Research Article  Open Access
A Delay Decomposition Approach to the Stability Analysis of Singular Systems with Interval TimeVarying Delay
Abstract
This paper investigates delaydependent stability problem for singular systems with interval timevarying delay. An appropriate LyapunovKrasovskii functional is constructed by decomposing the delay interval into multiple equidistant subintervals, where both the information of every subinterval and timevarying delay have been taken into account. Employing the LyapunovKrasovskii functional, improved delaydependent stability criteria for the considered systems to be regular, impulsefree, and stable are established. Finally, two numerical examples are presented to show the effectiveness and less conservativeness of the proposed method.
1. Introduction
During the last few decades, there has been a growing research interest in the analysis and synthesis of timedelay 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 [2–6]. In practice, however, the delay range may have a nonzero lower bound, and such systems are referred to as interval timevarying delay systems. For this reason, the stability of the systems with such interval timevarying 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 freeweighting matrices, [7, 8] present some stability conditions for systems with interval timevarying delays. In [9], via a LyapunovKrasovskii 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 timevarying delay systems. New classes of LyapunovKrasovskii functionals and augmented LyapunovKrasovskii functionals were introduced in [11–15], where some multiintegral terms were introduced in LyapunovKrasovskii functionals. In [16–18], combining with a decomposition approach, the upper bound of the derivative of LyapunovKrasovskii 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) [20–30] or causality (for discrete singular systems) [31–34]. Recently, more and more attention has been paid to singular systems with delay. To obtain delaydependent conditions, many efforts have been made in the literature. Via different LyapunovKrasovskii functionals, some stability criteria were obtained in [20, 21], and the results in [20, 21] were improved in [22, 23] using the freeweighting matrices and discretized LyapunovKrasovskii functional, respectively. However, the involved time delays in [20–23] are all time invariant, which limits the scope of applications of the given results. In the case where timevarying delays appear in singular systems, some stability results were proposed in [24–27]. The range of timevarying delay considered in [24–27] is from zero to an upper bound. For singular systems with interval timevarying delay, there are fewer results [28–30] 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 timevarying delay. An appropriate LyapunovKrasovskii functional is constructed by using the idea of “delay decomposition.” Employing this LyapunovKrasovskii functional, some new delaydependent 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, denotes the dimensional Euclidean space, while 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 positivesemidefinite (resp., positivedefinite). is the identity matrix with appropriate dimensions. refers to the Euclidean norm of the vector ; that is, .
2. Problem Formulation and Preliminaries
Consider the following singular system with interval timevarying delay described by where is the state vector and the initial condition is a continuously differentiable vectorvalued function. The matrix may be singular and ranks , are known real constant matrices. is the timevarying delay satisfying where and are known constant scalars.
In this paper, our objective is to establish new delaydependent stability conditions for singular timedelay system (1). The following definitions and lemma will be used in the proof of our main results.
Definition 1 (see [19]). (i) The pair is said to be regular if det is not identically zero. (ii) The pair is said to be impulsefree if deg(det rank.
Definition 2 (see [28, 30]). (i) The singular timedelay system (1) is said to be regular and impulsefree if the pair is regular and impulsefree. (ii) The singular timedelay system (1) is said to be stable if, for any , there exists a scalar such that, for any compatible initial conditions satisfying , the solution of system (1) satisfies for any ; moreover, .
Lemma 3 (see [1]). For any symmetric positive defined matrix , scalars , and vector function such that the following integrations are well defined, the following inequality holds
3. Main Results
In this section, we consider the stability problem of singular timedelay system (1). We first present a delaydependent stability criterion for singular timedelay system (1) as follows.
Theorem 4. Given scalars and for any delay satisfying (2), singular timedelay system (1) is regular, impulsefree, and stable if there exist the following matrices: and , such that the LMIs hold for any , where and is any fullcolumn rank matrix satisfying are block entry matrices, and .
Proof. The proof is divided into two parts. The first one deals with the regularity and impulsefree property, and the second one treats the stability property of the studied class of systems. Let us first of all show that the singular timedelay system (1) is regular and impulsefree. Using (5), it is easy to see that the following inequality holds:
Using the fact that , and , from (8), we have
It follows from (9) and Lemma 3.5 in [32] that the pair is regular and impulsefree. Thus, according to Definition 2, singular timedelay system (1) is regular and impulsefree.
Next, we will show the stability of the singular timedelay system (1). To the end, the following LyapunovKrasovskii functional for system (1) is considered:
where
with
Define . Now calculating the derivative of along the trajectory of the system (1), we derive
By Lemma 3, it can be shown that
When , we have
Using Lemma 3 again, we get
Define
with the condition (6); then, premultiplying and postmultiplying by and , respectively, we can obtain
hence,
Note that when or , we have or , respectively. So relation (20) still holds.
When , (15) can be rewritten as
From condition (6) and Lemma 3, proceeding in a similar manner as above, we can get
Noting , we can deduce
where is any matrix with appropriate dimensions.
Combining (10)–(14), (20)–(23) yield
Therefore, we can see if LMIs (5) and (6) hold for any ; then holds for any . Hence, there exist a sufficiently small scalar , such that
By (25), the following steps are similar to the proofs of Theorem 3.1 in [25] and Proposition 1 in [28]; we deduce that singular timedelay system (1) is stable. This completes the proof.
Remark 5. Based on the LyapunovKrasovskii functional (10), Theorem 4 proposed a new delaydependent criterion guaranteeing the singular timedelay system (1) to be regular, impulsefree, and stable. LyapunovKrasovskii functional (10) is constructed by using the idea of “delay decomposition” [28–30]. We decompose the delay intervals and into two equidistant subintervals, respectively, such that the information of delay states and is all taken into account. Compared with [29], in which the LyapunovKrasovskii functional was constructed by decomposing delay intervals and into two equidistant subintervals, respectively, our LyapunovKrasovskii functional contains the delay state . Thus, by checking the variation of for the case when or , respectively, we introduce a new estimation on the upper bound of . That is to say, the delay state plays a key role in the further reduction of conservatism. Furthermore, our LyapunovKrasovskii functional can be easily extended to the case as the number of delay partitions grows, which can be seen from the following discussion.
In the following, by decomposing the delay intervals and into equidistant subintervals, respectively, we will derive a more general stability result than Theorem 4. For this purpose, the following LyapunovKrasovskii functional for singular timedelay system (1) is considered: where
Using LyapunovKrasovskii functional (26), the following theorem can be obtained.
Theorem 6. Given an integer and scalars , , for any delay satisfying (2), singular timedelay system (1) is regular, impulsefree, and stable if there exist matrices and , such that LMIs hold for any , where and is any fullcolumn rank matrix satisfying are block entry matrices, and .
Proof. The proof of Theorem 6 can be carried out using methods in the proof of Theorem 4. Hence, it is omitted.
Remark 7. Based on the LyapunovKrasovskii functional (26), Theorem 6 proposed an improved delaydependent criterion guaranteeing the singular timedelay system (1) to be regular, impulsefree, and stable. LyapunovKrasovskii functional (26) is constructed by decomposing the delay intervals and into equidistant subintervals, respectively, such that the information of delay states and is all taken into account. Compared with [28, 30], the augmented vectors and were introduced in LyapunovKrasovskii functional (26), so (26) is more general than the ones in [28, 30].
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 and , we can obtain less conservative and simple results. This can be seen from the simulation results in the sequel.
For system (1) with the routine delay case described by where and are known constant scalars, the corresponding LyapunovKrasovskii reduces to where
Similar to the proof of Theorem 4, we can obtain the following delaydependent stability criterion for singular timedelay system (1) with satisfying (32).
Corollary 9. Given an integer and scalars , , for any delay satisfying (32), singular timedelay system (1) is regular, impulsefree, and stable if there exist matrices and , such that the LMI (30) and LMI hold for any , where and is any fullcolumn rank matrix satisfying are block entry matrices, and .
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 Corollary 9 are both delaydependent and ratedependent. 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, delaydependent and rateindependent criteria can be derived by choosing in Theorem 6 and Corollary 9.
For the case when satisfies and is unknown, the following result can be obtained from Theorem 6 by setting .
Corollary 11. Given an integer and scalars , for any delay satisfying , singular timedelay system (1) is regular, impulsefree, and stable if there exist matrices and , such that the LMI (30) and LMI hold for any , where and are the same as those defined in Theorem 6.
For the case when satisfies and is unknown, the following result can be obtained from Corollary 9 by setting .
Corollary 12. Given an integer and scalar , for any delay satisfying , singular timedelay system (1) is regular, impulsefree, and stable if there exist matrices and , such that the LMI (30) and LMI hold for any , where and are the same as those defined in Corollary 9.
Remark 13. Compared with the results in [20–25], 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.
When the matrix is nonsingular, the stability problem of singular timedelay system (1) is reduced to analyzing the stability of the following regular system: This problem has been widely studied in the recent literature (see, e.g., [7–15]). Choose the following LyapunovKrasovskii functional: where vectors and matrices are defined the same as those defined in (26).
By employing the LyapunovKrasovskii functional (45) and using the similar proof of Theorem 4, we can obtain the following delaydependent stability criterion for timedelay system (44).
Corollary 14. Given an integer and scalars , , for any delay satisfying (2), timedelay system (44) is stable if there exist matrices such that the LMI (30) and LMI (47) hold for any , where and are the same as those defined in Theorem 6.
Based on Corollary 9, we can get the following stability criterion for system (44) with delay satisfying (32).
Corollary 15. Given an integer and scalars , , for any delay satisfying (32), timedelay system (44) is stable if there exist matrices such that the LMI (30) and LMI hold for any , where and are the same as those defined in Corollary 9.
When the information of is unknown, the following results can be obtained directly from Corollaries 11 and 12 for system (44) for two cases: and .
Corollary 16. Given an integer and scalars , for any delay satisfying , timedelay system (44) is stable if there exist matrices such that the LMI (30) and LMI hold for any , where and are the same as those defined in Theorem 6.
Corollary 17. Given an integer and scalar , for any delay satisfying , timedelay system (44) is stable if there exist matrices such that the LMI (30) and LMI hold for any , where and are the same as those defined in Corollary 9.
4. Numerical Examples
In the section, numerical examples are given to illustrate the effectiveness and the less conservatism of obtained results in this paper.
Example 1. Consider the singular timedelay system (1) with For , the allowable upper bound , which guarantees regular, impulsefree, 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 timevarying delay, the allowable upper bound , which guarantees regular, impulsefree, and stable systems (1) with given lower bound 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 , the result in [28] is not feasible while the allowable upper bound can also be obtained from Theorems 4 and 6 in this paper.
