Reachability and Controllability Analysis of the State Covariance for Linear Stochastic Systems

TL;DR

This paper characterizes the set of terminal state covariances reachable in finite time for linear stochastic systems, providing complete results for discrete-time and bounds for continuous-time cases, along with controllability conditions.

Contribution

It offers a complete characterization for discrete-time systems and bounds for continuous-time systems regarding the reachability of state covariances, including controllability conditions.

Findings

Complete characterization for discrete-time systems

Upper bounds for continuous-time systems

Necessary and sufficient controllability conditions

Abstract

This paper studies the set of terminal state covariances that are reachable over a finite time horizon from a given initial state covariance for a linear stochastic system with additive noise. For discrete-time systems, a complete characterization of the set of reachable state covariances is given. For continuous-time systems, we present an upper bound on the set of reachable state covariances. Moreover, for both linear discrete-time and continuous-time systems, necessary and sufficient conditions are provided for the controllability of the state covariance over a finite horizon.