c
=
constant in Equation 26 or 27
variance, defined by Equation 25 central F st atistic defined by Equation 23 F,+ 1 f,f ( X ) ,fa, f, = dependent variable g(X) = x 1 matrix defined by Equation 31 g’(X) = 1 X lvmatrix, transpose of g(X) h = spacing in equal-spaced independent variable, = = =
Em+ F
YI
+
x,,.-
i, 1. k
summation indices factorial notation, defined by Equations 16 and 17 L defined by Equation 13 m degree of polynomial 1V number of (data points included in polynomial n- ’ = odd -V ’YLS = number of smoothings P3(X) = least squares polynomial of degreej, general case, in Equation 1 P = level of significance 43 = orthogonal least squares polynomial of degree j, defined for the general case by Equation 8, for the equal-spaced independent variable case by Equation 15 S = translated independent variable, of Equation 12 S = d g r n X ’ x ) - l g(X)2 t ( p / 2 , n - m - 1) = Student’s t statistic, at p/2 level of m - 1 degrees significance with n of- freedom X , X,, X , = independent variable (X’X) = &VX N m a t r i x defined by Equation 4 (X’X)-’ = inverse of (X’X) (X’Y) = N X 1 matrix, defined by Equation 5 = = = = =
(k)
-
Y
m
Y
=
$,Qj(X)
approximation o f f , calculated from ~
j=O
in the case of equally spaced data and from m
X7 for nonequally spaced data 3=0
= defined by Equation 34 = j , h coefficient of generalized
(Y’Y)
least squares polynomial model in Equation 1
aj
j t h least squares power function coefficient of model in Equation 2 = best linear unbiased estimate (least squares) of P j , calculated from Equation 3 = j t h least squares orthogonal coefficient of model in Equation 7 = best linear unbiased estimate of yj, calculated from Equation 11 or 18 = difference, i 1 point minus i point = random error = variance = estimated variance, defined by Equation 33 = m t h degree least squares coefficients in Equation 22 = m l t h degree least squares coefficients in Equation 21 =
= constant in Equation 26 or 27
D
+
literature Cited
Bright, J. W., Dawkins, G. S.,IND.END.CHEM.FUNDAMENTALS 4, 93 (1965). Forsythe, G. E., J . SOC.Ind. Appl. M a t h . 5 , 74 (1957). Graybill, F. A , , “Introduction to Linear Statistical Models,” Vol. I, McGraw-Hill, New York, 1961. “Handbook of Chemistry and Physics,” 37th ed., p. 1972, Chemical Rubber Publishing Co., Cleveland, Ohio, 1955. Hildebrand, F. B., “Introduction to Numerical Analysis,” McGraw-FIill, New York, 1956. Hildebrand, F. B., “Methods of Applied Mathematics,” PrenticeHall, Englewood Cliffs, N. J., 1963. Lanczos, C., “Applied Analysis,” Prentice-Hall, Englewood Cliffs, N. J., 1956. Lapidus, L., “Digital Computation for Chemical Engineers,” McGraw-Hill, New York, 1962. Orden, A,, “Mathematical Methods for Digital Computers,” Ralston IVilf, ed., iViley, New York, 1962. Ralston, A., “First Course in Numerical Analysis,” McGraw-Hill, Yew York, 1965. Sasuly, hiax, “Trend Analysis in Statistics,” Brookings Institute, Washington, D. C., 1934. Simons, H. P., Ind. Eng. Chem., Anal. Ed. 13, 536 (1941). Zakin, J. L.. Simha,. R.,. Hershev, . . H. C., J . Abbl. ._ Polymer Sci. 10,1455 (1966). RECEIVED for review August 22, 1966 ACCEPTED April 14, 1967
DESIGN AND EFFECTIVENESS OF FEEDFORWARD CONTROL §YSTEMS FOR M U LTICOM PON ENT D lSTl LLATION CO LU M NS T. W. CADMAN Department of Chemical Engineering, L‘niuersity of Maryland, College Park, Md. 20740 R. R. ROTHFUS AND R. I. HERMODE1 Department of Chemical Engineering, Carnegie Institute of Technology, Pittsburgh, P a . 15213 The design and simulation of feedforward and feedback control systems for a five-tray, single-feed distillation column have been carried out for the multicomponent system benzene-toluene-xylene. For changes in feed composition, linear feedforward compensation may range from nearly perfect to detrimental, depending on col!umn operating conditions.
the past few decades, the increasing interest in higher product quality, safety of operation, and optimum economic performance has been the motivation for a revolution in the theory of process dynamics and control. Unfortunately, ITHIN
Present address, University of Kentucky, Lexington, Ky. 40506
the complexity of chemical and petroleum processing has severely hampered the parallel development of potential applications. Such is particularly the case for one of the most common of the unit operations, multicomponent distillation. I n this paper, emphasis is placed on the evaluation of the potentials and application of the linear theory of feedforward and feedback control design to multicomponent distillation. VOL. 6
NO. 3
AUGUST 1967
421
literature Review
Several excellent review articles and bibliographies by Williams (1960, 1963), Archer and Rothfus (1961), and Rosenbrock (1962c) discuss various aspects of the dynamics and control of distillation columns. Experimental studies of column models and dynamics are presented by Boyd (1957), Huckaba (1963), Parkins (1959), Wilkinson and Armstrong (1957), and Wood and Armstrong (1960). Analytical dynamic response expressions have been obtained for the case of linear equilibrium by Armstrong and Wood (1961) and Wood and Armstrong (1960). Rosenbrock (1963) has studied the effect of nonlinearities on the stability of binary distillation column dynamics. Analog and digital simulations have been carried out by Rose and Williams (1955), Jajfi et al. (1965), Peiser and Grover (1962), Rademacher and Rijnsdorp (1959), Rosenbrock (1958), Rosenbrock et al. (1961), and Williams and Harnett (1957). Peiser and Grover (1962) present one of the most complete dynamic models available in the literature. Recent advances made in the application of linearization theory by Lamb et al. (1961) to a binary distillation column have been substantiated experimentally by Baber et al. (1961) for rather large changes in operating conditions. The model has been used for feedforward control design by Anderson (1964), Luyben (1963), Luyben and Gerster (1964), and Rippin and Lamb (1960) with satisfactory results. Models for reboilers and condensers have been made available by Peiser and Grover (1962) and Sproul and Gerster (1963), along with experimental verification by Sproul and Gerster (1963) of their efficacy. Empirically designed control systems for feedback control of temperature have dominated conventional methods for composition control in distillation, as indicated by Harriott (1964). Articles by Powell (1958), Williams (1958), and Woods (1933) discuss feedback controller design from dynamic information. Trial and error methods for controller design have been tried in experimental as well as computer simulation studies by Jajfi et al. (1965), Peiser and Grover (1962), Rose and Williams (1955), and Williams and Harnett (1957). Analytical attempts at control analysis which bypass the intermediate step of determining column dynamics have been presented by Rosenbrock (196213). The use of disturbance functions has been considered as well as the problem of interaction in multivariable feedback control. Within the past few years there has been considerable interest in the feedforward control of distillation columns. Internal reflux and feed enthalpy controllers, which contain an element of feedforward control, have been developed to improve column control. The practical incorporation of feedforward control has been presented by Lupfer and Johnson (1964) and MacMullan and Shinskey (1964). Rippin and Lamb (1960) have shown how the linear model of Lamb et al. (1961) can be solved to obtain the frequency transfer functions for a binary column. Luyben and Gerster (1964) experimentally determined the effectiveness of steady-state feedforward controllers and theoretically investigated the control of the linear model by feedforward control. Anderson (1964) incorporated dynamic feedforward controllers experimentally and made further theoretical investigations of the linear model. These studies have indicated the validity and feasibility of using the linear model approach for feedforward control design for distillation. Several areas remain to be investigated. One important extension is to the case of multicomponent systems. Another is the comparison of feedback control alone with the combina422
l&EC FUNDAMENTALS
tion of feedback and feedforward control. It is of particular interest to establish the effect of nonlinearities on the dynamic performance of the systems incorporating feedforward control. Techniques Used in This Study
Dynamic Models of Multicomponent Distillation. The dynamic models used in this study are based on over-all and competent mass balances analogous to those used by Rippin and Lamb (1960). The equations obtained for the various parts of the column (reboiler, condenser, etc.) are presented in detail by Cadman (1965) along with the linearized forms which are applicable to multicomponent systems. The nonlinear balances have been used to determine the steady-state and dynamic nonlinear effects. The steady-state form of the nonlinear model was solved using the method reported by Hanson et al. (1962); the dynamic form, by the Runga-Kutta method of numerical integration. T o complete the derivation of the linear model, it is necessary to assume an equilibrium relationship between the y 3s;, and the xj,z)s. For the special case of Raoult's law employed in this study, the derivation is as indicated below.
where
K , ., a .T
=
P io (TjT)/PT
(2)
and T j T = f ( x 1 , l T , . , , , x , , . ~ ~such ) that N
N Yj,aT
a=l
=
Xj,aT a=l
=
1
(3)
Linearization and Laplace transformation of Equation 1 yield the general equilibrium relationship used in this study.
(4) where
(5) and
The relationship shown in Equation 4 accounts for linear deviations in the tray temperature with composition changes and thus may be used for temperature control studies. Similar relationships may be developed for cases requiring nonideal equilibrium and tray efficiencies. Once the column and system to be studied are specified, the linear model reduces to a set of linear algebraic equations relating column flow rates, column compositions, and the Laplace transform variable. Numerical coefficients are determined by initial steady-state flows and compositions and by a holdup relationship incorporating R,and 61'. The model may be solved in the frequency domain to yield the frequency transfer functions relating the variables to be controlled to the manipulative and load variables. An iterative technique well suited for digital computation is presented by Cadman (1965) which permits any liquid composition, vapor composition, or tray temperature to be chosen as controlled variable when the reflux and boilup rates are the
principal manipulative ,variables and the feed characteristics are the loads. Any number of trays, components, feed streams, or product steams may be haridled using this technique. For those cases in which only specified transfer functions are desired, iterative techniques can be developed to solve the set of linear equations describing column dynamics for only the desired transfer functions with a net reduction in the amount of computation required. The “stepping procedure” developed by Rippin and Lamb (1960) for binary distillation is one such example. Since the equations are linear, the transfer functions arising from the dependent variables-namely, the load and the manipulative variables--may be calculated by successively assuming that all but one are zero. This fact can be used to clarify any proposed technique for solving the equations and to facilitate programming, particularly if storage is so limited that all load and manipulative variables cannot be considered simultaneously. Feedforward Controller Design. Once the column transfer functions are obtained, either in the form of frequency response or in subsequent approximation of the frequency response by transfer functions expressed in terms of the Laplace transform variable, feedforward control design can be undertaken using techniques described by Bollinger and Lamb (1962). When x D , L and x w , x are to be perfectly controlled by manipulating Lo and V,+l in. response to changes in F, q, 21. . . . . zN, the general solution is represented in matrix form by:
(7) where F11. . . . . F I ( ~ +and ~ ) F ~ I. .. . . F2(2+,,.) are the required feedforward controller transfer functions. P11. . . . . P 1 ( 4 + N ) are the functions relating x D , L to Lo, V k + l , F, q, 2 1 . . . . . z N , respectively. P z l . , . . . P 2 ( 4 + N ) are analogous transfer functions for x ~ , ~As. in the case of the column transfer functions, calculations may be done in the frequency domain in order to obtain a Bode plot for each feedforward controller. Practical incorporation of the derived characteristics is then achieved by synthesizing hardware which will approximate the theoretical results to the desired degree. I n order to control only X D , L or x ~ at its , initial ~ value, either Lo or v k + 1 can be fixed and the other variable manipulated to control the linear model perfectly. I n the case of set point changes, x D , L and x w , H represent transformed deviations of the set points and would not be set equal to zero. The characteristics of feedforward controllers can also be calculated directly from the original set of linear equations. I n this case, the variables which are to be controlled are set to zero in the equations, and the equations are solved to yield the required relationship between the manipulative and the load variables directly. Luyben (1 963) presents an example of this technique for binary distillation. Care must be exercised, however, to ensure that a unique solution exists. This is generally guaranteed by choosing the number of manipulative variables equal to the number of controlled variables. Feedback Control Design. I n order to examine the possible potentials of feedback control, it is assumed in this study
that continuous and instantaneous analysis of the controlled compositions is available. I n addition, the dynamics of control values, etc., are neglected. Pure time delays in the product lines have been included, however. Several methods for the specification of feedback controller settings are available for the design of a single-loop system (Harriott, 1964). Although the problem of multivariable feedback control has been considered by Rosenbrock (1962a), no methods for design approaching the simplicity of those developed for the single-loop design have been presented. I n general, however, x D , L is more dependent upon Lo than on v k + 1 , and x ~ is more , ~ dependent upon V k + 1 than on Lo. This suggests that a satisfactory control system, although perhaps not optimum in design, may be obtained by manipulating Lo according to errors only in x ~ and, manipulating ~ V,+i according to errors only in x ~ , ~I n . this case the corrective fluctuations in Lo and V k + l can be looked upon as being load fluctuations influencing x W V H and x D , L , respectively, and stability of the feedback system should be ensured by using only the two controllers. This method of feedback control design is used in this study. Numerical Example
Column Chosen for Study. The five-tray column chosen for this study is shown in Figure 1. T h e ternary system chosen consists of benzene, toluene, and p-xylene, which are referred to as components 1, 2, and 3, respectively. Also presented in Figure 1 are the initial steady-state values which are assumed. Five reflux ratios between 0.2 and 10.0 were chosen for study. The initial steady-state product compositions, obtained from the steady-state balances, are presented in Table I. For the control studies, x D , l and/or xw,3 are chosen as the controlled variables, LT and/or V B as the manipulative variables, and F, q, 21, 22, and 23 as the load variables.
D TOTAL C OND ENS ER
F q
fI f2 z3
TRAY TRAY TRAY TRAY
I 2 3 4
TRAY 5
REBOILER
Figure 1 . for study
x .. w 2 w3
Schematic diagram of column chosen
P r = 1 atm. q = 1.0 ?I, i ~ i,3 = 0.5, 0.3, 0.2 mole fraction LT/D = 10, 4, 2, 1, or 0.2 $6 = 2.0 V B = 0.2 mole/(unit time) VOL. 6
NO. 3
AUGUST 1967
423
Table 1. f D ,1 fD,2 XD.3 RW,l
XW,Z ZW,3
Initial Steady-State Values of Product Compositions
10
4
2
7
0.2
0.94105 0,05827 0.0006774 0,05895 0,54173 0.39932
0.92709 0.07146 0.001449 0,07291 0.52854 0.39855
0.90608 0.09098 0.002934 0.09391 0.50902 0.39707
0.87340 0.12017 0.006432 0.12660 0.47983 0.39357
0.79256 0.17774 0.02970 0.20744 0.42226 0.37030
LT/D =
Table II.
Lr/b P11 P12 p13 p14 p15 p16
PZl Pz 2 p23 Pz4 p25 P26
10
=
24.334 -24.281 11.877 0,88403 1.0532 0.10387 - 21 ,673 21 ,676 10.851 -0.78814 0.0094556 1.9962
-
Zero-Frequency Values of Transfer Functions
4 10.407 - 10.266 4.8516 0.82711 1 ,0422 0.13991 -9.7425 9,7510 -4.8926 -0.77962 0.015180 1 ,9924
Column Transfer Functions. Using the initial steadystate composition, temperature, and flow profiles and selecting values for the PJ’s and the HJ’s, the column transfer functions were calculated from the linear dynamic model. Sample results are presented in Figure 2. In Figure 2, the magnitude and phase angle of P11 and P P Iare presented as a function of the frequency for the extreme reflux ratios studied and for specified values of the P i s and H3’s, Analogous results for P I S and P25 are given by Cadman (1963). I n Table I1 the zero-frequency values of the transfer functions are presented for the five reflux ratios examined. These results indicate that the magnitudes of the transfer functions vary monotonically with frequency and may be closely approximated by a small series of linear leads and lags. The functions show a marked dependence on reflux ratio. As the ratio increases, the corner frequency decreases and the zero-frequency values of the transfer functions for the flow rates increase. Over the reflux range studied, the zero-frequency values for the feedcomposition transfer functions exhibited considerably less change. Increasing the p3’s and H3’s resulted in a decreasing of the corner frequency. Changes in H i s were more critical than changes in p i s . I n addition, adding a time delay of up to 1 unit in the liquid flow to each tray resulted in negligible change in the transfer functions.
2
7
0.2
5.7962 - 5.5049 2.4611 0.75211 1 ,0341 0.18995 -5.7282 5.7521 -2.9000 -0.76527 0.023377 1 ,9847
3.5410 -3.0004 1.2299 0.64433 1 ,0269 0.25306 - 3.6283 3.7026 - 1.8885 -0.73490 0.037525 1 ,9655
1.9071 -0.83621 0.31102 0.39233 0.99204 0.29502 - 1 ,3453 1.8717 -0.98848 -0.59043 0.075526 1.8287
Controller Design. The eight feedforward controllers required to control x D , l and xtF,3 for the chosen column are defined in Equation 8, where 22 has been eliminated because %l L2 f 2 3 = 0.
+
L.3J Bode diagrams for Fi3, F11, F 2 3 , and F 2 4 are presented for the extreme reflux ratios examined in Figure 3. In Table I11 the zero-frequency values of the controller transfer functions are given for the reflux ratios examined. Two characteristics are particularly evident. The controllers are considerably more complex than the column transfer functions and would be more difficult to approximate with a small series of linear leads and lags, particularly near unit frequency. I n addition, as the reflux ratio increases there is a marked increase in the gains of the controllers which operate on changes in the feed composition. The four feedforward controllers required to control x D , l or by manipulating LT or V , may be calculated directly from the column transfer functions. They are less dynamically
9
0 0.001
0.01
0.1
FREQUENCY
Figure 2. 424
I&EC FUNDAMENTALS
1.0
10.0
W
Transfer functions for
X D . ~and XIFJ
100.0
Zero-Frequency Values of Feedforward Controllers
Table 111. LT/D =
70
4
2
7
0.2
Fi 1 Fi 2
4.9799 -0.021034 -19.037 -41.879 5.4786 0.015342 -19.035 -41.965
1.9992 -0.042233 -7.0672 - 14.944 2,4992 0.037757 - 7.0626 -15.135
1.000 -0,0627933 - 3.3624 - 6.6494 1.500 0.070509 - 3.3525 - 6.9667
0,5000 -0,081233 - 1.7598 - 3,0721
0.1000 -0.098426 -0.78542 - 0.85144 0.6000 0,24471 -0.60487
F13 F14
Fzi F22
F23 4
FP
complex and generally have smaller steady-state gains than the ~ xry,3 control. controllers for x D , and Assuming time delays in the product sampling line of u p to 5.0 time units feedback controller settings were obtained using the method described by Harriott (1964). Settings for the feedback control of x g , l by manipulation of LT were obtained from Equation 9. G11
= PI1 (cos wl
- j sin wl)
1,000
0.11888 - 1 ,7346 -3.5412
Table IV.
Transfer Function Characteristics for Feedback Control Design Time ;GI< G = - 1350 Identlficatton L,/D Delay -7 3 5 O
Proportional Control 0 0.085 0.5 0.259 GI < G = - 180' Three-Term Feedback Control 0.2 0.5 0.08 0.2 2.0 0.305 0.2 2.5 0.368 5 .O 0.64 0.2 0.5 0.085 0.2 2.0 0.330.2 2.5 0.405 0.2 5 .O 0.71 0 2 0.5 0.13 10.0 10.0 2.0 0.49 2.5 0.58 10.0 5.0 1.12 10.0 10.0 0.5 0.081 2.0 0,409 10.0 10.0 2.5 0.52 10.0 5 .O 1.04 0.2 0.2
GI L GI,
(9) GI L G11
A similar transfer function, ( 2 2 2 , for the control of x T v , 3 by manipulation of V , is obtained by replacing P11 with P22 in Equation 9. I n Table IV, the characteristics of G11 and G22 necessary to calculate the settings used in subsequent dynamic evaluations are presented. Effectiveness of Designed Controllers. Up to this point, primary emphasis has been placed on the derivation and solution of the linear model and on the calculation of feedforward and feedback controller characteristics from the resulting transfer functions. The practical applicability of the
0.001
Figure 3.
0.01
0.1 FRI!QUENCY
- 1 ,5890
GI1
G11
G.? __ Gn2 G,? G22
GI, GI1
G .. ,t G11 G22
G?2 G22 G22
1.0
w,
1.56 0.615 0,528 0.32 2.98 0.84 0.69 0.38 0.44 0.24 0.22 . -~ 0.15 2.73 0.70 0.57 0.30
100.0
10.0
Y
Feedforward controllers for control of
and xw.3
X D , ~
VOL 6
NO. 3 A U G U S T 1 9 6 7
425
theory to control design cannot, however, be obtained from the linear model. With regard to feedforward control design in particular, three important areas remain to be examined.
Using the system chosen for study, these aspects of linear feedforward control design have been investigated for step changes in the load variables and the feed characteristics, using the nonlinear model representing the chosen system.
Although the designed feedforward controllers will perfectly control a column represented by the linear model, the dynamics of distillation are nonlinear. The effect of these nonlinearities on the performance of the feedforward controllers must be examined. The transfer functions for the feedforward controllers may contain more dynamic information than can be incorporated in practice. Consequently, the effect of further simplifications is important. Since incomplete process knowledge and unknown loads prevent the practical achievement of perfect control, a system controlled with only feedforward control will be subject to drift. Some feedback action is necessary to reduce the drift. The effectiveness of this combination should be compared with feedback control alone.
0.941 I
Steady-State Effectiveness of Feedforward Controllers for Step Changes
By comparing the steady-state response of the uncontrolled system with that of the feedforward controlled system, an indication is obtained of the possible benefits of feedforward control. I n addition, the range of load changes for which the zero-frequency values of the controllers represents the system is established. Figure 4 summarizes the results for feed composition changes when x D , ] was controlled by manipulation of LT. The step
rn
0.927 I
ll-
;;
X
0.9409 -0.05
0
0.9270
0.9269 -0.05
0.05
0
0.05
ZI 8)
0.9063
0,8746
0.9062
0.8740 IC p'
IC ;
E T ~ 4.0 E ~
Zg.-Z1
z2.-$
X
X
0.9061
22'0
0.8735
-
0.9060 q.05
0.8730
0
0.05
ZI
CI
-c,/a.
2.0
1
L'.gZ
z2.-2
0 .?9 4
zp.
IC p' X
0
0.793
0.782 -0.05
Figure 4. 426
l&EC FUNDAMENTALS
0
0.05
Steady-state effectiveness of feedforward controllers for
xD,l
control
Although only three ratios are indicated, the others examined were intermediate. The control of X D J by manipulation of V B gave similar results, although control was slightly poorer. The control of x ~ 'by, manipulation ~ of V B or LT was noticeably better. In Figure 5 analogous results for X D , ~ ?are given for the case ~ controlled by both Lr and in which both x D , l and X W , were V,. In general, the control was significantly better at the lower reflux ratios than at the higher ratios. The compensation was frequently so large that an impossible, negative reflux rate or boilup rate was predicted. The feedforward control can, as indicated by the more favorable response of the uncontrolled column, be detrimental in some cases. The con-
changes made in composition are 21, 22, and z 3 , respectively, with z1 z2 z 3 = 0. The final steady-state value attained by x ~ is ,indicated ~ by X D , l T . If the feedforward controllers were perfect, the various solid lines would be horizontal through the point of no change in the composition. The uncontrolled cases were essentially vertical through the point and, hence, are not indicated. In general, the feedforward controllers were nearly perfect over the range of composition changes examined, with some degradation as the reflux ratio decreased. A marked improvement over the uncontrolled system was evident. I t was generally observed that the smaller the predicted compensation, the better the control. The ratio of 22/21 had a :noticeable effect on the controllability.
+ +
-
-0.06
0
---
CONTROLLED
UNCONTROLLED
0.05
0.94
01.9 2 0.92
n- a 0.90
IC. 0
X
X
0.90
0.88
0.8 8
Z2. 0
0.8 6
-0.05
0
0.05
zI
fi) Z T / ~ 4. . 0
22. 0
- 0.0 6
0
0.05
Zl
E) Figure! 5.
~ T / B 10.0 =
Steady-state effectiveness of feedforward controllers for
X D , ~ and
VOL. 6
xw,acontrol
NO, 3
AUGUST
1967
427
trolled responses show a marked dependence on the ratio of 12 = 0 and 22 = -11 were generally the extremes, while z2 = -2J2 gave the most favorable response. Other ratios gave intermediate results. The control obtained for X W J was significantly better than that obtained for xD,1 in each case examined. Typical responses are given by Cadman (1965). I n Figure 6, results are presented for changes in both the feed rate and the feed composition, with z2 = z1/2, when both x ~ and , ~xIv,3 were controlled. These suggest that better performance could be obtained if the z d z 1 ratio were maintained near -0.5 by selectively mixing product and available feed, thereby changing the column feed rate, than if no modification were made. z2 to zl. I t was observed that
Dynamic Effectiveness of Controllers
0.79!
0.790
+= X
0.781
I n order to examine the dynamic effectiveness of the feedforward controllers, two extreme approximations of the predicted dynamic characteristics were made. The first was to approximate the characteristics by only the zero-frequency value, termed steady-state feedforward control. The second was to approximate them as closely as possible; this is called dynamic feedforward control. I n Figure 7 , a comparison of proportional feedback control and the combination of proportional feedback and steady-state feedforward control is presented for the control of x D , l by manipulation of LT. The comparison illustrates that the steady-state effectiveness of the feedforward system largely determines the possible benefits obtainable by adding feedforward control to a feedback system exhibiting offset. For the case presented, the steady-state effectiveness of the feedforward system is satisfactory, as can be seen in Figures 4 and 7. The benefits of feedforward control addition are most marked for the larger measurement delay, where the smaller permissible feedback proportionality constant yields a larger offset when only feedback control is used. I n Figure 8, the control of x D , l by manipulation of L T using steady-state feedforward and three-term feedback control a t L T / D = 0.2 is given. I n this case the performance of feedback control is severely hampered by the presence of time delays, and steady-state feedforward control can be used to improve control markedly if the addition of only feedforward control is an improvement over the uncontrolled response. For rapid feedback systems, however, steady-state feedforward
! I 0
5
2,.-0.05
ZfO
IO
IS
I PO
TIME
Figure 7. Dynamic cornparison of steady-state feedforward and proportional feedback control of xD.1
, --THREE
TERM F.B.
-THREE
TERM F.B.
0.800
0.95
0
5
IO
I5
20
TIME
F*JT
*.osv
Figure 8. Dynamic comparison of steady-state feedforward and three-term feedback control of X D J
*.05r
lkZO.94
0.795 ' 0
*-.05F
*-.I7 -0.05
0
0.05
-
0.05
21
Figure 6. Steady-state effectiveness 7 , ~ controllers for x D , ~and ~ ~control 428
*;IT
0,790
0.93
I&EC FUNDAMENTALS
0
0.05
*I
of
feedforward
control may be marginal or detrimental. The latter arises if the short-time response obtained using only feedforward control does not represent an improvement over the uncontrolled case, The possible benefits of steady-state feedforward control addition are qualitatively related to the steady-state effectiveness for other feed composition changes and reflux ratios examined. Analogous results were calculated for the control of both x D , I and xIv , 3 . The response obtained using feedforward control supported the generalizations which were made for the control of only x D , l . The damped cyclic response observed
~
~~~~
Table V.
Dynamic Feedforward Controller Characteristics
Control of
xD,l
at
(6.98 s
+ ( 0 . 4 0 3 s + 1 ) ( 1 . 6 5 s + 1 ) ( 1 . 6 4 s + 1 ) ( 4 . 3 6 s + 1) ( 0 . 7 1 5 ~ + 1 ) ( 2 . 2 4 s + 1)(3.31 s + 1)(0.159s + 1 ) +
pis/pii
Control Fl3
(m +
(4.27 s 1)(0.431 s
F24
-+
1)(0.921 s
F23
-+
and
(0.640s
+ 1)(4.93 s + 1)
+ 1)(0.872s+ w . 1 1 + w . 1 2 + 1 ) s
(6.98 s
s
+ 1)(2.77s + 1 )
( 0 . 7 2 2 s + 1 ) ( 0 . 7 2 3 s f 1 ) ( 2 . 3 4 s 4- 1 ) ( 6 . 5 0 s
xw.3
+ 1)
at i r / b = 0.2
+ 1 ) ( 0 . 3 2 s + 1)
+ 1)(0.404 s + 1)(0.231 s + 1 ) +
+
(3.03 s 1)(1.03 s 1) + 1)(0.291 s 1)(0.280 s
[&I2 +
+
1)(0.124s
ofxD,l
0.2
t T / b =
(0.403s+ 1)(1.65s+ 1)(1.64s+1)(4.36s+l) p 1 6 / p 1 1 ( 0 . 7 1 5 r + 1)(2.24s 1)(3.31 s 1)(0.159s 4- 1)
+
2 s (0.024) (2.24) + 1 1)(1.97s 1)(0.100s
+
+ 1)(0.140 s + I ) 2 s (2.41)
+ 1)
L, lE.0.2
+ 1)(0.11s + 1) ( I .19 s -I-1) ( O . 109 s + 1 )
I n Figure 10, the control of x D . 1 and x w , 3 at Lr/D = 0.2 is presented. As in the control of only x D , ~ , the addition of dynamic feedforward characteristics is beneficial and can be used to improve the performance of the feedback system markedly for large as well as small time delays. The improvement depends markedly on an improvement over the uncontrolled response within the settling time of the feedback system when only feedforward control is used. Analogous results are presented for t , / D = 10.0 in Figure 11. Although dynamic feedforward control is better than steady-state feedforward control, the slow response of the controlled variables (predictable from the column transfer functions) enables good control to be obtained with only feedback control. Consequently, little, if any, improvement in control upon incorporating feedforward control is obtained for the time delays studied.
in the former case is, however, replaced by a less regular response. This is attributed to the interaction which occurred when both LT and V , were manipulated. Table V summarizes the dynamic characteristics for z T / B = 0.2 obtained by approximating the phase angles of the feedforward controllers. Others are presented by Cadman (1965). Those for the control of x D , l by manipulating LT were approximated to less than O.iio: those for the control of x D . 1 and x w , 3 , to about 5" with most of the error being near unit frequency. I n Figures 9 to 11, sarmple results obtained by incorporating dynamic feedforward control are presented. (Note changes in ordinate scale when comparing Figures 4 through 11.) For the control of x D , l only (Figure 9), the addition of dynamic feedforward control characteristics significantly reduces the relatively small errors which existed when only steady-state feedforward control was used. I n addition, the use of dynamic feedforward control remits only in a n improvement over the uncontrolled response throughout the time range examined. A comparison of the results for z2 = 0 to 22 = 0.05 indicates the same qualitative trends as the steady-state effectiveness of the feedforward controllers.
-
(1.66 s
$1
Conclusions
A linear, dynamic model of multicomponent distillation has been developed. Application of the linear model to the design of feedforward and feedback control systems for multicomponent distillation has been examined.
-THREE TERM F.B. a DYNAMIC F.F.
z ,*-o.os
0.794
0
6
IO TIME
I6
eo
0
6
IO
13
90
TIME
Figure 9. Dynamic comparison of dynamic feedforward and three-term feedback control of x D . 1 VOL. 6
NO. 3
AUGUST 1967
429
-THREE
22 * 0.025
TERM F.B.
RF.
& DYNAMIC
0.374
21.-0.05
I
I
I
0.372
0
5
IO
I5
PO
-
0
1
I
1
5
10
15
TIME
20
TIME
Figure 10. Dynamic comparison of dynamic feedforward and three-term feedback control of X D , ~and X I V , ~ -THREE TERM F.B. e DYNAMIC F.F.
0
5
IO
IS
20
TIYE
0
Z2.0 Z1=-0.05
8
IO
15
20
TIYE.
Figure 1 1. Dynamic comparison of dynamic feedforward and three-term feedback control of X D , I and X W J
The nonlinear dynamic effectiveness of the predicted controller characteristics has been examined for a five-tray ternary column. Results for the control of one and two product compositions have indicated: The control of only one product composition is more easily achieved than the control of two product compositions. I n the latter case, control interaction is evident in the feedback systems and the feedforward controller gains may be large. For the column chosen for study the steady-state effectiveness of feedforward controllers is markedly better: if one composition is controlled than if two compositions are controlled, a t higher reflux ratios if one composition is controlled, and at lower reflux ratios if two compositions are controlled. In addition a marked dependence upon the distribution of the feed composition changes was noted. Maximum improvement was generally obtained when the light and the heavy components in the feed were changed in the ratio of 2 to - 1. The dynamic effectiveness of the designed feedforward controllers and the benefits to be derived from feedforward control addition to the feedback systems exhibit the same qualitative 430
I&EC FUNDAMENTALS
trends as the steady-state effectiveness of the feedforward controllers. The benefits of feedforward control addition to feedback systems with large time delays or an offset are largely dependent upon the steady-state effectiveness of the feedforward system. Dynamic compensation is still beneficial, however. Dynamic feedforward compensation is generally required in order to improve the control performance of a feedback system having small time delays. Dynamic characteristics may be specified by approximation of the frequency transfer functions predicted from the linear model. Generally, the predicted results contain too much detailed information to warrant inclusion of all of it. I n approximating, primary emphasis should be placed on those dynamic characteristics which are most important during the settling time of the feedback system. Acknowledgment
The computer time for this study was made possible through the facilities of the Computer Science Centers at the University of Maryland and Carnegie Institute of Technology.
Nomenclature = denotes functional dependence = number of trays in column or bottom tray = = = =
= = = = = = = = = = = = = =
= = = = =
time delay in composition flow quality of ferd time composition of component i in liquid from tray j composition of component i in vapor from tray j composition of component i in feed distillate flow rate feed flow rate feedforward icontroller transfer functions; defined in Equation ’7 transfer funiztion of open-loop feedback system; defined in Equation 9 phase angle of G,, liquid hold-u p on tray j equilibrium constant for component i on tray j first derivative of K j , %with respect to T , liquid flow rate from tray j liquid flow in enriching section vapor pressure of component i column pressure column transfer functions; defined in Equation 7 temperature on tray j vapor flow rate from tray j vapor flow rate in stripping section bottoms flow rate
GREEKSYMBOLS linearized relationship of holdup on tray j to liquid flow across tray j = sum over index a
pi’
=
0
= frequency, radiandunit time = ultimate frequency
w
wu
SUBSCRIPTS.Special C!omponents
H L
= =
A’
=
1, 2, 3 = Special
D
=
k W
= =
heaviest component lightest component number of components in system benzene, toluene, p-xylene, respectively, in column chosen for study Location distillate pro’duct bottom tray bottom product
SUPERSCRIPTS
T
-
total instantaneous value; used for dynamic conditions = total final steady-state value = overscore, denotes initial steady-state value ; no overscore ‘or other superscript denotes deviation from initial steady-state value =
literature Cited
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