Discrete-time Flatness and Linearization along Trajectories

TL;DR

This paper establishes that linearization of discrete-time flat systems along trajectories preserves flatness, providing a necessary condition for flatness in discrete-time systems and illustrating the results with examples.

Contribution

It proves that linearized discrete-time systems along trajectories remain flat, extending continuous-time flatness properties to the discrete domain.

Findings

Linearized systems are flat if the original system is flat.

Flat outputs of linearized systems are linearizations of original flat outputs.

The parameterization maps of linearized systems are linearizations of the original maps.

Abstract

The paper studies the relation between a nonlinear time-varying flat discrete-time system and the corresponding linear time-varying systems which are obtained by a linearization along trajectories. It is motivated by the continuous-time case, where it is well-known that the linearization of flat systems along trajectories results in linear time-varying systems which are again flat. Since flatness implies controllability, this constitutes an important verifiable necessary condition for flatness. In the present contribution, it is shown that this is also true in the discrete-time case: We prove that the linearized system is again flat, and that a possible flat output is given by the linearization of a flat output of the nonlinear system. Analogously, the map that describes the parameterization of the system variables of the linear system by this flat output coincides with the linearization of the corresponding map of the nonlinear system. The results are illustrated by two examples.