Control design for bilinear systems with a guaranteed region of stability: An LMI-based approach

This paper deals with the problem of stabilizing a bilinear system with unstable open-loop part by means of state feedback control. The implicit objective is to provide an estimate of the region of stability of the closed-loop system. The proposed procedure can be decomposed into two convex optimization problems described in terms of LMIs: i) Given a polytope which bounds the values of the state, containing the origin, find a stabilizing state feedback control law and an associate region of stability as large as possible inside the polytope. ii) For a solution of the first problem, find the largest polytope containing the ellipsoid such that the stability conditions hold. By iterating these two steps, constructive conditions are given to compute a state feedback control that maximizes the estimate of the region of stability. The results are illustrated by means of examples.