Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

Drmač, Zlatko; Mezić, Igor; Mohr, Ryan

doi:10.3390/math9172075

Cited by 6 publications

(2 citation statements)

References 30 publications

(55 reference statements)

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“…Other methods include computing the matrix logarithm of the Koopman operator (Mauroy & Goncalves 2020, Drmač et al 2021, approximating the Koopman operator family, and using finitedifferences to compute the Lie derivative of the Koopman operator (Giannakis 2021, Sechi et al 2020) approach the problem of approximating both the Koopman and its generator as a manifold-learning problem on a space-time manifold. This challenge was successfully addressed for ergodic dynamical systems, such as those evolving on a chaotic attractor.…”

Section: Infinitesimal Generatorsmentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

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Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

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Section: Infinitesimal Generatorsmentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

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“…Ref. [23] proposed the method of lifting the data snapshots into a suitable finite-dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. Ref.…”

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confidence: 99%

Koopman Approximator based Adaptive Model Predictive Control of Continuous Nonlinear Systems

Zheng¹,

Zhang

Li³

et al. 2022

Preprint

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This paper considers the real-time control of a class of complex continuous nonlinear systems with an increasing requirement on control accuracy and unknown dynamics at the start time due to their complex dynamics and uncertainties. A Koopman approximate model-based adaptive MPC design using the Lyapunov technique is explored. Specifically, the nonlinear system is modeled/transformed into a linear model in a lifting space with the Koopman operator. A recursive update of the approximator is provided by which the parameters set of the approximator is in a nested form and a shrinking of the boundary of the approximator’s mismatch from the real system is obtained. Also, based on the Koopman model and its mismatch boundary, a sufficient condition that ensures the states of the nonlinear system eventually converge to a small neighborhood of the origin is deduced. An FCCU example is employed to show the effectiveness of the proposed control law.

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The multiverse of dynamic mode decomposition algorithms

Colbrook

2024

Handbook of Numerical Analysis

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Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

Cited by 6 publications

References 30 publications

Residual dynamic mode decomposition: robust and verified Koopmanism

Residual dynamic mode decomposition: robust and verified Koopmanism

Koopman Approximator based Adaptive Model Predictive Control of Continuous Nonlinear Systems

The multiverse of dynamic mode decomposition algorithms

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