# Bridging direct & indirect data-driven control formulations via regularizations and relaxations

arXiv: https://arxiv.org/abs/2101.01273 · HTML: https://mlsanity.com/paper/2101.01273/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jan 2021.

## Built on by (most cited first, 11 of 11)

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- [A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2605.07052/) (2026)
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- [PAC-Bayesian Certificates for Quadratic Closed-Loop Control](https://mlsanity.com/paper/2606.28281/) (2026)

## Abstract (arXiv)

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 multi-criteria 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 non-parametric 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.