Frequency Theorem for discrete time stochastic system with multiplicative noise
For control theorists working on stochastic systems with multiplicative noise, this provides a frequency-domain characterization of optimal control, but the results are theoretical and incremental.
The paper derives necessary and sufficient conditions for optimal control of discrete-time linear stochastic systems with multiplicative noise, formulated as frequency-domain matrix inequalities, and shows that the optimal control reduces to solving a deterministic LQ problem. No concrete numerical results are provided.
In this paper we consider the problem of minimizing a quadratic functional for a discrete-time linear stochastic system with multiplicative noise, on a standard probability space, in infinite time horizon. We show that the necessary and sufficient conditions for the existence of the optimal control can be formulated as matrix inequalities in frequency domain. Furthermore, we show that if the optimal control exists, then certain Lyapunov equations must have a solution. The optimal control is obtained by solving a deterministic linear-quadratic optimal control problem whose functional depends on the solution to the Lyapunov equations. Moreover, we show that under certain conditions, solvability of the Lyapunov equations is guaranteed. We also show that, if the frequency inequalities are strict, then the solution is unique up to equivalence.