Equivalent Markov models for filtering and smoothing of mining vehicle positions

A linear state‐space model is described whose second‐order moments match that of a hidden Markov chain (HMC). This model enables a modified transition probability matrix to be employed within minimum‐variance filters and smoothers. However, the ensuing filter/smoother designs can exhibit suboptimal performance because a previously reported transition‐probability‐matrix modification is conservative, and identified models can lack observability and reachability. This article describes a less‐conservative transition‐probability‐matrix modification and a model‐order‐reduction procedure to enforce observability and reachability. An optimal minimum‐variance predictor, filter, and smoother are derived to recover the Markov chain states from noisy measurements. The predictor is asymptotically stable provided that the problem assumptions are correct. It is shown that collapsing the model improves state‐prediction performance. The filter and smoother recover the Markov states exactly when the measurement noise is negligible. A mining vehicle position tracking application is discussed in which performance benefits are demonstrated.