Stabilization of Networked Delay Systems

In this chapter, we study the stabilization of networked feedback systems in the presence of stochastic uncertainties and time delays. We model the stochastic uncertainty as a random process in a multiplicative form, and we assess the stability of system based on mean-square criteria. Based on the mean-square small-gain theorem, Theorem 2.5, we develop fundamental conditions of mean-square stabilizability, which ensure that an open-loop unstable system can be stabilized by output feedback. For SISO systems, a general, explicit stabilizability condition is obtained. This condition, both necessary and sufficient, provides a fundamental limit imposed by the system’s unstable poles, nonminimum phase zeros, and time delay. This condition answers to the question: What is the exact largest range of delay such that there exists an output feedback controller mean-square stabilizing all plants under a stochastic multiplicative uncertainty for delays within that range?  For MIMO systems, we provide a solution for minimum phase systems possibly containing time delays, in the form of a generalized eigenvalue problem. Limiting cases are also showing how the directions of unstable poles may affect mean-square stabilizability of MIMO minimum phase systems.