Numerical Methods for Scenario Tree Nonlinear Model Predictive Control

In this thesis we propose new methods in the field of numerical mathematics and stochastics for a model-based optimization method to control dynamical systems under uncertainty. In model-based control the model-plant mismatch is often large and unforeseen external influences on the dynamics must be taken into account. Therefore we extend the dynamical system by a stochastic component and approximate it by scenario trees. The combination of Nonlinear Model Predictive Control (NMPC) and the scenario tree approach to robustify with respect to the uncertainty is of growing interest. In engineering practice scenario tree NMPC yields a beneficial balance of the conservatism introduced by the robustification with respect to the uncertainty and the controller performance. However, there is a high numerical effort to solve the occuring optimization problems, which heavily depends on the design of the scenario tree used to approximate the uncertainty. A big challenge is then to control the system in real-time. The contribution of this work to the field of numerical optimization is a structure exploiting method for the large-scale optimization problems based on dual decomposition that yields smaller subproblems. They can be solved in a massively parallel fashion and additionally their discretization structure can be exploited numerically. Furthermore, this thesis presents novel methods to generate suitable scenario trees to approximate the uncertainty. Our scenario tree generation based on quadrature rules for sparse grids allows for scenario tree NMPC in high-dimensional uncertainty spaces with approximation properties of the quadrature rules. A further novel approach of this thesis to generate scenario trees is based on the interpretation of the underlying stochastic process as a Markov chain. Under the Markovian assumption we provide guarantees for the scenario tree approximation of the uncertainty. Finally, we present numerical results for scenario tree NMPC. We consider dynamical systems from the chemical industry and demonstrate that the methods developed in this thesis solve optimization problems with large scenario trees in real-time