Approximate Kalman-Bucy filter for continuous-time semi-Markov jump linear systems

TL;DR

This paper introduces a numerical approximation for the Kalman-Bucy filter tailored for semi-Markov jump linear systems, enabling pre-computation and improved efficiency through optimal quantization techniques.

Contribution

It develops a new approximation method based on optimal quantization, with proven convergence and error bounds, for filtering in semi-Markov jump systems.

Findings

The approximation converges under general conditions.

Error bounds relate to quantization and discretization errors.

Performance comparison shows advantages over traditional filters.

Abstract

The aim of this paper is to propose a new numerical approximation of the Kalman-Bucy filter for semi-Markov jump linear systems. This approximation is based on the selection of typical trajectories of the driving semi-Markov chain of the process by using an optimal quantization technique. The main advantage of this approach is that it makes pre-computations possible. We derive a Lipschitz property for the solution of the Riccati equation and a general result on the convergence of perturbed solutions of semi-Markov switching Riccati equations when the perturbation comes from the driving semi-Markov chain. Based on these results, we prove the convergence of our approximation scheme in a general infinite countable state space framework and derive an error bound in terms of the quantization error and time discretization step. We employ the proposed filter in a magnetic levitation example with markovian failures and compare its performance with both the Kalman-Bucy filter and the Markovian linear minimum mean squares estimator.