Ellipsoidal reachable sets of linear time-varying continuous and discrete systems in control and estimation problems

Abstract

The paper is devoted to reachable sets of linear time-varying continuous or discrete 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/sum 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 or difference equation. This result is used to synthesize optimal observers and estimators providing the minimal ellipsoidal sets as the estimates of the system state and unknown parameters, respectively, 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 relationships between the optimal ellipsoidal observer and the Kalman filter as well as between the optimal ellipsoidal estimator and the recursive least weighted squares algorithm are established. Numerical modeling with the Mathieu equation for parametric vibrations of a linear oscillator illustrates the results.