Finite-time stability analysis for linear time-varying singular impulsive systems

1 School of Business Administration, Northeastern University, Shenyang, 110819, Peoples Republic of China. 2 State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang, 110819, Peoples Republic of China. 3 Institute of Systems Science, Northeastern University, Shenyang, 110819, Peoples Republic of China. 4 CNPC Liaoyang Petrochemical Company Machinary Factory, Liaoyang, 111003, Peoples Republic of China.


INTRODUCTION
Singular systems are also referred to as descriptor, semistate, implicit, constrained, differential-algebraic equation, or generalized state-space systems and arise naturally in many practical applications (Campbell, 1980).In the past several decades, many fundamental system theories developed for standard state-space systems have been successively generalized to its counterparts for singular systems (Ishihara and Terra, 2002;Cobb, 1984).
On the other hand, since impulsive behaviours which are characterized by abrupt changes of states at certain instants occur in many practical systems (Yang, 2001), impulsive systems have attracted particular interest (Zhang and Sun, 2005;Cheng et al., 2010;Wang and Liu, 2007).Recently, singular impulsive systems, that is, singular systems subject to impulsive effects, have been proposed and studied (Guan et al., 1995;2001;2005;Wang and Lia, 2001;Yao, 2006).The problems of stability and stabilization of singular impulsive systems *Corresponding author.E-mail: chunyuyang@yahoo.cn.
are studied in the framework of Lyapunov stability and the proposed methods are shown to be useful and efficient.
It is known that Lyapunov stability is concerned with the behavior of the system over an infinite time interval.In practice, a stable system might be useless because the stable domain or the domain of the desired attractor is not large enough and on the other hand, sometimes an unstable system may be acceptable since the system oscillates sufficiently near the desired state on a predefined finite time interval.This boundedness on a finite time interval is referred to as the notion of finite-time stability (or short-time stability) which is firstly introduced in La Salle and Lefschetz (1961).Different versions of finite-time stability for various systems have been proposed by Weiss and Infante (1967), Amato and Ariola (2001) and Liu and Sun (2008).
In Zhao et al. (2008), finite-time stability of linear timevarying singular impulsive systems is defined and some sufficient conditions are derived.The proposed methods make some restrictions on the systems under consideration including: (1) (3) The involved disturbance signal is timeinvariant.In this paper, we aimed to study the finite-time stability of linear time-varying singular impulsive systems and try to remove the above mentioned restrictions.Thus, the development of new methods can be applied to more general systems.

PRELIMINARIES
The notations used in this paper are the same as those in Zhao et al. (2008) We now recall the definitions and results of Zhao et al. (2008).
Definition 1: Time-varying matrix ( ) E t is singular on time interval J , if there exists a t J ∈ % such that det( ( )) 0 E t = % .
Definition 2: Time-varying matrix Consider the following singular impulsive system: is said to be finite-time stable with respect to { } The basic assumptions of Zhao et al. (2008) are stated as follows: (H1) Time-varying matrices ( ) A t and ( ) E t are continuous on J .

MAIN RESULTS
It is known that one of the fundamental characteristics of singular systems is that the derivative matrix ( ) E t is singular.However, assumption (H5) requires ( ) E t to be nonsingular at the impulse instant k τ .Assumption (H4) is also a restrictive condition.The objective of this paper is to release the assumptions (H4) and (H5).We first give the following lemma.
be a symmetric and semi positive definite matrix and be a symmetric matrix.Assume and there exists a symmetric matrix Then the generalized eigenvalues of ( , ) ： Since Q is a symmetric and semi-positive definite, there exists an orthogonal matrix M such that 0 0 0 .
This completes the proof.
Remark 1: From Equation (3), the generalized eigenvalues of ( , ) As shown in Zhao et al. (2008), under the assumptions (H4) and (H5) and choosing k r as (H7), one has: Which plays an important role in the proofs for Theorems 1 and 2. Let We now introduce the following assumptions: We now define k r as: max ( , ).
From Lemma 3, it can be seen that Inequality (4) holds for k r defined by Equation ( 7) if (H11) holds.Then, similarly to the proofs for Theorems 1 and 2 given in Zhao et al. (2008), one can prove the following Theorems 4 and 5.

∑
where k r is defined by Equation ( 7), Y X , the r defined in Theorem 1 is the same to that defined by Equation ( 7).Hence, Theorems 4 and 5 reduce to Theorems 1 and 2, respectively, when assumption (H5) holds.
In practice, the disturbance signal is usually timevarying.However, the assumptions in Theorems 2 and 5 require the disturbance signal to be time-invariant.We now consider the finite-time stability of singular impulsive systems with time-varying disturbance.To do this, we make the following assumption: (H6 ') Signal ( ) t ω is time-varying and the set S is defined as Theorem 6: Suppose (H1)-( H3), (H6') and (H11) hold, ( ) ( ) 0 such that (H8) and (H12) T T

V t x t x t E t P t E t x t =
Consider the situation on the interval 0 1 ( , ) From (H8), it follows that: By Equations ( 8) and ( 9), we have: Then, using comparison principle (Pachpatte, 1998) implies: which shows We now consider the case Using comparison principle again, we have: From (H1), (H11), (H13) and Lemma 3, we have: Thus it follows from Inequalities ( 13) and ( 14) that: By the same progress, for 2,3, , , k m = L , we have: Then by (H13), we have

Conclusion
This paper has investigated the finite-time stability of linear time-varying singular impulsive systems.New sufficient conditions for the system to be finite-time stable have been derived by a newly developed inequality and the well-known comparison principle.Compared with the existing results, the proposed methods remove some basic assumptions and therefore can be applied to more general systems.The presented numerical example has illustrated the obtained results.
the same as that in Theorem 4.Remark 2： ： ：：The newly developed Theorems 4 and 5 are less restrictive than Theorems 1 and 2, respectively.Assume that (H5) holds.Then k X is nonsingular, which implies that . Let maximum eigenvalue of X and is abbreviated as max ( ) Y λ .
This completes the proof. 