An Improved Iterative Approach for Static Output Feedback Control of Discrete-Time Hidden Markov Jump Linear Systems

This paper features new results for the synthesis of <inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{2}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{\infty }$ </tex-math></inline-formula>, and mixed <inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{2}$ </tex-math></inline-formula>/<inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{\infty }$ </tex-math></inline-formula> static output feedback controllers for discrete-time Markov jump linear systems, in a scenario of partial information on the Markov chain. We assume that the mode of operation of the system is not available to the controller, but only an estimate provided by a detector, so that the joint process of the Markov chain and the detector is modeled as a hidden Markov model. In this context, we propose sufficient conditions in terms of bilinear matrix inequalities for the <inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{2}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{\infty }$ </tex-math></inline-formula>, and mixed <inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{2}$ </tex-math></inline-formula>/<inline-formula> <tex-math notation="LaTeX">${\mathcal {H}}_{\infty }$ </tex-math></inline-formula> control design cases. We also present an iterative approach, characterized by an LMI-based coordinate descent algorithm, which is employed to solve the proposed conditions and reduces the guaranteed cost from a feasible initial solution. Unlike existing methods, the proposed approach does not require the computation of stochastic stabilizing state-feedback gains for algorithm initialization. Three numerical examples are presented to illustrate the effectiveness of the proposed results and to demonstrate their advantages compared with existing methods in the literature.