State-feedback control of Markov jump linear systems with hidden-Markov mode observation

Abstract

SUMMARYIn this paper, we study state-feedback control of Markov jump linear systems with partial information. In particular, we assume that the controller can only access the mode signals according to a hidden-Markov observation process. Our formulation generalizes various relevant cases previously studied in the literature on Markov jump linear systems, such as the cases with perfect information, no information, and cluster observations of the mode signals. In this context, we propose a Linear Matrix Inequalities (LMI) formulation to design feedback control laws for (stochastic) stabilization, H 2 , and H∞ control of discrete-time Markov jump linear systems under hidden-Markovian observations of the mode signals. We conclude by illustrating our results with some numerical examples.

An iterative approach for the discrete‐time dynamic control of Markov jump linear systems with partial information

“…Alternatively, Ogura et al considered an observation process in which the Markov chain is accessed only when some modes of operation of a different Markov process are visited. It was discussed by Ogura et al that this model also generalizes the mode‐dependent, cluster, and mode‐independent formulations, along with the detector approach of Costa et al…”

“…As for the discrete‐time case, Liu et al presented design conditions for the control, but apparently the controller matrices also depended on θ . The design of state‐feedback controllers was considered by Ogura et al through the hidden model previously described, in which the controller depends on the most recent value of the cluster of the observed variable and the time elapsed since the last observation, with the results given in the LMI framework. According to Remark 18 of Ogura et al, even though the detector approach can be retrieved via the observation model presented in that paper, the proposed design conditions for the control can lead to a potential underperformance with respect to the ones presented by Costa et al The separation method has been used in dealing with the coupling problem in BMI conditions by several authors (see, for instance, the works of Song et al and Oliveira et al for a small sample of studies under this formulation).…”

“…The design of state‐feedback controllers was considered by Ogura et al through the hidden model previously described, in which the controller depends on the most recent value of the cluster of the observed variable and the time elapsed since the last observation, with the results given in the LMI framework. According to Remark 18 of Ogura et al, even though the detector approach can be retrieved via the observation model presented in that paper, the proposed design conditions for the control can lead to a potential underperformance with respect to the ones presented by Costa et al The separation method has been used in dealing with the coupling problem in BMI conditions by several authors (see, for instance, the works of Song et al and Oliveira et al for a small sample of studies under this formulation). The paper by Song et al studied the design of static output feedback sliding mode controllers subject to a round‐robin protocol in the context of networked control system (NCS), with a separation strategy employed for avoiding nonlinear terms in the design conditions.…”

“…This model is also known as asynchronous control as presented by Song et al 11,12 in which the problems of static output feedback control and sliding mode control of MJLS with hidden observations were considered. Alternatively, Ogura et al 13 considered an observation process in which the Markov chain is accessed only when some modes of operation of a different Markov process are visited. It was discussed by Ogura et al 13 that this model also generalizes the mode-dependent, cluster, and mode-independent formulations, along with the detector approach of Costa et al 10 In this work, we study the design of  2 ,  ∞ , and mixed  2 ∕ ∞ dynamic output feedback controllers within the hidden MJLS formulation of Costa et al 10 considering that both x(k) and (k) are not completely available.…”

“…Oliveira et al 14,15 presented a similar separation procedure for the hidden MJLS as the one used in this work, but neither the H ∞ and the mixed H 2 ∕H ∞ control was fully studied, nor an iterative method was presented. Moreover, the works of Ogura et al 13 and Song et al 11,22 are different with respect to our results in the sense that: (1) we deal with dynamic output feedback control in the hidden MJLS or detector approach, following the formulation by Costa et al, 10 and (2) our iterative separation method employs a state-feedback controller as an input to the algorithm so that the dynamic controller by the end of the iterations is composed by the state-feedback gain and a filter structure.…”