  The Willems' fundamental lemma, which characterizes linear time-invariant
(LTI) systems using input and output trajectories, has found many successful
applications. Combining this with receding horizon control leads to a popular
Data-EnablEd Predictive Control (DeePC) scheme. DeePC is first established for
LTI systems and has been extended and applied for practical systems beyond LTI
settings. However, the relationship between different DeePC variants, involving
regularization and dimension reduction, remains unclear. In this paper, we
first introduce a new bi-level optimization formulation that combines a data
pre-processing step as an inner problem (system identification) and predictive
control as an outer problem (online control). We next discuss a series of
convex approximations by relaxing some hard constraints in the bi-level
optimization as suitable regularization terms, accounting for an implicit
identification. These include some existing DeePC variants as well as two new
variants, for which we establish their equivalence under appropriate settings.
Notably, our analysis reveals a novel variant, called DeePC-SVD-Iter, which has
remarkable empirical performance of direct methods on systems beyond
deterministic LTI settings.
