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Improved confidence in design through controllability analysis
Copyright e IFAC InIegration of Procca. DeIign IIJId Ca1tro1, Baltimore, Maryllnd, USA, 1994
IMPROVED CONFIDENCE IN DESIGN THROUGH CONTROLLABILITY ANALYSIS BY
C.M.CALLAN(l) and T.I.MALIK(2) (1) ICI Chemicals and Polymers Ltd, Runcorn Heath, Cheshire, England.
(2) ICI Engineering, Brunner House, Northwich, Cheshire, England. This paper concerns the interaction between process design and process control in that a steady state model used for the process design has been further exploited to give controllability indicators. These were used to improve confidence in a pre-selected control structure. ICI's sequential modular package FLOWPACK was used for process design and for generating the G(O) matrix that can then be used for obtaining controllability indicators. Several sets of control inputs and outputs were tested both to develop confidence in control system design as well as overcoming inherent shortcomings in the choice of control inputs and outputs within the flowsheeting package.
INTRODUCTION
LIMITATIONS TO CONTROL INPUTS AND OUTPUTS ~CH CAN BE SPECIFIED IN FLOWPACK
This is an example of the application of controllability analysis at a late stage in a project. An extension on a plant had already been designed and the control structure (illustrated in Fig. 1) had been specified by the project control engineer. It was desired to improve confidence in this design by an independent means for example a dynamic model. This would have required translation of the steady state FLOWP ACK model used for process design into a dynamic model in another package. There was not sufficient time or resource available to do this. (With an Equation Based Package, as opposed to sequential modular as in FLOWPACK, this transfer from steady state to dynamic mode should be easier. See for example Garlick et al. 1993).
FLOWPACK, as a steady state package, calculates a number of outputs (results) given a number of inputs (specifications, for example, feed stream flow rates and conditions or reactor conversions). In order to obtain the 0(0) matrix it is required to specify control inputs (measured variables) and control outputs (the manipulated variables). There are restrIctIons in FLOWPACK, as there would be in any sequential modular package, on what these can be:
LIMITATION 1 Not all the variables are available at the problem input language level where the 0(0) calculations are requested. The variables that can be selected as control inputs or control outputs must either exist in the problem input file or be relatable to the same via simple additional equations that are permitted in FLOWPACK. Thus, detailed tray to tray temperature calculations, although they are carried out, are not passed on to the next unit in the calculation sequence. They do not need to exist as variables in the input file (with the exception of top and bottom temperatures) and hence cannot be specified as control outputs.
Controllability analysis was possible, however, as FLOWPACK does enable 0(0) matrices to be obtained by only a few additions to the input problem file. These can then be analysed by PCTB (Process Controllability ToolBox), (see Fararooyet al. 1992 and Oglesby et al. 1992), from which a number of controllability indicators are obtained. This analysis can be completed quite quickly and it was decided to pursue this as a means of getting further insight and confidence in the selected control structure.
17
LIMITATION 2 Not all combinations of inputs are legal, for example, in distillation unit operations, only certain combinations of inputs such as reflux ratio, reboiler ratio, product purity, product rates and reboiler and condenser heat duties are allowed. This may mean that a desired control input can not be used in the model. STRATEGY In order to build confidence in the analysis and the pre-defined control structure we chose to carry out analysis on a range of related control structures rather than only the pre-defined control structure. This also provided a way of handling the limitations mentioned above. Outputs that are closest to the actual desired ones are selected if limitation (I) applies.
Fig. 1 The Process and Control Scheme Diagram Key :
In order to carry out a number of related runs as well as to overcome limitation (2) a number of different steady state runs with different variables specified are carried out that otherwise give the same results. This is illustrated schematically in Fig. 2 for a model with 3 inputs and 4 outputs. Fig. 2A uses inputs A, B and C. For controllability we want to use F as an input but F and B are not legal combinations. To overcome this the value of F obtained from Fig. 2a is used as an input in Fig. 2b while the values of A and C are identical. Then the effect on the control outputs can be observed. If G was the control output of interest then the G(O) element between G and F can be obtained by perturbing F. THE PROCESS AND CONTROL SCHEME
F:C : Flow Controller LC : Level Controller TC : Temperature Controller PC : Pressure Controller RR : Ratio Element CONTROLLABILITY INDICATORS SET 1 (See Figure 3) The first run used two control outputs and two manipulated variables. The control outputs are the product purity in the second column (COL2) top product and the impurity in the first column (COLl) bottoms stream. The two manipulated variables selected are the reflux ratio in the second column and the total feed rate to the first column. The choice of control objectives is based upon two of the most important variables desired to be controlled (directly or indirectly) and the ease with which the problem can be specified. Both manipulated variables are already allowed for by the base case input file. The G(O) matrix as obtained from the FLOWP ACK run and the controllability indicators that were obtained from running PCTB are shown in Figure 3.
PROPOSED
The process consists of a pair of distillation columns whereby the bottom product from the first column is the feed to the second column. The main product is obtained from the top of the second column. It is essential that this product should be very pure (of order 99.99%). The objective of the second column is to separate two close boiling components whereas the first column removes the 'lights'. The concentration of one of the 'light' components must be less than PPM levels at the bottom of the first column. The proposed control scheme is as shown in (Fig. 1). Five sets of inputs and outputs have been used, i.e. five sets of controllability indicators have been obtained, as follows :
An observation of the G(O) matrix elements shows that the gains for the first control objective are much smaller than for the second objective. The rga shows very little interaction between COL2 reflux ratio and the COL 1 bottoms concentration.
18
The minimal Euclidean Condition Number (mecn) has lower bound Ob) at 1.0 and an upper bound (ub) at 1.1590. This indicates that the process is not badly directional. However, the condition number, the ratio of the two singular values is quite high, mainly because of the rather small value of the smallest singular value.
CONTROLLABll...ITY INDICATORS SET 3 (See Figure 5) This is a two by two system. Compared to set I the outputs are the same but the inputs have been made more representative of the pre-defined control scheme. The new inputs are COL2 reboiler duty rather than the reflux ratio and COL 1 boilup ratio rather than the feed rate.
The Niederlinski's indices are both poSItIve, although the second combination is quite large. The first combination (i.e. connecting the first output to the first input - the COU reflux ratio with COLl top purity) is obviously the best combination in this case.
The indicators obtained are shown in Figure 5 and are similar to set 1. The mecn is still low and the rga indicates a nearly non-interactive system. The minimum singular value is still rather large and leading to a high condition number. In the G(O) matrix there is still a very low gain between the COL2 top concentration and the inputs. The Niederlinski's index shows that the second combination will lead to instabilities which is not very surprising considering the other indicators. The obvious and only correct connection between the loops would be the COL2 reboiler duty to COL2 top concentration and COLl boilup ratio to COLl bottom concentration.
In the proposed scheme it is the steam to the reboiler of the second column that is controlled, not the reflux ratio. Intuitively one feels however that if the indicators are good with respect to the reflux ratio they will also be good with respect to the reboiler heat load, especially as the feed rate is fixed. To test this we need to carry out another FLOWP ACK run with the modified manipulated variable. The impurity in the bottom of the first column is sensitive to and controllable by the feed flow rate. The pre-defined scheme proposed to control impurity in the bottoms by changing the steam rate to the boiler. This is effectively the boilup ratio as the input to the steam flow controller is ratioed to the feed rate into the column. The output from the sump level controller sets the feed flow controller setpoint. Level control onto reflux ensures that the mass balance on COL I is closed. The manipulated variable needs to be the boilup ratio while the control objective remains the same.
The original intuition at the end of case 1 has been confirmed here to be correct. CONTROLLABll...ITY INDICATORS SET 4 (See Figure 6) This has the same outputs as set 2 but the first two inputs have been modified to reboiler duty and reboiler ratio in the second and first columns respectively. The biggest gains in the G(O) matrix are between the vent rate and product rate as inputs and the product concentration in the recycle from bottom of the second column as the output.
CONTROLLABll...ITY INDICATORS SET 2 (See Figure 4)
The rga matrix looks very good and shows that the diagonal elements are all close to one. The four loops therefore do not show much interaction with each other. The right connections are in the same order as the inputs and the outputs. Thus the first output should be connected to the first input and so on. The mecn is much improved compared to set 2. This shows the first two inputs used here which are closer to the real scheme than set 2 are much better than the previous two inputs.
This is a four by four system. The control outputs selected are, as before, the top product purity in the second column and the bottom impurity in the first column. In addition, there is the product concentration in the vent stream and the product in the bottom of the second column. The manipulated variables selected are, as before, the reflux ratio in the second column and the feed rate. In addition, the vent rate in the first column and the product rate in the second column are chosen to be the manipulated variables. Therefore the four elements in the top left corner of the G(O) matrix will be the same as that for set I above.
19
CONTROLLABILITY INDICATORS SET 5 (See Figure 7)
ELEMENTS:
The G(O) matrix indicates that there are higher gains for the second output than ~or the first one. The rga is quite perfect showmg that the loops would be independent of each other. ~e mecn is excellent and the condition number IS small. One would connect the boilup ratio to the bottoms concentration control loop and the feed rate to the top loss of product. This analysis has suggested that the boilup ratio is in .fact a good way to control the bottom concentration. INPlITS
OlITPlITS
Al BI Cl
DI El FI Gl
RUN I STEADY STATE MODEL
COLlFR COLlBC
RGA - RELATIVE GAIN ARRAY
0.9936 0.0064
0.0064 0.9936
MECN(UB) AND MECN(LB) UPPER AND LOWER BOUNDS ONMnrnMUM EUCLIDEAN CONDITION NO:
1.159 1.0
D2 E2 B2 G2
RUN 2 STEADY STATE MODEL
COL2RR COLlBC
0.I06E-3 0.268E-3 -O.249E-3 0.984EI
OlITPlITS
A2 F2 C2
COLlFR COL2TC
G(O):
Fig 2a : Original Model INPlITS
COL2RR COL2TC
SINGULAR VALUES:
9.8419 0.0001
IF Al = A2 THEN DI = D2 FI = F2 El = E2 Cl = C2 BI = B2 ANDGI=G2
CONDITION NUMBER:
9.2302E4 NIEDERLINSKI'S INDEX:
1.0064
Fig 2b : Transformed Model
OUTPlITS
INPlITS A3 DELTF3 C3 G(O) IF F3 A3 C3
= = = =
RUN 3 STEADY STATE MODEL
157.2944
Fig 3. Controllability Indicators for Set 1
DELTD3 DELTE3 DELT F3 DELT G3
KEY OF ELEMENT VARIABLE NAMES: RR : REFLUX RATIO FR : FEED RATE RD : REBOILER DUTY BR : BOILUP RATIO PR : PRODUCT RATE VR : VENT RATE BC : CONCENTRATION OF IMPURITIES IN THE BOTIOM PRODUCT STREAM TC : CONCENTRATION OF PRODUCT IN THE TOP PRODUCT STREAM VC : CONCENTRATION OF PRODUCT IN VENT RC : CONCENTRATION OF PRODUCT IN RECYCLE FROM BOTIOM OF COL2
DELT B3/ DELT F3 FI THEN G(O) ELEMENT Al CORRESPONDS TO Cl ORIGINAL STEADY STATE
Fig 2e: Perturbed Runs
Fig 2. Obtaining G(O) Matrices from FLOW PACK with F as a desired manipulated variable and B as a desired control objective.
20
ELEMENTS:
ELEMENTS : COLlVR COL2PR COL2TC COL2TC
COL2RD COL2TC
COLlBR COL2TC
COL2RR COLlFR COLlVR COLlVR COLlBC COLlBC COLlBC COLlBC
COL2RD COLlBC
COLlBR COLlBC
COL2RR COLlFR COL2TC COL2TC
G(O):
COL2RR COL2RR COLlVR COL2PR COLI VC COLlVC COLlVC COLlVC COL2RR COL2RR COL2RC COL2RC
0.116E-3 0. IIOE-3 -O.131E-l -O.119E2
COLlVR COL2PR COL2RC COL2RC
RGA - RELATIVE GAIN ARRAY: l.0011 -0.0011
G(O) : 0.0001 0.0003 -0.0249 9.8418 0.000 -3 .0235 -0.0057 7l.725
0.0000 -8.7727 3.0238 -21 .2424
-0.0003 -0.5197 -0.0002 -50.4829
MECN(UB) AND MECN(LB) UPPER AND LOWER BOUNDS ON MINIMUM EUCLIDEAN CONDITION NO: l.0698
RGA - RELATIVE GAIN ARRAY:
l.0698
SINGULAR VALUES:
l.0100 0.1148 -0.0137 -0.1111 -0.0103 18.0793 -16.1143 -0.9546 0.0000 -15.6784 16.6792 0.0008 -0.0003 -l.5157 0.4489 2.0665
10.941
0.0001
CONDmON NUMBER: 9.437E4
MECN(UB) AND MECN(LB) UPPER AND LOWER BOUNDS ON MINIMUM EUCLIDEAN CONDITION NO: 70.7985
-0.0011 l.0011
NIEDERLINSKI'S INDEX:
70.7623
0.9989
-877.517
SINGULAR VALUES : 90.88 8.8311 0.1049 0.0001
Fig 5. Controllability Indicators for Set 3
CONDITION NUMBER: 8.6896E5 NIEDERLINSKI'S INDEX: l.OE7* 0.0000 0.0000 -0.0002 0.1000 0.0000 0.0012 -0.0001 -0.0989 -0.0163
-0.4235 3.1647 0.0031
Fig 4. Controllability Indicators for
Set 2
21
ELEMENTS:
ELEMENTS:
COL2RD COLlBR COLlVR COL2PR COL2TC COL2TC COL2TC COL2TC
COLlFR COLIVC
COLlBR COLlVC
COL2RD COLlBR COLlVR COLlBC COLlBC COLlBC
COLlFR COLlBC
COLlBR COLIBC
COL2PR COLlBC
COL2RD COLlBR COLl VR COL2PR COLlVC COLlVC COLl VC COLl VC COL2RD COLlBR COL2RC COL2RC
G(O): -O.302El 0.396E-3 -O.134El -O.1035E2
COLlVR COL2PR COL2RC COL2RC
RGA - RELATIVE GAIN ARRAY:
G(O):
1.00 0.00
0.0001 0.0004 -0.0002 -0.0003 -0.0179 -10.9351 1.7757 0.0000 0.000 0.0022 3.0225 -0.0001 -0.0061 -0.0071 -21.2107 -50.4367
MECN(UB) AND MECN(LB) UPPER ANDLOWERBOUNDSONMnrnMUM EUCLIDEAN CONDITION NO:
RGA - RELATIVE GAIN ARRAY: 1.0049 -0.0052 0.0000 -0.0003
-0.0052 1.0051 0.0001 0.0000
0.0000 0.0001 0.9999 0.0000
0 .00 1.00
1.0082 1.00
0.0003 0.0000 0.0000 0.9997
SINGULAR VALUES: 10.4484
2.9961 MECN(UB) AND MECN(LB) UPPER AND LOWERBOUNDSONMnmMUM EUCLIDEAN CONDITION NO:
CONDmON NUMBER:
1.1653 1.1550
3.4873
NIEDERLINSKI'S INDEX: SINGULAR VALUES:
1.0
54.7323 11.0641 2.7531 0.0001
Fig 7. Controllability Indicators for Set 5 GENERAL COMMENTS AND CONCLUSIONS
CONDITION NUMBER: 4.7233E5
1. The analysis based upon a smaller number of inputs and outputs is easier to follow and understand. An analysis based on the four by four system is necessarily more complex. Larger dimensions can not always be avoided. list element.
NIEDERLINSKI'S INDEX: 1.0E15* 0.0000 0.0000 0.0000
0.0000 0 .0000 -0.0001
5.88E4
0.0000 0.0000 5.8043
0.0000 0.0000 -0.0148
2. Problem decomposition is sometimes possible and should be used if appropriate. In our example, the manipulated variables in COL 1 will affect the control outputs in COL2 .
Fig 6. Controllability Indicators for Set 4
22
However, the only effect of manipulated variables in COL2 on control outputs in COLl will be that due to numerical noise. (Of course, if there was a physical recycle then COL I outputs would be influenced by COL2 inputs).
3. Carrying out a number of related controllability analyses is a very effective way to gain insight into the process and the potential for control of the process. It also ensures that control objectives are clearly defined. Although there are limitations in the deductions possible based only on steady state models significant insight can be obtained even without dynamic models. 4 . It is desirable to carry out controllability analysis as early as possible in process design so that any changes or issues can be highlighted early. If this has not happened, it is nevertheless useful even at later stages to apply the analysis to confirm designs and/or highlight control issues that can subsequently be investigated further.
REFERENCES
1. Garlick S., Malik T I and Tyrrell MJ (1993). Application of Same Equation Based Model from Stcad~ State Design to Control System Analysis and Operator Training ESCAPE-3 meeting, Graz, Austna. Supplement to Comp. Chem. Engg
2. Fararooy, S., Perkins 1.0 .. Malik T.I (1992). A Software package for Controllability analysis. ESCAPE-I meeting, Elsinore, Denmark, Supplement to Comp. Chem. Engg.
3. Oglesby M .l., Malik TI, Williams S. (1992). Early Stage Process Controllability Assessment. IF AC Workshop, London, UK.
23
IMPROVED CONFIDENCE IN DESIGN THROUGH CONTROLLABILITY ANALYSIS BY
C.M.CALLAN(l) and T.I.MALIK(2) (1) ICI Chemicals and Polymers Ltd, Runcorn Heath, Cheshire, England.
(2) ICI Engineering, Brunner House, Northwich, Cheshire, England. This paper concerns the interaction between process design and process control in that a steady state model used for the process design has been further exploited to give controllability indicators. These were used to improve confidence in a pre-selected control structure. ICI's sequential modular package FLOWPACK was used for process design and for generating the G(O) matrix that can then be used for obtaining controllability indicators. Several sets of control inputs and outputs were tested both to develop confidence in control system design as well as overcoming inherent shortcomings in the choice of control inputs and outputs within the flowsheeting package.
INTRODUCTION
LIMITATIONS TO CONTROL INPUTS AND OUTPUTS ~CH CAN BE SPECIFIED IN FLOWPACK
This is an example of the application of controllability analysis at a late stage in a project. An extension on a plant had already been designed and the control structure (illustrated in Fig. 1) had been specified by the project control engineer. It was desired to improve confidence in this design by an independent means for example a dynamic model. This would have required translation of the steady state FLOWP ACK model used for process design into a dynamic model in another package. There was not sufficient time or resource available to do this. (With an Equation Based Package, as opposed to sequential modular as in FLOWPACK, this transfer from steady state to dynamic mode should be easier. See for example Garlick et al. 1993).
FLOWPACK, as a steady state package, calculates a number of outputs (results) given a number of inputs (specifications, for example, feed stream flow rates and conditions or reactor conversions). In order to obtain the 0(0) matrix it is required to specify control inputs (measured variables) and control outputs (the manipulated variables). There are restrIctIons in FLOWPACK, as there would be in any sequential modular package, on what these can be:
LIMITATION 1 Not all the variables are available at the problem input language level where the 0(0) calculations are requested. The variables that can be selected as control inputs or control outputs must either exist in the problem input file or be relatable to the same via simple additional equations that are permitted in FLOWPACK. Thus, detailed tray to tray temperature calculations, although they are carried out, are not passed on to the next unit in the calculation sequence. They do not need to exist as variables in the input file (with the exception of top and bottom temperatures) and hence cannot be specified as control outputs.
Controllability analysis was possible, however, as FLOWPACK does enable 0(0) matrices to be obtained by only a few additions to the input problem file. These can then be analysed by PCTB (Process Controllability ToolBox), (see Fararooyet al. 1992 and Oglesby et al. 1992), from which a number of controllability indicators are obtained. This analysis can be completed quite quickly and it was decided to pursue this as a means of getting further insight and confidence in the selected control structure.
17
LIMITATION 2 Not all combinations of inputs are legal, for example, in distillation unit operations, only certain combinations of inputs such as reflux ratio, reboiler ratio, product purity, product rates and reboiler and condenser heat duties are allowed. This may mean that a desired control input can not be used in the model. STRATEGY In order to build confidence in the analysis and the pre-defined control structure we chose to carry out analysis on a range of related control structures rather than only the pre-defined control structure. This also provided a way of handling the limitations mentioned above. Outputs that are closest to the actual desired ones are selected if limitation (I) applies.
Fig. 1 The Process and Control Scheme Diagram Key :
In order to carry out a number of related runs as well as to overcome limitation (2) a number of different steady state runs with different variables specified are carried out that otherwise give the same results. This is illustrated schematically in Fig. 2 for a model with 3 inputs and 4 outputs. Fig. 2A uses inputs A, B and C. For controllability we want to use F as an input but F and B are not legal combinations. To overcome this the value of F obtained from Fig. 2a is used as an input in Fig. 2b while the values of A and C are identical. Then the effect on the control outputs can be observed. If G was the control output of interest then the G(O) element between G and F can be obtained by perturbing F. THE PROCESS AND CONTROL SCHEME
F:C : Flow Controller LC : Level Controller TC : Temperature Controller PC : Pressure Controller RR : Ratio Element CONTROLLABILITY INDICATORS SET 1 (See Figure 3) The first run used two control outputs and two manipulated variables. The control outputs are the product purity in the second column (COL2) top product and the impurity in the first column (COLl) bottoms stream. The two manipulated variables selected are the reflux ratio in the second column and the total feed rate to the first column. The choice of control objectives is based upon two of the most important variables desired to be controlled (directly or indirectly) and the ease with which the problem can be specified. Both manipulated variables are already allowed for by the base case input file. The G(O) matrix as obtained from the FLOWP ACK run and the controllability indicators that were obtained from running PCTB are shown in Figure 3.
PROPOSED
The process consists of a pair of distillation columns whereby the bottom product from the first column is the feed to the second column. The main product is obtained from the top of the second column. It is essential that this product should be very pure (of order 99.99%). The objective of the second column is to separate two close boiling components whereas the first column removes the 'lights'. The concentration of one of the 'light' components must be less than PPM levels at the bottom of the first column. The proposed control scheme is as shown in (Fig. 1). Five sets of inputs and outputs have been used, i.e. five sets of controllability indicators have been obtained, as follows :
An observation of the G(O) matrix elements shows that the gains for the first control objective are much smaller than for the second objective. The rga shows very little interaction between COL2 reflux ratio and the COL 1 bottoms concentration.
18
The minimal Euclidean Condition Number (mecn) has lower bound Ob) at 1.0 and an upper bound (ub) at 1.1590. This indicates that the process is not badly directional. However, the condition number, the ratio of the two singular values is quite high, mainly because of the rather small value of the smallest singular value.
CONTROLLABll...ITY INDICATORS SET 3 (See Figure 5) This is a two by two system. Compared to set I the outputs are the same but the inputs have been made more representative of the pre-defined control scheme. The new inputs are COL2 reboiler duty rather than the reflux ratio and COL 1 boilup ratio rather than the feed rate.
The Niederlinski's indices are both poSItIve, although the second combination is quite large. The first combination (i.e. connecting the first output to the first input - the COU reflux ratio with COLl top purity) is obviously the best combination in this case.
The indicators obtained are shown in Figure 5 and are similar to set 1. The mecn is still low and the rga indicates a nearly non-interactive system. The minimum singular value is still rather large and leading to a high condition number. In the G(O) matrix there is still a very low gain between the COL2 top concentration and the inputs. The Niederlinski's index shows that the second combination will lead to instabilities which is not very surprising considering the other indicators. The obvious and only correct connection between the loops would be the COL2 reboiler duty to COL2 top concentration and COLl boilup ratio to COLl bottom concentration.
In the proposed scheme it is the steam to the reboiler of the second column that is controlled, not the reflux ratio. Intuitively one feels however that if the indicators are good with respect to the reflux ratio they will also be good with respect to the reboiler heat load, especially as the feed rate is fixed. To test this we need to carry out another FLOWP ACK run with the modified manipulated variable. The impurity in the bottom of the first column is sensitive to and controllable by the feed flow rate. The pre-defined scheme proposed to control impurity in the bottoms by changing the steam rate to the boiler. This is effectively the boilup ratio as the input to the steam flow controller is ratioed to the feed rate into the column. The output from the sump level controller sets the feed flow controller setpoint. Level control onto reflux ensures that the mass balance on COL I is closed. The manipulated variable needs to be the boilup ratio while the control objective remains the same.
The original intuition at the end of case 1 has been confirmed here to be correct. CONTROLLABll...ITY INDICATORS SET 4 (See Figure 6) This has the same outputs as set 2 but the first two inputs have been modified to reboiler duty and reboiler ratio in the second and first columns respectively. The biggest gains in the G(O) matrix are between the vent rate and product rate as inputs and the product concentration in the recycle from bottom of the second column as the output.
CONTROLLABll...ITY INDICATORS SET 2 (See Figure 4)
The rga matrix looks very good and shows that the diagonal elements are all close to one. The four loops therefore do not show much interaction with each other. The right connections are in the same order as the inputs and the outputs. Thus the first output should be connected to the first input and so on. The mecn is much improved compared to set 2. This shows the first two inputs used here which are closer to the real scheme than set 2 are much better than the previous two inputs.
This is a four by four system. The control outputs selected are, as before, the top product purity in the second column and the bottom impurity in the first column. In addition, there is the product concentration in the vent stream and the product in the bottom of the second column. The manipulated variables selected are, as before, the reflux ratio in the second column and the feed rate. In addition, the vent rate in the first column and the product rate in the second column are chosen to be the manipulated variables. Therefore the four elements in the top left corner of the G(O) matrix will be the same as that for set I above.
19
CONTROLLABILITY INDICATORS SET 5 (See Figure 7)
ELEMENTS:
The G(O) matrix indicates that there are higher gains for the second output than ~or the first one. The rga is quite perfect showmg that the loops would be independent of each other. ~e mecn is excellent and the condition number IS small. One would connect the boilup ratio to the bottoms concentration control loop and the feed rate to the top loss of product. This analysis has suggested that the boilup ratio is in .fact a good way to control the bottom concentration. INPlITS
OlITPlITS
Al BI Cl
DI El FI Gl
RUN I STEADY STATE MODEL
COLlFR COLlBC
RGA - RELATIVE GAIN ARRAY
0.9936 0.0064
0.0064 0.9936
MECN(UB) AND MECN(LB) UPPER AND LOWER BOUNDS ONMnrnMUM EUCLIDEAN CONDITION NO:
1.159 1.0
D2 E2 B2 G2
RUN 2 STEADY STATE MODEL
COL2RR COLlBC
0.I06E-3 0.268E-3 -O.249E-3 0.984EI
OlITPlITS
A2 F2 C2
COLlFR COL2TC
G(O):
Fig 2a : Original Model INPlITS
COL2RR COL2TC
SINGULAR VALUES:
9.8419 0.0001
IF Al = A2 THEN DI = D2 FI = F2 El = E2 Cl = C2 BI = B2 ANDGI=G2
CONDITION NUMBER:
9.2302E4 NIEDERLINSKI'S INDEX:
1.0064
Fig 2b : Transformed Model
OUTPlITS
INPlITS A3 DELTF3 C3 G(O) IF F3 A3 C3
= = = =
RUN 3 STEADY STATE MODEL
157.2944
Fig 3. Controllability Indicators for Set 1
DELTD3 DELTE3 DELT F3 DELT G3
KEY OF ELEMENT VARIABLE NAMES: RR : REFLUX RATIO FR : FEED RATE RD : REBOILER DUTY BR : BOILUP RATIO PR : PRODUCT RATE VR : VENT RATE BC : CONCENTRATION OF IMPURITIES IN THE BOTIOM PRODUCT STREAM TC : CONCENTRATION OF PRODUCT IN THE TOP PRODUCT STREAM VC : CONCENTRATION OF PRODUCT IN VENT RC : CONCENTRATION OF PRODUCT IN RECYCLE FROM BOTIOM OF COL2
DELT B3/ DELT F3 FI THEN G(O) ELEMENT Al CORRESPONDS TO Cl ORIGINAL STEADY STATE
Fig 2e: Perturbed Runs
Fig 2. Obtaining G(O) Matrices from FLOW PACK with F as a desired manipulated variable and B as a desired control objective.
20
ELEMENTS:
ELEMENTS : COLlVR COL2PR COL2TC COL2TC
COL2RD COL2TC
COLlBR COL2TC
COL2RR COLlFR COLlVR COLlVR COLlBC COLlBC COLlBC COLlBC
COL2RD COLlBC
COLlBR COLlBC
COL2RR COLlFR COL2TC COL2TC
G(O):
COL2RR COL2RR COLlVR COL2PR COLI VC COLlVC COLlVC COLlVC COL2RR COL2RR COL2RC COL2RC
0.116E-3 0. IIOE-3 -O.131E-l -O.119E2
COLlVR COL2PR COL2RC COL2RC
RGA - RELATIVE GAIN ARRAY: l.0011 -0.0011
G(O) : 0.0001 0.0003 -0.0249 9.8418 0.000 -3 .0235 -0.0057 7l.725
0.0000 -8.7727 3.0238 -21 .2424
-0.0003 -0.5197 -0.0002 -50.4829
MECN(UB) AND MECN(LB) UPPER AND LOWER BOUNDS ON MINIMUM EUCLIDEAN CONDITION NO: l.0698
RGA - RELATIVE GAIN ARRAY:
l.0698
SINGULAR VALUES:
l.0100 0.1148 -0.0137 -0.1111 -0.0103 18.0793 -16.1143 -0.9546 0.0000 -15.6784 16.6792 0.0008 -0.0003 -l.5157 0.4489 2.0665
10.941
0.0001
CONDmON NUMBER: 9.437E4
MECN(UB) AND MECN(LB) UPPER AND LOWER BOUNDS ON MINIMUM EUCLIDEAN CONDITION NO: 70.7985
-0.0011 l.0011
NIEDERLINSKI'S INDEX:
70.7623
0.9989
-877.517
SINGULAR VALUES : 90.88 8.8311 0.1049 0.0001
Fig 5. Controllability Indicators for Set 3
CONDITION NUMBER: 8.6896E5 NIEDERLINSKI'S INDEX: l.OE7* 0.0000 0.0000 -0.0002 0.1000 0.0000 0.0012 -0.0001 -0.0989 -0.0163
-0.4235 3.1647 0.0031
Fig 4. Controllability Indicators for
Set 2
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ELEMENTS:
ELEMENTS:
COL2RD COLlBR COLlVR COL2PR COL2TC COL2TC COL2TC COL2TC
COLlFR COLIVC
COLlBR COLlVC
COL2RD COLlBR COLlVR COLlBC COLlBC COLlBC
COLlFR COLlBC
COLlBR COLIBC
COL2PR COLlBC
COL2RD COLlBR COLl VR COL2PR COLlVC COLlVC COLl VC COLl VC COL2RD COLlBR COL2RC COL2RC
G(O): -O.302El 0.396E-3 -O.134El -O.1035E2
COLlVR COL2PR COL2RC COL2RC
RGA - RELATIVE GAIN ARRAY:
G(O):
1.00 0.00
0.0001 0.0004 -0.0002 -0.0003 -0.0179 -10.9351 1.7757 0.0000 0.000 0.0022 3.0225 -0.0001 -0.0061 -0.0071 -21.2107 -50.4367
MECN(UB) AND MECN(LB) UPPER ANDLOWERBOUNDSONMnrnMUM EUCLIDEAN CONDITION NO:
RGA - RELATIVE GAIN ARRAY: 1.0049 -0.0052 0.0000 -0.0003
-0.0052 1.0051 0.0001 0.0000
0.0000 0.0001 0.9999 0.0000
0 .00 1.00
1.0082 1.00
0.0003 0.0000 0.0000 0.9997
SINGULAR VALUES: 10.4484
2.9961 MECN(UB) AND MECN(LB) UPPER AND LOWERBOUNDSONMnmMUM EUCLIDEAN CONDITION NO:
CONDmON NUMBER:
1.1653 1.1550
3.4873
NIEDERLINSKI'S INDEX: SINGULAR VALUES:
1.0
54.7323 11.0641 2.7531 0.0001
Fig 7. Controllability Indicators for Set 5 GENERAL COMMENTS AND CONCLUSIONS
CONDITION NUMBER: 4.7233E5
1. The analysis based upon a smaller number of inputs and outputs is easier to follow and understand. An analysis based on the four by four system is necessarily more complex. Larger dimensions can not always be avoided. list element.
NIEDERLINSKI'S INDEX: 1.0E15* 0.0000 0.0000 0.0000
0.0000 0 .0000 -0.0001
5.88E4
0.0000 0.0000 5.8043
0.0000 0.0000 -0.0148
2. Problem decomposition is sometimes possible and should be used if appropriate. In our example, the manipulated variables in COL 1 will affect the control outputs in COL2 .
Fig 6. Controllability Indicators for Set 4
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However, the only effect of manipulated variables in COL2 on control outputs in COLl will be that due to numerical noise. (Of course, if there was a physical recycle then COL I outputs would be influenced by COL2 inputs).
3. Carrying out a number of related controllability analyses is a very effective way to gain insight into the process and the potential for control of the process. It also ensures that control objectives are clearly defined. Although there are limitations in the deductions possible based only on steady state models significant insight can be obtained even without dynamic models. 4 . It is desirable to carry out controllability analysis as early as possible in process design so that any changes or issues can be highlighted early. If this has not happened, it is nevertheless useful even at later stages to apply the analysis to confirm designs and/or highlight control issues that can subsequently be investigated further.
REFERENCES
1. Garlick S., Malik T I and Tyrrell MJ (1993). Application of Same Equation Based Model from Stcad~ State Design to Control System Analysis and Operator Training ESCAPE-3 meeting, Graz, Austna. Supplement to Comp. Chem. Engg
2. Fararooy, S., Perkins 1.0 .. Malik T.I (1992). A Software package for Controllability analysis. ESCAPE-I meeting, Elsinore, Denmark, Supplement to Comp. Chem. Engg.
3. Oglesby M .l., Malik TI, Williams S. (1992). Early Stage Process Controllability Assessment. IF AC Workshop, London, UK.
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