Adaptive estimation and identification for discrete systems with Markov jump parameters

Linear discrete-time stochastic dynamical systems with parameters which may switch among a finite set of values are considered. The switchings are modeled by a finite state ergodic Markov chain whose transition probability matrix is unknown and is assumed to belong to a compact set. A novel scheme, called truncated maximum likelihood estimation, is proposed for consistent estimation of the transition probabilities given noisy observations of the system output variables. Conditions for strong consistency are investigated assuming that the measurements are taken after the system has achieved a statistical steady state. The case when the true transition matrix does not belong to the unknown transition matrix set is also considered. The truncated maximum likelihood procedure is computationally feasible, whereas the standard maximum likelihood procedure is not, given large observation records. Finally, using the truncated ML algorithm, a suboptimal adaptive state estimator is proposed and its asymptotic behavior is analyzed.