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On convergence of the unscented Kalman-Bucy filter using contraction theory. (English) Zbl 1333.93243

Summary: Contraction theory entails a theoretical framework in which convergence of a nonlinear system can be analyzed differentially in an appropriate contraction metric. This paper is concerned with utilizing 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 linearization 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.

MSC:

93E11 Filtering in stochastic control theory
93E10 Estimation and detection in stochastic control theory
93B18 Linearizations
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[1] DOI: 10.1002/9781118032428 · doi:10.1002/9781118032428
[2] Bonnabel S., A contraction theory-based analysis of the stability of the extended Kalman filter (2012)
[3] DOI: 10.1109/9.566674 · Zbl 0876.93089 · doi:10.1109/9.566674
[4] Brown R.G., Introduction to random signals and applied Kalman filtering: With MATLAB exercises and solutions, 3. ed. (1997) · Zbl 0868.93002
[5] Gelb A., Applied optimal estimation (1974)
[6] DOI: 10.1080/0020772021000046225 · Zbl 1029.93008 · doi:10.1080/0020772021000046225
[7] DOI: 10.1109/TSP.2007.911295 · Zbl 1390.94218 · doi:10.1109/TSP.2007.911295
[8] DOI: 10.4173/mic.2010.3.2 · doi:10.4173/mic.2010.3.2
[9] DOI: 10.1109/JPROC.2003.823141 · doi:10.1109/JPROC.2003.823141
[10] DOI: 10.2514/3.56665 · Zbl 0838.93080 · doi:10.2514/3.56665
[11] DOI: 10.1080/00207721.2011.604737 · Zbl 1307.93402 · doi:10.1080/00207721.2011.604737
[12] DOI: 10.1021/i260073a028 · doi:10.1021/i260073a028
[13] DOI: 10.1016/j.automatica.2012.02.014 · Zbl 1246.93121 · doi:10.1016/j.automatica.2012.02.014
[14] DOI: 10.1109/TAC.1979.1101943 · Zbl 0399.93054 · doi:10.1109/TAC.1979.1101943
[15] DOI: 10.1016/S0005-1098(98)00019-3 · Zbl 0934.93034 · doi:10.1016/S0005-1098(98)00019-3
[16] DOI: 10.1002/aic.690460317 · doi:10.1002/aic.690460317
[17] Pham Q.C., Stochastic contraction in Riemannian metrics (2013)
[18] DOI: 10.1109/TAC.2008.2009619 · Zbl 1367.60073 · doi:10.1109/TAC.2008.2009619
[19] DOI: 10.1016/S0005-1098(98)00053-3 · Zbl 0938.93504 · doi:10.1016/S0005-1098(98)00053-3
[20] DOI: 10.1109/78.978374 · doi:10.1109/78.978374
[21] DOI: 10.1109/TAC.2007.904453 · Zbl 1366.93660 · doi:10.1109/TAC.2007.904453
[22] DOI: 10.1002/0470045345 · doi:10.1002/0470045345
[23] Van der Merwe R., Sigma-point Kalman filters for probabilistic inference in dynamic state-space models (PhD thesis) (2004)
[24] DOI: 10.1016/j.automatica.2005.10.004 · Zbl 1103.93045 · doi:10.1016/j.automatica.2005.10.004
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