Structured stochastic uncertainty

We consider linear time invariant systems in feedback with structured stochastic uncertainties. This setting encompasses linear systems with both additive and multiplicative noise. We provide a purely input-output treatment of these systems without recourse to state space models, and thus our results are applicable to certain classes of distributed systems. We derive necessary and sufficient conditions for mean square stability in terms of the spectral radius of a linear matrix operator whose dimension is that of the number of uncertainties, rather than the dimension of any underlying state space models. Our condition is applicable to the case of correlated uncertainties, and reproduces earlier results for uncorrelated uncertainties.