Stability And Stabilization For Systems With Time Delay

Abstract: Stabilization is the premise of practical systems normal operation and the stabilization is a significant issue for systems control theory. The time delay is an ubiquitous phenomenon in nature. It brings more complexity and trouble in analysis and synthesis, also it result in instability of systems. Thus, it is essential that analysis the stability of systems with time delay.There are two main types of method that can be used to examine stability of systems with time delay. One is using the relationship between roots of the generalized characteristic equation in the complex plane and the stability of linear systems with time delay. However, this criteria is unfeasible and impractical. The other is Lyapunov's method that was addressed by Krasovskii in 1959. It judge the stability of systems through select the various Lyapunov functional. The conservation of criteria is diverse according to choice the different Lyapunov functional. The aim of applying Lyapunov theory is estimate the time derivative of Lyapunov functional along the solution of systems. And to make sure the time derivative is negative definite ultimately. Generally, the time derivative is very complicated. In order to gain the maneuverable condition, it will enlarge the time derivative to ensure the negative definite. Thus, we discuss stability based on this theory in this paper. And construct a new Lyapunov functional. Utilize linear matrix inequality (LMI) that has advantages in computer to study the stability of linear systems with time delay.The main conclusions in paper are:(1) Consider linear systems with time delay. A new Lyapunov-Krasovskii functional is given to verdict stability of the systems. Though select positive definite matrix R and inequalityx^TA^TBy + y^TB^TAx ≤ x^TA^TRAX+y^TB^TR^-1Byto enlarge the time derivative of V(x(t)) along the solution of systems with time delay. And ensure this time derivative is negative definite. This consult have improved the conclusions before on normal linear systems with time delay. Furthermore, it expands this conclusion to generalized linear systems with time delay. Though enlarge v{x(t)) to verdict stability of the generalized linear systems with time delay.