Separation principle for stochastic control of continuous-time Markov jump linear systems under partial observations

Abstract

In MJLS literature the separation principle between filtering and control has been established in case the Markov mode switching process {θt} is fully observed, and the Euclidean state process {xt}is partially observed. In case the exact {θt}remains hidden, the separation principle has only been established under a linear filtering restriction. Since nonlinear filters can provide significant better estimates, the desire to extend the separation principle to MJLS with hidden {θt}is a long-standing challenge. The objective of this paper is to resolve this long-standing challenge in three steps. The first step is to transform the MJLS stochastic control problem into control under a quadratic performance criterion of a linear system driven by a martingale which is influenced by the control. The certainty equivalence (CE) condition known in literature applies to stochastic control of a linear system that is driven by a control independent martingale. Therefore, the second step is to relax this known CE condition such that it allows this control influence on the martingale. The third step is to prove that the relaxed CE condition is satisfied for the general MJLS control problem considered. The overall achievement is a CE control law for a partially observed MJLS, which assures the Separation Principle between filtering and control. The paper also shows that for the case that {xt}is fully observed and the exact{θt}remains hidden, that the novel CE control law differs significantly from the in literature well-developed Averaging MJLS control policy.