A Trajectory-Based Approach to Discrete-Time Flatness

TL;DR

This paper extends the concept of discrete-time flatness by incorporating backward-shifts, providing a more comprehensive framework that preserves key properties like controllability and linearizability.

Contribution

It introduces an extended approach to discrete-time flatness that includes backward-shifts, enhancing the theoretical understanding and practical applicability.

Findings

Flat systems with backward-shifts are still controllable and reachable.

The extended flatness allows straightforward trajectory planning.

Such systems can be linearized by specific dynamic feedbacks.

Abstract

For discrete-time systems, flatness is usually defined by replacing the time-derivatives of the well-known continuous-time definition by forward-shifts. With this definition, the class of flat systems corresponds exactly to the class of systems which can be linearized by a discrete-time endogenous dynamic feedback as it is proposed in the literature. Recently, verifiable necessary and sufficient differential-geometric conditions for this property have been derived. In the present contribution, we make an attempt to take into account also backward-shifts. This extended approach is motivated by the one-to-one correspondence of solutions of flat systems to solutions of a trivial system as it is known from the continuous-time case. If we transfer this idea to the discrete-time case, this leads to an approach which also allows backward-shifts. To distinguish the classical definition with forward-shifts and the approach of the present paper, we refer to the former as forward-flatness. We show that flat systems (in the extended sense with backward-shifts) still share many beneficial properties of forward-flat systems. In particular, they still are reachable/controllable, allow a straightforward planning of trajectories and can be linearized by a certain subclass of dynamic feedbacks.