researchers have fo c used o n the delay-dependent stability. Numbers of significant results have been reported
in recent litera tur e [1]- [25]. For example, a novel Lyapunov-Krasovskii functional was introduced in [11].
Augmented Lyapunov-K rasovskii functional appro ach was developed in [13] and [17]. Juensen integral
inequality approach was employed in [12], [15], [16], [18] and [24]. A novel piecewise analysis method was
proposed in [22]. Delay-range-dependent stability was inves tig ated in [8] by using the free-weighting matrix
approach [9], [20]. The stability problem of discrete-time systems with interval time-varying delay was
studied in [4] a nd [23].
In prac tice, real systems usually present some uncertainties due to environmental noise, uncerta in
or slowly varying parameters, e tc. Therefore, the stability pro blem of time-delay sy stems with nonlinear
perturbations has received increasing attention (see e.g., [1], [7], [25], [26]). A model transfo rmation method
was used in [1]. Bounding technique for some cross terms was proposed in [14]. A descriptor model
transformation together with a decomposition technique of the delay term matrix wa s employed in [7].
Recently, a less cons e rvative delay-dependent stability criterion was provided in [25] by employing the
free-weighting matrix approach. Robust stabilization for nonlinear discrete-time systems was studied in
[26]. In the above references, reducing the conservatism of the existing stability c riteria is a central issue.
As we know, b ounding technique [14] or model transformation [2] may increase the conservatism. The
free-weighting matrix method, by contrast, is helpful to reduce the conser vatism of stability criteria [20].
On the other hand, choosing appropriate Lyapunov-Krasovskii functional and estimating the upper bound
of its time derivative are very important in deriving the stability criteria.
In this paper, we deal with the delay-dependent stability problem for a class of linear systems with
nonlinear perturbations and interval time-varying delay. We first introduce a new Lyapunov-Krasovskii
functional by taking the range information of the delay into account. The delay-dependent stability of
systems is then analyzed by using the functional. An approach is proposed in estimating the upp e r bound
of the time derivative of the functional. New delay-range-dependent stability criteria are obtained by
intr oducing fre e-weighting matrices a nd free-weighting parameters. The propo sed stability criteria are
formulated in terms of a set of linear matrix inequalities (LMIs). Finally, two numerical examples are given
to show the effectiveness of the proposed approach.
Notations: R
n
denotes the n-dimensional Euclidean space. The superscript “T ” stands for matrix
transposition. X > Y (respectively, X ≥ Y ), where X and Y are real symmetric matrices, means that
the matrix X − Y is positive definite (respectively, positive semi-definite). I is an identity matrix with
appropriate dimension. In symmetric block matrices, we use an asterisk (∗ ) to represent a term that is
induced by symmetry. Matrices, if not explicitly stated, are assumed to have compatible dimensions.
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