ABSTRACT
This chapter aims to outline and interrelate a progression of results on approximation of systems of vector fields and functions, using nilpotent and graded structures; and to give an intrinsic setting for an approximation method. It explains the result by constructing realizations of higher order truncations of the Volterra/Chen series expansion of the input-output map, in such a way that the author can compare the resulting approximating systems with the original system. The chapter makes two important observations concerning the linear approximation. First, the state space dimension of the linearized system is the same as that of the original system, making the comparison of the two systems especially easy; corresponding to Taylor series expansion. Second, to write down the approximation, a choice of local coordinates has been made.
