Stabilization and Optimization of Discrete-Time Markovian Jump Linear Systems via Mode Feedback Control

This article is devoted to a specific kind of discrete-time switched linear systems, where the switching signal is governed by a Markov chain, i.e., Markovian jump linear systems. For such systems, a mode feedback control mechanism is adopted to adjust the mode transition probability matrix, which is referred to as the switching law design, and the optimal mode feedback controller is sought to minimize a quadratic performance index containing both the system state and the mode feedback control input. First, the admissible set of the mode feedback control is investigated, and a sufficient condition is derived to ensure the desired stochastic stability. Second, under the assumption that the concerned system is mode-cost sequential, the original performance index is approximated into a new tractable one and a suboptimal mode feedback controller is then derived within the admissible set. Third, we consider a general situation where no specific requirements are imposed on the Markovian jump linear systems' dynamics, derive the optimal mode feedback controller via a value-iteration-based algorithm, and prove the convergence of the obtained optimal controller. Finally, simulation results are provided to illustrate the validity of the proposed mode feedback control mechanism.