Hierarchical conditions for exponential stabilisation of time-varying delay systems using Bessel–Legendre inequalities and Finsler's lemma

Abstract

This paper addresses the exponential stabilisation of time-varying delay systems by means of Lyapunov–Krasovskii approaches, Bessel–Legendre inequalities and Linear Matrix Inequalities (LMIs). The key aspect of the proposed control synthesis method is that the LMI conditions are proved for the first time to be hierarchical under time-varying delays, that is to say, a progressive conservatism reduction is achieved as long as the degree of Legendre polynomials is increased. To this end, Finsler's lemma has been properly applied to decouple the Lyapunov matrices from the controller gain. Finally, simulation results are provided to illustrate the effectiveness of the proposed method.