A Global Geometrical Input{Output Linearization Theory

Input{output linearization is a method that uses nonlinear state feedback control to obtain linear equations between the input and output of a nonlinear control system. Its success depends crucially on the ability of the control to steer the state across the level sets of the output function. It is clear that this is essentially a global geometrical problem. In this work, we present a linearization theory that is geometrical. This means that the relation of the control vector elds to the output function is considered from a global geometrical perspective. We argue for general position, generic vector elds and output functions and linearization for typical trajectories. We also prove that the relative degree is generically one. This approach succeeds in resolving the problem of singularities and is thus more complete than the existing literature. The unavoidable cost is that since the concept of strong relative degree cannot, in general, be deened globally, linearization may be piecewise. A simple example is included with a common type of output function and linearization is worked out in some detail as an illustration of some of the features of this novel approach.