Stability Analysis Of Uncertain Time-Delays Systems

The present paper focuses on the investigation of stability analysis of uncertain time-delay systems. New Lyapunov functionals and Lyapunov-Krasovskii functionals are designed for several classes of uncertain time-varying delay systems. In addition sufficient stable condition of the systems is given. As the preliminary knowledge of time-delay systems, the basic concepts of time-delay system stability are summarized. Two basic theorem of asymptotic stability for time delay systems, Lyapunov-Krasovskii theorems and Raumikhin theorem, are also given. Then some common lemmas are introduced for inequality processing. The main contents of this paper are composed of the following parts:(I) Stability analysis of time delay based on the stability of normal systemIn this part, based on the existing works and the stability of nominal system, robust stability of linear systems with uncertain time varying delay is considered, where the nominal value of the delay is constant and nonzero. Such delay may be considered as uncertain delay with a zero nominal value and a delay perturbation in [0,μ]. The case of delay often appears in different phenomena such as internet networks and biological systems. But only a few rusults have been obtained up to now. Here we introduce a new complete Lyapunov-Krasovskii functional. Based on Lyapunov functional approach and linear matrix inequality technique, the new more practical stability criteria is given.(II) Stability analysis for systems with time-varying delayIn this part, a class of uncertain systems with interval time varying delay are studied. In the existing papers, the model transformation technique is often used to solve such problem. Recently, the free weighting matrices method is developed, and has been shown to be less conservative than previous methods. But this method introduces too many free variables which complicate the system synthesis and lead to a significant increase in the computational demand. Here, we improve the free weighting matrices method by introducing a new Lyapunov functionals and adding new matrices. So, it is more simpler than the existing ones. And it makes use of a tighter bounding technique, leading to much less conservative stability criteria.(III) Time-dependent stability criteria for uncertain delay system In this part, delay-dependent stability is studied for a class of uncertain delay system. The key features of the approach include introduction of a new Lyapunov functional and the use of inequation technology. By magnifying vector product and introducing a new integral inequation lemma, the time delay-dependent asymptotical stability criteria are given. Numerical examples show the effectiveness of our theoretical results.