Novel Stabilization Conditions for Uncertain Singular Systems with Time-Varying Delay

Abstract

The problem of delay-dependent robust stabilization for continuously singular time-varying delay systems with norm-bounded uncertainties is investigated in this paper. First, based on some mathematical transform, the uncertain singular system is described in a form which involves the time-delay integral items. Then, in terms of the delay-range-dependent Lyapunov functional and the LMI technique, the improved delay-dependent LMIs-based conditions are established for the uncertain singular systems with time-varying delay to be regular, causal, and stable. Furthermore, by solving these LMIs, an explicit expression for the desired state feedback control law can be obtained; thus, the regularity, causality, and stability of the closed-loop system are guaranteed. In the end, numerical examples are given to illustrate the effectiveness of the proposed methods.

1. Introduction

During the past several decades, singular systems, which are known as descriptor systems, implicit systems, and differential-algebraic systems, received a considerable attention because of their applications in many areas, such as engineering systems, social systems, economic systems, network analysis, biological systems, and time-series analysis [1, 2]. As we all know, it is required to consider not only stability but also regularity and absence of impulses (for continuous singular systems) or causality (for discrete singular systems) simultaneously for singular systems; thus, the study on singular systems is much more complicated than that on the regular ones [3, 4]. Recently, many scholars have applied themselves to the research of singular system and many stability and stabilization conditions have been established for singular systems; see, for example, [5–11] and the references therein.

On the other hand, much attention has been paid to the study of time-delay systems in recent years, because time delays inevitably exist in a variety of practical systems, such as chemical processes, nuclear reactors, and biological systems, and lead to the instability and poor performance of systems [12–15]. Generally speaking, the existing results can be classified into two types: delay-independent results (see, e.g., [16, 17] and the references therein) and delay-dependent results (see, e.g., [18, 19] and the references therein). Furthermore, the delay-independent case is regarded as more conservative than the delay-dependent case, especially when the time delay is comparatively small. Thus, the delay-dependent stability and stabilization conditions for singular time-delay systems have received increasing attention during the past years. For example, by utilizing model transformation and bounding technique for cross-terms, Zhu et al. [20, 21] investigated the delay-dependent robust stabilization problem for uncertain singular time-delay systems. References [22, 23] also discussed the problem, and neither model transformation nor bounding technique for cross-terms is needed in the development of the results. Based on an improved Lyapunov functional, which includes some nonpositive items, Weng and Mao [24] discussed the delay-dependent robust stability and stabilization for uncertain singular time-delay systems, and some LMIs-based results were obtained. However, in the practical systems, most of those delays are time-varying because the external perturbances and uncertainties are always existing [25]. Thus, proposing some time-varying delay-tolerant results for the singular system is obviously more meaningful. In terms of Lyapunov stability theory and LMI technique, some results about the admissibility and dissipativity for discrete-time singular systems with mixed time-varying delays were proposed in [26]. Based on the probability idea, Weng and Mao [27] presented some delay-range-dependent and delay-distribution-independent stability criteria for discrete-time singular systems with time-varying delay, and several sufficient results were obtained. However, to the best of our knowledge, the stabilization conditions for singular time-varying delay systems still have not been fully investigated, and there is still much room for improvement.

This paper is concerned with the problem of robust stabilization for continuously singular time-varying delay systems with norm-bounded and time-varying parametric uncertainties. The focus of this paper is to design a state feedback controller such that the closed-loop system is regular, causal, and stable for all admissible uncertainties. The proposed sufficient robust stabilization conditions of the considered system are described in terms of strict LMIs, which are formulated in terms of all the coefficient matrices of the original system. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed methods.

Notation 1. Throughout this paper, for real matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is semipositive definite (resp., positive definite). 𝐼 is the identity matrix with appropriate dimensions; a superscript “𝑇” represents transpose. ‖𝑥‖ refers to the Euclidean norm of the vector 𝑥. For a symmetric matrix, ∗ denotes the symmetric terms. 𝑀+𝑀^𝑇 is denoted as {𝑀}^𝐻 for simplicity.

2. Problem Formulation and Dynamic Models

Consider an uncertain singular system with time-varying delay described by Wu and Zhou [28]: 𝐸𝑥̇_(𝑡)=(𝐴+Δ𝐴)𝑥_(𝑡)+𝐴_𝑑+Δ𝐴_𝑑𝑥_{(𝑡−𝑑(𝑡))}(𝑡),𝑥+𝐵𝑢_(𝑡)=Φ_(𝑡),𝑡∈−𝑑₂,0, (1) where 𝑥_(𝑡)∈𝑅^𝑛 is the state variable and 𝑑_(𝑡) is a time-varying delay satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂ and 𝑑̇_(𝑡)≤𝜇. Φ_(𝑡) is a compatible initial value at 𝑡. The matrix 𝐸∈𝑅^𝑛×𝑛 may be singular and rank𝐸≤𝑛 is assumed.𝐴, 𝐴_𝑑, and 𝐵 are real constant matrices with appropriate dimensions. Δ𝐴 andΔ𝐴_𝑑 are norm-bounded parametric matrices and are assumed to be of the following form: Δ𝐴Δ𝐴_𝑑=𝑀𝐹_(𝑡)𝑁₁𝑁₂, (2) where 𝐹_(𝑡)∈𝑅^𝑘×𝑠 is an unknown parameter matrix satisfying 𝐹^𝑇_(𝑡)𝐹_(𝑡)≤𝐼.  𝑀, 𝑁₁, and 𝑁₂ are known constant matrices with appropriate dimensions.

The nominal unforced singular time-delay system of (1) can be described as 𝐸𝑥̇_(𝑡)=𝐴𝑥_(𝑡)+𝐴_𝑑𝑥_{(𝑡−𝑑(𝑡))},𝑥_(𝑡)=Φ_(𝑡),𝑡∈−𝑑₂,0. (3)

The following definitions and lemmas will be used in the proof of the main results.

Definition 1 (see [29]). (i) The pair(𝐸,𝐴)is regular if det(𝑆𝐸−𝐴)is not identically zero; (ii) the pair(𝐸,𝐴)is said to be impulse free if it is regular and deg{det(𝑆𝐸−𝐴)}=rank𝐸.

Definition 2 (see [29]). (i) The system (3) is said to be regular and impulse free if 𝑑_(𝑡) satisfies 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂; the pair (𝐸,𝐴) is regular and impulse free. (ii) The system (3) is said to be stable if, for any 𝜀>0, there exists a scalar 𝜇(𝜀), such that, for any compatible initial conditions sup_{−𝑑2≤𝑡≤0}‖Φ_(𝑡)‖²<𝜇(𝜀), when 𝑡>0, the solution 𝑥(𝑡) of the system (3) satisfies ‖𝑥(𝑡)‖≤𝜀. Furthermore, lim_𝑡→0𝑥(𝑡)=0. (iii) The system (3) is said to be admissible if it is regular, impulse free, and stable.

Definition 3. (i) The uncertain singular time-varying delay system (1) is said to be robustly admissible if the system (1) with 𝑢(𝑡)=0 is regular, impulse free, and stable for all admissible uncertainties satisfying (2) and any time-delay 𝑑_(𝑡) satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂. (ii) The singular time-varying delay system (1) is said to be stabilizable if there exists state feedback controller𝑢(𝑡)=𝐾𝑥(𝑡) such that the closed-loop system is admissible for any time delay 𝑑_(𝑡) satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂. (iii) The uncertain singular time-varying delay system (1) is said to be robustly stabilizable if there exists state feedback controller 𝑢(𝑡)=𝐾𝑥(𝑡) such that the closed-loop system is robustly admissible for all admissible uncertainties satisfying (2) and any time-delay 𝑑_(𝑡) satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂.
After some mathematical transform, the systems (1) and (3) can be described in the following forms: 𝐸𝑥̇_(𝑡)=𝐴+Δ𝐴+𝐴_𝑑+Δ𝐴_𝑑𝑥_(𝑡)−𝐴_𝑑+Δ𝐴_𝑑^𝑚−1_𝑖=0^{𝑡−(𝑖/𝑚)𝑑1}_{()𝑑𝑡−(𝑖+1)/𝑚1}𝑥̇_(𝑠)−𝐴𝑑𝑠_𝑑+Δ𝐴_𝑑^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇_(𝑠)𝑥𝑑𝑠,_(𝑡)=Φ_(𝑡),𝑡∈−𝑑₂,0,𝐸𝑥̇_(𝑡)=𝐴+𝐴_𝑑𝑥_(𝑡)−𝐴_𝑑^𝑚−1_𝑖=0^{𝑡−(𝑖/𝑚)𝑑1}_{()𝑑𝑡−(𝑖+1)/𝑚1}𝑥̇_(𝑠)𝑑𝑠−𝐴_𝑑^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇_(𝑠)𝑥𝑑𝑠,_(𝑡)=Φ_(𝑡),𝑡∈−𝑑₂,0. (4)(5) For description in brevity, we define 𝛾=𝑑₁/𝑚  and  𝑑₁₂=𝑑₂−𝑑₁. This section is concluded by presenting a lemma, which will be used in the proof of our main results.

Lemma 4 (see [30, 31]). Given matrices𝜒, 𝜇, and 𝜈 with appropriate dimensions and with𝜒symmetrical, then 𝜒+𝜇𝐹_(𝑡)𝜈+𝜈^𝑇𝐹^𝑇_(𝑡)𝜇^𝑇<0 holds for any 𝐹_(𝑡)∈𝑅^𝑘×𝑠 satisfying 𝐹^𝑇_(𝑡)𝐹_(𝑡)≤𝐼, if and only if there exists a scalar 𝜎>0such that 𝜒+𝜎𝜇𝜇^𝑇+𝜎⁻¹𝜈^𝑇𝜈<0.

3. Main Results

In this section, the delay-dependent conditions for system (1) to be stabilizable and robustly stabilizable are presented. As a basis, we first study system (5) and obtain the following Theorem 5.

Theorem 5. For the prescribed scalars satisfying 0<𝑑₁≤𝑑₂, the singular time-delay system (5) is admissible for any time-delay 𝑑_(𝑡) satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂ if there exist positive symmetric matrices 𝑃, 𝑈_𝑖(𝑖=1,2,3,4), 𝑄₁, 𝑄₂, matrices 𝐻_𝑖(𝑖=1,2), 𝑆_𝑖(𝑖=1,2,3,4,5,6,7), 𝐺_𝑖(𝑖=1,2,3,4,5,6,7,8), and 𝐿_𝑖(𝑖=1,2,3,4,5) satisfying the following LMIs:Ψ_𝑖=Ψ₁₁₁Ψ₁₁₂𝐺₁−𝐺₁Ψ₁₁₅Ψ₁₁₆−𝐿₁+𝐿^𝑇₂𝐿^𝑇₅∗Ψ_𝑖22𝐺₇−𝐺₇Ψ₁₂₅Ψ₁₂₆−𝐿^𝑇₂−𝐿₄−𝐿^𝑇₅∗∗Ψ₁₃₃Ψ₁₃₄Ψ₁₃₅Ψ₁₃₆𝐺^𝑇₂𝐺^𝑇₈∗∗∗Ψ₄₄Ψ₁₄₅Ψ₁₄₆−𝐺^𝑇₂−𝐺^𝑇₈∗∗∗∗Ψ₁₅₅Ψ₁₅₆𝐿𝑆^𝑇₂−𝐿₃𝐿𝑆^𝑇₇∗∗∗∗∗Ψ₁₆₆−𝐺^𝑇₂−𝐿^𝑇₂Ψ₁₆₈∗∗∗∗∗∗Ψ_𝑖77Ψ_𝑖78∗∗∗∗∗∗∗Ψ_𝑖88<0,𝑖=1,2, (6)where 𝐿∈𝑅^{𝑛×(𝑛−𝑟)} is any matrix with full columns and satisfies 𝐸^𝑇𝐿=0 and Ψ₁₁₁=𝑈₁+𝑈₂+𝑈₃+𝑈₄+𝐿₁+𝐻₁𝐴+𝐴_𝑑^𝐻,Ψ₁₁₂=−𝐻₁𝐴_𝑑−𝐿₁+𝐿^𝑇₄,Ψ₁₁₆=−𝑚𝐻₁𝐴_𝑑−𝐺₁−𝐿₁,Ψ₁₁₅=𝐸^𝑇𝑃−𝐻₁+𝐻₂𝐴+𝐴_𝑑^𝑇+𝑆₁𝐿^𝑇+𝐿^𝑇₃,Ψ₁₂₂=−3𝐸^𝑇𝑄₂𝐸−𝐿₄−𝐿^𝑇₄,Ψ₁₂₆=−𝐺₇−𝐿₄,Ψ₁₂₅=−𝐻₂𝐴_𝑑^𝑇+𝑆₆𝐿^𝑇−𝐿^𝑇₃,Ψ₁₃₃=−𝑈₁+𝐺₃^𝐻,Ψ₁₃₄=−𝐺₃+𝐺^𝑇₄,Ψ₁₃₅=𝑆₃𝐿^𝑇+𝐺^𝑇₅,Ψ₁₃₆=−𝐺₃+𝐺^𝑇₆,Ψ₁₄₄=−𝑈₂−𝐺₄^𝐻,Ψ₁₄₅=𝑆₄𝐿^𝑇−𝐺^𝑇₅,Ψ₁₄₆=−𝐺₄−𝐺^𝑇₆,Ψ₁₅₅=𝛾²𝑄₁+𝑑²₁₂𝑄₂−𝐻₂^𝐻,Ψ₁₅₆=−𝑚𝐻₂𝐴_𝑑+𝐿𝑆^𝑇₅−𝐺₅−𝐿₃,Ψ₁₆₆=−𝐸^𝑇𝑄₁𝐸−𝐺₆^𝐻,Ψ₁₆₈=−𝐺^𝑇₈−𝐿^𝑇₅,Ψ₁₇₈=−𝐿^𝑇₅+𝐸^𝑇𝑄₂Ψ𝐸,₁₇₇=−1−𝜇𝑈₃−𝐿₂^𝐻−𝐸^𝑇𝑄₂Ψ𝐸,₁₈₈=−𝑈₄−𝐸^𝑇𝑄₂Ψ𝐸,₂₂₂=−𝐸^𝑇𝑄₂𝐸−𝐿₄^𝐻,Ψ₂₇₇=−1−𝜇𝑈₃−𝐿₂^𝐻−3𝐸^𝑇𝑄₂Ψ𝐸,₂₇₈=−𝐿^𝑇₅+3𝐸^𝑇𝑄₂Ψ𝐸,₂₈₈=−𝑈₄−3𝐸^𝑇𝑄₂𝐸. (7)

Proof. Under the conditions of Theorem 5, it is first shown that the system (5) is regular and impulse free for any time-delay 𝑑_(𝑡) satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂. Define 𝐻=𝐼_𝑛𝐼_𝑛00𝐴^𝑇000. Then, by pre- and postmultiplying (6) by 𝐻 and 𝐻^𝑇, respectively, it is possible to obtain −𝐸^𝑇𝑄₂𝐸+𝐴^𝑇𝑃𝐸+𝐸^𝑇𝑃𝐴+𝐴^𝑇𝐿𝑆₁+𝑆₆^𝑇+𝑆₁+𝑆₆𝐿^𝑇𝐴<0. (8) Since rank𝐸=𝑟≤𝑛, there exist two nonsingular matrices 𝑀 and 𝑁 such that 𝑀𝐸𝑁=^𝐼𝑟0₀₀. Accordingly, denote 𝑀𝐴𝑁=^𝐴1𝐴2_𝐴3𝐴4, 𝑁^𝑇(𝑆₁+𝑆₆)=^𝑆11_𝑆12, 𝑀^−𝑇𝐿=⁰_𝐿1, where 𝐿₁∈𝑅^{(𝑛−𝑟)×(𝑛−𝑟)} is any nonsingular matrix and 𝐼_𝑟∈𝑅^𝑟×𝑟 is  an identity matrix. Then, by pre- and postmultiplying inequalities (8) by 𝑁^𝑇and 𝑁, respectively, it is possible to obtain ⋄⋄∗𝑆₁₂𝐿^𝑇₁𝐴₄+𝐴^𝑇₄𝐿₁𝑆^𝑇₁₂<0. (9) Here, “⋄” representing the matrix blocks are irrelevant to the following discussion; the real expression of these two variables is omitted here. From (9), it is possible to obtain that 𝑆₁₂𝐿^𝑇₁𝐴₄+𝐴^𝑇₄𝐿₁𝑆^𝑇₁₂<0. (10) It can be shown that 𝐴₄ is nonsingular. Thus, the pair (𝐸,𝐴) is regular and impulse free [29]; that is to say, system (5) is regular and impulse free.
Then, we are in a position to show that system (5) is stable under the conditions of Theorem 5. Choose a Lyapunov-Krasovskii functional candidate as 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (11) where 𝑉₁(𝑡)=𝑥^𝑇_(𝑡)𝐸^𝑇𝑃𝐸𝑥_(𝑡)+^𝑚−1_𝑖=0^𝑡_{𝑡−𝑖𝛾}𝑥^𝑇_(𝑠)𝑈₁𝑥_(𝑠)+𝑑𝑠_{𝑚−1𝑖=0𝑡()𝛾𝑡−𝑖+1}𝑥^𝑇_(𝑠)𝑈₂𝑥_(𝑠)𝑑𝑠+^𝑡_{𝑡−𝑑𝑡}𝑥^𝑇_(𝑠)𝑈₃𝑥_(𝑠)+𝑑𝑠_{𝑡𝑡−𝑑2}𝑥^𝑇_(𝑠)𝑈₄𝑥_(𝑠)𝑉𝑑𝑠,₂(𝑡)=𝛾^𝑚−1_𝑖=0⁰_−𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖𝛾+𝜀}𝑥^𝑇_(𝑠)𝑄₁𝑥_(𝑠)𝑉𝑑𝑠𝑑𝜀,₃(𝑡)=𝑑₁₂^−𝑑1_−𝑑2^𝑡_𝑡+𝜀𝑥̇^𝑇_(𝑠)𝐸^𝑇𝑄₂𝐸𝑥̇_(𝑠)𝑑𝑠𝑑𝜀. (12) The derivative of 𝑉(𝑡) along the trajectories of (5) satisfies 𝑉̇(𝑡)=𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡), (13) where 𝑉̇₁(𝑡)=𝑥̇^𝑇_(𝑡)𝐸^𝑇𝑃𝐸𝑥_(𝑡)+𝑚𝑥^𝑇_(𝑡)𝑈₁𝑥_(𝑡)−^𝑚−1_𝑖=0𝑥^𝑇_{𝑡−𝑖𝛾}𝑈₁𝑥_{(𝑡−𝑖𝛾)}+𝑚𝑥^𝑇_(𝑡)𝑈₂𝑥_(𝑡)−^𝑚−1_𝑖=0𝑥^𝑇_{()𝛾𝑡−𝑖+1}𝑈₂𝑥_{(𝑡−(𝑖+1)𝛾)}+𝑥^𝑇_(𝑡)𝑈₃𝑥_(𝑡)−1−𝜇𝑥^𝑇_{((𝑡𝑡−𝑑))}𝑈₃𝑥_{(𝑡−𝑑(𝑡))}+𝑥^𝑇_(𝑡)𝑈₄𝑥_(𝑡)−𝑥^𝑇_𝑡−𝑑2𝑈₄𝑥_{(𝑡−𝑑2)},𝑉̇₂(𝑡)≤𝛾²𝑥̇^𝑇_(𝑡)𝐸^𝑇𝑄₁𝐸𝑥̇_(𝑡)−^𝑚−1_𝑖=0()𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖+1}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₁𝐸_()𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖+1}𝑥̇_(𝑠)𝑉̇𝑑𝑠,₃(𝑡)=𝑑²₁₂𝑥̇^𝑇_(𝑡)𝐸^𝑇𝑄₂𝐸𝑥̇_(𝑡)−𝑑₁₂^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝐸^𝑇𝑄₂𝐸𝑥̇_(𝑠)𝑑𝑠−𝑑₁₂^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝐸^𝑇𝑄₂𝐸𝑥̇_(𝑠)𝑑𝑠≤𝑑²₁₂𝑥̇^𝑇_(𝑡)𝐸^𝑇𝑄₂𝐸𝑥̇_(𝑡)−𝑑₁₂𝑑_(𝑡)−𝑑₁^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)−𝑑𝑑𝑠₁₂𝑑₂−𝑑_(𝑡)^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑𝑡}_𝑡−𝑑2𝑥̇_(𝑠)𝑑𝑠. (14) We have −𝑑₁₂/(𝑑_(𝑡)−𝑑₁)−𝑑₁₂/(𝑑₂−𝑑_(𝑡))≤4; thus, when 𝑑₁≤𝑑_(𝑡)≤(𝑑₁+𝑑₂)/2, it is easy to obtain −𝑑₁₂𝑑_(𝑡)−𝑑₁^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)≤𝑑𝑑𝑠−4+¹²𝑑₂−𝑑_(𝑡)^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)𝑑𝑠. (15) By defining 𝜃₁(𝑡)=2−𝑑₁₂/(𝑑₂−𝑑_(𝑡)) and according to 𝑑₁≤𝑑_(𝑡)≤(𝑑₁+𝑑₂)/2, we have 0≤𝜃₁(𝑡)≤1. Thus, based on the convex theory, it is easy to obtain that 𝑑−4+¹²𝑑₂−𝑑_(𝑡)^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)−𝑑𝑑𝑠₁₂𝑑₂−𝑑_(𝑡)^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)=1𝑑𝑠2+𝜃₁(𝑡)2×−3^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)−𝑑𝑠_{𝑡−𝑑(𝑡)𝑡−𝑑2}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)+1𝑑𝑠1−2+𝜃₁(𝑡)2×−^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)𝑑𝑠−3^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)𝑑𝑠. (16) When (𝑑₁+𝑑₂)/2≤𝑑_(𝑡)≤𝑑₂, we have −𝑑₁₂𝑑₂−𝑑_(𝑡)^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)≤𝑑𝑑𝑠−4+¹²𝑑_(𝑡)−𝑑₁^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)𝑑𝑠. (17) By defining 𝜃₂(𝑡)=2−𝑑₁₂/(𝑑_(𝑡)−𝑑₁) and according to (𝑑₁+𝑑₂)/2≤𝑑_(𝑡)≤𝑑₂, we have 0≤𝜃₂(𝑡)≤1. Based on convex theory, we achieve −𝑑₁₂𝑑_(𝑡)−𝑑₁^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)+𝑑𝑑𝑠−4+¹²𝑑_(𝑡)−𝑑₁^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)=1𝑑𝑠2+𝜃₂(𝑡)2×−^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)𝑑𝑠−3^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)+1𝑑𝑠1−2+𝜃₂(𝑡)2×−3^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)−𝑑𝑠_{𝑡−𝑑(𝑡)𝑡−𝑑2}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)𝑑𝑠. (18) It is easy to obtain 0≤(1+𝜃₁(𝑡))/2≤1 and 0≤(1+𝜃₂(𝑡))/2≤1. Then, based on the analysis mentioned above, we can obtain that when 𝜃(𝑡) satisfies 0≤𝜃(𝑡)≤1, there exists −𝑑₁₂𝑑_(𝑡)−𝑑₁^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)−𝑑𝑑𝑠₁₂𝑑₂−𝑑_(𝑡)^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)(𝑡)𝑑𝑠≤𝜃−3^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)−𝑑𝑠_{𝑡−𝑑(𝑡)𝑡−𝑑2}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)+((𝑡×𝑑𝑠1−𝜃))−^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇_(𝑠)𝑑𝑠−3^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇^𝑇_(𝑠)𝑑𝑠𝐸^𝑇𝑄₂𝐸^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑥̇_(𝑠)𝑑𝑠. (19) By considering (5), it is obvious that 𝑥^𝑇_(𝑡)𝐻₁+𝑥̇^𝑇_(𝑡)𝐸^𝑇𝐻₂×^𝑚−1_𝑖=0𝐴+𝐴_𝑑𝑥_(𝑡)−𝑚𝐴_𝑑^{𝑡−(𝑖/𝑚)𝑑1}_{()𝑑𝑡−(𝑖+1)/𝑚1}𝑥̇_(𝑠)𝑑𝑠−𝐴_𝑑^𝑡−𝑑1_{(𝑡)𝑡−𝑑}𝑥̇_(𝑠)𝑑𝑠−𝐸𝑥̇_(𝑡)=0. (20) Noting 𝐸^𝑇𝐿=0, we obtain 2^𝑚−1_𝑖=0𝑥^𝑇_(𝑡)𝑆₁+𝑥^𝑇_{(𝑡)𝑡−𝑑}𝑆₂+𝑥^𝑇_{𝑡−𝑖𝛾}𝑆₃+𝑥^𝑇_{()𝛾𝑡−𝑖+1}𝑆₄+_()𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖+1}𝑥̇^𝑇_(𝑠)𝑑𝑠𝑆₅+^𝑡−𝑑1_{𝑡−𝑑𝑡}𝑥̇^𝑇_(𝑠)𝑑𝑠𝑆₆+𝑥^𝑇_𝑡−𝑑2𝑆₇𝐿^𝑇𝐸𝑥̇_(𝑡)=0. (21) Furthermore, according to the free-weighting-matrix method, we have 𝑥^𝑇_(𝑡)𝐺₁+𝑥^𝑇_{(𝑡)𝑡−𝑑}𝐺₂+𝑥^𝑇_{𝑡−𝑖𝛾}𝐺₃+𝑥^𝑇_{()𝛾𝑡−𝑖+1}𝐺₄+𝑥̇^𝑇_(𝑡)𝐸^𝑇𝐺₅+_()𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖+1}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐺₆+^𝑡−𝑑1_{𝑡−𝑑𝑡}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐺₇+𝑥^𝑇_𝑡−𝑑2𝐺₈×𝑥_{(𝑡−𝑖𝛾)}−𝑥_{(𝑡−(𝑖+1)𝛾)}−_()𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖+1}𝑥̇_(𝑠)𝑥𝑑𝑠=0,_𝑇(𝑡)𝐿₁+𝑥^𝑇_{(𝑡)𝑡−𝑑}𝐿₂+𝑥̇^𝑇_(𝑡)𝐸^𝑇𝐿₃+^𝑡−𝑑1_{𝑡−𝑑𝑡}𝑥̇^𝑇_(𝑠)𝑑𝑠𝐿₄+𝑥^𝑇_𝑡−𝑑2𝐿₅×𝑥_(𝑡)−𝑥_{(𝑡−𝑑𝑡)}−^𝑡−𝑑1_{𝑡−𝑑𝑡}𝑥̇_(𝑠)𝑑𝑠−^𝑚−1_𝑖=0()𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖+1}𝑥̇_(𝑠)𝑑𝑠=0. (22) Then, combining manipulations (13)–(22) yields 𝑉̇(𝑡)≤^𝑚−1_𝑖=0𝜃(𝑡)𝜉^𝑇_(𝑡)Ψ₁𝜉_(𝑡)+(1−𝜃(𝑡))𝜉^𝑇_(𝑡)Ψ₂𝜉_(𝑡), (23) where𝜉_(𝑡)=𝑥^𝑇_(𝑡)^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑥̇^𝑇_(𝑠)𝑑𝑠𝑥^𝑇_{𝑡−𝑖𝛾}𝑥^𝑇_{()𝛾𝑡−𝑖+1}𝑥̇^𝑇_(𝑡)𝐸^𝑇_()𝛾^{𝑡−𝑖𝛾}_{𝑡−𝑖+1}𝑥̇^𝑇_(𝑠)𝑑𝑠𝑥^𝑇_{(𝑡−𝑑𝑡)}𝑥^𝑇_{(𝑡−𝑑2)}^𝑇. (24)Then, we can obtain 𝑉̇(𝑡)<0 from (6). Thus, we can deduce that 𝑉̇(𝑡)≤−𝜆₂𝜁_(𝑡)²≤−𝜆₂𝑥_(𝑡)², (25) where 𝜆₂=−max{𝜆_max(Ψ₁),𝜆_max(Ψ₂)}>0. Therefore, the system (5) is stable based on Definition (2). This completes the proof.

Theorem 6. For the prescribed scalars satisfying 0<𝑑₁≤𝑑₂, the singular system (1) is stabilizable for any time-delay 𝑑_(𝑡) satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂ if there exist positive symmetric matrices 𝑃, 𝑈_𝑖(𝑖=1,2,3,4), 𝑄₁, 𝑄₂ matrices 𝐻₁=𝑍^−1𝐻1110_{𝐻121𝐻22}𝑍, 𝐻₂=𝛽𝑍^{−1𝛽−1𝐻2110}_{𝛽−1𝐻221𝐻22}𝑍, 𝑆_𝑖(𝑖=1,2,3,4,5,6,7), 𝐺_𝑖(𝑖=1,2,3,4,5,6,7,8), 𝐿_𝑖(𝑖=1,2,3,4,5), 𝑇∈𝑅^𝑝×𝑛, and scalar 𝛽 satisfying the following LMIs:Ψ_𝑖=Ψ₁₁₁Ψ₁₁₂𝐺₁−𝐺₁Ψ₁₁₅Ψ₁₁₆−𝐿₁+𝐿^𝑇₂𝐿^𝑇₅∗Ψ_𝑖22𝐺₇−𝐺₇Ψ₁₂₅Ψ₁₂₆−𝐿^𝑇₂−𝐿₄−𝐿^𝑇₅∗∗Ψ₁₃₃Ψ₁₃₄Ψ₁₃₅Ψ₁₃₆𝐺^𝑇₂Ψ₁₃₈∗∗∗Ψ₄₄Ψ₁₄₅Ψ₁₄₆−𝐺^𝑇₂−𝐺^𝑇₈∗∗∗∗Ψ₁₅₅Ψ₁₅₆𝐿𝑆^𝑇₂−𝐿₃𝐿𝑆^𝑇₇∗∗∗∗∗Ψ₁₆₆−𝐺^𝑇₂−𝐿^𝑇₂Ψ₁₆₈∗∗∗∗∗∗Ψ_𝑖77Ψ_𝑖78∗∗∗∗∗∗∗Ψ_𝑖88<0,𝑖=1,2, (26)where 𝐿∈𝑅^{𝑛×(𝑛−𝑟)}  is any matrix with full columns and satisfies 𝐸^𝑇𝐿=0, 𝐻₂₂∈𝑅^𝑝×𝑝 is any nonsingular matrix, 𝑍∈𝑅^𝑛×𝑛is any nonsingular constant matrix satisfying 𝑍𝐵=0𝐵^𝑇₁^𝑇, where 𝐵₁∈𝑅^𝑝×𝑝 is nonsingular, and Ψ₁₁₁=𝑈₁+𝑈₂+𝑈₃+𝑈₄+{𝐿₁+𝐻₁(𝐴+𝐴_𝑑)+𝐵𝑇}^𝐻, Ψ₁₁₅=(𝐻₂(𝐴+𝐴_𝑑)+𝛽𝐵𝑇)^𝑇+𝐸^𝑇𝑃−𝐻₁+𝑆₁𝐿^𝑇+𝐿^𝑇₃. Then, a suitable state feedback controller is described as 𝐾=(𝐻₂₂𝐵₁)⁻¹𝐵₁𝑇.

Proof. By replacing 𝐴 with 𝐴+𝐵𝐾 and choosing 𝐾=(𝐻₂₂𝐵₁)⁻¹𝐵₁𝑇, 𝐻₁=𝑍^−1𝐻1110_{𝐻121𝐻22}𝑍, and 𝐻₂=𝛽𝑍^{−1𝛽−1𝐻2110}_{𝛽−1𝐻221𝐻22}𝑍, it is easy to obtain 𝐻₁𝐵𝐾=𝐵𝑇 and 𝐻₂𝐵𝐾=𝛽𝐵𝑇. Thus, we can obtain (6) from LMIs (26). This completes the proof.

Theorem 7. For the prescribed scalars satisfying 0<𝑑₁≤𝑑₂, the singular system (1) is robustly stabilizable for any time-delay 𝑑_(𝑡) satisfying 0<𝑑₁≤𝑑_(𝑡)≤𝑑₂ if there exist positive symmetric matrices 𝑃, 𝑈_𝑖(𝑖=1,2,3,4), 𝑄₁, 𝑄₂, matrices 𝐻₁=𝑍^−1𝐻1110_{𝐻121𝐻22}𝑍, 𝐻₂=𝛽𝑍^{−1𝛽−1𝐻2110}_{𝛽−1𝐻221𝐻22}𝑍, 𝑆_𝑖(𝑖=1,2,3,4,5,6,7), 𝐺_𝑖(𝑖=1,2,3,4,5,6,7,8), 𝐿_𝑖(𝑖=1,2,3,4,5), 𝑇∈𝑅^𝑝×𝑛, and scalars 𝛽, 𝜆>0 satisfying the following LMIs: Ψ_𝑖Π^𝑇₂𝜆Λ^𝑇₂∗−𝜆𝐼0∗∗−𝜆𝐼<0,𝑖=1,2, (27) where Ψ₁,Ψ₂, and 𝐿 follow the same definitions as in Theorem 6 and 𝐻₂₂∈𝑅^𝑝×𝑝 is any nonsingular matrix, 𝑍∈𝑅^𝑛×𝑛 is any nonsingular constant matrix satisfying 𝑍𝐵=0𝐵^𝑇₁^𝑇, where 𝐵₁∈𝑅^𝑝×𝑝 is nonsingular, and Π₂=𝑀^𝑇𝐻^𝑇₁000𝑀^𝑇𝐻^𝑇₂000, Λ₂=𝑁₁+𝑁₂−𝑁₂000−𝑚𝑁₂00. Then, a suitable state feedback control law is described as 𝐾=(𝐻₂₂𝐵₁)⁻¹𝐵₁𝑇.

Proof. Replacing 𝐴 and 𝐴_𝑑 with 𝐴+Δ𝐴 and 𝐴_𝑑+Δ𝐴_𝑑, respectively, (26) can be expressed as Ψ_𝑖+Λ^𝑇₂𝐹_(𝑡)Π₂+Π^𝑇₂𝐹^𝑇_(𝑡)Λ₂<0,𝑖=1,2. (28) By Lemma 4, LMIs (28) hold for any 𝐹_(𝑡) satisfying 𝐹^𝑇_(𝑡)𝐹_(𝑡)≤𝐼 if and only if there exists scalar 𝜆>0 such that Ψ_𝑖+𝜆Λ^𝑇₂Λ₂+𝜆⁻¹Π^𝑇₂Π₂<0,𝑖=1,2. (29) Applying the Schur complement, (27) is equivalent to (29). This completes the proof.

4. Illustrative Examples

Example 1. Consider that the continuous singular time-varying delay system (5) has the system matrices of 𝐸=⁹³₆₂, 𝐴=^{−13.1−13.7}_{−15.4−23.8}, and 𝐴_𝑑=^{−18.6−10.4}_{−25.2−16.8}. Set 𝜇=0.2 and select 𝐿=[2−3]^𝑇. For the deferent lower bounds 𝑑₁, the upper bounds of 𝑑₂ for the system to be admissible are shown in Table 1. It is obvious that the delay-dependent stability results obtained in this paper are better than those in [32–34].

Table 1 

Comparisons of the upper bounds of 𝑑₂ with different 𝑑₁ of the system without uncertainties.

Example 2. Consider that the continuous singular time-varying delay system has the following system matrices [25]: ,𝐴𝐸=1101−11201,𝐴=211−1010.501_𝑑=.−1.50.5−0.8110.50.70.51,𝐵=121.5001 (30) Set 𝜇=0 and select 𝐿=−112^𝑇; Theorem 5 yields that the system is stable for any constant delay 𝑑 satisfying 0≤𝑑≤5, which has less conservatism than 0≤𝑑≤3.1 which was obtained in [28]. Then, set 𝜇=0.2 and select 𝐿=−112^𝑇. For any time-varying delay 𝑑_(𝑡) satisfying 0.5≤𝑑_(𝑡)≤2, the LMIs (26) are feasible, and a controller gain law can be obtained as follows: 𝐾=.−0.6466−0.1275−0.26810.20720.5078−0.1450 (31)

Now, we consider the system uncertainties, and the uncertain system matrices have the following forms of 𝑀=0.40.30.1^𝑇, 𝑁₁=0.20.40.5, and 𝑁₂=0.30.70.5. Set 𝜇=0.2  and choose 𝐿=−112^𝑇. For any time-varying delay 𝑑_(𝑡) satisfying 0.5≤𝑑_(𝑡)≤2, the LMIs (27) are feasible, and a robust controller can be obtained as 𝐾=.−2.2790−0.4928−0.94420.76541.6355−0.4224 (32)

5. Conclusion

In this research, the robustly delay-dependent stabilization for continuously singular time-varying delay systems with norm-bounded uncertainties is investigated. Based on Lyapunov stability theory and LMI technique, the new delay-dependent LMIs-based conditions are established for the singular time-varying delay system to be regular, impulse free, and stable. By solving these LMIs, the desired state feedback control law can be obtained, and the regularity, causality, and stability of the closed-loop system are guaranteed. Finally, simulation results are given to show the effectiveness of the proposed method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation (nos. 61305019 and 51365017), Jiangxi Provincial Natural Science Foundations (nos. GJJ13430 and GJJ13385), and the Natural Science Foundation of Jiangxi University of Science and Technology (no. NSFJ2014-K16) of China.