Ind. Eng. Chem. Process Des. Dev. 1984, 23, 854-857
854
kinetic expression, eq 3. The assumption of a simple first-order rate equation and constant usage ratio is valid to high degrees of conversion if the initial (H,/CO) ratio is quite low (e.g., 0.55). When this ratio is high (e.g., 1.8) the rate deviates increasingly from first order at high conversions and the usage ratio likewise changes significantly.
c
Registry No. CO, 630-08-0; magnetite, 1309-38-2.
//
HzKO a 0 5 5
Literature Cited
iI
/ 0
1
I
42
0
CO
I
I
I
l
~
0.4 0.6 0.0 CONVERSION I N A PFR
!
I
1
1.0
Figure 9. Comparison of CO conversion in a CSTR and a PFR for the same contact time: 263 O C , 0.79 MPa, T3H7* product.
Tropsch synthesis on iron-based catalysts. We cannot expect it to apply precisely quantitatively to other types of iron-based catalyses or even necessarily to reduced fused magnetite catalysts of different composition or reduced in a different manner. However, in the absence of specific and precise data on an iron catalyst of interest, eq 2 provides a basis for design estimates and interpretation of experimental data. Figures 1,2, and 8 show representative sets of conditions for which eq 2 should be used instead of the first-order
Anderson, R. B. In "Catalysis", Voi. 4; Emmett, P. ti., Ed.; Reinhold: New York, 1956. Deckwer, W.-D.; Serpemen, Y.; Ralek, M.; Schmldt, 8. Chem. Eng. Sci. 1981, 38,765. Deckwer, W.-D.; Serpemen, Y.; Ralek, M.; Schmidt, B. I d . Eng. Chem. Process Des. Dev. 1982, 21, 231. Huff, G. A., Jr.; Satterfleld, C. N. Ind. Eng. Chem. Process Des. Dev. 1984, in press. Rossinl, F. D.; Pltrer, K. S.; Taylor, W. J.; Ebert, J. P.; Kllpatrlck, J. E.; Beckett, C. W.; Williams, M. G.; Werner, H. G. A.P.I. Research Project 44. Satterfield, C. N.; Huff, G. A., Jr. Chem. Eng. Sci. 1980. 35, 195. Sanedieid. C. N., Huff, G. A,, Jr. Can. J . Chem. Eng. 1982, 6 0 , 159.
Department of Chemical Engineering George A. Huff, Jr. Charles N.Satter€ield* Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Received for review September 27, 1983 Accepted January 24,1984
Multivariable Control of Non-Square Systems A ternary distillation column is used as an example of how Rosenbrock's Direct Nyquist Array method can be wed to design a multivariable control scheme for processes with Don-square transfer function matrices. A pre-compensator, which is analogous to the inverse gain array for square systems, is used to square and decouple the system. The form of the compensator also permits constraint of actuator movement.
Introduction The problem of controlling a process with an unequal number of manipulated and controlled variables arises fairly often. The usual solution when designing a feedback control scheme is to square the system by eliminating or adding variables. This can lead to an increase in hardware expense if an actuator is to be added or poor performance if a controlled variable is to be dropped. Among the formal approaches to solution of this problem is to apply linear programming if a steady-state control scheme is adequate or schemes such as Dynamic Matrix Control due to Cutler and Ramaker (1979). These methods require considerable computing power and may be inappropriately complex for many problems. MacFarlane et al. (1978) and Kouvaritakis and Edmunds (1978) developed a method to design a compensator which squares down systems with more measured than manipulated variables. The method is based on frequency domain and state space design, and focuses on placement of the multivariable zeros generated in the squaring down process. The method developed here avoids the need to knock out controlled variables or square up by adding actuators. In contrast to the work of Kouvaritakis and Edmunds (1978) it retains the relative simplicity of square multivariable control schemes, using only a proportional controller for each controlled variable and a constant gain pre-compensator derived from the quadratic inverse or pseudo-inverse of the non-square gain matrix. A ternary
distillation control problem suggested by Doukas and Luyben (1978) is used here to illustrate the method and simulation results are presented as support for the technique. Process Description The process, shown in Figure 1, was suggested by Doukas and Luyben (1978) as the most economical configuration for separating a mixture of benzene, toluene, and xylene. The problem posed was to control the concentration of four impurities in the three product streams. Since there were only three manipulated variables, reboil duty, reflux ratio, and side stream flow rate, they decided to square the system by adding the position of the sidedraw to the set of manipulated variables. The distillation column was configured so that the side stream could be taken from any one of five trays. This is a very common configuration for side draw columns; however, side draw position is normally controlled by manually operated valves. It would be necessary to increase the number of remotely actuated valves, from 3 to 8, in order to realize this design in the field. This is clearly a very expensive scheme. As an alternative the scheme that will be proposed here controls the four impurities with only three manipulated variables. Compensator Design The transfer function matrix shown in Table I is that given by Doukas and Luyben except that their column
Ol96-43O5l84Ill23-O8548Ol.5OlO 0 1984 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
855
Table I. Process Transfer Function Matrix reboil duty toluene in bottom
toluene in tops
benzene in side draw
side draw
0.374e-7.756
-11 .3e-3.79s
(11.36s t 1)
(22.2s t 1)’
(21.74s t l ) a
5.984e-2*a4S
- 1.986e
(14.29s + 1 )
(66.67s t 1)’
(400s + 1)
2.38e- O . 42s
0.0204e-0*s9S
- 0.33e-0.686
(1.43s t 1)’
(7.14s t 1)’
( 2 . 3 8 s + 1)’
-11 .67e-1.916
xylene in side draw
reflux ratio
-9.8 1 l e - l *s96
(12.19s t 1 )
5.24e-605
o.
-0.176e-0.48s
4.48e-0.s2s
(6.9s + 1)’
(11.11s + 1)
.....
BENZ = 5 % = 89% = 6%
TOL XYL BEN2 = 10% TOL = 45%
XYL
= 45%
CONTROLLER
PROCESS
Figure 2. Control loop configuration.
Ihl-
Figure 1. Distillation column of Doukas and Luyben.
relating to side-draw position has been removed, resulting in a 4 X 3 transfer function matrix. For systems with square transfer function matrices the inverse gain matrix is a good starting point in the compensator design procedure. There is no inverse for non-square systems; however, by analogy the pseudo-inverse gain matrix may prove to be attractive. The pseudo-inverse of a non-square matrix, K, is defined as
(K’K)-’ K’ (1) where the prime indicates transposition. A more flexible form of the pseudo-inverse compensator may be taken from the work of Boyle (1978),who developed a quadratic optimization technique to solve steady-state problems. For a system which has a process model of the form e(k + 1) = e(k) + Ku (2) where e (k + 1) is next error, e (k)is present error, u is change in control input, and K is the process gain matrix. Boyle showed that a performance index
e’(k + 1)Ce(k + 1) + u’Bu is minimized by the following control law
(3)
+ B)-’K’C]e(k)
(4)
u = [(K’CK
COMPENSATOR
where C is a diagonal matrix of error penalty weights, B is a diagonal matrix of penalty weights on control moves, u is the vector of control changes, and e is the error vector. Boyle successfullyapplied this algorithm to the control of a system of 11 measurements with only 5 actuators. The algorithm takes no account of process dynamics. The control interval must be selected so that the process is at equilibrium at the start of each control interval. The form of the control law in eq 4 will also be recognized as the modified formulation of the Dynamic Matrix Control (DMC) law as suggested by Marchetti et al. (1981),
except that in the DMC formulation each element of the gain matrix, K, is replaced by a matrix of step response weights. The formulation using the matrix of transfer function gains only is attractive in that it will involve relatively small matrices. Boyle’s idea can be extended very simply so that the dynamic properties of the closedloop control system can be studied using traditional frequency domain design methods. Rather than using eq 4 as the control law the control matrix
D = (K’CK
+ B)-’K’C
(5)
is used as a decoupling compensator. Then, applying this non-square compensator to the non-square transfer function matrix, Rosenbrock’s Direct Nyquist Array method (Rosenbrock, 1974) can be used to modify the compensator so as to achieve dominance, to examine the closed-loop stability of the system, and to design simple controllers for the compensated system.
The Direct Nyquist Array Method The Direct Nyquist Array method is applied here in the usual way to generate the array for the compensated system, Q, where
Q ( s ) = G(s)D
(6)
G is the process transfer function matrix and (s) indicates the Laplace formulation. In the example here, Q, will be a 4 X 4 matrix permitting design of four proportional controllers. Treiber (1981) demonstrated the application of the Direct Nyquist Array method to a 2 X 2 distillation column control problem. The same criteria, for dominance and stability, apply here to the compensated system, Q, and its return difference matrix, F. The concept of dominance is not applicable to the uncompensated system, G , as a non-square system has no determinant. Upon achieving dominance in F a set of proportional controllers can be designed from the Nyquist plots of the diagonals of Q. The resulting control scheme has the structure shown in the block diagram of Figure 2. The control scheme will minimize a quadratic performance index like that shown in eq 3 which allows constraint of
856
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 4, 1984
actuator movement through the penalty matrix, B, and penalizing error through the matrix, C. In effect, these matrices become the tuning constants of the control scheme. A note of caution is warranted with respect to the particular system to be examined here; one with fewer manipulated than controlled variables. Despite the fact that the compensator squares up the process, the system is still underspecified. There is no guarantee that all errors can be reduced to zero, and the controllers should be proportional only. In fact, the compensated system matrix, Q, will be rank deficient and the resulting Nyquist array will be singular at all frequencies. This is expressed, in the Nyquist array of Q,by very broad Gershgorin bands. In the most extreme case for the 4 X 3 system to be discussed here, the Gershgorin band about one of the diagonals of Q will be very broad while the other bands will be relatively tight. In this case the loop with the broad Gershgorin band will be poorly controlled regardless of controller gain and will be unstable if integral action is applied. Manipulation of the compensator, D, can be used to shift the bandwidth among the loops so as to distribute the poor control among the loops. The designer can decide which loops require the tightest control. This difficulty will not arise in systems which have more manipulated than controlled variables.
Distillation Column Example The problem here is to control four measured impurities with only three manipulated variables, as described earlier. The steady-state gain matrix as taken from Table I is
K=
-9.8 6.0 2.4 -12.0
0.37 -2.0 0.02 -0.18
-11.0 5.2 -0.33 4.5
TERNARY DlSTILLATION CONTROL 8 HOUR TREND (2 MINI MEASURED VARIABLES TOLUENE IN TOLUENE IN BENZENE IN BOTTOM TOPS SIDE DRAW 4 87 80 0 97 100 4 98 0 0 100 00 100
WEB183 131438 XYLENE IN SIDE DRAW 0.0 5.88
10.0 I
1300
I 100
1000
900
800
100
600
I
I
I
I 00
TERNARY DISTILLATION CONTROL MANIPULATED VARIABLES REBOIL ,
DUTY
REFLUX
mno
1
495 91 4 0
550
,
,
mI I
I
100
8 HOUR TREND (2 M N SIDE DRAW
4905 4 50
W E B 1 8 3 131236
I
now
55;O
,
49 10 450
5Sl0
45.0
55.0
1
,
1300
la00
I100
loo0
900 800
(7)
700
600
The error and control penalty weight matrices, C and B, are the identity matrix, I, for the base case. Solving eq 5 for the compensator matrix, D
D=
-0.023 -0.20 -0.07
0.0047 -0.40 -0.0075
0.011 0.015 -0.0065
-0.061 -0.035 (8) 0.053
On application of the above pre-compensator the resultant system was found to be dominant with relatively tight Gershgorin bands about the first two loops, a somewhat broader band about the fourth loop, and an extremely wide band about the third loop. This means that the third loop, benzene in the side draw, would be poorly controlled. The control scheme would do little more than steer this variable. Assuming that this situation is tolerable the following proportional controllers were selected to achieve a gain margin of 3: toluene in bottoms, k = 4.13;toluene in tops, k = 15.0;benzene in side draw, k = 17.0; xylene in side draw, k = 1.7. With the exception of the toluene in tops controller these gains are one to two orders of magnitude larger than those selected by Doukas and Luyben. The distillation column, as described by the model in Table I, and the control scheme, in Figure 2, were simulated on a digital computer. Figure 3 shows the closed-loop response of the process to a setpoint change for toluene in bottoms from 5% to 10%. The ordinate in the figure shows the time in hours. The setpoint change was made at 850 h and it can be seen that the process variables were close to their steady-state values within 30 min. There was a small but significant offset in the toluene in bottoms and the other controlled variables were only slightly disturbed. Figure 4 shows the behavior of the manipulated variables during the test. The reboil duty, reflux ratio, and side draw
I
I
45.0
TOLUENE IN
55.0 8.91
/
I
45.0
55.0
I
TOLUENE IN
1
1
I
4.98
Figure 4. Closed loop behavior of manipulated variables.
flow are shown as percent of full scale rather than in engineering unit values. These values were intialized at 50% of scale. It can be seen that the reflux ratio was driven over almost 10% of its range while finally settling at a point less than 1%from the starting value. In some circumstances this radical movement of the actuator might not be acceptable, although otherwise the performance is good. Taking advantage of the compensator form in eq 4, movement of reflux ratio was constrained by changing the effort penalty matrix B to penalize movement of reflux ratio
B=
1.0 0.0 0.0
0.0 3.0 0.0
0.0 0.0 1.0
(9)
This triples the penalty on reflux change. Substituting the new value of B into eq 5 yields D=
70.021 -0.13 -0.065
0.0082 -0.26 0.0031
0.011 0.0096 -0.0068
-0.061 -0.023 (10) 0.054
The Direct Nyquist Array generated using the above compensator shows that the system is dominant, and that while the gain margins shift somewhat, the closed loop performance should be satisfactory without changing the controller tuning constants. Upon substituting the new compensator in the simulation and reinitializing the simulation at the same starting points as before, the test shown
Ind. Eng. Chem. Process Des. Dev. 1984, 23, 857-859 TERNARY DISI’ILLATION CONTROL MEASURED VARIABLES TRIPLE PENALTY ON REFLUX TOLUENE IN ,TLUENEIN BOTTOM 8.91 4 96
1
1300
1200
I 100 1000
900
800
700
600
I
I 0.0 REFLUX
I
I
10.0
0.0
I
10.0
I
0.0 REBOIL
49.27
1
10.0
I 00
10.0
49.92
Figure 5. Closed loop response of controlled variables: reflux constrained. TERNARY DlSTlLLAION CONTROL -TED VARIABLES TRIPLE PENALTY ON REFLUX REBOIL REFLUX RATIO DUTY 49 92
,
i :?
450
550
!3/FEB/83 130902
SIDE DRAW
now
,
49 71 45 0
55.0
857
ditional actuators and acceptance of the fact that this particular system is uncontrollable. Strict comparison of the performance of this control scheme to that of Doukas and Luyben is not possible here as they applied their designed linear control scheme to a nonlinear process model. In addition, they performed feed rate and composition disturbance tests, but they did not provide the process causality which would allow replication of their tests. Conclusions An application of the Direct Nyquist Array method to the design of control schemes for non-square systems has been demonstrated. The compensator design technique is borrowed from optimization theory and the resulting control scheme minimizes a quadratic performance index. It has been shown that good control performance is achieved and that actuator movement can be constrained. Nomenclature e = error vector u = control output vector (changes) K = process gain matrix C = error penalty matrix B = control penalty matrix D = pre-compensating decoupler matrix G = process transfer function matrix Q = compensated process transfer function matrix F = compensated process return-difference matrix Registry No. Benzene, 71-43-2; toluene, 108-88-3; xylene, 1330-20-7.
Literature Cited
I
45.0
TOLUENE IN
I
1
55.0
45.0
8.91
I
55.0
I
45.0
TOLUENE IN
I
55.0 4.96
Figure 6. Closed loop response of manipulated variables: reflux constrained.
in Figures 3 and 4 was repeated. Figure 5 shows the process response to the same 5% change in setpoint, of toluene in bottoms, as before. The performance was essentially the same as before but the variables equilibrated at slightly different values. Figure 6 shows that the reflux ratio was exercised over only 5% of its range in this case, a significant improvement in the performance. In comparing this approach to that of Doukas and Luyben, a trade-off must be made between the cost of ad-
Boyle, T. J. TAPPZ 1978, 61 (1). 77-8. Cutler, C. R.; Ramaker, B. L. American Institute of Chemical Engineers, 86th National Meeting, Houston, 1979. Doukas, N.; Luyben. W. Anal. Insfrum. 1978, 76, 51-8. Kouvaritakis, B.; Edmunds, J. M. “Alternatives for Linear Multivariable Control”; National Engineering Consortium, Inc.: Chicago, 1978; Paper 4.2, p 229. MacFarlane, A. G. J.; Kouvaritakis, B.; Edmunds, J. M. “Alternatives for Linear Multivariable Control”; National Engineering Consortium Inc.: Chicago, 1978; Paper 4.1, p 189. Marchetti, J. L.; Mellichamp, D. A.; Seborg, D. E. American Institute of Chemical Engineers, 74th Annual Meeting, New Orleans, 1981. Rosenbrock. H. H. “Computer-aided Control System Design”; Academic Press: London, 1974. Treiber, S. “Canadian Conference on Industrial Computer Systems”; Hamilton, Ont., May 3-5, 1981.
Shell Canada Limited Process Computer Applications Department Toronto, Ontario, M5S-2H8,Canada
Steven Treiber*
Received for review May 5, 1983 Accepted November 22,1983 Closed Loops Inc., 65 Queen Street West, Suite 1105. Toronto, Ontario M5H 2M5, Canada.
Modeling of Hydrogenation of Glucose in a Continuous Slurry Reactort A model for a continuous slurry reactor has been developed for the case of Langmuir-Hinshelwood type kinetics applicable to hydrogenation of glucose. The model predictions have been compared with pilot plant data on this system which agree well with the predicted results assuming that liquid flows in a plug flow.
Introduction Catalytic hydrogenation of glucose is an industrially important reaction as the productaorbitol-is a versatile NCL Communication No. 2856.
0 196-4305/84/ 1123-0857$01.50/0
chemical intermediate. A summary of the literature of this reaction is presented by Brahme (1972) and a detailed kinetics is reported by Brahne and Doraiswamy (1976). Generally, this process employs a slurry reactor operated either in a batch or a continuous manner. Continuous 0 1984 American Chemical Society
