Section: New Results

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

Participants : Benoîte de Saporta, Eduardo Costa.

We 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. This work was presented at the international conference [37] and is submitted to an international journal [50] .