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

TL;DR

This paper establishes a theoretical connection between direct and indirect data-driven control methods for linear systems, showing how regularizations and relaxations unify these approaches and explaining their empirical success.

Contribution

It formulates a unified framework linking system identification and control via behavioral systems theory and convex relaxations, revealing new insights into direct data-driven control.

Findings

Direct control methods can be derived as convex relaxations of indirect methods.

Regularizations implicitly perform system identification within direct control approaches.

The framework explains the strong empirical performance of direct methods on nonlinear systems.

Abstract

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.