Robust Optimization of Dynamic Systems

This thesis is about robust optimization, a class of mathematical optimization problems which arise frequently in engineering applications, where unknown process parameters and unpredictable external influences are present. Especially, if the uncertainty enters via a nonlinear differential equation, the associated robust counterpart problems are challenging to solve. The aim of this thesis is to develop computationally tractable formulations together with efficient numerical algorithms for both: finite dimensional robust optimization as well as robust optimal control problems. The first part of the thesis concentrates on robust counterpart formulations which lead to “min-max” or bilevel optimization problems. Here, the lower level maximization problem must be solved globally in order to guarantee robustness with respect to constraints. Concerning the upper level optimization problem, search routines for local minima are required. We discuss special cases in which this type of bilevel problems can be solved exactly as well as cases where suitable conservative approximation strategies have to be applied in order to obtain numerically tractable formulations. One main contribution of this thesis is the development of a tailored algorithm, the sequential convex bilevel programming method, which exploits the particular structure of nonlinear min-max optimization problems. The second part of the thesis concentrates on the robust optimization of nonlinear dynamic systems. Here, the differential equation can be affected by both: unknown time-constant parameters as well as time-varying uncertainties. We discuss set-theoretic methods for uncertain optimal control problems which allow us to formulate robustness guarantees with respect to state constraints. Algorithmic strategies are developed which solve the corresponding robust optimal control problems in a conservative approximation. Moreover, the methods are extended to open-loop controlled periodic systems, where additional stability aspects have to be taken into account. The third part is about the open-source optimal control software ACADO which is the basis for all numerical results in this thesis. After explaining the main algorithmic concepts and structure of this software, we elaborate on fast model predictive control implementations for small scale dynamic system as well as on an inexact sequential quadratic programming method for the optimization of large scale differential algebraic equations. Finally, the performance of the algorithms in ACADO is tested with robust optimization and robust optimal control problems which arise from various fields of engineering.