Robust-Control By 2 Lyapunov Functions

A new method of designing a robust control law is proposed for a general class of non-linear or linear systems with bounded uncertainties. The method uses the property that the Lyapunov function is not unique for a stable or stabilizable system. It is shown that the proposed control law normally guarantees the stability of the system if there are two Lyapunov functions whose null sets have a trivial intersection. The null set of a Lyapunov function is defined to be the set in state space in which the product of the transpose of the system input matrix and the gradient of the Lyapunov function is equal to zero. The robust control results impose no restriction on the structure and size of the input-unrelated uncertainties. Moreover, it is shown that asymptotic stabilization of the nominal system is not necessarily required in this method