Although many powerful control laws can be developed for linear systems, the vast majority of systems that occur in the physical world fail to be linear. Any differential models that do not fit (13.37) or (13.40) are called nonlinear systems. All of the models given in Section 13.1.2 are nonlinear systems for the special case in which .
One important family of nonlinear systems actually appears to be linear in some sense. Let be a smooth -dimensional manifold, and let . Let for some . Using a coordinate neighborhood, a nonlinear system of the form
For a control-affine system it is not necessarily possible to obtain zero velocity because causes drift. The important special case of a driftless control-affine system occurs if . This is written as
Many nonlinear systems can be expressed implicitly using Pfaffian constraints, which appeared in Section 13.1.1, and can be generalized from C-spaces to phase spaces. In terms of , a Pfaffian constraint is expressed as
Both holonomic and nonholonomic models may exist for phase spaces, just as in the case of C-spaces in Section 13.1.3. The Frobenius Theorem, which is covered in Section 15.4.2, can be used to determine whether control-affine systems are completely integrable.
Steven M LaValle 2012-04-20