new stability criteria for discrete-time systems with interval ... - SciELO

Latin American Applied Research

40:119-124 (2010)

NEW STABILITY CRITERIA FOR DISCRETE-TIME SYSTEMS WITH INTERVAL TIME-VARYING DELAY AND POLYTOPIC UNCERTAINTY W. ZHANG† , Q.Y. XIE† , X.S. CAI‡ , and Z.Z. HAN† †School of electronic information and electrical engineering, Shanghai Jiao Tong University 200240, Shanghai, China [email protected], [email protected], [email protected] ‡College of Mathematics, Physics, and Information Engineering, Zhejiang Normal University 321004, Jinhua, Zhejiang, China [email protected]

Abstract— This paper is considered with the robust stability problem for linear discretetime systems with polytopic uncertainty and an interval time-varying delay in the state. On the basis of a novel Lyapunov-Krasovskii functional, new delay-range-dependent stability criteria are established by employing the freeweighting matrix approach and a Jensen-type sum inequality. It is shown that the newly proposed criteria can provide less conservative results than some existing ones. Numerical examples are given to illustrate the effectiveness of the proposed approach. Keywords— Delay-range-dependent stability; Lyapunov-Krasovskii functional; Discretetime systems; Time-varying delay; Linear matrix inequality (LMI) I. INTRODUCTION Time-delays are frequently encountered in many fields of engineering systems such as long transmission lines in pneumatic systems, nuclear reactors, rolling mills, communication networks and manufacturing processes (Gu et al., 2003; Hale and Lunel, 1993; Su and Zhang, 2009). In general, the existence of delays in system models may induce instability or poor performance of the closed-loop schemes. Therefore, the stability problem of time-delay systems has attracted much attention during the past decades. Numbers of stability criteria have been established for various types timedelay systems. These criteria can be classified into two types: delay-dependent and delay-independent stability conditions; the former includes the information on the size of the delay, while the latter does not (Xu and Lam, 2008). Usually, delay-dependent stability conditions are less conservative than the delayindependent ones especially in the case when the delay is small. Therefore, in recent years many researchers have devoted to investigating delay-dependent stabil-

ity of time-delay systems (see e.g., Gu et al., 2003; Xu and Lam, 2008; and the references therein). Surveying in the literature, various approaches have been proposed to derive the delay-dependent stability conditions (Xu and Lam, 2008). For instance, the discretized Lyapunov-Krasovskii functional approach (Gu et al., 2003) and the descriptor system approach (Fridman and Shaked, 2002) together with the bounding techniques (Park, 1999 and Moon et al., 2001). Recently, the free-weighting matrix method (He et al. 2004a, 2004b) has been extensively used in deriving the delay-dependent criteria, which is very helpful to reduce the conservatism in existing stability criteria (He et al., 2007; Peng and Tian, 2008; Li et al., 2008). In Jiang and Han (2008), new stability criteria for uncertain systems with interval time-varying delay are proposed by introducing new Lyapunov-Krasovskii functional and employing an integral inequality (Han, 2005). However, it is worth mentioning that most of the delay-dependent stability results in the existing literature are concerned with norm-bounded uncertain continuous-time systems, while little attention has been paid to discrete-time case with polytopic uncertainty (Liu et al., 2006). Recently, the delay-dependent stability problem for discrete-time systems with interval time-varying delay has been studied in Gao et al. (2004), Fridman and Shaked (2005), Jiang et al. (2005), and Gao and Chen (2007). Some delay-dependent stability criteria are established by employing the free-weighting matrix approach (Gao and Chen, 2007) or the descriptor system approach (Fridman and Shanked, 2005). Very recently, Zhang et al. (2008) presented an improved stability criterion by considering the useful terms ignored in the Lyapunov-Krasovskii functional of the previous literature. However, there is still room for further investigation. For example, in Zhang et al. (2008), the term AT P AT is involved in the stability criterion. Therefore, it is not easy to extend the proposed criterion to polytopic-type systems. Moreover,

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W. ZHANG, Q. Y. XIE, X. S. CAI, Z. Z. HAN

the criterion in Liu et al. (2006) for polytopic systems is actually not a delay-dependent stability condition since it only depends on the size of the interval. In this note, we consider the delay-dependent stability of discrete-time systems with polytopic uncertainty and an interval time-varying delay in the state. Firstly, a novel Lyapunov-Krasovskii functional, which makes use of the information of both the lower and upper bounds of the interval time-varying delay, is introduced. Then, based on this functional, a new delayrange-dependent stability criterion is established for the nominal system in terms of linear matrix inequalities (LMIs). The criterion is easily adapted for the stability analysis of polytopic systems since it exhibits a kind of decoupling between the Lyapunov and the system matrices. Finally, numerical examples show that the proposed criteria can provide less conservative results than some existing ones. Notations: Rn denotes the n-dimensional Euclidean space and the notation P > 0 (≥ 0) means that P is real symmetric and positive definite (semi-definite). The superscript “T ” stands for matrix transposition. In symmetric block matrices, we use an asterisk (∗) 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. rM Moreover, for convenience, in the sum notation rm (·), if rM < rm , we denote rM rm

(·) = 0.

et al., 2004b; Jiang and Han, 2008). For discrete timedelay systems with polytopic uncertainty, we assume that the matrices A and Ad in (2) can be expressed as the form of r A Ad = λi A(i)

rM

T x(i)

W

rM

x(i)

≤

i=rm

(rM − rm + 1)

(3)

where dm and dM are nonnegative integers representing the lower and upper bounds of the delay, respectively. Note that the lower bound dm may not be equal to 0. It is worth mentioning that the assumption on the delay d(k) characterizes the real situation in many practical applications. A typical example containing interval-like delay (3) is the network control systems, which have been widely investigated in recent literature. See e.g., Gao et al. (2004), Gao and Chen (2007), Jiang and Han (2008) for further details. It is known that continuous time-delay systems with polytopic uncertainty have been extensively studied in the existing literature (Fridman and Shaked, 2002; He

(4)

Lemma 1 (Jiang et al. 2005). For any constant matrix W ∈ Rn×n , W = W T > 0, two integers rM and rm satisfying rM ≥ rm , vector function x : [rm , rM ] → Rn , the following inequality holds:

II. PROBLEM FORMULATION AND PRELIMINARIES

0 ≤ dm ≤ d(k) ≤ dM ,

,

r where i=1 λi = 1, 0 ≤ λi ≤ 1, i = 1, 2, . . . , r, and (i) A(i) , Ad are all known constant matrices that characterize the vertexes of the convex polytopic set. The main objective of this paper is to develop new delay-range-dependent stability conditions for system (2) with interval time-varying delay satisfying (3) and polytopic uncertainty (4). We end this section by introducing a Jensen-type sum inequality, which can be viewed as a discrete-time counterpart of the Jensen-type integral inequality proposed by Gu (2000) for continuous-time systems (see also Zhu and Yang, 2008). This inequality is helpful to derive the stability criteria.

i=rm

where x(k) ∈ Rn is the state; A and Ad are n × n system matrices. {φ(k) : k = −dM , −dM + 1, . . . , 0} is a given sequence of initial condition. The state delay d(k) is time varying and satisfies

i=1

(1)

Consider the following discrete-time system with a time-varying delay in the state x(k + 1) = Ax(k) + Ad x(k − d(k)) (2) x(k) = φ(k), k = −dM , −dM + 1, . . . , 0.

(i)

Ad

rM

xT (i)W x(i). (5)

i=rm

III. MAIN RESULTS In this section, we first consider the stability for the nominal system described by (2), i.e., A and Ad are both known matrices. By introducing a new Lyapunov-Krasovskii functional, which makes use of the range information of the time-varying delay, we can establish the following result. Theorem 1. The discrete time-delay system (2) is asymptotically stable for any time delay d(k) satisfying (3), if there exist matrices P > 0, Q1 > 0, Q2 > 0, T LT1 LT2 LT3 , R1 > 0, R > 0, T , T , T , L = 2 1 2 3 M T = M1T M2T M3T and N T = N1T N2T N3T of appropriate dimensions such that the following LMI holds: ⎡ ⎤ Ω M −L −N −L −M −N ⎢ ∗ Q2 − Q1 0 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ⎥ ∗ −Q 0 0 0 2 ⎢ ⎥ < 0, Θ=⎢ ⎥ ∗ ∗ ∗ −R 0 0 1 ⎢ ⎥ ⎣∗ ∗ ∗ ∗ −R2 0 ⎦ ∗ ∗ ∗ ∗ ∗ −R2 (6)

120


where ρ = dM − dm , ⎡ Ω11 Ω=⎣ ∗ ∗

40:119-124 (2010)

From (2), we have ⎤ Ω13 Ω23 ⎦ , Ω33

Ω12 Ω22 ∗

ψ1 (k) := (A − I)x(k) + Ad x(k − d(k)) − η(k) ≡ 0. (7)

Denote

with

k−1

α(k) :=

i=k−dm

Ω11 = T1 A + AT T1T − T1 − T1T + L1 + LT1 + Q1 , Ω13 = Ω22 = Ω23 = Ω33 =

+ ATd T2T ,

(8)

V1 (k) = xT (k)P x(k), V2 (k) =

x (i)Q1 x(i) +

i=k−dm

V3 (k) = dm

−1

k−d m −1

xT (i)Q2 x(i),

i=k−dM k−1

ψ2 (k) : = x(k) − x(k − dm ) − α(k) ≡ 0, ψ3 (k) : = x(k − dm ) − x(k − d(k)) − β(k) ≡ 0, ψ4 (k) : = x(k − d(k)) − x(k − dM ) − γ(k) ≡ 0. Denote ξ T (k) = ξ1T (k) ξ2T (k) ξ3T (k) , where ξ1T (k) = xT (k) xT (k − d(k)) η T (k) , ξ2T (k) = xT (k − dm ) xT (k − dM ) , ξ3T (k) = αT (k) β T (k) γ T (k) . Then, we have

η T (i)R1 η(i),

ΔV1 (k) = η T (k)P η(k) + η T (k)P x(k) + xT (k)P x(k)

j=−dm i=k+j

V4 (k) = ρ

−d m −1 k−1

+ 2ξ1T (k)T ψ1 (k) + 2ξ1T (k)Lψ2 (k)

η T (i)R2 η(i),

+ 2ξ1T (k)M ψ3 (k) + 2ξ1T (k)N ψ4 (k)

j=−dM i=k+j

η(k) = x(k + 1) − x(k),

¯ = ξ T (k)Θξ(k),

and P > 0, Q1 > 0, Q2 > 0, R1 > 0, R2 > 0 are matrices to be determined. Let us define for i = 1, . . . , 4, ΔVi (k) = Vi (k + 1) − Vi (k). Note that x(k + 1) = η(k) + x(k). Then, along the solution of system (2), we have ΔV1 (k) =η T (k)P η(k) + η T (k)P x(k) + xT (k)P η(k), ΔV2 (k) =xT (k)Q1 x(k) − xT (k − dM )Q2 x(k − dM ) + xT (k − dm )(Q2 − Q1 )x(k − dm ), ΔV3 (k) =d2m η T (k)R1 η(k) − dm

k−1

(9)

ΔV4 (k) =ρ η (k)R2 η(k) − ρ

k−d m −1

with ¯ 11 = T1 A − AT T T − T1 − T T + L1 + LT , Ω 1 1 1

η T (i)R1 η(i),

T

η (i)R2 η(i).

(10)

where T , L, M and N are free weighting matrices, ⎤ ⎡ ¯ M −L Ω −N −L −M −N ⎢ ∗ Q2 − Q1 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢∗ ∗ −Q 0 0 0 2 ¯ ⎥, ⎢ Θ=⎢ ⎥ ∗ ∗ ∗ −R 0 0 1 ⎥ ⎢ ⎣∗ ∗ ∗ ∗ −R2 0 ⎦ ∗ ∗ ∗ ∗ ∗ −R2 ⎡ ⎤ ⎡ ⎤ ¯ 11 Ω12 Ω13 T1 Ω ¯ =⎣ ∗ Ω22 Ω23 ⎦ , T = ⎣T2 ⎦ , Ω ¯ 33 T3 ∗ ∗ Ω

¯ 33 = P − T3 − T3T , Ω

i=k−dm 2 T

η(i).

According to (1), we have α(k) = 0 when dm = 0, β(k) = 0 when d(k) = dm , and γ(k) = 0 when d(k) = dM . Then, it is obvious that

where

T

i=k−dM

Proof. Choose a Lyapunov-Krasovskii functional candidate for the system (2) as follows

k−1

η(i)

k−d(k)−1

γ(k) :=

V (k) = V1 (k) + V2 (k) + V3 (k) + V4 (k),

k−d m −1 i=k−d(k)

and

Ω12 = T1 Ad − M1 + N1 + LT2 − T2T + AT T2T , P − T1 + LT3 − T3T + AT T3T , −M2 − M2T + N2 + N2T + T2 Ad −T2 − M3T + N3T + ATd T3T , P − T3 − T3T + d2m R1 + ρ2 R2 .

η(i), β(k) :=

and Ω12 , Ω13 , Ω22 , Ω23 are define in (7). On the other hand, by applying Lemma 1, we can obtain ΔV3 (k) ≤ d2m η T (k)R1 η(k) − αT (k)R1 α(k),

i=k−dM

121

(11)


(i)

ΔV4 (k) =ρ η (k)R2 η(k) − ρ

T

η (i)R2 η(i)

i=k−dM

−ρ

k−d m −1

η T (i)R2 η(i)

i=k−d(k)

Θi < 0,

≤ρ2 η T (k)R2 η(k)

− (dM − d(k))

η T (i)R2 η(i)

i=k−dM k−d m −1

η T (i)R2 η(i)

i=k−d(k)

≤ρ2 η T (k)R2 η(k) − γ T (k)R2 γ(k) − β T (k)R2 β(k).

(12)

with ρ = dM − dm , ⎡ˆ Ω11 ˆ =⎣ ∗ Ω

It then follows from (9)-(12) that ΔV (k) ≤ ξ(k)T Θξ(k).

(13)

If Θ < 0, then (12) implies that there exists a suffi2 cient small scalar ε > 0 such that ΔV (k) ≤ −ε x(k) (Hale and Lunel, 1993). Therefore, the system in (2) is asymptotically stable when the interval time-varying delay satisfies (3). 2 Remark 1. In the proof of Theorem 1, we introduce a new Lyapunov functional (8) to derive the stability criterion. Compared with those in Zhang et al. (2008) and Gao and Chen (2007), this functional is simpler. More precisely, the following terms: V5 (k) =

k−1

xT (i)Q3 x(i)

V6 (k) =

−d m

k−1

ˆ 13 ⎤ Ω ˆ 23 ⎦ , Ω ˆ 33 Ω

ˆ 12 Ω ˆ 22 Ω

∗

∗

(15)

ˆ 11 = T1 (A(i) − I) + (A(i) − I)T T T + L1 + LT + Q(i) , Ω 1 1 1 ˆ 12 = T1 A(i) − L1 − M1 + N1 + LT2 + (A(i) − I)T T2T , Ω d ˆ 13 = P (i) − T1 + LT3 + (A(i) − I)T T3T , Ω T ˆ 22 = −M2 − M2T + N2 + N2T + T2 A(i) + T2 A(i) , Ω d d T ˆ 23 = −T2 − M3T + N3T + A(i) T3T , Ω d ˆ 33 = P (i) − T3 − T3T + d2m R(i) + ρ2 R(i) . Ω 1 2 Proof. Choose a parameter dependent LyapunovKrasovskii functional as follows

i=k−d(k)

and

(14)

hold for i = 1, 2, . . . , r, where Θi is given by ⎡ ⎤ ˆ Ω M −L −N −L −M −N ⎢ ⎥ (i) (i) 0 0 0 0 ⎥ ⎢ ∗ Q2 − Q1 ⎢ ⎥ (i) ⎢∗ ∗ −Q2 0 0 0 ⎥ ⎢ ⎥, (i) ⎢∗ ∗ ∗ −R1 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ (i) ∗ ∗ ∗ −R2 0 ⎦ ⎣∗ (i) ∗ ∗ ∗ ∗ ∗ −R2

k−d(k)−1

− (d(k) − dm )

(i)

if there exist matrices P (i) > 0, Q1 > 0, Q2 > (i) (i) 0, R1 > 0, R2 > 0, (i = 1, 2, . . . , r), T1 , T2, T3 , T T T T T M2T M3T and L = L1 L2 LT3 , M = M1 T T T T N = N1 N2 N3 of appropriate dimensions such that the following LMIs:

k−d(k)−1 2 T

Vλ (k) = V1λ (k) + V2λ (k) + V3λ (k) + V4λ (k), xT (i)Q3 x(i)

where

j=−dM +1 i=k+j

are employed in Zhang et al. (2008), where Q3 > 0. Moreover, V3 (k) and V4 (k) in (8) are a little different from those in Zhang et al. (2008). However, as indicated in Example 1, Theorem 1 can provide less conservative results than those in Zhang et al. (2008) and Gao and Chen (2007). In what follows, on the basis of Theorem 1, we consider the robust stability of the system described by (2) and (3) subject to polytopic uncertainty (4). Assume that the matrices A, Ad in (2) have the form of (4). Then, based on Theorem 1, we can obtain the following result. Theorem 2. The discrete time-delay system in (2) subject to polytopic uncertainty (4) is robustly stable for any time-varying delay d(k) satisfying (3),

122

V1λ (k) =

r

xT (k)λs P (s) x(k),

s=1

V2λ (k) =

k−1 r

(s)

xT (i)λs Q1 x(i)

s=1 i=k−dm

+

r k−d m −1

(s)

xT (i)λs Q2 x(i),

s=1 i=k−dM

V3λ (k) =dm

−1 r

k−1

(s)

η T (i)λs R1 η(i),

s=1 j=−dm i=k+j

V4λ (k) =ρ

r −d m −1 k−1 s=1 j=−dM i=k+j

(s)

η T (i)λs R2 η(i),


40:119-124 (2010)

Table 1: Calculated maximum dM for given dm dm Lemma 2 (Fridman et al., 2005) Proposition 1 (Jiang et al., 2005) Theorem 1 (Gao and Chen, 2007) Theorem 3 (Gao and Chen, 2007) Theorem 1 (Zhang et al., 2008) Theorem 1 in this paper

7 13 13 14 14 15 15

10 15 16 15 15 17 17

Table 2: Calculated maximum dM for given dm

15 19 19 18 18 20 21

20 dm 2 4 5 7 23 Theorem 3 (Liu et al., 2006 ) 2 4 5 7 24 Theorem 2 in this paper 12 13 14 15 22 22 V. CONCLUSION 24 25 We have addressed the stability problem of linear discrete-time systems with interval-like time-varying delay and polytopic uncertainty. A new Lyapunov(s) (s) (s) (s) and P (s) > 0, Q1 > 0, Q2 > 0, R1 > 0, R2 > 0, Krasovskii functional, which includes the range infors = 1, 2, . . . , r, are matrices to be determined. Then mation of the delay, is proposed to derive a new delaywe can easily deduce the LMIs condition (14) by folrange-dependent stability criterion. The advantage of lowing a similar line as that in the proof of Theorem the proposed criterion lies in its less conservativeness. 1. 2 Robust stability condition has also been established for systems with polytopic-type uncertainty. Numerical IV. NUMERICAL EXAMPLES examples show that the proposed criteria can provide In this section, we provide two numerical examples to less conservative results than some existing ones. show the comparison between several existing stability criteria proposed in recent literature and the results VI. ACKNOWLEDGMENT obtained in this paper. The first example is borrowed The authors would like to thank the anonymous refrom Zhang et al. (2008). views and the subject editor Prof. Jorge A. Solsona for their helpful suggestions and valuable comments. Example 1. Consider the system (2) with This work was supported by the National Natural Sci ence Foundation of China under Grants 60674024 and 0.8 0 −0.1 0 A= , Ad = . (16) 60774011. The work of X.S. Cai was also supported by 0.05 0.9 −0.2 −0.1 the Natural Science Foundation of Zhejiang Province, For given dm , we calculate the allowable maximum China (No. Y105141). value of dM that guarantees the asymptotic stability of REFERENCES system (2). By using different methods, the calculated results are presented in Table 1. From the table, we can see that Theorem 1 in this paper provides the less Fridman, E. and U. Shaked, “An improved stabilization method for linear time-delay systems,” IEEE conservative result. Trans. Automat. Contr., 47, 1931-1937 (2002). Example 2. Consider the following discrete system described by (2) and (3) subject to polytopic-type uncertainty (4) with 0.80 0 0.90 0 A(1) = , A(2) = , 0.01 0.6 0.05 0.9 (1)

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Received: November 29, 2008. Accepted: February 28, 2009. Recommended by Subject Editor Jorge Solsona.