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Physics-informed deep koopman operator for Lagrangian dynamic systems

Science China(Information Sciences) 2024年第9期67卷 146-163页

作者： Xuefeng WANG Yang CAO Shaofeng CHEN Yu KANG Department of Automation University of Science and Technology of China State Key Laboratory of Fire Science University of Science and Technology of China Institute of Advanced Technology University of Science and Technology of China

Accurate mechanical system models are crucial for safe and stable control. Unlike linear systems,Lagrangian systems are highly nonlinear and difficult to optimize because of their unknown system *** research thus used deep neural networks to generate linear models of original systems by mapping nonlinear dynamic systems into a linear space with a koopman observable function encoder. The controller then relies on the koopman linear model. However, without physical information constraints, ensuring control consistency between the original nonlinear system and the koopman system is tough, as the learning process of the koopman observation function is unsupervised. This paper thus proposes a two-stage learning algorithm that uses structural subnetworks to build a physics-informed network topology to simultaneously learn the koopman observable functions and the system energy representation. In the koopman matrix learning session, a quadratic-constrained optimization problem is solved to ensure that the koopman representation satisfies the energy difference matching hard constraint. The proposed energy-preserving deep Lagrangian koopman(EPDLK) framework effectively represents the dynamics of the Lagrangian system while ensuring control consistency. The effectiveness of EPDLK is compared with those of various koopman observable function construction methods in multistep prediction and trajectory tracking tasks. EPDLK achieves better control consistency by guaranteeing energy difference matching, which facilitates the application of the control law generated on the koopman system directly to the original nonlinear Lagrangian system.

关键词： koopman operator Lagrangian physics-informed nonlinear model predictive control (MPC)

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Deep koopman operator for Model Predictive Control

Deep Koopman Operator for Model Predictive Control

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第35届中国过程控制会议

作者： Tengfei Li Yuhong Wang College of Control Science and Engineering China University of Petroleum

The paper introduces a koopman operator-based method designed to expedite Model Predictive Control(MPC) for nonlinear systems, which are traditionally burdened by the need to solve complicated optimization problems. By leveraging the koopman operator theory, the research transforms nonlinear dynamics into a linear framework in a higher-dimensional space, even considering system inputs. A deep learning architecture is then proposed to infer the necessary koopman embeddings and operator, incorporating residual connections to refine the model's accuracy. The derived linear model underpins the development of an MPC controller, which is empirically tested on an anaerobic digestion reactor. Results confirm that the method markedly enhances computational efficiency in nonlinear MPC applications, demonstrating potential for broader implementation in complex control systems.

关键词： Nonlinear model predictive control koopman operator deep neural network

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Determination of Parameters of Time-delayed Embedding Algorithm Using koopman operator-based Model Predictive Frequency Control

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CSEE Journal of Power and Energy Systems 2021年第6期7卷 1140-1151页

作者： Xiawen Li Chetan Mishra Shichao Chen Yajun Wang Jaime De La Ree Dominion Energy RichmondVAUSA Thorlabs Quantum Electronics. JessupMDUSA Department of Electrical and Computer Engineering Virginia Polytechnic Institute and State UniversityBlacksburgVAUSA

Power systems around the world have been registering a degenerating inertial response in view of the growth of inverter-based resources along with the withdrawal of conventional coal ***,there is a need for swift frequency support and its control,preferably by means of power electronic-interfaced storage devices,owing to their beneficial *** being particularly efficient,pragmatically,the traditional model-based non-linear control techniques are not highly popular in power system control design,primarily due to the complications faced in obtaining accurately suitable models for certain power system ***,the modelfree koopman operator-based model predictive control(KMPC)has proven to be highly conducive for data-driven non-linear control *** principle behind KMPC is to change the coordinates in a manner to get an approximately linear model,which can then be controlled using a linear model predictive *** this study,we employed time-delayed embedding of measurements to reconstruct a new set of preferable coordinates,thereby suggesting an approach for finding the optimal number of time lags and the embedding dimensions which are the key parameters of this *** efficacy of this KMPC framework is established by adopting a decentralized frequency control problem through a decoupled synchronous machine system,which we proposed for both the Kundur two-area system as well as the IEEE 39-bus test system.

关键词： False nearest neighbors koopman operator model predictive control non-linear control time-delayed embedding

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Data-driven Estimation for a Region of Attraction for Transient Stability Using the koopman operator

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CSEE Journal of Power and Energy Systems 2023年第4期9卷 1405-1413页

作者： Le Zheng Xin Liu Yanhui Xu Wei Hu Chongru Liu the School of Electrical and Electronic Engineering North China Electric Power UniversityBeijing 102206China Department of Electrical Engineering Tsinghua UniversityBeijing 100084China

This paper presents a novel koopman operator based framework to estimate the region of attraction for power system transient stability *** koopman eigenfunctions are used to numerically construct a Lyapunov *** the level set of the function is utilized to estimate the boundary of the region of *** method provides a systematic method to construct the Lyapunov function with data sampled from the state space,which suits any power system models and is easy to use compared to traditional Lyapunov direct *** addition,the constructed Lyapunov function can capture the geometric properties of the region of attraction,thus providing useful information about the instability *** method has been verified by a simple illustrative example and three power system models,including a voltage source converter interfaced system to analyze the large signal synchronizing instability induced by the phase lock loop *** proposed method provides an alternative approach to understanding the geometric properties and estimating the boundary of the region of attraction of power systems in a data driven *** Terms-koopman operator,lyapunov function,power system transient stability,region of attraction.

关键词： koopman operator lyapunov function power system transient stability region of attraction.

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Data-driven prediction of temperature variations in an open cathode proton exchange membrane fuel cell stack using koopman operator

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Energy and AI 2023年第4期14卷 439-450页

作者： Da Huo Carrie M.Hall Mechanical Materialsand Aerospace Engineering DepartmentIllinois Institute of Technology10 West 35th StreetChicago60616ILUnited States

In this study,a novel application of the koopman operator for control-oriented modeling of proton exchange membrane fuel cell(PEMFC)stacks is *** primary contributions of this paper are:(1)the design of koopman-based models for a fuel cell stack,incorporating K-fold cross-validation,varying lifted dimensions,radial basis functions(RBFs),and prediction horizons;and(2)comparison of the performance of koopman-based approach with a more traditional physics-based *** results demonstrate the high accuracy of the koopman-based model in predicting fuel cell stack behavior,with an error of less than 3%.The proposed approach offers several advantages,including enhanced computational efficiency,reduced computational burden,and improved *** study demonstrates the suitability of the koopman operator for the modeling and control of PEMFCs and provides valuable insights into a novel control-oriented modeling approach that enables accurate and efficient predictions for fuel cell stacks.

关键词： Proton exchange membrane fuel cell(PEMFC) Data-driven modeling koopman operator Dynamic modeling Control-oriented modeling Physics-based modeling

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Neural network with GSINDy-koopman operators for Modeling and predictive Control of nonlinear processes

Neural network with GSINDy-Koopman Operators for Modeling an...

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第35届中国过程控制会议

作者： Zihang Xu Xiaodong Xu Central South University

In this paper, we introduce a data-driven predictive control approach leveraging koopman operator theory for addressing nonlinear dynamics within intricate systems. The methodology employs GSINDy algorithms to construct a repository of functions, which are subsequently utilized within a sparse neural network to select pertinent feature functions from processed data. This GSINDy function repository enables the transformation of the original nonlinear state space into a higher-dimensional linear space, facilitating the construction of a koopman-based linear model tailored for low-dimensional nonlinear systems. Subsequently, a highly accurate linear MPC framework is developed based on this model. Illustrating the efficacy of the proposed methodology for nonlinear system control, we present a case study involving a benchmark chemical process.

关键词： koopman operator GSINDy Sparse neural network Model predictive control Nonlinear process

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Information flow between stock markets:A koopman decomposition approach

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Chinese Physics B 2022年第1期31卷 711-719页

作者： Semba Sherehe Huiyun Wan Changgui Gu Huijie Yang Department of Systems Science University of Shanghai for Science and TechnologyShanghai 200093China Faculty of Science Dar es Salaam University College of EducationUniversity of Dar es SalaamDar es SalaamTanzania

Stock markets in the world are linked by complicated and dynamical relationships into a temporal *** works have provided us with rich findings from the topological properties and their evolutionary trajectories,but the underlying dynamical mechanism is still not in *** the present work,we proposed a technical scheme to reveal the dynamical law from the temporal *** index records for the global stock markets form a multivariate time *** separates the series into segments and calculates the information flows between the markets,resulting in a temporal market network representing the state and its *** the technique of the koopman decomposition operator is adopted to find the law stored in the information *** results show that the stock market system has a high flexibility,i.e.,it jumps easily between different *** information flows mainly from high to low volatility stock *** the dynamical process of information flow is composed of many dynamic modes distribute homogenously in a wide range of periods from one month to several ten years,but there exist only nine modes dominating the macroscopic patterns.

关键词： transfer entropy koopman operator stock markets

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GAUSSIAN WHITE NOISE CALCULUS OF GENERALIZED EXPANSION

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Acta Mathematica Scientia 2002年第3期22卷 359-368页

作者：陈泽乾 Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences Wuhan 430071 China

A new framework of Gaussian white noise calculus is established, in line with generalized expansion in [3, 4, 7]. A suitable frame of Fock expansion is presented on Gaussian generalized expansion functionals being introduced here, which provides the integral kernel operator decomposition of the second quantization of koopman operators for chaotic dynamical systems, in terms of annihilation operators partial derivative(t) and its dual, creation operators partial derivative(t)*.

关键词： Gaussian white noise generalized expansion functional Fock expansion chaotic dynamical systems koopman operator

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koopman analysis of nonlinear systems with a neural network representation

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Communications in Theoretical Physics 2022年第9期74卷 183-193页

作者： Chufan Li Yueheng Lan School of Science Beijing University of Posts and TelecommunicationsBeijing 100876China State Key Lab of Information Photonics and Optical Communications Beijing University of Posts and TelecommunicationsBeijing 100876China

The observation and study of nonlinear dynamical systems has been gaining popularity over years in different *** intrinsic complexity of their dynamics defies many existing tools based on individual orbits,while the koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits,which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the ***,it is difficult to identify and represent the most relevant eigenfunctions in ***,combined with the koopman analysis,a neural network is designed to achieve the reconstruction and evolution of complex dynamical *** invoking the error minimization,a fundamental set of koopman eigenfunctions are derived,which may reproduce the input dynamics through a nonlinear transformation provided by the neural *** corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.

关键词： deep learning autoencoder koopman operator Van der Pol equation coupled oscillator

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koopman-Based Optimal Tracking Control of Nonlinear Dynamical Systems

Koopman-Based Optimal Tracking Control of Nonlinear Dynamica...

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第40届中国控制会议

作者： Xuyang Lou Ling Ma Qian Ye School of IoT Engineering Jiangnan University Wuxi Institute of Technology

A data-driven procedure to approximate the koopman operator is a leading technology in researching the properties of nonlinear dynamical systems. The koopman operator is an infinite-dimensional linear operator that can be used to predict and control a nonlinear system in a linear manner. In this paper, the reformulation and control problems of complex nonlinear dynamical systems are explored. The extended dynamical mode decomposition method and the deep neural network method are applied, respectively, to obtain a finite approximation of the koopman operator. Then, we present koopman-based optimal tracking controller for nonlinear dynamical systems by using the linear quadratic tracking control strategy. Finally, simulation results demonstrate that the linear embedding of nonlinear systems by data-driven procedure can approximate the states of the nonlinear systems and the desired output of nonlinear systems can be optimally tracked based on the designed controller of the linear embedding system.

关键词： koopman operator data-driven control dynamical mode decomposition deep neural network linear quadratic tracking control

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