Robust Stability and Stabilizability Conditions for Time-Delay Systems Under Stochastic Uncertainties

This study first explores the mean-square robust stability problem of stable continuous-time linear time-invariant systems subject to stochastic multiplicative uncertainties with prescribed variance bounds. The internal structures of uncertainties, however, are not presumed to cope with diverse random noises and errors arising from networked channels. A necessary and sufficient mean-square stability condition is obtained involving a novel small-gain type characterization. Next, we consider the output feedback controller synthesis problem of networked control systems facing stochastic multiplicative uncertainties and intrinsic channel-wise time delays simultaneously. Based on the obtained mean-square stability condition, further, we develop a fundamental necessary and sufficient condition of mean-square stabilizability explicitly. Such a condition equivalently determines that an open-loop unstable system can be stabilized by output feedback in the mean-square sense. Finally, an analysis of delay robustness is elaborated, in which the mean-square stabilizability condition is extended to the case tolerating uncertain but upper bounded time delays. Overall, this study provides a comprehensive analysis of the robust mean-square stability and mean-square stabilizability of time-delay systems under stochastic uncertainties, which can have practical implications for networked control systems.