[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85394-en":3,"doc-seo-85394-105":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},85394,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Polynomial Chaos-based Stochastic Model Predictive Control: An Overview and Future Research Directions","The paper surveys polynomial chaos theory (PCT) as a computationally tractable tool for propagating uncertainty through smooth nonlinear dynamical systems and for accelerating stochastic model predictive control (SMPC) with time-invariant uncertainties. It explains how PCT can reduce the complexity of SMPC chance constraints. The discussion contrasts robust MPC’s worst-case conservatism with SMPC’s probabilistic uncertainty modeling, highlighting stochastic and inherent robustness considerations under uncertainty representation choices such as independent time-varying and dependent time-invariant cases.","arXiv :2406 . 10734v2 [ ee ss . SY] 12 Jul 2026  \nPolynomial  \nChaos-based Stochastic Model Predictive Control  \nAN OVERVIEW AND  \nFUTURE RESEARCH DIRECTIONS  \nPrabhat K. Mishra, Joel A. Paulson, and Richard D. Braatz  \nIndian Institute of Technology Kharagpur, University of Wisconsin-Madison, Massachusetts Institute of Technology  \nM  \nodel Predictive Control (MPC) has gained popularity due to its ability to handle constraints and incorporate model-based predictions [1] . While MPC exhibits inherent robustness through its receding horizon implementation and feedback mechanism  \n[2], [3], this robustness is often limited in practical settings where uncertainties such as model mismatch, unmodeled dynamics, and fast-varying disturbances are significant. As highlighted by [4], nominal stability guarantees do not automatically imply robustness under perturbations or constraint tightening, especially when feasibility and stability margins are narrow. Recent results on inherent robustness are applicable when states are within the robust positively invariant sets and assumptions on maximal increase in cost are satisfied in nominal MPC [5, Assumptions 6 & 7] and in economic MPC [6, Assumption 6] . The inherent stochastic robustness [7, Assumptions 3 & 4] is also based on similar assumptions. It has been observed in [8] that nominal MPC may lead to performance degradation or constraint violations when operating in uncertain or changing environments.  \nTherefore, robust MPC is developed by considering worst case bounds of uncertainties in the design stage and by ensuring desired properties for all possible realizations of uncertainties within those bounds [9] . However, the bounded description  \nDigital Object Identifier 10.1109/MCS.2020.000000 Date of current version: XXXXXX  \nof uncertainties lead to a potentially very conservative MPC formulation in the presence of outliers or uncertainties with small chances of occurrences. As an alternative, Stochastic MPC (SMPC) overcomes conservatism by associating a probability distribution with the uncertainty [10] .  \nSummary  \nModern technology, from self-driving cars to the rapid  \nproduction of life-saving vaccines, relies on the ability of machines to make intelligent decisions in a world full of uncertainty. Traditionally, engineers designed control systems using “worst-case” scenarios. While safe, this approach is often inefficient and limits the performance of high-tech devices. In the past decade, polynomial chaos theory (PCT) has been shown to provide a computationally tractable way to perform complete and accurate uncertainty propagation through (smooth) nonlinear dynamic systems. As such, it represents a very useful computational tool for accelerating the computations needed in stochastic model predictive control (SMPC) with time invariant uncertainties. It turns out that it can also be used to reduce complexity of chance constraints, which are an important component of SMPC. In this paper, we provide an overview of PCT and discuss how it can be applied in SMPC.  \narXiv version « 1  \nUncertainty representation  \nWhile  \ning  \nthe importance of accounting for uncertainty durfeedback control design has been long established,  \nthere is no consensus on the form in which the uncertainty should be represented. An expressive framework for modeling uncertainty can be a joint distribution of all uncertainties overtime, constituting a non-Markovian stochastic process where the distribution of future uncertainties can depend on the complete or finite history of past uncertainty realizations. We often consider two special cases: (1) the uncorrelated process, where the autocorrelation function is a Dirac delta function, and (2) the fully correlated process, where the autocorrelation function is constant. Case (1) is typically known as the independent timevarying (TV) case [11], and case (2) is the dependent timeinvariant (TI) case [12] . Most of the work on SMPC is concentrated on the TV case s","cbCaimlabvUmJcPk","https://ap.wps.com/l/cbCaimlabvUmJcPk","pdf",654142,3,1,23,"English","en",105,"# Summary\n# Uncertainty representation\n## Time-varying vs time-invariant uncertainty\n# SMPC foundations\n## Stochastic optimal control and chance-constrained optimization","[{\"question\":\"What limitation of robust MPC motivates the use of SMPC?\",\"answer\":\"Robust MPC guarantees properties using worst-case bounds, which can become overly conservative, especially when uncertainties have small occurrence probabilities or include outliers. SMPC improves performance by attaching a probability distribution to uncertainties.\"},{\"question\":\"How does polynomial chaos theory (PCT) contribute to stochastic MPC?\",\"answer\":\"PCT provides a tractable method to propagate uncertainty through smooth nonlinear dynamic systems. It can also reduce the computational complexity of chance constraints, which are central in SMPC.\"},{\"question\":\"What are the two main uncertainty representation cases discussed?\",\"answer\":\"The document contrasts an independent time-varying (TV) case, where autocorrelation is a Dirac delta, with a dependent time-invariant (TI) case, where autocorrelation is constant. The TV case supports Markov-based simplifications, while TI can be more realistic for static parameters.\"}]",1784203103,58,{"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},"polynomial-chaos-based-stochastic-model-predictive-control-an-overview-and-future-research-directions","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/polynomial-chaos-based-stochastic-model-predictive-control-an-overview-and-future-research-directions/85394/",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},"What limitation of robust MPC motivates the use of SMPC?","Question",{"text":75,"@type":76},"Robust MPC guarantees properties using worst-case bounds, which can become overly conservative, especially when uncertainties have small occurrence probabilities or include outliers. SMPC improves performance by attaching a probability distribution to uncertainties.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does polynomial chaos theory (PCT) contribute to stochastic MPC?",{"text":80,"@type":76},"PCT provides a tractable method to propagate uncertainty through smooth nonlinear dynamic systems. It can also reduce the computational complexity of chance constraints, which are central in SMPC.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the two main uncertainty representation cases discussed?",{"text":84,"@type":76},"The document contrasts an independent time-varying (TV) case, where autocorrelation is a Dirac delta, with a dependent time-invariant (TI) case, where autocorrelation is constant. The TV case supports Markov-based simplifications, while TI can be more realistic for static parameters.","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},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"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":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]