Ind. Eng. Chem. Res. 1987,26, 2076-2078
2076
Sensitivity of Distillation Relative Gain Arrays to Steady-State Gains William L. Luyben Process Modeling a n d Control Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
The values of the Relative Gain Array (RGA) for a 2 X 2 binary distillation column system are shown t o be extremely sensitive t o the size of the manipulated variable increment that is used to calculate the steady-state gain. RGA’s can vary from plus infinity t o minus infinity. Extremely small increments must be used, even in moderate purity columns, in order to get accurate RGA’s. Proponents of the Relative Gain Array (RGA) have cited the following as reasons for its use: (1)ease of calculation (only the steady-state gains are needed); (2) guidance it gives in variable pairing. Opponents have criticized the use of RGA’s to discriminate among alternative processes and control structures. McAvoy (1983) presented a good summary of the definition and properties of the RGA’s that have been developed over the years by Bristol, Shinskey, and McAvoy. The calculation of the RGA requires the process steady-state gains. These gains are typically determined experimentally for an existing process or by computer simulation for a new process. Either dynamic or steadystate models can be used.
System Studied Several different columns were studied, but results from simulation studies of just one very simple binary distillation column are presented in this paper. They are typical of the findings for all the columns studied. The process open-loop steady-state gains were calculated from a steady-state rating program. Design parameters are given in Table I. A conventional R-V structure (reflux-vapor boilup) was assumed: distillate composition was controlled by reflux, and bottoms composition was controlled by vapor boilup. Note that this column does not have high-purity products, nor is it nonideal. Equimolal overflow and constant relative volatility assumptions were used. Small positive changes of different magnitude were made in the two manipulative variables (reflux (R) and vapor boilup (V)). Increment size (“DEL” in Tables I1 and 111) was varied from 1%to 0.0001% . A Cyber 850 with 16 significant figures of accuracy was used. A convergence criterion of was used on the guessed vs. calculated distillate composition in the rating program.
Resu1t s Table I1 gives values for steady-state gains (Kij’s) and RGA’s for different increment sizes (the DEL’S are in percent) in the manipulative variables. The steady-state gains have been made dimensionless by using the composition and flow transmitter spans given in Table I. We want to determine the steady-state gains that one would obtain from a linearized version of the process. These are the gains that should be used for controller design since the closed-loop system will be held near this steady-state condition by the controller. In this paper, accuracy of the gains means closeness to the linear steady-state gains. The magnitude of the delta had to be reduced to about 0.1 % to get steady-state gains that were fairly constant 088g-5885/87/2626-2076$01.50~0
Table I. Design Parameters for Column feed flow rate, F = 100 mol/h feed composition, z = 0.50 mole fraction of light component feed thermal condition = saturated liquid total no. of trays = 21 feed tray no. (from bottom) = 10 partial reboiler total condenser saturated liquid reflux distillate composition, xD = 0.986 214 6 mole fraction light bottoms composition, xB = 0.023 706 9 mol fraction light re1 volatility, a = 2 reflux flow rate = 126.6804 mol/h vapor boilup = 176.165 mol/h iterative convergence criterion for assumed vs. calcd distillate composition in rating program = composition transmitter spans: xD = 0.05 mole fraction xB = 0.10 mole fraction flow transmitter spans = twice steady-state flow rates of manipulated variables tray holdups = 1 mol base holdup = 10 mol reflux drum holdup = 10 mol level controllers are “perfect”, i.e., no change in levels tray hydraulic time constant = 12 s composition measurement lags = two 30-s first-order lags
as the delta size was changed. This delta size would have to be even smaller for a more nonlinear column (nonideal vapor-liquid equilibrium or high-purity products). This requirement for very small changes means that the gains obtained from experimental plant tests may be quite inaccurate. Changes in manipulative variables of 5-1070 must often be used in actual plant tests because of signal noise. Disturbing as this finding is, it is minor compared to what was found for the RGA values. As seen in Table 11, RGA values varied from minus infinity to plus infinity as the magnitudes of R and V were changed from 0.05% to 0.048%! Magnitudes had to be reduced to the 0.001% level in order to get constant RGA values. The reason for the discontinuity in RGA between 0.05 and 0.048 is the change in the Rijnsdorp Index (Rijnsdorp, 1965) (RI in Table 11) from greater than one to less than one. RI =
[K12K211/ [KllK221
RGA = 1/[1 - RI]
(1)
(2)
Relatively small changes in the individual gains make the RI change slightly, which makes a tremendous change in the RGA, since RI has a value around one. This clearly brings into serious doubt the practical usefulness and reliability of the RGA. Computer simulations can achieve this kind of accuracy (requiring double 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2077 Table 11. Steady-State Gains and RGA’s (R-V Structure) DEL ( % Am) KH Kl2 1.000 0.5000 0.1000 0.0500 0.0480 0.0450 0.0400 0.0300 0.0200 0.0100 0.0050 0.0010 0.0005 0.0001
-83.0 534
20.5084 25.9192 32.2800 33.2227 33.2612 33.3191 33.4152 33.6084 33.8032 33.9985 34.0980 34.1557 34.1985 33.9840
-65.5417 -48.2422 -46.2519 -46.1743 -46.0572 -45.8634 - 4 5.4 795 -45.0968 -44.7198 -44.5380 -44.4242 -44.3567 -44.4332
K2l
K22
RGA
RI
37.5453 35.2994 32.4228 31.9825 31.9646 31.9376 31.8924 31.8014 31.7098 31.6173 31.5711 31.5284 31.5322 31.4712
-25.9062 -35.0682 -43.5692 -44.5136 -44.5505 -44.6055 -44.6970 -44.8788 -45.0581 -45.2361 -45.3257 -45.4051 -45.4012 -45.4318
-0.205 -0.647 -8.917 -3790.6 252.8 97.406 48.389 24.330 16.360 12.399 11.087 10.324 10.083 10.605
5.869 19 2.545 37 1.112 15 1.000 26 0.996 04 0.989 73 0.979 33 0.958 90 0.938 88 0.919 35 0.909 80 0.903 14 0.900 82 0.905 70
Table 111. Steady-State Gains and RGA’s (D-V Structure) DEL (70 Am) K11 K12 1.000 0.5000 0.1000 0.0500 0.0100 0.0050 0.0010 0.0005
3.0938 3.1520 3.1998 3.2058 3.2116 3.2163 3.2177 3.2179
-16.6511 -14.9486 -13.6631 -13.5097 -13.3895 -13.3734 -13.3628 -13.3586
KL
K21
K2z
RGA
RI
-10.7252 -11.5555 r12.1703 -12.2430 -12.3009 -12.3077 -12.3138 -12.3138
-1.5154 - 1.5439 -1.5673 -1.5702 -1.5724 -1.5716 -1.5716 -1.5716
0.432 0.388 0.355 0.351 0.348 0.347 0.346 0.346
-1.31504 -1.578 23 -1.81860 -1.850 19 -1.876 45 -1.88344 -1.88669 -1.88740
83
68
-
20 0 0
R E S E T 3 4 00
25 00
ZT50 0 0
X
98 5
1
98.g 0 5 T O O 6
130t
2
12c1.
R-V
LT
4 . m x
4
T
m
2 --
x
2
u .
U
l
2004 >
>
150~ 0
150 15
30 TIME
45
60
75
90
0
15
(MINUTES)
30 TIME
60
45
7 5 9 0
(MINLTES)
Figure 1. Response of R-V structure to feed composition disturbance.
Figure 2. Response of D-V structure to feed composition disturbance.
precision on many computers), but plant data will not be within several orders of magnitude of the required accuracy. If the D-V control structure is used (manipulating distillate flow rate (D)and vapor boilup), the RGA values are no longer sensitive as shown in Table 111. Note that the RGA suggests that the opposite pairing (control xD with V and X B with R ) would be better. The dynamics of this pairing would be quite poor. One might conclude that the D-V structure should be used since the RGA’s are not as sensitive. However, dynamic simulation studies of the nonlinear model with the R-V structure gave very satisfactory performance. Figures 1-4 compare the D-V and R-V structures for step disturbances in feed composition (0.50-0.60) and feed rate (100-110 mol/h).
Table IV. Controller Tuning Constants ZieglerNichols final settings structure gain reset detuning gain reset R- V xD-R 2.7 8.5 4 0.68 34 XB-V
3.3
5.0
4
0.83
20
5 5
1.45 0.67
50 25
D-V xD-D
xB-v
7.3 3.3
10 5
Each configuration was tuned by using the same technique (a modification of the BLT method of Luyben (1986)). First the ultimate gains and periods were obtained for each individual loop on the nonlinear simulation. Then the Ziegler-Nichols settings were calculated for each loop.
2078 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
Kc = KZN/F TI
=
[.ZNI[q
(3)
(4)
Table IV gives the Ziegler-Nichols settings and the final settings. Comparison of Figures 1 and 2 shows that the R-V structure gives much better regulator response for feed composition load disturbances. This is because the reflux flow rate does not have to change much as feed composition changes. The distillate flow rate has to change directly with feed composition. Figures 3 and 4 show that the two configurations respond about the same for feed flow rate disturbances.
Conclusions Distillation column steady-state gains cannot be determined with any high degree of accuracy from plant tests. Any analysis or synthesis tool that requires very precise knowledge of the steady-state gains will be unreliable. The RGAs calculated from plant test data may be meaningless. 150: 0
-4
:1
63 I M E ftIII'LIdTES)
33
45
75
33
Figure 3. Response of R-V structure to feed rate disturbance.
K:: : 45 9 E S E T x cc
x ffi
2
E'
25 c 2
- *
Nomenclature BLT = biggest log-modulus tuning D = distillate flow rate, mol/h DEL = size of change in manipulated variable (% of steady-state value) F = feed flow rate, mol/h F = BLT detuning factor K , = controller gain KZN = Ziegler-Nichols controller gain KI1 = steady-state gain between xD and R K I 2= steady-state gain between xD and V K,, = steady-state gain between xB and R Kzz = steady-state gain between xB and V R = reflux flow rate, mol/h RGA = relative gain array RI = Rijnsdorp Index V = vapor boilup, mol/h xB = bottoms composition, mole fraction of light component xD = distillate composition, mole fraction of light component z = feed composition, mole fraction of light component Greek Symbols Am = percentage change in manipulate variable, 70 steady
state q = controller integral constant = RESET, min rZN = Ziegler-Nichols integral constant, min
Literature Cited Figure 4. Response of D-V structure to feed rate disturbance.
Finally a detuning factor (F)was changed until a reasonable closed-loop damping coefficient was obtained on the nonlinear simulation model. All Ziegler-Nichols gains were divided by the detuning factor (F),and all reset times were multiplied by F.
Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 654. McAvoy, T. J. "Interaction Analysis"; 1983, Instrument Society of America: Research Triangle Park, NC, 1983. Rijnsdorp, J. E. Automatica 19'65, 1(15), 29.
Received for review November 10, 1986 Revised manuscript received June 8, 1987 Accepted June 13, 1987
