Linear identification of nonlinear systems: A lifting technique based on the Koopman operator

Mauroy, Alexandre

doi:10.1109/cdc.2016.7799269

Cited by 103 publications

(86 citation statements)

References 12 publications

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“…The work presented here can be considered an extension of the work on Koopman-based modeling and control of Mauroy and Goncalves [18] and Korda and Mezić [14]. The novel contributions of this work, as depicted in Fig.…”

Section: Introductionmentioning

confidence: 91%

“…Koopman operator theory offers an approach that can overcome the challenges of modeling and controlling soft robots. The approach leverages the linear structure of the Koopman operator to construct linear models of nonlinear controlled dynamical systems from input-output data [6,18], and to control them using established linear control methods [1,14]. In theory, this approach involves lifting the state-space to an infinite-dimensional space of scalar functions (referred to as observables), where the flow of such observables along trajectories of the nonlinear dynamical system is described by the linear Koopman operator.…”

Section: Introductionmentioning

confidence: 99%

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Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control

Bruder

Gillespie

Remy

et al. 2019

Robotics: Science and Systems XV

111

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Controlling soft robots with precision is a challenge due in large part to the difficulty of constructing models that are amenable to model-based control design techniques. Koopman operator theory offers a way to construct explicit linear dynamical models of soft robots and to control them using established model-based linear control methods. This method is data-driven, yet unlike other data-driven models such as neural networks, it yields an explicit control-oriented linear model rather than just a "black-box" input-output mapping. This work describes this Koopman-based system identification method and its application to model predictive controller design. A model and MPC controller of a pneumatic soft robot arm is constructed via the method, and its performance is evaluated over several trajectory following tasks in the real-world. On all of the tasks, the Koopman-based MPC controller outperforms a benchmark MPC controller based on a linear state-space model of the same system.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control

Bruder

Gillespie

Remy

et al. 2019

Robotics: Science and Systems XV

111

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“…The discovered eigenfunctions are then used for control, resulting in the so-called KRONIC framework. Another extension of SINDy was derived in [21], allowing for the identification of parameters of a stochastic system using Kramers-Moyal formulae.A different avenue towards system identification was taken in [22,23]. Here, the Koopman operator is first approximated with the aid of EDMD, and then its generator is determined using the matrix logarithm.…”

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confidence: 99%

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

Klus

Nüske

Peitz

et al. 2020

Physica D: Nonlinear Phenomena

187

160

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We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems. arXiv:1909.10638v1 [math.DS] 23 Sep 2019Most of the aforementioned techniques turn out to be strongly related, with the unifying concept being Koopman operator theory [16,17,18]. In what follows, we will focus mainly on the generator of the Koopman operator and its properties and applications. SINDy [5] constitutes a milestone for data-driven discovery of dynamical systems. Because of the close relationship between the vector field of a deterministic dynamical system and its Koopman generator, SINDy is a special case of the framework we will introduce in this study. In [19,20], the authors presented an extension of SINDy to determine eigenfunctions of the Koopman generator. The discovered eigenfunctions are then used for control, resulting in the so-called KRONIC framework. Another extension of SINDy was derived in [21], allowing for the identification of parameters of a stochastic system using Kramers-Moyal formulae.A different avenue towards system identification was taken in [22,23]. Here, the Koopman operator is first approximated with the aid of EDMD, and then its generator is determined using the matrix logarithm. Subsequently, the right-hand side of the differential equation is extracted from the matrix representation of the generator. The relationship between the Koopman operator and its generator was also exploited in [24] for parameter estimation of stochastic differential equations.A method for computing eigenfunctions of the Koopman generator was proposed in [25], where the diffusion maps algorithm is used to set up a Galerkin-projected eigenvalue problem with orthogonal basis elements. Two efficient methods for computing the generator of the adjoint Perron-Frobenius operator based on Ulam's method and spectral collocation were presented in [26]. Provided that a model of the system dynamics is available, the computation of trajectories can be replaced by evaluations of the right-hand side of the system, which is often orders of magnitude faster.The purpose of this study is to present a general framework to compute a matrix approximation of the Koop...

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“…Literature Review: The eigenfunctions of the Koopman operator [1], [2] evolve linearly in time, and hence its eigendecomposition can be used to analyze and predict the behavior of dynamical systems [3]- [5]. This can simplify identification [6] and control [7]- [11] of nonlinear systems. Traditional roadblocks to the widespread use of the Koopman operator have been its infinite-dimensional nature and the lack of practical methods to find representations for it.…”

Section: Introductionmentioning

confidence: 99%

Efficient Identification of Linear Evolutions in Nonlinear Vector Fields: Koopman Invariant Subspaces

Haseli

Cortés

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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This paper presents a data-driven approach to identify finite-dimensional Koopman invariant subspaces and eigenfunctions of the Koopman operator. Given a dictionary of functions and a collection of data snapshots of the dynamical system, we rely on the Extended Dynamic Mode Decomposition (EDMD) method to approximate the Koopman operator. We start by establishing that, if a function in the space generated by the dictionary evolves linearly according to the dynamics, then it must correspond to an eigenvector of the matrix obtained by EDMD. A counterexample shows that this necessary condition is however not sufficient. We then propose a necessary and sufficient condition for the identification of linear evolutions according to the dynamics based on the application of EDMD forward and backward in time. Due to the complexity of checking this condition, we propose an alternative characterization based on the identification of the largest Koopman invariant subspace in the span of the dictionary. This leads us to introduce the Symmetric Subspace Decomposition strategy to identify linear evolutions using efficient linear algebraic methods. Various simulations illustrate our results.

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Linear identification of nonlinear systems: A lifting technique based on the Koopman operator

Cited by 103 publications

References 12 publications

Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control

Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

Efficient Identification of Linear Evolutions in Nonlinear Vector Fields: Koopman Invariant Subspaces

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