A jet space approach to check Pfaffian systems for flatness

This contribution uses the representation of control systems, given by a set of ODEs, as Pfaffian systems. It is shown, that flatness for this class of systems is equivalent to the fact, that the unique decomposition of the pull back of the Pfaffian system to a certain jet space into a horizontal and vertical part, has a trivial horizontal part only. We present an algorithm to construct the required jet space, as well as the decomposition. It uses the fact, that a flat Pfaffian system must admit a certain affine input representation after a possible extension by contact forms. Since this representation is not unique, a branching point may appear and all branches have to be checked. This applies also to the decomposition of the system in AI representation into a horizontal and a vertical part. It is shown, how the number of branches can be reduced significantly. The algorithm requires the solution of non linear algebraic equations and the determination of flows generated by distributions. It is worth mentioning, that the presented approach degenerates to the well known one for input to state linearizable system by static feedback. Finally, the application of the algorithm to three examples, all are not input to state linearizable by static feedback, is shown.