Global Asymptotic Stabilization for a Nonlinear System on a Manifold via a Dynamic Compensator

The purpose of this paper is to solve a global asymptotic stabilization problem for a nonlinear control system on a manifold with a complete metric. As well known, a system on a noncontractible manifold is not globally asymptotically stabilizable via a continuous feedback law. This problem results from the existence of multiple singular points of such a controlled system. It is shown that if all singular points can be assigned to a subspace of the extended state space using a dynamic compensator and a continuous feedback except for at most one jump, then the augmented system becomes globally asymptotically stable. Moreover, a method for stabilization is developed using a dynamic compensator and a global control Lyapunov function for an input-affine system. Finally, we propose a method for constructing the control Lyapunov function for a system such that the dimension of the coefficient matrix of the input is equal to the dimension of the state.