Stochastic $H^\infty$

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 $H^{\infty}$-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 $\gamma 0$. In the deterministic case, the analogous conditions involve two uncoupled linear matrix inequalities, but in the stochastic setting we obtain coupled nonlinear matrix inequalities instead. The connection between $H^{\infty}$ 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.