Circuits Syst Signal Process (2008) 27: 447–460 DOI 10.1007/s00034-008-9037-8

Delay-Dependent Robust Control for Uncertain Stochastic Time-Delay Systems Yun Chen · Anke Xue · Shaosheng Zhou · Renquan Lu

Received: 8 January 2007 / Revised: 6 November 2007 / Published online: 15 May 2008 © Birkhäuser Boston 2008

Abstract This paper is concerned with controller design for uncertain stochastic systems with time-varying delays. The parametric uncertainties which appear in all system matrices are assumed to be norm bounded. Two cases of time-varying delays are investigated. Based on the slack matrix technique, delay-dependent stability criteria for the delayed stochastic systems are derived. By using the analysis results, linear matrix inequality-based controllers are designed. Three numerical examples are provided to demonstrate the effectiveness of the proposed method. Keywords Delay dependent · Lyapunov–Krasovskii functional · Stochastic systems · Time delay

This work was supported by the National Natural Science Foundation of China under Grants 60434020 and 60604003. Y. Chen () Institute of Operational Research & Cybernetics, Hangzhou Dianzi University, Hangzhou 310018, People’s Republic of China e-mail: [email protected] Y. Chen National Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou 310027, People’s Republic of China A. Xue · S. Zhou · R. Lu Institute of Information & Control, Hangzhou Dianzi University, Hangzhou 310018, People’s Republic of China A. Xue e-mail: [email protected] S. Zhou e-mail: [email protected] R. Lu e-mail: [email protected]

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1 Introduction Many practical systems can be described by differential equations with time delays, such as population ecology, chemical processes, communications systems, and network control systems [12, 26]. During the past decades, the study of time-delay systems has attracted much attention due to its theoretical and practical importance, see for example, [1, 2, 4, 6, 7, 10, 12–17, 21, 26–28, 37, 38], and the references therein. On the other hand, modeling and control design for stochastic systems play an important role in many industrial fields. In recent years, increasing efforts have been made to study stochastic systems with time delays. Many delay-dependent stability conditions for stochastic systems have been reported in [3, 5, 22–24, 34–36]. An exponential output feedback control for stochastic delay nonlinear systems has been designed using a Lyapunov-based recursive method [8]. Robust H∞ control of uncertain stochastic time-delay systems has been handled in terms of linear matrix inequalities (LMIs) [30, 31, 33]. L2 –L∞ filtering for stochastic systems with delays has been addressed in [9]. When dealing with systems with time-varying delays, one usually needs a conservative and artificial condition that the delays are differentiable and the upper bounds of the delay derivative are less than one, such as in [9, 11, 19, 30, 31]. It can be seen that the controller and filter synthesis methods of [8, 9, 11, 19, 30, 31, 33] are all delay independent. Generally, delay-dependent criteria are less conservative than delay-independent ones, especially when the delays are small. The delaydependent methods of [18, 20, 32] seem to be conservative due to the requirement that the system matrix “A” is assumed to be stable. To the best of our knowledge, the problem of delay-dependent stabilization for stochastic delayed systems has not been fully investigated, which motivates this study. In this paper, we will investigate stability analysis and control design for a class of uncertain stochastic time-delay systems. The parametric uncertainties in all system matrices are time varying and norm bounded. The time-varying delays under consideration will be treated as the following cases: one is differentiable uniformly bounded with an upper bound of the delay derivative, and the other is continuous uniformly bounded. Delay-dependent stability conditions are obtained by introducing some slack matrices. Using these stability criteria, the sufficient conditions for the existence of desired controllers are formulated. Furthermore, the results are extended to systems with nonlinear uncertainties. The presented results are in LMI form, which can be efficiently solved by using LMI Toolbox. The effectiveness of the proposed approach is verified by three illustrative examples. Notation Throughout this paper, the notation is fairly standard:  ·  denotes Euclidean norm; Rn is n-dimensional Euclidean space; Rn×m stands for the set of all n × m real matrices; P > 0 (P < 0) means that the matrix P is symmetric positive (negative) definite; trace{·} represents the trace of a matrix. (Ω, F, P) is a probability space, where Ω is the sample space, and F is a σ -algebra of subsets of Ω. The  Q  P Q symmetric term in a symmetric matrix is denoted as ∗, i.e. P∗ R = QT R .

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2 Problem Formulation and Preliminaries Consider the following uncertain stochastic time-delay system:         ⎧ + A1 + A1 (t) x t − h(t) + B + B(t) u(t) dt ⎪ ⎨ dx(t) = A+ A(t) x(t)      + C + C(t) x(t) + D + D(t) x t − h(t) dw(t), ⎪ ⎩ x(θ ) = ψ(θ ), ∀θ ∈ [−h, 0], (1) where x(t) ∈ Rn is the state vector; u(t) ∈ Rm is the control input; w(t) is a scalar Wiener process defined on the probability space (Ω, F, P) satisfying E{dw(t)} = 0, E{dw 2 (t)} = dt; h(t) is a time-varying delay satisfying 0 ≤ h(t) ≤ h < ∞, ∀t > 0; ψ(·) is the initial condition for all t ∈ [−h, 0]; A, A1 , B, C, D are known real constant matrices with appropriate dimensions; A(t), A1 (t), B(t), C(t), D(t) are time-varying parametric uncertainties, of norm-bounded uncertainty type 

A(t)

A1 (t) B(t) C(t) D(t)  = LF (t) E1 E2 E3 E4 E5 ,

(2)

where L, E1 , E2 , E3 , E4 , E5 are constant matrices with compatible dimensions, and F (t) is an unknown and time-varying matrix function satisfying F T (t)F (t) ≤ I . In this paper, we will handle the following two cases of the time-varying delay h(t) [13, 14]: Case I. h(t) is differentiable with an upper bound of the delay derivative, i.e. 0 ≤ h(t) ≤ h < ∞,

˙ ≤ μ < ∞, h(t)

∀t > 0.

Case II. h(t) is continuous uniformly bounded, i.e. 0 ≤ h(t) ≤ h < ∞,

∀t > 0.

Definition 1 ([30]) The uncertain stochastic time-delay system (1) with u(t) = 0 is said to be robustly stable in the mean-square sense for all admissible uncertainties (2), if for any scalar  > 0 there exists a scalar σ () > 0 such that 2

 E x(t) < , when

∀t > 0

2

 sup E ψ(s) < σ ().

−h≤s≤0

Additionally, system (1) with u(t) = 0 is said to be robustly asymptotically stable in the mean-square sense if 2

 lim E x(t) = 0 t→∞

holds for any initial condition.

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The objective of this paper is to design a state feedback controller u(t) = Kx(t) for system (1) such that the resulting closed-loop system is robustly asymptotically stable in the mean-square sense. To this end, we need the following lemma, which is important for deriving the main results. Lemma 1 ([12]) For any constant matrix 0 < Q ∈ Rn×n , scalar τ > 0, and the vector function x(t) ∈ Rn such that the following integrals are well defined, then

t

t

t −τ x T (s)Qx(s) ds ≤ − x T (s) dsQ x(s) ds. (3) t−τ

t−τ

t−τ

3 Main Results In this section, we first give robust stability analysis results for system (1) with u(t) = 0. Then based on the stability conditions, we will design controllers for system (1). 3.1 Robust Stochastic Stability Analysis Proposition 1 Under Case I, system (1) with u(t) = 0 is robustly asymptotically stable in the mean-square sense if there exist scalars ε1 > 0, ε2 > 0 and matrices P > 0, Q > 0, R > 0, M, N with appropriate dimensions such that ⎡ ⎤ Ω11 Ω12 M C T P hAT R P L 0 ⎢ ∗ 0 0 ⎥ Ω22 N D T P hAT1 R ⎢ ⎥ ⎢ ∗ ∗ −R 0 0 0 0 ⎥ ⎢ ⎥ (4) Ω =⎢ ∗ ∗ −P 0 0 PL ⎥ ⎢ ∗ ⎥ < 0, ⎢ ∗ ∗ ∗ ∗ −R hRL 0 ⎥ ⎢ ⎥ ⎣ ∗ 0 ⎦ ∗ ∗ ∗ ∗ −ε1 I ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I where Ω11 = P A + AT P + Q − M − M T + ε1 E1T E1 + ε2 E4T E4 , Ω12 = P A1 + M − N T + ε1 E1T E2 + ε2 E4T E5 , Ω22 = −(1 − μ)Q + N + N T + ε1 E2T E2 + ε2 E5T E5 . Proof For simplicity, the following notation is adopted in the rest of this paper:   y(t) = A(t)x(t) + A1 (t)x t − h(t) ,   g(t) = C(t)x(t) + D(t)x t − h(t) , where A(t) = A + A(t), A1 (t) = A1 + A1 (t), C(t) = C + C(t), D(t) = D + D(t).

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Choose the following Lyapunov–Krasovskii functional candidate: V (t, xt ) = V1 (t, xt ) + V2 (t, xt ) + V3 (t, xt ),

(5)

where V1 (t, xt ) = x T (t)P x(t),

t V2 (t, xt ) = x T (α)Qx(α) dα, t−h(t)

V3 (t, xt ) = h

0

t

−h t+β

y T (α)Ry(α) dα dβ,

with P > 0, Q > 0, R > 0. By the Itô differential formula [25], the stochastic differential dV (t, xt ) along the trajectories of system (1) with u(t) = 0 is dV (t, xt ) = LV (t, xt ) dt + 2x T (t)P g(t) dw(t),

(6)

LV (t, xt ) = 2x T (t)P y(t) + g T (t)P g(t) + LV2 (t, xt ) + LV3 (t, xt ).

(7)

where

Note that      T ˙ LV2 (t, xt ) = x T (t)Qx(t) − 1 − h(t) x t − h(t) Qx t − h(t)     ≤ x T (t)Qx(t) − (1 − μ)x T t − h(t) Qx t − h(t) . Use Lemma 1 to obtain

t

LV3 (t, xt ) = h y (t)Ry(t) − h 2 T

y T (α)Ry(α) dα

t−h

t

≤ h2 y T (t)Ry(t) − h(t)

≤ h y (t)Ry(t) − 2 T

y T (α)Ry(α) dα

t−h(t) t

T

y(α) dα.

t−h(t)

Notice that   x(t) − x t − h(t) =

t

t−h(t)

t

t−h(t)

for any matrices M, N ∈ Rn×n .

t

y(α) dα +

t−h(t)

we have    2 x T (t)M + x T t − h(t) N 

  × −x(t) + x t − h(t) +

t

y (α) dαR

g(α) dw(α),

(8)

 g(α) dw(α) = 0

(9)

t−h(t)

y(α) dα +

t

t−h(t)

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From (6) and (9), one obtains dV (t, xt ) = LV˜ (t, xt ) dt + 2x T (t)P g(t) dw(t)

   t + 2 x T (t)M + x T t − h(t) N

g(α) dw(α),

(10)

t−h(t)

where    LV˜ (t, xt ) = LV (t, xt ) + 2 x T (t)M + x T t − h(t) N  

t   × −x(t) + x t − h(t) + y(α) dα t−h(t)

≤ ξ (t)Θξ(t) T

with

(11)

⎡

⎤ Θ11 Θ12 M Θ22 N ⎦ , Θ =⎣ ∗ ∗ ∗ −R t  ξ T (t) = x T (t) x T (t − h(t)) t−h(t) y T (α) dα

(12)

and Θ11 = P A(t) + AT (t)P + Q + C T (t)P C(t) + AT (t)h2 RA(t) − M − M T , Θ12 = P A1 (t) + C T (t)P D(t) + AT (t)h2 RA1 (t) + M − N T , Θ22 = −(1 − μ)Q + D T (t)P D(t) + AT1 (t)h2 RA1 (t) + N + N T . If Θ < 0, then LV˜ (t, xt ) < 0, which means that the stochastic system (1) with u(t) = 0 is robustly asymptotically stable in the mean-square sense by Definition 1 and the stochastic stability theory in [25]. In view of Schur’s complements [12], Θ < 0 is equivalent to ⎡ ⎤ Λ11 Λ12 M C T (t)P hAT (t)R ⎢ ∗ Λ22 N D T (t)P hAT1 (t)R ⎥ ⎢ ⎥ ⎥ < 0, Λ=⎢ (13) ∗ −R 0 0 ⎢ ∗ ⎥ ⎣ ∗ ⎦ ∗ ∗ −P 0 ∗ ∗ ∗ ∗ −R where Λ11 = P A(t) + AT (t)P + Q − M − M T , Λ12 = P A1 (t) + M − N T , Λ22 = −(1 − μ)Q + N + N T . Applying the technique of handling the norm-bounded uncertainties (2) to Λ < 0 (see [9]), Ω < 0 will follow immediately. This completes the proof. 

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If the Lyapunov–Krasovskii functional is chosen as V (t, xt ) = V1 (t, xt )+V3 (t, xt ) in (5), then the following proposition can be deduced easily. Proposition 2 Under Case II, system (1) with u(t) = 0 is robustly asymptotically stable in the mean-square sense if there exist scalars ε1 > 0, ε2 > 0 and matrices P > 0, R > 0, M, N with appropriate dimensions such that ⎡

Σ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ Σ =⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

Ω12 Σ22 ∗ ∗ ∗ ∗ ∗

M N −R ∗ ∗ ∗ ∗

CTP DTP 0 −P ∗ ∗ ∗

hAT R hAT1 R 0 0 −R ∗ ∗

PL 0 0 0 hRL −ε1 I ∗

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ PL ⎥ ⎥ < 0, 0 ⎥ ⎥ 0 ⎦ −ε2 I

(14)

where Σ11 = P A + AT P − M − M T + ε1 E1T E1 + ε2 E4T E4 , Σ22 = N + N T + ε1 E2T E2 + ε2 E5T E5 . Remark 1 By similar lines as in the proof of Proposition 3 of [14], we can draw the following conclusions. (1) For μ ≥ 1, if there exist scalars ε1 > 0, ε2 > 0 and matrices P > 0, Q > 0, R > 0, M, N such that Ω < 0 if and only if there exist scalars ε1 > 0, ε2 > 0 and matrices P > 0, R > 0, M, N such that Σ < 0. (2) For μ < 1, if there exist scalars ε1 > 0, ε2 > 0 and matrices P > 0, R > 0, M, N such that Σ < 0, then there exist scalars ε1 > 0, ε2 > 0 and matrices P > 0, Q > 0, R > 0, M, N such that Ω < 0; however, the reverse is not necessarily true. Remark 2 The requirement that matrix “A” should be stable is not needed due to the introduction of the term “−M − M T ” in (4) and (14). Furthermore, the restriction that the upper bound of the delay derivative is less than one (i.e. μ < 1) is removed in Proposition 1 because of the introduction of “N + N T ”. t Remark 3 We point out that in [19] the negative-definite term − t−h(t) x T (α) × Qx(α) dα is neglected, which leads to the delay-independent results. If the term t 0 t − t−h(t) x T (α)Qx(α) dα is ignored, then −h(t) t+β x T (α)Qx(α) dα dβ is redundant [11]. The key point of our approach lies in the fact that the cross terms between t t x(t) and t−h(t) y(α) dα, as well as the terms between x(t − h(t)) and t−h(t) y(α) dα in (12), are considered when the slack matrices M = 0, N = 0, respectively. Remark 4 We avoid applying model transformations and cross term estimating techniques in Propositions 1 and 2, which are expected to be less conservative.

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In order to compare with some previous results on stochastic delay systems with nonlinear uncertainties, we consider the following system:        dx(t) = A(t)x(t) + A1 (t)x t − h(t) dt + g t, x(t), x t − h(t) dw(t), (15) x(θ ) = ψ(θ ), ∀θ ∈ [−h, 0], where g(t, x(t), x(t − h(t))) is a nonlinear function satisfying

      trace g T t, x(t), x t − h(t) g t, x(t), x t − h(t) 2    2 ≤ G1 x(t) + G2 x t − h(t)  ,

(16)

and G1 and G2 are matrix functions with compatible dimensions. Taking the Lyapunov–Krasovskii functional as (5) and following similar lines to those in this paper and [23], we can obtain the proposition as follows. Proposition 3 Under Case I, system (15) is robustly asymptotically stable in the mean-square sense if there exist scalars ε > 0, δ > 0 and matrices P > 0, Q > 0, R > 0, M, N with appropriate dimensions such that ⎤ ⎡ H11 H12 M hAT R P L ⎢ ∗ 0 ⎥ H22 N hAT1 R ⎥ ⎢ ⎥ < 0, H=⎢ ∗ ∗ −R 0 0 ⎥ ⎢ (17) ⎣ ∗ ∗ ∗ −R hRL ⎦ ∗ ∗ ∗ ∗ −εI P < δI, where H11 = P A + AT P + Q − M − M T + δGT1 G1 + εE1T E1 , H12 = P A1 + M − N T + εE1T E2 , H22 = −(1 − μ)Q + N + N T + δGT2 G2 + εE2T E2 . Proposition 4 Under Case II, system (15) is robustly asymptotically stable in the mean-square sense if there exist scalars ε > 0, δ > 0 and matrices P > 0, R > 0, M, N with appropriate dimensions such that ⎡ ⎤ G11 H12 M hAT R P L ⎢ ∗ N hAT1 R 0 ⎥ G22 ⎢ ⎥ ⎢ G=⎢ ∗ ∗ −R 0 0 ⎥ ⎥ < 0, (18) ⎣ ∗ ∗ ∗ −R hRL ⎦ ∗ ∗ ∗ ∗ −εI P < δI, where G11 = P A + AT P − M − M T + δGT1 G1 + εE1T E1 , G22 = N + N T + δGT2 G2 + εE2T E2 .

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3.2 Robust Stochastic Stabilization Having derived the stability results, we are now in a position to design stochastic controllers for system (1). For Case I, we have the following result. Proposition 5 Under Case I, for given scalars h > 0 and λ > 0, if there exist scalars ¯ > 0, M, ¯ N¯ , Y with appropriate dimensions δ1 > 0, δ2 > 0 and matrices X > 0, Q such that ⎤ ⎡ T T T T T ⎢ ⎢ ⎢ ⎢ Υ =⎢ ⎢ ⎢ ⎣

Υ11 ∗ ∗ ∗ ∗ ∗ ∗

Υ12 Υ22 ∗ ∗ ∗ ∗ ∗

M¯ N¯ −λX ∗ ∗ ∗ ∗

XC XD T 0 −X + δ2 LLT ∗ ∗ ∗

Υ15 hXAT1 0 0 − λ1 X + δ1 h2 LLT ∗ ∗

XE1 + Y E3 XE2T 0 0 0 −δ1 I ∗

XE4 XE5T ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ < 0, ⎥ 0 ⎥ 0 ⎦ −δ2 I

(19) where ¯ − M¯ − M¯ T + δ1 LLT , Υ11 = AX + XAT + BY + Y T B T + Q Υ12 = A1 X + M¯ − N¯ T , Υ22 = −(1 − μ)Q¯ + N¯ + N¯ T , Υ15 = hXAT + hY T B T + δ1 hLLT , then system (1) is robustly stochastically stabilizable with a state feedback controller u(t) = Kx(t), and the controller gain is given by K = Y X −1 . Proof Let R = λP , where λ > 0 is a tuning parameter. Replacing A and E1 by A + BK and E1 + E3 K in (4), respectively, we arrive at ⎤ ⎡ Ξ11 Ξ12 M C T P h(A + BK)T λP PL 0 ⎢ ∗ N DTP hAT1 λP 0 0 ⎥ Ω22 ⎥ ⎢ ⎢ ∗ ∗ −λP 0 0 0 0 ⎥ ⎥ ⎢ Ξ =⎢ ∗ ∗ −P 0 0 PL ⎥ ⎥ < 0, (20) ⎢ ∗ ⎥ ⎢ ∗ ∗ ∗ ∗ −λP hλP L 0 ⎥ ⎢ ⎣ ∗ 0 ⎦ ∗ ∗ ∗ ∗ −ε1 I ∗ ∗ ∗ ∗ ∗ ∗ −ε2 I where Ξ11 = P (A + BK) + (A + BK)T P + Q − M − M T + ε1 (E1 + E3 K)T (E1 + E3 K) + ε2 E4T E4 , Ξ12 = P A1 + M − N T + ε1 (E1 + E3 K)T E2 + ε2 E4T E5 . Pre- and post-multiplying (20) with diag{P −1 , P −1 , P −1 , P −1 , (λP )−1 , I, I }, and setting X = P −1 ,

¯ = XQX, Q

M¯ = XMX,

N¯ = XN X,

Y = KX, (21)

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yields ⎡

Π11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ Π =⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

Π12 Π22 ∗ ∗ ∗ ∗ ∗

M¯ N¯ −λX ∗ ∗ ∗ ∗

XC T XD T 0 −X ∗ ∗ ∗

h(AX + BY )T hXAT1 0 0 − λ1 X ∗ ∗

L 0 0 0 hL −ε1 I ∗

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ L ⎥ ⎥ < 0, 0 ⎥ ⎥ 0 ⎦ −ε2 I

(22)

where ¯ − M¯ − M¯ T Π11 = AX + XAT + BY + Y T B T + Q + ε1 (E1 X + E3 Y )T (E1 X + E3 Y ) + ε2 XE4T E4 X, Π12 = A1 X + M¯ − N¯ T + ε1 (E1 X + E3 Y )T E2 X + ε2 XE4T E5 X, ¯ + N¯ + N¯ T + ε1 XE2T E2 X + ε2 XE5T E5 X. Π22 = −(1 − μ)Q Taking δi = εi (i = 1, 2), then Π < 0 can be transformed into LMI (19) by some simple manipulations. Thus, the controller gain can be determined as K = Y X −1 by (21). This completes the proof.  Similar to the proof of Proposition 5, we have the following result for Case II. Proposition 6 Under Case II, for given scalars h > 0 and λ > 0, if there exist scalars ¯ N¯ , Y with appropriate dimensions such that δ1 > 0, δ2 > 0 and matrices X > 0, M, ⎤ ⎡ T T T T T ⎢ ⎢ ⎢ ⎢ Ψ =⎢ ⎢ ⎢ ⎣

Ψ11 ∗ ∗ ∗ ∗ ∗ ∗

Υ12 Ψ22 ∗ ∗ ∗ ∗ ∗

M¯ N¯ −λX ∗ ∗ ∗ ∗

XC XD T 0 −X + δ2 LLT ∗ ∗ ∗

Υ15 hXAT1 0 0 − λ1 X + δ1 h2 LLT ∗ ∗

XE1 + Y E3 XE2T 0 0 0 −δ1 I ∗

XE4 XE5T ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ < 0, ⎥ 0 ⎥ 0 ⎦ −δ2 I

(23) where Ψ11 = AX + XAT + BY + Y T B T − M¯ − M¯ T + δ1 LLT , Ψ22 = N¯ + N¯ T , then system (1) is robustly stochastically stabilizable with a state feedback controller u(t) = Kx(t), and the controller gain is given by K = Y X −1 . Remark 5 Propositions 5 and 6 resort to the tuning parameter λ. They can be solved by employing a numerical optimization algorithm, for instance fminsearch in MATLAB Optimization Toolbox [29], for the feasibility problems Υ ≤ tmin I and Ψ ≤ tmin I , respectively. The locally convergent solutions to these problems are found

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if tmin < 0. The tuning parameter method has been widely used in the literature, such as in [7, 38] and so on. Using a similar method as in Proposition 5, we can design a stochastic controller for the following system (24): ⎧      dx(t) = A(t)x(t) + A1 (t)x t − h(t) + B + B(t) u(t) dt ⎪ ⎪ ⎨    + g t, x(t), x t − h(t) dw(t), (24) ⎪ ⎪ ⎩ x(θ ) = ψ(θ ), ∀θ ∈ [−h, 0]. Due to page limitations, it is omitted.

4 Illustrative Examples In this section, three numerical examples are given to illustrate the effectiveness of the proposed method. Example 1 Consider system (1) with   −2 0 A= , 0 −0.9 L = I,



 −1 0 A1 = , −0.2 −1.2

(25)

E1 = E2 = E4 = E5 = 0.2I.

The maximal allowable bounds of delay for system (25) by Propositions 1 and 2 are listed in Table 1. However, for this example, using the results in [11, 20] no conclusion can be made. Example 2 Consider system (15) with     −1 0 −2 0 , A= , A1 = −0.5 −1 1 −1 L = I, E1 = E2 = 0.1I,     

T trace g t, x(t), x t − h(t) g t, x(t), x t − h(t) 2    2 ≤ 0.1x(t) + 0.1x t − h(t)  .

(26)

This system has been considered in [34]. The upper bounds of the delay for system (26) given by Proposition 3 and [34] are shown in Table 2. Table 1 The maximal admissible bound of h¯ for Example 1

μ

0

0.5

0.9

≥1

Any

h¯ by Proposition 1 h¯ by Proposition 2

1.7075

1.1398

0.7678

0.6769

–

–

–

–

–

0.6769

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Table 2 The maximal admissible bound of h¯ for Example 2

μ

0

0.5

0.9

h¯ by Yue and Han [34] h¯ by Proposition 3

1.1812

0.8502

0.4606

2.1491

1.2956

0.8180

It is shown by Examples 1 and 2 that the results in this paper are less conservative than those of [11, 20, 34]. Example 3 Consider system (1) with     1 0 −1 1 A= , , A1 = 0.8 1 1 2 L = 0.2I,



 1 0.5 B= , 0.2 0.2

(27)

E1 = E2 = E3 = E4 = E5 = I.

When μ = 0, λ = 0.1, we can obtain a maximal admissible upper bound of the delay h¯ = 0.8064 by Proposition 5. The corresponding controller can be determined as   −6.7716 −24.7368 u(t) = Kx(t) = x(t). (28) −2.4008 −17.9003 When μ = 0.5, λ = 0.1, system (27) can be stabilized for any h(t) ≤ h¯ = 1.0149 by Proposition 5. If h(t) is not differentiable or there is no information for μ, then system (27) can be stabilized for any h(t) ≤ h¯ = 0.2477 (λ = 1.5) by Proposition 6.

5 Conclusions We have considered delay-dependent robust control for a class of uncertain stochastic time-delay systems with norm-bounded uncertainties. Two cases of time-varying delays have been studied. The presented results have been derived based on the Lyapunov–Krasovskii method and the slack matrix technique. The applicability of the provided method has been verified by three illustrative examples. Acknowledgements The authors would like to give sincere thanks to Associate Editor Professor Peng Shi and to the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper.

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