Minimal Realization Time-Delay Koopman Analysis for Nonlinear System Identification

Minimal Realization Time-Delay Koopman Analysis for Nonlinear System Identification | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Minimal Realization Time-Delay Koopman Analysis for Nonlinear System Identification Biqi Chen, Ying Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6029043/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Data is increasingly abundant in fields such as biology, engineering, neuroscience, and epidemiology. However, developing accurate models that capture the dynamics of the underlying system while ensuring interpretability and generalizability remains a significant challenge. To address this, we propose a novel methodology called Minimal Realization Time-Delay Koopman (MRTK) analysis, which is capable of identifying the minimal degrees of freedom in linear systems and handling both full-state and sparse measurements, even in noisy environments. For full-state measurements, we demonstrate that MRTK is equivalent to the Dynamic Mode Decomposition (DMD) method. For sparse measurements, it employs time-delay embedding techniques and the Koopman operator to construct a minimal realization linear model that is diffeomorphic to the attractor of the original system, unveiling the system's physical dynamics from a differential topology perspective. We validate the proposed approach using simulated data from transitional channel flow and the Lorenz system, as well as real-world temperature and wind speed data from the Hangzhou Bay Bridge. Integrating the identified model with a Kalman filter enables accurate estimation and prediction of sparse data. The results demonstrate high predictive accuracy in both scenarios, with the maximum NMSE prediction error for the wind speed field at 1.911%, highlighting the advanced identification capacity of the method and its potential to advance prediction and control of complex systems. Koopman operator theory Dynamic mode decomposition Time delay embedding Nonlinear dynamical system Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-6029043","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":416500653,"identity":"9f345a43-88cf-4857-9e87-8cc22e3935a2","order_by":0,"name":"Biqi Chen","email":"","orcid":"","institution":"Harbin Institute of Technology, Shenzhen","correspondingAuthor":false,"prefix":"","firstName":"Biqi","middleName":"","lastName":"Chen","suffix":""},{"id":416500659,"identity":"8ecb08a5-07e5-4737-8d9f-3cc23ab65015","order_by":1,"name":"Ying Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYDACZijNL8HYgMQlRovkDMbGBuK0wIDBDQZG4rQYHOc9/Jqn5o7d5tvN7Q8YKqwTG9jPHsCv5TBfmjXPsWfJ2+4cBDrsTHpiA09eAgEtPGbGOWyHk81uJDY2MLYdTmyQ4DEgQsu/w8nGM0Ba/hGnxfhxbtthOwMJkJYGIrRIAm1h/tt3OEEC6LAZCcfSjdt4cvBr4Tt/xvjjjG+H7flnpD/48KHGWraf/Qx+LQoHGNgkgHRiA4iXAMRseNUDgXwDA/MHIG1PSOEoGAWjYBSMYAAAZIFK7GRA/2QAAAAASUVORK5CYII=","orcid":"","institution":"Harbin Institute of Technology, Shenzhen","correspondingAuthor":true,"prefix":"","firstName":"Ying","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2025-02-14 09:08:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6029043/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6029043/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":83173336,"identity":"6c5ff92f-a481-464e-906b-5c48408f120a","added_by":"auto","created_at":"2025-05-20 18:16:56","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":13007300,"visible":true,"origin":"","legend":"","description":"","filename":"MinimalRealizationTimeDelayKoopmanAnalysisforNonlinearSystemIdentification.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6029043/v1_covered_e3fce0b3-2f2e-4235-9717-6c57dccbf1cd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Minimal Realization Time-Delay Koopman Analysis for Nonlinear System Identification","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Koopman operator theory, Dynamic mode decomposition, Time delay embedding, Nonlinear dynamical system","lastPublishedDoi":"10.21203/rs.3.rs-6029043/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6029043/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eData is increasingly abundant in fields such as biology, engineering, neuroscience, and epidemiology. 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