State and mode feedback control strategy for discrete-time Markovian jump linear systems with time-varying controllable mode transition probability matrix

In this article, the state and mode feedback control strategy is investigated for the discrete-time Markovian jump linear system (MJLS) with time-varying controllable mode transition probability matrix (MTPM). This strategy, consisting of a state feedback controller and a mode feedback controller, is proposed to ensure MJLS's stability and meanwhile improve system performance. First, a mode-dependent state feedback controller is designed to stabilize the MJLS based on the time-invariant part of the MTPM such that it can still keep valid even if the MTPM is adjusted by the mode feedback control. Second, a generalized quadratic stabilization cost is put forward for evaluating MJLS's performance, which contains system state, state feedback controller, and mode feedback controller. To reduce the stabilization cost, a mode feedback controller is introduced to adjust each mode's occurrence probability by changing the time-varying controllable part of MTPM. The calculation of such mode feedback controller is given based on a value-iteration algorithm with its convergence proof. Compared with traditional state feedback control strategy, this state and mode feedback control strategy offers a new perspective for the control problem of general nonhomogeneous MJLSs. Numerical examples are provided to illustrate the validity of the proposed strategy.