Assessment and Future Directions of Nonlinear Model Predictive ...

314 H. Arellano-Garcia et al.The resulting trajectories of the reactor temperature concerning both strategiesare illustrated in Fig. 5. The figure shows, that the reactor temperatureresulted by the back-off strategy reaches very early a stationary value caused byfixed bounds of the temperature formulated in the corresponding optimizationproblem. The temperature curve of the stochastic approach shows more drasticalchanges with lower values of temperatures in earlier parts of the diagramand higher values later. This is caused by the fact, that with the considerationof uncertainties in advance, also the change of sensitivities of uncertain parameterstowards the reactor temperature can be taken into consideration by thestochastic approach. At the beginning in the diagram, the stochastic approachrealizes the matching of a more conservative strategy to higher sensitivities, andthus the operation achieves more robustness than the one achieved by the backoffstrategy. At the end of the curves, the decrease of sensitivities is used for acloser approach to the maximum temperature and thus leads to a better objectivevalue. Therefore, the strategy leads to an improvement of both, robustnessand the objective value.5 ConclusionsThe chance constrained optimization framework has been demonstrated to bepromising to address optimization and control problems under uncertainties.Feasibility and robustness with respect to input and output constraints havebeen achieved by the proposed approach. Thus, the solution of the problemhas the feature of prediction, robustness and being closed-loop. The resultingNMPC scheme embedded in the on-line re-optimization framework is viable forthe optimization of the reactor recipe while simultaneously guaranteeing theconstraints compliance, both for nominal operation as well as for cases of largedisturbances e.g. failure situation. In fact, the approach is relevant to all caseswhen uncertainty can be described by any kind of joint correlated multivariatedistribution function. The authors gratefully acknowledge the financial supportof the Deutsche Forschungsgemeinschaft (DFG).References[1] O. Abel and W. Marquardt. Scenario-integrated on-line optimization of batchreactors. Journal of Process Control, 13:703–715, 2003.[2] H. Arellano-G., W. Martini, M. Wendt, P. Li, and G. Wozny. Chance constrainedbatch distillation process optimization under uncertainty. In I.E. Grossmannand C.M. McDonald, editors, FOCAPO 2003 - fourth International Conferenceon Foundations of Computer-Aided Process Operations, pages 609–612. CACHE,CAST Division, AICHE, 2003.[3] H. Arellano-Garcia, W. Martini, M. Wendt, and G. Wozny. A New OptimizationFramework for Dynamic Systems under Uncertainty. Computer-Aided ChemicalEngineering, 18:553–559, 2004.[4] J. H. Lee and Z. H. Yu. Worst-case formulations of model predictive control forsystems with bounded parameters. Automatica, 33:765–781, 1997.