Assessment and Future Directions of Nonlinear Model Predictive ...

Receding-Horizon Estimation and Control of Ball Mill Circuits 4914 Numerical ResultsIn this section, the combined scheme (software sensor + NMPC) is evaluated insimulation, using a typical test run corresponding to a setpoint change. Point1 in Figure 2 (where y =[0.86 0.35] T ) is the initial operating condition, andpoint 2 (with y =[0.90 0.31] T ) represents the target (this point correspondsto a higher product fineness and near maximum product flow rate). The samplingperiod T s = 5 min and the prediction horizon is 80 min (Np = 16). Twomanipulated variable moves are used (Nu = 2) and the weighting matrix Q i ischosen as a constant identity matrix. Amplitude saturations are qCmaxand Reg max = 100. Limits for the rates of change are ∆qCmaxand ∆Reg max = 80. Limits on the mill flow rate are q min=60 tonhour= 15 tonhourtonM = 60hourandqMmax ton=90hour. The observer parameters are defined in Section 3.2.It is first assumed that the process model is accurate and that the measurementsare noise free. Figure 4 shows the controlled and the manipulated variables (solidlines). The performance is satisfactory, the controlled variables reach the setpointafter about 20 min and the steady state is obtained after 70 min. Moreover, constraintsare active in the first 5 min (first sample), as the maximum register displacementand the maximum rate of change of the input flow rate are required.The performance of the control scheme is then tested when measurementsare subject to a noise with a maximum absolute error of 0.02 tonm(around a 5%maximum relative error). The software sensor can efficiently take these stochasticdisturbances into account, and the performance of the control scheme remainsquite satisfactory.Finally, the influence of parametric uncertainties (plant-model mismatch dueto errors at the identification stage) is investigated. A parametric sensitivityanalysis is performed, and Figure 5 shows step responses corresponding to eitheran accurate model or to a −10% error in the fragmentation rate or the transportvelocity. Clearly, fragmentation parameters (which represents material hardness0.920.315w P;30.910.90.890.880.870.86w M;20.310.850 20 40 60 80 1000.3050 20 40 60 80 100100408035Reg (%)60q C(t/h)304025200 20 40 60 80 100time (min)200 20 40 60 80 100time (min)Fig. 4. NMPC and state observer; solid: noise-free measurements, dashed: measurementscorrupted by white noise