Elsevier

Automatica

Volume 50, Issue 5, May 2014, Pages 1497-1506
Automatica

Brief paper
On robustness of predictor feedback control of linear systems with input delays

https://doi.org/10.1016/j.automatica.2014.03.018Get rights and content

Abstract

This paper is concerned with the robustness of the predictor feedback control of linear systems with input delays. By applying certain equivalent transformations on the characteristic equation associated with the closed-loop system, we first transform the robustness problem of a predictor feedback control system into the stability problem of a neutral time-delay system containing an integral operator in the derivative. The range of the allowable input delay for this neutral time-delay system can be computed by exploring its delay dependent stability conditions. In particular, delay dependent stability conditions for the neutral time-delay system are established by partitioning the delay into segments. The conservatism of this method can be reduced when the number of segments in the partition is increased. Numerical examples are worked out to illustrate the effectiveness of the proposed method.

Introduction

In the past several decades, time-delay systems have attracted much attention because of their many applications. Many control problems for time-delay systems, especially, the stability and stabilization problems, have been extensively studied for years (see, for example, Cong & Yin, 2012, Gu & Niculescu, 2003, Gu, Kharitonov, & Chen, 2003, Gu, Zhang, & Xu, 2011, He, Wang, Lin, & Wu, 2007, Lam, Gao, & Wang, 2007, Li & De Souza, 1997, Xu, Lam, & Yang, 2002 and the references therein). Roughly speaking, studies of time-delay systems can be classified into two categories: those that deal with systems with state delays (Fridman, 2001, Li and De Souza, 1997, Xu et al., 2002, Zhou, Li, Zheng et al., 2012) and those that deal with systems with input/output delays (Yue and Lam, 2004, Zhou, Li, Lin, 2012, Zhou et al., 2010, Zhou et al., 2012).

Many memoryless controllers have been designed in the literature to deal with the input delayed systems (see, for instance, Kim, Jeung, & Park, 1996, Kolmanovskii & Myshkis, 1992, Lin & Fang, 2007, Yoon & Lin, 2013 and Zhou et al., 2012). Memoryless controllers are easy to implement. However, they may fail to control the systems that are open-loop unstable and/or when the delays are large. On the other hand, controllers with memory in general lead to better performances of the closed-loop systems than memoryless controllers. One of the most efficient approach for designing memory controllers is the so-called model reduction method, which is also known as predictor feedback and finite spectrum assignment (see Chen & Zheng, 2006, Cheres, Palmor, & Gutman, 1990, Krstic, 2010 and the references cited there). This class of controllers, however, may suffer some implementation problems. More information on the implementation and applications of this class of controllers can be found in Lozano, Castillo, and Dzul (2004), Mirkin and Raskin (2003), Zhou, Li, Lin (2012), Zhou et al. (2012) and the references therein.

Despite of the voluminous literature on stability analysis and stabilization, few results are available on the robustness of the predictor feedback control of time-delay systems. Krstic (2008) is probably the first paper that studies the robustness of the predictor feedback control systems with input delay with respect to small (constant) perturbations in the input delay. However, the allowable upper and lower perturbation bounds are only proven to exist while their computation method is quite conservative. Very recently, Karafyllis and Krstic (2013) presented a robustness analysis for the predictor feedback control of input delayed linear time-invariant systems with possibly time-varying perturbations in the input delay. The computation of the allowable upper and lower perturbation bounds remains conservative as norms of matrices are involved.

In this paper, we will study the robustness of the predictor feedback control of a linear system with input delay and constant perturbations in the delay. We will transform the problem into one of stability analysis for a neutral differential equation whose right-hand side is delay-free and left-hand side is recognized as an integral operator. A linear-matrix-inequalities (LMI)-based delay-dependent sufficient condition guaranteeing the stability of this neutral time-delay system is then established by combining the Lyapunov–Krasovskii functional approach and the delay partition technique. The allowable upper and lower bounds of the perturbation on the delay are obtained by testing the LMIs-based conditions with a linear search technique. Three numerical examples show that the obtained bounds are significantly less conservative than those obtained by the approaches reported in the literature and are very close to the exact bounds.

The remainder of this paper is organized as follows. The problem formulation and some preliminary results are presented in Section  2. Our main results are then given in Section  3 and a couple of numerical examples are worked out in Section  4. Section  5 concludes the paper. Finally, the proofs for several technical results and some technical lemmas are collected in the Appendix.

Notation: For a matrix A, we use AT and He{A} to denote, respectively, its transpose and the symmetric matrix A+AT. For a matrix P0, the symbols λmin(P) and λmax(P) present, respectively, its minimal and maximal eigenvalues. We use 0n×m to denote a zero matrix with dimensions n×m. The symbol || refers to the Euclidean norm and the spectrum norm. The symbol denotes the Kronecker product of two matrices. Let BRn×m with rank(B)=p<m. Then the matrix BRm×(mp) with rank mp denotes the right orthogonal complement of B. Finally, suppose r>0 is a given real number; we let Cr,n=C([r,0],Rn)denote the Banach space of continuous functions mapping the interval [r,0] into Rn with the topology of uniform convergence. Define the norm of an element ϕ in Cr,n by ϕ=suprθ0|ϕ(θ)|. Moreover, we denote xt(θ)=x(t+θ)Cr,n,rθ0.

Section snippets

Problem formulation and preliminaries

We consider the following linear system with input delay ẋ(t)=Ax(t)+Bu(tr), where ARn×n and BRn×m are constant matrices and r0 is a constant scalar that may be unknown. Let (A,B) be stabilizable. Assume that the nominal value of r is r which is exactly known. By using the nominal value r, the predictor feedback for the time-delay system (1) can be designed as (see, for example, Krstic, 2008) u(t)=K(eArx(t)+r0eAsBu(t+s)ds), where K is chosen such that A+BK is Hurwitz.

In this paper,

Main results

Throughout this section, we denote F=A+BK and G=BKeAr. For the neutral time-delay system (5), we first discuss, without loss of generality (see Remark 2 given later), the case that rr. Denote π(s)=[x(s)x(s1N(rr))x(sN1N(rr))], where N1 is a given integer. In addition, we define Γ1=[0Nn×nINn0Nn×n],Γ2=[0Nn×2nINn],Γ3=[In0n×(N+1)n],Γ4=[0n×nIn0n×Nn]. Then we can present the following lemma which gives conditions guaranteeing the stability of the integral delay system φ(t)=0 defined in (6)

Numerical examples

In this section, we present three examples to demonstrate the effectiveness of the proposed approaches.

Example 1

We consider a scalar time-delay system in the form of (1) with A=1,B=1,r=1 and K=p, where p>1. This example is taken from Karafyllis and Krstic (2013).

Example 2

We consider a planar time-delay system in the form of (1) with A=[0111],B=[01],KT=[13], and r=1. It follows that λ(A)={12(1±3j)} and λ(A+BK)={1±j}.

Example 3

We consider a third-order time-delay system in the form of (1) with r=1 and A=[23153010

Conclusions

In this paper, the robustness problem of the predictor feedback control of linear systems with input delays was studied. The basic idea is to transform the robustness problem of the predictor feedback into the stability problem of a kind of neutral time-delay systems by using the equivalent transformation on the characteristic equation of the time-delay systems. Then a delay dependent stability condition for the stability of the neutral time-delay systems was proposed by using a delay partition

Acknowledgments

The authors would like to thank Professor Keqin Gu for providing the Matlab function for Li and Gu (2010).

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 under Grant HIT.NSRIF.2011007, by Program for Innovation Research of Science in Harbin Institute of Technology (PIRS of HITA201407), and by the National Science Foundation of the United States

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 is a professor of the Department of Control Science and Engineering at the Harbin Institute of Technology. He 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. He joined the School of Astronautics, Harbin Institute of Technology in February 2009. His current research interests include constrained control, time-delay systems, nonlinear control, and control applications in astronautic engineering. In these areas, he has published about 100 papers, over 80 of which are in archival journals. 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 and an associate editor of Journal of System Science and Mathematical Science.

    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. His current research interests include nonlinear control, robust control, and control applications. He was an Associate Editor of the IEEE Transactions on Automatic Control (2001–2003), IEEE/ASME Transactions on Mechatronics (2006–2009) and IEEE Control Systems Magazine (2005–2012). He was an elected member of the Board of Governors of the IEEE Control Systems Society (2008–2010) and has served on the operating committees and program committees of several conferences. 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 Sciences, 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).

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Hitay Ozbay under the direction of Editor Miroslav Krstic.

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