Robust Path Integral Control on Stochastic Differential Games

Abstract

This paper develops a robust path integral (RPI) model predictive control using the Monte Carlo (MC) sampling to address the optimal control (OC) problem for the stochastic differential game (SDG). The two-player zero-sum differential game has been extensively investigated, mostly as its outcome indicates the $H_{\infty}$ optimality. The proposed path integral (PI) control framework provides an analytically sound method for building an algorithm of optimal control for this game based on stochastic trajectory sampling. This is achieved by using Feynman-Kac (F-K) lemma which transforms the value function of stochastic optimal control (SOC) problem into an expectation over all probable trajectories. This transformation makes it possible to solve SOC problems through MC sampling of stochastic processes. Finally, the RPI model predictive control using MC sampling is efficiently implemented for an inverted pendulum system. The RPI control has achieved good performance for changes in inverted pendulum weight and friction when the complete nonlinear swing-up is concerned while such environmental adjustments are not dealt with in a regular PI control.