A Delay Decomposition Approach to the Stability Analysis of Singular Systems with Interval Time-Varying Delay

This paper investigates delay-dependent stability problem for singular systems with interval time-varying delay. An appropriate Lyapunov-Krasovskii functional is constructed by decomposing the delay interval into multiple equidistant subintervals, where both the information of every subinterval and time-varying delay have been taken into account. Employing the LyapunovKrasovskii functional, improved delay-dependent stability criteria for the considered systems to be regular, impulse-free, and stable are established. Finally, two numerical examples are presented to show the effectiveness and less conservativeness of the proposed method.


Introduction
During the last few decades, there has been a growing research interest in the analysis and synthesis of time-delay systems, which widely exist in various practical systems such as biological systems, chemical systems, electronic systems, and network control systems [1].Usually, the range of delays considered in most of existing references is from zero to an upper bound [2][3][4][5][6].In practice, however, the delay range may have a nonzero lower bound, and such systems are referred to as interval time-varying delay systems.For this reason, the stability of the systems with such interval time-varying delays has attracted considerable attention.As we know, in order to reduce conservatism of the stability criteria, many approaches were developed; for example, by using free-weighting matrices, [7,8] present some stability conditions for systems with interval time-varying delays.In [9], via a Lyapunov-Krasovskii functional with fewer matrix variables whose derivative is estimated using the convex analysis method, a simple stability criterion was obtained; this result is improved in [10] by using the reciprocally convex approach.Recently, some new methods were proposed to the stability analysis of interval time-varying delay systems.New classes of Lyapunov-Krasovskii functionals and augmented Lyapunov-Krasovskii functionals were introduced in [11][12][13][14][15], where some multi-integral terms were introduced in Lyapunov-Krasovskii functionals.In [16][17][18], combining with a decomposition approach, the upper bound of the derivative of Lyapunov-Krasovskii functional was estimated tightly and new stability results were obtained.It has been shown that these new methods can be applied efficiently to derive less conservative stability results for systems with interval timevarying delay.
On the other hand, singular systems have been extensively studied in the past years due to the fact that singular systems can better describe the behavior of some physical systems than regular ones.Singular systems are also referred to as descriptor systems, differential algebraic systems, or semistate systems.A great number of fundamental results based on the theory of regular systems have been extended to the area of singular systems [19].It is well known that the stability analysis for singular systems is much more complicated than that for regular systems because it requires considering not only stability but also regularity and absence of impulse (for continuous singular systems) [20][21][22][23][24][25][26][27][28][29][30] or causality (for discrete singular systems) [31][32][33][34].Recently, more and more attention has been paid to singular systems with delay.To obtain delay-dependent conditions, many efforts have been made in the literature.Via different Lyapunov-Krasovskii functionals, some stability criteria were obtained in [20,21], and the results in [20,21] were improved in [22,23] 2 Mathematical Problems in Engineering using the free-weighting matrices and discretized Lyapunov-Krasovskii functional, respectively.However, the involved time delays in [20][21][22][23] are all time invariant, which limits the scope of applications of the given results.In the case where time-varying delays appear in singular systems, some stability results were proposed in [24][25][26][27].The range of time-varying delay considered in [24][25][26][27] is from zero to an upper bound.For singular systems with interval time-varying delay, there are fewer results [28][29][30] and there still exists some room for further investigation.
In this paper, our purpose is to present some new delaydependent stability criteria for singular systems with interval time-varying delay.An appropriate Lyapunov-Krasovskii functional is constructed by using the idea of "delay decomposition." Employing this Lyapunov-Krasovskii functional, some new delay-dependent sufficient conditions ensuring the stability for the considered systems are obtained in terms of LIMs.The main contribution of this paper is in two aspects.First, the new results will be less conservative than the existing ones, which will be demonstrated by two numerical examples.Second, the new results will involve fewer decision variables than some existing ones; hence, they are mathematically less complex and more computationally efficient.
Notations.Throughout this paper, R  denotes the dimensional Euclidean space, while R × refers to the set of all real matrices with  rows and  columns.  represents the transpose of the matrix .For real symmetric matrices  and , the notation  ⩾  (resp.,  > ) means matrix  −  is positive-semidefinite (resp., positive-definite). is the identity matrix with appropriate dimensions.‖‖ refers to the Euclidean norm of the vector ; that is, ‖‖ = √   .

Problem Formulation and Preliminaries
Consider the following singular system with interval timevarying delay described by where () ∈ R  is the state vector and the initial condition () ∈ R  is a continuously differentiable vector-valued function.The matrix  ∈ R × may be singular and ranks  =  ⩽ , ,   ∈ R × are known real constant matrices.() is the time-varying delay satisfying where 0 <  1 <  2 and 0 ⩽  < 1 are known constant scalars.In this paper, our objective is to establish new delaydependent stability conditions for singular time-delay system (1).The following definitions and lemma will be used in the proof of our main results.

Main Results
In this section, we consider the stability problem of singular time-delay system (1).We first present a delay-dependent stability criterion for singular time-delay system (1) as follows.
Let us first of all show that the singular time-delay system (1) is regular and impulse-free.Using (5), it is easy to see that the following inequality holds: Using the fact that   > 0,   > 0 ( = 1, 2), and  1 > 0, from (8), we have It follows from (9) and Lemma 3.5 in [32] that the pair (, ) is regular and impulse-free.Thus, according to Definition 2, singular time-delay system (1) is regular and impulse-free.
Remark 5. Based on the Lyapunov-Krasovskii functional (10), Theorem 4 proposed a new delay-dependent criterion guaranteeing the singular time-delay system (1) to be regular, impulse-free, and stable.Lyapunov-Krasovskii functional Mathematical Problems in Engineering ( 10) is constructed by using the idea of "delay decomposition" [28][29][30].We decompose the delay intervals [0,  1 ] and [ )) plays a key role in the further reduction of conservatism.Furthermore, our Lyapunov-Krasovskii functional can be easily extended to the case as the number of delay partitions grows, which can be seen from the following discussion.
Remark 8.If the integer  is set to 2, then Theorem 6 reduces to Theorem 4. That is, Theorem 6 is an extension of Theorem 4. It should be pointed out that the conservatism of Theorem 6 lies in the parameter , which refers to the number of delay partitioning; that is, the conservatism is reduced as the partitions grow.On the other hand, the computational complexity also depends on the partition number ; that is, the computational complexity is increased as the partitions become thinner.Generally taking the small value of  such as  = 2 and  = 3, we can obtain less conservative and simple results.This can be seen from the simulation results in the sequel.
Corollary 9. Given an integer  ⩾ 2 and scalars  > 0, 0 ⩽  < 1, for any delay () satisfying (32), singular timedelay system (1) is regular, impulse-free, and stable if there exist matrices and , such that the LMI (30) and LMI hold for any  ∈ {1, 2, . . ., }, where and Ψ ∈ R ×(−) is any full-column rank matrix satisfying are block entry matrices, and  0 = ( T 1 +    T +2 ) T .Remark 10.Theorem 6 and Corollary 9 give stability criteria of system (1) with () satisfying ( 2) and ( 32), respectively.It is noted that the conditions in Theorem 6 and Mathematical Problems in Engineering Corollary 9 are both delay-dependent and rate-dependent.However, the information of delay rate may not be known in many cases, or () even may not be differentiable; then Theorem 6 and Corollary 9 fail to work.Regarding these circumstances, delay-dependent and rate-independent criteria can be derived by choosing  = 0 in Theorem 6 and Corollary 9.
For the case when () satisfies 0 <  1 ⩽ () ⩽  2 and  is unknown, the following result can be obtained from Theorem 6 by setting  = 0.
Remark 13.Compared with the results in [20][21][22][23][24][25], the LMIs in Theorems 4 and 6 and Corollaries 9 and 11, 12 are all strict LMIs; thus, they can be directly solved by using any LMI toolbox like the one of Matlab or the one of Scilab.
Based on Corollary 9, we can get the following stability criterion for system (44) with delay () satisfying (32).
When the information of  is unknown, the following results can be obtained directly from Corollaries 11 and 12 for system (44) for two cases: 0 <  1 ⩽ () ⩽  2 and 0 ⩽ () ⩽ .

Numerical Examples
In the section, numerical examples are given to illustrate the effectiveness and the less conservatism of obtained results in this paper.(58) For  1 = 0, the allowable upper bound  2 , which guarantees regular, impulse-free, and stable systems (1) for different , is listed in Table 1.From Table 1, it can be seen that the stability criteria in Corollary 9 are less conservative than those in [20,22,24,28].
For systems with interval time-varying delay, the allowable upper bound  2 , which guarantees regular, impulse-free, and stable systems (1) with given lower bound  1 for different , is listed in Table 2. From Table 2, it can be seen that the stability criteria in Theorems 4 and 6 are less conservative than those in [28,29].Especially, when  1 = 1.1, the result in [28] is not feasible while the allowable upper bound  2 can also be obtained from Theorems 4 and 6 in this paper.
The purpose is to compare the allowable upper bound  2 which guarantees the stability of the considered system for given lower bound  1 .
For various , the allowable upper bound  2 is listed in Table 3.Moreover, the number of decision variables involved in the stability criteria is given in Table 4. From Tables 3 and  4, it can be seen that Corollary 14 ( = 2) in this paper has fewer decision variables and less conservatism than those results in [8,12,15].

Conclusion
This paper deals with the problem of stability for singular systems with interval time-varying delay.By using a delay decomposition approach, new stability criteria for singular time-delay systems to be regular, impulse-free, and stable are proposed in terms of LMIs.Based on the obtained criterion, some improved stability results for the regular systems with interval time-varying delay are also given.The obtained results in this paper have been shown to be less conservative than recently reported results.Moreover, the proposed method decreases the computational complexity comparable to some existing methods.Two numerical examples are given to illustrate the applicability of the results.

Example 18 .
Consider the singular time-delay system (

Table 2 :
Allowable upper bound  2 with given  1 for different .

Table 3 :
Allowable upper bound  2 with given  1 for different .

Table 4 :
Number of decision variables.

Table 5 :
Allowable upper bound  2 with given  1 for unknown .