Feedback stabilization of MIMO systems in the presence of stochastic network uncertainties and delays
The purpose of this paper is to study stabilization problem of linear time-invariant systems subject to stochastic multiplicative uncertainties and time delays. We consider a structured multiplicative perturbation which consists of static, zero-mean stochastic processes and we assess the stability of system based on mean-square criteria. The mean-square stabilization problem for multi-input multi-output systems generally requires solving an optimization problem involving the spectral radius of a certain closed loop transfer function matrix. This problem in general is non-convex and by and large unresolved, only approximate solutions are available based on numerical algorithms resembling to the D-K iteration for μ-synthesis. Our main contributions include the fundamental conditions, both necessary and sufficient, which insure that the multi-input multi-output minimum phase systems can be stabilized by output feedback in the mean-square sense. We provide a complete, computationally efficient solution in the form of a generalized eigenvalue problem readily solvable by means of linear matrix inequality optimization. For conceptual insights, limiting cases are analyzed in depth to characterize and quantify explicitly how the directions of unstable poles may affect the mean-square stabilizability of multi-input multi-output systems.