A relaxed Lyapunov function lemma and genetic algorithm applying to time-delay systems

In the analysis of systems with time-varying delays, positive definite quadratic forms are often introduced as Lyapunov functions, and the matrices in Lyapunov functions are usually positive definite matrices. This paper develops a new relaxed Lyapunov function, which introduces some relaxed matrices, helping to reduce the conservatism. Sufficient conditions of quadratic functions are often introduced to deal with square of time-varying delays τ 2 ( t ) , in which the selection of the parameter of these sufficient conditions has a great influence on the conservative results. In this paper, genetic algorithm are utilized to solve the problem of selecting the optimal parameter in sufficient conditions of quadratic functions. The stability conditions of linear time-delay systems are established by using the relaxed Lyapunov function lemma and genetic algorithm, and the advantage of the criterion is further validated through a numerical example. Ultimately, the proposed criterion is utilized to assess the stability of load frequency control approaches in power systems.