Bridging Direct and Indirect Data-Driven Control Formulations via Regularizations and Relaxations

In this article, we discuss connections between sequential system identification and control for linear time-invariant systems, often termed indirect data-driven control, as well as a contemporary direct data-driven control approach seeking an optimal decision compatible with recorded data assembled in a Hankel matrix and robustified through suitable regularizations. We formulate these two problems in the language of behavioral systems theory and parametric mathematical programs, and we bridge them through a multicriteria formulation trading off system identification and control objectives. We illustrate our results with two methods from subspace identification and control: namely, subspace predictive control and low-rank approximation, which constrain trajectories to be consistent with a nonparametric predictor derived from (respectively, the column span of) a data Hankel matrix. In both cases, we conclude that direct and regularized data-driven control can be derived as convex relaxation of the indirect approach, and the regularizations account for an implicit identification step. Our analysis further reveals a novel regularizer and a plausible hypothesis explaining the remarkable empirical performance of direct methods on nonlinear systems.