  This paper considers the extension of data-enabled predictive control (DeePC)
to nonlinear systems via general basis functions. Firstly, we formulate a basis
functions DeePC behavioral predictor and we identify necessary and sufficient
conditions for equivalence with a corresponding basis functions multi-step
identified predictor. The derived conditions yield a dynamic regularization
cost function that enables a well-posed (i.e., consistent) basis functions
formulation of nonlinear DeePC. To optimize computational efficiency of basis
functions DeePC we further develop two alternative formulations that use a
simpler, sparse regularization cost function and ridge regression,
respectively. Consistency implications for Koopman DeePC as well as several
methods for constructing the basis functions representation are also indicated.
The effectiveness of the developed consistent basis functions DeePC
formulations is illustrated on a benchmark nonlinear pendulum state-space
model, for both noise free and noisy data.
