Flatness of Two-Input Discrete-Time Systems and their Linearization
Johannes Schrotshamer, Bernd Kolar, Markus Schöberl
TL;DR
This work develops a rank-based framework for discrete-time flat nonlinear systems, focusing on the two-input case with an $(x,u)$-flat output. By analyzing the Jacobian of the flat parameterization, the authors relate rank defects to forward- and backward-flatness and then introduce simple dynamic extensions—prolongations for forward-flat systems and prelongations for backward-flat systems—to achieve exact linearization. They prove that two-input flat systems can be linearized via a combination of these extensions, even when neither pure forward- nor backward-flatness suffices, and illustrate the results with planar VTOL and wheeled-mobile-robot-style examples. The approach promises computationally feasible tests and practical dynamic-feedback designs for discrete-time nonlinear control tasks, bridging discrete-time flatness theory with tractable linearization procedures.
Abstract
In this contribution we discuss flat discrete-time nonlinear systems in a general setting including two special subclasses, namely, forward- and backward-flat systems. We relate rank conditions for certain submatrices of the Jacobian of the flat parameterization to the mentioned subclasses. Motivated by these rank conditions, for the case of two-input systems that possess an (x,u)-flat output, we derive a simple type of dynamic extension for the purpose of an exact linearization.