[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85028-en":3,"doc-seo-85028-105":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},85028,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Uniform High-Probability ISS Tubes for Sampled-Data State Estimation","State estimates used in sampled monitoring and automation require uncertainty bounds that remain valid between measurement updates. A finite-horizon input-to-state-stability tube and observer co-design framework is developed for continuous-time observers driven by sampled-and-held outputs. The error model separates process disturbance, sampled measurement noise, and intersample mismatch, then transfers a horizon-level disturbance probability envelope through an ISS estimate to ensure simultaneous containment of the full error trajectory. Quadratic dissipation produces ellipsoidal and componentwise tubes, while semidefinite co-design minimizes normalized tube width across channels, with a structured nonlinear extension preserving known nonlinear dynamics.","Uniform High-Probability ISS Tubes for Sampled-Data State Estimation⋆  \narXiv :2607 .07892v1 [ ee ss . SY] 8 Jul 2026  \nJerzy Baranowskia,∗  \na Department of Automatic Control and Robotics, AGH University of Krakow, al. Mickiewicza 30, Krakow, 30-059, Poland  \n\n| ARTICLE INFO |  | AB STRACT |  |  |\n| --- | --- | --- | --- | --- |\n| Keywords:\u003Cbr>state estimation sampled-data systems input-to-state stability high-probability bounds observer design nonlinear systems |  | State estimates used in sampled monitoring and automation need bounds that remain valid between measurements. We develop a finite-horizon input-to-statestability tube and observer co-design framework for continuous-time observers driven by sampled and held outputs. The sampled-data error model separates process disturbance, sampled measurement noise, and intersample mismatch. A horizon-level disturbance-envelope event is transferred through an ISS estimate to simultaneous containment of the complete error trajectory. Quadratic dissipation inequalities yield ellipsoidal and componentwise tubes, and semidefinite codesign minimizes normalized tube width across the three channels. A structured nonlinear extension preserves known nonlinear channels. Co-design reduces the worst normalized half-width by 31% in a linear compartment benchmark and by a factor of 22 .4 in a flexible-joint benchmark. |  |  |\n| 1. Introduction\u003Cbr>State estimation is a central component of monitoring and automation systems in which decisions are made from incomplete, noisy, and delayed information. In process plants, mechatronic devices, networked sensors, and embedded diagnostic modules, the physical state evolves continuously while measurements are acquired at discrete instants. The estimator must therefore propagate information between sensor updates. For alarms, constraint checking, supervision, and diagnostic escalation, the relevant uncertainty often concerns the whole continuous-time evolution over a finite interval, including the intersample parts of the trajectory.\u003Cbr>This paper uses the term tube for such an interval-wise uncertainty object. Formally, a tube is a time-indexed family of sets around the observer trajectory, for example |  |  |  | or, in componentwise form, intervals |􀁸􀁩 (􀁴) −􀁸̂ 􀁩 (􀁴)| ≤ 􀁲􀁩 (􀁴) . A trajectory is contained in the tube when the corresponding inclusion holds for every 􀁴 ∈ [0, 􀁔 ] . The probability statement studied here is therefore a simultaneous statement over the full horizon. This contrasts with marginal statements at individual time instants.\u003Cbr>The sampled-data setting gives a concrete reason to distinguish these viewpoints. The observer correction uses the most recent sampled output, while the plant state continues to evolve. The estimation error is then driven by process disturbance, sampled measurement noise, and the mismatch between the current output and the held output. A finite-horizon tube for such an observer has to account for these channels together, and it has to state how a disturbance-side probability budget is transferred to an estimation-error bound.\u003Cbr>This distinction is operational in measurement and automation systems. A pointwise covariance band may be appropriate for reporting local estimator uncertainty, but an alarm, supervisory constraint, or diagnostic escalation rule often asks whether the unmeasured state could have left an admissible region at any time since the last update. The proposed construction is therefore intended |\n| 􀁔 (􀁴) = {􀁸 ∶ (􀁸 − 􀁸̂ (􀁴))⊤ 􀁐 (􀁸 − 􀁸̂ (􀁴))≤ 􀀚 (􀁴)}, 􀁴 ∈ [0, 􀁔 ] . |  |  | (1) |  |\n| ⋆ Dedicated to Professor Wojciech Mitkowski on the occasion of his 80th birthday.\u003Cbr>∗ Corresponding author\u003Cbr> [jb@agh.edu.pl](jb@agh.edu.pl) (J. Baranowski)\u003Cbr>ORCID (s): 0000-0003-3313-581X (J. Baranowski) |  |  |  |  |\n\nBaranowski: Preprint submitted to Elsevier Page 1 of 32  \nHigh-probability ISS tubes for sampled-data estimation  \nas a robustness-oriented monitoring bound used alongside statistical estimator unc","cbCaihTf1ufL0x9c","https://ap.wps.com/l/cbCaihTf1ufL0x9c","pdf",2141090,1,32,"English","en",105,"# Introduction\n## Related work and positioning\n# Main tube concept and probability statement","[{\"question\":\"What does the “tube” represent in the sampled-data state estimation setting?\",\"answer\":\"The tube is a time-indexed family of sets around the observer trajectory that captures uncertainty over the entire finite horizon, including intersample behavior. Containment holds when the true trajectory stays within the set for every time in the interval.\"},{\"question\":\"How are the estimation-error uncertainty channels modeled?\",\"answer\":\"The framework separates three channels: process disturbance, sampled measurement noise, and intersample mismatch caused by holding the output. The probability budget is handled at the horizon level rather than only pointwise in time.\"},{\"question\":\"How is the observer/tube co-design performed and what is optimized?\",\"answer\":\"Quadratic dissipation inequalities generate ellipsoidal and componentwise tubes, and semidefinite co-design minimizes the normalized tube width across multiple channels. The method is extended to structured nonlinear systems while preserving known nonlinear channels.\"}]",1784200489,81,{"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},"uniform-high-probability-iss-tubes-for-sampled-data-state-estimation","",{"@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/uniform-high-probability-iss-tubes-for-sampled-data-state-estimation/85028/",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 does the “tube” represent in the sampled-data state estimation setting?","Question",{"text":75,"@type":76},"The tube is a time-indexed family of sets around the observer trajectory that captures uncertainty over the entire finite horizon, including intersample behavior. Containment holds when the true trajectory stays within the set for every time in the interval.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the estimation-error uncertainty channels modeled?",{"text":80,"@type":76},"The framework separates three channels: process disturbance, sampled measurement noise, and intersample mismatch caused by holding the output. The probability budget is handled at the horizon level rather than only pointwise in time.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the observer/tube co-design performed and what is optimized?",{"text":84,"@type":76},"Quadratic dissipation inequalities generate ellipsoidal and componentwise tubes, and semidefinite co-design minimizes the normalized tube width across multiple channels. 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