Quadratic Control Lyapunov Functions for Bilinear Systems

In this paper the existence of a quadratic control Lyapunov function for bilinear systems is considered. The existence of a control Lyapunov function ensures the existence of a control law which ensures the global asymptotic stability of the closed loop control system. In this paper we will derive conditions for the existence of a control Lyapunov function for bilinear systems. These conditions will be derived for the whole class of two dimensional bilinear systems with one control input. This will lead to a simple flow diagram representation for the controller design for bilinear systems using a quadratic control Lyapunov function. The controller design itself is carried out using Sontag’s universal control law [1] to obtain an asymptotically stable closed loop system. The Gutman control law [2] which, however, only ensures practical stability of the closed loop system but which is considerably simpler than Sontag’s control law will also be considered. An example will conclude the paper.