Control and Filtering Problems for Linear Time-Varying Systems Based on Ellipsoidal Reachable Sets

Abstract

The paper is devoted to reachable sets of linear time-varying continuous systems under uncertain initial states and disturbances with a bounded uncertainty measure. The uncertainty measure is the sum of a quadratic form of the initial state and the integral over the finite-time interval from a quadratic form of the disturbance. It is shown that the reachable set of the system under this assumption is an evolving ellipsoid with a matrix being a solution to the linear matrix differential equation. This result is used to synthesize the optimal observer providing the minimal ellipsoidal set as the estimate of the system state, as well as optimal controllers steering the system state into a final target ellipsoidal set or keeping the entire system trajectory in a prescribed ellipsoidal tube under all admissible initial states and disturbances. The relationship between the optimal ellipsoidal observer and the Kalman filter are established. Numerical modeling with the Mathieu equation for parametric vibrations of a linear oscillator illustrates the results.