Applications of Homogeneity to Nonlinear Adaptive Control

Abstract

This article is intended to bring together ideas from the geometric theory of nonlinear control systems, in particular the employment of homogeneity properties for asymptotic feedback stabilization, with a classical approach to adaptive control, in particular the regulation problem in the presence of unknown parameters. The control systems under consideration are of the form

$$\dot x\left( t \right) = f\left( {x\left( t \right),p} \right) + u\left( t \right)g\left( {x\left( t \right),p} \right)$$

((1.1))

The control u takes vaLues in R, x E R“ and the vectorfields f and g are explicitly linearly parametrized by the supposedly unknown parameter p E R, i.e.

$$f\left( {x,p} \right) = {f_0}\left( x \right) + p{f_1}\left( x \right)andg\left( {x,p} \right) = {g_0}\left( x \right) + p{g_1}\left( x \right)$$

((1.2))

where k and gi are smooth vectorfields on R“, and we suppose that f_°(0) _ f1(0) = O. (Note, that this notion of linear parametrization is very much coordinate dependent.)