Composite Energy Function Based Learning Control

Abstract

The iterative leaning control discussed in preceding Chapters are based on the contraction mapping method, which would encounter extreme difficulty when the local Lipschitz continuous nonlinear factors are involved, either in the system dynamics or in the control mechanism. In the presence of such nonlinearities, finite escape time phenomenon may occur and the contraction mapping method is no longer applicable. That is the reason why most iterative learning control schemes proposed hitherto were of simple linear-type, and limited to global Lipschitz continuous (GLC) nonlinear systems. The contraction mapping based ILC design allows us to completely ignore the system dynamics part, whereas the finite time divergence shows how strongly could be such a dynamic impact. It is necessary to widen the learning control framework under which ILC can handle broader classes of system nonlinearities, such as local Lipschitz continuous ones, and system uncertainties, such as time-varying parametric and non-parametric uncertainties. In this chapter we are going to exploit two aspects. The first is to study how the x-dynamics in state space can be incorporated in ILC. The second is to exploit the use of energy function approaches in ILC, such as the Lyapunov direct method. Indeed, looking into the recent advances in control theories and applications, remarkable progress was made in state space with the Lyapunov direct method [116]. It would be very meaningful to look into these control methods, henceforth derive the energy function based ILC.