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Consensus is reframed as stabilization of the agreement set, using a geometric approach based on convex-hull invariance and strict interiority. A key contribution shows terminal inclusion guarantees hull invariance but does not generally imply the strictness needed for convergence. Lexicographic tie-breaking selects interiority-maximizing MPC trajectories, restoring strictness whenever an interior feasible terminal state exists, with explicit feasibility horizon conditions for single- and double-integrator agents.",{"@graph":63,"@context":118},[64,80,101],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/distributed-multi-step-model-predictive-control-for-consensus/307655/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/distributed-multi-step-model-predictive-control-for-consensus/307655.png","ImageObject",442,249,{"name":88,"@type":89},"Olivia Brown","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-31","2026-09-19",true,{"@type":98,"interactionType":99,"userInteractionCount":15},"InteractionCounter",{"@type":100},"ViewAction",{"@type":102,"mainEntity":103},"FAQPage",[104,110,114],{"name":105,"@type":106,"acceptedAnswer":107},"How is consensus characterized in this distributed MPC framework?","Question",{"text":108,"@type":109},"Consensus is recast as stabilization of the agreement set. The analysis uses geometric conditions tied to convex-hull invariance and strict interiority properties of the induced update map.","Answer",{"name":111,"@type":106,"acceptedAnswer":112},"Why does enforcing terminal inclusion not always guarantee convergence?",{"text":113,"@type":109},"Terminal inclusion in local neighbor convex hulls ensures hull invariance, but it does not generally imply the strict relative-interior condition required for convergence.",{"name":115,"@type":106,"acceptedAnswer":116},"What does the lexicographic tie-breaking mechanism do?",{"text":117,"@type":109},"Among optimal (or near-optimal) MPC solutions, the secondary lexicographic criterion selects trajectories that maximize an interiority measure relative to the neighbor hull, ensuring the strictness condition whenever an interior feasible terminal state exists.","https://schema.org",{"og:url":78,"og:type":120,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":122,"canonical":78},"index,follow",{"doc_id":124,"site_id":56},307655,1791722452,{"code":4,"msg":5,"data":127},{"doc_id":124,"user_id":128,"nickname":88,"user_avatar":129,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":15,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":135,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":20},16904993612988,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Distributed Multi-Step Model Predictive Control for Consensus  \nNavid Noroozi  \narXiv :2602 . 14714v2 [math .OC] 17 Feb 2026  \nAbstract  \nThis paper studies consensus of discrete-time multi-agent systems under time-varying directed communication, state and input constraints using a distributed multi-step model predictive control (MPC) framework. Consensus is recast as stabilization of the agreement set, and a geometric viewpoint based on convex-hull invariance and strict interiority is adopted. Building on an existing geometric necessary and sufficient condition for agreement, we show that enforcing terminal inclusion in local neighbor convex hulls guarantees hull invariance but does not, in general, imply the strict relative-interior property required for convergence. An explicit counterexample demonstrates that strictness cannot be deduced from feasibility and contraction constraints alone.  \nTo resolve this issue without shrinking feasible sets or altering primary performance objectives, alexicographic tie-breaking mechanism is introduced. Among optimal (or near-optimal) MPC solutions, the proposed secondary criterion selects trajectories maximizing an interiority measure with respect to the neighbor hull. It is shown that whenever an interior feasible terminal state exists, this selection rule enforces the strictness condition required for asymptotic consensus. Explicit horizon conditions are derived for single- and double-integrator agents with bounded inputs, ensuring feasibility and automatic existence of interior feasible terminal points. The resulting scheme provides a distributed and implementable route to consensus via finite-step set-Lyapunov contraction. Numerical simulations with distributed inter-process communication illustrate monotone diameter decay and report per-agent computational complexity.  \nCONTENTS  \nI Introduction 3  \nI-A Motivation and gap .............................. 4  \nN. Noroozi is with SIGNON Deutschland GmbH, R¨udesheimer Str. 7, 80686 Munich, Germany (e-mail: [navid.n.noroozi@deutschebahn.com](navid.n.noroozi@deutschebahn.com)) .  \nFebruary 18, 2026 DRAFT  \nI-B Contributions .................................. 4  \nI-C Relation to prior DMPC consensus work .................. 5  \nI-D Organization .................................. 6  \nII Preliminaries 6  \nIII Problem Formulation 7  \nIII-A Graph theory and connectivity ........................ 7  \nIII-B Finite-step control Lyapunov function and barycenter tracking ....... 8  \nIII-C Consensus problem and its reformulation as set stabilization ........ 9  \nIV Necessary and Sufficient Geometric Conditions for Consensus 10  \nV Consensus via Distributed Multi-Step MPC 13  \nV-A Consensus under OCP (12) via enforced finite-step barycenter contraction 15  \nVI Impossibility of consensus under the current OCP formulation 15  \nVII DMPC with Lexicographic Selection 17  \nVII-A When OCP (12) does not automatically yield RI strictness ......... 18  \nVII-B Lexicographic (two-stage) selection among optimal solutions........ 19  \nVII-C Consensus theorem under OCP (12) with lexicographic RI selection ... 20  \nVII-D Distributed multi-step MPC algorithm with lexicographic selection .... 21  \nVIII Application to Agents with Linear Dynamics 22  \nVIII-A Explicit special cases: single- and double-integrators ............ 25  \nIX Numerical Simulations 28  \nIX-A Simulation setup ................................ 28  \nIX-B Single-integrator case ............................. 29  \nIX-C Double-integrator case ............................. 30  \nIX-D Discussion ................................... 31  \nIX-E Computational complexity and implementation remarks .......... 32  \nX Acknowledgment 34  \nFebruary 18, 2026 DRAFT  \nReferences 34  \nI. INTRODUCTION  \nConsensus and agreement in networked multi-agent systems are fundamental in distributed estimation, cooperative robotics, and large-scale cyber–physical systems. A classical geometric viewpoint interprets consensus as set stabi","cbCaihZ2qJf9ce5O","https://ap.wps.com/l/cbCaihZ2qJf9ce5O","pdf",2073598,39,"English","# I Introduction\n## I-A Motivation and gap\n## I-B Contributions\n## II Preliminaries\n## III Problem Formulation\n## III-A Graph theory and connectivity\n## IV Necessary and Sufficient Geometric Conditions for Consensus\n## V Consensus via Distributed Multi-Step MPC","[{\"question\":\"How is consensus characterized in this distributed MPC framework?\",\"answer\":\"Consensus is recast as stabilization of the agreement set. The analysis uses geometric conditions tied to convex-hull invariance and strict interiority properties of the induced update map.\"},{\"question\":\"Why does enforcing terminal inclusion not always guarantee convergence?\",\"answer\":\"Terminal inclusion in local neighbor convex hulls ensures hull invariance, but it does not generally imply the strict relative-interior condition required for convergence.\"},{\"question\":\"What does the lexicographic tie-breaking mechanism do?\",\"answer\":\"Among optimal (or near-optimal) MPC solutions, the secondary lexicographic criterion selects trajectories that maximize an interiority measure relative to the neighbor hull, ensuring the strictness condition whenever an interior feasible terminal state exists.\"}]","Distributed Multi-Step Model Predictive Control for Consensus | PDF",1789852603]