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Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics

Daniel Bruder, Xun Fu, Ram Vasudevan

arXiv:2010.09961v3cs.RO

TL;DR

Linear Koopman models may not support efficient control when their input dependence is not linear, and finite-dimensional models only approximate infinite-dimensional embeddings. The paper develops data-driven linear, bilinear, and nonlinear realizations and realization conditions; it shows bilinear realizations exist for control-affine systems and, in a simulated robot-arm MPC task, combine better prediction accuracy than linear models with greater computational efficiency than nonlinear models.

  • Problem

    Koopman linearity over observables does not guarantee linear dependence on control inputs, limiting compatibility with efficient linear control; finite-dimensional truncations only approximate infinite-dimensional embeddings.

    Source: §I · p. 1

  • Method

    The paper gives necessary and sufficient realization conditions and constructs approximate linear, bilinear, and nonlinear Koopman models from data.

    Source: §I · p. 2

  • Results

    Every control-affine system admits a bilinear realization but not necessarily a linear one; in simulated robot-arm MPC, bilinear realizations exceed linear-model prediction accuracy and nonlinear-model computational efficiency.

    Source: §I · p. 2, §V · pp. 7–8

  • Takeaways & Limitations

    For unknown dynamics, bilinear realizations built from generic basis functions improve as basis count increases, whereas approximate linear realizations may not.

    Source: §V · p. 8

Abstract

from arXiv · show

Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This paper describes how the Koopman operator can be used to generate approximate linear, bilinear, and nonlinear model realizations from data, and argues in favor of bilinear realizations for characterizing systems with unknown dynamics. Necessary and sufficient conditions for a dynamical system to have a valid linear or bilinear realization over a given set of observable functions are presented and used to show that every control-affine system admits an infinite-dimensional bilinear realization, but does not necessarily admit a linear one. Therefore, approximate bilinear realizations constructed from generic sets of basis functions tend to improve as the number of basis functions increases, whereas approximate linear realizations may not. To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm are constructed from data. In a trajectory following task, the bilinear realization exceeds the prediction accuracy of the linear realization and the computational efficiency of the nonlinear realization when incorporated into a model predictive control framework.

I. INTRODUCTION

The introduction frames a gap between the efficiency of linear control methods and the uncertain existence of linear Koopman realizations for controlled nonlinear systems. The theory section gives conditions for linear and bilinear realizations, while the preliminaries define observables and their Koopman evolution.

I. INTRODUCTION

  • Lifting a nonlinear system into observables gives a global linear representation, unlike local linearization, which predicts well only near equilibrium points.Local linearization uses a first-order Taylor approximation around equilibria.Source: §I · p. 1
  • Koopman models linear in observables may still be nonlinear in the control input, limiting compatibility with efficient linear control; finite-dimensional truncations also only approximate infinite-dimensional embeddings.A linear-input realization may not exist even with an infinite-dimensional observable subspace.Source: §I · p. 1
  • Bilinear Koopman realizations are presented as a compromise: they retain some computational benefits of linear models and are more likely to exist for arbitrary systems.The paper also describes constructing approximate linear, bilinear, and nonlinear realizations from data.Source: §I · pp. 1–2

II. THEORY

  • Necessary and sufficient conditions show that every control-affine system has a bilinear realization, but a linear realization is not guaranteed.The bilinear realization may be infinite-dimensional.Source: §II · p. 2, §I · p. 2

A. Koopman operator preliminaries

  • Observables are square-integrable real-valued functions whose evolution along system trajectories is described by Koopman operators.The preliminaries describe the flow for a constant input over a time interval and introduce the infinitesimal generator of the Koopman operators.Source: A. Koopman operator preliminaries · p. 2

B. Model Realizations

The paper defines linear and bilinear realizations over chosen observables and gives necessary and sufficient conditions for their validity. Every control-affine system admits a bilinear realization over an infinite basis, but a linear realization is not guaranteed.

B. Model Realizations

  • A model realization is a dynamical system that reproduces the original system’s state response under any input signal.Koopman realizations are constructed by choosing observables whose evolution describes the lifted state.Source: B. Model Realizations · p. 2
  • In these realizations, the lifted state contains only state-dependent observables, while the input appears in linear and bilinear terms.A realization is tied to its chosen observables, so different choices can produce different realizations.Source: B. Model Realizations · p. 3
  • Theorem II.1 gives necessary and sufficient conditions for a chosen observable set to define a valid linear or bilinear realization.The state must be recoverable from the observables; the theorem distinguishes the conditions for bilinear and linear realizations.Source: B. Model Realizations · p. 3
  • Every control-affine system has a valid bilinear realization over an infinite basis; finite basis approximations converge toward one, while linear realizations need not.Generic polynomial, Fourier, or radial basis functions can represent arbitrary state-dependent functions when taken as a basis.Source: B. Model Realizations · pp. 3–4

III. TECHNIQUES FOR DATA-DRIVEN REALIZATION

This section turns from model realizations to identifying finite-dimensional Koopman matrix approximations from data, with the aim of producing linear, bilinear, or nonlinear realizations.

III. TECHNIQUES FOR DATA-DRIVEN REALIZATION

  • The section focuses on identifying data-driven Koopman matrix approximations for constructing model realizations.It follows the preceding discussion of constructing realizations over observable sets.Source: §III · p. 4
  • The identified matrix approximations are used to produce linear, bilinear, or nonlinear realizations.Source: §III · p. 4
  • The section describes the identification of finite-dimensional approximations rather than the abstract realization definitions.Source: §III · p. 4

A. Approximation of the Koopman operator from data

The paper approximates the Koopman operator on a finite observable subspace by applying EDMD linear regression to experimental data. It then uses the discrete-time approximation to obtain a continuous-time operator approximation.

A. Approximation of the Koopman operator from data

  • EDMD identifies a finite-dimensional Koopman matrix approximation by linear regression on observed data.The approximation represents the Koopman operator restricted to a finite-dimensional subspace.Source: A. Approximation of the Koopman… · p. 4
  • The finite-dimensional subspace is the span of a chosen set of linearly independent observables.The observable map evaluates the selected functions and stacks their values into a vector.Source: A. Approximation of the Koopman… · p. 4
  • The experimental data consist of snapshots recording measured states and an input applied between measurements.The snapshots share an identical sampling period, though consecutive snapshots need not come from consecutive measurements.Source: A. Approximation of the Koopman… · p. 4
  • The discrete-time matrix approximation is obtained by least squares, and the continuous-time approximation is recovered using the principal matrix logarithm.The discrete-time Koopman operator advances observable values by one time step.Source: A. Approximation of the Koopman… · p. 4

B. Linear Model Realization

The linear realization uses state-dependent observables alongside input-component projections, placing the input projections in the final coordinates of the lifted vector. Its linear-realization coefficients are embedded in the identified Koopman matrix.

B. Linear Model Realization

  • The basis is arranged with state-dependent observables first and input-component projections in the final m coordinates.This arrangement ensures that the input appears only in the last m components of the lifted vector.Source: B. Linear Model Realization · p. 4
  • Coefficients for the linear realization are embedded in the first N rows of the transpose of the identified Koopman matrix.The passage specifies the coefficient blocks for the state and input terms.Source: B. Linear Model Realization · pp. 4–5
  • The first n basis functions can be chosen as projections onto the components of the state.Source: B. Linear Model Realization · p. 5

C. Bilinear Model Realization

The bilinear realization builds a lifted observable set from state functions, their products with input components, and input projections. The identified Koopman matrix contains the bilinear system coefficients, while the output equation is defined as in the linear realization.

C. Bilinear Model Realization

  • The lifted coordinates comprise state observables, products of those observables with each input component, and input projections.The state-input products occupy the middle block of the construction.Source: C. Bilinear Model Realization · p. 5
  • The bilinear system coefficients are embedded in the first N rows of the transposed Koopman matrix, including A, B, and the H_i blocks.The matrix is identified from data, and the H_i blocks are specified by their component entries.Source: C. Bilinear Model Realization · p. 5
  • As in the linear realization, the first n basis functions and output-equation coefficients are defined using the constructions in (38) and (39).Source: C. Bilinear Model Realization · p. 5

D. Nonlinear Model Realization

Choosing observables other than the paper’s specified linear and bilinear sets can produce a nonlinear realization, with coefficients for state outputs obtained from the identified Koopman matrix.

D. Nonlinear Model Realization

E. Model Predictive Control

Linear MPC has a convex optimization problem, whereas nonlinear MPC can be more difficult to solve efficiently. The paper’s K-BMPC approximates bilinear dynamics by fixing lifted-state values, trading prediction error for a convex quadratic program that can be updated at each time step.

E. Model Predictive Control

  • Linear-model MPC is convex, while nonlinear-model MPC has nonlinear constraints and potentially nonlinear costs that can make global optimization harder.The linear formulation has quadratic cost and linear constraints; nonlinear MPC algorithms typically require initialization and may struggle to find globally optimal solutions.Source: E. Model Predictive Control · p. 5
  • K-MPC, K-BMPC, and K-NMPC use linear, bilinear, and nonlinear Koopman realizations, respectively, to generate predictions.Source: E. Model Predictive Control · p. 5
  • K-BMPC fixes the lifted-state value in bilinear terms over the prediction horizon, making its dynamics constraints linear approximations around the initial lifted state.Source: E. Model Predictive Control · pp. 5–6
  • The K-BMPC approximation introduces prediction error but yields a convex quadratic program that can be solved quickly enough for every time-step update.Its solutions are not necessarily optimal with respect to the true system dynamics, since finite-dimensional data-derived Koopman realizations only approximate those dynamics.Source: E. Model Predictive Control · p. 6

IV. EXPERIMENTS AND RESULTS

The experiments identify Koopman models from simulated planar-arm data and compare their prediction accuracy as model type and basis-function count vary. Linear-model accuracy changes little with dimension, while bilinear and nonlinear model accuracy improves as dimension increases.

IV. EXPERIMENTS AND RESULTS

  • The test system is a simulated three-link planar arm with joint torques as inputs and six Cartesian coordinates for link-end outputs.Each link has mass 100 g and length 0.33 m; the system has 3 joints.Source: §IV · p. 6, Figure 1 · p. 6

A. Model Prediction Comparison

  • The models use monomial basis subspaces, with linear, bilinear, and nonlinear constructions varying in their inclusion of input terms.The linear set adds input projections to state monomials; bilinear sets also include products of state monomials and input components, while nonlinear sets use monomials of both state and input.Source: A. Model Prediction Comparison · p. 6
  • Prediction accuracy is evaluated by comparing model simulations with validation data and averaging Euclidean output error in R6.Source: A. Model Prediction Comparison · p. 6
  • Linear-model accuracy changes little as dimension grows, whereas bilinear and nonlinear model errors decrease as the basis-function count increases.Fig. 2 reports prediction error against Koopman-matrix dimension for models identified from the same data.Source: A. Model Prediction Comparison · pp. 6–7, Figure 2 · p. 7

B. Control Performance Comparison

The linear, bilinear, and nonlinear Koopman controllers were compared on the same trajectory-following task for a simulated 3-link planar arm. They used identical cost functions and a 10-step prediction horizon.

B. Control Performance Comparison

  • The simulated 3-link planar arm followed the same planar reference trajectory using controllers built from linear, bilinear, and nonlinear Koopman realizations.Each controller was constructed from the corresponding model realization identified for ρ = 3.Source: B. Control Performance Comparison · p. 7
  • Each controller used a 10-step horizon and identical costs penalizing end-effector tracking error and input magnitude.The input penalty distinguishes among robot configurations that achieve the desired end-effector placement.Source: B. Control Performance Comparison · p. 7
  • The task traced a block-letter M for 15 seconds; tracking error was Euclidean distance, and computation time measured MPC optimization only.The experiment ran in simulation with a 0.05-second step; all trials used a 64 GB RAM, 2.4 GHz CPU computer.Source: B. Control Performance Comparison · p. 7

V. DISCUSSION AND CONCLUSION

In the simulated arm task, K-BMPC combined tracking accuracy comparable to K-NMPC with computation time comparable to K-MPC. The discussion links this result to bilinear realizations’ improving prediction accuracy with more basis functions and to their theoretical availability for control-affine systems.

V. DISCUSSION AND CONCLUSION

  • Bilinear-model prediction error decreases as basis functions increase, while the linear model does not improve with more basis functions.The authors interpret this as progress toward a true infinite-dimensional bilinear realization and evidence that an infinite-dimensional linear realization over monomial basis functions probably does not exist.Source: §V · p. 7
  • More than 15× larger mean tracking error marks K-MPC versus the other controllers; K-NMPC mean computation time is more than 500× larger than the other two.Fig. 4 reports the mean tracking error and computation time; the K-NMPC computation time exceeds the 50 ms time step, making it incompatible with closed-loop operation in this experiment.Source: §V · p. 7, Figure 4 · p. 8
  • K-BMPC has mean tracking error comparable to K-NMPC and mean computation time comparable to K-MPC, making it the only viable closed-loop controller in this experiment.Despite its suboptimal solutions, K-BMPC’s mean tracking error is nearly equivalent to K-NMPC’s.Source: §V · p. 7, Figure 4 · p. 8
  • Control-affine systems have infinite-dimensional bilinear realizations but do not necessarily have linear ones, supporting improved bilinear approximations from generic basis sets as their size grows.Approximate linear realizations may not improve as the number of basis functions increases.Source: §V · p. 8
  • The paper’s bilinear MPC framework could be improved; the authors call for theoretical guarantees for bilinear controllers and new approaches to bilinear optimal control.The paper characterizes bilinear realizations as combining linear-realization computational efficiency with nonlinear-realization prediction accuracy.Source: §V · p. 8
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