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

Abstract

Summary The scriptH2, scriptH∞ and mixed scriptH2false/scriptH∞ dynamic output feedback control of Markov jump linear systems in a partial observation context is studied through an iterative approach. By partial information, we mean that neither the state variable x(k) nor the Markov chain θ(k) are available to the controller. Instead, we assume that the controller relies only on an output y(k) and a measured variable trueθ^false(kfalse) coming from a detector that provides the only information of the Markov chain θ(k). To solve the problem, we resort to an iterative method that starts with a state‐feedback controller and solves at each iteration a linear matrix inequality optimization problem. It is shown that this iterative algorithm yields to a nonincreasing sequence of upper bound costs so that it converges to a minimum value. The effectiveness of the iterative procedure is illustrated by means of two examples in which the conservatism between the upper bounds and actual costs is significantly reduced.

$H_2$ /$H_\infty$ Simultaneous Fault Detection and Control for Markov Jump Linear Systems With Partial Observation

“…Proof: The proof follows the similar reasoning as the one employed in the proof of Theorem 1. Similarly as presented in [4], [10], the structure of matricesQ i andQ −1 i of (18)- (21), andQ i andQ −1 i of (22)- (25), arẽ…”

“…Proof: The proof follows the similar reasoning presented in [4] and [10]. We set the structure of matrices P i and P −1 i of (7)-(8) as…”

“…Due to that we will focus on a procedure that is extensively used in the literature known as the Coordinate Descent Algorithm (CDA), as implemented in [20], or [23]. The specific approach implemented in the present paper was first introduced in [4]. By inspection, it is possible to observe that all the nonlinearities are ''caused'' by the state-feedback controller K .…”

Fault accommodation controller under Markovian jump linear systems with asynchronous modes

“…One similarity of the aforementioned works is the premise that the Markov chain modes are instantly accessible, which is not a realistic premise in most of practical applications. Regarding some works that do not assume that the Markov modes are directly accessible but, instead, there is a detector providing information about this parameter, we can refer to References 18‐21 that deal with the control and filtering problem of such systems considering a hidden Markov set up for the Markov and detector parameters. Along similar lines, the paper by Ogura et al 22 deals with a state feedback control for MJLS considering hidden Markov mode observations, while that by Cheng et al 23 presents an event‐based asynchronous approach for MJS under a hidden mode detection formulation and missing measurements.…”

“…Remark: As for the  ∞ case, the matrices for the FAC controller in (16) and satisfying (21) are directly obtained from the solution of the inequalities (33)-(36).…”