Further results on the maximal contractively invariant ellipsoid of discrete-time linear systems with multiple inputs subject to actuator saturation

Contractively invariant ellipsoids have been extensively used as estimates of the domain of attraction of a linear system under a saturated linear feedback. For a discrete-time 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. Based on this condition, the determination of the maximal contractively invariant ellipsoid can be formulated and solved as an LMI problem. In this paper, we develop a criterion to determine if a contractively invariant ellipsoid is the maximal one for discrete-time saturated linear systems with multiple inputs. This criterion is based on the solution of an LMI problem, which involves a generalized convex hull representation of saturated linear feedbacks, and includes the existing result for discrete-time saturated linear systems with a single input as a special case. Simulation results demonstrate the effectiveness of our results.