Exact Covariance Characterization for Controlled Linear Systems subject to Stochastic Parametric and Additive Uncertainties

TL;DR

This paper provides an exact mathematical description of covariance dynamics in linear systems with stochastic uncertainties, enabling more tractable control design methods and validated through numerical comparisons.

Contribution

It introduces a novel exact covariance characterization for systems with unbounded stochastic uncertainties, improving control design approaches.

Findings

The new covariance characterization is more accurate than empirical estimates.

Control design conditions are less conservative and more computationally feasible.

Numerical results validate the effectiveness of the proposed methods.

Abstract

This work addresses the exact characterization of the covariance dynamics related to linear discrete-time systems subject to both additive and parametric stochastic uncertainties that are potentially unbounded. Using this characterization, the problem of control design for state covariance dynamics is addressed, providing conditions that are conservative yet more tractable compared to standard necessary and sufficient ones for the same class of systems. Numerical results assess this new characterization by comparing it to the empirical covariance and illustrate the control design problem.