Effiziente Charakterisierung von Parameterunsicherheiten in dynamischen Prozessen für die robuste Optimierung

Abstract:
Robust optimization of batch processes is a challenging task in process engineering. When assuming parametric uncertainties, such problems can be formulated as a semi-infinite program (see e.g. Blankenship & Falk (1976)). In order to solve these problems efficiently, the infinite number of constraints has to be reduced. In this work we focus on the general concept of local reduction methods (as proposed by Hettich & Kortanek (1993)) for this purpose. Different methods for formulating robust optimization problems are presented (cf. Arellano-Garcia et al. (2003); Diehl et al. (2006); Mönnigmann et al. (2007); Diehl et al. (2008)). All of these methods require a defined set of the uncertain parameters. Therefore we present a novel method, which allows the approximation of the distribution of the uncertain parameters, even for non normally distributed parameters. The main advantage of this method is the robust approximation of the set of parameters. As most methods assume the parameters to be normally distributed, the robust optimization, if the parameters are not normally distributed and the (1 􀀀 )-interval is approximated incorrectly, may deliver systematically incorrect solutions. By updating the corresponding entry in the basis matrix, the new method enlarges the parameter space for the directions of the not normally distributed parameters. Thus, the (1􀀀)-interval is approximated more accurately and allows a robust meeting of the critical constraints. Applying the approximative worst-case formulation (Diehl et al., 2006, 2008) to industrial relevant examples shows, that the robust optimization can fail, even if the set of uncertain parameters is known. As the linearization approximates the uncertainties only locally, changes in the structure of the state trajectories due to nonlinear effects are not considered. Therefore, in this work, a new optimization method is presented, the sigmapoint approach, which uses so called sigmapoints to characterize the space of uncertain parameters. Due to the propagation of these sigmapoints through the model, this approach can handle the changes in the structure of the state trajectories due to nonlinearities. The main advantage of this approach is the low computing effort and its independence of the optimizer, as e.g. no derivatives are needed. Furthermore we present optimization results for different examples from process engineering.

Keywords:
Robust optimization, Sigmapoint, Unscented Transformation