Normal forms for flat control-affine systems linearizable via one-fold prolongation

We present normal forms for nonlinear control-affine systems that are the closest to static feedback linearizable ones, that is, for systems that become feedback linearizable via the simplest dynamic feedback, namely one-fold prolongation of a suitably chosen control. We also show that the dynamic feedback may create singularities in the control space depending on the state and we discuss them. We illustrate our results by two examples: the polymerization reactor and the quadrotor helicopter.