In this section, we give a novel delay-dependent stability condition for discrete-time systems with interval-like time-varying delays. Now, consider the following system:
where is the state vector, and are known constant matrices, and is a time-varying delay satisfying , where and are positive integers representing the lower and upper bounds of the delay. For (3.1), we have the following result.
Give integers and . Then, the discrete time-delay system (3.1) is asymptotically stable for any time delay satisfying , if there exist symmetric positive definite matrices satisfying the following matrix inequalities:
where , .
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 Fact 1 is utilized in (3.6), respectively.
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.
Theorem 3.1 gives a sufficient condition for stability criterion for discrete-time 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 . But Zhang et al. in  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 .