Strategies for robust dynamic optimization in real time
Final Report Abstract
In this project we were able to utilize known SIP approaches, which are specific bi-level problems, for robust dynamic optimization. This robust satisfaction of path constraints in dynamic optimization can be interpreted as SIP extensions in two different ways. One way is the interpretation of the path constraints itself as infinite, because they have to be satisfied at every point in time. Recently we utilized this successfully to adapt the SIP algorithm from Mitsos (2011) to dynamic optimization problems. The other way is to interpret the parametric uncertainties within the process model as the infinite constraints while satisfying path constraints. To realize path constraint satisfaction under parametric uncertainties, we investigated the known SIP concept based on local reduction and the concept based on discretization in order to extend both to the dynamic case. In the concept based on local reduction, called KKT-embedding, the lower-level problem is successfully replaced with its necessary conditions of optimality in the dynamic optimization problem. The work is an extension of Diehl et al. (2008), in a way that not only endpoint constraints but also path constraints can be considered. In addition, with the extension it is also possible to consider worst-case points lying within the uncertainty set and not only those which are lying at the bounds. The investigations concerning the second concept based on discretization was threefold. One concept, which is comparable to the KKT-embedding, is the method which succesfully extended the SIP approach from Mitsos (2011) to the dynamic case. It is based on iteratively solving an upper- and lower-level problem with an increasing set of worst-case points in the upper level. These approaches are both approximations of the rigorous solution, which could be obtained with global solvers. Beyond this discretization approach one further concept based on discretization is investigated, which is a heuristic approach for robust dynamic optimization Puschke et al. (2017). The advantage of this method compared to the more rigorous ones before, is that it is easy to apply to large-scale systems and can be recaluated in each sampling interval within an eNMPC. The disadvantage of this method is that constraint satisfaction can not be guaranteed, but constraint violation is less likely. Beyond that, the basis discretization approach was further analyzed and improved in the context of SIPs without considering dynamics. Namely, a hybrid of two existing discretization algorithms for the global solution of SIPs was investigated. The goal of our future work is the incorporation of different robust dynamic optimization approaches into the two-stage NMPC, such that the approaches benefit from the advantages of both methods. Therefore, we will utilize our results and the results of the other project partners from this project. The two-stage NMPC presented by Lucia et al. considers a scenario tree, branching after the first two sampling intervals, where feedback information is considered before branching again. After two sampling intervals, the scenarios are fixed and no further branching is done. The branches contain a nominal model and each edge of the uncertainty set of the parameter values. Instead of considering all edges of the uncertainty set, the aim of the future work is to consider the worst-case models, because it will make the controller more efficient while keeping the robustness or even increasing it. In doing so, two of the investigated methods within this project for dynamic optimization considering parametric uncertainties will be utilized. One method is the adapted SIP approach based on discretization of the uncertain parameter set. With this approach all worst-case models are found iteratively, even those which are lying within the uncertainty set and not only the ones at the edges. The other method is the heuristic approach, in which some edges with a low sensitivity are deleted from the considered set, such that fewer branches will be required. In addition, with the combination of the approaches the introduced conservatism of the methods for robust dynamic optimization without considering feedback can be overcome.
Publications
- (2014). “Robust Dynamic Real-Time Optimization of a Semi-Batch Emulsion- Copolymerization Process with Uncertainties in the Parameter Values”. In: AIChE annual meeting: November 16-21, 2014, Atlanta, GA
J. Puschke and A. Mitsos
- (2015). “Local Optimization of Dynamic Programs with Guaranteed Satisfaction of Path Constraints”. In: Automatica 62, pp. 184–192
J. Fu, J.M.M. Faust, B. Chachuat, and A. Mitsos
(See online at https://doi.org/10.1016/j.automatica.2015.09.013) - (2015). “Local Optimization of Dynamic Systems with Guaranteed Feasibility of Path Constraints”. In: AIChE annual meeting: November 08-13, 2015, Salt Lake City
J.M.M. Faust, J. Fu, B. Chachuat, and A. Mitsos
- (2016). “Optimization of Dynamic Systems with Rigorous Path Constraint Satisfaction”. In: 26th European Symposium on Computer Aided Process Engineering. Vol. 38. Computer Aided Chemical Engineering. Elsevier, pp. 643–648. isbn: 9780444634283
J.M.M. Faust, J. Fu, B. Chachuat, and A. Mitsos
(See online at https://doi.org/10.1016/B978-0-444-63428-3.50112-0) - (2016). “Robust Dynamic Optimization of a Semi-Batch Emulsion Polymerization Process with Parametric Uncertainties - A Heuristic Approach”. In: Preprints, 11th IFAC Symposium on Dynamics and Control of Process Systems, including Biosystems, pp. 907–912
J. Puschke and A. Mitsos
(See online at https://doi.org/10.1016/j.ifacol.2016.07.305) - (2017). “A Hybrid Discretization Algorithm with Guaranteed Feasibility for the Global Solution of Semi-Infinite Programs”. In: Journal of Global Optimization 68.2, pp. 227–253
H. Djelassi and A. Mitsos
(See online at https://doi.org/10.1007/s10898-016-0476-7) - (2017). “Multi-Model Approach based on Parametric Sensitivities – A Heuristic Approximation for Dynamic Optimization of Semi-Batch Processes with Parametric Uncertainties”. In: Computers & Chemical Engineering 98, pp. 161–179
J. Puschke, A. Zubov, J. Kosek, and A. Mitsos
(See online at https://doi.org/10.1016/j.compchemeng.2016.12.004)
