Learning structured linear operators improves control system modeling
Demystifying Linear Operator Learning for Control Systems
Machine Learning
Summary
Modeling control systems can be hard when the rules change over time. This paper shows how to learn these changing rules as mathematical operators using ideas from groups and inverse problems. The authors analyze how the learned models relate to the data and prove that their method converges to the true system over time. Their approach works under certain mathematical conditions that match common assumptions in control system modeling.
What this means in practice
- •For control system engineers: Improve the accuracy of time-varying system models by using structured learning algorithms based on semigroup theory and inverse problem techniques.
- •For signal processing teams: Create better estimators for dynamic processes that evolve continuously over time under bounded linear operators.
Authors
Max Beier, Nicolas Hoischen, Sandra Hirche, Petar Bevanda
Abstract
This paper proposes a structured approach to learning linear operators for control systems from data. We address both structural and learning-theoretic aspects of the problem. To derive structural assumptions, we propose using the well-established framework of (semi)groups for evolution equations, as operators in control systems are of the same type. Further, we propose analyzing learning algorithms through the lens of the inverse problems framework. This reveals how a learned model depends on the data via error decompositions, convergence guarantees, and optimal regularization -- enabling us to compare existing methods and derive provably advantageous algorithms. In order to obtain these results, we restrict our scope to bounded operators on Hilbert spaces. Although this may appear restrictive, existing approaches often make this assumption implicitly to obtain matrix-like representations. We demonstrate the power of using these frameworks by deriving a convergent estimator for time-varying systems.