Stochastic Detectability and Mean Bounded Error Covariance of the Recursive Kalman Filter with Markov Jump Parameters

Abstract

In this article, we study the error covariance of the recursive Kalman filter when the parameters of the filter are driven by a Markov chain taking values in a countably infinite set. We do not assume ergodicity nor require the existence of limiting probabilities for the Markov chain. The error covariance matrix of the filter depends on the Markov state realizations, and hence forms a stochastic process. We show in a rather direct and comprehensive manner that this error covariance process is mean bounded under the standard stochastic detectability concept. Illustrative examples are included.

Keywords:

• Detectability
• Kalman filtering
• Markov jump systems
• Stochastic systems

Mathematics Subject Classification:

• 93E03
• 93E11
• 93B99

Research supported in part by the EPSRC Research Grant EP/E057438, Nonlinear observation theory with applications to Markov jump systems, the FAPESP Grants 06/02004-0 and 06/04210-6 and the CNPq Grants 304429/2007-4 and 482386/2007-0.