Composite quadratic Lyapunov functions for constrained control systems

The composite quadratic function based on a group of quadratic functions was introduced in our earlier paper. Some important properties of this Lyapunov function were revealed. We showed that this function is continuously differentiable and its level set is the convex hull of a group of ellipsoids. In this paper, we use these results to study the set invariance properties of linear systems with input and state constraints. We show that for a system under a given saturated linear feedback, the convex hull of a group of invariant ellipsoids is also invariant. If each ellipsoid in a group can be made invariant with a bounded control of the saturating actuator, then their convex hull can also be made invariant by the same actuator. For a group of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.