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Data-Driven Model Predictive Control using Interpolated Koopman Generators

Sebastian Peitz, Samuel E. Otto, Clarence W. Rowley

arXiv:2003.07094v1math.OCmath.DS

TL;DR

Real-time MPC for high-dimensional nonlinear systems is difficult, while data-efficient Koopman switching restricts control to finite input levels. This paper interpolates Koopman generators to produce bilinear reduced models for continuous-input control; it reports no added interpolation error for control-affine systems and five-to-six-order speedups for 2D Navier–Stokes equations.

  • Problem

    Finite autonomous Koopman models are data-efficient but restrict inputs to fixed levels and yield a combinatorial switching problem, while high-dimensional nonlinear MPC is difficult to solve in real time.

    Source: §1 · pp. 1–2

  • Method

    The paper approximates Koopman generators at constant inputs and linearly interpolates them to form bilinear reduced-order models for continuous-input MPC.

    Source: §1 · p. 2, §6 · p. 18

  • Results

    For 2D Navier–Stokes equations, the reduced model runs five to six orders of magnitude faster; generator interpolation adds no error for control-affine systems.

    Source: §6 · p. 18

  • Takeaways & Limitations

    The bilinear reduced control problem can use efficient solution methods, while continuous inputs are supported without increasing training-data requirements.

    Source: §1 · p. 2, §6 · p. 18

    The paper leaves stronger error statements for the K-ROM approach, including basis-size and required-data bounds, for future work.

    Source: §6 · p. 18

Abstract

from arXiv · show

In recent years, the success of the Koopman operator in dynamical systems analysis has also fueled the development of Koopman operator-based control frameworks. In order to preserve the relatively low data requirements for an approximation via Dynamic Mode Decomposition, a quantization approach was recently proposed in [Peitz \& Klus, Automatica 106, 2019]. This way, control of nonlinear dynamical systems can be realized by means of switched systems techniques, using only a finite set of autonomous Koopman operator-based reduced models. These individual systems can be approximated very efficiently from data. The main idea is to transform a control system into a set of autonomous systems for which the optimal switching sequence has to be computed. In this article, we extend these results to continuous control inputs using relaxation. This way, we combine the advantages of the data efficiency of approximating a finite set of autonomous systems with continuous controls. We show that when using the Koopman generator, this relaxation --- realized by linear interpolation between two operators --- does not introduce any error for control affine systems. This allows us to control high-dimensional nonlinear systems using bilinear, low-dimensional surrogate models. The efficiency of the proposed approach is demonstrated using several examples with increasing complexity, from the Duffing oscillator to the chaotic fluidic pinball.

1. Introduction

Real-time MPC for high-dimensional nonlinear systems is difficult, while existing Koopman control approaches trade data efficiency against continuous input flexibility. The paper interpolates Koopman generators to build bilinear reduced models, enabling continuous-input MPC without additional training data and with no interpolation error for control-affine systems.

1. Introduction

  • Real-time MPC is challenging for complex nonlinear systems because its open-loop optimization must be solved quickly, motivating faster surrogate models such as Koopman-based reduced-order models.The introduction describes Koopman operators as linear operators on observables and notes DMD and EDMD as popular approximation algorithms.Source: §1 · p. 1
  • Earlier Koopman control approaches can require many nonlinear input features and large datasets, whereas finite autonomous-model approaches are data-efficient but restrict inputs to a finite set and create a combinatorial switching problem.The finite-model approach is associated with, and arbitrarily close approximation is possible with enough nonlinear features [28].Source: §1 · pp. 1–2
  • Generator interpolation is affine in the input without added error for control-affine systems, while finite-time Koopman-operator interpolation is accurate to first order in time.The resulting reduced models are bilinear and can predict dynamics with short zero-order holds on the input.Source: §1 · p. 2
  • The proposed models extend MPC to continuous inputs without increasing training-data requirements, and the examples range from the Duffing oscillator to chaotic fluidic pinball.The approach is presented as a relaxation of switching MPC that allows intermediate control values when control enters nonlinearly.Source: §1 · p. 2

2. Koopman operator-based optimal control

This section frames optimal control as a real-time problem and describes how EDMD-based Koopman reduced models turn it into a switching optimization over autonomous systems. The approach uses data to approximate low-dimensional observation dynamics, with low data requirements but a combinatorial control problem.

2. Koopman operator-based optimal control

  • MPC repeatedly solves a finite-horizon control problem and applies part of its solution, but high-dimensional nonlinear systems can make solving it within the application interval challenging.The control objective concerns state x(t), input u(t), and dynamics H; the formulation also allows partial differential equations by analogy.Source: §2 · p. 2

2.1. Koopman operator and EDMD

  • For an autonomous system, the Koopman operator advances observables linearly, although it generally acts on an infinite-dimensional observable space.DMD and its nonlinear-observable extension EDMD are popular numerical approximation methods.Source: §2.1 · p. 3
  • EDMD uses measured observable values at time-separated state pairs to compute a finite-dimensional matrix approximation of the Koopman operator in a chosen feature space.The observable vector may be lifted with features such as monomials, Hermite polynomials, or radial basis functions; the matrix construction uses the Moore-Penrose pseudoinverse.Source: §2.1 · p. 3

2.2. Switched system control

  • The switched-system strategy replaces the controlled dynamics with finitely many autonomous systems and constructs linear Koopman models for their low-dimensional observations.EDMD approximates the individual operators from observations, and the resulting dynamics replace the original system over the prediction horizon under a zero-order hold.Source: §2.2 · pp. 3–4
  • The resulting optimization selects which autonomous system to apply at each step, forming a mixed-integer control problem with combinatorial complexity.The discrete-time formulation expresses the objective using measurements at each time step.Source: §2.2 · pp. 3–4
  • Data-based Koopman approximations can describe observation dynamics when explicit dynamics are unavailable, and individual switched models can often be learned from fewer than 100 data points each.Observations may include part of the state or functions of it, such as a flow-field lift coefficient.Source: §2.2 · p. 4

3. Continuous control inputs

This section develops generator-based Koopman reduced models for continuous control, discussing when interpolation is exact and how these models relate to finite-time operator approximations. It evaluates the models on the forced Duffing equation and cylinder flow, reporting accurate predictions for interpolated inputs and computational speed-up for the cylinder-flow model.

3. Continuous control inputs

  • Generator-based models address the combinatorial optimization, restricted input levels, and reduced control frequency of switched discrete-time control by enabling continuous input interpolation.The section proposes approximating Koopman generators under piecewise-constant inputs; for control-affine systems, the generators are affine in the input.Source: §3 · p. 4

3.1. Koopman generator-based models for control

  • For control-affine dynamics and differentiable observables, Koopman generators are input-affine, yielding a bilinear observable model whose coefficients can vary with time-varying inputs.The input’s time variation is passed to the coefficients of an affine combination of fixed generators.Source: §3.1 · pp. 5–6

3.2. EDMD for generator-based modeling

  • Generator-based EDMD learns the bilinear reduced model from arbitrary input values, with data requirements comparable to approximating generators at n_c + 1 fixed input levels.The stated data requirement grows linearly with input dimension; a small, informative observable dictionary can make the reduced model more efficient than evolving the full state.Source: §3.2 · pp. 6–8

3.3. Discrete vs. Continuous Approximation Methods

  • Finite-time operator EDMD is equivalent to generator-based EDMD with forward differences and explicit Euler under the stated full-row-rank assumption, while input-affine finite-time models are generally only first-order accurate.The equivalence requires the specified finite-difference formula and approximations; for general control-affine systems, finite-time input-affine approximations are justifiable only at sufficiently small time steps.Source: §3.3 · p. 9
  • For linear systems with affine observables, finite-time Koopman operators are input-affine for any time interval; weakly nonlinear input dependence can also yield satisfactory short-horizon MPC results.In the weakly nonlinear case, accuracy loss depends on the degree of nonlinearity.Source: §3.3 · pp. 9–10

3.4. Example: forced Duffing Equation

  • The Duffing K-ROM used degree-five monomials in x1 and x2 for a 21-dimensional feature space, comparing endpoint-only operator construction with continuous-signal data collection.For endpoint inputs {−1, 1}, the control matrix was formed from the difference between the two corresponding Koopman operators.Source: §3.4 · p. 10
  • For most initial conditions, both Duffing K-ROM approaches predicted the system very accurately for about one second, including interpolated inputs u(t) = 0 and u(t) = sin(πt).The comparison included the data-collection inputs u(t) = ±1.Source: §3.4 · p. 10, Figure 1 · p. 11

3.5. Example: Cylinder flow

  • At Re = 100, cylinder rotation controls a reduced model of lift, drag, and six wake velocities; quadratic observables yield a 45-dimensional K-ROM trained on a long simulation with inputs {0, 2}.The flow is governed by the two-dimensional incompressible Navier–Stokes equations, and the uncontrolled system has a periodic von Kármán vortex street.Source: §3.5 · p. 11
  • Approximately five orders of magnitude speed-up (OpenFOAM vs. MATLAB) accompanied a 45-dimensional K-ROM that agreed remarkably well with a 22,000-cell discretization (66,000 unknowns per time step).For sinusoidal control, agreement remained satisfactory although the system was not control affine and the K-ROM was only first-order accurate.Source: §3.5 · p. 12, Figure 2 · p. 12

4. MPC solution using EDMD-based ROMs

The MPC formulation uses Koopman reduced-order models with control signals restricted to a finite-dimensional subspace. An adjoint-based gradient supports optimization, while the bilinear model structure enables a Newton-type solver.

4. MPC solution using EDMD-based ROMs

  • MPC uses a Koopman reduced-order model and restricts allowable input signals to a finite-dimensional subspace, such as piecewise-constant signals over zero-order-hold intervals.The method also allows other finite-dimensional choices, including a Fourier-series input representation for limited-bandwidth actuators.Source: §4 · pp. 12–13

4.1. Gradient and first-order optimality

  • An adjoint-based approach computes objective gradients with respect to the input under the reduced-order model's dynamical constraints.The adjoint equation is solved backward in time, yielding an explicit gradient with respect to the input signal.Source: §4 · p. 13, §4.1 · pp. 13–14
  • The bilinear structure of the Koopman reduced-order model enables an efficient Newton-type solver for the first-order optimality conditions.When additional input or state constraints are present, the derived gradient can also benefit sequential quadratic programming or interior-point methods.Source: §4 · p. 13
  • The gradient is derived by relating state and input perturbations through the linearized model and using an adjoint variable to express the objective change in terms of input perturbations.The initial perturbation is zero because the initial condition is fixed.Source: §4.1 · p. 13
  • For the first-order Koopman-operator interpolation, EDMD-based approximations over Δt are equivalent to Euler integration of the generator approximations over the same interval.The adjoint equation and gradient can also be discretized.Source: §4.1 · p. 14

5. Numerical examples

The numerical examples test the data-driven MPC framework on the Duffing equation, 1D Burgers equation, and fluidic pinball, including increasingly complex flow regimes. The reported results range from indistinguishable Duffing performance to high-speedup control and increased tracking oscillations in chaotic regimes.

5. Numerical examples

  • The examples evaluate the framework using the affine generator K-ROM and a first-order-accurate Koopman-operator version.The fluidic pinball is the high-complexity example and exhibits chaotic behavior at moderate Reynolds numbers.Source: §5 · p. 14, §5.3 · p. 15

5.1. Forced Duffing Equation

  • The Duffing controller stabilizes three stationary points, with results indistinguishable from a controller using the nonlinear dynamics.The MPC uses a piecewise-constant reference trajectory and prediction horizon ℓ = 5.Source: Figure 3 · p. 14, §5.1 · p. 14

5.2. 1D Burgers equation

  • The 15-dimensional bilinear surrogate controls the nonlinear Burgers equation, and its optimization runs many orders faster than the full-state problem.The setup tracks a sinusoidal reference using four equidistant state observations and a shape function scaled by the input.Source: Figure 4 · p. 15, §5.2 · p. 15

5.3. The fluidic pinball

  • For the fluidic pinball, the controller uses cylinder-force observations with delay coordinates and localized reduced models across the control domain.Five Koopman operators are approximated to construct four reduced models, each valid on a respective part of the control domain.Source: §5.3 · pp. 15–16, Figure 6 · p. 16
  • At Re = 100, the 12-dimensional bilinear surrogate achieves excellent control and more than six orders of magnitude speed-up over solving the full system.The uncontrolled dynamics are quasi-periodic, and real-time applicability is achieved.Source: Figure 7 · p. 17, §5.3 · p. 17
  • At Re = 140, fluctuations around the desired state increase, but control remains satisfactory with a 12-dimensional linear system replacing a PDE with ≈150,000 degrees of freedom.The authors attribute the increased fluctuations as possibly due to both mild chaos and lower K-ROM accuracy.Source: Figure 8 · p. 17, §5.3 · p. 17
  • At higher Reynolds numbers, tracking oscillations grow, and adding 1,000 radial basis functions does not improve control performance.The comparison indicates that inaccuracy is mainly due to chaotic dynamics.Source: §5.3 · pp. 17–18, Figure 9 · p. 18

6. Conclusion

The approach builds bilinear reduced models by interpolating Koopman generators, with no added interpolation error for control-affine systems; its scope and error guarantees still warrant further study.

6. Conclusion

  • Interpolating EDMD-approximated Koopman generators at constant inputs yields bilinear models for intermediate controls.For control-affine systems, this interpolation introduces no additional error; interpolating the Koopman operator instead is accurate to first order in the timestep.Source: §6 · p. 18
  • The approach produces very good results in several examples, including cases where the control-affine assumption does not hold.Source: §6 · p. 18
  • Five to six orders of magnitude faster solves are reported for 2D Navier–Stokes equations.The reduced model benefits from larger step sizes and K-ROM linearity.Source: §6 · p. 18
  • Stronger error statements, including how basis size and required data affect K-ROM error, remain a research direction.Source: §6 · p. 18

A. Koopman-Kolmogorov Theorem Proof

The proof uses the autonomous flow and the definitions of the Koopman operator and generator to establish the stated derivative identity and expansion.

A. Koopman-Kolmogorov Theorem Proof

B. Efficient Newton-solver

The appendix exploits the bilinear structure of the reduced model to formulate a Newton solver for the discretized control problem and its dynamics and adjoint equations.

B. Efficient Newton-solver

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