Absolute Stability of a Class of Nonlinear Singular Systems with Time Delay

and Applied Analysis 3 and a matrix S with appropriate dimensions, such that the following LMI holds:


Introduction
Since the concept of absolute stability and the Lur' e problem were introduced, the absolute stability of Lur' e control systems has received considerable attention and many rich results have been proposed during the last decades [1].Time delays widely exist in practical systems, which is a source of instability and deteriorated performance [2][3][4].Therefore, great efforts have been made to investigate the absolute stability of Lur' e systems with time delay and many results have been achieved [4][5][6][7][8][9].
Recently, an integral inequality approach was proposed to investigate the Lur' e system with time delay and new absolute stability criteria were obtained [7].In addition, as it is impossible to reduce the conservatism of the derived conditions by employing simple Lyapunov-Krasovskii functional, some other efforts are made to improve the delay-dependent conditions via introducing new Lyapunov-Krasovskii functionals.For example, improved results for time delay systems were obtained by introducing the augmented Lyapunov-Krasovskii functional [10] and the delay-partitioning Lyapunov-Krasovskii functional [5].By employing a discretized Lyapunov-Krasovskii functional, new absolute stability condition for a class of nonlinear neutral systems is derived in [11].Although [11] can achieve less conservative results, the condition was much more complicated than those based on simple Lyapunov-Krasovskii functionals.
On the other hand, singular systems have been extensively studied in the past few years due to the fact that singular systems describe physical systems better than statespace ones [12][13][14][15].Depending on the area of application, these models are also called descriptor systems, semistate systems, differential-algebraic systems, or generalized statespace systems.Therefore, the study of the absolute stability problem for the Lur' e singular system with time delay is of theoretical and practical importance [16].
In this paper, by employing the delay-partitioning approach proposed in [17], we construct a new Lyapunov-Krasovskii functional to investigate the absolute stability of Lur' e singular systems with time delay.Improved delaydependent absolute stability criteria are presented.The criteria are easy to follow, and those criteria obtained in [16] by using simple Lyapunov-Krasovskii functional are involved in our results.Numerical example is given to demonstrate the advantage of the proposed method.
Notation.Throughout this paper, R  denotes the ndimensional Euclidean space; R × is the set of all  ×  real matrices; for a real matrix ,  > 0 (resp.,  < 0) means that  is real symmetric and positive definite (resp., negative definite);  is an identity matrix of appropriate dimensions,

Problem Statement and Preliminaries
where () ∈ R  is the state vector of the system; () ∈ R  and () ∈ R  are input vector and output vector, respectively; , , , , ,  are constant matrices, where  may be singular and it is assumed that rank  =  ≤  and that the scalar ℎ > 0 is the delay of the system; the initial condition, (), is a continuous vector-valued function of  ∈ [−ℎ, 0].(, ()) ∈ R  is a nonlinear function, which is piecewise continuous in , globally Lipschitz in (), (, 0) = 0, and satisfies the following sector condition: where  1 and  2 are constant real matrices and  =  2 − 1 is a symmetric positive definite matrix.It is customary that such a nonlinear function (, ()) is said to belong to a sector In this paper, we also investigate the robust absolute stability of the following uncertain system: where the uncertainties are of the form where ,   , and   are constant matrices, and () is a timevarying matrix satisfying Next, the following definitions and lemmas are introduced, which will be used in the proof of the main results.

Main Results
Firstly, by means of the loop transformation suggested in [21], it can be concluded that the absolute stability of system (1) in the sector [ 1 ,  2 ] is equivalent to that of the following system in the sector [0,  2 −  1 ]: where Thus, for the absolute stability of system (1), we have the following result.Theorem 6.Given integer  and scalar  = ℎ/ > 0, the system (1) with nonlinear connection function satisfying (2) is absolutely stable in the sector [ 1 ,  2 ] if there exist a scalar  > 0, matrices Abstract and Applied Analysis 3 and a matrix  with appropriate dimensions, such that the following LMI holds: where and  ∈ R ×(−) is any matrix with full column rank and satisfying    = 0.
In what follows, we show that the nonlinear singular system (1) is regular and impulse-free.Since rank  =  ≤ , there exist two invertible matrices  and  ∈ R × such that Then,  can be parameterized as where Φ ∈ R (−)×(−) is any nonsingular matrix.Like in (20), we define Since   ( +   ) + (   +   ) +  11 −    1  < 0 and  11 ≥ 0, we can formulate the following inequality easily: Pre-and postmultiplying  < 0 by   and , respectively, yield As ψ 11 and ψ 12 are irrelevant to the results of the following discussion, the expressions about these two variables are omitted here.It is easy to deduce from (24) that and thus Ā22 is nonsingular.Otherwise, supposing  22 is singular, there must exist a nonzero vector  ∈ R − which ensures that Ā22  = 0.And then it can be concluded that ) = 0, and this contradicts (25).So Ā22 is nonsingular.Then, it can be shown that which implies that det( − ) is not identically zero and deg(det( − )) =  = rank .Then, the pair of (, ) is regular and impulse-free, which implies that system (1) is regular and impulse-free. Defining where λ1 =  min ( P Noting that ‖()‖ and ∫  0 ‖()‖ 2  are bounded, it follows that ‖()‖ and ∫  0 ‖()‖ 2  are bounded; from Lemma 3, one can conclude that lim  → ∞ () = 0; thus lim  → ∞ () = 0.According to Definition 2, the singular system ( 8) is globally uniformly asymptotically stable for (, ()) ∈ [0,  2 −  1 ].Thus the singular system ( 8) is absolutely stable in the sector [0,  2 −  1 ], which is equivalent to the absolute stability of system (1) in the sector [ 1 ,  2 ].This completes the proof.
Remark 8.It is worth mentioning that the conservatism is reduced with the increase of .At the same time, more matrix variables are involved in the corresponding LMI, which will increase the computing complexity.
Remark 9.In [5], some absolute stability conditions have been obtained for Lur' e system with time delay based on a delay-partitioning approach.However, the results proposed in this paper achieve some improvement and are more general than [5].Let  = ,  = 0, and   = 0 ( ̸ = ) in (30); Theorem 7 reduces to Theorem 3 in [5].

Numerical Example
In this section, we provide a numerical example to demonstrate the effectiveness of the proposed method.(32) In this example, we choose  = [0 1]  .For various , the maximum upper bounds of time delay obtained by Theorem 7 are listed in Table 1 in comparison with those obtained by [16].It is clear that our approach provides larger stability region than [16].Furthermore, it is concluded from the table that larger upper bounds of ℎ can be obtained as  increases.

Conclusions
The absolute stability problem has been investigated for time delay singular systems with sector-bounded nonlinearity.Some improved conditions have been derived based on the delay-partitioning approach.A numerical example has been given to verify the effectiveness of the proposed methods.

Example 10 .
Consider uncertain system (3) with the following parameters: