Stochastic Dynamic Optimization and Model Predictive Control based on Polynomial Chaos Theory and Symbolic Arithmetic

Polynomial chaos theory (PCT) quantifies the effects of probabilistic uncertainties on the states and outputs of linear and nonlinear dynamical systems using a model structure that is amenable for use in mathematical formulations for stochastic dynamic optimization and model predictive control. This article presents symbolic methods for automating the construction of exact PCT representations for dynamical systems described by ordinary differential equations (ODEs). The methods automatically construct exact PCT representations for both nonlinear dynamical systems and their linearizations with respect to the state. These methods are implemented using the Matlab Symbolic Toolbox and demonstrated in a case study on the stochastic model predictive control (SMPC) of an automated continuous-flow system for multi-step chemical synthesis. In the case study, SMPC based on a combination of PCT and symbolic algebra has an on-line computational time less than 12 milliseconds per time step, which is more than two orders of magnitude faster than required for real-time implementation.