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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 532912, 12 pages
Research Article

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

School of Mechanical & Electrical Engineering, Heilongjiang University, Harbin 150080, China

Received 8 August 2011; Revised 4 November 2011; Accepted 4 November 2011

Academic Editor: Kai Diethelm

Copyright © 2012 S. Cong and Z.-B. Sheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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, [24].

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, [613].

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]):̇𝑥1(𝑡)=𝐴𝑥1(𝑡)+𝐵11𝑥1(𝑡(𝑡))+𝐵12𝑥2(𝑡(𝑡)),(1.1)0=𝑥2(𝑡)+𝐵21𝑥1(𝑥(𝑡))+𝐵22𝑥2(𝑡(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)𝑅𝑟 and 𝑥2(𝑡)𝑅𝑛𝑟 as the slow and fast variables, respectively. In addition, (𝑡)[0,] is the delay with bounded varying rate |̇(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 of𝐸̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡(𝑡)),(1.3) where 𝑥(𝑡)=𝑥1𝑥(𝑡)2(𝑡), 𝐸=diag{𝐼𝑟,0}, 𝐴=diag{𝐴,𝐼𝑛𝑟}, and 𝐵=𝐵11𝐵12𝐵21𝐵22. 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 limsup𝑡ln|𝑥(𝑡;𝑥0)|/𝑡𝜖.

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 𝐴𝑃𝐴<𝛾𝑃, 𝛾>0; (ii) there are a symmetric matrix 𝑃 and a matrix 𝐺 such that 𝛾𝑃𝐴𝐺𝐺+𝐺𝑃>0.

To study the stability of system (1.1), we construct the Lyapunov functional as follows:𝑉𝑥𝑡=6𝑘=1𝑉𝑘𝑥𝑡(2.1) with the terms defined as𝑉1𝑥𝑡=𝑥(𝑡)𝐸𝑉𝑃𝐸𝑥(𝑡),2𝑥𝑡=2𝑥(𝑡)𝐸𝑄𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥𝑉(𝜃)d𝜃,3𝑥𝑡=𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑅𝑡𝑡𝑒𝛼(𝜃𝑡)𝑉𝑥(𝜃)d𝜃,4𝑥𝑡=𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥𝑉(𝜃)𝑇𝑥(𝜃)d𝜃,5𝑥𝑡=𝑡𝑡(𝑡)𝑒𝛼(𝜃𝑡)𝑥(𝑉𝜃)𝑆𝑥(𝜃)d𝜃,6𝑥𝑡=0d𝜃𝑡𝑡+𝜃𝑒𝛼(𝜎𝑡)𝑥(𝜎)𝐸̇𝑥(𝜎)𝑋𝑌𝑍𝑥(𝜎)𝐸̇𝑥(𝜎)d𝜎.(2.2) Therefore, we can havė𝑉1𝑥𝑡=2̇𝑥(𝑡)𝐸𝑈𝐹1𝑥̇𝑉+𝑃𝐸(𝑡),2𝑥𝑡+𝛼𝑉2𝑥𝑡=2̇𝑥(𝑡)𝐸𝑈𝐹2+𝑄𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+2𝑥(𝑡)𝐸𝑄𝑥(𝑡)𝑒𝛼𝑥,̇𝑉(𝑡)3𝑥𝑡+𝛼𝑉3𝑥𝑡=2𝑥(𝑡)𝑒𝛼𝑥(𝑡)𝑅𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝛼𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑅𝑡𝑡𝑒𝛼(𝜃𝑡)̇𝑉𝑥(𝜃)d𝜃,4𝑥𝑡+𝛼𝑉4𝑥𝑡=𝑥(𝑡)𝑇𝑥(𝑡)𝑒𝛼𝑥̇𝑉(𝑡)𝑇𝑥(𝑡),5𝑥𝑡+𝛼𝑉5𝑥𝑡=𝑥̇𝑒(𝑡)𝑆𝑥(𝑡)1(𝑡)𝛼(𝑡)𝑥̇𝑉(𝑡(𝑡))𝑆𝑥(𝑡(𝑡)),6𝑥𝑡+𝛼𝑉6𝑥𝑡=𝑥(𝑡)𝐸̇𝑥(𝑡)𝑋𝑌𝑍𝑥(𝑡)𝐸̇𝑥(𝑡)𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)𝐸̇𝑥(𝜃)𝑋𝑌𝑍𝑥(𝜃)𝐸̇𝑥(𝜃)d𝜃,(2.3) where 𝑈 is any square matrix satisfying 𝐸𝑈=0 and rank𝑈=𝑛𝑟, and, therefore, 𝐹1 and 𝐹2 can be any square matrices.

Theorem 2.2. Let 𝛾(0,1) and 𝛼=ln𝛾/>0. If there exist matrices 𝑃,𝑄,𝑅 with 𝑃𝑄𝑅>0, 𝑆>0, 𝑇>0, 𝑋,𝑌,𝑍, and 𝐿𝑞(𝑞=1,,6), 𝐻𝑞(𝑞=1,,4), 𝐺𝑞(𝑞=1,,6), 𝐹𝑞(𝑞=1,2), and 𝑀𝑝𝑞,𝑁𝑝𝑞(𝑝,𝑞=1,,4) such that the following matrix inequalities can be satisfied: Ξ+(1𝛾)Γ10,(2.4)Ξ+(1𝛾)Γ2Υ0,(2.5)1=𝛼Ω1Σ1ΥΠ0,(2.6)2=𝛼Ω2Σ2𝛼Π0,(2.7)𝑅+𝐸𝐿5+𝐸𝐿5𝐿5+𝛼𝐸𝐿6𝐿6+𝐿60,(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: ΞΞ=11Ξ12Ξ13Ξ14Ξ22Ξ23Ξ24Ξ33Ξ34Ξ44;Γ;Π=𝑋𝑌𝑍1=𝑀11𝑀12𝑀13+𝐿1𝐸𝑀14𝑀22𝑀23+𝐿2𝐸𝑀24𝑀33+𝐿3𝐸+𝐸𝐿3+(1𝜇)𝑆𝑀34+𝐸𝐿4𝑀44;Γ2=𝑁11𝑁12𝑁13+𝐻1𝐸𝑁14𝑁22𝑁23+𝐻2𝐸𝑁24𝑁33+𝐻3𝐸+𝐸𝐻3𝑁34+𝐸𝐻4𝑁44;Ω1=𝑀11𝑀12𝑀13𝑀14𝑀22𝑀23𝑀24𝑀33𝑀34𝑀44;Ω2=𝑁11𝑁12𝑁13𝑁14𝑁22𝑁23𝑁24𝑁33𝑁34𝑁44;Σ1=𝐴𝐺5+𝑅+𝐸𝐿5𝛼𝐿1𝐸𝐸𝐿6𝐿1𝐴𝐺6𝐵𝐺5+𝑈𝐹2+𝑄𝛼𝐿2𝐸𝐵𝐺6𝐿2𝛼𝐿3𝐸(1𝛾)𝐸𝐿5𝐿3(1𝛾)𝐸𝐿6𝛾𝑅+𝐸𝐿5𝛼𝐿4𝐸𝛾𝐸𝐿6𝐿4;Σ2=𝐴𝐺5+𝑅+𝐸𝐿5𝛼𝐻1𝐸𝐸𝐿6𝐻1𝐴𝐺6𝐵𝐺5+𝑈𝐹2+𝑄𝛼𝐻2𝐸𝐵𝐺6𝐻2𝛼𝐻3𝐸(1𝛾)𝐸𝐿5𝐻3(1𝛾)𝐸𝐿6𝛾𝑅+𝐸𝐿5𝛼𝐻4𝐸𝛾𝐸𝐿6𝐻4;Ξ11=𝐴𝐺1𝐺1𝐴+𝛼𝐸𝑃𝐸+𝐿1𝐸+𝐸𝐿1+𝐸𝑄+𝑄𝐸Ξ+𝑆+𝑇+𝑋;12=𝐸𝐿2+𝑈𝐹1+𝑃𝐸+𝐺1𝐴𝐺2Ξ+𝑌;13=𝛾𝐻1𝐸+𝐸𝐿3𝐸𝐿1𝐺1𝐵𝐴𝐺3;Ξ14=𝐸𝐿4𝐻𝛾1𝐸+𝐸𝑄𝐴𝐺4;Ξ22=𝐺2+𝐺2Ξ+𝑍;23=𝐿2𝐸+𝛾𝐻2𝐸+𝐺3𝐺2Ξ𝐵;24=𝛾𝐻2𝐸+𝐺4;Ξ33𝐻=𝛾3𝐸+𝐸𝐻3𝐿3𝐸𝐸𝐿3(1𝜇)𝑆𝐵𝐺3𝐺3Ξ𝐵;34=𝐸𝐿4+𝛾𝐸𝐻4𝛾𝐻3𝐸𝐵𝐺4;Ξ44=𝛾𝑇+𝐻4𝐸+𝐸𝐻4.(2.9)

Proof. The first thing we have to do is to note that 5𝑘=1𝑉𝑘(𝑥𝑡) is positive-definite, and the positiveness of 𝑉6(𝑥𝑡) 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 ̇𝑉𝑥𝑡𝑥+𝛼𝑉𝑡2̇𝑥(𝑡)𝐸𝑈𝐹1𝑥+𝑃𝐸(𝑡)+2̇𝑥(𝑡)𝐸𝑈𝐹2+𝑄𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+2𝑥(𝑡)𝑒𝛼𝑥(𝑡)𝑄𝐸𝑥(𝑡)+𝑅𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+𝑥(𝑡)(𝑆+𝑇)𝑥(𝑡)𝑒𝛼𝑥(𝑡)𝑇𝑥(𝑡)+(1𝜇)1𝑒𝛼(𝑡)𝑥(𝑡(𝑡))𝑆𝑥(𝑡(𝑡))(1𝜇)𝑥(𝑡(𝑡))𝑆𝑥(𝑡(𝑡))+𝑥(𝑡)𝐸̇𝑥(𝑡)𝑋𝑌𝑍𝑥(𝑡)𝐸̇𝑥(𝑡)𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)𝐸̇𝑥(𝜃)𝑋𝑌𝑍𝑥(𝜃)𝐸̇𝑥(𝜃)d𝜃+𝛼𝑥(𝑡)𝐸𝑃𝐸𝑥(𝑡)𝛼𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑅𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃.(2.10) In addition, partitioning the interval [,0] into the union of [,(𝑡)] and [(𝑡),0], by integrating by parts we therefore obtain that 𝐸𝑥(𝑡)𝑒𝛼(𝑡)𝑥(𝑡(𝑡))𝑡𝑡(𝑡)𝛼𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑡𝑡(𝑡)𝑒𝛼(𝜃𝑡)𝐸𝑒̇𝑥(𝜃)d𝜃=0,𝛼(𝑡)𝑥(𝑡(𝑡))𝑒𝛼𝑥(𝑡)𝑡(𝑡)𝑡𝛼𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑡(𝑡)𝑡𝑒𝛼(𝜃𝑡)̇𝑥(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:02𝑥(𝑡)+1𝑒𝛼(𝑡)𝑥(𝑡(𝑡))𝑥(𝑡(𝑡))𝑡𝑡(𝑡)𝛼𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑡𝑡(𝑡)𝑒𝛼(𝜃𝑡)̇𝑥(𝜃)d𝜃×𝐸𝐿1𝑥(𝑡)+𝐿2𝐸̇𝑥(𝑡)+𝐿3𝑥(𝑡(𝑡))+𝐿4𝑥(𝑡)+𝐿5𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+𝐿6𝑡𝑡𝑒𝛼(𝜃𝑡)𝑒𝐸̇𝑥(𝜃)d𝜃+2𝛼(𝑡)𝑒𝛼𝑥(𝑡(𝑡))+𝑒𝛼𝑥(𝑡(𝑡))𝑒𝛼𝑥(𝑡)𝑡(𝑡)𝑡𝛼𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑡(𝑡)𝑡𝑒𝛼(𝜃𝑡)̇𝑥(𝜃)d𝜃×𝐸𝐻1𝑥(𝑡)+𝐻2𝐸̇𝑥(𝑡)+𝐻3𝑥(𝑡(𝑡))+𝐻4𝑥(𝑡)+𝐿5𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+𝐿6𝑡𝑡𝑒𝛼(𝜃𝑡)𝐸̇𝑥(𝜃)d𝜃=2𝑥(𝑡)𝑥(𝑡(𝑡))𝑡𝑡(𝑡)𝛼𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑡𝑡(𝑡)𝑒𝛼(𝜃𝑡)̇𝑥(𝜃)d𝜃×𝐸𝐿1𝑥(𝑡)+𝐿2𝐸̇𝑥(𝑡)+𝐿3𝑥(𝑡(𝑡))+𝐿4𝑒𝑥(𝑡)+2𝛼𝑥(𝑡(𝑡))𝑒𝛼𝑥(𝑡)𝑡(𝑡)𝑡𝛼𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃𝑡(𝑡)𝑡𝑒𝛼(𝜃𝑡)̇𝑥(𝜃)d𝜃×𝐸𝐻1𝑥(𝑡)+𝐻2𝐸̇𝑥(𝑡)+𝐻3𝑥(𝑡(𝑡))+𝐻4𝑥(𝑡)+2𝑥(𝑡)1𝑒𝛼𝑥(𝑡(𝑡))𝑒𝛼𝑥(𝑡)𝐸×𝐿5𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+𝐿6𝑡𝑡𝑒𝛼(𝜃𝑡)𝐸𝐸̇𝑥(𝜃)d𝜃2𝑡𝑡𝛼𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+𝑡𝑡𝑒𝛼(𝜃𝑡)𝐸̇𝑥(𝜃)d𝜃×𝐿5𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+𝐿6𝑡𝑡𝑒𝛼(𝜃𝑡)𝐸̇𝑥(𝜃)d𝜃+21𝑒𝛼(𝑡)𝑥(𝑡(𝑡))𝐸𝐿1𝑥(𝑡)+𝐿2𝐸̇𝑥(𝑡)+𝐿3𝑥(𝑡(𝑡))+𝐿4𝑒𝑥(𝑡)+2𝛼(𝑡)𝑒𝛼𝑥(𝑡(𝑡))𝐸𝐻1𝑥(𝑡)+𝐻2𝐸̇𝑥(𝑡)+𝐻3𝑥(𝑡(𝑡))+𝐻4.𝑥(𝑡)(2.12) Furthermore, we can have additional slack matrices through the following identity 02𝐸̇𝑥(𝑡)𝐴𝑥(𝑡)𝐵𝑥(𝑡(𝑡))×𝐺1𝑥(𝑡)+𝐺2𝐸̇𝑥(𝑡)+𝐺3𝑥(𝑡(𝑡))+𝐺4𝑥(𝑡)+𝐺5𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)d𝜃+𝐺6𝑡𝑡𝑒𝛼(𝜃𝑡).𝐸̇𝑥(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 as𝑥𝜂(𝑡)=(𝑡)𝐸̇𝑥(𝑡)𝑥(𝑡(𝑡))𝑥(𝑡),𝜁𝑡𝜂(𝜃)=(𝑡)𝑥(𝑡+𝜃)𝐸̇𝑥(𝑡+𝜃)(2.14) yields ̇𝑉𝑥𝑡𝑥+𝛼𝑉𝑡𝜂(𝑡)Ξ𝜂(𝑡)+21𝑒𝛼(𝑡)𝑥(𝑡(𝑡))𝐸𝐿1𝑥(𝑡)+𝐿2𝐸̇𝑥(𝑡)+𝐿3𝑥(𝑡(𝑡))+𝐿4𝑒𝑥(𝑡)+2𝛼(𝑡)𝑒𝛼𝑥(𝑡(𝑡))𝐸𝐻1𝑥(𝑡)+𝐻2𝐸̇𝑥(𝑡)+𝐻3𝑥(𝑡(𝑡))+𝐻4𝑥(𝑡)0(𝑡)𝑒𝛼𝜃𝜁𝑡(𝜃)0Σ1𝜁Π𝑡(𝜃)d𝜃(𝑡)𝑒𝛼𝜃𝜁𝑡(𝜃)0Σ2𝜁Π𝑡(𝜃)d𝜃𝑡𝑡𝑒𝛼(𝜃𝑡)𝑥(𝜃)𝐸̇𝑥(𝜃)𝛼d𝜃𝑅+𝐸𝐿5+𝐸𝐿5𝐿5+𝛼𝐸𝐿6𝐿6+𝐿6𝑡𝑡𝑒𝛼(𝜃𝑡)+𝑥(𝜃)𝐸̇𝑥(𝜃)d𝜃1𝑒𝛼(𝑡)𝛼𝑡𝑡(𝑡)𝑒𝛼(𝜃𝑡)𝜂d𝜃(𝑡)Ω1+𝑒𝜂(𝑡)𝛼(𝑡)𝑒𝛼𝛼𝑡(𝑡)𝑡𝑒𝛼(𝜃𝑡)𝜂d𝜃(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 ̇𝑉𝑥𝑡𝑥+𝛼𝑉𝑡𝜂(𝑡)Ξ+1𝑒𝛼(𝑡)Γ1+𝑒𝛼(𝑡)𝑒𝛼Γ2𝜂(𝑡)0(𝑡)𝑒𝛼𝜃𝜁𝑡(𝜃)Υ1𝜁𝑡(𝜃)d𝜃(𝑡)𝑒𝛼𝜃𝜁𝑡(𝜃)Υ2𝜁𝑡(𝜃)d𝜃.(2.16) Thus, recalling (2.6) and (2.7), we have ̇𝑉𝑥𝑡𝑥+𝛼𝑉𝑡𝜂(𝑡)Ξ+1𝑒𝛼(𝑡)Γ1+𝑒𝛼(𝑡)𝑒𝛼Γ2𝜂(𝑡).(2.17) The right-hand side of (2.17) turns out to be convex in 𝑒𝛼(𝑡). It then can be eliminated through the boundary conditions (2.4) and (2.5), which correspond to the cases of 𝑒𝛼=𝛾 and 𝑒𝛼0=1, respectively. We then obtain ̇𝑉𝑥𝑡𝑥𝛼𝑉𝑡.(2.18) And hence, 𝑥1(𝑡)𝑃11𝑥1𝑥(𝑡)𝑉0𝑒𝛼𝑡,𝑡0,(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(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 ΦΦ=11Φ12Φ13Φ22Φ23Φ330,(2.20) where Φ11=𝐴𝐺1𝐺1𝐴+𝛼𝐸𝑃𝐸+𝐿1𝐸+𝐸𝐿1+𝐸𝑄+𝑄𝐸+𝑆, Φ12=𝐸𝐿2+(𝑈𝐹1+𝑃𝐸)+𝐺1𝐴𝐺2, Φ13=𝛾𝐻1𝐸+𝐸𝐿3𝐸𝐿1𝐺1𝐵𝐴𝐺3+(1𝛾)𝐿1𝐸, Φ22=𝐺2+𝐺2, Φ23=𝐿2𝐸+𝛾𝐻2𝐸+𝐺3𝐺2𝐵+(1𝛾)𝐿2𝐸, and Φ33=𝛾(𝐻3𝐸+𝐸𝐻3𝐿3𝐸𝐸𝐿3(1𝜇)𝑆)𝐵𝐺3𝐺3𝐵.
Pre- and postmultiplying (2.20) by the matrix 𝐼𝐴00𝐵𝐼 and its transpose, respectively, we get thatΨΨ=11Ψ12Ψ220,(2.21) where Ψ11=𝛼𝐸𝑃𝐸+𝐿1𝐸+𝐸𝐿1+𝐸𝑄+𝑄𝐸+𝑆+𝐴𝐿2𝐸+𝐸𝐿2𝐴+𝐴𝑃𝐸+𝐸𝑃𝐴+𝐴𝑈𝐹1+𝐹1𝑈𝐴, Ψ12=𝐸𝐿2+𝐸𝑃𝐵+𝐹1𝑈𝐵+𝛾𝐻1𝐸+𝐸𝐿3𝐸𝐿1+(1𝛾)𝐿1𝐸𝐴𝐿2𝐸+𝛾𝐴𝐻2𝐸+(1𝛾)𝐴𝐿2𝐸, and Ψ22=𝛾(𝐻3𝐸+𝐸𝐻3𝐿3𝐸𝐸𝐿3+𝐵𝐻2𝐸+𝐸𝐻2𝐵𝐵𝐿2𝐸𝐸𝐿2𝐵(1𝜇)𝑆).
In fact, the family of slack matrices 𝐺𝑞(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 that𝑆22+𝐹1,22+𝐹1,22𝐹1,22𝐵22𝛾(1𝜇)𝑆220,(2.22) where 𝑆22 and 𝐹1,22 represent the last (𝑛𝑟)×(𝑛𝑟)-order principle blocks of of the matrices 𝑆 and 𝐹1, respectively. According to Lemma 2.1, we have 𝐵22𝑆22𝐵22𝛾(1𝜇)𝑆22𝛾𝑆22.(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/(1𝛿) holds for all vectors 𝜉1,𝜉2 of same dimension and number 𝛿(0,1), from (2.19) and (2.23) we can estimate that𝑥2(𝑡)𝑆22𝑥21(𝑡)𝑥1𝛿1(𝑡(𝑡))𝐵21𝑆22𝐵21𝑥11(𝑡(𝑡))+𝛿𝑥2(𝑡(𝑡))𝐵22𝑆22𝐵22𝑥2𝑥(𝑡(𝑡))𝜒𝑉0𝑒1𝛿𝛼(𝑡(𝑡))+𝛾𝛿sup𝑡𝜃𝑡𝑥2(𝑡+𝜃)𝑆22𝑥2𝑥(𝑡+𝜃)𝜒𝑉0𝑒𝛾(1𝛿)𝛼𝑡+𝛾𝛿sup𝑡𝜃𝑡𝑥2(𝑡+𝜃)𝑆22𝑥2(𝑡+𝜃),𝑡0,(2.24) where 𝜒 is sufficiently large so that 𝐵21𝑆22𝐵21𝜒𝑃11 and 𝛿 is specified within (𝛾,1). Let 𝜓𝑘=sup(𝑘1)𝜃𝑘{𝑥2(𝑡+𝜃)𝑆22𝑥2(𝑡+𝜃)},𝑘=0,1,2,. Then, from (2.24) it follows that 𝜓𝑘𝑥𝜒𝑉0𝑒𝛾(1𝛿)𝛼(𝑘1)+𝛾𝛿𝜓𝑘1𝜓𝑘,𝑘1.(2.25) Therefore, 𝑒𝜖𝑘𝜓𝑘𝑥𝜒𝑉0𝑒𝛾(1𝛿)𝛼(𝛼𝜖)𝑘+𝛾𝛿𝑒𝜖𝑒𝜖(𝑘1)𝜓𝑘1𝑒𝜖𝑘𝜓𝑘,𝑘1,(2.26) where the positive number 𝜖 is sufficiently small for 𝛾𝑒𝜖<𝛿.
Noting 𝜖<(ln𝛿ln𝛾)/<𝛼, we deduce thatmax1𝑖𝑘𝑒𝜖𝑖𝜓𝑖𝑥𝜒𝑉0𝑒𝛾(1𝛿)𝛼+𝛾𝛿𝑒𝜖𝜓0+max1𝑖𝑘𝑒𝜖𝑖𝜓𝑖.(2.27) Hence, max1𝑖𝑘𝑒𝜖𝑖𝜓𝑖𝑥𝜎=𝜒𝑉0/(1𝛿)+(𝛾/𝛿)𝑒𝜖𝜓01(𝛾/𝛿)𝑒𝜖.(2.28) Thus, 𝜓𝑘𝜎𝑒𝜖𝑘,𝑘1.(2.29) But this implies that limsup𝑘ln𝜓𝑘𝑘𝜖.(2.30) And it follows that limsup𝑡||𝑥ln2||(𝑡)2𝑡𝜖.(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 [,0] is decomposed into the union of [,(𝑡)] and [(𝑡),0] and, moreover, the terms factored by 𝑒𝛼(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(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=0.50000.30001𝐵=100.2110.4100.3.(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(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.


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.


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