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Koopman based data-driven predictive control

Yingzhao Lian, Renzi Wang, Colin N. Jones

arXiv:2102.05122v2eess.SYcs.SYmath.OC

TL;DR

Data-driven prediction and control from the Willems’ fundamental lemma is established for LTI systems, while extending it to nonlinear systems and long-term prediction remains challenging. This paper learns a Koopman-based finite approximation with uncertainty handling and uses it for data-driven prediction and control. The reported experiments show improved Wasserstein-loss prediction under lifting-function uncertainty and reference tracking in the tested control examples.

  • Problem

    Koopman approaches often provide one-step predictions, while long-term prediction methods based on subspace identification face scalability and numerical limitations.

    Source: §I · pp. 1–2

  • Method

    The paper learns finite-order Koopman lifting functions through a data-driven predictive-control framework and accounts for approximation uncertainty using Wasserstein-distance prediction.

    Source: §I · p. 2, §VI · p. 9

  • Results

    Wasserstein loss outperforms the original quadratic loss when lifting-function uncertainty is considered, and the tested control methods track references while respecting constraints or computing inputs in acceptable time.

    Source: B. Prediction based on Wasserstein… · p. 8, 1) Control of a bilinear motor · p. 9, 2) Control of nonlinear Korteweg–de… · p. 9

  • Takeaways & Limitations

    The proposed framework extends data-driven predictive control to nonlinear systems and provides a Wasserstein-distance simulation scheme that considers lifting-function uncertainty.

    Source: §VI · p. 9

    The learning approximates a potentially infinite basis with a finite one and relaxes an all-state condition to sampled states, so the training data must represent that condition adequately.

    Source: B. Leaning the Koopman Basis Functions · p. 5

Abstract

from arXiv · show

Sparked by the Willems' fundamental lemma, a class of data-driven control methods has been developed for LTI systems. At the same time, the Koopman operator theory attempts to cast a nonlinear control problem into a standard linear one albeit infinite-dimensional. Motivated by these two ideas, a data-driven control scheme for nonlinear systems is proposed in this work. The proposed scheme is compatible with most differential regressors enabling offline learning. In particular, the model uncertainty is considered, enabling a novel data-driven simulation framework based on Wasserstein distance. Numerical experiments are performed with Bayesian neural networks to show the effectiveness of both the proposed control and simulation scheme.

I. INTRODUCTION

The introduction motivates data-driven control for nonlinear systems by connecting the Willems’ fundamental lemma with Koopman operator theory. It presents Koopman-based learning and control as a response to limitations in existing nonlinear extensions and long-term prediction methods.

A. Willems’ Fundamental Lemma

  • The Willems’ fundamental lemma characterizes LTI input-output trajectories from measured data under controllability and persistent-excitation assumptions.It underlies data-driven prediction and control schemes.Source: §I · p. 1, A. Willems’ Fundamental Lemma · p. 2

I. INTRODUCTION

  • Existing Koopman approaches often focus on one-step prediction, while long-term approaches based on subspace identification face scalability and numerical limitations.The introduction also notes theoretical difficulties in some nonlinear extensions of the fundamental lemma.Source: §I · pp. 1–2
  • The paper combines the Willems’ fundamental lemma with Koopman learning to develop data-driven control for nonlinear systems.Its proposed learning uses sensitivity analysis of a parametric programming problem and accounts for model uncertainty.Source: §I · p. 2

NOTATIONS

  • The paper’s notation distinguishes signal sequences from measurements using boldface and lightface, respectively.It also reserves subscript d for offline data and superscript * for an optimization solution.Source: NOTATIONS · p. 2

II. PRELIMINARY

  • The preliminaries introduce the fundamental lemma and Koopman theory, then present parametric-optimization sensitivity analysis as an enabler of learning.The lemma’s Hankel-matrix representation supports data-driven simulation and control.Source: §II · p. 2, A. Willems’ Fundamental Lemma · p. 2

B. Koopman Operator

The Koopman operator represents nonlinear dynamics through the evolution of observables, which is linear in function space. With control inputs, the paper augments the state by the input sequence and uses a basis to represent the resulting evolution and outputs.

B. Koopman Operator

  • The Koopman operator composes an observable with the system dynamics, yielding a linear operator even when the underlying dynamics are nonlinear.An observable is a function on the state space; its eigenfunctions evolve linearly along system trajectories.Source: B. Koopman Operator · p. 3
  • For controlled systems, the paper augments the state with the infinite control sequence and models its evolution using a left-shift operator.This makes the augmented system autonomous and infinite-dimensional.Source: B. Koopman Operator · p. 3
  • When the chosen basis is closed under the dynamics, its evaluations evolve linearly and output evaluations can be reconstructed from Koopman modes.The output modes are stacked in Cu.Source: B. Koopman Operator · p. 3

C. Differential Parametric Optimization

The section uses sensitivity analysis of parametric optimization to obtain derivatives of the solution and optimal value. It explains the calculation through a quadratic-programming example and the linear system formed by differentiating its KKT conditions.

C. Differential Parametric Optimization

  • Sensitivity analysis studies the smoothness of parametric optimization, applying the implicit function theorem to the KKT system.The paper uses the optimal value function because it is smoother than the solution map.Source: C. Differential Parametric Optimization · p. 4
  • A parametric quadratic program illustrates the derivative calculation, with a positive-definite objective matrix and parameters defining its constraints and objective.The exposition points to a general conic form for broader treatment.Source: C. Differential Parametric Optimization · p. 4
  • Differentiating the KKT conditions yields a linear system for derivatives of the optimizer with respect to problem parameters and function input.The resulting optimizer derivatives are used to calculate the gradient of the optimal value.Source: C. Differential Parametric Optimization · p. 4

III. KOOPMAN BASED DATA-DRIVEN PREDICTION

This section introduces the fundamental lemma in the Koopman setting and uses it to motivate learning a finite basis from data. The learned basis is intended to make function evaluations evolve linearly, while probabilistic models can support stochastic prediction.

III. KOOPMAN BASED DATA-DRIVEN PREDICTION

  • The section proposes learning a Koopman operator basis by minimizing prediction error on training data, with stochastic prediction compatible with probabilistic models.Bayesian neural networks and Gaussian processes are named as examples.Source: §III · p. 4

A. Koopman Operator with the Fundamental Lemma

  • The fundamental lemma provides the stated criterion for representing a dynamical system of order nx as a linear system.The section uses this result to connect Koopman basis learning with data-driven trajectory representations.Source: A. Koopman Operator with the… · p. 4
  • The learning target is a basis whose evaluations span the outputs and admit a data-based trajectory representation under persistent excitation.The input sequence is required to be persistently exciting of order nφu − nu + L.Source: A. Koopman Operator with the… · p. 4

B. Leaning the Koopman Basis Functions

The scheme learns finite Koopman basis functions by maximizing linearity over sampled input-output data, using a prediction problem whose differentiation supports training. It avoids learning the EDMD matrices and is designed for multi-step prediction, scalability, and parallelization.

B. Leaning the Koopman Basis Functions

  • The method assumes each underlying state can be determined from the preceding Tini-step input-output sequence.Source: B. Leaning the Koopman Basis Functions · p. 4
  • The method uses a differentiable learner, with neural networks and Gaussian processes recommended for learning the basis functions.The offline input-output data are partitioned, with one subset building Hankel matrices and the remaining subsets learning the basis functions.Source: B. Leaning the Koopman Basis Functions · pp. 4–5
  • The learned basis functions maximize the Koopman linearity condition over sampled states, relaxing both the infinite basis requirement and the condition’s all-state scope.The training data should be large enough to represent the condition over states.Source: B. Leaning the Koopman Basis Functions · pp. 4–5
  • Unlike general EDMD, the scheme avoids learning A, B, and Cu, whose solution is not unique and whose learning is ill-conditioned.Source: B. Leaning the Koopman Basis Functions · p. 5
  • The learning objective optimizes multi-step forward prediction, while the Willems’ fundamental lemma is used to provide numerical stability in training.The passage identifies numerical stability as a key challenge in fully connected recurrent neural-network training.Source: B. Leaning the Koopman Basis Functions · p. 5
  • The scheme is described as scalable and parallelizable, and its prediction problem does not require nonlinear mappings of future inputs and outputs.Source: B. Leaning the Koopman Basis Functions · p. 5

C. Stochastic Prediction

For probabilistic learners, the paper develops stochastic prediction methods to use distributional outputs in prediction and Koopman-operator training. One method uses Monte Carlo sampling to approximate the output distribution and expected learning cost.

C. Stochastic Prediction

  • Probabilistic learners produce distributions rather than deterministic points, and the section develops ways to use these outputs for prediction and Koopman-operator training.Source: C. Stochastic Prediction · p. 5

1) Monte-Carlo Prediction:

2) Wasserstein Distanced based Prediction:

The Wasserstein-based prediction method models probabilistic-learner outputs as Gaussian and minimizes a bound on their distance from the target. Its objective accounts for mean and covariance discrepancies, with a further relaxation used to address numerical ill-conditioning.

2) Wasserstein Distanced based Prediction:

IV. KOOPMAN-BASED DATA-DRIVEN PREDICTIVE CONTROL

The proposed bi-level DeePC formulation separates prediction from control to address failures associated with their interweaving. It converts the prediction subproblem to KKT conditions, but integrating Wasserstein-based prediction remains future work because that subproblem is non-convex.

IV. KOOPMAN-BASED DATA-DRIVEN PREDICTIVE CONTROL

  • Separating prediction from control addresses DeePC’s risk of using a non-optimal prediction, which can lead to control failure when λy is insufficient and the system starts away from the reference.In that setting, the algorithm may compensate with a relatively large σy.Source: §IV · p. 7
  • The bi-level formulation uses squared norms in its prediction objective to keep the lower-level problem smooth.Its upper level performs control, while the lower level performs prediction.Source: §IV · p. 7
  • Replacing the lower-level prediction problem with its KKT conditions yields an equivalent single-level formulation.The paper notes that this formulation is solvable with many optimization toolboxes.Source: §IV · p. 7
  • The paper leaves control integrated with Wasserstein-distance prediction for future work because that prediction problem is non-convex.Bi-level problems are generally NP-hard, and the paper notes no valid general solution when the lower-level problem is non-convex.Source: §IV · p. 7

V. SIMULATION RESULTS

The simulation-results section presents prediction experiments on a Van der Pol oscillator, control of a bilinear motor, and a large-scale control example using the nonlinear Korteweg–de Vries equation.

V. SIMULATION RESULTS

  • The experiments include prediction results for a Van der Pol oscillator.The section first illustrates Monte Carlo and Wasserstein-distance-based prediction.Source: §V · p. 7
  • A numerical experiment demonstrates control of a bilinear motor.Source: §V · p. 7
  • The section demonstrates the proposed scheme on a large-scale control problem using the nonlinear Korteweg–de Vries equation.Source: §V · p. 7

A. Stochastic prediction

The section describes stochastic prediction with a dropout neural network and evaluates a Wasserstein-based prediction loss on Van der Pol data. It reports that the Wasserstein loss outperforms the original quadratic loss when lifting-function uncertainty is considered.

A. Stochastic prediction

  • For stochastic prediction, the trained network is run 120 times on test data to estimate prediction means and standard deviations.The test uses 50 points from trajectories independent of training and Hankel-matrix data; the light-blue region indicates two standard deviations.Source: A. Stochastic prediction · p. 7

B. Prediction based on Wasserstein distance

C. Control with Koopman-based DeePC

The section evaluates Koopman-based DeePC on a bilinear motor and a nonlinear KdV equation, then summarizes the nonlinear data-driven predictive-control approach and its uncertainty-aware simulation scheme.

C. Control with Koopman-based DeePC

1) Control of a bilinear motor:

2) Control of nonlinear Korteweg–de Vries equation:

VI. CONCLUSION

  • The conclusion describes nonlinear lifting into an infinite-dimensional linear space, data-driven approximation of lifting functions, and uncertainty-aware simulation based on Wasserstein distance.Source: §VI · p. 9
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