Assessment and Future Directions of Nonlinear Model Predictive ...

NMPC of the Hashimoto Simulated Moving Bed Process 479(21). In order to guarantee that at least 70% of the mass of the components fedto the plant averaged over the prediction horizon leaves the plant in the extractproduct stream, an additional productivity requirement (19) is added. The deviationbetween the prediction of the model and the plant behavior is consideredby the error feedback term (23). The resulting mathematical formulation of theoptimization problem is:minβ I,β II,β III,β IV∑H PQ De,i + ∆βR∆β (15)i=1s.t. x i smb = x i smb,0 +x i+1∫ τt=0f smb (x smb (t),u(t),p)dt (16)smb,0 = Mxi smb,τ (17)H∗∑ PPur Ex,ii=1H PH∑Pm Ex,ii=1≥ ( PurEx,min ∗ − ∆P ur )Ex(18)H P≥ 0.7m Fe − ∆m Ex (19)Q I ≤ Q max (20)Q De ,Q Ex ,Q Fe ,Q Re ≥ 0, (21)where M is the shifting matrix, τ the period length. The extract purity, thepurity error, the mass output, and the mass error are evaluated according to:Pur Ex =τ∫t=0τ∫t=0c Ex,A dt(c Ex,A + c Ex,B )dt(22)∆P ur Ex = PurEx,plant,i−1 ∗ − Pur∗ Ex,model,i−1 (23)τ∫(c i,A + c i,B )Q i dt0m i =(24)τ∆m Ex = m Ex,plant,i−1 − m Ex,model,i−1 . (25)Since the plant is operated close to 100% extract purity, the purities are scaled(∗) according topurity ∗ =11 − purity , (26)that provides a large slope in the scaled purity for very high purities. The numericaltractability is improved by translating the degrees of freedom (period