Reduced-order H∞ filtering of phase-type semi-Markov jump linear systems

In this work, we present design conditions for robust reduced-order $H_{\infty }$ filters for a type of hybrid system, the so-called phase-type (PH) semi-Markov jump linear systems (S-MJLS), in a context of partial observation of the jump process. For this purpose, we recast the non-linear design problem for obtaining reduced-order $H_{\infty }$ filters into a convex problem written as linear matrix inequalities (LMI) by employing a new linearization technique based on slack variables. Furthermore, by restricting the filter structure to the plant’s order and reducing the semi-Markov jump system into a Markov jump linear system with perfect mode observation, we are able to design the optimal $H_{\infty }$ filter. In the case that the project can afford full-order filters, and by using mild assumptions on the system matrices, we also present conditions for clusterized model-based $H_{\infty }$ observers, which are robust with respect to uncertain transition rates and some system matrices. For showing the effectiveness of our results, we present an example in the context of networked control systems subject to packet dropouts.