1IS2
Znd. Eng. Chem. Res. 1991, 30, 1182-1186
A Disturbance Sensitivity Parameter and Its Application to ,Distillation Control Peter M.Sandelin,+ K u r t E.Haggblom, a n d K u r t V . Waller* Process Control Laboratory, Department of Chemical Engineering, Abo Akademi, 20500 Abo, Finland
The control properties of processes are generally different for different control structures; that is, they are dependent on the way controlled and manipulated variables are paired. In this paper the correlation between a disturbance sensitivity parameter and control quality is investigated for four commonly used control structures for distillation. The control structures, labeled after the primary manipulators, are (L, V), (D,V), ( D / ( L + D),V), and ( D / ( L + D),V / B ) . For a system where the disturbance transfer function is first order, the disturbance sensitivity parameter is defined as the absolute value of the ratio of the gain and the time constant of the transfer function. This ratio is the slope of the initial nonzero response and the maximum slope of the response to a step disturbance. At one-point control, the performance of the four structures strongly correlates with this parameter. Also in two-point control the control quality obtained correlates well with the disturbance sensitivity parameter for the two structures with RGA values close to unity. For the more interacting structures the results are more complicated, depending both on the amount of interaction and on the disturbance sensitivity. Introduction In recent times, much effort in process control research, especially in distillation, has been devoted to control structure selection, that is, the selection and pairing of controlled and manipulated variables. The interaction between the control loops and the sensitivity of the system to disturbances are two important control structure properties, generally different for different structures. Usually the interaction properties have been considered the most important, but it has also been shown both experimentally and analytically that different control structures can have very different disturbance sensitivity properties (Waller et al., 1986). According to Luyben (1988),the disturbance sensitivity properties are far more important than the interaction properties when choosing a control structure. A number of different disturbance sensitivity measures for evaluating control structures have been suggested in the literature. For example, Stanley et al. (1985) introduced the relative disturbance gain RDG, Shinskey (1985) defined a load-sensitivity factor, and Skogestad (1988) used the sensitivity of the ratio DIB as a disturbance sensitivity measure for distillation control structures. Disturbance properties of distillation control systems is also discussed by Tyreus (1987). A drawback with the disturbance sensitivity measures suggested is that they take into account only the steadystate behavior of the process. Of course, the steady-state gain between an output and a disturbance is an important disturbance sensitivity parameter, but the rate of change of the output, its dynamic behavior, is also important. If the disturbance function is-or can be approximated by-a finborder transfer function with the gain K, and the time constant T,, the simplest parameter that takes these effects into account is the ratio K,/T,. In open-loop operation, this ratio is, in fact, the slope of the initial nonzero response to a step disturbance and also the maximum slope of the step response. In this paper, the rationale behind using lKw/Twlas a disturbance sensitivity parameter and the relationship between the parameter lKw/Twland the performance of four different distillation control structures at both onepoint and two-point control are investigated. It is shown 'Present address: Neste Oil, Porvoo hfmery, SF-06101Porvoo,
Finland.
0888-5885/91/2630-1182$02.50/0
by simulations based on models experimentally obtained that there is a strong linear correlation between control performance, as measured by integrated absolute error, and this parameter when interaction effects are small. This implies that the parameter can be used as a disturbance sensitivity measure in distillation, as already suggested in Waller et al. (1988b). Some Simple General Observations It has often been noticed in distillation that the dynamic relation between a manipulator such as L or V and an output such as product composition is dominated by a large time constant. If this relation is modeled as a first-order plus dead-time transfer function, the time constant is in a typical case at least 10 times as large as the dead time, often much larger. If a feedback PI controller is tuned for the first-order dead-time system, the setpoint step response is not much affected by the time constant, as long as it is large, for instance, more than 10 times the dead time. For the system of Figure 1, an illustration is given in Figure 2, where the ratios between the time constant and dead time are 10,20, and 100, respectively. The PI controller is tuned according to the Ziegler-Nichols ultimate sensitivity settings. For these large ratios between constant and dead time, the critical frequency is determined by the dead time, the time constant contributing almost 90' phase lag at the critical frequency. In distillation the effective dead time is often determined by secondary effects such as flow lags, and it may be approximately the same for different control configurations. Note, however, that loosely tuned level control loops may significantly affect the column dynamics, as was recently illustrated by Yang et al. (1990). This also supports the result that the dominant time constant in distillation is not very interesting for control purposes, as long as it is large. This has been noticed in several previous studies, among them one by Hammarstrom et al. (1982). If the feedback loops for different structures are equivalently tuned, that is, if their setpoint responses are the same, the output after disturbances in w in Figure 1 is given by K W
Y =G
w
0 1991 American Chemical Society
mW
Ind. Eng. Chem. Res., Vol. 30,No. 6,1991 1183
Y
‘.Ob-I
-1.0
Figure 1. Simple feedback control system.
Y
0.0
u
10
I
20
t Figure 4. Step responses to w = -1 for system in Figure 1 with T = 10 and K, = T, = 0.5 K, T, = 10 (-), and K, T, 50 (*a*)
-
(- -).
Table I. Transfer Functions of Four Distillation Control Structures’
10
0
0
20
(I,, V)
-0,045e4.& O.O48eQ6 -O.OOle-l.@ 11s + 1 10s + 1 8.1s 1 -0.23e-l.& 0.55e4.b -0.16e-1.@ 10s 1 5.5s + 1 8.1s 1 0.074e4.& -0.052e-1.b -0.005e-’.@ 14s + 1 15s + 1 23s + 1 0.3&-1,b 0.03e4.& -0).1&-’” 1%+l 1b+1 7.58+1
t Figure 2. Setpoint step responses for system in Figure 1 with T = 5 T = 10 (-), and T = 50 (---). (e-),
0.004e-1.0
+ 8.5s + 1 -0.65e-l.@ + + 9.29 + 1
(D,V)
-0.076e-3.b 20s + 1 -1.06e-’.@ 1%+1
- - - -
(&,VI
-1.0 J
0
10
20
t Figure 3. Step responses to w = -1 for system in Figure 1 with T 10 and K, = T, = 2 and K, = T, 10 (-). (-8)
where G , expresses the equivalently tuned feedback loops. Quahon 1shows that at small times (large s) the output y is proportional to K,/T,, which also expresses the slope of the initial step response. At longer times the response is more complicated. However, if the initial response is the same for different, equivalently tuned systems, it is reasonable to expect the whole response of the control system not to vary very much, the PI controller taking the output back to zero in roughly the same way. Naturally, the smaller the difference between the Tw’s,the smaller the difference between the responses. An illustration of this hypothesis is shown in Figure 3 for K, = T, = 2 and K, = Tu= 10 (and T = 10). Although the responses do differ somewhat, the integrated absolute errors (IAE, as a measure of control quality) are very similar. To illustrate the range of validity for this observation, Figure 4 shows step responses for K, = T, = 0.5,10, and 50, respectively. For T, = 50 the initial response is roughly the same as for T , = 10, but the long tail in the response for T, = 50 increases the IAE. This result is to be expected, since the disturbance is close to a ramp, a case where other controllers than PI should be considered. For many distillation columns the T i s in different structures do not differ much from each other. Actually distillation columns possess such dynamic characteristics that the simple observations above can be utilized for comparison of control qualities obtained with different control structures. This is illustrated in the following example, based on experimental data from a pilot-plant column.
(&’E)
v
6.7e4% lls+1 34e-I.” 1&+l 6,4e4.& 23e-l.” 26s+ 1
0.010e4.b 1%+1 0.35e-”.& 1&+1 1.0e4.& 2%+l 34e4.& 1%+ 1
-0.003e-’” -0.026e-2b 2&+1 23s+1 -0.17e-l*@ -0.81e-1.h 4.%+1 13s+l O.OO1e-l.@ -0.029e-7.b 6.0~+1 1&+1 -0.05e-’.@ -0.89e-l.& 7.5s+ 1 7.5s+ 1
- - - -
OUnits, flow rate, kg/h; composition, w t %; temperature, O C ; time, min. Table 11. Nominal Steady-State Data feed flow rate F distillate flow rate D bottoms flow rate B feed composition z distillate composition bottom composition reflux flow rate L steam flow to reboiler V feed temp reflux temp
200 kg/h 60 kg/h 140 kg/h 30 w t % 87 w t % 5wt%
60 kg/h 72 kg/h 65 O C 62 O C
However, it should be emphasized that the results obtained and displayed in Figures 2-4 are general and by no
means restricted to distillation. Also, the choice of the firsborder dead-time transfer function for the process was made for the sake of convenience only; the results should be valid for any process transfer function as long as the feedback controllers are equivalently tuned, that is, as long as the setpoint responses are the same.
Distillation System Studied The distillation process studied is a binary 15-plate pilotiplant distillation column for which first-order transfer function models have been experimentally determined (Waller et al., 198th). These models have been reconciled by optimization subject to known consistency relationships (Hiiggblom, 1989). The reconciled models are given in Table I. Steady-state operating data are given in Table 11. The primary outputs are two temperatures, on plates 4 and 14 (counting from top). The studied control structures, labeled after the manipulators used for tem-
1184 Ind. Eng. Chem. Res., Vol. 30, No.6, 1991 0.02
Table 111. Relative Gains struct ( L , V) (D,V) ( D / ( L+ D),V) ( D / ( L + D),V I B ) 1.17 1.12 re1 gain 1.81 0.10
I
I
OC
Table IV. PI Controller Settings for Single-Point Control" top loop bottom loop struct K, Ti KO Ti 1.6 26 1.6 ( L , V) -260 1.6 480 1.6 (D,V) 270 1.6 41 1.6 ( D / ( L + D),V) 2.4 1.6 0.63 1.6 ( D / ( L + D),V I B ) 3.4 a
Units consistent with units of transfer functions, Table I. 0.2
-0.02
1
I
1
15 min
0
+t Figure 7. One-point control responses of the top temperature to a ' at time t = 0. step disturbance in feed composition by 6 wt % 0.3
"C
"C
AT4
0.0
0.1
t
AT14
t
-0.3
0.0 -0.6
15 min
0
-t Figure 8. One-point control responses of the bottom temperature to a step disturbance in feed composition by 6 wt % at time t = 0. - L.V
0.0
t' l'
tI
I
0
+ t' .-.\,
0.00
.
I 15 min
+t
-0.01
One-Point Control The controller settings used at one-point control in this study are the Ziegler-Nichols ultimate sensitivity settings. The settings are given in Table IV. Responses to setpoint changes obtained with so-tuned PI controllers are shown in Figures 5 and 6, illustrating that the controllers are equivalently tuned for setpoint changes in the different structures. This means that differences in control performance at one-point control are due to differences in the disturbance sensitivity. In Figures 7 and 8 the four structures are compared at one-point control for a step disturbance in the feed composition. In Figure 7 control of T4is shown when TI4is uncontrolled. In Figure 8 control of TI4is shown when the T4 loop is open. The corresponding results for a step disturbance in the feed flow rate are shown in Figures 9 and 10. The differences in the control quality obtained are in accordance with the differences in the disturbance sensitivity parameter for the different structures. The
I
I
I
0
Figure 6. One-point control responses of the bottom temperature to a setpoint change by 1.0 O C at time t = 0.
perature control, are (L, V), (D,V), ( D / ( L+ D),V), and ( D / ( L+ D),V/B). That the disturbance sensitivities of the structures are different can be seen in Table I, where the two right-hand columns contain the disturbance transfer functions. In Table I11 the (steady-state) relative gains for the four structures are given.
.
15 min
+t Figure 9. One-point control responses of the top temperature to a step disturbance in feed flow rate by 20 kg/h at time t = 0.
-0.6
I
0
I
I
+t
I 15 min
Figure 10. One-point control responses of the bottom temperature to a step disturbance in feed flow rate by 20 kg/h at time t = 0. Table V. Disturbance Sensitivity Parameter. IK,,,/T,,l X loL top loop bottom loop struct 2 F 2 F ( L , V) 0.47 0.10 70.7 29.1 24.0 0.22 70.7 (D,V) 3.80 37.8 0.11 62.3 ( D / ( L+ D),V) 1.13 6.67 0.17 119 ( D / ( L+ D ) , V / B ) 1.61 Units consistent with units of transfer functions, Table I,
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1185 Table VI. IAE/Aw
X
Id for Single-Point Controla top loop 2
F
2
F
(D/(L + D), V) (D/(L+ D),V / B )
0.583 5.383 1.800 2.083
0.135 0.330 0.185 0.250
89.17 104.3 86.67 173.5
40.15 31.15 53.25 9.70
(L,v)
0, V) a
bottom loop
struct
6m
Units in O C min/wt % or O C min/(kg/h). 0.4
0 0
6
0.0
0.0
0.4
-2.0
, 0
+t
50 min
Figure 13. Two-point control responses of the two temperaturesto a step distybance in feed composition by 6 w t % at time t = 0. 0
0
0
200
60
Figure 11. Normalized IAE values, describing the obtained onepoint control quality, aa a function of the parameter lKu/Tulfor four structures. Two outputs and two disturbances are studied.
Table VII. PI Controller Settings for Two-Point Control" top loop bottom loop struct KC Ti Kc Ti (L,v) -130 1.6 13 1.6 (D,v) 130 3.3 9.0 4.5 0.76 1.6 13 1.6 1.0 1.6 0.19 1.6
:2
0 a
0
0
cp 0
B
Figure 12. Correlation between IAE values and the disturbance Tul for two outputs, two dieturbmcea,and sensitivity parameter four control structurers.
ww/
values of the disturbance sensitivity parameter are given in Table V. A structure that is sensitive to one disturbance can be very insensitive to another. An example is given by the ( D / ( L+ D),V / B )structure. The output of the lower loop, TI4,is strongly affected by a disturbance in the feed composition but only slightly so by a disturbance in the feed flow rate (Table I). The control qualities are here measured as integrated absolute errors (IAE) up to t = 15 min after nonzero response. In Table VI normalized integrated absolute errors at one-point control are given for the two disturbances and for the two outputs. The correlation between one-point control performance and the ratio (K,/T,I is illustrated in Figure 11, where the normalized IAE values are plotted as a function of this ratio. The points (representing the different structures) lie close to a straight line in all four cases. The linear regression lines go approximately through the origin, and they all have approximately the same slope. Figure 12, where all points are included, covers two outputs, two
Units consistent with units of transfer functions, Table I.
disturbances, and four control structures. All points lie very close to a straight line. Thus the parameter lK,/ T,I seems to be useful in evaluating distillation control structures for one-point control.
Two-Point Control For two-point control of distillation the control quality often deteriorates due to the interaction between the two control loops. The amount of interaction is generally different for different control structures. Table I11 lists the relative gains, which express the steady-state interaction for the structures studied. In Figure 13, a comparison of control performance at two-point control is shown for a step disturbance in the feed composition. The controllers used for one-point control had to be detuned in order to give reasonable responses for two-point control. The settings are given in Table VII. Overall, the best control quality is obtained with the one-ratio scheme ( D / ( L+ D),V) and by far the worst with the (D,V) scheme. The results obtained with the two-ratio sheme ( D / ( L + D),V / B ) are poorer than could be expected on the basis of the relative gain values. The relatively poor control quality obtained for the lower loop with this scheme is explained by the relatively small T,value, which gives a large disturbance sensitivity parameter ~ K Z / T Ifz ~the . disturbance sensitivitieswere equal for the two schemes, in the sense that lK,/T,I were equal, the performance obtained of the bottom loop (T1Jwould be almost equal for the two structures ( D / ( L+ D),V) and ( D / ( L + D),V / B ) . For the top loop, on the other hand, there would still be a difference in the performance between these two schemes. Even if the relative gains and also the disturbance sensitivity parameters were almost the same for the
1186 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
ratio schemes, the scheme ( D / ( L + D),V) would give better control quality for the distillation system studied. This can be explained by differences in the dynamic interaction for these schemes. For the structure ( D / ( L + D),V / B ) the disturbance affects the two control loops through very different dead times. In this case this leads to difficulties when one tries to control the loops simultaneously.
Summary and Discussion The correlation between a disturbance sensitivity parameter and control performance, for both one-point and two-point control, has been investigated by simulation of four different structures for distillation control. The parameter is defined as the absolute value of the ratio between the gain and the time constant of the transfer functions from the disturbances to the outputs, here denoted IKw/T,I. This parameter expresses the maximum slope of the response and the slope of the initial nonzero response (for a first-order system) to a step disturbance. The parameter is generally different for different structures, disturbances, and outputs. A linear correlation between the parameter lKw/T,I and one-point control quality has been found. For two-point control, the control quality correlates with the disturbance sensitivity parameter when the steady-state interaction between the loops is small. In more interacting structures interaction effects have a larger and more unpredictable effect on control quality. Also dynamic interaction caused by a large difference in dead times in the disturbance transfer functions has been found to decrease control quality for two-point control. When considering different control structures for distillation, it would be helpful to be able to predict K , and T, for the structures from known values for K , and T, for one structure. For K, this can easily be done by use of the transformations discussed by Hiiggblom and Waller (1988). The same transformations are valid also for T, if the levels can be assumed to be perfectly controlled (much faster than the composition or temperature loops) or if the level loop manipulators are not changed with the change of composition control structure. For the cases when these assumptions are not valid, the dynamic transformations discussed by Yang et al. (1990) can be utilized. Acknowledgment Figures 1-4 have been calculated and plotted by Mr. Johan Pensar. The results reported have been obtained during a long-range project on multivariable process control supported by the Neste Foundation, the Academy of Finland, the Nordisk Industrifond, and Tekes. This support is gratefully acknowledged. The comments from the reviewers were very useful in the revision of the paper. Nomenclature B = bottoms flow rate
D = distillate flow rate
F = feed flow rate Gi,.= transfer function between an output variable i and an mput variable j (subecript notation applies ale0 for gain and time constant of a transfer function) Cj = disturbance transfer functions related to an input j = F, z , or w (subscript notation applies also for gain and time constant) K = gain of transfer function (Kij or Kj; see G) K , = controller gain L = reflux flow rate T = time constant of transfer function (Tij or Ti;see G) T4 = temperature on plate 4 T14= temperature on plate 14 Ti = controller integration time t = time V = rate of steam flow to reboiler w = arbitrary disturbance y = primary top output variable (T4) 1: = primary bottom output variable (T14) z = feed composition
Literature Cited Hiiggblom, K. E. Reconciliation of process gains for distillation control structures. Proc. IFAC Symp. on Dynamics and Control of
Chemical Reactors, Distillation Columns and Batch Processes;
Maastricht, The Netherlands, 1989; pp 241-248. Hiiggblom, K. E.; Waller, K. V. Transformations and Consistency Relations of Distillation Control Structures. AfChE J. 1988,34, 1634-1648. Hammarstram, L. G.; Waller, K. V.; Fagervik, K. C. On modelling accuracy for multivariable distillation control. Chem. Eng. Commun. 1982,19,11-90. Luyben, W. L. The concept of 'eigenstructure" in process control. Ind. Eng. Chem. Res. 1988,27,206-208. Shinskey, F. G. Disturbance-rejection capabilities of distillation control systems. Proc. ACC; Boston, 1985; pp 1072-1077. Skogestad, S. Disturbance rejection in distillation columns. CHEMDATA 88; Goteborg, Sweden 1988. Stanley, G.; Marino-Galarraga, M.; McAvoy, T. J. Shortcut operability analysis. 1. The Relative Disturbance Gain. Ind. Eng. Chem. Process Des. Dev. 1986,24, 1181-1188. Tyreus, B. D. Optimization and multivariable control of distillation columns. Adv. Instrum. 1987, 42, 26-44. Waller, K. V.; Finnerman, D. H.; Sandelin, P. M.; Hiiggblom, K. E. On the difference between distillation column control structures. Report 86-2, Procgss Control Laboratory, Department of Chemical Engineering, Abo Akademi, 1986. Waller, K. V.; Finnerman, D. H.; Sandelin, P. M.; Higgblom, K.E.; Gustafsson, S. E. An experimental comparison of four control structures for two-point control of distillation. Ind. Eng. Chem. Res. 1988a, 27,624-630. Waller, K. V.; Hiiggblom, K. E.; Sandelin, P. M.; Finnerman, D. H. Disturbance sensitivity of distillation control structures. N C h E J. 1988b, 34, 853-858. Yang, D. R.; Waller, K. V.; Seborg, D. E.; Mellichamp, D. A. Dynamic Structural Transformations for Distillation Control Configurations. AIChE J. 1990, 36, 1391-1402.
Received for review November 10,1989 Revised manuscript received November 5, 1990 Accepted November 21, 1990
