This paper is concerned with the problem of delaydependent stability analysis for discretetime systems with intervallike timevarying delays. The problem is solved by applying a novel Lyapunov functional, and an improved delaydependent stability criterion is obtained in terms of a linear matrix inequality.
1. Introduction
Recently, the problem of delaydependent stability analysis for timedelay systems has received considerable attention, and lots of significant results have been reported; see, for example, Chen et al. [1], He et al. [2], Lin et al. [3], Park [4], and Xu and Lam [5], and the references therein. Among these references, we note that the delaydependent stability problem for discretetime systems with intervallike timevarying delays (i.e., the delay satisfies ) has been studied by Fridman and Shaked [6], Gao and Chen [7], Gao et al. [8], and Jiang et al. [9], where some LMIbased stability criteria have been presented by constructing appropriate Lyapunov functionals and introducing freeweighting matrices. It should be pointed out that the Lyapunov functionals considered in these references are more restrictive due to the ignorance of the term Moreover, the term is also ignored in Gao and Chen [7] and Gao et al. [8]. The ignorance of these terms may lead to considerable conservativeness.
On the other hand, in the study of stabilization for the discretetime linear systems, traditional idea of the control schemes is to construct a control signal according to the current system state [10]. However, as pointed out by Xiong and Lam [11], in practice there is often a system that itself is not timedelayed but timedelayed may exist in a channel from system to controller. A typical example for the existence of such delays is the measurement and the network transmission of signals. In this case, a timedelayed controller is naturally taken into account. It is worth noting that the closedloop system resulting from a delayed controller is actually a timedelay system. Therefore, stability results of timedelay systems could be applied to design timedelayed controller.
The present study, based on a new Lyapunov functional, an improved delaydependent stability criterion for discretetime systems with timevarying delays is presented in terms of LMIs. It is shown that the obtained result is less conservative than those by Fridman and Shaked [6], Gao and Chen [7], Gao et al. [8], Jiang et al. [9], and Zhang et al. [12].
2. Preliminaries
Fact 1.
For any positive scalar and vectors and the following inequality holds:
Let us denote
Lemma 2.1 (see [13]).
The zero solution of difference system is asymptotic stability if there exists a positive definite function such that
along the solution of the system. In the case the above condition holds for all , say one that the zero solution is locally asymptotically stable.
Lemma 2.2 (see [13]).
For any constant symmetric matrix , scalar , vector function , one has
3. Improved Stability Criterion
In this section, we give a novel delaydependent stability condition for discretetime systems with intervallike timevarying delays. Now, consider the following system:
where is the state vector, and are known constant matrices, and is a timevarying delay satisfying , where and are positive integers representing the lower and upper bounds of the delay. For (3.1), we have the following result.
Theorem 3.1.
Give integers and . Then, the discrete timedelay system (3.1) is asymptotically stable for any time delay satisfying , if there exist symmetric positive definite matrices satisfying the following matrix inequalities:
where , .
Proof.
Consider the Lyapunov function , where
with being symmetric positive definite solutions of (3.2) and
Then difference of along trajectory of solution of (3.1) is given by
where
where Fact 1 is utilized in (3.6), respectively.
Note that
and hence
Then we have
Using Lemma 2.2, we obtain
From the above inequality it follows that
where , , and
By condition (3.2), is negative definite; namely, there is a number such that and hence, the asymptotic stability of the system immediately follows from Lemma 2.1. This completes the proof.
Remark 3.2.
Theorem 3.1 gives a sufficient condition for stability criterion for discretetime systems (3.1). These conditions are described in terms of certain diagonal matrix inequalities, which can be realized by using the linear matrix inequality algorithm proposed in [14]. But Zhang et al. in [12] proved that these conditions are described in terms of certain symmetric matrix inequalities, which can be realized by using the Schur complement lemma and linear matrix inequality algorithm proposed in [14].
4. Conclusions
In this paper, an improved delaydependent stability condition for discretetime linear systems with intervallike timevarying delays has been presented in terms of an LMI.
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