Acknowledgment
P. M . Aiken assembled the experimental equipment. literature Cited
Ark, R., Amundson, N. R., Chem. Eng. Sci. 7 , 131 (1958). Athans, M., Falb, P. L., “Optimal Control,” McGraw-Hill, New York, 1966. Biery, S. C., Boylan, D. R., IND.ENG.CHEM.FUNDAMENTALS, 2, 44 (1963). Coughanowr, D. R., Koppel, L. B., “Process Systems Analysis and Control,” McGraw-Hill, New York, 1965. Crowther, R. H., Pitrak, J. E., Ply, E. N., Chem. Eng. Progr. 57, No. 6, 39 (1961). Gibson, J . E., Johnson, C. D., I E E E Trans. Auto. Contr. AC-8, No. 1, 4 (1963). Gray, R . I., Prados, J. W., A.Z.Ch.E.J. 9,No. 2, 211 (1963). Grethlein, H. E., Lapidus, L., A.Z.Ch.E.J. 9, No. 2, 230 (1963). Harriott. Control.” McGraw-Hill. New York. 1964. ~ ~ . P.. . “Process ~ Hougen,’J. O., Chem. Eng. Progr. Monograph Sei. 4, 60 (1964). Hougen, J. O., Chem. Eng. Progr. 59,49 (1963). Kouuel. L. B., Latour, P. R., IND. ENG.CHEM.FUNDAMENTALS, k,‘463 (1965). Lapse, C. G., I S A J . 3, 134 (1956). Latour, P. R., M.S. thesis, Purdue University, M‘est Lafayette, Ind., June 1964. Latour, P. R., Ph.D. thesis, Purdue University, West Lafayette, Ind., June 1966. ~
~
~
Lefkowitz, I., Chem. Eng. Progr. Symp. Ser. 46, 59, 178 (1963). Lupfer, D. E., Oglesby, M. W., I S A Trans. 1, No. 1 , 72 (1962). Lupfer, D. E., Parsons, J. R., Chem. Eng. Progr. 58, No. 9, 37 (1962). Marquardt, D. LY., Chem. Eng. Progr. 5 5 , 65 (1959). Marr, G. R., Johnson, E. F., Chem. Eng. Progr. Symp. Ser., No. 36, 57, 109 (1961). Mayer, F. X., Rippel, G. R., Chem. Eng. Progr. Symp. Ser., No. 46, 59. 84 11963). Mocieck,‘J. S.’, Otto, R. E., LYilliams, T. J., Chem. Eng. Progr. Symp. Ser.: No. 55, 61, 136 (1965). Oldenbourg R. C., Sartorius, H., Trans. A . S . M . E . , 7 0 , 78 (1948). Orent, H. H., Ph.D. thesis, Purdue University, LYest Lafayette, Ind.. June 1965. Pontryagin, L. S., Boltyanskii, V. G., Gamkralidze, R. V., Mischchenko, E. F., “The Mathematical Theory of Optimal Processes,” Wiley, New York, 1962. Roquemore, K. G.. Eddey, E. E., Chem. Eng. Progr. 57, No. 9, 35 (1961) Siebenthal, C D., Ark, R.. Chem. Eng. Sci. 19, No. 10, 729 (1964). Sproul, J. S., Gerster, J . A., Chem. Eng. Progr. Symp, Ser., No. 46, 59, 21 (1963). Williams, T. J., ZSA J . 12, 9, 76 (1965). RECEIVED for review October 31, 1966 ACCEPTED March 27, 1967 A.1.Ch.E. Meeting, Atlantic City, N.J. Financial assistance received from Purdue University and the National Science Foundation.
NONINTERACTING PROCESS CONTROL SHEAN-LIN
L I U
Central Research Division Laboratory, Research Department, Mobil Oil Gorp., Princeton, N . J . 08540
A new technique for the design of noninteracting control systems can handle constraint conditions on the control variables and can be applied to nonlinear problems. Two examples illustrate the design method. The first concerns a nonisothermal chemical reactor. The second deals with the control of a plate-type absorption column. It is demonstrated that one state variable can be moved from one point to another without affecting the other state variables.
N MULTIVARIABLE
feedback control systems, a change in one
I reference variable will usually affect all output variables.
In some applications (temperature control in a chemical reactor, for example), one desires to design a noninteracting control-that is, a system in which a variation of any one reference input quantity will cause only the one corresponding controlled output variable (such as one state variable) to change. T h e design of such systems was considered by Boksenbom and Hood (Tsien, 1954). Using matrix algebra, Kavanagh (1957), Freeman (1958), and Morozovskii (1962) discussed the transfer matrix of noninteracting control systems. Chatterjee (1960) considered noninteracting process control using an analog computer and standard three-mode process controllers. I n all the above references only linear problems were discussed, and constraints on control variables were neglected. Petrov (1960) discussed a very special nonlinear problem without constraints. Although noninteracting control is potentially a powerful tool for reducing the complexity of control systems, it has several limitations, as discussed by Morgan (1958) and Mesarovic (1964). The present procedure requires a complete dynamic model of the system and a n on-line digital computer. This paper announces two new results: Certain nonlinear, unconstrained processes can (1) be made noninteracting over the entire state variable space in a manner analogous to that for linear systems, and (2) be controlled in a piecewise non460
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
interacting way even when there are constraints on the process input variables. Two examples illustrate the present method. The first deals with a nonisothermal chemical reactor in which a second-order irreversible chemical reaction, 2A -+ B, takes place. The concentration of component A and the temperature are to be controlled by manipulating the flow rates of reactant and coolant. Either state variable, temperature or concentration, can be moved from one point to another without affecting the other state Tiariable. In the second example, noninteracting control of a plate-type absorption column is considered. I t is assumed that there are seven plates in the absorption column and that one component in the gas phase is absorbed by liquid absorbent. I t is demonstrated that, by manipulating the liquid flow rate, one can maintain the gas outlet concentration a t a fixed value even if perturbations in the gas flow rate or the gas inlet concentration occur. Basic Theory
Before discussing the design of noninteracting control systems, the classical approach to control of linear multivariable processes (Freeman, 1958; Kavanagh, 1957) is reviewed, Since the process is linear, the control problem can be discussed in terms of Laplace transformed variables. In a closed loop system, as shown in Figure 1, if P represents the
G
?
P
?a
where
and B are n X 1 column matrices
Figure 1 . Block diagram of linear feedback control system
j r. . i . ;
I-{
COMPUTER
7
I
>
T h e ith component, Gi,of B is a function only o f t and the ith error signal, e t . I t is clear from Equation 3 that each output variable, yi, is affected only by changes in the corresponding reference input variable, Ti. From Equation 3, we have
Figure 2. Block diagram of a process computer control system
transfer function matrix of the process, D represents the transfer function matrix of the controller, and k , I?, k,and p are the column matrices for the reference values, error signals, inputs, and outputs, respectively. Then
p
(1a)
= PDE
P = (Z + PD)-' PDR
where Q and A are the plant state vector (or outputs) and the control vector (or inputs), respectively, and are defined by
=
The control process is subject to n control signals, which a t each moment must satisfy the inequality constraints. bi
2
mi
2
ci,
i
Obviously, there are many different functions that one can assign for G iand a simple one is
(1b)
where Z is the identity matrix. I n Equations l a and l b , it is assumed that the number of inputs equals the number of outputs. I n order to obtain noninteraction between p and A, it is necessary to design the controller so that PD is diagonal. I n Equation l b , it is assumed implicitly that any required values of the inputs, Ad, are available. If any constraint is encountered, Equation l b is no longer valid and noninteraction cannot be obtained. 'The inability t o handle constraints on control variables is one disadvantage of the classical approach. For actual process control, one must take into consideration the constraints on inputs. T h e technique described in this paper can handle constraints on input variables and can be applied to nonlinear problems. Since Formula l a - b is not valid for nonlinear differential equations, one cannot discuss nonlinear control problems using the concept of the Laplace transformation. As shown in Figure 2, consider a n nth-order control process characterized by
and
(4)
(5) where aiis a constant and ri denotes the reference value of yi. Equation 5 is stable for ai> 0. Now a n essential point is the manipulation of the control values A so that the process will move according to the prescribed Equation 3. From Equations 2 and 3, we get
If, a t each moment, we apply control values A that satisfy Equation 6, then the system will be noninteracting and each state variable y i will change according to the decoupled Equation 4. Equation 6 can usually be solved for iii analytically or numerically. If any constraint on control variable mi is encountered during the integration of Equations 3 and 6, it is necessary to modify the prescribed dynamic equations in some automatic fashion to force the control variables to satisfy their constraints. Such a situation may be defined as being under piecewise noninteracting control. I n this case, noninteracting control is still achieved locally, although interactions between state variables may arise a t a finite number of instants when the prescribed equations are modified. If Equations 5 are used as the prescribed equations and a t some instant no admissible control vector A can be found for a particular set of ai,one can often change the values of ai in some automatic fashion-for example, reduce some ai by a factor of 0.5-until an admissible control vector is found and then continue the integration of Equations 5 and 6. T h e technique described above is applicable t o both linear and nonlinear problems. If the dynamic Equation 2 is linear in the control variables, mi,one can find the analytical solution for A. Thus if the dynamic equation is
= 1, . . . n
A control vector, A, which satisfies the constraints, will be referred to as a n admissible control vector. One can obtain a noninteracting control system by manipulating the control values m i to force the process to move according t o the follo\i ing prescribed noninteracting dynamic equations:
where K, is an n X n matrix and L, is a n n X 1 column matrix, A, can be found even if the elements of these t\vo matrices are nonlinear functions of g and t , Lvhenever the inverse of K , exists. From Equation 6, one obtains
(3)
VOL. 6
NO. 4
OCTOBER 1 9 6 7
461
T h e solution of the above equation for the control vector f i is f i a = Ka-'(Ga
- La)
c, cc qc F
(9)
If we use f i aas defined by Equation 9, Equations 7 and 3 are identical and they are decoupled. If the number of control variables, r , is greater than that of the state variables, n, one can obtain a noninteracting system as follows: T h e dynamic equation can be written
where g and Lb are n X 1 column matrices, Kb is a n n X r matrix, and mb is a n r X 1 column matrix. We assume r > n. From Equations 3 and 10, one obtains a n equation similar to 8,
where Gb is a n n X 1 column matrix and is the prescribed function as used in Equation 3. I n Equation 1l a there are more unknowns, r, than equations, n. However, from the theory of algebra, if the rank of K b ( g , t ) 2 n, one can solve Equation 1 l a for n unknowns as a function of the other (r n) unknowns. Write & f i b as a sum of two matrices.
-
00
= specific heats of reactant and coolant
flow rate of coolant 2QcPcCc/hAl8 = q/V
=
=
Two dependent variables, concentration A and temperature T , are to be controlled in the system. The flow rate of the reactant, q, and the coolant flow rate, qc, are used as the controllable independent variables or input variables. T h e reference points are denoted by A , and T,, and the corresponding steady-state flow rates of reactant and coolant by qs and qcs, respectively. Two error signals can be defined as follows:
-A Ts - T
E1 = A , E2 =
Equations 12a and 12b can be transformed into a system of equations in which the steady state has the vector coordinates zero. Let
yi = A - A, y z = T - Ts I n the new coordinates, Equations 12a and 12b reduce to dyl = Bg(A, - y l
dt
- A,)
- kle-E/R(Y2
(yl
+ As)'
FIOII, Y 2 , q, qc)
=
(134
where K n > nis nonsingular. Then from Equation 11a we find the solution for f i n : where I n Equation l l b , it is seen that if one assigns (r - n) values of the control variables, fir-,, then the remaining n control variables, f i n , can be determined explicitly.
a1 = hAh,
= ccpc,
cy3
2ala2/V
=
I n order to have a nonintereacting system, one can prescribe the following noninteracting dynamic equations:
*
Nonirothermal Chemical Reactor
T h e stability and control of a nonisothermal stirred-tank reactor were considered by Aris and Amundson (1958). Consider a single stirred-tank reactor of volume V into which a stream of reactant flows a t a flow rate q. I t is assumed that the vessel is well stirred, so that the concentration and temperature of its contents are the same throughout the reactor. I n an exothermic second-order irreversible reaction, 2A + B, if A is the concentration of reactant A and T is the temperature of the reacting mixture in the reactor, the mass and heat balances become
a2
dt
=
alEl = -alyl
T h e values of a1 and a2 must be positive, a t least near the equilibrium point, in order to make the system described by Equations 14a and 14b stable. Since we wish the process to move according to Equations 14a and 14b, a t each instant, the right-hand sides of Equations 13a and 13b must be equal to those of Equations 14a and 14b, respectively. We have
FI(yl,.Y2,q,qc) = FZ(yl,Yz,q,qc) =
- a1y1 -azyz
(1 5 4 (15b)
Solving Equations 15a and 15b for q and q. simultaneously, one obtains where
4 =
inlet temperature of reactant inlet temperature of coolant = inlet concentration of reactant A A0 ( - A H ) = heat of reaction k = kl exp(-E/RT)
T* Tc,
462
= =
I&EC
PROCESS DESIGN A N D DEVELOPMENT
V Ao - y l - As
[kle-E/R(-1'2-k T8)011
+ A,)2 - alyl] (16 4
+ azy-yz] - 2a2[as + azyz] a1[a5
qo = CY6
(16b)
To
=
Ah
= 500 ~ 9ft. . = 1.0 Ib. mole/cu. ft.
where
A, E AH
= = =
k~
qmax = ff3
ff6
= CP
(yz
al,max =
+ ir, - T C J
a2,max =
T h e values of q and qc determined by Equations 16a and 16b must satisfy the constraints on these two flow rates. Since there are physical limitations upon the movement of control values, one has qmax
2q20
where q,,, and qc are the maximum flow rates of reactant and coolant, respectively. If, for a particular set of a1 and az, the solution obtained from Equations 16a and 16b violates the constraints on q and qc, one can change the values of a1 and a2 in some automatic fashion to seek an admissible solution for q and q c . An algorithm for the calculation of permissible a1 and a2 is shown in Figure 3. I n the diagram, al,msx and denote the upper limit$;of a1 and a2, respectively, and the superscripts of a1 and a2 denote the numbers of tries a t selecting a ] and az. I n Figure 3 it is seen that the initial values of a1 and az that give an admissible solution for q and qc are computed by the algorithm. If constraints o n the control variables are not encountered throughout the integration of Equations 14a-b, these satisfactory initial values of a1 and a2 are not changed. At some points, there may be no feasible solution for q and qc in Equations 16a and 16b with any positive values of a1 and a i . Then, negative values of a1 or a2 must be allowed. Numerical Example. Let us consider a reactor operating under the noninteracting control system above. T h e following values of the parameters have been fixed :
p
=
pc
=
T,,
= =
v
= = qomax =
C
c,
680' R. 44,700 B.t.u./lb. mole -20,000 B.t.u./lb. mole 2.7 X loll cu. ft./mole sec. 2.0 cu. ft./sec. 1.0 set.-' 1.0 set.-' 60 lb./cu. ft. 62.4 lb./cu. ft. 560' R. 100 cu. ft. 1.0 B.t.u./lb. mole 1.0 B.t.u./lb. mole 2.0 cu. ft./sec.
For given steady-state values of q and qc, the corresponding steady-state values of A and T can be obtained by solving Equations 12a and 12b with dA/dt and d T / d t simultaneously equal to zero. Suppose that the final desired condition is
q3 = 0.128 cu. ft./sec. qcs = 0.06 cu. ft./sec. A , = 0.1 lb. mole/cu. ft.
T,
=
790'R.
and the initial state is
A = 0.1 lb. mole/cu. ft.
T
=
720' R.
O u r concern is to move from the initial state, p , to the final state, c, without affecting the concentration. T h e response without control can be obtained by numerically integrating the transient Equations 13a and 13b with 43 and qcs held a t state, c, and with A and T initially a t state p. T h e solid curve in Figure 4 shows the trajectory of the system without control. If piecewise noninteracting controls are applied to q and qc according to Equations 14a-b, 16a-b, and 17, permissible control values of q and qc are found by starting with
CONSTRAINTS
INTEGRATE EOUAT I ONS
JYES
EXIT
Figure 3.
Flow diagram I for calculation of a1 and
a2
VOL.
6
NO. 4
OCTOBER 1 9 6 7
463
Control of Absorption Column a] =
1.0 sec.-l
a2 =
2.384 X 10-4 set.?
The above values give the complete trajectory. As shown by the broken line in Figure 4, the trajectory is a vertical line, and the temperature changes from 720' to 790' R., without disturbing the concentration of component A in the reactor (0.1 lb. mole/cu. ft.). T h e concentration responses are shown in Figure 5. As shown in Figure 6, the normal temperature response without control reaches a peak temperature of about 91 0' R., whereas by applying piecewise noninteracting controls, the temperature reaches the steady state monotonically. Figure 7 shows the changes in control variables q and qc. Although the error signal of concentration, El, is zero, control variables q and qc both change continuously. Such noninteracting controls can also be seen from Equations 16a and lbb, where q and qc are coupled functions ofyl, y ~ a,i , and as. Figure 8 shows the trajectories of normal and piecewise noninteracting control responses from various initial states. The trajectories from the initial states pl and p3 are vertical lines, which means that the temperature changes without affecting the Concentration. T h e trajectories from the initial states p2 and pb are horizontal, and the concentration changes without affecting the temperature. We can also move the temperature and concentration simultaneously, as illustrated by the trajectory from the initial statepl. Appropriate initial values of a1 and a2 (as obtained from the computer using the algorithm of Figure 3), which give admissible values of q and qc for the various initial states in Figure 8, are tabulated in Table I. Although values of a1 and a2 in Table I are satisfactory for obtaining the corresponding complete trajectories, one may be able to obtain piecewise noninteracting control only for some other initial states. I t is very difficult to determine in advance the region of state variable space and initial values of a ] and a2 that would give only piecewise noninteracting control.
950
The process control of a bubble cap absorption column was considered by Ceaglske (1961). H e analyzed the linearized equations by Laplace transform techniques and discussed the effects of three-mode controls upon the stability and transient behavior of an absorption column. Using Bellman's dynamic programming, Lapidus and Shapiro (1961) considered the dynamic control of an absorption column by varying inlet compositions. As they pointed out, however, from a physical point of view, one would normally maintain fixed compositions and vary the flow rates. T o show how piecewise noninteracting control can be applied to the control of a plate type of absorption to\ver, let us consider a column with ideal trays. The definition of ideal tray (or plate) and the necessary steady-state equations were presented by Treybal (1955) and the corresponding transient equations by Lapidus and Shapiro (1961). Suppose that gas and liquid stream flow in opposite directions and that in the process some component in the gas stream is transferred to the liquid stream. Assume that the outgoing gas composition is to be maintained a t some fixed composition by manipulating the liquid flow rate. The nonlinear dynamic equations will be treated without linearization. An unsteady-state material balance around the nth plate gives
where x , = composition of liquid leaving the nth plate y n = composition of vapor leaving the nth plate iT = number of plates h = liquid holdup on any plate H , = vapor holdup on any plate L = absorbent rate G = inert gas rate From the linear equilibrium relation (Henry's law), one gets
y , = axn
r
C;CONTROL POINT
Equation 18 becomes
PjINITIAL POINT
for n = 1, 2, . . . N where e = H,a
(19)
+h
The initial conditions are
and the feed conditions are x,(t)
= xo(t);
n
=
0
Writing out Equations 19 for the set (yl, y2, . . ,y,\.), one obtains
Ga
dY1--dt
600
Gff
e
+ 7 oq + ) ( .2
1.0
--
dyn dt
CONCENTRATION, LE. MOLE/FT.3
Figure 464
+L
4. Phase plane
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
L
-y n - 1 -
e
Ga
L
Gff
~n
- Yn+l
for n = 2, . . .K - 1
(21)
STEADY STATE VALUE OF 9.1 LB.MOLE/FT.3
NONINTERACTING CONTROL
1
~~
00
Figure 5.
I
I
10
20
i
I
I
30 4 0 t, T I M E I N SEC.
50
70
60
80
x 10-3 Noninteracting control and normal response of concentration
.__
1.0
*d dt
- L -e Y.v-1
-
3.0
2.0
4.0 5.0 t , T I M E IN SEC. X 10-3
70
6.0
Figure 6.
Noninteracting control and normal response of temperature
(--) +
y.v
Ga
L
+ Ga ys+1 -
T h e steady-state compositions (or equilibrium compositions) on the plates are determined by solving Equations 21 with dy,ldt = 0 (n = 1, . . ..V). Suppose that initially the absorption column is in a n equilibrium state corresponding to fixed values of G, L, xo, and yN+l. If the gas flow rate, G, or the gas inlet composition, y N + l , changes, without noninteracting control on the process, the composition of outlet gas, y l will change according to the transient Equation 21 until a new steady state is reached. If the purpose of the control is to maintain the outlet gas composition a t a reference value yls regardless of perturbations in G and yN+l, we can apply the technique of piecewise noninteracting control described above to manipulate the liquid flow rate L to move y l to 3 . 1 ~and maintain it a t y l s even if perturbations in G and Y . ~ + Ioccur.
14.0
8.0"
15.0
In order to have a noninteracting system for Equation 21 for n = 1, we must assign y l a dynamic equation that is independent of G andyx+1-for example, one can have
!?! dt
(22)
= alEl
where E1 is the error signal E1 = yis
- YI
and a1 should have positive values in order to make Equation 22 stable. From Equation 21 for n = 1 and Equation 22, one obtains -axo
-
(T)+
GCX - 3.2 = alEl
yl
(23)
T h e solution of Equation 23 for L is
L = Ga(yl - y J ax,
VOL. 6
+ ea&
- yl
NO, 4
OCTOBER 1 9 6 7
(24)
465
K3 TIME IN SEC x 10-3 Figure 7. Changes in Q and Q,
I
0.05
I 0.1
I
I
0.15
0.2
I 0.25
I
0.3
1 0 35
CONCENTRATION, LB. MOLE/FT3
Figure 8.
One must determine the values of a1 that give feasible solutions for an L that satisfies the constraint Lmax
1L 2
0
(25)
where L,,, is the maximum liquid flow rate allowed. From Equation 25, it is seen that if a x o = yl, L becomes 466
l & E C P R O C E S S D E S I G N AND DEVELOPMENT
Phase plane
infinite. One usually uses fresh absorbent, so xo = 0 and a solution for L exists wherever yl # 0. Using the liquid flow rate L determined by Equation 24, y l is found to move according to Equation 22. Moreover, when y l is at steady state, then El = 0 and y l remains the same regardless of perturbations in G or y.v+l. T h e necessary transient equations are
From these we find
Table I. Initial Values of a1 and a2 Control points q s = 0.128 cu. ft./sec., A , = 0.1 lb. mole/cu. ft. qca = 0.06 cu. ft./sec., T, = 790" R. A , Lb. Initial State $1
fiz fi3
p4 fi5
Cu.Ft.
Mole/
T, R.
al, Sec.-]
0.1 0.01 0.1 0.35 0.2
840 790 760 720 790
1.0 2.38 X 1.0 1.907 X 3.05 X 10-2
For fixed values of G,L , and J J . ~ + I , the equilibrium compositions (or steady states) on each plate can be determined by solving Equations 21 with dy,/dt = 0 by matrix methods, or by integrating the transient equations until the steady states arp reached. T h e latter method was used in this study. With the following inlet and flow conditions:
1.525 X 10-2 1.0 1 . 9 0 7 X 10-3 3.814 X 10-3 1.0
G L
= = y,v++1 =
a
H, h
L,,, ,G ,, xo
3.6
y3 =
=
3.6
=
= 1'7 =
NONINTERACTING CONTROL
I
I
t
5
10
15
Figure
42
y4
0.04346 0.07968 0.10985 0.13498 0.15592 0.17336 0.18789
(26)
Suppose that the absorption tower is initially a t the steady states ( a ) and that a t t = 0, a step change is applied to G or yN+l. If there is no control on the system, the values of y n (for n = 1, . . . 7 ) will change according to the transient Equations 21. If the purpose of the control is to maintain y l
I
I
=
~1 ~2
= 7 plates = 0.75 = 1.0 = 70 = 100.0 lb./min. = 80.0 lb./min. = 0.0 (pure absorbent)
4.4
68.5 lb./min. 42.8 lb./min. 0.2 lb. mole/lb.
the steady-state compositions \vi11 be
Equations 22, 25, and 21 for n = 2, 3, . . .Ar. I t is seen from Equation 24 that we need transient values of G, x,,, y, and y2 to compute L. Numerical Example. To illustrate the piecewise noninteracting control of a n absorbent tower, the following numerical parameters are chosen :
N
+ h = 70.75
e = H,a
9.
I
STEADY STATEVALUEOF 0.04346
I
20 25 t , T l M E IN MINUTES
I
I
I
30
35
40
Noninteracting control and normal response of yl
I
I
2
4
1 6
I
I
I
A1
I
8
10
12
1
20
I x)
40
1,TIME IN MINUTES
Figure 10.
Liquid flow rate vs. time VOL. 6
NO. 4
OCTOBER 1967
467
-NONINTERACTING CONTROL
45
---
WITHOUT CONTROL
40
3.5
-x
N
Figure 1 1. Noninteracting control and normal response of yl
30
h -
WITHOUT CONTROL
25
-------____ 1.0 STEADY STATE VALUE OF 0.02 1.5
2
4
6
8
IO
I2
14
IS
t , TIME IN MINUTES
loo
-NONINTERACTING CONTROL
i\
--- WITHOUT CONTROL,
Figure 12.
Liquid flow rate vs. time
WITHOUT CONTROL, STEADY STATE 50
I
I
10
12
I 14
I 16
1,TIME IN MINUTES
a t the initial value (0.04346), then we can apply the technique of piecewise noninteracting control described above to manipulate the liquid flow rate, L, according to Equation 24. Since E1 = 0, from Equation 24 L is independent of al and uniquely determined by the value of y2, which is obtained by solving Equations 24 and 21 for n = 2, . , .7. Figure 9 shows the transient responses ofyl, with and without noninteracting controls, for a step change in ys+l from 0.2 to 0.3. By using piecewise noninteracting controls the value ofy1 is unchanged. The solid line in Figure 10 shows that the liquid flow rate, L, changes continuously and reaches the steady-state value of 49.182 pounds per minute when the steady states on all plates are obtained. Although it is not illustrated here, if one uses conventional three-mode controls (proportional, derivative, and integral) on the liquid flow rate, one cannot hold y l unchanged during the transient. If the initial compositions are as defined by Equation 26 and the steady-state liquid flow rate changes from 42.8 to 60.321 pounds per minute, the value of ),>,+I from 0.2 to 0.3, and the set point of y l from 0.04346 to 0.02, then the transient responses obtained will be the dashed curve in Figure 11 for normal response without control and the solid curves in Figure 12 for piecewise noninteracting controls with various values of a1 in Equation 22. The larger the value of a],the faster yl reaches the new steady-state value (0.02). However, a1 = 468
l&EC PROCESS D E S I G N AND D E V E L O P M E N T
1.5 is about the maximum value that yields a feasible solution for L that satisfies constraint condition 25. The changes in the liquid flow rate L are shown in Figure 12. Acknowledgment
The author thanks E. Norman, R. F. Wheeling, and J. S. Hicks for their helpful discussions. literature Cited
Aris, R., Amundson, N. R., Chem. Eng. Sci.7, 121 (1958). Ceaglske, N. H., “International Federation of Automatic Control Proceedings,” Vol. 4, p. 288, Butterworth’s, London, 1961. Chatterjee, H. K., “International Federation of Automatic Control Proceedings,” Vol. 1, p. 132, Butterworth’s, London, IO m
Frddmyn, H., Trans. A.I.E.E. 7 7 , l(1958). Kavanagh, R. J., Trans. A.I.E.E. 76, 95 (1957). Lauidus. L.. Shauiro. E.. A.Z.Ch.E. J . 7, 288 (1961). Mdsarovic, M .D,,BirtaiL., Automatica 2, 15 (1964). Morgan, B. S., IEEEIntern. Cow. Rec. 6,87 (1958). T., Automation Remote Control 23, 1113 (1962). Morozovskii, Petrov, B. N., International Federation of Automatic Control,” Vol. 1, p. 117, Butterworth’s, London, 1960. Tsien, H. S., “Engineering Cybernetics,” McGraw-Hill, New York, 1954. Treybal, R. E., “Mass-Transfer Operations,” McGraw-Hill, New York, 1955. RECEIVED for review January 6, 1967 ACCEPTED June 12, 1967
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