A hybrid feedback linearizing-Kalman filtering control algorithm for a distillation column

ISA Transactions®

Volume 45, Number 1, January 2006, pages 87–98

A hybrid feedback linearizing-Kalman filtering control algorithm for a distillation column Amiya Kumar Janaa,* Amar Nath Samantab a

Department of Chemical Engineering, Birla Institute of Technology and Science-Pilani, Rajasthan-333 031, India b Department of Chemical Engineering, Indian Institute of Technology-Kharagpur, West Bengal-721 302, India

共Received 23 September 2004; accepted 17 February 2005兲

Abstract This paper studies the design of a discrete-time multivariable feedback linearizing control 共FLC兲 structure. The control scheme included 共i兲 a transformer 关also called the input/output 共I/O兲 linearizing state feedback law兴 that transformed the nonlinear u − y to a linearized ␷ − y system, 共ii兲 a closed-loop observer 关extended Kalman filter 共EKF兲兴, which estimated the unmeasured states, and 共iii兲 a conventional proportional integral 共PI兲 controller that was employed around the ␷ − y system as an external controller. To avoid the estimator design complexity, the design of EKF for a binary distillation column has been performed based on a reduced-order compartmental distillation model. Consequently, there is a significant process/predictor mismatch, and despite this discrepancy, the EKF estimated the required states of the simulated distillation column precisely. The FLC in conjunction with EKF 共FLC-EKF兲 and that coupled with a measured composition-based reduced-order open-loop observer 共FLC-MCROOLO兲 have been synthesized. The FLC structures showed better performance than the traditional proportional integral derivative controller. In practice, the presence of uncertainties and unknown disturbances are common, and in such situations, the proposed FLC-EKF control scheme ensured the superiority over the FLC-MCROOLO law. © 2006 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Distillation; global linearization; FLC-EKF; FLC—MCROOLO; PID

1. Introduction Since the mid-1980s, the area of nonlinear process control has received a considerable amount of research attention because of the realization that linear controllers are inadequate to control even the moderately nonlinear processes. So far, two major research directions have been pursued: the model predictive approach 关1–8兴 and the differential geometric approach 关9–13兴. However, many researchers have shown that the feedback lineariz*Corresponding author. Tel.: ⫹91-01596-245073-215; Fax: ⫹91-01596-244183; [email protected]

E-mail

address:

ing control structure and model predictive controller 共MPC兲 are very closely related 关14,15兴 and even identical 关11,12兴. The geometric-based feedback linearizing control 共FLC兲 technique has the ability 共i兲 to linearize the nonlinear process and 共ii兲 to provide the decoupled process response for the coupled system. The transformer requires information on state variables. In practice, on-line measurement of all the states is rarely possible. Hence, there is a requirement to estimate the unmeasured states. For linear systems, the observer theory has been well established and widely available 关16,17兴. For nonlinear systems, due to their distinct behavior, few methods are available with any generality. The exact

0019-0578/2006/$ - see front matter © 2006 ISA—The Instrumentation, Systems, and Automation Society.

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A. K. Jana, A. N. Samantha / ISA Transactions 45, (2006) 87–98

Nomenclature B dt D DSS F h k KC , KC1 , KC2 , KR , KV Ki LR LS Ln m mB ˙B m mD mDsp m Dx D mDxDsp mf M mR mS n ¯P i Pˆi q Qi R RSS Ri ¯S i Sˆi t u ␷

Bottom rate Derivative time Distillate rate Bias of PI controller whose output is D Feed rate State transition matrix Time index

V VR VS VSS V1

Gain of conventional linear controller 共PI and PID兲 Kalman gain at time step i , n ⫻ m matrix Internal liquid flow rate in the rectifying section Internal liquid flow rate in the stripping section Internal liquid flow rate leaving nth tray Number of measured variables Holdup in the reboiler Time derivative of mB Holdup in the reflux drum Set point value of mD Multiplication of reflux drum holdup and top composition Set point value of mDxD Liquid hold-up in the feed plate Vector of measured variables Liquid hold-up in the rectifying section Liquid hold-up in the stripping section Number of states a priori estimate error covariance at time step i共n ⫻ n兲 matrix a posteriori estimate error covariance at time step i共n ⫻ n兲 matrix Elements of process noise covariance matrix Process noise covariance at time step i共n ⫻ n兲 matrix Reflux flow rate Bias of PID controller whose output is R Measurement noise covariance at time step i共m ⫻ m兲 matrix a priori state estimate / Predicted states 共model兲 at time step i共n ⫻ 1兲 matrix a posteriori state estimate / Estimated states at time step i共n ⫻ 1兲 matrix Time at which the PI and PID controllers produce the outputs Manipulated input Input to the linearized ␷ − y system

V2 V1SS V2SS x xˆ xB x˙B xBsp xD x˙D xDsp xf x˙ f xR x˙R xS x˙S y yB yf yi yR yS y sp Z

Vapor boil-up rate Internal vapor flow rate in the rectifying section Internal vapor flow rate in the stripping section Bias of PID controller whose output is V External PI controller output corresponding to xD External PI controller output corresponding to xB Bias of PI controller whose output is V1 Bias of PI controller whose output is V2 Vector of state variables Vector of estimated state variables Bottoms composition Time derivative of xB Set point of xB Distillate composition Time derivative of xD Set point of xD Composition of liquid leaving feed plate Time derivative of x f Composition of liquid leaving rectifying section Time derivative of xR Composition of liquid leaving stripping section Time derivative of xS Output variable Composition of vapor leaving reboiler Composition of vapor leaving feed plate Measured output at time step i共m ⫻ 1兲 matrix Composition of vapor leaving rectifying section Composition of vapor leaving stripping section Set point of y Feed composition.

Greek Letters ␤10 , ␤11 , ␤20 , ␤21 ⌬t ␧ , ␧1 , ␧2 ␶ , ␶1 , ␶2 , ␶R , ␶V ␶D

Tuning parameters of the FLC transformer Sampling period Errors to the linear controllers Integral time for linear controller Derivative time for PID controller.

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process model-based open-loop observer 共EPMOLO兲 关9兴 is one which computes all the states using the full-order dynamic process model. The order of the open-loop observer may be reduced by the number of measured process outputs. For open-loop unstable processes 共where the error of an open-loop observer may grow without bound兲 or open-loop stable processes with slow dynamics 共where a significant initial-state error may decay quite slowly兲 or open-loop stable processes under uncertainty, the open-loop observer failed to estimate the states correctly and, consequently, the corresponding observer-based controller performance has been degraded significantly. Subsequently, the research interest has been shifted to design and application of the efficient closed-loop observers 关18–22兴 for state estimation. Distillation columns exhibit highly interactive strongly nonlinear dynamic behavior with many other complexities. Model-based feedback linearizing control structure was suitable to control the distillation processes coping with both nonlinearities and interactions 关23,24兴. Unfortunately, the model-based controllers are adversely affected by modeling errors. On the contrary, an accurate dynamic model for the large industrial distillation system includes a large number of balance equations, which offer complexity in estimator design. Moreover, the development of a perfect mathematical model of a complicated process is often difficult. So the significant challenge is to estimate the unavailable states using a simplified model within the closed-loop observer for large processes such as distillation column. The main contributions of the present work are outlined as follows. In the extended kalman filter 共EKF兲 estimator, the number of differential equations to be solved is much more than the number of differential equations representing the process model. Motivated by this fact, the EKF design of a severely nonlinear distillation column having 20 trays was performed using a compartmental distillation model, which has three trays 共rectifying section⫹feed section⫹stripping section兲. Additionally, no tray hydraulics and constant vapor flow rate throughout the column are assumed. Consequently, excessive process/predictor mismatch occurred, and even with this mismatch the proposed EKF predicted the complex distillation dynamics accurately using the estimated states. The guaranteed and fast convergence of the esti-

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mation error is a key point of this method. Next, the FLC structure coupled with the high-gain EKF observer has been designed and used to control the example process. The FLC measured compositionbased reduced-order open-loop observer 共MCROOLO兲 control structure was also developed for the same distillation process. The proposed FLC-EKF control scheme was then compared with the FLC-MCROOLO and traditional dual-loop PID controller. It is often challenging to maintain a desired product purity for a complex process under uncertainty and unmeasured disturbance. But in such a situation, the FLC-EKF structure showed superior performance over the FLC-MCROOLO. 2. Process description In order to simulate the column behavior, a distillation column simulator has been used as a real process. This column had 20 theoretical trays and the saturated liquid feed mixture of 1-propanol/ ethanol was introduced at the tenth plate. The overhead vapor was totally condensed in an overhead condenser and collected the reflux drum. At the base of the column, a liquid bottom product was removed and vapor boil-up was generated using a thermo-siphon reboiler. The holdup of the vapor was assumed to be negligible throughout the system. Tray hydraulics was accounted for by the Francis weir formula. In this process, top and bottom compositions of the distillation column were controlled by manipulating reflux flow and vapor boil-up rates, respectively. The complete model and the relevant information for this distillation process are available in the literature 关25兴. The steady-state design parameters are given in Table 1. 3. Controller synthesis Three different control structures, namely FLC-EKF, FLC-MCROOLO, and conventional PID controllers, have been tested on the example distillation process. The synthesis of these controllers is presented here. 3.1. FLC-EKF control structure The compartmental model has been used as a state predictor within the closed-loop EKF. The feedback linearizing control law in conjunction

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Table 1 Steady-state design parameters of the simulated column. Feed composition 共mol fract.兲, Z Distillate composition 共mol fract.兲, xD Bottoms composition 共mol fract.兲, xB Feed flow rate 共mol/ min兲 , F Distillate flow rate 共mol/ min兲 , D Reflux flow rate 共mol/ min兲 , R Bottoms flow rate 共mol/ min兲 , B Vapor boil-up rate 共mol/ min兲 , V Total number of trays, nT Feed tray location, NF Tray holdup 共mol兲, m Reflux drum holdup 共mol兲, mD Bottom holdup 共mol兲, mB Tray hydraulic time constant 共min兲, ␩

0.5 0.98 0.02 100 50 128.01 50 178.01 20 10 10 100 100 0.1

with EKF formed the FLC-EKF control structure. The block diagram of FLC structure and nonlinear process is shown in Fig. 1. The different parts of the control algorithm are briefly described in the following section.

3.1.1.1. Compartmental model The simplified column 共Fig. 2兲 had three theoretical stages 共rectifying section, feed section, and stripping section兲, with the feed stream entered as saturated liquid. The energy balance equations have been neglected on each tray by assuming constant vapor flow rates throughout the column and constant internal liquid flow rates in each section resulting in constant molar hold up on each tray 共mR = m f = mS = 10兲. Thus, in this model, only composition dynamics are being considered. The summary of the modeling equations is reported in Appendix A. Steady-state specifications for the model are shown in Table 2. The steady-state values of the input variables 共R, F, Z, and V兲, ␣, D, B, mD, and mB were the same with the values of these variables in the actual process. 3.1.1.2. Extended Kalman filter As mentioned in the Introduction, the stochastic nonlinear EKF has been used as an estimator within the FLC framework. A detailed description of the EKF can be found elsewhere 关26兴, hence only the key steps are highlighted. 共i兲

3.1.1. Observer design The EKF observer was required to estimate the unmeasured states, and those estimated states along with the measured process outputs were required for FLC transformer calculation. To avoid estimator design complexity and, additionally, to reduce computational load, the EKF predictor was designed using a dynamic compartmental model of the process

Prediction of states,

¯S = h 共Sˆ 兲 . i i i−1 共ii兲

共1兲

Prediction of the covariance matrix of states, T ¯P = ␦hi Pˆ + Pˆ ␦hi + Q . 共2兲 i i−1 i−1 i ␦Sˆi−1 ␦Sˆi−1

共iii兲

Kalman gain matrix,

Fig. 1. Block diagram for FLC structure.

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共6兲

lim Ki = M −1 i . →0

Ri

On the other hand, as the a priori estimate error Pi approaches zero, the gain Ki covariance ¯ weights the residual less heavily, i.e.,

共7兲

lim Ki = 0. ¯P→0 i

The gain matrix corrected the predicted 共model兲 output based on the error between the measured output and the value predicted by the short-cut model. Two sets of tuning parameters, process noise covariance 共Qi兲, and measurement noise covariance 共Ri兲 played an important role in the performance of EKF.

3.1.2. Transformer design

Fig. 2. Schematic diagram of the compartmental distillation column.

Ki = ¯PiM Ti 共 M i¯PiM Ti + Ri兲−1 . 共iv兲 共v兲

共3兲 R共k兲 =

Update of the state estimation,

Sˆi = ¯Si + Ki共y i − M i¯Si兲 .

共4兲

Update of the covariance matrix of states,

Pˆi = 共I − KiM i兲 ¯Pi .

共5兲

From Eq. 共3兲, if the measurement error covariance Ri approaches zero, the gain Ki weights the residual more heavily. Specifically, Table 2 Steady-state design parameters of the compartmental process model. Distillate composition 共mol fract.兲, xD Composition of liquid in rectifying section 共mol fract.兲, xR Composition of liquid in feed plate 共mol fract.兲, x f Composition of liquid in stripping section 共mol fract.兲, xS Bottoms composition 共mol fract.兲, xB

From the model equations, the relative order of xD and xB with respect to R and V, respectively, is unity. Two transformer equations of FLC have been derived based on the short-cut predictor model and can be expressed as follows 共details are given in Appendix B兲:

0.7504868 0.6006191 0.4735253 0.3665335 0.2495303

冋

mB ␤ V 共k兲 − 共1 ⌬t关xS共k兲 − xB共k兲兴 20 2 + ␤21兲xB共k兲 −

⌬t 兵F关xS共k兲 − xB共k兲兴 mB

册

+ V共k兲关xB共k兲 − y B共k兲兴其 ,

V共k兲 =

共8兲

mD 关␤ V 共k兲 ⌬t关y R共k兲 − xD共k兲兴 10 1 − 共1 + ␤11兲xD共k兲兴 .

共9兲

xD共k兲 and xB共k兲 were obtained through direct measurements. The other states as per the transformer requirements like xS共k兲 and xR共k兲 关or y R共k兲兴 were obtained from the observer. Hence, the above two transformer equations can be rewritten using superscript “hat” 共∧兲 for estimated variables as 共details are given in Appendix B兲

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R共k兲 =

冋

previous time step. Therefore, the observer and actual process have the same steady-state design parameters and operating conditions 共Table 1兲.

mB ␤20V2共k兲 − 共1 ⌬t关xˆS共k兲 − xB共k兲兴 + ␤21兲xB共k兲 −

⌬t 兵F关xˆS共k兲 − xB共k兲兴 mB

册

+ V共k兲关xB共k兲 − y B共k兲兴其 , V共k兲 =

3.2.2. Transformer design

共10兲

mD 关␤10V1共k兲 ⌬t关yˆ R共k兲 − xD共k兲兴 − 共1 + ␤11兲xD共k兲兴 .

共11兲

Note that the relative order of mDxD and xB with respect to R and V, respectively, is unity. The transformer equations of the FLC-MCROOLO control scheme have been derived based on the exact process model and can be expressed as 共details are given in Appendix B兲 R共k兲 =

共1 + ␤11兲mDxD共k兲 − ␤10V1共k兲 + ⌬t兵V共k兲yˆ 20共k兲 − D共k兲xD共k兲其 共⌬t兲xD共k兲

3.1.3. External controller

,

共14兲

Two conventional PI controllers of the form

V1 = V1SS + KC1

冉

1 ␧1 + ␶1

冕 冊

␧1 dt ,

V2 = V2SS + KC2

冉

V共k兲 =

0

where ␧1 = xDsp − xD , 1 ␧2 + ␶2

共1 + ␤21兲xB共k兲 − ␤20V2共k兲 +

t

冕 冊

mB

共12兲

t

mB

兵xˆ1共k兲 − xB共k兲其 .

兵y B共k兲 − xB共k兲其

共15兲 The superscript “hat” 共∧兲 has been used for the variables which were obtained from the observer.

␧2 dt ,

0

where ␧2 = xBsp − xB ,

⌬t

⌬tLˆ1共k兲

共13兲

have been used as the external controller for the ␷ − y system. The external controller outputs were bounded: 0.6612艋 V1 艋 1.2734; 0.74艋 V2 艋 1.065. 3.2. FLC-MCROOLO control structure The full-order model of the real process except the balance equations around the reflux drum and reboiler 共since the top and bottom compositions were assumed to be measurable兲 is termed here as the measured compositions-based reduced-order open-loop observer. This observer has been used as a state estimator within the FLC structure and the resulting algorithm is called the FLCMCROOLO control structure. The different elements of this control methodology are derived here.

3.2.3. External controller Two conventional single-variable PI controllers 关Eqs. 共12兲 and 共13兲兴 have been used as the external controller around the linearized ␷ − y system. 3.3. PID control structure The equations of a traditional dual-loop PID controller, which control the nonlinear process, are as

冉

R = RSS + KR ␧1 +

1 ␶iR

冕

t

␧1dt + ␶DR

0

冊

d␧1 , dt 共16兲

冉

V = VSS + KV ␧2 +

1 ␶iV

冕

t

0

␧2dt + ␶DV

冊

d␧2 . dt 共17兲

3.2.1. Observer design As stated, the exact dynamic process model excluding the balance equations around the reflux drum and reboiler is used to estimate the states by using the state and manipulated variables from the

3.4. Level controller A single-variable PI controller was employed in controlling the level of the reflux drum by manipulating the distillate flow rate 共D兲,

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Table 3 Controller tuning parameters. Controller 共1兲 FLC-EKF EKF

Tunable parameters

Transformer External Controller 共2兲 FLC-MCROOLO Transformer External Controller Level Controller 共PI兲 共3兲 Dual-loop PID

q11 = 0.0000048; q22 = q33 = q44 = 0.0005; q55 = 0.0001025; r11 = 0.002; r22 = 0.0004 ␤10 = 0.00000684; ␤11 = −0.998; ␤20 = 0.0015; ␤21 = −0.99 KC1 = 7000; ␶1 = 10; KC2 = −1000; ␶2 = 10

␤10 = 0.0784; ␤11 = −0.9992; ␤20 = 0.0002; ␤21 = −0.99 kC1 = 1.5; ␶1 = 5000; kC2 = 10; ␶2 = 1000 kC = −14.1; ␶ = 2.53 kR = 2250; ␶iR = 3.25; ␶DR = 4.75; kV = −25500; ␶iV = 0.12; ␶DV = 12.34

冉 冕 冊

D = DSS + KC ␧ +

1 ␶

t

␧dt ,

0

where ␧ = mDsp − mD .

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drum was held constant using a PI controller. The tuning of the level controller was tightened so that the variation of reflux drum holdup was significantly small and can be considered constant. Hence, ␧1 = mDxDsp − mDxD has been replaced by ␧1 = xDsp − xD in the external controller equation of FLC-MCROOLO. The input variables are bounded as follows: 50艋 R 艋 300; 100艋 V 艋 350.

4. Results and discussion The performance of the different proposed control structures was carried out through extensive simulations using the sample distillation column. The tuning of EKF has been carried out based on the guidelines proposed by Baratti et al. 关21兴, and the optimal tunable parameters of FLC and PID controllers have been determined by the use of integral square error 共ISE兲 performance criteria. The tuning parameters are listed in Table 3. EKF performance

共18兲

The transformer design for FLC-EKF control structure was straightforward and for ease of the complicated transformer design of FLCMCROOLO control law, the level of the reflux

To inspect the tracking performance of the EKF estimator, several simulated open-loop experiments have been performed. Figs. 3 and 4 show that the estimated values were in good agreement with the measured values of distillate as well as bottom compositions. In Fig. 3, the estimator

Fig. 3. Open-loop responses of EKF with step changes in R 共changed from 128.01 to 134.41 at time= 2.5 min and then from 134.41 to 128.01 at time= 12.5 min兲.

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Fig. 4. Open-loop responses of EKF with step changes in R 共changed from 128.01 to 134.41 at time= 2.5 min and then from 134.41 to 128.01 at time= 12.5 min兲 under process noise 共amp= ± 0.5兲 in R.

performance was tested for +5 % 共128.01 → 134.41兲 and, subsequently, −4.8% 共134.41 → 128.01兲 step changes in reflux flow rate. Fig. 4 shows the same step changes with the addition of random noise to the reflux flow rate. These figures also included the estimated values of xR and xS, which were required for the transformer calculations.

Start-up performance of FLC-EKF The most difficult task for any controller is to stabilize the system 共steady state兲 in the presence of an excessive process/model mismatch. From the start-up profile 共Fig. 5兲 it is observed that the quality control actions of the FLC structure helped to achieve the steady-state condition very quickly.

Fig. 5. Start-up performance of FLC-EKF control structure.

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Fig. 6. Comparative servo performance of FLC-EKF, FLC-MCROOLO, and PID control structures with set point step changes in xB 共changed from 0.02 to 0.01 and then from 0.01 to 0.02兲.

For this reason, the time scale in the subsequent plots has been started at a minimum 10 min. Comparison of observer-based FLC structures and PID The modeling error affects adversely the modelbased controller performance owing to the mismatch of estimator and process outputs. The dynamic compartmental model has been used as the state predictor within the closed-loop high-gain EKF observer and the full-order process model was used as the measured composition-based reduced-order open-loop observer. The corresponding observer-based controllers, FLC-EKF and FLC-MCROOLO, have been compared with a conventional PID controller in Fig. 6 for large set point step changes in the bottom composition 共changed from 0.02 to 0.01 and then from 0.01 to 0.02兲. From the simulation experiment, it is obvious that despite process/predictor mismatch, the FLC-EKF controller showed an improved response over the FLC-MCROOLO and traditional PID controllers.

presence of even small model parametric uncertainty, the estimation error within the MCROOLO grows continuously and sharply up to an unacceptable level. Consequently, the FLC-MCROOLO structure provided a poorer performance with permanent deviation in process outputs. On the other hand, the EKF provided fast convergence of the error between estimated and process outputs, and as a result, the FLC-EKF control scheme achieved an excellent performance. Under unmeasured disturbance The performance between the FLC-EKF and FLC-MCROOLO control strategies for unmeasured disturbance in feed composition 共changed from 0.5 to 0.52 at time= 12.5 min兲 is shown in Fig. 8. The FLC in conjunction with MCROOLO did little to reduce and control the estimation error. As a result, the process outputs deviated continuously resulting in a large offset. Due to the effective estimation error convergence capability of EKF, significantly improved performance has been observed for the FLC-EKF when compared to the FLC-MCROOLO controller.

Comparison of FLC-EKF and FLC-MCROOLO Under parametric uncertainty

5. Conclusions

Fig. 7 shows the comparative performance of FLC-EKF and FLC-MCROOLO control structures under uncertain relative volatility 共changed from 2.0 to 1.95 at time= 12.5 min兲. Due to the

The feedback linearizing control structure in conjunction with the closed-loop extended Kalman estimator has been designed for a highly nonlinear multivariable distillation column. The ex-

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Fig. 7. Comparative performance of FLC-EKF and FLC-MCROOLO under parametric uncertainty in ␣ 共changed from 2.0 to 1.95兲 at time= 12.5 min.

ample process had 20 theoretical trays. To avoid the design complexity of EKF, a compartmental dynamic process model having three trays 共rectifying section⫹feed section⫹stripping section兲 has been used for state predictions within the EKF. Despite excessive process/predictor mismatch, the EKF estimated the required process outputs with sufficient accuracy. The proposed FLC-EKF law showed satisfactory start-up performance. In the simulation experiment with large set point step

changes, the FLC-EKF controller provided a better performance when compared to that of the FLC-MCROOLO and traditional PID controllers. The excellent control of the distillation column under uncertainty and unmeasured disturbance is the major concern in industrial systems. The proposed FLC-EKF algorithm showed superior performance when compared to the FLC-MCROOLO due mainly to the exponential estimation error convergence capability of EKF.

Fig. 8. Comparative performance of FLC-EKF and FLC-MCROOLO under unmeasured disturbance in Z 共changed from 0.5 to 0.52兲 at time= 12.5 min.

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Appendix A Internal liquid flow rate in the rectifying section

LR = R.

x˙B =

1 关共R + F兲共xS − xB兲 + V共xB − y B兲兴 . mB

共A1兲

共B3兲

Internal liquid flow rate in the stripping section including feed plate,

In discretized form,

共A2兲

x B共 k + 1 兲 = x B共 k 兲 +

LS = R + F. Internal vapor flow rates,

VS = VR = V. The reflux drum component balance,

x˙D =

V 共y − x 兲 . mD R D

1 关R共xD − xR兲 + V共y f − y R兲兴 . mR

共A5兲

Substituting xB共k + 1兲 from Eq. 共B4兲 into Eq. 共B5兲 and simplifying, one obtains Eq. 共8兲. Only by replacing xS by xˆS 共estimated value of xS兲 in Eq. 共8兲 has Eq. 共10兲 been obtained.

The discretized form of Eq. 共A4兲,

1 关L 共x − x 兲 + F共Z − x f 兲 + V共y S − y f 兲兴 . mf R R f

x D共 k + 1 兲 = x D共 k 兲 +

⌬tV共k兲 关y R共k兲 − xD共k兲兴 . mD

共A6兲 The stripping section component balance,

x˙S =

1 关L 共x − x 兲 + V共y B − y S兲兴 . mS S f S

共A7兲

The reboiler component balance,

x˙B =

1 关L x − BxB − Vy B兴 . mB S S

共B5兲

Deduction of Eqs. (9) and (11)

The feed plate component balance,

x˙ f =

共B4兲

Since the relative order of xB with respect to V is 1, the linearized ␷ − y equation can be written as

xB共k + 1兲 + ␤21xB共k兲 = ␤20V2共k兲 . 共A4兲

The rectifying section component balance,

x˙R =

⌬t 兵关R共k兲 + F兴关xS共k兲 − xB共k兲兴 mB

+ V共k兲关xB共k兲 − y B共k兲兴其 . 共A3兲

97

共A8兲

共B6兲 The relative order of xD with respect to R is 1, and so the linearized ␷ − y equation can be written as

xD共k + 1兲 + ␤11xD共k兲 = ␤10V1共k兲 .

共B7兲

Substitution of xD共k + 1兲 from Eq. 共B6兲 into Eq. 共B7兲 and after rearrangement, Eq. 共9兲 has been obtained. Now replacing y R by yˆ R, Eq. 共9兲 can be rewritten as Eq. 共11兲. Deduction of Eq. (14) The discretized form of the component continuity equation around the reflux drum is

Appendix B

mDxD共k + 1兲 = mDxD共k兲 + ⌬t兵V共k兲y 20共k兲 − 关R共k兲

Deduction of Eqs. (8) and (10) Total continuity equation around the reboiler,

˙ B = LS − V − B = 0. m

共B1兲

+ D共k兲兴xD共k兲其 .

共B8兲

The linearized ␷ − y equation can be written

mDxD共k + 1兲 + ␤11mDxD共k兲 = ␤10V1共k兲 .

which means

共B2兲

共B9兲

Substituting LS共=R + F兲 and B from Eq. 共B2兲 into Eq. 共A8兲 and rearranging, the following equation has been obtained:

By substitution of mDxD共k + 1兲 from Eq. 共B8兲 into Eq. 共B9兲 with replacing the states by their estimated values, Eq. 共14兲 has been obtained.

B = LS − V = R + F − V.

98

A. K. Jana, A. N. Samantha / ISA Transactions 45, (2006) 87–98

Deduction of Eq. (15) The component continuity equation around the reboiler is

˙ B = L 1x 1 − B xB − V y B . mBx˙B + xBm

共B10兲

˙ B and simplifying, Substituting m x˙B =

L 1共 x 1 − x B兲 + V 兵 x B − y B其 . mB

共B11兲

The discretized form of Eq. 共B11兲,

x B共 k + 1 兲 = x B共 k 兲 +

⌬t 兵L 共k兲关x1共k兲 − xB共k兲兴 mB 1

+ V共k兲关xB共k兲 − y B共k兲兴其 .

共B12兲

The linearized closed-loop ␷ − y equation is

xB共k + 1兲 + ␤21xB共k兲 = ␤20V2共k兲 .

共B13兲

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