[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84203-en":3,"doc-seo-84203-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84203,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Implicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum","Estimating the region of attraction (RoA) of nonlinear systems is central to closed-loop stability assessment and safe controller operation. Lyapunov-certified methods provide formal guarantees but often produce conservative inner approximations. This work combines a certified Lyapunov-based positively invariant set with an implicit predecessor-based representation to compute a substantially less conservative RoA approximation. Robust positive invariance is further studied under bounded additive input disturbances. The approach is demonstrated on a nonlinear two-wheeled inverted pendulum using a saturated linear quadratic regulator, validated via Monte Carlo simulations and hardware experiments.","arXiv :2607 .07231v1 [ ee ss . SY] 8 Jul 2026  \nImplicit Predecessor-Based Region of Attraction Estimation and Robust Invariance Analysis for a Two-Wheeled Inverted Pendulum  \nLorenzo Fici 1 a and Guillaume Ducard 1 b *  \n1 Université Côte d‘Azur, i3S/CNRS, Sophia Antipolis, France  \nfici@i3s.unice.fr, ducard@i3s.unice.fr  \nKeywords:  \nRegion of Attraction, Robust Positive Invariance, Lyapunov Stability, Nonlinear Systems, Two-Wheeled Inverted Pendulum, Linear Quadratic Regulator.  \nAbstract:  \nEstimating the region of attraction (RoA) of nonlinear systems is fundamental for assessing closedloop stability and ensuring safe operation. While Lyapunov-based approaches provide certified stability guarantees, they often yield conservative inner approximations of the RoA. This paper combines a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to compute a significantly less conservative inner approximation of the RoA while preserving formal stability guarantees. In addition, the robust positive invariance of the initial certified Lyapunov-based invariant set is analyzed under bounded additive input disturbances, providing formal robustness guarantees. The proposed methodology is demonstrated on a nonlinear two-wheeled inverted pendulum stabilized by a saturated linear quadratic regulator. The resulting RoA approximation is compared with the initial Lyapunov-certified invariant set and validated through Monte Carlo simulations and hardware experiments, showing a substantially enlarged certified operating region that matches the empirical closed-loop behavior. These results demonstrate the practical applicability of combining certified Lyapunov analysis with predecessorbased set propagation for RoA approximation and robustness assessment of nonlinear systems.  \n1 INTRODUCTION  \nThe Region of Attraction (RoA) is a fundamental concept in nonlinear control, defined as theset of all initial conditions from which the closedloop system asymptotically converges to a desired equilibrium (Khalil, 2002) . As such, the RoA provides a measure of closed-loop stability, characterizing the range of operating conditions for which a controller can successfully recover the system. Despite its importance, computing the exact RoA of nonlinear systems is generally intractable, and practical approaches typically rely on either conservative analytical  \na [https://orcid.org/0009-0001-9450-1441](https://orcid.org/0009-0001-9450-1441)  \nb [https://orcid.org/0000-0002-7400-4915](https://orcid.org/0000-0002-7400-4915)  \n*This work was supported by the French government through the France 2030 investment plan managed by the National Research Agency (ANR), as part of the Initiative of Excellence Université Côted’Azur under reference number ANR-15-IDEX-01 .  \napproximations or computationally intensive numerical methods (Khalil, 2002; Gross et al., 2022; Tellez and Collado, 2013; Horibe et al., 2018) .  \nTwo-Wheeled Inverted Pendulums (TWIPs) constitute a representative class of underactuated nonlinear systems for which RoA analysis is particularly relevant. Due to their inherently unstable dynamics and actuator limitations, the ability of a controller to recover the upright equilibrium depends strongly on the initial condition of the system (Zhang and Mohamad Nor, 2025) . Consequently, the RoA provides a meaningful metric for assessing the closed-loop performance of balancing controllers and for quantifying the range of initial conditions from which balance can be successfully restored.  \nA variety of approaches have been proposed to approximate the RoA of nonlinear systems. Lyapunov-based methods provide certified inner approximations with formal stability guarantees but are often conservative, whereas numerical and  \nsampling-based techniques generally yield larger approximations at the expense of losing rigorous certification. Predecessor-set methods offer a complementary perspective by iteratively propagating i","cbCaimt3P0lNqVo2","https://ap.wps.com/l/cbCaimt3P0lNqVo2","pdf",360592,4,1,9,"English","en",105,"# Introduction\n## Related work","[{\"question\":\"What is the main goal of the paper?\",\"answer\":\"To estimate a less conservative region of attraction for nonlinear systems while preserving formal stability guarantees, and to analyze robustness of the certified invariant set under bounded additive input disturbances.\"},{\"question\":\"How does the proposed method improve on standard Lyapunov-based RoA estimation?\",\"answer\":\"It combines a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to enlarge the inner approximation of the RoA, reducing conservatism.\"},{\"question\":\"On what system is the methodology demonstrated and how is it validated?\",\"answer\":\"A nonlinear two-wheeled inverted pendulum stabilized by a saturated linear quadratic regulator. Results are compared to the initial Lyapunov-certified invariant set and validated using Monte Carlo simulations and hardware experiments.\"}]",1784193918,23,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"implicit-predecessor-based-region-of-attraction-estimation-and-robust-invariance-analysis-for-a-two-wheeled-inverted-pendulum","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/implicit-predecessor-based-region-of-attraction-estimation-and-robust-invariance-analysis-for-a-two-wheeled-inverted-pendulum/84203/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","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 is the main goal of the paper?","Question",{"text":75,"@type":76},"To estimate a less conservative region of attraction for nonlinear systems while preserving formal stability guarantees, and to analyze robustness of the certified invariant set under bounded additive input disturbances.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method improve on standard Lyapunov-based RoA estimation?",{"text":80,"@type":76},"It combines a certified Lyapunov-based positively invariant set with a predecessor-based implicit representation to enlarge the inner approximation of the RoA, reducing conservatism.",{"name":82,"@type":73,"acceptedAnswer":83},"On what system is the methodology demonstrated and how is it validated?",{"text":84,"@type":76},"A nonlinear two-wheeled inverted pendulum stabilized by a saturated linear quadratic regulator. Results are compared to the initial Lyapunov-certified invariant set and validated using Monte Carlo simulations and hardware experiments.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]