Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 532912 | 12 pages | https://doi.org/10.1155/2012/532912

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

Academic Editor: Kai Diethelm
Received08 Aug 2011
Revised04 Nov 2011
Accepted04 Nov 2011
Published18 Dec 2011

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, [2ā€“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, [6ā€“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]):Ģ‡š‘„1(š‘”)=š“š‘„1(š‘”)+šµ11š‘„1(š‘”āˆ’ā„Ž(š‘”))+šµ12š‘„2(š‘”āˆ’ā„Ž(š‘”)),(1.1)0=š‘„2(š‘”)+šµ21š‘„1(š‘„āˆ’ā„Ž(š‘”))+šµ22š‘„2(š‘”āˆ’ā„Ž(š‘”)).(1.2) This model is composed of the slow subsystem in (1.1) and the fast subsystem in (1.2). We refer to š‘„1(š‘”)āˆˆš‘…š‘Ÿ and š‘„2(š‘”)āˆˆš‘…š‘›āˆ’š‘Ÿ as the slow and fast variables, respectively. In addition, ā„Ž(š‘”)āˆˆ[0,ā„Ž] is the delay with bounded varying rate |Ģ‡ā„Ž(š‘”)|ā‰¤šœ‡<1. The matrices š“,šµ11,šµ12,šµ21, and šµ22 are of appropriate dimensions. To simplify typography, let us rewrite the system in (1.1) and (1.2) into the form ofšøĢ‡š‘„(š‘”)=š“š‘„(š‘”)+šµš‘„(š‘”āˆ’ā„Ž(š‘”)),(1.3) where ī‚ƒš‘„(š‘”)=š‘„1š‘„(š‘”)2(š‘”)ī‚„, šø=diag{š¼š‘Ÿ,0}, š“=diag{š“,š¼š‘›āˆ’š‘Ÿ}, and ī‚ƒšµ=šµ11šµ12šµ21šµ22ī‚„. 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 šœ–>0 such that limsupš‘”ā†’āˆžln|š‘„(š‘”;š‘„0)|/š‘”ā‰¤āˆ’šœ–.

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 š‘ƒ such that š“ī…žš‘ƒš“<š›¾š‘ƒ, š›¾>0; (ii) there are a symmetric matrix š‘ƒ and a matrix šŗ such that ī‚ƒš›¾š‘ƒš“ā€²šŗā€²ā‹†šŗ+šŗā€²āˆ’š‘ƒī‚„>0.

To study the stability of system (1.1), we construct the Lyapunov functional as follows:š‘‰ī€·š‘„š‘”ī€ø=6ī“š‘˜=1š‘‰š‘˜ī€·š‘„š‘”ī€ø(2.1) with the terms defined asš‘‰1ī€·š‘„š‘”ī€ø=š‘„ī…ž(š‘”)šøī…žš‘‰š‘ƒšøš‘„(š‘”),2ī€·š‘„š‘”ī€ø=2š‘„ī…ž(š‘”)šøī…žš‘„ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„š‘‰(šœƒ)dšœƒ,3ī€·š‘„š‘”ī€ø=ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„ī…žī€œ(šœƒ)dšœƒš‘…š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘‰š‘„(šœƒ)dšœƒ,4ī€·š‘„š‘”ī€ø=ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„ī…žš‘‰(šœƒ)š‘‡š‘„(šœƒ)dšœƒ,5ī€·š‘„š‘”ī€ø=ī€œš‘”š‘”āˆ’ā„Ž(š‘”)š‘’š›¼(šœƒāˆ’š‘”)š‘„ī…ž(š‘‰šœƒ)š‘†š‘„(šœƒ)dšœƒ,6ī€·š‘„š‘”ī€ø=ī€œ0āˆ’ā„Žī€œdšœƒš‘”š‘”+šœƒš‘’š›¼(šœŽāˆ’š‘”)āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦š‘„(šœŽ)šøĢ‡š‘„(šœŽ)ī…žāŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦š‘‹š‘Œā‹†š‘š‘„(šœŽ)šøĢ‡š‘„(šœŽ)dšœŽ.(2.2) Therefore, we can haveĢ‡š‘‰1ī€·š‘„š‘”ī€ø=2Ģ‡š‘„ī…ž(š‘”)šøī…žī€·š‘ˆš¹1ī€øš‘„Ģ‡š‘‰+š‘ƒšø(š‘”),2ī€·š‘„š‘”ī€ø+š›¼š‘‰2ī€·š‘„š‘”ī€ø=2Ģ‡š‘„ī…ž(š‘”)šøī…žī€·š‘ˆš¹2ī€øī€œ+š‘„š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„(šœƒ)dšœƒ+2š‘„ī…ž(š‘”)šøī…žš‘„ī€ŗš‘„(š‘”)āˆ’š‘’āˆ’š›¼ā„Žš‘„ī€»,Ģ‡š‘‰(š‘”āˆ’ā„Ž)3ī€·š‘„š‘”ī€ø+š›¼š‘‰3ī€·š‘„š‘”ī€øī€ŗ=2š‘„(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€»š‘„(š‘”āˆ’ā„Ž)ī…žš‘…ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī€œš‘„(šœƒ)dšœƒāˆ’š›¼š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„ī…žī€œ(šœƒ)dšœƒš‘…š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)Ģ‡š‘‰š‘„(šœƒ)dšœƒ,4ī€·š‘„š‘”ī€ø+š›¼š‘‰4ī€·š‘„š‘”ī€ø=š‘„ī…ž(š‘”)š‘‡š‘„(š‘”)āˆ’š‘’āˆ’š›¼ā„Žš‘„ī…žĢ‡š‘‰(š‘”āˆ’ā„Ž)š‘‡š‘„(š‘”āˆ’ā„Ž),5ī€·š‘„š‘”ī€ø+š›¼š‘‰5ī€·š‘„š‘”ī€ø=š‘„ī…žī€·Ģ‡ī€øš‘’(š‘”)š‘†š‘„(š‘”)āˆ’1āˆ’ā„Ž(š‘”)āˆ’š›¼ā„Ž(š‘”)š‘„ī…žĢ‡š‘‰(š‘”āˆ’ā„Ž(š‘”))š‘†š‘„(š‘”āˆ’ā„Ž(š‘”)),6ī€·š‘„š‘”ī€ø+š›¼š‘‰6ī€·š‘„š‘”ī€øāŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦=ā„Žš‘„(š‘”)šøĢ‡š‘„(š‘”)ī…žāŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦āˆ’ī€œš‘‹š‘Œā‹†š‘š‘„(š‘”)šøĢ‡š‘„(š‘”)š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦š‘„(šœƒ)šøĢ‡š‘„(šœƒ)ī…žāŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦š‘‹š‘Œā‹†š‘š‘„(šœƒ)šøĢ‡š‘„(šœƒ)dšœƒ,(2.3) where š‘ˆ is any square matrix satisfying šøī…žš‘ˆ=0 and rankš‘ˆ=š‘›āˆ’š‘Ÿ, and, therefore, š¹1 and š¹2 can be any square matrices.

Theorem 2.2. Let š›¾āˆˆ(0,1) and š›¼=āˆ’lnš›¾/ā„Ž>0. If there exist matrices š‘ƒ,š‘„,š‘… with ī€ŗš‘ƒš‘„ā‹†š‘…ī€»>0, š‘†>0, š‘‡>0, š‘‹,š‘Œ,š‘, and šæš‘ž(š‘ž=1,ā€¦,6), š»š‘ž(š‘ž=1,ā€¦,4), šŗš‘ž(š‘ž=1,ā€¦,6), š¹š‘ž(š‘ž=1,2), and š‘€š‘š‘ž,š‘š‘š‘ž(š‘,š‘ž=1,ā€¦,4) such that the following matrix inequalities can be satisfied: Īž+(1āˆ’š›¾)Ī“1ā‰¤0,(2.4)Īž+(1āˆ’š›¾)Ī“2Ī„ā‰¤0,(2.5)1=āŽ”āŽ¢āŽ¢āŽ¢āŽ£š›¼Ī©1Ī£1āŽ¤āŽ„āŽ„āŽ„āŽ¦Ī„ā‹†Ī ā‰„0,(2.6)2=āŽ”āŽ¢āŽ¢āŽ¢āŽ£š›¼Ī©2Ī£2āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£š›¼ī€·ā‹†Ī ā‰„0,(2.7)š‘…+šøšæī…ž5+šøī…žšæ5ī€øšæī…ž5+š›¼šøī…žšæ6ā‹†šæī…ž6+šæ6āŽ¤āŽ„āŽ„āŽ„āŽ¦ā‰„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)ā€“(2.8) are defined as follows: āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£ĪžĪž=11Īž12Īž13Īž14ā‹†Īž22Īž23Īž24ā‹†ā‹†Īž33Īž34ā‹†ā‹†ā‹†Īž44āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦;Ī“;Ī =š‘‹š‘Œā‹†š‘1=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š‘€11š‘€12š‘€13+šæī…ž1šøš‘€14ā‹†š‘€22š‘€23+šæī…ž2šøš‘€24ā‹†ā‹†š‘€33+šæī…ž3šø+šøī…žšæ3+(1āˆ’šœ‡)š‘†š‘€34+šøī…žšæ4ā‹†ā‹†ā‹†š‘€44āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦;Ī“2=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š‘11š‘12š‘13+š»ī…ž1šøš‘14ā‹†š‘22š‘23+š»ī…ž2šøš‘24ā‹†ā‹†š‘33+š»ī…ž3šø+šøī…žš»3š‘34+šøī…žš»4ā‹†ā‹†ā‹†š‘44āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦;Ī©1=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š‘€11š‘€12š‘€13š‘€14ā‹†š‘€22š‘€23š‘€24ā‹†ā‹†š‘€33š‘€34ā‹†ā‹†ā‹†š‘€44āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦;Ī©2=āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£š‘11š‘12š‘13š‘14ā‹†š‘22š‘23š‘24ā‹†ā‹†š‘33š‘34ā‹†ā‹†ā‹†š‘44āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦;Ī£1āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āˆ’=āˆ’š“ī…žšŗ5+š‘…+šøī…žšæ5āˆ’š›¼šæī…ž1šøšøī…žšæ6āˆ’šæī…ž1āˆ’š“ī…žšŗ6āˆ’šµī…žšŗ5+ī€·š‘ˆš¹2ī€ø+š‘„āˆ’š›¼šæī…ž2šøāˆ’šµī…žšŗ6āˆ’šæī…ž2āˆ’š›¼šæī…ž3šøāˆ’(1āˆ’š›¾)šøī…žšæ5āˆ’šæī…ž3āˆ’(1āˆ’š›¾)šøī…žšæ6ī€·āˆ’š›¾š‘…+šøī…žšæ5ī€øāˆ’š›¼šæī…ž4šøāˆ’š›¾šøī…žšæ6āˆ’šæī…ž4āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦;Ī£2āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āˆ’=āˆ’š“ī…žšŗ5+š‘…+šøī…žšæ5āˆ’š›¼š»ī…ž1šøšøī…žšæ6āˆ’š»ī…ž1āˆ’š“ī…žšŗ6āˆ’šµī…žšŗ5+ī€·š‘ˆš¹2ī€ø+š‘„āˆ’š›¼š»ī…ž2šøāˆ’šµī…žšŗ6āˆ’š»ī…ž2āˆ’š›¼š»ī…ž3šøāˆ’(1āˆ’š›¾)šøī…žšæ5āˆ’š»ī…ž3āˆ’(1āˆ’š›¾)šøī…žšæ6ī€·āˆ’š›¾š‘…+šøī…žšæ5ī€øāˆ’š›¼š»ī…ž4šøāˆ’š›¾šøī…žšæ6āˆ’š»ī…ž4āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦;Īž11=āˆ’š“ī…žšŗ1āˆ’šŗī…ž1š“+š›¼šøī…žš‘ƒšø+šæī…ž1šø+šøī…žšæ1+šøš‘„ī…ž+š‘„šøī…žĪž+š‘†+š‘‡+ā„Žš‘‹;12=šøī…žšæ2+ī€·š‘ˆš¹1ī€ø+š‘ƒšøī…ž+šŗī…ž1āˆ’š“ī…žšŗ2Īž+ā„Žš‘Œ;13=š›¾š»ī…ž1šø+šøī…žšæ3āˆ’šøī…žšæ1āˆ’šŗī…ž1šµāˆ’š“ī…žšŗ3;Īž14=šøī…žšæ4ī€·š»āˆ’š›¾ī…ž1šø+šøī…žš‘„ī€øāˆ’š“ī…žšŗ4;Īž22=šŗī…ž2+šŗ2Īž+ā„Žš‘;23=āˆ’šæī…ž2šø+š›¾š»ī…ž2šø+šŗ3āˆ’šŗī…ž2Īžšµ;24=āˆ’š›¾š»ī…ž2šø+šŗ4;Īž33ī€·š»=š›¾ī…ž3šø+šøī…žš»3ī€øāˆ’šæī…ž3šøāˆ’šøī…žšæ3āˆ’(1āˆ’šœ‡)š‘†āˆ’šµī…žšŗ3āˆ’šŗī…ž3Īžšµ;34=āˆ’šøī…žšæ4+š›¾šøī…žš»4āˆ’š›¾š»ī…ž3šøāˆ’šµī…žšŗ4;Īž44ī€·=āˆ’š›¾š‘‡+š»ī…ž4šø+šøī…žš»4ī€ø.(2.9)

Proof. The first thing we have to do is to note that āˆ‘5š‘˜=1š‘‰š‘˜(š‘„š‘”) is positive-definite, and the positiveness of š‘‰6(š‘„š‘”) 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 Ģ‡š‘‰ī€·š‘„š‘”ī€øī€·š‘„+š›¼š‘‰š‘”ī€øā‰¤2Ģ‡š‘„ī…ž(š‘”)šøī…žī€·š‘ˆš¹1ī€øš‘„+š‘ƒšø(š‘”)+2Ģ‡š‘„ī…ž(š‘”)šøī…žī€·š‘ˆš¹2ī€øī€œ+š‘„š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„ī€ŗ(šœƒ)dšœƒ+2š‘„(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€»š‘„(š‘”āˆ’ā„Ž)ī…žī‚øī€œš‘„ā€²šøš‘„(š‘”)+š‘…š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹š‘„(šœƒ)dšœƒ+š‘„ī…ž(š‘”)(š‘†+š‘‡)š‘„(š‘”)āˆ’š‘’āˆ’š›¼ā„Žš‘„ī…žī€·(š‘”āˆ’ā„Ž)š‘‡š‘„(š‘”āˆ’ā„Ž)+(1āˆ’šœ‡)1āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€øš‘„ī…ž(š‘”āˆ’ā„Ž(š‘”))š‘†š‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’(1āˆ’šœ‡)š‘„ī…žāŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦(š‘”āˆ’ā„Ž(š‘”))š‘†š‘„(š‘”āˆ’ā„Ž(š‘”))+ā„Žš‘„(š‘”)šøĢ‡š‘„(š‘”)ī…žāŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦āˆ’ī€œš‘‹š‘Œā‹†š‘š‘„(š‘”)šøĢ‡š‘„(š‘”)š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦š‘„(šœƒ)šøĢ‡š‘„(šœƒ)ī…žāŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦š‘‹š‘Œā‹†š‘š‘„(šœƒ)šøĢ‡š‘„(šœƒ)dšœƒ+š›¼š‘„ī…ž(š‘”)šøī…žī€œš‘ƒšøš‘„(š‘”)āˆ’š›¼š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„ī…žī€œ(šœƒ)dšœƒš‘…š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„(šœƒ)dšœƒ.(2.10) In addition, partitioning the interval [āˆ’ā„Ž,0] into the union of [āˆ’ā„Ž,āˆ’ā„Ž(š‘”)] and [āˆ’ā„Ž(š‘”),0], by integrating by parts we therefore obtain that šøī‚øš‘„(š‘”)āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€œš‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘”š‘”āˆ’ā„Ž(š‘”)š›¼š‘’š›¼(šœƒāˆ’š‘”)ī€œš‘„(šœƒ)dšœƒāˆ’š‘”š‘”āˆ’ā„Ž(š‘”)š‘’š›¼(šœƒāˆ’š‘”)ī‚¹šøī‚ƒš‘’Ģ‡š‘„(šœƒ)dšœƒ=0,āˆ’š›¼ā„Ž(š‘”)š‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘’āˆ’š›¼ā„Žāˆ«š‘„(š‘”āˆ’ā„Ž)āˆ’š‘”āˆ’ā„Ž(š‘”)š‘”āˆ’ā„Žš›¼š‘’š›¼(šœƒāˆ’š‘”)āˆ«š‘„(šœƒ)dšœƒāˆ’š‘”āˆ’ā„Ž(š‘”)š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚„Ģ‡š‘„(šœƒ)dšœƒ=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:ī‚øī€·0ā‰”2š‘„(š‘”)+1āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€øī€œš‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘”š‘”āˆ’ā„Ž(š‘”)š›¼š‘’š›¼(šœƒāˆ’š‘”)āˆ’ī€œš‘„(šœƒ)dšœƒš‘”š‘”āˆ’ā„Ž(š‘”)š‘’š›¼(šœƒāˆ’š‘”)ī‚¹Ģ‡š‘„(šœƒ)dšœƒī…žĆ—šøī…žī‚øšæ1š‘„(š‘”)+šæ2šøĢ‡š‘„(š‘”)+šæ3š‘„(š‘”āˆ’ā„Ž(š‘”))+šæ4š‘„(š‘”āˆ’ā„Ž)+šæ5ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„(šœƒ)dšœƒ+šæ6ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹ī‚øī€·š‘’šøĢ‡š‘„(šœƒ)dšœƒ+2āˆ’š›¼ā„Ž(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€øš‘„(š‘”āˆ’ā„Ž(š‘”))+š‘’āˆ’š›¼ā„Žš‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘’āˆ’š›¼ā„Žī€œš‘„(š‘”āˆ’ā„Ž)āˆ’š‘”āˆ’ā„Ž(š‘”)š‘”āˆ’ā„Žš›¼š‘’š›¼(šœƒāˆ’š‘”)āˆ’ī€œš‘„(šœƒ)dšœƒš‘”āˆ’ā„Ž(š‘”)š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹Ģ‡š‘„(šœƒ)dšœƒī…žĆ—šøī…žī‚øš»1š‘„(š‘”)+š»2šøĢ‡š‘„(š‘”)+š»3š‘„(š‘”āˆ’ā„Ž(š‘”))+š»4š‘„(š‘”āˆ’ā„Ž)+šæ5ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„(šœƒ)dšœƒ+šæ6ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹ī‚øī€œšøĢ‡š‘„(šœƒ)dšœƒ=2š‘„(š‘”)āˆ’š‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘”š‘”āˆ’ā„Ž(š‘”)š›¼š‘’š›¼(šœƒāˆ’š‘”)ī€œš‘„(šœƒ)dšœƒāˆ’š‘”š‘”āˆ’ā„Ž(š‘”)š‘’š›¼(šœƒāˆ’š‘”)ī‚¹Ģ‡š‘„(šœƒ)dšœƒī…žĆ—šøī…žī€ŗšæ1š‘„(š‘”)+šæ2šøĢ‡š‘„(š‘”)+šæ3š‘„(š‘”āˆ’ā„Ž(š‘”))+šæ4ī€»ī‚øš‘’š‘„(š‘”āˆ’ā„Ž)+2āˆ’š›¼ā„Žš‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘’āˆ’š›¼ā„Žš‘„ī€œ(š‘”āˆ’ā„Ž)āˆ’š‘”āˆ’ā„Ž(š‘”)š‘”āˆ’ā„Žš›¼š‘’š›¼(šœƒāˆ’š‘”)š‘„ī€œ(šœƒ)dšœƒāˆ’š‘”āˆ’ā„Ž(š‘”)š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹Ģ‡š‘„(šœƒ)dšœƒī…žĆ—šøī…žī€ŗš»1š‘„(š‘”)+š»2šøĢ‡š‘„(š‘”)+š»3š‘„(š‘”āˆ’ā„Ž(š‘”))+š»4ī€»ī€ŗī€·š‘„(š‘”āˆ’ā„Ž)+2š‘„(š‘”)āˆ’1āˆ’š‘’āˆ’š›¼ā„Žī€øš‘„(š‘”āˆ’ā„Ž(š‘”))āˆ’š‘’āˆ’š›¼ā„Žī€»š‘„(š‘”āˆ’ā„Ž)ī…žšøī…žĆ—ī‚øšæ5ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„(šœƒ)dšœƒ+šæ6ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹ī‚øšøī€œšøĢ‡š‘„(šœƒ)dšœƒāˆ’2š‘”š‘”āˆ’ā„Žš›¼š‘’š›¼(šœƒāˆ’š‘”)š‘„ī€œ(šœƒ)dšœƒ+š‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹šøĢ‡š‘„(šœƒ)dšœƒī…žĆ—ī‚øšæ5ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„(šœƒ)dšœƒ+šæ6ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹ī€·šøĢ‡š‘„(šœƒ)dšœƒ+21āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€øš‘„ī…ž(š‘”āˆ’ā„Ž(š‘”))šøī…žī€ŗšæ1š‘„(š‘”)+šæ2šøĢ‡š‘„(š‘”)+šæ3š‘„(š‘”āˆ’ā„Ž(š‘”))+šæ4ī€»ī€·š‘’š‘„(š‘”āˆ’ā„Ž)+2āˆ’š›¼ā„Ž(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€øš‘„ī…ž(š‘”āˆ’ā„Ž(š‘”))šøī…žī€ŗš»1š‘„(š‘”)+š»2šøĢ‡š‘„(š‘”)+š»3š‘„(š‘”āˆ’ā„Ž(š‘”))+š»4ī€».š‘„(š‘”āˆ’ā„Ž)(2.12) Furthermore, we can have additional slack matrices through the following identity ī‚ƒ0ā‰”2šøĢ‡š‘„(š‘”)āˆ’š“š‘„(š‘”)āˆ’ī‚„šµš‘„(š‘”āˆ’ā„Ž(š‘”))ī…žĆ—ī‚øšŗ1š‘„(š‘”)+šŗ2šøĢ‡š‘„(š‘”)+šŗ3š‘„(š‘”āˆ’ā„Ž(š‘”))+šŗ4š‘„(š‘”āˆ’ā„Ž)+šŗ5ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)š‘„(šœƒ)dšœƒ+šŗ6ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹.šøĢ‡š‘„(šœƒ)dšœƒ(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 asī‚ƒš‘„šœ‚(š‘”)=ī…ž(š‘”)šøĢ‡š‘„(š‘”)š‘„ī…ž(š‘”āˆ’ā„Ž(š‘”))š‘„ī…žī‚„(š‘”āˆ’ā„Ž)ī…ž,šœš‘”ī‚ƒšœ‚(šœƒ)=ī…ž(š‘”)š‘„ī…ž(š‘”+šœƒ)šøī…žĢ‡š‘„ī…žī‚„(š‘”+šœƒ)ī…ž(2.14) yields Ģ‡š‘‰ī€·š‘„š‘”ī€øī€·š‘„+š›¼š‘‰š‘”ī€øā‰¤šœ‚ī…žī€·(š‘”)Īžšœ‚(š‘”)+21āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€øš‘„ī…ž(š‘”āˆ’ā„Ž(š‘”))šøī…žī€ŗšæ1š‘„(š‘”)+šæ2šøĢ‡š‘„(š‘”)+šæ3š‘„(š‘”āˆ’ā„Ž(š‘”))+šæ4ī€»ī€·š‘’š‘„(š‘”āˆ’ā„Ž)+2āˆ’š›¼ā„Ž(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€øš‘„ī…ž(š‘”āˆ’ā„Ž(š‘”))šøī…žī€ŗš»1š‘„(š‘”)+š»2šøĢ‡š‘„(š‘”)+š»3š‘„(š‘”āˆ’ā„Ž(š‘”))+š»4ī€»āˆ’ī€œš‘„(š‘”āˆ’ā„Ž)0āˆ’ā„Ž(š‘”)š‘’š›¼šœƒšœī…žš‘”āŽ”āŽ¢āŽ¢āŽ¢āŽ£(šœƒ)0Ī£1āŽ¤āŽ„āŽ„āŽ„āŽ¦šœā‹†Ī š‘”ī€œ(šœƒ)dšœƒāˆ’āˆ’ā„Ž(š‘”)āˆ’ā„Žš‘’š›¼šœƒšœī…žš‘”āŽ”āŽ¢āŽ¢āŽ¢āŽ£(šœƒ)0Ī£2āŽ¤āŽ„āŽ„āŽ„āŽ¦šœā‹†Ī š‘”āˆ’ī€œ(šœƒ)dšœƒš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦š‘„(šœƒ)šøĢ‡š‘„(šœƒ)ī…žāŽ”āŽ¢āŽ¢āŽ¢āŽ£š›¼ī€·dšœƒš‘…+šøšæī…ž5+šøī…žšæ5ī€øšæī…ž5+š›¼šøī…žšæ6ā‹†šæī…ž6+šæ6āŽ¤āŽ„āŽ„āŽ„āŽ¦ī€œš‘”š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)āŽ”āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ¦+ī‚øī€·š‘„(šœƒ)šøĢ‡š‘„(šœƒ)dšœƒ1āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€øī€œāˆ’š›¼š‘”š‘”āˆ’ā„Ž(š‘”)š‘’š›¼(šœƒāˆ’š‘”)ī‚¹šœ‚dšœƒī…ž(š‘”)Ī©1+ī‚øī€·š‘’šœ‚(š‘”)āˆ’š›¼ā„Ž(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€øī€œāˆ’š›¼š‘”āˆ’ā„Ž(š‘”)š‘”āˆ’ā„Žš‘’š›¼(šœƒāˆ’š‘”)ī‚¹šœ‚dšœƒī…ž(š‘”)Ī©2šœ‚(š‘”).(2.15) By using (2.8) to drop the sixth term and arranging the remaining terms on the right-hand side, (2.15) becomes Ģ‡š‘‰ī€·š‘„š‘”ī€øī€·š‘„+š›¼š‘‰š‘”ī€øā‰¤šœ‚ī…žī€ŗī€·(š‘”)Īž+1āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€øĪ“1+ī€·š‘’āˆ’š›¼ā„Ž(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€øĪ“2ī€»āˆ’ī€œšœ‚(š‘”)0āˆ’ā„Ž(š‘”)š‘’š›¼šœƒšœī…žš‘”(šœƒ)Ī„1šœš‘”ī€œ(šœƒ)dšœƒāˆ’āˆ’ā„Ž(š‘”)āˆ’ā„Žš‘’š›¼šœƒšœī…žš‘”(šœƒ)Ī„2šœš‘”(šœƒ)dšœƒ.(2.16) Thus, recalling (2.6) and (2.7), we have Ģ‡š‘‰ī€·š‘„š‘”ī€øī€·š‘„+š›¼š‘‰š‘”ī€øā‰¤šœ‚ī…žī€ŗī€·(š‘”)Īž+1āˆ’š‘’āˆ’š›¼ā„Ž(š‘”)ī€øĪ“1+ī€·š‘’āˆ’š›¼ā„Ž(š‘”)āˆ’š‘’āˆ’š›¼ā„Žī€øĪ“2ī€»šœ‚(š‘”).(2.17) The right-hand side of (2.17) turns out to be convex in š‘’āˆ’š›¼ā„Ž(š‘”). It then can be eliminated through the boundary conditions (2.4) and (2.5), which correspond to the cases of š‘’āˆ’š›¼ā„Ž=š›¾ and š‘’āˆ’š›¼0=1, respectively. We then obtain Ģ‡š‘‰ī€·š‘„š‘”ī€øī€·š‘„ā‰¤āˆ’š›¼š‘‰š‘”ī€ø.(2.18) And hence, š‘„ī…ž1(š‘”)š‘ƒ11š‘„1ī€·š‘„(š‘”)ā‰¤š‘‰0ī€øš‘’āˆ’š›¼š‘”,š‘”ā‰„0,(2.19) where š‘ƒ11 is the first š‘ŸĆ—š‘Ÿ order principle block of the matrix š‘ƒ. 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 š‘„2(š‘”)+šµ22š‘„2(š‘”āˆ’ā„Ž(š‘”))=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 āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£Ī¦Ī¦=11Ī¦12Ī¦13ā‹†Ī¦22Ī¦23ā‹†ā‹†Ī¦33āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦ā‰¤0,(2.20) where Ī¦11=āˆ’š“ī…žšŗ1āˆ’šŗī…ž1š“+š›¼šøī…žš‘ƒšø+šæī…ž1šø+šøī…žšæ1+šøš‘„ī…ž+š‘„šøī…ž+š‘†, Ī¦12=šøī…žšæ2+(š‘ˆš¹1+š‘ƒšø)ī…ž+šŗī…ž1āˆ’š“ī…žšŗ2, Ī¦13=š›¾š»ī…ž1šø+šøī…žšæ3āˆ’šøī…žšæ1āˆ’šŗī…ž1šµāˆ’š“ī…žšŗ3+(1āˆ’š›¾)šæī…ž1šø, Ī¦22=šŗī…ž2+šŗ2, Ī¦23=āˆ’šæī…ž2šø+š›¾š»ī…ž2šø+šŗ3āˆ’šŗī…ž2šµ+(1āˆ’š›¾)šæī…ž2šø, and Ī¦33=š›¾(š»ī…ž3šø+šøī…žš»3āˆ’šæī…ž3šøāˆ’šøī…žšæ3āˆ’(1āˆ’šœ‡)š‘†)āˆ’šµī…žšŗ3āˆ’šŗī…ž3šµ.
Pre- and postmultiplying (2.20) by the matrix ī‚ƒš¼š“ā€²00šµā€²š¼ī‚„ and its transpose, respectively, we get thatāŽ”āŽ¢āŽ¢āŽ¢āŽ£ĪØĪØ=11ĪØ12ā‹†ĪØ22āŽ¤āŽ„āŽ„āŽ„āŽ¦ā‰¤0,(2.21) where ĪØ11=š›¼šøī…žš‘ƒšø+šæī…ž1šø+šøī…žšæ1+šøš‘„ī…ž+š‘„šøī…ž+š‘†+š“ī…žšæī…ž2šø+šøī…žšæ2š“+š“ī…žš‘ƒšø+šøī…žš‘ƒš“+š“ī…žš‘ˆš¹1+š¹ī…ž1š‘ˆī…žš“, ĪØ12=šøī…žšæ2+šøī…žš‘ƒšµ+š¹ī…ž1š‘ˆī…žšµ+š›¾š»ī…ž1šø+šøī…žšæ3āˆ’šøī…žšæ1+(1āˆ’š›¾)šæī…ž1šøāˆ’š“ī…žšæ2šø+š›¾š“ī…žš»ī…ž2šø+(1āˆ’š›¾)š“ī…žšæī…ž2šø, and ĪØ22=š›¾(š»ī…ž3šø+šøī…žš»3āˆ’šæī…ž3šøāˆ’šøī…žšæ3+šµī…žš»ī…ž2šø+šøī…žš»2šµāˆ’šµī…žšæī…ž2šøāˆ’šøī…žšæ2šµāˆ’(1āˆ’šœ‡)š‘†).
In fact, the family of slack matrices šŗš‘ž(š‘ž=1,ā€¦,6) plays a key role in getting (2.21). Furthermore, noting the special structures of the matrices šø, š‘ˆ, and š“, from (2.21) we can deduce thatāŽ”āŽ¢āŽ¢āŽ¢āŽ£š‘†22+š¹ī…ž1,22+š¹1,22š¹ī…ž1,22šµ22ā‹†āˆ’š›¾(1āˆ’šœ‡)š‘†22āŽ¤āŽ„āŽ„āŽ„āŽ¦ā‰¤0,(2.22) where š‘†22 and š¹1,22 represent the last (š‘›āˆ’š‘Ÿ)Ɨ(š‘›āˆ’š‘Ÿ)-order principle blocks of of the matrices š‘† and š¹1, respectively. According to Lemma 2.1, we have šµī…ž22š‘†22šµ22ā‰¤š›¾(1āˆ’šœ‡)š‘†22ā‰¤š›¾š‘†22.(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 |šœ‰1+šœ‰2|2ā‰¤|šœ‰1|2/š›æ+|šœ‰2|2/(1āˆ’š›æ) holds for all vectors šœ‰1,šœ‰2 of same dimension and number š›æāˆˆ(0,1), from (2.19) and (2.23) we can estimate thatš‘„ī…ž2(š‘”)š‘†22š‘„21(š‘”)ā‰¤š‘„1āˆ’š›æī…ž1(š‘”āˆ’ā„Ž(š‘”))šµī…ž21š‘†22šµ21š‘„11(š‘”āˆ’ā„Ž(š‘”))+š›æš‘„ī…ž2(š‘”āˆ’ā„Ž(š‘”))šµī…ž22š‘†22šµ22š‘„2ā‰¤ī€·š‘„(š‘”āˆ’ā„Ž(š‘”))šœ’š‘‰0ī€øš‘’1āˆ’š›æāˆ’š›¼(š‘”āˆ’ā„Ž(š‘”))+š›¾š›æsupš‘”āˆ’ā„Žā‰¤šœƒā‰¤š‘”š‘„ī…ž2(š‘”+šœƒ)š‘†22š‘„2ā‰¤ī€·š‘„(š‘”+šœƒ)šœ’š‘‰0ī€øš‘’š›¾(1āˆ’š›æ)āˆ’š›¼š‘”+š›¾š›æsupš‘”āˆ’ā„Žā‰¤šœƒā‰¤š‘”š‘„ī…ž2(š‘”+šœƒ)š‘†22š‘„2(š‘”+šœƒ),š‘”ā‰„0,(2.24) where šœ’ is sufficiently large so that šµī…ž21š‘†22šµ21ā‰¤šœ’š‘ƒ11 and š›æ is specified within (š›¾,1). Let šœ“š‘˜=sup(š‘˜āˆ’1)ā„Žā‰¤šœƒā‰¤š‘˜ā„Ž{š‘„ī…ž2(š‘”+šœƒ)š‘†22š‘„2(š‘”+šœƒ)},š‘˜=0,1,2,ā€¦. Then, from (2.24) it follows that šœ“š‘˜ā‰¤ī€·š‘„šœ’š‘‰0ī€øš‘’š›¾(1āˆ’š›æ)āˆ’š›¼(š‘˜āˆ’1)ā„Ž+š›¾š›æī€·šœ“š‘˜āˆ’1āˆØšœ“š‘˜ī€ø,š‘˜ā‰„1.(2.25) Therefore, š‘’šœ–š‘˜ā„Žšœ“š‘˜ā‰¤ī€·š‘„šœ’š‘‰0ī€øš‘’š›¾(1āˆ’š›æ)š›¼ā„Žāˆ’(š›¼āˆ’šœ–)š‘˜ā„Ž+š›¾š›æš‘’šœ–ā„Žī€·š‘’šœ–(š‘˜āˆ’1)ā„Žšœ“š‘˜āˆ’1āˆØš‘’šœ–š‘˜ā„Žšœ“š‘˜ī€ø,š‘˜ā‰„1,(2.26) where the positive number šœ– is sufficiently small for š›¾š‘’šœ–ā„Ž<š›æ.
Noting šœ–<(lnš›æāˆ’lnš›¾)/ā„Ž<š›¼, we deduce thatmax1ā‰¤š‘–ā‰¤š‘˜ī€·š‘’šœ–š‘–ā„Žšœ“š‘–ī€øā‰¤ī€·š‘„šœ’š‘‰0ī€øš‘’š›¾(1āˆ’š›æ)š›¼ā„Ž+š›¾š›æš‘’šœ–ā„Žī‚øšœ“0+max1ā‰¤š‘–ā‰¤š‘˜ī€·š‘’šœ–š‘–ā„Žšœ“š‘–ī€øī‚¹.(2.27) Hence, max1ā‰¤š‘–ā‰¤š‘˜ī€·š‘’šœ–š‘–ā„Žšœ“š‘–ī€øī€·ī€·š‘„ā‰¤šœŽāˆ¶=šœ’š‘‰0ī€ø/(1āˆ’š›æ)+(š›¾/š›æ)š‘’šœ–ā„Žšœ“0ī€øī€·1āˆ’(š›¾/š›æ)š‘’šœ–ā„Žī€ø.(2.28) Thus, šœ“š‘˜ā‰¤šœŽš‘’āˆ’šœ–š‘˜ā„Ž,š‘˜ā‰„1.(2.29) But this implies that limsupš‘˜ā†’āˆžlnšœ“š‘˜š‘˜ā„Žā‰¤āˆ’šœ–.(2.30) And it follows that limsupš‘”ā†’āˆž||š‘„ln2||(š‘”)2š‘”ā‰¤āˆ’šœ–.(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 [āˆ’ā„Ž,0] is decomposed into the union of [āˆ’ā„Ž,āˆ’ā„Ž(š‘”)] and [āˆ’ā„Ž(š‘”),0] and, moreover, the terms factored by š‘’āˆ’š›¼ā„Ž(š‘”) 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, š›¾, which deserves a brief discussion. In fact, as shown in (2.23), it is introduced to guarantee the difference equation š‘„2(š‘”)+šµ22š‘„2(š‘”āˆ’ā„Ž(š‘”))=0 to be Schur stable. Therefore, we use š›¾ to characterize the effect of the fast subsystem on the decay rate of the slow variables, š›¼. Indeed, this turns out to be a typical perturbation approach to prove stability. Besides, š›¾ 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: āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦,šø=100010000āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦,š“=0.50000.30001āŽ”āŽ¢āŽ¢āŽ¢āŽ¢āŽ¢āŽ£āŽ¤āŽ„āŽ„āŽ„āŽ„āŽ„āŽ¦šµ=āˆ’100.2āˆ’1āˆ’1āˆ’0.4100.3.(3.1) Specified š›¾=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 š›¾ with fixing šœ‡=0, which is shown in Table 2. In the light of the discussion on the parameter š›¾, it would become clear that there is a mutually constraint relation between the difference operator š‘„2(š‘”)+šµ22š‘„2(š‘”āˆ’ā„Ž(š‘”)) and the size of delay in guaranteeing stability.


šœ‡ 0.0 0.5 0.9
ā„Ž 2.130 1.741 1.336


š›¾ 0.6 0.7 0.8 0.9 0.99
ā„Ž 1.786 1.895 1.990 2.071 2.137

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.

References

  1. L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1989. View at: Publisher Site
  2. J. Y. Ishihara and M. H. Terra, ā€œOn the Lyapunov theorem for singular systems,ā€ IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1926ā€“1930, 2008. View at: Publisher Site | Google Scholar
  3. I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, ā€œHāˆž control for descriptor systems: a matrix inequalities approach,ā€ Automatica, vol. 33, no. 4, pp. 669ā€“673, 1997. View at: Publisher Site | Google Scholar
  4. E. Uezato and M. Ikeda, ā€œStrict LMI conditions for stability, robust stabilization, and Hāˆž control of descriptor systems,ā€ in Proceedings of the 38th IEEE Conference on Decision and Control, pp. 4092ā€“4097, Phoenix, Ariz, USA, 1999. View at: Google Scholar
  5. K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems, BirkhƤuser, Boston, Mass, USA, 2003.
  6. S. Zhu, C. Zhang, Z. Cheng, and J. Feng, ā€œDelay-dependent robust stability criteria for two classes of uncertain singular time-delay systems,ā€ IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 880ā€“885, 2007. View at: Publisher Site | Google Scholar
  7. S. L. Campbell and V. H. Linh, ā€œStability criteria for differential-algebraic equations with multiple delays and their numerical solutions,ā€ Applied Mathematics and Computation, vol. 208, no. 2, pp. 397ā€“415, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. E. Fridman, ā€œStability of linear descriptor systems with delay: a Lyapunov-based approach,ā€ Journal of Mathematical Analysis and Applications, vol. 273, no. 1, pp. 24ā€“44, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. A. Haidar and E. K. Boukas, ā€œExponential stability of singular systems with multiple time-varying delays,ā€ Automatica, vol. 45, no. 2, pp. 539ā€“545, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. Z. Wu and W. Zhou, ā€œDelay-dependent robust Hāˆž control for uncertain singular time-delay systems,ā€ IET Control Theory & Applications, vol. 1, no. 5, pp. 1234ā€“1241, 2007. View at: Publisher Site | Google Scholar
  11. S. Xu, P. van Dooren, R. Ştefan, and J. Lam, ā€œRobust stability and stabilization for singular systems with state delay and parameter uncertainty,ā€ IEEE Transactions on Automatic Control, vol. 47, no. 7, pp. 1122ā€“1128, 2002. View at: Publisher Site | Google Scholar
  12. D. Yue and Q. L. Han, ā€œDelay-dependent robust Hāˆž controller design for uncertain descriptor systems with time-varying discrete and distributed delays,ā€ IEE Control Theory and Applications, vol. 152, no. 6, pp. 628ā€“638, 2005. View at: Publisher Site | Google Scholar
  13. S. Zhu, Z. Li, and C. Zhang, ā€œDelay decomposition approach to delay-dependent stability for singular time-delay systems,ā€ IET Control Theory & Applications, vol. 4, no. 11, pp. 2613ā€“2620, 2010. View at: Publisher Site | Google Scholar
  14. M. C. de Oliveira, J. Bernussou, and J. C. Geromel, ā€œA new discrete-time robust stability condition,ā€ Systems & Control Letters, vol. 37, no. 4, pp. 261ā€“265, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright Ā© 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.


More related articles

1304Ā Views | 414Ā Downloads | 3Ā Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.