Assessment and Future Directions of Nonlinear Model Predictive ...

A Nonlinear Model Predictive Control Framework as Free Software 235Table 1. Model dataC A0 10. mol/lC p ,C pj 4184. Jkg −1 K −1F 0 ,F 4.0 l/minE a/R 10080. Kk 0 6.20 × 10 14 mol m −3 s −1T 0 21.0◦ C◦ CT j0 26.0U 900. Wm −2 K −1V j 0.014 m 3α j 7.0 × 10 5 J/K(−∆H r) 33488. J/molρ, ρ j 1000. kg/m 3Table 2. Typical steady statesSteady states lower upperh 0.30 0.30 mC A 7.82 4.60 mol/lT r 31.5 40.1 ◦ CT j 28.0 28.0 ◦ CF j 14.0 48.8 l/minThe reactor temperature dynamics is described bydT rdt = F 0V (T 0 − T r ) −UAρC p V (T r − T j )+ (−∆H r)k 0 e −Ea/(R Tr) , (15)ρC pand the jacket temperature dynamics is described bydT jdt = 1[]ρ j C pj F j (T j0 − T j )+ UA(T r − T j ) , (16)ρ j C pj V j + α jwhere C pj is the specific heat capacity of the coolant, and F j is the coolantflow rate. The heat transfer area is calculated from A = π(r 2 +2rh)withr =0.237 m. Finally, the coefficient α j in (16) stands for the contribution ofthe wall and spiral baffle jacket thermal capacitances. A summary of the datamodel is given in Table 1. Two typical steady states of this system, one stableat a lower temperature and one unstable at an upper temperature, are given inTable 2. Further details on this model are provided in [23, 24].3.1 Simulation ResultsThe output variables are the reactor level and the temperature, y T =[hT r ],and the controls are the coolant flow rate and the outlet flow rate, u T =[F j F ].The following operating limits on the outputs and the controls are considered:0.08 h 0.41 m; T r 0; 0 F j 76 l/min; and 0 F 12 l/min.The results presented in Figure 2 were obtained assuming that the modelis perfect and that all the state variables are measured. The output terminalconstraints, integral action, control move rate constraints and constraintrelaxation were turned off. These results were obtained using predictive horizons(p, m) =(20, 5), a sampling time of 30 s, and diagonal weighting matricesQ yk = diag(5 × 10 2 , 10 5 )andQ uk = diag(10 −1 , 10 −3 ), k =1, ··· ,p.