Control and estimation in linear time-varying systems based on ellipsoidal reachability sets

Linear continuous or discrete time-varying systems in which the sum of a quadratic form of the initial state and the integral or sum of quadratic forms of a disturbance on a finite horizon is bounded above by a given value are considered. It is demonstrated that the reachability set of such a continuous- or discrete-time system is an evolving ellipsoid, and its ellipsoid matrix satisfies a linear matrix differential or difference equation, respectively. The optimal ellipsoidal observer and identification algorithm that yield the best ellipsoidal estimates of the system’s state and unknown parameters are designed. In addition, the optimal controllers ensuring that the system’s state will fall into a target set or that the system’s trajectory will stay within the ellipsoidal tube are designed. A connection between the optimal ellipsoidal observer and the Kalman filter is established. Some illustrative examples for the Mathieu equation, which describes the parametric oscillations of a linear oscillator, are given.