Abstract
This paper addresses an H2 optimal control problem for a class of discrete-time stochastic systems with Markov jump parameter and multiplicative noises. The involved Markov jump parameter is a uniform ergodic Markov chain taking values in a Borel-measurable set. In the presence of exogenous white noise disturbance, Gramian characterization is derived for the H2 norm, which quantifies the stationary variance of output response for the considered systems. Moreover, under the condition that full information of the system state is accessible to measurement, an H2 dynamic optimal control problem is shown to be solved by a zero-order stabilizing feedback controller, which can be represented in terms of the stabilizing solution to a set of coupled stochastic algebraic Riccati equations. Finally, an iterative algorithm is provided to get the approximate solution of the obtained Riccati equations, and a numerical example illustrates the effectiveness of the proposed algorithm.
Highlights
Markov jump systems belong to a class of multimodal stochastic dynamical models with regime switching governed by a Markov chain, and have found wide-spread applications ranging from manipulator robot [1], biology of viruses [2], portfolio selection [3]
According to the number of states included in the state space of Markov chain, finite and infinite Markov jump systems are classified and investigated, respectively
After a half century’s research, the control theory for finite Markov jump systems has reached a remarkable degree of maturity, such as stability analysis [5], observability and detectability [6], linear quadratic (LQ) optimal control [7], filter design [8] and finite-time control [9]
Summary
Markov jump systems belong to a class of multimodal stochastic dynamical models with regime switching governed by a Markov chain, and have found wide-spread applications ranging from manipulator robot [1], biology of viruses [2], portfolio selection [3]. When the state space of Markov chain is expanded to an infinite set, there will arise some properties essentially different from the finite case. In a recent paper [21], an H2 control problem was studied for discrete-time Markov jump systems where the Markov chain has a Borel state space. It was recognized that the multiplicative noise perfectly depicts the stochastic fluctuation of physical parameter caused by uncertain environment; on the other hand, Markov chain with a Borel-measurable state space can provide substantial benefit for real applications, e.g., the networked control systems analyzed in [22]. Among all nc -dimensional dynamic stabilizing controllers, the optimal H2 control strategy is achieved by a zero-order controller with feedback gain determined by the stabilizing solution to a set of coupled stochastic algebraic Riccati equations. Rn : n-dimensional real Euclidean space; Rm×n : the normed linear space of all m by n real matrices; k · k: the Euclidean norm of Rn or the operator norm of Rm×n ; A0 : the transpose of a matrix (or vector) A; Tr ( A): the trace of a square matrix A; Sn : the set of n × n real symmetric matrices; A > 0 (≥ 0): A is a positive (semi-)definite symmetric matrix; In : the n × n identity matrix; δ(·) : the Kronecker functional; Z+ := {0, 1, 2, · · · }
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