Fuzzy Control of Nonlinear Time-delay Systems: Stability and Design Issues Yongru Gu Hua O. Wang1
Kazuo Tanaka
Dept. of Elec. and Comp. Eng. Duke University
Dept. of Mech. Eng. and Intell. Syst. University of Electro-Communications
Durham, NC 27708 USA
1-5-1 Chofugaoka, Chofu, Tokyo 182 Japan
Linda Bushnell Dept. of Elec. Eng. University of Washington Seattle, WA 98195 USA
Abstract In this pape, a class of nonlinear time-delay systems based on the Takagi-Sugeno (T-S)fuzzy model is defined. We investigate the delay-independent stability of this model. A model-based fuzzy stabilization design utilizing the concept of the so-called “parallel distributed compensation” (PDC) is employed. The main idea of the controller design is to derive each control rule to compensate each rule of a fuzzy system. Also, the problems of H1 control is considered. The associated control synthesis problems are formulated as linear matrix inequality (LMI) problems. Keywords — Nonlinear time-delay systems, T-S fuzzy model, PDC (parallel distributed compensation), LMI, stabilization, disturbance rejection.
using a Lyapunov function approach ([2], [3], [4], [5], [6]). A further and significant step has also been taken to utilize Lyapunov-function based control design techniques to the control synthesis problem for T-S models. The so-called parallel distributed compensation (PDC) [4] is one such control design framework that has been proposed and developed over the last few years. The PDC control structure ([2] and [4]) utilizes a nonlinear state feedback controller which mirrors the structure of the associated T-S model. The gains of the controller can be determined automatically using an LMI formulation. It has also been shown that within the framework of T-S fuzzy model and PDC control design, design conditions for the stability and performance of a system can be stated in terms of the feasibility of a set of linear matrix inequalities (LMIs)([2]). This is a significant finding in the sense that there exist very efficient numerical algorithms for determining the feasibility of LMIs, so even large scale analysis and design problems are computationally tractable.
1 Introduction As the interest in fuzzy systems has increased, researchers have considered the stability analysis of these systems within various nonlinear frameworks. However, because there are a variety of different approaches to implement a fuzzy controller, it is difficult to identify a single framework which can deal with every application. One of the fuzzy modeling techniques which has attracted recent attention is the approach of Takagi and Sugeno [1]. In this approach, local dynamics in different state space regions are represented by linear models and the overall system is represented as the fuzzy interpolation of these linear models. The appeal of the T-S (Takagi-Sugeno) model is that the stability and performance characteristics of the system can be verified 1 Corresponding author. E-mail:
[email protected]; Tel: (919) 660-
5273; Fax: (919) 660-5293.
In traditional T-S fuzzy models, there is no delay in the control and state. However, time delays often occur in many dynamical systems such as biological systems, chemical systems, metallurgical processing systems and network systems. Their existences are frequently a cause of instability and poor performance. The study of stability and stabilization for linear time-delay systems has received considerable attentions ([7], [8], [9], [10]). But these efforts were mainly restricted to linear time-delay systems. Thus, it is important to extend the stability and stabilization issues to nonlinear time-delay systems. In this paper, a particular class of nonlinear time-delay systems is considered based on Tagaki-Sugeno fuzzy model. This kind of nonlinear system is represented by a set of linear time-delay systems. We will call this model T-S model with time delays (T-SMTD). In the literature, the problem of stability and stabilization of
time-delay systems has been dealt with a number of different ways. There are some results that are independent of the size of the time delays in [7], [8], [9], and the stability is satisfied for any value of the time delays. There are also some delay-dependent results, in which the stability is guaranteed up to some maximum value for the time delays (([10]). This paper is concerned with the problems of delay- independent stability and stabilization of T-S fuzzy model with time delays. Particularly, we will employ the concept of parallel distributed compensation to study these problems. Several new results concerned with the stability and stabilization of T-SMTD are derived. Also, a sufficient condition for H 1 control of this model is given. All the synthesis problems are formulated as LMIs, thus they are numerically efficient. The paper is structured as follows. In section 2, we define the T-S fuzzy model with time delays. A sufficient delayindependent stability condition for the open loop system is derived. In section 3, the problem of stability of the closedloop system is investigated. Based on PDC, in section 4 we study the delay-independent stabilization of our proposed model. In section 5, a sufficient condition for H 1 control is given. In section 6, a numerical example is presented. Finally, we give a brief conclusion. Throughout the paper, the notation M > 0 will mean that M is positive definite symmetric matrix.
2.1 T-S Fuzzy Model with Delays To begin with, we represent a given nonlinear plant by the Takagi-Sugeno fuzzy model. Then, we will define a new kind of model, the Takagi-Sugeno fuzzy model with time delays. The main feature of the T-S fuzzy model is to express the joint dynamics of each fuzzy implication (rule) by a linear system model. Specifically, the Takagi-Sugeno fuzzy systems is described by fuzzy IF-THEN rules, which locally represent linear input-output relations of a system. The fuzzy system is of the following form: Dynamic Part: Rule i: IF p1 (t) is Mi1 , , and pl (t) is Mil THEN
Output Part: Rule i: IF p1 (t) is Mi1 , , and pl (t) is Mil THEN
y(t) = Ci x(t)
!i (p(t)) =
l Y j =1
Mij (pj (t)):
Throughout this paper, we will assume that each !i is a nonnegative function and that the truth value of at least one rule is always nonzero. The second function is called the firing probability. The firing probability for the ith rule is defined by the equation
hi (p(t)) = Pr!i (!p(t())p(t)) i
i=1
where r denotes the number of rules in the rule base. Under the previously stated assumptions, this is alway a well defined function taking values between 0 and 1, and the sum of all the firing probabilities is identically equal to 1. Now, we introduce time delays into the above T-S fuzzy model. Here, we assume there are time delays in both the state and control of the dynamic part. Then, the ith rule of the dynamic part of T-S fuzzy model becomes:
2 T-S Fuzzy Model with Delays and Stability Conditions
x_ (t) = Ai x(t) + Bi u(t); i = 1; 2; ; r:
where x(t), u(t), y (t), and p(t) respectively denote the state, input, output, and parameter vectors. The jth component of p(t) is denoted by p j (t), and the fuzzy membership function associated with the ith rule and jth parameter component is denoted by M ij . Each pj (t) is a measurable time-varying quantity. In general, these parameters may be functions of the state variables, external disturbances, and /or time. There are two functions of p(t) associated with each rule. The first function is called the truth value. The truth value for the ith rule is defined by the equation
(1)
Rule i: IF p1 (t) is Mi1 , , and pl (t) is Mil THEN
x_ (t) = Ai0 x(t) + Aid x(t ; 1 ) +Bi0 u(t) + Bid u(t ; 2 ); i = 1; 2; ; r (2) where 0 1 < 1 and 0 2 < 1 are the size of time delays. The initial condition is x(t) = 0, where t < 0. We would like to call this model T-S model with time delays(TSMTD). In the following we will investigate the stability and design issues, such as delay-independent stabilization and H1 control, of this system. The dynamics described by our model evolve according to the system of equations
x_ (t) =
r X i=1
hi (p)fAi0 x(t) + Aid x(t ; 1 ) B ut
+ i0 ( ) +
y(t) =
Bid u(t ; 2 )g
r X i=1
hi (p)Ci x(t)
(3)
mation
The open-loop system is of the form
x_ (t) =
r X i=1
hi (p)fAi0 x(t) + Aid x(t ; 1 )g
u(t) = ;
(4)
r X i=1
hi (p)Fi x(t):
(7)
Note that the controller (10) is nonlinear in general. Remark: Our proposed model description can also be viewed as parameter-dependent interpolation between linear models; however, the exact classification of the resultant system depends on the nature of the parameters. For example, if each pi is a known function of time, then the T-S model describes a linear time-varying system. If, on the other hand, each pi is a function of the state variables, then the T-S model describes an autonomous nonlinear system. 2.2 Stability Analysis via Lyapunov Approach A sufficient delay-independent stability condition for the open-loop system (4) is given as follows: Theorem 1 The open-loop T-S fuzzy system with timedelays (4) is asymptotically stable in the large if there exist two common positive definite matrices P and R such that
PAi0 + ATi0 P + PAid R;1 ATid P +R < 0; i = 1; 2; ; r
Z t
;
t 1
x(s)T Rx(s)ds:
x_ (t) =
(6)
Then we can get V_ (x) 0. 2. Remark: The system (4) is also said to be quadratically stable and the function V (x) is called a quadratic Lyapunov function. Theorem 1 thus presents a sufficient condition for quadratic stability of the open-loop system (4). 2.3 Parallel Distributed Compensation Control In [2], Wang et al. utilized the concept of parallel distributed compensation (PDC) to design fuzzy controllers to stabilize fuzzy system (1). The idea is to design a compensator for each rule of the fuzzy model. The resulting overall fuzzy controller, which is nonlinear in general, is a fuzzy blending of each individual linear controller. The fuzzy controller shares the same fuzzy sets with the fuzzy system (1). Here, we will apply the same controller structure to the T-SMTD, so the ith control rule is Control Rule i: IF p1 (t) is Mi1 and, , and pl (t) is Mil (t), THEN u(t) = ;Fi x(t); i = 1; ; r: The output of the PDC controller is determined by the sum-
r X i=1
;Bid Fi x(t ; 2 )g + 2
h2i (p)fGii x(t) + Aid x(t ; 1 )
X i 0, R1 > 0, and R2 > 0 such that the following matrix inequalities are satisfied, the closed-loop system (11) is quadratically stable.
PGii + GTii P + PAid R1;1 ATid P + R1 ;1 R;1P ;1 F T B T P +PBid Fi P i id 2 +PR2 P < 0; i = 1; ; r
(9)
P ( Gij +2 Gji ) + ( Gij +2 Gji )T P
+
1 2
P (Aid R1;1 ATid + Ajd R1;1 ATjd )P + R1
P (Bid Fj P ;1 R2;1 P ;1 FjT BidT ;1 R;1P ;1 F T B T ) + PR2 P 0 +Bjd Fi P i jd 2 +
1
2
(10)
Proof: Define the following Lyapunov function for the closed-loop system:
V (x) = x(t)T Px(t) +
Z t
Z t +
;
t 2
;
t 1
x(s)T R1 x(s)ds
x(s)T PR2 Px(s)ds:
(11)
Taking derivate of V (x) along the closed-loop system and using the fact that for any vector x 1 and x2 and matrix Y
xT1 Y x2 + xT2 Y T x1 xT1 Y R;1 Y T x1 + xT2 Rx2
(12)
where R is a positive definite matrix, we have
V_ (x) =
r X
holds, where
Zii = Ai0 X + XATi0 + Aid W1 ATid ; Bi0 Mi ; MiT BiT0
h2i (p)x(t)T fPGii
i=1 T ii +
G P PAid R1;1 ATid P ;1 R;1P ;1 F T B T P +R1 + PBid Fi P i id 2 X G + G ij +PR2 P gx(t) + 2 hi hj x(t)T fP ( 2 ji ) +
i 0 and
P
V_ (x) < 0;
i 0, W1 > 0, W2 > 0, and Mi , 1 i r, such that the following two LMI conditions hold: (a) For every 1 i r, the following equation is satisfied; 2 4
Zii + W2 X Bid Mi X ;W1 0 MiT BidT 0 ;W2
3 5
(b) For every pair of indices satisfying 1 equation 2 6 6 4
Furthermore, if the matrices exist which satisfies these inequalities, then the feedback gains Fi = Mi X ;1 will provide a quadratically stabilizing PDC controller. Proof Let P = X ;1 , W1 = R1;1 , and W2 = R2 , then we can get the above results following Theorem 2. 2
0, W1 > 0, W2 > 0, and Mi , 1 i r, such that the following two LMI conditions hold: (1) For every 1 i r, the equation 2
3
H1 Control
6 6 6 6 4
Zii + W2 X Bid Mi Di XEiT X ;W1 0 0 0 MiT BidT 0 ;W2 0 0 DiT 0 0 ; I 0 Ei X 0 0 0 ; I
3 7 7 7 7 5
