Mean Square Stability Analysis of Stochastic Continuous-time Linear Networked Systems

TL;DR

This paper develops a mean square stability analysis framework for stochastic continuous-time linear systems with multiplicative uncertainties, extending discrete-time results and providing practical LMI-based tools and stability limitations.

Contribution

It introduces necessary and sufficient conditions for stability using mean square system norms, generalizes stability analysis from discrete to continuous systems, and offers an LMI-based computational approach.

Findings

Stability conditions expressed via mean square system norm.

LMI-based method for stability analysis.

Fundamental limitations for single input channel uncertainty.

Abstract

In this technical note, we study the mean square stability-based analysis of stochastic continuous-time linear networked systems. The stochastic uncertainty is assumed to enter multiplicatively in system dynamics through input and output channels of the plant. Necessary and sufficient conditions for mean square exponential stability are expressed in terms of the input-output property of deterministic or nominal system dynamics captured by the {\it mean square} system norm and variance of channel uncertainty. The stability results can also be interpreted as a small gain theorem for continuous-time stochastic systems. Linear Matrix Inequalities (LMI)-based optimization formulation is provided for the computation of mean square system norm for stability analysis. For a special case of single input channel uncertainty, we also prove a fundamental limitation result that arises in the mean square exponential stabilization of the continuous-time linear system. Overall, the contributions in this work generalize the existing results on stability analysis from discrete-time linear systems to continuous-time linear systems with multiplicative uncertainty. Simulation results are presented for WSCC $9$ bus power system to demonstrate the application of the developed framework.