Delay-range-dependent Robust Stability for Uncertain Singular Systems with Interval Time-varying Delays

Abstract In this paper, we consider the delay-rangedependent robust stability problem of uncertain singular system with interval time-varying delays. Stability criteria, which guarantee the concerned singular system is regular, free and stable, is derived in terms of linear matrix inequalities (LMIs) by introducing some identical equations with integral inequality approach (IIA) and using the Leibniz-Newton formula to reduce the conservatism of the results. The upper bound of time-delay can be obtained by using the modified generalized eigenvalue minimization problem (GEVP) technique such that the system can be stabilized for all time-delays. Finally, numerical examples show the method presented in our paper is more effective and less conservative than the existing ones.


Introduction
Engineering processes often involve both nonlinear and time-delay models. In many physical and biological phenomena, the rate of variation in the system state depends on past states. Time delay phenomena were first discovered in biological systems and were later found in many engineering systems, such as mechanical transmissions, fluid transmissions, metallurgical processes, and networked control systems. They are often a source of instability and poor control performance. Therefore, many efforts have been made for the stability problems for various delayed systems [1-2, 4-14, 16-31]. Moreover, because of unavoidable factors, such as modeling error, external perturbation and parameter fluctuation, the time delay systems certainly involve uncertainties such as perturbations and component variations, which will change the stability of time delay systems. And in recent years, the stability analysis issues for time delay systems in the presence of parameter uncertainties perturbations have stirred some initial research attention [4,5,8,12,16,19,20,[22][23][24][25]31].
Singular systems have found numerous practical applications: e.g., engineering systems, social systems, economic systems, biological systems. Unlike classical state space representation via a set of ordinary differential equations, singular system can be viewed as a composite formed by several interconnected systems with two layers: dynamic property described by differential equation and interconnection property expressed by algebraic equation. Therefore, some results on the stability of singular systems are achieved [1, 5-8, 12-13, 16-18, 20-25, 27-30] and the references therein. The existing stability criteria for singular time-delay systems can be classified into two types: delayindependent [5,24,25,30] and delay-dependent [1, 6-8, 12-13, 16-19, 21-23, 26, 28-29, 31]. Generally, delaydependent conditions are less conservative than the delayindependent ones, especially when the time delay is small. For singular systems with delays, several kinds of simple Lyapunov-Krasovskii functionals, i.e. functionals parameterized with constant matrices, have been proposed, which lead to different levels of conservatism due to the different model transformations and the bounding techniques for some cross-terms [7,8]. A tighter bounding for cross-terms can reduce the conservatism. However, there are no obvious ways to obtain less conservative results, even if one is willing to expend more computational effort on the problem, and to find a tighter bound for the cross-terms. This is the serious limitation for these criteria. To overcome this limitation, one has to find some more general Lyapunov-Krasovskii functional (LKF) for handling the delay-range-dependent robust stability problem for singular systems. To the best of our knowledge, this delay-range-dependent robust stability problem has not been fully investigated for singular systems with time-varying delay, which motivates the present study.
On the other hand, the range of time-varying delay systems considered in [17,18,20,27] is from 0 to an upper bound. In practice, a time-varying interval delay is often encountered, that is, the range of delay varies in an interval for which the lower bound is not restricted to 0. In this case, the stability criteria for time-varying delay systems in [17,18,20,27] are conservative because they do not take into account the information of the lower bound of delay. To the best of authors' knowledge, there have been few results on the delay-range-dependent robust stability of the singular systems with time-varying interval delays, which remain as an interesting research topic.
Motivated by the above discussions, we propose the improved delay-range-dependent robust stability for singular systems with time-varying interval delay and uncertainties. First, by defining a novel Lyapunov function, a delay-range-dependent stability criterion for the nominal singular time-delay system is established in terms of LMIs. Next, less conservative result is obtained by considering some useful terms when estimating the upper bound of the derivative of Lyapunov functional and introducing the additional terms into the proposed Lyapunov functional, which includes the information of the range. The maximum allowable value of the time delay can be obtained by solving a set of linear matrix inequalities (LMIs) and the modified generalized eigenvalue minimization problem (GEVP) algorithm. Finally, Numerical examples demonstrate that the results obtained in this paper are effective and are a significant improvement over previous ones.

Stability Description and Preliminaries
Consider the following uncertain singular system with an interval time-varying delay as follows: with the initial condition where , , a M N and b N are known real constant matrices with appropriate dimensions, and ( ) F t is an unknown, real, and possibly time-varying matrix with Lebesguemeasurable elements satisfying The main objective is to find the range of 1 Lemma 3 [2]. The following matrix inequality: where ( ) ( ), ( ) ( ) and ( ) x is equivalent to Universal Journal of Control and Automation 1(1): 1-9, 2013 3 and 1 ( ) Lemma 4 [2]. Given matrices , , , In this paper, a new Lyapunov functional is constructed, which contains the information of the lower bound of delay 1 h and upper bound 2 . h The nominal unforced time delay singular system (1) can be written as follows: The following Theorem 1 presents a delay-range-dependent result in terms of LMIs and expresses the relationships between the terms of the Leibniz-Newton formula. Theorem 1: For three given positive scalars 1 2 , , h h and , d h the time-delay singular system (9) is asymptotically stable if there exist matrices 0, 0, with the following constraint 0.
Consider the time-varying delay singular system (9), using the Lyapunov-Krasovskii functional candidate in the following form, we can write: Calculating the derivative of (11) with respect to 0 t > along the trajectories of (9) leads to 1 2 3 Using the Leibniz-Newton formula ( ) ( ) ( ) , To obtain an easily solvable LMI, we introduce the matrix ( ) n n r Z R × − ∈ satisfying 0 T Z E = and rank Z n r = − [18,19,22].Then, we have Substituting the above equations (13)- (20) into (12) (21) Therefore, the interval time-varying delay system (9) is asymptotically stable if (10) is true. Thus, the proof is complete.

Robust for Uncertain Singular Interval Time-varying Delay System
In the section, extending Theorem 1 to uncertain singular system (1) with interval time-varying delays yields the following Theorem 2.
Theorem 2: For three given positive scalars 1 2 , , h h and , d h the uncertain singular time-varying delay system (1) is asymptotically stable if there exist symmetry positive-definite matrices 0, ε > S with appropriate dimensions, positive semi-definite matrices 11 12 13 12 22 23 with the following constraint 0.
Based on that, a convex optimization problem is formulated to find the bound on the allowable delay time h which maintains the delay-dependent stability of the time delay system (9).
Proof: Consider the singular time-varying delay system (9), using the Lyapunov-Krasovskii functional candidate in the following form, we can rewrite as where Z follows the same definition as that in Theorem 1, and ( , 1, 2,3; 3) ij i j i j = < ≤ Ψ are defined in (25). Remark 1: It is interesting to note that 1 2 , , h h appear linearly in (10a) and (22a). Thus a generalized eigenvalue problem as defined in [2] can be formulated to solve the minimum acceptable 2 1/ h (or 1 1/ h ) and therefore the maximum 2 h (or 1 h ) to maintain robust stability as judged by these conditions. In this way, our optimization problem becomes a standard generalized eigrnvalue problem, then which can be solved using GEVP technique. From this discussion, we have the following Remark 2.
Remark 2: Theorem 2 provides delay-dependent asymptotic stability criteria for the time-varying delay singular systems (1)

Illustrative Examples
Example 1: Consider the following time delay singular systems: where 1 0 0.5 0 1.1 1 , , to guarantee the above system (29) to be asymptotically stable.   Table 2, in which "-" means that the results are not applicable to the corresponding cases. For comparison, the Table 2 also lists the upper bounds obtained from the criteria in [1, 6-8, 17, 19, 20, 22, 23, 26, 28, 29, 31]. It can be seen that our methods are less conservative. The simulation of the system (26) for h =1.06 is depicted in Fig.1 (30), we are able to find a feasible solution for the set of LMIs for any 1 2 [ , ] [0,1.1547]. h h ∈ This means that the maximum allowable delay bound (MADB) under which the system is uniformly asymptotically stable is 1.1547. Table 3 lists the maximum allowable delay bound (MADB) as judged by the criteria in [4,6,7,21]. We can see from this table that there is still room for reducing the conservativeness by comparing with the numerical solution, but we know that with fewer matrix variables the stability results obtained in Corollary 1 is less conservative than the one in [4,6,7] and the same as [21]. Using these data, a simulation program has been written in Matlab. As Fig. 2 shows, the simulation of the above system (30)

Conclusion
In this paper, the improved delay-range-dependent robust stability criterion for uncertain singular time-delay systems with time-varying interval delays has been investigated. By defining a novel Lyapunov function, a delay-rangedependent stability criterion is established in terms of LMIs, which guarantees the nominal singular time-delay systems to be regular, impulse free and asymptotically stable. Less conservative result is obtained by considering some useful terms when estimating the upper bound of the derivative of Lyapunov functional and introducing the additional terms into the proposed Lyapunov function which includes the information of the range. The robust stability problem is also investigated and the obtained results are expressed in terms of strict LMIs, which can be easily solved by using a convex optimization algorithm. By comparing our results with others through numerical examples, it has been shown that the derived criterion is less conservative than those in the literature.