On Exponential Stability Conditions of Descriptor Systems with Time-Varying Delay

Abstract

We are interested in the exponential stability of the descriptor system, which is composed of the slow and fast subsystems with time-varying delay. In computing a kind of Lyapunov functional, we employ a necessary number of slack matrices to render the balance and to yield the convexity condition for reducing the conservatism and dealing with the case of time-varying delay. Therefore, we can get the decay rate of the slow variables. Moreover, we characterize the effect of the fast subsystem on the derived decay rate and then prove the fast variables to decay exponentially through a perturbation approach. Finally, we provide an example to demonstrate the effectiveness of the method.

1. Introduction

Descriptor systems are also referred to as singular systems, generalized systems, differential-algebraic systems, and so on. This kind of systems turns out to be precise to describe some practical systems that may undergo some extremal conditions, such as lossless transmission lines. Therefore, it has received considerable attentions to characterize the dynamics of such systems and develop the fundamental control theory in parallel with that of regular ones. In this respect, it has been proven to be a useful approach to decompose a descriptor system into slow and fast subsystems; see [1] and the references therein. Moreover, the existence of this kind of decomposition can be implied by some Lyapunov equality or Lyapunov inequality. Therefore, the Lyapunov method provides an efficient tool for both stability and stabilization problems; see, for example, [2–4].

Meanwhile, there have been great efforts dedicated to the study of time-delay systems since hysteresis is regarded as the important element in modeling many natural and artificial systems and it can be the source of instability and poor performance; see [5] and the references therein. In particular, in the presence of time-delay, the dynamics of a descriptor system can become rather complex; for example, the decomposition according to slow and fast subsystems no longer guarantees the absence of impulsive behavior; see [6]. For the stability problem of delayed descriptor systems, some sophisticated matrix transformation techniques have been posed and even extended to robust control problems so as to apply Lyapunov functional method and the associated convex optimization method under the constraint condition of reduced rank of matrix; see, for example, [6–13].

In this paper, we consider a class of descriptor systems with time-varying delay. The starting point is that the system under consideration satisfies some mild conditions so that it can be converted into the following differential-algebraic equations (see, e.g., [8, 11]):𝑥̇₁0(𝑡)==𝑥₂𝐴𝑥₁(𝑡)+𝐵₁₁𝑥₁(𝑡−ℎ(𝑡))+𝐵₁₂𝑥₂(𝑡−ℎ(𝑡)),(1.1)(𝑡)+𝐵₂₁𝑥₁(𝑥−ℎ(𝑡))+𝐵₂₂𝑥₂(𝑡−ℎ(𝑡)).(1.2) This model is composed of the slow subsystem in (1.1) and the fast subsystem in (1.2). We refer to 𝑥₁(𝑡)∈𝑅^𝑟 and 𝑥₂(𝑡)∈𝑅^𝑛−𝑟 as the slow and fast variables, respectively. In addition, ℎ(𝑡)∈[0,ℎ] is the delay with bounded varying rate |ℎ̇(𝑡)|≤𝜇<1. The matrices 𝐴,𝐵₁₁,𝐵₁₂,𝐵₂₁, and 𝐵₂₂ are of appropriate dimensions. To simplify typography, let us rewrite the system in (1.1) and (1.2) into the form of𝐸𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−ℎ(𝑡)),(1.3) where 𝑥(𝑡)=^{𝑥1𝑥(𝑡)}_2(𝑡), 𝐸=diag{𝐼_𝑟,0}, 𝐴=diag{𝐴,𝐼_𝑛−𝑟}, and 𝐵=^𝐵11𝐵12_𝐵21𝐵22. The goal of this paper is to establish conditions guaranteeing the zero solution of such a system to be exponentially stable.

Definition 1.1. System (1.3) is said to be exponentially stable if there exists an 𝜖>0 such that limsup_𝑡→∞ln|𝑥(𝑡;𝑥₀)|/𝑡≤−𝜖.

The most of the existing results on the stability problem of descriptor systems with delay only pertain to the case of constant delay. In short, as pointed out in [9], this is due to that time-varying delay makes it become hardly possible to explicitly express the fast variables. In [9], to tackle this problem, some terminologies have been borrowed from graph theory to model the dependency of the fast variables on past instants and express them in terms of the slow variables. This approach, however, is rather complicated for application and further improvement.

In this paper, we will focus on the case of time-varying and address the stability problem in such a way that we first get the decay rate of the slow variables by using Lyapunov functional approach and prove the stability of the fast subsystem through a perturbation approach. More precisely, we drop out the idea of expressing the fast variables but use them to perturb the derived decay rate and, therefore, get the conditions guaranteeing their convergence. To this end, we present a necessary number of slack matrices to produce some balance and convexity conditions, which can play a key role for reducing the conservatism caused by delay itself.

2. Main Results

In what follows we need the following fact; see [14].

Lemma 2.1. The following statements are equivalent: (i) there is a positive-definite matrix 𝑃 such that 𝐴^𝑃𝐴<𝛾𝑃, 𝛾>0; (ii) there are a symmetric matrix 𝑃 and a matrix 𝐺 such that ^{𝛾𝑃𝐴′𝐺′}_⋆𝐺+𝐺^′_−𝑃>0.

To study the stability of system (1.1), we construct the Lyapunov functional as follows:𝑉𝑥_𝑡=⁶_𝑘=1𝑉_𝑘𝑥_𝑡(2.1) with the terms defined as𝑉𝑉₁𝑥_𝑡=𝑥^(𝑡)𝐸^𝑃𝐸𝑥(𝑡),₂𝑉𝑥_𝑡=2𝑥^(𝑡)𝐸^𝑄^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃,₃𝑉𝑥_𝑡=^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥^(𝜃)d𝜃𝑅^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃,₄𝑉𝑥_𝑡=^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥^(𝜃)𝑇𝑥(𝜃)d𝜃,₅𝑉𝑥_𝑡=^{𝑡𝑡−ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}𝑥^(𝜃)𝑆𝑥(𝜃)d𝜃,₆𝑥_𝑡=^0−ℎd𝜃^{𝑡𝑡+𝜃}𝑒^{𝛼(𝜎−𝑡)}⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜎(𝜎))⎤⎥⎥⎥⎦^⎡⎢⎢⎢⎣𝑋⋆𝑍𝑌⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜎(𝜎))⎤⎥⎥⎥⎦d𝜎.(2.2) Therefore, we can have𝑉̇𝑉₁̇𝑥_𝑡=2𝑥̇^(𝑡)𝐸^𝑈𝐹₁+𝑃𝐸𝑥(𝑡),₂𝑉̇𝑥_𝑡+𝛼𝑉₂𝑥_𝑡=2𝑥̇^(𝑡)𝐸^𝑈𝐹₂+𝑄^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃+2𝑥^(𝑡)𝐸^𝑄𝑥(𝑡)−𝑒^−𝛼ℎ𝑥(𝑡−ℎ)₃,𝑥_𝑡+𝛼𝑉₃𝑥_𝑡=−2𝛼𝑥(𝑡)_{𝑡𝑡−ℎ}−𝑒^−𝛼ℎ𝑥(𝑡−ℎ)^𝑅^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃𝑉̇𝑒^{𝛼(𝜃−𝑡)}𝑥^(𝜃)d𝜃𝑅^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃,₄𝑉̇𝑥_𝑡+𝛼𝑉₄𝑥_𝑡=𝑥^(𝑡)𝑇𝑥(𝑡)−𝑒^−𝛼ℎ𝑥^(𝑡−ℎ)𝑇𝑥(𝑡−ℎ),₅𝑉̇𝑥_𝑡+𝛼𝑉₅𝑥_𝑡=𝑥^(𝑡)𝑆𝑥(𝑡)−1−ℎ̇(𝑡)𝑒^{−𝛼ℎ(𝑡)}𝑥^(𝑡−ℎ(𝑡))𝑆𝑥(𝑡−ℎ(𝑡)),₆𝑥_𝑡+𝛼𝑉₆𝑥_𝑡=ℎ⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝑡()𝑡)⎤⎥⎥⎥⎦^−⎡⎢⎢⎢⎣𝑋⋆𝑍𝑌⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝑡()𝑡)_{𝑡𝑡−ℎ}⎤⎥⎥⎥⎦𝑒^{𝛼(𝜃−𝑡)}⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜃(𝜃))⎤⎥⎥⎥⎦^⎡⎢⎢⎢⎣𝑋⋆𝑍𝑌⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜃(𝜃))⎤⎥⎥⎥⎦d𝜃,(2.3) where 𝑈 is any square matrix satisfying 𝐸^𝑈=0 and rank𝑈=𝑛−𝑟, and, therefore, 𝐹₁ and 𝐹₂ can be any square matrices.

Theorem 2.2. Let 𝛾∈(0,1) and 𝛼=−ln𝛾/ℎ>0. If there exist matrices 𝑃,𝑄,𝑅 with ^{𝑃𝑄⋆𝑅}>0, 𝑆>0, 𝑇>0, 𝑋,𝑌,𝑍, and 𝐿_𝑞(𝑞=1,…,6), 𝐻_𝑞(𝑞=1,…,4), 𝐺_𝑞(𝑞=1,…,6), 𝐹_𝑞(𝑞=1,2), and 𝑀_𝑝𝑞,𝑁_𝑝𝑞(𝑝,𝑞=1,…,4) such that the following matrix inequalities can be satisfied: ΞΞ++((11−−𝛾𝛾))ΓΓ₁₂≤0,(2.4)Υ≤0,(2.5)₁=⎡⎢⎢⎢⎣𝛼Ω¹Σ₁Υ⋆Π⎤⎥⎥⎥⎦≥0,(2.6)₂=⎡⎢⎢⎢⎣𝛼Ω²Σ₂⎡⎢⎢⎢⎣𝛼𝑅+𝐸𝐿_5⋆Π⎤⎥⎥⎥⎦≥0,(2.7)+𝐸^𝐿₅𝐿^5+𝛼𝐸^𝐿₆⋆𝐿^6+𝐿₆⎤⎥⎥⎥⎦≥0,(2.8) then the system in (1.3) is exponentially stable. Here, the matrices, the matrix blocks, and the matrix elements presented in (2.4)–(2.8) are defined as follows: Ξ=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣Ξ¹¹Ξ₁₂Ξ₁₃Ξ₁₄⋆Ξ₂₂Ξ₂₃Ξ₂₄⋆⋆Ξ₃₃Ξ₃₄⋆⋆⋆Ξ₄₄Γ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;Π=⎡⎢⎢⎢⎣𝑋⋆𝑍₁𝑌⎤⎥⎥⎥⎦;=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑀¹¹𝑀₁₂𝑀₁₃+𝐿^1𝐸𝑀₁₄⋆𝑀₂₂𝑀₂₃+𝐿^2𝐸𝑀₂₄⋆⋆𝑀₃₃+𝐿^3𝐸+𝐸^𝐿₃+(1−𝜇)𝑆𝑀₃₄+𝐸^𝐿₄⋆⋆⋆𝑀₄₄Γ₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑁¹¹𝑁₁₂𝑁₁₃+𝐻^1𝐸𝑁₁₄⋆𝑁₂₂𝑁₂₃+𝐻^2𝐸𝑁₂₄⋆⋆𝑁₃₃+𝐻^3𝐸+𝐸^𝐻₃𝑁₃₄+𝐸^𝐻₄⋆⋆⋆𝑁₄₄Ω₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑀¹¹𝑀₁₂𝑀₁₃𝑀₁₄⋆𝑀₂₂𝑀₂₃𝑀₂₄⋆⋆𝑀₃₃𝑀₃₄⋆⋆⋆𝑀₄₄⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;Ω₂=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑁¹¹𝑁₁₂𝑁₁₃𝑁₁₄⋆𝑁₂₂𝑁₂₃𝑁₂₄⋆⋆𝑁₃₃𝑁₃₄⋆⋆⋆𝑁₄₄Σ₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;=−⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝐴^𝐺₅+𝑅+𝐸^𝐿₅−𝛼𝐿^1𝐸𝐸^𝐿₆−𝐿^1−𝐴^𝐺₆−𝐵^𝐺₅+𝑈𝐹₂+𝑄−𝛼𝐿^2𝐸−𝐵^𝐺₆−𝐿^2−𝛼𝐿^3𝐸−(1−𝛾)𝐸^𝐿₅−𝐿^3−(1−𝛾)𝐸^𝐿₆−𝛾𝑅+𝐸^𝐿₅−𝛼𝐿^4𝐸−𝛾𝐸^𝐿₆−𝐿^4Σ₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;=−⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝐴^𝐺₅+𝑅+𝐸^𝐿₅−𝛼𝐻^1𝐸𝐸^𝐿₆−𝐻^1−𝐴^𝐺₆−𝐵^𝐺₅+𝑈𝐹₂+𝑄−𝛼𝐻^2𝐸−𝐵^𝐺₆−𝐻^2−𝛼𝐻^3𝐸−(1−𝛾)𝐸^𝐿₅−𝐻^3−(1−𝛾)𝐸^𝐿₆−𝛾𝑅+𝐸^𝐿₅−𝛼𝐻^4𝐸−𝛾𝐸^𝐿₆−𝐻^4Ξ₁₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦;Ξ=−𝐴^𝐺₁−𝐺^1𝐴+𝛼𝐸^𝑃𝐸+𝐿^1𝐸+𝐸^𝐿₁+𝐸𝑄^+𝑄𝐸^+𝑆+𝑇+ℎ𝑋;₁₂Ξ=𝐸^𝐿₂+𝑈𝐹₁+𝑃𝐸^+𝐺^1−𝐴^𝐺₂+ℎ𝑌;₁₃Ξ₁₄=𝛾𝐻^1𝐸+𝐸^𝐿₃−𝐸^𝐿₁−𝐺^1𝐵−𝐴^𝐺₃;Ξ₂₂=𝐸^𝐿₄−𝛾𝐻^1𝐸+𝐸^𝑄−𝐴^𝐺₄;Ξ=𝐺^2+𝐺₂+ℎ𝑍;₂₃Ξ=−𝐿^2𝐸+𝛾𝐻^2𝐸+𝐺₃−𝐺^2𝐵;₂₄Ξ₃₃=−𝛾𝐻^2𝐸+𝐺₄;Ξ=𝛾𝐻^3𝐸+𝐸^𝐻₃−𝐿^3𝐸−𝐸^𝐿₃−(1−𝜇)𝑆−𝐵^𝐺₃−𝐺^3𝐵;₃₄Ξ₄₄=−𝐸^𝐿₄+𝛾𝐸^𝐻₄−𝛾𝐻^3𝐸−𝐵^𝐺₄;=−𝛾𝑇+𝐻^4𝐸+𝐸^𝐻₄.(2.9)

Proof. The first thing we have to do is to note that ∑^5𝑘=1𝑉_𝑘(𝑥_𝑡) is positive-definite, and the positiveness of 𝑉₆(𝑥_𝑡) is guaranteed simultaneously by (2.6) and (2.7). The subsequent proof is rather long; so for the clarity we will proceed in two steps. The first one concentrates on getting the decay rate for the slow variables, while the second one turns the focus on proving the fast variables eventually fallen into decay.Step 1 (Present the decay rate of the slow variables). Combining (2.3) gives 𝑉̇≤𝑥_𝑡2𝑥+̇_𝛼𝑉𝑥_𝑡+2(𝑥𝑡)(𝑡𝐸)^−𝑈𝑒𝐹_1−𝛼ℎ+𝑃𝐸𝑥(𝑡)+2𝑥̇^(𝑡)𝐸^𝑈𝐹₂+𝑄^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃𝑥(𝑡−ℎ)^𝑄′𝐸𝑥(𝑡)+𝑅^{𝑡𝑡−ℎ}+𝑥_𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃+(1(−𝑡)𝜇(𝑆)+1𝑇−)𝑒𝑥(𝑡)_{−𝛼ℎ(𝑡)}−𝑒^−𝛼ℎ𝑥^(𝑡−ℎ)𝑇𝑥(𝑡−ℎ)+ℎ⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝑡()𝑡)⎤⎥⎥⎥⎦_𝑥^(𝑡−ℎ(𝑡))𝑆𝑥(𝑡−ℎ(𝑡))−(1−𝜇)𝑥^(𝑡−ℎ(𝑡))𝑆𝑥(𝑡−ℎ(𝑡))⎡⎢⎢⎢⎣𝑋⋆𝑍𝑌⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝑡()𝑡)⎤⎥⎥⎥⎦−_{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}+𝛼𝑥_⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜃(𝜃))⎤⎥⎥⎥⎦^⎡⎢⎢⎢⎣𝑋⋆𝑍𝑌⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜃(𝜃))⎤⎥⎥⎥⎦d𝜃(𝑡)𝐸^𝑃𝐸𝑥(𝑡)−𝛼^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥^(𝜃)d𝜃𝑅^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃.(2.10) In addition, partitioning the interval [−ℎ,0] into the union of [−ℎ,−ℎ(𝑡)] and [−ℎ(𝑡),0], by integrating by parts we therefore obtain that 𝐸𝑒𝐸𝑥(𝑡)−𝑒^{−𝛼ℎ(𝑡)}𝑥(𝑡−ℎ(𝑡))−^{𝑡𝑡−ℎ(𝑡)}𝛼𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃−^{𝑡𝑡−ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}𝑥̇(𝜃)d𝜃=0,_{−𝛼ℎ(𝑡)}𝑥(𝑡−ℎ(𝑡))−𝑒^−𝛼ℎ𝑥(𝑡−ℎ)−∫^{𝑡−ℎ(𝑡)𝑡−ℎ}𝛼𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃−∫^{𝑡−ℎ(𝑡)𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥̇(𝜃)d𝜃=0.(2.11) It is known that inserting some slack matrices into computing the constructed Lyapunov functional can produce some balance and convexity conditions. Doing this, we combine (2.11) into the identity as follows:0≡2𝑥(−𝑡)+1−𝑒^{−𝛼ℎ(𝑡)}𝑥(𝑡−ℎ(𝑡))−𝑥(𝑡−ℎ(𝑡))−^{𝑡𝑡−ℎ(𝑡)}𝛼𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃_{𝑡𝑡−ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}𝑥̇(𝜃)d𝜃^×𝐸^𝐿¹+𝐿𝑥₅(𝑡)+𝐿₂𝐸𝑥̇(𝑡)+𝐿₃𝑥(𝑡−ℎ(𝑡))+𝐿₄𝑥(𝑡−ℎ)^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃+𝐿₆^{𝑡𝑡−ℎ}+2𝑒_{−𝛼ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}𝐸𝑥̇(𝜃)d𝜃−−𝑒^−𝛼ℎ𝑥(𝑡−ℎ(𝑡))+𝑒^−𝛼ℎ𝑥(𝑡−ℎ(𝑡))−𝑒^−𝛼ℎ𝑥(𝑡−ℎ)−^{𝑡−ℎ(𝑡)𝑡−ℎ}𝛼𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃_{𝑡−ℎ(𝑡)𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥̇(𝜃)d𝜃^×𝐸^𝐻¹+𝐿₅𝑥(𝑡)+𝐻₂𝐸𝑥̇(𝑡)+𝐻₃𝑥(𝑡−ℎ(𝑡))+𝐻₄𝑥(𝑡−ℎ)^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃+𝐿₆^{𝑡𝑡−ℎ}=2𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))−_{𝑡𝑡−ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}𝐸𝑥̇(𝜃)d𝜃𝛼𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃−^{𝑡𝑡−ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}𝑥̇(𝜃)d𝜃^×+𝐸2^𝑒𝐿_1−𝛼ℎ𝑥(𝑡)+𝐿₂𝐸𝑥̇(𝑡)+𝐿₃𝑥(𝑡−ℎ(𝑡))+𝐿₄𝑥(𝑡−ℎ)𝑥(𝑡−ℎ(𝑡))−𝑒^−𝛼ℎ𝑥(𝑡−ℎ)−^{𝑡−ℎ(𝑡)𝑡−ℎ}𝛼𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃−^{𝑡−ℎ(𝑡)𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥̇(𝜃)d𝜃^×+𝐸2^𝑥𝐻(𝑡)₁−𝑥(𝑡1)+−𝐻𝑒₂𝐸_−𝛼ℎ𝑥̇(𝑡)+𝐻₃𝑥(𝑡−ℎ(𝑡))+𝐻₄𝑥(𝑡−ℎ)𝑥(𝑡−ℎ(𝑡))−𝑒^−𝛼ℎ𝑥(𝑡−ℎ)^𝐸^×𝐿₅^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃+𝐿₆^{𝑡𝑡−ℎ}−2𝐸_{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝐸𝑥̇(𝜃)d𝜃𝛼𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃+^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝐸𝑥̇(𝜃)d𝜃^×𝐿₅^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃+𝐿₆^{𝑡𝑡−ℎ}+21−𝑒_{−𝛼ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}𝐸𝑥̇(𝜃)d𝜃+2𝑒_{−𝛼ℎ(𝑡)}𝑥^(𝑡−ℎ(𝑡))𝐸^𝐿₁𝑥(𝑡)+𝐿₂𝐸𝑥̇(𝑡)+𝐿₃𝑥(𝑡−ℎ(𝑡))+𝐿₄𝑥(𝑡−ℎ)−𝑒^−𝛼ℎ𝑥^(𝑡−ℎ(𝑡))𝐸^𝐻₁𝑥(𝑡)+𝐻₂𝐸𝑥̇(𝑡)+𝐻₃𝑥(𝑡−ℎ(𝑡))+𝐻₄𝑥(𝑡−ℎ).(2.12) Furthermore, we can have additional slack matrices through the following identity 0≡2𝐸𝑥̇(𝑡)−𝐴𝑥(𝑡)−𝐵𝑥(𝑡−ℎ(𝑡))^×𝐺₁𝑥(𝑡)+𝐺₂𝐸𝑥̇(𝑡)+𝐺₃𝑥(𝑡−ℎ(𝑡))+𝐺₄𝑥(𝑡−ℎ)+𝐺₅^{𝑡𝑡−ℎ}+𝐺₆𝑒^{𝛼(𝜃−𝑡)}𝑥(𝜃)d𝜃^{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}𝐸𝑥̇(𝜃)d𝜃.(2.13)
Therefore, substituting (2.12) and (2.13) into the right-hand side of (2.10) and rearranging the obtained terms according to the augmented system variables as𝜂(𝑡)𝜁=_𝑡𝑥^(𝑡)𝐸𝑥̇(𝑡)𝑥^(𝑡−ℎ(𝑡))𝑥^(𝑡−ℎ)^,(𝜃)=𝜂^(𝑡)𝑥^(𝑡+𝜃)𝐸^𝑥̇^(𝑡+𝜃)^(2.14) yields 𝑉̇≤𝑥_𝑡𝜂_+𝛼𝑉𝑥_𝑡+2(𝑡)𝑒Ξ𝜂(𝑡_{−𝛼ℎ(𝑡)})+21−𝑒^{−𝛼ℎ(𝑡)}𝑥^(𝑡−ℎ(𝑡))𝐸^𝐿₁𝑥(𝑡)+𝐿₂𝐸𝑥̇(𝑡)+𝐿₃𝑥(𝑡−ℎ(𝑡))+𝐿₄𝑥(𝑡−ℎ)−−𝑒^−𝛼ℎ𝑥^(𝑡−ℎ(𝑡))𝐸^𝐻₁𝑥(𝑡)+𝐻₂𝐸𝑥̇(𝑡)+𝐻₃𝑥(𝑡−ℎ(𝑡))+𝐻₄𝑥(𝑡−ℎ)_{0−ℎ(𝑡)}𝑒^𝛼𝜃𝜁^𝑡(𝜃)⎡⎢⎢⎢⎣0Σ¹⋆Π⎤⎥⎥⎥⎦𝜁^𝑡(𝜃)d𝜃−^{−ℎ(𝑡)−ℎ}𝑒^𝛼𝜃𝜁^𝑡(𝜃)⎡⎢⎢⎢⎣0Σ²⋆Π⎤⎥⎥⎥⎦𝜁^𝑡−(𝜃)d𝜃_{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜃(𝜃))⎤⎥⎥⎥⎦^d𝜃⎡⎢⎢⎢⎣𝛼𝑅+𝐸𝐿^5+𝐸^𝐿₅𝐿^5+𝛼𝐸^𝐿₆⋆𝐿^6+𝐿₆⎤⎥⎥⎥⎦_{𝑡𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}+1−𝑒_{−𝛼ℎ(𝑡)}⎡⎢⎢⎢⎣𝐸𝑥𝑥(̇𝜃(𝜃))⎤⎥⎥⎥⎦d𝜃+𝑒−𝛼^{𝑡𝑡−ℎ(𝑡)}𝑒^{𝛼(𝜃−𝑡)}d𝜃𝜂^(𝑡)Ω₁𝜂(𝑡)_{−𝛼ℎ(𝑡)}−𝑒^−𝛼ℎ−𝛼^{𝑡−ℎ(𝑡)𝑡−ℎ}𝑒^{𝛼(𝜃−𝑡)}d𝜃𝜂^(𝑡)Ω₂𝜂(𝑡).(2.15) By using (2.8) to drop the sixth term and arranging the remaining terms on the right-hand side, (2.15) becomes 𝑉̇𝑥_𝑡+𝛼𝑉𝑥_𝑡≤−𝜂^(𝑡)Ξ+1−𝑒^{−𝛼ℎ(𝑡)}Γ₁+𝑒^{−𝛼ℎ(𝑡)}−𝑒^−𝛼ℎΓ₂𝜂(𝑡)_{0−ℎ(𝑡)}𝑒^𝛼𝜃𝜁^𝑡(𝜃)Υ₁𝜁_𝑡(𝜃)d𝜃−^{−ℎ(𝑡)−ℎ}𝑒^𝛼𝜃𝜁^𝑡(𝜃)Υ₂𝜁_𝑡(𝜃)d𝜃.(2.16) Thus, recalling (2.6) and (2.7), we have 𝑉̇𝑥_𝑡+𝛼𝑉𝑥_𝑡≤𝜂^(𝑡)Ξ+1−𝑒^{−𝛼ℎ(𝑡)}Γ₁+𝑒^{−𝛼ℎ(𝑡)}−𝑒^−𝛼ℎΓ₂𝜂(𝑡).(2.17) The right-hand side of (2.17) turns out to be convex in 𝑒^{−𝛼ℎ(𝑡)}. It then can be eliminated through the boundary conditions (2.4) and (2.5), which correspond to the cases of 𝑒^−𝛼ℎ=𝛾 and 𝑒^−𝛼0=1, respectively. We then obtain 𝑉̇𝑥_𝑡≤−𝛼𝑉𝑥_𝑡.(2.18) And hence, 𝑥^1(𝑡)𝑃₁₁𝑥₁(𝑡)≤𝑉𝑥₀𝑒^−𝛼𝑡,𝑡≥0,(2.19) where 𝑃₁₁ is the first 𝑟×𝑟 order principle block of the matrix 𝑃. Therefore, it is proven that the slow subsystem is exponentially stable.Step 2 (Prove the fast variables fallen into decay exponentially). We proceed in such a way that we first conclude the Schur stability of the difference equation 𝑥₂(𝑡)+𝐵₂₂𝑥₂(𝑡−ℎ(𝑡))=0, and, therefore, prove the fast variables eventually fallen into decay via evaluating their effect on the decay rate derived as in (2.19) From (2.4) or (2.5), it is straightforward to see Φ=⎡⎢⎢⎢⎢⎢⎣Φ¹¹Φ₁₂Φ₁₃⋆Φ₂₂Φ₂₃⋆⋆Φ₃₃⎤⎥⎥⎥⎥⎥⎦≤0,(2.20) where Φ₁₁=−𝐴^𝐺₁−𝐺^1𝐴+𝛼𝐸^𝑃𝐸+𝐿^1𝐸+𝐸^𝐿₁+𝐸𝑄^+𝑄𝐸^+𝑆, Φ₁₂=𝐸^𝐿₂+(𝑈𝐹₁+𝑃𝐸)^+𝐺^1−𝐴^𝐺₂, Φ₁₃=𝛾𝐻^1𝐸+𝐸^𝐿₃−𝐸^𝐿₁−𝐺^1𝐵−𝐴^𝐺₃+(1−𝛾)𝐿^1𝐸, Φ₂₂=𝐺^2+𝐺₂, Φ₂₃=−𝐿^2𝐸+𝛾𝐻^2𝐸+𝐺₃−𝐺^2𝐵+(1−𝛾)𝐿^2𝐸, and Φ₃₃=𝛾(𝐻^3𝐸+𝐸^𝐻₃−𝐿^3𝐸−𝐸^𝐿₃−(1−𝜇)𝑆)−𝐵^𝐺₃−𝐺^3𝐵.
Pre- and postmultiplying (2.20) by the matrix ^{𝐼𝐴′00}_𝐵^′_𝐼 and its transpose, respectively, we get thatΨ=⎡⎢⎢⎢⎣Ψ¹¹Ψ₁₂⋆Ψ₂₂⎤⎥⎥⎥⎦≤0,(2.21) where Ψ₁₁=𝛼𝐸^𝑃𝐸+𝐿^1𝐸+𝐸^𝐿₁+𝐸𝑄^+𝑄𝐸^+𝑆+𝐴^𝐿^2𝐸+𝐸^𝐿₂𝐴+𝐴^𝑃𝐸+𝐸^𝑃𝐴+𝐴^𝑈𝐹₁+𝐹^1𝑈^𝐴, Ψ₁₂=𝐸^𝐿₂+𝐸^𝑃𝐵+𝐹^1𝑈^𝐵+𝛾𝐻^1𝐸+𝐸^𝐿₃−𝐸^𝐿₁+(1−𝛾)𝐿^1𝐸−𝐴^𝐿₂𝐸+𝛾𝐴^𝐻^2𝐸+(1−𝛾)𝐴^𝐿^2𝐸, and Ψ₂₂=𝛾(𝐻^3𝐸+𝐸^𝐻₃−𝐿^3𝐸−𝐸^𝐿₃+𝐵^𝐻^2𝐸+𝐸^𝐻₂𝐵−𝐵^𝐿^2𝐸−𝐸^𝐿₂𝐵−(1−𝜇)𝑆).
In fact, the family of slack matrices 𝐺_𝑞(𝑞=1,…,6) plays a key role in getting (2.21). Furthermore, noting the special structures of the matrices 𝐸, 𝑈, and 𝐴, from (2.21) we can deduce that⎡⎢⎢⎢⎣𝑆₂₂+𝐹^1,22+𝐹_1,22𝐹^1,22𝐵₂₂⋆−𝛾(1−𝜇)𝑆₂₂⎤⎥⎥⎥⎦≤0,(2.22) where 𝑆₂₂ and 𝐹_1,22 represent the last (𝑛−𝑟)×(𝑛−𝑟)-order principle blocks of of the matrices 𝑆 and 𝐹₁, respectively. According to Lemma 2.1, we have 𝐵^22𝑆₂₂𝐵₂₂≤𝛾(1−𝜇)𝑆₂₂≤𝛾𝑆₂₂.(2.23) This constraint condition does imply that if the varying rate of delay exceeds 1, the system in (1.1) would no more be able to retain the stability.
By the fact that |𝜉₁+𝜉₂|²≤|𝜉₁|²/𝛿+|𝜉₂|²/(1−𝛿) holds for all vectors 𝜉₁,𝜉₂ of same dimension and number 𝛿∈(0,1), from (2.19) and (2.23) we can estimate that𝑥^2(𝑡)𝑆₂₂𝑥₂(𝑡)≤11−𝛿𝑥^1(𝑡−ℎ(𝑡))𝐵^21𝑆₂₂𝐵₂₁𝑥₁(𝑡−ℎ(𝑡))+1𝛿𝑥^2≤𝜒𝑉𝑥₀(𝑡−ℎ(𝑡))𝐵^22𝑆₂₂𝐵₂₂𝑥₂(𝑡−ℎ(𝑡))1−𝛿𝑒^{−𝛼(𝑡−ℎ(𝑡))}+𝛾𝛿sup_{𝑡−ℎ≤𝜃≤𝑡}≤𝜒𝑉𝑥₀𝑥^2(𝑡+𝜃)𝑆₂₂𝑥₂(𝑡+𝜃)𝛾(1−𝛿)𝑒^−𝛼𝑡+𝛾𝛿sup_{𝑡−ℎ≤𝜃≤𝑡}𝑥^2(𝑡+𝜃)𝑆₂₂𝑥₂(𝑡+𝜃),𝑡≥0,(2.24) where 𝜒 is sufficiently large so that 𝐵^21𝑆₂₂𝐵₂₁≤𝜒𝑃₁₁ and 𝛿 is specified within (𝛾,1). Let 𝜓_𝑘=sup_{(𝑘−1)ℎ≤𝜃≤𝑘ℎ}{𝑥^2(𝑡+𝜃)𝑆₂₂𝑥₂(𝑡+𝜃)},𝑘=0,1,2,…. Then, from (2.24) it follows that 𝜓_𝑘≤𝜒𝑉𝑥⁰𝛾(1−𝛿)𝑒^{−𝛼(𝑘−1)ℎ}+𝛾𝛿𝜓^𝑘−1∨𝜓_𝑘,𝑘≥1.(2.25) Therefore, 𝑒^𝜖𝑘ℎ𝜓_𝑘≤𝜒𝑉𝑥⁰𝛾(1−𝛿)𝑒^{𝛼ℎ−(𝛼−𝜖)𝑘ℎ}+𝛾𝛿𝑒^𝜖ℎ𝑒^{𝜖(𝑘−1)ℎ}𝜓_𝑘−1∨𝑒^𝜖𝑘ℎ𝜓_𝑘,𝑘≥1,(2.26) where the positive number 𝜖 is sufficiently small for 𝛾𝑒^𝜖ℎ<𝛿.
Noting 𝜖<(ln𝛿−ln𝛾)/ℎ<𝛼, we deduce thatmax_{1≤𝑖≤𝑘}𝑒^𝜖𝑖ℎ𝜓_𝑖≤𝜒𝑉𝑥⁰𝛾(1−𝛿)𝑒^𝛼ℎ+𝛾𝛿𝑒^𝜖ℎ𝜓⁰+max_{1≤𝑖≤𝑘}𝑒^𝜖𝑖ℎ𝜓_𝑖.(2.27) Hence, max_{1≤𝑖≤𝑘}𝑒^𝜖𝑖ℎ𝜓_𝑖≤𝜎∶=𝜒𝑉𝑥⁰/(1−𝛿)+(𝛾/𝛿)𝑒^𝜖ℎ𝜓₀1−(𝛾/𝛿)𝑒^𝜖ℎ.(2.28) Thus, 𝜓_𝑘≤𝜎𝑒^{−𝜖𝑘ℎ},𝑘≥1.(2.29) But this implies that limsup_𝑘→∞ln𝜓_𝑘𝑘ℎ≤−𝜖.(2.30) And it follows that limsup_𝑡→∞ln||𝑥₂(𝑡)||²𝑡≤−𝜖.(2.31) This together with (2.19) then completes the proof.

In arranging the augmented system variables, we insert a necessary number of slack matrices to render some balance and flexibility. Also, with the aid of slack matrices, the interval [−ℎ,0] is decomposed into the union of [−ℎ,−ℎ(𝑡)] and [−ℎ(𝑡),0] and, moreover, the terms factored by 𝑒^{−𝛼ℎ(𝑡)} are reformulated into a form of convex combination. In this way, the Lyapunov functional in (2.1) is computed almost without loss of its generality.

In the derived stability conditions, there only is a parameter to be specified, namely, 𝛾, which deserves a brief discussion. In fact, as shown in (2.23), it is introduced to guarantee the difference equation 𝑥₂(𝑡)+𝐵₂₂𝑥₂(𝑡−ℎ(𝑡))=0 to be Schur stable. Therefore, we use 𝛾 to characterize the effect of the fast subsystem on the decay rate of the slow variables, 𝛼. Indeed, this turns out to be a typical perturbation approach to prove stability. Besides, 𝛾 is one of boundary conditions of the convex combination on the right-hand side of (2.17).

3. An Example

In this section, we use a numerical example to demonstrate the theoretical results.

Example 3.1. Consider a system in the form of (1.3) with the following parameters: 𝐸=⎡⎢⎢⎢⎢⎢⎣100010000⎤⎥⎥⎥⎥⎥⎦,𝐴=⎡⎢⎢⎢⎢⎢⎣000.5000.3001⎤⎥⎥⎥⎥⎥⎦,𝐵=⎡⎢⎢⎢⎢⎢⎣−−111−001−000..23.4⎤⎥⎥⎥⎥⎥⎦.(3.1) Specified 𝛾=0.98, the calculated stability margins for various varying-rate of delay are presented in Table 1. On the other hand, we compute the upper bound of size of delay for various specified 𝛾 with fixing 𝜇=0, which is shown in Table 2. In the light of the discussion on the parameter 𝛾, it would become clear that there is a mutually constraint relation between the difference operator 𝑥₂(𝑡)+𝐵₂₂𝑥₂(𝑡−ℎ(𝑡)) and the size of delay in guaranteeing stability.

Table 1 

Calculated upper bound of size of delay for various varying rates.

Table 2 

Calculated upper bound of size of delay for various specified 𝛾.

4. Conclusion

We considered a class of descriptor systems with time-varying delay. We developed a Lyapunov technique to investigate the exponential stability of such a system, which combines a necessary number of slack matrices, convexity condition, and matrix transformation. Therefore, after getting the decay rate for the slow variables, through a perturbation approach we came to the conclusion that the fast variables eventually fall into decay exponentially. A numerical example was given to illustrate the theoretical results.

Acknowledgments

The authors would like to thank the anonymous reviewers for the detailed and constructive comments that helped in improving this paper. This work is supported by the National Natural Science Foundation of China under Grant 60974027.