Linearization of control systems: A Lie series approach

We address the problem of static state linearization of multi-input nonlinear control systems via coordinate transformation. Necessary and sufficient geometric conditions, in terms of certain set of vector fields associated with the system, were obtained in the early eighties stating the fact that such set of vector fields should be commutative and of constant rank. The state linearization problem, i.e., the finding of linearizing coordinates, was thus reduced to solving a set of partial differential equations. The objective of this paper is to provide an algorithm allowing to compute explicitly the linearizing state coordinates. The algorithm is performed using a maximum of n−1n−1 steps (nn being the dimension of the system) and is made possible by extending the explicit solvability of the Flow-Box Theorem to a commutative set of vector fields. Examples are provided to illustrate the results. An extension of the method to dynamic feedback linearization is also outlined.