On the Equivalence of Direct and Indirect Data-Driven Predictive Control Approaches

Recently, several direct Data-Driven Predictive Control (DDPC) methods have been proposed, advocating the possibility of designing predictive controllers from historical input-output trajectories without the need to identify a model. In this letter, we show their equivalence to a (relaxed) indirect approach, allowing us to reformulate direct methods in terms of estimated parameters and covariance matrices. This allows us to provide further insights into how these direct predictive control methods work, showing that, for unconstrained problems, the direct methods are equivalent to subspace predictive control with a reduced weight on the tracking cost, and analyzing the impact of the data length on tuning strategies. Via a numerical experiment, we also illustrate why the performance of direct DDPC methods with fixed regularization tends to degrade as the number of training samples increases.