Every control-affine system must be one or the other (not both) of the following:
The notion of integrability used here is quite different from that required for (14.1). In that case, the state transition equation needed to be integrable to obtain integral curves from any initial state. This was required for all systems considered in this book. By contrast, complete integrability implies that the system can be expressed without even using derivatives. This means that all integral curves can eventually be characterized by constraints that do not involve derivatives.
To help understand complete integrability, the notion of an integral curve will be generalized from one to dimensions. A manifold is called an integral manifold of a set of Pfaffian constraints if at every , all vectors in the tangent space satisfy the constraints. For a set of completely integrable Pfaffian constraints, a partition of into integral manifolds can be obtained by defining maximal integral manifolds from every . The resulting partition is called a foliation, and the maximal integral manifolds are called leaves .
The task in this section is to determine whether a system is completely integrable. Imagine someone is playing a game with you. You are given an control-affine system and asked to determine whether it is completely integrable. The person playing the game with you can start with equations of the form and differentiate them to obtain Pfaffian constraints. These can then be converted into the parametric form to obtain the state transition equation (15.53). It is easy to construct challenging problems; however, it is very hard to solve them. The concepts in this section can be used to determine only whether it is possible to win such a game. The main tool will be the Frobenius theorem, which concludes whether a system is completely integrable. Unfortunately, the conclusion is obtained without producing the integrated constraints . Therefore, it is important to keep in mind that ``integrability'' does not mean that you can integrate it to obtain a nice form. This is a challenging problem of reverse engineering. On the other hand, it is easy to go in the other direction by differentiating the constraints to make a challenging game for someone else to play.
Steven M LaValle 2020-08-14