Stochastic H^∞

We consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations. Our objective is to develop an H∞-type theory for such systems. We prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations. This enables us to obtain necessary and sufficient conditions for the existence of a stabilizing compensator which keeps the effect of the perturbations on the to-be-controlled output below a given threshhold γ > 0. In the deterministic case, the analogous conditions involve two uncoupled linear matrix inequalities, but in the stochastic setting we obtain coupled nonlinear matrixinequalities instead. The connection between H∞ theory and stability radii is discussed and leads to a lower bound for the radii, which is shown to be tight in some special cases.