Reachability Analysis of the State Transition and State Covariance Matrices for an LTV System

TL;DR

This paper analyzes the reachability of state transition and covariance matrices in linear time-varying systems over finite intervals, using Riccati differential equations to characterize possible end states.

Contribution

It provides a novel characterization of the reachable sets of these matrices in LTV systems, including non-controllable cases, via solutions to matrix Riccati differential equations.

Findings

Characterized the set of reachable closed-loop state transition matrices.

Determined the set of reachable state covariance matrices from a given initial covariance.

Extended reachability analysis to non-controllable LTV systems.

Abstract

In this paper, we study the reachability of two closely related matrices appearing in the analysis of linear time-varying (LTV) systems over a finite time interval, namely, its closed-loop state transition matrix via a state feedback control and its state covariance matrix starting from some given initial state covariance matrix. Under a mild assumption, we first characterize the set of closed-loop terminal state transition matrices reachable from the identity matrix using controls of the state feedback form. Then, we provide the set of terminal state covariance matrices reachable from any given positive definite initial state covariance matrix when the LTV system is not necessarily controllable. Both results are based on the solutions of corresponding matrix Riccati differential equations (RDE).