[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84116-en":3,"doc-seo-84116-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84116,687197207057,"Sage","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Input-to-State Stability Certification via Projection Residuals for Koopman Learning Control of Nonlinear Repetitive Systems","This paper studies input-to-state stability (ISS) certification for data-driven Koopman learning control of unknown discrete-time nonlinear repetitive systems over finite trial horizons. Instead of introducing a new learning law, it certifies when a fixed Koopman-assisted constrained update yields a practical stability bound on the selected tracking error along the trial axis. Prediction accuracy alone is insufficient: a positive selected-channel margin and an accounted projection residual for unreachable output increments are required. The approach models an ISS learning-axis system, treats Koopman residuals and deployment shifts as ISS inputs, and derives a deterministic ultimate error band estimate with finite-sample implementation and numerical verification.","arXiv :2607 .06459v1 [ ee ss . SY] 7 Jul 2026  \nHighlights  \nInput-to-State Stability Certification via Projection Residuals for Koopman Learning Control of Nonlinear Repetitive Systems  \nYue Wu, Ye Cao, Jianfu Cao  \n• Develops a trial-axis input-to-state stability certificate for Koopman learning control.  \n• Separates prediction residuals from selected-channel learning certifiability.  \n• Quantifies the projection-residual contribution to the ultimate error band.  \n• Combines episode-level residual calibration with channel, projection, and shift margins.  \n• Evaluates residual scaling, weak-channel rejection, projection closure, and band coverage.  \nInput-to-State Stability Certification via Projection Residuals for Koopman Learning Control of Nonlinear  \nRepetitive Systems Yue Wua,b,∗, Ye Caoa , Jianfu Caoa  \na School of Automation, Xi’an Jiaotong University, Xi’an, 710049, China b Xinjiang Cigarette Factory, Hongyun Honghe Tobacco (Group) Co. ,  \nLtd., Urumqi, 830000, China  \nAbstract  \nThis paper studies input-to-state stability (ISS) certification for data-driven Koopman learning control of unknown discrete-time nonlinear repetitive systems over finite trial horizons. Rather than proposing a new learning law, we certify when a fixed Koopman-assisted constrained update yields a practical stability bound for the selected tracking error along the trial axis. Prediction accuracy alone is insufficient for this purpose: the selected finite-horizon input-output channel must have a positive margin, and the unreachable component of the requested output increment must be accounted for through a projection residual. Thus, a Koopman predictor with small held-out prediction residuals may still fail the learning-stability certificate if its selected channel is weak. We formulate the selected stacked tracking error as the state of a discrete-time learning-axis system and treat Koopman residuals, reset mismatch, channel uncertainty, projection residuals, deployment shifts, and numerical tolerances as ISS inputs. The deterministic result gives a practical ISS estimate from the initial learning error to an explicit ultimate band. A finite-sample implementation constructs an episode-level residual bound under a fixed controller and combines it with reported channel, projection, shift, and numerical margins. Numerical checks on nonlinear repetitive systems support the predicted residual-to-band scaling, weak-channel rejection, projection closure, and ultimate-band coverage.  \n∗ Corresponding author.  \nEmail address: [wuyue0619@stu.xjtu.edu.cn](wuyue0619@stu.xjtu.edu.cn) (Yue Wu)  \nKeywords: Input-to-state stability, practical ISS, iterative learning control, Koopman operator, data-driven control, nonlinear repetitive systems, finite-sample certificates  \n1. Introduction  \nStability is a central issue in learning control. In a repetitive task, the controller uses data from previous trials to update the input for future trials; the same mechanism that improves tracking can also amplify modelling errors, nonrepetitive disturbances, reset mismatch, sensor noise, and actuator constraints. Classical iterative learning control (ILC) therefore developed lifted-domain, two-dimensional repetitive-system, norm-optimal, robust, and monotonic-convergence analyses that clarify when tracking errors converge, when learning transients remain bounded, and how uncertainty affects final tracking performance [9–13] .  \nThe stability question addressed here is different from classical zero-error convergence or nominal monotonic convergence. In data-driven Koopman learning control, the finite-dimensional predictor is learned from data and is used through a selected finite-horizon input-output channel. A small prediction residual does not imply that this selected channel can generate the requested output correction under input-update constraints. Moreover, reset mismatch, nonrepetitive disturbances, channel-identification errors, deployment shifts, and numerical t","cbCaidT3ISoLQCtR","https://ap.wps.com/l/cbCaidT3ISoLQCtR","pdf",1366376,1,29,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What does the paper certify in Koopman learning control?\",\"answer\":\"It certifies a practical input-to-state stability bound for the selected tracking error along the trial axis under a fixed Koopman-assisted constrained update over finite trial horizons.\"},{\"question\":\"Why are small prediction residuals not enough for certification?\",\"answer\":\"Because the selected finite-horizon input-output channel must have a positive margin and the unreachable component of the requested output increment must be represented via a projection residual; otherwise learning may fail despite good prediction.\"},{\"question\":\"How is the certification derived and validated?\",\"answer\":\"The selected stacked tracking error is modeled as the state of a discrete-time learning-axis ISS system, treating Koopman residuals, reset mismatch, channel uncertainty, projection residuals, deployment shifts, and numerical tolerances as ISS inputs; finite-sample residual bounds are then combined with channel/projection/shift margins and checked numerically on nonlinear repetitive systems.\"}]",1784192973,73,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"input-to-state-stability-certification-via-projection-residuals-for-koopman-learning-control-of-nonlinear-repetitive-systems","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/input-to-state-stability-certification-via-projection-residuals-for-koopman-learning-control-of-nonlinear-repetitive-systems/84116/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the paper certify in Koopman learning control?","Question",{"text":75,"@type":76},"It certifies a practical input-to-state stability bound for the selected tracking error along the trial axis under a fixed Koopman-assisted constrained update over finite trial horizons.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are small prediction residuals not enough for certification?",{"text":80,"@type":76},"Because the selected finite-horizon input-output channel must have a positive margin and the unreachable component of the requested output increment must be represented via a projection residual; otherwise learning may fail despite good prediction.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the certification derived and validated?",{"text":84,"@type":76},"The selected stacked tracking error is modeled as the state of a discrete-time learning-axis ISS system, treating Koopman residuals, reset mismatch, channel uncertainty, projection residuals, deployment shifts, and numerical tolerances as ISS inputs; finite-sample residual bounds are then combined with channel/projection/shift margins and checked numerically on nonlinear repetitive systems.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]