Preliminary Sketch of Possible Fixed Point Transformations for Use in Adaptive Control

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 it is based on simple geometric considerations on the basis of which the control task can be formulated as a Fixed Point Problem for the solution of which a Contractive Mapping is created that generates an Iterative Cauchy Sequence for Single Input - Single Output (SISO) systems. Consequently it converges to the fixed point that is the solution of the control task. In the formerly developed approaches for monotone increasing or monotone decreasing systems the proper fixed points had only a finite basin of attraction outside of which the iteration might become divergent. The here sketched potential solutions apply a special function built up of the “response function” of the excited system under control and of a few parameters. This function has almost constant value apart from a finite region in which it has a “wrinkle” in the vicinity of the desired solution that is the “proper” fixed point of this function. By the use of an affine approximation of the response function around the solution it is shown that at one of its sides this fixed point is repulsive, while at the opposite side it is attractive. It is shown, too, that at the repulsive side another, so called “false” fixed point is present that is globally attractive, with the exception of the basin of attraction of the “proper” one. This structure is advantageous because a) no divergence can occur in the iteration, b) the convergence to the “false” value can easily be detected, and c) by using some ancillary tricks in the most of the cases the solution can be kicked from the wrong fixed point into the basin of attraction of the “proper one”. In the paper preliminary calculations are presented.N/

Robust Fixed Point Transformations in Adaptive Control Using Local Basin of Attraction

A further step towards a novel approach to adaptive nonlinear control developed at Budapest Tech in the past few years is reported. This approach obviates the use of the complicated Lyapunov function technique that normally provides global stability of convergence at the costs of both formal and essential restrictions, by applying Cauchy sequences of local, bounded basin of attraction in an iterative control that is free of such restrictions. Its main point is the creation of a robust iterative sequence that only slightly depends on the features of the controlled system and mainly is determined be the control parameters applied. It is shown that as far as its operation is considered the proposed method can be located between the robust Variable Structure / Sliding Mode and the adaptive Slotine-Li control in the case of robots or other Classical Mechanical Systems. The operation of these method is comparatively analyzed for a wheel + connected mass system in which this latter component is “stabilized” along one of the spokes of the wheel in the radial direction by an elastic spring. The robustness of these methods is also investigated againts unknown external disturbances of quite significant amplitudes. The numerical simulations substantiate the superiority of the robust fixed point transformations in the terms of accuracy, simplicity, and smoothness of the control signals applied.N/