Stochastic Stability of Nonlinear MPPI via Contraction Theory and Control Lyapunov Functions
For practitioners of nonlinear control, this work offers a theoretical stability guarantee for MPPI, which previously lacked such certificates despite its algorithmic flexibility.
This paper provides a closed-loop stability certificate for Model Predictive Path Integral (MPPI) control on nonlinear systems, showing that finite-sample MPPI inherits contraction properties from a nominal deterministic MPC policy under a small-gain condition, with approximation error decomposing into a bias floor and a Monte Carlo term vanishing at an inverse square-root rate.
Model Predictive Path Integral (MPPI) control is directly implementable on nonlinear systems because its online update requires only forward rollouts of the dynamics, not gradients, linearizations, or convex optimization. However, this algorithmic flexibility does not by itself provide a closed-loop stability certificate. This paper establishes such a certificate through a stability-inheritance argument. We assume that there exists a deterministic nonlinear MPC policy whose disturbance-free closed loop is certified by a Control Lyapunov Function terminal cost and a contraction metric, and we show that finite-sample MPPI inherits the nominal contraction when its sampling-based update approximates this reference policy with sufficient accuracy. The approximation error decomposes into a finite-temperature bias floor and a Monte Carlo term that vanishes at the inverse square-root rate in the sample count. Under an explicit small-gain condition, the resulting MPPI closed loop satisfies a finite-horizon, high-probability localized mean practical stability bound with residual floors due to MPPI approximation error, Gaussian process noise, and bad sampling events. The paper also gives an ISS-type restatement and a finite-horizon design procedure for choosing the localization set, temperature, and sample count.