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

J. Deutscher

TL;DR

This work develops a data-driven framework for stabilizing linear boundary-controlled parabolic PDEs using the Koopman operator. It solves a Koopman eigenstructure assignment to place a finite set of closed-loop eigenvalues via state feedback built from Koopman eigenfunctionals, with the remaining spectrum unchanged. A Krylov-DMD-based extension to parabolic systems enables data-driven identification of the dominant Koopman eigenpairs from finite spatial-temporal samples, yielding a finite invariant subspace for control. The authors prove exponential stability of the data-driven closed loop under small estimation errors and demonstrate the method on an unstable diffusion–reaction PDE, highlighting practical viability and potential extensions to other PDE classes and nonlinear settings.

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.

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

TL;DR

This work develops a data-driven framework for stabilizing linear boundary-controlled parabolic PDEs using the Koopman operator. It solves a Koopman eigenstructure assignment to place a finite set of closed-loop eigenvalues via state feedback built from Koopman eigenfunctionals, with the remaining spectrum unchanged. A Krylov-DMD-based extension to parabolic systems enables data-driven identification of the dominant Koopman eigenpairs from finite spatial-temporal samples, yielding a finite invariant subspace for control. The authors prove exponential stability of the data-driven closed loop under small estimation errors and demonstrate the method on an unstable diffusion–reaction PDE, highlighting practical viability and potential extensions to other PDE classes and nonlinear settings.

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.
Paper Structure (7 sections, 6 theorems, 55 equations, 2 figures)

This paper contains 7 sections, 6 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. State Feedback Design
  4. Krylov-DMD for Parabolic Systems
  5. Stability of the Data-Driven Closed Loop
  6. Example
  7. Concluding Remarks

Key Result

Theorem 1

Assume that $\lambda_{\text{max}} = \max_{\lambda \in \sigma(\Lambda_n-B_nK)}\operatorname{Re}\lambda < 0$, let $\epsilon > 0$ be such that $\alpha = \lambda_{\text{max}} + \epsilon$ satisfies $\lambda_i < \alpha < 0$, $i > n$. Then, the closed-loop system resulting from appyling sf to plant is expo for $M_x \geq 1$.

Figures (2)

  • Figure 1: Plots of the eigenvalue error (\ref{['m1']}), the Koopman mode error (\ref{['m2']}) and the relative residual (\ref{['m3']}) for the open-loop system in the first row. The second row shows the errors for the closed-loop eigenvalue (\ref{['m4']}) and the related eigenvectors $\hat{\tilde{\psi}}_i$ (\ref{['m5']}) of $\tilde{\mathcal{A}}^*$ defining the assigned closed-loop eigenfunctionals \ref{['clefktl']}.
  • Figure 2: Solution $x(z,t)$ of the closed-loop system for the initial condition used to collect the data.

Theorems & Definitions (17)

  • Remark 1
  • Remark 2
  • Theorem 1: Stabilizing state feedback controller
  • proof
  • Theorem 2: Parametric state feedback gain
  • proof
  • Lemma 1: Riesz basis assignment
  • proof
  • Theorem 3: Companion-matrix DMD
  • proof
  • ...and 7 more