Brazilian Journal of Chemical Engineering
Print version ISSN 01046632Online version ISSN 16784383
Braz. J. Chem. Eng. vol.17 n.47 São Paulo Dec. 2000
http://dx.doi.org/10.1590/S010466322000000400031
SENSOR LOCATIONS AND NOISE REDUCTION IN HIGHPURITY BATCH DISTILLATION CONTROL LOOPS
R.M.Oisiovici and S.L.Cruz
Departamento de Engenharia de Sistemas Químicos/FEQ/UNICAMP
Cidade Universitária. Zeferino Vaz, 13083970, Campinas  SP, Brasil
Fone: +55197883948; +55192313346
Email: ronia@desq.feq.unicamp.br
(Received: September 30, 1999 ; Accepted: April 6, 2000)
Abstract  The influence of the sensor locations on the composition control of highpurity batch distillation columns has been investigated. Using concepts of the nonlinear control theory, an inputoutput linearizing controller was implemented to keep the distillate composition constant at a desired value by varying the reflux ratio. An Extended Kalman Filter was developed to estimate the compositions required in the control algorithm using temperature measurements. In the presence of measurement noise, the control performance depended greatly on the sensor locations. Placing the sensors further from the top stages reduced the detrimental effects of noise but increased the inference error. To achieve accurate composition control, both noise reduction and composition estimate accuracy should be considered in the selection of the sensor locations.
Keywords: Batch distillation, Nonlinear control, Kalman filter, Sensor location, Highpurity
INTRODUCTION
Batch distillation is the most frequent separation method in batch processes and has been widely used in fine chemical plants. Fine chemicals are generally high addedvalue products which must be manufactured according to high and welldefined standards of purity. Therefore, composition control has an essential role in batch distillation operations.
According to Betlem et al. (1998), if the degree of separation difficulty or the degree of exhaustion remains within bounds, then constant quality control appears to be the best control strategy. Control strategies for constantdistillatecomposition operations of batch distillation columns usually demand the knowledge of instantaneous distillate composition. Online analyzers are usually complex, expensive and introduce a time delay in the control loop. Therefore, compositions are commonly inferred from temperature measurements, allowing for inferential control of the product quality.
For ideal mixtures, Raoult's law provides a relationship among composition, temperature and total pressure. In the case of ideal binary mixtures, instantaneous compositions at a stage j can be inferred using real time measurements of the temperature at stage j. For systems which depart significantly from ideality, the relationship among these variables can be determined using equilibrium constants. In such cases, the composition is usually obtained using a trial and error procedure.
However, as it will be shown in this paper, lack of sensitivity of temperature sensors is a drawback in the control of the distillate composition in highpurity batch distillation columns. Variations in temperature are small towards the top stages of highpurity columns and it may be difficult to distinguish real variations from measurement noise. Control actions which are based on compositions inferred from temperature measurements in that region may be very corrupted by noise.
Digital filters (such as exponential and moving average filters) may attenuate the detrimental effect of noise on the control performance but they tend to introduce a time delay in the control loop, as it has been discussed by Oisiovici et al. (1999).
In order to obtain reliable and accurate estimates of compositions at the top stages of highpurity batch distillation columns, the temperature sensors should preferably be placed away from the top of the column. In this case, a state estimation algorithm must be employed to infer compositions at points which are different from the sensor locations.
Very few papers address the issue of state estimation of batch distillation columns. QuinteroMarmol et al. (1991) applied an extended Luenberger observer (ELO) to predict compositions in multicomponent systems from temperature measurements. The authors have reported that it was necessary to "turnoff" the gains of the ELO by the end of the batch when tests with measurement errors were made. Barolo and Berto (1998) used an ELO in the control of constantdistillatecomposition operations of batch distillation columns. The separation of an ethanol/water mixture in an 8tray column and the separation of an ideal ternary mixture with constant relative volatilities in a 10tray column were considered. The maximum composition setpoint adopted was x_{D,SP} = 0.97. The authors have found that increasing the number of trays makes the ELO harder to tune and pointed out the need for choosing a stochastic estimator (like a Kalman filter) when a large degree of noise is expected.
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. In the stochastic formulation of the nonlinear state estimation problem, a covariance matrix of estimation errors is minimized in spite of perturbations on the state dynamics, measurement noise and uncertainty about initial conditions. The Kalman Filter is one of the most widely used stochastic estimation algorithm and its extension for nonlinear systems, the Extended Kalman Filter (EKF), has been applied successfully in a number of chemical engineering applications (polymerization reactors, batch fermentation). Nonetheless, the EKF has not been applied to batch distillation systems yet.
In this work, an EKF has been developed to estimate compositions of highpurity binary batch distillation columns. The EKF was combined with a control law to keep the distillate composition constant at a desired value by varying the reflux ratio. The control strategy was derived using the inputoutput linearization technique. In a more realistic approach, the presence of measurement noise and plant/model mismatch have been considered. The influence of the location of temperature sensors on a highpurity batch distillation control loop has been investigated. In the presence of measurement noise, the results have shown that selecting appropriate sensor locations is very important to achieve tight composition control in highpurity batch distillation systems.
BATCH DISTILLATION NONLINEAR MODEL
The nonlinear model of the batch distillation column used in the EKF algorithm and in the design of the inputoutput linearizing controller will be presented in this section.
Due to complexity and computational burden, it is often impractical to use rigorous models for online state estimation. So, it is necessary to develop models which describe the essential elements of the dynamics.
The following assumptions were made: equimolal overflow; theoretical stages; negligible vapor holdup; constant liquid holdup; constant pressure; negligible reflux drum holdup and total condenser.
The separation of binary mixtures has been considered. If the state variables are the liquid composition of the light component in every stage (reboiler and trays), the state vector is given by: x =[x_{1}... x_{j} ... x_{NP+1}]^{T}, where j is the stage ('1' is the top stage and 'NP+1' is the reboiler). Considering that a total condenser was assumed, there was no need to include the condenser in the statespace model because the condenser state (x_{o}) is related to the top stage vapor composition (y_{1}) by the equation x_{o} = y_{1}. The liquid flow rate (L) and the vapor flow rate (V) are the elements of the input vector: u = [L V]^{T}.
Then, under the above assumptions, the nonlinear model of the batch distillation column in the statespace representation is:
(1) 
where y is the vapor composition of the light component and S is the liquid holdup.
The equilibrium relationship is given by the Raoult's law:
, where a_{j} = P_{1}^{vap}(T_{j})/P_{2}^{vap}(T_{j})  (2) 
The vapor pressure (P_{i}^{vap}) is determined using the Antoine equation, , 1£ i £NC. In equation 2, substituting P_{1}^{vap} for the Antoine equation of the light component, rearranging and isolating T, the sensor model on a stage p where a sensing element is placed has the following form:
(3) 
CONTROLLER DESIGN
Feedback linearization is a nonlinear control technique that can produce a linear model that is an exact representation of the original nonlinear model over a large set of operating conditions. It is based on two operations: nonlinear change of coordinates and nonlinear state feedback (Henson and Seborg, 1997). In the inputoutput linearization approach, the objective is to linearize the map between a transformed input (n) and the actual output (Y). A linear controller is then designed for the linearized inputoutput model.
Some concepts from the differential geometry that are relevant to the nonlinear analysis and synthesis problems will be presented here. Nonlinear SISO systems described by the following statespace model are considered:

(4) 
where f and g are ndimensional vectors of nonlinear functions, U is the manipulated input variable and q is a nonlinear function.
The Lie derivative is defined as:
(5) 
which is the directional derivative of the function q(x) in the direction of the vector f(x).
Using the previous definition,
(6) 
The relative degree of a system is the least positive integer r for which . The relative degree represents the number of times that the output Y must be differentiated with respect to time to recover the input U.
The control strategy developed for constantdistillatecomposition operation of batch distillation column was based on the Globally Linearizing Control (GLC) structure proposed by Kravaris and Chung (1987). Henson and Seborg (1997) have shown that the GLC is an inputoutput linearization technique for processes of arbitrary relative degree.
The GLC control law is described by the following equation:
(7) 
where b_{i} are controller tuning parameters. A PI controller is designed for the feedback linearized system:
(8) 
If R is the reflux ratio, then . Choosing , then equation (1) can be rewritten as:
(9) 
where:
For the batch distillation system described by equation (9), the relative degree is r = 1. Considering b_{1} = 1, b_{o} = 0 and Y = x_{D} = x_{o} = y_{1} (total condenser and negligible condenser holdup), the following control law was obtained:
(10) 
THE EXTENDED KALMAN FILTER
Kalman filtering is a method for recursively estimating the state of a system by optimally combining predictions of a system model with available measurement data. Many references discuss the Kalman filter techniques in details (for instance, Brown and Hwang, 1992), so the EKF will be presented briefly here.
It is assumed that a discrete nonlinear system can be represented by the following equations:
(11) 
(12) 
where z is the measurement vector, is the discrete system function, is the discrete measurement function, w and v are, respectively, the process and measurement noise vectors.
In the EKF algorithm, the nonlinear system is linearized about the current state estimate at each sampling time. The process noise and the measurement noise are assumed to be Gaussiandistributed zeromean variables with covariance Q and R, respectively.
The EKF is first initialized with _{0} and P_{0}, and then it operates recursively performing a single cycle each time a new set of measurements becomes available. The EKF algorithm consists of two stages: update and prediction.
Update Stage
(13) 

(14) 
(15) 
Prediction Stage
(16) 

(17) 
The EKF algorithm was applied to the batch distillation model (equation 1) and sensor model (equation 3) presented previously. The simplifying assumptions and the discretization of the system model introduce modeling errors for which the EKF is expected to compensate.
RESULTS AND DISCUSSION
The batch distillation column considered in this study was operated as follows:
A batch of liquid is charged to the reboiler and the column is brought to equilibrium under total reflux. Then, the distillate product starts to be collected (t = 0) at a constant reflux ratio. When the production phase starts, the distillate composition is usually above the setpoint. When the distillate composition is approaching the setpoint, the controller is switched on to keep the product quality at the desired value by manipulating the reflux ratio. At each sampling period, composition values are inferred from the available process temperatures and the control action is calculated using the current estimates.
In order to simulate the column behavior, a rigorous batch distillation simulator was employed. A 5second sampling period was considered in all runs. The example column has 30 ideal stages (including the reboiler) and a total condenser. The operating conditions adopted in the runs and the controller settings are presented in Table 1. For the sake of comparison, a highpurity product specification (x_{D,SP} = 0.99) as well as a lowpurity product specification (x_{D,SP} = 0.80) have been considered.
Before presenting the results obtained with the EKF, the need of using a state estimator in highpurity batch distillation control loops will be discussed. The operating conditions adopted in the following discussion are shown in Table 1.
The GLC controller (equation 10) demands the knowledge of instantaneous compositions at stages 1 and 2. For a binary system, the Gibbs phase rule indicates that there is a onetoone relationship between temperature and composition if pressure is constant. If a VLE relationship is available and the system pressure is known, one would promptly suggest that binary compositions at stages 1 and 2 could be easily determined by placing one temperature sensor at stage 1 and another sensor at stage 2. In order to show the limitations of this simple inference procedure, Figures 1 and 2 present batch distillation runs in which the required composition values were inferred using the system thermodynamic equations and temperature measurements of stages 1 and 2. Figures 1 (a) and (b) show that when the temperature sensors were assumed to be perfect (no measurement noise), tight composition control was obtained. Since perfect measurements are not found in practice, the presence of temperature noise was considered in the rest of the runs. When the sensors were corrupted by a Gaussdistributed white noise with standard deviation of 0.1 K, the controller was able to keep the distillate composition at 0.80 (Figure 2a) but failed to achieve the control objective when a highpurity distillate was to be produced (Figure 2b). The reflux ratio profiles in Figures 2 (a) and (b) indicate that the GLC controller was more sensitive to the presence of measurement noise as the product purity increased. This fact can be understood by examining the instantaneous column temperature profiles presented in Figure 3. As it is shown in Figures 3 (a) and (b), variations in temperature are relatively small towards the top stages in the highpurity batch distillation run. Since temperature sensors are usually not sensitive enough to detect the small temperature changes at the top stages of highpurity distillation columns, it is difficult to distinguish real variations from measurement noise. For inferential control of highpurity batch distillation columns the temperature sensors should preferably be placed away from the top stages. In such cases, a state estimator must be employed to estimate instantaneous compositions at stages 1 and 2 using temperature values of stages removed from the top.
From the preceding discussion, an important question to be posed is how far from the top of the column the measuring elements should be placed. Some runs were then carried out using the EKF to estimate compositions every sampling time. The EKF parameters adopted in the runs presented in Figures 47 are shown in Table 1.
Theoretically, if NC is the number of components, a distillation column is observable if at least (NC1) temperature sensors are used. QuinteroMarmol et al. (1991) found that the ELO needed at least NC sensors to be effective. For robust convergence, the authors recommend the use of (NC+2) measurements. In this work, two temperature sensors were employed.
When the sensors were placed at stages 4 and 8 (Figure 4), the EKF estimates were good and the GLC controller was able to keep the distillate composition at the setpoint. As the sensors were removed from the top of the column (Figure 57), smoother manipulated and controlled variable profiles were obtained. On the other hand, the inference error (difference between the actual compositions and the estimated compositions) has increased.
Therefore, the selection of the measurement locations must be a tradeoff between obtaining noise reduction and guaranteeing accurate composition estimates. For the examples considered here, Figures 47 indicate that actual tight distillate composition control was obtained when the sensors were located at stages 4 and 8.
When the sensors are appropriately located, 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 an important feature because the initial composition profile is usually not known. Furthermore, the initial conditions may often vary significantly from batch to batch.
CONCLUSIONS
In this work, the influence of the sensor locations on a highpurity batch distillation control loop has been investigated. The presence of temperature measurement noise and plant/model mismatch have been considered in the analysis.
An inputoutput controller based on the GLC structure was implemented to keep the highpurity distillate composition constant at a desired value by varying the reflux ratio. The results have shown that the GLC controller becomes more sensitive to the presence of measurement noise as the product purity increases. For highpurity batch distillation columns, the controlled and manipulated variable profiles presented a very oscillatory behavior and the controller was not able to keep the product composition at the setpoint when the sensors were located at the top stages.
In order to obtain reliable and accurate estimates of compositions at the top stages of highpurity batch distillation columns, an Extended Kalman Filter was developed to provide the top composition estimates which are needed in the control algorithm. The combined estimator/controller was tested using a rigorous batch distillation simulator.
In the presence of measurement noise, the control performance depended greatly on the sensor locations. The detrimental effect of noise on the controller was reduced and smoother profiles were obtained by locating the sensors further removed from the top of the column. On the other hand, the inference error has increased.
Therefore, the selection of the sensors locations must be a tradeoff between obtaining noise reduction and guaranteeing that the EKF will provide acceptable estimate accuracy.
ACKNOWLEDGEMENT
Ronia M. Oisiovici wishes to thank FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) for the financial support.
NOMENCLATURE
A, B, C  Antoine constants 
discrete system function  
f, g  vectors of nonlinear functions 
discrete measurement function  
K_{c}  controller gain, s^{1} 
L  liquid flow rate, mol s^{1} 
NC  number of components 
NP  number of trays 
P  error covariance matrix 
P_{0}  initial estimate of the error covariance matrix 
P  system pressure, Pa 
P_{i}^{vap}  vapor pressure of component i, Pa 
Q  process noise covariance matrix 
q  nonlinear function 
R  measurement noise covariance matrix 
R  reflux ratio 
r  relative degree 
S  liquid holdup, mol 
T  temperature, K 
t  time, s 
U  manipulated input variable 
u  input vector 
V  vapor flow rate, mol s^{1} 
v  measurement noise vector 
w  process noise vector 
x  liquid composition of the light component, mole fraction 
x  state vector 
_{0}  initial estimate of the system state 
estimate of the state at sample time m given output measurements up to sample time k  
Y  controlled output variable 
y  vapor composition of the light component, mole fraction 
z  measurement vector 
Greek letters
a  relative volatility 
b  controller tuning parameter 
n  transformed input variable 
s  temperature measurement standard deviation, K 
t_{I}  integral time, s 
Subscripts
D  distillate 
j  stage 
k  current values 
NP+1  reboiler 
o  condenser 
SP  setpoint 
REFERENCES
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