Possible adaptive control by tangent hyperbolic fixed point transformations used for controlling the -⁶-type van der pol oscillator

Abstract

In this paper a further step towards a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years is reported. Its main advantage in comparison with the complicated Lyapunov function based techniques is that its fundament is some simple geometric consideration allowing to formulate the control task as a Fixed Point Problem for the solution of which various Contractive Mappings can be created that generate Iterative Cauchy Sequences for Single Input - Single Output (SISO) systems. These sequences can converge to the fixed points that are the solutions of the control tasks. Recently alternative potential solutions were proposed and sketched by the use of special functions built up of the "response function" of the excited system under control. These functions have almost constant values apart from a finite region in which they have a "wrinkle" in the vicinity of the desired solution that is the "proper" fixed point of these functions. It was shown that at one of their sides these fixed points were repulsive, while at the opposite side they were attractive. It was shown, too, that at the repulsive side another, so called "false" fixed points were present that were globally attractive, with the exception of the basins of attraction of the "proper" ones.