Regularization in Data-driven Predictive Control: A Convex Relaxation Perspective

This paper explores the role of regularization in data-driven predictive control (DDPC) through the lens of convex relaxation. Using a bi-level optimization framework, we model system identification as an inner problem and predictive control as an outer problem. Within this framework, we show that several regularized DDPC formulations, including l1-norm penalties, projection-based regularizers, and a newly introduced causality-based regularizer, can be viewed as convex relaxations of their respective bi-level problems. This perspective clarifies the conceptual links between direct and indirect data-driven control and highlights how regularization implicitly enforces system identification. We further propose an optimality-based variant, A-DDPC, which approximately solves the inner problem with all identification constraints via an iterative algorithm. Numerical experiments demonstrate that A-DDPC outperforms existing regularized DDPC by reducing both bias and variance errors. These results indicate that further benefits may be obtained by applying system identification techniques to pre-process the trajectory library in nonlinear settings. Overall, our analysis contributes to a unified convex relaxation view of regularization in DDPC and sheds light on its strong empirical performance beyond linear time-invariant systems.