Robust Disturbance Rejection by the Attractive Ellipsoid Method – Part I: Continuous-time Systems

Abstract

Abstract This paper develops sufficient conditions for the constrained robust stabilization of continuous-time polytopic linear systems with unknown but bounded perturbations. The attractive ellipsoid method (AEM) is employed to determine a robustly controllable invariant set, known as attractive ellipsoid, such that the state trajectories of the system asymptotically converge to a small neighborhood of the origin despite the presence of non-vanishing perturbations. To solve the stabilization problem, we employ the Finsler’s lemma and derive new linear matrix inequality (LMI) conditions for robust state-feedback control design, ensuring convergence of state trajectories of the system to a minimal size ellipsoidal set. We also consider the state and control constrained problem and derive extended LMI conditions. Under certain conditions, the obtained LMIs guarantee that the attractive ellipsoid is nested inside the bigger ellipsoids imposed by the control and state constraints. Finally, we extend our AEM approach to the gain-scheduled state-feedback control problem, where the scheduling parameters governing the time-variant system are unknown in advance but can be measured in real-time. Two examples demonstrate the feasibility of the proposed AEM and its improvements over previous works.