Finite horizonH₂/H_∞control of time‐varying stochastic systems with Markov jumps and (x,u,v)‐dependent noise

Abstract

This study investigates the finite horizon H 2 /H ∞ control of time-varying stochastic Markov jump systems (SMJSs) with state, control and disturbance-dependent noise. Firstly, the stochastic bounded real lemma of SMJSs is established, which by itself has theoretical importance. Secondly, several necessary and sufficient conditions for H 2 /H ∞ control of SMJSs are proposed by means of coupled generalised differential Riccati equations. Finally, two numerical examples are given to show the effectiveness of the obtained results.

Mixed H 2 / H ∞ control of time‐varying stochastic discrete‐time systems under uniform detectability

“…Unlike the state transition matrix of time-varying deterministic discrete-time systems in [14], φ l,k for the stochastic case is much more complicated. Therefore an iterative form instead of an analytical form of the state transition matrix for system (4) is presented in (5). Moreover, we can give another iterative form for the state transition matrix of system (4)…”

On mean‐square H∞ control for discrete‐time nonlinear stochastic systems with (x, u, v)‐dependent noises

“…Another kind of noises, the control‐dependent noises, have also received some research interest in the past decade as the noises, though possibly tiny, are usually inevitable during the controller implementation that might lead to the fragility of the controlled system (see other works). Recently, the simultaneous consideration of the state‐, control‐, and disturbance‐dependent noises ( (x, u, v) ‐dependent noises for short) has started to draw some initial research attention because of their general engineering insight (see related works for some recent results). For example, a sufficient condition has been given in the work of Zhang et al for the H ∞ control problem of general nonlinear stochastic systems with (x, u, v) ‐dependent noises by means of an HJI (instead of the three coupled HJEs as in the work of Zhang and Feng).…”

“…For example, a sufficient condition has been given in the work of Zhang et al for the H ∞ control problem of general nonlinear stochastic systems with (x, u, v) ‐dependent noises by means of an HJI (instead of the three coupled HJEs as in the work of Zhang and Feng). In the work of Gao et al, the finite‐horizon H 2 / H ∞ control problem has been discussed for time‐varying linear stochastic systems with Markov jump and (x, u, v) ‐dependent noises. Note that most reported results have been concerned with the continuous‐time systems, and the corresponding results for the discrete‐time counterparts have been very relatively few despite their potential in practical applications, and the main motivation of this paper is therefore to shorten such a gap by providing a rather general framework.…”

Stochastic H2/H∞ control of nonlinear systems with time-delay and state-dependent noise

“…Owing to the fact that the mixed H 2 /H ∞ control can minimize a desired control performance and eliminate the effect of disturbance, it is more attractive than the sole H ∞ control in engineering practice [11][12][13][14]. References [12] and [13] solved the nonlinear stochastic H 2 /H ∞ control problems by means of coupled Hamilton-Jacobi equations (HJEs) in the finite-and infinitehorizon cases, respectively.…”

“…References [12] and [13] solved the nonlinear stochastic H 2 /H ∞ control problems by means of coupled Hamilton-Jacobi equations (HJEs) in the finite-and infinitehorizon cases, respectively. Based on coupled Riccati equations, reference [14] designed the H 2 /H ∞ controller for time-varying linear stochastic systems. However, most of existing work was concerned with stochastic H 2 /H ∞ control for delay-free systems, while little attention was paid on delayed systems.…”