Sigma-point and stochastic gradient descent approach to solving global self-optimizing controlled variables

Abstract

Direct numerical optimization for the global self-optimizing control (gSOC) problem has been recently attempted in the rigorous nonlinear programming (NLP) framework. Compared with the previous perturbation-based SOC approaches, the global scheme is of potential to obtain solutions with better performances, as the economics are evaluated via the rigorous nonlinear process model, rather than approximations using the Taylor expansion. The main obstacles for solving the NLP are, however, difficulties for the statistical computations for the cost and constrained variables. In this paper, we firstly introduce the sigma-point approach, which generates less and more efficient sampling points with linear complexity with respect to the uncertain variables, such that the computational load is eased. Furthermore, we incorporate the stochastic gradient descent algorithm to accelerate the search of optimal combination matrix, which can be carried out upon evaluations of only a few, rather than all, sampling points. The scheme, therefore, makes it possible to deal with problems that have high dimensional uncertain parameters and/or when a single evaluation of the cost is time-consuming. A batch reactor and a batch distillation column are investigated to show the usefulness of the presented ideas.