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

• Published in: IEEE transactions on automatic control Vol. 68; no. 2; pp. 883 - 897
• Main Authors: Dorfler, Florian, Coulson, Jeremy, Markovsky, Ivan
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Aerospace electronics; Complexity theory; Control systems; Data models; Empirical analysis; Hankel matrices; Linear systems; Mathematical analysis; Mathematical programming; Multiple criterion; Nonlinear systems; Optimal control; Optimization; Pareto optimization; Predictive control; System identification; System theory; Systems theory; Time invariant systems; Trajectory
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3148374

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.