Nonlinear stochastic receding horizon control: stability, robustness and Monte Carlo methods for control approximation

TL;DR

This paper analyzes the stability of nonlinear stochastic receding horizon control with approximate controllers, demonstrating that errors do not accumulate and proposing a Monte Carlo-based approximation method that ensures practical stabilization.

Contribution

It introduces a stability analysis for approximate controllers in stochastic receding horizon control and proposes a Monte Carlo simulation method based on the Feynman-Kac formula for control approximation.

Findings

Controller errors do not accumulate over time.

The process converges exponentially to a neighborhood of the origin.

Monte Carlo method practically stabilizes the nonlinear process.

Abstract

This work considers the stability of nonlinear stochastic receding horizon control when the optimal controller is only computed approximately. A number of general classes of controller approximation error are analysed including deterministic and probabilistic errors and even controller sample and hold errors. In each case, it is shown that the controller approximation errors do not accumulate (even over an infinite time frame) and the process converges exponentially fast to a small neighbourhood of the origin. In addition to this analysis, an approximation method for receding horizon optimal control is proposed based on Monte Carlo simulation. This method is derived via the Feynman-Kac formula which gives a stochastic interpretation for the solution of a Hamilton-Jacobi-Bellman equation associated with the true optimal controller. It is shown, and it is a prime motivation for this study, that this particular controller approximation method practically stabilises the underlying nonlinear process.