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

Descriptor systems, also known as generalized state systems, have beenwidely 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 byWei [4] and Takaba et al. [5]; Hsiung and Lee [6]; [7–10] presented necessary and sufficient conditions of the stability for timevarying 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 stability and impulse-free for the time-varying descriptor system. Section 3 gives the design of the state feedback controller of the time-varying descriptor system based on Lyapunov and Riccati equations. In Section 4, a numerical example illustrates the content of this paper studies in-line with the actual demand; themethod also is more simple.


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][8][9][10] presented necessary and sufficient conditions of the stability for timevarying 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][15][16][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 stability and impulse-free for the time-varying descriptor system.Section 3 gives the design of the state feedback controller of the time-varying descriptor system based on Lyapunov and Riccati equations.In Section 4, a numerical example illustrates the content of this paper studies in-line with the actual demand; the method also is more simple.
From Definition 1, the regularity of the system (1) is equivalent to the analytical solvability in the sense of Campbell [7].Put the system into the following decomposition.
For system (1), suppose the constant invertible matrices  and  such that (2) then the system (1) is restricted equivalent to Obviously, the necessary and sufficient condition of impulsefree a for the system (1) is that  22 () is invertible.
Then time-varying descriptor systems ( 4) is said asymptotically stable.
The proof is completed.

Asymptotic Stabilization of Time-Varying Descriptor Systems
Definition 5 (see [12]).The system (1) is called strongly asymptotically stable, if the system (1) is impulse-free and asymptotical stable.
Theorem 7. Time-varying descriptor system (1) is said to be stabilized, if for all  ∈ [0, ]; system (12) is uniformly regular, and the following conditions hold at the same time: (1) ‖()‖ ≤ ,  > 0; (2) for all  ≥ 0,  22 () is invertible; (3) if there exist  > 0,  is a symmetric matrix, so that Riccati equation, has a positive definite symmetric solution , and there exists a state feedback Proof.By Schur complement, it is easy to check that inequality (13) holds if and only if the following inequality holds: Taking the same line as Theorem 4, proved.

Conclusions
In this paper, a sufficient condition of asymptotical stability and impulse-free was presented, and a state feedback controller was designed in terms of Riccati equation to make the time-varying descriptor system asymptotically stable and impulse-free.Finally, a numerical example was presented to illustrate the main results.