A Detector-Based Approach for the <inline-formula> <tex-math notation="TeX">$H_{2} $</tex-math></inline-formula> Control of Markov Jump Linear Systems With Partial Information

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

“…Oliveira et al 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 and Song et al 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, 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.…”

“…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.…”

“…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…”

“…Considering some observation assumptions on θ ( k ), the most known approaches are the mode‐dependent control , in which θ ( k ) can be perfectly measured, such as that by Geromel et al; the cluster case , where the controller has access only to some sets in which the system is operating, such as that by Val et al, and Fioravanti et al; and the mode‐independent case, in which there is no information available to the controller, such as that by Morais et al On the other hand, a modeling approach that has been receiving a great deal of attention is the so‐called detector approach or hidden MJLS , in which a hidden Markov chain (HMC) is employed for the analysis and design of the controllers. In the work of Costa et al, it is discussed that the aforementioned mode‐dependent, cluster, and mode‐independent cases can be obtained from the detector approach framework, and also that the observed variable could be viewed as a failure detector . This model is also known as asynchronous control as presented by Song et al in which the problems of static output feedback control and sliding mode control of MJLS with hidden observations were considered.…”

“…In this work, we study the design of , , and mixed dynamic output feedback controllers within the hidden MJLS formulation of Costa et al considering that both x ( k ) and θ ( k ) are not completely available. For that, we present a new iterative method based on the so‐called separation procedure studied by Oliveira et al for hidden MJLS, that is formulated in terms of bilinear matrix inequalities (BMIs).…”