Sufficient Conditions of Asymptotic Stability of the Time-Varying Descriptor Systems

We discuss the time-varying descriptor systems. Firstly, a sufficient condition of asymptotic stability and impulse-free is derived based on Riccati equation. Secondly, we design a state feedback controller to make the close-loop system asymptotically stable and impulse-free. Finally, a numerical example demonstrates the proposed results. 1. Introduction Descriptor systems, also known as generalized state systems, have been widely studied in the past several decades. Descriptor systems have some special properties, such as regularity and impulse behavior. Asymptotic stability is one of the properties for this control systems. Many researchers have focused on the problems of asymptotic stability. Lyapunov approach plays an important role in the analysis of control systems. Lyapunov approach has become a powerful method for standard state space systems. In recent years, many researchers are trying to apply this approach to descriptor systems. Lyapunov equation of descriptor systems were firstly presented by Campell and Petzold [1], and the results were promoted by Lewis [2]; Lyapunov equation was used to study the structural stability of the descriptor systems by Khargonekar et al. [3], Wei [4], and Takaba et al. [5]; Lyapunov inequality and Riccati equation are introduced by Wei [4] and Takaba et al. [5]; Hsiung and Lee [6]; [7–10] presented necessary and sufficient conditions of the stability for time-varying descriptor systems; in [11], admissibility analysis for time-varying periodically descriptor systems was given; [12] presents new results for a generalized Lyapunov approach of linear time-varying periodically descriptor systems; a linear matrix differential inequality approach of linear descriptor systems was obtained in [13]; [14–17] discussed the stabilization of several class of descriptor systems. Up to now, scholars introduced the necessary and sufficient condition of asymptotic stability and used the Lyapunov equation for solving the problems of asymptotic stability, stabilization; there is no effective method for solving the stabilization via a Riccati equation. In this paper, we mainly consider sufficient conditions of asymptotic stability and impulse-free based on Riccati equation; the fast subsystem can be determined by the slow subsystem according to the second limited equivalence transformation on the previous basis; then we give a state feedback controller to make the time-varying descriptor systems asymptotically stable and impulse-free. The paper is organized as follows. Section 2 presented the sufficient condition of asymptotic