Robust multivariable control of abinary distillation column

G.W.M. Coppus, S.L. Shah and R.K. Wood

Indexing terms: Multivariable control systems

Abstract: Davison's robust, multivariable, feedback-feedforward controller (DAVISON, E.J.: IEEE Trans.,1976, AC-21, pp. 35-47) is applied to the control of a pilot-scale binary distillation column, as a master loop ina cascaded configuration. The inner or slave loop consists of the original plant with multiloop proportionalcontrol. This cascaded configuration is shown to have a number of advantages for practical implementation.The performance of the controller is evaluated by experimental application to a computer-controlled distillationcolumn. The robust controller gives improved control performance compared to that achieved using conven-tional multiloop proportional plus integral plus derivative (PID) control.

1 Introduction

A technique for the synthesis of robust multivariablefeedback-feedforward controllers for an unknown plantoffers significant advantages in an industrial environmentwhere state-space descriptions of many processes are oftenvery difficult to obtain. This paper is concerned with theexperimental application to a computer-controlled pilot-plant distillation column, of a technique proposed byDavison [1].

Most of the modern control synthesis techniquesrequire a mathematical model of the process that is to becontrolled. In many instances, the modelling of complex,often poorly understood processes, is a substantial andcrucial task and by no means routine. A promising alterna-tive for the control of such processes is to employ designmethods that require only plant input/output data and notan explicit mathematical model of the process. Self-tuningmethods, based mainly on the work of Astrom andcoworkers [2, 3] and of Clarke and Gawthrop [4, 5],which employs an assumed model and uses input/outputdata for parameter identification, have been used suc-cessfully to control a number of industrial processes. Amore recent technique introduced by Davison [1] reliesonly on plant input/output data and is based on thenotion of compensator identification as opposed to plantidentification. It is to be noted that Davison's controllerand Francis and Wonham's [6] controller based on theinternal model principle are very similar in structure, inthat both require an internal model. A good comparison ofthe two methodologies is described by Kwatny and Kal-nitsky [7].

In order to apply Davison's technique, the plant char-acteristics should be such that it can be described by alinear, time-invariant model and be open-loop asymp-totically stable. However, unlike the self-tuning methods,Davison's robust feedback-feedforward technique does notrequire any assumptions regarding the plant model order,minimum phase characteristics etc. Secondly, the techniqueguarantees a robust (with respect to small parametervariations), closed-loop, asymptotically stable, multi-variable system.

The present paper describes the application andexperimental evaluation of Davison's robust controller toa computer-controlled pilot-plant distillation column. Thedistillation process is the most common separation unit inthe chemical process industry, and hence its control is a

Paper 2659D, first received 14 July 1982 and in revised form 20th July 1983

The authors are with the Department of Chemical Engineering, University ofAlberta, Edmonton, Alberta, Canada T6G 2E1

subject of considerable interest [8-10]. The set-pointtracking and regulatory control properties of the robustfeedback-feedforward controller are evaluated and com-pared with the regulatory performance of well tuned multi-loop PID control.

2 Davison's robust feedback-feedforwardcontroller

Let a linear, time-invariant, multivariable plant bedescribed by the following state-space or input/outputmodel:

x(t) = Ax(t) + Bu(t) + Ew(t)

y(t) = Cx(t)

y(s) = C(sl - Ay'Buis)

£ G(s)u(s) + D(s)w(s)

C{sl -

(1)

(2)

where x e Rn, u e Rm, y e Rr, and w e Rd are state, input,output, and disturbance vectors, respectively. For mostprocess systems of moderate complexity, the matrices A, B,C, and E are seldom accurately known. In addition, thelinear plant descriptions (eqn. 1 or 2) are only valid for asmall operating range. Outside a specified operatingregion, the assumption of linearity is invalid. A distillationcolumn is typical of such nonlinear processes. Davison'srobust design technique, as presented in Reference 1, istherefore of limited use in its application to such processesbecause it requires the plant to be linear and time invari-ant. Experimental results obtained from the pilot-plant dis-tillation column presented in Section 4 illustrate this point.A rather simple, but practically useful modification, is sug-gested to extend the applicability of the robust design tech-nique to a larger class of processes.

2.1 Robust control of cascaded plantUtilisation of simple multiloop proportional or pro-portional plus derivative control for a typical plantachieves two very important plant characteristics. With thecorrect choice of gains, the plant may be made stable and,secondly, the assumption of linearity with respect to set-point and/or load changes is valid over a wider range ofoperating conditions than the open loop plant. Thus, therobust controller can be implemented as a master loop in acascaded scheme, as shown in Fig. 1. The slave or innerloop consists of the original plant with multiloop pro-portional (P) control with feedback gain K. Proportionalplus derivative (PD) control action may also be employedin the slave loop. However, it is not possible to use anyintegral control action in the slave loop, because this

IEE PROCEEDINGS, Vol. 130, Pt. D, No. 5, SEPTEMBER 1983 201

would eliminate steady-state offset in the outputs, suggest-ing nonexistence of the robust controller (cf. Davison'stheorem II following).

w(t)

Fig. 1 Block diagram for distillation-column control using master robustfeedback-feedforward controller with slave multiloop proportional controller

Use of the cascaded scheme has three advantages.First, the proportional feedback action renders the plantmore linear, i.e. the excursions due to set-point interactionsand disturbances are reduced. The linearity assumptioncan now be considered to be valid, and theorems I and IIof Davison and the resulting algorithms can be applied tothe augmented 'linear' plant. Secondly, with the cascadeconfiguration it is possible to design the robust controllerfor an industrial unit without performing experiments Iand II on the open loop plant [1]. The normal steady-stateinput/output data would be sufficient for the controllerdesign. Thirdly, the plant inherently has analog backup inthe form of P or PD feedback control in the event of acomputer failure. The two latter features are very impor-tant for field implementation of advanced control stra-tegies on industrial plant units. For a plant described byeqn. 1, the augmented plant with controller,u = Ky1 — Ky, becomes

x = (A - BKC)x 4- BKy1 + Ew

y = Cx

(3)

where yl is the set point for the slave loop. Thus, themodified plant is characterised by matrices (A — BKC),BK, and E which will be denoted by A, B and E, respec-tively. To guarantee existence of a robust controller for themodified system, Davison's results in terms of A, B and Ecan be restated as follows:

Theorem /[V]: If the disturbance w is measurable, thenthere exists a feedforward controller for the augmentedsystem (eqns. 3), so that e(t) = (yref — y)—> 0 as t—* oo forall constant disturbances, w e Rd and constant referenceinputs yref e Rr, if and only if

rank G'(0) = rank C{-A)~lB = r

Theorem / / [ / ] : There exists a robust controller for theaugmented system (eqns. 3), such that e(t)—> 0 as t—> oo forall constant disturbances vv e Rd and all constant referenceinputs yref e Rr, and the closed-loop system is asymp-

totically stable, if and only if

rank G'(0) = rank C(-A)~iB = r

Theorems I and II are elementary extensions of Davison'sresults [1]. Having ascertained the existence of a robustfeedback-feedforward controller, the final controller forconstant set points and disturbances is given by:

u(t) = T~lyref -T~lDw-sT1 [\y-ynf)dt

Jo

where T = G'(0) and D = C{-A)'lE and e > 0.The tuning of this controller is done by the choice of

just one parameter, e. This is an important advantage overclassical multiloop PI or PID control, in which the task ofselecting controller parameters is nontrivial.

It is interesting to note that block diagram manipula-tion of this scheme shows that the overall 'robust' controlscheme is in fact nothing more than a multivariable PIscheme, as can be seen from Fig. 2. Notice also that a

w(t)

Fig. 2 Modified block diagram of cascaded structure shown in Fig. 1 toshow multivariable PI control action

different amount of proportional action is applied to theyref and y signals. From a frequency-domain viewpoint,the controller T'1 is simply a steady-state decouplingmatrix.

3 Distillation-column equipment

The 22.8 cm diameter, 8-plate, pilot-scale distillationcolumn used in this study has been used in a number ofprevious control studies [10, 11]. Each plate at a spacingof 30.5 cm is fitted with four bubble caps. Operating condi-tions, with feed to the fourth tray, are adjusted to providea separation of the 50% by weight methanol/watermixture into top and bottom compositions of about 95%and 5% by weight methanol, respectively. A detailedsummary of typical steady-state operating conditions isgiven in Table 1.

A schematic diagram of the column, which is com-pletely instrumented with conventional industrial controlvalves, controllers and transmitters, is shown in Fig. 3. Ascan be appreciated, only the principal control instrumen-tation pertinent to the objective of this control study hasbeen shown. Since the objective is to maintain top andbottom compositions at their specified values, bothstreams are equipped with instrumentation for composi-tion analysis. Top composition, although continuously

Table 1 : Typical steady-state operating conditions

Feed-flow rateReflux-flow rateSteam-flow rateTop-product flow rate

18.0 g/s8.8 g/s

13.6 g/s9.0 g/s

Bottom-product flow rateFeed compositionTop compositionBottom composition

9.0 g/s50.0 g/s95.0 g/s

5.0 g/s

202

Note. Compositions expressed as % methanol by weight

1EE PROCEEDINGS, Vol. 130, Pt. D, No. 5, SEPTEMBER 1983

monitored using a temperature-compensated in-line capac-itance probe, is sampled at the same rate as the bottomcomposition signal. Bottom composition is measured using

coolingwater

feed HP 1000-DEC 11/03distributed computersystem

steam

bottomproduct

Fig. 3 Schematic diagram of pilot-plant distillation column

CR, Analyses recorder GC, Gas chromatographFRC, Flow recorder/controller LC, Level controller

a HP 5722A gas chromatograph (GC). Liquid sampling ofthe bottom-product stream is under the control of one ofthree HP 1000s of the Department's distributed computernetwork of mini- and micro-computers. Analysis of thechromatogram, on a 180 s cycle, is also performed by thisnode of the distributed network, which then transmits asignal to a 'local' node of the network, a DEC 11/03 micro-computer. The control sample interval for both loops is setequal to 3 min, as a unit integer multiple of the 3 minmeasurement delay time that occurs in the GC analysis.The schematic configuration of the pilot-plant units andthe distributed computer network is shown in Fig. 4. Ascan be seen from the diagram, implementation of thecontrol algorithm and data acquisition, except for the GCcomposition measurement, is performed using the DEC11/03. Operator communication to the DEC 11/03 is bymeans of a CRT terminal. Detailed documentation of the

bulk storage

Fig. 4 Schematic diagram of distributed network of mini- and micro-computers and interface with computer-controlled distillation column

system is given by Coppus [12]. Besides data transfer, theexisting links between the higher node(s) (HP 1000) andlower level node(s) (e.g. DEC 11/03) are designed to facili-tate future applications of advanced computer-controlstudies.

4 Robust controller design

The design of Davison's robust feedback-feedforward con-troller requires that a series of simple experiments be per-formed on the plant to be controlled. The input/outputdata obtained as a result of these experiments are thenused for the controller design. To design a robustfeedback-feedforward controller for asymptotic set-pointtracking and regulation, two series of experiments are con-ducted.

Experiment 1The first series of experiments is concerned with finding thesteady-state gain matrix of the open-loop system. The con-trolled outputs of interest are yx = top-product composi-tion, and y2 — bottom-product composition; manipulativeor control variables are: ux = reflux-flow rate and u2 =steam-flow rate. To determine the steady-state gain matrixT, the flow rate was changed from its steady-state value by10% in the positive direction, Fig. 5, and then by 10% in

960

s~ 95.0

en

1* 94.0

QX

8.5

o 7.0c

CD

5.5

A.O

6°

30 60 90time , s x 102

120 150

Fig. 5 Open-loop response of product compositions to + 10% stepchange in reflux flow rate (u,)

the negative direction, Fig. 6. Similarly, the steam-flow ratewas changed by ±8% from its steady-state values, asshown in Figs. 7 and 8.

The resulting steady-state gain matrices T, were foundto be significantly different for positive and negative inputchanges:

fO.83 -1.40]1 L3 0 0 -4.40j

= [1.08 -0.54]2 \_237 -6.15J

positive stepchange in inputs

negative stepchange in inputs

The substantial difference between corresponding elementsof T\ and T'2 for nominal changes in ut and u2 clearlydemonstrates the asymmetric dynamics of the distillationcolumn. It was also observed that the signs of the determi-

IEE PROCEEDINGS, Vol. 130, Pt. D, No. 5, SEPTEMBER 1983 203

nant of T\ and T'2 are different. This would suggest that,for some nonzero operating conditions, det T = 0, i.e. therobust controller may not be locally existent for some setsof operating conditions. The second series of experiments

96.0 r

95.0

E« 94.0

I 93.0

oo

5.5

A.O

!? 2.5

CO i.oli30 60 90

timers x102120 150

Fig. 6 Open-loop response of product compositions to —70% stepchange in reflux flow rate (u,)

to determine D(0) were not repeated because of the nonlin-ear dynamic behaviour of the column. Instead, as sug-gested in Section 2, the robust controller was designed as amaster loop, using multiloop proportional control as theslave loop. The slave loop, as can be seen from Fig. 2, hasthe same outputs as the original distillation column, butthe manipulative inputs for the master loop are the setpoints for the top and bottom composition loops.

96.5

oo 95.5

en'5

o"93.5x

7.00

1.0030 60 90

time ,s x iO2120 150

Fig. 7 Open-loop response of product compositions to + 8% stepchange in steam-flow rate (u2)

Experiment 1 with distillation column under multiloopproportional controlMultiloop proportional gains of 2.50 and —0.40 wereimplemented on the distillation column. These gains corre-

96.0

^ 9 5 . 5

a 95.0

15.0

oo"S 11.0E

CDX

7.0

3.030 60

time ,s x 10'90 120 150

Fig. 8 Open-loop response of product compositions to — 8% stepchange in steam-flow rate (u2)

sponded to an arbitrary fraction (0.7) of the actual pro-portional gains implemented on a PID scheme that isdiscussed later. The first series of experiments were re-peated on this closed-loop system by changing the top andbottom composition setpoints by ±1.0%. The resultingclosed-loop 'steady-state' gain matrices were, respectively,found to be:

10.68 -0.04.041 = [0.68 0.04]

-6 J 2 L 0 7 2 °-54JIn comparison to the previous T\ and T'2 for the open-loop case, the corresponding elements of Tx and T2 havethe same order of magnitude gains. The closed-loop plant(with multiloop proportional control) has reduced inter-action terms in the steady-state gain matrix and can nowbe considered to be linear over the operating region ofinterest. The above results serve to confirm remarks madein Section 2 on the utility of implementing the robust con-troller in a cascaded structure.

Experiment 2To design the feedforward element of the robust controller,an additional series of experiments were conducted toobtain the steady-state load transfer function gains. Theresponses obtained by making step changes of ± 10% inthe main load variable, feed flow rate, to obtain the gains,are shown in Fig. 9. The resulting gains for positive andnegative step disturbances denoted by Dt and D2 respec-tively, were

D\= [-0.026 0.935]

D\= [-0.042 0.744]

From these two series of experiments, the final robust plusmultiloop proportional controller evaluated in the experi-ments performed to study the control performance was

204 IEE PROCEEDINGS, Vol. 130, Pt. D, No. 5, SEPTEMBER 1983

u =

where

yref - - 3W(0] dt

T2.50 0 "I~ [O -0.40j

1.4761.969

- Ky{t)

0.017]1.706J

0.

i 3 r

0 15 30 45sample time (Ts=3min)

60

Fig. 9 Column response under multiloop proportional control to stepchanges of ± 10% in feed-flow rate

+ 10% feed disturbance— 10% feed disturbance

where T is the arithmetic average of Tx and T2 (i.e. T =0.57; + 0.5T2), similarly, D = O.SDy + 0.5D2, and s is atuning parameter [1].

Experimental evaluation of robust controllerThe major advantage in implementing such a scheme onan operating process unit is that, for asymptotic trackingand regulation of step-function changes, the final tuning isdone by the choice of just one parameter e. The selectioncriterion used in tuning e was the absolute magnitude ofthe error (IAE = J | e(t) \ dt). Table 2 summarises some ofthe initial runs performed with different values of e, for a— 25% feed-flow disturbance.

From the sum of the top and bottom composition IAEvalues, it was found that e = 0.10 gives the best results forthe feed-flow disturbance to the column.

The performance of the robust feedback plus feed-forward (FB + FF) controller for a —25% step change infeed-flow rate is shown in Fig. 10. It is clear that asymp-

Table 2: Tuning of robust feedback-feedforward controllerfor -25% feed-flow disturbance

IAE

y^

0.050.100.150.200.25

5.73.02.42.22.2

4.96.29.0

10.510.9

5s 93

QX

13

1

CD15 30 45

sample time (Ts= 3 min )60

Fig. 10 Closed-loop response of product compositions under robust FBand FB + FF control for —25% step change in feed-flow rate

robust FB and FF control robust FB control

totic regulation of output variables yt and y2 is satisfac-torily achieved. The effect of feedforward action, Dw(t), onthe control performance is significant, as can be seen fromthe difference in the responses with and withoutfeedforward-control action.

In order to assess the regulator characteristics of therobust controller, compared to the control performancethat can be achieved using conventional control, thecolumn was operated under well tuned multiloop PIDcontrol. The controlled responses, using the two PID con-trollers, with and without feedforward control, for a— 25% step change in feed-flow rate, are shown in Figs. 11

16

I 13Eo

* 10

x

CD

10 20 30

10 20 30sample time ( T s r 3 m i n )

40

I 9

S 7

oI 96

93

90

10 20 30 40

10 20sample time

30 40= 3 min)

Fig. 11 Closed-loop responses under multiloop PID control for —25% step change in feed-flow rate

IEE PROCEEDINGS, Vol. 130, Pt. D, No. 5, SEPTEMBER 1983 205

and 12. The feedforward controller used with the multi-loop control scheme was identical to that employed for therobust controller results in Fig. 10. The PID controller

changes in feed, steam and reflux flow rates (±25%, ~±20%, ~ ±23%, respectively) are all significantly outsidethe regime in which the open-loop tests were conducted

5

E "O"35

ocos 9a

Fig. 12

15 30 45 60I

75

| 7

oo

1 96r

o> 93

15 30 60 75

15 30 45sample time (T, = 3 min )

60 75 0 15 30 45 60 75sample time (Ts = 3 min)

Closed-loop responses under multiloop PID plus feedforward control for -25% step change in feed-flow rate

Table 3: Multiloop PID constants(Sample time 3 mins)

Top composition controllerController setting:

Proportional gain, 3.571Integral time constant, 695 sDerivative time constant, 42 s

Bottom composition controllerController setting:

Proportional gain, -0.576Integral time constant, 771 sDerivative time constant, 40 s

settings that yielded these responses are given in Table 3and the IAE values for the two controllers, computed for90 min of operation, are summarised in Table 4.

The column responses and the corresponding manipu-lative variable change to a ±25% step changes in feed-flow rate, with the column under robust feedback plusfeedforward, are shown in Fig. 13. For these tests, the

Table 4: Comparison of controller performance for load dis-turbance

Disturbance

-25% stepchange infeed-flow rate

Controller

Robust FB onlyPID FB onlyRobust FB + FFPID + FF

Ki

4.42.13.01.7

IAE

Vi

35.321.76.2

12.6

Figure

10111012

(±10%, ±8% and ± 10%). This controller performance isa clear indication of the robust nature of Davison's controlscheme.

As can be seen from the responses in Figs. 10-12 andthe IAE values, the robust FB + FF controller provides

15 30 45

X

* 50 15 30 45 60

- 5

I0 15 30 45

sample time (Ts = 3 m i n )60

oco.c

£

- 93

900 15 30 45

sample time (Ts= 3 min)60

Fig. 1 3 Closed-loop response of product compositions and corresponding manipulative variables under robust FB + FF control for ±25"A> step change infeed-flow rate

increase decrease

206 IEE PROCEEDINGS, Vol. 130, Pt. D, No. 5, SEPTEMBER 1983

better control of the product compositions than can beachieved with conventional control. However, besides theperformance, a significant advantage of the robustFB + FF controller is its ease of tuning. The tuning of amultiloop PID control is a nontrivial task, as six par-ameters must be tuned. In addition, the tuning alsodepends on the sequence in which the loops are tuned. Adifferent set of gains would be obtained if loop 1 weretuned first with loop 2 open, in comparison to loop 2 beingtuned first with loop 1 open. In addition, no guarantee ofstability and 'robustness' is obtained.

To evaluate the set-point tracking characteristics ofthe robust controller versus the use of conventional con-trollers, a set-point change of +1 % step in the top com-position, with the feed-flow rate maintained constant, wasmade for the column operated under robust feedbackcontrol and PID control. The control performance isshown in Fig. 14. Despite the required changes in steamand reflux flow rate of +15% and +29%, respectively,well beyond the range of the open loop tests (cf. Figs. 5-8),the robust controller demonstrated excellent asymptoticset-point tracking characteristics, and outperformed thePID controller, as evidenced by the IAE values sum-marised in Table 5.

Inspection of Fig. 14 reveals only small interactionamong the two variables. This is expected, since the robustcontroller has incorporated within it a steady-state decou-pler (T~l). This matrix tends to eliminate most of thelow-frequency interaction in the process.

5 Conclusions

This paper has been concerned with the design, applicationand experimental evaluation of Davison's robust, multi-variable, feedback-feedforward controller for the control ofa binary distillation column. It was demonstrated byexperiment that the nonlinear nature of the distillationcolumn made direct application of the robust controllerimpractical. Use of the robust controller in a master loop,cascaded in series to the distillation column under multi-loop proportional control, was implemented and evalu-ated. The main conclusions of this study are:

17 , -

r 14

1 3 , -

10

O

- 9

3

S 720

Table 5: Comparisonchange

Set-point change

+ 1 % step changein top-productcomposition

of controller

Controller

RobustPID

performance

IAE

Yy

3.12.4

y2

5.710.3

for set-point

Figure

14

(i) Excellent control of the distillation column wasobtained with robust feedback and feedforward controlcascaded to a simple multiloop proportional controller.This was despite little information on the distillationcolumn model, significant measurement time delay (3 min),process time lags, and nonlinear characteristics.

(ii) Excellent set-point tracking and regulatory controlperformance was observed, despite large upsets in processoperating conditions, e.g. —25% step change in feed-flowrate. The controller proved to be robust and relativelyinsensitive to changes in parameters caused by such largeupsets. It is important to note that both the set-point anddisturbance tests conducted on the robust controllerinvolved changes in the disturbance and manipulated vari-ables significantly beyond the range of change for theopen-loop tests, which formed the basis for controllerdesign. The robust nature is further illustrated by the factthat, despite the cascaded structure of the final controller,some asymmetricity in column dynamics is still in evi-dence. These results are thus a strong indication of therobust nature of Davison's controller. The interactionbetween the two outputs during set-point tracking wasminimal. Thus the objective of minimal low frequencyinteraction was also achieved.

(iii) The effort involved in tuning the controller (throughone parameter e) is significantly less than in tuning a multi-loop PID controller.

(iv) With the suggested cascaded structure, design datafor the robust controller can be easily collected duringregular operation of the unit with the conventional PI orPID controllers operated only in proportional mode.

30 50 10 20 30 50

oo

E 9

CD 1

0 10 20 30 40sample time (T. = 3 min )

50

* 96

O.93

£9010 20 30

sample time (T, = 3 min )50

Fig. 14 Closed-loop response of product compositions under robust FB and multiloop PID control for + 1% step change in top composition set pointrobust control PID control

1EE PROCEEDINGS, Vol. 130, Pt. D, No. 5, SEPTEMBER 1983 207

(v) In the final scheme, integrity with respect to digitalcomputer failure is high if the slave-loop proportionalaction is implemented by conventional analog instrumen-tation. These two latter features are especially importantfrom an industrial viewpoint.

6 Acknowledgments

One of the authors (GWMC) would like to thank theSocial Sciences and Humanities Research Council ofCanada for financial assistance. Financial support from theNatural Sciences and Engineering Research Council ofCanada is also gratefully acknowledged.

7 References

1 DAVISON, E.J.: 'Multivariable tuning regulators: the feedforwardand robust control of a general servomechanism problem', IEEETrans., 1976, AC-21, pp. 35-47

2 ASTROM, K.J., BORISSON, U., LJUNG, L., and WITTENMARK,B.: 'Theory and applications of self-tuning regulators', Automatica,1977, 13, pp. 457-476

3 ASTROM, K.J., and WITTENMARK, B.: 'On self-tuning regulators',ibid., 1973, 9, pp. 185-199

4 CLARKE, D.W., and GAWTHROP, P.J.: 'Self-tuning controller',Proc. IEE, 1975, 122, (9), pp. 929-934

5 CLARKE, D.W., and GAWTHROP, P.J.: 'Self-tuning control', ibid.,1979,126, (6), pp. 633-640

6 FRANCIS, B.A., and WONHAM, W.M.: 'The internal modes prin-ciple of control theory', Automatica, 1976,12, pp. 457-465

7 KWATNY, H.G., and KALNITSKY, K.C.: 'On alternate method-ologies for the design of robust linear multivariable regulators', IEEETrans., 1978, AC-23, pp. 930-933

8 SCHWANKE, CO., EDGAR, T.F., and HOUGEN, J.O.: 'Develop-ment of multivariable control strategies for distillation columns', ISATrans., 1978, 16, pp. 69-81

9 SHINSKEY, F.G.: 'Distillation control for productivity and energyconservation'(McGraw-Hill, 1977)

10 MEYER, C.B.G., WOOD, R.K., and SEBORG, D.E.: 'Experimentalevaluation of analytical and Smith predictors for distillation columncontrol', A.I.Chem. J., 1979, 25, pp. 24-32

11 SASTRY, V.A., SEBORG, D.E., and WOOD, R.K.: 'Self-tuning regu-lator applied to a binary distillation column', Automatica, 1977, 13,pp. 417-424

12 COPPUS, G.W.M.: 'Robust control of distillation column'. M.Sc.thesis, Dept. of Chemical Engineering, Univ. of Alberta, Edmonton,Canada, 1980

208 IEE PROCEEDINGS, Vol. 130, Pi. D, No. 5, SEPTEMBER 1983