Mixed Reduced-Order Filtering for Discrete-Time Markov Jump Linear Systems With Partial Information on the Jump Parameter

This article deals with the
$H_{2}$
,
$H_{\infty }$
, and mixed
$H_{2} / H_{\infty }$
reduced-order filters for discrete-time Markov jump linear systems, within the context of partial observation of the jump parameter. It is considered that the Markov chain parameter, denoted by
$\theta (k)$
, is not observable but, instead, only an estimation, represented by
$\hat \theta (k)$
, is available for the filter design. Sufficient synthesis conditions for the filter design, based on linear matrix inequalities, are provided. These conditions for the
$H_{2}$
and
$H_{\infty }$
filters are not conservative in the sense that, for the full-order case and perfect information of the Markov parameter, they become also necessary. Simplified conditions are derived for the Bernoulli case. This article is concluded with an illustrative example in the context of networked control systems, in which we study the effects of reducing the order of the filter on the estimation performance.