Ind. Eng. Chern. Res. 1988,27, 1450-1460
1450
Design of Robust Noninteracting Controllers for High-Purity Binary Distillation ColuDlns Triantafillos J. Mountziaris* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544
Apostolos Georgiou t Process Modeling and Control Research Center, Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015
The high nonlinearity of high-purity distillation columns makes the derivation of a low-order linear dynamic model by no means a trivial task. However, various steady-state control design criteria, estimated by using the material balances and design equations of the column, can be employed to guide the synthesis of multivariable control structures with desirable steady-state properties. On the basis of such criteria, a general strategy for synthesis of robust noninteracting control schemes for dual-composition control of high-purity binary distillation columns is presented. The derived controller is a nonlinear function of the distillate and bottom purities and it meets the requirements of robustness and integral controllability from a steady-state point of view. The required process information for the proposed scheme is the relative volatility, product and feed specifications, and reflux ratio. The performance of the nonlinear control scheme has been tested in rigorous nonlinear dynamic simulation of two high-purity binary distillation columns and found to be better than that of conventional controllers. More effective control was obtained by a temperature/composition cascade-type controller derived from the proposed nonlinear scheme. This control configuration significantly improved the performance of the system and was able to handle higher disturbances of the feed concentration (up to ±40%). 1. Introduction High-purity distillation columns are common multivariable operations in the process industry with unique characteristics, such as complex dynamics, high nonlinearity, and interaction between the control loops. More particularly, the nonlinearity of the column generally increases as the purity of the product increases. Tsogas and McAvoy (1982) divided distillation processes into three categories based on product purity specifications. Lowpurity columns usually exhibit linear behavior; mediumpurity columns have gains that change nonlinearly, but their time constants and dead times remain fairly constant; finally, in high-purity columns the gains as well as the time constants and dead times change nonlinearly. Dual-composition control of high-purity columns is being considered as one of the most challenging problems in process control, since, besides the situation where both products are valuable, it also minimizes the energy requirements of the column (Luyben, 1975). The research work on dual-composition control has been reviewed by Edgar and Schwanke (1977), Tolliver and Waggoner (1980), Waller (1981), and McAvoy and Wang (1984). The most popular method for dual-composition control implements parallel single-input single-output (8180) loops to control the compositions of both products. Bristol's (1966) relative gain array has been used extensively in the pairing selection of manipulated and controlled variables for distillation columns, and its applications to dual-composition control have been discussed by McAvoy (1983). A noteworthy paper on the design of multiple 81S0 controllers has been recently published by
*Author to whom correspondence should be addressed. Present address: Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455. t Present address: Department of Chemical Engineering, Princeton University, Princeton, NJ 08544. 0888-5885/88/2627-1450$01.50/0
Yu and Luyben (1986). This paper presents a systematic approach for the design of conventional control structures for multivariable systems. The vast majority of the existing studies, however, has been concentrated on low or moderate product purities, and only a few researchers have included high-purity columns in their analyses, despite their industrial importance. A double differential control scheme to maintain overhead purities of about 10 ppm in a benzene/toluene separation was proposed by Boyd (1975), while Tyreus and Luyben (1976) reported the highly nonlinear behavior of a methanol/water column with product purities in the range of 1000 ppm. Fuentes and Luyben (1983) studied the effects of product purity, relative volatility, composition analyzer sampling time, and magnitude of disturbance on the performance of highpurity distillation columns and observed a highly nonlinear dynamic response. They reported that for high-purity columns not only the process gains change nonlinearly but also the time constants and dead times. Most of the attempts toward the design of multivariable controllers for distillation columns involved the use of one-way or two-way decouplers or variable transformations between column flow rates and combinations of controlled variables. However, very few applications of such controllers on high-purity columns exist in the literature, and only recently some applications of more advanced control techniques (dynamic compensation, dynamic matrix control, etc.) on such columns have been reported (McDonald and McAvoy, 1985; Georgiou et al., 1987). One of the most serious problems associated with these multivariable techniques is that their performance depends on the quality of a reduced linear model of the process, which is not always achievable for high-purity columns. In a recent work, Maurath et ale (1985) applied multivariant predictive control to a two-point composition control of a moderate purity distillation column and reported that the performance depends strongly on the accuracy of the model. They also showed that the performance of the predictive con© 1988 American Chemical Society
Ind. Eng. Chern. Res., Vol. 27, No.8, 1988 1451 troller is as good as that of a dual-loop PID control system with one-way decoupling and better than that of a standard dual-loop PID control system. The objective of this paper is twofold: (1) to address the controller design problem for dual composition control of high-purity distillation columns and to present a systematic procedure for synthesis of multivariable control structures based on steady-state control design tools; (2) to show that the resulting nonlinear controllers can also be implemented in a nonlinear temperature/composition cascade configuration, which can handle larger disturbances and can significantly improve the overall performance of the control system. It has been pointed out by many researchers (e.g., Rosenbrock (1962), McAvoy (1983), Shinskey (1984» that, in general, interaction between the control loops leads to performance deterioration and instability. A systematic control design procedure, based on minimization of the steady-state interaction between the control loops, is discussed first. Since this approach leads to more than one control structures, other control design criteria, such as the minimum singular value and integral controllability indexes, are used in the selection of the best possible control configuration. Then, the dynamic performance of the derived nonlinear control scheme is tested in rigorous nonlinear dynamic simulation experiments of two highpurity binary columns and the performance of the nonlinear controller is compared to that of conventional control schemes. It is well-known that the controller performance becomes worse as the composition analyzer dead times increase. On the other hand, temperature sensors have faster dynamic behavior and can be used as part of a temperature/composition cascade control system to improve the performance of the controller (Fuentes and Luyben, 1983; Chiang, 1985). The transformation of the derived nonlinear multivariable control structure into a temperature/composition cascade controller is finally presented. The performance of this nonlinear cascade control system in dynamic simulation experiments has been found to be superior to that obtained by direct composition control. 2. Controller Design Procedure 2.1. Theoretical Analysis. In the derivation of the new control scheme, we start with the familiar material balance control configurations (xn-D / F,XB- V / F) and xn-L/F,XB-B/F). If we successively substitute one controlled variable, Xn or XB, in each of the above schemes with an unknown function ~(Xn,XB)' we can derive four new control schemes: (I) (xn-D/F,~-V/F), (II) (~-D/F,XBV/F), (III) (xn-L/F,~-B/F), and (IV) (~-L/F,XB-B/F). The objective is to find proper functions of ~, which minimize steady-state interactions between the two control loops of the column and, at the same time, allow the control system to handle effectively model/plant mismatches, changes in operating conditions, and input uncertainties. The matrix-vector representation of the relationship between the vector of controlled variables c and that of manipulated variables m can be written as follows: (1)
c= Gm
where G is the gain matrix whose elements are defined by gij
= [iJciliJmj]mk
with i, j, and k = 1 or 2 and k
~
j.
(2)
The Bristol number or interaction measure (Bristol, 1966) is defined as 1
A=---1 _ gl~21 gllg22
(3)
Starting with the control scheme (xn-D/F,~-V/F), the process gains related to the unknown controlled variable ~ can be written as g2j = [o~/omj]mk = [iJ~1 iJxn]xB[oxnl iJmj]mk
+
[iJ~1 OXB]XD[iJXBIiJmj]mk (4)
with j, k = 1 or 2 and k ~ j. Substituting (4) into (3) and after some algebraic manipulations, we find that the interaction measure, AJ, of the new scheme is related to the interaction measure, Ab of the parent material balance control scheme (xn-D / F,XB-V / F) through the following equation: AI = (1
+ K)AI
(5)
where [iJ~1oXn]xB[oxnliJxB]D/F
K=--------[o~/OXB]XD
(6)
Following the same procedure, the interaction measure of the scheme (~-D / F,XB- V / F) can be written as All = [1 + (I/K)]AI - (11K) (7) while for the schemes
(xn-L/F,~-B/F)
and
(~-L/F,XB
B/F), which are derived from the other material balance
scheme, the corresponding interaction measures are AlII = (1 + K)A2 - K (8) and AIV = [1
+ (IIK)]A2
(9)
Here, A2 is the interaction measure of the parent material balance scheme (xn-L/F,XB-B/F). Jafarey et a1. (1979) presented accurate analytical expressions for the interaction measure of the material balance control schemes (AbA2) for dual-composition control of binary distillation columns. They also proved that the sum (AI + A2) is always close to unity for high-purity columns. As a result, the sums (AI + AlII) and (All + AIV) are also close to unity for high-purity columns. Fagervik et ale (1981) have shown that, for dual-composition control, one-way decoupling always results in better (or at least equal) performance than two-way decoupling. One-way decoupling of the two control loops of a distillation column cannot be achieved under all possible operating conditions by using the material balance control schemes (i.e., in general, Al ~ 1 and A2 ~ 1). However, for the four new control structures, one-way steady-state decoupling can be achieved under all operating conditions, since it can be shown that for each case there is at least one ~ such that [iJ~/iJmj]mk = 0
j ~ k
(10)
If we require AI to be always equal to unity, in order to assure at least one-way decoupling (Le., [a~/aD]V/F = 0), then AIV will also become close to unity, since [o~/iJD]VIF ~ -[iJ~/iJB]L/F for high-purity columns, and from the summation properties we find that both All and AlII will be close to zero. On the other hand, if All is unity, then AlII will also be unity, and for high-purity columns both AI and AIV will be close to zero. Case A. Control Schemes II (~D / F ,xB- V / F) and III (xn-L / F,~-B / F). From eq 7 and 8, we find that All
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Ind. Eng. Chern. Res., Vol. 27, No.8, 1988
and AlII become unity when K = -1. If we eliminate the bottom flow rate B between the equations of the overall and the component material balances of the column, we get (11)
Two simple functions, which satisfy eq 13, are ~I=
(21)
[ ::: ]DIF = -::-=-:-: = -1, from eq 6 and 12 we get [a~ I aXn]XB XF - XB [a~ I aXB]XD
Xn - XF
(12)
(13)
which is a condition to be fulfilled by the unknown controlled variable, ~, in order to minimize steady-state interactions between the two control loops of any binary distillation column (Le., to give "A = 1), if either control scheme II or III is employed. Case B. Control Schemes I (xn-D / F ,~- V / F) and IV (~-L / F,XB-B / F). Georgakis and Papadourakis (1982) have shown that for high-purity columns Al can be accurately approximated by the following formula: "A =_C_ 1 1+C
(14)
where (15) and 1- xn w=-xB
Here w is the separation symmetry factor, () is the ratio of the actual reflux ratio to the minimum one, and a is the average relative volatility of the binary mixture in the column. If we require AI to be unity, from eq 5 and 14 we obtain the following condition: K
= 1/C
(17)
For high-purity columns, we can approximate eq 12 by [axnl aXB]D/F
= -(1 -
XF) I XF
(18)
Then, from eq 6 and 15-18, we obtain [a~/axn]XB [a~/aXB]XD
(20)
and
From the above formula we have
Then, if K
Xn - XF]n -[ XF - XB
(19)
where bi and b2 are defined in (16). This is another condition to be fulfilled by the unknown controlled variable, ~, in order to minimize steady-state interactions between the two control loops of any highpurity binary distillation column, if either control scheme I or IV is employed. 2.2. Possible Noninteracting Control Structures. The possible noninteracting control structures should satisfy either eq 13 or 19. The solutions of these equations with respect to ~ are not unique. However, simple functional forms of ~, which satisfy the above equations, can always be derived.
where n is an arbitrary exponent, whose value has little effect on the steady-state performance criteria and it is taken to be equal to one. An inspection of eq 19 reveals that the unknown function, ~, cannot be a linear combination of the variables Xn and XB. Two simple nonlinear functions, which satisfy eq 19, are and (23) There are also other more complicated functional forms of ~, which satisfy eq 13 or 19, but those presented here give the most practical controllers. The candidate noninteracting control structures are summarized as follows: (~l-D / F,XB- V / F), (~2-D/ F,XBV/ F), (xn-D / F'~3- V/ F), (xn-D / F'~4- V / F), (xn-L / F'~I
B/ F), (xn-L/ F'~2-B/F), (~3-L/F,XB-B/F), (~4-LIF,XBB/F).
Since there is more than one control structure, which has a Bristol number equal to one, additional steady-state control design criteria are used in the selection of the best possible configuration. 2.3. Steady-State Control Design Criteria. Before worrying about process dynamics, it is necessary to make sure that the system meets some steady-state control design criteria. A steady-state model for binary distillation columns, which includes the overall and component material balances and Underwood's column design equations (King, 1971), has been used for computing the process gains. This approximate model provides a simple alternative to a numerical stage-to-stage solution of the energy and mass balance equations. The estimated steady-state process gains are subsequently used for evaluating the following design criteria: relative gain array (RGA), static resiliency index (SRI), Niederlinski index (NI), Morari integral controllability indices (MIC). The RGA gives an indication of the interaction between the various control loops of a process. It is a square matrix whose elements are the different interaction measures resulting from all possible pairings between the manipulated and controlled variables. For a (2 X 2) system, its diagonal elements are both equal to A and its off-diagonal elements are both equal to (1 - "A). The desirable form of the RGA is the identity matrix (Le., A = 1), which guaranties at least one-way decoupling. For a given transfer function matrix, G, the magnitude of the minimum singular value O"min[G(iw)], called Morari resiliency index (MRI), is a measure of the inherent ability to the process (control structure) to handle disturbances, model-plant mismatches, and changes in operating conditions (Grosdidier et aI., 1985). The larger the value of the MRI, the more resilient the control structure is (Yu and Luyben, 1986). Since we are interested in the steady-state properties of the control structures, we examine the MRI at steady state (w = 0) and we call it static resiliency index (SRI). The Niederlinski index (NI) is defined as [det (G) / (gllg22)], while the Morari integral controllability
Ind. Eng. Chern. Res., Vol. 27, No.8, 1988 1453 Table I. Steady-State Design Specifications of the Methanol/Water Distillation Columns Studied column column design parameters 1 2 feed composition, mole fraction distillate composition, mole fraction bottoms composition, mole fraction feed rate, mol/min feed temp, °C distillate rate, mol/min bottoms rate, moll min operating pressure, mmHg relative volatility no. of trays feed tray location tray efficiency column diameter, m reflux ratio reboiler heat duty, 103 kcal/min reboiler temp, °C reflux drum temp, °C
0.3 0.999 0.001 45000 57 13482 31518 760 2.45-7.58 38 10 0.75 3.2 1.19 275.6 99.5 64.7
0.5 0.999 0.001 45000 57 22500 22500 760 2.45-7.58 42
D
9
0.75 3.2 1.03 405.5 99.5 64.7
Table II. Steady-State Control Design Indexes control scheme A SRI NI MIC 1.00 1.00 1.00 1.00 1.04 0.99 1.04 0.99 33.2 0.26 0.75
Column 1 0.008 0.019 0.008 0.019 0.39 0.79 1.9 X 10-5 2.0 X 10-5 0.014 0.02 0.02
1.00 1.00 1.00 1.00 0.96 1.01 0.96 1.01 0.03 3.83 1.34
1.00 1.00 (xn-LI F'~l-B/F) 1.00 (~2-D/F,XB-V/F) 1.00 (xn-L / F'~2-B / F) 1.02 (xo-D/F'~3-V/F) 1.00 (~3-LIF,XB-B/F) 1.02 (xn-D / F'~4- V/ F) 1.00 (~4-L/ F,XB-B/ F) (xn-L / F,XB-V/ F) 32.5 0.31 (xn-D/F,XB-V/F) 0.70 (xn-LI F,XB-B/ F)
Column 2 0.012 0.013 0.012 0.013 0.49 1.33 3.2 X 10-6 3.2 X 10-6 0.012 0.017 0.017
1.00 4.00,0.013 1.00 4.00,0.013 1.00 4.00, 0.013 1.00 4.00, 0.013 0.99 ·30.5, 0.62 1.00 30.6, 1.40 0.99 0.61,3.2 X 10-6 1.00 1.40, 3.2 X 10-6 0.03 1.00 ± 0.82i 3.27 0.58, 0.045 1.43 1.39, 0.019
(~l-DI F,XB-VIF) (xn-LI F'~l-BIF) (~2-DIF,XB- VIF)
(xn-LI F'~2-B/F) (xn-D / F'~3- V/ F) (~3-L/F,XB-B/F) (xn-D / F'~4- V/ F) (~4-L / F,XB-B/ F) (xn-L / F,XB- V/ F) (xo-D IF,XB- V/ F) (xo-L IF,XB-B / F) (~l-D/F,XB-VIF)
11.14, 0.008 11.14, 0.019 4.77,0.008 4.77,0.019 30.01,0.91 30.05, 1.07 0.87, 1.9 X 10-5 1.06, 2.0 X 10-5 0.97 ± 0.94i 0.84,0.033 1.05,0.026
indices (MIC) are the eigenvalues of the steady-state gain matrix with its diagonal elements replaced by their absolute values. If all the SISO controllers contain integral action and have positive loop gains, negative values of the NI or values of the MIC in the open left-half complex plane will produce an unstable closed-loop system. If the value of the interaction measure is positive, it can be proved that both the NI and the real part of the MIC will also be positive. Therefore, the RGA and the SRI are the only independent indices which can be used to evaluate the steady-state properties of the proposed control schemes. 2.4. Systems Studied. The controller design problem for two high-purity methanol/water distillation columns is studied in this work. In both cases the product purities are assumed to be 99.90/0, andthe only difference between the two columns is in the feed composition, which is low (XF = 0.3) for column 1 and moderate (XF = 0.5) for column 2. The steady-state design of these columns has been studied by Chiang (1985), and a summary of their specifications is listed in Table I. 2.5. Selection of the Control Structure. The previously discussed steady-state control design criteria have been evaluated for each of the eight nonlinear and the
Figure 1. Schematic representation of the proposed nonlinear control scheme (NLS: xn-D/F'~3-V/F).
three traditional control schemes [i.e., (xn-D / F,XB-V j F),(xn-LjF,xB-B/F), and (xn-L/F,XB-VjF)], when applied to columns 1 and 2. The results are listed in Table II. Although it has been proved that, from the four criteria, only the RGA and the SRI are independent, the results for the NI and the MIC have also been included for reference purposes. As expected, all nonlinear schemes have an interaction measure close to one. The values of the SRI for the traditional structures are almost the same, while the SRI of the control schemes (xn-DjF'~3-VjF) and (~3-LjF,XB B j F) is at least 1 order of magnitude larger than that of the other schemes, which makes them the most attractive. Another observation is that the nonlinear schemes related to ~4' although they have an interaction measure close to one, have SRI close to zero, implying potential difficulties when implementing feedback control. This analysis also leads to the conclusion that the most attractive controllers are likely to belong to the group of nonlinear schemes using an internal flow (L or V) to control the value of the nonlinear function~. Finally, since the interaction measure of all schemes is positive, the corresponding NI values are also positive and the MIC values lie in the open right-half complex plane. Thus, all 11 schemes will produce stable closed-loop systems, if both SISO controllers contain integral action and have positive loop gains, but the ~3- based nonlinear controllers will be the most resilient at conditions close to steady state. The two most attractive nonlinear control schemes have comparable indexes and are expected to give comparable performance. In this paper we concentrate on the control structure (xn-D j F'~3- V j F). A schematic representation of the proposed nonlinear structure (hereafter called NLS) is shown in Figure 1. In this structure the controlled variables are the nonlinear function ~3(Xn,XB) and the top concentration Xn, while the manipulated variables are the vapor boilup V and the distillate flow D, respectively. The
1454
Ind. Eng. Chern. Res., Vol. 27, No.8, 1988
0.05
,...--,.----,----,----,r------,r--,-----.---,.-----....... 14
(a)
50/0 0.04 12
..,.
0.03
~ X
10 ........-\-----------===--~
--- ---
\
\
/ \
,
'\
0.02
8
0.01
---
I
--~ (D,V)
I ",/
\_,
6
o
100
200
o. o
100
50
150
250
200
300
400
Time (min)
Time (min)
60
Figure 2. Open-loop step response of column 2. Effect of different step changes in reflux flow on bottoms concentration.
(b)
50
Table III. Steady-State Interaction Data for Several Column Design Specifications Using the Control Scheme (xD-DIF'~3-V IF). In All Cases, It Is q = 1.0 and 0 = 1.2 error %a X(exact) a W XO-XB XF
_
..,.
40
o
2
30
itt,-
1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5
0.1 0.1 0.1 1.0 1.0 1.0 10.0 10.0 10.0 0.1 0.1 0.1 1.0 1.0 1.0 10.0 10.0 10.0
aError %
0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75 0.25 0.50 0.75
0.9999-0.0010 0.9999-0.0010 0.9999-0.0010 0.9999-0.0001 0.9999-0.0001 0.9999-0.0001 0.9990-0.0001 0.9990-0.0001 0.9990-0.0001 0.9999-0.0010 0.9999-0.0010 0.9999-0.0010 0.9999-0.0001 0.9999-0.0001 0.9999-0.0001 0.9990-0.0001 0.9990-0.0001 0.9990-0.0001
= ([X(exact) -
1.047 1.023 1.010 1.003 1.000 0.998 0.999 0.998 0.997 1.072 1.051 1.037 1.005 1.001 0.996 0.999 0.995 0.990
4.50 2.23 0.96 0.31 0.00 -0.22 -0.11 -0.23 -0.35 6.80 4.82 3.58 0.54 0.12 -0.39 -0.12 -0.49 -1.02
1.0l/X(exact)}100.
controller part consists of two PI controllers. Although this scheme is multivariable in nature, since the measurement of Xn affects both inputs, it still retains the attractive advantages of parallel SISO control loops, which are used extensively in industry. We must also point out that the controller has no feedforward action and, hence, the feed concentration, XF, in the nonlinear function ~3 retains its original steady-state value. The exact Bristol number of the NLS has been computed for a variety of high-purity column designs, and the results can be found in Table III. Notice that, as the purity of the products increases, the Bristol number comes closer to unity, and one-way decoupling is achieved in all cases. Moderate feed compositions give values of the Bristol number close to unity even for moderate and lowpurity symmetric columns (w = 1.0). 3. Nonlinear Dynamic Simulation
The performance of the proposed control scheme was tested in nonlinear dynamic simulation experiments of the two high-purity methanol/water distillation columns. The trays were numbered starting from the bottom of the column. The dynamic simulation was based on rigorous
20
...-- (L,V)
10 O'--'""""-......._..L..-......L----L_"'--....L----L---J
o
100
200
300
400
Time (min)
Figure 3. Comparison between the (L, V) and (D, V) control schemes in closed-loop dynamic simulation of column 1 for a +100/0 step change in feed concentration (0.3 ---+ 0.33). (a) Distillate concentration, (b) bottoms concentration.
tray-to-tray calculations by making the assumption of negligible pressure drop in the column, but by taking into account the effects of nonequal molal overflow and nonideal vapor-liquid equilibrium. The tray efficiency was assumed to be 7S%, the actual reflux ratio was 1.2 times the minimum one, and the tray hydraulics were taken from the Francis weir formula. It was also assumed that the reboiler was partial and the condenser was total, while the holdup time constants for the reflux drum and the base of the column were both taken to be equal to S min. Finally, the composition analyzer dead time, t d , was assumed to be 6 min and was treated as process time delay. The scaled open-loop step response (0/0 change in composition transmitter readings) of the two columns to different step changes of the various manipulated variables can be used to demonstrate their nonlinear nature. Figure 2 shows the response of the bottom composition, XB, of column 2 to O.S%, 20/0, and S% changes in the reflux flow, with constant vapor boilup. The observed differences in the time constants are of an order of magnitude or more. Such a highly nonlinear behavior makes the derivation of a nominal low-order transfer function matrix by no means a trivial task. However, steady-state gains for such columns can still be estimated from the material balances and design equations, and they can be used in the selection of the appropriate control scheme. Because of the high nonlinearity of the systems studied, the dynamic performance of the derived controller is evaluated by digital simulation of the detailed mathematical model.
Ind. Eng. Chern. Res., Vol. 27, No.8, 1988 1455 20
r
I
I
14 (a)
(a)
15
12 ~
~
~
~
x
x
---
Q
10
10
Q
~
~
"\../
~
0
DMC/
~
8
5
6 I
0 0 100
0
200
300
400
100
I
I
I
200
300
I
400
Time (min)
Time (min)
20 (b)
50
15
(b)
40 ~
S
;-
=:
x
30
=
~
=
~
10
~
20
5
10
o .............I~-..I_......L_--L_~_ ......- ~ _... 200 300 100 400 o 0
Time (min) 0
100
200
300
400
Time (min)
Figure 4. Comparison between the (L,V) and DMC control schemes (Georgiou et aI., 1988) in closed-loop dynamic simulation of column 1 for a +10% step change in feed concentration (0.3 -0.33). (a) Distillate concentration, (b) bottoms concentration.
Figure 6. Performance of the NLS in closed-loop dynamic simulation of column 1 for a -500/0 step change in XB set point (0.001 -0.0005). (a) Distillate concentration, (b) bottoms concentration.
40
24
30 ~
Q ~
x
20
..,.
20
\
~ x
\
',/(L,V)
16 l:Q
\
~
NLS \
/" 12
,,
10
,
O ....._0010-_....-_"""'-_......_~_--a._.-..A._.-..A._ 100 400 o 300 200
'" "- " 100
200
.......
Time (min)
8
0
F-------..-..-~~-_-.-..-...-=::lI_---___I
300
400
Time (min)
Figure 5. Comparison between the NLS and (L, V) control schemes in closed-loop dynamic simulation of column 1 for a +50/0 step change in feed concentration (0.3 -- 0.315). Response of bottoms concentration.
3.1. Direct Composition Control. Pulse testing was initially tried in an attempt to get transfer function models for all control configurations. This approach did not work very well, due to the high nonlinearity of the systems. An
Figure 7. Comparison between the NLS and (L,V) control schemes in closed-loop dynamic simulation of column 2 for a +5% step change in feed concentration (0.5 -. 0.525). Response of bottoms concentration.
experimental approach was used to tune all the control loops in a consistent manner (Finco and Luyben, 1987). First, the ultimate gain and frequency for each loop of the nonlinear model were found by increasing the gain of a proportional controller until a sustained oscillation occurred. Then, the Ziegler-Nichols settings for each loop were calculated for a +5% load disturbance (Luyben, 1973). Finally, a single detuning factor, f, was used for both
1456
Ind. Eng. Chern. Res., Vol. 27, No.8, 1988 10 .....- . . - - - - - . . - - -.......- . . . . - - - . - - - . - - - -....
11 /(L,V) ,- "",.--------
8 10 "?
~
6
X \ ,I ' \
(L,V)
/
..., ,r
Q
""',
~= 4
~
9
........
.................. ......
_-
(a)
---- ---
2
8 0
200
100
300
400
Time (min)
O
......_
a.-.._""'-_~_..a.__
o
....._
......_
300
200
100
......_
_'
400
10
Time (min)
(b)
Figure 8. Comparison between the NLS and (L,V) control schemes in closed-loop dynamic simulation of column 2 for a +100/0 step change in feed concentration (0.5 -+ 0.55). Response of bottoms concentration.
8 \ \ \
6
..,.
25
\
\
~ x
\ \ \
=:l
""'-
4
~
.......
..........~(L,V)
---
2
"?
~ 15 x
""'-- NLS
0 100
0
-
10
200
300
Time (min)
Figure 10. Comparison between the NLS and (L, V) control schemes in closed-loop dynamic simulation of column 2 for a -850/0 step change in XB set point (0.001 - 0.00015). (a) Distillate concentration, (b) bottoms concentration.
5 (a)
O .........a-.--'---l'--....--L.---I_~.......----......... 400 500 100 2()() 300 o
loops to increase the reset times and decrease the gains until the closed-loop response to the nonlinear model showed reasonable damping coefficients (about 0.4 or greater). The above procedure is outlined in the following:
Time (min)
25 (b)
K el
= Kelo If
TIl
= Tnof
20
K e2 15
I
--
I I
o ..\..o;;:;..a.-........_ o
100
-_-~-_"-- ......._
....
200
300
............_
400
............
500
Time (min)
Figure 9. Comparison between the NLS and (L,V) control schemes in closed-loop dynamic simulation of column 2 for a -0.1 % step change in Xn set point (0.999 - + 0.998). (a) Distillate concentration, (b) bottoms concentration.
= K e20If
TI2
= TI20f
(24)
where f is a detuning parameter and KeP and Tlio (i = 1 and 2) are the Ziegler-Nichols settings. This procedure required a considerable amount of computer time, but it gave very reliable results and fair comparisons among the alternative control schemes. The tuning constants for each column design case are listed in Table IV. Column 1. The response of the conventional control scheme (L,V) to a +100/0 step change in feed concentration is compared to that of the material balance control scheme (D, V) in Figure 3. The (L, V) structure gives slightly better performance (smaller overshoot). In Figure 4, the response of the (L, V) structure is compared to that of the dynamic matrix control (DMC) structure for a + 100/0 step change in feed concentration (Georgiou et aI., 1988). The response of the DMC structure is very underdamped, because the large load disturbance took the system away from its nominal steady state. Since in all cases examined the (L, V) structure gave the same or better performance than the material balance control schemes, we will con-
Ind. Eng. Chern. Res., Vol. 27, No.8, 1988 1457 13
-
- - •
12
.:; ~
td
= 3 min
td
= 6 min
11
(a)
11
10
Q lol!
9 (a)
8
7 0
100
200
300
400
Time (min)
- - -: t d
60
= 3 min
- - : t d =6min
SO
40
.:; ~
30
= 20
(b)
lol!
10 0
0
100
O .... 200
300
400
Time (min)
o
..I.o---a,_.......---.lI--....._..L...---&_....
100
200
300
400
Time (min)
Figure 11. Closed-loop dynamic response of column 1, with NLS control scheme, to a + 100/0 step change in feed concentration (0.3 - + 0.33) for two different values of the composition analyzer dead time. (a) Distillate concentration, (b) bottoms concentration.
Figure 13. Comparison between the NLS and NLCS control schemes in closed-loop dynamic simulation of column 1 for a + 10% step change in feed concentration (0.3 - + 0.33). (a) Distillate concentration, (b) bottoms concentration.
Figure 12. Schematic representation of the proposed temperature/composition nonlinear cascade control scheme (NLCS) for column 1.
centrate upon the more challenging comparison between (L, V) and the proposed nonlinear scheme (NLS). The responses of the NLS and (L, V) control structures to a +5% step change in feed concentration are shown in Figure 5. The nonlinear scheme has faster response and smaller overshoot than the conventional scheme. Similar results were obtainedfor a higher load disturbance (10%). The effectiveness of the decoupling is demonstrated by the servo response of the NLS for a -500/0 step change in XB set point (from 0.001 to 0.0005), which is shown in Figure 6. The oscillations observed at short times were expected, since the controller has been designed to minimize interactions between the two loops at steady state (w = 0), which is approached at long times. Column 2. The responses of both the (L, V) and NLS schemes to a +50/0 step change in feed concentration are shown in Figure 7. The performance of the NLS is again much better than that of the conventional scheme (faster response and lower overshoot). For a +100/0 step change in feed concentration, the response of the conventional structure is very slow (chronic offset), while the response of the proposed nonlinear scheme is faster and has less oscillations (Figure 8). The servo response of the NLS is compared to that of the (L, V) scheme in Figures 9 and 10 for a small (-0.1 %) step change in Xn set point (from 0.999 to 0.998) and a large (-850/0) step change in XB set point (from 0.001 to 0.00015), respectively. In both cases the NLS performed better than the conventional (L, V)
1458 Ind. Eng. Chern. Res., Vol. 27, No.8, 1988 20 ......- -.......- -.....- -.....~-......
ccs
(a)
..,.
x
(a)
---- ---
15
-::
20
15
..,. 10
~
~~~---.....;;:I-----___t
10
~
\ \
\..... ......
Q ~
5
....
" ....... --._---_..--..--. ...... --............. CCS
0
0 ......- -......- -......._ _..........._--""
o
50
100
150
50
0
200
100
150
200
Time (min)
Time (min) 20 .....- -......- -......- -......~ -......
25
(bl
(b)
20
15
,-
..,.
~
x
-
10 w.-;...,-------~===-----t
IS
..,.
~
I \
" ',,-
x ....,
--------
=
10
~
- - - - - - - - - - CCS
0 ....._ _......_ _......_ _......
o
50
100
___
150
200
Time (min)
0'----...-.--...-.--"'---..-..
Figure 14. Comparison between the (L, V)-based conventional cascade control scheme, CCS (Chiang, 1985), and the proposed NLCS cascade scheme in closed-loop dynamic simulation of column 1 for a +330/0 step change in feed concentration (0.3 -+ 0.4). (a) Distillate concentration, (b) bottoms concentration.
structure, which was very slow. Finally, it should be noted that, for high-purity columns, the disturbance rejection capability of a controller is much more important in practice than its servo response. 3.2. Temperature/Composition Cascade Control. It is well-known that time delays, which are being introduced to the control system by the composition analyzers, degrade its performance. Parts a and b of Figure 11 show the difference in the performance of the nonlinear scheme, applied to column 1, for a +100/0 step change in feed concentration and two different analyzer dead times, 6 and 3 min. Temperature sensors, due to their faster dynamic response, can be used as part of a temperature/composition cascade control scheme to improve the controllability of the system (Chiang, 1985). The slave temperature controller (PI) is used for fast correction of disturbances that might go undetected for a long time by a slow composition analyzer. The composition controller (PI) is used to bring the product compositions to their steady-state specifications. Since for binary distillation columns there is a one-to-one correspondence between temperatures and compositions, given by the thermodynamic equilibrium relation, we can rewrite the proposed nonlinear scheme as a function of temperatures. The nonlinear controlled variable, ~3' whose value is controlled by the vapor flow rate, V, can be written as ~3
= b I In (1 - T R*) - b2 In (Ts *)
(25)
where bI and b2 are defined in (16), T denotes temperature, the subscripts Rand S denote the best plates to locate
o
50
100
150
200
Time (min)
Figure 15. Comparison between the (L, V)-based conventional cascade control scheme, CCS (Chiang, 1985), and the proposed NLCS cascade scheme in closed-loop dynamic simulation of column 1 for a -330/0 step change in feed concentration (0.3 -+ 0.2). (a) Distillate concentration, (b) bottoms concentration.
temperature sensors in the rectifying and stripping section, respectively, and the superscript * denotes that the temperature is normalized by the temperature transmitter span (temperature transmitter span, 120°C; it is always
o < T*
