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

Alexandre Mauroy, Jorge Goncalves
2016 arXiv   pre-print
We exploit the key idea that nonlinear system identification is equivalent to linear identification of the socalled Koopman operator.  ...  Instead of considering nonlinear system identification in the state space, we obtain a novel linear identification technique by recasting the problem in the infinite-dimensional space of observables.  ...  The authors acknowledge support from the Luxembourg National Research Fund (FNR 14/BM/8231540).  ... 
arXiv:1605.04457v2 fatcat:6s2lbaodardmjfywqek47skj7e

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

Alexandre Mauroy, Jorge Goncalves
2016 2016 IEEE 55th Conference on Decision and Control (CDC)  
We exploit the key idea that nonlinear system identification is equivalent to linear identification of the socalled Koopman operator.  ...  Instead of considering nonlinear system identification in the state space, we obtain a novel linear identification technique by recasting the problem in the infinite-dimensional space of observables.  ...  The authors acknowledge support from the Luxembourg National Research Fund (FNR 14/BM/8231540).  ... 
doi:10.1109/cdc.2016.7799269 dblp:conf/cdc/MauroyG16 fatcat:dlies2nt5vbsplrbdxbhokempi

Nonlinear System Identification of Soft Robot Dynamics Using Koopman Operator Theory [article]

Daniel Bruder, C. David Remy, Ram Vasudevan
2019 arXiv   pre-print
This paper implements and evaluates a system identification method based on Koopman operator theory in which models of nonlinear dynamical systems are constructed via linear regression of observed data  ...  by exploiting the fact that every nonlinear system has a linear representation in the infinite-dimensional space of real-valued functions called observables.  ...  By providing a linear representation of a system with a one-to-one correspondence to its nonlinear representation, Koopman operator theory enables linear system identification of nonlinear systems.  ... 
arXiv:1810.06637v2 fatcat:7tzppaiysfapxjycb3nf2byoea

Deep Identification of Nonlinear Systems in Koopman Form [article]

Lucian Cristian Iacob, Gerben Izaak Beintema, Maarten Schoukens, Roland Tóth
2021 arXiv   pre-print
The present paper treats the identification of nonlinear dynamical systems using Koopman-based deep state-space encoders.  ...  The encoder represents the lifting function to the space where the dynamics are linearly propagated using the Koopman operator.  ...  One such embedding technique is based on the Koopman framework, where the concept is to lift the nonlinear state space to a (possibly) infinite dimensional space through socalled observable functions.  ... 
arXiv:2110.02583v1 fatcat:yuj2tucfknbwvhezvodpnmyeka

Koopman-based lifting techniques for nonlinear systems identification [article]

Alexandre Mauroy, Jorge Goncalves
2019 arXiv   pre-print
We develop a novel lifting technique for nonlinear system identification based on the framework of the Koopman operator.  ...  The key idea is to identify the linear (infinitedimensional) Koopman operator in the lifted space of observables, instead of identifying the nonlinear system in the state space, a process which results  ...  This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science  ... 
arXiv:1709.02003v4 fatcat:xurw52i2x5crtlaimsrbiwyufu

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control

Milan Korda, Igor Mezić
2018 Automatica  
In this work, we extend the Koopman operator to controlled dynamical systems and compute a finite-dimensional approximation of the operator in such a way that this approximation has the form a linear controlled  ...  In an uncontrolled setting, this procedure amounts to a numerical approximation of the Koopman operator associated to the nonlinear dynamics.  ...  , then these must best estimated from the available data, either using one of the nonlinear estimation techniques or using the Koopman operator-based estimator proposed in [20, 19] .  ... 
doi:10.1016/j.automatica.2018.03.046 fatcat:rcvdnla225g5zpfc4nie4graam

Advantages of Bilinear Koopman Realizations for the Modeling and Control of Systems with Unknown Dynamics [article]

Daniel Bruder, Xun Fu, Ram Vasudevan
2020 arXiv   pre-print
Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator.  ...  To demonstrate the advantages of bilinear Koopman realizations for control, a linear, bilinear, and nonlinear Koopman model realization of a simulated robot arm are constructed from data.  ...  One such technique, based on Koopman operator theory, achieves this by lifting the system into a higherdimensional space of scalar-valued functions called observables.  ... 
arXiv:2010.09961v3 fatcat:xxdupz4n3vfjphi3kehimmxbfm

Koopman operator framework for spectral analysis and identification of infinite-dimensional systems [article]

Alexandre Mauroy
2021 arXiv   pre-print
We consider Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals.  ...  This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results.  ...  We have also developed a novel identification method, which allows to identify nonlinear PDEs although it relies solely on linear techniques.  ... 
arXiv:2103.12458v3 fatcat:ljs5a2vklrblnppjfcsycdimgi

System Norm Regularization Methods for Koopman Operator Approximation [article]

Steven Dahdah, James Richard Forbes
2022 arXiv   pre-print
In particular, the H-infinity norm is used as a regularizer to penalize the input-output gain of the linear system defined by the Koopman operator.  ...  Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered.  ...  Koopman operator theory [18, 26, 6, 25] allows a nonlinear system to be exactly represented as a linear system in terms of an infinite set of lifting functions  ... 
arXiv:2110.09658v2 fatcat:bk3dvlxtajh5vnwtrxjv3uuj6e

Koopman Operator Framework for Spectral Analysis and Identification of Infinite-Dimensional Systems

Alexandre Mauroy
2021 Mathematics  
We consider the Koopman operator theory in the context of nonlinear infinite-dimensional systems, where the operator is defined over a space of bounded continuous functionals.  ...  This approach yields a linear method for nonlinear PDE identification, which is complemented with theoretical convergence results.  ...  Data Availability Statement: Data is contained within the article. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/math9192495 fatcat:hvuofmc2jnf3pbtd3n3ztujmfq

Data-driven Identification of Nonlinear Power System Dynamics Using Output-only Measurements [article]

Pranav Sharma, Venkataramana Ajjarapu, Umesh Vaidya
2021 arXiv   pre-print
The ESI method is suitable for system identification, capturing nonlinear modes, computing participation factor of output measurements in system modes and identifying system parameters such as system inertia  ...  In this paper, we propose a novel approach for the data-driven characterization of power system dynamics.  ...  For such a system, we can define a linear infinite dimension Koopman operator K, such that [22] , [23] , [28] : KΨ(x) := Ψ • F (x). (6) Koopman operator lifts a general nonlinear system into an infinite  ... 
arXiv:2110.01469v1 fatcat:iajjgleopvflhf7hfgbrldu2oe

Data Driven Online Learning of Power System Dynamics [article]

Subhrajit Sinha, Sai Pushpak Nandanoori, Enoch Yeung
2020 arXiv   pre-print
In this work, a systematic approach based on data-driven operator theoretic methods involving Koopman operator is proposed for the online identification of power system dynamics.  ...  The efficiency of the proposed algorithm is illustrated on an IEEE 9 bus system using synthetic data from the nonlinear model and on IEEE 39 bus system using synthetic data from the linearized model.  ...  The linearity of the operator facilitates the use of linear control techniques for the analysis and control of nonlinear systems.  ... 
arXiv:2003.05068v1 fatcat:oxz7qjl3u5htfjmlspbmp34gqa

Koopman Operators for Generalized Persistence of Excitation Conditions for Nonlinear Systems [article]

Nibodh Boddupalli, Aqib Hasnain, Sai Pushpak Nandanoori, Enoch Yeung
2019 arXiv   pre-print
We use the input-Koopman operator method to model nonlinear systems and derive identifiability conditions for open-loop systems initialized from a single initial condition.  ...  We show that nonlinear identifiability is intrinsically tied to the rank of a given dataset's power spectral density, transformed through the lifted Koopman observable space.  ...  The authors would also like to thank Igor Mezic  ... 
arXiv:1906.10274v2 fatcat:qm3re76ahjerpiv3kkw6fq7i2y

Data-driven Identification and Prediction of Power System Dynamics Using Linear Operators

Pranav Sharma, Bowen Huang, Venkatramana Ajjarapu, Umesh Vaidya
2019 2019 IEEE Power & Energy Society General Meeting (PESGM)  
In this paper, we propose linear operator theoretic framework involving Koopman operator for the datadriven identification of power system dynamics.  ...  We explicitly account for noise in the time series measurement data and propose robust approach for data-driven approximation of Koopman operator for the identification of nonlinear power system dynamics  ...  In this work, we proposed a robust algorithm for approximation of Koopman operator for the identification of nonlinear power system dynamics.  ... 
doi:10.1109/pesgm40551.2019.8973724 fatcat:6dcznx6mcbcxjk5ywrgb2u7eey

Data-driven Identification and Prediction of Power System Dynamics Using Linear Operators [article]

Pranav Sharma, Bowen Huang, Umesh Vaidya, Venkatramana Ajjarapu
2019 arXiv   pre-print
In this paper, we propose linear operator theoretic framework involving Koopman operator for the data-driven identification of power system dynamics.  ...  We explicitly account for noise in the time series measurement data and propose robust approach for data-driven approximation of Koopman operator for the identification of nonlinear power system dynamics  ...  In this work, we proposed a robust algorithm for approximation of Koopman operator for the identification of nonlinear power system dynamics.  ... 
arXiv:1903.06828v1 fatcat:g5go3caom5hdjl7joddid47tri
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