Analytic normal forms and symmetries of strict feedforward control systems

Abstract

This paper deals with the problem of convergence of normal forms of control systems. We identify an n‐dimensional subclass of control systems, called special strict feedforward form, shortly (SSFF), possessing a normal form which is a smooth (resp. analytic) counterpart of the formal normal form of Kang. We provide a constructive algorithm and illustrate by several examples including the Kapitsa pendulum and the cart–pole system. The second part of the paper is concerned about symmetries of single‐input control systems. We show that any symmetry of a smooth system in SSFF is conjugated to a scaling translation and any 1‐parameter family of symmetries is conjugated to a family of scaling translations along the first variable. We compute explicitly those symmetries by finding the conjugating diffeomorphism. We illustrate our results by computing the symmetries of the cart–pole system. Copyright © 2009 John Wiley & Sons, Ltd.