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

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.

Keywords:

• stochastic contraction
• unscented Kalman–Bucy filter
• virtual–actual framework
• exponential convergence
• statistical linearisation

Notes

1. For brevity sake, we will omit time-varying variable t for the remainder of this proof.

Additional information

Funding

The work is supported by the Strategic University Program on Control, Information and Communication Systems for Environmental and Safety Critical Systems.

Notes on contributors

J.P. Maree

Johannes Maree received the MEng degree in electronic engineering from Department of Electrical, Electronic and Computer Engineering, University of Pretoria, South Africa, in 2009. He is currently working towards his PhD degree at Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU), Trondheim, Norway. For a part of his PhD studies, he was a visiting scholar at Department of Chemical and Biological Engineering, University of Wisconsin-Madison, WI, USA. His research interests include contraction theory, model predictive control with the emphasis on control performance and optimality and set-theoretic methods.

L. Imsland

Lars Imsland received the PhD degree in electrical engineering from Department of Engineering Cybernetics at the Norwegian University of Science and Technology (NTNU) in Trondheim, in 2002. For a part of his PhD studies, he was a visiting scholar at the Institute for Systems Theory in Engineering at the University of Stuttgart, Germany. After his PhD studies, he has worked as a post-doctoral researcher at NTNU, a research scientist at SINTEF and as a specialist for Cybernetica AS, before becoming a professor in control engineering at NTNU in 2009. His main research interests are in the theory and application of nonlinear and optimising control and estimation.

J. Jouffroy

Jerome Jouffroy received the Diplome d’Etudes Approfondies in applied control systems and the PhD degree in control systems from the University of Savoie, Annecy, France, in 1999 and 2002, respectively. From 2002 to 2003, he worked as a research engineer at the French Institute for Ocean Research (IFREMER), Toulon, France, on navigation algorithms for underwater vehicles. Between 2003 and 2006, he was a post-doctoral fellow at the Center for Ships and Ocean Structures (CeSOS), from the Norwegian University of Science and Technology (NTNU), Trondheim, Norway. Since 2007, he is an associate professor in control at the Mads Clausen Institute, University of Southern Denmark (SDU). His research interests include nonlinear control and estimation, cooperative control, with applications to autonomous vehicles, sailing systems, and power electronics.