Convex Hull of Ellipsoids: A New Tool for Constrained Control of
Uncertain Time-Varying Linear Discrete-Time Systems
Abstract
In this paper we address the problem of determining convex conditions
for the convex hull of ellipsoids to be robustly invariant. The
motivation for the study is from the vertex control. Based on the convex
hull of a set of vertices, necessary and sufficient conditions were
suggested in the literature for the stability analysis and controller
design problems of a constrained uncertain time-varying linear
discrete-time system. However it is not trivial to compute the vertices,
especially for high dimensional systems. In the present paper, we
propose to replace the convex hull of vertices by the convex hull of
ellipsoids. The associated ellipsoids are not required to be robustly
invariant. The conditions are given in terms of linear matrix
inequalities (LMIs). Hence the main drawback of vertex control is
removed. The conditions are losslessly extended to stabilization,
providing then an LMI solution to the non-linear state feedback control
design problem of constrained uncertain time-varying linear discretetime
systems. Several examples are included with comparison to earlier
solutions from the literature to illustrate the results.