Comparison And Improvement Of Time-delay System Stability Research Methods

Time-delay phenomenon extensively occurs in various fields. Its existence often results in the system instability and uncertainty. Therefore, with the development of science and technology, a huge of methods and theories are proposed to study the stability conditions of the time-delay system in these decades, as the model transformation approach, the free-weighting-matrix approach, and the discretized Lyapunov functional method.In this paper, we focus on some types of time-delay systems, and review the methods and results on these time-delay systems in the literature with illustration by some examples. The main contents of this paper are outlined as follows:(1) The descriptor system approach has been attracted increasing interest of many researchers in these decades. In this article, we study the linear time-invariant delay-time system, the linear time-varying delay-time system, and the neural time-delay system etc., and review the results obtained in this applied field.(2) Many model transformation approaches are based on the Newton-Leibniz formula; however, most of these methods have a limitation on the conservative of stability conditions. To obtain the less conservative, a free-weighting-matrix approach was proposed in the latest years. We reviewed the results and applications of the free-weighting-matrix method on the single time-delay system, multiple time-delay system, and neural time-delay systems. The performance of the free-weighting-matrix method was illustrated by some examples.(3) the discretized Lyapunov functional method is another approach to obtain the less conservative of stability condition of time-delay system. It can efficiently escape the checking of the existence of the quadratic Lyapunov functional. However, a drawback is its intensive computational burden. In this article, we also reviewed some results of this method. And by using this method to improve and promote time-varying time-delay system stability,the advantage of the method is applied to time-varying delay system,change its boundedness that this method is applied to time-invariant system.