Stabilization of ε-coupled bilinear systems using state feedback control

Abstract

The state feedback stabilization problem of ε-coupled bilinear systems is considered. It is shown, via Lyapunov stability theory, that the stabilizing control is quadratic in the state. The scalar control case is considered first, and the Lyapunov matrices P 1 and P 2 which stabilize the decoupled subsystems, in which the system is split when ε = 0, are determined. The overall stabilizing matrix P for the ε≠0 case is obtained by employing a suitable smiilarity state transformation T (T≠0). In the particular case of strictly bilinear systems it is shown that, under certain conditions, the stabilizing control law is decoupled with respect to the subsystem states. Finally the class of multiparameter ε-coupled bilinear systems is considered. Two examples are worked out. The paper closes by demonstrating how the vectorial control case can be treated by the scalar controller results.