A spectral operator-theoretic framework for global stability

Mauroy, Alexandre; Mezić, Igor

doi:10.1109/cdc.2013.6760712

Cited by 48 publications

(59 citation statements)

References 13 publications

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“…Thus, the Lyapunov function decays everywhere under action of the adjoint Liouville operator and is related to its properties and those of the Koopman operator family. In [113], it has been shown that global stability of a fixed point can be established through the existence of a set of C 1 eigenfunctions of the Koopman operator associated with the eigenvalues of the Jacobian of the vector field, and that Koopman eigenfunctions can be used to define a Lyapunov function and contracting metrics. A numerical scheme based on a Taylor expansion is proposed to compute stability properties including the domain of attraction [113].…”

Section: U Q F Z 1 5 K E Z W O W V F S U Z a W Q C I 4 D Q O O Q C mentioning

confidence: 99%

“…In [113], it has been shown that global stability of a fixed point can be established through the existence of a set of C 1 eigenfunctions of the Koopman operator associated with the eigenvalues of the Jacobian of the vector field, and that Koopman eigenfunctions can be used to define a Lyapunov function and contracting metrics. A numerical scheme based on a Taylor expansion is proposed to compute stability properties including the domain of attraction [113]. These ideas have been further extended to examine global stability properties of hyperbolic fixed points and limit cycles and also for non-analytical eigenfunctions [112].…”

Section: U Q F Z 1 5 K E Z W O W V F S U Z a W Q C I 4 D Q O O Q C mentioning

confidence: 99%

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Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.

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Section: U Q F Z 1 5 K E Z W O W V F S U Z a W Q C I 4 D Q O O Q C mentioning

confidence: 99%

Section: U Q F Z 1 5 K E Z W O W V F S U Z a W Q C I 4 D Q O O Q C mentioning

confidence: 99%

Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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“…Also, using approximative linear operators enables the application of linear systems algorithms and theory to nonlinear systems. Applying linear stability analysis to establish global stability is one example (Mauroy and Mezic, 2013). Sootla et al (2016) leverage the Koopman operator in the design of temporal pulse control of bistable monotone systems.…”

Section: Literature Review On the System Identification And Finite LImentioning

confidence: 99%

Efficient Infrastructure Restoration Strategies Using the Recovery Operator

González

Chapman

Dueñas‐Osorio

et al. 2017

Computer aided Civil Eng

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Infrastructure systems are critical for society's resilience, government operation, and overall defense. Thereby, it is imperative to develop informative and computationally efficient analysis methods for infrastructure systems, which reveal system vulnerabilities and recoverability. To capture practical constraints in systems analyses, various layers of complexity play a role, including limited element capacities, restoration resources, and the presence of interdependence among systems. High‐fidelity modeling such as mixed integer programming and physics‐based modeling can often be computationally expensive, making time‐sensitive analyses challenging. Furthermore, the complexity of recovery solutions can reduce analysis transparency. An alternative, presented in this work, is a reduced‐order representation, dubbed a recovery operator, of a high‐fidelity time‐dependent recovery model of a system of interdependent networks. The form of the operator is assumed to be a time‐invariant linear dynamic model apt for infrastructure restoration. The recovery operator is generated by applying system identification techniques to numerous disaster and recovery scenarios. The proposed compact representation provides simple yet powerful information regarding systemic recovery dynamics, and enables generating fast suboptimal recovery policies in time‐critical applications.

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“…We will call the family of operators U t indexed by time t the Koopman operator of the continuous-time system (1). This family was defined for the first time in [3], for Hamiltonian systems.…”

Section: Introductionmentioning

confidence: 99%

On applications of the spectral theory of the Koopman operator in dynamical systems and control theory

Mezić

2015

2015 54th IEEE Conference on Decision and Control (CDC)

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Recent contributions have extended the applicability of Koopman operator theory from dynamical systems to control. Stability theory got reformulated in terms of spectral properties of the Koopman operator [1], providing a nice link between the way we treat linear systems and nonlinear systems and opening the door for the use of classical linear e.g. pole placement theory in the fully nonlinear setting.New concepts such as isostables proved useful in the context of optimal control. Here, using Kato Decomposition we develop Koopman expansion for general LTI systems. We also interpret stable and unstable subspaces in terms of zero level sets of Koopman eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions to extend these results to globally stable systems. In conclusion, we discuss how the classical Hamilton-Jacobi-Bellman setting for optimal control can be reformulated in operator-theoretic terms and point the applicability of spectral operator theory in max-plus algebra to it. Geometric theories such as differential positivity have been also related to spectral theories of the Koopman operator [2], in cases when the attractor is a fixed point or a limit cycle, pointing the way to the more general case of quasiperiodic and chaotic attractors.

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A spectral operator-theoretic framework for global stability

Cited by 48 publications

References 13 publications

Data-Driven Approximations of Dynamical Systems Operators for Control

Data-Driven Approximations of Dynamical Systems Operators for Control

Efficient Infrastructure Restoration Strategies Using the Recovery Operator

On applications of the spectral theory of the Koopman operator in dynamical systems and control theory

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