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The aim of this paper is to propose a new numerical approximation of theKalman-Bucy filter for semi-Markov jump linear systems. This approximation isbased on the selection of typical trajectories of the driving semi-Markov chainof the process by using an optimal quantization technique. The main advantageof this approach is that it makes pre-computations possible. We derive aLipschitz property for the solution of the Riccati equation and a generalresult on the convergence of perturbed solutions of semi-Markov switchingRiccati equations when the perturbation comes from the driving semi-Markovchain. Based on these results, we prove the convergence of our approximationscheme in a general infinite countable state space framework and derive anerror bound in terms of the quantization error and time discretization step. Weemploy the proposed filter in a magnetic levitation example with markovianfailures and compare its performance with both the Kalman-Bucy filter and theMarkovian linear minimum mean squares estimator.

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