Adaptive MPPI Control with Online Disturbance Covariance Estimation Tightens Stability

Researchers Hyung-Jin Yoon and Hunmin Kim address a fundamental challenge in Model Predictive Path Integral (MPPI) control: unknown, spatially varying process disturbances that degrade stability guarantees. Standard MPPI assumes a fixed disturbance covariance, but when that assumption is mismatched, the closed-loop stability certificate suffers a persistent penalty. The authors propose an online adaptive approach using a cell-wise recursive covariance estimator enhanced with spatial diffusion. They derive a finite-horizon error bound that explicitly separates three error sources: stochastic-approximation noise, 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 a weighted Lyapunov framework—a clever mathematical trick that enables provable convergence.

The paper's main contribution is a payoff theorem: after a computable crossover time, the adaptive controller always achieves a strictly tighter certified stability bound compared to any fixed covariance choice whose mismatch exceeds the residual smoothing and drift allowance. This means the controller actively learns and adapts to unknown disturbances, tightening its safety guarantees over time. Numerical experiments confirm the estimator's convergence and demonstrate the practical stability-tightening effect. The work is supported by two companion papers (arXiv:2607.04006 and arXiv:2607.06945) and open-source simulation code. For robotics and autonomous systems operating in uncertain environments, this method could significantly improve reliability and safety without requiring precise pre-calibrated disturbance models.