Abstract
This paper is concerned with a kind of discrete-time stochastic systems with Markov jump parameters taking values in a Borel measurable set. First, both strong exponential stability and exponential stability in the mean square sense are introduced for the considered systems. Based on generalized Lyapunov equation and inequality, necessary and sufficient conditions are derived for the strong exponential stability. By use of the given stability criteria, it is shown that strong exponential stability can lead to exponential stability and further to stochastic stability. Moreover, strong exponential stability can guarantee the so-called $l_{2}$ input-state stability, which characterizes the asymptotic behavior of system state influenced by exogenous disturbance with finite energy. Second, $H_{\infty }$ performance is analyzed for the perturbed dynamic models over finite and infinite horizons, respectively. For a prescribed disturbance attenuation level, stochastic bound real lemmas are presented in terms of Riccati equations or linear matrix inequalities. As a direct application, the infinite-horizon $H_{\infty }$ control problem is settled and the state-feedback controller is constructed. Numerical simulations are conducted to illustrate the validity of the proposed results.
Highlights
Markov jump systems have attracted considerable attention over the past half century, primarily due to their widespread applications ranging from fault detection techniques [1], to genetic regulator network [2] and portfolio selection [3]
Most researchers are concentrated on finite Markov jump systems, where the state space of Markov chains consists of finite possible values [4]–[12]
In the monographs [13] and [14], stability analysis and control synthesis have been elaborately addressed for continuous-time finite Markov jump systems
Summary
Markov jump systems have attracted considerable attention over the past half century, primarily due to their widespread applications ranging from fault detection techniques [1], to genetic regulator network [2] and portfolio selection [3]. From the viewpoint of theoretical research, it is well known that all states of a finite and irreducible Markov process are recurrent and admit a stationary probability distribution This property does not hold in general for the Borel-measurable Markov chain. A numerical example is presented to demonstrate when Borel-measurable state space description is isolated as a point-wise set, VOLUME 8, 2020 the desired control objective can not be achieved even if the control law remains same. This justifies the necessity to study Borel-measurable Markov jump systems. C: the set of all complex numbers; Rn: n-dimensional real Euclidean space; Rm×n: the normed linear space of all m by n real matrices; · : the Euclidean norm of Rn or the operator norm of Rm×n; A : the transpose of a matrix (or vector) A; Tr(A): the trace of a square matrix A; Sn: the subset of Rn×n composed of all n × n symmetric matrices; A > 0 (≥ 0): A is a positive (semi-)definite symmetric matrix; In: the n × n identity matrix; δ(·): the Kronecker functional; diag(·): the (block-)diagonal matrix; Z+: the set of all nonnegative integers
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