Optimal Control of Stochastic Singularly Perturbed Systems with Exponentiated Quadratic Cost

Abstract

The optimal control of singularly perturbed stochastic linear systems with exponentiated quadratic cost is studied under perfect state measurements. Both finite and infinite horizon formulations are addressed, and particular focus is given to the issues of robustness of the solution to unmodeled fast dynamics, and robustness to the precise value of a singular perturbation parameter (say, ∊). An ∊-independent control is constructed by solving two separate (slow and fast) optimal control problems, and combining the optimal controls of these two problems. It is shown that this composite controller achieves a performance level close to the optimal one whenever the full-order problem has a solution. The slow controller, on the other hand, achieves a finite performance level, with some degradation from the optimal one, whenever the fast subsystem is open-loop stable. It is further shown that if the intensity of the noise in the system dynamics decreases to zero, the slow controller also yields a performance close to the optimal one. The paper also delineates the link between this problem with a positively exponentiated cost and a class of deterministic singularly perturbed H∞-optimal control problems.