Mixed 𝓗/𝓗 filtering for Markov jump linear systems

We study in this work the mixed $ \mathcal {H}_{2}/\mathcal {H}_{\infty } $ H2/H∞-filtering problem for Markov jump linear systems in a partial observation context. Rather than the Markov chain $ \theta (k) $ θ(k), we consider that only an estimation $ \hat {\theta }(k) $ θˆ(k) coming from a detector is available to the filter. We present a sufficient condition for the synthesis of a filter depending only on $ \hat \theta (k) $ θˆ(k) such that the estimation errors for the $ \mathcal {H}_{2} $ H2 and $ \mathcal {H}_{\infty } $ H∞ filtering problems are bounded by given scalars. The robust mixed $ \mathcal {H}_{2}/\mathcal {H}_{\infty } $ H2/H∞ filtering considering that the transition and detection probabilities are uncertain, as well as the complete observation, cluster and mode-independent cases, are also studied. The results are given in terms of linear matrix inequalities and are illustrated by a numerical example.