# Applications of Homogeneity to Nonlinear Adaptive Control

• Matthias Kawski
Part of the Progress in Systems and Control Theory book series (PSCT, volume 11)

## 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.)

## Keywords

Filter Design Nonlinear Control System Feedback Stabilization Lyapunov Function Versus Open Neighbourhood Versus
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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