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Observability and nonlinear filtering
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  • Published: 17 June 2008

Observability and nonlinear filtering

  • Ramon van Handel1 

Probability Theory and Related Fields volume 145, pages 35–74 (2009)Cite this article

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Abstract

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability is explored. Finally, the results are partially extended to non-compact signal state spaces.

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Authors and Affiliations

  1. Physical Measurement and Control 266-33, California Institute of Technology, Pasadena, CA, 91125, USA

    Ramon van Handel

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  1. Ramon van Handel
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Correspondence to Ramon van Handel.

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Cite this article

van Handel, R. Observability and nonlinear filtering. Probab. Theory Relat. Fields 145, 35–74 (2009). https://doi.org/10.1007/s00440-008-0161-y

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  • Received: 27 August 2007

  • Revised: 26 May 2008

  • Published: 17 June 2008

  • Issue Date: September 2009

  • DOI: https://doi.org/10.1007/s00440-008-0161-y

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Keywords

  • Nonlinear filtering
  • Asymptotic stability
  • Observability
  • Detectability
  • Controllability
  • Hidden Markov models

Mathematics Subject Classification (2000)

  • 93E11
  • 60J25
  • 62M20
  • 93B05
  • 93B07
  • 93E15
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