Overview
This study investigates the stochastic stability of Model Predictive Path Integral (MPPI) control for nonlinear systems. It aims to provide a stability certificate for MPPI, which is directly implementable on nonlinear systems without requiring gradients, linearizations, or convex optimization for its online update. The research employs a stability-inheritance argument, linking MPPI's behavior to a deterministic nonlinear Model Predictive Control (MPC) policy.
Research Context
MPPI control is notable for its direct applicability to nonlinear systems due to its reliance on forward rollouts of dynamics. This characteristic bypasses the need for gradient computations, system linearizations, or convex optimization typical in some control methodologies. However, this algorithmic flexibility, by itself, does not inherently guarantee closed-loop stability. The absence of a formal stability certificate for MPPI is the primary gap addressed by this work.
Approach
The research methodology is based on a stability-inheritance argument. It posits the existence of a reference deterministic nonlinear MPC policy. This reference policy's disturbance-free closed loop is assumed to be certified by a Control Lyapunov Function (CLF) terminal cost and a contraction metric. The core of the approach involves demonstrating that finite-sample MPPI can inherit this nominal contraction under specific conditions. This inheritance is contingent on MPPI's sampling-based update accurately approximating the behavior of the reference MPC policy.
The approximation error inherent in MPPI is decomposed into two components: a finite-temperature bias floor and a Monte Carlo term. The Monte Carlo term is observed to diminish at an inverse square-root rate with respect to the sample count. The analysis then proceeds to establish a stability bound for the resulting MPPI closed loop. This bound is achieved under an explicit small-gain condition. It describes a finite-horizon, high-probability localized mean practical stability, accounting for residual floors. These residual floors are attributed to MPPI approximation error, Gaussian process noise, and the occurrence of bad sampling events.
Additionally, the paper provides an Input-to-State Stability (ISS)-type restatement of the findings. It also outlines a finite-horizon design procedure. This procedure is intended for selecting key parameters such as the localization set, the temperature parameter (related to sampling), and the required sample count in MPPI implementation.
Findings
- A stability certificate for MPPI control in nonlinear systems has been established through a stability-inheritance argument.
- Finite-sample MPPI inherits the nominal contraction of a reference deterministic nonlinear MPC policy, provided its sampling-based update sufficiently approximates this reference policy.
- The MPPI approximation error is composed of a finite-temperature bias floor and a Monte Carlo term, with the Monte Carlo term vanishing at the inverse square-root rate in sample count.
- Under an explicit small-gain condition, the MPPI closed loop satisfies a finite-horizon, high-probability localized mean practical stability bound.
- Residual floors in the stability bound are due to MPPI approximation error, Gaussian process noise, and bad sampling events.
- The research includes an ISS-type restatement and a finite-horizon design procedure for selecting localization set, temperature, and sample count.