Multilayer minimum projection method for nonsmooth strict control Lyapunov function design

Abstract

Asymptotic stabilization on noncontractible manifolds is known as a difficult control problem. To address this problem, we had proposed the minimum projection method to design nonsmooth control Lyapunov functions. This method, however, has some problems: difficult étale-surjection design, undesirable resulting control Lyapunov functions, etc. In this paper, we propose a new nonsmooth control Lyapunov function design method called the ‘ Multilayer minimum projection method’ for nonsmooth control Lyapunov function design on general manifolds. The method considers many simple-structured smooth manifolds associated with the original manifold by étale mappings, and then a function on the original manifold is obtained by projecting control Lyapunov functions defined on the simple-structured manifolds onto the original manifold. In this paper, we prove that the resulting function by the proposed method is a nonsmooth control Lyapunov function on the original manifold. Moreover, we prove that if all control Lyapunov functions defined on simple-structured manifolds are strict, the control Lyapunov function on the original manifold is a strict control Lyapunov function. Finally, the effectiveness of the proposed method and the advantage over the conventional minimum projection method are confirmed by an example.