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.
Theorem 3.1.
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
, 
.
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 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 [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].