Predictive Control of Discrete Time Stochastic Nonlinear State Space Dynamical Systems: A Particle Nonparametric Approach

Abstract

This paper presents the main principles of a stochastic nonlinear model predictive control (NMPC) novel approach for imperfectly observed discrete time systems described by time varying nonlinear non-Gaussian state space models with unknown parameters. A convergent particle estimator of the conditional expectation of a chosen cost function on a given receding control horizon is built, leading to an almost sure (a.s.) epi-convergent estimator of the NMPC cost-to-go criterion. The estimator of the expected cost function relies upon simulations and on a recently developed nonparametric convergent particle estimator of a multi-step ahead conditional probability density function (pdf) of the state variables. The theory of stochastic epi-convergence is applied to the estimated cost-to-go criterion to prove the almost sure convergence of the optimal solutions of the approximated NMPC problem to their true counterparts, when both simulation and particle numbers grow to infinity.