Properties of Flat Systems with regard to the Parameterization of the System Variables by the Flat Output

For flat systems, by definition the state- and input variables can be parameterized by the flat output and its time derivatives. In this paper, we study the properties of the corresponding map. First, we present a proof of the known result that for flat outputs that only depend on state- and input variables, the maximum order of the time derivatives of the flat output that occur in this map is bounded by the state dimension. Our proof is based on the observation that a certain codistribution, which can be associated to a flat output, has a very special structure. This approach also allows to derive smaller bounds for flat outputs which only depend on the state, as well as for configuration-flat outputs of mechanical systems. Furthermore, for flat outputs that depend on time derivatives of the input up to some fixed order, we derive bounds that are smaller than the ones that can be found in the literature. Second, we examine the special structure of the Jacobian matrix of the parameterizing map, and show how this Jacobian matrix is connected to the existence of certain input transformations, where as special case we get the well-known ruled manifold necessary condition.