Assessment and Future Directions of Nonlinear Model Predictive ...

Close-Loop Stochastic Dynamic Optimization 309Fig. 2. Back-off strategy within moving horizon; back-off from active constraintsthe set-point trajectory would be infeasible. By introducing a back-off from theactive constraints in the optimization, the region of the set-point trajectory ismoved inside the feasible region of the process to ensure, on the one hand, feasibleoperation, and to operate the process, on the other hand, still as closelyto the true optimum as possible. By this means, the black-marked area in Fig.2 illustrates the corrected bounds ỹ max of the hard constraints. Here, it shouldhowever be noted that due to the severe bound at the computation of the previoushorizon, the initial value at t 0 is rather far away from the constraint limitin the feasible area. Thus, in the first intervall of the current moving horizon,the bound is set at the original physical limit to avoid infeasibility. The back-offadjustment starts from the second interval, i.e. from the time point on, where thenext re-optimization begins. The size of ỹ max strongly depends on parametricuncertainty, disturbances, and the deviation by measurement errors. Thus, theconstraints in (8) within the moving horizon (8 intervals) are now reformulatedas follows with j =2, ..., 8, α =0.5, ˜T max =4K and ˜T ad, max =3K:T (j) ≤ 356 K − ˜T max · α (j−2) ; T ad (j) ≤ 500 K − ˜T ad, max · α (j−2) (7)The decision variable is the cooling flow rate. In order to test robustnesscharacteristics of the controler, the performances of the open-loop nominal solution,the nominal NMPC, and the NMPC with the proposed adaptive back-offapproach are compared under different disturbances, namely: catalyst activitymismatch and fluctuations of the reactor jacket cooling fluid temperature. Additionally,all measurements are corrupted with white noise e.g. component amount8% and temperature 2%.3.1 Dynamik Real–Time OptimizationThe size of the dynamic operating region around the optimum (see Fig. 2 right)is affected by fast disturbances. These are, however, efficiently buffered by theproposed regulatory NMPC-based approach. On the other hand, there are, infact, slowly time-varying non-zero mean disturbances or drifting model parameterswhich change the plant optimum with time. Thus, a online re-optimizationi.e. dynamic real-time optimization (D-RTO) may be indispensable for an optimaloperation. When on-line measurement gives access to the system state,