Convergence of discrete-time Kalman filter estimate to continuous time estimate

ABSTRACT

This article is concerned with the convergence of the state estimate obtained from the discrete-time Kalman filter to the continuous time estimate as the temporal discretisation is refined. The convergence follows from Martingale convergence theorem as demonstrated below; however, surprisingly, no results exist on the rate of convergence. We derive convergence rate estimates for the discrete-time Kalman filter estimate for finite and infinite dimensional systems. The proofs are based on applying the discrete-time Kalman filter on a dense numerable subset of a certain time interval [0, T].

KEYWORDS:

• Kalman filter
• infinite dimensional systems
• temporal discretisation
• sampled data

Acknowledgements

The author thanks Dr. Jarmo Malinen for valuable comments on the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author was financially supported by the Finnish Graduate School in Engineering Mechanics.