[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83325-en":3,"doc-seo-83325-105":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},83325,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Cost of Sensing in Optimal Control: Basic Formulations, Examples, and Applications","Cost of sensing is integrated into the optimal control framework to better model systems where sensing duration and sensor-related expenses significantly influence total performance. The work develops multiple approaches that add an integral sensing-cost term for linear time-invariant (LTI) systems, balancing sensing against standard control and stabilization costs. Optimal sensing intervals are derived using Pontryagin’s Minimum Principle, with extensions for nonlinear formulations and validation via numerical simulations. Reduced infinite-horizon expressions, a closed-form solution for a first-order case, a Shrinking Horizon implementation, and a wastewater treatment plant case study establish practical relevance.","Advanced Control for Applications  \narXiv :2607 .08119v1 [ ee ss . SY] 9 Jul 2026  \n ORIGINAL ARTICLE  OPEN ACCESS   \nCost of Sensing in Optimal Control: Basic Formulations, Examples, and Applications  \nDung Tran | Tri Ngo | Tuhin Das  \nDepartment of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, Florida, USA | Correspondence: Dung Tran ([Dung.Tran@ucf.edu](Dung.Tran@ucf.edu)) | Tuhin Das ([Tuhin.Das@ucf.edu](Tuhin.Das@ucf.edu))  \nReceived: Added at production | Revised: Added at production | Accepted: Added at production  \nAcademic Editor:   | Guest Editor:    \nKeywords: Sensing-Cost, Optimal Control, Two-Point Boundary Value Problem (TPBVP), Pontryagin’s Minimum Principle, Shrinking Horizon  \nABSTRACT  \nIncorporating a notion of cost of sensing, or sensing-cost, within the optimal control framework is beneficial in controlling systems where the duration of sensing, and/or the cost of sensors themselves, have a considerable impact on the overall cost. In this regard, this paper presents multiple methods for incorporating an integral sensing-cost into the optimal control framework for Linear Time-Invariant (LTI) systems. Sensing-cost is traded off against the conventional costs of control and stabilization. Optimal sensing intervals are derived by applying the Pontryagin’s Minimum Principle. Other formulationsofthe sensing-cost problem, and extension to nonlinear systems, are possible. The theoretical developments ofthis paper are validated through numerical solutionsand demonstrated through simulations. A reduced-form expression for the infinite-horizon multi-dimensional case with single switching point is derived, and a closed-form solution is obtained for the infinite-horizon first-order case. Additionally, a Shrinking Horizon method is demonstrated for practical implementation of the proposed theory and as a means to address uncertainties. A practical case study of a wastewater treatment plant is introduced to examine the applicability of sensing-cost considerations in areal-world setting.  \n1  Introduction  \nIn conventional optimal control problems, such as in LQR, the performance index typically weighs transient performance vs. control effort. An additional consideration can be the cost of sensing. Cost of sensing, or alternately sensing-cost, plays a pivotal role in shaping the overall design, implementation, and performance of control systems. It encompasses various aspects, including the price of sensors, the computational overhead associated with sensing and processing, energy consumption, and impacts on system complexity and reliability. This paper aims to explore the implications of incorporating sensing-cost within the optimal control framework, summarizing basic formulations, key methodologies, and findings.  \nIncorporation of sensing-cost within optimal control has not been reported in the literature. However, due to their ubiquitousness and practicality in control systems, numerous works have addressed various facets of sensing, such as sensor placement,  \nscheduling, sampling, etc. While such works are not directly relevant to this paper, we discuss selected ones from literature to put this work in context. For instance,[1] propose optimization techniques for sensor placement, employing greedy algorithms to achieve cost-effective configurations that maintain desired performance. In [2], the authors propose the concept of sharedsensing for reversible transducers that are continuously switched between actuator and sensor modes. The authors in [3] introduce a method to reduce the cost and power requirements of sensing while streamlining data storage and processing invibration-based monitoring and diagnostics using compressive sensing. In[4], the authors address collaborative sensing with the goal of determining a sensor schedule that minimizes the error covariance. In [5], the authors address a sensor scheduling problem that involves estimating the state of an uncertain process usin","cbCaicsjiwzaX9Le","https://ap.wps.com/l/cbCaicsjiwzaX9Le","pdf",2369395,2,1,15,"English","en",105,"# Introduction\n# Sensing-Cost in Optimal Control Framework\n## Integral Sensing-Cost Formulations for LTI Systems\n## Pontryagin’s Minimum Principle and Optimal Sensing Intervals\n## Extensions to Nonlinear Systems\n# Validation Through Numerical Solutions and Simulations\n## Infinite-Horizon Reduced-Form and Closed-Form Results\n## Shrinking Horizon Implementation\n# Case Study: Wastewater Treatment Plant Applications","[{\"question\":\"What problem does sensing-cost address in optimal control?\",\"answer\":\"It accounts for sensing duration and sensor expenses that materially affect the overall objective, alongside conventional control and stabilization costs.\"},{\"question\":\"How are optimal sensing intervals determined?\",\"answer\":\"Optimal sensing intervals are derived by applying Pontryagin’s Minimum Principle to the sensing-cost-augmented optimal control problem.\"},{\"question\":\"What results and practical demonstrations support the proposed theory?\",\"answer\":\"The paper validates the developments with numerical solutions and simulations, provides infinite-horizon reduced-form and closed-form results for specific cases, demonstrates a Shrinking Horizon method, and applies the ideas in a wastewater treatment plant case study.\"}]",1784186743,38,{"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},"cost-of-sensing-in-optimal-control-basic-formulations-examples-and-applications","",{"@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/cost-of-sensing-in-optimal-control-basic-formulations-examples-and-applications/83325/",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","2026-07-24","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 problem does sensing-cost address in optimal control?","Question",{"text":75,"@type":76},"It accounts for sensing duration and sensor expenses that materially affect the overall objective, alongside conventional control and stabilization costs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are optimal sensing intervals determined?",{"text":80,"@type":76},"Optimal sensing intervals are derived by applying Pontryagin’s Minimum Principle to the sensing-cost-augmented optimal control problem.",{"name":82,"@type":73,"acceptedAnswer":83},"What results and practical demonstrations support the proposed theory?",{"text":84,"@type":76},"The paper validates the developments with numerical solutions and simulations, provides infinite-horizon reduced-form and closed-form results for specific cases, demonstrates a Shrinking Horizon method, and applies the ideas in a wastewater treatment plant case 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