On feedback linearization of multi-input nonlinear control systems via time scaling and prolongation

Abstract

Currently, there are no necessary and sufficient conditions for linearizability of multi-input nonlinear control systems of general form in a class of transformations that include not only state and input changes, but also time scaling. In the present paper, we show that the problem of feedback linearizing a multi-input nonlinear control system via time scaling can be solved by constructing the Pfaffian system J naturally associated with the control system, followed by eliminating the time differential from J. We prove that the Pfaffian system I obtained in this way is diffeomorphic to an extended Goursat normal form if and only if the control system is feedback linearizable via time scaling. We also prove that if a control system is not feedback linearizable via time scaling, but there exist coordinates in which the Pfaffian system I becomes an extended Goursat normal form, then the control system can be feedback linearized via time scaling and prolongation. We show that either a one-fold or a total prolongation may be required in this case. We give an example of a two-input control system that is not flat but is feedback linearizable via time scaling and a total prolongation.