[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81891-en":3,"doc-seo-81891-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81891,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Finite-Sample Closed-Loop Stability of Model Predictive Path Integral Control for Linear Time-Invariant Systems","Finite-sample closed-loop stability guarantees are developed for Model Predictive Path Integral (MPPI) control on discrete-time linear time-invariant systems with additive Gaussian process disturbances. For unconstrained LTI/quadratic setups using the DARE terminal cost, the finite-horizon MPC action matches the infinite-horizon LQR first move for any planning horizon, enabling MPPI analysis as a stochastic perturbation of LQR. High-probability LQR feedback approximation is quantified by a Monte Carlo term and a finite-temperature bias. Lyapunov perturbation yields practical exponential stability in expectation, with explicit residual floors tied to noise, MPPI approximation, and sampling failure confidence, and a computable sample threshold. In the limit of infinite samples and vanishing temperature bias, the stochastic LQR stability bound is recovered.","Finite-Sample Closed-Loop Stability of Model Predictive Path Integral Control for Linear Time-Invariant Systems  \nHyung-Jin Yoon† and Hunmin Kim‡  \narXiv :2607 .04006v1 [math .OC] 4 Jul 2026  \nAbstract—We establish finite-sample closed-loop stability guarantees for Model Predictive Path Integral (MPPI) control applied to discrete-time Linear Time-Invariant (LTI) systems with additive Gaussian process disturbances. The key observation is that, for unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law has the same first control action as the infinite-horizon LQR law for every planning horizon. Thus, finite-sample MPPI can be analyzed as a stochastic perturbation of LQR. First, we show that the MPPI control law approximates the LQR feedback with high probability. The approximation error decomposes into a Monte Carlo term that decreases with the sample count and an infinite-sample temperature bias that persists at finite temperature but vanishes as the temperature is reduced. The resulting constants are written in terms of the horizondependent stacked cost matrices, making explicit that the finitesample certificate is parametrized by the selected planning horizon. Second, we use a Lyapunov perturbation argument to prove practical exponential stability in expectation. On sample paths that remain in a compact Lyapunov sublevelset over a finite operating horizon, the expected state norm decays exponentially up to three residual floors: a processnoise floor, an MPPI approximation floor, and a confidence floor from the per-step sampling failure probability. The sufficient sample threshold is explicit and computable from the DAREsolution, LQR stability margin, MPPI sampling parameters, temperature, and planning horizon. In the joint limit of infinite samples and vanishing temperature bias, the result recovers the stochastic LQR stability bound.  \nI. INTRODUCTION  \nModel Predictive Path Integral (MPPI) control [1], [2] is a sampling-based receding-horizon method that has achieved strong empirical performance across robotics and autonomous systems, including off-road navigation [3], legged locomotion, and aerial vehicles. Its central appeal is that it requires no gradient of the cost or dynamics: at each time step it draws M random control perturbations, rolls them out in parallel (GPU-accelerated), and forms an importance-weighted average that approximates the information-theoretic optimal. This gradient-free, massively parallel structure makes MPPI uniquely attractive for nonlinear and non-smooth problems where classical gradientbased MPC solvers struggle.  \nDespite a growing body of work on MPPI theory, the question of closed-loop stability under receding-horizon execution remains largely unresolved. In particular, it  \n†H.-J. Yoon is with the Department of Mechanical and Nuclear Engineering, Tennessee Technological University, Cookeville, TN, USA.  \n‡H. Kim is with the School of Engineering, Department of Electrical and Computer Engineering, Mercer University, Macon, GA, USA.  \nThis work was supported by internal funding at Tennessee Technological University.  \nis unclear under what conditions the state xk remains bounded and converges when MPPI is implemented with a finite sample count M and subjected to persistent process disturbances. This is not merely an academic concern: without stability guarantees, practitioners cannot reason systematically about how many samples are sufficient or how performance degrades as M decreases. The difficulty is that finite-sample approximation errors arise at every control update and interact with stochastic disturbances over the horizon, making one-step optimization guarantees insufficient for establishing long-term closed-loop behavior.  \nA. Prior Work and the Remaining Gap  \nMPPI foundations. The original MPPI derivation [1] frames control as minimization of a KL-divergence between a controlled and an uncontrolled trajectory distribution, with the impor","cbCaiaMmt7ZMS3I1","https://ap.wps.com/l/cbCaiaMmt7ZMS3I1","pdf",637227,9,1,11,"English","en",105,"# Introduction\n## MPPI foundations\n## Approximation error and optimizer convergence\n## Robust MPPI and performance bounds\n## Contraction theory and CLF-based MPC","[{\"question\":\"What stability property is proved for MPPI under finite samples and disturbances?\",\"answer\":\"Practical exponential stability in expectation is proved via a Lyapunov perturbation argument, showing exponential decay of the expected state norm up to residual floors caused by process noise, MPPI approximation, and confidence from per-step sampling failure probability.\"},{\"question\":\"How does the paper relate finite-horizon MPC in this setting to infinite-horizon LQR?\",\"answer\":\"For unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law applies the same first control action as the infinite-horizon LQR law for every planning horizon.\"},{\"question\":\"What factors determine the required number of samples M?\",\"answer\":\"A sufficient sample threshold is given as an explicit computable quantity depending on the DARE solution, the LQR stability margin, MPPI sampling parameters, the temperature, and the planning horizon.\"}]","Finite-Sample Closed-Loop Stability of Model Predictive Path Integral Control for Linear Time-Invariant Systems | PDF",1784176898,28,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"finite-sample-closed-loop-stability-of-model-predictive-path-integral-control-for-linear-time-invariant-systems","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/finite-sample-closed-loop-stability-of-model-predictive-path-integral-control-for-linear-time-invariant-systems/81891/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-08-04","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What stability property is proved for MPPI under finite samples and disturbances?","Question",{"text":77,"@type":78},"Practical exponential stability in expectation is proved via a Lyapunov perturbation argument, showing exponential decay of the expected state norm up to residual floors caused by process noise, MPPI approximation, and confidence from per-step sampling failure probability.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How does the paper relate finite-horizon MPC in this setting to infinite-horizon LQR?",{"text":82,"@type":78},"For unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law applies the same first control action as the infinite-horizon LQR law for every planning horizon.",{"name":84,"@type":75,"acceptedAnswer":85},"What factors determine the required number of samples M?",{"text":86,"@type":78},"A sufficient sample threshold is given as an explicit computable quantity depending on the DARE solution, the LQR stability margin, MPPI sampling parameters, the temperature, and the planning horizon.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,117,122,125,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & 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