Realizations of higher-order nonlinear differential equations

We discuss the realization problem for nonlinear systems which are not given by a nonlinear input-output map, but as a set of smooth higher-order differential equations involving the inputs and outputs. We show that under general conditions the existence of realizations involving auxiliary driving variables is guaranteed, but that for input-output realizations extra integrability conditions have to be imposed. The realization procedure uses the concept of zero dynamics which is briefly discussed.