Dynamic Boundary Conditions based Optimization Approach for Efficient Model Order Reduction of Complex Systems

Dynamic Boundary Conditions based Optimization Approach for Efficient Model Order Reduction of Complex Systems | 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 Dynamic Boundary Conditions based Optimization Approach for Efficient Model Order Reduction of Complex Systems Anuj Goel, Amit Kumar Manocha This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4480682/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Feb, 2025 Read the published version in Electrical Engineering → Version 1 posted You are reading this latest preprint version Abstract This paper proposed a new methodology to address the key problem in model order reduction methods of setting random values of lower & upper bounds and initial values of the parameters in optimization-based approaches. The moth flame optimization (MFO) method is utilized for the model order reduction process wherein the search space boundaries are found using a novel strategy with the classical balanced truncation technique. Both the numerator and denominator coefficients of the desired reduced-order system are found using the proposed optimization approach. The integral square error (ISE) is employed as the objective function in the optimization of SISO systems while a novel objective function is framed using ISE for the MIMO systems. The key advantage of using balanced truncation-based search space boundaries ensures targeted search with potential solutions and stability of the reduced order model. Further, the disadvantage of steady-state error of the balanced truncation is overcome using a gain adjustment factor. The overall methodology takes very less simulation time while keeping all the necessary parameters of the reduced-system close to those of the original system. To test the efficacy of the proposed methodology, five real-world high-order systems with two SISO systems, two MIMO systems and one discrete-time system are considered and compared with existing methods through several error indices and time and frequency-domain specifications. It has been found that the proposed methodology results in significant reduction of ISE and improvement in matching of step responses, preserving stability of the reduced-order models. Model order reduction moth flame optimization balanced truncation search space boundaries gain adjustment objective functions SISO and MIMO systems continuous-time and discrete-time systems Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Full Text Additional Declarations No competing interests reported. Tables 1 to 16 are available in the Supplementary Files section. Supplementary Files Tables1.docx Cite Share Download PDF Status: Published Journal Publication published 03 Feb, 2025 Read the published version in Electrical Engineering → 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-4480682","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":309816290,"identity":"a2b361cf-2174-4c0c-adab-38411106ddd7","order_by":0,"name":"Anuj Goel","email":"","orcid":"","institution":"Maharaja Ranjit Singh Punjab Technical University","correspondingAuthor":false,"prefix":"","firstName":"Anuj","middleName":"","lastName":"Goel","suffix":""},{"id":309816291,"identity":"797c9034-f02d-4904-8612-e9d675485634","order_by":1,"name":"Amit Kumar Manocha","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA6klEQVRIiWNgGAWjYDACHgiVwMDegCSaQJQWngMg2oAULRIJSFrwAf6e04mfCyrs8vhnvk77+HPHn8QG9sMPGB7uwK1F4mzvZukZZ5KLJW7nbp7Ne8YgsYEnzYAh8Qwea87zbpDmbTuQ2ADUwszYBtTCkMPAkNiGW4f8ed7Nv0Fa5t88u5nxJ0gL/xv8WgzO9m4D27LhBu9mBl6QFgkCthieObvNmudMcuLGM0CH8Z4xNm6TeGZwAJ8WOaDK2zwVdonzjoMctkNOtp8/+eHDn3i0oALGBgYGNiB9gFgNEC2jYBSMglEwCtABAG7nVYHzxfDHAAAAAElFTkSuQmCC","orcid":"","institution":"Maharaja Ranjit Singh Punjab Technical University","correspondingAuthor":true,"prefix":"","firstName":"Amit","middleName":"Kumar","lastName":"Manocha","suffix":""}],"badges":[],"createdAt":"2024-05-26 15:44:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4480682/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4480682/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00202-024-02929-0","type":"published","date":"2025-02-04T00:00:00+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":58048314,"identity":"f2e98a2a-5921-47a2-a717-25e5c5b4c370","added_by":"auto","created_at":"2024-06-10 12:05:57","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":37552,"visible":true,"origin":"","legend":"\u003cp\u003eFlowchart showing steps involved in moth flame optimization algorithm\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/bacb3f3bfd39a892a7b701f0.jpg"},{"id":58047738,"identity":"26203956-c7be-45c1-8d96-8152665265ba","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":38751,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e(a)\u003c/strong\u003e Concept of parameter-specific search space based on its value \u003cstrong\u003e(b)\u003c/strong\u003e Scaling of search space for all parameters using scaling constant 'ε'\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/2dbd90d6ad03f20e37035313.jpg"},{"id":58047741,"identity":"c0ba51c2-ed8e-4d9a-893c-db2cbef194c7","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":62192,"visible":true,"origin":"","legend":"\u003cp\u003eFlowchart of the proposed methodology for SISO \u0026amp; MIMO systems\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/02b14cf2c473f48644b5ca2a.jpg"},{"id":58047747,"identity":"0808ed07-9e03-4048-a9f2-864d8f39e189","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":80163,"visible":true,"origin":"","legend":"\u003cp\u003e(a). Step response comparison for example 1 (b) Enlarged view of step response\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/17e07a93a78ba3e50dbffa93.jpg"},{"id":58047744,"identity":"1af296de-a2f9-4468-94c4-3a33d407e860","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":131541,"visible":true,"origin":"","legend":"\u003cp\u003eBode plot comparison for example 1\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/73c63515697863bc8b910572.jpg"},{"id":58047748,"identity":"79d4865d-430a-4eb2-8abd-f108cc56e0f0","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":126503,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e(a)\u003c/strong\u003e Step response comparison for example 2 and (b) Bode plot comparison\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/aaa3d7b6260e5b154eee3590.jpg"},{"id":58047746,"identity":"fb653e6c-c065-439b-b3a0-23496e1880b7","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":122391,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Step response comparison for example 3 and (b) Bode plot comparison\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/7f447d666731714aad9fed59.jpg"},{"id":58047745,"identity":"52051c68-9789-4394-a35f-4a7c22322626","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":219953,"visible":true,"origin":"","legend":"\u003cp\u003eStep response comparison of the 2\u003csup\u003end\u003c/sup\u003e order mdoels obtained by proposed method and Vasu \u003cem\u003eet al.\u003c/em\u003e (2020a) with the original 6\u003csup\u003eth\u003c/sup\u003e order system for example 4 (a) subsystem g\u003csub\u003e11\u003c/sub\u003e(s), (b) subsystem g\u003csub\u003e12\u003c/sub\u003e(s), (c) Subsystem g\u003csub\u003e21\u003c/sub\u003e(s) and (d) subsystem g\u003csub\u003e22\u003c/sub\u003e(s)\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/21a200d71b431ade5efd0b1b.jpg"},{"id":58048318,"identity":"0eea5944-8683-4694-b2d2-c8a5444da77c","added_by":"auto","created_at":"2024-06-10 12:05:57","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":289035,"visible":true,"origin":"","legend":"\u003cp\u003eBode plot comparison of the 2\u003csup\u003end\u003c/sup\u003e order models obtained by proposed method and Vasu \u003cem\u003eet al.\u003c/em\u003e (2020a) with the original 6\u003csup\u003eth\u003c/sup\u003e order system for example 4. (a) subsystem g\u003csub\u003e11\u003c/sub\u003e(s), (b) subsystem g\u003csub\u003e12\u003c/sub\u003e(s), (c) Subsystem g\u003csub\u003e21\u003c/sub\u003e(s) and (d) subsystem g\u003csub\u003e22\u003c/sub\u003e(s)\u003c/p\u003e","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/bbb00679c10353ca659550b7.jpg"},{"id":58047743,"identity":"c44657fc-8297-47cc-888f-439cff58395f","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":141765,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Step response comparison of the 3\u003csup\u003erd\u003c/sup\u003e order mdoels obtained by proposed method, Gupta \u0026amp; Manocha (2021) and Vasu \u003cem\u003eet al. \u003c/em\u003e(2020b) with the original 7\u003csup\u003eth\u003c/sup\u003e order system for subsystems g\u003csub\u003e11\u003c/sub\u003e(s) and g\u003csub\u003e22\u003c/sub\u003e(s) of example 5 and (b) subsystems g\u003csub\u003e12\u003c/sub\u003e(s) \u0026amp; g\u003csub\u003e21\u003c/sub\u003e(s) of example 5\u003c/p\u003e","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/ebe5d86a205b460963fde4b1.jpg"},{"id":58047749,"identity":"9b8c75fd-9c28-4aa2-8870-cf63f3c0b206","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":265002,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Bode plot comparison of the 3\u003csup\u003erd\u003c/sup\u003e order models obtained for subsystems g\u003csub\u003e11\u003c/sub\u003e(s) and g\u003csub\u003e22\u003c/sub\u003e(s) for example 5 and (b) subsystems g\u003csub\u003e12\u003c/sub\u003e(s) \u0026amp; g\u003csub\u003e21\u003c/sub\u003e(s) for example 5\u003c/p\u003e","description":"","filename":"11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/f9c414075fdd1d9d0a93f43b.jpg"},{"id":78701961,"identity":"df38354e-fe3a-43de-af81-50b86fdfcad0","added_by":"auto","created_at":"2025-03-17 19:36:25","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2017164,"visible":true,"origin":"","legend":"","description":"","filename":"ManuscriptDynamicBoundaryConditionsbasedOptimizationApproachforEfficientModelOrderReductionofComplexSystems.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1_covered_0bc55238-316d-46f3-bcb9-26e031bdaa11.pdf"},{"id":58047740,"identity":"b32bebcc-3a36-4e6b-800b-e66ac5ff62f3","added_by":"auto","created_at":"2024-06-10 11:57:57","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":38121,"visible":true,"origin":"","legend":"","description":"","filename":"Tables1.docx","url":"https://assets-eu.researchsquare.com/files/rs-4480682/v1/b03df172b7769dc915141ed6.docx"}],"financialInterests":"\u003cp\u003eNo competing interests reported.\u003c/p\u003e\n\u003cp\u003eTables 1 to 16 are available in the Supplementary Files section.\u003c/p\u003e","formattedTitle":"Dynamic Boundary Conditions based Optimization Approach for Efficient Model Order Reduction of Complex Systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"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":"Model order reduction, moth flame optimization, balanced truncation, search space boundaries, gain adjustment, objective functions, SISO and MIMO systems, continuous-time and discrete-time systems","lastPublishedDoi":"10.21203/rs.3.rs-4480682/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4480682/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper proposed a new methodology to address the key problem in model order reduction methods of setting random values of lower \u0026amp; upper bounds and initial values of the parameters in optimization-based approaches. The moth flame optimization (MFO) method is utilized for the model order reduction process wherein the search space boundaries are found using a novel strategy with the classical balanced truncation technique. Both the numerator and denominator coefficients of the desired reduced-order system are found using the proposed optimization approach. The integral square error (ISE) is employed as the objective function in the optimization of SISO systems while a novel objective function is framed using ISE for the MIMO systems. The key advantage of using balanced truncation-based search space boundaries ensures targeted search with potential solutions and stability of the reduced order model. Further, the disadvantage of steady-state error of the balanced truncation is overcome using a gain adjustment factor. The overall methodology takes very less simulation time while keeping all the necessary parameters of the reduced-system close to those of the original system. To test the efficacy of the proposed methodology, five real-world high-order systems with two SISO systems, two MIMO systems and one discrete-time system are considered and compared with existing methods through several error indices and time and frequency-domain specifications. It has been found that the proposed methodology results in significant reduction of ISE and improvement in matching of step responses, preserving stability of the reduced-order models.\u003c/p\u003e","manuscriptTitle":"Dynamic Boundary Conditions based Optimization Approach for Efficient Model Order Reduction of Complex Systems","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-06-10 11:57:52","doi":"10.21203/rs.3.rs-4480682/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"b8cf568a-7d2f-4af7-92bc-a8aa73016f10","owner":[],"postedDate":"June 10th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-03-17T19:36:18+00:00","versionOfRecord":{"articleIdentity":"rs-4480682","link":"https://doi.org/10.1007/s00202-024-02929-0","journal":{"identity":"electrical-engineering","isVorOnly":false,"title":"Electrical Engineering"},"publishedOn":"2025-02-04 00:00:00","publishedOnDateReadable":"February 4th, 2025"},"versionCreatedAt":"2024-06-10 11:57:52","video":"","vorDoi":"10.1007/s00202-024-02929-0","vorDoiUrl":"https://doi.org/10.1007/s00202-024-02929-0","workflowStages":[]},"version":"v1","identity":"rs-4480682","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4480682","identity":"rs-4480682","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}