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Multimodel identification for control of an ill-conditioned distillation column
J. Proc. Cont. Vo]. 8, No. 3, pp. 209 218, 1998 ,c: 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0959-1524/98 $19.00 + (I.[)0
ELSEVIER
PII: S0959-1524(97)00040-1
Multimodel identification for control of an ill-conditioned distillation column K. E. H~iggblom and J. M. B61ing Process Control Laboratory, Department of Chemical Engineering, Abo Akademi Universi~, FIN-20500 Abo, Finland o
Identification for control of an ill-conditioned system requires special techniques. The directionality of such a system should be taken into account in the design of identification experiments. In distillation, information about the directionality properties can be obtained from certain flow gains, which are easy to determine in practice. Based on such information, the high- and low-gain directions of the plant can be explicitly excited. In this paper, a pilot-scale distillation column is identified by this approach at two different operating points. At each operating point, a nominal second-order plus time-delay model with logarithmic outputs is determined. This model structure makes it possible to capture the dynamic directionality of the plant. In addition, models describing variations and uncertainties in the high- and low-gain directions are determined by a special technique. The models obtained are superior to models determined via traditional step tests. The former satisfy integral controllability requirements, while the latter tend to violate them. ~; 1998 Elsevier Science Ltd. All rights reserved Keywords: distillation; identification for control; directionality; ill-conditioned systems
When a model for control design is determined through identification it is important to excite the plant sufficiently in both directions. It is particularly important to excite the plant in the low-gain direction to limit the relative uncertainty associated with this direction 4,5,7,~. The excitation of the plant in both high- and low-gain directions tends to correlate model errors and uncertainties in such a way that integral controllability requirements can be satisfied 5"6"~. This is usually not achieved by standard identification procedures where one input at a time is perturbed or where all inputs are perturbed simultaneously, but independently of each other. In both cases, the low-gain direction tends to be insufficiently excited. One solution is to apply closedloop identification with setpoint changes in different directions 7. Another possibility is to carry out the experiments in open loop with simultaneous perturbations in all inputs such that the relevant directions are sufficiently excited 5"8. Obviously, both approaches require some prior knowledge of the plant. The latter approach requires information about the directionality properties. It has been proved 4'9 that the high- and low-gain input directions of a distillation column can be estimated with good accuracy from flow gains, which are easy to determine experimentally. This makes it relatively easy to design identification experiments where the two directions are excited separately. In this paper, a
A process model for control design should possess certain control-relevant properties. A good nominal model with given uncertainties is not necessarily sufficient for design of a robust high-performance controller. It is important that the model uncertainty is small at frequencies significant for control, while the uncertainty may be larger at other frequencies. Furthermore, a multivariable plant has unique properties that require particular attention, especially if the plant is ill-conditioned. An ill-conditioned plant is a plant whose gain matrix has a high condition number. The gain matrix is almost singular and its determinant is close to zero in the sense that the sign of the determinant may be affected by quite small model errors. If the determinant of the gain matrix for the model and for the plant have different signs, there is no controller with integral action that can stabilize both the model and the plant I 3. Control design for an ill-conditioned plant is thus very sensitive to unstructured model uncertainty. If the uncertainties are structured (correlated) in a favourable way, larger uncertainties can be tolerated 46. An ill-conditioned plant is also characterized by a strong directionality. In geometrical terms this means that the length of the vector of outputs depends strongly on the direction of the vector of inputs. The maximum and minimum gains are obtained for changes in the socalled high-gain and low-gain directions, respectively.
209
210
Identification of an ill-conditioned distillation column: K, E. Haggblom and J. M. Boling
pilot-scale distillation column is identified by such an approach at two different operating points. At each operating point a nominal second-order plus time-delay model with logarithmic outputs is determined. In addition, models describing variations and uncertainties in the high- and low-gain directions around both operating points are determined. Such models are suitable for robust multimodel control designs 1°,11. It is known that a distillation column operated with the so-called L V-structure has a relative gain array (RGA) ~2 with diagonal entries larger than one 13. The models obtained have this property, which also means that they satisfy integral controllability requirements. In a related study made for comparison 8, models were determined from experiments where step changes were applied to the two inputs (reflux flow rate and flow rate of steam to the reboiler) one at a time. These models predict changes in the low-gain direction poorly and they do not satisfy integral controllability requirements. The models obtained by the use of traditional step tests are thus inadequate for control design in this case.
Gain directionality in distillation The conventional way of operating a distillation column is to control the product qualities by the reflux flow rate L and a variable V, which affects the boilup (e.g. the flow rate of steam to the reboiler), and to control the column inventories (condenser and reboiler holdups) by the product flow rates D and B. This control structure is known as the L V-structure. Figure 1 illustrates the L Vstructure of a distillation column with only inventory control loops closed. In the LV-structure, the product qualities are essentially controlled by the internal flows of the distillation column. Typically, this results in an ill-conditioned system characterized by a strong directionality and large relative gains ~4-j6. In particular, this applies to high-purity distillation. Figure 2 illustrates the gain directionality of a 15-tray pilot-scale distillation column operated with
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Figure 2 Contour plots of L(--), V(--) and D(- - -) in the composition plane the L V-structure. The product qualities are the distillate composition y and the bottoms composition x. The figure shows contour lines of constant reflux flow rate L, steam flow rate V and distillate flow rate D in the composition plane. The results have been obtained by means of a simulator, which was calibrated with actual plant data 14. In Figure 2 changes of the operating point from point A to points B, C and D are all of equal length in terms of the Euclidean norm of the output vector [AyAx] r. However, the required changes of the inputs L and V vary considerably. For a change from point A to D, the length of the input vector is 75 times the length of the input vector for a change from A to B. This means that the plant gains in these two directions differ by a factor of 75, which also indicates that the condition number of the plant is at least 75 (with unchanged variable scaling). It can be seen that the change from A to B is close to the high-gain direction while the change from A to D is close to the low-gain direction. This difference in gains is a consequence of the small angle between the contour lines for constant L and V, not due to the nonlinearity of the plant, which affects the curvature of the contour lines. Figure 2 also shows that the contour lines for constant distillate flow rate are essentially orthogonal to the contour lines for constant reflux and steam flow rate. This means that the product flow rates are almost unaffected by changes in the low-gain direction of the product compositions, whereas changes in the high-gain direction cause relatively large changes in the product flow rates. This is in agreement with previous observationslS 18 and it has also been proved mathematically 4,9. This fact makes it possible to estimate the high- and low-gain input directions of a distillation column with good accuracy from the gains between the product flow rates and the inputs L and V.
Estimation of high- and low-gain input directions Figure 1 L V-structure of a distillation column
A distillation column operated with the L V-structure has a static R G A with diagonal entries larger than
Identification of an ill-conditioned distillation column. K. E. Haggblom and J. M. Boling
one 13. For such a system, simple bounds on the highand low-gain input directions can be derived 4'9. By means of this result it can be shown that the corresponding input directions in distillation can be estimated with good accuracy from flow gains. Bounds on input directions
211
Remark 1: For ill-conditioned systems with X >> 1 the bounds in Equations (6) and (7) are tight since ~ - ~2 - $~X *
(g)
%~21 -- %~1I = ~21~. -1
(9)
Consider a 2 x 2 system described by the static model y=Ku,
K=
I Kll K21
K12] K22J
Estimation of input directions (1)
where u = [uj u2] T and y = [yl Y2] T a r e vectors of inputs and outputs, respectively, and K is a matrix of steadystate gains. The relative gain array for this system is given by
A=
,~ =
l
(2)
Z
K l l K22
K l l K22
(KI1K22 - KI2K21)
- - -
det(K)
(3)
The complete R G A for a 2 x 2 system is thus defined by a single parameter X, which will be referred to as the relative gain for the system. The system gain induced by the Euclidean vectornorm is defined IlYlI2 - [lull2 -
~
+ y2
V~
+ u2
(4)
The maximum gain ~ is obtained for some input fi, 11~[12 = 1, producing the output y, and the minimum gain a, is obtained for an input _u, I]_ul]2= 1, producing the output y. The condition number for the system is ~, = ~/_~. The input direction ~ and the parameters fl and ~2 are defined Kll ~2 K 2 1 = U~u2~' = ~2~2' &:
(5)
Without loss of generality it can be assumed that ~1 ~ ~2 since this can always be achieved by suitable permutations of the inputs and the outputs. Furthermore, it is assumed that each column and row K contains at least one non-zero element. Then the following theorem h o l d s 4,9.
Theorem 1: The high- and low-gain input directions and se. respectively, are bounded by ~l _< ~ _< ~2
(6)
-~/1 < ~ < -~21
(7)
The static behaviour of a distillation column operated with the L V-structure (Figure 1) can be described by the model
where y and x denote product compositions, L and V denote flow rates of reflux and steam flow to the reboiler, respectively. Under weak assumptions it can be shown by physical considerations that the system has a relative gain ,k > 1 (Ref. 13). When the inventory control loops are closed, the product flow rates D and B become dependent variables, and there exist static relationships AD
By means of material balances it can be shown that the composition gains in Equation (10) and the flow gains in Equation (11) are related as t~ KDI~ = -KBz~ =
1.
DKvL + BK, L "
(12)
3' -- X
KDV
KBV =
DKvv + BK,t )'
X
(13)
where D, B, y and x denote nominal steady-state values. The following theorem can be proved by means of Equations (12) and (13). Theorem 2: For a two-product distillation column with Z > 1, the product composition and flow gains satisfy the inequalities KvL
KDL
m < - - < Krv Kov
K~L ~<0 Kxv
(14)
The following Corollary of Theorems 1 and 2 implies that the high- and low-gain input directions can be estimated with good accuracy from the flow gains Kin, and KD v 4"9. Corollar.v: Estimates of the high- and low-gain input directions according to "
if and only i f Z < _ 0 o r Z >
KDV]
KDL ~__ -- KD V' -
KDV KDL
(15)
212
Identification of an ill-conditioned distillation column: K. E. Haggblom and J. M. BOling
respectively, are within the bounds stated by Theorem 1. The more ill-conditioned the distillation column is, the smaller are the margins of error. Remark 2: A change in the direction A L / A V = ~ does not change the product flow rates (cf. Equation (11)). This explains why the contour lines for constant D are essentially orthogonal to the contour lines for constant L and V in Figure 2.
Fundamental problems in identification of ill-conditioned plants In this section some important issues in identification for control of ill-conditioned plants are considered. In particular, it is demonstrated that the experimental design is of outmost importance. It is shown that an inadequate design easily results in a model that misrepresents the integral controllability properties of the plant. Experiments with a pilot-scale distillation column are used for illustration s . In the next main section, the identification of the distillation column is treated in more detail.
Identification from step changes in individual inputs The conventional way of exciting a distillation column is to make separate step changes in the reflux and steam flow rates. Such an experiment with the pilot column is shown in Figure 3. The figure shows step responses of the distillate composition y, the bottoms composition x, the distillate flow rate D as well as the sequence of step changes applied to the setpoints of the reflux and steam flow rates L and V, respectively. The composition responses are essentially of first order with some nonlinearity. Consequently, first-order plus time-delay models were fitted to the experimental data. The major part of the nonlinearities was captured 91.8
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Figure 3
C o n v e n t i o n a l step e x c i t a t i o n o f d i s t i l l a t i o n c o l u m n
by the use of logarithmic outputs. Figure 4, graphs (a) and (b), shows the composition responses from Figure 3 as well as the model fit (see B61ing and H/iggblom 8 for details). In another experiment (Figure 5), the distillation column was excited in the (approximate) low- and highgain directions. Graphs (c) and (d) in Figure 4 show the responses of this experiment and the responses predicted by the model determined from the previous experiment. As can be seen, the prediction is poor in the low-gain direction (t = 50-140 min). The model determined from step changes in the individual inputs has a relative gain k = -1.2 for diagonal variable pairing 8. However, it is known that a two-product distillation column operated with the LV-structure should have a relative gain ,k > 1 (Ref. 13) (note that the logarithmization does not affect the relative gain). Since this means that the gain matrix of the model has a determinant with the wrong sign (cf. Equation (3)), there is no controller with integral action that can stabilize both the model and the plant. A similar result was obtained when the column was identified from step changes in the individual inputs at another operating point 8 (operating point B, see below). It seems that this is not an uncommon problem in the identification of ill-conditioned plants. In a simulation study of a distillation column, Luyben 2° found that it was necessary to use step changes as small as 0.01% of the nominal values of the inputs to obtain a model whose relative gain has the correct sign. The main results of Luyben 2° are summarized in Table 1. The gains given in physical units in Table 1 are determined from the normalized gains given in Luyben 2°.
Identification from step changes in high- and low-gain directions As mentioned in the introduction and also indicated by Figure 4, it is particularly important to excite the lowgain direction in the identification of an ill-conditioned plant. In distillation, the directions of the input changes required for such an excitation can be estimated with good accuracy by Equation (15). For the pilot column, the flow gains KDL ~--0.65 and KDv ~ 1.3 can be estimated from the experiment shown in Figure 3 (operating point A, see below). ~ccording to Equation (15), this gives the estimates = - 0 . 5 and ~ = 2 of the high- and low-gain input directions, respectively. Figure 5 shows an experiment, where the inputs are changed almost simultaneously (L is changed 0.5min prior to V in order to compensate for the hydraulic delays in the column) according to the ratios ~ and ~ to excite the distillation column in the low- and high-gain directions. The sizes of the step changes were chosen to give outputs of suitable magnitude on the basis of the composition gains that can be estimated from Figure 3. Note the small input steps applied in the high-gain direction ( t = 140-250min). From the data of this
Identification of an ill-conditioned distillation column. K. E. Hdggblom and J. lid. Boling
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Figure 4
Model determined from individual step changes. Model fits (a) and (b) and predictions of low- and high-gain responses (c) and (d). Experiments ( ) and model outputs (...)
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Figure 5 Excitation of distillation column in low- and high-gain directions
Table 1 Composition gains and relative gains for different incremental sizes. Adapted from Luyben 2° Au%
K,.l.
Kvv
Kx/.
K~:v
k
1.000 0.500 0.100 0.050 0.010 0.005 0.001
0.4047 0.5115 0.6370 0.6556 0.6710 0.6729 0.6741
-1.1786 -0.9301 -0.6846 -0.6564 -0.6346 -0.6320 -0.6304
1.4819 1.3932 1.2797 1.2623 1.2479 1.2461 1.2444
-0.7353 --0.9953 -1.2366 -1.2634 -1.2839 -1.2865 -1.2887
0.2054 -0.6471 -8.9140 -3790.6 12.399 11.087 10.324
experiment, a model with the same structure as the previous model was determined. Graphs (a) and (b) in Figure 6 show the fit between the model and the experimental data. The ability of the model to predict the responses of the experiment in Figure 3 are shown in graphs (c) and (d). As can be seen, the predictions are nearly as good as the model fits in Figure 4, graphs (a) and (b). The results clearly indicate that the column should be explicitly excited in the high- and low-gain directions in control-relevant identification. Furthermore, they show that the corresponding input directions can be estimated with good accuracy from the flow gains Koc and Ko~,. This is also supported by the example by Luyben 2°. Table 2 shows the flow gains that can be calculated by means of Equations (12) and (13) from the composition gains in Table 1 and the steady-state data given by kuyben 2°. The condition number y and the high-gain input direction ~ determined from the gain matrix are also given. As can be seen, the flow gain ratio K m J K m provides an excellent estimate of the true high-gain input direction (~ = -0.9865) regardless of the step size used. In particular, it can be noted that this also applies when the relative gain of the model has the wrong sign (cf. Table 1).
Identification for control In this section a more thorough identification of the pilot-scale distillation column is made. The column is excited in its high- and low-gain directions as described in the previous section. A model structure that allows not only static but also dynamic directionality is used 17.
21 4
Identification of an ill-conditioned distillation column." K. E. Haggblom and J. M. BOling 92.0
92.0 a
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91.5
91.5
91.0 50 4.0
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2.0
2.0 x
0.0 50
1O0
150 t (min)
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0.0 50
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150 t (min)
200
250
Figure 6 Model based on excitation in low- and high-gain directions. Model fits (a) and (b) and predictions of responses to individual step changes (c) and (d). Experiments (--) and model outputs (...)
Table2
Condition number, high-gain input direction and flow gains for different incremental sizes Au%
y
~
Koc
Kov
KoL/Kov
1.000 0.500 0.100 0.050 0.010 0.005 0.001
2.5715 4.9568 45.710 14030. 58.313 51.933 48.186
-1.1466 -1.0975 -1.0113 -0.9991 -0.9893 -0.9880 -0.9865
-0.9858 -0.9942 -0.9991 -0.9996 -0.9999 -0.9999 -0.9996
0.9919 1.0006 1.0010 1.0005 1.0001 1.0001 1.0005
-0.9939 -0.9936 -0.9982 -0.9991 -0.9998 -0.9998 -0.9992
By a special technique, several models describing variations and uncertainties about an operating point can be estimated from the same experiment. Two different operating points are considered. The column, which separates a mixture of ethanol and water, has 15 bubble-cap trays and is 0.30m in diameter. It is equipped with a thermosyphon-type reboiler and a total condenser, and it operates under atmospheric pressure. An operability and controllability study of the column was made by Hfiggblom and Lehtinen 14, where a region of ill-conditioned operating points was determined. The column is further described by Waller 21. Model structure
It is well known that the step responses of a distillation column to changes in the internal flows are characterized by a large dominant time constant implying that the dynamics might well be described by first order transfer functions. On the other hand, it has also been noted in
simulation studies that the plant dynamics depend on the input direction resulting from simultaneous changes in the internal flows. The dynamics are relatively slow in the high-gain direction but tend to be much faster in the low-gain direction 15"17. Such a behaviour cannot be described by a model where the transfer functions are of first order only; it is required that the transfer functions are of second order (or higher). In accordance with this observation, a model structure consisting of second-order plus time-delay transfer functions is used. The experiment in Figure 5 does not indicate a particularly fast response in the low-gain direction, but a reason for this is that the time delays associated with the two inputs are different. In that case, the dominant time constant will always influence the observed responses to some extent. The composition responses in the identification experiments are nonlinear, especially for the bottoms composition. As noted in the previous section, part of the nonlinearities can be captured by using logarithms of the product impurity fractions as model outputs. The use of such logarithmic impurities as a means for linearizing the plant has been suggested previously 15,22. Another nice feature of the use of logarithmic impurities is that the parameter estimation results in a minimization of relative errors instead of absolute errors in the true impurities. This is useful when the impurities vary over some range which includes very small impurities. The selected model structure is
[ Aln(y~-y) ] Aln(x-xe)]
[G,*,L = LG*L
G*vv][AL] Gxv j
AV
(16)
Identification of an ill-conditioned distillation column: K. E. Haggblom and J. M. Boling
(. KyL,1
K~vL,2 )
G~*L' " = \TvL, ls + 1 + TvL,2S" + 1 e-°~Ls
(. /~vV,I
/~yV,2 )
G~*'v - = \ T y v , ls + 1 + 77v,2s + f e O,.vs Gxt =
(\T~L, IS + 1 +
T~L,2S + 1
)
(17)
e°x ,
(K>'_ G*~v = \Txv, is
+
1
7~,-v,2s+ 1
where the superscript * denotes that the gains apply for logartithmetic outputs instead of true compositions as in Equation (10). The offset parameters y,. and x~ should, in principle, have the values Yc = 0.96 (ethanol and water form an azeotropic mixture at this composition) and x,. = 0. However, the values y~ = 0.94 and x,. = -0.02 were found to give better model fits to the experimental data and are therefore used. The partitioning of the transfer functions as indicated in Equation (17) makes it easy to analyse the directionality properties and, if desired, to enforce constraints associated with these properties. The analysis which follows applies both for true compositions and logarithmic compositions. This follows from the fact that G~*,L/G~*.v = GytJG.,.v and GxL/G.*vv = GxL/Gxv, where GIRL,Grv, G.,-L and Gxv denote transfer functions with true compositions as outputs, which can be determined by a linearization of the logarithms in Equation (16). Assume that the time constant TI is larger than the time constant T2 in the transfer functions. If TI is large compared to T2 in all transfer functions, the known properties of distillation columns suggest that the Tls are approximately equal. Assume therefore that TvL,1 = Tvv, i (and T~c,l = Tvv,I). Neglecting for the moment the time delays, there is an input direction = A L / ~ x V = -K~.v.I/K~,L. 1 (~ = -K*~-v,I/K*~L,1) which cancels out the part of the transfer functions containing 7~.,.,1 and T~.v.i (T~L,I and Txv.i), resulting in a fast response for 3' (and x). If also T,,L,2= Z,.v,2 (7~L,2 = T~v,2), there is an input direction = z X L / A V = -ICvv?_/K~vL, 2 (~ = -/C~v,2//CxL,2) which cancels out the part of ihe transfer functions containing 77~.L.2 and TyV,2 (T~L,2 and T~v,2), resulting in a slow first-order response for y (and x). If 0,'L ¢ 0yv (Oxc ¢ Oxv), the input associated with the shorter time delay should be delayed relative to the other input by the difference between the time delays. In practice, it may not be possible to obtain fast responses in both outputs simultaneously if0v L Ovv¢ 0xL - 0xV. Sfigfors and Waller 17 suggest a model structure where the Tl s are equal and the ~ s are equal, that is, a model structure with only two different time constants T~ and T2. In addition, they assume that the slow and the fast dynamics are exactly aligned with changes in the external flows, that is, that KvL, l/Kvv, i = K ~ L I / K ~ v . J = - Kvr,2/ KrL,2 = --Kxv, z/ KvL,2 = KDL/ KDv. This assumption appears more realistic than an assumption that the dynamics are exactly aligned with the true high- and low-gain directions, which are scaling-dependent mathematical abstractions.
21 5
The assumptions above obviously result in a model with very ideal properties, which may be too unrealistic. In practice, a plant with slow and fast dynamics, depending on the input direction, can be described by a model with TvL, I ~ Tvv.l, 7~vL,I "~ T~j.L, and the Tjs significantly larger than the ~ s . Consequently, the model parameters do not have to satisfy any precise constraints. However, in the following some constraints are enforced so as to reduce the number of free parameters and to make the parameter estimation more robust. Estimation q f nominal model
In the first part of the identification experiment shown in Figure 5 (t = 5(>140min), the inputs are changed according to the ratio A L / A V = - - K m ' / K D L . If it is assumed that K~L.1/K~.r.I = K*~c.l/K*.,.~,l = KDL/KDv, 77~.C,1 = 7~rV,j and Z~L,I = T,-r,l, the latter part of the partitioned transfer functions in Equation (17) can be estimated from this part of the experiment. If it is further assumed that K~.L,2/Kvv. 2 = K]L,e/K*,.j,,2 = - K i n . /KDL, T~'L,2 = T~.v.2 and T~L2 = 7~v.2, there is only one independent parameter, in addition to the time delay, to estimate for the latter part of each transfer function. The first part of the transfer functions can be estimated in a similar way from the latter part of the experiment. A drawback of this approach is that the flow gains were determined from another experiment. As can be inferred from Figure 5, it was possible to excite, at least approximately, the high- and low-gain directions of the compositions by knowledge of these flow gains, but it can also be seen that the distillate flow rate is varying to some extent in the first part of the experiment. This means that applied input ratio A L / A V is not exactly equal to - K m , / K m for the actual experiment. Some of the underlying assumptions are therefore violated if the partitioning of the transfer functions is based on the flow gain ratio obtained from the previous experiment. To circumvent this problem, it is here only assumed that the slow and fast dynamics are excited by orthogonal input directions. It is not specified whether these directions correspond to the true high- and low-gain directions, to approximate high- and low-gain directions based on flow gains, or to some other orthogonal input directions. This is determined by the experimental data. The following constraints are therefore imposed on the nominal model:
K;L,I
K~'V,2
,
Tvr.I = T~.r,i,
T~.r.2 = Trv.2
K*~L,I
K*vv,2
K~xr,1 --
KvL. 2, ,
T,-L,I= T,)'.I,
T~I..2 = T,-),2
K ~
vr. 1
-
K)L.2
(18) The time delays were basically determined from step tests where one input at a time was changed as shown in Figure 3. Some minor adjustments of these initial values were made in order to improve the model fit. Only multiples of the sampling rate, 0.5 min, were considered.
216
Identification of an ill-conditioned distillation column."K. E. Haggblom and J. M. Boling
The column was identified at two different operating points, A and B, which are defined in Tables 3 and 4. The experiments made at operating point A are shown in Figures 3 and 5. A similar set of experiments were made at operating point B. Figure 7 shows the model fits obtained by minimization of the sum of squares of the output errors. As can be seen, the fits are reasonably good. The remaining discrepancies between the experiments and the model fits are mainly due to nonlinearities not captured by the use of logarithmic outputs. Numerical values of the model parameters are given in Tables 3 and 4 (model 'Nom.'). The gains K~ are clearly larger in magnitude than the gains K~ and the time constants T1 are also larger than T2. These differences are especially large for operating point B, implying strong directionality properties. As shown in Table 5, the relative gain for the nominal model is k = 7.73 at operating point A and )~ = 14.42 at operating point B. The condition numbers are ~,* = 120.0 and },* = 124.6, respectively. The low-gain input direction ~* determined from the gain matrix for the nominal model is 2.35 at operating point A and 2.36 at operating point B. These values differ from the input ratio A L / A V = 2 used to excite the column in the assumed low-gain direction based on the flow gains estimated from the experiment in Figure 3. The experiment in Figure 5 (operating point A) and the corresponding experiment at operating point
B gave slightly different flow gains. According to these experiments, the flow gain ratio --KDv/KDL is 2.38 at operating point A and 2.44 at operating point B. This experimental verification supports the claim that the high- and low-gain input directions can be estimated with sufficient accuracy from the flow gains. It can further be noted that the ratio -K~yv,1/K;yL, l is 2.94 at operating point A and 2.60 at operating point B, whereas the ratio --I~xv, I/IC~L, l is 2.47 at A and 2.48 at B. This means that the slow and the fast dynamics for the bottoms composition are well aligned with the highand low-gain directions, respectively. (This is not seen in the experiments because of the different time delays and the fact that the inputs were not changed exactly in the high- and low-gain directions.) For the distillate composition the alignment is less perfect, especially at operating point A. Here, the difference between the time constants is also smallest.
Table 3 Model parameters for operating point A (L = 140kghr i V = 9 6 k g h r i F = 1 5 0 k g h r t z=25%)
Table 4 Model parameters for operating point B (L = 160kghr ~, V= 1 0 5 k g h r i F = 1 5 0 k g h r I z = 2 5 % )
Multimodel estimation
For a robust control design it is desirable to have information about parameter variations and uncertainties in a model. One way to obtain such information is to determine several models that capture relevant phenomena in the desired operating region. There are control design techniques for which such a multimodel description is particularly well suited 1°,1~.
Model
K*~.L.I
K*,'t..e
TvL.I
7~,'L.2
OyL
Model
K*yt,.l
K*.,.L.2
TvL.1
TyL.2
OyL
Nora. l 2 3 4 5 6
-0.0175 -0.0175 -0.0175 -0.0175 -0.0154 -0.0192 0.0272
-0.00650 -0.00532 -0.00627 -0.00697 -0.00650 -0.00650 -0.00650
17.91 17.91 17.91 17.91 19.14 23.42 29.43
9.89 8.56 10.36 11.82 9.89 9.89 9.89
1.5 1.5 1.5 1.5 1.5 1.5 3.0
Nom. 1 2 3 4 5 6
-0.0368 -0.0368 -0.0368 -0.0368 -0.0370 -0.0369 -0.0560
-0.00549 0.00404 -0.00533 -0.00576 0.00549 -0.00549 -0.00549
25.16 25.16 25.16 25.16 33.16 25.32 47.82
5.55 5.23 5.49 7.79 5.55 5.55 5.55
1.5 1.5 1.5 1.5 1.5 4.0 6.0
Model
K*yv, l
K*~,r.:
7"~,v.l
T~4".2
Oyv
Model
K*yv.~
K*yv.:
Tvv.l
Tvv.2
Ova.
Nom. 1 2 3 4 5 6
0.0515 0.0515 0.0515 0.0515 0.0451 0.0564 0.0798
-0.00221 -0.00181 -0.00213 -0.00237 -0.00221 -0.00221 0.00221
17.91 17.91 17.91 17.91 19.14 23.42 29.43
9.89 8.56 10.36 11.82 9.89 9.89 9.89
1.5 1.5 1.5 1.5 1.5 1.5 3.0
Nom. 1 2 3 4 5 6
0.0956 0.0956 0.0956 0.0956 0.0960 0.0959 0.1453
-0.00211 -0.00156 -0.00206 -0.00221 -0.00211 -0.00211 -0.00211
25.16 25.16 25.16 25.16 33.16 25.32 47.82
5.55 5.23 5.49 7.79 5.55 5.55 5.55
1.5 1.5 1.5 1.5 1.5 4.0 6.0
Model
K",.L.I
K*vL,2
TtL,1
T~L,2
O.vL
Model
K*,.L.z
K*vL,2
T,-L.I
T,L.2
O.~L
Nom. 1 2 3 4 5 6
0.1256 0.1256 0.1256 0.1256 0.1158 0.1198 0.1466
0.00507 0.00339 0.00514 0.00468 0.00507 0.00507 0.00507
22.67 22.67 22.67 22.67 18.01 22.92 31.57
2.75 1.40 3.48 1.48 2.75 2.75 2.75
3.5 3.5 3.5 3.5 3.5 3.5 3.5
Nom. 1 2 3 4 5 6
0.1128 0.1128 0.1128 0.1128 0.1680 0.1286 0.1112
0.00453 0.00340 0.00506 0.00567 0.00453 0.00453 0.00453
30.98 30.98 30.98 30.98 45.61 30.59 38.11
3.60 1.35 6.23 1.38 3.60 3.60 3.60
3.5 3.5 3.5 3.5 5.5 3.5 3.5
Model
K*,-r./
K*xv.:
T,:v3
T~v.:
O~.r
Model
K*xv,l
K*xv,2
T~v,l
T~v.2
O~v
Nom. 1 2 3 4 5 6
-0.3100 -0.3100 -0.3100 -0.3100 -0.2856 -0.2956 0.3616
0.00205 0.00137 0.00209 0.00190 0.00205 0.00205 0.00205
22.67 22.67 22.67 22.67 18.01 22.92 31.57
2.75 1.40 3.48 1.48 2.75 2.75 2.75
3.0 3.0 3.0 3.0 3.0 3.0 3.0
Nom. 1 2 3 4 5 6
-0.2804 -0.2804 -0.2804 -0.2804 -0.4176 0.3196 -0.2765
0.00182 0.00137 0.00204 0.00228 0.00182 0.00182 0.00182
30.98 30.98 30.98 30.98 45.61 30.59 38.11
3.60 1.35 6.23 1.38 3.60 3.60 3.60
3.0 3.0 3.0 3.0 5.0 3.0 3.0
Identification of an ill-conditioned distillation column. K. E. Haggblom and J. M. B61ing
21 7
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Experiment (
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200
250
0.0 50
100
150 t (min)
200
250
) and nominal model fit (...) at two operating points A, (a) and (b), and B, (c) and (d)
Figure 7 shows that the nominal model fit is not perfect. Additional models can be obtained by fitting a separate model to the step responses of each simultaneous change in L and V (Figure 5). When a new model is fitted, the remaining effects of previous input changes can be taken into account by the previously fitted models. (Note that steady state is not reached between input changes.) A problem with this approach is that the effects of the two inputs cannot be separated. Furthermore, the number of parameters of each transfer function is probably too large to allow unique parameter estimates.
The latter problem could be solved by using a simpler model structure (e.g. first-order transfer functions), but here it is desired to retain the original model structure. This is accomplished as follows. Consider the experiment shown in Figure 5. The first three simultaneous step changes mainly excite the lowgain direction. This means that the first parts of the partitioned transfer functions defined in Equation (17) are almost completely cancelled out. Since the remaining effects of these parts are small, it is assumed that they can be described by the corresponding nominal parameters (K~s and Tls). Furthermore, the constraints 91.8
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C
i
91.5
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i
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200
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100
150
200
250
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200
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Figure 8
Experiment (
1O0
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200
250
50
) and multimodel fit (...) at two operating points A, (a) and (b), and B, (c) and (d)
218
Table 5
Identification of an ill-conditioned distillation column: K. E. Haggblom and J. M. B Oling
a multimodel description can be used for a robust control design.
Directionality properties of estimated models Operating point A
Operating point B
Model
2.
F*
se*
,k
•*
se*
Nom. 1 2 3 4 5 6
7.73 11.01 8.39 6.57 6.11 9.72 35.70
120.0 179.4 131.6 100.1 96.0 135.8 470.2
2.35 2.39 2.35 2.35 2.33 2.34 2.37
14.42 24.05 16.52 15.21 12.11 13.49 55.56
124.6 214.8 143.4 130.9 146.3 129.6 372.2
2.36 2.39 2.35 2.33 2.40 2.37 2.36
defined in Equation (18) are enforced. In addition to the time delays, the model for each output then contains only two independent parameters, which can easily be estimated. The original time delays are also adjusted if this improves the fit. The last three simultaneous step changes excite the high-gain direction of the compositions. Here it can be assumed that the uncancelled effects of the second parts of the partitioned transfer functions, which are related with the low-gain direction, can be described by their nominal parameters,/C2s and T2s. When Equation (18) is taken into account, the parameters in the first parts of the transfer functions can be estimated. As above, the time delays are adjusted if this improves the fit. The model fits obtained in this way are shown in Figure 8. The estimated parameters of the seven models at operating points A and B are given in Tables 3 and 4, respectively. Table 5 shows that the relative gain and the condition number vary for the models, but that the lowgain input direction is almost unaffected.
Conclusions Control-relevant models for an ill-conditioned distillation column operated with the L V-structure have been determined through identification. Fundamental knowledge of directionality properties of distillation columns was employed in the design of identification experiments. In this way it was possible to explicitly excite the (approximate) high- and low-gain directions of the column. It was shown that excitation of the lowgain direction is particularly important in control-relevant identification of ill-conditioned plants. The models consist of second-order plus time-delay transfer functions. This model structure makes it possible to capture known (ideal) properties of distillation columns with regard to slow and fast dynamics in different input directions. The estimated parameters are in good agreement with theoretical expectations. The models also satisfy known integral controllability requirements. As shown, this is not a trivial achievement for an ill-conditioned nonlinear system. The design of the identification experiment made possible the estimation of multiple models by a special technique. The set of models describe variations and uncertainties associated with the high- and low-gain directions as well as with slow and fast dynamics. Such
References 1. Grosdidier, P., Morari, M. and Holt, B. R., Closed-loop properties from steady-state gain information. Industrial & Engineering Chemistry Fundamentals, 1985, 24, 221-235. 2. Skogestad, S. and Morari, M., Implications of large RGA elements on control performance. Industrial & Engineering Chemistry Research, 1987, 26, 2323 2330. 3. Yu, C.-C. and Luyben, W. L., Robustness with respect to integral controllability. Industrial & Engineering Chemistry Research, 1987, 26, 1043 1045. 4. H~iggblom, K. E., Static directionality of 2 × 2 systems: a control relevant property with application to distillation. Proceedings of the European Control Conference, Rome, 1995, pp. 3117 3122. 5. Koung, C.-W. and MacGregor, J. F., Design of identification experiments for robust control. A geometric approach for bivariate processes. Industrial & Engineering Chemistry Research, 1993, 32, 1658 1666. 6. McDonald, K. A., Palazoglu, A. and Bequette, B. W., Impact of model uncertainty descriptions for high-purity distillation control. A IChE Journal, 1988, 34, 1996-2004. 7. Andersen, H. W. and Kfimmel, M., Evaluating estimation of gain directionality, Parts 1 and 2. Journal of Process Control, 1992, 2, 59-66, 67 86. 8. B61ing, J. M. and H/iggblom, K. E., Control-relevant identification of an ill-conditioned distillation column. Proceedings of the IEEE Conference on Control Applications, Dearborn, MI, 1996, pp. 570~575. 9. H~iggblom, K. E., Proof of the relation between singular directions and external flows in distillation. Proceedings of the IEEE Conference on Decision and Control, New Orleans, LA, 1995, pp. 2502 2504. 10. Pensar, J. A. and Toivonen, H. T., A multimodel mixed H2/H~ problem for plants with structured uncertainty. Preprint of IFAC Symposium on Advanced Control of Chemical Processes (ADCHEM'94), Kyoto, Japan, 1994, pp. 289-294. 11. Sandelin, P. M., Toivonen, H. T., Oster~s, M. and Waller, K. V., Robust multiobjective linear quadratic control of distillation using low-order controllers. Chemical Engineering Science, 1991, 46, 2815 2827. 12. Bristol, E. H., On a new measure of interactions for multivariable process control. IEEE Transactions on Automatic Control, 1996, AC-11, 133 134. 13. H~iggblom, K. E., Analytical relative gain expressions for distillation control structures. In Consistent control structure modelling with application to distillation control. Dr. Tech. thesis, Part V, Abo Akademi University, Finland. 14. H~iggblom, K. E. and Lehtinen, K., Investigation of operability and controllability properties of a pilot-scale distillation column. Preprint of IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes (DYCORD + "95), Helsingor, Denmark, 1995, pp. 339-344. 15. Skogestad, S. and Morari, M., Understanding the dynamic behaviour of distillation columns. Industrial & Engineering Chemistry Research, 1988, 27, 1848 1862. 16. Skogestad, S., Morari, M. and Doyle, J. C., Robust control of illconditioned plants: high-purity distillation. IEEE Transactions on Automatic Control, 1988, 33, 1092 1105. 17. S~tgfors, M. F. and Waller, K. V., Dynamic low-order models for capturing directionality in non-ideal distillation. Industrial & Engineering Chemist O, Research, 1995, 34, 2038-2050. 18. Shinskey, F. G., Distillation Control, 2nd edn. McGraw Hill, New York, 1984. 19. H~iggblom, K. E. and Waller, K. V., Transformations and consistency relations of distillation control structures. AIChE Journal, 1988, 34, 1634-1648. 20. Luyben, W. L., Sensitivity of distillation relative gain arrays to steady-state gains. Industrial & Engineering Chemistry Research, 1987, 26, 2076-2078. 21. Waller, K. V., Experimental comparison of control structures. In Practical Distillation Control, ed. W. L. Luyben. Van Nostrand Reinhold, New York, 1992, pp. 313-330. 22. Ryskamp, C. J., Explicit versus implicit decoupling in distillation control. In Chemical Process Control 2, ed. D. E. Seborg and T. F. Edgar. United Engineering Trustees, New York, 1982, pp. 361-375.
ELSEVIER
PII: S0959-1524(97)00040-1
Multimodel identification for control of an ill-conditioned distillation column K. E. H~iggblom and J. M. B61ing Process Control Laboratory, Department of Chemical Engineering, Abo Akademi Universi~, FIN-20500 Abo, Finland o
Identification for control of an ill-conditioned system requires special techniques. The directionality of such a system should be taken into account in the design of identification experiments. In distillation, information about the directionality properties can be obtained from certain flow gains, which are easy to determine in practice. Based on such information, the high- and low-gain directions of the plant can be explicitly excited. In this paper, a pilot-scale distillation column is identified by this approach at two different operating points. At each operating point, a nominal second-order plus time-delay model with logarithmic outputs is determined. This model structure makes it possible to capture the dynamic directionality of the plant. In addition, models describing variations and uncertainties in the high- and low-gain directions are determined by a special technique. The models obtained are superior to models determined via traditional step tests. The former satisfy integral controllability requirements, while the latter tend to violate them. ~; 1998 Elsevier Science Ltd. All rights reserved Keywords: distillation; identification for control; directionality; ill-conditioned systems
When a model for control design is determined through identification it is important to excite the plant sufficiently in both directions. It is particularly important to excite the plant in the low-gain direction to limit the relative uncertainty associated with this direction 4,5,7,~. The excitation of the plant in both high- and low-gain directions tends to correlate model errors and uncertainties in such a way that integral controllability requirements can be satisfied 5"6"~. This is usually not achieved by standard identification procedures where one input at a time is perturbed or where all inputs are perturbed simultaneously, but independently of each other. In both cases, the low-gain direction tends to be insufficiently excited. One solution is to apply closedloop identification with setpoint changes in different directions 7. Another possibility is to carry out the experiments in open loop with simultaneous perturbations in all inputs such that the relevant directions are sufficiently excited 5"8. Obviously, both approaches require some prior knowledge of the plant. The latter approach requires information about the directionality properties. It has been proved 4'9 that the high- and low-gain input directions of a distillation column can be estimated with good accuracy from flow gains, which are easy to determine experimentally. This makes it relatively easy to design identification experiments where the two directions are excited separately. In this paper, a
A process model for control design should possess certain control-relevant properties. A good nominal model with given uncertainties is not necessarily sufficient for design of a robust high-performance controller. It is important that the model uncertainty is small at frequencies significant for control, while the uncertainty may be larger at other frequencies. Furthermore, a multivariable plant has unique properties that require particular attention, especially if the plant is ill-conditioned. An ill-conditioned plant is a plant whose gain matrix has a high condition number. The gain matrix is almost singular and its determinant is close to zero in the sense that the sign of the determinant may be affected by quite small model errors. If the determinant of the gain matrix for the model and for the plant have different signs, there is no controller with integral action that can stabilize both the model and the plant I 3. Control design for an ill-conditioned plant is thus very sensitive to unstructured model uncertainty. If the uncertainties are structured (correlated) in a favourable way, larger uncertainties can be tolerated 46. An ill-conditioned plant is also characterized by a strong directionality. In geometrical terms this means that the length of the vector of outputs depends strongly on the direction of the vector of inputs. The maximum and minimum gains are obtained for changes in the socalled high-gain and low-gain directions, respectively.
209
210
Identification of an ill-conditioned distillation column: K, E. Haggblom and J. M. Boling
pilot-scale distillation column is identified by such an approach at two different operating points. At each operating point a nominal second-order plus time-delay model with logarithmic outputs is determined. In addition, models describing variations and uncertainties in the high- and low-gain directions around both operating points are determined. Such models are suitable for robust multimodel control designs 1°,11. It is known that a distillation column operated with the so-called L V-structure has a relative gain array (RGA) ~2 with diagonal entries larger than one 13. The models obtained have this property, which also means that they satisfy integral controllability requirements. In a related study made for comparison 8, models were determined from experiments where step changes were applied to the two inputs (reflux flow rate and flow rate of steam to the reboiler) one at a time. These models predict changes in the low-gain direction poorly and they do not satisfy integral controllability requirements. The models obtained by the use of traditional step tests are thus inadequate for control design in this case.
Gain directionality in distillation The conventional way of operating a distillation column is to control the product qualities by the reflux flow rate L and a variable V, which affects the boilup (e.g. the flow rate of steam to the reboiler), and to control the column inventories (condenser and reboiler holdups) by the product flow rates D and B. This control structure is known as the L V-structure. Figure 1 illustrates the L Vstructure of a distillation column with only inventory control loops closed. In the LV-structure, the product qualities are essentially controlled by the internal flows of the distillation column. Typically, this results in an ill-conditioned system characterized by a strong directionality and large relative gains ~4-j6. In particular, this applies to high-purity distillation. Figure 2 illustrates the gain directionality of a 15-tray pilot-scale distillation column operated with
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Figure 2 Contour plots of L(--), V(--) and D(- - -) in the composition plane the L V-structure. The product qualities are the distillate composition y and the bottoms composition x. The figure shows contour lines of constant reflux flow rate L, steam flow rate V and distillate flow rate D in the composition plane. The results have been obtained by means of a simulator, which was calibrated with actual plant data 14. In Figure 2 changes of the operating point from point A to points B, C and D are all of equal length in terms of the Euclidean norm of the output vector [AyAx] r. However, the required changes of the inputs L and V vary considerably. For a change from point A to D, the length of the input vector is 75 times the length of the input vector for a change from A to B. This means that the plant gains in these two directions differ by a factor of 75, which also indicates that the condition number of the plant is at least 75 (with unchanged variable scaling). It can be seen that the change from A to B is close to the high-gain direction while the change from A to D is close to the low-gain direction. This difference in gains is a consequence of the small angle between the contour lines for constant L and V, not due to the nonlinearity of the plant, which affects the curvature of the contour lines. Figure 2 also shows that the contour lines for constant distillate flow rate are essentially orthogonal to the contour lines for constant reflux and steam flow rate. This means that the product flow rates are almost unaffected by changes in the low-gain direction of the product compositions, whereas changes in the high-gain direction cause relatively large changes in the product flow rates. This is in agreement with previous observationslS 18 and it has also been proved mathematically 4,9. This fact makes it possible to estimate the high- and low-gain input directions of a distillation column with good accuracy from the gains between the product flow rates and the inputs L and V.
Estimation of high- and low-gain input directions Figure 1 L V-structure of a distillation column
A distillation column operated with the L V-structure has a static R G A with diagonal entries larger than
Identification of an ill-conditioned distillation column. K. E. Haggblom and J. M. Boling
one 13. For such a system, simple bounds on the highand low-gain input directions can be derived 4'9. By means of this result it can be shown that the corresponding input directions in distillation can be estimated with good accuracy from flow gains. Bounds on input directions
211
Remark 1: For ill-conditioned systems with X >> 1 the bounds in Equations (6) and (7) are tight since ~ - ~2 - $~X *
(g)
%~21 -- %~1I = ~21~. -1
(9)
Consider a 2 x 2 system described by the static model y=Ku,
K=
I Kll K21
K12] K22J
Estimation of input directions (1)
where u = [uj u2] T and y = [yl Y2] T a r e vectors of inputs and outputs, respectively, and K is a matrix of steadystate gains. The relative gain array for this system is given by
A=
,~ =
l
(2)
Z
K l l K22
K l l K22
(KI1K22 - KI2K21)
- - -
det(K)
(3)
The complete R G A for a 2 x 2 system is thus defined by a single parameter X, which will be referred to as the relative gain for the system. The system gain induced by the Euclidean vectornorm is defined IlYlI2 - [lull2 -
~
+ y2
V~
+ u2
(4)
The maximum gain ~ is obtained for some input fi, 11~[12 = 1, producing the output y, and the minimum gain a, is obtained for an input _u, I]_ul]2= 1, producing the output y. The condition number for the system is ~, = ~/_~. The input direction ~ and the parameters fl and ~2 are defined Kll ~2 K 2 1 = U~u2~' = ~2~2' &:
(5)
Without loss of generality it can be assumed that ~1 ~ ~2 since this can always be achieved by suitable permutations of the inputs and the outputs. Furthermore, it is assumed that each column and row K contains at least one non-zero element. Then the following theorem h o l d s 4,9.
Theorem 1: The high- and low-gain input directions and se. respectively, are bounded by ~l _< ~ _< ~2
(6)
-~/1 < ~ < -~21
(7)
The static behaviour of a distillation column operated with the L V-structure (Figure 1) can be described by the model
where y and x denote product compositions, L and V denote flow rates of reflux and steam flow to the reboiler, respectively. Under weak assumptions it can be shown by physical considerations that the system has a relative gain ,k > 1 (Ref. 13). When the inventory control loops are closed, the product flow rates D and B become dependent variables, and there exist static relationships AD
By means of material balances it can be shown that the composition gains in Equation (10) and the flow gains in Equation (11) are related as t~ KDI~ = -KBz~ =
1.
DKvL + BK, L "
(12)
3' -- X
KDV
KBV =
DKvv + BK,t )'
X
(13)
where D, B, y and x denote nominal steady-state values. The following theorem can be proved by means of Equations (12) and (13). Theorem 2: For a two-product distillation column with Z > 1, the product composition and flow gains satisfy the inequalities KvL
KDL
m < - - < Krv Kov
K~L ~<0 Kxv
(14)
The following Corollary of Theorems 1 and 2 implies that the high- and low-gain input directions can be estimated with good accuracy from the flow gains Kin, and KD v 4"9. Corollar.v: Estimates of the high- and low-gain input directions according to "
if and only i f Z < _ 0 o r Z >
KDV]
KDL ~__ -- KD V' -
KDV KDL
(15)
212
Identification of an ill-conditioned distillation column: K. E. Haggblom and J. M. BOling
respectively, are within the bounds stated by Theorem 1. The more ill-conditioned the distillation column is, the smaller are the margins of error. Remark 2: A change in the direction A L / A V = ~ does not change the product flow rates (cf. Equation (11)). This explains why the contour lines for constant D are essentially orthogonal to the contour lines for constant L and V in Figure 2.
Fundamental problems in identification of ill-conditioned plants In this section some important issues in identification for control of ill-conditioned plants are considered. In particular, it is demonstrated that the experimental design is of outmost importance. It is shown that an inadequate design easily results in a model that misrepresents the integral controllability properties of the plant. Experiments with a pilot-scale distillation column are used for illustration s . In the next main section, the identification of the distillation column is treated in more detail.
Identification from step changes in individual inputs The conventional way of exciting a distillation column is to make separate step changes in the reflux and steam flow rates. Such an experiment with the pilot column is shown in Figure 3. The figure shows step responses of the distillate composition y, the bottoms composition x, the distillate flow rate D as well as the sequence of step changes applied to the setpoints of the reflux and steam flow rates L and V, respectively. The composition responses are essentially of first order with some nonlinearity. Consequently, first-order plus time-delay models were fitted to the experimental data. The major part of the nonlinearities was captured 91.8
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Figure 3
C o n v e n t i o n a l step e x c i t a t i o n o f d i s t i l l a t i o n c o l u m n
by the use of logarithmic outputs. Figure 4, graphs (a) and (b), shows the composition responses from Figure 3 as well as the model fit (see B61ing and H/iggblom 8 for details). In another experiment (Figure 5), the distillation column was excited in the (approximate) low- and highgain directions. Graphs (c) and (d) in Figure 4 show the responses of this experiment and the responses predicted by the model determined from the previous experiment. As can be seen, the prediction is poor in the low-gain direction (t = 50-140 min). The model determined from step changes in the individual inputs has a relative gain k = -1.2 for diagonal variable pairing 8. However, it is known that a two-product distillation column operated with the LV-structure should have a relative gain ,k > 1 (Ref. 13) (note that the logarithmization does not affect the relative gain). Since this means that the gain matrix of the model has a determinant with the wrong sign (cf. Equation (3)), there is no controller with integral action that can stabilize both the model and the plant. A similar result was obtained when the column was identified from step changes in the individual inputs at another operating point 8 (operating point B, see below). It seems that this is not an uncommon problem in the identification of ill-conditioned plants. In a simulation study of a distillation column, Luyben 2° found that it was necessary to use step changes as small as 0.01% of the nominal values of the inputs to obtain a model whose relative gain has the correct sign. The main results of Luyben 2° are summarized in Table 1. The gains given in physical units in Table 1 are determined from the normalized gains given in Luyben 2°.
Identification from step changes in high- and low-gain directions As mentioned in the introduction and also indicated by Figure 4, it is particularly important to excite the lowgain direction in the identification of an ill-conditioned plant. In distillation, the directions of the input changes required for such an excitation can be estimated with good accuracy by Equation (15). For the pilot column, the flow gains KDL ~--0.65 and KDv ~ 1.3 can be estimated from the experiment shown in Figure 3 (operating point A, see below). ~ccording to Equation (15), this gives the estimates = - 0 . 5 and ~ = 2 of the high- and low-gain input directions, respectively. Figure 5 shows an experiment, where the inputs are changed almost simultaneously (L is changed 0.5min prior to V in order to compensate for the hydraulic delays in the column) according to the ratios ~ and ~ to excite the distillation column in the low- and high-gain directions. The sizes of the step changes were chosen to give outputs of suitable magnitude on the basis of the composition gains that can be estimated from Figure 3. Note the small input steps applied in the high-gain direction ( t = 140-250min). From the data of this
Identification of an ill-conditioned distillation column. K. E. Hdggblom and J. lid. Boling
92.0
213
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91.5 91.5 >,
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Figure 4
Model determined from individual step changes. Model fits (a) and (b) and predictions of low- and high-gain responses (c) and (d). Experiments ( ) and model outputs (...)
91.8 F ~91.4 >, 91.0 4.0
--
k
i
50
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i
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110" 1100E_
I
I
I
I
50
lOO
150
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~ 800
, 50
10o
150
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>
J
t (min)
Figure 5 Excitation of distillation column in low- and high-gain directions
Table 1 Composition gains and relative gains for different incremental sizes. Adapted from Luyben 2° Au%
K,.l.
Kvv
Kx/.
K~:v
k
1.000 0.500 0.100 0.050 0.010 0.005 0.001
0.4047 0.5115 0.6370 0.6556 0.6710 0.6729 0.6741
-1.1786 -0.9301 -0.6846 -0.6564 -0.6346 -0.6320 -0.6304
1.4819 1.3932 1.2797 1.2623 1.2479 1.2461 1.2444
-0.7353 --0.9953 -1.2366 -1.2634 -1.2839 -1.2865 -1.2887
0.2054 -0.6471 -8.9140 -3790.6 12.399 11.087 10.324
experiment, a model with the same structure as the previous model was determined. Graphs (a) and (b) in Figure 6 show the fit between the model and the experimental data. The ability of the model to predict the responses of the experiment in Figure 3 are shown in graphs (c) and (d). As can be seen, the predictions are nearly as good as the model fits in Figure 4, graphs (a) and (b). The results clearly indicate that the column should be explicitly excited in the high- and low-gain directions in control-relevant identification. Furthermore, they show that the corresponding input directions can be estimated with good accuracy from the flow gains Koc and Ko~,. This is also supported by the example by Luyben 2°. Table 2 shows the flow gains that can be calculated by means of Equations (12) and (13) from the composition gains in Table 1 and the steady-state data given by kuyben 2°. The condition number y and the high-gain input direction ~ determined from the gain matrix are also given. As can be seen, the flow gain ratio K m J K m provides an excellent estimate of the true high-gain input direction (~ = -0.9865) regardless of the step size used. In particular, it can be noted that this also applies when the relative gain of the model has the wrong sign (cf. Table 1).
Identification for control In this section a more thorough identification of the pilot-scale distillation column is made. The column is excited in its high- and low-gain directions as described in the previous section. A model structure that allows not only static but also dynamic directionality is used 17.
21 4
Identification of an ill-conditioned distillation column." K. E. Haggblom and J. M. BOling 92.0
92.0 a
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91.5
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Figure 6 Model based on excitation in low- and high-gain directions. Model fits (a) and (b) and predictions of responses to individual step changes (c) and (d). Experiments (--) and model outputs (...)
Table2
Condition number, high-gain input direction and flow gains for different incremental sizes Au%
y
~
Koc
Kov
KoL/Kov
1.000 0.500 0.100 0.050 0.010 0.005 0.001
2.5715 4.9568 45.710 14030. 58.313 51.933 48.186
-1.1466 -1.0975 -1.0113 -0.9991 -0.9893 -0.9880 -0.9865
-0.9858 -0.9942 -0.9991 -0.9996 -0.9999 -0.9999 -0.9996
0.9919 1.0006 1.0010 1.0005 1.0001 1.0001 1.0005
-0.9939 -0.9936 -0.9982 -0.9991 -0.9998 -0.9998 -0.9992
By a special technique, several models describing variations and uncertainties about an operating point can be estimated from the same experiment. Two different operating points are considered. The column, which separates a mixture of ethanol and water, has 15 bubble-cap trays and is 0.30m in diameter. It is equipped with a thermosyphon-type reboiler and a total condenser, and it operates under atmospheric pressure. An operability and controllability study of the column was made by Hfiggblom and Lehtinen 14, where a region of ill-conditioned operating points was determined. The column is further described by Waller 21. Model structure
It is well known that the step responses of a distillation column to changes in the internal flows are characterized by a large dominant time constant implying that the dynamics might well be described by first order transfer functions. On the other hand, it has also been noted in
simulation studies that the plant dynamics depend on the input direction resulting from simultaneous changes in the internal flows. The dynamics are relatively slow in the high-gain direction but tend to be much faster in the low-gain direction 15"17. Such a behaviour cannot be described by a model where the transfer functions are of first order only; it is required that the transfer functions are of second order (or higher). In accordance with this observation, a model structure consisting of second-order plus time-delay transfer functions is used. The experiment in Figure 5 does not indicate a particularly fast response in the low-gain direction, but a reason for this is that the time delays associated with the two inputs are different. In that case, the dominant time constant will always influence the observed responses to some extent. The composition responses in the identification experiments are nonlinear, especially for the bottoms composition. As noted in the previous section, part of the nonlinearities can be captured by using logarithms of the product impurity fractions as model outputs. The use of such logarithmic impurities as a means for linearizing the plant has been suggested previously 15,22. Another nice feature of the use of logarithmic impurities is that the parameter estimation results in a minimization of relative errors instead of absolute errors in the true impurities. This is useful when the impurities vary over some range which includes very small impurities. The selected model structure is
[ Aln(y~-y) ] Aln(x-xe)]
[G,*,L = LG*L
G*vv][AL] Gxv j
AV
(16)
Identification of an ill-conditioned distillation column: K. E. Haggblom and J. M. Boling
(. KyL,1
K~vL,2 )
G~*L' " = \TvL, ls + 1 + TvL,2S" + 1 e-°~Ls
(. /~vV,I
/~yV,2 )
G~*'v - = \ T y v , ls + 1 + 77v,2s + f e O,.vs Gxt =
(\T~L, IS + 1 +
T~L,2S + 1
)
(17)
e°x ,
(K>'_ G*~v = \Txv, is
+
1
7~,-v,2s+ 1
where the superscript * denotes that the gains apply for logartithmetic outputs instead of true compositions as in Equation (10). The offset parameters y,. and x~ should, in principle, have the values Yc = 0.96 (ethanol and water form an azeotropic mixture at this composition) and x,. = 0. However, the values y~ = 0.94 and x,. = -0.02 were found to give better model fits to the experimental data and are therefore used. The partitioning of the transfer functions as indicated in Equation (17) makes it easy to analyse the directionality properties and, if desired, to enforce constraints associated with these properties. The analysis which follows applies both for true compositions and logarithmic compositions. This follows from the fact that G~*,L/G~*.v = GytJG.,.v and GxL/G.*vv = GxL/Gxv, where GIRL,Grv, G.,-L and Gxv denote transfer functions with true compositions as outputs, which can be determined by a linearization of the logarithms in Equation (16). Assume that the time constant TI is larger than the time constant T2 in the transfer functions. If TI is large compared to T2 in all transfer functions, the known properties of distillation columns suggest that the Tls are approximately equal. Assume therefore that TvL,1 = Tvv, i (and T~c,l = Tvv,I). Neglecting for the moment the time delays, there is an input direction = A L / ~ x V = -K~.v.I/K~,L. 1 (~ = -K*~-v,I/K*~L,1) which cancels out the part of the transfer functions containing 7~.,.,1 and T~.v.i (T~L,I and Txv.i), resulting in a fast response for 3' (and x). If also T,,L,2= Z,.v,2 (7~L,2 = T~v,2), there is an input direction = z X L / A V = -ICvv?_/K~vL, 2 (~ = -/C~v,2//CxL,2) which cancels out the part of ihe transfer functions containing 77~.L.2 and TyV,2 (T~L,2 and T~v,2), resulting in a slow first-order response for y (and x). If 0,'L ¢ 0yv (Oxc ¢ Oxv), the input associated with the shorter time delay should be delayed relative to the other input by the difference between the time delays. In practice, it may not be possible to obtain fast responses in both outputs simultaneously if0v L Ovv¢ 0xL - 0xV. Sfigfors and Waller 17 suggest a model structure where the Tl s are equal and the ~ s are equal, that is, a model structure with only two different time constants T~ and T2. In addition, they assume that the slow and the fast dynamics are exactly aligned with changes in the external flows, that is, that KvL, l/Kvv, i = K ~ L I / K ~ v . J = - Kvr,2/ KrL,2 = --Kxv, z/ KvL,2 = KDL/ KDv. This assumption appears more realistic than an assumption that the dynamics are exactly aligned with the true high- and low-gain directions, which are scaling-dependent mathematical abstractions.
21 5
The assumptions above obviously result in a model with very ideal properties, which may be too unrealistic. In practice, a plant with slow and fast dynamics, depending on the input direction, can be described by a model with TvL, I ~ Tvv.l, 7~vL,I "~ T~j.L, and the Tjs significantly larger than the ~ s . Consequently, the model parameters do not have to satisfy any precise constraints. However, in the following some constraints are enforced so as to reduce the number of free parameters and to make the parameter estimation more robust. Estimation q f nominal model
In the first part of the identification experiment shown in Figure 5 (t = 5(>140min), the inputs are changed according to the ratio A L / A V = - - K m ' / K D L . If it is assumed that K~L.1/K~.r.I = K*~c.l/K*.,.~,l = KDL/KDv, 77~.C,1 = 7~rV,j and Z~L,I = T,-r,l, the latter part of the partitioned transfer functions in Equation (17) can be estimated from this part of the experiment. If it is further assumed that K~.L,2/Kvv. 2 = K]L,e/K*,.j,,2 = - K i n . /KDL, T~'L,2 = T~.v.2 and T~L2 = 7~v.2, there is only one independent parameter, in addition to the time delay, to estimate for the latter part of each transfer function. The first part of the transfer functions can be estimated in a similar way from the latter part of the experiment. A drawback of this approach is that the flow gains were determined from another experiment. As can be inferred from Figure 5, it was possible to excite, at least approximately, the high- and low-gain directions of the compositions by knowledge of these flow gains, but it can also be seen that the distillate flow rate is varying to some extent in the first part of the experiment. This means that applied input ratio A L / A V is not exactly equal to - K m , / K m for the actual experiment. Some of the underlying assumptions are therefore violated if the partitioning of the transfer functions is based on the flow gain ratio obtained from the previous experiment. To circumvent this problem, it is here only assumed that the slow and fast dynamics are excited by orthogonal input directions. It is not specified whether these directions correspond to the true high- and low-gain directions, to approximate high- and low-gain directions based on flow gains, or to some other orthogonal input directions. This is determined by the experimental data. The following constraints are therefore imposed on the nominal model:
K;L,I
K~'V,2
,
Tvr.I = T~.r,i,
T~.r.2 = Trv.2
K*~L,I
K*vv,2
K~xr,1 --
KvL. 2, ,
T,-L,I= T,)'.I,
T~I..2 = T,-),2
K ~
vr. 1
-
K)L.2
(18) The time delays were basically determined from step tests where one input at a time was changed as shown in Figure 3. Some minor adjustments of these initial values were made in order to improve the model fit. Only multiples of the sampling rate, 0.5 min, were considered.
216
Identification of an ill-conditioned distillation column."K. E. Haggblom and J. M. Boling
The column was identified at two different operating points, A and B, which are defined in Tables 3 and 4. The experiments made at operating point A are shown in Figures 3 and 5. A similar set of experiments were made at operating point B. Figure 7 shows the model fits obtained by minimization of the sum of squares of the output errors. As can be seen, the fits are reasonably good. The remaining discrepancies between the experiments and the model fits are mainly due to nonlinearities not captured by the use of logarithmic outputs. Numerical values of the model parameters are given in Tables 3 and 4 (model 'Nom.'). The gains K~ are clearly larger in magnitude than the gains K~ and the time constants T1 are also larger than T2. These differences are especially large for operating point B, implying strong directionality properties. As shown in Table 5, the relative gain for the nominal model is k = 7.73 at operating point A and )~ = 14.42 at operating point B. The condition numbers are ~,* = 120.0 and },* = 124.6, respectively. The low-gain input direction ~* determined from the gain matrix for the nominal model is 2.35 at operating point A and 2.36 at operating point B. These values differ from the input ratio A L / A V = 2 used to excite the column in the assumed low-gain direction based on the flow gains estimated from the experiment in Figure 3. The experiment in Figure 5 (operating point A) and the corresponding experiment at operating point
B gave slightly different flow gains. According to these experiments, the flow gain ratio --KDv/KDL is 2.38 at operating point A and 2.44 at operating point B. This experimental verification supports the claim that the high- and low-gain input directions can be estimated with sufficient accuracy from the flow gains. It can further be noted that the ratio -K~yv,1/K;yL, l is 2.94 at operating point A and 2.60 at operating point B, whereas the ratio --I~xv, I/IC~L, l is 2.47 at A and 2.48 at B. This means that the slow and the fast dynamics for the bottoms composition are well aligned with the highand low-gain directions, respectively. (This is not seen in the experiments because of the different time delays and the fact that the inputs were not changed exactly in the high- and low-gain directions.) For the distillate composition the alignment is less perfect, especially at operating point A. Here, the difference between the time constants is also smallest.
Table 3 Model parameters for operating point A (L = 140kghr i V = 9 6 k g h r i F = 1 5 0 k g h r t z=25%)
Table 4 Model parameters for operating point B (L = 160kghr ~, V= 1 0 5 k g h r i F = 1 5 0 k g h r I z = 2 5 % )
Multimodel estimation
For a robust control design it is desirable to have information about parameter variations and uncertainties in a model. One way to obtain such information is to determine several models that capture relevant phenomena in the desired operating region. There are control design techniques for which such a multimodel description is particularly well suited 1°,1~.
Model
K*~.L.I
K*,'t..e
TvL.I
7~,'L.2
OyL
Model
K*yt,.l
K*.,.L.2
TvL.1
TyL.2
OyL
Nora. l 2 3 4 5 6
-0.0175 -0.0175 -0.0175 -0.0175 -0.0154 -0.0192 0.0272
-0.00650 -0.00532 -0.00627 -0.00697 -0.00650 -0.00650 -0.00650
17.91 17.91 17.91 17.91 19.14 23.42 29.43
9.89 8.56 10.36 11.82 9.89 9.89 9.89
1.5 1.5 1.5 1.5 1.5 1.5 3.0
Nom. 1 2 3 4 5 6
-0.0368 -0.0368 -0.0368 -0.0368 -0.0370 -0.0369 -0.0560
-0.00549 0.00404 -0.00533 -0.00576 0.00549 -0.00549 -0.00549
25.16 25.16 25.16 25.16 33.16 25.32 47.82
5.55 5.23 5.49 7.79 5.55 5.55 5.55
1.5 1.5 1.5 1.5 1.5 4.0 6.0
Model
K*yv, l
K*~,r.:
7"~,v.l
T~4".2
Oyv
Model
K*yv.~
K*yv.:
Tvv.l
Tvv.2
Ova.
Nom. 1 2 3 4 5 6
0.0515 0.0515 0.0515 0.0515 0.0451 0.0564 0.0798
-0.00221 -0.00181 -0.00213 -0.00237 -0.00221 -0.00221 0.00221
17.91 17.91 17.91 17.91 19.14 23.42 29.43
9.89 8.56 10.36 11.82 9.89 9.89 9.89
1.5 1.5 1.5 1.5 1.5 1.5 3.0
Nom. 1 2 3 4 5 6
0.0956 0.0956 0.0956 0.0956 0.0960 0.0959 0.1453
-0.00211 -0.00156 -0.00206 -0.00221 -0.00211 -0.00211 -0.00211
25.16 25.16 25.16 25.16 33.16 25.32 47.82
5.55 5.23 5.49 7.79 5.55 5.55 5.55
1.5 1.5 1.5 1.5 1.5 4.0 6.0
Model
K",.L.I
K*vL,2
TtL,1
T~L,2
O.vL
Model
K*,.L.z
K*vL,2
T,-L.I
T,L.2
O.~L
Nom. 1 2 3 4 5 6
0.1256 0.1256 0.1256 0.1256 0.1158 0.1198 0.1466
0.00507 0.00339 0.00514 0.00468 0.00507 0.00507 0.00507
22.67 22.67 22.67 22.67 18.01 22.92 31.57
2.75 1.40 3.48 1.48 2.75 2.75 2.75
3.5 3.5 3.5 3.5 3.5 3.5 3.5
Nom. 1 2 3 4 5 6
0.1128 0.1128 0.1128 0.1128 0.1680 0.1286 0.1112
0.00453 0.00340 0.00506 0.00567 0.00453 0.00453 0.00453
30.98 30.98 30.98 30.98 45.61 30.59 38.11
3.60 1.35 6.23 1.38 3.60 3.60 3.60
3.5 3.5 3.5 3.5 5.5 3.5 3.5
Model
K*,-r./
K*xv.:
T,:v3
T~v.:
O~.r
Model
K*xv,l
K*xv,2
T~v,l
T~v.2
O~v
Nom. 1 2 3 4 5 6
-0.3100 -0.3100 -0.3100 -0.3100 -0.2856 -0.2956 0.3616
0.00205 0.00137 0.00209 0.00190 0.00205 0.00205 0.00205
22.67 22.67 22.67 22.67 18.01 22.92 31.57
2.75 1.40 3.48 1.48 2.75 2.75 2.75
3.0 3.0 3.0 3.0 3.0 3.0 3.0
Nom. 1 2 3 4 5 6
-0.2804 -0.2804 -0.2804 -0.2804 -0.4176 0.3196 -0.2765
0.00182 0.00137 0.00204 0.00228 0.00182 0.00182 0.00182
30.98 30.98 30.98 30.98 45.61 30.59 38.11
3.60 1.35 6.23 1.38 3.60 3.60 3.60
3.0 3.0 3.0 3.0 5.0 3.0 3.0
Identification of an ill-conditioned distillation column. K. E. Haggblom and J. M. B61ing
21 7
91.8 92.0 l
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>,
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Experiment (
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0.0 50
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) and nominal model fit (...) at two operating points A, (a) and (b), and B, (c) and (d)
Figure 7 shows that the nominal model fit is not perfect. Additional models can be obtained by fitting a separate model to the step responses of each simultaneous change in L and V (Figure 5). When a new model is fitted, the remaining effects of previous input changes can be taken into account by the previously fitted models. (Note that steady state is not reached between input changes.) A problem with this approach is that the effects of the two inputs cannot be separated. Furthermore, the number of parameters of each transfer function is probably too large to allow unique parameter estimates.
The latter problem could be solved by using a simpler model structure (e.g. first-order transfer functions), but here it is desired to retain the original model structure. This is accomplished as follows. Consider the experiment shown in Figure 5. The first three simultaneous step changes mainly excite the lowgain direction. This means that the first parts of the partitioned transfer functions defined in Equation (17) are almost completely cancelled out. Since the remaining effects of these parts are small, it is assumed that they can be described by the corresponding nominal parameters (K~s and Tls). Furthermore, the constraints 91.8
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C
i
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>, 91.5:
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i
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, 91.2 ~ 5O 1.5!
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1.0i 04
2.0
x
0.0 50
Figure 8
Experiment (
1O0
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200
250
50
) and multimodel fit (...) at two operating points A, (a) and (b), and B, (c) and (d)
218
Table 5
Identification of an ill-conditioned distillation column: K. E. Haggblom and J. M. B Oling
a multimodel description can be used for a robust control design.
Directionality properties of estimated models Operating point A
Operating point B
Model
2.
F*
se*
,k
•*
se*
Nom. 1 2 3 4 5 6
7.73 11.01 8.39 6.57 6.11 9.72 35.70
120.0 179.4 131.6 100.1 96.0 135.8 470.2
2.35 2.39 2.35 2.35 2.33 2.34 2.37
14.42 24.05 16.52 15.21 12.11 13.49 55.56
124.6 214.8 143.4 130.9 146.3 129.6 372.2
2.36 2.39 2.35 2.33 2.40 2.37 2.36
defined in Equation (18) are enforced. In addition to the time delays, the model for each output then contains only two independent parameters, which can easily be estimated. The original time delays are also adjusted if this improves the fit. The last three simultaneous step changes excite the high-gain direction of the compositions. Here it can be assumed that the uncancelled effects of the second parts of the partitioned transfer functions, which are related with the low-gain direction, can be described by their nominal parameters,/C2s and T2s. When Equation (18) is taken into account, the parameters in the first parts of the transfer functions can be estimated. As above, the time delays are adjusted if this improves the fit. The model fits obtained in this way are shown in Figure 8. The estimated parameters of the seven models at operating points A and B are given in Tables 3 and 4, respectively. Table 5 shows that the relative gain and the condition number vary for the models, but that the lowgain input direction is almost unaffected.
Conclusions Control-relevant models for an ill-conditioned distillation column operated with the L V-structure have been determined through identification. Fundamental knowledge of directionality properties of distillation columns was employed in the design of identification experiments. In this way it was possible to explicitly excite the (approximate) high- and low-gain directions of the column. It was shown that excitation of the lowgain direction is particularly important in control-relevant identification of ill-conditioned plants. The models consist of second-order plus time-delay transfer functions. This model structure makes it possible to capture known (ideal) properties of distillation columns with regard to slow and fast dynamics in different input directions. The estimated parameters are in good agreement with theoretical expectations. The models also satisfy known integral controllability requirements. As shown, this is not a trivial achievement for an ill-conditioned nonlinear system. The design of the identification experiment made possible the estimation of multiple models by a special technique. The set of models describe variations and uncertainties associated with the high- and low-gain directions as well as with slow and fast dynamics. Such
References 1. Grosdidier, P., Morari, M. and Holt, B. R., Closed-loop properties from steady-state gain information. Industrial & Engineering Chemistry Fundamentals, 1985, 24, 221-235. 2. Skogestad, S. and Morari, M., Implications of large RGA elements on control performance. Industrial & Engineering Chemistry Research, 1987, 26, 2323 2330. 3. Yu, C.-C. and Luyben, W. L., Robustness with respect to integral controllability. Industrial & Engineering Chemistry Research, 1987, 26, 1043 1045. 4. H~iggblom, K. E., Static directionality of 2 × 2 systems: a control relevant property with application to distillation. Proceedings of the European Control Conference, Rome, 1995, pp. 3117 3122. 5. Koung, C.-W. and MacGregor, J. F., Design of identification experiments for robust control. A geometric approach for bivariate processes. Industrial & Engineering Chemistry Research, 1993, 32, 1658 1666. 6. McDonald, K. A., Palazoglu, A. and Bequette, B. W., Impact of model uncertainty descriptions for high-purity distillation control. A IChE Journal, 1988, 34, 1996-2004. 7. Andersen, H. W. and Kfimmel, M., Evaluating estimation of gain directionality, Parts 1 and 2. Journal of Process Control, 1992, 2, 59-66, 67 86. 8. B61ing, J. M. and H/iggblom, K. E., Control-relevant identification of an ill-conditioned distillation column. Proceedings of the IEEE Conference on Control Applications, Dearborn, MI, 1996, pp. 570~575. 9. H~iggblom, K. E., Proof of the relation between singular directions and external flows in distillation. Proceedings of the IEEE Conference on Decision and Control, New Orleans, LA, 1995, pp. 2502 2504. 10. Pensar, J. A. and Toivonen, H. T., A multimodel mixed H2/H~ problem for plants with structured uncertainty. Preprint of IFAC Symposium on Advanced Control of Chemical Processes (ADCHEM'94), Kyoto, Japan, 1994, pp. 289-294. 11. Sandelin, P. M., Toivonen, H. T., Oster~s, M. and Waller, K. V., Robust multiobjective linear quadratic control of distillation using low-order controllers. Chemical Engineering Science, 1991, 46, 2815 2827. 12. Bristol, E. H., On a new measure of interactions for multivariable process control. IEEE Transactions on Automatic Control, 1996, AC-11, 133 134. 13. H~iggblom, K. E., Analytical relative gain expressions for distillation control structures. In Consistent control structure modelling with application to distillation control. Dr. Tech. thesis, Part V, Abo Akademi University, Finland. 14. H~iggblom, K. E. and Lehtinen, K., Investigation of operability and controllability properties of a pilot-scale distillation column. Preprint of IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns, and Batch Processes (DYCORD + "95), Helsingor, Denmark, 1995, pp. 339-344. 15. Skogestad, S. and Morari, M., Understanding the dynamic behaviour of distillation columns. Industrial & Engineering Chemistry Research, 1988, 27, 1848 1862. 16. Skogestad, S., Morari, M. and Doyle, J. C., Robust control of illconditioned plants: high-purity distillation. IEEE Transactions on Automatic Control, 1988, 33, 1092 1105. 17. S~tgfors, M. F. and Waller, K. V., Dynamic low-order models for capturing directionality in non-ideal distillation. Industrial & Engineering Chemist O, Research, 1995, 34, 2038-2050. 18. Shinskey, F. G., Distillation Control, 2nd edn. McGraw Hill, New York, 1984. 19. H~iggblom, K. E. and Waller, K. V., Transformations and consistency relations of distillation control structures. AIChE Journal, 1988, 34, 1634-1648. 20. Luyben, W. L., Sensitivity of distillation relative gain arrays to steady-state gains. Industrial & Engineering Chemistry Research, 1987, 26, 2076-2078. 21. Waller, K. V., Experimental comparison of control structures. In Practical Distillation Control, ed. W. L. Luyben. Van Nostrand Reinhold, New York, 1992, pp. 313-330. 22. Ryskamp, C. J., Explicit versus implicit decoupling in distillation control. In Chemical Process Control 2, ed. D. E. Seborg and T. F. Edgar. United Engineering Trustees, New York, 1982, pp. 361-375.