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Best Linear Filter with Unknown (x(t),θ(t))

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Continuous-Time Markov Jump Linear Systems

Part of the book series: Probability and Its Applications ((PIA))

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Abstract

It is a well-known fact that the optimal nonlinear filter for continuous-time MJLS, in the general case in which both the state variable and the jump parameter are not known, cannot be given in terms of a closed finite system of stochastic differential equations (it is not a finite filter). The aim of this chapter is to derive the best linear mean-square estimator for continuous-time MJLS in the scenario described above, i.e., assuming that only an output is available. The idea is to derive a filter which bears the desirable property of the Kalman filter, a recursive scheme suitable for computer implementation which allows some offline computation that alleviates the computational burden. The filter is derived as a function of the error covariance matrix whose dynamics is governed by two matrix differential equations: one associated with the second moment of the state variable and the other one associated with the second moment of the estimator. Both the finite- and infinite-horizon cases are considered.

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Authors and Affiliations

  1. Polytechnic School, Department of Telecommunications, University of São Paulo, Av. Prof. Luciano Gualberto, Trav. 3 158, São Paulo, 05508-900, São Paulo, Brazil

    Oswaldo L.V. Costa

  2. Department of Systems and Control, LNCC/MCT, Nat. Lab. for Scientific Computing, Av. Getúlio Vargas, Quitandinha 333, Petrópolis, 25651-070, Rio de Janeiro, Brazil

    Marcelo D. Fragoso

  3. Department of Systems and Control, LNCC/MCT, Nat. Lab. for Scientific Computing, Av. Getúlio Vargas, Quitandinha 333, Petrópolis, 25651-070, Rio de Janeiro, Brazil

    Marcos G. Todorov

Authors
  1. Oswaldo L.V. Costa
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  2. Marcelo D. Fragoso
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  3. Marcos G. Todorov
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Costa, O., Fragoso, M., Todorov, M. (2013). Best Linear Filter with Unknown (x(t),θ(t)). In: Continuous-Time Markov Jump Linear Systems. Probability and Its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34100-7_7

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