Assessment and Future Directions of Nonlinear Model Predictive ...

462 P. Kühl et al.4.1 Min-Max NMPC Simulation ResultsFor the numerical solution of (5) the direct multiple shooting approach was used.The NMPC settings are the same as described in Section 3.2. The confidencefactor γ was slowly increased from zero to the desired level. For each optimizationwith a respective γ, the previous result has been used to initialize the states andcontrol. In the case presented here, the initial amount of catalyst K 1 is assumedto be uncertain. The standard deviation of K 1 is 0.17 g. The confidence levelhas been chosen to be 99.7 %, i.e. we have γ = 3 to obtain the 3σ-interval.Figure 2 shows the simulation results for the robust version of the NMPC for thenominal amount of catalyst and an increased amount. The solution is comparedto the solution of the nominal NMPC for the nominal catalyst amount chargedto the reactor. One can see that the robust solution strongly resembles thenominal solution. Only, the dosing is stopped earlier. This ensures that lessA is accumulating in the reactor and leaves a safety margin to the adiabatictemperature as can be seen in the lower right graph. When the robust NMPCcontroller is confronted with a plant-model mismatch it reacts by dosing A morecarefully. The singular sub-arc becomes flatter than computed with the nominalNMPC. Eventually, less A is present in the reactor and the temperature peakgets lower. This also leads to a slower consumption of B which, however, isaccounted for by a higher reactor temperature at the end of the batch so thatthe final productivity losses are very small.The safety margins for the reactor temperature and the adiabatic temperatureare the main feature of the robust NMPC scheme. The fact that they are also[mol/s][K]43210330320310300x 10 −3Dosing rate u(t)0 1000 2000 3000time [s]Reactor temperatureSafetymargin2900 1000 2000 3000time [s][mol][K]76543212−butanol00 1000 2000 3000time [s]360350340330320310300NMPC: 5.01 gRNMPC: 5.01 gRNMPC: 5.10 gAdiabatic temperatureSafetymargin2900 1000 2000 3000time [s]Fig. 2. Simulation results of robust (RNMPC) compared with nominal NMPC. TheRNMPC is shown for the nominal and a perturbed initial amount of catalyst K 1.Dashed lines denote constraints. Note the RNMPC safety margins.