Controlled invariance of ellipsoids: linear vs. nonlinear feedback

Several equivalent conditions or statements for set invariance were obtained for systems with one saturating actuator in a recent paper. In particular, it was shown that the existence of a nonlinear feedback that makes an ellipsoid invariant is equivalent to the existence of a feedback linear inside the ellipsoid that makes it invariant. In this paper, we will show that this equivalence property holds conditionally for systems with multiple saturating actuators. We will provide a criterion to check if the largest ellipsoid made invariant by nonlinear feedback can also be made invariant by a feedback linear inside the ellipsoid. Numerical examples reveal that this criterion is usually satisfied. The equivalence of other set invariance conditions will also be investigated.