- Email: [email protected]
Identification for Control of High-Purity Distillation Columns - A Benchmark Problem
Copyright © rFAC Dynamics and Control of Chemical Reactors (DYCORD+'95), Copenhagen, Denmark, 1995
IDENTIFICATION FOR CONTROL OF HIGH-PURITY DISTILLATION COLUMNS - A BENCHMARK PROBLEM ELLING W. JACOBSEN" S3 - Automatic Control Royal Institute of Technology - KTH , 100 44 Stockholm, Sweden
Abstract. The note presents a high-purity distillation column which is proposed as a benchmark problem for identification of multivariable dynamic models intended for control applications . The column is ill-conditioned, i.e., has strong gain directionality, and is also strongly interactive. This poses a specific problem in identification in that it is difficult to obtain a reasonable model for the low-gain direction of the plant . The low-gain direction is usually hardly visible in open-loop responses but becomes important under feedback control. Furthermore , the column has a single slow pole (time-constant) which easily may become repeated in the identified model , causing a poor prediction of the control behavior of the column , in particular under partial feedback control. The note includes a detailed description of the example column and the "experimental" conditions available for identification. A specification of the intended use of the identified model is also included and it is stressed that an important control objective is disturbance rejection . Thus, a disturbance model is desired in addition to the usually considered input model. At the end of the note an example is presented which demonstrates that a classical identification method involving independent PRBS signals and ARMAX-type MISO models yields a multi variable model which is poor for feedback controi applications. Keywords. Identification , multivariable, ill-conditioned, interactive, distillation .
1.
INTRODUCTION
Tight composition control of distillation columns usually requires some kind of model-based control scheme and thus the availability of a dynamic model , typically linear . For distillation columns operating with reflux L and boilup \/ as independent inputs , i.e., LVconfiguration, the desired linear model is usually written
The main difference between SISO and MIMO plants is the presence of "interactions" or "directions" in the latter case, i.e ., the gain of the plant depends on the direction of the input vector. The degree of directionality and in~eractions is usually evaluated through the Singular Value Decomposition (SVD) and the Relative Gain Array (RGA) (see e.g. , Skogestad and Morari, 1988) . Plants for which the ratio of the largest to smallest singular value, i.e., the condition number , is large are noted ill-conditioned while those with large values of the RGA elements are denoted strongly interactive. Note that strongly interactive plant always are ill-conditiolil.ed while the opposite does not necessarily apply.
ae-
dYD) =CL V ( s) (dL) ( dXB dV
(1)
where YD and XB denote product purities. In practice, the model C LV (s) must , at least partially, be obtained through experiments on the plant. The traditional approach has been to obtain the individual transfer-matrix elements gt v (s) independently, e.g., by fitting openloop responses of the plant to first-order-plus-deadtime transfer-functions. However , several authors (e.g., Skogestad and Morari , 1988) have pointed out that this may be a poor approach if the model is intended for control applications. The reason is that one through this approach usually is unable to capture specific multivariable properties of importance for control. "E-mail : [email protected] .se
High-purity distillation columns operating with the L"'configuration are usually strongly interactive. Typically, increasing L and V such that the product flows remain unchanged increases the purity of both products and the corresponding gain is generally small. This is usually denoted changes in the internal flows and corresponds to the low-gain direction of the plant . Similarly, changing L and V such that there is a maximum change in the product flows makes one product purer and the other less pure and the corresponding gain is generally large. This is usually denoted changes in the external flows and corresponds to the high-gain direction of the
317
plant. A multi variable model intended for control purposes should capture these effects in a reasonable way. However, it is well known that if the interactions are strong then this may be a difficult task. One reason is that the open-loop responses of the plant tend to be dominated by the high-gain direction which renders it difficult to obtain a reasonable model for the low-gain direction . Although often hardly visible in open-loop responses, the low-gain direction becomes important under feedback control and it is therefore crucial to obtain a good model also for this direction of the plant. We start the note with a brief review of some of the works published on identification of strongly interactive plants in general and high-purity distillation columns in particular. We then present a linear distillation column which is proposed as a benchmark problem for which different methods may be tested and compared. The motivation behind using a linear model as the "plant" is that the problem of capturing a plants directionality is present both in the linear and nonlinear plant, and we do not want to confuse this problem with the specific problems caused by nonlinearities. In addition to presenting the model we specify the experimental conditions under which the identification should be performed. The identified model is intended for control applications and it is stressed that an important control objective is disturbance rejection. Thus, a disturbance model is desired in addition to the usually considered input model G LV (s) (1). At the end of the note we apply a standard identification method involving independent PRBS signals and MISO ARMAX models to the benchmark problem . The resulting model is shown to be poor for control applications.
2.
A BRIEF REVIEW
Over the years a number of papers discussing the problem of identifying dynamic models for strongly interactive plants in general, and distillation columns in particular, have appeared. We do not intend to provide a complete review of these here, but mention some of the more recent works only. All the papers mentioned either directly address the problem of identifying distillation columns or employ a distillation column as an example l . As mentioned above , the traditional approach to identification of distillation columns has been to obtain the transfer-matrix elements independently, i.e., through SISO identification . The main reason for this is undoubtedly that the general literature on system identification has been focused on identification of SISO systems while identification of MIMO systems has been considered a difficult problem. However, Skogestad and Morari (1988) argue that one should be careful about identifying the individual SISO elements of the 1 In fact , we have not found any papers addressing this problem that do not mention distillation.
transfer-matrix of a high-purity distillation column independently as this easily may lead to a poor model for the low-gain direction of the plant . In particular , the identified model will easily have the wrong sign of the determinant at steady state and will in that case be useless for control. Andersen et al. (1989) demonstrate through an example that SISO identification yields a poor model of the plants low-gain direction . Andersen and Kiimmel (1991) consider identifying two MISO models and find that the resulting MIMO model also in this case yields a poor prediction of the plants low-gain direction . They also consider using closed-loop identification with a single loop closed combined with MISO identification but find similar results as with open-loop identification. However, by low-pass filtering the closedloop data prior to fitting the model they obtain a reasonable model for the low-gain direction of the plant . Jacobsen et al. (1991) propose to use SISO identification and fit the initial responses (high-frequency behavior) only. The motivation behind this suggestion is that it is the initial response which usually is most important for feedback control and that distillation columns generally are significantly less interactive at high and intermediate frequencies than at steady state. To avoid problems with the wrong sign of the determinant of the steady state gain matrix they propose to correct the steady state gains to match an estimated value of the Relative Gain Array. They find that this approach yields models reasonable for multi variable control applications with relatively high bandwidth. Several authors (e.g., Skogestad, 1988; Alsop and Edgar, 1990; Andersen and Kiimmel, 1991) suggest that, for identification, it may be better to use the DV -configuration instead of the LV-configuration. The main motivation behind this is that with the DVconfiguration the low-gain direction is closely aligned with changes in the input V only, and this may be utilized to obtain a better model for the low-gain direction. In particular, the determinant of the steady state gain matrix is less likely to have the wrong sign when obtained with the DV-configuration. Koung and MacGregor (1993 , 1994) extend this idea to systems where the low-gain direction is not well aligned with a single input, e.g., LV-configuration . The main idea is to utilize a priori knowledge of the plants directionality to obtain better information for the low-gain direction . Koung and MacGregor (1993 , 1994) consider the problem of obtaining a model with the correct sign of the determinant of the steady state gain matrix only. Jacobsen (1994) considers several different approaches to identification of strongly interactive plants and stress that, in order to ensure a model suitable for control applications, both reasonable experimental data for all plant directions as well as a proper multi variable algorithm for fitting the data to a model are required. In cases where knowledge of the plant directionality is available he proposes to use an open-loop experimental design similar to that proposed by Koung and MacGre-
318
gor (1993) . It is demonstrated that this yields satisfactory data for the low-gain direction of the plant , not only at steady state but also at intermediate and high frequencies , even when the knowledge of directionality is somewhat uncertain. In cases where knowledge of the plant directionality is poor or even lacking , Jacobsen (1994) considers using closed-loop identification with a single proportional controller as proposed by Andersen and Kiimmel (1991) and shows that any controller gain will yield a closed-loop plant which is less interactive than the open-loop plant . This may be utilized to ensure a sufficient excitation of all plant directions , including the low-gain direction of the open-loop plant . Jacobsen (1994) suggests that the poor results obtained by Andersen and Kiimmel (1991) is due to the fact that they fit the obtained data using two MISO models. By fitting the obtained data to a MIMO model Jacobsen (1994) find that a good model is obtained both for the high- and low-gain direction of the plant. A subspace based identification algorithm (e.g. , Van Overschee and de Moor, 1994), which requires no a priori knowledge of the model structure, is proposed for fitting the data to a model.
have been proposed in the literature . However, most authors demonstrate the goodness of their method through some specific example and it is difficult to make a comparison of different methods. Furthermore, most papers include little or no information on the experimental conditions , e.g., noise levels , used in applying their method for demonstration. Finally, few papers actually demonstrate the quality of their model for feedback control. The purpose of the present note is to propose a benchmark problem for which different proposed methods may be tested and compared. A similar benchmark problem, based on a simplified heat-exchanger , has previously been presented in Jacobsen and Skogestad (1994b). However, the previous problem was less interactive than most high-purity distillation columns and furthermore included predefined PRBS signals to be used in the experiments, thereby not allowing for specific experimental designs to be applied. In this note we present a more "flexible" problem for which experimental designs , closed-loop identification schemes etc . may be tested.
3. Varga and J0rgensen (1994) apply the experimental design method suggested by Koung and MacGregor (1993, 1994) to a high-purity column . They use the obtained data to fit a MIMO Output Error model with a prespecified structure and find that the low-gain direction is reasonably captured in the resulting model. Li and Lee (1994) propose to identify both the openloop plant as well as its inverse and then combine these into a single model. The motivation behind this approach is that the inverse of the plant generally will yield good information about the low-gain direction of the open-loop plant . In order to identify the approximate inverse of the plant they employ four separate closed-loop experiments using four different controllers. Both the open-loop plant and its inverse are fitted using SISO identification methods. All the papers mentioned above discuss the problem of identifying a reasonable model for the low-gain direction of the plant. A different , although related, problem in identification of strongly interactive plants is discussed in Jacobsen and Skogestad (1994a). A strongly interactive plant will usually contain a single slow pole (time-constant) which tends to dominate all open-loop responses of the plant. Fitting each SISO transferfunction independently, such that they all contain the slow pole , will generally result in a MIMO model with this pole repeated, and hence an inconsistent model. This inconsistency results in a poor prediction of the plants control behavior, in particular under partial control. Jacobsen (1994) find that the only way to avoid this problem is to employ a true MIMO method for fitting the experimental data to a model. As seen from the above, several different methods for identification of dynamic models for distillation columns
PROBLEM DESCRIPTION
The column we consider is separating a binary mixture of ethylene and ethane (C2-splitter) , has purities of 99% in both products and is operated with mass reflux Lw [kg/min] and reboiler heat input Q B [MJ /min] as independent inputs, i.e., LwQ B-configuration. This configuration is similar to the LV -configuration. Data for the column are summarized in Table 1. In the nonlinear dynamic model we assume constant relative volatility Q , vapor-liquid equilibrium, perfect mixing on each tray, constant pressure, negligible vapor holdups and saturated reflux and feed flows. The liquid flow dynamics are modeled using a linearized Francis Weir formula. A simplified static energy-balance is included in the model. We neglect changes in liquid enthalpy with composition (pure components as saturated liquids at column pressure are chosen as reference of state) and assume a linear relation for the vapor enthalpy. With these assumptions we obtain a model with two states per tray, expressed as fraction of light component and total liquid holdup. As the "plant" we employ a linear model obtained through linearization of the full non linear model around the nominal operating point . The linearized model. which contains a total of 52 states, is available from the author via e-mail. The linear model is wTitten (2)
where G(s) is a 2 x 3 transfer-matrix expressing the effect of small changes in the manipulated variables Lu.' and QB and disturbance Fw on the product compositions YD and XB· The motivation behind employing a linear model as the "plant" is that we mainly are concerned with linear aspects here and do not want to con-
319
25
13
0.5
29
69 .99
33.23
XB
0.99
om
- The manipulated inputs are uncertain such that dL w
=
0.95 dL w c 8+1
where subscript c denotes command signal . This "uncertainty" applies during identification only, that is, it should not be considered an inherent part of the plant .
in mole fractions of light component . • L w and Fu' in [kg/ min] . QB in [MJ/min] . • Reboiler is tray 1. • Feed is saturated liquid. • Total condenser with saturated reflux. • Liquid holdups are Mi = 0.5 kmol , including condenser and reboiler . • Flow dynamics included , hydraulic time constant TL = 0.067 min . • Constant pressure P = 12 atm . • Liquid enthalpy H dxd = O. Reference state: pure components as saturated liquids at column pressure . • Vapor enthalpy H v (x,) = 12.45 - 1.39x, . • ZF ,
YD and
1.75
YD
XB
- The feed flow Fw may be manipulated during identification and the maximum allowed perturbation is 5.0 kg/min. The disturbance model may be obtained through a separate experiment if desired. - The distillate flow Dw [kg/min] is measured with uncertainty dD = 0.95 dD wm
8
+1
w
where subscript m denotes measured. - The temperature on tray 20 is measured in DC and contains normally distributed zero mean white noise with standard deviation (Je = 0.I5 D C .
Table 1. Data for example column separating a binary mixture of ethylene and ethane.
- If feedback control of the product compositions is used during identification then a measurement delay of 5 min . should be assumed . fuse matters. Furthermore, most identification methods aim at identifying linear models . Undoubtedly, identifying a linear model from a nonlinear plant will prove harder than from a linear plant , but we believe the latter problem needs to be solved before attacking the former problem . However , for those interested , a nonlinear model is also available from the author upon request . The singular values and the 1, I-element of the RGA for the input model , Le., with Lw and Q B as inputs, are shown as functions of frequency in Figures 2 and 3 (solid lines). At steady state the condition number is 214 and the 1, I-element of the RGA is 37. Thus, the plant is ill-conditioned and also strongly interactive. However , as seen from Figure 2 and 3, the plant is less ill-conditioned and interactive at intermediate and high frequencies than at steady state. This is typical for high-purity distillation columns . The "experimental" conditions available for identification are: - The two product compositions YD and XB are sampled with a period of 1 min . The total number of samples should not exceed 200 . - The composition measurements contain normally distributed zero mean white noise with standard deviation (J e = 0.0005. - The maximum allowed deviation in both product compositions is 0.01 (the main purpose of this specification is to limit the signal-to-noise-ratio and may, for instance, be satisfied by simply scaling the obtained noise-free outputs and inputs accordingly.). - The maximum allowed perturbation in the manipulated inputs are 5 kg/min for the reflux Lu' and 4 MJ /min for the heat input QB
An example file for generating experimental data with Matlab according to the above specifications is available from the author via e-mail. The challenge is to identify a model (2) based on the obtained experimental data, possibly combined with the knowledge that the object is a distillation column separating ethane and ethylene at the given pressure . The intended use of the model is for feedback control applications. Set point changes as well as feed flow disturbance rejection are considered important . The maximum expected setpoint change is 0.01 for both product compositions and the maximum expected disturbance in the feed flow rate is 5 kg/min. Both product compositions are measured with a delay of 5 minutes. Control of a single product composition (one-point control) as well as control of both product compositions (twopoint control) are of interest . In both cases the predicted responses in both product compositions should be considered and compared with those obtained v.:ith the "plant". To allow for a judgement of the ability of a proposed method to identify a reasonable model of the disti.Jlation column it is desired that some kind of uncertainty bounds are presented along with the nominal model. The uncertainty bounds may for instance be estimated through Monte Carlo type simulations.
4.
MISO-IDEKTIFICATION USING AN ARMAX-TYPE MODEL
In this section we employ a fairly standard identification technique to the distillation column presented above. We employ the Matlab System Identification Toolbox (Ljung, 1991) and use MISO-identification with an ARMAX-type model structure. We choose to identify
320
the input model and disturbance model using a single experiment . Independent PRBS signals are applied to the three inputs Lwe , QBe and Fw with amplitudes 2, 1 and 2, respectively, a minimum switching time of 10 min. and a switching probability of 0.5. The measurements of the product flow D1JJ and the tray temperature are not utilized. For each output we fit a strictly proper fifth order model which is determined close to optimal based on the Final Prediction Error (e.g., Ljung , 1991). Figure 1 shows the resulting fit of the magnitude for the transfer-function from Lw to YD and we see that the experimental conditions allows for a reasonable fit of this transfer-function. Although not shown here, the other five transfer-functions of the 2 x 3 transfer-matrix (2) are fitted with similar precision.
inconsistent in terms of the number of slow poles and will yield a poor prediction of the plant under one-point control (Jacobsen and Skogestad, 1994a) .
'0'
'0'
,o~
,o~
10· t
W
10'
[rad/min]
Fig. 3. Magnitude of 1, I-element of RGA for fitted input model (dashed) and plant (solid) . 1D~
10·, r-----~----~----_,
104
..
,,
,o ~
,,
, ,o~
,o~
,o~
10. 1
'0'
,o~
w [rad/min] Fig. 1. Magnitude of transfer-function from Lw to YD for fitted model (dashed) and plant (solid).
'O~ 'O~
'O~
10· '
'0'
w [rad/min] Fig. 4. Magnitude of Closed-Loop Disturbance Gain from Fw to YD for fitted model (dashed) and plant (solid). ,o~
,o~
Figure 4 shows the resulting fit of the Closed Loop Disturbance Gain (Hovd and Skogestad, 1994) which takes into account the relative directions of the disturbance model and the input model and we see that we also have a relatively poor multi variable model for the effect of disturbances.
.(J ..
,o~L-----.......,...----_------' 04 10
10-a
10- '
W
10'
[rad/min]
Fig. 2. Singular values for fitted input model (dashed) and plant (solid) . Figure 2 shows the fit of the singular values for the input model (Lw and Q B as inputs) and we see that , while the high-gain direction is well fitted , the low-gain direction is poorly fitted, in particular at low and intermediate frequencies . This is reflected in the poor fit of the RGA shown in Figure 3. The poor fit of the low-gain direction is at least partially explained by the use of two MISO models in fitting the data (Jacobsen , 1994) . From Figure 2 it is also seen that the low-gain direction of the identified model contains a slow pole at approximately 0.01 rad/min , which in the plant belongs exclusively to the high-gain direction. Thus, the identified model is 321
Based on the model analysis presented above we would expect that the model is of poor quality for control applications. Indeed , using the identified model to design a two-point controller using H (Xl-optimal design we predict the set point response shown by the dashed lines in Figure 5. The solid lines show the corresponding response when applying the same controller to the "plant", i.e., the 52 state full linear model , and we see that the identified model yields a poor prediction of the plant behavior also under two-point control. In particular, the controller when applied to the plant yields a very slow settling towards steady-state. Similar results are found for responses to disturbances in F", . Note that we have not presented the requested error bounds, e.g., for the singular values, here. The reason is that even the nominal model is of such a poor quality
Andersen, H.W . and M. Kummel (1991) "Identifying Gain Directionality of Multivariable Processes", Proc. of ECC'91, Grenoble, France, July 1991.
0.012
., ,, , ,
0.01
YD
,
0._
Hovd, M. and S. Skogestad (1994) "Simple Frequency Dependent Tools for Control-System Analysis, Structure Selection and Design" , Automatica, 28 , 5, 989-99
0.006
0.004
Jacobsen, E .W . (1994) "Identification for Control of Strongly Interactive Plants", Preprints NATO ASI on Model Based Control, Antalya, Turkey, August 1994. Also: paper 226ah , 1994 AIChE Annual Meeting, San Francisco, November 1994.
0.002
- - -.::r;.B 0.00
-.002
--- -,. --.- -- ... - - --
,, ,
-.004
100
50
Jacobsen , E .W., P . Lundstrom and S. Skogestad (1991) "Modelling and Identification for Robust Control of IIIConditioned Plants - A Distillation Case Study", Proceedings of 1991 American Control Conference, Boston, MA , 242-248.
150
time [min] Fig . 5. Closed-loop response for set-point change in YD with fitted model (dashed) and plant (solid).
Jacobsen, E .W. and S. Skogestad (1994a) "Inconsistencies in Dynamic Models for Ill-Conditioned Plants: Application to Low-Order Models of Distillation Columns", Ind.Eng. Chem .Res, 33, 3, 631-640.
that error bounds are of little interest in this case.
5.
CONCLUSIONS
• We have presented a high-purity distillation column which is proposed as a challenge problem for identification of multi variable models intended for feedback control applications. • A detailed description of the experimental conditions is provided along with a model of the column which is available from the author via e-mail. • A standard identification method involving independent PRBS signals and MISO ARMAXmodels were shown to yield a model poor for control applications.
FURTHER INFORMATION
Jacobsen, E .W . and S. Skogestad (1994b) "Identification of Dynamic Models for Ill-Conditioned Plants - a Benchmark Problem" , in the Modeling of Uncertainty in Control Systems, Eds. R. Smith and M. Dahleh, Springer-Verlag, London. Koung, C .W. and J.F. MacGregor (1993) "Design of Identification Experiments for Robust Control. A Geometric Approach for Bivariate Processes", Ind.Eng.Chem.Res ., 32, 1658-1666. Koung, C.W. and J.F. MacGregor (1994) "Identification for Robust Multivariable Control: the Design of Experiments", Automatica, 30, 10, 1541-1554. Li, W. and J.H. Lee (1994) "Closed-Loop Identification of Ill-Conditioned Multivariable Systems", Paper 226ag, 1994 AIChE Annual Meeting, San Francisco, November 1994~
The following is available from the author upon request (e-mail: [email protected]) : - A linear state space model (in ASCII format) for generating noise-free data.
Ljung, L. (1992) System Identification Toolbox Version 3.0a, Math Works Inc . Skogestad, S (1988) "Disturbance Rejection in Distillation Columns", Preprints CHEMDATA '88, Gothenburg , Sweden , 365-371. Published by IVA, Stockholm, Sweden.
- An example file for generating experimental data according to the specified experimental conditions using the linear model and Matlab .
Skogestad, S. and M. Morari (1988) "Understanding the Dynamic Behavior of Distillation Columns", Ind. fj Eng. Chem . Res, 27, 10, 1848-1862 .
- A non linear model for use with Matlab/Simulink. REFERENCES Alsop , A.W. and T.F . Edgar (1990) "Nonlinear Control of a High-Purity Distillation Column by the use of Partially Linearized Control Variables" , Comp. Chem.Eng., 14, 665-678. Andersen, H .W ., M. Kummel, and S.B. Jorgensen (1989) "Dynamics and Identification of a Binary Distillation Column", Chem .Eng.Sci. , 44 , 2571-2581. 322
Van o verschee , P. and B. de Moor (1994) "N4SID: Subspace Algorithms for the Identification of Combined Deterministic-Stochastic Systems" , Automatica, 30 , 1, 75-94. Varga, E.!. and S.B . J0rgensen (1994) "Multivariable Process Identification: Estimating Gain Directions" , Paper 229f, 1994 AIChE Annual Meeting, San Francisco, November 1994.
IDENTIFICATION FOR CONTROL OF HIGH-PURITY DISTILLATION COLUMNS - A BENCHMARK PROBLEM ELLING W. JACOBSEN" S3 - Automatic Control Royal Institute of Technology - KTH , 100 44 Stockholm, Sweden
Abstract. The note presents a high-purity distillation column which is proposed as a benchmark problem for identification of multivariable dynamic models intended for control applications . The column is ill-conditioned, i.e., has strong gain directionality, and is also strongly interactive. This poses a specific problem in identification in that it is difficult to obtain a reasonable model for the low-gain direction of the plant . The low-gain direction is usually hardly visible in open-loop responses but becomes important under feedback control. Furthermore , the column has a single slow pole (time-constant) which easily may become repeated in the identified model , causing a poor prediction of the control behavior of the column , in particular under partial feedback control. The note includes a detailed description of the example column and the "experimental" conditions available for identification. A specification of the intended use of the identified model is also included and it is stressed that an important control objective is disturbance rejection . Thus, a disturbance model is desired in addition to the usually considered input model. At the end of the note an example is presented which demonstrates that a classical identification method involving independent PRBS signals and ARMAX-type MISO models yields a multi variable model which is poor for feedback controi applications. Keywords. Identification , multivariable, ill-conditioned, interactive, distillation .
1.
INTRODUCTION
Tight composition control of distillation columns usually requires some kind of model-based control scheme and thus the availability of a dynamic model , typically linear . For distillation columns operating with reflux L and boilup \/ as independent inputs , i.e., LVconfiguration, the desired linear model is usually written
The main difference between SISO and MIMO plants is the presence of "interactions" or "directions" in the latter case, i.e ., the gain of the plant depends on the direction of the input vector. The degree of directionality and in~eractions is usually evaluated through the Singular Value Decomposition (SVD) and the Relative Gain Array (RGA) (see e.g. , Skogestad and Morari, 1988) . Plants for which the ratio of the largest to smallest singular value, i.e., the condition number , is large are noted ill-conditioned while those with large values of the RGA elements are denoted strongly interactive. Note that strongly interactive plant always are ill-conditiolil.ed while the opposite does not necessarily apply.
ae-
dYD) =CL V ( s) (dL) ( dXB dV
(1)
where YD and XB denote product purities. In practice, the model C LV (s) must , at least partially, be obtained through experiments on the plant. The traditional approach has been to obtain the individual transfer-matrix elements gt v (s) independently, e.g., by fitting openloop responses of the plant to first-order-plus-deadtime transfer-functions. However , several authors (e.g., Skogestad and Morari , 1988) have pointed out that this may be a poor approach if the model is intended for control applications. The reason is that one through this approach usually is unable to capture specific multivariable properties of importance for control. "E-mail : [email protected] .se
High-purity distillation columns operating with the L"'configuration are usually strongly interactive. Typically, increasing L and V such that the product flows remain unchanged increases the purity of both products and the corresponding gain is generally small. This is usually denoted changes in the internal flows and corresponds to the low-gain direction of the plant . Similarly, changing L and V such that there is a maximum change in the product flows makes one product purer and the other less pure and the corresponding gain is generally large. This is usually denoted changes in the external flows and corresponds to the high-gain direction of the
317
plant. A multi variable model intended for control purposes should capture these effects in a reasonable way. However, it is well known that if the interactions are strong then this may be a difficult task. One reason is that the open-loop responses of the plant tend to be dominated by the high-gain direction which renders it difficult to obtain a reasonable model for the low-gain direction . Although often hardly visible in open-loop responses, the low-gain direction becomes important under feedback control and it is therefore crucial to obtain a good model also for this direction of the plant. We start the note with a brief review of some of the works published on identification of strongly interactive plants in general and high-purity distillation columns in particular. We then present a linear distillation column which is proposed as a benchmark problem for which different methods may be tested and compared. The motivation behind using a linear model as the "plant" is that the problem of capturing a plants directionality is present both in the linear and nonlinear plant, and we do not want to confuse this problem with the specific problems caused by nonlinearities. In addition to presenting the model we specify the experimental conditions under which the identification should be performed. The identified model is intended for control applications and it is stressed that an important control objective is disturbance rejection. Thus, a disturbance model is desired in addition to the usually considered input model G LV (s) (1). At the end of the note we apply a standard identification method involving independent PRBS signals and MISO ARMAX models to the benchmark problem . The resulting model is shown to be poor for control applications.
2.
A BRIEF REVIEW
Over the years a number of papers discussing the problem of identifying dynamic models for strongly interactive plants in general, and distillation columns in particular, have appeared. We do not intend to provide a complete review of these here, but mention some of the more recent works only. All the papers mentioned either directly address the problem of identifying distillation columns or employ a distillation column as an example l . As mentioned above , the traditional approach to identification of distillation columns has been to obtain the transfer-matrix elements independently, i.e., through SISO identification . The main reason for this is undoubtedly that the general literature on system identification has been focused on identification of SISO systems while identification of MIMO systems has been considered a difficult problem. However, Skogestad and Morari (1988) argue that one should be careful about identifying the individual SISO elements of the 1 In fact , we have not found any papers addressing this problem that do not mention distillation.
transfer-matrix of a high-purity distillation column independently as this easily may lead to a poor model for the low-gain direction of the plant . In particular , the identified model will easily have the wrong sign of the determinant at steady state and will in that case be useless for control. Andersen et al. (1989) demonstrate through an example that SISO identification yields a poor model of the plants low-gain direction . Andersen and Kiimmel (1991) consider identifying two MISO models and find that the resulting MIMO model also in this case yields a poor prediction of the plants low-gain direction . They also consider using closed-loop identification with a single loop closed combined with MISO identification but find similar results as with open-loop identification. However, by low-pass filtering the closedloop data prior to fitting the model they obtain a reasonable model for the low-gain direction of the plant . Jacobsen et al. (1991) propose to use SISO identification and fit the initial responses (high-frequency behavior) only. The motivation behind this suggestion is that it is the initial response which usually is most important for feedback control and that distillation columns generally are significantly less interactive at high and intermediate frequencies than at steady state. To avoid problems with the wrong sign of the determinant of the steady state gain matrix they propose to correct the steady state gains to match an estimated value of the Relative Gain Array. They find that this approach yields models reasonable for multi variable control applications with relatively high bandwidth. Several authors (e.g., Skogestad, 1988; Alsop and Edgar, 1990; Andersen and Kiimmel, 1991) suggest that, for identification, it may be better to use the DV -configuration instead of the LV-configuration. The main motivation behind this is that with the DVconfiguration the low-gain direction is closely aligned with changes in the input V only, and this may be utilized to obtain a better model for the low-gain direction. In particular, the determinant of the steady state gain matrix is less likely to have the wrong sign when obtained with the DV-configuration. Koung and MacGregor (1993 , 1994) extend this idea to systems where the low-gain direction is not well aligned with a single input, e.g., LV-configuration . The main idea is to utilize a priori knowledge of the plants directionality to obtain better information for the low-gain direction . Koung and MacGregor (1993 , 1994) consider the problem of obtaining a model with the correct sign of the determinant of the steady state gain matrix only. Jacobsen (1994) considers several different approaches to identification of strongly interactive plants and stress that, in order to ensure a model suitable for control applications, both reasonable experimental data for all plant directions as well as a proper multi variable algorithm for fitting the data to a model are required. In cases where knowledge of the plant directionality is available he proposes to use an open-loop experimental design similar to that proposed by Koung and MacGre-
318
gor (1993) . It is demonstrated that this yields satisfactory data for the low-gain direction of the plant , not only at steady state but also at intermediate and high frequencies , even when the knowledge of directionality is somewhat uncertain. In cases where knowledge of the plant directionality is poor or even lacking , Jacobsen (1994) considers using closed-loop identification with a single proportional controller as proposed by Andersen and Kiimmel (1991) and shows that any controller gain will yield a closed-loop plant which is less interactive than the open-loop plant . This may be utilized to ensure a sufficient excitation of all plant directions , including the low-gain direction of the open-loop plant . Jacobsen (1994) suggests that the poor results obtained by Andersen and Kiimmel (1991) is due to the fact that they fit the obtained data using two MISO models. By fitting the obtained data to a MIMO model Jacobsen (1994) find that a good model is obtained both for the high- and low-gain direction of the plant. A subspace based identification algorithm (e.g. , Van Overschee and de Moor, 1994), which requires no a priori knowledge of the model structure, is proposed for fitting the data to a model.
have been proposed in the literature . However, most authors demonstrate the goodness of their method through some specific example and it is difficult to make a comparison of different methods. Furthermore, most papers include little or no information on the experimental conditions , e.g., noise levels , used in applying their method for demonstration. Finally, few papers actually demonstrate the quality of their model for feedback control. The purpose of the present note is to propose a benchmark problem for which different proposed methods may be tested and compared. A similar benchmark problem, based on a simplified heat-exchanger , has previously been presented in Jacobsen and Skogestad (1994b). However, the previous problem was less interactive than most high-purity distillation columns and furthermore included predefined PRBS signals to be used in the experiments, thereby not allowing for specific experimental designs to be applied. In this note we present a more "flexible" problem for which experimental designs , closed-loop identification schemes etc . may be tested.
3. Varga and J0rgensen (1994) apply the experimental design method suggested by Koung and MacGregor (1993, 1994) to a high-purity column . They use the obtained data to fit a MIMO Output Error model with a prespecified structure and find that the low-gain direction is reasonably captured in the resulting model. Li and Lee (1994) propose to identify both the openloop plant as well as its inverse and then combine these into a single model. The motivation behind this approach is that the inverse of the plant generally will yield good information about the low-gain direction of the open-loop plant . In order to identify the approximate inverse of the plant they employ four separate closed-loop experiments using four different controllers. Both the open-loop plant and its inverse are fitted using SISO identification methods. All the papers mentioned above discuss the problem of identifying a reasonable model for the low-gain direction of the plant. A different , although related, problem in identification of strongly interactive plants is discussed in Jacobsen and Skogestad (1994a). A strongly interactive plant will usually contain a single slow pole (time-constant) which tends to dominate all open-loop responses of the plant. Fitting each SISO transferfunction independently, such that they all contain the slow pole , will generally result in a MIMO model with this pole repeated, and hence an inconsistent model. This inconsistency results in a poor prediction of the plants control behavior, in particular under partial control. Jacobsen (1994) find that the only way to avoid this problem is to employ a true MIMO method for fitting the experimental data to a model. As seen from the above, several different methods for identification of dynamic models for distillation columns
PROBLEM DESCRIPTION
The column we consider is separating a binary mixture of ethylene and ethane (C2-splitter) , has purities of 99% in both products and is operated with mass reflux Lw [kg/min] and reboiler heat input Q B [MJ /min] as independent inputs, i.e., LwQ B-configuration. This configuration is similar to the LV -configuration. Data for the column are summarized in Table 1. In the nonlinear dynamic model we assume constant relative volatility Q , vapor-liquid equilibrium, perfect mixing on each tray, constant pressure, negligible vapor holdups and saturated reflux and feed flows. The liquid flow dynamics are modeled using a linearized Francis Weir formula. A simplified static energy-balance is included in the model. We neglect changes in liquid enthalpy with composition (pure components as saturated liquids at column pressure are chosen as reference of state) and assume a linear relation for the vapor enthalpy. With these assumptions we obtain a model with two states per tray, expressed as fraction of light component and total liquid holdup. As the "plant" we employ a linear model obtained through linearization of the full non linear model around the nominal operating point . The linearized model. which contains a total of 52 states, is available from the author via e-mail. The linear model is wTitten (2)
where G(s) is a 2 x 3 transfer-matrix expressing the effect of small changes in the manipulated variables Lu.' and QB and disturbance Fw on the product compositions YD and XB· The motivation behind employing a linear model as the "plant" is that we mainly are concerned with linear aspects here and do not want to con-
319
25
13
0.5
29
69 .99
33.23
XB
0.99
om
- The manipulated inputs are uncertain such that dL w
=
0.95 dL w c 8+1
where subscript c denotes command signal . This "uncertainty" applies during identification only, that is, it should not be considered an inherent part of the plant .
in mole fractions of light component . • L w and Fu' in [kg/ min] . QB in [MJ/min] . • Reboiler is tray 1. • Feed is saturated liquid. • Total condenser with saturated reflux. • Liquid holdups are Mi = 0.5 kmol , including condenser and reboiler . • Flow dynamics included , hydraulic time constant TL = 0.067 min . • Constant pressure P = 12 atm . • Liquid enthalpy H dxd = O. Reference state: pure components as saturated liquids at column pressure . • Vapor enthalpy H v (x,) = 12.45 - 1.39x, . • ZF ,
YD and
1.75
YD
XB
- The feed flow Fw may be manipulated during identification and the maximum allowed perturbation is 5.0 kg/min. The disturbance model may be obtained through a separate experiment if desired. - The distillate flow Dw [kg/min] is measured with uncertainty dD = 0.95 dD wm
8
+1
w
where subscript m denotes measured. - The temperature on tray 20 is measured in DC and contains normally distributed zero mean white noise with standard deviation (Je = 0.I5 D C .
Table 1. Data for example column separating a binary mixture of ethylene and ethane.
- If feedback control of the product compositions is used during identification then a measurement delay of 5 min . should be assumed . fuse matters. Furthermore, most identification methods aim at identifying linear models . Undoubtedly, identifying a linear model from a nonlinear plant will prove harder than from a linear plant , but we believe the latter problem needs to be solved before attacking the former problem . However , for those interested , a nonlinear model is also available from the author upon request . The singular values and the 1, I-element of the RGA for the input model , Le., with Lw and Q B as inputs, are shown as functions of frequency in Figures 2 and 3 (solid lines). At steady state the condition number is 214 and the 1, I-element of the RGA is 37. Thus, the plant is ill-conditioned and also strongly interactive. However , as seen from Figure 2 and 3, the plant is less ill-conditioned and interactive at intermediate and high frequencies than at steady state. This is typical for high-purity distillation columns . The "experimental" conditions available for identification are: - The two product compositions YD and XB are sampled with a period of 1 min . The total number of samples should not exceed 200 . - The composition measurements contain normally distributed zero mean white noise with standard deviation (J e = 0.0005. - The maximum allowed deviation in both product compositions is 0.01 (the main purpose of this specification is to limit the signal-to-noise-ratio and may, for instance, be satisfied by simply scaling the obtained noise-free outputs and inputs accordingly.). - The maximum allowed perturbation in the manipulated inputs are 5 kg/min for the reflux Lu' and 4 MJ /min for the heat input QB
An example file for generating experimental data with Matlab according to the above specifications is available from the author via e-mail. The challenge is to identify a model (2) based on the obtained experimental data, possibly combined with the knowledge that the object is a distillation column separating ethane and ethylene at the given pressure . The intended use of the model is for feedback control applications. Set point changes as well as feed flow disturbance rejection are considered important . The maximum expected setpoint change is 0.01 for both product compositions and the maximum expected disturbance in the feed flow rate is 5 kg/min. Both product compositions are measured with a delay of 5 minutes. Control of a single product composition (one-point control) as well as control of both product compositions (twopoint control) are of interest . In both cases the predicted responses in both product compositions should be considered and compared with those obtained v.:ith the "plant". To allow for a judgement of the ability of a proposed method to identify a reasonable model of the disti.Jlation column it is desired that some kind of uncertainty bounds are presented along with the nominal model. The uncertainty bounds may for instance be estimated through Monte Carlo type simulations.
4.
MISO-IDEKTIFICATION USING AN ARMAX-TYPE MODEL
In this section we employ a fairly standard identification technique to the distillation column presented above. We employ the Matlab System Identification Toolbox (Ljung, 1991) and use MISO-identification with an ARMAX-type model structure. We choose to identify
320
the input model and disturbance model using a single experiment . Independent PRBS signals are applied to the three inputs Lwe , QBe and Fw with amplitudes 2, 1 and 2, respectively, a minimum switching time of 10 min. and a switching probability of 0.5. The measurements of the product flow D1JJ and the tray temperature are not utilized. For each output we fit a strictly proper fifth order model which is determined close to optimal based on the Final Prediction Error (e.g., Ljung , 1991). Figure 1 shows the resulting fit of the magnitude for the transfer-function from Lw to YD and we see that the experimental conditions allows for a reasonable fit of this transfer-function. Although not shown here, the other five transfer-functions of the 2 x 3 transfer-matrix (2) are fitted with similar precision.
inconsistent in terms of the number of slow poles and will yield a poor prediction of the plant under one-point control (Jacobsen and Skogestad, 1994a) .
'0'
'0'
,o~
,o~
10· t
W
10'
[rad/min]
Fig. 3. Magnitude of 1, I-element of RGA for fitted input model (dashed) and plant (solid) . 1D~
10·, r-----~----~----_,
104
..
,,
,o ~
,,
, ,o~
,o~
,o~
10. 1
'0'
,o~
w [rad/min] Fig. 1. Magnitude of transfer-function from Lw to YD for fitted model (dashed) and plant (solid).
'O~ 'O~
'O~
10· '
'0'
w [rad/min] Fig. 4. Magnitude of Closed-Loop Disturbance Gain from Fw to YD for fitted model (dashed) and plant (solid). ,o~
,o~
Figure 4 shows the resulting fit of the Closed Loop Disturbance Gain (Hovd and Skogestad, 1994) which takes into account the relative directions of the disturbance model and the input model and we see that we also have a relatively poor multi variable model for the effect of disturbances.
.(J ..
,o~L-----.......,...----_------' 04 10
10-a
10- '
W
10'
[rad/min]
Fig. 2. Singular values for fitted input model (dashed) and plant (solid) . Figure 2 shows the fit of the singular values for the input model (Lw and Q B as inputs) and we see that , while the high-gain direction is well fitted , the low-gain direction is poorly fitted, in particular at low and intermediate frequencies . This is reflected in the poor fit of the RGA shown in Figure 3. The poor fit of the low-gain direction is at least partially explained by the use of two MISO models in fitting the data (Jacobsen , 1994) . From Figure 2 it is also seen that the low-gain direction of the identified model contains a slow pole at approximately 0.01 rad/min , which in the plant belongs exclusively to the high-gain direction. Thus, the identified model is 321
Based on the model analysis presented above we would expect that the model is of poor quality for control applications. Indeed , using the identified model to design a two-point controller using H (Xl-optimal design we predict the set point response shown by the dashed lines in Figure 5. The solid lines show the corresponding response when applying the same controller to the "plant", i.e., the 52 state full linear model , and we see that the identified model yields a poor prediction of the plant behavior also under two-point control. In particular, the controller when applied to the plant yields a very slow settling towards steady-state. Similar results are found for responses to disturbances in F", . Note that we have not presented the requested error bounds, e.g., for the singular values, here. The reason is that even the nominal model is of such a poor quality
Andersen, H.W . and M. Kummel (1991) "Identifying Gain Directionality of Multivariable Processes", Proc. of ECC'91, Grenoble, France, July 1991.
0.012
., ,, , ,
0.01
YD
,
0._
Hovd, M. and S. Skogestad (1994) "Simple Frequency Dependent Tools for Control-System Analysis, Structure Selection and Design" , Automatica, 28 , 5, 989-99
0.006
0.004
Jacobsen, E .W . (1994) "Identification for Control of Strongly Interactive Plants", Preprints NATO ASI on Model Based Control, Antalya, Turkey, August 1994. Also: paper 226ah , 1994 AIChE Annual Meeting, San Francisco, November 1994.
0.002
- - -.::r;.B 0.00
-.002
--- -,. --.- -- ... - - --
,, ,
-.004
100
50
Jacobsen , E .W., P . Lundstrom and S. Skogestad (1991) "Modelling and Identification for Robust Control of IIIConditioned Plants - A Distillation Case Study", Proceedings of 1991 American Control Conference, Boston, MA , 242-248.
150
time [min] Fig . 5. Closed-loop response for set-point change in YD with fitted model (dashed) and plant (solid).
Jacobsen, E .W. and S. Skogestad (1994a) "Inconsistencies in Dynamic Models for Ill-Conditioned Plants: Application to Low-Order Models of Distillation Columns", Ind.Eng. Chem .Res, 33, 3, 631-640.
that error bounds are of little interest in this case.
5.
CONCLUSIONS
• We have presented a high-purity distillation column which is proposed as a challenge problem for identification of multi variable models intended for feedback control applications. • A detailed description of the experimental conditions is provided along with a model of the column which is available from the author via e-mail. • A standard identification method involving independent PRBS signals and MISO ARMAXmodels were shown to yield a model poor for control applications.
FURTHER INFORMATION
Jacobsen, E .W . and S. Skogestad (1994b) "Identification of Dynamic Models for Ill-Conditioned Plants - a Benchmark Problem" , in the Modeling of Uncertainty in Control Systems, Eds. R. Smith and M. Dahleh, Springer-Verlag, London. Koung, C .W. and J.F. MacGregor (1993) "Design of Identification Experiments for Robust Control. A Geometric Approach for Bivariate Processes", Ind.Eng.Chem.Res ., 32, 1658-1666. Koung, C.W. and J.F. MacGregor (1994) "Identification for Robust Multivariable Control: the Design of Experiments", Automatica, 30, 10, 1541-1554. Li, W. and J.H. Lee (1994) "Closed-Loop Identification of Ill-Conditioned Multivariable Systems", Paper 226ag, 1994 AIChE Annual Meeting, San Francisco, November 1994~
The following is available from the author upon request (e-mail: [email protected]) : - A linear state space model (in ASCII format) for generating noise-free data.
Ljung, L. (1992) System Identification Toolbox Version 3.0a, Math Works Inc . Skogestad, S (1988) "Disturbance Rejection in Distillation Columns", Preprints CHEMDATA '88, Gothenburg , Sweden , 365-371. Published by IVA, Stockholm, Sweden.
- An example file for generating experimental data according to the specified experimental conditions using the linear model and Matlab .
Skogestad, S. and M. Morari (1988) "Understanding the Dynamic Behavior of Distillation Columns", Ind. fj Eng. Chem . Res, 27, 10, 1848-1862 .
- A non linear model for use with Matlab/Simulink. REFERENCES Alsop , A.W. and T.F . Edgar (1990) "Nonlinear Control of a High-Purity Distillation Column by the use of Partially Linearized Control Variables" , Comp. Chem.Eng., 14, 665-678. Andersen, H .W ., M. Kummel, and S.B. Jorgensen (1989) "Dynamics and Identification of a Binary Distillation Column", Chem .Eng.Sci. , 44 , 2571-2581. 322
Van o verschee , P. and B. de Moor (1994) "N4SID: Subspace Algorithms for the Identification of Combined Deterministic-Stochastic Systems" , Automatica, 30 , 1, 75-94. Varga, E.!. and S.B . J0rgensen (1994) "Multivariable Process Identification: Estimating Gain Directions" , Paper 229f, 1994 AIChE Annual Meeting, San Francisco, November 1994.