Ind. Eng. Chem. Fundam. 1980, 79, 109-117
Literature Cited Baumgartner, F., Finsterwalder, L., J . Phys. Chem., 74, 108 (1970). Burger, L. L,, "The Transfer of Uranyl Nitrate A~~~~~ the Water-Tributyl Phosphate Interface", HW-62087 (1959). Chen, S.J., "Drop Formation of Low Viscosity Fluids in the Static Mixer Unit", KTEK-5, Kenics COrp. (July 1972) (available from Kenics Corp., Kenics Park, North Andover, Mass. 01845). Farbu. L., McKay, H. A. C., Wain, A. G..Proc. Int. Soh. Extr. Conf. (Lyon). . . . 2427 (1974): Healy, T. V., "Effect of Diluents on the Solvent Extraction of Nitrates. Part 11. Extraction of Uranyl Nitrate into Phosphate Esters", AERE-C/R-1772, Harwell, Berkshire, England, 1956. Keisch, B., "Exchange Studies of the Uranium-Tributyl Phosphate Reaction", IDO-14490 (1959). Knoch, K., Lindner, R., Z . Elektrochem., 84, 1020 (1960). Martin, G. C., Wain, A. G., AERE Harwell, personal communication, 1975.
109
McDowell, W. J., Oak Ridge National Laboratory, personal communication, 1978. Moszkowiczv p., Kikanda,j,T.9 c. R . Acad. SCi. Paris, Ser. c, 321 (1975). Petrich, G., Kolarik, Z., Distribution of U(VI), Pu(IV) and Nitric Acid in the System Uranyl NRrate-Plutonium(1V) Nitrate-Nitric Acid-WaterlSO % TBP in Aliphatic Diluents: A Compilation and Critical Evaluation of Equilibrium Data", KFK-2576 (1977). c' "' Scriven9L. E.pAIChE J . , 5, 514 (Ig5').
Received for review May 2 , 1979 Accepted November 8, 1979 ~~~~~~~hsponsored by the ~ i ~of Chemical i ~ i sciences, ~ ~ D ~ partment of Energy, under Contract W-7405-eng-26 with the Union Carbide Corporation.
Multivariable Control System for Two-Bed Reactors by the Characteristic Locus Method Alan S. Foss,' John M. Edmunds, and Basil Kouvarltakis Engineering Department, Control and Management Systems, University of Cambridge, Cambridge CB2 IRX, England
A threeinput multivariable control system is developed for the regulation of the product concentration and temperature of a two-bed exothermic catalytic reactor. Two of the inputs are the flow rate and temperature of a quench stream injected between the beds; the third input is the feed temperature. Through the use of the characteristic locus method of control system analysis, a cascade of dynamic compensators is developed by which interaction among variables is suppressed and the effects of feed concentration disturbances are controlled. In this development, an analysis is made of the potential improvements in performance accruing from the use of internal bed temperature measurements. The system also has the feature of resetting the quench Row rate to its nominal value upon sustained system disturbances.
Introduction A multivariable control system for a two-bed catalytic reactor is synthesized here by a method that tailors the frequency-dependent eigenproperties of a transfer-function matrix to achieve control of both concentration and temperature of the reactor product stream. Such a control objective extends earlier investigations of single-output control, and the application of such a design technique complements earlier applications of the Linear-Quadratic-Gaussian technique to this reactor control problem (Silva et al., 1979; Wallman et al., 1979). The best possible control is sought, of course, with either design method, but here, instead of seeking an optimum of a quadratic criterion, the suppression of interaction between the two controlled variables is addressed directly. To achieve control of both concentration and temperature upon sustained disturbances in feed concentration, a control system configuration such as that shown in Figure 1is proposed. An exothermic chemical reaction whose rate depends upon both temperature and concentration takes place in both beds of the reactor shown in this figure. It is this coupling of variables through the reaction rate that suggests the control configuration shown. Measurements of the controlled variables Cout and Tout are assumed available, and measurements of fluid temperatures at three
* On sabbatical leave from the Department of Chemical Engineering, University of California, Berkeley, Calif. 94720.
points internal to each bed may also be considered available for use in the control system should they prove beneficial. In industrial circumstances, measurement of Coutmay be impossible, in which case a reconstruction of this variable from temperature measurements may be attempted as demonstrated by Silva and Wallman. Here, for purposes of system analysis, concentration is considered measurable. Three manipulatable inputs (main feed temperature T f ,quench temperature TQ,and quench flow rate Q) are to be used. Such a control configuration is applicable in the frequently encountered industrial circumstance where conditions only at the entrance of each bed can be adjusted. With the exception of the input T f ,such a configuration has been investigated in the recent work of Silva and Wallman. Following the results of that work, it is proposed to employ the quench flow rate only for short-term corrective action, forcing the quench flow rate to return to its nominal value as quickly as practicable after the transient effects of the disturbance have been suppressed. Such a stipulation leaves inputs Tfand TQto eliminate steady-state offsets in Cd and Tow Temperature inputs are best suited for long-term corrections owing to their relatively slow rate of propagation through the reactor and their relatively high sensitivity. It can be argued on physical grounds that T f and TQ determine steady-state concentration and temperature at the entrance of Bed I1 and thus are sufficient to eliminate steady-state offsets in the variables Coutand Toutprovided that the ranges of the manipulated inputs
0019-7874/80/1019-0109$01.00/00 1980 American Chemical Society
110
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 Main Feed
Main Feed Heater
$j
~
-;
"O--J+fi Mixing Chamber
-1
Qwnch stream
-
~
L---
F
I1
Figure 2. Multivariable feedback control configurations: a, simple loop using controlled output measurements only; b, double loop using extra process measurements in inner loop.
objectives set for the reactor. The reactor dynamic characteristics are represented by G(s) in this figure and may be derived from the state-space model (eq 1) according to G(s) = C(SZ - A)-l B
Figure 1. Diagram of two-bed reactor and proposed control configuration for the regulation of C ,,, and To,,.
are adequate for the size of the feed disturbance. However, it is not so evident without some analysis that this three-input configuration will be sufficient for the control of dynamic conditions in the reactor. There are questions of the relative effect that the inputs have on the outputs over the important frequency range and the benefits accruing from the creation and placement of zeros through the use of temperature measurements at the three internal points in each bed. The synthesis of dynamic compensators for this control system configuration is carried out here in two stages. First, because Bed I1 is obviously the important dynamic element of this configuration, a two-input two-output control system for this bed is synthesized and analyzed. Bed I and the associated third input Tf is then added serially to controlled Bed I1 with the additional requirement that the quench flow rate is to be driven by integral action to its nominal value a t long times. All of these design stages are carried out with the use of locally linear seventh-order state-space models of the reactor beds of the form x = Ax + Bu (la) y =
Cx
+ Du
Ob) The approach to the design, however, was not rigidly proscribed by these models but was influenced by the knowledge that high-order process modes were lacking in the models. To investigate the robustness of the designs, simulations of closed-loop performance were made using 14th-order linear models of the reactor, thus introducing the potentially troubling phase lag of unmodeled modes. The design technique used is the characteristic locus method (MacFarlane and Kouvaritakis, 1977). The calculations were conveniently carried out on a minicomputer with an interactive program specially developed for linear system analysis (Edmunds, 1978). Outline of t h e Design Method The designer's task is to specify elements of the controller matrix K ( s ) in the closed-loop system shown in Figure 2a that accomplish in an acceptable way the control
+D
(2) The characteristic locus method of design makes use of the frequency-dependent properties of the eigenvalues of the open-loop matrix Q(s) = G(s)K(s)as s traverses the standard Nyquist contour. In particular, the designer can ensure closed-loop stability by selecting K(s) such that the net sum of counter-clockwise encirclements of the point (-1,Oj) by the loci of these eigenvalues (called characteristic loci) is equal to the number of open-loop unstable poles (MacFarlane and Postlethwaite, 1977), a situation analogous to the single-variable case. Further, the shaping of these characteristic loci and the reorientation of their associated eigenvectors through the choice of K ( s ) constitute the means of meeting control objectives. The basis for the use of Q(s) in the analysis of the closed-loop system is the congruence of the closed- and open-loop eigenframes and a simple relation between the open- and closed-loop eigenvalues. These relations are most clearly seen from the dyadic expansion of the open-loop and closed-loop (under unity feedback) transfer function matrices Q(s) and
R(s). m
Q(s) =
C q,(s)w,(s)u,W
k=1
(3) (4)
The (qi(s)Jare the eigenvalues of Q(s), and the wi(s) and uiT(s) are the corresponding eigenvectors and dual eigenvectors. The suppression of interaction among controlled variables may be accomplished at low frequencies by the simple expedient of endowing the characteristic loci with large moduli through the introduction of integral action (MacFarlane and Kouvaritakis, 1977). At high frequencies, such a strategy is impossible because high power would be demanded of the control inputs and because stability conditions force the characteristic loci to have small moduli at high frequency. Further, because the characteristic loci of G(s)tend to small moduli at high frequencies, one finds that the closed-loop transfer function tends to Q ( s ) at high frequencies. Thus to effect a sup-
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
pression of interaction in the closed-loop system at high frequencies, one must select K(s)such that the eigenvectors of Q(s) = G(s)K(s) are as nearly as possible coincident with the unit base vectors (MacFarlane and Kouvaritakis, 1977). This alignment can be done approximately over a band of frequencies (Edmunds and Kouvaritakis, 1979). If the relative sizes of the elements of the frequency response matrix do not change too rapidly for increasing frequency such a controller would produce approximate alignment over a range of frequencies, and thus for ease of calculation the alignment is usually carried out at only one high frequency, Wh, specified by the designer. As detailed by MacFarlane and Kouvaritakis, a real matrix Kh is determined such that G(&)Kh approximates a diagonal matrix. The shaping of the characteristic loci over the rest of the frequency range is generally an involved problem because the effect of changes in K(s) on the eigenproperties of the product G(s)K(s) is complicated. The situation is simplified, however, if the eigenvectors of K(s) are made coincident with those of G(s); then the eigenvalues of the product are simply the product of the eigenvalues of G(s) and K(s),and the eigenvectors remain unchanged. Under such conditions, G(s) and K(s) commute. The proposal of MacFarlane and Kouvaritakis (1977) is to construct a compensator K(s) of the form K(s) = WA(s)V (6) that approximately commutes with G(s) at a specified frequency. The real matrices W and V are determined by an alignment procedure, similar to that mentioned above, that obtains real eigenvectors and dual eigenvectors for K ( s )that approximate the complex eigenvectors of GGw) at a specified frequency. By taking A(s) diagonal, one may therefore easily modify each locus of Q(s)independently by making appropriate choices of the diagonal elements of A(s). A K(s)so constructed is called an approximately commutative controller because K ( s ) is made to nearly commute with G(s) at a particular frequency. The approximately commutative controller may be employed at any of several frequencies at which modification of the characteristic loci appears necessary or desirable. The diagonal elements of A(s) are chosen in the upper frequency range to improve the gain and phase margins of the characteristic loci in an attempt to reduce oscillatory and overshooting behavior in the closed loop. In the low and intermediate frequency range, the elements of A(s) are chosen with the aim of improving the system accuracy to tracking, minimizing steady-state offsets, and suppressing low-frequency interaction. Closing the loop around G(s) by selection of K ( s ) as described above can result in acceptable system performance if enough feedback gain can be introduced. In some instances, however, the gain is restricted to unacceptably low values by early crossing of the root loci into the right half of the complex plane. Zeros of G(s) in the right-half plane are well known to be a source of this difficulty. There are ways, however, by which root loci may be altered favorably by making use of process measurements in addition to the measurements of the controlled variables. Specifically, if the extra process measurements, y e shown in Figure 2b, are "squared down" (combined) through a constant matrix F and used to form an inner loop, one is afforded the opportunity to create and place the zeros of the inner loop system FG(s) (MacFarlane et al., 1977). The idea is to place these zeros judiciously in the left half plane so that by closing the inner loop, the poles may be conducted to locations that are more favorable as starting locations for closure of the outer loop. The zeros are placed by the choice of F.
111
Disturbance
I
I
Q
To
Control inputs
Figure 3. Reactor system components and input-output variables. Notation: T = temperature; C = concentration; Q = flow rate; X = state vector.
Two-Bed Reactor Model The reactor model used in these design calculations derives from a set of nonlinear partial differential equations representing one-dimensional material and heat balances of the packed bed. The equations and methods of their reduction to linear state-space form have been detailed by Silva et al. (1979) and Michelson et al. (1973) and are not repeated here. Rather, it is sufficient for purposes of this study to state the A, B, C, and D matrices for each bed and the coupling between the beds. A sketch showing the input-output configuration of the two-bed system is given in Figure 3. The models associated with each block of this figure are as follows. feed heater
1
To = --To 7
+
1 -T, 7
bed I
=
C'X'
+ D'
Ej
mixing chamber
bed I1
xn
i] To ut
B=
DU
cout
The relations for the mixing chamber are linearizations of the steady-state material and heat balances at that point. Reactor models of two different orders are used here. The design is carried out using a seventh-order model for each bed. These models were obtained from 14th-order models by a reduction technique in which the fast modes are discarded but the steady-state gains are preserved (Wallman, 1977); catalyst temperatures were retained as the state variables. The 14th-order model is used here in closed-loop simulations to test the performance of the control system designed on the basis of the lower order model. The 14th-order model was not used for the design calculations since the software available at the time the design was being carried out could not handle all 28 states.
112
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
I
O~
To
Table I hatrlc-es f o r t h e heactor S v s t e m Moa&
I e e o neater
1=
0.2956
(221
1
-2
+
Figure 4. Nyquist array for the second bed of the reactor: outputs (Tout,Cod; inputs (Q,TQ).
-2.36j2 0.9280 0.5462 AI= -0,;blj -0.j797 -0,0805 0.0265
-0.1717 -1.868b 1.8614 0.4916 -0.7645 -0.8bdU -0.7549
2.5136 1.5082
0.0175 0.0645
0.2840 0.6519 0.6540
0.1001 0.1258 O.ljjb
I1
36
0.1561 -0.1bO6 0.1802 -0.~527 0.1550 -0.jO25 0.1642 -0.2818 0.3885 -0.2315 -1.4966 -0.4014 0.5689 -0.464b 0.2Yj8 2.jj99 -1.4191 -0.4404 0.485'1 -0.2892 0.1716 0.b6jl 2.5iLj -1.7568 -0.5569 -0.6112 1.7lbj 2.018j -2.2067 -0.1147 -2.7725 -0.6771 1.412, i.Oj61 0.6079
r
1
0.1160 -0.1088 0.0450
0.0000 0.0000 0.0000
0.0000
0.7789
1
J
(24,251
-0.j155 0.8200 0.489 -0.1658 0.1229 -0.1427 0.0840 0.2118 -0.2499 0.5766 0.ilYj -0.2745 0.2158 -0.1585 -0.0855 0.0860 -0,1199 O.jOY7 0.95j4 -0.4021 0.2112 -0.0050 0.0048 -0.0059 0.0098 -0.0259 0.2917 0.7260 -0,0846 -0,1912 -0.2604 -0.2298 -0.jO12 -0,1621 0.0000
Figure 5. Characteristic loci of Bed 11.
Elements of the matrices of the seventh-order reactor model and the mixing matrix are given in Table I. The numerical values are identical with those determined for the experimental conditions of Silva and represent reactor dynamic behavior in the vicinity of the nominal steady state of those experiments. Matrices for the 14th-order model are given by Silva (1978). Control System Design for Bed I1 The control system for Bed I1 will employ the inputs Q and TQshown in Figure 3 to regulate effluent variables Tout and C,. One of the first matters to be investigated in such a multivariable system is the sensitivity of the outputs to the inputs. To achieve independent control over Toutand Gout, one would prefer the transfer function matrix G(s) relating outputs to inputs to be well conditioned over all frequencies. Regulation of Toutand C,,, to the setpoint, for example, requires that G(s = 0) be invertible. The situation for this reactor transfer function matrix is reflected in the open-loop Nyquist array, the characteristic loci, and the eigenvectors of G(s). The first two of these are displayed in Figures 4 and 5 and reveal that we begin the design under inauspicious circumstances. The close similarity of the columns of the Nyquist array (Figure 4) and the large disparity of the eigenvalues of G(s) (Figure 5) would both seem to imply a near linear dependence of the inputs Q and T Qfor this set of outputs. However, the determinant of the steadystate gain matrix G(0) is only 10 to 20 times smaller than the individual elements and thus suggests that control to setpoint is feasible. A further indicator of the feasibility of this 2 X 2 control configuration for Bed I1 is the angle between the two eigenvectorsof G(s). Significant deviation from orthogonality of the eigenvectors implies the need for inordinately large control inputs. The angle between the two vectors for Bed I1 was found to be about 70' in the range of frequencies 0.1 to 3, the range above which model accuracy cannot be assured. The situation seems to be favorable; angles at the intermediate and high frequencies appear significantly large, and it is these that are relevant because Q is not used for low-frequency corrective action.
1
(23)
1
(2b)
Muum Chamber M:
[
-O.>jIj -0.7226 1.0000
0.2j70 11.0000 0.0000
0.0000 0.7630 0.0000
0.7630 0.0000 0.0000
1
(211
kLLX -5,5108 -0,1285 0.1029 -0.127~, 0.1581 -0.2~07 O.lj46 0.!675 -2.1256 -0.2OOj 0.2004 -0.2jj5 O.jl94 -0.1940 1.7995 -1.8801 -0.2849 0.2694 -0.3464 0.2080 1.5039 2,4101 -1.8j52 -0.2149 0.2818 -0.1626 1.6907 1.iOlb -2.0525 -0.2j75 0.1205 -0,6847 -0.2164 O.jj44 1.9009 1.4282 -2.3437 -0,1116 -1.0197 -1.4581 1.44Oj 0.6254 -2.8285 -1.0121 -1.6634 -0.05OU 1.1249
:::::;
BII=
[
i.8j14 2.jbj9 0.8489 -1.4594 -1,5940 -0.18j7 0.2538
0.Ob79 0.2812 0.5514 0.5891 0.j488 0.2054 0.1872
-11.0266 -0.1255 -0.2026 -0.0781 0.2194 O.jUO9 0.4044
]
DII=
[
0.0471 -0,0941 0.060Y 0.0026 0.0000
1
(28)
0.0000 -0.0029 0.0000 -0.0035
0.0000 -0.0035 0.0000 -0.0019 0.24jj 0.2759
-0,1104 1.OOYb 0.0706 -0.Oj19 0.0244 -0,0289 0.0111 0.8659 0.j817 -0.1912 0.501'1 -0,1166 0.19Oj -0.2400 -0,1168 0.1262 -0.1865 0.6320 0.6820 -0.437 0.2546 -0.0045 0.0045 -0.0060 0.0100 -0.026j 0.j416 0.6759 -0.0465 -0.llj5 -0.1909 -0,261Y -0.26j4 -0.1422 -0.0002
Inner-Loop Zeros. Before proceeding with the design of the 2 x 2 system, an investigation is made of the benefits that might accrue by use of the intermediate temperature measurements T5,T,, and T , shown in Figure 3 in the inner loop described in an earlier section. By "squaring down" from the four measurements to either one or two inputs, one creates zeros in the inner loop and is afforded the opportunity to place these zeros to enhance the use of high gains in the outer loop. That there is an incentive to consider such a configuration may be seen from the presence of righ-half-plane zeros in the transmittances (Q to T,3 and (TQto T,3. Figure 6a shows a right-half-plane zero in each transmittance close to the origin, and, because the poles are clustered not far away, this zero will assuredly exert a strong attraction on the root locus when single loops are closed from To, to either Q or TQ. The gains that can be tolerated in such loops would therefore be so small that feedback control would be ineffective. The situation is much more favorable, however, for the control of both Cout and Toutwith Q and TQ,as may be seen from the location of the zeros in Figure 6b. There is only one right-half-plane zero, and that is far from the origin.
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 113
-10
+99,4)
-10
:-17,36)
9
-4 x
x Reactor poles o Zeros : 0 -Tfut
x
o
a
Zeros : T~--T.”,, Frequency
-
Figure 7. Misalignment angles of eigenvectors of G(s)Khwith the basis vectors.
-I
0 : -102 1
I
I -4
0
I
IO:
IL1-
45.41-
0
-
Invariant Zeros
b
0 8 To
T i t & Czu,
Figure 6. Poles and zeros of open-loop configurations for Bed 11; 7th-order reactor model; a, reactor poles and zeros of single-input single-output systems; b, zeros of 2x2 system with inputs Q and TQ and with outputs To, and C,,,.
The number of zeros capable of being placed and the most desirable location of these zeros is a matter that seems to be highly problem dependent. In the present case, only three of seven zeros were found mobile under a “squaring down” operation that employed all four temperature measurements in Bed II; four zeros remained near the real axis in the region -2 to -3 for several different linear combinations of the measurements. Zeros with such properties have been encountered in other systems and have been termed “latent zeros” (Kouvaritakis and Edmunds, 1977). A similar clustering of four zeros was found in the transfer functions relating each of the temperature measurements individually to inputs Q and T p Such a characteristic would seem to be associated with the “cascade” nature of the reactor. The close proximity of the latent zeros to the zeros of the individual elements of C(s) implies the near singularity of G(s) at the latent zeros. Thus, any attempts to shift the latent zeros will require a system model of unrealistically high accuracy. For the three zeros that can be shifted, one may contemplate three regions of the complex plane of Figure 6 relative to the open-loop poles shown there. The three general regions are: (1)near the origin, (2) far to the left of the clustered poles, and (3) in the vicinity of the pole cluster. Unfortunately, none of these offers any significant enhancement to the control system performance in this case. Zeros near the origin are undesirable because they lead to small gains at low frequencies, resulting in slow approaches to setpoints. Locating zeros far to the left of the poles is ineffective in influencing the phase of the characteristic loci since several of the zeros have to be left near the cluster of poles, thus yielding an impotent inner loop. The placement of all the zeros near the pole clusters and the subsequent use of high inner loop feedback gains would make the system less sensitive to parameter changes but does not improve the speed of response of the outer loop since it leaves all the poles in the same region. Thus, the decision was made to abandon the prospect of an inner loop in the control configuration. The intransigence of the process can be blamed in this case rather than a stumbling of the general idea of zero placement in an inner loop. The characteristic loci of the two-input two-output reactor given in Figure 5 are thus considered the starting point for the introduction of compensators.
Figure 8. Characteristic loci of reactor with constant pre-compensator Kh to suppress interaction at w = 2.
Even though it was not practicable to shift zeros in an inner loop, the zero configuration of the 2x2 system shown in Figure 6b is not unmanageable because the single right-half-plane zero at +45 will have very little influence on the root loci at useable gains. Compensator Design. The addition of compensators for purposes of forging the characteristic loci into desired shapes proceeds though four distinguishable stages: (1) suppression of interaction at high frequency, (2) attenuation of the loci in the very high frequency range for which the model is poor, (3) boosting of gain at intermediate frequencies, and (4) addition of integral action to enhance steady-state accuracy and to suppress low-frequency interaction. Suppression of interaction at high frequencies is accomplished as described above by introducing a constant precompensator Kh that approximately aligns the eigenvectors of G(s)Kh with the unit vectors at a selected frequency. Examination of the misalignments between the eigenvectors and the basis vectors at several frequencies suggested that the alignment would be best accomplished at w = 2. The alignment angles so obtained are shown in Figure 7 where it is seen that alignment only to within about 3 5 O was achieved and that the alignment of one of the vectors worsens rapidly as frequency is decreased. Alignment a t higher frequencies was obviated by model uncertainties; alignment at lower frequencies yielded nearly singular Kh matrices. The Kh matrix obtained for the alignment at w = 2 is
r-1.161.041 Kh=
1
-3.35
i
2.34
and the characteristic loci of the reactor with this precompensator are given in Figure 8. Upon comparison with the uncompensated system (Figure 5), there is seen to be a considerable reduction in the disparity between the eigenvalues of the system. The larger locus in Figure 8 exhibits appreciable magnitude at frequencies of 3 and above, the range of discarded and neglected modes in the reactor model. Because of the uncertainties of the characteristic loci in this frequency range, one is not justified in designing on the basis of the loci at these frequencies. We therefore propose to atten-
114
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 30
2 01
1, I1
Figure 9. Characteristic loci: align at 2, and approximate commutative controller at 2; Kh and K,(s) given by eq 13 and 14.
uate and lag the large locus at frequencies in the region of w = 2 to bring its magnitude and phase into closer correspondence with the smaller locus. This is accomplished by calculating an approximate commutative controller a t w = 2 by which the large locus is lagged over a 10-fold range of frequencies from 0.5 and the smaller locus is simply increased in magnitude by a factor of 2. This compensator is
Figure 10. Characteristic loci; align at 2, approximate commutative controller at 2, approximate commutative controller at 0.3; Kh, Kl(s), and K,(s) from eq 13, 14, and 15. 1.5 t
X
-0.571
1
The choice of the lag frequency 0.5 is influenced by the desire to decrease appreciably the magnitude of the locus at w = 2 but to maintain gains as high as possible at low and intermediate frequencies. The loci resulting from the introduction of Kl(s) as a precompensator are shown in Figure 9. The phases of the loci are now seen to be in much closer correspondence. The loci magnitudes are also more nearly equal. Compensation is now introduced to further modify these loci for the purpose of boosting the magnitudes of each at intermediate and low frequencies without affecting the phase at high frequencies. Lags at w = 0.2 and 0.05, each with a 10-fold frequency range, are applied to the large and small loci, respectively. These lags are introduced through an approximate commutative controller calculated at w = 0.3, the approximate center of the frequency range of the two new lags. This precompensator is
KJs) =
[
1.031 L 0 3 j
[-
0
s s
-0.033 -0.657
+ 0.5 + 0.05 1.019
I
]
-0.C49
L
1.6031 -1.599 J
The resulting loci are shown in Figure 10. These loci seem balanced well enough to attempt closure to the loop. A scalar proportional-integral controller is introduced at a frequency of 0.05; this produces high gains in both loci at low frequency with only a few degrees of phase shift in the slowest locus near the critical point. This compensator is taken to be s
+ 0.05
K&) = 4 S
1
Figure 11. Closed-loop control of Bed I1 with compensators Kh, K,(s),K,(s), and K3(s)using the 7th- and 14th-order models. Step increase in set point of Gout: -, 7th-order model; ---,14th-order model.
The gain of 4 was selected to give a 45" phase margin to the locus having the largest phase lag near the critical point. To test the performance of this cascade of compensators, transient responses to step changes in set point were calculated using both the seventh- and 14th-order reactor models. It was essential that the control system be tested on the higher order model because the reactor modes neglected in the design model could easily lead to instability if care had not been exercised in the design sequence. The closed-loop response of these two reactor models to a step increase in the setpoint Coutis shown in Figure 11. The response of the seventh-order model displays an acceptable degree of stability, but the 14th-order model, simulated with the same control system, shows an undesirable resonance. Similar behavior is also observed upon a step change in the Toutsetpoint as well. This resonance apparently arises owing to inattention to the higher frequency process modes and may be cured by further modifying the characteristic loci at high frequencies as follows. To better balance the loci of Figure 9, compensator Kl(s) is modified by the introduction of further attenuation of the fast locus (the locus possessing the highest frequency near the critical point) at w = 0.5 and a boosting of the low-frequency part of the slower locus. The first modification should aid in reducing the resonance, the second in achieving a faster approach to the set point. These loci modifications, if successful, would seem more desirable than the mere reduction of the scalar gain. These modifications are accomplished with lags so that the modified compensator is
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
(s + 0.5)(s
+
0.5)
The characteristic loci resulting from this modification are shown in Figure 12a, and when the approximate commutative controller K2,(s) has been recalculated and added, the loci take the forms of Figure 12b. The modified controller is
:1 r
1.223
K,nl(s) =
115
0.811
s
s
-0.446 -0.651
J
0
s
+
+
[ 1.495
1
-1.023
I
@i \
@I
@@i
Id-
1
+ 0.2
20 t
r‘
316
tp.
0.5 0.05
2@!
1.8581
0 25
Q’ D
-2.807
An improved balance is evident. Finally, the scalar PI controller is added with the same gain as before but with twice the integration rate of that used previously. That is
Figure 12. Characteristic loci of Bed I1 with modified compensators: a, characteristic loci with modified approximate commutative controller a t w = 2; Kh and Kl,(s) given by eq 13 and 17; b, characteristic loci with &, K,,(s), and &&).
? 0l
The response to a setpoint change in Cout with these modified compensators is shown in Figure 13. When compared with the responses shown in Figure 11, it is seen that the resonance has been eliminated, the recovery rate enhanced, and the steady state accuracy improved. There is, however, considerable interaction present in the mid frequencies. Design of the Two-Bed System. The incorporation of Bed I into the multivariable design just described for Bed I1 is accomplished in a simple way with simple objectives. Feed temperature Tf is the new input variable associated with Bed I and is known (Silva et al., 1979) to have little potential for regulating dynamic conditions in the reactor. Its sole function in this design is therefore relegated to driving the quench flow rate to its nominal value after the quench has accomplished the major portion of its short-term corrective action. The control system configuration can therefore be viewed as sketched in Figure 14. The Bed I1 compensator just developed is the product of the individual compensators Kh,K1, K2,and K3 and is employed to regulate the two variables Toutand Gout; the quench flow rate Q is considered the third output variable and is linked to input Tf by a single loop. The design of this simple loop between Tf and Q is straightforward. The Nyquist locus, which is shown in Figure 15, has a high-order character. The proportional-integral controller
K ~ ( s=) 0.07-
s
+S 0.2
(20)
when introduced into the loop yields a gain margin of about 3 and an integration rate that would reset Q in about 10 time units. Three time units correspond approximately
Figure 13. Response of system to step change in the setpoint for Gout: -, 7th-order model with controllers &, Kl,(s), Kzm(s), K3,,,(s); - - -,14th-order model with controllers &,, K&), Kzrn(8),K3,h); 14th-order model with controllers Kh, K,,(s), K,(s), Kbm(s),K ~ ( s ) . e..,
I
I
Figure 14. Control system configuration for two-bed reactor.
to one “time constant” of the uncontrolled two-bed reactor. In the course of the investigation of the performance of the complete system, it was found that some additional tuning could be made of the scalar PI controller K3,,,(s)of Bed 11. The integration rate was doubled to give the modified scalar controller s K‘3m(~) =4
+S 0.2
(21)
116
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980
'T -I 10
/
042
00
v
i
I/o
& 7;
49
35 -I
0
I. I
Figure 18. Characteristic loci for final design. Figure 15. Nyquist locus of Q(s)/Tf(s)of Figure 14 (bottom loop open, top loops closed). ...-.- ~.......... . out
I
'.
Lncontralled
Caul
Figure 16. Transient behavior for a unit step disturbance in Co using both beds of the 14th-order model.
4
4
I
-2.6
1
Figure 17. Control action while suppressing a unit step disturbance in C, for the 14th-order model.
System Performance and Properties. System performance was evaluated by simulation of the reactor under the controls devised above upon a step disturbance in the feed concentration CW Fourteenth-order models of both beds were used in these simulations. A comparison of the controlled and uncontrolled transients in Toutand Coutis given in Figure 16. Both temperature and concentration are seen to be held by the controls to small values relative to the uncontrolled case, and both are returned to the vicinity of the set point with reasonable speed. The control action required to achieve this performance is shown in Figure 17. The response to a set point change is shown by the dotted lines in Figure 13. Mid-frequency interaction is evident in both set-point and disturbance responses. The magnitude of the control inputs is an important matter in multivariable systems; it is certain to be larger than that of a single-output system because of the extra demands on the controller. For example, in this case independent control has been demanded over output temperature and concentration. In practice, the range of the inputs is limited, and thus an equivalent concern is the
magnitude of the disturbances that can be accommodated without saturation of the control inputs. The results of Figure 17 imply, for example, that this control system could accommodate feed concentration steps of up to 2.5% before saturation would occur on the quench flow rate of the experimental reactor used in earlier studies. For larger disturbances the transient interaction would increase, but steady-state regulation could still be achieved until one of the heaters saturates. In the single-output systems of Silva and Wallman concentration disturbances of 10 to 20 % could be handled before the quench flow rate reached its limits. The three characteristic loci of the final 3x3 system are shown in Figure 18. One of the three is considerably slower than the other two and is undoubtedly associated with the more or less independent PI controller linking Q to Tf.The two faster loci exhibit gain and phase characteristics near the critical point that would contribute to some slight overshoot of the controlled variables, and indeed this is observed in the response of TWtand C, shown in Figure 16. Quench flow rate Q, however, responds in an overdamped manner, as would be suggested by the shape of the slower locus. It is noticed that the seven poles introduced through the compensators K,,(s), Kz,(s), K3,(s), and K4(s)are clustered in the region 0 to -0.5, about four times slower than the reactor poles (cf. Figure 6a). Further, there is an excess of poles over zeros introduced in that region. This compensator pole-zero configuration is typical of that usually used for high-order cascaded processes. The implementation of the set of compensators would be straightforward in either analog or digital realizations. In a digital implementation, a seventh-order state-space realization of the compensator transfer functions would be required, a size that is easily accommodated in a process control computer.
Conclusion This investigation of multivariable control system design for a two-bed reactor has given new insight into the control problems of these reactors. First, the investigation of zero creation and placement through the use of internal bed temperature measurements has given a more penetrating explanation of the ineffectiveness of these measurements on control system performance. Further, through the characteristic loci and the associated eigenvectors one has a much more appealing and geometrically interpretable method of dealing with interaction than is contained in the Linear Quadratic design approach, for example. Attempts to align eigenvectors with the reactor output basis vectors have revealed a limited frequency range in which interaction can be effectively suppressed. In the present case, this range has been centered around w = 2, which is seen from Figure 18 to be close to the critical frequency
Ind. Eng. Chem. Fundam. 1980, 79, 117-121
of one of the characteristic loci. The technique used here to suppress interation between the two controlled variables is perhaps not the final solution of this aspect of the design procedure. A minimum in the interaction was imposed at a single (high) frequency. More appropriately, one may want to specify a frequency dependent degree of suppression to accommodate any physical insights that one has about the feasibility of noninteraction. Here, for example, such a gradation in suppression of controlled temperature interaction might have proved beneficial, particularly since there is a fairly clear frequency separation in the control authority of Q and TQ.Such a modification would likely result in reduced control effort, a desirable outcome. The methods used to shape the characteristic loci by selection of dynamic compensators for the commutative controllers have in this study relied heavily on the designer's appreciation of the relationship between the shape of the characteristic loci and the transient response of the closed loop. There is an art in this aspect of the design even for single-loop systems, and without a keen ability at it, one can spend a considerable effort in trial and error. What seems to be needed are easily understood quantitative relationships between the desired closed-loop transient and frequency-dependent eigenproperties of the system matrix that can be used as guides in selecting the compensators. Such relationships will differ, of course, for cases of input tracking and disturbance rejection.
117
One of the major objectives of this investigation was to determine the feasibility of simultaneous control of product concentration and temperature. It was found that there is indeed a chance of accomplishing this but at a cost of control efforts that are about eight times those needed for concentration control alone. These control efforts probably can be reduced by a relaxing of the degree of noninteraction, but most of the additional control effort may be attributed to a near singular system matrix. Such circumstances prompt one to search for new sets of control input variables.
Literature Cited Edmunds, J. M., CUED/F-CAMS/TR 170, Engineering Department, University of Cambridge, 1978. Edmunds, J. M. Kouvarltakis, B., Int. J. Control, to be published, 1979. Kouvarltakis, B., Edmunds, J. M., National Engineering Consortium Symposium on Multivariable System Design, Chicago, Ill., 1977. MacFarlane, A. G. J., Kouvaritakis, B., Int. J. Control, 25(E), E37 (1977). MacFarlane, A. G. J., Kouvaritakis, B., Edmunds, J. M., National Englneerlng Corsortium Symposium on Multivariable System Design, Chicago, Ill., 1977. MacFarlane, A. G. J., Postlethwaite, I., Int. J. Control, 25(1), 81 (1977). Michelsen, M. L., Vakil, H. B., Foss, A. S., Ind. Eng. Chem. Fundam., 12, 323 (1973). Silva, J. M., Ph.D. Thesis, University of California, Berkeley, 1978. Silva, J. M., Wallman, P. H., Foss, A. S., I n d . Eng. Chem. Fundam., 18, 383 (1979). Wallman, P. H., Ph.D. Thesis, University of California, Berkeley, 1977. Wallman. P. H., Silva, J. M., Foss. A. S., Ind. Eng. Chem. Fundam., 18, 392 (1979).
Received for review April 16, 1979 Accepted October 11, 1979
Succinic Acid Crystal Growth Rates in Aqueous Solution J. W. Mullin' and M. J. L. Whiting Department of Chemical and Biochemical Engineering, University College London, London, WC 1E 7J€, England
The effects of supersaturation, temperature, solution velocity, and crystal orientation_on the individual face growth rates of succinic acid crystals in aqueous solution have been measured. For the (111) and (010) faces, the growth process is diffusion controlled and approximately first order with respect to supersaturation. The growth order of the (001) face is about 1.5. Overall growth and dissolution rates of single crystals, located within a well defined hydrodynamic environment, have been measured by a semicontinuous weighing technique.
Introduction The 0 form of succinic acid [(CH2COOH)2,mol wt = 118.09, crystal density 1572 kg/m3] crystallizes from aqueous solution in the monoclinic prismatic class. The unit cell parameters and general crystallography are well established (Broadley et al., 1959). A typical symmetrically formed crystal of succinic acid grown from aqueous solution a t room temperature and low supersaturation, illustrated in Figure 1,resembles a six-sided prism with a large predominant basal plane (001)bounded by smaller faces of the type (OlO], Illi), and (110). The solubility of succinic acid in water over the temperature range 20 < 8 < 40 "C may be represented by the equation c* = 5.502 X
-
1.157 X 10-38+ 9.4307
X
(1)
with a standard deviation of 5.7 x kg of succinic acid/kg of water. This second-orderpolynomial represents the best least-squares fit of 18 data points and is in good 0019-7874/80/1019-0117$01.00/0
agreement with other recorded measurements (ICT, 1926; Seidell, 1941). The densities of saturated aqueous solutions of succinic acid over the temperature range 20-40 "C, measured by the pycnometric technique suggested by Findlay (1955), were correlated by the equation = 1.01780 x 103 - 2.51262 x 10-18 + 1.59298 x 10-202 (2) with a standard deviation of 0.171 kg/m3. The viscosities of saturated aqueous solutions of succinic acid at 18, 25, and 27.3 OC are 1.17, 1.03, and 1.00 Pa s, respectively. Face Growth Rate Measurements Individual face growth rates of single crystals of succinic acid were measured by a technique previously described (Mullin and Amatavivadhana, 1967; Mullin and Garside, 1967). Briefly, a crystal was mounted on a wire in a glass cell through which a solution flowed under carefully controlled conditions. Any chosen face or edge could be ob0 1980 American Chemical Society
