Improvements to the linear differential inclusion approach to stability analysis of linear systems with saturated linear feedback

Ellipsoids, as level sets of quadratic Lyapunov functions, and the convex hull of ellipsoids, as a level set of a certain composite quadratic Lyapunov function, have both been extensively used as estimates of the domain of attraction of a linear system under saturated linear feedback. By expressing the saturated linear feedback law on the convex hull of a group of linear feedback laws, which in turn expresses the linear system under this saturated linear feedback in a linear differential inclusion, conditions have been established under which an ellipsoid or the convex hull of a group of ellipsoids are contractively invariant sets and are thus estimates of the domain of attraction. These conditions are usually less conservative for single input systems than for multiple input systems. In this paper, we consider multiple input systems and establish conditions for contractive invariance of the convex hull of ellipsoids that are less conservative than the existing conditions.