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Analysis of different control possibilities for the divided wall column: feedback diagnoal and dynamic matrix control
European Symposiumon ComputerAided ProcessEngineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
283
Analysis of different control possibilities for the Divided Wall Column: feedback diagonal and Dynamic Matrix Control M. Serra a, M. Perrier b, A. Espuna a and L. Puigjaner a aDepartment of Chemical Engineering, Universitat Polit6cnica de Catalunya, Diagonal 647, Barcelona 08028, Spain bDepartment of Chemical Engineering, t~cole Polytechnique de Montr6al, C.P.6079, succ. Centre-ville, Montr6al H3C3A7, Canada This work addresses the control of the Divided Wall Column (DWC). Different control structures of diagonal feedback control are compared using MIMO linear analysis tools in the frequency domain. A controllability analysis of the process is done for the separation of different mixtures and for different operating conditions, including optimal operation. Application of Dynamic Matrix Control (DMC) to the DWC is evaluated. Through simulation, the ability of DMC for disturbance rejection and setpoint tracking is studied and compared to that of the feedback diagonal control. 1. I N T R O D U C T I O N The DWC is a non-conventional distillation arrangement for the separation of ternary mixtures. Its interest is based on its potential to save energy and reduce investment costs. Its design was proposed almost 50 years ago [ 1]. Since then, many authors have addressed design aspects [2], but operation and control have received much less attention [3,4]. In this work, different control strategies are compared, and the influence of operating conditions on the controllability are analysed. 2. C O N T R O L L A B I L I T Y STUDY OF T H R E E D I F F E R E N T S E P A R A T I O N S In order to develop the controllability study, a specific DWC design has been selected. It has 13 trays in the prefractionator and 33 trays in the main column. Counting trays from the bottom, the feed tray is tray 7 of the prefractionator. In the main column, the side product tray is 17, the last common tray before the wall is 8 and the first common tray after the wall is 26. The separation of three ternary mixtures (components called A, B and C) into 0.99 pure products has been studied. Products are liquid saturated flows and feeds are liquid saturated equimolar flows. The relative volatility and Easy of Separation Indexes (ESI) of mixture 1 are ct=(1 : 1.85 : 4.65) and ESI=l.36. For mixture 2, or=(1 : 2.15 : 4.65) and ESI=I. For mixture 3, c~=(1 : 2.45 : 4.65) and ESI=0.77. In a DWC with the three product compositions controlled, two extra degrees of freedom remain for optimisation. For the three studied separations, the nominal steady state operations have been optimised to minimise the boilup rate. Linear analysis tools are used to compare different composition control structures. The considered manipulated variables are L, V, D, B, S, SPLITD and SPLITB, where L is the
284 reflux, V the boilup, D the distillate, B the bottom flowrate, S the side product flowrate, SPLITD the liquid split at the top of the wall and SPLITB the vapour split at the bottom of the wall. With closed inventory control loops, the system is linearised. Four inventory control structures are considered, which are called "DB", "LB", "DV" and "LV". The first letter of the name is the manipulated variable that controls the condenser level and the second letter is the manipulated variable that controls the reboiler level. In Tables 1, 2 and 3, the best composition control structures for the different stabilised columns and for the different mixtures are shown. Morari Resilency Index (MRI) and Condition Number (CN) values indicated correspond to a frequency of 0.04 rad/min. This frequency is the one corresponding to the main open-loop time constant divided by ten. Intersivity Index (II=MRI/CN) is used to classify the structures. The structure with largest II is the preferred one. Table 1. Preferred structures for mixture 1 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" DBS LBS VDS LVS MRI=0.12 MRI=0.14 MRI=0.14 MRI=0.16 CN=14 CN=35 CN=34 CN=264 Table 2. Preferred structures for mixture 2 (analysis at s=0.04 rad/min) "LV" "DB . . . . LB . . . . DV" DBS LBS VDS LVS MRI=0.34 MRI=0.28 MRI=0.29 MRI=0.25 CN=4.6 CN=I 1 CN=12 CN=91 Table 3. Preferred structures for mixture 3 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" DBS LBS VDS LVS MRI=0.17 MRI=0.19 MRI=0.18 MRI=0.20 CN=10 CN=18 CN=28 CN = 184 It can be observed from tables 1, 2 and 3 that for all inventory controls, the best set of manipulated variables does not depend on the mixture. For all mixtures, the preferred structure with "DB" is the worst of the four preferred structures, and the preferred structure with "LV" is the best of the four preferred structures. Specifically CN is very large for control structures with "DB" inventory controls. In none of the cases, the preferred control structures include SPLITD or SPLITB as manipulated variables. The analysis of controllability indexes has been done at frequency 0.04 rad/min but it should be done at the closed-loop bandwidth frequency. The preferred set of manipulated variables is found to have a small dependence on the analysis frequency but MRI and CN can vary considerably with the frequency.
3. C O N T R O L L A B I L I T Y AT DIFFERENT OPERATING CONDITIONS In this section, the controllability of the same distillation process at three operating conditions has been studied. The separation of mixture 2 described in the previous section has
285 been chosen. Optimal operation has been compared with two non-optimal operations, indicated as operation 1 and operation 2. Optimal operation has SPLITD=0.634 and SPLITB=0.500. Operation 1 was found fixing SPLITD at 0.614 and SPLITB at 0.500. Boilup increased by 3%. Operation 2 was found fixing SPLITD at 0.654 and SPLITB at 0.500. Boilup increased by 10%. In Tables 4 and 5, the preferred control structures for the different stabilised columns at the different operations are shown. Since SPLITD will be easier to manipulate in practice, when controllability with SPLID and SPLITB are similar, SPLITD is chosen. Comparing Tables 2, 4 and 5, it can be noticed that at optimal operation, the preferred sets of manipulated variables do not include the split variables. On the contrary, at the non-optimal operations, the preferred sets of manipulated variables include the split variables. For all inventory controls, nonoptimal operation preferred control structures have better controllability indexes. Table 4. Preferred structures at operation 1 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" B S SPLITD B S SPLITB D S SPLITD L S SPLITD MRI=0.67 MRI-0.61 MRI=0.69 MRI=0.70 CN=2.11 CN=3.86 CN=3.82 CN-23.18 Table 5. Preferred ;tructures at operation 2 (analysis at s=0.04 rad/min) "LV" "DV" "LB" "DB" B S SPLITD D S SPLITB D S SPLITD L S SPLITD MRI=I.03 MRI=0.71 MRI=0.91 MRI =1.07 CN=2.61 CN=2.04 CN=2.95 CN = 16.09 3.1. "DB" inventory control Relative Gain Array (RGA) of the preferred structures indicate L SPLITD S as the best pairing for the non-optimal operations and L S V for the optimal operation. According to RGA, L SPLITD S for operation 2 has better controllability than L S V for the optimal operation. PI controllers with P=I and I-0.0125 are implemented on each control loop and the bandwidths are determined. Indexes at the bandwidth indicate the same preferred structures indicated at frequency 0.04 rad/min. The CN of the optimal operation at the bandwidth frequency is very high. According to the results, controllability of the non-optimal operations is better. Simulations show good performance of the L SPLITD S control structure. Thus, it is possible to make a trade off between controllability and energy optimality. 3.2. "LV" inventory control RGA plots indicate that S SPLITD B, D S B and D SPLITD S are the best pairings of the preferred structures. With PI controllers of P-1 and I=0.0125 to all loops, the bandwidths are determined. At the bandwidth frequency, the preferred structure for operation 1 is D B S (MRI=2.7 and CN=2.4). For operation 2, it is D S SPLITD (MRI-1.3 and CN=2.7). At different frequencies, singular value decomposition indicates different preferred structures because of similar controllability indexes of the different control structures. For optimal operation, the best structure is D S B (MRI=0.7 and CN=I 8). Controllability indexes for nonoptimal operations are better. A stability analysis through w~*T~ maximum singular value indicate robust stability for the three operations with their preferred structures (w~ is the uncertainty in input channels and TI the closed-loop transfer function at the input) [5].
286 4. DYNAMIC MATRIX CONTROL Assuming that inventory control is solved at a lower control level, three manipulated variables are used for the control of the three product compositions. A 3x3 system has to be solved by the DMC. The tuning parameters are: At (sampling period), n (identification horizon), p (prediction horizon), m (control horizon) and k2 (move suppression factor). In some cases treated later, k2 is substituted by )~l, )~2 and )~3 (suppression factors for the individual inputs) [6]. System identification is performed applying step changes to the manipulated variables in open loop. The sign and the size of the steps have been found to have a very large influence over the identification of the DWC and over the DMC system derived from this identification. The main reason is the DWC non-linearity. Depending on the size and sign of the step change used for identification, the control converges or diverges. The observed inverse responses for several identification profiles are not responsible for control divergence, but changes in steady state gains. Small errors in the identification profiles can give the model the knowledge that when L increases the same amount than V, the purity of B decreases. But in the linear region, when L increases the same amount than V, B purity increases. This wrong sign makes the system diverge. Identification within the linear region is needed, which is not typically feasible in a real plant. In the previous sections, the "LV" D S B control structure was found to be advantageous. However, to apply DMC with this control structure has a major problem due to the open-loop instablility of the inventory control structure. The problem appears at the early stage of identification. Because of the instability, final responses do not exist. In this sense, PI control strategy is advantageous because it can be applied to an open-loop unstable system. To compare PI control and DMC strategies, discrete PI are considered and simulated. "DB" stabilised columns with L S V control structures are considered. At= 1 has been imposed to both control strategies. DMC could be in principle a better control approach because it takes into account interactions. However, the DMC depends highly on the identification of a non-linear system into a simple model. PI may have the advantage that interactions favour naturally the rejection of disturbances. Some distillation examples will help to compare these two control strategies.
4.1. Setpoint tracking The same separation described as mixture 2 is studied. With tuning parameters At=l, n=600, p=300, m=6, k2=100, a set point change of +0.001 in A purity is simulated. It is seen in Fig. 1 that convergence is extremely slow. Different move suppression factors in the different inputs are implemented. However, very small differences have been found. All the other parameters have been changed and a tuning achieving faster control and smaller overshoots has not been found. To compare DMC and PI control, the same setpoint change is simulated with PI control. Tuning of loops for composition of A, B and C is P=15, I=0.5; P=-2, I=-0.03; P=15, I=0.5, respectively. As seen in Fig. 2, at time=500 min, the set points of all three outputs are achieved (much faster than the DMC). On the other hand, overshoot of output 3 is larger than that of DMC. A tuning reducing this overshoot has not been found.
287 x 10 .4
X 10 .4
8
A
_o 4
B
-8 0
400
Time (min) 1400
2000
0
Fig. 1. Compositions with DMC
200
Time (min)
800
1000
Fig. 2. Compositions with PI
For set point changes in B and C purity, similar results are found: the PI can reach set points much faster but with larger deviation for the outputs that are not asked to be changed. Therefore, if the main objective is to achieve setpoint changes quickly, PI is better. If the main objective is to keep the other outputs constant when one output is changed, DMC is better. 4.2. D i s t u r b a n c e rejection
Rejection of a disturbance in A feed composition by the two control strategies is compared through simulation. The DMC tuning is At=l, n=600, p=300, m-6, )~1=14, L2=14, )~3=60. Reducing more the move suppression factors, the overshoots can still be reduced, but profiles begin to be very irregular. PI tuning for A, B and C purity loops is P=8, I=0.1; P=-8, I=-0.1; P=8, I=0.1, respectively. Greater overshoots are found with DMC and input variables vary more. As found for setpoint changes, the time response is shorter with PI. In Figures 3 and 4, the input profiles of the DMC and PI described simulations are shown. A similar behaviour is found for a disturbance in B feed composition: deviation is smaller and response time shorter with PI control. For the rejection of a feed flowrate disturbance, a greater overshoot is found with DMC, but it gives faster response. L and V increase with the same ratio. 0.3 0.02
0.2
0.01
0.1 -0.01
L S 1000
Time (min)
4000
6000
Fig. 3. Input changes with DMC
-0.02 0
100 Time (min) 400 Fig. 4. Input changes with PI
500
288 Contrary to what happened for set point changes, for disturbance rejection, the overshoots of the variables are larger with DMC than with PI. Both strategies are affected by the directionality of the DWC system and have slow responses when L and V increase with the same ratio. However, PI deals better with the problem. DMC certainly uses the information of what is the influence of L, V and S over all the outputs when L, V and S play alone. When L is the only input variable that changes, B and D also change and the purity of the products is affected by these changes in D and B. However, when L and V both change at the same time, D and B can remain almost unchanged and the purity of the products is only affected by the change of the internal variables (L and V) and not by the external variables (D and B). In this case, much larger changes in L are needed to increase the purity of product A. The DWC non-linearity makes DMC limited. Non-linear models for Model Predictive Control should be compared to DMC in future work. 5. CONCLUSIONS Different composition control structures of diagonal feedback control are compared through MRI, CN and RGA. The separation of three different mixtures at optimal operation has been studied. The preferred control structures do not include the split variables and are the same for the three mixtures. DWC with "DB" stabilisation present very high CN. Simulations show that "LV" stabilisation and D S B composition control is a good control structure for the DWC. Three operating conditions including the optimal one have been compared. At optimal operation, split variables are not in the set of preferred manipulated variables but they are for non-optimal operations. Controllability of non-optimal operations is better, indicating a possible trade off between controllability and energy optimisation. With "DB', stabilisation, robust stability of non-optimal operations should be further studied. However, with "LV" stabilisation, all operation conditions present robust stability. A preliminary study is done applying DMC to the DWC. Some conclusions are obtained from the studied examples. "DB", L S V control structure is considered. For setpoint tracking, the DMC presents smaller deviations but longer response time. For disturbance rejection, PI presents smaller deviations and better response time. In general, DMC has been found to be quite limited for the control of the DWC. REFERENCES
1. Wright, R. O. U. S. Patent 2, 471,134, May 24, 1949. 2. C. Triantafyllow and R. Smith, The Design and Optimisation of Fully Thermally Coupled Distillation Columns, Trans. Inst. Chem. Eng. 70, 118-132, 1992. 3. E. A. Wolff and S. Skogestad, Operation of Integrated Three-Product (Petlyuk) Distillation Columns, Ind. Eng. Chem. Res. 34, 2094-2103, 1995. 4. M. Serra, A. Espuna and L. Puigjaner, Control and Optimisation of the Divided Wall Column, Chem. Eng. Process. 38, 549-562, 1999. 5. W.L. Luyben, Practical Distillation Control, Van Nostrand Reinhold, New York, 1992. 6. B. A. Ogunnaike and W. H. Ray, Process Dynamics, Modelling, and Control, Oxford University Press, New York, 1994.
283
Analysis of different control possibilities for the Divided Wall Column: feedback diagonal and Dynamic Matrix Control M. Serra a, M. Perrier b, A. Espuna a and L. Puigjaner a aDepartment of Chemical Engineering, Universitat Polit6cnica de Catalunya, Diagonal 647, Barcelona 08028, Spain bDepartment of Chemical Engineering, t~cole Polytechnique de Montr6al, C.P.6079, succ. Centre-ville, Montr6al H3C3A7, Canada This work addresses the control of the Divided Wall Column (DWC). Different control structures of diagonal feedback control are compared using MIMO linear analysis tools in the frequency domain. A controllability analysis of the process is done for the separation of different mixtures and for different operating conditions, including optimal operation. Application of Dynamic Matrix Control (DMC) to the DWC is evaluated. Through simulation, the ability of DMC for disturbance rejection and setpoint tracking is studied and compared to that of the feedback diagonal control. 1. I N T R O D U C T I O N The DWC is a non-conventional distillation arrangement for the separation of ternary mixtures. Its interest is based on its potential to save energy and reduce investment costs. Its design was proposed almost 50 years ago [ 1]. Since then, many authors have addressed design aspects [2], but operation and control have received much less attention [3,4]. In this work, different control strategies are compared, and the influence of operating conditions on the controllability are analysed. 2. C O N T R O L L A B I L I T Y STUDY OF T H R E E D I F F E R E N T S E P A R A T I O N S In order to develop the controllability study, a specific DWC design has been selected. It has 13 trays in the prefractionator and 33 trays in the main column. Counting trays from the bottom, the feed tray is tray 7 of the prefractionator. In the main column, the side product tray is 17, the last common tray before the wall is 8 and the first common tray after the wall is 26. The separation of three ternary mixtures (components called A, B and C) into 0.99 pure products has been studied. Products are liquid saturated flows and feeds are liquid saturated equimolar flows. The relative volatility and Easy of Separation Indexes (ESI) of mixture 1 are ct=(1 : 1.85 : 4.65) and ESI=l.36. For mixture 2, or=(1 : 2.15 : 4.65) and ESI=I. For mixture 3, c~=(1 : 2.45 : 4.65) and ESI=0.77. In a DWC with the three product compositions controlled, two extra degrees of freedom remain for optimisation. For the three studied separations, the nominal steady state operations have been optimised to minimise the boilup rate. Linear analysis tools are used to compare different composition control structures. The considered manipulated variables are L, V, D, B, S, SPLITD and SPLITB, where L is the
284 reflux, V the boilup, D the distillate, B the bottom flowrate, S the side product flowrate, SPLITD the liquid split at the top of the wall and SPLITB the vapour split at the bottom of the wall. With closed inventory control loops, the system is linearised. Four inventory control structures are considered, which are called "DB", "LB", "DV" and "LV". The first letter of the name is the manipulated variable that controls the condenser level and the second letter is the manipulated variable that controls the reboiler level. In Tables 1, 2 and 3, the best composition control structures for the different stabilised columns and for the different mixtures are shown. Morari Resilency Index (MRI) and Condition Number (CN) values indicated correspond to a frequency of 0.04 rad/min. This frequency is the one corresponding to the main open-loop time constant divided by ten. Intersivity Index (II=MRI/CN) is used to classify the structures. The structure with largest II is the preferred one. Table 1. Preferred structures for mixture 1 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" DBS LBS VDS LVS MRI=0.12 MRI=0.14 MRI=0.14 MRI=0.16 CN=14 CN=35 CN=34 CN=264 Table 2. Preferred structures for mixture 2 (analysis at s=0.04 rad/min) "LV" "DB . . . . LB . . . . DV" DBS LBS VDS LVS MRI=0.34 MRI=0.28 MRI=0.29 MRI=0.25 CN=4.6 CN=I 1 CN=12 CN=91 Table 3. Preferred structures for mixture 3 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" DBS LBS VDS LVS MRI=0.17 MRI=0.19 MRI=0.18 MRI=0.20 CN=10 CN=18 CN=28 CN = 184 It can be observed from tables 1, 2 and 3 that for all inventory controls, the best set of manipulated variables does not depend on the mixture. For all mixtures, the preferred structure with "DB" is the worst of the four preferred structures, and the preferred structure with "LV" is the best of the four preferred structures. Specifically CN is very large for control structures with "DB" inventory controls. In none of the cases, the preferred control structures include SPLITD or SPLITB as manipulated variables. The analysis of controllability indexes has been done at frequency 0.04 rad/min but it should be done at the closed-loop bandwidth frequency. The preferred set of manipulated variables is found to have a small dependence on the analysis frequency but MRI and CN can vary considerably with the frequency.
3. C O N T R O L L A B I L I T Y AT DIFFERENT OPERATING CONDITIONS In this section, the controllability of the same distillation process at three operating conditions has been studied. The separation of mixture 2 described in the previous section has
285 been chosen. Optimal operation has been compared with two non-optimal operations, indicated as operation 1 and operation 2. Optimal operation has SPLITD=0.634 and SPLITB=0.500. Operation 1 was found fixing SPLITD at 0.614 and SPLITB at 0.500. Boilup increased by 3%. Operation 2 was found fixing SPLITD at 0.654 and SPLITB at 0.500. Boilup increased by 10%. In Tables 4 and 5, the preferred control structures for the different stabilised columns at the different operations are shown. Since SPLITD will be easier to manipulate in practice, when controllability with SPLID and SPLITB are similar, SPLITD is chosen. Comparing Tables 2, 4 and 5, it can be noticed that at optimal operation, the preferred sets of manipulated variables do not include the split variables. On the contrary, at the non-optimal operations, the preferred sets of manipulated variables include the split variables. For all inventory controls, nonoptimal operation preferred control structures have better controllability indexes. Table 4. Preferred structures at operation 1 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" B S SPLITD B S SPLITB D S SPLITD L S SPLITD MRI=0.67 MRI-0.61 MRI=0.69 MRI=0.70 CN=2.11 CN=3.86 CN=3.82 CN-23.18 Table 5. Preferred ;tructures at operation 2 (analysis at s=0.04 rad/min) "LV" "DV" "LB" "DB" B S SPLITD D S SPLITB D S SPLITD L S SPLITD MRI=I.03 MRI=0.71 MRI=0.91 MRI =1.07 CN=2.61 CN=2.04 CN=2.95 CN = 16.09 3.1. "DB" inventory control Relative Gain Array (RGA) of the preferred structures indicate L SPLITD S as the best pairing for the non-optimal operations and L S V for the optimal operation. According to RGA, L SPLITD S for operation 2 has better controllability than L S V for the optimal operation. PI controllers with P=I and I-0.0125 are implemented on each control loop and the bandwidths are determined. Indexes at the bandwidth indicate the same preferred structures indicated at frequency 0.04 rad/min. The CN of the optimal operation at the bandwidth frequency is very high. According to the results, controllability of the non-optimal operations is better. Simulations show good performance of the L SPLITD S control structure. Thus, it is possible to make a trade off between controllability and energy optimality. 3.2. "LV" inventory control RGA plots indicate that S SPLITD B, D S B and D SPLITD S are the best pairings of the preferred structures. With PI controllers of P-1 and I=0.0125 to all loops, the bandwidths are determined. At the bandwidth frequency, the preferred structure for operation 1 is D B S (MRI=2.7 and CN=2.4). For operation 2, it is D S SPLITD (MRI-1.3 and CN=2.7). At different frequencies, singular value decomposition indicates different preferred structures because of similar controllability indexes of the different control structures. For optimal operation, the best structure is D S B (MRI=0.7 and CN=I 8). Controllability indexes for nonoptimal operations are better. A stability analysis through w~*T~ maximum singular value indicate robust stability for the three operations with their preferred structures (w~ is the uncertainty in input channels and TI the closed-loop transfer function at the input) [5].
286 4. DYNAMIC MATRIX CONTROL Assuming that inventory control is solved at a lower control level, three manipulated variables are used for the control of the three product compositions. A 3x3 system has to be solved by the DMC. The tuning parameters are: At (sampling period), n (identification horizon), p (prediction horizon), m (control horizon) and k2 (move suppression factor). In some cases treated later, k2 is substituted by )~l, )~2 and )~3 (suppression factors for the individual inputs) [6]. System identification is performed applying step changes to the manipulated variables in open loop. The sign and the size of the steps have been found to have a very large influence over the identification of the DWC and over the DMC system derived from this identification. The main reason is the DWC non-linearity. Depending on the size and sign of the step change used for identification, the control converges or diverges. The observed inverse responses for several identification profiles are not responsible for control divergence, but changes in steady state gains. Small errors in the identification profiles can give the model the knowledge that when L increases the same amount than V, the purity of B decreases. But in the linear region, when L increases the same amount than V, B purity increases. This wrong sign makes the system diverge. Identification within the linear region is needed, which is not typically feasible in a real plant. In the previous sections, the "LV" D S B control structure was found to be advantageous. However, to apply DMC with this control structure has a major problem due to the open-loop instablility of the inventory control structure. The problem appears at the early stage of identification. Because of the instability, final responses do not exist. In this sense, PI control strategy is advantageous because it can be applied to an open-loop unstable system. To compare PI control and DMC strategies, discrete PI are considered and simulated. "DB" stabilised columns with L S V control structures are considered. At= 1 has been imposed to both control strategies. DMC could be in principle a better control approach because it takes into account interactions. However, the DMC depends highly on the identification of a non-linear system into a simple model. PI may have the advantage that interactions favour naturally the rejection of disturbances. Some distillation examples will help to compare these two control strategies.
4.1. Setpoint tracking The same separation described as mixture 2 is studied. With tuning parameters At=l, n=600, p=300, m=6, k2=100, a set point change of +0.001 in A purity is simulated. It is seen in Fig. 1 that convergence is extremely slow. Different move suppression factors in the different inputs are implemented. However, very small differences have been found. All the other parameters have been changed and a tuning achieving faster control and smaller overshoots has not been found. To compare DMC and PI control, the same setpoint change is simulated with PI control. Tuning of loops for composition of A, B and C is P=15, I=0.5; P=-2, I=-0.03; P=15, I=0.5, respectively. As seen in Fig. 2, at time=500 min, the set points of all three outputs are achieved (much faster than the DMC). On the other hand, overshoot of output 3 is larger than that of DMC. A tuning reducing this overshoot has not been found.
287 x 10 .4
X 10 .4
8
A
_o 4
B
-8 0
400
Time (min) 1400
2000
0
Fig. 1. Compositions with DMC
200
Time (min)
800
1000
Fig. 2. Compositions with PI
For set point changes in B and C purity, similar results are found: the PI can reach set points much faster but with larger deviation for the outputs that are not asked to be changed. Therefore, if the main objective is to achieve setpoint changes quickly, PI is better. If the main objective is to keep the other outputs constant when one output is changed, DMC is better. 4.2. D i s t u r b a n c e rejection
Rejection of a disturbance in A feed composition by the two control strategies is compared through simulation. The DMC tuning is At=l, n=600, p=300, m-6, )~1=14, L2=14, )~3=60. Reducing more the move suppression factors, the overshoots can still be reduced, but profiles begin to be very irregular. PI tuning for A, B and C purity loops is P=8, I=0.1; P=-8, I=-0.1; P=8, I=0.1, respectively. Greater overshoots are found with DMC and input variables vary more. As found for setpoint changes, the time response is shorter with PI. In Figures 3 and 4, the input profiles of the DMC and PI described simulations are shown. A similar behaviour is found for a disturbance in B feed composition: deviation is smaller and response time shorter with PI control. For the rejection of a feed flowrate disturbance, a greater overshoot is found with DMC, but it gives faster response. L and V increase with the same ratio. 0.3 0.02
0.2
0.01
0.1 -0.01
L S 1000
Time (min)
4000
6000
Fig. 3. Input changes with DMC
-0.02 0
100 Time (min) 400 Fig. 4. Input changes with PI
500
288 Contrary to what happened for set point changes, for disturbance rejection, the overshoots of the variables are larger with DMC than with PI. Both strategies are affected by the directionality of the DWC system and have slow responses when L and V increase with the same ratio. However, PI deals better with the problem. DMC certainly uses the information of what is the influence of L, V and S over all the outputs when L, V and S play alone. When L is the only input variable that changes, B and D also change and the purity of the products is affected by these changes in D and B. However, when L and V both change at the same time, D and B can remain almost unchanged and the purity of the products is only affected by the change of the internal variables (L and V) and not by the external variables (D and B). In this case, much larger changes in L are needed to increase the purity of product A. The DWC non-linearity makes DMC limited. Non-linear models for Model Predictive Control should be compared to DMC in future work. 5. CONCLUSIONS Different composition control structures of diagonal feedback control are compared through MRI, CN and RGA. The separation of three different mixtures at optimal operation has been studied. The preferred control structures do not include the split variables and are the same for the three mixtures. DWC with "DB" stabilisation present very high CN. Simulations show that "LV" stabilisation and D S B composition control is a good control structure for the DWC. Three operating conditions including the optimal one have been compared. At optimal operation, split variables are not in the set of preferred manipulated variables but they are for non-optimal operations. Controllability of non-optimal operations is better, indicating a possible trade off between controllability and energy optimisation. With "DB', stabilisation, robust stability of non-optimal operations should be further studied. However, with "LV" stabilisation, all operation conditions present robust stability. A preliminary study is done applying DMC to the DWC. Some conclusions are obtained from the studied examples. "DB", L S V control structure is considered. For setpoint tracking, the DMC presents smaller deviations but longer response time. For disturbance rejection, PI presents smaller deviations and better response time. In general, DMC has been found to be quite limited for the control of the DWC. REFERENCES
1. Wright, R. O. U. S. Patent 2, 471,134, May 24, 1949. 2. C. Triantafyllow and R. Smith, The Design and Optimisation of Fully Thermally Coupled Distillation Columns, Trans. Inst. Chem. Eng. 70, 118-132, 1992. 3. E. A. Wolff and S. Skogestad, Operation of Integrated Three-Product (Petlyuk) Distillation Columns, Ind. Eng. Chem. Res. 34, 2094-2103, 1995. 4. M. Serra, A. Espuna and L. Puigjaner, Control and Optimisation of the Divided Wall Column, Chem. Eng. Process. 38, 549-562, 1999. 5. W.L. Luyben, Practical Distillation Control, Van Nostrand Reinhold, New York, 1992. 6. B. A. Ogunnaike and W. H. Ray, Process Dynamics, Modelling, and Control, Oxford University Press, New York, 1994.