Introduction to the Koopman Operator in Dynamical Systems and Control Theory

Mauroy, Alexandre; Susuki, Yoshihiko; Mezić, Igor

doi:10.1007/978-3-030-35713-9_1

Cited by 54 publications

(101 citation statements)

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“…In this section, we review the definition of the asymptotic phase and amplitude for deterministic limit-cycle oscillators and discuss their relationship with the Koopman eigenfunctions [10][11][12][13][14]19,26]. We consider a deterministic dynamical systeṁ…”

Section: Classical Definition Of the Asymptotic Phase And Amplitudementioning

confidence: 99%

“…Because the backward Liouville operator L + X coincides with the infinitesimal generator of the Koopman operator A in the deterministic case given in Equation ( 8), the Koopman eigenfunction Ψ 0 (X) of A in Equation ( 10) is an eigenfunction of L + X with an eigenvalue iω. Thus, the definition of the asymptotic phase for stochastic oscillators in Equation ( 30) can be considered a natural generalization of the definition of the asymptotic phase for deterministic oscillators in Equation (11). Similarly, the Koopman eigenfunction R 0 (X) of A in Equation ( 12) is an eigenfunction of L + X with an eigenvalue Λ 2 = λ 1 , so the definition of the amplitude for stochastic oscillators in Equation (37) also corresponds to that for deterministic oscillators in Equation (12).…”

Section: Limit Of Vanishing Noise Intensitymentioning

confidence: 99%

“…Recently, the asymptotic phase and isochrons (level sets of the asymptotic phase), classical notions in the theory of nonlinear oscillations since Winfree [8] and Guckenheimer [9], have been studied from a viewpoint of the Koopman operator theory by Mauroy, Mezić, and Moehlis [10], and their relationship with the Koopman eigenfunction associated with the fundamental frequency of the oscillator has been clarified [10][11][12][13][14]. Moreover, they have shown that the (asymptotic) amplitude and isostables, which characterize deviation of the system state from the limit cycle and extend the Floquet coordinates [13,15,16] to the nonlinear regime, can be introduced naturally in terms of the Koopman eigenfunctions associated with the Floquet exponents with non-zero real parts [10][11][12][13][14]17]. By using the asymptotic phase and amplitude functions, we can obtain a reduced description of limitcycle oscillators, which is useful for the analysis and control of synchronization dynamics of limit-cycle oscillators [18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

et al. 2021

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The asymptotic phase is a fundamental quantity for the analysis of deterministic limit-cycle oscillators, and generalized definitions of the asymptotic phase for stochastic oscillators have also been proposed. In this article, we show that the asymptotic phase and also amplitude can be defined for classical and semiclassical stochastic oscillators in a natural and unified manner by using the eigenfunctions of the Koopman operator of the system. We show that the proposed definition gives appropriate values of the phase and amplitude for strongly stochastic limit-cycle oscillators, excitable systems undergoing noise-induced oscillations, and also for quantum limit-cycle oscillators in the semiclassical regime.

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Section: Classical Definition Of the Asymptotic Phase And Amplitudementioning

confidence: 99%

Section: Limit Of Vanishing Noise Intensitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

et al. 2021

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“…The Koopman operator is a linear operator defined on the lifted infinite-dimensional state-space and is utilized for analyzing complex dynamical systems. Motivated by its high ability to express nonlinearity, data-driven finite-dimensional approximation of the operator has been studied and applied not only in system analysis [13][14][15] but also in control system design [16][17][18][19][20][21][22][23][24][25][26]. In this paper, a nonlinear dynamical system is described using the Koopman operator, and its data-driven approximation is addressed.…”

Section: Introductionmentioning

confidence: 99%

Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design

Hara

Inoue

2021

Mathematics

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In this paper, we address the data-driven modeling of a nonlinear dynamical system while incorporating a priori information. The nonlinear system is described using the Koopman operator, which is a linear operator defined on a lifted infinite-dimensional state-space. Assuming that the L2 gain of the system is known, the data-driven finite-dimensional approximation of the operator while preserving information about the gain, namely L2 gain-preserving data-driven modeling, is formulated. Then, its computationally efficient solution method is presented. An application of the modeling method to feedback controller design is also presented. Aiming for robust stabilization using data-driven control under a poor training dataset, we address the following two modeling problems: (1) Forward modeling: the data-driven modeling is applied to the operating data of a plant system to derive the plant model; (2) Backward modeling: L2 gain-preserving data-driven modeling is applied to the same data to derive an inverse model of the plant system. Then, a feedback controller composed of the plant and inverse models is created based on internal model control, and it robustly stabilizes the plant system. A design demonstration of the data-driven controller is provided using a numerical experiment.

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“…Koopman operator theory has been shown to provide insight into system behavior [16][17][18][19][20][21][22][23] through the analysis of the spectrum of a nonlinear system's Koopman operator (see Section 3 for a detailed introduction). Specifically, for biological applications, Ref.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

Harrison

Yeung²

2021

Mathematics

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The genetic toggle switch is a well known model in synthetic biology that represents the dynamic interactions between two genes that repress each other. The mathematical models for the genetic toggle switch that currently exist have been useful in describing circuit dynamics in rapidly dividing cells, assuming fixed or time-invariant kinetic rates. There is a growing interest in being able to model and extend synthetic biological function for growth conditions such as stationary phase or during nutrient starvation. As cells transition from one growth phase to another, kinetic rates become time-varying parameters. In this paper, we propose a novel class of parameter varying nonlinear models that can be used to describe the dynamics of genetic circuits, including the toggle switch, as they transition from different phases of growth. We show that there exists unique solutions for this class of systems, as well as for a class of systems that incorporates the microbial phenomena of quorum sensing. Further, we show that the domain of these systems, which is the positive orthant, is positively invariant. We also showcase a theoretical control strategy for these systems that would grant asymptotic monostability of a desired fixed point. We then take the general form of these systems and analyze their stability properties through the framework of time-varying Koopman operator theory. A necessary condition for asymptotic stability is also provided as well as a sufficient condition for instability. A Koopman control strategy for the system is also proposed, as well as an analogous discrete time-varying Koopman framework for applications with regularly sampled measurements.

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Introduction to the Koopman Operator in Dynamical Systems and Control Theory

Cited by 54 publications

References 54 publications

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

Asymptotic Phase and Amplitude for Classical and Semiclassical Stochastic Oscillators via Koopman Operator Theory

Gain-Preserving Data-Driven Approximation of the Koopman Operator and Its Application in Robust Controller Design

Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

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