Adaptive stabilization without high-gain

Abstract

During the last few years there has been a very intense discussion on the applicability of adaptive control and on ‘standard assumptions’ made in the traditional theory. Some years ago, the question of what is really the relevant information needed for successful adaptive control was starting to receive some attention. The present work belongs to this tradition.

A very brief introduction to the concept of adaptive control is first given. The prototype problem of stabilizing an unstable, unknown plant is studied. The main result is the complete characterization of necessary and sufficient a priori knowledge needed for adaptive stabilization, namely knowledge of the order of any stabilizing controller. The concept of switching function controller is introduced, and some properties stated. ‘The Turing Machine of Universal Controllers’ is then presented. As the title indicates, this adaptive controller possessed the greatest stabilizing power a smooth, non-linear controller can have. The preceding works in this field have all dealt with variations on the theme of high-gain stabilization. This paper deals only with adaptive stabilization algorithms not requiring high-gain-stabilizability. Finally, the problem of stabilization to a possibly non-zero reference value is solved.