  In this paper we study the state-feedback stabilization of a discrete-time
Markov jump linear system when the observation of the Markov chain of the
system, called the Markov state, is time-randomized by another Markov chain.
Embedding the Markov state into an extended Markov chain, we transform the
given system with time-randomized observations to another one having the
enlarged Markov-state space but with so-called cluster observations of Markov
states. Based on this transformation we propose linear matrix inequalities for
designing stabilizing state-feedback gains for the original Markov jump linear
systems. The proposed method can treat both periodic observations and many of
renewal-type observations in a unified manner, which are studied in the
literature using different approaches. A numerical example is provided to
demonstrate the obtained result.
