Approximate feedback linearization: approximate integrating factors

Under the assumption of linear controllability a single input nonlinear system is feedback linearizable if and only if its characteristic distribution is involutive. We define a characteristic one-form to be any one-form annihilating the characteristic distribution. A characteristic one-form is defined uniquely up to scaling by a nonzero smooth function. It is well-known that there exist a scaling function (an integrating factor) that makes a given characteristic form exact if and only if the system is exactly feedback linearizable. For the systems that are not linearizable, we consider the problem of choosing an optimal scaling factor (best approximate integrating factor) for a given characteristic one form. The approximate integrating factors may be applied to approximate feedback linearization and to obtain some measures of noninvolutivity of the characteristic distribution.