MAP estimation of target manoeuvre sequence with the expectation-maximization algorithm

Two algorithms are derived for the problem of tracking a manoeuvring target based on a sequence of noisy measurements of the state. Manoeuvres are modeled as unknown input (acceleration) terms entering linearly into the state equation and chosen from a discrete set. The expectation maximization (EM) algorithm is first applied, resulting in a multi-pass estimator of the MAP sequence of inputs. The expectation step for each pass involves computation of state estimates in a bank of Kalman smoothers tuned to the possible manoeuvre sequences. The maximization computation is efficiently implemented using the Viterbi algorithm. A second, recursive estimator is then derived using a modified EM-type cost function. To obtain a dynamic programming recursion, the target state is assumed to satisfy a Markov property with respect to the manoeuvre sequence. This results in a recursive but suboptimal estimator implementable on a Viterbi trellis. The transition costs of the latter algorithm, which depend on filtered estimates of the state, are compared with the costs arising in a Viterbi-based manoeuvre estimator due to Averbuch, et al. (1991). It is shown that the two criteria differ only in the weighting matrix of the quadratic part of the cost function. Simulations are provided to demonstrate the performance of both the batch and recursive estimators compared with Averbuch's method and the interacting multiple model filter.