Control System

Reliable Control of Fuzzy Descriptor Systems

with Time-Varying Delay

Yuhao Yuan1, Zhonghu Yuan2, Qingling Zhang1, Daqing Zhang1,

and Bing Chen3

1

Institute of Systems Science, Northeastern University, Shenyang 110004

2

School of Information Engineering, Shenyang University, Shenyang 110044

3

Institute of Complexity Science, Qingdao University, Shandong 266071

******@****.***, *******@****.***.***.**

Abstract. The reliable fuzzy controller design problem of T-S fuzzy de-

scriptor systems with time-varying delay is introduced. Based on linear

matrix inequality approach, a less conservative reliable controller design

method is presented. The resulting fuzzy control systems are reliable in

the sense that asymptotic stability is achieved not only when all control

components are operating well, but also in the presence of some compo-

nent failures. Moreover, the result is extended to the case of observer-

based reliable fuzzy control.Two numerical examples are also given to

illustrate the design procedures and their e ectiveness.

1 Introduction

Reliable control is an e ective approach to improve system reliability. The kernel

idea of this approach is to design a xed controller such that the closed-loop can

maintain stability and performance, not only when all control components are

operational, but also in the case of some admissible control component outages.

In the past two decades, reliable control problems have been extensively studied

by many researchers [1, 2, 3, 4].

On the other hand, many complex nonlinear systems can be expressed in a

certain form of mathematical models locally. Takagi and Sugeno have proposed

a fuzzy model to describe the complex systems [5]. In this T-S fuzzy model,

local dynamics in di erent state space regions are represented by local linear

systems. The overall model is obtained by blending these linear models through

membership functions. As a common belief, the control technique based on the

T-S fuzzy model is conceptually simple and e ective for the control of complex

systems.

As to reliable control of T-S fuzzy systems, progress has been made in most

recent years too [6, 7, 8]. In literature [6, 7], reliable controller design are based

on a assumption that control component failures are modeled as outages, i.e.,

when a failure occurs, the signal (in the case of sensors) or the control action (in

the case of actuators) simply becomes zero. In [8], a more general failure model

Supported by Natural Science Foundation of Liaoning Province (No.20042001).

L. Wang et al. (Eds.): FSKD 2006, LNAI 4223, pp. 169 178, 2006.

c Springer-Verlag Berlin Heidelberg 2006

170 Y. Yuan et al.

is adopted for actuator failures, which studied problem that control components

being failure to some extent, i.e., the failure coe cients take value in the interval

[0, 1]. When the actuator is invalid but kept in the admissible area, the controller

will stabilizes the system.

In 1999, Tanniguchi and Tanaka et al extended the T-S fuzzy model to de-

scriptor nonlinear systems [9, 10]. They brought the concept of T-S fuzzy de-

scriptor systems forward. But so far, the reliable control problems for T-S fuzzy

descriptor system has scarcely been studied.

Time-delays often occur in many dynamic systems, it has been shown that

the existence of delays usually becomes the source of instability and deteriorates

performance of systems. So, it is worth to study a system with time-delay both

theoretically and practicality.

In practical situations, failure of actuators often occurs. Thus, from a safety

point as well as a performance point of view, an important requirement is to

have a reliable control such that the stability and performance of the closed-

loop system can tolerate actuator failures. In this paper, we will consider the

reliable control problem of T-S fuzzy descriptor systems with time-delay.

The paper is organized as follows. Firstly, the problem is formulated. In sec-

tion 3, based on the solvability of LMIs, taking account of a ects of all subsys-

tems, which gives the design method of the reliable state feedback controller. In

section 4, the result obtained in section 3 is extended to the case of observer-

based reliable fuzzy control. In section 5, numerical examples are used to illus-

trate the results. Finally, concluding remarks are made in section 6.

Notations : Matrix X > 0 (X 0) denotes thatX is a positive (semi-positive)

de nite matirx, A > B denotes A B > 0. Symbol I stands for the unit

matrix with appropriate dimensions.

2 Problem Formulation and Failure Model

In this section, if the uncertain system parameter information is considered, the

nonlinear descriptor system can be presented as an uncertain fuzzy descriptor

model with time-varying delay. The i th fuzzy rule is of the following form:

Ri : IF z1 is Ni1 and zp is Nip, THEN

E x (t) = (Ai + Ai (t)) x (t) + (A1i + A1i (t)) x (t (t))

+ (Bi + Bi (t)) u (t), (1)

y (t) = Ci x (t),

x (t) = (t), t [ 0, 0] . i = 1, 2, . . ., r .

Where Nij are the fuzzy sets, z1, z2,, zp are premise variables. Scalar r is

the number of IF-THEN rules. E Rn n may be singular. x (t) Rn is the

state, u (t) Rm is the control input and y (t) Rl is the output. Ai, A1i

Rn n, Bi Rn m, Ci Rl n, i = 1, 2,, r. The (t) is the time-varying delay

and satis es 0 0,

M F (t) E + E T F T (t) M T M M T + 1 E T E .

3 Reliable Control Via State Feedback for T-S Model

First, the reliable fuzzy controller will be designed to stabilize system (2). The

fuzzy controller shares the same fuzzy sets with the fuzzy model in the premise

parts and has local linear controller in the consequent parts. The i th fuzzy rule

of the fuzzy controller is of the following form

Ri : IF z1 is Ni1 and zp is Nip, THEN

u = Ki x, i = 1, 2,, r .

For fuzzy reliable control problems, the following actuator fault model is used.

Ri : IF z1 is Ni1 and zp is Nip, THEN

172 Y. Yuan et al.

u = i Ki x, i = 1, 2,, r .

Where Ki, i = 1, 2,, r are the local linear feedback gains. Hence, the overall

fuzzy controller is given by

r

u = i i Ki x, (3)

i=1

where i = diag [ i (1), i (2),, i (m)], i (j ) [0, 1], i = 1, 2,, r,

j = 1, 2,, m. Matrix i describes the fault extent. i (j ) = 0 means that

the j th component in the i th local actuator is invalid, i (j ) (0, 1) implies

that the j th component is at fault in some extent and i (j ) = 1 denotes that

the j th component operates properly. Thus, for a given diagonal matrix i, i =

r

1, 2,, m. the set = {u = i=1 i i Ki x, and i i, i = 1, 2,, m}

is called an admissible set of actuator fault. Namely, symbol i, i = 1, 2,, m

in set describes the worst status of the scaling factor i, i = 1, 2, m. Once

the scaling factor extent become smaller than i, the reliable controller can not

work properly anymore .

Remark 1. It is obvious that when i =, i = 1, 2,, r, and (j ) takes

only the values of 0 and 1, the actuator failure model is just the same as that

in [6,7] and the references cited therein.From this point,the problem to be solved

here is more general.

For the case of u, the closed-loop system is given by

r r

Ex =

i j [(Ai + Ai ) x + (A1i + A1i ) x + (Bi + Bi ) j Kj x] .

i=1 j =1

(4)

Theorem 1. Consider system (4), if there exist nonsingular matrices P, S >

0, Ki, Xij, where Xii = Xii, Xij = Xji, i = j, (i, j = 1, 2,, r), such that

T T

ET P = P T E 0, (5)

ii P T (A1i + A1i )

ii = 0. Di erentiating V (t) along the trajectory of

system (4) gives

T

X11 X1r

1 1

r r r

. . . .

i j T ij . . . . 0, ij > 0, nonsingular

matrix X, Y > 0, Zij, where Zii = Zii, Zij = Zji, i = j, i, j = 1, 2,, r such

T T

that the following LMIs are satis ed:

X T E T = EX 0, (9)

Ai X + X T AT Bi i Bi

T

A1i Y Zii2 0 X T Ei

T

XT

i

Zii1 + i M M

T

Y Zii3 T

0 Y E1i 0

0, Y2 > 0, Mi, Zii11

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