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Koopman operator-based model reduction for switched-system control of PDEs
Sebastian Peitz, Stefan Klus
TL;DR
The paper addresses the computational difficulty of optimal and feedback control for nonlinear PDEs. It transforms finite-valued control into switching among autonomous systems and replaces each with a low-dimensional linear K-ROM built from Koopman approximations. The framework reports accelerated computation with accuracy and an optimality result under the stated convergence setting.
Problem
Repeated optimization for nonlinear PDEs can be infeasible with expensive high-fidelity models, motivating faster reduced models for control.
Method
Finite-valued controls create autonomous subsystems whose dynamics are approximated from observations by EDMD-based Koopman linear systems.
Results
The numerical results show excellent accuracy and computing-time performance, while the control objectives coincide under the stated assumptions.
Takeaways & Limitations
The approach provides finite-dimensional linear surrogate models for controlling infinite-dimensional nonlinear systems and can yield optimal solutions when the Koopman operator is computed exactly.
Takeaways & Limitations
The paper identifies stability properties of K-ROM-based MPC as a direction for further research and notes time-grid and continuous-time objective mismatches in the convergence analysis.
Abstract
from arXiv · showhide
We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. In situations where the Koopman operator can be computed exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach yields optimal control inputs. Furthermore, a recent convergence result for EDMD suggests that the approach can be applied to more complex dynamics as well. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier--Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.
1 Introduction
Nonlinear PDE control is difficult because standard discretizations make repeated optimization too slow, while existing POD-based reduced models can become infeasible as dynamics grow complex. The paper proposes K-ROMs that transform finite-valued control into switching among autonomous Koopman models for faster open- and closed-loop control.
- Nonlinear PDEs make the short-horizon optimization required by MPC generally infeasible with standard finite-element or finite-volume discretizations.
- POD-based Galerkin surrogates reduce computational cost, but their required mode count can grow rapidly with system-dynamics complexity.Calibration or specially constructed modes may also be needed.
- Koopman ROMs provide linear reduced models of observable dynamics and can work with sensor measurements, including when the underlying dynamics are unknown.
- The proposed framework restricts controls to finitely many constant values, replaces the control system with autonomous systems, and switches among their Koopman approximations.Open-loop control becomes a switching-time optimization problem, while closed-loop control uses MPC.
- The framework is evaluated on an ODE problem, the 1D Burgers equation, and the 2D Navier–Stokes equations.
2 Preliminaries
The framework formulates PDE-constrained control with function-space states and continuous controls, then approximates it through switching among autonomous systems and linear observable dynamics. EDMD supplies finite-dimensional Koopman approximations whose convergence supports the resulting control analysis.
- The target problem controls a function-valued PDE state y in a space Y using a constrained control u and dynamics G: Y × U → Y.The objective is written as depending explicitly on y, with extension to control-dependent objectives noted as straightforward.
- Control techniques: STO is used for open-loop control, whereas MPC repeatedly solves finite-horizon problems and applies only the first control segment.The prediction horizon moves forward by one sample time after each application.
- Switching Time Optimization: The switching formulation uses a predetermined cyclic sequence of autonomous right-hand sides, with switching times determining when the active system changes.
- Switching Time Optimization: Restricting u to finitely many constant values replaces G with autonomous systems and converts the control problem into optimization over switching instants.The fixed controls may be chosen equidistantly or determined problem-specifically offline.
- Koopman Operator and EDMD: EDMD computes finite-dimensional Koopman approximations from measurements or simulations using arbitrary basis functions and lifted feature coordinates.DMD is recovered when the dictionary satisfies ψ(z) = z.
- Koopman Operator and EDMD: EDMD convergence is stated as convergence of the approximating matrix toward the Koopman operator as basis size and measurement count increase.The supplied result specifies strong convergence in L2(µ).
3 Open- and closed-loop control using K-ROMs
The framework replaces controlled PDE dynamics with switched autonomous systems and low-dimensional Koopman reduced models, enabling faster open- and closed-loop optimization. Its convergence guarantees depend on assumptions, while discrete sampling and switching grids constrain the open-loop formulation.
- Motivation: Observations are used to approximate Koopman operators when PDE simulations are expensive or system dynamics are unknown.The resulting K-ROM approach uses an offline data-collection phase and an online optimization phase.
- Switched K-ROM construction: The method computes separate Koopman operators for the finite set of autonomous systems created by restricting controls to fixed values.This avoids representing all state-control combinations with one Koopman operator.
- Switched K-ROM construction: EDMD approximations define discrete linear systems in low-dimensional observables that replace the original differential equations and accelerate computation.The approximation is generally only approximate at each step because the Koopman matrices are obtained from an overdetermined system.
- Open-loop control: Switching-time optimization is adapted to the sampling step h, yielding an integer formulation whose switching instants lie on the sample-time grid.The reduced objective is evaluated with respect to observations, under an assumption that the full objective can be represented from those observations.
- Closed-loop control: Identity of the full and reduced MPC objective functions holds asymptotically for every admissible control and almost every initial state under the stated assumptions.The result follows as the basis size and number of sampled data points tend to infinity, together with EDMD convergence and the observation-based objective assumption.
- Closed-loop control: The approach achieves real-time applicability only if the reduced MPC problem is solved within the sample time h.The paper evaluates all possible controls for its introductory setting, while larger prediction horizons require more scalable optimization methods.
4 Results
The numerical results show that K-ROMs substantially accelerate switching-time optimization and accurately support MPC for PDE control using severely restricted control inputs. Across the ODE, 1D Burgers, and 2D Navier–Stokes examples, reduced models achieve close agreement with full-system objectives or successfully track reference trajectories.
- Switching Time Optimization: The K-ROM reproduces the switching-time optimization behavior increasingly closely as the number of switches grows.The distance between full-problem and K-ROM trajectories in y2 decreases with more switching instants.
- Switching Time Optimization: A factor of approximately 50 speed-up is achieved for switching-time optimization.The acceleration is attributed to K-ROM linearity and an eightfold larger time step.
- 1D Burgers Equation: K-ROM-based MPC controls the 1D Burgers equation accurately using observations at four spatial points and only three autonomous systems.The observed state is z = (y(0, t), y(0.5, t), y(1, t), y(1.5, t))⊤, and the control uses three time-independent shape functions.
- 1D Burgers Equation: For 1D Burgers control, the K-ROM MPC objective is almost equal in quality to the full PDE-constrained problem.The comparison uses identical MPC setups with three restricted control inputs.
- 2D Navier–Stokes Equations: For 2D Navier–Stokes flow around a cylinder, K-ROM MPC successfully tracks the desired lift trajectory using three control inputs and a linear reduced model.The cylinder is controlled through rotation, with constant angular velocities u0 = 0, u1 = 2, and u2 = −2.
- 2D Navier–Stokes Equations: The Navier–Stokes results show deviations for large reference values because the prescribed trajectory may be infeasible under the box constraints [−2, 2], with insufficient data also cited as a possible explanation.The authors suggest adapting control bounds, adding controls, or updating Koopman approximations with additional operational data.
5 Conclusion
The framework uses K-ROMs to control nonlinear infinite-dimensional systems with finite-dimensional linear surrogates, achieving strong accuracy and computing-time performance. Future work includes stability analysis, streaming-data updates, and maintaining real-time applicability with multiple control inputs.
- The framework transforms control systems into switching-time problems over autonomous systems with fixed control inputs.This enables open- and closed-loop control using K-ROMs.
- The approach enables control of infinite-dimensional nonlinear systems using finite-dimensional, linear surrogate models.
- Using a recent EDMD convergence result, the framework can prove optimality of the obtained solution.
- The numerical results show excellent accuracy and computing-time performance.
- Future research should examine stability properties of K-ROM-based MPC and the effects of regular streaming-data updates.
- With multiple control inputs, maintaining real-time applicability may become challenging, motivating the use of relaxation techniques.