Optimal filtering on continuous-time systems with Markovian communication delays and packet dropouts

This paper is concerned with the optimal filter problems for networked systems with random transmission delays, while the delay process is modeled as a multi-state Markov chain which incorporates the data losses naturally. By defining an indicator function of the random delay, the optimal filter problems are transformed into the ones of the standard Markov jumping parameter measurement system. We first present an optimal Kalman filter, which is with time-varying, path-dependent filter gains, and the number of the paths grows exponentially in time delay. Thus an alternative optimal Markov jump linear filter is presented, in which the filter gains just depend on the present value of the Markov chain, and as a result, the obtained filter is again a Markov jump linear system. It can be further shown that the proposed Markov jump linear filter converges to a constant-gain filter under appropriate assumptions.