$H_2$-Control of Continuous-Time Hidden Markov Jump Linear Systems

This technical note addresses the <inline-formula><tex-math notation="LaTeX">$H_2$</tex-math></inline-formula> -control problem for a continuous-time Markov Jump Linear System (MJLS) with partial information on the mode of operation. The jumps of the system depend on a continuous-time hidden Markov model (CT-HMM) where the hidden process represents the dynamics of the real system, while the emitted signal represents the information available to the controller. We start by analyzing the stochastic stability control problem with the goal to design a state feedback linear controller that stochastically stabilizes the closed-loop system, relying on the information from the detector. Two dual formulations, based on LMI problems, are derived as a solution for this problem. These results are then extended to the <inline-formula><tex-math notation="LaTeX">$H_2$</tex-math></inline-formula> guaranteed cost control problem. The technical note is concluded with a numerical example which illustrates the derived results.

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