Observed-Mode-Dependent State Estimation of Hidden Semi-Markov Jump Linear Systems

This paper is concerned with state estimation for a class of hidden semi-Markov jump linear systems governed by a two-layer stochastic process in the discrete-time context. A semi-Markov chain and an observed-mode sequence constitute the lower and upper layer of the process, respectively. With the aid of the emission probability, a novel filter, which is dependent both on the elapsed time within the activated mode and on the observed mode instead of the system mode, is constructed and called observed-mode-dependent (OMD) filter. A modified $\sigma$-error mean square stability ($\sigma$-MSS) is proposed by considering the weight of expected operation time in each actual system mode. Based on the new $\sigma$-MSS, together with a class of Lyapunov functions depending on both the system modes and the corresponding observed ones, numerically checkable conditions on the existence of the OMD filter are presented such that the estimation error system is $\sigma$-MSS with a prescribed $\mathcal {H}_{\infty }$ disturbance attenuation level. A numerical example is presented to demonstrate the theoretical findings.