On Triangular Forms for x-Flat Control-Affine Systems With Two Inputs
Georg Hartl, Conrad Gstöttner, Markus Schöberl
TL;DR
The article addresses the problem of identifying flat outputs for $x$-flat, two-input control-affine systems by introducing and leveraging the general triangular form (GTF). It proves that any $x$-flat system can be transformed to the GTF after a finite number of input prolongations, effectively unifying many known normal forms under a single framework. A refined, distribution-based algorithm is proposed to identify components of flat outputs, improving upon prior triangular-form methods and handling cases where existing approaches fail. Through concrete examples, including a VTOL-like system, the authors illustrate both the theoretical reach of the GTF and the practical utility of the refined algorithm. The work advances systematic flatness testing and flat-output synthesis, with potential impact on control design for complex nonlinear, multi-input systems.
Abstract
This paper examines a broadly applicable triangular normal form for x-flat control-affine systems with two inputs. First, we show that this triangular form encompasses a wide range of established normal forms. Next, we prove that any x-flat system can be transformed into this triangular structure after a finite number of prolongations of each input. Finally, we introduce a refined algorithm for identifying candidates for x-flat outputs. Through illustrative examples, we demonstrate the usefulness of our results. In particular, we show that the refined algorithm exceeds the capabilities of existing methods for computing flat outputs based on triangular forms.