A Closed-Form Model Predictive Control Framework for Nonlinear Noise Corrupted Systems

Abstract: In this paper, a framework for Nonlinear Model Predictive Control (NMPC) that explicitly incorporates the noise influence on systems with continuous state spaces is introduced. By the incorporation of noise, which results from uncertainties during model identification and the measurement process, the quality of control can be significantly increased. Since NMPC requires the prediction of system states over a certain horizon, an efficient state prediction technique for nonlinear noise-affected systems is required. This is achieved by using transition densities approximated by axis-aligned Gaussian mixtures together with methods to reduce the computational burden. A versatile cost function representation also employing Gaussian mixtures provides an increased freedom of modeling. Combining the prediction technique with this value function representation allows closed-form calculation of the necessary optimization problems arising from NMPC. The capabilities of the framework and especially the benefits that can be gained by considering the noise in the controller are illustrated by the example of a mobile robot following a given path. NOTATION x variable x random variable x vector-valued random variable ˜f x (x) probability density function of x f x (x) approximate of ˜f x (x) N (x − µ;σ 2) Gaussian density with mean µ and standard deviation σ Ex{x} expected value of x Jk(x k) value function Vk(x k,u k) input dependent value function gn(x n,u n) cost function k time index n time index of prediction horizon