An Input-Output Approach to Structured Stochastic Uncertainty

We consider linear time-invariant systems with exogenous stochastic disturbances, and in feedback with structured stochastic uncertainties. This setting encompasses linear systems with both additive and multiplicative noise. Our concern is to characterize second-order properties such as mean-square stability (MSS) and performance. A purely input–output treatment of these systems is given without recourse to state-space models, and, thus, the results are applicable to certain classes of distributed systems. We derive necessary and sufficient conditions for MSS 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. For cases where state-space realizations are given, linear matrix inequality equivalents of the input-output conditions are given.