Data-Driven Control of Linear Parabolic Systems using Koopman Eigenstructure Assignment

TL;DR

This paper introduces a data-driven method for stabilizing linear parabolic PDEs using Koopman eigenstructure assignment, leveraging extended Krylov-DMD with finite data samples to design controllers ensuring exponential stability.

Contribution

It develops a novel data-driven control approach for parabolic PDEs by extending Krylov-DMD to eigenstructure assignment, requiring only finite output and input samples.

Findings

Successfully stabilizes a diffusion-reaction system.

Demonstrates robustness to small Krylov-DMD errors.

Provides a practical data-driven control design for distributed systems.

Abstract

This paper considers the data-driven stabilization of linear boundary controlled parabolic PDEs by making use of the Koopman operator. For this, a Koopman eigenstructure assignment problem is solved, which amounts to determine a feedback of the Koopman open-loop eigenfunctionals assigning a desired finite set of closed-loop Koopman eigenvalues and eigenfunctionals to the closed-loop system. It is shown that the designed controller only needs a finite number of open-loop Koopman eigenvalues and modes of the state. They are determined by extending the classical Krylov-DMD to parabolic systems. For this, only a finite number of pointlike outputs and their temporal samples as well as temporal samples of the inputs are required resulting in a data-driven solution of the eigenstructure assignment problem. Exponential stability of the closed-loop system in the presence of small Krylov-DMD errors is verified. An unstable diffusion-reaction system demonstrates the new data-driven controller design technique for distributed-parameter systems.