PID Stabilization of Nonlinear Systems: Integrating Stabilizing Sets, Lyapunov Analysis, and Gain Scheduling

PID Stabilization of Nonlinear Systems: Integrating Stabilizing Sets, Lyapunov Analysis, and Gain Scheduling | 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 PID Stabilization of Nonlinear Systems: Integrating Stabilizing Sets, Lyapunov Analysis, and Gain Scheduling Yun-Chen Lee This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7661582/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 This research investigates the use of PID controllers to stabilize nonlinear systems by combining stabilizing set analysis with Lyapunov-based verification techniques. The proposed method begins with linearizing the nonlinear system around equilibrium points, computing the PID stabilizing set, and verifying closed-loop stability via Lyapunov functions. For systems with multiple operating points, a gain-scheduled PID controller is constructed, and a common Lyapunov function is introduced to assess the possibility of global stability. Choose an appropriate weighting function to establish a composite Lyapunov function, which ensures the stability among all operating points. Case studies on the Van der Pol oscillator and the pendulum equation demonstrate the approach, showing consistent regulation, improved transient performance, and robustness to moderate disturbances. MATLAB simulations illustrate that appropriate PID gain scheduling, combined with Lyapunov verification, can extend local linear designs to wider nonlinear scenarios. This framework combines traditional PID design with modern nonlinear stability analysis and offers a systematic strategy to extend local linear controller performance to broader nonlinear conditions. PID control Nonlinear systems Lyapunov stability Gain scheduling 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-7661582","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":520167613,"identity":"bbc0f4bc-19a0-4500-a645-de356703892d","order_by":0,"name":"Yun-Chen Lee","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAw0lEQVRIiWNgGAWjYBACNhDxgIFBDspnJlJLAgODMQ/RWhigWhJ7iNbCx3/G8ENChU36fvbTaRIMFdaJDQQdJpFjLJFwJi23hyd3mwTDmXRitPCYMSS2Hc7tYQBqYWw7TIQW/jNALf/+p/PwvwVq+UeMFoYcoJaGAwk8EiBbGojRIpFWLJFwLNmw58bbzRYJx9KNCWqR7z+88cOHGjt59v7cjTc+1FjLEtTCwMBhgGAnEFYOAuwPiFM3CkbBKBgFIxcAAH7ROT2iEFNTAAAAAElFTkSuQmCC","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Yun-Chen","middleName":"","lastName":"Lee","suffix":""}],"badges":[],"createdAt":"2025-09-20 00:53:05","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7661582/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7661582/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":103617787,"identity":"0fbc1ebf-3871-46ee-bf47-a17427ff08f5","added_by":"auto","created_at":"2026-02-27 17:11:07","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":801050,"visible":true,"origin":"","legend":"","description":"","filename":"PIDStabilizationofNonlinearSystems.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7661582/v1_covered_23eb738f-6ca7-4b63-a25d-77d2a64f8d7b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"PID Stabilization of Nonlinear Systems: Integrating Stabilizing Sets, Lyapunov Analysis, and Gain Scheduling","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":"PID control, Nonlinear systems, Lyapunov stability, Gain scheduling","lastPublishedDoi":"10.21203/rs.3.rs-7661582/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7661582/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This research investigates the use of PID controllers to stabilize nonlinear systems by combining stabilizing set analysis with Lyapunov-based verification techniques. 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