Delay-Dependent Stability Criteria for Singular Systems with Interval Time-Varying Delay

Abstract

This paper is concerned with stability analysis for singular systems with interval time-varying delay. By constructing a novel Lyapunov functional combined with reciprocally convex approach and linear matrix inequality (LMI) technique, improved delay-dependent stability criteria for the considered systems to be regular, impulse free, and stable are established. The developed results have advantages over some previous ones as they involve fewer decision variables yet less conservatism. Numerical examples are provided to demonstrate the effectiveness of the proposed stability results.

1. Introduction

It is well known that time delays frequently occur in many practical systems, such as biological systems, chemical systems, electronic systems, and network control systems. The time delays are regarded as the major source of oscillation, instability, and poor performance of dynamic systems. During the last two decades, there has been some remarkable theoretical and practical progress in stability, stabilization, and robust control of linear time-delay systems [1, 2]. Currently, the results of stability for time-delay systems mainly focus on time-varying delay with range zero to an upper bound. However, in practice, the delay range may have a nonzero lower bound, and such systems are referred to interval time-varying delay systems. Typical examples for interval time-delay systems are networked control systems [3]. With rapid advancement in the networked control systems technology, a number of significant results have been reported in the recent past for the stability of interval time-delay systems [3–14]. For example, in [3], a discretized Lyapunov functional approach is employed to obtain stability criteria for linear uncertain systems with interval time-varying delays. By using free-weighting matrices, [4, 5] present some less conservative stability conditions. The free-weighting matrices method was further improved in [6, 7] by constructing augmented Lyapunov functionals. The free-weighting matrices method is regarded as an effective way to reduce the conservatism of the stability results; however, one chief shortcoming is that too many free-weighting matrices introduced in the theoretical derivation sometimes cannot reduce the conservatism of the obtained results, on the contrary, they make criteria mathematically complex and computationally less effective. In [8, 9], via different Lyapunov functionals with fewer matrix variables whose derivative is estimated using Jensen inequality, some simple stability criteria were obtained, these results were improved in [10] using the convex analysis method, and the result in [10] was further improved in [11] using the reciprocally convex approach. Recently, by introducing some integral terms in the augmented vector and using the Lyapunov functionals with triple-integral terms, some less conservative results were obtained in [12–14].

Singular systems, which are also referred to as descriptor systems, differential algebraic systems, or semistate systems whose behaviors are described by differential equations (or difference equations) and algebraic equations. Singular systems have strong practical relevance in a variety of physical processes such as power systems, social economic systems, and circuit systems [15]. For this reason, singular systems have attracted a lot of researches from mathematics and control communities. A great number of fundamental results based on the theory of regular systems have been extended to the area of singular systems [16]. Recently, more and more attention has been paid to singular systems with delay. Singular time-delay systems can preserve the structure of practical systems and have extensive applications in various engineering systems, including aircraft attitude control, flexible arm control of robots, large-scale electric network control, chemical engineering systems, and lossless transmission lines [17]. It is well known that the stability analysis for singular systems is much more complicated than that for regular systems because it requires to consider not only stability, but also regularity and absence of impulse (for continuous singular systems) [18–28] or causality (for discrete singular systems) [29–32]. In order to obtain stability conditions of singular time-delay systems, many efforts have been made in the literature, among which the model transformation and bounding technique for cross-terms are often used [18–20]. However, it is well known that these two kinds of methods are the main source of conservatism. Without using model transformation and bounding technique for cross-terms, some improved stability conditions with less conservatism have been provided by introducing free-weighting matrices [21, 22], integral inequality [23, 24], delay decomposition [25], and parameterized Lyapunov functional [26]. However, the involved time delays of [18–26] 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 [27, 28]. The range of time-varying delay considered in [27, 28] is from zero to an upper bound. In the case of the lower bound of delay is not restricted to be zero, the stability criteria in [27, 28] are conservative because they do not take into account the information of the lower bound of delay. Very recently, singular systems with time-varying delay in a range are studied in [33–38]. Nevertheless, there still exists some room for deriving less conservative as well as computationally less expensive stability criteria, which has motivated this paper.

In this paper, we will construct a novel Lyapunov functional and extend the reciprocally convex approach inspired by Park et al. [11] to analyze the stability of singular systems with interval time-varying delay. Some improved results for the considered systems to be regular, impulse free, and stable are established in terms of LMIs. The obtained stability criteria involve fewer decision variables comparable to those based on the free-weighting matrices method; hence they are mathematically less complex and computationally more efficient. Meanwhile, the new criteria are less conservative than existing ones, which will be demonstrated by some numerical examples.

Notations. Throughout this paper, 𝐑^𝑛denotes the 𝑛-dimensional Euclidean space, while 𝐑^𝑚×𝑛 refers to the set of all real matrices with 𝑚 rows and 𝑛 columns. 𝐴^𝑇 represents the transpose of the matrix 𝐴, while 𝐴⁻¹ denotes the inverse of 𝐴. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means matrix 𝑋−𝑌 is positive semidefinite (resp., positive-definite). 𝐼 is the identity matrix with appropriate dimensions. ‖𝑥‖ refers to the Euclidean norm of the vector 𝑥, that is, ‖𝑥‖=𝑥^𝑇𝑥.

2. Problem Formulation and Preliminaries

Consider the singular system with interval time-varying delay described by: 𝐸𝑥̇(𝑡)(𝑡)((𝑡,𝑥(𝜃)(𝜃)=𝐴𝑥+𝐵𝑥𝑡−𝑑))=𝜑,𝜃∈−𝑑₂,0, (2.1) where 𝑥(𝑡)∈𝐑^𝑛 is the state vector, and 𝜑(𝜃)∈𝐑^𝑛 is a continuous vector-valued initial function of 𝜃∈[−𝑑₂,0]. The matrix 𝐸∈𝐑^𝑛×𝑛 may be singular, and it is assumed that rank𝐸=𝑟≤𝑛, 𝐴,𝐵∈𝐑^𝑛×𝑛 are known real constant matrices with appropriate dimensions. 𝑑(𝑡) is the time-varying delay and is assumed to satisfy 𝑑₁≤𝑑(𝑡)≤𝑑₂,𝑑̇(𝑡)≤𝜇, (2.2) where 0<𝑑₁<𝑑₂ and 0≤𝜇<1 are known constants; 𝑑₁ and 𝑑₂ represent the lower and upper bounds of the time-varying 𝑑(𝑡), respectively, 𝜇 is the bound on the delay derivative.

The purpose of this paper is to formulate new delay-dependent criteria to check the stability of singular time-delay system (2.1). Let us give the following definitions and lemmas, which will play an indispensable role in deriving our criteria.

Definition 2.1 (see [16]). (i) The pair (𝐸,𝐴) is said to be regular if det(𝑠𝐸−𝐴) is not identically zero. (ii) The pair (𝐸,𝐴) is said to be impulse free if deg(det(𝑠𝐸−𝐴))=rank𝐸.

Definition 2.2 (see [35]). (i) The singular time-delay system (2.1) is said to be regular and impulse free if the pairs (𝐸,𝐴) and (𝐸,𝐴+𝐵) are regular and impulse free. (ii) The singular time-delay system (2.1) is said to be stable if for any 𝜀>0, there exists a scalar 𝛿(𝜀)>0 such that, for any compatible initial conditions 𝜑(𝑡) satisfying sup_{−𝑑2≤𝑡≤0}‖𝜑(𝑡)‖≤𝛿(𝜀), the solution 𝑥(𝑡) of system (2.1) satisfies ‖𝑥(𝑡)‖≤𝜀 for any 𝑡≥0, more over lim_𝑡→∞𝑥(𝑡)=0.

Definition 2.3 (see [11]). Let 𝜙₁,𝜙₂,…,𝜙_𝑁∶𝐑^𝑚↦𝐑^𝑛 be a given finite number of functions such that they have positive values in an open subset 𝐃 of 𝐑^𝑚. Then, a reciprocally convex combination of these functions over 𝐃 is a function of form 1𝛼₁𝜙₁+1𝛼₂𝜙₂1+⋅⋅⋅+𝛼_𝑁𝜙_𝑁∶𝐃↦𝐑^𝑛, (2.3) where the real numbers 𝛼_𝑖 satisfy 𝛼_𝑖>0 and ∑_𝑖𝛼_𝑖=1.

Lemma 2.4 (see [1]). For any symmetric positive define matrix 𝑅>0, scalars 𝛾₂>𝛾₁>0 and vector function 𝑥∶[𝛾₁,𝛾₂]↦𝐑^𝑛 such that the integrations concerned are well defined, the following inequality holds −𝛾₂−𝛾₁^𝛾2_𝛾1𝑥^T(𝑠)𝑅𝑥(𝑠)d𝑠≤−^𝛾2_𝛾1𝑥^T(𝑠)d𝑠𝑅^𝛾2_𝛾1𝑥(𝑠)d𝑠. (2.4)

Lemma 2.5 (see [11]). Let 𝑓₁,𝑓₂,…,𝑓_𝑁∶𝐑^𝑚↦𝐑 have positive values in an open subset 𝐃of𝐑^𝑚. Then, the reciprocally convex combination of 𝑓_𝑖 over 𝐃 satisfies min_{𝛼𝑖∣𝛼𝑖>0,∑𝑖𝛼𝑖=1𝑖}1𝛼_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖,𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖,𝑗(𝑡)𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜𝑔_𝑖,𝑗∶𝐑^𝑚↦𝐑,𝑔_𝑗,𝑖(𝑡)≜𝑔_𝑖,𝑗(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖,𝑗(𝑡)𝑔_𝑖,𝑗(𝑡)𝑓_𝑗(𝑡)≥0. (2.5)

3. Main Results

In this section, we consider the stability of singular time-delay system (2.1). For simplicity, we define that 𝜉(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−(𝑑(𝑡)/2))𝑥^𝑇(𝑡−𝑑(𝑡))𝑥^𝑇(𝑡−(𝑑₁/2))𝑥^𝑇(𝑡−𝑑₁)𝑥^𝑇(𝑡−(𝑑₂/2))𝑥^𝑇(𝑡−𝑑₂)]^𝑇, 𝑒_𝑖(𝑖=1,2,…,7) are block entry matrices, for example, 𝑒₃=[00𝐼0000]^𝑇 and 𝑒₈=(𝐴𝑒^𝑇₁+𝐵𝑒^𝑇₃)^𝑇. Now, we provide a novel delay-dependent stability criterion for singular time-delay system (2.1) as follows.

Theorem 3.1. Given scalars 0<𝑑₁<𝑑₂ and 0≤𝜇<1, for any delay 𝑑(𝑡) satisfying (2.2), singular time-delay system (2.1) is regular, impulse free, and stable if there exist matrices 𝑃>0, 𝑄_𝑖=^{𝑄𝑖1𝑄𝑖2}_{𝑄𝑇𝑖2𝑄𝑖3}≥0(𝑖=1,2), 𝑄₃≥0, 𝑄₄≥0, 𝑅₁≥0, 𝑅₂≥0, 𝑅₃≥0, 𝑍₁, 𝑍₂ and 𝑆, such that the following LMIs (3.1)–(3.3) hold Υ=𝑒₁𝐸^𝑇𝑃𝑒^𝑇₈+𝑒₈𝑃𝐸𝑒^𝑇₁+𝑒₁𝑒₄𝑄₁𝑒₁𝑒₄^𝑇−𝑒₄𝑒₅𝑄₁𝑒₄𝑒₅^𝑇+𝑒₁𝑒₆𝑄₂𝑒₁𝑒₆^𝑇−𝑒₆𝑒₇𝑄₂𝑒₆𝑒₇^𝑇+𝑒₁𝑄₃𝑒^𝑇₁−𝜇1−2𝑒₂𝑄₃𝑒^𝑇₂+𝑒₅𝑄₄𝑒^𝑇₅−1−𝜇𝑒₃𝑄₄𝑒^𝑇₃+𝑒₈𝑅𝑒^T₈−𝑒₁−𝑒₄𝐸^𝑇𝑅₁𝐸𝑒₁−𝑒₄^𝑇+𝑒₁𝑆Ψ^𝑇𝑒^𝑇₈+𝑒₈Ψ𝑆^𝑇𝑒^𝑇₁−𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂^𝑇𝐸^𝑇𝑅₂𝐸𝐸^𝑇𝑍₁𝐸𝐸^𝑇𝑍^𝑇₁𝐸𝐸^𝑇𝑅₂𝐸𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂−𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃^𝑇𝐸^𝑇𝑅₃𝐸𝐸^𝑇𝑍₂𝐸𝐸^𝑇𝑍^𝑇₂𝐸𝐸^𝑇𝑅₃𝐸𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃𝑅<0₂𝑍₁𝑍^𝑇₁𝑅₂𝑅≥0₃𝑍₂𝑍^𝑇₂𝑅₃≥0, (3.1)(3.2)(3.3) where 𝑅=(𝑑₁/2)²𝑅₁+((𝑑₂−𝑑₁)/2)²𝑅₂+(𝑑₂−𝑑₁)²𝑅₃, and Ψ∈𝐑^{𝑛×(𝑛−𝑟)} is any full-column rank matrix satisfying 𝐸^𝑇Ψ=0.

Proof. The proof is divided into two parts. The first part deals with the regularity and impulse-free properties, and the second part treats the stability property of the studied class of systems. First of all, we show that the singular time-delay system (2.1) is regular and impulse free for any time-delay 𝑑(𝑡) satisfying (2.2). From LMI (3.1), it follows that Ξ₁₁𝐸^𝑇𝑃𝐵+𝑆Ψ^𝑇𝐵𝑄₁₂+𝐸^𝑇𝑅₁𝐵𝐸0_𝑇𝑃𝐸+𝐵^𝑇Ψ𝑆^𝑇Ξ₂₂0𝐸^𝑇𝑅₃𝐸−𝐸^𝑇𝑍₂𝐸𝑄^𝑇₁₂+𝐸^𝑇𝑅₁𝐸0Ξ₃₃−𝑄₁₂0𝐸^𝑇𝑅₃𝐸−𝐸^𝑇𝑍^𝑇₂𝐸−𝑄^𝑇₁₂−𝑄₁₃+𝑄₄−𝐸^𝑇𝑅₃𝐸<0, (3.4) where Ξ₁₁=𝐸^𝑇𝑃𝐴+𝐴^𝑇𝑃𝐸+𝑄₁₁+𝑄₂₁+𝑄₃−𝐸^𝑇𝑅₁𝐸+𝑆Ψ^𝑇𝐴+𝐴^𝑇Ψ𝑆^𝑇,Ξ₂₂=−1−𝜇𝑄₄−2𝐸^𝑇𝑅₃𝐸+𝐸^𝑇𝑍₂𝐸+𝐸^𝑇𝑍^𝑇₂Ξ𝐸,₃₃=𝑄₁₃−𝑄₁₁−𝐸^𝑇𝑅₁𝐸−𝐸^𝑇𝑅₂𝐸. (3.5) From LMI (3.4), it easy to see that Ξ₁₁<0, using the fact that 𝑄₁₁≥0,𝑄₂₁≥0 and 𝑄₃≥0, we have 𝐸^𝑇𝑃𝐴+𝐴^𝑇𝑃𝐸−𝐸^𝑇𝑅₁𝐸+𝑆Ψ^𝑇𝐴+𝐴^𝑇Ψ𝑆^𝑇<0. (3.6) Since rank𝐸=𝑟≤𝑛, there must exist two invertible matrices 𝐺,𝐻∈𝐑^𝑛×𝑛 such that 𝐼𝐺𝐸𝐻=^𝑟000. (3.7) Set 𝐴𝐺𝐴𝐻=¹¹𝐴₁₂𝐴₂₁𝐴₂₂,𝐺^−𝑇0ΨΨ=,𝐻^𝑇𝑆𝑆=¹𝑆₂, (3.8) where Ψ∈𝐑^{(𝑛−𝑟)×(𝑛−𝑟)} is a nonsingular matrix. Pre-multiplying and post-multiplying (3.6) by 𝐻^𝑇 and 𝐻, respectively, we can easily formulate the following inequality: Θ₁₁Θ₁₂Θ^𝑇₁₂𝐴^𝑇₂₂Ψ𝑆^𝑇₂+𝑆₂Ψ^𝑇𝐴₂₂<0, (3.9) where Θ₁₁ and Θ₁₂ are not relevant in the following discussion; the real expression of these two matrices are omitted here. From (3.9), it is easy to see that 𝐴^𝑇₂₂Ψ𝑆^𝑇₂+𝑆₂Ψ^𝑇𝐴₂₂<0, (3.10) which implies that matrix 𝐴₂₂ is nonsingular. Otherwise, supposing that 𝐴₂₂ is singular, there must exist a nonzero vector 𝜂∈𝐑^𝑛−𝑟, which ensures that 𝐴₂₂𝜂=0. And then we can conclude that 𝜂^𝑇(𝐴^𝑇₂₂Ψ𝑆^𝑇₂+𝑆₂Ψ^𝑇𝐴₂₂)𝜂=0, and this contradicts (3.10). So 𝐴₂₂ is nonsingular, which implies that the pair (𝐸,𝐴) is regular and impulse free [16].
On the other hand, Pre-multiplying and post-multiplying (3.4) by [𝐼𝐼𝐼𝐼] and [𝐼𝐼𝐼𝐼]^𝑇, respectively, yields 𝐸^𝑇𝑃(𝐴+𝐵)+(𝐴+𝐵)^𝑇𝑃𝐸−𝐸^𝑇𝑅₂+𝑅₃𝐸+𝑆Ψ^𝑇(𝐴+𝐵)+(𝐴+𝐵)^𝑇Ψ𝑆^𝑇+𝑄₂₁+𝑄₃+𝜇𝑄₄<0. (3.11) From (3.11), taking conditions 𝑄₂₁≥0,𝑄₃≥0,𝑄₄≥0 and 𝜇≥0 into account, we obtain 𝐸^𝑇𝑃(𝐴+𝐵)+(𝐴+𝐵)^𝑇𝑃𝐸−𝐸^𝑇𝑅₂+𝑅₃𝐸+𝑆Ψ^𝑇(𝐴+𝐵)+(𝐴+𝐵)^𝑇Ψ𝑆^𝑇<0. (3.12) Proceeding in a similar manner as above, we can find (3.12) implies that the pair (𝐸,𝐴+𝐵) is regular and impulse free. Thus, according to Definition 2.2, singular time-delay system (2.1) is regular and impulse free for any time-delay 𝑑(𝑡) satisfying (2.2).
In the following, we will prove that singular delay-delay system (2.1) is stable. Construct a new class Lyapunov functional for system (2.1) as follows: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (3.13) where 𝑉₁(𝑡)=𝑥^𝑇(𝑡)𝐸^𝑇(𝑡),𝑉𝑃𝐸𝑥₂(𝑡)=^𝑡_{𝑡−𝑑1/2}𝜉^𝑇₁(𝑠)𝑄₁𝜉₁(𝑠)𝑑𝑠+^𝑡_{𝑡−𝑑2/2}𝜉^𝑇₂(𝑠)𝑄₂𝜉₂(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝑑/2}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑑𝑠+^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₄𝑥(𝑠)𝑉𝑑𝑠,₃(𝑡)=𝑑₁2⁰_−𝑑1/2^𝑡_𝑡+𝛼𝑥̇^𝑇𝛽𝐸^𝑇𝑅₁𝐸𝑥̇𝛽𝑑𝑑𝛽𝑑𝛼+²−𝑑₁2^−𝑑1/2_−𝑑2/2^𝑡_𝑡+𝛼𝑥̇^𝑇𝛽𝐸^𝑇𝑅₂𝐸𝑥̇𝛽+𝑑𝑑𝛽𝑑𝛼₂−𝑑₁^−𝑑1_−𝑑2^𝑡_𝑡+𝛼𝑥̇^𝑇𝛽𝐸^𝑇𝑅₃𝐸𝑥̇𝛽𝑑𝛽𝑑𝛼, (3.14) with 𝜉₁(𝑠)=𝑥^𝑇(𝑠)𝑥^𝑇𝑑𝑠−¹2^𝑇,𝜉₂(𝑠)=𝑥^𝑇(𝑠)𝑥^𝑇𝑑𝑠−²2^𝑇. (3.15) It is easy to see that 𝑉̇(𝑡)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡). (3.16) The time derivative of each 𝑉_𝑖(𝑡)(𝑖=1,2,3) along trajectories of the singular time-delay system (2.1) can be processed as 𝑉̇₁(𝑡)=2𝑥^𝑇(𝑡)𝐸^𝑇𝑃𝐸𝑥̇(𝑡)=2𝜉^𝑇(𝑡)𝑒₁𝐸^𝑇𝑃𝑒^𝑇₈𝜉(𝑡),𝑉̇₂(𝑡)=𝜉^𝑇₁(𝑡)𝑄₁𝜉₁(𝑡)−𝜉^𝑇₁𝑡−𝑑₁2𝑄₁𝜉₁𝑑𝑡−¹2+𝜉^𝑇₂(𝑡)𝑄₂𝜉₂(𝑡)−𝜉^𝑇₂𝑑𝑡−²2𝑄₂𝜉₂𝑑𝑡−²2+𝑥^𝑇(𝑡)𝑄₃𝑥(𝑡)−𝑑̇(𝑡)1−2𝑥^𝑇𝑑(𝑡)𝑡−2𝑄₃𝑥𝑑(𝑡)𝑡−2+𝑥^𝑇𝑡−𝑑₁𝑄₄𝑥𝑡−𝑑₁−1−𝑑̇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₄𝑥(𝑡−𝑑(𝑡))≤𝜉^𝑇(𝑡)𝑒₁𝑒₄𝑄₁𝑒₁𝑒₄^𝑇−𝑒₄𝑒₅𝑄₁𝑒₄𝑒₅^𝑇+𝑒₁𝑒₆𝑄₂𝑒₁𝑒₆^𝑇−𝑒₆𝑒₇𝑄₂𝑒₆𝑒₇^𝑇+𝑒₁𝑄₃𝑒^𝑇₁−𝜇1−2𝑒₂𝑄₃𝑒^𝑇₂+𝑒₅𝑄₄𝑒^𝑇₅−1−𝜇𝑒₃𝑄₄𝑒^𝑇₃𝜉(𝑡),𝑉̇₃(𝑡)=𝑥̇^𝑇(𝑡)𝐸^𝑇𝑅𝐸𝑥̇(𝑡)−𝑑₁2^𝑡_{𝑡−𝑑1/2}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₁𝐸𝑥̇(𝑠)−𝑑𝑑𝑠₂−𝑑₁2^{𝑡−𝑑1/2}_{𝑡−𝑑2/2}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₂𝐸𝑥̇(𝑠)𝑑𝑠−𝑑₂−𝑑₁^𝑡−𝑑1_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₃𝐸𝑥̇(𝑠)=𝑥̇𝑑𝑠_𝑇(𝑡)𝐸^𝑇𝑥̇(𝑡)−𝑑𝑅𝐸¹2^𝑡_{𝑡−𝑑1/2}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₁𝐸𝑥̇(𝑠)−𝑑𝑑𝑠₂−𝑑₁2^{𝑡−𝑑(𝑡)/2}_{𝑡−𝑑2/2}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₂𝐸𝑥̇(𝑠)𝑑𝑑𝑠−²−𝑑₁2^𝑡−𝑑1_(𝑡)^/2_{𝑡−𝑑/2}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₂𝐸𝑥̇(𝑠)−𝑑𝑑𝑠₂−𝑑₁^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₃𝐸𝑥̇(𝑠)𝑑𝑠−𝑑₂−𝑑₁^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝐸^𝑇𝑅₃𝐸𝑥̇(𝑠)𝑑𝑠. (3.17)(3.18)(3.19) Applying Lemma 2.4 to the last five integral terms, we can obtain 𝑉̇₃(𝑡)≤𝜉^𝑇(𝑡)𝑒₈𝑅𝑒^𝑇₈𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒₁−𝑒₄𝐸^𝑇𝑅₁𝐸𝑒₁−𝑒₄^𝑇𝜉(𝑡)−1𝛼₁𝑓₁(𝑡)−1𝛼₂𝑓₂(𝑡)−1𝛼₁𝑔₁(𝑡)−1𝛼₂𝑔₂(𝑡), (3.20) where 𝛼₁=𝑑₂−𝑑(𝑡)𝑑₂−𝑑₁,𝛼₂=𝑑(𝑡)−𝑑₁𝑑₂−𝑑₁,𝑓₁(𝑡)=𝜉^𝑇(𝑡)𝑒₂−𝑒₆𝐸^𝑇𝑅₂𝐸𝑒₂−𝑒₆^𝑇𝜉(𝑡),𝑓₂(𝑡)=𝜉^𝑇(𝑡)𝑒₄−𝑒₂𝐸^𝑇𝑅₂𝐸𝑒₄−𝑒₂^𝑇𝜉(𝑡),𝑔₁(𝑡)=𝜉^𝑇(𝑡)𝑒₃−𝑒₇𝐸^𝑇𝑅₃𝐸𝑒₃−𝑒₇^𝑇𝜉(𝑡),𝑔₂(𝑡)=𝜉^𝑇(𝑡)𝑒₅−𝑒₃𝐸^𝑇𝑅₃𝐸𝑒₅−𝑒₃^𝑇𝜉(𝑡). (3.21) Pre-multiplying and post-multiplying LMI (3.2) by diag{𝜉^𝑇(𝑡)(𝑒₂−𝑒₆)𝐸^𝑇,𝜉^𝑇(𝑡)(𝑒₄−𝑒₂)𝐸^𝑇} and diag{𝐸(𝑒₂−𝑒₆)^𝑇𝜉(𝑡),𝐸(𝑒₄−𝑒₂)^𝑇𝜉(𝑡)}, respectively, we have 𝑓₁(𝑡)𝑓_1,2(𝑡)𝑓_2,1(𝑡)𝑓₂(𝑡)≥0, (3.22) where 𝑓_1,2(𝑡)=𝜉^𝑇(𝑡)𝑒₂−𝑒₆𝐸^𝑇𝑍₁𝐸𝑒₄−𝑒₂^𝑇𝜉(𝑡),𝑓_2,1(𝑡)=𝜉^𝑇(𝑡)𝑒₄−𝑒₂𝐸^𝑇𝑍^𝑇₁𝐸𝑒₂−𝑒₆^𝑇𝜉(𝑡). (3.23) By using Lemma 2.5, we have −1𝛼₁𝑓₁(𝑡)−1𝛼₂𝑓₂(𝑡)≤−𝑓₁(𝑡)−𝑓₂(𝑡)−𝑓_1,2(𝑡)−𝑓_2,1(𝑡)=−𝜉^T(𝑡)𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂^𝑇𝐸^𝑇𝑅₂𝐸𝐸^𝑇𝑍₁𝐸𝐸^𝑇𝑍^𝑇₁𝐸𝐸^𝑇𝑅₂𝐸𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂𝜉(𝑡). (3.24) Similarly, from LMI (3.3), we can get −1𝛼₁𝑔₁(𝑡)−1𝛼₂𝑔₂(𝑡)≤−𝜉^𝑇(𝑡)𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃^𝑇𝐸^𝑇𝑅₃𝐸𝐸^𝑇𝑍₂𝐸𝐸^𝑇𝑍^𝑇₂𝐸𝐸^𝑇𝑅₃𝐸𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃𝜉(𝑡). (3.25) Combining (3.20)–(3.25), we can furthermore get 𝑉̇₃(𝑡)≤𝜉^𝑇(𝑡)𝑒₈𝑅𝑒^𝑇₈𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒₁−𝑒₄𝐸^𝑇𝑅₁𝐸𝑒₁−𝑒₄^𝑇𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂^𝑇𝐸^𝑇𝑅₂𝐸𝐸^𝑇𝑍₁𝐸𝐸^𝑇𝑍^𝑇₁𝐸𝐸^𝑇𝑅₂𝐸𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂𝜉(𝑡)−𝜉^𝑇(𝑡)𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃^𝑇𝐸^𝑇𝑅₃𝐸𝐸^𝑇𝑍₂𝐸𝐸^𝑇𝑍^𝑇₂𝐸𝐸^𝑇𝑅₃𝐸𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃𝜉(𝑡). (3.26) Note that when 𝑑(𝑡)=𝑑₁ or 𝑑(𝑡)=𝑑₂, we have 𝜉^𝑇(𝑡)(𝑒₄−𝑒₂)=𝜉^𝑇(𝑡)(𝑒₅−𝑒₃)=0 or 𝜉^𝑇(𝑡)(𝑒₂−𝑒₆)=𝜉^𝑇(𝑡)(𝑒₃−𝑒₇)=0, respectively. So relation (3.26) still holds.
Noting 𝐸^𝑇Ψ=0, we have 0=2𝑥^𝑇(𝑡)𝑆Ψ^𝑇𝐸𝑥̇(𝑡)=2𝜉^𝑇(𝑡)𝑒₁𝑆Ψ^𝑇𝑒^𝑇₈𝜉(𝑡), (3.27) where 𝑆 is any matrix with appropriate dimensions.
Adding (3.27) to the right of (3.16) and substituting (3.17),(3.18), and (3.26) into (3.16), we have 𝑉̇(𝑡)≤𝜉^𝑇(𝑡)Υ𝜉(𝑡). (3.28) From LMI (3.1), it is easy to see that 𝑉̇(𝑡)<0 for any 𝜉(𝑡)≠0. Hence, there exists a sufficiently small positive scalar 𝜀>0, such that 𝑉̇(𝑡)≤−𝜀‖𝑥(𝑡)‖². (3.29) By (3.29), the following steps are similar to the proof of Proposition 1 in [35] and Theorem 1 in [36], we can deduce that singular time-delay system (2.1) is stable. This completes our proof.

Remark 3.2. Based on the new Lyapunov functional in (3.13), together with the reciprocally convex approach and LMI technique, Theorem 3.1 proposed a delay-dependent criterion guaranteeing the considered singular time-delay system to be regular, impulse free, and stable. Lyapunov functional (3.13) is constructed by using the idea of “delay-partitioning” [25, 37, 39]. We consider the lower bound 𝑑₁, upper bound 𝑑₂, and time-varying delay 𝑑(𝑡) in our Lyapunov functional by dividing them into two equal segments, such that the information of delayed states 𝑑₁/2, 𝑑₂/2 and 𝑑(𝑡)/2 are all taken into account. Therefore, the criterion in Theorem 3.1 is expected to be less conservative than some previous ones, which will be demonstrated in the sequel.
If the matrix 𝐸 is nonsingular, then the stability problem of singular system (2.1) is reduced to analyzing the stability of the regular system: 𝑥̇(𝑡)(𝑡)((𝑡,𝑥(𝜃)(𝜃)=𝐴𝑥+𝐵𝑥𝑡−𝑑))=𝜑,𝜃∈−𝑑₂,0. (3.30) This problem has been widely studied in the recent literature (see, e.g., [3–14]). We choose Lyapunov functional: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (3.31) where 𝑉₁(𝑡)=𝑥^T(𝑡)(𝑡),𝑉𝑃𝑥₃(𝑡)=𝑑₁2⁰_−𝑑1/2^𝑡_𝑡+𝛼𝑥̇^𝑇𝛽𝑅₁𝑥̇𝛽𝑑𝑑𝛽𝑑𝛼+²−𝑑₁2^−𝑑1/2_−𝑑2/2^𝑡_𝑡+𝛼𝑥̇^𝑇𝛽𝑅₂𝑥̇𝛽+𝑑𝑑𝛽𝑑𝛼₂−𝑑₁^−𝑑1_−𝑑2^𝑡_𝑡+𝛼𝑥̇^𝑇𝛽𝑅₃𝑥̇𝛽𝑑𝛽𝑑𝛼, (3.32)(3.33) and 𝑉₂(𝑡) is defined in (3.13).

By employing the Lyapunov functional (3.31) and using the similar proof of Theorem 3.1, we can obtain the following delay-dependent stability criterion for time-delay system (3.30).

Corollary 3.3. Given scalars 0<𝑑₁<𝑑₂ and 0≤𝜇<1, for any delay 𝑑(𝑡) satisfying (2.2), time-delay system (3.30) is stable if there exist matrices 𝑃>0, 𝑄_𝑖=^{𝑄𝑖1𝑄𝑖2}_{𝑄𝑇𝑖2𝑄𝑖3}≥0(𝑖=1,2), 𝑄₃≥0, 𝑄₄≥0, 𝑅₁≥0, 𝑅₂≥0, 𝑅₃≥0 and 𝑍₁, 𝑍₂, such that LMIs (3.2), (3.3), and following LMI (3.34) hold 𝑒₁𝑃𝑒^𝑇₈+𝑒₈𝑃𝑒^𝑇₁+𝑒₁𝑒₄𝑄₁𝑒₁𝑒₄^𝑇−𝑒₄𝑒₅𝑄₁𝑒₄𝑒₅^𝑇+𝑒₁𝑒₆𝑄₂𝑒₁𝑒₆^𝑇−𝑒₆𝑒₇𝑄₂𝑒₆𝑒₇^𝑇+𝑒₁𝑄₃𝑒^𝑇₁−𝜇1−2𝑒₂𝑄₃𝑒^𝑇₂+𝑒₅𝑄₄𝑒^𝑇₅−1−𝜇𝑒₃𝑄₄𝑒^𝑇₃+𝑒₈𝑅𝑒^𝑇₈−𝑒₁−𝑒₄𝑅₁𝑒₁−𝑒₄^𝑇−𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂^𝑇𝑅₂𝑍₁𝑍^𝑇₁𝑅₂𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂−𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃^𝑇𝑅₃𝑍₂𝑍^𝑇₂𝑅₃𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃<0, (3.34) where matrix 𝑅 is defined in (3.1).

Remark 3.4. As mentioned in the introduction, through the use of free-weighting matrices [5] or the introduction of the Lyapunov functional with triple-integral terms [13, 14], we can derive less conservative stability criteria for system (3.30), but it makes the criteria mathematically complex and computationally less effective. In this paper, the Lyapunov functional (3.31) does not contain any triple-integral terms, and when estimating 𝑉̇(𝑡), we have not introduced free-weighting matrix. From a mathematical point of view, it is simple. Meanwhile, Corollary 3.3 in this paper is less conservative than the results in [5, 13, 14], which will be demonstrated in the sequel.
Theorem 3.1 and Corollary 3.3 give new stability criteria of system (2.1) and system (3.30) with 𝑑(𝑡) satisfying (2.2), respectively. They can be applied to both slow and fast time-varying delays only if 0≤𝜇<1 is known. In many circumstances, the information of the time derivative of delay 𝜇 is unknown or the time derivative of delay is known but 𝜇≥1. Regarding this case, the delay-dependent and rate-independent criteria can be derived by choosing 𝑄₃=𝑄₄=0 in Theorem 3.1 and Corollary 3.3, respectively. Therefore, we have the following Corollaries 3.5 and 3.6.

Corollary 3.5. Given scalars 0<𝑑₁<𝑑₂, for any delay 𝑑(𝑡) satisfying 𝑑₁≤𝑑(𝑡)≤𝑑₂, singular system (2.1) is regular, impulse free, and stable if there exist matrices 𝑃>0, 𝑄_𝑖=^{𝑄𝑖1𝑄𝑖2}_{𝑄T𝑖2𝑄𝑖3}≥0,𝑍_𝑖(𝑖=1,2), 𝑅_𝑗≥0(𝑗=1,2,3) and 𝑆, such that LMIs (3.2), (3.3), and following LMI (3.35) hold 𝑒₁𝐸^𝑇𝑃𝑒^𝑇₈+𝑒₈𝑃𝐸𝑒^𝑇₁+𝑒₁𝑒₄𝑄₁𝑒₁𝑒₄^𝑇−𝑒₄𝑒₅𝑄₁𝑒₄𝑒₅^𝑇+𝑒₁𝑒₆𝑄₂𝑒₁𝑒₆^𝑇−𝑒₆𝑒₇𝑄₂𝑒₆𝑒₇^𝑇+𝑒₈𝑅𝑒^𝑇₈−𝑒₁−𝑒₄𝐸^𝑇𝑅₁𝐸𝑒₁−𝑒₄^𝑇+𝑒₁𝑆Ψ^𝑇𝑒^𝑇₈+𝑒₈Ψ𝑆^𝑇𝑒^𝑇₁−𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂^𝑇𝐸^𝑇𝑅₂𝐸𝐸^𝑇𝑍₁𝐸𝐸^𝑇𝑍^𝑇₁𝐸𝐸^𝑇𝑅₂𝐸𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂−𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃^𝑇𝐸^𝑇𝑅₃𝐸𝐸^𝑇𝑍₂𝐸𝐸^𝑇𝑍^𝑇₂𝐸𝐸^𝑇𝑅₃𝐸𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃<0, (3.35) where matrices 𝑅 and Ψ are defined in (3.1).

Corollary 3.6. Given scalars 0<𝑑₁<𝑑₂, for any delay 𝑑(𝑡) satisfying 𝑑₁≤𝑑(𝑡)≤𝑑₂, time-delay system (3.30) is stable if there exist matrices 𝑃>0, 𝑄_𝑖=^{𝑄𝑖1𝑄𝑖2}_{𝑄𝑇𝑖2𝑄𝑖3}≥0,𝑍_𝑖(𝑖=1,2) and 𝑅_𝑗≥0(𝑗=1,2,3), such that LMIs (3.2), (3.3), and following LMI (3.36) hold 𝑒₁𝑃𝑒^𝑇₈+𝑒₈𝑃𝑒^𝑇₁+𝑒₁𝑒₄𝑄₁𝑒₁𝑒₄^𝑇−𝑒₄𝑒₅𝑄₁𝑒₄𝑒₅^𝑇+𝑒₁𝑒₆𝑄₂𝑒₁𝑒₆^𝑇−𝑒₆𝑒₇𝑄₂𝑒₆𝑒₇^𝑇+𝑒₈𝑅𝑒^𝑇₈−𝑒₁−𝑒₄𝑅₁𝑒₁−𝑒₄^𝑇−𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂^T𝑅₂𝑍₁𝑍^𝑇₁𝑅₂𝑒^𝑇₂−𝑒^𝑇₆𝑒^𝑇₄−𝑒^𝑇₂−𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃^𝑇𝑅₃𝑍₂𝑍^𝑇₂𝑅₃𝑒^𝑇₃−𝑒^𝑇₇𝑒^𝑇₅−𝑒^𝑇₃<0, (3.36) where matrix 𝑅 is defined in (3.1).

Remark 3.7. By employing similar method and choosing 𝑄₁=0,𝑅₁=0 in Lyapunov functionals (3.13) and (3.31), all the above results obtained in this paper can easily extend to delay-dependent stability results for systems (2.1) and (3.30) with 𝑑₁=0. As for example, we can obtain a delay-dependent stability criterion using similar method of Theorem 3.1 for system (2.1) with 𝑑₁=0, and 𝑑(𝑡)=𝑑 is a constant time delay. The obtained stability criterion for this case is shown in the following Corollary 3.8.

Corollary 3.8. Given scalar 𝑑₂>0, for any constant time delay 𝑑(𝑡)=𝑑 satisfying 0≤𝑑≤𝑑₂, singular time-delay system (2.1) is regular, impulse free, and stable if there exist matrices 𝑃>0, 𝑄₂=^𝑄21𝑄22_{𝑄𝑇22𝑄23}≥0, 𝑄₃≥0, 𝑄₄≥0, 𝑅₂≥0, 𝑅₃≥0, 𝑍₁, 𝑍₂ and 𝑆, such that LMIs (3.2),(3.3), and following LMI (3.37) hold 𝑒₁𝐸^𝑇𝑃𝑒^𝑇₆+𝑒₆𝑃𝐸𝑒^𝑇₁+𝑒₁𝑒₄𝑄₂𝑒₁𝑒₄^𝑇−𝑒₄𝑒₅𝑄₂𝑒₄𝑒₅^𝑇+𝑒₁𝑄₃+𝑄₄𝑒^T₁−𝑒₂𝑄₃𝑒^𝑇₂−𝑒₄𝑄₄𝑒^𝑇₄+𝑒₆𝑅𝑒^𝑇₆+𝑒₁𝑆Ψ^𝑇𝑒^𝑇₆+𝑒₆Ψ𝑆^𝑇𝑒^𝑇₁−𝑒^𝑇₂−𝑒^𝑇₄𝑒^𝑇₁−𝑒^𝑇₂^𝑇𝐸^𝑇𝑅₂𝐸𝐸^𝑇𝑍₁𝐸𝐸^𝑇𝑍^𝑇₁𝐸𝐸^𝑇𝑅₂𝐸𝑒^𝑇₂−𝑒^𝑇₄𝑒^𝑇₁−𝑒^𝑇₂−𝑒^𝑇₃−𝑒^𝑇₅𝑒^𝑇₁−𝑒^𝑇₃^𝑇𝐸^𝑇𝑅₃𝐸𝐸^𝑇𝑍₂𝐸𝐸^𝑇𝑍^𝑇₂𝐸𝐸^𝑇𝑅₃𝐸𝑒^𝑇₃−𝑒^𝑇₅𝑒^𝑇₁−𝑒^𝑇₃<0, (3.37) where 𝑅=(𝑑₂/2)²𝑅₂+𝑑²₂𝑅₃, Ψ∈𝐑^{𝑛×(𝑛−𝑟)} is any full-column rank matrix satisfying 𝐸^TΨ=0, and 𝑒_𝑖(𝑖=1,2,…,5) are block entry matrices, for example, 𝑒₂=[0𝐼000]^𝑇, and 𝑒₆=(𝐴𝑒^𝑇₁+𝐵𝑒^𝑇₃)^𝑇.

Remark 3.9. It is worth pointing out that the obtained results in this paper are formulated in terms of LMIs, they can be easily solved using any LMI toolbox like one of Matlab or the one of Scilab.

4. Numerical Examples

In this section, we use three examples and compare our results with the previous ones to show the effectiveness of ours.

Example 4.1. Consider the singular time-delay system (2.1) with 𝐸=1000,𝐴=0.500−1,𝐵=−1.1100.5. (4.1)
For various 𝜇, the allowable upper bounds 𝑑₂, which guarantee regular, impulse free, and stable of system (2.1) for given lower bounds 𝑑₁, are listed in Table 1. From Table 1, it can be seen that the stability criterion in Theorem 3.1 is less conservative than that in [35]. Especially, when 𝑑₁=1.1, the result in [35] is not feasible while the allowable upper bounds 𝑑₂ can also be obtained from Theorem 3.1 in this paper.

Table 1 

Allowable upper bound 𝑑₂ with given 𝑑₁ for different 𝜇.

Example 4.2. Consider the time-delay system (3.30) with 𝐴=−200−0.9,𝐵=−10−1−1. (4.2)
For various 𝜇, the allowable upper bounds 𝑑₂, which guarantee the stability of system (3.30) for given lower bounds 𝑑₁, are listed in Table 2. Moreover, the number of decision variables involved in the stability criteria are given in Table 3. From Tables 2 and 3, it can be seen that Corollary 3.3 in this paper has fewer decision variables and less conservatism than those results in [5, 13, 14].

Table 2 

Allowable upper bound 𝑑₂ with given 𝑑₁ for different 𝜇.

Table 3 

Number of decision variables.

For unknown 𝜇, the allowable upper bounds 𝑑₂, which guarantee the stability of system (3.30) for given lower bounds 𝑑₁, are listed in Table 4. From Table 4, it can be seen that Corollary 3.6 in this paper give larger upper bounds of time delay than ones in [4, 5, 8–11, 13, 14].

Table 4 

Allowable upper bound 𝑑₂ with given 𝑑₁ for unknown 𝜇.

Example 4.3. Consider the singular time-delay system (2.1) with 𝐸=1000,𝐴=−0.30120.12570.2351−1.0998,𝐵=−0.5𝑐00−0.1𝑐, (4.3) where 𝑐 is a scalar. It is assumed that 𝑑₁=0 and 𝑑(𝑡)=𝑑 are constant delays satisfying 0≤𝑑≤𝑑₂. For various 𝑐, the allowable upper bounds 𝑑₂, which guarantee regular, impulse free, and stable of system (2.1), are listed in Table 5. From Table 5, it is clear that the Corollary 3.8 in this paper gives better results than those in [18, 22–24, 26].

Table 5 

Allowable upper bound 𝑑₂ for different 𝑐.

5. Conclusion

This paper deals with the problem of stability for singular systems with interval time-varying delay. A new stability criterion for singular systems to be regular, impulse free, and stable is 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. Three numerical examples are given to illustrate the applicability of the results.

Acknowledgments

The author is grateful to the academic editor and four anonymous referees whose comments have helped to improve the final version of the present work. This work was supported by the scientific research program funded by Shaanxi provincial education department (nos. 2010JK400 and 11JK0509) and the scientific research project funded by Baoji University of Arts and Sciences (no. 11044).