Generalized criterions for determining the maximal ellipsoidal invariant set of linear systems under saturated linear feedback

Ellipsoids have been extensively used as estimates of the domain of attraction of a linear system under a saturated linear feedback. For a linear system with a single input subject to actuator saturation, based on a convex hull representation of the saturated linear feedback, a necessary and sufficient condition for an ellipsoid to be contractively invariant was previously established, which, through the solution of an LMI problem, leads to the maximal ellipsoidal invariant set. For a linear system with multiple inputs subject to actuator saturation, it has also been proven that the optimal ellipsoid resulting from the optimization problem is the maximal one only under additional conditions. In this paper, we develop a complete characterization of the maximal ellipsoidal invariant set of a linear system with multiple inputs subject to actuator saturation, which is summarized as a comprehensive algorithm to determine if an invariant ellipsoid is the maximal one.