On the triangular canonical form for uniformly observable controlled systems

TL;DR

This paper investigates the transformation of uniformly observable controlled systems into a triangular canonical form, highlighting conditions under which Lipschitz continuity may be lost and linking this to uniform infinitesimal observability.

Contribution

It establishes the possibility of transforming such systems into a (partial) triangular canonical form with non-Lipschitz functions and characterizes where Lipschitzness fails.

Findings

Triangular canonical form can be achieved for these systems.

Lipschitzness may be lost at certain points.

Connection between Lipschitzness loss and uniform infinitesimal observability.

Abstract

We study controlled systems which are uniformly observable and differentially observable with an order larger than the system state dimension. We establish that they may be transformed into a (partial) triangular canonical form but with possibly non locally Lipschitz functions. We characterize the points where this Lipschitzness may be lost and investigate the link with uniform infinitesimal observability.