Static Nonsmooth Control Lyapunov Function Design via Dynamic Extension

In this paper, we propose a method to obtain a control Lyapunov function (CLF) by the reduction of a CLF of an augmented system with a dynamic compensator. For asymptotically stabilizing control, dynamic compensators are not necessary in most of the cases. However, in some cases, we can easily design a stabilizing control law using a dynamic compensator. Therefore, a constructive design method using a static controller via a dynamic controller has advantages and is preferable in practice. In this paper, we assume that a CLF has been designed on an extended state space with a dynamic compensator, and show that taking minimum values of the CLF on the extended state space gives a nonsmooth CLF on the original state space. This method can be considered as an extension of the minimum projection method[1], [2]. We also show that the obtained CLF fulfills Lipschitz continuity and local semiconcavity if the original CLF on the extended state space is Lipschitz continuous and locally semiconcave. The effectiveness of the proposed method is demonstrated by an example.