On the Disturbance Attenuation Problem for a Wide Class of Time Invariant Linear Stochastic Systems

In this paper we extend the results of 9] on the stochastic disturbance attenuation problem to a wider class of stochastic systems. We consider time-invariant stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic parameter perturbations. The aim is to develop an H 1 {type theory for such systems. The Ito equations considered in 9] contained two (not necessarily independent) scalar Wiener processes, one for the state dependent noise term and one for the input dependent noise term. In this paper the two scalar Wiener processes are replaced by (not necessarily independent) vector valued Wiener processses. For this wider class of systems a bounded real lemma is derived which provides the basis for an LMI approach towards the stochastic disturbance attenuation problem. Necessary and suucient conditions are derived for the existence of a stabilizing controller reducing the norm of the closed loop perturbation operator to a level below a given threshold. These conditions take the form of coupled rational Riccati inequalities. They generalize the conditions given in 9] and in the absence of stochastic terms reduce to the Riccati inequalities or LMI's, well known from deterministic H 1-theory. Finally, the results are applied to obtain lower estimates for (supremal) stability radii of a given stochastic system.