Elsevier

Automatica

Volume 44, Issue 11, November 2008, Pages 2849-2856
Automatica

Brief paper
Non-fragile H filter design for linear continuous-time systems

https://doi.org/10.1016/j.automatica.2008.03.018Get rights and content

Abstract

This paper studies the problem of non-fragile H filter design for linear continuous-time systems. The filter to be designed is assumed to include additive gain variations, which result from filter implementations. A notion of structured vertex separator is proposed to approach the problem, and exploited to develop sufficient conditions for the non-fragile H filter design in terms of solutions to a set of linear matrix inequalities (LMIs). The designs guarantee the asymptotic stability of the estimation errors, and the H performance of the system from the exogenous signals to the estimation errors below a prescribed level. A numerical example is given to illustrate the effect of the proposed method.

Introduction

H estimation has been attracting considerable attention in the past decades. Several approaches have been proposed to solve the H filtering problem (Nagpal and Khargonekar, 1991, Yaesh and Shaked, 1992). The H filtering approach was adopted for systems with norm-bounded type of parameter uncertainty using a Riccati equation approach in Fu, de Souza, and Xie (1992) and a convex optimization approach in Li and Fu (1997). The H filtering for systems with a polytopic type of parameter uncertainty has been addressed in Geromel and de Oliviera (2001) and Palhares and Peres (1999). The recent work Gao, Lam, and Wang (2005) provides complete results on the induced l2 and generalized H2 filtering problem for a class of nonlinear discrete-time systems.

All the above works are based on an implicit assumption that the filter will be implemented exactly. However, inaccuracies or uncertainties do occur in the implementation of a designed filter or controller. Such uncertainties can be due to, among other things, roundoff errors in numerical computation during the filter or controller implementation and the need to provide practicing engineers with safe-tuning margins. In Keel and Bhatacharyya (1997), by means of numerical examples, it has shown that the controllers designed by using weighted H, μ and L1 synthesis techniques may be very sensitive, or fragile, with respect to relatively small perturbations in controller parameters. So a significant issue is how to design a filter or controller for a given plant such that the filter or controller is insensitive to some amount of errors with respect to its gain, i.e., the designed filter or controller is resilient or non-fragile. This issue has received some attention from the control systems community, and some relevant results have appeared in the last decade (Dorato, 1998, Famularo et al., 2000, Haddad and Corrado, 2000, Ho et al., 2001, Jadbabaie et al., 1998, Peaucelle et al., 2004, Takahashi et al., 2000, Yang et al., 2000, Yang and Wang, 2001, Yang and Wang, 2003). A robust non-fragile Kalman filtering problem is addressed in Yang and Wang (2001a). In Mahmoud (2004), the problem of designing robust resilient linear filtering for a class of continuous-time systems with norm-bounded uncertainty was investigated. Recently, an approach of designing the optimal filter transfer function and its realization is developed (De Oliveira & Geromel, 2006). In Yaz, Jeong, and Yaz (2006), a novel stochastic design approach for resilient observer is presented.

Noting that in the above mentioned works on the non-fragile problem, the gain uncertainties considered are all of the norm-bounded type. However, this type of uncertainty cannot reflect the uncertain information due to the FWL effects exactly. Correspondingly, the interval type of uncertainty (Li, 1998) is more exact than the former type to describe the uncertain information, but up to now, there is no work on the non-fragile controller or filter design problem taking account of interval gain uncertainty. Motivated by this point, this paper is concerned with the problem of non-fragile H filter design for continuous-time systems with the consideration of additive interval gain variations. Firstly, an LMI-based sufficient condition is given for solving the non-fragile H filtering problem, which requires checking all of the vertices of the set of uncertain parameters that grows exponentially with the number of uncertain parameters. It will be very difficult to apply the result to systems with high orders. To overcome the difficulty, a notion of structured vertex separator is proposed to approach the problem, and exploited to develop sufficient conditions for the non-fragile H filter design in terms of solutions to a set of LMIs. The structured vertex separator-based method can significantly reduce the number of the LMI constraints involved in the design condition. The designs guarantee the asymptotic stability of the estimation errors, and the H performance of the system from the exogenous signals to the estimation errors below a prescribed level. It should be pointed out that the existing method given in Mahmoud (2007) and Yang and Wang (2001a), for the non-fragile problem with norm-bounded gain variations, is also applicable for the non-fragile H filtering problem considered here. But this method is more conservative than our new proposed one, which will be shown in Lemma 10 of Section 3.

Notation: For a matrix E, ET and E1 denote its transpose and inverse if it exists, respectively. For a column-rank deficient matrix H, NH denotes a matrix whose columns form a basis for the null space of H. I denotes the identity matrix with an appropriate dimension. 0i×j represents zero matrix of i rows and j columns. The symbol within a matrix represents the symmetric entries.

Section snippets

Problem statement

Consider a linear time-invariant model described by ẋ(t)=Ax(t)+B1ω(t),z(t)=C1x(t),y(t)=C2x(t)+D21ω(t), where x(t)Rn is the state, y(t)Rp is the measured output and z(t)Rq is the regulated output, respectively. A,B1,C1,C2 and D21 are known constant matrices of appropriate dimensions.

To formulate the filtering problem, consider a filter with gain variations of the following form: ξ̇(t)=(AF+ΔAF)ξ(t)+(BF+ΔBF)y(t),zF(t)=(CF+ΔCF)ξ(t), where ξ(t)Rn is the filter state, zF(t) is the estimation of z

Non-fragile H filter design with additive uncertainty

In this section, LMI-based methods for designing H filters with respect to additive uncertainties are presented, and further, a comparison with the existing method is given.

Firstly, new proposed methods for designing non-fragile H filters are presented in the following subsection.

Example

Here, an example is given to illustrate the effectiveness of the proposed non-fragile H filter design method.

Consider system (1) with A=[01000112.51],B1=[1.501020],C1=[321],C2=[212],D21=[00.9]. As indicated in Remark 6, for the case that the designed filter contains no gain variations, by Theorem 5, the optimal H performance index of the standard closed-loop system is achieved as γopt=4.6536.

Conclusions

The non-fragile H filtering problem for linear continuous-time systems has been addressed, where the filter to be designed is assumed to be with additive gain variations of interval type. Firstly, an LMI-based sufficient condition is given for the solvability of the non-fragile H filtering problem, but it requires checking all of the vertices of the set of uncertain parameters that grows exponentially with the number of uncertain parameters, which results in the difficulty of applying the

Acknowledgments

This work was supported in part by the Funds for Creative Research Groups of China (No. 60521003), the State Key Program of National Natural Science of China (Grant No. 60534010), the Funds of National Science of China (Grant No. 60674021), the Funds of Ph.D. program of MOE, China (Grant No. 20060145019) and the 111 Project (B08015).

Guang-Hong Yang received the B.S. and M.S. degrees in mathematics from Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research

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    Guang-Hong Yang received the B.S. and M.S. degrees in mathematics from Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault tolerant control, fault detection and isolation, non-fragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Systems Science (IJSS) and the International Journal of Control, Automation, and Systems (IJCAS), and an Associate Editor of the Conference Editorial Board of the IEEE Control Systems Society.

    Wei-Wei Che received the B.S. degree in mathematics and applied mathematics in 2002 from Jinzhou Normal University, China, and the M.S. degree in applied mathematics in 2005 from Bohai University, China. Currently, she is pursuing the Ph.D. degree in Northeastern University, China. Her research interest includes non-fragile control as well as quantization control and their applications to networked control system design.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Antonio Vicino under the direction of Editor Torsten Söderström.

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