An Input-Output Approach to Structured Stochastic Uncertainty in Continuous Time

TL;DR

This paper develops an input-output framework to analyze mean-square stability of continuous-time LTI systems with stochastic uncertainties, broadening applicability to infinite-dimensional systems and delays.

Contribution

It introduces a purely input-output approach for MSS analysis, avoiding state-space models, and connects stochastic block diagrams with stochastic integral equations across interpretations.

Findings

MSS characterized by spectral radius of a matrix operator

Applicable to infinite-dimensional systems and delays

Provides conversion schemes between stochastic interpretations

Abstract

We consider the continuous-time setting of linear time-invariant (LTI) systems in feedback with multiplicative stochastic uncertainties. The objective of the paper is to characterize the conditions of Mean-Square Stability (MSS) using a purely input-output approach, i.e. without having to resort to state space realizations. This has the advantage of encompassing a wider class of models (such as infinite dimensional systems and systems with delays). The input-output approach leads to uncovering new tools such as stochastic block diagrams that have an intimate connection with the more general Stochastic Integral Equations (SIE), rather than Stochastic Differential Equations (SDE). Various stochastic interpretations are considered, such as It\=o and Stratonovich, and block diagram conversion schemes between different interpretations are devised. The MSS conditions are given in terms of the spectral radius of a matrix operator that takes different forms when different stochastic interpretations are considered.