Data-Enabled Gradient Flow as Feedback Controller: Regulation of Linear Dynamical Systems to Minimizers of Unknown Functions

TL;DR

This paper introduces a data-driven feedback control method that combines gradient flow dynamics with learning to regulate linear systems towards unknown optimization problem solutions, ensuring stability despite estimation errors.

Contribution

It develops a novel control scheme integrating learning with gradient flow for unknown cost functions in linear systems, with theoretical stability guarantees.

Findings

Error bounds depend on learning accuracy and disturbance variability.

System achieves exponential input-to-state stability.

Method is validated through numerical simulations.

Abstract

This paper considers the problem of regulating a linear dynamical system to the solution of a convex optimization problem with an unknown or partially-known cost. We design a data-driven feedback controller - based on gradient flow dynamics - that (i) is augmented with learning methods to estimate the cost function based on infrequent (and possibly noisy) functional evaluations; and, concurrently, (ii) is designed to drive the inputs and outputs of the dynamical system to the optimizer of the problem. We derive sufficient conditions on the learning error and the controller gain to ensure that the error between the optimizer of the problem and the state of the closed-loop system is ultimately bounded; the error bound accounts for the functional estimation errors and the temporal variability of the unknown disturbance affecting the linear dynamical system. Our results directly lead to exponential input-to-state stability of the closed-loop system. The proposed method and the theoretical bounds are validated numerically.