Stability of Time-Delay Systems

Abstract

When feedback controls are used for time-delay control systems, the resulting closed-loop systems become homogeneous time-delay systems without external variables. Thus the stability of homogeneous time-delay systems is very important. This chapter is devoted to various stability and robust stability test methods for homogeneous time-delay systems. After general time-delay systems are briefly introduced, then linear time-delay systems are followed. The existence and uniqueness of solutions of general nonlinear time-delay systems are dealt with. The stability, uniform stability, asymptotic stability, and uniform asymptotic stability for time-delay systems are defined. Two important stability results such as the Krasovskii theorem and the Razumikhin theorem are introduced. Linear and bilinear matrix inequalities are introduced together with recent integral inequalities for quadratic functions such as in the integral inequality lemma and the Jensen inequality lemmas with and without reciprocal convexity, which are useful when the Krasovskii and Razumikhin theorems are utilized. Various stability test methods, either delay-independent or delay-dependent, for single state delayed systems are developed based on the Razumikhin theorem, the Krasovskii theorem, and the discretized state description. Some results are extended to systems with time-varying delays with and without bounded derivatives. Robust stability test methods for time-delay systems with model uncertainties are developed based on the Krasovskii theorem and the discretized state description. Some results based on the Krasovskii theorem are extended to systems with both model uncertainties and time-varying delays with and without bounded derivatives. Some stability and robust stability test methods for distributed state delayed systems are briefly introduced since distributed delays are often appeared in optimal controls in later chapters.