Transforming a Nonlinear system to a p-normal form via coordinates transformation

Using the differential geometric control theory, we presented the necessary and sufficient conditions under which smooth affine nonlinear systems with single inputs are equivalent to, via state transformation, a given p-normal form. The presented verifiable conditions involve the vector fields obtained by some suitably chosen iterative Lie brackets of the vector fields which define the system. The proof is based on a new technique for applying the Frobenius Theorem in a more convenient way by converting a set of quasilinear partial differential equations with the same main portion to a set of homogeneous linear partial differential equations. Finally, an example is given.