662
Ind. Eng. Chem. Res. 1991,30,662-670
Reference System Decoupling for Multivariable Control Peruvemba R. Krishnaswamy,' Neelkant V. Shukla, and Pradeep B. Deshpande* Department of Chemical Engineering, University of Louisville, Louisville, Kentucky 40292
Mohammed N. Amrouni Exxon Chemical Company, Baton Rouge, Louisiana 70821-0241
A new method for controlling multivariable processes is presented. The method, which we call reference system decoupling, features a single tuning constant per loop, complete interaction compensation (when there are no nonminimum phase elements present), and offset free performance in the presence of modeling errors. In the absence of modeling errors the tuning constants specify the closed-loop set-point trajectories, while in the presence of modeling errors they are adjusted to maintain robustness. The method is illustrated with two simulation examples. One is a distillation system studied widely in the literature, and the second involves multiple flash drums operating in parallel at an industrial site. A comparison of the results with PI (proportional-integral) control, with decoupling, shows the excellent capability of reference system decoupling for servo and regulatory control of multivariable processes. Introduction Automatic control of multivariable systems poses interesting challenges to the designer owing to the presence of loop interactions, nonlinearities, and nonminimum phase elements. With the availability of powerful microprocessors, hardware realizability has ceased to be an issue in control system development. Indeed, a number of powerful computer-based control techniques have appeared in the literature in recent years that take full advantage of the high speed and large memory capacity of modern machines. Some examples include model algorithmic control (Richalet et al., 1978), dynamic matrix control (Cutler and Ramaker, 1979),internal model control (Garcia and Morari, 1985a,b), and simplified model predictive control (Arulalan and Deshpande, 1987). Techniques that take process nonlinearities into account in a direct way include global linearization (Kravaris and Chung, 1987), a nonlinear system controller approach (Boye and Brogan, 1986),generic model control (Lee and Sullivan, 1988),and reference system structure (Bartusiak et al., 1988). In the traditional approach to achieving interaction compensation of an n X n multivariable system (e.g., see Luyben, 1970) one designs a matrix of decouplers such that one input to the decoupler matrix affects only one controlled variable. The system thus decoupled is controlled with n PI controllers. A shortcoming of this approach is that tuning of these n controllers poses problems in the presence of modeling errors since 2n interacting parameters are involved. In what follows, a new approach, which we call reference system decoupling (RSD), is described and applied to two simulated multivariable systems. For comparison the simulation results for PI control, with decoupling, have also been evaluated. In the new approach, traditional decoupling concepts are employed but the P I controllers are replaced by digital controllers having desirable properties as we will see.
* To whom correspondence should be addressed. 'Visiting professor. Permanent addreas: Chemical Engineering Department, National University of Singapore, Kent Ridge, Singapore 0511.
Met hod The block diagram of an open-loop multivariable system is shown in Figure 1. In the following derivation ztransforms are assumed throughout but the z-transform operator has been omitted for brevity. We begin with the open-loop pulse transfer function matrix relating the controlled and the manipulated variables for a general n X n system:
where
Equation 1 can be compactly written as
Y=GM
(2)
The goal of decoupling is to insert a matrix of decouplers D ahead of G as shown in Figure 2a such that each input to the decoupler affects only one controlled variable. Mathematically,this means that the product of D and G must be a diagonal matrix, which we denote as H. Thus
M=DU
(34
Y=GDU=HU
(3b)
Therefore
where
H=
...
Solving (3b) for D gives
D = G-lH
0888-58~5/9~/2630-0662~02.50/0 0 1991 American Chemical Society
(4)
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 663
R-Y-+=T-y (b)
Figure 1. Open-loop multivariable systems: (a) general n system; (b) 2 X 2 example.
M
U
X
n
-I
I
Y
(bl Figure 3. Reference system decoupling for multivariable systems.
set equal to 1,thus allowing for the off-diagonal elements H2, ..., H,, to be computed. Although either of D and HI, approach to decoupling can be used in RSD, simplified decoupling is used in the following analysis for illustration of the method. With Dll, DZ2,..., D,,, set equal to 1, ( 5 ) gives Dij = gij/gjj for i # j (64 (b)
"I
HI
u2
Hl
* Yl
-
* Yl
(C)
Figure 2. Traditional decoupling concepts.
Denoting the elements of G-' by the symbol {gij),(4) can be written in the expanded form as
In the traditional approach to decoupling the open-loop decoupled system is controlled with n PID-type controllers whose tuning constants are determined on the basis of the elements of the matrix H. A block diagram of the control system based on RSD is shown in Figure 3. The objective is to design the controller G, such that the multivariable system yields desired closed-loop responses. The closed-loop pulse transfer function matrix of the system in Figure 3 to a change in set point R is P = (I + GDGJ-lGDG, (7) Solving (7) for G, gives G, = (GD)-'P(1 - P)-'
(::Ff ii; ::)c' ;;ii ) Bnl Bn2
***
gnn
...
Hn
(5) This set has n2 + n unknowns (n2D'sand nPs) but only n2 equations. Thus the system is underspecified by n equations. Of these, n unknowns must somehow be fixed so that the other n2 elements can be determined. This is accomplished in the traditional approach to decoupling in one of two ways. In ideal decoupling it is specified that the response of each loop (with all loops on automatic control) be the same as the response one would get if the other loops were on manual (thus fixing the other manipulated variables). This allows the specification of HI, H2, ...,H,,, and thus the n2 elements of D can be specified. In s i m p l i f i e d decoupling the diagonal elements of D are
(8)
or
G, = H-'P(1 - P)-'
(9) Note that since the system is decoupled, P is a diagonal matrix and thus G, is a diagonal matrix also, as expected. We are free to choose the matrix P as long as G,remains realizable. If the plant transfer function matrix G contains fmt order with dead time (FODT) elements, then the same form can be specified for the elements of P giving
P=
i, \
. ...- I-A
l - h d Z
(10) where Ni ( i = 1,..., n) represent the dead times (in terms of number of sampling periods) in Hi (i = 1,...,n ) and 0;s
664 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 Table I. Process Parameters for Wood-Berry Column process K" 7, ed G11 12.8 16.7 1.0 Gl2 -18.9 21.0 3.0 GZl 6.6 10.9 7.0 G22 -19.4 14.4 3.0 load Kl. 11. $1. GL1 3.8 14.9 8.9 GL2 4.9 13.2 3.4
Table 11. Pulse-Transfer Functions for Wood-Berry Column pulse-transfer pulse-transfer process function proceas function
0.2957~-'~
Controller Parameters BLT Parameters for PI controllers 7i
0.375 8.29
B
0.8
Kc
G21(2)
-0.075 23.60
RSD Parameters 0.9
determine the speed of the closed-loop response of Yi.p's lie between 0 and 1. In the absence of modeling errors, j3's can be tightly tuned. In the presence of modeling errors, j3's may be adjusted to maintain robustness. Larger values of p's closer to 1 make the system responses more sluggish and vice versa. The solution of (9) for G, is
...
O
\
or
1.0 - 0.9552f'
GL&)
0.18212" 1.0 - 0.96282-'
The process-transfer functions and the steady-state operating data are shown in Table I. The pulse transfer function matrix whose elements are calculated according to (la) are shown in Table 11. For a 2 X 2 system, the decouplers may be calculated according to
with
where hij are impulse response coefficients of GiP Cross multiplication and inversion gives the decoupler outputs in the time domain. It may be noted that these decouplers will be realizable only if the minimum dead time in each row of the process transfer function matrix occurs on the diagonal (i.e., dead time in Gll I dead time in G12;dead time in GZ2Idead time in GZ1). If this condition is not satisfied, the positive dead times in D12and DZl have to be omitted and in such a case H will not be a diagonal matrix, implying that complete decoupling will not be feasible (see (3b)) (Garcia and Morari, 1982). The manipulated variables can now be calculated according to
Mi = U1 +
U12
(14a)
M2 = U2 +
U21
(14b)
and where Hi's are given in (6b) and Pi's are given in (10). Cross multiplying the terms in (12) and inverting gives the controller outputs Vi in the time domain. The algorithm in (12) is based on linear control concepts. If a nonlinearity is introduced into the control system by manipulated variable saturation, it will become necessary to detune the controllers by choosing higher values of Bi or by using a separate anti (reset) windup scheme. Applications We consider two simulation examples to demonstrate the capabilities of reference system decoupling. The first involves a 2 X 2 distillation system widely studied in the literature. This example will show how to apply RSD to general n X n systems. The second is an industrial process involving multiple flash drums. All simulation work was carried out on a VAX cluster at the University of Louisville. Example 1 (Wood a n d Berry, 1973). This example involves a binary distillation column that separates a mixture of methanol and water into two relatively pure products in a column that is equipped with a total condenser and a reboiler. The goal of the control effort is to maintain the two product compositions at set point in the presence of disturbances.
Now, working with (134 and the elements of the pulse transfer function matrix shown in Table 11, the elements H1 and H2can be calculated. These elements are shown in Table 111. Finally, the controller outputs Ul and U2 can be calculated as per (12). These equations are shown in Table IV. For simulation purposes the process outputs at the kth sampling instant are calculated according to rill Y1,k
=
n12
ChilM1,k-i + Zh,"M2,k-i
i=l
i=l
(154
The sampling period selected for this application is T = 0.5. The system equations were programmed on the digital computer according to the arrangement shown in Figure 3, and the following tests were conducted to assess the various aspects of RSD. 1. Servo Responses and Loop Shaping. The RSD system in the absence of modeling errors was subjected to a step change in the set point of each loop. The resulting responses are shown in Figures 4 and 5. To demonstrate
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 665 Table 111. Open-LoopDecoupled Transfer Function Matrices I
I
*
N
E
0.6
2
-
0
K
Substitution for G,'s from Table I1 gives Kpll(I- a l 1 ) . d N 1 1 + ' ) Hik) =
1 - (Y112-1
-I -I
KPJl
0.3
- (Y&-(N12+1)
1 - (Y212-1
- (Y&-(N22+1) 1 - a222-'
H2(z) =
X
0.0
0
(1 - a222-')
Km(l - (Y&-(N22+')
KPI2( 1 - ,lz)2-(~1rc')
Km(l
X
1 - alg1
- (~21)~-(~21+')
1 - (Yz12-'
(1 - al1z-1)
I I
10
30
20
-
5.132) 12.341)
50
40
Time, minutes
b 0.00 N
I
KPJ1 - all)2-(N~1+')
where aij = e-T/rij and Njj = int [Oij/Z'l. Using the process parameters from Table I with T = 0.5 min, these expressions simplify to
-0.20 4
0
+
0.37762-'(1 - 1.9317f' 0.93272-') Hi(z) = (1 - 0.9705~-')(1- 0.9765~-')(1- 0.95522-') 0.1985~-'~(1 - 1.93642-' + 0.9374f2) (1 - 0.9705~-')(1 - 0.97652-')( 1 - 0.95522-')
Hzb) =
PID (IS€
*-•
1 - (Y122-1
KmJ1 - ,21)2-~21+')
Km(l
- - RSD (IS€
I
10
I
30
20
40
Time, minutes
Figure 5. (a)Servo response of Wood-Berry column to a unit step change in set point of loop 2. (b) Manipulated variable movements associated with the tests in (a).
-0.66152-'(1 - 1.93172-' + 0.9327f2) (1 - O.96592-')( 1 - 0.97652-') (1 - 0,95522-') 0.3478~-'~(1 - 1.93642-' 0.9374f2)
+
+
(1 - 0.9659~-')(1- 0.9765~-')(1- 0.95522-') a
1.2,
0.9
--
-
0-0
= 0.5
p
-
(IS€ 1.1 67) p
0.9 (ISE
p
3.132)
)-
t
e
K
0.6-
- - RSD (IS€ 0.3-
0.0
4J 0
-
PID (IS€
0-0
0.0Y 0
1.889)
-
b 20
30
40
30
20
1
40
Time, minutes
2.079)
x10
10
1.2
i
- - p 1= 0.5
I,
I
50
Time, minutes
I
- - RSD 0-0
PID
0.04
0
f
I
10
20
30
40
50
Time, minutes
Figure 6. (a) Loop shaping with RSD in the absence of modeling errors: responses to a unit step change in set point of loop 1. (b) Manipulated variable movements associated with the testa in (a).
0
10
20
30
40
I
50
Time, minutes
Figure 4. (a) Servo response of Wood-Berry column to a unit step change in set point of loop 1. (b) Manipulated variable movements associated with the testa in (a). the loop-shaping capabilities of RSD in the absence of modeling errors, a unit step change was introduced in one
of the loops. The resulting responses for three values of 0 are shown in Figures 6 and 7. As expected, smaller values of 0 give responses that are closer to deadbeat control but the manipulated variable movementa increase, too. 2. Regulatory Performance and Effect of Modeling Errors. To assess the regulatory performance of RSD in the absence of modeling errors, a step change of 0.34
- - RSD (ISE PID (ISE
0-0
5n
--
0.954)
0.761)
0.1
t E
-0.1
0
20
40
60
80
100
80
100
Time, minutes
0.10,
I
-0.104 0
20
40
60
80
100
-0.10
0
20
Time, minutes
n
PID (ISE
0-0
-0.1
4
0
-
Figure 9. (a) Regulatory response of Wood-Berry column (loop 1) in the presence of a 50% modeling error in each parameter. (b) Manipulated variable movements associated with the tests in (a). a
I.11 Jn 0.9
- - RSD (ISE
t
40
60
80
1.1
04.886)
23.978)
-0.1
0-0
20
E I
I 20
40
60
40
60
80
100
80
100
- - RSD
.-n
PI0
-0.104
0
31.550)
Time, minutes
- - RSD
z6 -0.05
06.599)
PID (ISE
i 0
100
Time, minutes
.-
- - RSD (ISE 0-0
I 20
60
Time, minutes
Figure 7. (a) Regulatory response of Wood-Berry column (loop 1) to a step change of 0.34lb/min in feed flow rate in the absence of modeling errors. (b) Manipulated variable movements associated with the tests in (a). a
40
80
100
Time, minutes
Figure 8. (a) Regulatory response of Wood-Berry column (loop 2) to a step change of 0.34 lb/min in feed flow rate in the absence of modeling errors. (b) Manipulated variable movements associated with the testa in (a).
lb/min in feed flow is made. The resulting load responses are shown in Figure 8. To assess the effect of modeling errors, a +50% modeling error is introduced in each parameter of the multivariable system and a step change of 0.34 lb/min is introduced in the feed flow rate for a specific choice of &
-0.05
0-0
PID
-0.104
0
20
40
60
Time, minutes
Figure 10. (a) Regulatory response of Wood-Berry column (loop 2) in the presence of a 50% modeling error in each parameter. (b) Manipulated variable movements associated with the testa in (a).
and pZ. The resulting load responses are shown in Figure 9. For comparison purposes the load responses with and without modeling errors under PI control with decoupling are also evaluated. In this instance the decouplers remain the same as in RSD, but the RSD controllers are replaced by PI controllers. The BLT (biggest log modulus) tuning constants reported by Luyben (1990) for this application
Y
Table V. Process Parameters for Example 2
I monomers
'U e
Steam
.
.
Steam
Polymer Slurry
Figure 11. Schematic of the industrial process (3 X 3 system used for illustration purposes).
have been used. The resulting load responses are shown in Figure 10. A comparison of the results with RSD and decoupling PI control indicates that RSD is capable of providing nearly deadbeat responses for set-point changes in the absence of modeling errors. Furthermore, the regulatory performance of RSD in the presence of modeling errors is considerably better than that with PI control. Example 2. This example involves multiple flash drums operated in parallel a t an industrial plant site. A threedrum operation has been selected for the illustration of the method. The polymer solution, "cement", leaving a reactor is contacted with high-pressure steam and hot water in the flash drums, causing the solvent and the unreacted monomers to vaporize. The polymer product is recovered as crumb particles in a water slurry. The feed valves to the flash drums are used to regulate the reactor pressure and control the cement split between flash drums to produce consistent polymer crumb. A schematic of the process is shown in Figure 11. Pseudo random binary sequence (PRBS) testing in the field was employed to identify the process dynamics by time series analysis (Box and Jenkins, 1976). The resulting transfer functions are shown in Table V. The corresponding pulse-transfer functions are shown in Table VI. The system in the field
process G11 Gl2 G13
G21 G22 G23 G31
G32 G33 load GL1 GL2 GL3
KP
TP
ed
-0.10 -0.10 -0.10 -0.17 0.34 -0.17 -0.17 -0.17 0.34 KI. 1.0 1.0 1.0
0.0 0.0 0.0 5.1 5.1 5.1 5.1 5.1 5.1
0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0
71.
01.
0 0 0
0 0 0
is on PI control and it is desired to assess the suitability of RSD for this application. For the choice of Dll = D22= DS3= 1.0, (4) gives
/-0.32-'
0
0
(1
+ 0.961G-1) 0
0.0lstL~
H=[O
\o
0
Now, the controllers can be calculated according to
where Pi are the elements of the diagonal closed-looppulse transfer function matrix of the type given in (10). Substitution for Pi and Hiinto (18) followed by cross multi-
668 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 1.1,
plication and inversion gives the controller outputs in the time domain (1 - 01) U,(k) = U,(k- 1) El(k) (19a) 0.3 Uz(k) = &Uz(k - 1) + (1 - &)U2(k - Nz - 1) + (1 - Pz) [E&) - (~2Ez(k- I)] (1%) 0.51(1 - ( ~ 2 ) U3(k) = 83U3(k - 1) + (1 - &)U3(k - N3 - 1) (1 - P 3 ) [E,@) - w?33(k - 1)1 (194 0.51(1 - ( ~ 3 ) where a2 = a3 = = 0.9615 T = 0.2 min; r = r 2 = r 3 = 5.1 and Nz= N3= int [ O / q = 5 with e = e2 = e3 = 1.0
-
0.7
0.3
+
*-•
PID (ISE
--
RSD (ISE = 0.040)
9
0.247)
0.1
-0.1
0
2
0
6
4
10
Time, minutes
-0.5
Time, minutes
t
5
0.8 *-•
--
0
4
8
PID (ISE
-
1.504)
RSO (IS€ = 1.388)
12
16
20
16
20
Time, minutes
d
’1 *-.
--
PI0 RSD
Figure 12. Block diagram of the 3 X 3 RSD scheme. Table VI. Pulae-Transfer Functions for E ~ ~ ~ l n 2p l e process
pulee-transfer function
1 - 0.96162-1
0.013072” 1 - 0.96152-’ -0.006642” 1 - 0.9615~-’
process
pulse-transfer function -0.006642“ 1- 0.96152-’ -0).006642+ 1 - 0.96162-1
0.01307~“ I
- 0.9616~-’ 2-1
2-1 2-1
04 0
4
8
12
lime, minutes
Figure 13. (a) Servo responses of the 3 X 3 system in the absence of modeling errors: set-point change in loop 1. (b) Manipulated variable movements associated with the testa in (a). (c) Servo responses of the 3 X 3 syetem in the absence of modeling errors: set-point change in loop 2. (d) Manipulated variable movements associated with the testa in (c).
Table VII. Regulatory Responses for Example 20 modeling decoupled PI load disturbance error, % ISEl ISEZ ISES controller (81, 82, le,) ISEl 1, 1, 1 0 0.25 1.30 1.30 2.333,0.5,0.5 0.04 1, 1, 1 40 0.13 2.04 2.04 2.333,0.5,0.5 0.01 1, 1, 1 80 0.08 4.02 4.02 2.333,0.5,0.5 0.00 0, 0,1 80 0.00 0.00 4.02 2.333,0.5,0.5 0.00 1, 1, 1 150 0.06 unstable unstable 2.333,0.5,0.5 0.00 1, 1, 1 150 2.333,0.7, 0.7 0.00 1, I, 1 150 2.333,0.85, 0.85 0.00 1, 1, 1 150 2.333,0.95,0.95 0.00 1 P I controller settings: loop 1, K , = -0.600,71 = 0.20; loop 2,K , = 5.767,T I = 6.23; loop 3,K , = 5.767,T~ = 6.23.
RSD ISEZ 1.39 1.67 2.25 0.00 4.42 4.25 4.32 5.57
ISES 1.39 1.67 2.25 2.25 4.42 4.25 4.32 5.57
Ind. Eng. Chem. Res., Vol. 30, No. 4,1991 669 PID (ISE
.-e
--
V.,
6
-
I
1.11
~
-
0.247)
*-•
0.91
--
0.040)
-0.1
i
1
0
RSD (ISE
--
9
12
0
1
2
4
5
40
50
I *-•
--
’
20
3
Time, minutes
b 301
a
RSD (ISE = 0.000)
4
Time, minutes
._ 0
PID (ISE = 0.057)
PID
RSD
-20I
04
0
3
6
1 2 1 5
9
-30 I 0
10
Time, minutes
1.1
*-a
--
20
30
Time, minutes
PID (ISE RSD (ISE
--
1.304) 1.388)
\ \ \
\
0.1
-0.1 0
3
6
9
1 2 1 5
3
6
9
10
20
30
40
Time, minutes
Time, minutes
0
0
1 2 1 5
Time, minutes
Figure 14. (a) Regulatory performance of the 3 X 3 system to a unit step change in load affecting all loops in the absence of modeling errors: loop 1. (b) Manipulated variable movements associated with the tests in (a). (c) Regulatory performances in the absence of modeling errors: loop 2 or 3. (d) Manipulated variable movements associated with the testa in (c).
A block diagram of the RSD system for this process is shown in Figure 12. The following tests were conducted to assess the performance of RSD for this application. 1. Servo Responses. The responses of RSD and decoupling PI control for two combinations of set-point changes have been evaluated in the absence of modeling errors. Figure 13 shows the closed-loopresponses for these tests for both control strategies. 2. Regulatory Performance and Effect of Modeling Errors. The responses of RSD and decoupling PI control for a unit step change in load affecting all loops have been
Figure 15. (a) Regulatory performance in the presence of 150% modeling error in each parameter of the 3 X 3 system: loop 1. (b) Manipulated variable movements associatedwith the testa in (a). (c) Regulatory performance in the presence of 150% modeling error in each parameter: loop 2 or 3. (d) Manipulated variable movements associated with the testa in (c).
evaluated in the absence of modeling errors. The tabular results for these tests are shown in Table VII, and the corresponding responses are shown in Figure 14. The regulatory performance of RSD and decoupling PI control in the presence of a modeling errors in each of the parameters ranging from 0 to 150% has been evaluated, and the tabular results are’ also shown in Table VII. Representative plots for one of the tests are shown in Figure 15. The foregoing results show that for servo control the performance of RSD and decoupling PI control is similar for comparable valve movements. However, RSD is ca-
670 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991
pable of providing deadbeat control in the absence of modeling errors. The regulatory performance of both strategies in the absence of modeling errors is also similar, but the tabular data in Table VI1 show that RSD exhibits a remarkable level of robustness vis-&vis decoupling PI control. It may be noted that as the modeling errors were increased to 150%, PI control became unstable. RSD was stable in the presence of 150% modeling errors although the control valve movements showed a tendency toward ringing. By increasing the 8)s, ringing could be completely eliminated at a small cost in terms of ISE (integral square error). Discussion a n d Conclusions A new method for multivariable control has been presented. The design is built around traditional decoupling concepts, but RSD allows the user to specify reference trajectories. In the applications studied, first order with dead-time trajectories have been used but this specification is not unique; higher order trajectories could be used if warranted. We have shown through simulation results that RSD yields excellent servo and regulatory responses, offset-free performance in the presence of modeling errors, and a high level of robustness. Complete decoupling is not always the best answer, as in high-purity distillation, but when it is deemed desirable, RSD has been shown to provide a good framework for achieving noninteracting control of multivariable systems. Nomenclature DI2= decoupler transfer function E = error input to the controller G, = zero-order hold transfer function Gij = plant transfer function relating ith controlled variable to jth manipulated variable G , = controller transfer function for loop i CL = load transfer function = plant transfer function = open-loop decoupled transfer function matrix hi, = impulse response coefficients relating ith controlled variable to jth manipulated variable Kc, 71, TD = PID constants Li = load disturbance in loop i Mi = manipulated variable for loop i n = dimension of multivariable system
2
nii = settling time of ijth element of G matrix (in number of sampling instants) Nii = dead time for iith element of € and I P matrices Nij = dead time for ijth element of G matrix (in number of sampling instants) P = closed-loop decoupled transfer function matrix R = set-point matrix Vi = controller output for loop i Y, = output for loop i Pi = tuning parameter for ith loop Bi = dead time for loop i in matrix P T~ = time constant for loop i in matrix P
Literature Cited Arulalan, G. R.; Deshpande, P. B. Simplified Model Predictive Control. Znd. Eng. Chem. Res. 1987,26, 347. Bartusiak, R. D.; Georgakis, C.; Reilly, M. J. Designing Nonlinear Control Structures by Reference System Synthesis. h o c . Am. Control Conf. 1988,2, 1585. Box, G. E. P.; Jenkins, G. M. Time Series Analysis; Forecasting and Control; Holden-Day: San Francisco, 1976. Boye, J. A,; Brogan, W. L. A Nonlinear System Controller. Znt. J. Control 1986, 44 (51,1209. Cutler, C. R.; b a k e r , B. L. Dynamic Matrix Control-A Computer Control Algorithm. Abstract of Papers, 86th AZChE National Meeting, Houston, T X , 1979; AIChE: New York, 1979; p 51b. Garcia, C. E.; Morari, M. Internal Model Control l-A Unifying Review and Some New Results. Znd. Eng. Chem. Process Des. Dev. 1982,21,308-323. Garcia, C. E.; Morari, M. Internal Model Control 2-Design Procedures for Multivariable Systems. Znd. Eng. Chem. Process Des. Dev. 1985a, 24,472-484. Garcia, C. E.; Morari, M. Internal Model Control 3-Multivariable Control Law Computation and Tuning Guidelines. Znd. Eng. Chem. Process Des. Dev. 198513,24,484-494. Kravaris, C.; Chung, C. B. Nonlinear State Feedback Synthesis by Global Input/Output Linearization. AZChE J. 1987, 33, 592. Lee, P. L.; Sullivan, G. R. Generic Model Control (GMC). Comput. Chem. Eng. 1988,12, 573. Luyben, W. L. Distillation Decoupling. AZChE J. 1970, 16, 2. Luyben, W. L. Process Modeling, Simulation, and Control for Chemical Engineers, 2nd ed.; McGraw-Hill: New York, 1990. Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Model Predictive Heuristic Control: Application to Industrial Processes. Automatica 1978,14,413-428. Wood, R.; Berry, M. W. Terminal Composition Control of Binary Distillation Column. Chem. Eng. Sci. 1973, 28, 1707.
Received for review July 23, 1990 Revised manuscript received October 18, 1990 Accepted October 24, 1990
