New stability and stabilization methods for nonlinear systems with time-varying delays

This paper establishes new robust delay-dependent stability and stabilization methods for a class of nominally linear continuous-time systems with time-varying delays. The parameter uncertainties are convex-bounded and the unknown nonlinearities are time-varying perturbations satisfying Lipschitz conditions in the state and delayed state. An appropriate Lyapunov functional is constructed to exhibit the delay-dependent dynamics via descriptor format. Delay-dependent stability analysis is performed to characterize linear matrix inequalities (LMIs)-based conditions under which the nominally linear delay system is robustly asymptotically stable with a γ-level ℒ-gain. Then we design delay-dependent feedback stabilization schemes: a static one based on state-measurements and a dynamic one based on observer-based output feedback. In both schemes, the closed-loop feedback system enjoys the delay-dependent asymptotic stability with a prescribed γ-level ℒ-gain. The feedback gains are determined by convex optimization over LMIs. All the developed results are tested on a representative example.