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

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 letter, this conjecture is disproved by means of a counterexample, which illustrates the difficulties one may run into when studying flatness. 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 a dim(x)-fold prolongation of one of its (new) inputs after a suitable input transformation has been applied.