Fuzzy-Model-Based ${{\cal D}} $-Stability and Nonfragile Control for

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 4, AUGUST 2014

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Short Papers Fuzzy-Model-Based D-Stability and Nonfragile Control for Discrete-Time Descriptor Systems With Multiple Delays Fanbiao Li, Peng Shi, Ligang Wu, and Xian Zhang Abstract—This paper is concerned with the problems of D-stability and nonfragile control for a class of discrete-time descriptor Takagi–Sugeno (T–S) fuzzy systems with multiple state delays. D-stability criteria are proposed to ensure that all the poles of the descriptor T–S fuzzy system are located within a disk contained in the unit circle. Furthermore, a sufficient condition is presented such that the closed-loop system is regular, causal, and D-stable, in spite of parameter uncertainties and multiple state delays. The corresponding solvability conditions for the desired fuzzy-ruledependent nonfragile controllers are also established. Finally, examples are given to show the effectiveness and advantages of the proposed techniques. Index Terms—D-stability, descriptor system, multiple time-delays, nonfragile control, Takagi–Sugeno (T–S) fuzzy systems.

I. INTRODUCTION In order to achieve the various aspects of optimum system performance, the dynamic response of a linear time-invariant system can be modified by means of placing the poles in predetermined locations [14]. However, due to the existence of uncertainties, poles cannot be placed at a specific location. Therefore, it is satisfactory in practice to assign all the poles of closed-loop system in a specified region. A typical region for linear systems can be chosen as a disk D(α, r) centered at α with radius r, with |α| + r < 1. The considerable amount of results on this issue have been reported in the literature; see, for example, [3] and [11], and the references therein. Meanwhile, descriptor systems have attracted a lot researchers due to the fact that descriptor systems can better describe the behavior of some physical systems than state space ones. A great number of fundamental results that are based on the theories of state-space systems have been extended to the area of descriptor systems [1], [2], [5], [21], [24], [25]. Recently, the problems of D-stability and D-stabilization for discrete descriptor systems with time-delay and parameter uncertainty have been studied [16], [26], [27]. In many engineering areas of the real world, most physical systems and processes are nonlinear, which introduce serious difficulties in system analysis and design. It is well known that there has been an effective method with the advent of the Takagi–Sugeno (T–S) fuzzy model to

Manuscript received January 14, 2013; revised April 19, 2013; accepted June 5, 2013. Date of publication July 10, 2013; date of current version July 31, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61174126 and Grant 61222301, in part by the Fundamental Research Funds for the Central Universities under Grant HIT.BRETIV.201303, in part by the National Key Basic Research Program, China under Grant 2012CB215202, and in part by the 111 Project under Grant B12018. F. Li and L. Wu are with the Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, China (e-mail: [email protected]; [email protected]). P. Shi is with the School of Electrical and Electronic Engineering, the University of Adelaide, Adelaide, S.A. 5005, Australia, and also with the College of Engineering and Science, Victoria University, Melbourne, Vic. 8001, Australia (e-mail: [email protected]). X. Zhang is with the School of Mathematical Science, Heilongjiang University, Harbin 150080, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2013.2272647

analyze and synthesize nonlinear systems [4], [6], [22], [23]. Hence, researchers have been paying remarkable attention to the problems of analysis and synthesis for T–S fuzzy systems. A T–S fuzzy model consists of a set of local linear models in different premise variable-space regions, which are blended through fuzzy-membership functions. It has a convenient dynamic structure so that some well-established linear system theories can be easily applied for the theoretical analysis and design of nonlinear systems. Recently, a great number of results on the analysis and the control of T–S fuzzy systems have been presented (see, for example, [7]–[10], [13], [18], [19], [28]–[30], and the references therein). On the other hand, the issue of D-stability has been taken into account in T–S fuzzy systems by some researchers, for example, In [17], the problem of H∞ filtering for fuzzy dynamical systems with D-stability constraints was studied. The filter is quadratically stable in a prespecified linear matrix inequality D-stability region and the controller’s premise variable is allowed to be different from the fuzzy model’s premise variable. In [2], a fuzzy output feedback controller that not only guarantees the L2 -gain of the mapping from the exogenous input noise to the regulated output to be less than some prescribed value, but also ensures the closed-loop poles of each subsystem are in a prespecified region has been proposed. However, it should be pointed out that the D-stability constraints in these literatures are under some restricted constraints which makes their results somewhat conservative. Thus, it is highly desirable to use the direct approach to avoid those constraints. In this paper, it should be noted that the D-stability is derived by one direct approach, but not in the form of D-stability constraints as proposed in [2] and [17]. To the authors’ knowledge, there are few results that are reported on the D-stability for discretetime descriptor T–S fuzzy systems with multiple state delays. Many open questions still remain unsolved. Motivated by the previous discussion, in this paper, we investigate the problems of robust D-stability and nonfragile control for the discrete-time descriptor T–S fuzzy system with multiple state delays. Sufficient conditions for robust D-stability and nonfragile controller design are obtained in terms of linear matrix inequality (LMIs). Specifically, the problems to be studied can be formulated as follows. 1) Under one direct approach, effort is made toward investigating directly the D-stability and D-stabilization of discrete-time descriptor systems instead of state augmentation. Thus, the proposed D-stability condition is much less conservative than the existing results (see Example 1 for details). 2) The nonfragile control problem is investigated in our study, and the designed controller allows the uncertainties that exist in the control input gain. In addition, the controller can guarantee that the closed-loop system is regular, causal, and D-stable, in spite of parameter uncertainties and multiple state delays. 3) The controller design is cast into solving a convex optimization problem, which can be solved by using standard softwares such as LMI Toolbox in MATLAB. Notations: The notations used throughout the paper are standard. Rn and Rn ×m denote the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. For real symmetric matrices P and Q, P > Q(P < Q) means that the matrices P − Q is positive (negative) definite. The superscripts “T ” and “H” represent the transpose and Hermitian transpose, respectively. sym{A} is the shorthand notation for A + AT , and ρ(X) denotes the spectral radius of matrix X. Let a and |a| denote the conjugate and modulus of a complex number,

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respectively. In symmetric block matrices or long-matrix expressions, we use a star “” to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. II. SYSTEM DESCRIPTION AND PRELIMINARIES Consider the following descriptor T–S fuzzy systems with multiple time-delays  Plant Form: Rule j: IF θ1 (k) is Mj 1 , θ2 (k) is Mj 2 , . . . , and θp (k) is Mj p , THEN ⎧ N  ⎪ ⎨ Ex(k + 1) = A(j ) x(k) + Ai (j ) x(k − hi ) + B(j ) u(k) ⎪ ⎩

i= 1

x(k) = φ(k),

ˆ . . . , 0, j = 1, . . . , L k = −h,

where x(k) ∈ R is the state vector, u(k) ∈ R is the control input vec¯ i , and h ˆ= tor; h1 , . . . , hN are positive integers that satisfy 0 < hi  h ¯ i }. The matrix E may be singular, that is, rankE = maxi = 1 , 2 , . . . , N {h r  n. φ(k) denotes the initial condition. Mj s is the fuzzy set, L is the number of IF-THEN rules, and θ(k) = [θ1 (k), θ2 (k), . . . , θp (k)] is the premise variables vector, and it is assumed that the premise variables do not depend on the input variables u(t). A(j ) , Ai (j ) , and B(j ) are known constant matrices with appropriate dimensions. The fuzzy basis functions are given by p = 1 M j s (θj (k)) hj (θ(k)) = L s  p j=1 s = 1 M j s (θj (k)) n

m

with Mj s (θj (k)) representing the grade of membership of θj (k) in M . Therefore, for all k, we have hj (θ(k))  0, j = 1, 2, . . . , L, and jLs j = 1 hj (θ(k)) = 1. Given a pair of (x(k), u(k)), a more compact presentation of the descriptor T–S fuzzy system can be given by ⎧  L  ⎪ ⎪ Ex(k + 1) = hj (θ(k)) A(j ) x(k) ⎪ ⎪ ⎪ j=1 ⎪ ⎨ N  (1) ⎪ + Ai (j ) x(k − hi ) + B(j ) u(k) ⎪ ⎪ i = 1 ⎪ ⎪ ⎪ ⎩ ˆ −h ˆ + 1, . . . , 0. x(k) = φ(k), k = −h, Moreover, we define 

A(θ) =

L

hj (θ(k))A(j ) ,



Ai (θ) =

j=1 

B(θ) =

L

L

hj (θ(k))Ai (j )

j=1

hj (θ(k))B(j ) .

j=1

Then, system (1) can be written as ⎧ N  ⎪ ⎨ Ex(k + 1) = A(θ)x(k) + Ai (θ)x(k − hi ) + B(θ)u(k) ⎪ ⎩

then system (2) with u(k) = 0 is said to be D(α, r)-stable if all solutions of the characteristic equation F (z) = 0 lie in D(α, r), where 

D(α, r) = {z : |z − α| < r}, |α| + r < 1. Definition 3: System (2) is said to be D(α, r)-admissible if it is regular, causal, and D(α, r)-stable. III. MAIN RESULTS A. Robust D-Admissible In this section, we will discuss the problem of D(α, r)-admissible for the unforced system of (2), which will play an essential role in this paper. Our first result in this paper is as follows. Theorem 1: The unforced system of (2) is D(α, r)-admissible if there exist matrices X(j ) > 0, j = 1, 2, . . . , L and Q such that A H (θ)X(θ)A (θ) − r 2 E T X(θ)E + QS T A (θ) +A H (θ)SQT < 0 where 

X(θ) =

ˆ −h ˆ + 1, . . . , 0. k = −h,

(2) Definition 1: [27] System (2) is said to be regular if det(F (z)) is not ˆ + rankE, identically zero, and if, in addition, its degree is equal to nh the system is further named causal. Definition 2: The characteristic polynomial F (z) of the unforced system of (2) is defined by  N

ˆ+1 ˆ ˆ −h i h h h E − z A(θ) − Ai (θ)z F (z) = det z i= 1

L

hj (θ(k))X(j )

j=1 

A (θ) = A(θ) − αE +

N

Ai (θ)(rv + α)−h i

∀|v|  1

i= 1

and S ∈ Rn ×(n −r ) is any matrix with full column satisfies E T S = 0. 

. Then, |v|  1. Assume Proof: For any z ∈ / D(α, r), let v = z −α r that there exist matrices X(θ) and Q such that (3) is satisfied. We first show that the pair (E, A (θ)) is regular and causal. To this end, we choose two nonsingular matrices M and N such that

 I 0 E=M N (4) 0 0 

A1 (θ) A2 (θ) N. (5) A (θ) = M A3 (θ) A4 (θ)   0 −T Then, S can be given as S = M H, where H is any nonsingular I matrix. Setting

 X1 (θ) X2 (θ) T (6) M X(θ)M = X2T (θ) X4 (θ)

 Q1 N −T QHT = . (7) Q2 Now, substituting (4)–(7) into (3) gives

   

i= 1

x(k) = φ(k),

(3)

A(θ)

0, it is easy to see that (10) is a contradiction since ξ T A2T (θ)X1 (θ)A2 (θ)ξ  0

  A1 (θ) F (z) = 0 ⇔ det vIq − = 0. r

(13)

Correspondingly, let

˜ −T X(θ)M ˜ −1 = M

X1 (θ)

X2 (θ)

X2T

X3 (θ)

(θ)

 .

(14)

Substituting (11), (12), and (14) into (3), we obtain that

A1 (θ) X1 (θ)A1 (θ) − r X1 (θ) < 0 ∀|v| ≥ 1   which implies that ρ A1 (θ) < r, and hence   A1 (θ) det vIq − = 0 ∀|v|  1. r 2

⎢ ⎢ ⎢ ⎢ ⎣

−Q(j )

W



0

T

0

∗

Ξ(j )

∗

∗

P(j ) − GT − G

Φ(j )

∗

∗

∗

−Ψ(j )

A(j ) − αE

G

Θ(j )

⎥ ⎥ ⎥ 0, and scalars λi (j ) > 0(i = 1, 2, . . . , N ; j = 1, 2, . . . , L) such that

⎤

0 ˆH G⎥ ⎥ M (j ) ⎦

0 ⎡ ⎤⎡ ⎤T 0 0 ⎢ ⎥⎢ ⎥ ⎣ I ⎦ ⎣ −(W T − QS T )Ai (j ) ⎦ T

0 ⎡

G Ai (j )

⎤ ⎡ ⎤T 0 0 ⎢ ⎥ + (rv + α)−h i ⎣ −(W T − QS T )Ai (j ) ⎦ ⎣ I ⎦ 0 GT Ai (j ) ⎡ ⎤ ⎡ ⎤T ⎡ ⎤ 0 0 0 1 −2 h i ⎢ ⎢ ⎥ ⎥  λi (j ) ⎣ I ⎦ ⎣ I ⎦ + τ ⎣ −(W T − QS T )Ai (j ) ⎦ λi (j ) 0 0 GT Ai (j ) ⎤T ⎡ 0 ⎥ ⎢ × ⎣ −(W T − QS T )Ai (j ) ⎦ T

G Ai (j ) we have (18); thus, the proof is completed.  Remark 1: Note that the conditions in Theorems 1 and 2 are all of strict LMI form. However, the conditions in [16] are not all of strict LMI form due to matrix equality constraint of E T P E  0. This may cause big trouble with checking the conditions numerically, since the matrix equality constraint is fragile and usually not satisfied perfectly. Therefore, the strict LMI conditions in this brief are more desirable than [16] from the numerical point of view.

where



Gj s = ⎡ ⎢ ⎢ G1 j s = ⎢ ⎢ ⎣ 

⎡

The overall state feedback control law is inferred by

=



G2 j s

∗

G3 j s



−Q(j )

W

0

∗

Ω(j )

Υ(j )

∗

∗

P(j ) − GT − G

∗

∗

∗



⎤

0

⎥ Θ(j ) ⎥ ⎥ Φ(j ) ⎥ ⎦ −Ψ(j )

0

0

UT

N T

0

0

0

0

!

⎤

0

⎥ ⎥ ⎥ T G B(j ) M ⎥ ⎦ ˜ (j ) Υ 0

"

G3 j s = diag − V(s ) , −V(s ) , − I, − I  L

 T T B(j ) B(j Ω(j ) = Ξ(j ) − ) (W − SQ ) j=1 L

T B(j )

j=1 



Υ(j ) = A(j ) − αE

T

 G+



L

B(j )

T B(j )G

j=1  ˜ (j ) = Υ − (W T − QS T )B(j ) M

and Ξ(j ) , Θ(j ) , Φ(j ) , Ψ(j ) , and τ are defined as in Theorem 2. Moreover, when (20) and (21) are satisfied, a nonfragile feedback controller is given by

K(s ) = V(s−1) U +

L

T B(j ).

j=1

  ˜ (s ) x(k) hj (θ(k))hs (θ(k)) A(j ) − B(j ) K

j = 1 s= 1

i= 1

G1 j s

hj (θ(k)) K(j ) + ΔK(j ) (k) x(k)

L L



+

(21)

with



where ΔK(j ) (k) is a perturbed matrix with the following form ΔK(j ) (k) = Mj F (k)Nj , where Mj and Nj are known matrices with appropriate dimension. F (k) is an unknown continuous function and satisfying F T (k)F (k)  I. The aim is to determine the local feedback gain K(j ) , such that the following closed-loop system is regular, causal, and D(α, r)-stable:

N

1j