[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85909-en":3,"doc-seo-85909-105":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},85909,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Tracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization","Input–output feedback linearization breaks down at decoupling singularities where the decoupling matrix loses rank, relative degree is lost, and the linearizing control becomes unbounded. A singularity-robust trajectory-tracking controller for square nonlinear control-affine systems tracks through isolated decoupling singularities with bounded control. The method reformulates tracking as real-time arc-length homotopy continuation, using the least-norm Moore–Penrose solution of an augmented matrix A=[Λ|b] under a transversality condition to maintain full row rank. The resulting flow matches feedback linearization away from the singular set, achieves O(1/k) tracking error, and re-locks after crossing, with theory covering Whitney folds and Filippov sliding behavior.","arXiv :2607 . 10436v1 [ ee ss . SY] 11 Jul 2026  \nTracking Through Decoupling Singularities: A Singularity-Robust Homotopy-Continuation Extension of Feedback Linearization  \nAlex Borisevich  \nJuly 14, 2026  \nAbstract  \nInput–output feedback linearization fails at decoupling singularities, where the decoupling matrix loses rank, the relative degree is lost, and the linearizing control becomes unbounded. This paper develops a singularity-robust trajectory-tracking controller for square nonlinear control-affine systems that tracks through isolated decoupling singularities with bounded control. The method recasts tracking as real-time arc-length homotopy continuation, equivalently a continuous-time Newton/Davidenko flow, and replaces the inverse decoupling matrix by the least-norm Moore–Penrose solution of an augmented matrix A = [Λ | b], where b is the homotopy direction. A transversality condition w Tb  0, with w in the left null space of the decoupling matrix, keeps the augmented matrix full row rank through a generic rank-one loss. The resulting flow agrees with feedback linearization away from the singular set, tracks with O(1/k) error, and re-locks after each crossing. The theory also characterizes the reflection-versusbranch-crossing dichotomy at Whitney folds and relates the reflection case to a Filippov sliding mode. Extensions cover dynamic relative-degree-one minimum-phase systems and arbitrary relative degree via filtered-error reduction. Simulations include a redundant 2-DOF manipulator, relative-degree-one and relative-degree-two plants, and a dual-active-bridge series-resonant DC/DC converter, where the method performs bounded inversion across buck/boost and resonance singularities while preserving zero-voltage soft switching.  \nKeywords: feedback linearization; loss of relative degree; decoupling matrix; decoupling singularity; nonlinear trajectory tracking; homotopy continuation; arc-length continuation; continuous-time Newton (Davidenko) method; augmented Jacobian; least-norm inverse; transversality; singularity-robust control; damped least squares; singularity-robust inverse kinematics; kinematic singularity; redundant manipulators; resolved-rate control; Type-2 parallel-mechanism singularity crossing; Filippov sliding mode; Whitney fold; monodromy; minimum phase; series resonant converter; dual-active-bridge; DC–DC converter; zero-voltage switching (ZVS); soft switching.  \n1 Introduction  \nFeedback linearization is the standard route to tracking control for nonlinear systems with a well-defined relative degree [1, 2, 3] . For a square control-affine plant of relative degree one it inverts the decoupling  \nmatrix Λ(x) = Dh(x)g (x) to cancel the nonlinearity and impose linear output dynamics. The construction is, however, intrinsically partial: the linearizing law u = Λ −1(v − a) is defined only off the decoupling singularity (equivalently, the locus of relative-degree loss) D = {detΛ(x) = 0}, and the commanded control grows without bound as the trajectory approaches D. The same obstruction appears kinematically as the manipulator-Jacobian singularity of resolved-rate control, and dynamically as the Type-2 (parallel) singularity of closed-chain mechanisms.  \nThis paper develops a controller that tracks a reference through isolated decoupling singularities with bounded effort, using one formula and no case analysis. The idea is to embed the tracking problem inan arc-length homotopy continuation and to replace the partial inverse Λ −1 by the least-norm (Moore– Penrose) solution of the augmented matrix A = [Λ | b], where b is the homotopy direction. Augmenting Λ with the extra column b repairs an isolated rank drop precisely when a transversality condition holds, and the resulting flow crosses the singularity at finite velocity while retaining the accuracy of feedback linearization elsewhere.  \nContributions.  \n1. A kernel structure lemma (Lemma 1) characterizing exactly when the augmentation repairs a rank drop, th","cbCaimHWLbS1Vwu9","https://ap.wps.com/l/cbCaimHWLbS1Vwu9","pdf",1826036,4,1,50,"English","en",105,"# Introduction\n# Related work\n# Method overview and system setup\n# Continuation construction\n## Kernel structure lemma\n## Tracking theorem\n# Reflection-versus-crossing at Whitney folds\n# Extensions and higher relative degree\n# Simulations and converter application\n# Conclusion","[{\"question\":\"Why does feedback linearization fail at decoupling singularities?\",\"answer\":\"The decoupling matrix loses rank, the relative degree is lost, and the linearizing control becomes unbounded as the trajectory approaches the singular set.\"},{\"question\":\"How does the proposed controller keep control bounded while tracking through decoupling singularities?\",\"answer\":\"It embeds trajectory tracking in an arc-length homotopy continuation and replaces the inverse decoupling matrix Λ⁻¹ with the least-norm Moore–Penrose solution of an augmented matrix A=[Λ|b], using a transversality condition to preserve full row rank through the isolated rank-one loss.\"},{\"question\":\"What distinguishes reflection from branch crossing at Whitney folds in the theory?\",\"answer\":\"The method resolves a dichotomy where the same kernel direction cannot both advance the homotopy parameter and change branch. At a generic fold, reflection corresponds to the canonical least-norm lift realized as a Filippov sliding mode, linked to a Z/2 monodromy image.\"}]",1784207105,126,{"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},"tracking-through-decoupling-singularities-a-singularity-robust-homotopy-continuation-extension-of-feedback-linearization","",{"@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/tracking-through-decoupling-singularities-a-singularity-robust-homotopy-continuation-extension-of-feedback-linearization/85909/",{"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","2026-07-25","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},"Why does feedback linearization fail at decoupling singularities?","Question",{"text":75,"@type":76},"The decoupling matrix loses rank, the relative degree is lost, and the linearizing control becomes unbounded as the trajectory approaches the singular set.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed controller keep control bounded while tracking through decoupling singularities?",{"text":80,"@type":76},"It embeds trajectory tracking in an arc-length homotopy continuation and replaces the inverse decoupling matrix Λ⁻¹ with the least-norm Moore–Penrose solution of an augmented matrix A=[Λ|b], using a transversality condition to preserve full row rank through the isolated rank-one loss.",{"name":82,"@type":73,"acceptedAnswer":83},"What distinguishes reflection from branch crossing at Whitney folds in the theory?",{"text":84,"@type":76},"The method resolves a dichotomy where the same kernel direction cannot both advance the homotopy parameter and change branch. At a generic fold, reflection corresponds to the canonical least-norm lift realized as a Filippov sliding mode, linked to a Z/2 monodromy image.","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},[93,97,101,105,110,114,119,122,127,130,134],{"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":22,"slug":113},6,"Technology","technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"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"]