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Filtering of switching systems via a singular minimax approach

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This paper considers the problem of state estimation for discrete-time systems whose dynamics switches within a finite set of linear stochastic behaviors. The solution of the filtering problem depends on the a priori informations available on the
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  A. Germani, C. Manes, P. PalumboFILTERING OF SWITCHING SYSTEMSVIA A SINGULAR MINIMAX APPROACHR. 552 Luglio 2001 Alfredo Germani  { Dipartimento di Ingegneria Elettrica, Universitµa degli Studi dell'Aquila,67040 Monteluco (L'Aquila), Italy and Istituto di Analisi dei Sistemi ed Informatica delCNR, Viale Manzoni 30, 00185 Roma, Italy. Email:  germani@iasi.rm.cnr.it . Costanzo Manes  { Dipartimento di Ingegneria Elettrica, Universitµa degli Studi dell'Aquila,67040 Monteluco (L'Aquila), Italy and Istituto di Analisi dei Sistemi ed Informatica delCNR, Viale Manzoni 30, 00185 Roma, Italy. Email:  manes@ing.univaq.it . Pasquale Palumbo  { Istituto di Analisi dei Sistemi ed Informatica del CNR, Viale Manzoni30, 00185 Roma, Italy. Email:  palumbo@iasi.rm.cnr.it . ISSN: 1128{3378  Collana dei Rapportidell'Istituto di Analisi dei Sistemi ed Informatica, CNRviale Manzoni 30, 00185 ROMA, Italytel. ++39-06-77161fax ++39-06-7716461email:  iasi@iasi.rm.cnr.it URL:  http://www.iasi.rm.cnr.it  Abstract This paper considers the problem of state estimation for discrete-time systems whose dynamicsswitches within a ¯nite set of linear stochastic behaviors. In recent years such systems arereceiving a growing attention because of their importance from an applicative point of view, inthat switching phenomena are normally present in many engineering problems. The solution of the ¯ltering problem depends on the amount of the  a priori   information about the switchingprocess. In most papers the switching process is modeled by a discrete Markov chain, witha known transition matrix. For this problem the exact computation of the optimal ¯lter iscumbersome, and most papers deal with the problem of computing approximate ¯lters.In this paper a switching process, that is not statistically characterized, is considered. Thesystem is regarded as an uncertain regular system and it is transformed into a singular systemwith uncertainties only on the second order statistics of noises. This allows us to develop aminimax linear ¯lter, that is the ¯lter that gives the minimum error variance in the worst caseof noise statistics. Key words:  State estimation, switching systems, hybrid systems, minimax ¯ltering, singularsystems.  3. 1. Introduction Switching systems, also denoted hybrid systems or variable structure systems, are receivinga growing attention in recent years because of their importance from an applicative point of view, in that switching phenomena are normally present in many engineering problems (for asurvey on hybrid systems control and applications see [7,2,14]). Many authors investigated theproblem of state estimation for switching systems. Most papers in literature deal with systemswith a stochastically driven switching sequence, modeled as a ¯nite-state Markov Chain (seee.g. [1, 3, 12, 9, 16, 8] for the discrete-time case and [13, 20, 19] for the continuous-time case). This paper considers the case in which no statistical information is available on the switchingprocess. We will show that in this case it is often still possible to estimate the state by consideringthe switching process as an additional state, whose dynamics is unknown, and by consideringthe noise processes with unknown and bounded second order statistics. One key tool to solvethis problem is the theory of singular systems [6]. Such systems are characterized by a lackof information about some components of the dynamic equation [17]. In this paper, by usingsome new results on the ¯ltering for stochastic singular systems [10] and following the minimaxapproach for ¯ltering of uncertain stochastic linear systems [4, 15, 18] it will be shown how to overcome the lack of statistical informations. 2. Stochastic switching systems We consider discrete-time stochastic systems whose dynamics switches between a ¯nite numberof linear functioning modes. Such systems can be described as follows: x (0) =  x 0 ;x ( k + 1) =  A ¹ ( k ) x ( k )+ B ¹ ( k ) u ( k )+ F  ¹ ( k ) N  1 ( k ) ; k  ¸ 0 ;y ( k ) =  C  ¹ ( k ) x ( k ) + D ¹ ( k ) u ( k ) + G ¹ ( k ) N  2 ( k ) ; (2.1)where  x ( k )  2  IR n is the system state,  y ( k )  2  IR q is the measured output and  u ( k )  2  IR  p is aknown deterministic input.  N  1 ( k ) and  N  2 ( k ) are uncorrelated standard white noise sequences(identity covariance matrix),  x 0  is a random vector with given mean ¹ x 0  and covariance ª x 0 .The sequence  f ¹ ( k ) g  commands the switching between the ¯nite modes.Most papers in the literature model the sequence  f ¹ ( k ) g  as a ¯nite-state Markov Chain. Inthis case system (2.1) is also called a  jump Markov linear system  . One of the ¯rst papers in the¯eld is [1], in which only the switching of the noise covariance matrices F  ¹ ( k ) F  T ¹ ( k )  and G ¹ ( k ) G T ¹ ( k ) is considered. A widely appreciated approximate ¯ltering approach is the one proposed in[3], while suboptimal ¯xed-interval smoothing was presented in [12]. In [9] exact ¯lters are computed, whose complexity grows with time. Comparisons among di®erent approximate stateestimation algorithms are performed in [16] and in [8]. The case of continuous-time jump Markov linear systems has been considered in [13, 20, 19]. In this paper we study the ¯ltering problem in the case of absence of any  a priori   statisticalinformation on the switching sequence: nothing can be said on  when   a switch will occur andon  how long   it will last. With this assumption the switching system appears as a linear time-varying partially known system. In particular we consider the case of a binary unknown sequence ¹ ( k )  2 f 0 ; 1 g . The dependence of the system matrices on the binary parameter  ¹ ( k ) can beexpressed either as A ¹ ( k )  =  A 0 ¡ 1 ¡ ¹ ( k ) ¢ + ¹ ( k ) A 1  (2.2)
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