An LMI approach for H2 and H∞ reduced-order filtering of uncertain discrete-time Markov and Bernoulli jump linear systems

Abstract

Abstract This paper proposes a new approach based on parameter-dependent linear matrix inequality (LMI) conditions associated with a scalar parameter that are sufficient to provide robust H 2 and H ∞ reduced-order mode-dependent, partially mode-dependent or mode-independent filters for discrete-time Markov jump linear systems (MJLS) with time-invariant uncertain transition probabilities. Time-invariant uncertainties in the state–space matrices of the modes can be handled as well. As main difference with respect to the existing approaches in the literature, the filter matrices are obtained directly from the slack variables introduced in the conditions. Moreover, the proposed conditions become also necessary for a particular choice of the scalar parameter when mode-dependent full-order filters are designed for systems without uncertainties. Additionally, for precisely known generalized Bernoulli jump systems (i.e., the case where all the rows of the transition probability matrix are equal), optimal solutions are obtained for both mode-dependent and mode-independent full-order filters. Examples (including one motivated by a practical application) are presented to illustrate the proposed approach.