Mean-Square Stabilizability Under Unstructured Stochastic Multiplicative Uncertainties: A Mean-Square Small-Gain Perspective

Abstract

In this paper we study the stability and stabilizability problems of multi-input, multi-output linear time-invariant systems subject to stochastic multiplicative uncertainties under the mean-square criterion. We consider the matrix-valued unstructured perturbations, which consist of static, zero-mean stochastic processes. We first obtain a necessary and sufficient condition to ensure the stability of the open-loop stable system against uncertainties in the mean-square sense. Based on the obtained mean-square stability condition, we further answer the question: How can an open-loop unstable system be stabilized by output feedback in the mean-square sense despite the presence of such stochastic uncertainties? The complete and explicit stabilizability conditions are derived, which reveal how the locations and directions associated with unstable poles and nonminimum phase zeros of the plant coupled together affect the mean-square stabilizability.