ChemicalEngineering
Science,
1972, Vol. 27, pp. 13251335.
Pergamon
Press.
Printed inGreat
Britain
Sampleddata noninteracting control for distillation columns J. P. SHUNTA and W. L. LUYBEN Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, U.S.A. (Received
27 May 197 1; accepted
9 September
197 1)
AbstractThe problem of interaction in the sampleddata control of both ends of a binary distillation column is investiaated. A dualaleorithm control concept is proposed that effectively handles both load and setpoint disturbances sim&neously. INTRODUCTION
between control loops can occur when feedback control is used to regulate the compositions of product streams in both the rectifying section and the stripping section of a distillation column simultaneously. A change in vapor boilup, for example, causes the rectifying section controlled variable to deviate from its setpoint, thereby initiating a change in the reflux flow, which in turn affects the controlled variable in the stripping section. The control loops are “coupled”. An action in one loop causes a reaction in the opposite loop and this can lead to instability. Luyben[ 11 points out several brute force techniques to circumvent this problem such as overdesigning a column by providing more trays than required to achieve the desired separation and/or running the column at an excessively high reflux rate. He approaches the problem of interaction in columns by employing continuous compensators to “decouple” the loops. Buckley [2] suggested a simple design for decoupling elements that cancel the direct effect of one manipulative variable on a controlled variable by the correct change in the other manipulative variable. Luyben [ 11 showed that this technique is effective. The decoupling of control loops in distillation columns by digital compensators has not appeared in the literature. However, Fitzpatrick [3] designed decoupling controllers for setpoint disturbances in multivariable sampleddata systems in general. This design method is effective for set point changes but is not apINTERACTION
plicable for other types of disturbances encountered in distillation columns such as load disturbances in feed composition and feed rate. This paper applies Fitzpatrick’s design to setpoint interaction compensators and introduces a new method to cope with load disturbances. These sets of compensators are only adequate for the particular disturbance they are designed for. That is, setpoint interaction compensators do not perform well for feed composition disturbances and vice versa. Therefore, a design method is presented which eliminates interaction when either type of disturbance is encountered, separately or together. SYSTEM
A twenty tray binary distillation column was simulated on Lehigh’s CDC 6400 digital computer. The simplified model assumed equimolal overtlow, ideal trays, negligible vapor holdup, and constant relative volatility. The nonlinear tray equations were numerically integrated to obtain the time response. The steady state conditions and parameters for the simulation are listed in Table 1. It is desired to control the compositions on tray 18 and tray 2 by manipulating the reflux flow and vapor boilup. The individual control loops, void of interaction compensators, are shown in Figs. 1 and 2. For purposes of illustration, the sensor is shown to be a composition transmitter. One can assume that this device is a temperature transmitter and the analog signal is multiplied by a gain to indicate
1325 CES Vol. 27 No. 61
and W. L. LUYBEN
J. P. SHUNTA Table 1. Steadystate conditions and parameters
Feed 
Total number of trays Feed tray Relative volatility Feed rate Overhead product rate Feed composition Reflux drum holdup Tray holdup Reboiler holdup Hydaulic constant Reflux rate Vapor boilup Bottoms product composition Overhead product composition
20
Xf
10
2 100 moleslmin 50 moleslmin 0.5 mole fraction 100 moles 10 moles lOOmoles O1 min 128.01 moleslmin 17801 moleslmin O02 mole fraction 0% molefraction
va
Steam
BottonlscL’
Distillate
Fig. 2. Sampleddata
Feed
feedback control system in stripping section.
x‘
Fig. 1. Sampleddata
feedback control system in rectifying section.
a deviation in composition. H is a zeroorder hold which represents the interface between the computer and the process.
D represents the digital computer algorithm. The continuous flow loop is assumed to be ideal, that is, the dynamics are fast enough in comparison with the composition control loop to be neglected. The digital controller thus falls into the category of supervisory control because it supplies a setpoint signal to the flow loop. The Bode plots of the transfer functions required to design the sampleddata compensators were determined from the linearized tray equations using the “stepping” technique of Rippin and Lamb[4]. The transfer functions were approximated in the Laplace domain by fitting the Bode plots with the leadlag and dead time models as shown in Table 2. INTERACTION
H(s)
=
1w
(We s
T is the sampling period Laplace transform variable.
(1)
(min) and s is the
IN DISTILLATION CONTROL
COLUMN
Setpoint disturbance Figure 3 illustrates interaction when only setpoint changes are made and both ends of the column are on automatic control.
1326
Sampleddata noninteracting control for distillation columns
Dsmp[5]. The sampling period is one minute or about 10 per cent of the process time constant. Minimal prototype design specifies that the closed loop response of the controlled variable, to a specified input disturbance, reach the setpoint in the minimum time without having intersample oscillations or “ripple”[6]. For first order processes, the minimum time without ripple is one sampling period. For a second order process, it is two sampling periods. The specified response with minimal prototype digital controllers can be obtained when one loop is on control and the manipulative variable at the other end of the column is fixed. However, when both loops operate simultaneously, interaction results as indicated in Fig. 3. Curves a and b show that when the Dsmp controllers are used, both controlled variables xl8 and nz become offset from their respective desired steady state values following a setpoint change at one end of the column. Instability occurs when both setpoints are changed simultaneously (not shown). Curves c and d are the responses with continuous controllers designed for a + decay ratio. The top loop contains a PI controller with a gain of 600 and a reset time of 0.5 min. The bottom loop contains a proportional controller with a gain of 2000. The interaction with these continuous controllers is less than with the digital controllers.
Table 2. Transfer functions
Gtis) %lR
O+M13(1~19s+ l)e“‘MJ’ (12.6s+l)(2s+l)
X18/V
0.0394 13*4s+ 1 O.(J27e1”’
X,lR
12.5s+ 1
G(s) G&f
4. 195e1’08S 13.3s + 1 2.94e0.41
af

13.3s + 1
Disturbances setpoint
5
load
$
4
2 Time,
6 min
Fig. 3. Interaction with set point disturbance.
The digital control algorithm is a minimal prototype controller designed to compensate for a step change in setpoint and is denoted
Load disturbance Interaction with a step load disturbance (Ax,= 0.05 m.f.) is substantial for digital control also. Figure 4 illustrates this when only one loop is on control and when both loops are on control. The digital controller used in this case was designed to obtain minimal prototype response in the face of a step change in feed composition and is denoted Dlmp[5]. This digital controller drives the controlled variable to the desired value in the minimum amount of time without ripple when only one loop is in operation
1327
J. P. SHUNTA
and W. L. LUYBEN
(curve c). The load response of the continuous PI and P controllers is quite good (curve d). NONINTERACTING
min
Fig. 4. Interaction with x, load disturbance.
Design methodfor
(curves a and b). However, when both ends of the column are controlled simultaneously with the Dlmp algorithm, the process is not stable
GPH(z)
LOOPS
The design presented to obtain noninteraction involves the same criterion used for minimal prototype algorithms, that is, the controlled variables and the disturbance variables are specified at the outset. Minimal prototype controllers Dsmp and Dlmp designed for single loops are adequate only for the particular type of disturbance specified in the design equations [5]. Similarly, the noninteracting controllers presented in the next two sections are satisfactory for only one type of disturbance. Thus, it is necessary to design one set of noninteracting controllers for setpoint changes and a different set for load changes. The following section describes the design technique for setpoint changes. It is essentially the method presented by Fitzpatrick [ 31.
e Time,
CONTROL
setpoint
disturbances
The multivariable control system is shown in Fig. 5. The process transfer function matrix GpH(z) is:
= O0413( 1*19s+ l)e0’063s (12*6s+ 1) (2s+ 1)
0.002~1( 1

+
H(s)*
(2)
0.00127~~( 1 + 0*638z‘) ( 1  0*923z’ ) loop equation
1)
0*428z’ ) (1 038z’ ) ) ( 1  0*607z’ )
( 10.924~~
The zdomain
0.0281(1*33s+ (12*5s+l)(s+l)
for the controlled
variable Jr,(z) is: 1328
Sampleddata noninteracting control for distillation columns X~(z) = 6 p I  I ( z ) O , ( z ) [ x ~ ( z )  X ~ ( z ) ] .
(3)
T h e superscript a r r o w d e n o t e s a v e c t o r and bold face type d e n o t e s a matrix. Specify ~c(Z)= ~ t ( z ) z 1, that is, xc r e a c h e s the setpoint in one sampling period. (T = 1 min.) Substituting for ~c (z) in Eq. (3):
~e~(z)z~ =
Ds21 (z) = 240z 1 ( 1 t0"638z 1 ) ( 1  0"924z ~ ) X ( 1  0"607z 1 ) ( 1  0"368z ~ ) × (1  0928z 1 ) / D E N
Ds22 (z) = 379(1  0"428z ~ ) (1 + 0"08z 1 ) x (1  0"923z ~ ) (1  0"368z 1 ) × (1  0928Z1)/DEN.
G p H ( z ) a s (z)[x~°t(z)  xse~(z)zq. (4)
T h e controller matrix D s (z) is obtained b y canceiling ~set, premultiplying by G p H ( z ) 1 and postmultiplying by [I Iz~] 1: Ds(z ) = GpH~(z)zI[IIz~]L
(5)
E a c h element o f D, (z) is:
Dsij (z) =
GpHo1 (z)z 1 1  z 1
(6)
GpH~(Z) is the ijth e l e m e n t o f the inverse o f GpH(z) where
T h e d e n o m i n a t o r o f Eq. (8) is: D E N = (1  0,381z 1) (1 + 0.429z ~ ) (1   z ~ )
× (1z1)(11.5zl+O.431z2).
(9)
T h e controllers can be applied to any form o f setpoint change but ~c is specified to be equal to ~set(z)z~, so that Xe will always lag X set by one sampling period. T h e closed loop r e s p o n s e s to setpoint dist u r b a n c e s in xls and x2 on the nonlinear simulation are s h o w n in Fig. 6. C o m p l e t e decoupling o f the control loops is achieved. T h e process transfer functions in G p H ( s ) w e r e also a p p r o x i m a t e d with a simple first o r d e r
0"00286z1 (1 0"928z ~)
I_ 0"00264z~ ( 1   0 . 9 2 3 z  ~ ) ( 1   0 . 3 6 8 z ~)
)]
GpH(z)I =
0.002z ~ ( 1  043z x ) (1 + 0"08z 1 ( 1  0924z ~ ) ( 1  0"607z 1 )
0 . 0 0 1 2 7 z  2 ( l + 0 . 6 4 z ~) ( 1   0 . 9 2 3 z ~)
x
( 1  0.924z ~ ) (1  0.607z 1 ) ( 1  0.923z ~ ) 1 . 9 x 10 ~ z_2(l_O.381z_~)(1+O.429z_l)(l_z_l)
x ( 1  0368z 1 ) ( 1  0"928z 1 ) (1  l'5z1 + 0"43 lz  z ) J"
T h e elements o f D, (z) are:
Ds~ (z) = 500(1  0"48z ~ ) (1  0.924z 1 ) x ( 1  0.607z ~ ) (1  0 . 9 2 8 z  ~ ) / D E N (8)
Dsa2 (z) = 540 ( 1  0"928z 1 ) (1  0"607z 1 )
(7)
lag and deadtime model. T h e nonlinear closed loop setpoint r e s p o n s e with the simplified controllers (not shown) was m o r e oscillatory than with the m o r e e x a c t controllers but still satisfactory.
Design method for load disturbances
x ( 1  0"923z 1) ( 1  0.368z  1 ) / D E N
T h e block diagram for only load changes is 1329
J. P. SHUNTA Matrix bbck diagram
GP(s)
H (5)
and W. L. LUYBEN
shown in Fig. 7. The zdomain loop equation is:
s;, (s)
K(z) =
G,L(z)GPH(z)WZ)%(Z).
(lo)
Rearranging: &(z)&(z)=
GPWZI’
[a(z)
G(z)l.
(11)
DL(z) cannot be obtained explicitly from Eq. (11). However, the following shows how the controller matrix can be determined implicitly. The left side of Eq. (11) is: Scalar block dicqmm
XSM
2
+
1 ?my
a
L(s)
SIZ
Fig. 5. Multivariable control syx
MilWiX
(L(s) = 0).
aD Y
______b
SCCJk Mock N
x
0.06
I
I
2 Time,
I
4
I
6
min
Setpoint change in X18
Curve aa’ bb’ cc’ BZI Fig. 6. Closed loop response with noninteracting D, for setpoint changes.
L*J control Fig. 7. Multivariable control system (Pet = 0).
1330
Sampleddata noninteracting control for distillation columns
The right side of Eq. (11) expands to the following if xi is specified to be the numerator of G&:
1
GpH,,'
1
GpH,,'
GpHn'
GPH~~'
(1l > x1
Ql
:(
k1 2
1
I
(13)
x.2
Qi is the denominator of G&i. For this system QI = Q2. Define a new variable Y: Y= (kl)=
(k1).
(14)
Multiplying the terms in Eq. (13) we obtain: (GPHIIlx1 + G~Hnlxz)
y
( GPH~~~x~+ GpHz2‘x2) 1 *
(15)
Equating the elements in Eq. (12) with the corresponding elements in Eq. (15): DLij (Z) = GPHIJ’ Y
(16)
where 1.928zl(1O48zi) Y= (1zl)(l0928z‘)’
(17)
The elements of D, (z) are: DLll (z) = 964( 1  0*48z1) ( l0*482’ ) x (I 0.924z+) (1  0.607~~‘)/DEN DL12(Z) =1040(10924z1)(1O~~7z‘) x (1  0.48~~l) (1  0.368~~‘)/DEN DLzI (z) = 464z’ (I+ 0.638~~’ ) (1  0.482l ) x ( I 0.607~~’ ) ( 1  0.368~~’ ) x (1  0924z‘)/DEN DL22(z) = 730( 1  0.428~~l) (1 + @O~Z~) x ( I @&’ ) (1  0.368~~’ ) x (I0923z‘)/DEN. (18) Note the similarity between 4(z) and Ddz). The denominator of the D, (z) and D,(z) elements are equal. The gain constant of D,(z) is 1.928 times as large as the gain in D, (z) . D,(z)
contains a zero in each element (z. = 0.48) that is about half the value of the corresponding zero (z = 0928) in D,(z). The closedloop response of x18 and x2 to a step change in feed composition with DL(z) (curve e) is compared to interacting control in Fig. 4. The controlled variables respond as they would if the control loop contained a Dlmp controller and the opposite manipulative variable was fixed. A simple process model GpH(s) led to D,(z) controllers that gave an unstable response to the load disturbance. However, it was noted in the previous section that the simplified model yielded satisfactory D, (z) controllers. The explanation is that the DL(z) controllers have larger gain constants than D,(z) controllers and therefore are more sensitive to process nonlinearities. Dual noninteracting control scheme The control schemes described in the previous two sections are satisfactory for the particular disturbance they are designed for. However, when the other type of disturbance is encountered, the closed loop response is not satisfactory. It is therefore necessary to design a control system, as shown in Fig. 8, in which there are algorithms for both setpoint and load disturbances. The load algorithm matrix DL(z) can be determined by setting the setpoint vector to zero. 4(z) is the same as the algorithm matrix in the previous section. The setpoint algorithm matrix D, (z) is determined from the matrix equation for the block diagram in Fig. 8 when L(s) is set to zero. K(z)
= GpH(z)[D,(z)XZSe’(z)
DL(z)Xc(z)l (19)
Specify: X,(z) = Pt(Z)Zl. Substitute Pt(z).
1331
EC’=
(20)
the above into Eq. (19) and cancel
GpH(z)D,(z)
GpH(z)DL(z)z‘.
(21)
J. P. SHUNTA
and W. L. LUYBEN
The elements of this controller do not contain the zero (z = O928) present in the controller D, developed for setpoint disturbances. Closed loop responses The closed loop responses of x2 and xl8 to step setpoint and x, disturbances are shown in Figs. 9a and 9b. Curve a in Fig. 9a shows that the minimal prototype response of x2 and xl8 is achieved without interaction when an x, disturbance and setpoint changes in both control loops occur simultaneously. The oscillatory response in x2 indicates the inaccuracy in the linear models to totally represent the nonlinear behavior of the process, and that only one sampling period was specified to reach the setpoint. Curves b and c in Fig. 9a compare the response with the dual digital noninteraction controllers to the continuous controllers for a load disturbance. Digital control occurs without interaction. However, the continuous controllers achieve better control even without any sort of decoupling mechanism. Figure 9b compares digital noninteracting Fig. 8. Dual non interacting control system.
Premultiply
by GpH(z)’
and solve for D, (2).
D,(z) = GpH(z)‘[I+GpH(z)D,(z)]z1.
B x
(22)
Each element of D,(z) is:
Dtij (Z) = G~ffij
(z)'[I
+ GpHtj (Z)DLij (z)]zl. (23)
The elements of D, (z) are: DSll(z)
=500(1O~48z1)(10924z1) x (l0.607~~‘)/DEN Dsrz (z) = 540( 1 0.607z’ ) ( I O9232’ ) x (1  0.368~~‘)/DEN Dszl (z) = 24Ozl( 1 + 0.638~~‘) (1  0*924z‘) x (l0~607~~)(10~368z‘)/DEN DSz2 (z) = 379( l 0*428z‘) (1 +O.O8z‘) x (l0924z‘)(I0.368~~‘)/DEN.
Curve
Ax%’
A.$’
Ax,
b” CPIP
0.01 0 0
0.01 0 0
0.05 0.05
Fig. 9a. Closedloop
(24) 1332
responses with dual noninteracting control.
Sampleddata noninteracting control for distillation columns
Time,
min
Fig. 9b. Setpoint change in both ends.
control with continuous control for setpoint changes in both loops. Interaction with the continuous controllers is not a problem. Interaction with digital control is eliminated by the design method presented. Figure 9c shows the stability of digital noninteracting control when large process upsets like a feed rate change from 100 to 140 moles/ min occur.
Fig. 10. Simplified digital decoupling scheme.
continuous systems. The extension control is quite simple. The effects of both manipulative V(z) and R (z) on xz (z) are:
x,(z) =
GpH,,(z)V(z)
to digital variables
+GP&(z)R(z).
(25)
If R is changed, I/ can be changed to counteract the effect and keep x2 constant. The interaction compensator D,(z) is determined from Eq. (25) by setting x,(z) (perturbation variable) equal to zero.
V(z) D,(z) = =R(z)
Time,
(z) (z)
= 0.481z‘( 1+ 064.P) (l 0.3682l) (l0.48~~‘)
min
Fig. 9c. Feed rate disturbance.
Simplified digital decoupling
GP& GPHB
*
(26)
scheme
The dual noninteracting control scheme can Similarly, D,(z) is determined from Eq. (27). be simplified by replacing the algorithms DL12, (27) X~Z) = GP&(z)~(z) +GPfbdZ)R(Z) D SE!9 Q,zl, Dszl in Fig. 8 by two compensators, D, and D2, as shown in Fig. 10. D, receives its input signal from the output of the control R(z) _ zpd;; Ds(z) = v(z) algorithms Dsll and DLII and D, receives its 11 signal from Dszz and D,,,. Dsll, DUE, D.w, 1.43(10~924~~)(10~607~‘) and DLzz are the same as shown in the previous = (l0.~8z‘)(~O~428~‘)(1+O~~z1)’ section. The concept for this type of noninter(28) acting system was introduced by Buckley[2] for 1333
J. P. SHUNTA
and W. L. LUYBEN
The closed loop responses with D, and D, are identical to those for the noninteracting control system described in the previous section. The sampling instants of D, and D2 coincide with DSlly DLlly Dsz2 and DLz2. The advantages of this configuration are that there are half as many interaction compensators as the previous scheme and D, and D2 are less complex than the algorithms they replace. However, the feedback portion of the noninteracting system, DLll, DLz2, Dsll and Dsz2 are still determined from the matrix Eqs. (16) and (22). An alternative to this dual combination of load and setpoint interaction compensators is to utilize the load algorithms during normal plant operation and then switch to the setpoint algorithms when a setpoint change is made. This concept was proposed by Mosler[7] for implementing minimal prototype algorithms in single loops but can be applied to the interaction problem as well. It was shown, however, that the dual scheme for implementing minimal prototype control is superior to the “switching” method [5].
CONCLUSIONS
Interaction is a significant problem when controlling both the rectifying section and stripping section tray compositions with digital controllers. Noninteracting digital control algorithms were designed to eliminate the interaction when setpoint and load disturbances are encountered. Controllers designed specifically to eliminate interaction for setpoint disturbances are not satisfactory when load disturbances are encountered and vice versa because the controlled variable has to be specified differently in the design equations. A dual combination of these algorithms is presented which achieves noninteracting control for both types of disturbances.
NOTATION
composition transmitter digital controller simplified decoupling algorithms minimal prototype algorithm for load disturbances Dsmp minimal prototype algorithm for setpoint disturbances controller for load DL noninteracting disturbances D, noninteracting controller for setpoint disturbances denominator of DLand D, DEN sampled error signal E*(s) F feed rate, moleslmin FC flow controller CL(S) load transfer function GLL (z) ztransform of combined load transfer function and load variable process transfer function GP(s) Ztransform of combined process GPHW transfer function and hold device zero order hold H(s) I identity matrix load variable Us) sampled manipulative variable M*(s) m.f. mole fraction NC control tray NT top tray P proportional controller PI proportional integral controller Qi denominator of GLLi R reflux flow rate, moleslmin Laplace transform variable ; sampling period, min V vapor boilup, moles/min x composition of liquid phase, m.f. controlled variable, m.f. XC feed composition, m.f. Xf XSet setpoint variable, m.f. Y variable defined by Eq. (14) Z Ztransform variable CT D(z) DI,Dz Dlmp
REFERENCES [I] LUYBEN W. L.,A.I.Ch.E.J11970 16. [2] BUCKLEY P. S., Chemical Engineering Seminar Presented at Ohio University 133 FITZPATRICKT. J. and LAW V. J., Chem. Engng Sci. 1970 25.
1334
1967.
Sampleddata noninteracting
control for distillation columns
[4] RIPPIN D. W. T. and LAMB D. E., A Theoretical Study of the Dynamics and Control of Binary Distillation presented at A.1.Ch.E. Meeting, Washington, DC. 1960. [5] SHUNTA J. P., Ph.D. Dissertation, Lehigh University, Bethlehem. Penna. 1971. 161 TOU J. T., Digital andSampledData Control Systems. McGrawHill, New York 1959. [7] MOSLERH.A.etal.,A.I.Ch.E.JIl96713. R&mm6 Les auteurs analysent le problbme de l’interaction dans le systeme de contile B chaque extremitb dune colorme de distillation binaire. Les auteurs proposent un concept de contr$le B double algorithme, qui traite avec efficacid et simultanCment les perturbations de la charge de la colonne et des points limites du contrele. Zusammenfassung  Das Problem der Wechselwirkung bei der Abtastregelung beider Enden einer binlren Destillationskolonne wird untersucht. Es wird ein DualAlgorithmus Regelbegti vorgeschlagen, der aufwirksame Art sowohl Last als such Sollwertstiirungen gleichzeitig behandelt.
1335