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 ldquoresponse functionrdquo 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 ldquowrinklerdquo in the vicinity of the desired solution that is the ldquoproperrdquo 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 ldquofalserdquo fixed point is present that is globally attractive, with the exception of the basin of attraction of the ldquoproperrdquo one. This structure is advantageous because (a) no divergence can occur in the iteration, (b) the convergence to the ldquofalserdquo 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 ldquoproper onerdquo. In the paper preliminary calculations are presented.