Comparison of Nonlinear Controllers for Distillation Startup andOperation†

Chimmiri Venkateswarlu and Kota Gangiah*

Process Dynamics and Control Group, Chemical Engineering Sciences, Indian Institute of ChemicalTechnology, Hyderabad 500 007, India

A control algorithm which has been acclaimed as the best algorithm for startup control andalso operation control of a real system may not be the best algorithm for a different real system.Therefore, a nonlinear internal model control (NIMC) strategy supported by an on-linedeterministic estimator is presented for startup control and also operation control of a continuousdistillation column. The performance of the NIMC strategy is evaluated by comparing withglobally linearizing control (GLC) and generic model control (GMC) strategies. The results showthat NIMC, GLC, and GMC have exhibited nearly the same performance for startup andoperation control of a continuous distillation column. NIMC strategy is recommended for startupand operation control of a continuous distillation column due to easier tuning of one controllerparameter and best transition from total reflux to steady-state operation.

Introduction

Startup of chemical processes especially the continu-ous distillation is a challenging control problem whichinvolves complex heat- and mass-transfer operationsand encounters a wide range of operating conditionsduring startup period. The commonly used strategy inindustry is to switch on to a conventional PI controllerto maintain the desired tray temperature.The dynamic behavior of distillation columns during

startup has been studied and analyzed by Ruiz et al.(1988) by simulation studies. The highly nonlineartransition from total reflux to specified reflux in thepresence of several disturbances is to be effected opti-mally and conventional controllers are usually lesseffective. Yasuoka et al. (1987) proposed a semiempiri-cal characteristic function for determining the optimalswitching time from total reflux to steady-state opera-tion. The characteristic function is to be calculatedusing the on-line composition and temperature mea-surements of the column. The optimal switching timeis determined as the time corresponding to the mini-mum of the characteristic function. This strategy lacksrobustness. A model-based controller, viz., the globallylinearizing control (GLC) framework, using a nonlineartransformation that transforms a nonlinear input/output system into a linear input/output system isproposed by Kravaris and Chung (1987). The genericmodel control (GMC) introduced by Lee and Sullivan(1988) allows the implementation of a nonlinear processmodel directly into the controller structure. Further,Henson and Seborg (1991) proposed a nonlinear internalmodel control (NIMC) approach which is different fromthe GMC of Lee and Sullivan (1988) and Kravaris andChung (1987).Barolo et al. (1993, 1994) presented a simple model

for control of distillation by grouping a number ofcomponent dynamic balance equations into one dynamicequation, in which the controlled variable is consideredas the inventory of multiple tray temperatures. First,Barolo et al. (1993) applied the GMC-based strategy forstartup of a binary distillation column. Further, the

GLC-based strategy was applied by Barolo et al. (1994)for startup and operation control of a distillationcolumn.In this study, a nonlinear internal model control

(NIMC) based strategy supported by an on-line estima-tor is presented for startup and operation control of adistillation column. The performance of the proposedstrategy is evaluated by applying it to a continuousdistillation column separating a methanol-water mix-ture. Further, comparison of the NIMC-based strategyis made with the GMC-based strategy of Barolo et al.(1993) and also with the GLC-based strategy of Baroloet al. (1994).

Control Algorithms Description

The general form of a control affine single-inputsingle-output (SISO) system with a state-space descrip-tion is

where x is a vector of n states, u is the manipulatedinput, y is the measured output, f(x) and g(x) are vectorfunctions, and h(x) is a scalar function. The relativeorder r of a system described by eqs 1 and 2 is definedby the following equations:

where Lfh(x) is the Lie derivative of h(x) with respectto f(x), Lgh(x) is the Lie derivative of h(x) with respectto g(x), and LgLfh(x) is the Lie derivative of h(x) firstwith respect to f(x) and then with respect to g(x).A general form of the control law is written as

where v is the new input which is defined in terms ofthe recent advanced process model based controllers.

* To whom correspondence should be addressed. E-mail:root@csiict.ren.nic.in. Phone: (040)7173626. Fax: (040)-7173387.

† IICT Communication No. 3809.

x ) f(x) + g(x) u (1)

y ) h(x) (2)

LgLfkh(x) ) 0; k < r - 1 (3)

LgLfr-1h(x) * 0 (4)

u ) p(x) + q(x) v (5)

5531Ind. Eng. Chem. Res. 1997, 36, 5531-5536

S0888-5885(97)00285-6 CCC: $14.00 © 1997 American Chemical Society

Many control laws can be derived based on the newinput, v.Globally Linearizing Control. The GLC approach

has been proposed by Kravaris and Chung (1987) forcontrol affine systems and is based on an input-outputlinearization through the use of suitable static statefeedback and coordinate transformations. An externalPI loop

can be used to force the output y to track a given desiredtrajectory, yd. The closed-loop transfer function of thesystem is given by

According to this approach, the control law for a systemis given by

The values of â0, â1, ..., âr, kc, and τI are chosen so thatthe roots of the characteristic equation have negativereal parts.Generic Model Control. Lee and Sullivan (1988)

proposed GMC which forces the process output rate tomatch a reference rate. The reference rate for theprocess is generated from the setpoint deviation as

where v is the desired process output rate, yd is thesetpoint, and k1 and k2 are the generic control loopconstants. The reference rate is proportional to thedistance from the setpoint and includes integral actionto eliminate offset. The control signal is obtained bysetting

and solving for the manipulated input, u.The closed-loop transfer function of the GMC which

is restricted to relative order one system is

The control law for GMC is expressed as

The generic loop constants k1 and k2 are selected usingthe tuning map of Lee and Sullivan (1988).Nonlinear Internal Model Control. The NIMC

approach has been proposed by Henson and Seborg(1991) for nonlinear systems. This approach includesan implicit integral action by using the difference

between the plant and model outputs as a feedbacksignal:

where y is the the process model output. The feedbacksignal simplifies to e ) yd with perfect model assump-tion. A nonlinear filter is employed in NIMC whichprovides a tuning parameter that can be adjusted forprocess/model mismatch. The new input v for NIMC isdefined as

where τi are controller tuning parameters, r is relativeorder, and e ) yd. The closed-loop transfer function ofthe system is

where ε is a tuning parameter. According to thisapproach, the control law for a system is given by

The performances of various nonlinear process modelbased control algorithms presented above are evaluatedfor distillation startup and operation control.

Distillation System

The use of model-based controllers for startup of adistillation column needs the determination of unmea-sured variables which are to be determined on-line.Hence, startup and operation control of distillation withthe support of an on-line estimator for the unmeasuredvariables can be a suitable example for critical evalu-ation of the nonlinear controllers.Instead of a real distillations system, a rigorous

dynamic model-based distillation package is resorted toas a test case to compare the performance of thecontrollers. A binary distillation column for methanol-water separation (Wood and Berry, 1973) is used in thisstudy. The rigorous nonlinear model considers thesimultaneous effects of heat- and mass-transfer opera-tions and fluid flow on the plates. The model is derivedfrom first principles involving dynamic material andcomponent, and algebraic energy equations supportedby vapor-liquid equilibrium and physical properties.The details of the distillation column are given in Table1. The distillation dynamics package has the majorcomputation subroutines like tray hydraulics, enthalpycalculations, average molecular weight, and densitycalculations, and vapor-liquid equilibrium calculations.This dynamic model-based distillation package is notrequired for the case of a real distillation system.Inference of Compositions. Distillation column

temperatures can be measured on-line. Liquid andvapor compositions have to be inferred from the tem-perature measurements. Polynomial equations relatingX vs T and Y vs T are developed by using the actual

v ) kc[(yd - y) + 1τI∫0t(yd - y) dσ] (6)

y(s)yd(s)

)kc(s + 1/τI)

ârsr+1 + âr-1s

r + ... + â1s2 + (â0 + kc)s + kc/τI

(7)

u )

v - ∑k)0

r

âkLfkh(x)

ârLgLfr-1h(x)

) p(x) + q(x) v (8)

v ) k1(yd - y) + k2∫0t(yd - y) dσ (9)

v ) yd ) h(x) (10)

y(s)yd(s)

)k1s + k2

s2 + k1s + k2(11)

u )v - Lfh(x)

Lgh(x)) p(x) + q(x) v (12)

e ) yd - (y - y) (13)

v ) -τrLfr-1h(x) - τr-1Lf

r-2h(x) - ... - τ1h(x) + τ1e(14)

y(s)yd(s)

)τ1

sr + τrsr-1 + ... + τ2s + τ1

) 1(εs + 1)r

(15)

u )v - Lf

rh(x)

LgLfr-1h(x)

) p(x) + q(x) v (16)

5532 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997

vapor-liquid equilibrium data of the methanol-watersystem:

where X is the liquid composition of methanol, Y is thevapor composition of methanol, and T is the traytemperature. Thus, the compositions are computedfrom temperature measurements using a subroutinewhich is also made available for this test case of thedistillation dynamics package. The X, Y, and T dataare used to compute the enthalpies needed for a on-lineestimator which supports the controllers.

Distillation Nonlinear Controllers

The strategy of distillation startup and operationcontrol is described as follows.Inventory Model for Controlling. The controlled

variable is assumed to be the inventory of the lightcomponent from the top tray to any convenient tray inthe enriching section of the column. The reflux rate isconsidered as the manipulated variable that tracks thedesired composition profiles in the column. Barolo etal. (1993) presented a very simple model for control bygrouping a number of component dynamic balanceequations into one dynamic equation

where NP is the control tray, NT is the top tray, D isthe distillate rate, V is the vapor mole rate, L is theliquid mole rate, L0 is the reflux mole rate, M is theliquid holdup, X is the liquid composition, and Y is thevapor composition. The location of the plate NP is tobe chosen according to the requirement of controllersensitivity. The simplified dynamic model (eq 18) ismore advantageous as it is in control affine form: arelative order one with respect to the manipulated

input. Both feed and steam flow rates can act asunmeasured disturbances in the model.Derivation of Control Algorithms. Different non-

linear model-based control laws are obtained as follows.(i) GLC. Substitution of eq 18 into eq 8 yields the

state feedback law as

The external PI controller of the form

is employed around the input/output linearized system.The control law can be finally obtained as

(ii) GMC. The control law for the reference trajectoryis of the form

On substituting eq 18 into eq 12, the manipulativereflux rate is obtained as

(iii) NIMC. The input v is obtained from eq 14 as

The control law is obtained according to eq 16 as

On-Line Estimation of Unmeasurable Variables.The practical implementation of control laws requiresthe knowledge of unmeasurable states. Distillationcolumns are usually equipped with thermocouples fromwhich the temperature profiles along the column canbe known at any instant of time. Moreover, the bottompressure, condenser duty, and top and bottom holdupsare measured on-line. Based on these measurements,a procedure for on-line estimation of the unmeasurablestates such as internal flow rates and compositions hasbeen given by Barolo et al. (1993).

Table 1. Details of the Distillation Column

number of trays 8feed plate position 4column pressure (atm) 1components methanol-waterfeed propertiescomposition (more volatile mole fraction) 0.330flow rate (kmol/h) 2.941temperature (K) 349.99

distillate propertiescomposition (more volatile mole fraction) 0.9334flow rate (kmol/h) 1.031temperature (K) 338.80

bottom product propertiescomposition (more volatile mole fraction) 0.0042flow rate (kmol/h) 1.910temperature (K) 372.23

steady-state operating conditionsreboiler heat duty (kJ/h) 1.026 × 105reflux flow rate (kmol/h) 1.71

setpoints of level controllersreboiler (kmol) 0.780reflux drum (kmol) 0.181

X ) -0.254805 × 10-3 + 0.160080 × 102T -0.392840T2 + 0.474115 × 10-2T3 - 0.282603 ×

10-4T4 + 0.667288 × 10-7T5

Y ) -0.212577 × 102 + 1.22564T - 0.2454252 ×10-1T2 + 0.213576 × 10-3T3 - 0.694292 × 10-6T4

(17)

d

dt[ ∑j)NP

NT

(MX)j] ) (VY)NP-1 + L0XD - (VY)NT - (LX)NP(18)

u ) [v - â0 ∑j)NP

NT

(MX)j - â1{(VY)NP-1 - (VY)NT -

(LX)NP}]/â1XD (19)

v ) kc(δ + 1/τI∫0t δ dσ),

δ ) ∑j)NP

NT

(MX)j,d - ∑j)NP

NT

(MX)j (20)

L0 )(VY)NT - (VY)NP-1 + (LX)NP

XD

+

kc(δ + 1/τI∫0t δ dσ) - â0 ∑j)NP

NT

(MX)j

â1XD

(21)

d

dt[ ∑j)NP

NT

(MX)j] ) k1δ + k2∫0tδ dσ (22)

L0 ) [(VY)NT - (VY)NP-1 + (LX)NP + k1δ +

k2∫0tδ dσ]/XD (23)

v ) ( ∑j)NP

NT

(MX)j,d - ∑j)NP

NT

(MX)j)τ1 (24)

L0 ) [(VY)NT - (VY)NP-1 + (LX)NP + ( ∑j)NP

NT

(MX)j,d -

∑j)NP

NT

(MX)j)τ1]/XD (25)

Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5533

The internal flow rates are calculated using deter-ministic on-line balances:

Condenser duty, Qc, is obtained by measuring thetemperature rise of the cooling water across the con-denser and multiplying it by the cooling water flow rate.Total condensation is assumed for the condenser. Thederivative of the reflux drum holdup is estimated fromthe direct measurements of MD. A three-term dif-ferentiation is used to reduce the instability problemsin determining the derivative of MD.

The overhead composition, XD, is obtained by the on-line integration of top mass balance,

Procedure of Startup and Operation Control.Each of the nonlinear controllers, viz., GLC or GMC orNIMC, supported by an on-line estimator is employedfor the distillation system. The sequence of actionswhich forms the basis for startup and operation controlof distillation column is described as follows:1. The reboiler is initially filled with feed mixture,

which is then heated up. The cooling system or con-denser is also switched on. Vapor boilup is heldconstant.2. When the reflux drum is full, feed is introduced

and reflux is allowed to flow down the column. Bottomflow rate is then controlled by the reboiler level.3. In the early stages of operation, the top product

flow is zero and the column is under the total reflux.4. Operation is continued in total reflux mode until

all the plates have enough liquid holdup.5. When all the plates have been filled with the

liquid, the computation of reflux rate according to GLCstrategy or GMC strategy or NIMC strategy is started.6. As time goes on, the light component concentrates

in the enriching section of the column and the refluxdemand decreases. The setpoint of the reflux flowcontrol loop is updated at each sampling instant by thecontroller, and the distillate flow is established by thedistillate drum level control.7. Once the steady state is reached, the same control-

ler is used for operation control of the column.

Results and Discussion

The controlled variable, ∑j)68 (MX)j, is the inventory

of 6, 7, and 8 trays in the rectifying section of thecolumn. Steady-state plate holdups and light-compo-nent compositions are computed using the dynamicmodel of the distillation column. On the 6th, 7th, and8th trays, steady-state holdups are 0.0187, 0.0171, and0.0160 kmol, whereas steady-state methanol composi-tions in mole fractions are 0.640, 0.771, and 0.866,respectively. The steady-state value of ∑(MX)j,d is thusdetermined as 0.039 kmol. Desired trajectories during

operation control are generated by considering stepchanges in ∑(MX)j,d. An initial XD value of 0.55 is usedfor the estimator. Controller and estimator calculationsare carried out at each sampling instant using a sampletime of 0.02 h. In this study, a deterministic stateestimator is employed for on-line estimation of unmea-surable states for the nonlinear controller. Processnoise in the measurements is not considered for thestate estimation.The initial tuning parameters for the GMC are

derived using the dynamic distillation model. Thetuning parameters are further adjusted as k1 ) 17.5 h-1

and k2 ) 8.0 h-2 so as to obtain satisfactory startupperformance. The preliminary tuning parameters forthe GLC are chosen to obtain the desired closed-looppole specifications of eq 7. These tuning parameters arefurther adjusted as â0 ) 0.1, â1 ) 0.4 h, τI ) 0.4 h, andkc ) 7.5, respectively. The initial tuning parameter forthe NIMC is obtained by using eq 15 and choosing εsuch that the characteristic polynomial has negativereal roots. The finally adjusted tuning parameter is τ1) 9.5 h-1.The temperature responses of trays and reboiler for

GMC and GLC have shown a sharp trend duringstartup operation and rapidly attained the steady-statecondition, whereas temperature responses of NIMC hasexhibited a less sharp trend and quickly approached thesteady-state condition. The responses of controlledvariable and estimated distillate composition for thethree controllers are shown in Figure 1. The corre-sponding reflux flow rate and distillate flow rate areshown in Figure 2. The results of Figure 1a show thatthe responses of GLC and GMC exhibit a little overshootduring the transition period from total reflux condition

LNP ) [D(zD - ZNP-1) + Qc - (dMD/dt)ZNP-1]/(ZNP-1 - zNP) (26)

VNT ) Qc/(ZNT - zD) (27)

VNP-1 ) LNP + D + dMD/dt (28)

dMD/dt ) (3MD[k] - 4MD

[k-1] + MD[k-2])/(2∆t) (29)

(MX)D[k+1] ) (MX)D

[k] + [(VY)NT - (L0 + D)XD][k]∆t

(30)

Figure 1. Startup and operation control: (a) controlled variableprofile; (b) estimated distillate composition profile.

5534 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997

to steady-state reflux condition. The controlled variableresponse of NIMC shows a smooth trend and quicklyattained the steady-state condition. The results ofFigure 2 show that the reflux flow rate is very highduring the initial startup period and drastically reducedby the controllers so as to allow the light component tomove to the top of the column. The distillate flow ratesare also slightly high during the initial withdrawalperiod and quickly settle down to their nearest actualvalues. The responses of the controlled variable andestimated top product composition are shown in Figure3 for a step change in controlled variable from 0.039 to0.045 kmol during steady-state operation. The corre-sponding reflux flow rate and distillate flow rate areshown in Figure 4. The results show the performanceof GMC, GLC, and NIMC strategies for the step changein controlled variable. In this study, setpoint changeonly is considered for startup control and also operationcontrol similar to Barolo et al. (1993, 1994). Both feedand steam flow rates as unmeasured disturbances arenot included in the inventory model for the test cases.In this study, the initial controller parameters for

GLC, GMC, and NIMC are selected based on the desiredclosed-loop pole specifications and further tuned suchthat the responses track the desired trajectories. Thisapproach is generally followed for tuning differentnonlinear controllers. Tuning of GMC is involved withtwo parameters, k1 and k2. GLC has four parameters,â0, â1, kc, and τI. Tuning of NIMC is involved with asingle parameter, ε. The controller parameters chosenfor GMC, GLC, and NIMC have shown reasonablybetter performances for startup and operation controlof binary distillation column. Recently, Barolo (1994)have shown conditions that assure equivalence between

Figure 2. Startup and operation control: (a) manipulated vari-able; (b) top product.

Figure 3. Responses of (a) controlled variable and (b) estimateddistillate composition for step change in controlled variable.

Figure 4. Responses of (a) manipulated variable and (b) topproduct for step change in controlled variable.

Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997 5535

the GMC and GLC closed-loop responses for relativeorder one systems. Then the two parameters â0 and â1of the GLC can be chosen arbitrarily, the third param-eter τI becomes a combination of â0 and â1, and thefourth parameter kc becomes a tuning parameter. Thetuning constants kc and τI found in the GLC formulationare identical with 2ê/τ and 2êτ in the GMC formulation.Thus, a GLC closed-loop response which is exactlyequivalent to the GMC controller can be induced bysetting kc ) â1k1 and τI ) k1/k2. In this study, with suchan equivalence, GLC and GMC has produced identicalresults which are not reported for brevity. Withoutfollowing such an equivalence approach, the arbitrarytuning parameters chosen for GMC and GLC of thisstudy has also produced almost nearer results, Figures1-4. In this study, the comparison of three controllersinstead of three pairs of controllers, viz., (i) GMC andGLC, (ii) GLC and NIMC, and (iii) GMC and NIMC, iscarried out which is only possible on the basis of (moreor less) arbitrary tuning of controllers. There is nosignificant difference in the control performance ofNIMC and produced almost identical results to thoseof GLC and GMC. NIMC is employed with a singletuning parameter, the tuning of which is quite easy. Thecontrolled variable response of NIMC exhibits a best andsmooth trend during the transition period which occursnear the end of the total reflux condition.

Conclusions

A nonlinear internal model control (NIMC) strategysupported by an on-line estimator is presented forstartup and operation control of a continuous distillationcolumn. The performance of the control strategy isevaluated by applying it to the Wood and Berry binarydistillation column with methanol-water separation. Inorder to compare the performance of the NIMC strategywith the advanced model-based controllers, two morenonlinear control strategies, namely, a globally linear-izing control (GLC) and a generic model control (GMC),are also applied for startup and operation control of thedistillation column. The results show that NIMC, GLC,and GMC have produced nearly the same performancefor startup and operation control of a continuous distil-lation column. The controlled variable response ofNIMC exhibits best and smooth transition from totalreflux condition to steady-state operating condition.NIMC strategy is recommended for startup and opera-tion control of a continuous distillation column due toeasier tuning of a single controller parameter and besttransition from total reflux to steady-state operaton.

Nomenclature

D ) distillate flow rate, kmol/he ) errorf(x) ) vector function of a process modelg(x) ) vector function of a process modelh(x) ) scalar functionkc ) proportional gain of the PI controllerk1 ) tuning parameter of GMCk2 ) tuning parameter of GMCL ) liquid flow rate, kmol/hL0 ) reflux flow rate, kmol/hM ) liquid holdup, kmolp(x) ) transfer function of the linearized process model

Qc ) condenser duty, kJ/hq(x) ) transfer function of the linearized process modelT ) temperature, Kt ) timeu ) manipulated inputV ) vapor flow rate, kmol/hv ) new manipulated inputx ) vector of state variablesX ) liquid mole fraction of the light componenty ) process outputY ) vapor mole fraction of the light componentZ ) vapor enthalpy, J/kmolz ) liquid enthalpy, J/kmol

Greek Symbols

âi ) tuning parameters of GLCδ ) error function∆t ) sampling time, hε ) tuning parameter of NIMCτi ) tuning parameters of NIMCτI ) reset time of the external PI controller in the GLCcontrol law

Subscripts and Superscripts

d ) desired valuej ) tray numberNP ) control trayNT ) top trayr ) relative order of the output with respect to themanipulated input

Literature Cited

Barolo, M. On the Equivalence between the GMC and the GLCcontrollers. Comput. Chem. Eng. 1994, 18, 769-772.

Barolo, M.; Guarise, G. B.; Rienzi, S.; Trotta, A. On-Line Startupof a Distillation Column Using Generic Model Control. Comput.Chem. Eng. 1993, 17(S), 349-354.

Barolo, M.; Guarise, G. B.; Rienzi, S. A.; Trotta, A. NonlinearModel-Based Startup and Operation Control of a DistillationColumn: An Experimental Study. Ind. Eng. Chem. Res. 1994,33, 3160-3167.

Henson, M. A.; Seborg, D. E. An Internal Model Control Strategyfor Nonlinear Systems. AIChE J. 1991, 37, 1065-1081.

Kravaris, C.; Chung, C. B. Nonlinear State Feedback Synthesisby Global Input/Output Linearization. AIChE J. 1987, 33, 592-603.

Lee, P. L.; Sullivan, G. R. Generic Model Control (GMC). Comput.Chem. Eng. 1988, 12, 573-580.

Ruiz, C. A.; Cameron, I. T.; Gani, R. A Generalized Dynamic Modelfor Distillation ColumnssIII. Study of Startup Operations.Comput. Chem. Eng. 1988, 12, 1-14.

Wood, R. K.; Berry, M. W. Terminal Composition Control of aBinary Distillation Column. Chem. Eng. Sci. 1973, 28, 1707-1717.

Yasuoka, H.; Nakanishi, E.; Kunigita, E. Design of an On-LineStartup System for a Distillation Column Based on a SimpleAlgorithm. Int. Chem. Eng. 1987, 27, 466-472.

Received for review April 14, 1997Revised manuscript received September 30, 1997

Accepted October 2, 1997X

IE970285X

X Abstract published in Advance ACS Abstracts, November1, 1997.

5536 Ind. Eng. Chem. Res., Vol. 36, No. 12, 1997