Improved results for non-linear discrete-time systems with time-varying delays

In this paper, complete results for delay-dependent stability, feedback stabilization and linear filtering for a class of non-linear discrete-time systems are developed. The system under consideration has time-varying delays with Lipschitz-type non-linearities and subject to real convex bounded parametric uncertainties in all system matrices. A major thrust of the analysis is the constructive use of an appropriate Lyapunov functionals coupled with 'Finsler's lemma' and free-weighting parameter matrices. We establish a linear matrix inequality (LMI) characterization of delay-dependent conditions under which the non-linear discrete delay system is robustly asymptotically stable with an ℒ₂ gain smaller than a prescribed constant level. Feedback stabilization schemes, based on state, static output or by using dynamic output feedback, are designed to guarantee that the corresponding closed-loop system enjoys the delay-dependent asymptotic stability with an ℒ₂ gain smaller than a prescribed constant level. Finally, the developed approach is applied to linear filtering to design both ℋ_∞ and ℒ₂ - ℒ_∞ filters. All the developed results are expressed in terms of convex optimization over LMIs and tested on several representative examples.