Chemical Engineering Science 55 (2000) 4667}4680
State estimation of batch distillation columns using an extended Kalman "lter R. M. Oisiovici*, S. L. Cruz Dept. Engenharia de Sistemas Qun& micos/FEQ/UNICAMP 13083-970, Campinas, SP, Brazil Received 9 April 1999; accepted 14 January 2000
Abstract Composition monitoring and control play an essential role during a batch distillation cycle, but on-line composition analyzers are expensive, di$cult to maintain and give delayed responses. Considering the need and lack of a stochastic estimator for batch distillation columns, a discrete extended Kalman "lter (EKF) for binary and multicomponent systems has been developed and tested. The aim of the EKF was to provide reliable and real-time column composition pro"les from few temperature measurements and easily available information. Accurate composition estimates and fast convergence were obtained, and the EKF has con"rmed its ability to incorporate the e!ects of noise (from both measurement and modeling). The number of sensors and the observation frequency have shown to be important variables in the design of the EKF, especially for systems with fast dynamics. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Batch distillation; Kalman "ltering; Stochastic estimator; Discrete systems; Inference; Nonlinear
1. Introduction Batch distillation is the most frequent separation method in batch processes (Lucet, Charamel, Chapuis, Guido & Loreau, 1996) and is widely used in "ne chemical plants. Composition monitoring and composition control play an essential role in these plants because "ne chemicals must be manufactured according to high and well-de"ned standards of purity. During a batch cycle, important steps, such as the withdrawal of a product or the removal of a slop cut, are taken based on composition data. Instantaneous composition values are also relevant in automatic closed-loop control scheme to drive the process following a desired operating strategy. Despite advances in on-line composition analyzers, direct composition measurement is often expensive and di$cult to maintain. It can also introduce time-delay into the control loop. So, the purpose of applying on-line state estimation techniques to batch distillation columns is to
* Correspondence address. FEQ-UNICAMP, CP 6066, CampinasSP, Brazil, 13083-970. Tel.: #55-19-788-3948. E-mail address:
[email protected] (R. M. Oisiovici).
provide reliable and real-time composition estimates from readily available temperature measurements. Most of the works regarding the application of state estimation methods to distillation systems are devoted to continuous distillation columns. Joseph and Brosilow (1978) were pioneers in the development of optimal and suboptimal estimators to infer the product compositions in a continuous distillation column using temperature and #ow measurements. Since then, several authors have studied this topic. For example, Mejdell and Skogestad (1991a, b) discussed the use of the static principal-component-regression (PCR) and partialleast-squares (PLS) estimators and implemented them on a pilot-plant column. The Kalman}Bucy "lter, Brosilow's inferential estimator and the PCR estimator were compared by Mejdell and Skogestad (1993). Baratti, Bertucco, Da Rold, and Morbidelli (1995, 1998) applied a nonlinear extended Kalman "lter to infer the compositions of the streams leaving a binary (1995) and a ternary (1998) continuous distillation column from temperature values. Very few papers address the issue of state estimation of batch columns. Batch distillation, as well as continuous distillation, is a complex, nonlinear and high-order system. However, the batch operation mode presents one additional feature which makes the state estimation
0009-2509/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 0 8 8 - 9
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a more challenging task: batch distillation is an intrinsically dynamic process. During the course of the entire operation, composition pro"les and operating conditions can change over a wide range of values, and the state estimator must be designed to deal with the time-varying nature of the batch column. In the batch distillation area, Quintero-Marmol, Luyben and Georgakis (1991) applied extended Luenberger observers (ELO) to predict compositions in multicomponent systems from temperature measurements. The authors used a linearized state-space model with average values of compositions to calculate the openloop eigenvalues. Some guidelines were given to select the closed-loop eigenvalues which ensure that the estimate error decays within the batch cycle. According to the algorithm proposed by the authors, the matrix of gains of the observer is obtained o!-line. If the designed observer does not converge, it must be redesigned by changing some closed-loop eigenvalues or increasing the number of measurements. Such procedures may sometimes be very time consuming. The main drawback of the Luenberger observer is that it is a deterministic estimator and may not work properly in the presence of plant/model mismatch and process and/or measurement noise. Wilson and Martinez (1997) applied the ELO and the extended Kalman "lter (EKF) to estimate reagent compositions in a combined reactor/still vessel. They concluded that the EKF was more robust than the ELO. Barolo and Berto (1998) used an ELO in a batch distillation control loop and pointed out the need for choosing a stochastic estimator (like a Kalman "lter) when a large degree of noise is expected. Deterministic processes are rare. Model uncertainties and presence of unmeasured process disturbances are common in industry and the current state of a dynamic system must be estimated despite the availability of only few and noisy measurements. In such cases, the extended Kalman "lter (EKF) has shown to provide good results in chemical process applications (polymerization reactors, batch fermentation). However, it has not yet been applied to conventional batch distillation systems. Considering the need and lack of an on-line stochastic estimator for batch distillation columns, a discrete EKF was developed and tested. The aim of the "lter is to infer from temperature measurements not only the product compositions but also the composition pro"le along the column, which is time-varying in the case of batch distillation. Unlike the ELO developed by Quintero-Marmol et al. (1991), the gain of the EKF is calculated and updated on-line using measurable and easily available information. Important issues such as the number of sensors, the presence of temperature noise and the sampling rate are also addressed in this work.
2. The extended Kalman 5lter (EKF) Very good references discuss the Kalman "lter techniques in detail (Gelb, 1974; Brown & Hwang, 1992; Jacobs, 1993), so the EKF will be here presented brie#y. It is assumed that the process has a state vector x(t)3RL and is described by the continuous-time process model x (t)"f(x(t), u(t))#w(t).
(1)
The process noise vector w contains all the perturbations which act on the system and which are not described deterministically. These include unmodeled dynamics and noise corrupted (or unmeasured) input. The corresponding discrete-time process model is x "f (x , u )#w . (2) I> I I I It is assumed that the process noise is a Gaussian-distributed, zero-mean random variable, with covariance Q and is independent of the process noises or the states I of the system which have occurred at any previous time: p(w )&N(0, Q ), I I Q , i"k, E[w w2]" I I G 0, iOk,
(3) (4)
E[w x2]"0, ∀i, k. (5) I G The measurements acquired at time k are aggregated into the p-dimensional observation vector z , which is related I to the state of the system by z "h (x )#v . (6) I I I The vector v represents the measurement noise and is I assumed to be a zero-mean Gaussian distributed random variable with variance R : I (7) p(v )&N(0, R ). I I The vector v is independent of the measurement noises I at all previous time steps and is independent of the process noise:
R , i"k, E[v v2]" I I G 0, iOk,
(8)
E[w v2]"0, ∀i, k. (9) I G The a priori and a posteriori estimate errors are de"ned, respectively, as e ,x !x , (10) II\ I II\ e ,x !x . (11) II I II The a priori estimate error covariance and the a posteriori estimate error covariance are then P "E[e e2 ], II\ II\ II\ P "E[e e2 ]. II II II
(12) (13)
R. M. Oisiovici, S. L. Cruz / Chemical Engineering Science 55 (2000) 4667}4680
The EKF is "rst initialized with x "x and P "P , and then it operates recursively performing a single cycle each time a new set of measurements becomes available. Each iteration propagates the estimate from the time the last measurement was obtained to the current time. The propagation process consists of two stages: update and prediction. 2.1. Update stage The update equations are responsible for the feedback, i.e. for incorporating a new measurement set into the a priori estimate to obtain an improved a posteriori estimate. The a posteriori state estimate x is computed as II a linear combination of an a priori estimate x and II\ a weighted di!erence between an actual measurement z and a measurement prediction: I x( "x( #L [z !h (x( )], (14) II II\ I I II\ where L "P H 2(H P H 2#R )\, (15) I II\ I I II\ I I *h (x) . (16) H " I *x x x( II\ The matrix L is chosen to be the gain that minimizes the I a posteriori error covariance (13). For more details, see Brown and Hwang (1992) or Jacobs (1993). The covariance matrix is updated by
P "(I!L H )P . II I I II\
(17)
2.2. Prediction stage The prediction equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain a priori estimates for the next time step. The state and covariance matrix in the next sampling instant are estimated by x( "f (x( , u ), I>I II I P "F P F 2#Q , I>I I II I I where
F " I
*f (x, u) *x
x
(18) (19)
(20) II x(
u
u I
A rigorous dynamic model for batch distillation columns consists of a large number of nonlinear di!erential equations and demands much information about the system (compositions, vapor and liquid #ow-rates, liquid hold-ups, in all stages every instant, tray hydraulics, energy balances, liquid}vapor equilibrium data). Due to complexity and computational burden, it is often impractical to use rigorous models for on-line state estimation. So, it is necessary to develop models which capture the essential elements of the dynamics. The assumptions made in the model of the batch distillation system are as follows: E equimolal over#ow (¸ "¸, 0)j)NP; < "<, H H 1)j)NP#1); E theoretical stages; E negligible vapor hold-up; E constant liquid hold-up (S , 1)j)NP); H E constant pressure; and E negligible re#ux drum hold-up and total condenser: x "y , 1)i)NC. G G The state variables are the liquid compositions in every stage (still and trays): x"[x 2 x 2 x 2x 2x 2 H ,.> G GH x 2x 2x 2x ]2 G,.> ,!\ ,!\H ,!\,.> 1)i)NC!1; 1)j)NP#1.
In this section, the nonlinear model of the batch distillation column used in the EKF algorithm will be presented.
(21)
For a system with NC components, (NC!1) state variables are considered in every stage. The composition of the NCth component is obtained by di!erence: x "1.0! ,!\x , 1)j)NP#1. ,!H G GH The input vector is given by
¸
u"
<
.
(22)
Then, under the above assumptions, the nonlinear model of the batch distillation column in the state-space representation is
x $
f
x G $
f (x, u) G $
x GH $
x
3. Batch distillation nonlinear model
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(x, u) $
f (x, u) GH $
"f(x, u)"
f
G,.> $
x ,!\,.>
f
G,.> $
(x, u)
,!\,.>
1)i)NC!1; 1)j)NP#1,
(x, u)
(23)
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where f "(¸x !¸x #
GH H 2)j)NP, f "(¸x !¸x # G,. G,.> G,.> ! ,.> The simpli"cations made in the model introduce modeling errors for which the EKF is expected to compensate. The corresponding discrete-time model is given by the Euler integration method: x "x #¹a f(x ,u )"f (x ,u ). (25) I> I I I I I The Euler method, which usually works well at high sampling rates, provided good results in this study. 3.1. Sensor model In Eq. (6), the measurement vector is obtained as a function of the state vector. As it has been assumed that the stages are ideal, the relationship between temperature and composition is given by VLE equations. In many cases, it is hard to "nd a function which explicitly relates temperature to compositions. However, since the system pressure and the a priori composition estimates are known, the elements of vector h can be obtained by solving the bubble-point temperature problem for each stage where a sensor is located.
4. Implementation of the EKF The EKF is initialized with xL and P . A batch of liquid is charged to the still and the system is "rst brought to steady-state under total re#ux. During the total re#ux period, a steady-state temperature pro"le is established and the prediction of the "lter tends to x( "x( . Then, the withdrawal of the overhead I>I II product starts (t is set equal to zero at this moment) in accordance with a desired re#ux policy. At each sampling period, a set of process temperatures is available and the updated state estimate is obtained. To calculate f (x( ,u ) and F in the prediction stage, II I I the EKF for batch distillation columns demands the knowledge of other variables, which are determined as follows: E ¸ and < : from the current estimate of the still comI I position, the latent heat of vaporization of the still content is calculated and, as the heating power is known, the vapor-#ow rate is given by:
< "Pot/*H , and ¸ "[R /(R #1)]< , where I I I I I R is the current re#ux ratio. I E S : (S ) ,.> ,.> I ,. I "S ! S ! (
5. Observability Using a degree of freedom argument, Yu and Luyben (1987) concluded that a distillation column is observable if the number of measurements is at least equal to (NC!1). Quintero-Marmol et al. (1991) found that even though the linear observer in theory needs only (NC!1) temperature measurements to be observable, the extended Luenberger observer needed at least NC sensors to be e!ective. For robust convergence, the authors recommend the use of (NC#2) measurements. The discrete nonlinear system represented by Eqs. (2) and (6) is observable at x if there exists a neighborhood of x and a p-tuple of integers k *k *2 *k *0, N k "n such that the observability N G G matrix O (Eq. (26)) for the discrete nonlinear system is full rank (Lee & Nam, 1991).
h (x) G O * h G (F (x)) O " $ , O " , G *x $ O h (F IG \(x)) G
(26)
where F G is the ith composite of the system function f . The observability of discrete nonlinear systems depends on the operating point x and it can also be a function of the sampling rate because O depends on f . Using the linear algebra theory, it is possible to show that if the entries of a matrix A are subject to errors dA (such as measuring or modeling errors), the perturbed matrix A#dA may have a rank less than the rank of A. If A is a nonsingular matrix with rank n, dA is a small perturbation of A, c(A) is the condition number of A, and let "" ) "" denote any matrix norm, then E A#dA is nonsingular (rank(A#dA)"n) as long as ""dA"" 1 ( , ""A"" c(A) E A#dA is singular (rank(A#dA)(n) as long as ""dA"" 1 * , ""A"" c(A)
R. M. Oisiovici, S. L. Cruz / Chemical Engineering Science 55 (2000) 4667}4680
where c(A)"""A"" ""A\"". If the condition number is de"ned in terms of the 2-norm of the matrix, then it is possible to show that c(A) is the ratio between the maximum and minimum singular values of A. The observability matrix of stochastic systems may be subjected to errors (dO ) which may cause the observability test to fail (rank(O #dO )(n) and the "lter to diverge. So, from the above discussion, the sampling rate and the presence of noise are relevant issues to be considered in the design of discrete estimators for stochastic systems.
6. Results and discussion In order to simulate the column behavior, a rigorous batch distillation simulator was employed. At each sampling period, selected stage temperatures were used as the input to the EKF algorithm. Most of the runs have considered the presence of a normally distributed white temperature noise with standard deviation p. The batch distillation model of the
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EKF algorithm was not perfect (some simplifying assumptions have been made), so modeling noise was present in all tests. Considering the operating conditions in Table 1, the mismatch between the process and the model used in the EKF design (Eqs. (23) and (24)) is presented in Fig. 1. The model was initialized with the actual initial composition pro"le. As the composition pro"le along the column is time-varying, some instantaneous pro"les (at
Table 1 Operating conditions and EKF parameters adopted in runs 1}5 System Pressure (mm Hg) Stages Pot (W) x /x 1 1 R Q P x( R(3C)
ethanol (1)/1-propanol (2) 760 30 (29 plates#still) 1250 0.60/0.40 1.0 diag(1;10\, 2, 1;10\) diag(1;10\, 2, 1;10\) 0.80 diag(p, 2, p)
Fig. 1. Mismatch between the process and the model used in the EKF design. (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
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t"0.05t , 0.25t , 0.50t and 0.75t ) have been 2-2 2-2 2-2 2-2 chosen to be analyzed. The total batch time (t ) corres2-2 ponds to the time period which starts at the beginning of the distillate withdrawal (t"0) and "nishes when the instantaneous distillate composition meets a preset value. Fig. 1(b) shows that the process/model mismatch varies considerably along the batch and from stage to stage. When t"0.05t and 0.75t the model predictions 2-2 2-2 are good. When t"0.25t and t"0.50t the model 2-2 2-2 predictions are accurate for some stages and inaccurate
for other stages. In such cases, a noise adaptive Kalman "lter could have been considered. However, good results were obtained using a constant diagonal process covariance matrix (Q). An important point to be mentioned is that even if a perfect batch distillation model was available, a composition estimator would still be needed because a perfect model cannot provide accurate predictions when the initial conditions are not known exactly. In batch distillation systems, the initial composition pro"le is usually not
Table 2 Parameters which were varied in the runs
Run Run Run Run Run
1 2 3 4 5
p
Sensor locations
S (mol)
S H (mol)
¹a (s)
Temperature noise
7 7 4 4 4
2, 8, 13, 17, 21, 26, 30 2, 8, 13, 17, 21, 26, 30 1, 10, 20, 30 1, 10, 20, 30 1, 10, 20, 30
50.0 50.0 50.0 50.0 250.0
0.2 0.2 0.2 0.2 1.0
10 10 10 5 10
Ideal sensors p"$0.13C p"$0.13C p"$0.13C p"$0.13C
`1a corresponds to the top plate.
Fig. 2. Run 1: (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
R. M. Oisiovici, S. L. Cruz / Chemical Engineering Science 55 (2000) 4667}4680
known. Furthermore, the initial conditions may often vary signi"cantly from batch to batch. Therefore, a composition estimator must be designed to be able to converge to the actual column state in spite of being initialized with only guessed initial conditions. The EKF algorithm for batch distillation has been thoroughly tested and some runs were chosen to illustrate the main issues. Table 1 presents the operating conditions which were kept constant in the binary runs 1}5, while Table 2 presents the parameters which were changed at each of these runs. A constant re#ux ratio policy was adopted. During the batch, each column stage followed a trajectory along a wide range of composition values. In the "rst run, the temperature sensors were assumed to be perfect (no measurement noise). Fig. 2(a) shows the distillate and still composition pro"les along the batch and Fig. 2(b) gives some instantaneous column composition pro"les when seven noiseless sensors were used. The agreement between the actual compositions and the estimates was very good, and at t"0.05t the "lter had 2-2 already converged to the actual column pro"le.
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Figs. 3(a) and (b) present the actual and estimated compositions when the seven sensors were corrupted by a Gauss-distributed white noise with a standard deviation of $0.13C. Even in the presence of noise, the estimates are good and the convergence was as fast as when there was no measurement error. Quintero-Marmol et al. (1991) reported that it was necessary to `turn-o!a the gains of the ELO by the end of the batch when tests with measurement errors were made. The gains of the EKF are updated on-line and, unlike the ELO, the EKF has con"rmed its ability to incorporate the e!ects of noise (from both measurement and modeling). Since perfect measurements are not found in practice, the presence of temperature noise were always considered in the rest of the runs. Reducing the number of sensors from 7 to 4 (run 3), the column pro"les estimated by the EKF were less accurate, as it is shown in Fig. 4(b). The convergence was slower and the EKF completely converged only at t"0.75t . 2-2 An interesting result was obtained when the same number of sensors was kept unchanged (4 sensors) but
Fig. 3. Run 2: (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
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Fig. 4. Run 3: (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
the sampling period was reduced from 10 to 5 s (run 4). In this case, Figs. 5(a) and (b) show that the agreement of the estimates with the actual values and the convergence time have considerably improved. As it was expected, the sampling period has a great in#uence on the accuracy of the "lter estimates and its value must be chosen depending on the system dynamics. All variables that a!ect the dynamic behavior of the batch column may a!ect the minimum sampling frequency that guarantees a good estimator performance. This will be illustrated by varying the hold-up on the plates (S ), which is a variable that plays an important H role in the column dynamics. Run 5 was carried out using the same conditions of run 3 but changing the hold-up on the plates from 0.2 to 1.0 mol. For the sake of comparison, the initial charge was also changed so that the initial column composition pro"le was the same as the initial pro"le in run 3. Comparing run 5 (Fig. 6) with the run 3 (Fig. 4), the EKF provided better results when S was equal to 1.0 mol H (convergence at t"0.25t ) than when it was equal to 2-2
0.2 mol (convergence at t"0.75t ). So, in these exam2-2 ples, a 10 s sampling period is a good choice when S "1.0 mol while for S "0.2 mol it is not. H H When S "0.2 mol the system presents a relatively H faster dynamics. For such plate hold-up value and in the presence of temperature noise, the EKF provided better composition estimates in run 2 (p"7; ¹a"10 s) and in run 4 (p"4; ¹a"5 s). For the same operating conditions in a batch distillation column, the smaller the hold-up on the plates, the greater are the changes in composition per unit time. If the changes in composition are fast, the EKF seems to require higher information rates about the system. And this can be accomplished, for instance, by increasing the number of measurements as in run 2, or by increasing the observation frequency (sampling frequency) as in run 4. At this point, one must be curious as to whether it is possible to infer the column composition pro"le using a single temperature measurement. Tests made with only one sensor are shown in Fig. 7. The number of stages were varied (10, 20 and 30 stages were considered) and
R. M. Oisiovici, S. L. Cruz / Chemical Engineering Science 55 (2000) 4667}4680
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Fig. 5. Run 4: (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
the single sensor was placed at the middle stage (stage 5, 10 and 15, respectively). The top stage and the still are the furthest stages from the sensor. So, the still and distillate composition pro"les were chosen to be analyzed. Figs. 7(a)}(c) show that the distillate composition predicted by the EKF converged faster to the actual value than the still composition estimate. As the number of stages increased, the convergence of the still composition estimate was slower and the prediction of the overhead product composition was less accurate. The results have demonstrated that it is possible to use one single temperature measurement to infer binary composition pro"les of batch distillation columns, but one should be aware of some disadvantages: with one sensor, the EKF demands faster sampling rates, the convergence is slower and the predictions are less accurate than when more sensors are used. The EKF algorithm was also applied to multicomponent batch distillation systems. The system ethanol/1propanol/1-butanol was considered. The ternary batch distillation runs are shown in Figs. 8}10, and the corre-
sponding operating conditions and EKF parameters are presented in Table 3. In Figs. 8}10, the total batch time (t ) corresponds to the time period in which the 2-2 three components are present in the column. This period starts at the beginning of the distillate withdrawal (t"0) and "nishes when the more volatile component (ethanol) is exhausted. After the withdrawal of the more volatile component, the rest of the batch is a binary batch distillation problem, which has already been discussed. Fig. 8 compares the EKF estimates with the actual compositions when "ve noisy sensors evenly spaced were used. The EKF converged at t"0.20t and provided 2-2 accurate composition estimates of the multicomponent mixture in all stages. Keeping the same sampling period and reducing the number of sensors from 5 to 3, the convergence time was longer, as it is shown in Fig. 9. However, when more sensors were added (Fig. 10), no improvement in the convergence time of the EKF was observed. Some general comments on the implementation of the EKF to estimate compositions of batch distillation columns can be made:
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Fig. 6. Run 5: (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
Fig. 7. EKF estimates using a single temperature sensor at the middle of the column.
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Fig. 8. EKF estimates for the system ethanol (1)/1-propanol (2)/1-butanol (3) using "ve sensors at stages 1,5,9,13 and 17. (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
The EKF has shown to be able to converge to the actual column state even when it is initialized with only guessed initial conditions. For batch distillation systems, this is an important feature because initial compositions at all stages of the column are hardly known. The EKF will only converge to the actual state if an acceptable mathematical model is available. The extent of the model uncertainty and the sensitivity of the "lter performance to such uncertainty vary substantially from one problem to another. The design of a well-tuned EKF is an iterative process of system modeling and trading o! between the performance capabilities and the computational constraints and requirements. For practical applications, it is important to validate the model with experimental data. In the implementation of the EKF for batch distillation systems, accurate description of the vapor}liquid equilibrium has shown to be relevant to achieve a good "lter performance. This fact has also been reported by Baratti et al. (1995, 1998) in the development of an EKF for continuous columns.
The binary and ternary examples indicate that the EKF performance usually improves when measurements are added and/or the sampling frequency is increased. Nonetheless, above a certain number of sensors and above a certain sampling frequency value the improvements in terms of convergence time and estimate accuracy may be negligible. The measurement noise covariance R can be obtained from the measurement data and knowledge of sensor characteristics. The process noise covariance Q is usually selected through a trial-and-error procedure using computer simulation or experimental data. In many cases, a well-tuned Kalman "lter can be designed by assuming a diagonal and time-invariant process noise matrix. If the required "lter performance is not obtained using these procedures for selecting the covariance matrices, an alternative is to use a noise adaptive Kalman "lter (Dimitratos, Georgakis, ElAasser & Klein, 1991). In an adaptive "lter algorithm, the Q and R matrices and the state variables are simultaneously estimated.
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Fig. 9. EKF estimates for the system ethanol (1)/1-propanol (2)/1-butanol (3) using three sensors at stages 1,9 and 17. (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
Table 3 Operating conditions and EKF parameters adopted in the multicomponent runs System Pressure (mm Hg) Stages Pot (W) x /x /x 1 1 1 R S (mol) S (mol) H ¹a (s) p Q
P xL R(3C)
ethanol (1)/1-propanol (2)/1-butanol (3) 760 17 (16 plates#still) 1500 0.25/0.35/0.40 1.0 200.0 0.85 5 $0.13C q "1;10\, 1)m)NP#1#NP
q "1;10\, m"NP#1#NP#1
q "0, mOr
diag(1;10\, 2, 1;10\) 0.30 diag(p, 2, p)
7. Conclusions A discrete nonlinear EKF estimator for binary and multicomponent distillation columns has been developed and tested. The EKF for batch distillation is an on-line stochastic estimator which infers instantaneous column composition pro"les from few temperature measurements and readily available information. Unlike the time-consuming o!-line design of the ELO, the gain of the EKF is calculated and updated on-line. The composition estimates were accurate, the convergence was fast and the EKF has con"rmed its ability to incorporate the e!ects of measurement and modeling noises. The number of sensors and the observation frequency (sampling frequency) have shown to be important
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Fig. 10. EKF estimates for the system ethanol (1)/1-propanol (2)/1-butanol (3) using seven sensors at stages 1,4,7,10,13,15 and 17. (a) distillate and still composition pro"les; (b) instantaneous column composition pro"les.
variables in the design of the EKF, especially for systems with fast dynamics. Faster convergence and more accurate estimates were obtained by increasing the number of sensors and/or increasing the sampling frequency. However, above a certain number of sensors the improvement in the EKF performance may not be signi"cant and may not justify the cost of adding more measurements. For the same EKF performance in terms of convergence velocity and accuracy of estimates, fast dynamic systems, in general, require more measurements and faster sampling rates than slow dynamic systems.
Notation
e f f
estimate error vector continuous nonlinear system function discrete nonlinear system function
h *H ¸ L n NC NP p P Pot Q R R S H S ¹a t t 2-2 u v <
discrete measurement function latent heat of vaporization, J mol\ liquid #ow rate, mol s\ Kalman "lter gain matrix system order number of components number of trays number of sensors error covariance matrix heating power, W process noise covariance matrix re#ux ratio measurement noise covariance matrix liquid tray hold-up, mol initial charge, mol sampling period, s time, s total batch time, s input vector measurement noise vector vapor #ow rate, mol s\
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w x GH x G1 x x( HI y GH z
R. M. Oisiovici, S. L. Cruz / Chemical Engineering Science 55 (2000) 4667}4680
process noise vector liquid composition of component i in stage j, mole fraction composition of component i in the initial charge, mole fraction state vector estimate of the state at sample time j given the output measurements up to sample time k vapor composition of component i in stage j, mole fraction measurement vector
Greek letters p
standard deviation
Subscripts k current values NP#1 still 0 condenser
Acknowledgements Ronia M. Oisiovici wishes to thank FAPESP (Fundac7 a o de Amparo a` Pesquisa do Estado de Sa o Paulo) for the "nancial support.
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