$\mathscr{H}_{\infty}$ state-feedback gain-scheduled control for MJLS with non-homogeneous Markov chains*

This paper deals with discrete-time Markov jump linear systems (MJLS) whose stochastic process that rules the jumps between the operation modes is non-homogeneous (time-varying transition probabilities). This class of systems is specially useful in networked control system problems where the Markov chain can be associated with time-varying parameters that can be measured or estimated, for example, packet loss variant with distance, network congestion, protocol communication, among others. With this motivation, it is proposed a design procedure for $\mathscr{H}_{\infty}$ state-feedback controllers scheduled by the time-varying probabilities, potentially providing improved performance with respect to the standard robust controllers (in-dependent of the probabilities). Synthesis conditions are given in terms of parameter-dependent linear matrix inequalities (LMIs), that can be numerically solved in terms of polynomial approximations of fixed degrees for the decision variables. As a physically motivated example, it is investigated a problem of car pursuit, where the packet loss rate of control signal varies with time, depending on the distance of the vehicle to the control unit. Additionally, the accuracy of the proposed conditions is evaluated through numerical comparisons with techniques from the literature in the case of homogeneous MJLS (time-invariant probabilities).