Reachability Analysis of the State Transition and State Covariance Matrices for an LTV System
This work provides theoretical foundations for control and estimation in LTV systems, but it is incremental as it extends known results from time-invariant to time-varying systems.
The paper characterizes the reachable sets of closed-loop state transition matrices and state covariance matrices for linear time-varying systems over a finite interval, using solutions of matrix Riccati differential equations. It provides explicit descriptions of these sets under mild assumptions, including cases where the system is not controllable.
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).