Adaptive MPPI with Online Disturbance Covariance Estimation: Provable Stability Tightening via Spatial Smoothing

2026-07-09

AI Summary

This work addresses the degradation of stability guarantees in Model Predictive Path Integral (MPPI) control for nonlinear systems due to mismatch between the assumed and the true additive process noise covariance, which is unknown, spatially varying, and slowly time-varying. To mitigate this issue, the authors propose a block-wise recursive covariance estimator augmented with a spatial diffusion mechanism that continuously updates the disturbance covariance online and embeds it into the MPPI sampling distribution. By integrating a weighted Lyapunov analysis, the approach provides adaptive stability guarantees. The method innovatively incorporates spatial smoothing and an invertible diffusion kernel to disentangle stochastic approximation error, smoothing bias, and time-varying drift effects. Theoretically, it is shown that after a finite transient period, the proposed scheme yields a strictly tighter stability bound than any fixed covariance choice. Numerical experiments corroborate the estimator’s convergence and its efficacy in enhancing closed-loop stability.

Abstract

We study Model Predictive Path Integral (MPPI) control for nonlinear systems with additive process disturbances whose covariance is unknown, spatially varying, and slowly time-varying. A mismatched disturbance covariance produces a persistent penalty in closed-loop stability certificates, while online estimation can reduce this penalty as data are collected. We propose a cell-wise recursive covariance estimator with spatial diffusion and prove a finite-horizon error bound that separates stochastic-approximation error, spatial-smoothing bias, and temporal-drift effects. The diffusion kernel is chosen to be reversible with respect to the stationary visitation measure, making the diffusion operator dissipative in the weighted Lyapunov analysis. We then substitute the resulting covariance estimate into the MPPI sampling distribution and derive an adaptive stability certificate with an explicit learning penalty. The main result is a payoff theorem: after a computable crossover time, the adaptive controller achieves a strictly tighter certified stability bound than any fixed covariance choice whose mismatch exceeds the residual smoothing and drift allowance. Numerical experiments illustrate the estimator convergence and the resulting stability-tightening effect.

Problem

• MPPI control
• disturbance covariance estimation
• adaptive control
• stability certification
• spatially varying disturbances

Innovation

• adaptive MPPI
• online covariance estimation
• spatial smoothing
• stability certificate
• diffusion kernel