[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83003-en":3,"doc-seo-83003-105":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},83003,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Commutator-Driven Stability Bounds for Periodic Switching","Averaged models are widely used for analyzing periodically switched linear systems, but averaged-flow stability does not automatically transfer to the true switched dynamics. The gap is driven by noncommutativity among subsystem generators, requiring stability certificates that quantify this dependence in a Lyapunov-compatible contraction metric. The work derives an explicit operator-norm bound for the one-period mismatch between switched and averaged propagators. The leading-order error depends on pairwise commutator norms, yielding a computable switching-period threshold for uniform exponential stability.","Commutator-Driven Stability Bounds for Periodic Switching  \nDebanjan Mallik1 , Nikhil Chopra 1  \narXiv :2607 .05829v1 [math .OC] 7 Jul 2026  \nAbstract—Averaged models are widely used to analyze periodically switched linear systems, yet the stability of the averaged flow does not automatically guarantee the stability of the true switched dynamics. The discrepancy arises from noncommutativity among the subsystem generators, so stability certificates benefit from bounds that expose this dependence in a form compatible with Lyapunov contraction metrics. We derive an explicit operator-norm bound for the one-period mismatch between the switched and averaged propagators, in which the leading-order error depends explicitly on the pairwise commutator norms of the scaled mode generators, with a closedform prefactor depending only on the generator norms. This bound yields a computable threshold for the switching period below which the switched system inherits exponential stability from its averaged model, uniformly certified over admissible duty fractions. The analysis extends to an arbitrary number of switching modes via telescoping induction, and a semidefinite program provides sampled duty-dependent Lyapunov metrics for implementing the certificate.  \nI. INTRODUCTION  \nPeriodically switched linear systems admit two complementary descriptions: the one-period state-transition (monodromy) matrix, given by a product of exponentials, and the averaged flow, determined by convex combinations of the subsystem matrices. We consider the switched linear system  \nx˙(t) = Aσ (t)x (t), x (t) ∈ Rn ,  \nunder periodic switching. In this letter, the switching signal is restricted to a fixed cyclic schedule: over each period τ , the modes are visited in the prescribed order 1 ,..., m, and mode i is active for time αi τ, where  \nm  \nα ∈ ∆m−1 := {α ∈ R0 : Xαi = 1} .  \ni=1  \nThus, cyclic switching is the structured periodic subclass considered here. For fast switching, the averaged model often captures the dominant stability behavior, but the inheritance of stability by the true system is obstructed by noncommutativity among the modes. This motivates the search for explicit bounds that quantify the conditions under which averaged contraction dominates the splitting error. In this work, we  \n1The authors are with Institute for Systems Research, University of Maryland, College Park, MD 20742, USA. (e-mail: [dmallik@umd.edu](dmallik@umd.edu) ; [nchopra@umd.edu](nchopra@umd.edu)).  \nThis article has been accepted for publication in IEEE Control Systems Letters. DOI: 10 . 1109/LCSYS.2026.3708744. © 2026 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.  \nderive such bounds and obtain uniform exponential stability certificates for admissible duty-cycle sets.  \nStability of periodically switched linear systems has long been studied through direct analysis of finite products of exponentials [1] and Floquet-theoretic characterizations of the monodromy [2] . Related existence results show that ifa stable convex combination of subsystem matrices exists, then a periodic switching law can be constructed to asymptotically stabilize the overall system [3] . In the fast-switching regime, Porfiri, Roberson and Stilwell [4] combine classical averaging methods [5] with exponential splitting to derive computable switching-period bounds that expose the role of commutators. Averaging-based stability results without commutativity assumptions have also been developed in related settings, including switched-DAE formulations [6], [7] . Fora broad survey of stability criteria for switched and hybrid systems, including common Lyapunov functions, dwell-time constr","cbCaic2DN7609F5T","https://ap.wps.com/l/cbCaic2DN7609F5T","pdf",303969,2,1,6,"English","en",105,"# Introduction\n## Periodically switched linear systems: monodromy vs averaged flow\n## Cyclic periodic switching schedule and duty fractions\n## Motivation: noncommutativity and computable bounds\n## Prior work and commutator-based stability criteria","[{\"question\":\"Why doesn’t stability of the averaged model guarantee stability of the true switched system?\",\"answer\":\"Because subsystem generators generally do not commute, noncommutativity creates a mismatch between the switched propagator and the averaged flow. The discrepancy appears as an error that must be explicitly bounded to certify stability.\"},{\"question\":\"What main result is derived for periodic cyclic switching?\",\"answer\":\"An explicit operator-norm bound on the one-period mismatch between the switched and averaged propagators. The leading-order term depends on pairwise commutator norms of scaled mode generators, with a closed-form prefactor determined by generator norms.\"},{\"question\":\"How does the bound translate into a stability guarantee for the switched system?\",\"answer\":\"The derived estimate provides a computable switching-period threshold such that, for sufficiently fast switching, the switched system inherits exponential stability from its averaged model. The certificate is uniform over admissible duty fractions.\"}]",1784184597,15,{"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},"commutator-driven-stability-bounds-for-periodic-switching","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/commutator-driven-stability-bounds-for-periodic-switching/83003/",4,{"url":51,"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","2026-07-23","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 doesn’t stability of the averaged model guarantee stability of the true switched system?","Question",{"text":75,"@type":76},"Because subsystem generators generally do not commute, noncommutativity creates a mismatch between the switched propagator and the averaged flow. The discrepancy appears as an error that must be explicitly bounded to certify stability.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What main result is derived for periodic cyclic switching?",{"text":80,"@type":76},"An explicit operator-norm bound on the one-period mismatch between the switched and averaged propagators. The leading-order term depends on pairwise commutator norms of scaled mode generators, with a closed-form prefactor determined by generator norms.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the bound translate into a stability guarantee for the switched system?",{"text":84,"@type":76},"The derived estimate provides a computable switching-period threshold such that, for sufficiently fast switching, the switched system inherits exponential stability from its averaged model. The certificate is uniform over admissible duty fractions.","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":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":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"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":22,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},"Technology",50,"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"]