Elsevier

Automatica

Volume 42, Issue 11, November 2006, Pages 1919-1926
Automatica

Brief paper
Robust H2 and H filtering for uncertain linear systems

https://doi.org/10.1016/j.automatica.2006.06.004Get rights and content

Abstract

This paper extends the existing results on the robust H2 and H filtering problems with respect to polytopic uncertainty. A new structure of slack variables is introduced to provide extra free dimensions in the solution space for the H2 and H optimization. This directly leads to performance improvement and reduction of conservativeness in the filtering solution. The method is applicable to both discrete and continuous time systems. Performance of the proposed solutions in comparison with that of the existing methods is illustrated by several examples.

Introduction

The analysis and design of robust filtering and control systems have been investigated extensively over the last two decades and it has been well known that existing conditions in the literature for the analysis and design of robust filtering and control systems are only sufficient. Thus, many efforts have been made in the direction of reducing the conservativeness of the analysis and design methods for improving the systems performance, see Geromel (1999), Geromel et al. (2000), Grigoriadis and Watson (1997), Xie et al. (1998) and references therein. In order to reduce the conservativeness of traditional Lyapunov function methods, Feron et al. (1996), Gahinet et al. (1994) and de Oliveira et al. (1999) proposed a new approach, known as the parameter-dependent Lyapunov method, to the study of robust stability of systems with parametric uncertainties. By introducing a new instrumental matrix variable, the stability condition is relaxed to a new LMI (linear matrix inequality) in de Oliveira et al. (1999) where Lyapunov matrix is independent of the state matrix of systems. With this new degree of freedom, a parameter-dependent Lyapunov function can be built to analyze and design robust filters/controllers. The conservativeness is further reduced in Peaucelle et al. (2000) by introducing another degree of freedom.

There has been a considerable amount of works on extensions and applications of parameter-dependent Lyapunov method to the analysis and design of robust filters and controllers, see Barbosa et al. (2004), Geromel et al. (2002), de Oliveira et al. (2002) and Xie et al. (2004) and references therein. In the filter design, Tuan et al. (2001) and Tuan et al. (2002) studied robust and reduced-order filtering based on the method of (de Oliveira et al., 1999). Using the technique of Peaucelle et al. (2000), Xie et al. (2004) presented an improved robust filtering method for discrete-time systems. Further, Barbosa et al. (2004) introduced a new robust H2 filtering method for continuous-time systems to reduce the filtering conservativeness. A common feature of Xie et al. (2004) and Barbosa et al. (2004) is the introduction of extra (scalar) parameters to circumvent the NMIs (nonlinear matrix inequalities). When these parameters are preset constants, the NMIs become LMIs which can be easily solved for suboptimal solutions. When these extra parameters are also searched, beyond the LMI searching, for better solutions, the algorithms become nonconvex. This nonconvexity nature is a major difficulty associated with these methods. It is noted that these methods are all derived from some sufficient conditions and there has been no quantitative analysis available about how far they are from the necessary condition.

This paper is a technical extension of the results of Xie et al. (2004) and Peaucelle et al. (2000) for discrete time and continuous time systems. The main contribution of the paper is a new structure of the key slack variable matrix. This new structure provides extra free dimensions in the solution space for the H2 and H optimization. It directly leads to performance improvement and reduction of conservativeness in the filtering solution and performance evaluation.

The rest of this paper is organized as follows. Section 2 presents an improved robust H2 and H filtering method for uncertain discrete-time systems and derives effective convex algorithms. Sections 3 and 4 generalizes the results of Section 2 to the H2 and H filtering and performance analysis of continuous-time systems. Section 5 demonstrates examples and compares the results of this paper with that of Tuan et al. (2001), Xie et al. (2004), Barbosa et al. (2004) and Shaked (2001).

Section snippets

Robust H2 and H filtering of discrete time systems

Consider the following discrete-time stable systemxk+1=Axk+Bwk,yk=Cxk+Dwk,zk=Lxk,where xkRn is the system state, ykRq is the measurement, zkRp is the signal to be estimated, and wkRm is the noise input. The matrices A,B,C,D and L are appropriately dimensioned with partially unknown parameters belonging to the following uncertainty polytope (Xie et al., 2004):Ω=(A,B,C,D,L)|(A,B,C,D,L)=i=1Nαi(A(i),B(i),C(i),D(i),L(i)),αi0,i=1Nαi=1.Assume that wk is Gaussian white noise with zero-mean and

Robust H2 and H filtering of continuous time systems

The results of Theorems 1 and 2 can be easily extended to the H2 and H filtering of continuous time systems. Consider the following continuous time linear stable system:x˙=Ax+Bw,y=Cx+Dw,z=Lx,where xRn is the system state vector, yRq is the measurement, zRp is the signal to be estimated, wRm is the noise input. Assume that w is a zero-mean Gaussian white noise with unit power spectrum density matrix. The standard H2 filtering problem is to find a stable filter of the formx˙F=AFxF+BFy,zF=LFxF

Robust performance analysis of continuous time systems

A result on evaluation of robust performance for discrete time systems has been presented in Xie et al. (2004). We can extend it to continuous time systems using the technique of Lemmas 3 and 4 in the following.

Theorem 5

The continuous time system(19)satisfiesE[zclTzcl]<γ2if and only if there exist matrices(F,G,P)with compatible dimensions andP>0such thatPCclTCclZ>0,tr(Z)<γ2,-G-GTdiag(P,0.5I)-F+A˜TGTFA˜+A˜TFT<0.

Proof

Let A˜=[Acl0Bcl-I], then (25) is equivalent to diag(P,0.5I)A˜+A˜Tdiag(P,0.5I)<0. This is the

Examples

Example 1

Consider the discrete-time system used in Geromel et al. (2002) and Xie et al. (2004). It is in the form of (1) withA=0.90.1+0.06α0.01+0.05β0.9,B=100010,C=10,D=001.414,L=11,where |α|1 and |β|1. This is a two-block structured uncertainty which can be described by a four-vertex polytope. Following from Xie et al. (2004), we also adopt a strictly proper filter by letting DF=0. The values of the H2 guaranteed cost obtained by different methods are shown in Table 1.

Example 2

This is the discrete-time system

Conclusion

This paper extends the existing results on robust H2 and H filtering to present a new structure for the relaxation variable F for both discrete time and continuous time systems. Differently from the existing work, where F is constrained by other variables, our new structure introduces free slack variables which guarantee to provide extra free dimensions in the solution space for the H2 and H optimization. This directly leads to performance improvement and reduction of conservativeness in the

Zhisheng Duan was born in Inner Mongolia, China in 1972. He received the M. S. degree in mathematics from Inner Mongolia University and the Ph. D. degree in control theory from Peking University in 1997 and 2000, respectively. From 2000 to 2002, he worked as a post-doctor in Peking University. Since 2003, he has been an associate professor with the Department of Mechanics and Engineering Science, Peking University. From June 2004 to May 2005, he worked at the Monash University as a visiting

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Zhisheng Duan was born in Inner Mongolia, China in 1972. He received the M. S. degree in mathematics from Inner Mongolia University and the Ph. D. degree in control theory from Peking University in 1997 and 2000, respectively. From 2000 to 2002, he worked as a post-doctor in Peking University. Since 2003, he has been an associate professor with the Department of Mechanics and Engineering Science, Peking University. From June 2004 to May 2005, he worked at the Monash University as a visiting scholar. He has authored and coauthored more than 20 papers in robust control and interconnected systems and he received 2001 Chinese Control Conference Guan-ZhaoZhi Award. His research interests include robust control and filtering, stability of interconnected systems, frequency–domain method of nonlinear systems, analysis and control of chaotic systems.

Jingxin Zhang received M.Eng. and Ph.D. degrees in Electrical Engineering, in 1983 and 1988, from Northeastern University, China. Between 1988 and 1992, he was with Northeastern University, China, as associate professor. Since 1989, he has held research positions in University of Florence, Italy, University of Melbourne, and Cooperative Research Center for Sensor Signal and Information Processing, Australia, and senior lecturer position in University of South Australia and Deakin University, Australia. He is currently with the Department of Electrical and Computer Systems Engineering, Monash University, Australia. He is the author and coauthor of many research papers in diverse research areas such as adaptive and predictive control, time varying systems, robust filtering, multirate signal processing and medical imaging. He is recipient of 1989 Fok Ying Tong Educational Foundation (Hong Kong) for the outstanding Young Faculty Members in China and 1992 China National Education Committee Award for the Advancement of Science and Technology. His current research interests are in control and signal processing and their applications to industrial and medical systems.

Cishen Zhang received the B.Eng. degree from Tsinghua University, China, in 1982 and Ph.D. degree in Electrical Engineering from Newcastle University, Australia, in 1990. Between 1971 and 1978, he was an Electrician with Changxindian (February Seven) Locomotive Manufactory, Beijing, China. He carried out research work on control systems at Delft University of Technology, The Netherlands, from 1983 to 1985. After his Ph.D. study from 1986 to 1989 at Newcastle University, he was with the Department of Electrical and Electronic Engineering at the University of Melbourne, Australia as a Lecturer, Senior Lecturer and Associate Professor and Reader till October 2002. He is currently with the School Electrical and Electronic Engineering and School of Chemical and Biomedical Engineering at Nanyang Technological University, Singapore. His research interests include signal processing, medical imaging and control.

Edoardo Mosca obtained his Dr. Eng. degree in 1963 in Electronics Engineering from the University of Rome “La Sapienza”, Italy. He then spent four years in the aerospace industry where he worked on the research and development of advanced radar systems: particularly, optimal signal synthesis and processing, and phased-array radar systems. Thereafter, from 1968 to 1972, he held academic positions at the University of Michigan, Ann Arbor, Michigan, and McMaster University, Ontario, Canada. Since 1972, he has been with the Engineering Faculty, University of Florence, Italy: from 1972 to 1974 as an Associate Professor, and since 1975 as a full Professor of Control Engineering. In the latter capacity, he founded in 1981 the Department of Systems and Computer Science and Engineering, of which he was the first chair until 1987, and which he conceived and set up as a department including all the academic staff and the related teaching and research activities in computer and control of the University. Dr. Mosca has been a visiting professor in universities and research centers in many different countries. From 1995 to 1998 he has been the President of the Italian Association of Researchers in Automatic Control, and from 1983 to present the coordinator of several national research projects in the field of automatic control. He is the author of many research papers spanning various diversified fields such as radar signal synthesis and processing, radio communications, system identification, adaptive, predictive, switching supervisory control, and detection of performance degradation in feedback-control systems. He is the author of a book, Optimal, Predictive, and Adaptive Control, Prentice Hall, 1995. He is an editor of the following journals: European Journal of Control; International Journal of Adaptive Control and Signal Processing, Wiley; and IEE Proceedings—Control Theory and Applications. He is the Italian NMO representative in IFAC (International Federation of Automatic Control). He has been a Council member of EUCA (European Union Control Association) until 1998, and from 1996 to 2002 a Council member of IFAC . In 2001, he was awarded “honoris causa” the Doctor in Information Engineering degree from the Universidade Tecnica de Lisboa, Lisbon, Portugal, and in 1997 elected to the grade of Fellow of the IEEE (Institute of Electrical and Electronics Engineers) for his contributions to adaptive and predictive control.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hitay Ozbay under the direction of Editor Ian Petersen. Work supported by the National Science Foundation of China under Grant 60334030 and Australian Research Council under Grant DP03430457.

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