4452
Ind. Eng. Chem. Res. 2003, 42, 4452-4460
Consistency of Property Estimators in Multicomponent Distillation Control Gabriele Pannocchia and Alessandro Brambilla* Department of Chemical Engineering, University of Pisa Via Diotisalvi, 2-56126 Pisa, Italy
This paper addresses the problem of input selection in inferential control of multicomponent distillation columns by introducing the concept of consistency. An estimator is consistent when a feedback (multivariable) control system that uses the estimates of the controlled variables guarantees low closed-loop steady-state offset in the true unmeasured controlled variables when disturbances enter the system (and/or set-point changes are considered). A definition of this property is given, and the relations between the estimator consistency and the closed-loop steadystate offset are derived for both single-input-single-output and multi-input-multioutput systems. A multicomponent distillation column case study is presented to show that the selection of the most “precise” inputs does not necessarily guarantee the lowest closed-loop offset in the presence of disturbances, whereas the use of less precise but more “consistent” inputs leads to a well-designed estimator that guarantees a lower closed-loop steady-state offset. 1. Introduction Achieving tight and efficient composition control is one of the main problems in distillation column control because on-line composition analyzers are generally expensive and suffer from large measurement delays and frequent maintenance needs. In fact, the use of single-tray temperature-control schemes is a very popular, inexpensive, and fast alternative. In this context, the temperature of a chosen tray (often called the “pilot tray”) is kept at a constant set point. The location of this measurement is critical because it has a significant influence on the achievable quality-control performance. However, the use of a single temperature to infer the product composition is often not reliable, especially for multicomponent distillation columns, for several reasons.1 Indeed, many studies have pointed out that the use of multiple temperatures improves the quality of the estimates and provides a more efficient inferential control scheme. Weber and Brosilow2 proposed a static estimator that makes use of several temperatures as well as the values of the manipulated variables. Morari and Stephanopoulos3 presented several criteria for selecting the measurements in the presence of nonstationary disturbances. Yu and Luyben4 presented a rigorous composition estimator based on a number of temperatures chosen with a singular value decomposition (SVD) technique. The use of multiple temperatures requires that sensitivity problems related to the “almost” collinearity of the temperature changes in different trays be addressed. Joseph and Brosilow5 outlined an iterative procedure for choosing the set of temperatures that gives the best estimate while keeping the condition number of the corresponding estimation matrix below a given value. More refined multivariate regression techniques, such as principal component regression (PCR) and partial least-squares (PLS) regression can be used to overcome these problems and improve the robustness of the * To whom correspondence should be addressed. E-mail address:
[email protected]. Tel.: +39 050 511243. Fax: +39 050 511266.
estimators, as suggested by Mejdell and Skogestad.6 These methods also provide useful insights into the temperature distribution and indications for weighting and choosing the appropriate temperatures.1 Kano et al.7 presented an inferential distillation quality-control technique based on dynamic PLS estimators. In this paper, the problem of designing estimators for composition control is addressed from a slightly different point of view. In particular, the main interest of the present work is understanding the closed-loop steadystate behavior of a multicomponent distillation column controlled by an inferential regulator. Assuming that integral action is used, the controller removes offset in the estimated product compositions, but in general, this approach does not lead to zero offset in the actual product compositions. A desirable property of this control scheme would be to guarantee the lowest steadystate offset in the presence of disturbances (and/or setpoint changes). As shown in this work, this property (referred to as “consistency”) is not necessarily related to the precision of the estimator in fitting the training data. Thus, the choice of the most precise estimator among several candidates might not be appropriate in terms of obtaining the lowest closed-loop steady-state offset. In this paper, two-product multicomponent distillation columns are considered, and a procedure for designing consistent estimators is proposed. However, the concept of closed-loop consistency applies to any kind of inferential applications to process control. The present paper is organized as follows. In section 2, the basic principles of multivariate regression techniques used in this work are reviewed. Then, in section 3, the concept of consistency for estimators is introduced, and relations for evaluating this property are derived. Next, in section 4, several criteria for selecting the estimator inputs (i.e., the auxiliary measurements) are outlined. These concepts and the selection criteria are extensively discussed in section 5, where a multicomponent distillation case study is presented. Finally, the main contributions of this paper are summarized in section 6.
10.1021/ie020576s CCC: $25.00 © 2003 American Chemical Society Published on Web 01/25/2003
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4453
2. Short Review of Multivariate Regression Techniques 2.1. Problem Statement. Let θ ∈ Rm be the vector of the measured variables (e.g., the temperatures of several trays) used to infer the product compositions y ∈ Rp. A linear static estimator is considered
yˆ ) Kθ
(1)
in which yˆ ∈ Rp is the vector of the product composition estimates; K ∈ Rp×m is the estimator; and both yˆ and θ are in deviation from a reference steady state, so that no constant term appears in eq 1. Given a set of n calibration runs, the corresponding composition vectors (y) and measured variable vectors (θ) are put as rows in the matrices Y ∈ Rn×p and Θ ∈ Rn×m, respectively. Thus, the estimation equation becomes
Y ˆ ) ΘK
T
(2)
where Y ˆ ∈ Rn×p contains, as rows, the estimates of the product compositions of each calibration run. The least-squares solution of the estimation problem is the well-known pseudo-inverse estimator (see, for example, Golub and Van Loan8) T
T
-1
KLS ) Y Θ(Θ Θ)
(3)
which is defined if and only if the columns of Θ are linearly independent. Such an estimator minimizes the variance of Y - Y ˆ. 2.2. Partial Least-Squares Regression. It is wellknown5 that, when the number of measured variables increases, the condition number of the matrix Θ increases significantly, while the residual error (i.e., the variance of Y - Y ˆ ) decreases. In a distillation column, for instance, this occurs because the temperature changes on different trays are related to each other, and this ill conditioning makes the estimator very sensitive to the noise in both the independent variables (i.e., the temperatures) and the dependent variables (i.e., the compositions). To overcome this problem, several multivariate regression techniques can be used, and in this work, the partial least-squares (PLS) regression method is used. PLS (which also means projection to latent structures) was developed a few decades ago9,10 and has received significant attention in the areas of chemometrics11 and, more recently, process engineering1,7,12 PLS generates a “few” (k in number) Θ scores as a linear combination of the original variables
T ) ΘW
(4)
where T ∈ Rn×k is the Θ-score matrix and W ∈ Rm×k is an appropriate weight matrix. These weights are computed so that each one maximizes the covariance between the response variables (Y) and the Θ scores. The Θ scores, T, multiplied by an appropriate loading matrix P ∈ Rm×k, are good “summaries” of Θ, that is
Θ ) TPT + E
(5)
where E ∈ Rn×m contains the Θ residuals. Then, a linear regression model for Y is carried out, leading to
Y ) TQT + F ) ΘWQT + F
(6)
where Q ∈ Rp×k is an appropriate matrix and F ∈ Rn×p contains the Y residuals. Finally, the PLS regression estimator can be written as
Y ˆ ) ΘKPLST
(7)
in which the estimator gain is given by
KPLS ) QWT
(8)
The number of factors (k) is typically chosen by means of the so-called “explained variance” of the dependent variables. Let yi,j be the actual value of the jth product composition in the ith calibration run, and let yˆ i,j(k) be the corresponding estimate obtained by an estimator with k factors. Then, the mean square error (MSE) for the jth product composition is given by
MSEj(k) )
1
n
[yi,j - yˆ i,j(k)]2 ∑ ni)1
(9)
Thus, the explained variance (EV) for the jth product composition is
[
EVj(k) ) 100 1 -
]
MSE(k) MSE(0)
(10)
Usually, one starts with k ) 1 and then increases the number of factors until the improvement in EV is not significant. Often, the above-defined quantities are computed on a subset of data not used in the calibration, and they are called mean square error of prediction (MSEP) and explained prediction variance (EPV), respectively. In general, EPV does not necessarily increase with the number of factors, because the least important factors include mostly noise. The use of least important factors makes the estimator suitable only to fit the training data but not to make reliable predictions of new data (see, for example, Mejdell and Skogestad1). Qin and Dunia13 addressed the problem of choosing the number of principal components by making use of the variance of reconstruction error. Many PLS algorithms exist, of which the SIMPLS14 technique is used in the present work and described in Appendix A. 3. Consistency of an Estimator 3.1. The Concept. In the previous section, the performance of an estimator is considered in terms of its precision in fitting the training data and its numerical robustness. However, it is crucial to keep in mind that the estimator is meant to be inserted into a (multivariable) control loop. Thus, a desirable property of an estimator is that it guarantees the smallest closedloop steady-state offset possible in the presence of disturbances (and/or set-point changes). From a qualitative point of view, an estimator inserted in a control loop is consistent if the steady-state changes of the manipulated variables (due to a disturbance or a set-point variation) are close to the corresponding changes of the manipulated variables obtained if the actual controlled variables had been measured. 3.2. Basic Definition. Consider the generic inferential control scheme reported in Figure 1, where u ∈ Rp is the vector of the manipulated variables, y ∈ Rp is the vector of the controlled variables (unmeasurable), θ ∈
4454 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
from which the consistency matrix as in eq 11 is a scalar parameter given by
∂u ( ∂d) ξ) (∂d∂u)
yˆ )0
)
(KGdθ)G
(13)
(KGθ)Gd
y)0
Hence, the steady-state offset of the unmeasurable controlled variable y is Figure 1. Inferential control scheme.
Rm is the vector of auxiliary variables (measurable), yˆ ∈ Rp is the vector of the estimates of the controlled variables, and d ∈ Rq is the vector of the disturbance inputs. The feedback controller C operates on the estimate of the controlled variables, i.e., on yˆ . For a generic disturbance d, one can define the consistency matrix15 ξ ) {ξi,j} whose generic elements are given by
( ) ( )
∂ui ∂dj yˆ ξi,j ) ) ∂ui ∂dj y (variation of ui when rejecting the effect of dj on yˆ ) . (variation of ui when rejecting the effect of dj on y) (11) It is clear that an estimator will be consistent for the rejection of the disturbance d if the elements of the matrix ξ are close to 1. A similarly straightforward definition can be made for set-point changes. 3.3. Consistency and Closed-Loop Steady-State Offset: The SISO Case. The consistency matrix can be related to the steady-state offset by using a linear description of the plant, which, in general, will be accurate in a neighborhood of the nominal steady state. The SISO (single-input-single-output) case in the presence of a scalar disturbance is considered first, i.e., the case in which p ) q ) 1. In a neighborhood of the nominal steady state, the relations between the manipulated variable and disturbance changes and the controlled and measured variable changes can be written as
(12)
θ ) Gθu + Gdθd
where G ∈ R and Gd ∈ R are the steady-state gains from the manipulated variable and the disturbance, respectively, to the controlled variable. Gθ ∈ Rm×1 and Gdθ ∈ Rm×1 are the steady-state gain vectors from the manipulated variable and the disturbance, respectively, to the measured variables. If the controller C has integral action, then the controlled variable estimate yˆ ) Kθ, with K ∈ R1×m, will, at steady state, be equal to its set point, assumed to be r ) 0 for simplicity of presentation. Thus, one can write
( )
d
∂u G )∂d y)0 G
( )
]
[
-Gdd 1 -
(KGdθ)G (KGθ)Gd
]
Because -Gdd would be the offset without control action (i.e., in open-loop operation), one finally obtains that
eCL )1-ξ eOL
(14)
That is, the consistency parameter is the complement to 1 of the ratio between the open-loop and closed-loop steady-state offsets. It is clear that, if ξ ≈ 1, then the closed-loop offset will be small. 3.4. Consistency and Closed-Loop Steady-State Offset: The MIMO Case. For MIMO (multi-inputmultioutput) systems, it is more convenient to derive directly the relations between the steady-state offset and the disturbance change (and/or the set-point change). In a neighborhood of the steady state, one can write
y) Gu + Gdd
(15)
θ ) Gθu + Gdθd
where G ∈ Rp×p and Gθ ∈ Rm×p are the steady-state gain matrices from the manipulated inputs to the controlled and auxiliary variables, respectively. Gd ∈ Rp×q and Gdθ ∈ Rm×q are the steady-state gain matrices from the disturbance inputs to the controlled and auxiliary variables, respectively. Given the set-point reference r, and assuming that the controller C has integral action, at steady state, one can write
r ) yˆ ) Kθ ) KGθu + KGdθd
y ) Gu + Gdd
KGdθ ∂u , )∂d yˆ )0 KGθ
[ ( )
KGdθ d + Gdd ) eCL ) - y ) - -G KGθ
Thus, the corresponding steady-state input vector is
u ) (KGθ)-1(r - KGdθd)
(16)
The steady-state closed-loop offset can be written as
eCL ) r - y ) [I - G(KGθ)-1]r + [G(KGθ)-1KGdθ - Gd]d For convenience of notation, let r and d be the following matrices
r ) I - G(KGθ)-1
(17a)
d ) G(KGθ)-1KGdθ - Gd
(17b)
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4455
Thus, the steady-state offset is given by
eCL ) rr + dd
(18)
The matrices r and d can be regarded as the gain matrices from the set-point reference and the disturbance, respectively, to the steady-state offset. It is clear that, in the presence of a disturbance d, the closed-loop steady-state error is small if the norm of d is small. In fact, if d * 0 (and r ) 0), one can write
||eCL||2 e ||d||2 ) σ1(d) ||d||2
(19)
where σ1(‚) indicates the largest singular value. It is important to note that the amplitude of the closed-loop steady-state error varies with the direction of the disturbance, i.e., the amplitude of the error depends on the disturbance alignment with the principal directions of d. Similarly, for a set-point reference r * 0 (and d ) 0), one can write
||eCL||2 e ||r||2 ) σ1(r) ||r||2
(20)
Thus, the smaller the norm of d (or r), the more consistent the estimator K in rejecting a generic disturbance (or accomplishing a set-point change). 3.5. Remarks. It is important to point out that the consistency of an estimator is not necessarily related to its precision. To clarify this point, consider again an SISO system in which a single auxiliary measurement θ is used to infer the scalar controlled variable y. For instance, consider controlling a single product composition by keeping the temperature of a tray at a given set point. For such systems, the consistency parameter in eq 13 becomes
ξ)
GdθG GθGd
so that the steady-state offset in the presence of a disturbance is independent of the estimator gain K. Thus, for the distillation column example, the consistency of the estimator depends on the choice of the pilot tray but not on the estimator gain. This property also holds whenever the estimator is square, i.e., whenever the number of auxiliary measurements is equal to the number of controlled variables. In plant applications, it is common to have data from step tests of the manipulated variables. These tests permit one to build the gain matrices G and Gθ and, hence, to evaluate the consistency of a given estimator for a set-point change (i.e., the matrix r). On the other hand, many disturbances are unmeasurable, and the gain matrices for these disturbances (i.e., Gd and Gdθ) can be difficult to construct. Thus, the estimator consistency for disturbance rejection is more difficult to evaluate from plant data. However, when designing an inferential control scheme for an actual plant, it is possible to use rigorous simulators to obtain a detailed process model that can be used to evaluate the most consistent locations of the auxiliary measurements. Once the auxiliary measurements are chosen and installed, one can compute the estimator gain K from plant data.
Finally, it is important to point out that these consistency parameters are based on a linear analysis of the process. Thus, in practice, the consistency parameters are strictly valid in a limited range around the nominal steady state. However, as shown in the case study, this analysis permits one to choose inputs that actually guarantee low steady-state offset even if the column behavior is nonlinear. 4. Input Selection Criteria Multivariate regression techniques, such as PLS, can be used to infer the product compositions by means of all tray temperatures. Even though this design might improve the quality of the estimation, in general, it is not applicable from a practical point of view. Indeed, it is possible to use information from the PLS regression to select a smaller number of tray temperatures as inputs to the composition estimator. Mejdell and Skogestad1 suggested choosing as inputs the tray temperatures in which the peaks of the KPLS matrix are located. Fisichella15 proposed choosing the tray temperatures in which peaks of the first latent variable occur. It is also possible to select the measurements that give the largest explained variance (best precision) or the best consistency for disturbances (or set-point changes). Two different criteria compared in the case study are summarized below. Criterion 1 (Best Precision). Start with one measurement (m ) 1). 1. Test all possible combinations with m inputs by repeating the following steps for each combination. (Of course, the set of considered inputs can be limited to selected sections of the column.) (a) Perform PLS regression with a suitable number of factors k e m. (b) Compute the explained variance EV. 2. Choose the combination of m inputs that gives the largest EV. 3. If EV is adequate or the increase in EV from the previous iteration (i.e., with m - 1 inputs) is not significant, stop; otherwise, go to step 4. 4. Increase m and go to step 1. Criterion 2 (Best Consistency for Disturbance Rejection). Start with one measurement (m ) 1). 1. Test all possible combinations with m inputs by repeating the following steps for each combination. (Of course, the set of considered inputs can be limited to selected sections of the column.) (a) Perform PLS regression with a suitable number of factors k e m. (b) Compute the gain matrix from the disturbance to the steady-state offset, i.e., d, from eq 17b. (c) Determine the norm of d as
Φd )
||d|| dim d
(21)
where the symbol ||‚|| represents a matrix norm (e.g., the 1-norm or 2-norm). 2. Choose the combination of m inputs that gives the smallest Φd. 3. If Φd is adequate or Φd did not decrease significantly from the previous iteration (i.e., with m - 1 inputs), stop; otherwise, go to step 4. 4. Increase m and go to 1.
4456 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 1. Nominal Composition of Feed and Products (Mole Basis) ID
name
feed
top product
LLK1 LLK2 LK HK HHK1 HHK2
i-butane 2-methylpentane n-hexane 2,2-dimethylpentane 2,3-dimethylpentane 2,2-dimethyleptane
0.05 0.10 0.30 0.30 0.15 0.10
0.112 0.223 0.645 0.020 0.000 0.000
Table 3. Training Set Synopsis
bottom product 0.000 0.000 0.020 0.527 0.272 0.181
case
number of runs
run(s)
range
base varying D varying Q varying F varying p varying feed composition
1 10 10 10 10 18
1 2-11 12-21 22-31 32-41 42-59
(0.8-1.2)Dnom (0.7-1.3)Qnom (0.9-1.1)Fnom (0.9-1.2)pnom -
Table 2. Nominal Operating Parameters parameter
ID
units
value
feed rate distillate rate reboiler duty top reflux ratio condenser pressure reboiler pressure
F D Q R ptop pbot
kmol/h kmol/h MMkcal/h atm atm
100.0 44.79 1.692 3.97 1.1 1.4
It is important to remark that, in eq 21, one can use weight factors for different disturbances to assign a higher priority to the most frequent disturbances that might affect the plant. Moreover, if set-point changes occur frequently, one can modify eq 21 as follows
Φdr ) Rd||d|| + Rr||d||
(22)
where Rd and Rr are nonnegative weight factors, defined by the user, to assign a relative importance between consistency for disturbances and consistency for setpoint changes. 5. Case Studies In this section, two case studies on a multicomponent distillation column are presented, and the problem of designing property estimators is considered from a consistency point of view. In the first case study, only the top-product composition is controlled by manipulating the distillate rate (SISO system), whereas in the second case study, both the top- and bottom-product compositions are controlled by varying the distillate rate and the reboiler duty (MIMO system). A two-product six-component distillation column, simulated by means of the rigorous simulator Aspen Plus 10.2, is considered. The column has 60 ideal stages, a total condenser, and a Kettle reboiler. The saturated liquid feed enters the column at the 31st column stage (counted from top to bottom). Feed and product compositions under nominal conditions (on a mole basis) are reported in Table 1. Components, ranging from C4 to C9, are denoted as LK and HK (light and heavy key components, respectively), LLK1 and LLK2 (lighter than light key components), or HHK1 and HHK2 (heavier than heavy key components). All operating parameters under nominal conditions are reported in Table 2. To deal with the process nonlinear behavior, pseudobinary logarithmic compositions are used as controlled variables, as suggested by Mejdell and Skogestad.1 That is, the top- and bottom-product pseudo-binary mole fractions are defined as
xDLK
xBLK
, yB ) B yD ) D x LK + xDHK x LK + xBHK
(23)
respectively, and the corresponding logarithmic compositions are given by
YD ) log(1 - yD), YB ) log(yB).
(24)
The training set consists of 59 runs obtained by varying, one by one, the manipulated variables (distillate rate and reboiler duty) and some disturbance variables (feed rate, feed composition, and operating pressure). Details of the training sets are given in Table 3. 5.1. SISO Case. 5.1.1. Introduction. In the first case study, only the top-product composition, as in eq 24, is controlled by varying the distillate rate, with the reboiler duty kept constant. Different tray temperatures are used to estimate the top-product composition, and the condenser pressure is also included as additional input to capture the effect of operating pressure changes. More precisely, the estimator has the form
Y ˆ D ) c + Kθ + Kp log ptop where θ is the vector of tray temperatures (whose selection is discussed in the next subsection) and ptop is the condenser pressure. A logarithmic dependence of the product composition on the operating pressure is assumed because of physical considerations (e.g., Antoine’s law). The standard linear notation as in eq 1 can be recovered by appending to the vector θ an element that contains the logarithm of the condenser pressure and then centering both the composition and the auxiliary measurements with respect to their steady-state value. 5.1.2. Input Selection. The problem of choosing the estimator inputs is addressed in this subsection. Note that the condenser pressure is always chosen as an input, so that the estimators differ from each other only in the number and location of the tray temperature measurements. The problem is first considered from a precision point of view, that is, following criterion 1. One can start by searching for the single tray temperature that maximizes the explained variance of the top-product composition. Then, one can search for the pair of trays that maximizes the explained variance of the top-product composition. Let P1 denote the most precise singletemperature estimator, and let P2 denote the most precise two-temperature estimator. The estimator coefficients are found using the PLS regression, in which the number of latent variables is chosen in such a way that the last variable covers at least 5% of the overall explained variance. All estimator characteristics are reported in Table 4, where the consistency parameter Φd is computed for the disturbances discussed below. Then, the problem is considered from a consistency point of view, that is, following criterion 2. Typical disturbances for distillation columns are changes in feed flow rate, feed composition, and operating pressure.
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4457
Figure 2. Training data fitting for single-temperature estimators (SISO case).
Figure 3. Training data fitting for two-temperature estimators (SISO case)
Table 4. Estimator Characteristics (SISO Case)
manipulated variable change or disturbance change as described in Table 3. However, Table 5 reports the openloop (OL) steady-state offset and the closed-loop steadystate offset computed with the rigorous simulator for disturbance rejections (negative and positive variations of feed rate, operating pressure, and feed composition, i.e., d1, d2, and d3) when an SISO control loop is closed to the logarithmic composition estimate obtained from P1 and C1. It is clear from Table 5 that C1 is more consistent than P1. In fact, the mean absolute steadystate offset obtained with C1 is 0.054, whereas that obtained with P1 is 0.089. Next, the two-temperature estimators P2 and C2 are compared (Figure 3). From Table 4, it is clear that P2 is slightly more precise than C2 (in fact, the explained variance for P2 is 96.1, whereas that for C2 is 95.9). However, C2 is more consistent than P2. In fact, the mean absolute steady-state offset obtained with C2 is 0.046, whereas that obtained with P2 is 0.070, as shown in Table 5. It is also interesting to notice that estimator C1, which uses a single temperature, is more consistent than P2, which uses two temperatures, for disturbances in the operating pressure and in the feed composition. Finally, from a comparison between C1 and C2, it is clear that, even though the use of two temperatures increases the estimator precision significantly, it does not improve the consistency very much. 5.1.4. Dynamic Responses. In this subsection, a comparison of the column dynamic response obtained using PI (proportional-integral) composition controllers based on C1 and P1 is presented. These dynamic responses were obtained using Aspen Dynamics 11.1, and the PI controllers were tuned using the “process reaction curve” tuning method (see, for example, Stephanopoulos16). In Figure 4, the closed-loop and open-loop responses of the “true” top-product HK mole fraction and the distillate rate are shown for the case in which, at time 0.5 h, the feed composition changes. In particular, the feed LK mole fraction becomes 0.32, and the HK mole fraction becomes 0.28 (i.e., the ratio LK/HK increases). In the transient, the controller based on P1 is more aggressive than that based on C1, because the corresponding pilot tray is more sensitive to feed composition changes. However, the dominant effect is the steady-state behavior: it is clear that the controller based on P1 increases the manipulated variable much more than necessary, and the mole fraction of HK increases even if it would have decreased under open-loop operation. On the other hand, the controller based on C1 increases
ID
inputs
tray(s)
PLS factors
P1 C1 P2 C2
2 2 3 3
18 4 20, 50 19, 45
2 2 3 3
EV
consistency parameter (Φd)
92.4 65.6 96.1 95.9
1.746 1.257 0.656 0.534
More precisely, the following normalized disturbances are considered
1 d1 ) nom∆F F 1 d2 ) nom∆ptop ptop and
( ) ( )
d3 )
1
xFLK
∆ nom
xFLK
xFHK
xFHK
As discussed in sections 3 and 4, the following parameter is used to measure the consistency of the estimator for disturbance rejection
Φd )
||d||1 dim d
where d is defined in eq 17b and the gain matrices G, Gθ, Gd, and Gdθ are identified from data by leastsquares technique. One can start by searching for the single tray that minimizes Φd. Then, one can search for the pair of trays that minimizes Φd. Let C1 denote the most consistent single-temperature estimator, and let C2 denote the most consistent two-temperature estimator. All estimator characteristics are reported in Table 4. 5.1.3. Data Fitting and Closed-Loop Consistency. First, the single-temperature estimators P1 and C1 are compared. From Table 4, it is clear that P1 is significantly more precise than C1 (in fact, the explained variance for P1 is 92.4, whereas the explained variance for C1 is “only” 65.6). In Figure 2, the estimators P1 and C1 are compared in terms of fitting the training data, and P1 clearly appears to be more precise than C1. Note that, in Figure 2, each run corresponds to a
4458 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 5. Open-Loop and Closed-Loop Steady-State Offsets (SISO Case) ID
d1 ) +0.05
d1 ) -0.05
d2 ) +0.045
d2 ) -0.045
d3 ) 0.069
d3 ) -0.065
mean
OL P1 C1 P2 C2
0.550 -0.148 -0.115 0.077 0.037
-0.934 0.170 0.107 0.060 0.021
-0.056 -0.015 0.019 -0.007 -0.008
0.061 0.022 0.001 0.022 0.022
0.607 -0.093 0.042 -0.134 -0.100
-0.555 0.084 -0.042 0.120 0.090
0.461 0.089 0.054 0.070 0.046
Table 6. Estimator Characteristics (MIMO Case) ID P11 C11 C1C1
Figure 4. Open-loop and closed-loop dynamic responses for a feed composition step disturbance (SISO case).
the distillate rate almost to the value actually needed to reject the disturbance. Several other disturbances were simulated, and dynamic results similar to those presented here were obtained. That is, if the controllers are properly tuned, the dynamic behavior does not appear to be particularly significant with respect to the steady-state behavior, which is dominated by the estimator consistency. 5.2. MIMO Case. 5.2.1 Introduction. In the second case study, top- and bottom-product compositions, as in eq 24 are controlled by varying the distillate rate and the reboiler duty (MIMO system). Because the main interest of this work is in the steady-state behavior of such a controlled system, the type of controller used (decentralized or multivariable) is not important, provided that it is designed for integral action. For each product composition, an estimator of the same form as used in the SISO case is designed, that is
Y ˆ D ) cD + KDθ + KD p log ptop Y ˆ B ) cB + KBθ + KBp log ptop 5.2.2. Input Selection. Here, the problem of choosing the temperature location for each product estimator is addressed and, for simplicity of presentation, this search is limited to single-temperature estimators. Moreover,
top product tray EV 18 8 4
92.4 74.2 65.6
bottom product tray EV 45 51 51
94.1 87.0 87.0
consistency parameter (Φd) 0.929 0.422 0.699
it is important to point out that the top- and bottomproduct estimators cannot use the same tray temperature as input since this may lead to an infeasible control system. For each product composition, the single-tray estimator that maximizes the explained variance is found, i.e., criterion 1 is followed. Clearly, for the top-product composition, the same estimator as in the SISO case is found, that is, the 18th-stage temperature is used as input along with the condenser pressure. For the bottom-product estimator, the 45th-stage temperature is found to be optimal from a precision point of view. Let P11 denote this pair of top- and bottom-product estimators, whose characteristics are reported in Table 6. Then, the top- and bottom-product single-temperature estimators that minimize the 1-norm of d are found, i.e., criterion 2 is followed. It is important to point out that the closed-loop consistency depends on both inferential control loops, so that all possible pairs of top- and bottom-product single-temperature estimators need to be tested. Another important observation is that, if both product compositions are controlled, any estimator will be consistent for disturbances in the feed flow rate. In fact, if the feed flow rate changes and both product estimates are controlled, the steady-state temperature profile of the column will be the same as in the nominal case because both the distillate rate and the reboiler duty change proportionally to the feed-rate change. Hence, no offset in the actual product compositions occurs. Therefore, in building the matrix d, only the operating pressure and the feed composition (i.e., d2 and d3) are considered as possible disturbances. It is found that the most consistent pair of estimators uses the 8thstage temperature as input to the top-product estimator and the 51st-stage temperature as input to the bottomproduct estimator. Let C11 denote this pair of top- and bottom-product estimators, whose characteristics are also reported in Table 6. Another pair of top- and bottom-product estimators (C1C1) is also tested, which uses the 4th-stage temperature as input to the top-product estimator and the 51ststage temperature as input to the bottom-product estimator. This pair of estimators uses the most consistent inputs for the top- and bottom-product composition control loops considered as SISO loops (i.e., as in the previous case study). This pair of estimators is tested to show that inputs that were the most consistent ones for the SISO case might not be the most consistent ones for the MIMO case. 5.2.3. Data Fitting and Closed-Loop Consistency. From Table 6, it is clear that P11 is significantly more precise than C11 in fitting the training data. In fact,
Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4459 Table 7. Open-Loop and Closed-Loop Steady-State Offsets (MIMO Case) d2 ) +0.045
d2 ) -0.045
d3 ) +0.069
d3 ) -0.065
mean
ID
YD
YB
YD
YB
YD
YB
YD
YB
YD
YB
OL P11 C11 C1C1
-0.056 0.013 0.016 0.093
-0.046 0.016 0.022 0.032
0.061 -0.003 -0.002 -0.070
0.049 -0.031 -0.036 -0.041
0.607 -0.084 0.002 0.057
-0.429 -0.045 0.012 0.020
-0.555 0.078 -0.002 -0.056
-0.362 -0.049 0.012 -0.015
0.320 0.045 0.006 0.069
0.222 0.035 0.021 0.027
rather smaller than the corresponding offset obtained with P11, especially in the top-product composition. Therefore, also in the MIMO case, it follows that, despite being less precise in fitting the training data, the estimators found using criterion 2 are more consistent for disturbances than the ones obtained with criterion 1. Moreover, from Table 7, it follows that C1C1 is significantly less consistent than C11, and these results show that, if both compositions are controlled, the input selection must take the multivariable nature of the inferential control system into account. 6. Conclusions
Figure 5. Training data fitting for single-temperature estimators (MIMO case).
the explained variance of P11 is 92.4 for the top-product estimator and 94.1 for the bottom-product estimator, whereas for C11, the explained variance is 74.2 for the top-product estimator and 87.0 for the bottom-product estimator. This is also shown in Figure 5, where P11 and C11 are compared in terms of fitting the training data. Note that, in Figure 5, each run corresponds to a manipulated variable change or disturbance change as described in Table 3. Table 7 shows the open-loop (OL) steady-state offset and the closed-loop steady-state offset of each product composition computed with the rigorous simulator for disturbance rejections (negative and positive variations of the operating pressure and the feed composition, i.e., d2 and d3) when the two control loops are closed to the top- and bottom-product logarithmic composition estimates obtained from P11, C11, and C1C1. As remarked previously, the final closed-loop steady state does not depend on the type of controller used, provided that integral action is used and the closed-loop system is stable. From these results, it is clear that P11 and C11 are almost equally consistent for disturbances in the operating pressure whereas C11 is significantly more consistent for disturbances in the feed composition, especially for the top-product composition. Thus, the mean absolute steady-state offset obtained with C11 is
In this work, the problem of designing composition estimators for distillation columns has been addressed. First, the concept of estimator consistency for disturbance rejection (or set-point change) was introduced. An estimator is consistent if the inferential controller based on this estimator guarantees low closed-loop steadystate offset in the presence of disturbances (or set-point changes). It was shown that this property is not necessarily related to the estimator precision in fitting the training data. In fact, the consistency of the estimator depends on the choice of the inputs (i.e., the chosen trays), and measurements that make the estimator precise do not necessarily guarantee low closed-loop steady-state offset in the presence of disturbances. A procedure for choosing the most consistent inputs was outlined that is based on a linear analysis of the closed-loop steady-state offset. A simulated multicomponent distillation case study was presented: a single-product quality-control scheme was considered first (SISO system), and then, a top- and bottom-product quality-control scheme was considered (MIMO system). The results show that the proposed method actually provides consistent estimators even in the presence of nonlinearity (as in the case of distillation columns) and also that precision does not necessarily imply low steady-state offset. The results also show that inputs that were optimal from a consistency point of view in an SISO control scheme might not be the most consistent ones when an MIMO control scheme is considered. Therefore, when designing a property estimator, it is important to evaluate and maximize its consistency for the particular control scheme used. Dynamic simulations show that the transient behavior of different estimators exhibits characteristics of lesser importance than the steady-state consistency properties. Finally, it is important to remark that the choice of the most consistent inputs depends on the disturbances considered and, hence, one should design an estimator that is consistent for the most frequent disturbances that might affect the plant (and/or for set-point changes if they occur often). Acknowledgment The authors acknowledge Claudia Fisichella who started to work on this subject for her MS Thesis.
4460 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003
Appendix A. SIMPLS Algorithm For each h ) 1, ..., k (where A1 ) ΘTY, M1 ) ΘTΘ, and C1 ) I), repeat the following steps: 1. Compute qh, the dominant eigenvector of AhTAh. 2. Set wh ) Ahqh, ch ) whTMhwh, and wh r wh/xch, and store wh in W as a column. 3. Set ph ) Mhwh, and store ph in P as a column. 4. Set qh ) AhTwh, and store qh in Q as a column. 5. Set vh ) Chph and vh r vh/||vh||2. 6. Set Ch+1 ) Ch - vhvhT and Mh+1 ) Mh - phphT. 7. Set Ah+1 ) Ch+1Ah. Literature Cited (1) Mejdell, T.; Skogestad, S. Estimation of Distillation Composition from Multiple Temperature Measurements Using PartialLeast Squares Regression. Ind. Eng. Chem. Res. 1991, 30, 25432555. (2) Weber, R.; Brosilow, C. The Use of Secondary Measurements to Improve Control. AIChE J. 1972, 18, 614-623. (3) Morari, M.; Stephanopoulos, G. Optimal Selection of Secondary Measurements with the Framework of State Estimation in the Presence of Persistent Unknown Disturbances. AIChE J. 1980, 26, 247-258. (4) Yu, C.-C.; Luyben, W. L. Control of Multicomponent Distillation Columns Using Rigorous Composition Estimators. In Distillation and Adsorption; IChemE Symposium Series; The Institution of Chemical Engineers: Brighton, U.K., 1987; pp A29-A69. (5) Joseph, B.; Brosilow, C. B. Inferential Control of Processes. AIChE J. 1978, 24, 485-509. (6) Mejdell, T.; Skogestad, S. Estimators for Ill-Conditioned Plants: High Purity Distillation. In Proceedings of Advanced
Control of Chemical ProcessessADCHEM ‘91; Pergamon Press: Tarrytown, NY, 1991; pp 227-232. (7) Kano, M.; Miyazaki, K.; Hasebe, S.; Hoshimoto, I. Inferential Control System of Distillation Compositions Using Dynamic Partial Least Squares Regression. J. Process Control 2000, 10, 157-166. (8) Golub, G. H.; Van Loan, C. F. Matrix Computations; The Johns Hopkins University Press: Baltimore, MD, 1996. (9) Horst, P. Relation among m sets of measures. Psychometrica 1961, 26. (10) Wold, S. Cross-Validatory Estimation of the Number of Components in Factor and Principal Component Models. Technometrics 1978, 20, 397-404. (11) Wold, S.; Sjo¨stro¨m, M.; Eriksson, L. PLS Regression: A Basic Tool of Chemometrics. Chemom. Intell. Lab. Syst. 2001, 58, 109-130. (12) Kresta, J. V.; Marlin, T. E.; MacGregor, J. F. Development of Inferential Process Models Using PLS. Comput. Chem. Eng. 1994, 18, 597-611. (13) Qin, S. J.; Dunia, R. Determining the Number of Principal Components for Best Reconstruction. J. Process Control 2000, 10, 245-250. (14) de Jong, S. An Alternative Approach to Partial Least Squares Regression. Chemom. Intell. Lab. Syst. 1993, 18, 251263. (15) Fisichella, C. Consistency of Property Estimators in Model Predictive Control. M.S. Thesis in Chemical Engineering, University of Pisa, Pisa, Italy, 2001 (in Italian). (16) Stephanopoulos, G. Chemical Process Control: An Introduction to Theory and Practice; Prentice-Hall: Englewood Cliffs, NJ, 1984.
Received for review August 5, 2002 Revised manuscript received November 20, 2002 Accepted November 25, 2002 IE020576S
