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

TL;DR

This paper disproves the conjecture that all differentially flat systems have an (x,u)-flat output by providing a counterexample, showing some flat systems depend on derivatives of inputs and cannot be simplified to (x,u)-flat form.

Contribution

The paper presents a counterexample of a flat system that depends on input derivatives and lacks an (x,u)-flat output, challenging previous assumptions in nonlinear control theory.

Findings

Counterexample of a flat system without (x,u)-flat output

Dependence on input derivatives can prevent (x,u)-flatness

Linearization approach relates to the absence of (x,u)-flat outputs

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.