[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83824-en":3,"doc-seo-83824-105":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},83824,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","On the Modulating Function Method for Control Problems","The modulating function method is an algebraic framework previously used for state and parameter estimation and fault detection in linear, fractional-order, distributed, and some nonlinear systems. This work introduces dual modulating functions via auxiliary systems and duality, showing the framework also supports controller design for LTV systems. Necessary and sufficient existence conditions for the resulting control laws are provided. Known controllers appear as special cases, and a new fixed-time law is proposed with transient behavior estimates, validated by simulations benchmarked against LQR under unmatched disturbances.","arXiv :2607 .04544v1 [ ee ss . SY] 5 Jul 2026  \nOn the Modulating Function Method for Control Problems  \nDavi G. Accioli a , Jerome Jouffroy a  \naSDU Mechatronics, University of Southern Denmark (SDU), Alsion 2, 6400, Sønderborg, Denmark  \nAbstract  \nThe modulating function method is an algebraic framework that, thus far, has been used for state and parameter estimation, as well as fault detection, of linear, fractional-order, distributed, and some nonlinear systems. At the core of the method lies the modulating function, which can either be selected directly or be obtained as a solution to an auxiliary system. By introducing the notion of dual modulating functions and dual modulations using auxiliary systems and duality, this paper shows that this framework is not only an estimation framework, but also a controller design framework for LTV systems. In particular, necessary and sufficient conditions for the existence of the associated control laws are introduced; the well-known state feedback law is obtained as a particular case of the dual modulation approach, along with output feedback, LTI sliding mode control, the reachability gramian, and the state-transition matrix; and a new fixed-time control law is proposed for both LTI and LTV systems, including an estimate of the transient behavior. Moreover, numerical simulations of the newly proposed control law are performed, indicating similar performance levels to a benchmark LQR even when handling unmatched disturbances.  \nKey words: asymptotic stabilization; fixed-time control; modulating function method; time-varying systems.  \n1 Introduction  \nControl theory developments are historically linked to integral transform methods, in particular because they allow for rephrasing differential equations into algebraic ones. Although this is well-known for linear control problems due to the ubiquitous Laplace transform and PID control (Astrom & Hagglund, 1995), integral transform methods also play a crucial role in different estimation tasks for linear and even some nonlinear systems.  \nThe modulating function method (MFM) is an algebraic estimation framework initially proposed in M. Shinbrot, 1954 as a method for parameter estimation tasks. Since then, it has been extended to state estimation and fault detection of linear and some nonlinear systems (D. Liu, T. Laleg-Kirati, Perruquetti, & Gibaru, 2014; Jouffroy & J. Reger, 2015; F. Fischer, Todorovski,& J. Deutscher, 2021; Matti Noack, Rueda-Escobedo, Johann Reger, & Moreno, 2016), as well as different estimation tasks for PDEs, fractional-order, and distributed systems (Friedrich, Fokken, Matti Noack, & Johann Reger, 2025; Zhang, D.-Y. Liu, Boutat, & Wu, 2024; Belkhatir & T.-M. Laleg-Kirati, 2018) .  \n⋆ The material in this paper was not presented at any conferences.  \nEmail addresses: [davi@sdu.dk](davi@sdu.dk) (Davi G. Accioli), [jerome@sdu.dk](jerome@sdu.dk) (Jerome Jouffroy).  \nThe main idea behind the method consists of multiplying the differential equation by a user-defined function, called the modulating function (MF), integrating both sides of the resulting equation, and applying integration by parts: a process that can be elegantly formulated using linear operators (Ungarala & Co, 2000) . By then selecting this function such that it, and its derivatives, satisfy a particular set of boundary conditions, different estimation problems can be tackled. In particular, MFs can be divided into three classes: total modulating functions, if both left and right boundaries are zero up to a certain derivative-order; left modulating functions, if their left boundary is zero while the right boundary is not identically zero; and right modulating functions, if their right boundary is zero while the left boundary is not identically zero (Jouffroy & J. Reger, 2015) . Through this approach, the modulating function method allows for fixed-time estimation, while also being robust to additive disturbances (Korder, Matti Noack, & Johann Reger, 202","cbCailuv1l01Aztv","https://ap.wps.com/l/cbCailuv1l01Aztv","pdf",2448173,1,16,"English","en",105,"# Introduction\n## Modulating function method background\n## Modulating functions and estimation via boundary conditions\n## Families of modulating functions\n## Auxiliary-system approach and duality connection\n## Motivation for control applications","[{\"question\":\"What is the core idea of the modulating function method in this paper?\",\"answer\":\"It multiplies the governing differential equation by a user-defined modulating function, integrates both sides, and applies integration by parts so boundary-condition choices shape the resulting algebraic relations for estimation and control synthesis.\"},{\"question\":\"How does the paper extend the modulating function method from estimation to control?\",\"answer\":\"By introducing dual modulating functions and dual modulations using auxiliary systems and duality, the framework is shown to support controller design for LTV systems, with existence conditions for the associated control laws.\"},{\"question\":\"What new control contribution is proposed, and how is it evaluated?\",\"answer\":\"A new fixed-time control law is proposed for both LTI and LTV systems, including an estimate of transient behavior. Numerical simulations show comparable performance to a benchmark LQR even with unmatched disturbances.\"}]",1784190667,40,{"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},"on-the-modulating-function-method-for-control-problems","",{"@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/on-the-modulating-function-method-for-control-problems/83824/",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 is the core idea of the modulating function method in this paper?","Question",{"text":75,"@type":76},"It multiplies the governing differential equation by a user-defined modulating function, integrates both sides, and applies integration by parts so boundary-condition choices shape the resulting algebraic relations for estimation and control synthesis.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper extend the modulating function method from estimation to control?",{"text":80,"@type":76},"By introducing dual modulating functions and dual modulations using auxiliary systems and duality, the framework is shown to support controller design for LTV systems, with existence conditions for the associated control laws.",{"name":82,"@type":73,"acceptedAnswer":83},"What new control contribution is proposed, and how is it evaluated?",{"text":84,"@type":76},"A new fixed-time control law is proposed for both LTI and LTV systems, including an estimate of transient behavior. Numerical simulations show comparable performance to a benchmark LQR even with unmatched disturbances.","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},[93,97,101,105,110,115,119,122,127,130,134],{"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":28,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]