Rough Linearization By One-Dimensional Rough Paths

Abstract

In this paper, we propose a linearization method of nonlinear control systems by using rough path analysis. The linearization is achieved by the addition of rough signals, which are designed by functions with unbounded variations such as a sinusoidal wave having infinitely-large frequency Therefore, the proposed method is considered to be an extension of the analysis of Lie bracket motions such as the approximation algorithms by Liu and Sussmann in 1990’s, or, the transverse function approach by Morin et al. in 2000’s. Note, however, that, the characteristic feature of the method is to derive controllable linear approximate models for nonholonomic systems even if no Lie bracket motion is generated. Instead, another motions appear based on the similar effects of Wong-Zakai correction terms in stochastic analysis, provided that any stochastic signals are never dealt with in this paper. The validity of the method is confirmed by a few simple examples of linearization problems for nonlinear control systems.