Ellipsoidal Reachability Sets of Linear Time-Varying Systems in Estimation and Control Problems

We consider a linear time-varying system with an initial state and disturbance that are known imprecisely and satisfy a common constraint. The constraint is the sum of a quadratic form of the initial state and the time integral of a quadratic form of the disturbance, and these quadratic forms are allowed to be degenerate. We obtain a linear matrix differential Lyapunov equation describing the evolution of the ellipsoidal reachability set. In the problem of estimating the state based on output observations, this result is used to find the minimum-size ellipsoidal set of admissible system states, which is determined by the optimal observer and by the reachability set of the corresponding observation error equation. A method for control law synthesis ensuring that the system state reaches the target set or the system trajectory remains in a given ellipsoidal tube is proposed. Illustrative examples are given for the Mathieu equation, which describes parametric oscillations of a linear oscillator.