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arXiv · 2206.03845

A Flat System Possessing no (x,u)-Flat Output

Conrad Gstöttner·Bernd Kolar·Markus Schöberl
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Abstract

In general, flat outputs of a nonlinear system may depend on the system's state and input as well as on an arbitrary number of time derivatives of the latter. If a flat output which also depends on time derivatives of the input is known, one may pose the question whether there also exists a flat output which is independent of these time derivatives, i.e., an (x,u)-flat output. Until now, the question whether every flat system also possesses an (x,u)-flat output has been open. In this contribution, this conjecture is disproved by means of a counterexample. We present a two-input system which is differentially flat with a flat output depending on the state, the input and first-order time derivatives of the input, but which does not possess any (x,u)-flat output. The proof relies on the fact that every (x,u)-flat two-input system can be exactly linearized after an at most dim(x)-fold prolongation of one of its (new) inputs after a suitable input transformation has been applied.

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Conrad Gstöttner, Bernd Kolar, Markus Schöberl. 2022-09-20. A Flat System Possessing no (x,u)-Flat Output. https://doi.org/10.1109/lcsys.2022.3229863

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