Mean-Square Stability Radii for Stochastic Robustness Analysis: A Frequency-Domain Approach

The purpose of this paper is to develop a general framework of mean-square robust control in the presence of stochastic multiplicative uncertainties. We consider block diagonally structured stochastic uncertainties of a variety of structures including diagonal scalar, element-by-element, repeated scalar and full-block stochastic processes with prescribed variance bounds, which may arise from multiple sources and can be used to model various communication losses. In an important distinction from the previous work, we allow the stochastic uncertainties to be statistically correlated. A general mean-square robustness measure, termed mean-square stability radius (MSSR), is introduced as the metric to quantify stability robustness under the mean-square criterion. Explicit expressions of the MSSR are derived, and a small-gain type necessary and sufficient condition is obtained for mean-square robust stability. Based on the MSSR expressions, mean-square stabilization problems are studied by minimizing the MSSR over all possible stabilizing controllers, leading to stabilizability conditions for mean-square feedback stabilization. Moreover, as an analogy to its counterpart in the robust control theory, a mean-square optimal performance problem is shown to be equivalent to one of mean-square stabilization, by introducing a fictitious stochastic uncertainty and augmenting the system appropriately.