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. This is essentially a global controllability problem. In this work, we present a linearization theory that is geometrical. This means that the relation of the control vector fields to the output function is considered from a global geometrical perspective. Our argument involves generic vector fields and output functions and linearization for typical trajectories. We prove that the relative degree is generically one. This approach also succeeds in resolving the problem of control past the singular set, and thus goes further than the existing literature. The unavoidable cost is that linearization may be piecewise, since the concept of strong relative degree cannot (in general) be defined globally. 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.