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## Latin American applied research

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*versión On-line* ISSN 1851-8796

### Lat. Am. appl. res. vol.40 no.2 Bahía Blanca abr. 2010

**ARTICLES**

**New stability criteria for discrete-time systems with interval time-varying delay and polytopic uncertainty**

**W. Zhang ^{1}, Q. Y. Xie^{1}, X. S. Cai^{2}, and Z. Z. Han^{1}**

^{1} *School of electronic information and electrical engineering, Shanghai Jiao Tong University 200240, Shanghai, China wizzhang@gmail.com, qyxie@sjtu.org, zzhan@sjtu.edu.cn*

^{2}

*College of Mathematics, Physics, and Information Engineering, Zhejiang Normal University 321004, Jinhua, Zhejiang, China*

xiushan@zjnu.cn

xiushan@zjnu.cn

*Abstract* This paper is considered with the robust stability problem for linear discrete-time 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 free-weighting 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; Discrete-Time 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 time-delay 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 delay-independent ones especially in the case when the delay is small. Therefore, in recent years many researchers have devoted to investigating delay-dependent stability 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 exising 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 *A ^{T} PA^{T}* is involved in the stability criterion. Therefore, it is not easy to extend the proposed criterion to polytopic-type systems. Moreover, 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 delay-range-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*: *R ^{n}* 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. Moreover, for convenience, in the sum notation , if

*r*<

_{M}*r*, we denote

_{m}(1) |

**II. PROBLEM FORMULATION AND PRELIMINARIES**

Consider the following discrete-time system with a time-varying delay in the state

(2) |

where *x*(*k*) ∈ *R ^{n}* is the state;

*A*and

*A*are

_{d}*n*×

*n*system matrices. {(

*k*) :

*k*= -

*d*, -

_{M}*d*+ 1, . . . , 0} is a given sequence of initial condition. The state delay

_{M}*d*(

*k*) is time varying and satisfies

0 = *d _{m}* =

*d*(

*k*) =

*d*, (3)

_{M}where *d _{m}* and

*d*are nonnegative integers representing the lower and upper bounds of the delay, respectively. Note that the lower bound

_{M}*d*may not be equal to 0. It is worth mentioning that the assumption on the delay

_{m}*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 *et al.*, 2004b; Jiang and Han, 2008). For discrete time-delay systems with polytopic uncertainty, we assume that the matrices *A* and *A _{d}* in (2) can be expressed as the form of

(4) |

where ?* _{i}* = 1, 0 = ?

*= 1,*

_{i}*i*= 1, 2, . . . ,

*r*, and

*A*

^{(i)}, 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.

**Lemma 1** (*Jiang et al. 2005*). For any constant matrix *W* ∈ *R ^{n×n}*,

*W = W*> 0, two integers

^{T}*r*and

_{M}*r*satisfying

_{m}*r*≥

_{M}*r*, vector function

_{m}*x*: [

*r*,

_{m}*r*] ⇒

_{M}*R*, the following inequality holds:

^{n}(5) |

**III. MAIN RESULTS**

In this section, we first consider the stability for the nominal system described by (2), i.e., A and A d 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, *Q _{1}* > 0,

*Q*> 0,

_{2}*R*> 0,

_{1}*R*> 0,

_{2}*T*

_{1},

*T*

_{2},

*T*

_{3},

*L*= ,

^{T}*M*= and

^{T}*N*= of appropriate dimensions such that the following LMI holds:

^{T}(6) |

where *? = d _{M} - d_{m}*,

(7) |

with

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

*V*(*k*) = *V*_{1}(*k*) + *V*_{2}(*k*) + *V*_{3}(*k*) + *V*_{4}(*k*), (8)

where

and *P* > 0, *Q*_{1} > 0, *Q*_{2} > 0, *R*_{1} > 0, *R*_{2} > 0 are matrices to be determined.

Let us define for *i* = 1, . . . , 4,

?*V _{i}*(

*k*) =

*V*(

_{i}*k*+ 1) -

*V*(

_{i}*k*).

Note that *x*(*k* + 1) = *?*(*k*) + *x*(*k*). Then, along the solution of system (2), we have

(9) |

From (2), we have

Denote

and

According to (1), we have *a*(*k*) = 0 when *d _{m}* = 0,

*ß*(

*k*) = 0 when

*d*(

*k*) =

*d*, and

_{m}*?*(

*k*) = 0 when

*d*(

*k*) =

*d*. Then, it is obvious that

_{M}Denote where

Then, we have

(10) |

where *T*, *L*, *M* and *N* are free weighting matrices,

with

and O_{12}, O_{13}, O_{22}, O_{23} are define in (7). On the other hand, by applying Lemma 1, we can obtain

(11) | |

(12) |

It then follows from (9)-(12) that

?*V*(*k*) = ?(*k*)* ^{T}* T?(

*k*). (13)

If T < 0, then (12) implies that there exists a sufficient small scalar *e* > 0 such that ?*V*(*k*) = -*e* ||*x*(*k*)||^{2} (Hale and Lunel, 1993). Therefore, the system in (2) is asymptotically stable when the interval time-varying delay satisfies (3).

**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:

and

are employed in Zhang *et al.* (2008), where Q 3 > 0. Moreover, *V*_{3}(*k*) and *V*_{4}(*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*, *A _{d}* 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), if there exist matrices *P*^{(i)} > 0, > 0, > 0, > 0, > 0, (*i* = 1, 2, . . . , *r*), *T*_{1}, *T*_{2}, *T*_{3}, *L ^{T}* = ,

*M*= and

^{T}*N*= of appropriate dimensions such that the following LMIs:

^{T}T* _{i}* < 0, (14)

hold for *i* = 1, 2, . . . , *r*, where T* _{i}* is given by

with *?* = *d _{M}* -

*d*,

_{m}(15) |

*Proof*. Choose a parameter dependent Lyapunov-Krasovskii functional as follows

*V*_{?}(*k*) = *V*_{1?}(*k*) + *V*_{2?}(*k*) + *V*_{3?}(*k*) + *V*_{4?}(*k*),

where

and *P*^{(s)} > 0, > 0, > 0, > 0, > 0, *s* = 1, 2, . . . , *r*, are matrices to be determined. Then we can easily deduce the LMIs condition (14) by following a similar line as that in the proof of Theorem 1.

**IV. NUMERICAL EXAMPLES**

In this section, we provide two numerical examples to show the comparison between several existing stability criteria proposed in recent literature and the results obtained in this paper. The first example is borrowed from Zhang *et al.* (2008).

**Example 1.** Consider the system (2) with

(16) |

For given *d _{m}* , we calculate the allowable maximum value of dM that guarantees the asymptotic stability of 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 conservative result.

**Table 1**: Calculated maximum *d _{M}* for given

*d*

_{m}**Example 2.** Consider the following discrete system described by (2) and (3) subject to polytopic-type uncertainty (4) with

For given *d _{m}* , we computer the maximum allowable value of dM such that the polytopic system is robustly asymptotically stable. Table 2 shows the computation results by using the stability criteria given in (Liu et al., 2006) and Theorem 2 in this paper. From the table, it is easy to see that the stability criterion obtained in this paper gives a much less conservative result than those in (Liu

*et al.*, 2006). In fact, the stability criterion proposed in (Liu

*et al.*, 2006) for polytopic-type systems is actually not a delay-dependent stability condition since it only depends on the size of the interval, i.e.,

*d*-

_{M}*d*.

_{m}**Table 2**: Calculated maximum *d _{M}* for given

*d*

_{m}**V. CONCLUSION**

We have addressed the stability problem of linear discrete-time systems with interval-like time-varying delay and polytopic uncertainty. A new Lyapunov-Krasovskii functional, which includes the range information of the delay, is proposed to derive a new delay-range-dependent stability criterion. The advantage of the proposed criterion lies in its less conservativeness. Robust stability condition has also been established for systems with polytopic-type uncertainty. Numerical examples show that the proposed criteria can provide less conservative results than some existing ones.

**VI. ACKNOWLEDGMENT**

The authors would like to thank the anonymous reviews and the subject editor Prof. Jorge A. Solsona for their helpful suggestions and valuable comments. This work was supported by the National Natural Science Foundation of China under Grants 60674024 and 60774011. The work of X.S. Cai was also supported by the Natural Science Foundation of Zhejiang Province, China (No. Y105141).

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