  This paper contributes a theoretical framework for data-driven feedback
linearization of nonlinear control-affine systems. We unify the traditional
geometric perspective on feedback linearization with an operator-theoretic
perspective involving the Koopman operator. We first show that if the
distribution of the control vector field and its repeated Lie brackets with the
drift vector field is involutive, then there exists an output and a feedback
control law for which the Koopman generator is finite-dimensional and locally
nilpotent. We use this connection to propose a data-driven algorithm Koopman
Generator-based Feedback Linearization (KGFL) for feedback linearization.
Particularly, we use experimental data to identify the state transformation and
control feedback from a dictionary of functions for which feedback
linearization is achieved in a least-squares sense. We also propose a
single-step data-driven formula which can be used to compute the linearizing
transformations. When the system is feedback linearizable and the chosen
dictionary is complete, our data-driven algorithm provides the same solution as
model-based feedback linearization. Finally, we provide numerical examples for
the data-driven algorithm and compare it with model-based feedback
linearization. We also numerically study the effect of the richness of the
dictionary and the size of the data set on the effectiveness of feedback
linearization.
