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Flow primitives and infinitesimal generators of Perron-Frobenius and Koopman operators

Phanindra Tallapragada

TL;DR

The paper addresses how Perron-Frobenius and Koopman operators evolve when the underlying dynamical vector field is perturbed or augmented by another field. It proposes a framework that uses the exponentials of infinitesimal generators and their Lie brackets to approximate the operators for the combined field, with error scaling as $O(t^3)$ and dominated by nested Lie-bracket terms such as $[(g-h), [g,h]]$. By expressing vector-field changes as sums or perturbations, the authors derive practical approximations: $\mathcal P^t \approx \tfrac{1}{2}(\mathcal P_1^t \mathcal P_2^{\epsilon t} + \mathcal P_2^{\epsilon t} \mathcal P_1^t)$ and $\mathcal K^t \approx \tfrac{1}{2}(\mathcal K_1^t \mathcal K_2^{\epsilon t} + \mathcal K_2^{\epsilon t} \mathcal K_1^t)$. They illustrate the method with flow-primitive examples and parameter-perturbation cases using EDMD to construct finite-dimensional operators, showing good agreement with direct simulations for short horizons and substantial computational savings. The framework is agnostic to the operator-approximation method, enabling broad applicability in robotics, flow control, and related nonlinear systems where model updates are frequent.

Abstract

The Koopman and the Perron-Frobenius operators are increasingly becoming popular in the control of complex nonlinear systems such as in a wide variety of robotics problems and flow control. This is in addition to the wide interest in the application of operator methods for better understanding of fluid flows. A particular problem of relevance to all such applications is, how does the Koopman or the Perron-Frobenius (PF) operator change if the underlying vector field of the dynamical system undergoes small changes or if two vector fields are added. The current numerical methods rely on significant computations and model or parameter changes to the dynamical system often require all the computations to be repeated. This paper reports on a novel approach to the computation of the approximate PF and Koopman operators in such cases. The approach makes use of the exponentials of the infinitesimal generators of these operators. It is shown that this approximation depends on the Lie bracket of the vector field and the perturbation vector field. Examples are described where the Koopman and PF operators are constructed from operators of primitive flows and for cases where the model parameters undergo perturbations.

Flow primitives and infinitesimal generators of Perron-Frobenius and Koopman operators

TL;DR

The paper addresses how Perron-Frobenius and Koopman operators evolve when the underlying dynamical vector field is perturbed or augmented by another field. It proposes a framework that uses the exponentials of infinitesimal generators and their Lie brackets to approximate the operators for the combined field, with error scaling as and dominated by nested Lie-bracket terms such as . By expressing vector-field changes as sums or perturbations, the authors derive practical approximations: and . They illustrate the method with flow-primitive examples and parameter-perturbation cases using EDMD to construct finite-dimensional operators, showing good agreement with direct simulations for short horizons and substantial computational savings. The framework is agnostic to the operator-approximation method, enabling broad applicability in robotics, flow control, and related nonlinear systems where model updates are frequent.

Abstract

The Koopman and the Perron-Frobenius operators are increasingly becoming popular in the control of complex nonlinear systems such as in a wide variety of robotics problems and flow control. This is in addition to the wide interest in the application of operator methods for better understanding of fluid flows. A particular problem of relevance to all such applications is, how does the Koopman or the Perron-Frobenius (PF) operator change if the underlying vector field of the dynamical system undergoes small changes or if two vector fields are added. The current numerical methods rely on significant computations and model or parameter changes to the dynamical system often require all the computations to be repeated. This paper reports on a novel approach to the computation of the approximate PF and Koopman operators in such cases. The approach makes use of the exponentials of the infinitesimal generators of these operators. It is shown that this approximation depends on the Lie bracket of the vector field and the perturbation vector field. Examples are described where the Koopman and PF operators are constructed from operators of primitive flows and for cases where the model parameters undergo perturbations.

Paper Structure

This paper contains 10 sections, 3 theorems, 64 equations, 4 figures.

Table of Contents

  1. Introduction
  2. PF operator and infinitesimal generator
  3. Exponentials and Commutativity
  4. Sum of vector fields
  5. Sum of divergence free vector fields and the PF operator
  6. Perturbation of parameters
  7. Examples
  8. Flow primitives and PF operator
  9. Examples - Perturbation of parameters
  10. Conclusion

Key Result

Lemma 1

The PF operator associated with the dynamical system is $\mathcal{P}^t = \mathcal{P}_1^t\mathcal{P}_2^t$ if $\mathcal{A}_1\mathcal{A}_2 = \mathcal{A}_2\mathcal{A}_1$. The Koopman operator associated with the dynamical system is $\mathcal{K}^t = \mathcal{K}_1^t\mathcal{K}_2^t$ if $\mathcal{C}_1\mathc

Figures (4)

  • Figure 1: The first three columns of figures show the evolution of a blob of tracers. (a) The first column shows the evolution of a blob of tracers due to the vector field $g$ produced by a single rotlet at $(1,0)$, (b) the second column shows the evolution of a blob of tracers due to a vector field $h$ produced by a single rotlet at $(-1,0)$, (c) the third column shows the evolution of a blob of tracers due to a vector field $g+h$ produced by two rotlets at $(-1,0)$ and $(1,0)$ respectively. (d) The fourth column shows the evolution of a (gaussian) density function with its center initially located at the initial center of the blob. This density is propagated by $\frac{1}{2}(P_1^tP_2^t + P_2^tP_1^t)$
  • Figure 2: (a) The first column shows the evolution of a blob of tracers due to the exact the vector field $f(x;p)$\ref{['eq:pen_approx']}. (b) The second column shows the propagation of a gaussian density function by the PF operator $P^t$ due to the vector field $g+h$. (c) The third column shows the propagation of a gaussian density function by the PF operator $\frac{1}{2}(P_1^tP_2^t + P_2^tP_1^t)$ where the operator $P_1^t$ is due to vector field $g$ and the operator $P_2^t$ due to the vector field $h$. The parameter values are $p_0=1$ and $\epsilon = 0.1$.
  • Figure 3: (a) The first column shows the evolution of a blob of tracers due to the vector field $g+h$\ref{['eq:duffing']}. (b) The second column shows the propagation of a gaussian density function by the PF operator $P^t$ due to the vector field $g+h$. (c) The third column shows the propagation of a gaussian density function by the PF operator $\frac{1}{2}(P_1^tP_2^t + P_2^tP_1^t)$ where the operator $P_1^t$ is due to vector field $g$ and the operator $P_2^t$ due to the vector field $h$.
  • Figure 4: The propagation of the density functions shown in fig. \ref{['fig:rotlet']} and fig. \ref{['fig:pendulum']} at $t=1$ by $P^t = e^{(A_1+A_2)t}$. (a) Density propagated by the generators due to the two rotlets vector fields, $g$ and $h$ in \ref{['eq:rotlet']}, (b) Density propagated by the generators due to the vector fields $g$ and $h$ in \ref{['eq:pen_approx']}.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • Definition 2
  • Corollary 1
  • Example 1
  • Example 2