Sampling-based Model Predictive Control (MPC) has been a practical and effective approach in many domains, notably model-based reinforcement learning, thanks to its flexibility and parallelizability. Despite its appealing empirical performance, the theoretical understanding, particularly in terms of convergence analysis and hyperparameter tuning, remains absent. In this paper, we characterize the convergence property of a widely used sampling-based MPC method, Model Predictive Path Integral Control (MPPI). We show that MPPI enjoys at least linear convergence rates when the optimization is quadratic, which covers time-varying LQR systems. We then extend to more general nonlinear systems. Our theoretical analysis directly leads to a novel sampling-based MPC algorithm, CoVariance-Optimal MPC (CoVo-MPC) that optimally schedules the sampling covariance to optimize the convergence rate. Empirically, CoVo-MPC significantly outperforms standard MPPI by 43-54% in both simulations and real-world quadrotor agile control tasks. Videos and Appendices are available at \url{https://lecar-lab.github.io/CoVO-MPC/}.

Sampling-based Model Predictive Control (MPC)在许多领域中具有实际和有效的方法，尤其是基于模型的强化学习，由于其灵活性和可并行化性。本文对一种广泛使用的基于采样的MPC方法，Model Predictive Path Integral Control (MPPI)的收敛性进行了表征。理论分析直接导致了一种新颖的基于采样的MPC算法，CoVariance-Optimal MPC (CoVo-MPC)，它通过优化收敛速度来最优调度采样协方差。在模拟和实际的四旋翼敏捷控制任务中，CoVo-MPC在性能上显著优于标准的MPPI，提升了43-54%。

CoVO-MPC：采样模型预测控制和最优协方差设计的理论分析