[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82103-en":3,"doc-seo-82103-105":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},82103,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Adaptive MPPI with Online Disturbance Covariance Estimation","Adaptive MPPI control is developed for systems with additive process disturbances whose covariance is unknown and spatially varying. An incorrect disturbance covariance yields a persistent mismatch penalty inside the closed-loop stability certificate, while online estimation reduces this penalty over time. A cell-wise disturbance-covariance estimator with spatial diffusion is proposed and proven to converge to a smoothed fixed point, with an explicit error bound separating stochastic approximation, spatial smoothing bias, and temporal drift effects.","Adaptive MPPI with Online Disturbance Covariance Estimation: Provable Stability Tightening via Spatial Smoothing  \nHyung-Jin Yoon† and Hunmin Kim‡  \narXiv :2607 .08942v1 [ ee ss . SY] 9 Jul 2026  \nAbstract—We address Model Predictive Path Integral (MPPI) control for systems subject to additive process disturbances whose covariance is unknown and spatially varying. An incorrect disturbance covariance induces a persistent mismatch penalty in the closed-loop stability certificate, whereas online covariance estimation can reduce this penalty over time. We propose an online cell-wise disturbancecovariance estimator with spatial diffusion and prove that it converges to a smoothed fixed point, with an error bound separating three effects: a stochastic-approximation term that decreases as more closed-loop data are collected, a spatialsmoothing bias caused by diffusion across neighboring cells, and a drift term caused by slow temporal variation of the disturbance field. Choosing the diffusion kernel proportional to the stationary visitation measure makes the spatial diffusion term dissipative in the weighted norm used in the Lyapunov analysis. Substituting the resulting covariance estimate into the MPPI sampling distribution yields a time-varying adaptation penalty that quantifies the transient cost of learning in the closed-loop bound; this penalty decreases with the stochasticapproximation component and, over any prescribed finite horizon, is bounded by the sum of a residual smoothing bias and an accumulated drift allowance. The main result is a payoff theorem: although the adaptive controller may initially pay a larger transient penalty while learning, for any fixed mismatched disturbance covariance whose mismatch exceeds this residual smoothing-bias-plus-drift-allowance bound, there exists a finite, computable crossover time, within the prescribed horizon, after which the adaptive stability certificate is strictly tighter. In contrast to heuristic covariance adaptation, the proposed estimator treats the disturbance covariance as a statistical estimand and updates the MPPI sampling covariance and corresponding control penalty together, preserving the path-integral structure throughout adaptation.  \nI. INTRODUCTION  \nModel Predictive Path Integral (MPPI) control [1] solves finite-horizon stochastic optimal control by drawing M parallel sample trajectories and returning an importanceweighted control update. Because it requires no gradient of the dynamics or cost, MPPI has been applied to off-road navigation [1], legged locomotion, and aerial vehicles.  \nCompanion work has begun to develop formal closedloop stability guarantees for MPPI. For linear time-invariant systems, one companion paper [2] establishes exponential stability in expectation up to residual floors caused by process noise, finite-sample MPPI approximation error, and  \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.  \nsampling-confidence effects, via a finite-sample MPPI–LQR approximation bound and a Lyapunov perturbation argument. A second companion paper extends the analysis to nonlinear systems using contraction theory and Control Lyapunov Functions (CLFs), yielding a finite-horizon, high-probability localized mean practical stability bound with the same qualitative structure [3] .  \nBoth certified bounds share a common shape: an exponentially decaying nominal term plus additive residual terms. Among these residuals, the process-noise contribution depends explicitly on the covariance Σw , whereas increasing the MPPI sample count mainly reduces the Monte Carlo approximation error and does not remove the disturbance floor. So once the finite-sample MPPI error is controlled, i","cbCairZ59KEy8vLX","https://ap.wps.com/l/cbCairZ59KEy8vLX","pdf",854723,2,1,19,"English","en",105,"# Abstract\n# Introduction\n## MPPI control and existing stability guarantees\n## Limitations from unknown disturbance covariance\n## Prior RL-based covariance tuning vs statistical estimation\n## Paper contribution and assumptions","[{\"question\":\"What problem does the paper address in MPPI control?\",\"answer\":\"It addresses MPPI control for systems with additive disturbances whose covariance is unknown and varies across space.\"},{\"question\":\"Why does an incorrect disturbance covariance harm stability certificates?\",\"answer\":\"A mismatch between assumed and true covariance creates a persistent penalty term in the closed-loop stability certificate, limiting how tight the guarantee can be.\"},{\"question\":\"How does the proposed method improve covariance knowledge during control?\",\"answer\":\"It uses an online cell-wise covariance estimator with spatial diffusion, updates the MPPI sampling covariance and control penalty together, and proves convergence to a smoothed fixed point with a bounded transient 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problem does the paper address in MPPI control?","Question",{"text":75,"@type":76},"It addresses MPPI control for systems with additive disturbances whose covariance is unknown and varies across space.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does an incorrect disturbance covariance harm stability certificates?",{"text":80,"@type":76},"A mismatch between assumed and true covariance creates a persistent penalty term in the closed-loop stability certificate, limiting how tight the guarantee can be.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed method improve covariance knowledge during control?",{"text":84,"@type":76},"It uses an online cell-wise covariance estimator with spatial diffusion, updates the MPPI sampling covariance and control penalty together, and proves convergence to a smoothed fixed point with a bounded transient 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