Stability Analysis And Synthesis Of Linear Systems With Time-Varying Delays

Time-delay is a phenomenon in many practical realities and engineering systems, mechanical transmission system, metallurgical industrial process control systems and network system. From the perspective of system theory, the past state of any time delay system is effect on the current state of the time delay system, so time delay phenomenon is often encountered in practical engineering systems. Thus, the analysis and synthesis analysis have become a hot issue sector of time-delay systems.This paper is concerned with the systems with interval time-varying delays. By applying the Lyapunov second method, constructing a suitable Lyapunov–Krasovskii functional and using the linear matrix inequalities(LMI) theory, the stability criteria and the controller-designing has been derived on the linear system with interval time-varying delays. The main results as following:Firstly, through constructing a new augmented Lyapunov–Krasovskii functional, using Jensen integral inequality and a new relevant inequality, in terms of linear matrix inequalities, the improved delay-dependent stability criteria is derived for the the linear systems with interval time-varying delays.Secondly, a new criteria for delay-dependent robust stability analysis of discrete systems with interval time-varying delay is obtained, by building a new augmented Lyapunov–Krasovskii functional, applying the relaxed inequality, using Jensen integral inequality, the employment of a delay partitioning and a reciprocally convex combine technique. Furthermore, the H∞ stability criterion for discrete systems with time-varying delay has been achieved by applying the same method.Thirdly, a sufficient method for designing feedback gains of time-delayed controllers of discrete systems with time-varying delay is put forward. By constructing a suitable Lyapunov–Krasovskii, based on the new relaxed inequality and Jensen integral inequality, the employment of a delay partitioning and a reciprocally convex combine technique, a new criteria for delay-dependent stabilization analysis of discrete systems with time-varying delay is obtained.Examples are given to demonstrate the effectiveness and feasibility of the proposed approach. In addition, the work which has been done is summarized in this paper, and the proposals for the future work is put forward.