Online Bayesian estimation of transition probabilities for Markovian jump systems

Markovian jump systems (MJSs) evolve in a jump-wise manner by switching among simpler models, according to a finite Markov chain, whose parameters are commonly assumed known. This paper addresses the problem of state estimation of MJS with unknown transition probability matrix (TPM) of the embedded Markov chain governing the jumps. Under the assumption of a time-invariant but random TPM, an approximate recursion for the TPMs posterior probability density function (PDF) within the Bayesian framework is obtained. Based on this recursion, four algorithms for online minimum mean-square error (MMSE) estimation of the TPM are derived. The first algorithm (for the case of a two-state Markov chain) computes the MMSE estimate exactly, if the likelihood of the TPM is linear in the transition probabilities. Its computational load is, however, increasing with the data length. To limit the computational cost, three alternative algorithms are further developed based on different approximation techniques-truncation of high order moments, quasi-Bayesian approximation, and numerical integration, respectively. The proposed TPM estimation is naturally incorporable into a typical online Bayesian estimation scheme for MJS [e.g., generalized pseudo-Bayesian (GPB) or interacting multiple model (IMM)]. Thus, adaptive versions of MJS state estimators with unknown TPM are provided. Simulation results of TPM-adaptive IMM algorithms for a system with failures and maneuvering target tracking are presented.