Feedback control of nonlinear PDEs using data-efficient reduced order models based on the Koopman operator

TL;DR

This paper introduces two data-efficient Koopman operator-based control strategies for nonlinear PDEs, enabling real-time model predictive control with proven convergence and adaptability, demonstrated on fluid dynamics equations.

Contribution

It presents novel Koopman-based control methods for nonlinear PDEs, including a switching control approach and a bilinear surrogate model, with convergence guarantees and online adaptivity.

Findings

Effective control strategies for nonlinear PDEs demonstrated on Burgers and Navier-Stokes equations.

Proven convergence to the true optimum using EDMD approximation.

Enhanced data efficiency and reduced complexity in control optimization.

Abstract

In the development of model predictive controllers for PDE-constrained problems, the use of reduced order models is essential to enable real-time applicability. Besides local linearization approaches, Proper Orthogonal Decomposition (POD) has been most widely used in the past in order to derive such models. Due to the huge advances concerning both theory as well as the numerical approximation, a very promising alternative based on the Koopman operator has recently emerged. In this chapter, we present two control strategies for model predictive control of nonlinear PDEs using data-efficient approximations of the Koopman operator. In the first one, the dynamic control system is replaced by a small number of autonomous systems with different yet constant inputs. The control problem is consequently transformed into a switching problem. In the second approach, a bilinear surrogate model, is obtained via linear interpolation between two of these autonomous systems. Using a recent convergence result for Extended Dynamic Mode Decomposition (EDMD), convergence to the true optimum can be proved. We study the properties of these two strategies with respect to solution quality, data requirements, and complexity of the resulting optimization problem using the 1D Burgers Equation and the 2D Navier-Stokes Equations as examples. Finally, an extension for online adaptivity is presented.