Possible Adaptive Control by Tangent Hyperbolic Fixed Point Transformations Used for Controlling the Φ6-Type Van der Pol Oscillator - Tar, József K. (2008)
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. This structure seemed to be advantageous because no divergences could occur in the iterations, the convergence to the “false” values could easily be detected, and by using some ancillary tricks in the most of the cases the solutions could be kicked from the wrong fixed points into the basins of attraction of the “proper ones”. It was expected that via adding simple rules to the application of these transformations good adaptive control can be developed. However, due to certain specialties of these functions practical problems arose. In the present paper novel transformations are presented that seem to evade these difficulties. Their applicability is illustrated via simulations in the adaptive control of the popular nonlinear paradigm, the Φ6 Van der Pol oscillator.