Computers chem. EngngVol. 20, No. 8, pp. 979988, 1996 Copyright© 1996ElsevierScienceLtd Printed in Great Britain. All rights reserved 00981354(95)002138 00981354/96$15.00+ 0.00
Pergamon
T U N I N G GMC C O N T R O L L E R S USING THE ATV PROCEDURE SCOTT E. FLATHOUSEand JAMES B. RIGGS Department of Chemical Engineering, Texas Tech University, Lubbock, TX 794093121, U.S.A. (Received 15 November 1994;final revision received 30 May 1995) AbstractA procedure has been developed for applying auto tune variation (ATV) tuning [Astrom and Hagglund, Automatic Tuning of PID Controllers, ISA (1988)] to generic model control (GMC) controllers [Lee and Sullivan, Generic model control, Computers Chem. Engng 12, 573 (1988)]. A nonlinear two input/two output test problem, which resembles an exothermic CSTR, and a distillation column test case have been used to evaluate the effectiveness of this new tuning procedure. For each case, the GMC controller tuned with the ATV tuning procedure was compared with a GMC controller with optimal tuning. It was typically found that the optimally tuned controller had an integral absolute error (IAE) that was approximately 20o lower than the IAE for the ATV tuned controller. In addition, the CSTRIike case was used to evaluate the effect of process/model mismatch on the effectiveness of the proposed tuning procedure.
INTRODUCTION
a 4D search for an optimum. Trialanderror online selection of the four tuning parameters is usually a slow and costly process and typically leads
Regardless of the type of controller being implemented, controller tuning is a critical step. Since each control application has its own specific objectives (e.g. priority rating of the control objectives, limitations on the movement of manipulated variables, etc.) and each process has its own characteristics (e.g., dynamic response, coupling, disturbance levels, etc.), online tuning is essential. There are two major issues involved in controller tuning. First, the controller should be tuned for robustness. That is, if it is to remain in service in an industrial setting, it must be able to handle the full range of upsets faced by the process. If a controller is tuned for optimum rejection of low level disturbances, it can go unstable when faced with large upsets. Since set point changes are almost always easier to implement than disturbances, setpoint changes are usually convenient for controller tuning, In general, the larger the setpoint change used for tuning a controller, the more robust the controller, but the poorer the disturbance rejection performance. Using setpoint changes over the expected range of operation is usually a robust but conservative basis for tuning, The second major factor involved with controller tuning is selecting tuning parameters for the controller once the basis for tuning has been determined. For example, for diagonal PI controllers applied to a two input/two output process, there are four unknown tuning parameters (i.e. two gains and two reset times). The tuning objective is to optimize performance for the tuning test which involves
to inferior control performance since processes are many times slow responding and continuously subjected to a variety of disturbances. This paper is concerned with the application of the ATV testing procedure (Astrom and Hagglund, 1988) to the tuning of GMC controllers (Lee and Sullivan, 1988).
PI TUNING There are a number of methods for tuning PI single loop controllers, most notably the Cohen and Coon method (Cohen and Coon, 1953) and the ZieglerNichols (ZN)method (Ziegler and Nichols, 1942). The Cohen and Coon method is based upon open loop responses which are difficult to obtain for processes which have relatively slow response times and are susceptible to significant error due to disturbances that occur during the identification periods. The ZN method is based upon measuring the ultimate controller gain and ultimate period, but can result in excessive variations in the controlled variable. Astrom and Hagglund (1988) use a relay feedback approach to measure the ultimate controller gain and the ultimate period. Then the classical ZN settings can be used directly. Figure 1 shows a portion of such a test and is referred to here as auto tune variation (ATV) tuning. The ATV procedure is as follows:
979
980
S.E. FLATHOUSEand J. B. R]6Gs
.¥
rr~ "Amplitude ~.~
/
"Period ( T )
0.,
.Relay Height
Time Fig. 1. Results of a typical ATV test.
1. The process should be operating at relatively constant conditions, i.e. at Y0 (controlled variable) and u, (manipulated variable). 2. Choose a change in the manipulated variable, Au, such that an adequate change in y results. Typically, Au is in the range of 110% of u. 3. Make a change in u from u0 to u0 + Au. After y has departed significantly from Y0, switch u to u 0  Au.
this occurs, average values for P, and a should be used. The ultimate controller gain Kc, is given by: 4Au Kc,(1) Jra Then the ZN settings are given by: KczN= 0.45Kc, (2)
4. When y crosses Y0, switch to u0 + Au. 5. When y crosses Y0, switch to u 0  Au. Return
If these controller settings are applied to a multivariable problem, poor controller performance will
to Step 4.
r zN= 0.83P,.
(3)
usually result due to the coupling present in the
After several cycles, a standing wave will form. The amplitude, a, and the period, P,, can be read directly (see Fig. 1). With the appropriate choice of Au (the relay height, h), a proper level for the amplitude, a, can be obtained (i.e. small enough not
process. Luyben (1986) proposed the biggest log modulus (BLT) method for tuning PI controllers. The BLT method is based upon decreasing each controller gain and increasing the reset times by a single detuning factor, For (Toijala and Fagervik,
to upset operation of the process, but large enough that an accurate estimate of a can be obtained). Since the ATV procedure measures the process response at the crossover frequency, ATV testing is not possible for a pure firstorder process or an integrating process, neither of which has a crossover frequency. It should be pointed out that when the ATV identification procedure is applied to a process with an analyzer delay (i.e. discrete values are available only at the analyzer frequency), variations in the amplitude and period can result. When
1972): K,,= KZN/Fm
(4)
rl = r zN x FDr. (5) The BLT method uses a multivariable Nyguist plot to determine the value of For that meets a preset stability criteria. Therefore, a linear, stationary process subjected to low disturbance levels would likely be sluggishly tuned using the BLT method while a highly nonlinear, nonstationary process subject to large disturbance levels could be
GMC controller tuning using ATV procedure
981
tuned too aggressively using the BLT procedure, Since the BLT procedure uses linear, fixedgain models and does not consider the magnitude of the disturbances that a process faces, there can be a substantial difference between the value of For it calculates and the optimal value of For. The procedure proposed here is to adjust For by trialanderror in order to optimize the tuning performance, The 1D search required to adjust Fro. should be relatively straightforward in an industrial setting using an analysis of setpoint changes and/or disturbance rejection performance since the user only has to decide if the controller is too aggressive or too sluggish. In addition, the value of For typically ranges between 2 and 6. This procedure has been shown (Kulkarni, 1994) to be relatively insensitive to variations in disturbances, to nonstationary
it can provide complete decoupling. As the model mismatch increases, the degree of coupiing in a multivariable GMC controller will increase. The amount and type model mismatch will directly affect the GMC controller settings. • Measurement deadtime. As the ratio of measurement deadtime to process time constant increases, the aggressiveness of the GMC controller must be decreased in order to maintain stable controller performance. • Dynamic factors. Inverse action and highorder dynamics can also severely limit the degree of aggressiveness of a GMC controller. In addition, controller tuning can be significantly affected when a process responds much faster to one manipuluated variable and to another.
behavior and to measurement noise. A significant part of the lack of sensitivity to those factors is due to the online determination of Fo> When tuning diagonal PI controllers for a (2 x 2) multivariable distillation process, it has been observed that optimal values of For for the overhead control loop and for the bottoms control loop were generally found to be with 0.20.3 of each other (Kulkaini, 1994). These results point out the advan
As a result of these processdependent factors, GMC controller tuning remains a challenging problem, particularly for multivariable problems. Similar to the problem associated with diagonal PI tuning, the trialanderror selection fo tuning parameters can lead to suboptimal tuning parameters. The GMC control law (Lee and Sullivan, 1988) is given by: dy ~t =f(Y0, u, d, p)
tage of this tuning procedure. In summary, the A T V procedure provides relatively fast identification for slow processes (almost an order of magnitude faster than open loop tests) while not excessively upsetting the process. The A T V procedure has been shown not to be highly sensitive to changes in disturbances and nonstationary behaviour. The 1D trialanderror tuning of FDr provides near optimal tuning in an industrially relevant manner,
for a SISO process where y is the controlled variable, u is the manipulated variable, d is the measured disturbance, p is the model parameter, K~ and K2 are the controller tuning parameters and f is the nonlinar dynamic model function. Converting Kt
GMC tuning
and/£2 into K,. and r~ yields:
The GMC was derived and presented (Lee and Sullivan, 1988; Lee, 1993) as a nonlinear controller based upon a reference trajectory of the plant. That is, the shape and speed of the response to a setpoint change are chosen by the user. The tuning procedure proposed by Lee and Sullivan (1988) is to first choose the desired shape of the response and then set the "timing of the response in relation to known or estimated plant speed of response" (Lee, 1993). Unfortunately, several factors can undermine the effectiveness of this approach to GMC controller tuning:
• Process~model mismatch. As the amount of process model mismatch increases, the degree of aggressiveness of the GMC controller must be decreased. If the controller model is perfect,
/', = Kl(y~pyo)+ K: J (y~py) dt
(6)
0
dy [ If, ] ~t = Kc (YspY0) + 0 (y~py) d t .
(7)
The RHS of equation (7) is a PI control law. The PI control law sets the value of dy/dt and the dynamic model equation If(y, u, d, p)] is solved for the control action, u. Figure 2 shows this version of the GMC controller schematically. If the dynamic model inverse and the process are considered as a single process (denoted with the dashed line box), this schematic can be considered to be identital to a PI feedback control loop. In that case, by making relay changes in the value of dy/dt, an ATV test can be conducted on the combination of the model inverse and the process in order to identify the ZN values of Kc and rz. That is, the relay values of dy/dt would be used with the dynamic model to determine the manipulated variable value, u, which would be
982
S.E. FLATHOUSEand J. B. RIGGS
applied to the process. The relay value of dy/dt would be switched when y crossed its nominal value as with A T V tests for PI controllers. Note that the initial value of dy/dt is assumed to be zero since the A T V test is conducted at relatively steadystate conditions. Also, the relay value of dy/dt would switch from positive to negative values of dy/dt during the A T V test. For a multivariable GMC controller, an ATV test would be conducted for each controlled variable, Then the controller settings for the control laws would each be detuned by the same factor, F D T , using setpoint changes and/or disturbance rejection tests, Now consider how GMC is applied using steadystate nonlinear models (Lee, 1993). The GMC control law of a SISO process is given by: , y~S=y0+ K]~(YspYo)+K~'~; (YspY) dt (8) 0 f ~ ( y ~, u, d, p) = 0 (9)
where Yi is the value of y when the controller is tuned on. Then the combination of equations (9) and (11) would be the new GMC control law using a steadystate model and is shown sch, natically in Fig. 3. It should be pointed out that equation (11) would be expected to be superior to equation (10) since equation (10) suffers from proportional kick when setpoint changes are made. In fact, it is usually necessary to filter setpoint changes when using equation (10) in order to stabilize the steadystate GMC controller for setpoint changes. Figure 3 shows the schematic of the new GMC controller [equations (9) and (11)] for using steadystate models. Considering the dashed box as a single process, it is again seen that ATV tuning can be used to identify the tuning parameters of the PIcontroller in Fig. 3. This would be done by using relay changes to y~S based upon when y crosses its nominal value. Under normal conditions (i.e. not excessive model mismatch), the change in y ss (Ay ~s) used in the A T V test should result in measured changes in y(Ay) such
where y~ is the steadystage target calculated by the GMC control law for the nonlinear steadystate model, K] s and K~~ are the controller tuning parameters and )~s is the nonlinear steadystate model function. Note that equation (8) is used to calculate y s~and then equation (9) is used along with the value of y ~ to determine the manipulated variable level, u. Using the following definitions:
that:
f
K~s = 1 + KS,?
AySS>Ay.
(12)
This result should allow the values of Ay Ss to be chosen with confidence that the process will not be unduly upset. It is informative to consider the application of GMC with a dynamic model and with a steadystate model to the following linear process: dy
K~s = KSS/r~s
dt dy+cu
(13)
the following results:
y~S=ysp+KSS[ (y~py0)+r}lf'o(YsPy)dt ~ ]
(10)
Note that equation (8) is not, in general, a PI control law. As long as the controller setpoint, Ysp, remained unchanged, it would be equivalent to a P! controller, but if Ysp changes, it would be distinctly different. By changing equation (8) to the form of a PI controller, the following results: y~=yi+
K~.~ (Y~pYo)+r}~
Ysp~

(11)
where c and d are constants. The application of equations (7) and (11) with equation (13) for the case of Ponly control (i.e. r~,~) and assuming that u from each GMC controller (dynamic and steadystate) is equivalent results in the following relation between K~ and K~S: KS~=
K~ d
1.
(14)
This indicates that both controllers will determine the same control action if their controller gains maintain the relation given by equation (14). Also note that K,~ and K~ can have different signs.
dy I
ControelrPI ~
f'l()
tI ~  ~
I P
[I ,I I
Fig. 2. Schematic of GMC based upon using dynamic controller models.
y ~.~
GMC controller tuning using ATV procedure
I Controller
I
I
983
I l
Fig. 3. Schematic of GMC based upon steadystate controller models.
Now considering both proportional and integral action in a dynamic and steadystate GMC controller, equating calculated control action, and assuming the relationship given in equation (14) results in the following relationship between rl and r~S: 1 1 ![ Kcd ] r~ rll_Kc+d j (15)
changes faster than the concentration. In this case the coefficients were arbitrarily chosen to be unity in order that Yt and Y2 would have roughly the same dynamic response. Open loop responses for the test case showed that both Yl and Y2 have approximately first order dynamics. Table 1 lists the approximate transfer function models at the base conditions (y1=0.602802, y2=2.397198, u l = l . 0 , u2=l.0). The steadystate
For the linear case, these results indicate that dynamic and steadystate GMC can yield the same control action although the values of the tuning parameters will be different. One can easily see that for a nonlinear model no such correspondence between dynamic and steadystate GMC is likely to exist.
R G A (Bristol, 1966) is 1.24 at the base case conditions. From an analysis of the gains of this system, it was found that all the gains are nonlinear, particularly for the two gains involving Y2.
CONTROL RESULTS FOR TEST CASE I TEST CASE I (CSTR CONTROL)
The following two input/two output functions are used to test and study the ATV tuning of GMC: dyl   = ul(1 Yl) Yl e x p {  l/y2} dt
(16)
Equations (16) and (17) represent the process model in this case. In an actual application, one never has a perfectly accurate controller model. Therefore, equations (16) and (17) were modified to include model mismatch. The models for the GMC controllers are given by:
dy2 = u~(1 Y2) + Y] exp{  l/y2} + u2
(17)
where ya and Y2 are the controlled variables and u~ and u2 are the manipulated variables. The measured values of Yl and Y2 are the result of an exponential first order filter on y~ and Y2, respectively, using a single filter factor of 0.1. Note that these equations have the same general form as the equations describing an exothermic CSTR where y~ and Y2 correspond to reactant concentration and reactor temperature, respectively, and u~ and u2 correspond to reactor feed rate and heat removal rate from the reactor, respectively. In a typical exothermic CSTR, the temperature Table
1. T r a n s f e r
function
models
of test case at basecase conditions
G(s) =
0.276 1.30 0.85s+ t 1.4s+1  0.0432 1.04 1.82s+1
1.15s+l
dy~  u~(1  y J b ) y~ e x p {  b/y2}
(18)
dy2= bu](1Y2) +Y] e x p {  b / y 2 } + u2
(19)
dt ~
where b is a model mismatch parameter. Note that when b = 1, there is no model mismatch. Remember that the measured values of y~ and Y2 are filtered while the controller model contains no filtering ony~ or Y2. The base case control test involves simultaneous setpoint changes in y~ and Y2 That is, Yl is reduced by 25% from the base case condition and Y2 is reduced by 5%. The integral absolute error (IAE) of y~ and Y2 was used as the index of control performance for all tests. Each dynamic simulation consisted of numerically integrating equations (16) and (17) from t = 0 to t = 2.0 using a Euler integrator (Riggs, 1994) with a step size At of 0.001.
984
S . E . FLATHOUSE a n d J. B. RIGGS Table 2. Tuning parameters for ATV and optimal tuning for different values of b b 1.0
Yl
For
7.0
K,.
6.55 1.60
r/
1.1
1.2
4.6
1.33
5.0
1.5
5.3
1.67
5.7
5.2
13.0 0.718
13.9 0.747
12.9 0.836
18.6 0.750
11.8 0.950
9.99 0.755
9.17 0.830
8.62 0.880
8.11 0.938
8.29 0.885
35.3 0.731
8.78 1.96
8.4l 1.55
8.05 1.35
10.2 0.507
30.5 1.52
31.2 1.25
33.9 1.19
ATV
K,.
9.99 0.930
Y2
rt
Y~
K~ r~
15.1 7.5 × 10~
Y2
K, rt
5.47 1.6 x 106
27.1 9.13
OFT 9.74 0.789
First consider GMC using a dynamic approximate model [equation (6)]. Application of equations (18) and (19) to equation (6) yields: ul
(dyl/dt ) + Yl exp(  b/y2)
setpoint changes in Yl (25% reduction) and simultaneous setpoint change in Y2 (5% reduction). The size of the setpoint change and whether it represents a reduction or an increase were found to affect the value of the optimum detuning factor, For. The optimum value of FDT was found to be
(20)
1y~/b u2 = (dy2/dt)bul(1 Y2) Yl e x p (  b/y2) (21)
larger (58) for the setpoint reduction cases while it was lower (24) for the setpoint increase. The I A E for the setpoint changes generally had a fairly fiat minimum. As a result, a value of FDTof 4.5 would be expected to provide the best overall performance in the range 0 . 3 < y 1 < 0 . 9 . In order to assess the effectiveness of the ATV tuning of GMC, optimum controller settings were determined. The optimum controller settings were
where dyJdt and dy2/dt are given by equation (7). Consider the case for which b = 1.2. The A T V tests were conducted for this case and the resulting controller settings are shown in Table 2. The optimum detuning factor, For, was found to be 6.2; yl and Y2 are shown in Figs 4 and 5, respectively, for values of For of 4.2, 6.2 and 8.2. These curves are based upon
0.65
06 =
0 . 5 5
,z

........
~
~
.....................
~. . . . . . . . . . . . . . . . . . . .
~ .....................
'; . . . . . . . . . . . . . . . .
" .............
............. )i ........................................ i ............... i ..........
0.45
0 . 4
":''!~.,.,...~
................
.
0.35 0
0.2
.....
: i. . . . . .
L 0.4
~ ................
........
....... 
...... i = :
.......
l 0.6
{ = ' .
~ .................
.
0.8
.
.
: = I  . . . . . . .
.
1.2
1.4
I 1.6
TIME
4.2 Fig. 4. E f f e c t o f
6.2
~.................
............... i ............. i .............
i
.
~. . . . . . . . . . . . . . . .
8.2
FDT o n r e s p o n s e f o r Yt f o r b = 1.2 ( d y n a m i c G M C ) .
: : ~
. . . . . .
1.8
2
GMC controller tuning using ATV procedure 2.4
985
'i
2.3i
>"
2,25
.
i "~
"..
~
•
:
!
=
!
!,,
i
=
:
i
2 . 2 . . . . . . . . . . . i.x . . . . . ,,f.............. i............... i. . . . . . . . . L__. . . . . , _ . ! ~ ./ i
...............
~
i .....................i.......................... ! ........... i .....
i
i
i
i
i i
i i
J ............ i ....... i ........
2.1 0
0.2
0,4
0.6
0.8
1.2
1.4
1.6
1.8
2
TIME
4.2
6.2'
8.2
Fig. 5. Effect of F~T on response of y2 for b = 1.2 (dynamic GMC). obtained using a N e l d e r  M e a d optimizer (Riggs, 1994) to search the 4D space (two gains and two reset times) for the m i n i m u m cumulative IAE. Figure 6 compares the I A E obtained for the A T V tuned G M C controller with the results of the optimally tuned G M C controller as a function of the model mismatch parameter, b, for the case of a setpoint 25°/0 reduction in y, and 5% reduction in Y2. Note that until b > 1.1, the cumulative I A E for the A T V tuned controller is not significantly affected by mismatch in this case. In the range of b = l. 11.2, which is a reasonable range for model mismatch, the A T V tuned G M C controller differs from the optimally tuned controller by about 20% based upon I A E
0.65 0.6
0.5s ~ ~1~I ~ 0.5. 0.45\ / ~/~v,0.4 0.3s  0
0~2
L ~ ]
0.4
0.6
I
0.8
For
1
TIME
1.2
OPT,
1.4
1.6
1.8
/
Fig. 7. Response for y] for ATV and optimally tuned controllers for b = 1.2 (dynamic GMC). 11 1o 9 ~
8
/
s~ . . , . j ~ ~ c  ~ : 1
values. Figures 7 and 8 show the response for y~ and for b = 1,2 for the optimally tuned and A T V tuned controllers for the base case test, Although both G M C controllers used negative values of the manipulated variables, both showed smooth and consistent use of the manipulated variables. When the controller model is perfect (b  1 ) , the optimally tuned controller has an I A E that is 30% lower than the A T V tuned controller. U n d e r these conditions, the optimally tuned controllers use essentially no integral action (see Table 2) since there is no steadystate model mismatch. Also, note Y2
1.1
~
1.2 1.a 1.4 1,s Model Mismatch Parameter (b)
. 1.6
1.~
Fig. 6. Control performance for ATV and optimally tuned GMC controllers as a function of models mismatch (dynamic GMC).
986
S . E . FLATHOUSEand J. B. Rzoos
that the controller gain on y~ is increased by factor of
0.55 l
i
two and the controller gain on Y2 is approximately
0 6 ,,
i
;
~ ;i
~ !
I "i
i
i
l
i t
J I
cut in half. Figures 9 and 10 show the response for Y' and Y2 for b = 1 for the optimally tuned and A T V tuned controllers. Note that the largest difference between these controllers is for the control of Y2 where the optimally tuned response shows considerably less overshoot. Even when b = 1.67, the optimally tuned controller has an I A E that is only 38% lower than the A T V tuned controller. The major difference between the A T V and optimally tuned G M C controller occurred for the control performance of Y2The optimum
tuning
results
for Y2 show
more
behavior but less overshoot than the A T V tuned controller, If the values of K~ and r/ for the A T V tuned controller (listed in "Fable 2) are converted in ~ (damping coefficient) and r (closed loop time constant) (Lee and Sullivan, 1988), the values of ~ are found to vary between 1.4 and 1.8 while r varies from 0.2 to 0.5 both of which are quite reasonable for this case. Now consider the application of G M C to the test case using steadystate models. Application of G M C using steadystate models results in:
My]S)
y]* exp( 
~i ~ 0.5
'
i
0.45! i 0.4~ ] 0356
i
i
i
i
\:
i
' 016 0.8
014
i
/
~ 0.2
i
1.2
; i = i 114 1.6 1.8
1y]Vb u2=_bu~(l_y~2~)_y]~exp(_b/y)~) where y]~ and y~ are given by equation (11). The base case control test [step reductions in the setpoint of Yl (  2 5 % ) and Y2 (  5 % ) ] was used to study the effect of model mismatch on a G M C controller that was tuned using the A T V procedure with online detuning. It was obvserved that the
I
FDT 
OPT. 1
Fig. 9. Response of Yt for ATV and optimally tuned controllers for b = 1.0 (dynamic GMC). I A E for the A T V tuned controllers remained constant at approximately 21.2 as b varied from 1.0 to 1.45. In addition, the I A E for the optimally tuned G M C controller based upon steadystate models was found to remain relatively constant at 13.1 as b varied from 1.0 to 1.45. The I A E for the optimally tuned controller was 38% lower than the I A E for the A T V tuned controller. Note that the I A E for the A T V and optimally tuned controllers based upon dynamics models were substantially lower than the corresponding results for the controllers based upon steadystate models (see Fig. 6). Results were also obtained for step a 25% increase in y~ and a 5% increases in Y2. For the controllers based upon steadystate models, the A T V tuned controller yielded a constant I A E of 11.0 while the optimally tuned controller had a constant I A E of 4.7 as b was varied from 1.0 to 1.45. For this case, the optimally tuned controllers resulted in [ A E values that were 58% lower than A T V tuned controllers.
2.35 .
.
.
.
.
.
.
.
.
2.362
: 2.3
'
i
2.25
2.34~  
i
"
I
1
'~
l I
1
1
~
2.324
[
~
2.28t
' 2.2
;
'
'
0.2
014
o'6
0.1ai
1'.2
i
1'.4
2.244 2.224
1'.5
t\ I
. /

1.8
2
TIME
I FDT OPT. ] Fig. 8. Response for Y2 for ATV and optimally tuned controllers for b = 1.2 (dynamic GMC).
2.2~0
1
~
Ii / 0'.4
"
T
[
.
.
.
.
•
~
~

,"
•
r
,
1.6
1.8
.
"
0.2
'
. . . .
2.264
j
2.1~
2
TIME
oscil
latory
u~ 
0.55
ill/~t/\ t
!
0'.6
0'.8
"1 TIME
1.2
1.4
I FDT   OPT. I Fig. 10. Response of Y2 for ATV and optimally tuned controllers for b = 1.0 (dynamic GMC).
GMC controller tuning using ATV procedure Table 3. Design specification for distillation case study Relative volatility Number of ideal stages Feed tray location (from bottom) Feed Composition (Mole % light key) Feed rate Design factor times minimum reflux ratio Bottom purity Overhead purity Reflux ratio Boilup ratio Feed quality Composition control configuration Hydraulic time constant for each tray Reboiler and accumulator level control Residence time in reboiler Residence time in accumulator
987
Table 6. Tuning parameters for ATV and optimal tuning for distillation case study
2.0 50 25 50% 1.0 kg mole/s 1.2 1.0% 99.0% 1.9548 2.9548 Saturated
ATV tuned
Optimally tuned
x
K~ r~
4.5 58. l 298
145 449
Y
K~ r? IAE
34.6 713 8.97
145 937 7.62
Detuning factor
(L/D,V/B) 6s P only 3.3 min 2.2 min
0.998 0'9971
0.996]
E 0.994] / Table4. Modelling assumptions for distillation case study
~ 0.99.3
Liquid dynamics
Yes
Analyzer delays on product composition Constant relative volatility Heat transfer dynamics Saturated liquid feed Subcooled reflux Pressure dynamics modeled Equimolar overflow
l0 s Yes No Yes No No Yes
Negligible vaporholdup Valve dynamic on flows
Yes No
.
~~ 0.9921 /
......
0.99~ 0.969 0
t
. . . . . . 1000 2000 3000 4000 6000 6000 7000 8000 Time,
seconds
I I
OptimumGMC
ATV/GMC
I J
Fig. 11. Overhead composition responses for ATV and optimally tuned controllers for setpoint changes (steadystage GMC). TEST CASE I1 (DISTILLATION CONTROL) Test Case II involves applying the A T V t u n i n g p r o c e d u r e to the steadystate version of G M C for a distillation column. W h i l e t h e distillation c o l u m n m o d e l does n o t r e p r e s e n t a specific industrial colu m n , it c o n t a i n s the key dynamics a n d n o n l i n e a r aspects of a n i n d u s t r i a l d i s t i l l a t i o n c o l u m n (e.g. traytotray m o d e l , tray dynamics, a c c u m u l a t o r and r e b o i l e r level control a n d analyzer delays), with a simplified t h e r m o d y n a m i c description (i.e. c o n s t a n t m o d e l overflow a n d c o n s t a n t relative volatility). T h e c o l u m n has a relative volatility of 2.0 with p r o d u c t impurities of 1.0%. Since tray t e m p e r a t u r e s could be used for this case to infer p r o d u c t c o m p o sition, an analyser delay of 10 s (the r e s p o n s e time of a t h e r m o c o u p l e ) was used. T h e c o l u m n uses the
CONTROL RESULTS FOR TEST CASE II T h e G M C controller was b a s e d u p o n the a p p r o a c h used by Riggs (1993) in which t h e controller model was a steadystate traytotray binary model. In fact, the controller m o d e l is t h e steadystate version of the c o l u m n stimulator. T h e only m i s m a t c h b e t w e e n the d y n a m i c c o l u m n m o d e l
et al.
0.011 0.0091/ .~ °'°°811 ~ o.o071\
. .
. .
. .
.
.
. . . . .
. .
.
.
d°ublerati°c°nfigurati°n(L/D'V/B) f°rc°mp°E°°'°°6]=~ sition control. T h e design specifications for the colu m n are listed in T a b l e 3 while the d y n a m i c modelling a s s u m p t i o n s are listed in T a b l e 4.
Table5. Settings for Ponly level controllers Level controller gain (kg mol/s kg mole) Reboiler Accumulator
°'°°5 / °'°°41 O.O03J O.002
0
b
1600 2000 3(J00 4000 5600 6000 7000 8000 Time,
I ~_ 0.005 0.0025
seconds
OptimumGMC  
A'D//GMC
I I
Fig. 12. Bottoms composition response for ATV and optimally tuned controllers for setpoint changes (steadystate GMC).
988
S.E. FLATHOUSEand J. B. Rm6s
and the steadystate controller model is the result of dynamic mismatch; therefore, the specifications of the steadystate controller model can be found in Table 3.
manipulated variables than the optimally tuned controller.
The column is equipped with Ponly level controls for the accumulator and reboiler. The level controller settings are listed in Table 5. These settings represent relatively tight level control, The G M C  A T V tests were conducted for each product composition control loop. Then the detuning factor, For, was determined based upon simultaneous setpoint changes in the overhead and bottom product composition. The bottom product setpoint was reduced from 1.0 to 0.5% at the same time that the overhead setpoint was increased from 99.0 to
An industrially relevant means of tuning GMC controllers (using either steadystate or dynamic models) based upon the ATV procedure has been developed and demonstrated to be effective on two test cases. For the case studies considered, the optimally tuned controllers had IAEs that were typically 20% lower or less than the corresponding ATV tuned controllers.
99.5%. The detuning factor was adjusted to yield the minimum IAE for this setpoint change. The tuning factor and the controller settings are listed in Table 6. In a manner similar to the procedure used for the first test case, a NelderMead optimizer was interfaced with the simulations and controller and allowed to adjust the four controller parameters until the minimum IAE was obtained for the previously described setpoint test. The optimal controller settings are also listed in Table 6. Note that the optimally tuned GMC controller had an IAE that was 15% lower than the ATV tuned GMC controller. Figures 11 and 12 show the control cornparisons for the setpoint change. The ATV tuned and optimally tuned controllers had essentially equivalent performance for the overhead composition. For the bottom composition, both showed the effects of coupling. The optimally tuned controller showed faster settling time than the ATV tuned controller for the bottom composition, but also showed more high frequency variations. As a result, the ATV tuned controller had a smoother use of
CONCLUSIONS
REFERENCES Astrom K. J. and T. Hagglund. Automatic Tuning o f PID Controllers. ISA: Research Triangle Park (1988). Bristol E. H., On a new measure of interaction for multivariable process control. IEEE Trans. Auto. Control ACI1, 133 (1966). Cohen, G. H. and G. A. Coon, Theoretical considerations of retarded control. Trans. A S M E 75, 827 (1953). Kulkarni S., Modeling and control of a vacuum binary distillation column. M.S. Thesis, Texas Tech University (1994). Lee, P. L., Generic model control   the basics. In Nonlinear Process Control: Application o f Generic Model Control, Lee, P. L., Ed. SpringerVerlag, New
York (1993). Lee, P. L. and G. R. Sullivan, Generic model control (GMC). Computers Chem. Engng. 12, 573 (1980). Luyhen W. L., Simple method for tuning SISO controllers in multivariable system. Ind Engng Chem. Res. 25, 654 (1986). Riggs J. B., An Introduction to Numerical Methods for Chemical Engineers, 2nd Edn. Texas Tech University Press (1994). RiggsJ. B., M. Beauford and J. Watts, Using traytotray models for distillation control. In Advances in Industrial Control (Lee, P. L., Ed.) SpringerVerlag, New York (1993). Toijala K. and K. Fagervik, A digital simulation study of twopoint control of distillation columns. Kemian Teollisuis 29, 5 (1972). Ziegler J. G. and N. B. Nichols, Optimum settings for automatic controllers. Trans. A S M E 64, 759 (1942).