Elsevier

Automatica

Volume 44, Issue 4, April 2008, Pages 937-948
Automatica

H2 robust filter design with performance certificate via convex programming

https://doi.org/10.1016/j.automatica.2007.08.010Get rights and content

Abstract

In this paper a new approach to H2 robust filter design is proposed. Both continuous- and discrete-time invariant systems subject to polytopic parameter uncertainty are considered. After a brief discussion on some of the most expressive methods available for H2 robust filter design, a new one based on a performance certificate calculation is presented. The performance certificate is given in terms of the gap produced by the robust filter between lower and upper bounds of a minimax programming problem where the H2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all linear, rational and causal filters. The calculations are performed through convex programming methods developed to deal with linear matrix inequality (LMI). Many examples borrowed from the literature to date are solved and it is shown that the proposed method outperforms all other designs.

Introduction

Over the past years, many researchers have devoted a great deal of attention to the design of robust filters for LTI systems subject to parameter uncertainty. Two classes of parameter uncertainty appeared namely, norm bounded and convex bounded uncertainties, the last one originating the well known class of polytopic systems to be treated in this paper. The main difficulty for robust filter design stems from the necessity to determine an unique linear filter able to cope with different models generated by a set of uncertain parameters, keeping the estimation error norm below some guaranteed level, (Jain, 1975). Many contributions are available dealing with continuous-time (Geromel, 1999, Li et al., 2002, Scherer and Köse, 2006, de Souza and Trofino, 1999, Tuan et al., 2001, Xie and Soh, 1994) and discrete-time (Geromel et al., 2002, Shaked et al., 2001, Theodor and Shaked, 1996, Xie et al., 1999) systems, among others. All of them, but (Scherer & Köse, 2006), share the following basic characteristics:

  • (a)

    the order of the robust filter is equal to the order of the plant,

  • (b)

    performance is not certificated.

Indeed, (a) is imposed as an instrumental condition to keep the design problem convex. In addition, the robust filter is determined from the minimization of a guaranteed cost, actually an upper bound of the true performance index, which does not provide any information about the distance to the true optimal filter. In other words, no information about the degree of sub-optimality is given. Hence, optimality is not theoretically certificated. Concerning (b), each new proposal is compared to the previous ones by means of simulation and performance determination for specific selected examples. Very recently a new important result on this area appeared (Scherer & Köse, 2006). The authors have developed a new H2 robust filter design method which eliminates both drawbacks we have just discussed. The robust filter is not restricted to have the same order of the plant and global optimality is assured. The method follows from an adequate description of the uncertainty by means of a class of multipliers. This very interesting theoretical result has, however, a difficulty which consists on how to fix the multiplier dynamics and its order for a particular design problem to be solved. As (Scherer & Köse, 2006) shows, the optimal filter performance depends on these multiplier characteristics.

For systems with known parameters, the minimization of the H2 norm of the estimation error provides the celebrated Kalman filter which is linear and, as a direct consequence of the optimality conditions, has the same order of the plant (Anderson and Moore, 1979, Colaneri et al., 1997). To deal with parameters uncertainty, the optimal filter is characterized by the equilibrium solution of a minimax optimization problem which can be interpreted as a Man–Nature game (for more detail on this aspect the reader is requested to see Martin & Mintz, 1983). The Nature selects the uncertain parameter by maximizing the H2 norm of the estimation error produced by a filter fixed by Man which has been selected by minimizing the same norm of the estimation error produced by a parameter fixed by Nature. The equilibrium solution (if any) provides the best robust filter and the worst uncertainty. Only in some especial cases the best filter is a Kalman filter associated to the worst uncertainty, (Geromel and Regis, 2006, Poor, 1980). Unfortunately, in the general case, such an equilibrium solution is extremely difficulty to calculate and only recently its existence has been proven for a particular class of polytopic parameter uncertainty (Geromel & Regis, 2006). Due to this, in the general case, it is not yet known the order of the optimal filter and it is not even known if it is finite but, the results of Geromel and Regis (2006) suggest that the order of the optimal filter is greater than the order of the plant.

In this paper, continuous- and discrete-time systems with parameter uncertainty of polytopic type are considered. The equilibrium solution of the already mentioned Man–Nature game is not exactly calculated but lower and upper bounds of the equilibrium H2 cost are provided as a way to certify the optimality gap and, by consequence, the distance from a particular filter to the optimal robust filter. The lower bound is optimized and yields a filter of order, prior to eventual zeros and poles cancellations, much greater than the order of the plant. Based on the result of this first step, a robust filter is determined. An upper bound and, consequently, the optimality gap are determined to certify the performance of the proposed robust filter with respect to the optimal one.

As a result, the order of the robust filter is, putting aside eventual poles and zeros cancellations, equal to the order of the plant times the number of vertices of the convex polytopic domain. With this respect two important points should be noticed. To our best knowledge, the first method available in the literature able to design a higher order filter (comparing to the order of the plant) from the solution of a convex programming problem expressed in terms of pure LMIs was proposed in Geromel and Regis (2006). The present paper generalizes the results of Geromel and Regis (2006) to cope with general polytopic systems. Second, the greater order of the proposed filter with respect to that of the plant appears to be essential to reduce conservatism yielding more accurate results when compared to the previous robust filter design procedures. As the examples solved indicate in many cases we obtain the optimal or, at least, a near-optimal robust filter.

The paper is organized as follows. In the next section the problem to be dealt with is stated and previous results on H2 robust filtering are discussed. In Section 3 a lower bound on the equilibrium solution of the Man–Nature game is proposed and its determination by means of LMIs is analyzed. Section 4 is devoted to determine a robust filter and an upper bound of the equilibrium cost. In Section 5 a great number of examples borrowed from the literature are solved and performances are compared. Both continuous- and discrete-time systems are considered. In Section 6 a more realistic practical application consisting on the estimation of the displacement of a tapered bar is presented. Models of increasing orders are considered to evaluate the proposed robust filter performance. Finally, Section 7 contains the conclusion and final remarks.

The notation used throughout is standard. Capital letters denote matrices and small letters denote vectors. For scalars, small Greek letters are used and N={1,,N}. For real matrices or vectors () indicates transpose. For square matrices Tr(X) denotes the trace function of X being equal to the sum of its eigenvalues and, for the sake of easing the notation of partitioned symmetric matrices, the symbol () denotes generically each of its symmetric blocks. For matrices or transfer functions Uλ denotes the linear parameter dependence Uλ=iλiUi. Finally, the same notationis used either for transfer functions of continuous- or discrete-time systems, where the real matrices A, B, C and D of compatible dimensions define a possible state space realization. With no ambiguity, for continuous-time systems, G(ω) denotes G(ζ) calculated at ζ=jω and for discrete-time systems G(ω) denotes G(ζ) calculated at ζ=ejω where, in both situations, ωR. For any real signal ξ, defined in the continuous- or discrete-time domain, ξ^ denotes its Laplace or Z transform, respectively.

Section snippets

Preliminaries and problem statement

Fig. 1 shows the basic structure of the filtering design problem in terms of the indicated transfer functions. From the exogenous signal w^, the transfer function H(ω) generates the transmitted signal y^ and, simultaneously, the signal z^ to be estimated. Adopting the partitionH(ω)=T(ω)S(ω)the filter transfer function F(ω) has to be designed in such a way that its output z^F is the best estimate of z^ that can be obtained from the available data contained in y^. For robust filter design, this

Optimistic performance

In the general case of uncertain polytopic systems, as already mentioned, the determination of the global equilibrium solution of problem (8) is not a simple task. For this reason, our purpose here is to calculate a lower bound to the equilibrium cost of (8). We want to stress that the optimal filter associated to the proposed lower bound is not restricted to have the same order of the plant, a fact that naturally allows more accurate results. A lower bound of (8) is determined fromJ*minFFmaxi

Robust performance

In the previous section we have determined a filter FLF associated to the minimum lower bound of the filter design problem (8). Clearly, FL(ω) is not a robust filter in the sense that the performance level JL cannot be guaranteed for all λΛ. In this section our goal is to determine a filter FHF associated to a minimum robust performance JH, that is, a filter for which the performance level JH is guaranteed for all λΛ. One of the simplest way to generate an upper bound is to solve the

Examples and comparisons

In this section the performance of the proposed robust filter is compared with several designs available in the literature. Both continuous- and discrete-time systems are considered. The proposed robust filter and the associated performance certification have been determined by the following algorithm:

  • From the optimal solution of the convex programming problemJL=infσ,YΦ{σ:Tr(Wi)<σ,iN}determine the optimistic performance level JL, the filter transfer functionand extract its minimal

Practical application

In this section, the displacement of a tapered bar subject to external perturbations as indicated in Geromel (1989) and Meirovitch, Baruh, and Oz (1983) is analyzed. Considering that the actuators and sensors are colocated at positions p1,,pm the model with ν vibration modes, valid for all t0 assuming the bar is at rest at t=0, can be expressed asx¨i(t)+ωi2xi(t)=j=1mφi(pj)uj(t),yj(t)=i=1νφi(pj)xi(t)with zero initial conditions xi(0)=x˙i(0)=0 and where uj(t) is the intensity of the force

Conclusion

In this paper a new approach to H2 robust filter design for continuous- and discrete-time polytopic systems has been proposed. It is based on the determination of lower and upper bounds of the equilibrium solution of a minimax problem. A robust filter is constructed from the optimal solution of the problems defined by both bounds. The most interesting characteristic of the design method proposed is that these problems are expressed in terms of linear matrix inequalities without the limitation

José C. Geromel was born in Itatiba, Brazil in 1952. He received the B.S. and M.S. degrees in electrical engineering from the University of Campinas (UNICAMP), Campinas, Brazil, in 1975 and 1976, respectively. He received the Doctorat d’ État degree from the University Paul Sabatier, Toulouse, France, in 1979.

In 1975, he joined the School of Electrical and Computer Engineering, UNICAMP, where he was the Dean for Graduate Studies from 1998 to 2002 and is presently Professor of Control Theory and

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  • Cited by (0)

    José C. Geromel was born in Itatiba, Brazil in 1952. He received the B.S. and M.S. degrees in electrical engineering from the University of Campinas (UNICAMP), Campinas, Brazil, in 1975 and 1976, respectively. He received the Doctorat d’ État degree from the University Paul Sabatier, Toulouse, France, in 1979.

    In 1975, he joined the School of Electrical and Computer Engineering, UNICAMP, where he was the Dean for Graduate Studies from 1998 to 2002 and is presently Professor of Control Theory and Design. In 1987, he was a Visiting Professor at the Milan Polytechnic Institute, Milan, Italy. He is coauthor with P. Colaneri and A. Locatelli, of the book Control Theory and Design: AnRH2andRHViewpoint (New York: Academic Press, 1997), and coauthor, with A.G.B. Palhares, of the book Análise Linear de Sistemas Dinâmicos : Teoria, Ensaios Práticos e Exercícios, in Portuguese, (São Paulo, Brazil: Edgard Blucher Ltda, 2004). His current research interests include convex programming theory, control systems design, robust control and filtering and switched systems.

    Prof. Geromel is Associate Editor of the International Journal of Robust and Nonlinear Control and European Journal of Control. He was awarded, in 1994 the Zeferino Vaz Award for his teaching and research activities at UNICAMP and, in 2007 the Scopus Award, jointly awarded by Elsevier and Capes/Brazil. Since 1991, he has been a Fellow of the CNPq–the Brazilian Council for Research and Development. In 1999 he was named for Chevalier dans l’ Ordre des Palmes Academiques by the French Minister of National Education. Since 1998 he has been a member of the Brazilian Academy of Science. In 2003 he was elected member of the IFAC Council for the term 2003–2005.

    Rubens H. Korogui was born in Jundiaí, Brazil, in 1982. He received the B.S. and M.S. degrees in electrical engineering from the School of Electrical and Computer Engineering, University of Campinas (UNICAMP), Campinas, Brazil, in 2004 and 2006, respectively, where he is now working toward his Ph.D. degree. His research interests include convex programming, robust control and filtering with application to electrical and mechanical engineering.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrey V. Savkin under the direction of Editor Ian Petersen. This research was supported by grants from “Conselho Nacional de Desenvolvimento Científico e Tecnológico—CNPq” and “Fundação de Amparo à Pesquisa do Estado de São Paulo—FAPESP”, Brazil.

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