LMI-based design of ℋ∞ dynamic output feedback controllers for MJLS with uncertain transition probabilities

Abstract

This paper addresses the problem of ℋ∞ dynamic output feedback control design for discrete-time Markov jump linear systems (MJLS) with uncertainties affecting the transition probability matrix. Robustly stabilizing controllers are obtained through a two-step procedure based on linear matrix inequalities (LMIs) and polynomially parameter-dependent decision variables. In the first step, matrix inequalities with a scalar parameter provide a stabilizing parameter-dependent state feedback gain. The conditions, that become LMIs for fixed values of the scalar, also include a linear transformation to prevent the occurrence of blocks of zero in the gain. This parameter-dependent gain is used as input in the second step, where sufficient LMI conditions provide a fixed order dynamic controller that assures an upper bound to the ℋ∞ norm of the closed-loop system. Differently from other approaches, the synthesized controller does not depend explicitly on the matrices of the system, what makes the method specially adequate to handle reduced order requirements and partial or complete unavailability of modes for feedback. Moreover, the obtained dynamic output feedback controller can be used as a new input for the second step, giving rise to a procedure where less conservative ℋ∞ guaranteed costs can be achieved iteratively. Numerical examples illustrate that reduced order controllers providing smaller ℋ∞ guaranteed costs can be obtained with the proposed methodology.