Elsevier

Automatica

Volume 41, Issue 6, June 2005, Pages 999-1007
Automatica

Brief paper
Network-based robust H control of systems with uncertainty

https://doi.org/10.1016/j.automatica.2004.12.011Get rights and content

Abstract

This paper is concerned with the design of robust H controllers for uncertain networked control systems (NCSs) with the effects of both the network-induced delay and data dropout taken into consideration. A new analysis method for H performance of NCSs is provided by introducing some slack matrix variables and employing the information of the lower bound of the network-induced delay. The designed H controller is of memoryless type, which can be obtained by solving a set of linear matrix inequalities. Numerical examples and simulation results are given finally to illustrate the effectiveness of the method.

Introduction

Considering the disturbance attenuation problem, robust H control for uncertain systems under the assumption that the controller dynamics is continuous has been investigated by many researchers (Khargonekar et al., 1990, Niculescu, 1998, Petersen, 1987, Xie and de Souza, 1992). However, in many practical systems, such as computer-based control systems, the system is controlled by a discrete-time controller with sample and hold devices. In this case, the system can be expressed as a class of sampled-data models. Recently, much research work has been done on the robust H control for linear sampled-data systems (Bamieh and Pearson, 1992, Toivonen, 1992) or nonlinear sampled-data systems (Nguang and Shi, 2001, Oriov and Acho, 2000). It is worth pointing out that all the aforementioned results are under the assumption that no transmitting delay exists in the controller signal.

As is well known, in modern industrial systems, sensors, controllers and plants are often connected over a network medium (Chow & Tipsuwan, 2001), which are called networked control systems (NCSs). There are many advantages in NCSs, such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. Thus, increasing research interests have recently been paid to the study of the stability and stabilization of NCSs (Chow & Tipsuwan, 2001; Nilsson, Bernhardsson, & Wittenmark, 1998; Walsh, Ye, & Bushnell, 2002; Zhang, Branicky, & Phillips, 2001). However, since the sampling data and controller signals are transmitted through a network, network-induced delays and data dropout in NCSs are always inevitable. For NCSs with different scheduling protocols, the network-induced delay may be constant, time-varying, or even random (Zhang et al., 2001). More recently, the stability analysis and stabilization controller design for NCSs have been investigated by many researchers when the effects of network-induced delay and/or data dropout are taken into account. In these works, analysis and synthesis methods are provided based on discrete-time model (Hu and Zhu, 2003, Nilsson et al., 1998), continuous-time model (Walsh et al., 2002) or hybrid system model (Zhang et al., 2001). Considering the effects of the external disturbance on the system, stability analysis and disturbance attenuation analysis are carried out by Lin, Zhai, and Antsaklis (2003) based on a framework of discrete-time switched systems. However, only the case when there are no parameter uncertainties in the system is considered and no controller synthesis method is given by Lin et al. (2003). Moreover, the effects of controller-to-actuator delay is neglected by Lin et al. (2003). To the best of the authors’ knowledge, the disturbance attenuation problem for NCSs has not been fully investigated to date. Especially for the H control synthesis of NCSs, no results have been available in the literature so far, which motivates the present study.

In this paper, we are concerned with the design of robust H controllers for uncertain networked control systems. Both network-induced delay and data dropout are considered in the model. The network-induced delay considered in the model is composed of sensor-to-controller delay and the controller-to-actuator delay, as well as the computation delay. In our method, the discretization of the system model and the assumption that the controller dynamics is continuous are not needed for the controller design. In contrast with controller design methods based on discrete-time models, our method is formulated in the continuous-time domain, that is, the inter-sampling behavior is taken into account. Through introducing some slack matrix parameters, the derived criteria can lead to less conservative results than those in some existing references even for stability analysis. Moreover, since the lower bound of the network-induced delay is employed to derive the criteria, considerably less conservative results can be obtained by using the criteria in this paper especially for the case where the lower bound of the network-induced delay is nonzero. The criteria for both H performance analysis and H control synthesis are derived based on a linear matrix inequality (LMI) approach. The designed H controller is of memoryless type, which can be obtained by solving a set of LMIs. To illustrate the effectiveness of the method, numerical examples and simulation results are given.

Notation

Rn denotes the n-dimensional Euclidean space, Rn×m is the set of n×m real matrices, I is the identity matrix of appropriate dimensions, · stands for the Euclidean vector norm or the induced matrix 2-norm as appropriate. The notation X>0 (respectively, X0), for XRn×n means that the matrix X is a real symmetric positive definite (respectively, positive semi-definite). λmax(P) (λmin(P)) denotes the maximum (minimum) of eigenvalue of a real symmetric matrix P. For an arbitrarily real matrix B and two real symmetric matrices A and C,A*BC denotes a real symmetric matrix, where * denotes the entries implied by symmetry.

Section snippets

System description and preliminaries

Consider the following system with parameter uncertainties given byx˙(t)=[A+ΔA(t)]x(t)+[B+ΔB(t)]u(t)+Bww(t),x(t0)=x0,z(t)=Cx(t)+Du(t),where x(t)Rn, u(t)Rm and z(t)Rq are the state vector, control input vector and controlled output, respectively; x0Rn denotes the initial condition; A, B, Bw,C and D are some constant matrices of appropriate dimensions; ΔA(t) and ΔB(t) denote the parameter uncertainties satisfying the following condition:[ΔA(t)ΔB(t)]=GF(t)[EaEb],where G,Ea and Eb are constant

H performance analysis

Define τ0=η+τm2,δ=η-τm2.For any matrices Ni,Si and Mi(i=1,2,3,4) of appropriate dimensions, it can be seen that[xT(t)N1+xT(ikh)N2+xT(t-τ0)N3+x˙T(t)N4]x(t)-x(ikh)-ikhtx˙(s)ds=0,[xT(t)S1+xT(ikh)S2+xT(t-τ0)S3+x˙T(t)S4]x(ikh)-x(t-τ0)-t-τ0ikhx˙(s)ds=0andxT(t)M1+xT(ikh)M2+xT(t-τ0)M3+x˙T(t)M4[-[A+ΔA(t)]x(t)-[B+ΔB(t)]Kx(ikh)-Bww(t)+x˙(t)]=0.Next, based on a Lyapunov–Krasovskii functional and combining (13)–(15), we conclude the following result.

Lemma 8

For given scalarsτm,ηandγand a matrixK,if there exist

Robust H controller design

Based on Theorem 10, we are now in a position to design the feedback gain K, which can make system (1)–(3) robustly exponentially stable with an H norm bound γ.

Theorem 12

For given scalarsρl(l=2,3,4), τm, ηandγ,if there exist matricesP˜k(k=1,2,3), T˜j>0, R˜j>0(j=1,2), a nonsingular matrix X and any matricesN˜iandS˜i(i=1,2,3,4)of appropriate dimensions and a scalarμ>0such thatΦ11*Φ21Φ22<0,P˜1P˜2P˜2TP˜3>0,whereΦ11=Σ11***Σ21Σ22**Σ31Σ32Σ33*Σ41Σ42Σ43Σ44,Φ21=τ0P˜30-τ0P˜3τ0P˜2TηN˜1TηN˜2TηN˜3TηN˜4TδS˜1TδS˜2TδS˜3

Numerical examples

Example 13

Consider the following system borrowed from Zhang et al. (2001):x˙(t)=010-0.1x(t)+00.1u(t).When considering the effect of the external perturbation on the system, (47) can be expressed asx˙(t)=010-0.1x(t)+00.1u(t)+0.10.1w(t),z(t)=01x(t)+0.1u(t).For this example, we will employ the same feedback controller as in Zhang et al. (2001), that is, u(t)=-3.75-11.5x(t). This controller is designed without considering the presence of the network. Under an assumption that the controller dynamics in (47)

Conclusion

The disturbance attenuation problem for NCSs has been investigated based on a Lyapunov–Krasovskii functional method. The criteria for H performance analysis and H control synthesis have been derived by introducing some free-weighting matrices and exploiting the information concerning the lower bound of variation of the network-induced delay, which has been shown by the examples to be effective.

Acknowledgements

The author would like to thank the associate editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality and the presentation of the paper.

The research work of D. Yue and Q.-L. Han was partially supported by National Natural Science Foundation of China (60474079) and Central Queensland University for the 2004 Research Advancement Awards Scheme Project “Analysis and Synthesis of Networked Control Systems”. The research work of D. Yue was also

Dong Yue was born in China in 1964. He received his B.Sc. from Guilin Institute of Electrical Engineering in 1985, M.Sc. from Anhui University in 1991 and Ph.D. from South China of Technology in 1995, respectively. From 1995 to 1997, he was a Post-doctoral Research Fellow at China University of Mining and Technology. He was a full Professor from 1997 to 2001 at China University of Mining and Technology. From June 1999 to September 1999, he was a Research Associate at City University of Hong

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Dong Yue was born in China in 1964. He received his B.Sc. from Guilin Institute of Electrical Engineering in 1985, M.Sc. from Anhui University in 1991 and Ph.D. from South China of Technology in 1995, respectively. From 1995 to 1997, he was a Post-doctoral Research Fellow at China University of Mining and Technology. He was a full Professor from 1997 to 2001 at China University of Mining and Technology. From June 1999 to September 1999, he was a Research Associate at City University of Hong Kong. From August 2000 to August 2001, he worked at Pohang University of Science and Technology as a Senior Scientist. From June 2002 to September 2002, he was a Research Associate at Hong Kong University. From August 2003 to October 2003, he was a Visiting Professor at Central Queensland University. From March 2004 to March 2005, he was a Research Fellow at Central Queensland University. He is currently a full Professor at Nanjing Normal University and Director of Research Centre for Information and Control Engineering Technology. His research interests include networked control systems, time delay systems, robust control.

Qing-Long Han was born in Shandong, China, in 1963. He received the B.S. degree in Mathematics from the Shandong Normal University, Jinan, China, in 1983, and the M.S. and Ph.D. degrees in Information Science (Electrical Engineering) from the East China University of Science and Technology, Shanghai, China, in 1992 and 1997, respectively. From September 1997 to December 1988, he was a Post-Doctoral Researcher Fellow at LAII-ESIP, Université de Poitiers, France. From January 1999 to August 2001, he was a Research Assistant Professor in the Department of Mechanical and Industrial Engineering, Southern Illinois University at Edwardsville, USA. In September 2001 he joined the Faculty of Informatics and Communication, Central Queensland University, Australia, where he is currently a Senior Lecturer. He is serving as an Associate Editor for Dynamics of Continuous, Discrete & Impulsive Systems—Series B: Applications & Algorithms, and a Guest Editor for the Special Issue on Time-Delay Systems in Asian Journal of Control. His research interests include time-delay systems, robust control, networked control systems, complex systems and software development processes.

James Lam received a first class BSc degree in Mechanical Engineering from the University of Manchester in 1983. He was awarded the Ashbury Scholarship, the A.H. Gibson Prize and the H. Wright Baker Prize for his academic performance. From the University of Cambridge, he obtained the MPhil and PhD degrees in the area of control engineering in 1985 and 1988 respectively. His postdoctoral research was carried out in the Australian National University between 1990 and 1992. Dr. Lam is a Scholar (1984) and Fellow (1990) of the Croucher Foundation.

Dr. Lam has held faculty positions at now the City University of Hong Kong and the University of Melbourne. He is an Associate Professor in the Department of Mechanical Engineering, the University of Hong Kong, and is holding a Concurrent Professorship at the Northeastern University, Guest Professorship at the Huazhong University of Science and Technology, Consulting Professorship at the South China University of Technology, and Guest Professorship of Shandong University. Dr. Lam is a Chartered Mathematician, a Fellow of the Institute of Mathematics and Its Applications (UK), a Senior Member of the Institute of Electrical & Electronics Engineers (US), a Member of the Institution of Electrical Engineers (UK). He is an Honorary Editor of the IEE Proceedings: Control Theory and Applications, an Associate Editor of the Asian Journal of Control, the International Journal of Applied Mathematics and Computer Science, the International Journal of Systems Science, and the Conference Editorial Board of IEEE Control System Society. His research interests include model reduction, delay systems, descriptor systems, stochastic systems, multidimensional systems, robust control and filtering, fault detection, and reliable control.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor J. Peterson.

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