Znd.Eng. C h e m . Res. 1996,34,2038-2050
2038
PROCESS DESIGN AND CONTROL Dynamic Low-Order Models for Capturing Directionality in Nonideal Distillation Mats F. SBgfors and Kurt V. Waller* Process Control Laboratory, Department of Chemical Engineering, Abo Akademi, FZN-20500 Abo, Finland
The common way to obtain low-order models of distillation columns by identifying the transfer function elements independently may result in models that are inconsistent with the physical reality. The description of the process directionality may be poor, and the model is likely to contain a n excessive number of slow poles. In this paper, the directionality of nonideal (real) distillation columns is studied with the help of a number of published experimental and simulated models. Based on the results, a multivariable model structure for capturing process directionality in continuous nonideal distillation is proposed. Further, a method is given for how to avoid multiple representation of single states in the model, when flow lags are explicitly included. 1. Introduction
In a recent paper by Jacobsen and Skogestad (1994), the problem of how to obtain consistent low-order models of distillation columns was addressed. It was found that the common way to obtain low-order models of multivariable processes by identifying the transfer functions between inputs and outputs independently easily leads to multiple representation of single dominant states of the process. This is particularly true of processes that are ill-conditioned, i.e., that have a high degree of directionality. Models that are inconsistent in terms of number of slow states will yield a poor prediction of the closed loop behavior of the process, particularly under one-point control. A multivariable process showing directionality means that a vector of inputs is differently amplified according to its direction. Accurate modeling of the process directionality is of crucial importance for the design of control systems for multivariable processes (Skogestad et al., 1988;Andersen and Kummel, 1992a,b;Koung and McGregor, 1993). Seemingly accurate knowledge of the individual transfer function elements between input and output variables does not guarantee sufficient knowledge of the directionality. A model with an incorrect description of the process directionality may be quite useless for controller design (Andersen and Kiimmel, 1992bj. More effort should be put on the modeling and identification of the true multivariable characteristics of strongly coupled processes, such as process directionality. Under restrictive and idealizing assumptions it has been found that the process directionality of the continuous distillation process seems to be connected with the fundamental difference between the internal flows ( L and V) and the external flows (Dand B ) (Skogestad and Morari, 1988; Andersen et al., 1989). On the basis of these observations, Skogestad and Morari (1988) suggested a simple model structure to capture this behavior. A similar approach is used in the present study, but with the intention to get a model structure
* Author to whom correspondence should be addressed. E-mail:
[email protected].
useful also for real, nonideal distillation. The goal is to construct a model structure as simple as possible to describe directionality in distillation and as accurate as necessary for control applications. Consistency relations based on external material balances are useful for this purpose (Haggblom and Waller, 1988,1992). A number of experimental and simulation models from the literature are utilized to validate that the internal and external flow changes closely correspond to the lowest and the highest gain of the process, respectively, also for nonideal (real) separations. A rigorous 15-plate column simulator (Lehtinen, 19941 adapted t o experimental data from the pilot plant at Ab0 Akademi is used to illustrate the results. Further, a method is given for how explicitly t o include flow dynamics in the model without getting a model with an excessive number of slow poles. This was found to be particularly difficult in work by Jacobsen and Skogestad (1994).
2. Preliminaries The concept of scalar gain for single input-single output (SISO) systems can be extended t o multiple input-multiple output (MIMO)systems by considering the set of inputs and the set of outputs as a vector of inputs and as a vector of outputs, respectively. The gain of the multivariable system is operating on the length of the input vector, i.e., a “MIMO gain” could be defined as Ilylldllmlln,where m is the vector of inputs and y the vector of outputs. This gain is dependent on the direction of the vectors. The s i n g u l a r v a l u e d e c o m p o s i t i o n (SVD)is the mathematical tool normally used for this directionality analysis, For a linearized multivariable process model of the form
y(s>= G(s) m(s>
(1)
the singular value decomposition of the process gain matrix at steady state is given by (Golub and Van Loan, 1983)
G(O)= UZVT
0888-5885/95/2634-2038$09.QQIQ 0 1995 American Chemical Society
(2)
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2039 A.
m2
t
B.
ALzAV
0.004 i-
0.12
,
AL=-AV
I
o.mLz
-0.004 0
10
20
Time
= 2.55.
A.q = 3.61 . Ar3 = 0.30.10‘’
I
...._... ..”
’).
= -2.s3.10-3
Figure 1. Geometric illustration of the singular value decomposition.
where, for a 2 x 2 matrix,
U = [u 21,
X
= diag(B,g)
V = [V yl
(3)
The maximum singular value ?t can be viewed as the largest possible gain of G. This “high gain” is obtained when the input vector m is changed parallel to 0; the output vector is then parallel to ii. Similarly, a change in inputs aligned with 1 moves the outputs in the direction of y with the gain g, which is the smallest gain of G(0) D i i = G(0)V
(4a) (4b)
The singular vectors 0 and y span the input space. Any set of inputs can be expressed as a weighted sum of the input singular vectors. The matrices U and V are orthogonal and can be expressed by trigonometric functions, e.g.
V=
a coda - 90”) [cos sin a sin(a - 90”)1
(5)
where a is the angle of the anticlockwise rotation with respect to the input ml of the input direction giving the largest possible gain (Figure 1). The U matrix can then be expressed in a similar fashion cos p cos@ f 90”) U = [sin p sin@ f 90”)
1
(6)
Since the singular values (the “gains”) are positive by definition, the signs in the matrix in eq 6 are dependent on the signs in G(O1, expressed by f 9 0 ” in the equation above. A matrix is said to be ill-conditionedif there is a large difference between the maximum and minimum singular values. This is expressed by the condition number y = dg
(7)
A large condition number means that the gain of the model is strongly dependent on the direction of the input vector. However, since the singular value decomposition is dependent on the scaling choice (choice of units) of the inputs and the outputs in the model, a process is considered ill-conditioned only if it has a gain which is strongly dependent on the direction of the input vector regardless of scaling (Waller and Waller, 1995). This holds for 2 x 2 processes with large values in the relative gain array. Numerical results indicate that this result is valid for larger systems as well (Grosdidier et al., 1985).
Art = -3.93.
Time
A I S = 7 . 1 2 , IO-’ Az, = 9.55.10-2 Ara = 8.25. lo-‘ A q = 10.7, lo-’ Azl = 10.8. lo-’
EM,Az, = 0
M,Az, = 0.464
Figure 2. Response of the five-stage distillation model: common simplifying assumptions, e.g., AD = -AL + AV and constant holdup; Mi = 1(Mi= 0) on all stages; constant relative volatility a = 2; no flow dynamics. Feed on third stage F = 1, ZF = 0.5, L = 1, V = 1.5. Index i plate number. (A) Fast response and small changes in product compositions when &l4ihr, = 0 (AL= 0.1, AV = 0.101 094). (B) Slow response and large changes in product compositions for &14jhrz,* 0 (AL= -AV = 0.1).
3. Directionality in Distillation In this section, the (L,W structure in distillation is considered. This structure always has a relative gain A11 (Bristol, 1966)greater than unity (Haggblom, 19881, which means that this control structure has a gain which is always dependent on the direction of the input vector. It is easily shown that 2 x 2 processes with A11 > 1always have a misalignment of the input and output singular vectors to the single inputs and single outputs, respectively (Appendix A). Andersen and Kummel (1992b) found that modeling and identification of the directionality is particularly difficult for processes where this kind of misalignment occurs. Already in 1962, Rosenbrock noted a basic difference between the response of distillation columns to changes in internal and external flows (Rosenbrock, 1962). By using some simplifying assumptions, more recent research (Skogestad and Morari, 1988; Andersen et al., 1989) has concluded that the directionality and the dynamics of the distillation process are physically related t o the difference between external and internal flows in the following way: (1)Changes in the internal flows (L and V) such that the external flows (Dand B ) remain unchanged result in fast responses where the composition profile in the column becomes more or less steep. The product compositions ( X B andyD) are not very sensitive to these changes. Changes in internal flows only correspond to the low-gain direction. (2) Changes in the external flows (Dand B ) move the concentration profile up or down the column. The response is slow. Quite small changes may have a large effect on the product compositions. An input giving maximum change in the external flows corresponds to the high-gain direction. In Figure 2 these observations are illustrated for a strongly idealized nonlinear five-stage distillation model. As can be seen, the response is strongly dependent on the direction of the input vector. The dominating slow response appearing in Figure 2B can be approximated by a single time constant given by (Skogestad and Morari, 1987)
2040 Ind. Eng. Chem. Res., Vol. 34, No. 6,1995 z1 =
change in holdup of one component imbalance in supply of this component
This dominant time constant is extensively treated in the literature (see, e.g., Waller, 1979; Skogestad and Morari, 1987), whereas the fast behavior observed for perturbations in the internal flows has received only little attention (e.g., Skogestad and Morari, 1988; Andersen et al., 1989). The fast behavior in Figure 2A believed to be connected to the internal flows is, according to Skogestad and Morari (19881,occurring when the denominator of eq 8 is zero (AD = AB = 0 as well as A(FzF) = 0). However, the example in Figures 2 and 3 indicates that the dominating slow pole is not completely canceled when AD = AB = A(FzF) = 0 but rather when the numerator of eq 8 is zero. This can be seen in Figure 3, where the transients after an input vector forcing the numerator of eq 8 to zero reach steady state faster than the response resulting from a set of inputs that gives a zero denominator of the same equation. This result gives an intuitively more appealing explanation of the fast transient in Figure 2A: The speed of response after a disturbance is fast when the total holdup of each component remains unchanged. The components are only redistributed between the plates during the fast transient. Thus, there does not seem to be any fundamental principle connecting changes in internal flows only with fast dynamics. The speed of response is determined by the rate of change in the accumulation (holdup) of the states. However, since we in practice do not have complete state information and the misalignment between the directions forcing either the numerator or denominator to zero in eq 8 seems to be small, the difference between external and internal flows may provide a good practical estimate of the directionality in distillation. The difference in dynamic behavior depending on input direction is also in accordance with results previously reported on distillation dynamics (Waller, 1979): The difference in the speed of response depending on input direction can be interpreted in terms of the basic principle of conservation. The response is fast for changes that make both ends of the column more pure or less pure, without altering the total holdup of each component in the column. The holdup of each component in the column is rearranged by the internal flows, and the average concentration change on the stages is often small. For changes that move the whole concentration profile up or down, the total component holdup is changed. The average concentration changes are often quite large and the response is slow, since the imbalance in component holdup has to be removed by the external flows. The example in Figure 2 was modeled under quite common assumptions, such that the external flows are related to the internal flows in the following way (9) However, this assumption is often far from reality. Equation 9 is a consequence of the following assumptions, “constant (molar) flows” in the column by neglecting energy balance equations, non-subcooled reflux and feed, no column-environment heat exchange, etc. Note,
0
10 Time
0.0
20
I
I
I
0
10
20
Time
Figure 3. Step responses of the five-stage distillation model of Figure 2. The responses are normalized with respect to the final values (t -1. Full line: AL = 0.1, AV = 0.101 094 giving W , k , = 0 (cf. Figure 2A). Dashed line: AL = AV = 0.1 resulting in AD =hB=o.
-
however, that eq 9 does not have to imply an assumption of perfect level control in the condenser and reboiler, merely the assumption that the level controllers can stabilize the levels since eq 9 is concerned with steadystate values only. In general, eq 9 is not valid for real columns and its usefulness is restricted to simple models of the kind illustrated in Figure 2. For a more thourough discussion, see Haggblom and Waller (1992) pp 195-196. For the general case, the relation replacing eq 9 can be written
(10) The gains in eq 10, which can suitably be called “flow gains”, can be either measured or calculated by the use of consistency relations based on external material balances (Haggblom and Waller, 1988, 1992)
In real columns the flow gains may be very different from the values in eq 9. Values of KDLranging from -0.25 to -0.7 and KDVranging from 0.49 to 1.35 for experimental columns are reported by Haggblom and Waller (1989). In the literature we have not found any experimental column models where KDLand KDVwould be even close to -1 and f l , respectively. The subject has been treated more extensively elsewhere (Haggblom and Waller, 1988, 1989, 1992; Haggblom, 1991). In previous investigations (Skogestad and Morari, 1988; Andersen et al., 1989) it has been found that the smallest possible process gain in distillation is obtained when both reflux and boilup are equally much increased (or decreased) ([ALAUTII[1 UT), which would not
Ind. Eng. Chem. Res., Vol. 34,No. 6,1995 2041 perturb the external flows at all if the assumptions of eq 9 were valid. Similarly, the highest process gain was found to be obtained for an increase in one input and a decrease in the other input of the same size ([ALAUTII [l -1IT), which would give a maximal change in the external flows. The example in Figure 2 is in quite close agreement with these results. The input vector in Figure 2A is almost aligned with the low-gain direction, and in Figure 2B the inputs are in close alignment with the high-gain direction. The simple model has a (linearized) gain matrix at the operating condition studied given by
["I
- [0.372 -0.340][A.L] (12) k, 0.523 -0.557 AV This gain matrix can be decomposed by SVD into -0.8351 U = [0.550 0.835 0.550
= p.915 0
1
0 0.0321 = [0.701 -0.7131 (13) -0.713 -0.701 Thus, it is seen from the V matrix that the extreme gains of eq 12 are obtained for perturbations of almost equal size in both inputs. However, to see whether they are of any importance in practical applications, these observations should be checked when KDLand KDVin eq 10 are significantly different from -1 and 1, respectively. Equation 10 shows that input vectors (changes in L and V) parallel with the vector [KDV-KDLP give AL)= AB = 0, i.e., no changes in the external flows. For input vectors of a given length, the maximum change in the external flows is obtained for input vectors parallel with the vector [KDLK D v I ~this , vector being orthogonal to the previously mentioned vector [KDV-KmIT. The (steady-state) low-gain and high-gain directions are obtained by singular value decomposition of the process gain matrix, which for the (L,V)structure can be written
Table 1. Four Models Identified from Experimental Separations. The Directions of the Input Singular Are Vector t and the Vector of Flow Gains -[KDL Compared by Their Angle to the Single Input Vector [AL 01T
1
-0.6926 1.3332
-62.5
-63.6
0.0458 -0.6154 1.3523 0.5432
-65.5
-67.2
-0.2459 0.4915
-63.4
-62.7
-0.3360 0.6217
-61.6
-64.1
-0.1346 -0.8113
2
-0.0435 -0.2260
3
12.8000 -18.900 6.600 -19.400
4
1.0081 2.3375
-1.1161 -5.0712
Model 1,Haggblom, 1988; model 2, Waller et al., 198810; model 3, Wood and Berry, 1973; model 4, Morris et al., 1982. The flow gains are calculated in Haggblom and Waller, 1989. Angles in degrees.
Table 2. The Same Comparison as in Table 1, but for Five Simulated Columns
5
0.0150 0.0267
-0.0093 -0.0317
-0.5635 0.6056
-47.1
-47.2
6
13.3433 14.0433
-0.3683 -0.4041
-0.7013 0.0198
-1.6
-1.6
7
0.5800 0.3500
-0.4500 -0.4800
-0.8100 0.8100
-45.0
-44.1
8
0.5620 0.3440
-0.5160 -0.3940
-0.7100 0.7100
-45.0
-44.6
9
0.4710 0.7490
-0.4950 -0.8320
-0.9300 1.0200
-47.6
-47.6
Model 5 , Haggblom and Waller, 1989; model 6, Haggblom, 1988; models 7-9, McAvoy and Weishedel, 1981. The flow gains are given in Haggblom and Waller, 1988,1989. Angles in degrees.
*"i
(14) In the literature one can find a number of experimental separations (pilot plant) as well as simulation examples of the (L,V) structure for which the gain matrix in eq 14 as well as flow gains according to eq 10 are given or can be calculated (Haggblom and Waller, 1988,1989). We may now compare the input directions given by SVD of the gain matrix and the directions given by the previously mentioned vectors consisting of flow gains. If the observations mentioned above (concerning lowgain and high-gain directions in distillation) are of any practical value, these directions should be closely aligned. The comparison is made in Tables 1 and 2 and expressed as the angles that the high-gain singular vector ii and the flow-gain vector -[KDL KDvI~, respectively, form with respect to the single input vector [AL OIT (Figure 4). As seen, there is a very close resemblance between the directions calculated by the two methods. Thus, these examples indicate that the observation that the low-gain direction is in close alignment with changes in internal flows may be useful in the nonideal case. (Since the high-gain direction is orthogonal to the low-gain direction the comparison could equally well have been made for the latter one.) These findings may be useful for a number of different purposes. Koung and McGregor (1992)found that the
0.1637 0.3818
AL
Figure 4. Schematic illustration of the angles compared in Tables 1and 2.
degree of robustness in model-based control is dependent on the accuracy of the V matrix. In a later paper (Koung and McGregor, 1993)the same authors present a method for iterative identification of models for robust control, which would gain significantly from prior knowledge of the V matrix. In this study it has been found that the V matrix may be estimated from the flow gains, which should be fairly easy to measure. For the experimental separations in Table 1 we also note that all distillation columns have their highest gain for approximately 2hL = -AV. Since all experimental examples in Table 1are alcohol-water separations (ethanol-water and methanol-water), it may be that this finding is a separation-specific prop-
2042 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995
erty, i.e., specific for the mixture distilled and independent of the operation point and the distillation column in which the separation is performed. It is not very farfetched to expect that other separations have similar properties. The simulated model 6 in Table 2, which is a methanol-ethanol separation, could indicate that methanol-ethanol separations have their highest gain when AL, x -35AV. Provided that such a relation holds and is known for the separation at hand, the use of consistency relations valid for continuous distillation (Haggblom and Waller, 1988, 1992) means that fewer measurements are needed to gain a complete steadystate model of the process. Note that these results for nonideal distillation columns are very different from the results that would be obtained if eq 9 was valid. In the latter case the high-gain direction would be obtained for AL. x -AV. Thus, for alcohol-water separations, we may expect the V matrix, eq 5, to be approximately given by COS(-65") ~0~(-155") 0.42 -0.911 = [sin(-65") sin(-155")] = -0.91 -0.42
[
Then the following model is obtained
(18) where t l characterizes the response to changes in external flows and t 2 the response to changes in internal flows.
+
Equations 10 and 18 show that if (KDLAL. KDvAV) AB = 0 and that the dynamics of the response is characterized by t2 alone. This is achieved for input vectors [AL. AVITIIIK~v -KDLIT. Similarly, for [AL.AUT/~[KDL &VIT, the maximum change in the external flows is obtained when (KDvAL,- KDLAV)= 0 and the dynamics of the response is characterized by t l only. = 0 then it follows that AD =
Equation 18 can be written
r
(15)
An estimate of the V matrix can be obtained by the use of the flow gains with the equation
Kll
[TIS
K12
+1
t2S
1
+ 11
Equations 14 and 19 give
On the basis of first principles, Haggblom (1994) has recently derived bounds showing that the flow gains in eq 16 provide a very accurate estimate of the V matrix for ill-conditioned distillation columns with large values in the relative gain array. Further, we may use the flow gains to construct a model structure utilizing the observation that the response in the internal flows and the response in the external flows are associated with considerably different dynamics (Skogestad and Morari, 1988; Bequette and Edgar, 1989; Andersen and Kummel, 1989). This is done below.
where
Zil
=
KDLKilz2
KDLKil
+ KDVKi2tl + KDVKiZ
(22a)
- KDLKi'2Zl - KDLKi2
(22b)
4. A Dynamic Model Structure for Capturing
Process Directionality Unless care is taken, it is likely that a dynamic model of continuous distillation for a column identified in an open loop is going to contain a description of the dominant dynamics only, e.g., expressed as
This model has two slow states, something that according to Jacobsen and Skogestad (1994) is inconsistent with most real distillation columns. Further, the fast transient connected to the internal flows is unmodeled. Below we follow the appealing approach of Skogested and Morari (1988) t o model the response to perturbations in external and internal flows separately, using different dynamics for the two. However, we extend the approach t o cover nonideal distillation by using the relations in eq 10. If we further, for simplicity,use firstorder dynamics for the two directions, we get the same modeling assumptions as Skogestad and Morari: The response to changes in the external flows is first order with a time constant t1. The response t o changes in the internal flows is first order with time constant tp.
and
212
=
KDVKilt2 KDVKil
The model structure in eq 21 is similar to that of the low-order model obtained by Jacobsen and Skogestad (1994) by mathematical reduction of a rigorous model of a 41-plate column. As such, the model in eq 21 has 10 parameters. However, since the transfer function zeroes are given by eqs 22, there are not more than 6 independent parameters in the model left to be identified: the gains and the two time constants. It should be emphasized that the flow gains in eq 10 are not independent of the process gains in eq 14; identified gains should be reconciled to satisfy the external material balances valid for continuous distillation, cf. eqs l l a and l l b (Haggblom and Waller, 1988, 1992). In case there are no measurements of the external (product) flow rates, a model similar to that of eq 19 can be obtained by assigning different dynamics to the maximum and minimum gain directions (cf. eqs 16 and
Ind. Eng. Chem. Res., Vol. 34,No. 6,1995 2043
Kll
K12 'KyL + Kyv
[z]
+1 = KxL +K,v Z$ + 1 z+
-zls + 1 tfl 1
+
The transfer function matrix of the model in eq 23 can be written as
KYB TIS
KYV
+ 1 zls + 1
"
'
K,v K,, [AV] (30) zls 1 zls 1
+
+
mvT UUVT
+ Tfl+ Z1S+1
G ( s )= -
(24)
1
where 0,u, and v are obtained by SVD on the process gain matrix. In case of perfect alignment between and the input direction that results in AD = AB = 0, the models in eqs 21 and 24 are identical. Comparison with One of the Models of Skogestad and Morari (1988). A popular distillation model (e.g., Brambilla and D'Elia, 1992;Figueroa et al.,19931, suggested by Skogestad and Morari (1988),which is based on the assumption O f KDL= -KDV = -1, is given by
c
1 lg4'
0.878 -0.86412.1s 15s + 2 S O . : ' -1.096
+ 1'
+
which, in the approach suggested in this paper, becomes
+1
[TIS
t2S
+1
TIS
E:]
+ 1J
It can be rewritten
[TIS
+1
1 zls + 1
Tfl
+1
t2S
J
+1
J
The difference between these two models is that the fast pole in the Skogestad-Morari model (eq 25)is canceled for inputs parallel with [l OIT whereas in eq 27, z2 is canceled for inputs aligned with [l -UT. The reason for this difference is that Skogestad and Morari connect the time constants ~1and 72 with changes in the external flow D and in the internal flow V,respectively (eq 22 in Skogestad and Morari (1988))
If Skogestad's and Morari's (1988) approach is used, an equally well-motivated model is obtained by connecting z1 and 72 with, e.g., AB and AL,respectively, [ hA Y D B]
[Kyi +
= KxL + KK,, yV
1
+ 1)(15s + 1) + 1) -0.864(13.5~- 1) [0.878(16.4~ 1.082(13.8~ + 1) -1.096(16.1~+ 1)
+ + '1) I] (29)
-K,v A B / ( T ~ s -KyV][u'(r#
= (194s
I[ 1 AI,
AV (32) The models in eqs 31 and 32 look quite similar. There is, however, one important difference between the two modeling approaches, and it lies in the fact that the model structure suggested in this paper is applicable for real columns and not restricted t o the idealized assumptions of eq 9. Example: A Nonlinear Simulator. To illustrate the results presented above, a nonlinear simulator (Lehtinen, 1994) of an ethanol-water separation is used. The simulator is modeled to describe the 15-plate pilot-plant column at Ab0 Akademi. A brief presentation of the simulator is given in Appendix B. The simulator has been adapted t o experimental data from the real column. It should be noted that the column has now been equipped with sensors measuring the product compositions (Gustafsson, 19941, and this is why the models given below have product compositions as outputs. In previous studies (Waller et al., 1988a; Waller, 1992) the column has had two tray temperatures as outputs and operation points with quite low relative gain array (RGA) have been studied. In this work an operating point with quite a high RGA is used in order to put more emphasis on the problem of directionality and ill-conditionedness. At the operation conditions given in Table 3 the following gains are obtained [KYi KYv]= [0.254 -0.5281 K x K,v ~ 0.241 -0.534
(33)
]
KDL KDV = -0.405 0.881 (34) [KBL KBvI 0.405 -0.881 The gain matrix in eq 33 has 111 = 16.2 and y = 81.8. A singular value decomposition of the gain matrix yields
[
2044 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 Table 3. Parameters and Variable Values for the Nonlinear Simulator at the Studied Steady State separation number of trays feed on tray tray efficiencies feed (F) distillate flow (D) composition of feed (ZF) steam flow to reboiler (V) reflux flow ( L ) distillate composition DO) bottom product composition ( X B O )
~1
[AL AVIT = 10.881 0.405IT
A.
90.980 90.975
ethanol-water 15 12 (counting from top) 62-73% 150 k g h 40.9 k g h 25 mass % 90 kgih 130 k g h 90.9695 mass % 0.2753 mass %
E. [AL AV]' = 1-0.405 0.881JT.lo-'
90.970 90.966
90.970 90.965
n
20
80
100
40 60 80 Time (run)
100
40
60
90.962
0
20
0
20
T i (min)
0.270
n
20
0.268
40 60 Time (min)
40
60
80
100
80
100
T ~ (min) K
Figure 5. Open loop responses to simultaneous step changes in both reflux and boilup. Full line: response of nonlinear simulator. Dashed line: response of low-order model in eq 39. (A) [ALAUT = [OB810.405IT,Le., a change which does not change the external i.e., flows a t steady state. (B) [ALAUT= [-0.405 0.881ITx an input direction giving maximal change in the external flows. Very small perturbations are used so as to supress the nonlinearity of the simulator.
giving AD = AB = 0, and the modeling assumption for the dynamic low-order model structure previously used seems reasonable also for this non-ideal distillation. A fairly good low-order approximation is obtained by use of eqs 20-22, when z1= 20 min and t2 = 6 min and the gains in eqs 33 and 34 are used (Figure 5)
121 1
1
+ 1)(6s + 1) 0.254(6.51~+ 1) -0.528(5.89~+ 1) 0.241(5.80~+ 1) -0.534(6.04~+ 1) = (20s
I[ 1 AL
AV (39) This low-order model is consistent in terms of the number of slow poles and it captures quite well the difference in dynamic behavior related to the external flows and internal flows, respectively. Note that it would not be possible to get a reasonably accurate description of the low-gain direction of this system with the models in eqs 25 or 27. Although the simple model in eq 17 may be a reasonable open loop approximation of the distillation process, it may be an inadequate description of the process for feedback control purposes (Skogestad and Morari, 1988; Jacobsen, 1991; Jacobsen and Skogestad, 1994). This is illustrated by the following example. Two inverse-based (IMC) controllers with first-order filters (Morari and Zafiriou, 1989) are designed and applied to the nonlinear simulator. One of the controllers is based on the model in eq 17, the other on the model in eq 21. The values T I = 20 min and z2 = 6 min and the gains in eqs 33 and 34 ared used. Both controllers can be written (cf. eqs 19 and 20)
= [0.423
-0.9061 (35) -0.906 -0.423 Thus, the angle a of the input giving the largest gain is
a = -65.0' and the angle 4 of the vector -[KDL = -65.3'
(36) is (37)
The alignment of a and 4 is within half a degree. Thus, the V matrix could be estimated from the flow gains by eq 16 with a fairly good accuracy, the estimate being = [0.418
-0.9091 (38) -0.909 -0.418 Again, the highest gain is given for 2A.L. % -AV. This was only t o be expected since the simulator in question is €or an alcohol-water separation. In Figure 5 two simulations with the detailed simulator are shown, one for such changes in the internal flows that do not change the external flows, the other for such changes in the internal flows that maximally change the external flows with respect to the internal flows. The simulations in Figure 5 reveal that there is an input direction canceling the slow behavior of the process which dominates the response for all other perturbations. This also suggests that this process should be modeled as having only one slow state, which is in accordance with the results of Jacobsen and Skogestad (1994). The fast transient in Figure 5A is obtained for an input vector aligned with the direction
If the design is based on eq 17, then tl = t2. C ( s )can be regarded as two PI controllers with pre- and postcompensators of the inputs and outputs, respectively. The IMC design was chosen since it provides an easy way to specify the desired (nominal) closed loop response. The intention is not to argue about the benefits and shortcomings of this particular design method but to illustrate the difference resulting from the use of two different models in the solution of the control problem. A similar analysis but for a distillation column controlled by multiloop SISO controllers has been performed by Jacobsen (1991). Both controllers in this example are designed for a nominal closed loop response of first order with a time constant of ZCL = l l k l for any perturbance direction. In Figure 6A, k1 = 0.05 min-l, which results in ZCL = 20 min for both controllers. When applied to the nonlinear process, it can be seen that the design based upon the two-time-constant model in eq 21 results in a response much closer to the one desired. In Figure 6B, the design objective is a nominal response of first order with TCL = 5 min (k1 = 0.2 min-l). Also in this case, the design based upon the model in eq 21 leads to a controller performance closer to the specifications than the performance of the controller designed on the model in eq 17. As expected, the high-frequency dynamics of the simulator, unmodeled in both the low-order models, is
Ind. Eng. Chem. Res., Vol. 34, No. 6 , 1995 2046 B. 0.012
0.012
........ .....
........ .......
:
0.w
0
20
40 60 Time (min)
100
80
0
10
,
4.W8 -0,012
0
,'
20 30 Time (min)
40
SO
20 30 Time (min)
40
SO
.
,< 10
Time (min)
Figure 6. Response of nonlinear simulator controlled by IMC controllers. [ A ~ A D x~ B , ~=~LO.01 -0.01IT. Full line: IMC design based on the model in eq 21. Dotted line: IMC design based on the model in eq 17. Dashed line: Nominal closed loop response for both controllers. (A) Nominal closed loop response: ~ C = L 20 min. (B)ZCL = 5 min. 90.980
.......................................
90.975 90.970 90.965
20
0
40 60 Time (min)
80
100
0.272 0.270 0.268
20
0
131.0 130.8 130.6 130.4 130.2
40 60 Time (min)
80
100
D .......
130.0 0
20
Time 40 (min) 60
80
100
controllers is at its largest. Seemingly, both controllers perform equally well for changes in the high-gain direction. Any other setpoint direction can be regarded as a combination of these high-gain and low-gain directions. The experimental identification of the smaller time constant z2 is a difficult task since, in practice, it is not possible to simultaneously perturb the inputs with perfect accuracy in the direction of the perturbation, as was done in the simulations shown in Figures 2 and 5. Even for a small misalignment z1 will dominate. This is probably why the small time constant is not generally present in distillation models obtained from experiments (e.g., Wood and Berry, 1973; Waller et al., 1988a). However, by relay feedback in combination with knowledge of a model for the high-gain direction, encouraging results in the identification of the low-gain direction have been achieved (Sbgfors et al., 1994). A brief presentation of the idea of this identification method is given in Appendix C. The flow dynamics, although present in the nonlinear simulations, is not explicitly modeled in eq 39. The flow dynamics can be observed in X B in Figure 5A as a fast initial response and a slight overshoot (cf. Haggblom, 1989). Since x g is initially decoupled from L due to the liquid lag through the column, this means that ZB initially acts as if V alone had been perturbed. The high amplification from V t o X B is suppressed as soon as the effect of the change in reflux reaches ZB, the overall response in X B showing a fast initial response which suddenly stops. Thus, in the low-gain direction the effect of the flow lags results in a very fast initial response, contrary to what might be expected. The flow dynamics is generally quite hard t o observe in open loop operation, since the dominant pole - l / q dominates the open loop responses for most input signals. The problem of how to include flow dynamics in low-order models is treated later in this paper (section 5). The Material Balance Structure (0,V). By referring to the observations that perturbations in the internal flows as well as in the external flows have different dynamics a t the same time as they are differently amplified in the outputs, it has been suggested that one should model, identify, and control the material balance (D,W structure instead of the (L,W structure (Skogestad and Morari, 1988; Andersen et al., 1989; Bequette and Edgar, 1989). The argument is that in the (D,V structure we have one external flow and one internal flow as input signals, and this is why we may expect an overall behavior of the process that could be adequately described by a model of the form (cf. eq 28)
r
KVD
Kvv
1
Figure 7. Open loop responses of the material balance (D,V) structure to a step change in boilup: AV = 0.405, AD = 0. The are shown (the initial product compositions as well as the reflux (L) value of L being 130 k g h ) . Full line: response of nonlinear simulator. Dashed line: response of the model in eq 41 under the assumption of t2 = 6 min.
getting more important when the performance specifications are tightened from ZCL = 20 min to TCL = 5 min. The setpoint changes in Figure 6 are in close alignment with the worst disturbance (low-gain) direction, i.e., the direction which requires the most control action. This is the direction where the difference between the
where z1 > t2. This applies for extremely simplified models of the kind illustrated in Figure 2, where a perfect level control of the condenser is assumed and the flow lags are neglected. Under such assumptions, a (step) change in V (while keeping D unchanged) will cause an immediate change in L, so that D and B also initially remain constant. Under these circumstances, the fast transient connected with the internal flows can be observed by perturbing only V since the reflux (L) will also respond immediately.
2046 Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995
However, this is never true in reality where flow lags and the condenser level controller are delaying the response of the reflux and where the flow dynamics are coupled with the composition dynamics. This is illustrated in Figure 7 where a step disturbance in V of the same size as in Figure 5A is applied to the 15-plate nonlinear simulator. Even though the lags caused by the liquid holdups and the condenser are very small compared to both 21 and 22, the response of L will be delayed long enough to make 22 invisible. (The level controller is tightly tuned so as to respond with a time constant of 15 s, and the total liquid lag in the column is approximately 10 s.) To make t2 visible, simultaneous perturbations of L and V such that D and B remain constant would be needed. This cannot be accomplished in the (D,W structure, except for strongly simplified and unrealistic models similar to the one illustrated in Figure 2. At steady state, L will eventually settle on the same value as in Figure 5A. The (D,V) structure may thus be useful for gain modeling and identification, but dynamic modeling is more complicated because the level controller in the condenser becomes an internal part of the model. See also Koggersbal and Jargensen (19941, where frequency-dependent RGA is used in an analysis of a particular system operated in the material balance configuration, and Yang et al. (1990).
y = limb - Ai)G(s) S-L,
(43) To ensure a single pole at A1 = -UTI, which according to Jacobsen and Skogestad (1994) is consistent with the physical reality of most distillation columns, K1 has to be singular, i.e., rank K1 = 1. With the flow lag gL(s) included in Gzl in the proposed model structure (eq 21)
1
the multiplicity of the pole
rank
K1
KYL
Kx. [2]=
KyL
+ Kyv -~KYL '
z2s
+1
tls
+1
K x+~Kxv -- %L
" [AV]
is given by
= rank
(45) for s = -1h1. If the model is rewritten according to eq 19 in the following form
G ( s )=
rls + 1
-1h1
lKDLK21
gL(s) KDVK21 1
tlS
+
1
+
it is seen that ifgL(s) = 1 (no flow dynamics) there is only one slow state in the model, since K1 becomes singular. On the other hand, if a dynamic description of the liquid lag is included, there will be multiple slow poles since K1 then has full rank. However, by small changes in the model parameters K1 can also be made singular when the flow dynamics is included. From eq 46 it is seen that the elements in the first row of K1 are proportional by the factor KDL/ KDV. K1 can be made singular by moving the transfer function zero 2 2 1 , making the elements in the second row of K1 proportional by the same factor. By changing 2 2 1 to
HIbecomes singular. One of the slow poles is canceled by a zero, and there is only one slow state in the model again. This single slow state is canceled by the input direction giving ALI = AB = 0, i.e., for inputs parallel with [KDV-KDLP, as stated in the modeling assumptions in section 4. Suppose that we want to augment the model in eq 32 by the flow lags gL(s) approximated by (Jacobsen and Skogestad, 1994)
(48)
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2047 O.oo00
-0.0006
-0.0012
200 Time (min)
0
400
Figure 8. One point control. The response to a step disturbance in V. The distillate composition yo is controlled by L with the use of a PI controller. Full line: reference model in eq 31. Dashed line: model in eq 49. Dotted line: model in eq 50.
The process model then becomes
[2]
1
+ 1)(15s + 1) 0.878(16.4s + 1) -0.864(13.5~ + 1) 1.082(13.8~+ l)gL(S) -1.096(16.1~+ 1) = (194s
(49) Comparing the response of this model under partial control t o the model in eq 31, which, according to Jacobsen and Skogestad (19941, is a good process approximation under one-point control, shows the slow settling in the uncontrolled output caused by double slow poles, Figure 8 (cf. Figure 9 in Jacobsen and Skogestad (1994)). However, if one of the slow poles is canceled by the use of eq 47, we get zil = 16.1 (for both expressions in eq 48),and the resulting model is
[2]
+ + +
1
+
= (194s 1)(15s 1) 0.878(16.4s 1) -0.864(13.5~ 1) 1.082(16.1~ l)gL(S) -1.096(16.1~ 1)
+ +
(50) which does not have the slow setting in the uncontrolled output, Figure 8. This model should be a reasonable process approximation for the system when it is operated as an open loop, when it is operated with one-point control, and when it is operated with two-point control. This study shows how extremely sensitive fundamental properties of a strongly coupled process model may be to small parameter changes. Conclusions as regards disturbance rejection based on the models in eqs 49 or 50 could be very different, even though the models seem almost identical. This supports the claim that the identification and modeling of the multivariable properties of the process are much more important than is satisfactory estimation of the individual transfer functions (Andersen and Kummel, 199213). In fact, it can be considered an almost impossible task to identify all the parameters with the accuracy required if the individual transfer functions are independently identified, regardless of identification method. This highlights the significance of a multivariable model structure of the kind proposed, which defines certain relationships between the parameters on the basis of physical knowledge. 6. Conclusions
The directionality of the continuous distillation process has been studied. Previous analyses based on
strongly simplifying assumptions indicate that the directionality of the distillation process is connected with the difference in perturbations affecting the internal and external flows, respectively. The validity of these findings for real distillation processes has been examined by the use of a number of experimental and simulated models. The results connecting the external flows (Dand B ) and the internal flows ( L and V) with the maximum and the minimum gains of the distillation process, respectively, have been found to be a useful approximation for real columns. The gains relating the internal flows ( L and VI to the external flows (Dand B ) can thus provide an estimate of the directionality of the distillation process. A low-order model structure for nonideal distillation is proposed. The model structure is based on the results concerning the directionality in distillation and on a modeling assumption that assumes different dynamics for the maximum and minimum gain directions. The model structure is analyzed by the use of a nonlinear simulator adapted to experimental data. The problem discussed by Jacobsen and Skogestad (1994) concerning multiple representation of single dominant states is also addressed. The extra slow pole occurring when flow lags are included in the model can be removed by small adjustments of the model parameters. A method of doing this is given.
Acknowledgment The results reported have been obtained during a long-range project on multivariable process control supported by Tekes, the Academy of Finland, Nordisk Industrifond, the Neste Foundation, and Neste Oy. Mr Kari Lehtinen is gratefully acknowledged for several stimulating discussions and for help with the simulations.
Nomenclature B = bottom product rate D = distillate product rate F = feed rate G(s)= transfer function matrix g&) = liquid lag K, = matrix defined in eq 34 Ku = scalar gain kl = IMC filter parameter L = reflux flow rate Mi = holdup on tray i m = manipulated variable vector U = left decomposition matrix in SVD ti = output singular vector associated with a u = output singular vector associated with g = right decomposition matrix in SVD V = estimated right decomposition matrix V = boilup flow rate ii = input singular vector associated with a v = input singular vector associated with Fi = composition of light component on stage i xg = bottom product composition (light component) y = vector of outputs yo = distillate composition (light component) ZF = feed composition (light component) zu = transfer function zeros zil = modified transfer function zero Greek Letters
a = rotation angle of ii (Figure 1) /3 = rotation angle of u (Figure 1) A = deviation operator y = condition number
2048 Ind. Eng. Chem. Res., Vol. 34,No. 6,1995
Ail = 1,1element in the relative gain array 4 = rotation angle of the vector -[&L &VI (Figure 4) Z = diagonal matrix containing the singular values ij = the maximum singular value -u = the minimum singular value z1 = dominant (large) time constant tp = smaller time constant T C L = closed loop time constant Subscripts 0 = initial steady state s = setpoint values
~
0
5
~~
10
Time ( m h )
Appendix A Directionality of 2 x 2 Processes and 1211. Consider a nonsingular 2 x 2 gain matrix of a coupled system (GQf 0)
The gain matrix can be decomposed by scaling matrices, SI and SZ,such as
Figure 9. Relay feedback of TCwith qc on the simple nonlinear heat exchanger model. Full line shows output Tc; dashed line shows input signal qc.
uneven scaling of the inputs (condition number of S2 large), the input singular vectors can be made arbitrarily close to the single input directions. A similar mapping from the inputs, llmll2 = 1, to the outputs, reveal the output singular vectors.
Appendix B (52) where lull = lazl, lbll = Ibzl, and SI,s2 > 0. The linear operator in y = Gm maps all input vectors of the same magnitude (say llmliz = 1)as an ellipse in the output space (c.f. Figure 1). Similarly, all output vectors I IyI 12 = 1 are mapped by an ellipse in the input space. The longest and shortest input vectors resulting in I IyI 12 = 1are then defined by the principal axes of the ellipse in the input space. The equation of the ellipse can be written
where Izi are constants. The principal axes are aligned with the actuators ml and 171.2 only if the cross multiplication term of the ellipse is zero (k3 = 0). With the matrix G expressed as in eq 52 the system in eq 51 can be written lalS2ml
Y2
+ Slblm2
= b2s2m1+ (121112
(54)
The unitary length outputs are defined through lIY1122 =Y12 +Y22 = 1
(55)
Squaring y1 and y2 in eq 55, using eq 54,and solving for k s = 0 in the resulting ellipse equation reveal that the cross multiplication term in eq 53 is zero only when
2s2(sl2a,b,+ a2bJ = 0
(56)
Since lalbll = la2bz1, eq 56 is satisfied only when S I = 1 and there is an odd number of negative elements in the coupled gain matrix G , i.e., albl = -a2b2. (Note that the scaling in eq 52 does not affect the number of negative elements; they are the same in G as in G . ) For 2 x 2 matrices, this means that 1A111 1. Consequently, processes with elements in the relative gain array greater than unity always have the input singular vectors misaligned with the single input vectors. By
The &Plate Nonlinear Distillation Model. The nonlinear 15-plate distillation column simulator (Lehtinen, 1994)used as an example in section 4 is based on the pilot plant distillation column at Ab0 Akademi. The components taking part in the separation are ethanol and water. Physical dimensions, e.g., number of trays, plate dimensions, etc. are taken from the real column. Some parameters and steady-state values are given in Table 3. The dynamics is modeled using dynamic mass and energy balances on each stage. The liquid overflow is modeled using the Francis weir formula. The inputs are the reflux ( L )and the steam flow to the reboiler (V). Both the reflux and feed are subcooled. The purity of the two products is quite high at the operating point studied, the ethanol-water mixture having an azeotropic point close to 94 wt % modeled by a strongly nonlinear relationship of the relative volatility. The simulator is adapted to experimental data by tuning the tray efficiencies and the enthalpy of the steam flow to the reboiler. A good correlation between the simulator and validation data not used in the model adaptation was obtained. A rigorous description of the simulator is given in Lehtinen (1994). For illustrative purposes, no dynamics is assumed in the heat exchanger of the reboiler in the simulations since simultaneous step responses in both vapor flow and reflux flow inside the column would then be impossible to perform. Appendix C Relay Identification of ZZ. Applying relay feedback to a process G(s) may make the system oscillate. By sampling the system twice a period at the instants where the relay switches, the system can be regarded as a sampled system with a piecewise constant input. If thFre is a periodic oscillation, the period P is given by (Astrom and Hagglund, 1984) HG(PI2,-1) = - f d
(57)
where HG(T,,z) is the pulse transfer function with the sampling time T,.Equation 57 can be used to identify
Ind. Eng. Chem. Res., Vol. 34, No. 6, 1995 2049 model parameters (Chang et al., 1992). By changing the hysteresis E (or relay output d),process information can be obtained at different frequencies. To illustrate the method of identifying the small time constant t 2 associated with the low-gain direction by relay feedback, a simple heat exchanger model given in Jacobsen and Skogestad (1994) is used. At the operating point studied an analytically linearized model is given by Jacobsen and Skogestad as
+
1%
][h4']
Kll(4.76~ 1) K12 Kzz(4.76~ 1) Aq,
+
(58)
where qc and qH are the cold and hot inlet flows, respectively, and Tc and THare the temperatures of the flows out of the exchanger. Numerical values given are z1 = 100, z2 = 2.44, K11 = -K22 = -1874, and K12 = -Kzl= 1785 (the time unit is minutes). The small time constant t 2 is difficult t o observe through open loop step responses. The linear model can also be written as eq 24 mvT
+
O W T
G(s)= t,s+l t#+1
(59)
Assume that perfect knowledge of the gains and the dynamics in the high-gain direction is available and apply relay feedback from Tc to qc. The transfer function from TCto qc can be written
g1
+
g2
G(s)= - t,s+ 1 Z # + 1
(60)
where only t 2 is unknown. For this transfer function, eq 57 gives
where o is the oscillation frequency. Tc has been fed back to qc in the nonlinear model with a relay with a hysteresis E = f0.02 and a relay output of d = FO.001. A period P PZ 2.3 min is observed (Figure 9). Solving for t 2 in eq 61 with the assumption of perfect knowledge of the other parameters gives z2 = 2.66 min, the correct value being 2.44 min.
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Received for review October 14, 1994 Revised manuscript received March 7, 1995 Accepted March 27, 1995@ IE940597P
@Abstract published in Advance A C S Abstracts, May 1, 1995.
