On the necessity of a recent set invariance condition under actuator saturation

A sufficient condition for an ellipsoid to be invariant was obtained recently and an LMI approach was developed to find the largest ellipsoid satisfying the condition. This condition was later shown to be necessary for the single input case. This paper is dedicated to the multi-input case. We will examine when this condition is also necessary for multi-input systems. Our investigation is based on studying the optimal solution to a related LMI problem. A criterion is presented to determine when the condition is necessary and when the largest invariant ellipsoid has been obtained by using the LMI method.