On the Exact Linearization and Control of Flat Discrete-time Systems

TL;DR

This paper develops methods for the exact linearization of flat nonlinear discrete-time systems using generalized feedbacks, enabling simplified control design and illustrating the approach with robotic system models.

Contribution

It introduces systematic procedures for selecting forward-shifts of flat outputs as new inputs and constructing suitable feedbacks for exact linearization of discrete-time flat systems.

Findings

Derived verifiable conditions for feasible input selection.

Presented a systematic method for minimal forward-shift construction.

Demonstrated the approach on robotic system models.

Abstract

The paper addresses the exact linearization of flat nonlinear discrete-time systems by generalized static or dynamic feedbacks which may also depend on forward-shifts of the new input. We first investigate the question which forward-shifts of a given flat output can be chosen in principle as a new input, and subsequently how to actually introduce the new input by a suitable feedback. With respect to the choice of a feasible input, easily verifiable conditions are derived. Introducing such a new input requires a feedback which may in general depend not only on this new input itself but also on its forward-shifts. This is similar to the continuous-time case, where feedbacks which depend on time derivatives of the closed-loop input - and in particular quasi-static ones - have already been used successfully for the exact linearization of flat systems since the nineties of the last century. For systems with a flat output that does not depend on forward-shifts of the input, it is shown how to systematically construct a new input such that the total number of the corresponding forward-shifts of the flat output is minimal. Furthermore, it is shown that in this case the calculation of a linearizing feedback is particularly simple, and the subsequent design of a discrete-time flatness-based tracking control is discussed. The presented theory is illustrated by the discretized models of a wheeled mobile robot and a 3DOF helicopter.