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On convergence of the unscented Kalman–Bucy filter using contraction theory

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  • J.P. Maree
  • L. Imsland
  • J. Jouffroy

Abstract

Contraction theory entails a theoretical framework in which convergence of a nonlinear system can be analysed differentially in an appropriate contraction metric. This paper is concerned with utilising stochastic contraction theory to conclude on exponential convergence of the unscented Kalman–Bucy filter. The underlying process and measurement models of interest are Itô-type stochastic differential equations. In particular, statistical linearisation techniques are employed in a virtual–actual systems framework to establish deterministic contraction of the estimated expected mean of process values. Under mild conditions of bounded process noise, we extend the results on deterministic contraction to stochastic contraction of the estimated expected mean of the process state. It follows that for the regions of contraction, a result on convergence, and thereby incremental stability, is concluded for the unscented Kalman–Bucy filter. The theoretical concepts are illustrated in two case studies.

Suggested Citation

  • J.P. Maree & L. Imsland & J. Jouffroy, 2016. "On convergence of the unscented Kalman–Bucy filter using contraction theory," International Journal of Systems Science, Taylor & Francis Journals, vol. 47(8), pages 1816-1827, June.
    • Handle: RePEc:taf:tsysxx:v:47:y:2016:i:8:p:1816-1827
    DOI: 10.1080/00207721.2014.953799
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    References listed on IDEAS

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    1. Z. Kowalczuk & M. Domżalski, 2012. "Optimal asynchronous estimation of 2D Gaussian–Markov processes," International Journal of Systems Science, Taylor & Francis Journals, vol. 43(8), pages 1431-1440.
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