⎦, ×(I2 ⊗ X −T )(Ωij + Ωji )(I2 ⊗ X −1 ) ⎣ x(t − τ (t)) where ⊗ denotes the Kronecker product. So, if the LMI’s in (10.20) and (10.21) hold, it is easy to see that there exists α > 0 such that V˙ (xt ) ≤ −αxt . By Lemma 10.3.2, the proof is completed. It is noted that for the special case of Bi = B, i = 1, 2, . . . , r (i.e., the input matrices in all the r fuzzy plant rules are equal), Theorem 10.4.1 reduces to the following result. Corollary 10.4.1. Suppose that Bi = B, i = 1, 2, . . . , r. Then the closed-loop fuzzy descriptor system (10.18) under fuzzy control law (10.17) is regular, impulse free and stable if there exist matrices X, Z > 0 and Yi with X invertible satisfying LMI’s in (10.19) and (10.20). In this case, the local feedback gain Fi is given by (10.23). Proof. The proof follows immediately by noting that, under the condition of Bi = B, i = 1, 2, . . . , r, the closed-loop fuzzy descriptor system (10.18) is simpliﬁed as E x(t) ˙ =

r

hi (θ)[(Ai − Bi Fi )x(t) + Di x(t − τ (t))],

(10.26)

i=1

x(t) = φ(t), t ∈ [−τ0 , 0],

Theorem 10.4.1 provides a suﬃcient condition in terms of LMI’s for fuzzy controller design of fuzzy time-delay descriptor systems. We remark that the stabilization result for the augmented system considered in [123] is valid only for the case that all the local pairs (E ∗ , A∗ik ) are regular and impulse-free (due to invertible Ek ’s implied by the given results). The result of Theorem 10.4.1 does not require such constraints. In the next section, we will show by an illustrative example that even for some local pair to be not impulse-free the present result still works well.

10.5 Illustrative Examples

183

Remark 10.4.1. When restricted to standard systems, i.e., E = I, Theorem 10.3.1 coincides with the result of Theorem 2 in [9]. Remark 10.4.2. The analysis and design methods in this chapter can deal with uncertain fuzzy descriptor systems of the form (10.5) with Ai , Di and Bi subjected to norm-bounded uncertainties, A∆i , D∆i and B∆i , of the form [A∆i D∆i B∆i ] = Mi ∆i [NAi NDi NBi ] ,

i = 1, 2, . . . , r,

where Mi , NAi , NDi and NBi are real constant matrices with appropriate dimensions, and ∆i are the uncertainties satisfying ∆Ti ∆i ≤ I. Results for robust stability and robust fuzzy controller design can be established similarly, and are omitted here. We would like to claim that a recent paper [141] studies exactly the same problem as considered in this chapter and the stability and stabilization results are quite similar to the present results. However, the proofs provided therein need further discussion. In a more detail, the stability cannot be ensured by simply choosing a function V (t) as in [141] and showing V˙ < 0. This fact can be easily seen that, for standard systems with time-delay, the main tools of stability theory are Lyapunov-Krasovskii Theorem and Razumikhin Theorem (see [44], or the Appendix). However, so far, there has been no eﬀective tool to test the stability of time-delay descriptor systems. In this circumstance, there are mainly two ways to study the stability for time-delay descriptor systems: (i) one is through decomposition method and adopting the stability theory for standard systems, see for instance the method in [148]; (ii) the other is to ﬁnd an appropriate tool parallel to those for standard systems with time-delay as in [44]. Recently, a new method is reported in [31] which indeed can be used as a descriptor version of Lyapunov-Krasovskii Theorem. The result in [31] is summarized in Lemma 10.3.2 with a slight modiﬁcation. Based on Lemma 10.3.2, in this chapter we give rigorous and detailed proofs for the stability and stabilization results.

10.5 Illustrative Examples In this section, we use two examples to illustrate the present results. The ﬁrst example is borrowed from [123] to illustrate the merit of the results in case of no delay terms. Example 10.5.1. Consider a nonlinear system ¨ = −bθ˙ 3 (t) + cθ(t) + du(t), (1 + a cos θ(t))θ(t) ˙ is assumed to satisfy |θ(t)| ˙ where the range of θ(t) < φ. The purpose is to stabilize the system using fuzzy control approach. In [123], it is assumed that |a| < 1. It is shown therein that the method in [123] involves ﬁve LMI’s while the approach in [121] has to solve twelve LMI’s. However, both methods in [121] and [123] are not applicable if |a| ≥ 1. Next, we use descriptor

184

10. Descriptor Method for T-S Fuzzy Control Systems

expressions to achieve the stabilization for the example using fuzzy control approach. The descriptor method in this chapter does not require |a| < 1. Furthermore, fewer terms in the expressions are involved and fewer LMI computations are needed compared with the standard fuzzy control approach [121] and the method in [123]. Introduce new variables x(t) = [x1 (t), x2 (t), x3 (t)]T with x1 (t) = θ(t), ˙ and x3 (t) = θ(t). ¨ x2 (t) = θ(t) Then the system is described by ⎡ ⎡ ⎤ ⎡ ⎤ ⎤ 0 1 0 0 100 ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ ˙ = ⎢0 ⎢ 0 1 0 ⎥ x(t) ⎥ x(t) + ⎢ 0 ⎥ u(t). 0 1 ⎣ ⎣ ⎦ ⎣ ⎦ ⎦ 2 c −bx2 (t) −1 − a cos x1 (t) d 000 The above can be expressed exactly by the following fuzzy descriptor form E x(t) ˙ =

3

hi (Ai x(t) + Bi u(t)),

(10.27)

i=1

where

⎡

⎤ 100

⎡

⎤ 0

1

0

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ E = ⎢ 0 1 0 ⎥ , A1 = ⎢ 0 0 1 ⎥, ⎣ ⎦ ⎣ ⎦ 000 c −b(φ2 + 2) a − 1 ⎡ ⎡ ⎤ ⎤ 01 0 01 0 ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ A2 = ⎢ 0 0 ⎥ , A3 = ⎢ 0 0 1 ⎥ , 1 ⎣ ⎣ ⎦ ⎦ c 0 −a − 1 − aφ2 c 0 a−1 ⎡ ⎤ 0 ⎢ ⎥ x2 (t) ⎢ ⎥ B1 = B2 = B3 = ⎢ 0 ⎥ , h1 = 22 , ⎣ ⎦ φ +2 d φ2 − x22 (t) + 1 − cos x1 (t) . φ2 + 2 3 It is seen that 0 ≤ hi ≤ 1, i=1 hi = 1, and system (10.27) is of the form (10.5) without delay. This fuzzy model admits arbitrary scalar a, and the stabilization problem cannot be achieved by the method in [121] and [123] due to rank (E) = 2 < 3. System (10.27) may not be impulse-free for |a| ≥ 1 (for instance, not impulsefree for a = 1). We next use Theorem 10.4.1 to design fuzzy control law to stabilize system (10.27). For simulation purpose we set a = b = c = d = 1 and φ = 2. Since Bi shares a common matrix, Corollary 10.4.1 is valid for this example. This leads to that we need only solve the following four LMI’s with respect to X and Yi , i = 1, 2, 3. h2 =

1 + cos x1 (t) , φ2 + 2

h3 =

10.5 Illustrative Examples

EX = X T E ≥ 0, Ai X + X T ATi − Bi Yi − YiT BiT < 0,

185

i = 1, 2, 3.

One set of solutions to the above LMI’s is given by ⎡ ⎤ 2 −1 0 ⎢ ⎥ ⎢ ⎥ X = ⎢ −1 1 0 ⎥ , Y1 = [8 − 6 1] , ⎣ ⎦ 0 −1 1 Y2 = [2 6 − 5] ,

Y3 = [2 0 1] ,

which concludes that the system (10.27) is stabilizable under a certain fuzzy control law of the form (10.17). From (10.23), the local feedback gains Fi are given by F1 = [3 − 2 1] ,

F2 = [3 4 − 5] ,

F3 = [3 4 1] .

(10.28)

Figure 10.1 shows the perfect convergence result under the overall control law u(t) = −

r

hi Fi x(t)

i=1

with initial condition x0 = [0.5, − 1, 0.9]T . The next example is a time-delay one which illustrates the use of the present time-delay results. Example 10.5.2. Consider a nonlinear time-delay system which is formed from Example 10.5.1 with consideration of time delay in variable θ(t), i.e., ¨ = −bθ˙ 3 (t) + cθ(t) + cτ θ(t − τ (t)) + du(t), (1 + a cos θ(t))θ(t) where cτ = 0.8, τ (t) = 1 + 0.2 sin(t) (thus, τ˙ (t) ≤ 0.2 < 1) and other parameters are as in Example 10.5.1. With the same treatment as in Example 5.1, the timedelay system can be expressed exactly by the following fuzzy descriptor form with time-delay E x(t) ˙ =

3

hi (Ai x(t) + Bi u(t)) + Dx(t − τ (t)),

i=1

where E, Ai , Bi and hi remain unchanged, and ⎡ ⎤ 0 00 ⎢ ⎥ ⎢ ⎥ D = ⎢ 0 0 0⎥. ⎣ ⎦ cτ 0 0

(10.29)

186

10. Descriptor Method for T-S Fuzzy Control Systems 1.5

x (t) 1

1

x2(t)

x3(t) 0.5

0

−0.5

−1

0

1

2

3 Time t

4

5

6

(a) 2.5

2

u(t) 1.5

1

0.5

0

0

1

2

3 Time t

4

5

6

(b) Fig. 10.1. Simulation results for system (10.27)

In using Theorem 10.4.1 (or Corollary 10.4.1 due to equal Bi ’s), one set of feasible solutions is computed as ⎡ ⎤ 18.9506 −28.8585 0 ⎢ ⎥ ⎢ ⎥ X = ⎢ −28.8585 73.9112 ⎥, 0 ⎣ ⎦ −67.6845 −51.8620 −618.7948

10.5 Illustrative Examples

187

2.5

x1(t)

2

x2(t) 1.5

x (t) 3

1

0.5

0

−0.5

−1

0

1

3 Time t

2

4

5

6

5

6

(a) 1.5

1

u(t) 0.5

0

−0.5

−1

−1.5

−2

0

1

3 Time t

2

4

(b) Fig. 10.2. Simulation results for time-delay system (10.29)

⎡

33.0292 −6.2267 −0.0000

⎤

⎢ ⎥ ⎢ ⎥ Z = ⎢ −6.2267 59.9175 0.0000 ⎥ , ⎣ ⎦ −0.0000 0.0000 51.4940 Y1 = 103 · [0.1921 − 1.0911 0.0552] , Y2 = 103 · [0.4251 − 0.3365 3.7680] , Y3 = [18.9506 − 647.6533 55.2492] , which concludes that the system (10.29) is stabilizable under a control law of the form (10.17). From (10.23), the local feedback gains Fi are given by

188

10. Descriptor Method for T-S Fuzzy Control Systems

F1 = [−31.4694 − 27.1124 − 0.0893] , F2 = [−31.4694 − 21.1124 − 6.0893] , F3 = [−31.4694 − 21.1124 − 0.0893] .

(10.30)

Figure 10.2 shows the perfect convergence result of the closed-loop system under initial condition φ(t) = [0.5, − 1, 2.4]T , t ≤ 0.

10.6 Conclusion and Notes This chapter studies a class of fuzzy time-delay descriptor systems. Suﬃcient conditions for the stability and stabilization problems are obtained by using appropriate analysis methods for descriptor systems. The present results are in terms of LMI’s and can be viewed as extensions of some existing developments. As in other control issues for descriptor systems, if the derivative matrices Ei ’s do not share a common rank, the corresponding problems are very complicated and still remain open.

Appendix A: Stability Theory for Time-Delay Systems

We brieﬂy introduce the stability theory for general time-delay systems. For more details, refer to [44]. Consider the retarded functional diﬀerential equation x(t) ˙ = f (t, xt ),

x(σ) = φ(σ), σ ∈ [−τ, 0],

(A.1)

where f : R × Cn,τ → R is continuous and is assumed to be smooth enough to ensure that the solution x(t0 , φ)(t) through (t0 , φ) is continuous in (t0 , φ, t) in the domain of deﬁnition of the function (Here, Cn,τ denotes the Banach space of continuous functions mapping [−τ, 0] to Rn with the topology of uniform convergence); f (t, 0) = 0 for all t ∈ R; xt = x(t + σ), σ ∈ [−τ, 0]. Let ψ ∈ B(0, δ) stand for ψ ∈ {ψ(σ) ∈ Cn,τ : supσ∈[−τ,0] ψ(σ) < δ}. n

Deﬁnition A.1. The solution x ≡ 0 of (A.1) is said to be stable if for any t0 ∈ R and > 0, there is a δ = δ(t0 , ) > 0 such that φ ∈ B(0, δ) implies xt (t0 , φ) ∈ B(0, ) for all t ≥ t0 . It is said to be asymptotically stable if it is stable and there is a b0 = b0 (t0 ) > 0 such that φ ∈ B(0, b0 ) implies x(t0 , φ)(t) → 0 as t → ∞. It is said to be uniformly stable if the above δ is independent of t0 . It is said to be uniformly asymptotically stable if it is uniformly stable and there is a b0 > 0 independent of t0 such that for every η > 0 there is a T (η) such that φ ∈ B(0, b0 ) implies xt (t0 , φ) ∈ B(0, η) for all t ≥ t0 + T (η) and x(t0 , φ)(t) → 0 as t → ∞. If a functional V : R × Cn,τ → R is continuous and x(t, φ) is a solution to (A.1), we deﬁne 1 V˙ (t, φ) := lim sup (V (t + h, xt+h (t, φ) − V (t, φ)). h→0+ h Theorem A.1. Suppose f : R × Cn,τ → Rn takes R× bounded sets of Cn,τ into bounded sets of Rn , and u, v, w : R+ → R+ are continuous, nondecreasing functions satisfying u(s) > 0, v(s) > 0 and w(s) > 0 for s > 0, u(0) = v(0) = 0. If there exist a continuous function V : R × Cn,τ → R such that u(|φ(0)|) ≤ V (t, φ) ≤ v(|φ|), V˙ (t, φ) ≤ −w(|φ(0)|), then the solution x ≡ 0 of (A.1) is uniformly asymptotically stable. Theorem A.2 (Razumikhin Theorem). Suppose f : R × Cn,τ → Rn takes R× bounded sets of Cn,τ into bounded sets of Rn , and u, v, w : R+ → R+ are

190

Appendix A

continuous, nondecreasing functions satisfying u(s) > 0, v(s) > 0 and w(s) > 0 for s > 0, u(0) = v(0) = 0, and v strictly increasing. If there exist a continuous function V : R × Rn → R and a continuous nondecreasing function p(s) > s for s > 0 such that u(|x|) ≤ V (t, x) ≤ v(|x|), ∀t ∈ R, x ∈ Rn , V˙ (t, x) ≤ −w(|x|), if V (t + σ, x(t + σ)) ≤ p(V (t, x)), ∀σ ∈ [−τ, 0], then the solution x ≡ 0 of (A.1) is uniformly asymptotically stable. In this book, we simply use asymptotically stable instead of uniformly asymptotically stable.

Appendix B: Schur Complement

Let A, B, C and D be matrices of sizes p × p, p × q, q × p and q × q, respectively, and consider the block matrix ⎡ ⎤ AB ⎦. M =⎣ CD Suppose that A is invertible. Then we have ⎡ ⎤⎡ ⎤⎡ ⎤ ⎡ ⎤ AB A 0 I A−1 B I 0 ⎦⎣ ⎦⎣ ⎦=⎣ ⎦. M =⎣ −1 −1 CD 0 I 0 D − CA B CA I The q × q matrix, D − CA−1 B is called the Schur complement of the block A of the matrix M . Analogously if D is invertible, the Schur complement of the block D of the matrix M is the q × q matrix, A − BD−1 C. In the ﬁrst case, when A is invertible, the Schur complement arises as the result of performing a partial Gaussian elimination by multiplying the matrix M from the left with the lower triangular block matrix, ⎡ ⎤ A−1 0 ⎣ ⎦. −CA−1 I Analogously, in the second case, when D is invertible, the Schur complement is taken by multiplying the matrix M from the right with the lower triangular block matrix, ⎡ ⎤ I 0 ⎣ ⎦. −1 −1 −D C D Throughout this book, the following application of the Schur complements is frequently used. Let A, B and D be appropriately dimensional matrices with A and D symmetric. Then, ⎡ ⎤ A B ⎣ ⎦>0 BT D if and only if any of the following holds: (i) A > 0 and D − B T A−1 B > 0; (ii) D > 0 and A − BD−1 B T > 0.

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Index

H2 /H∞ ﬁltering, 150 H∞ control, 4, 6 H∞ ﬁlter, 4, 7, 149, 150 H∞ fuzzy estimation, 150 H∞ output tracking control, 7, 125 H∞ performance, 77, 108, 152 aﬃne fuzzy model, 3 artiﬁcial intelligence, 1 asymptotically stable, 11, 17, 33, 40, 58, 67, 78, 189, 190 augmented fuzzy weighting-dependent Lyapunov-Krasovskii functional, 17 augmented Lyapunov-Krasovskii functional, 18–20 backstepping, 125 bilinear matrix inequalities, 47 bounded real lemma, 86 bounding method, 42, 46, 48 chattering, 67 closed-loop system, 40, 48 complex nonlinear system, 9 computational burden, 25 cone complementarity problem, 106, 118 controller gain, 77, 80, 85, 128 Controller Rule, 77, 107, 127 convex optimization problem, 5, 119 crisp, 1 defuzziﬁcation, 3 degree of membership, 1 delay-dependent, 6, 9–11, 23 delay-independent, 7, 9, 15, 23 derivative matrix, 173, 174

descriptor method, 7, 173 descriptor systems, 7, 173 descriptor type Lyapunov-Krasovskii functional, 174, 178 diﬀerential algebraic method, 125 disturbance attenuation, 78, 95, 108 dynamic output-feedback control, 4 equilibrium, 152 external disturbance, 6 feedback linearization, 125 ﬁlter error system, 152 ﬂoating point, 25 free-weighting matrix, 6, 7, 10, 27, 46 fuzzy blending, 3, 5, 10, 76 fuzzy clustering, 2 fuzzy control, VI, 1, 2, 6, 40, 44, 79, 95, 98, 117, 132, 137, 180 fuzzy descriptor systems, 174 fuzzy implication, 1 fuzzy large-scale system, 28 fuzzy large-scale systems, 6, 27 fuzzy logic, 1 fuzzy modeling, V, 2 fuzzy observer, 75 fuzzy relation model, 3 fuzzy set, 1, 10 fuzzy system, 1 Fuzzy theory, 1 fuzzy tracking control, 5 fuzzy weighting-dependent LyapunovKrasovskii functional, 6, 10, 12, 14, 25 fuzzy weighting function, 4, 14, 152 fuzzy weighting functions, 11

204

Index

grade of membership, 4 IF-THEN rules, 1, 2 impulse-free, 176 initial condition, 128 initial function, 10, 28, 39, 56, 151, 174 input delay, 6, 39 interior-point method, 5 iteration, 118

parallel distributed compensation, 4, 39, 77, 107, 127 parameter perturbation, 6 perturbation, 56 plant rule, 10, 28, 56, 150 polynomial-time, 5 premise variable, 10 quadratic Lyapunov-Krasovskii functional, 6, 10, 14

Kronecker product, 182 least squares, 2 linguistic fuzzy model, 3 linguistic variable, 1 LMI, 5 LMI toolbox, 5, 22, 25 local feedback gain, 40 lumped uncertainties, 57 Lyapunov-Krasovskii functional, 6, 9 matched uncertainties, 55, 57, 61 matching condition, 57 matrix decoupling technique, 75, 76 matrix inequalities, 79 membership, 1 membership function, 1, 3, 4, 11 mismatched uncertainties, 55, 62, 69 model following control, 125 multiple time-delay, 25 neural networks, 2 neutral delay systems, 47 Newton-Leibniz formula, 12, 18 Nonlinear ﬁltering, 149 observer gain, 77, 80, 85, 128 observer-based H∞ control, 75, 105 observer-based fuzzy control, 4, 5, 75, 78 observer-based guaranteed cost control, 4 observer-based reliable control, 4 observer-based stabilization, 4 overall fuzzy model, 3, 5, 10, 28, 39, 56, 76, 106, 127, 151, 175

rate-independent, 7, 15, 157 Razumikhin Theorem, 6, 43, 189 reference signal, 127 regular, 176 relaxed LMI technique, 16 rule-based fuzzy systems, 2 Schur complement, 14, 32, 156, 191 slack variable, 10, 25 sliding surface, 60, 61, 63, 65 stability, 6, 189 stabilization, 6 stable, 189 state feedback control, 40 static output-feedback control, 4 switching surface, 60 T-S fuzzy model, 3, 4, 9, 10 T-S fuzzy systems, 2–4, 6 Takagi-Sugeno fuzzy model, 7, 9 time-delay, 4, 10, 189 time-delay systems, 4 time-varying delay, 17 trace, 118 uncertainties, 56 uniformly asymptotically stable, 189, 190 uniformly globally bounded, 127, 129, 131, 136 uniformly stable, 189 unmeasurable state, 127 variable structure control, 4, 6, 55, 58

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