Elsevier

Automatica

Volume 49, Issue 11, November 2013, Pages 3368-3376
Automatica

Brief paper
On exponential stability of integral delay systems

https://doi.org/10.1016/j.automatica.2013.08.004Get rights and content

Abstract

This paper is concerned with exponential stability of a class of integral delay systems with a prescribed decay rate. First, by carefully exploring the literature on this topic, a delay decomposition approach is established to reduce the conservatism in the existing sufficient conditions by constructing new Lyapunov–Krasovskii (LK) functionals. It is proven that the proposed sufficient conditions are less conservative than a recently established set of sufficient conditions. Second, by analyzing the characteristic equation of the considered integral delay system, necessary and sufficient conditions for the stability are obtained by computing the right-most zeros of a certain auxiliary point-delay linear system, for which stability criteria that are easy to test are also established based on this method. Numerical examples illustrate the effectiveness of the obtained results.

Introduction

Time delay arises frequently in many engineering systems such as nuclear reactors, long transmission lines in pneumatic systems, rolling mills, sampled-data control, manufacturing processes and networked control systems (Fridman, 2010, Gu et al., 2003, Hale, 1977). Because of their infinite dimensional nature, control problems, especially, the problems of asymptotic stability analysis and stabilization, for time-delay systems have been recognized to be very difficult. As a result, during the past several decades, control of time-delay systems has received considerable attentions from the researchers and a large number of results have been reported in the literature (see Du, Lam, and Shu (2011), He, Wang, Lin, and Wu (2007), Lam, Gao, and Wang (2007), Xie, Fridman, and Shaked (2001), Xu, Lam, and Yang (2001), Yakoubi and Chitour (2007), Zhang, Zhang, and Xie (2004) and Zhou, Lin, and Duan (2012) and the references therein).

Among the existing methods for carrying out asymptotic stability analysis and stabilization of time-delay systems, the Lyapunov–Krasovskii (LK) functional based methods are probably the most efficient ones. In these methods, one generally needs to start with certain system transformations that bring into the original delay system additional dynamics that can be described by the so-called integral delay system (Gu and Niculescu, 2001, Kharitonov and Melchor-Aguilar, 2003). The stability of such an integral delay system is thus important in the stability analysis of the original delay system. Another efficient approach to dealing with the stabilization of time-delay systems is the predictor feedback, which is especially effective for input delayed systems (Artstein, 1982, Krstic, 2010, Mirkin, 2004). However, it is now clear that the resulting infinite dimensional controllers can be safely implemented if and only if certain integral delay systems are asymptotically stable (Mondie et al., 2001, Mondie and Melchor-Aguilar, 2012, Van Assche et al., 1999).

Besides the two examples mentioned above, integral delay systems appear also in some other problems associated with delay systems, for example, delay approximations of the partial differential equations for describing the propagation phenomena in excitable media (Niculescu, 2001), and stability analysis of some difference operators in neutral type functional differential equations (Hale, 1977). For more introduction on integral delay systems, the reader may refer to Melchor-Aguilar (2010), Melchor-Aguilar, Kharitonov, and Lozano (2010) and Mondie and Melchor-Aguilar (2012) and the references cited there. Because of their many applications, integral delay systems have received much attention in recent years. LK theorems for integral delay systems have recently been introduced in Melchor-Aguilar (2010) and Melchor-Aguilar et al. (2010). With the help of this LK theorem, some sufficient conditions in terms of linear matrix inequalities (LMIs) were recently established in Mondie and Melchor-Aguilar (2012) for the exponential stability of some classes of integral delay systems with analytic kernels, which include the integral delay systems encountered in the model transformation and predictor feedback for time-delay systems as special cases.

In this paper, with the aid of the delay decomposition technique (Gouaisbaut & Peaucelle, 2006), we find that the LK functional proposed in Mondie and Melchor-Aguilar (2012) can be written as an integral quadratic form of some generalized state vector obtained by the fractionizing of the delay intervals of the state. This motivates us to propose a more general LK functional in terms of this generalized state vector. Delay dependent LMI conditions guaranteeing the exponential stability of an integral delay system are then obtained by using this new LK functional. It is proven that these conditions are less conservative than those proposed in Mondie and Melchor-Aguilar (2012). Moreover, by analyzing the characteristic equation of the considered integral delay system, necessary and sufficient conditions for stability are obtained in terms of the right-most (unstable) zeros of a certain auxiliary point-delay linear system. Consequently, for the special case that a parameter matrix in the kernel of the integral delay system is anti-stable, the stability of the integral delay system is equivalent to the stability of the auxiliary point-delay system, for which stability criteria that are easy to test are also presented. Numerical examples are worked out to illustrate the effectiveness of the obtained results.

The remainder of this paper is organized as follows. The problem formulation and some preliminary results are presented in Section  2. The LMIs based sufficient conditions are given in Section  3 while the characteristic equation based results are given in Section  4. Numerical examples are presented in Section  5 to show the effectiveness of the proposed approach and Section  6 concludes the paper.

Notation

The notation used in this paper is fairly standard. For a matrix ARn×n, we use AT,rank(A),λ(A),det(A) and He(A) to denote its transpose, rank, eigenvalue set, determinant, and the symmetric matrix A+AT. We use diag{A1,A2,,Ap} to denote a diagonal matrices whose diagonal elements are Ai,i=1,2,,p and AB to denote the Kronecker product of matrices A and B. For a semi-positive definite matrix P, we use λmin(P) and λmax(P) to denote, respectively, its minimal and maximal eigenvalues. For a positive scalar h and an integer m, let Cm,h=C([h,0],Rm) denote the Banach space of continuous vector functions mapping the interval [h,0] into Rm with the topology of uniform convergence, and let xtCm,h denote the restriction of x(t) to the interval [th,t] translated to [h,0], that is, xt(θ)=x(t+θ),θ[h,0]. For any φCm,h, the norm of φ is defined as φh=supθ[h,0]φ(θ).

Section snippets

Problem formulation and preliminaries

In this paper we are interested in the following integral delay system: x(t)=h0GB(θ)x(t+θ)dθ,t0, where h>0 is a constant, GRm×n, and B(θ):[h,0]Rn×m. Furthermore, we assume that B(θ) is continuously differentiable for all θ[h,0] and satisfies (Mondie & Melchor-Aguilar, 2012) Ḃ(θ)=MB(θ),θ[h,0], with MRn×n being a constant matrix. For notational simplicity, we denote B(0)=B. Moreover, according to Mondie and Melchor-Aguilar (2012), we should assume that there exists a γ>0 such that min

Construction of the LK Functional

In order to reduce the conservatism of the LK functional approach, motivated by the work in Gouaisbaut and Peaucelle (2006), we consider the fractions hNi,i=1,2,,N, of the delay h, where N1 is a given integer. For an sR, we define π(s)=[B1(st)x1(s)B2(st)x2(s)BN(st)xN(s)], in which, for i=1,2,,N, xi(s)=x(shN(i1)),Bi(s)=B(shN(i1)). We notice that π(s) is also dependent on t. Let θ+hN(i1)=s,i=1,2,,N. Then it follows from (7) that Vh(xt)=i=1NhNihN(i1)xT(t+θ)BT(θ)e2βθR(θ)B(θ)x(t+θ)

Characteristic equation based necessary and sufficient conditions

For the integral delay system (1), we provide in this section necessary and sufficient conditions for stability by analyzing its associated characteristic equation. We rewrite (5) as B(θ)=eMθB(0)=eAθB,θ[h,0], where A=M. The integral delay system (1) can then be rewritten as x(t)=h0GeAθBx(t+θ)dθ, whose characteristic eigenvalue set can be obtained as λID={s:det(Imh0GeAθBeθsdθ)=0}. It is known that the integral delay system (40) is asymptotically stable if and only if λIDC (Hale,

Numerical examples

In this section, we present some examples to validate the effectiveness of the proposed approaches.

Example 1

Example 1 in Mondie and Melchor-Aguilar (2012)

Consider an integral delay system in the form of (1) with G=B=I2 and M=[0123]. Notice that in this case the matrix M is Hurwitz since λ(M)={1,2}. For different values of β, by applying different approaches, the corresponding maximal values of the delay h are recorded in Table 1. From this table we can observe the following facts.

  • (1)

    Corollary 1 is less conservative than Lemma 2 (Theorem 7 in Mondie

Conclusions

This paper proposes some delay-dependent sufficient conditions for the exponentially stability of a class of integral delay systems which are frequently encountered in studying stability problems of time-delay systems. The basic idea is to divide the delay interval into N small intervals so that more information of the delayed state can be utilized to construct the LK functional, which in turns can reduce the conservatism of the resulting sufficient conditions expressed in linear matrix

Zhao-Yan Li was born in Hebei Province, PR China, on August 13, 1982. She received her B.Sc. Degree from the Department of Information Engineering at North China University of Water Conservancy and Electric Power, Zhengzhou, PR China, in 2005, and her M.Sc. and Ph.D. Degrees in Department of Mathematics, Harbin Institute of Technology, PR China, in 2007 and 2010, respectively. She is a Research Associate at the Department of Electrical and Computer Engineering, University of Virginia from July

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    Zhao-Yan Li was born in Hebei Province, PR China, on August 13, 1982. She received her B.Sc. Degree from the Department of Information Engineering at North China University of Water Conservancy and Electric Power, Zhengzhou, PR China, in 2005, and her M.Sc. and Ph.D. Degrees in Department of Mathematics, Harbin Institute of Technology, PR China, in 2007 and 2010, respectively. She is a Research Associate at the Department of Electrical and Computer Engineering, University of Virginia from July 2012 to August 2013. She is now a lecturer in the Department of Mathematics at Harbin Institute of Technology, PR China. Her research interest includes stochastic system theory and time-delay systems.

    Bin Zhou was born in Luotian County, Huanggang, Hubei Province, PR China, on July 28, 1981. He received the Bachelor’s degree, the Master’s Degree and the Ph.D. degree from the Department of Control Science and Engineering at Harbin Institute of Technology, Harbin, China, in 2004, 2006 and 2010, respectively. He was a Research Associate at the Department of Mechanical Engineering, University of Hong Kong from December 2007 to March 2008, a Visiting Fellow at the School of Computing and Mathematics, University of Western Sydney from May 2009 to August 2009, and a Visiting Research Scientist at the Department of Electrical and Computer Engineering, University of Virginia from July 2012 to August 2013. In February 2009, he joined the School of Astronautics, Harbin Institute of Technology, where he has been a Professor since December 2012. He is a reviewer for American Mathematical Review and is an active reviewer for a number of journals and conferences. He was selected as the “New Century Excellent Talents in University”, the Ministry of Education of China in 2011. He received the “National Excellent Doctoral Dissertation Award” in 2012 from the Academic Degrees Committee of the State Council and the Ministry of Education of PR China. He is currently an associate editor on the Conference Editorial Board of the IEEE Control Systems Society.

    Zongli Lin is a Professor of Electrical and Computer Engineering at University of Virginia. He received his B.S. degree in Mathematics and Computer Science from Xiamen University, Xiamen, China, in 1983, his Master of Engineering degree in Automatic Control from Chinese Academy of Space Technology, Beijing, China, in 1989, and his Ph.D. degree in Electrical and Computer Engineering from Washington State University, Pullman, Washington, USA, in 1994. He was an Associate Editor of the IEEE Transactions on Automatic Control (2001–2003), IEEE/ASME Transactions on Mechatronics (2006–2009), IEEE Control Systems Magazine (2005–2012). He has served on the operating committees and program committees of several conferences and was an elected member of the Board of Governors of the IEEE Control Systems Society (2008–2010). He currently chairs the IEEE Control Systems Society Technical Committee on Nonlinear Systems and Control and serves on the editorial boards of several journals and book series, including Automatica, Systems & Control Letters, Science China: Information Science, and Springer/Birkhauser book series Control Engineering. He is a Fellow of the Institute of Electrical and Electronics Engineers (IEEE), the International Federation of Automatic Control (IFAC) and the American Association for the Advancement of Science (AAAS).

    This work was supported in part by the National Natural Science Foundation of China under grant numbers 61104124, 61273028 and 61322305, by the Fundamental Research Funds for the Central Universities (HIT.NSRIF.2011007 and HIT.BRETIV.201305), and by the National Science Foundation of the United States under grant number CMMI-1129752. The material in this paper was presented at the 2013 American Control Conference (ACC2013), June 17–19, 2013, Washington, DC, USA. This paper was recommended for publication in revised form by Associate Editor Akira Kojima under the direction of Editor Ian R. Petersen.

    1

    Tel.: +86 451 87556211; fax: +86 451 86415647.

    2

    Zhao-Yan Li and Bin Zhou were on leave with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904-4743, USA, from July 2012 to August 2013.

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