Identification and Control of a Distillation Column

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IDENTIFICATION AND CONTROL OF A DISTILLATION COLUMN B. Dahhou, H. Youlal, A. Hmamed and A. Majdoul 1.. t:L.S .A .. Fuodt, dt'.' Sr;fIIrn. HY .

Abstract:

f()/4

Ra/lilt ..l/u,.,,((u

The paper presents theoritical and experimental results on the modelisation

and identifi cation of a distillation column.

First, a physical model based on heat and

mass balances is derived for the hole pilot plant.

A linearised input output model for

the top temperatures is considered for identification purpose. from on line identification are included.

Experimental results

Control trials using an adaptive control

scheme for the top composition via top temperature control are reported. Keywords

Dlstlllation Column; Modellsation; parameter estimatlon; adaptive control; computer control.

couled total condenser, and a reflux drum that is

INTRODUCTION

open to the atmosphere . The major obstacle to the utillzation of linear

twenty trays.

The column consists of

The liquid levels in the reboiler,

state space model is the non-linear behaviour of

in the column base and in the reflux drum are

distillation columns.

controlled by overflow weirs.

Another difficulty that

limits this approach is the large number of state

The distillation unit also is provided with a large

variables that are involved.

buffer tank in which the distillate and bottom

Theoritical models

can contain several hundred state variables but in

product are collected.

practice the number of informations on the process

Methanol-water mixture is pumped from this tank and

that are available is quite limited.

Thus by neces-

The feed, consisting of a

introced on the 10th tray .

sity reduced state space models must be employed.

In order to obtain a

uniform liquid composition, the tank is equipped

On the other hand the physical models of the distil-

with a circulation loops.

lation column is described by a set of non linear

The feed rate is adjusted by a manual valve.

differential equations.

The main parts of the distillation unit are (Fig. I):

However, it is alway s difficult to handel differen-

dosing pump 5 ... 70 l/h (pos.l)

tial equations models. Therefore, the input output

feed pre-heater made of glass with stainless

representation is considered for the identification and control design.

stell immersion hester, 3 kw (pos.3)

The paper will be organized as

bubble cap tray column with 10 plater (pos.4)

follow: in paragraph 11, a brief description of

reflux meter for the determination of the column

the plant and equipement is provided.

load.

The problem of physical modeling is investigated

circulating evaporator with electric immersion

in paragraph Ill. Paragraph IV deals with input output model identification.

heater 6 kw made of stainless stell (pos.6)

An adaptive control

safety cooler (pos . 7)

strategy is considered for top temperature control in paragraph V.

pressure dsop manometer for display, control and

Results of on-line experiments

limitation (pos.B)

will be reported i n para graph VI.

withdrow coder (pos.9) sample take off device for the bottom (pos.IO/II)

PROCESS DESCRIPTION

bubble cap tray column with 10 plates (pos.12) The pilot distillation unit consists 0f a sieve tray

column head for the automatic reflux dividing

column, a steam heated kettle type reboiler, a water

(pos.13) 1719

B. Dahhou e t ciz.

1720 main cooler (pos. 14/15)

(feed stage).

distillate cooler (pos. 16)

This equation has two parameters

distillate sample take-off (pos . 17) stirring motor with support

- Mi (see

(pos.l~)

mixing receptacle with lines

-

(pos.l~)

~_

assu~ption

4

which also can be assumed to have the same

1

value for i=1 ••. 10 and

i~l

1 ••• LO respectively.

set of clamp connections. lines and sealings. Notice DERIVATION OF MODELS

o(i's are important for the steady state profile. Mi's give the "time scale" i.e. the speed of dyna-

The model that describe the dynamic

of the distil-

mic response.

lation column is derived from unsteady material balances.

The derivation is based on the following

assumptions :

dx

-

O

dt

- Constant pressure in the column l

Reboiler

Uniform liquid composition on each tray

.!.r ~ M"l:L 1x 1-(L 1-V 0 )x 0 -V 0 l(x 0 0 ~

=

(3)

yI(x . ) = in the reboiler equilibrium is assumed o

3 - Negligible vapor holdup

1

The vapor steam

4 - Constant liquid holdup on each tray

Vo is a function of heat input:

Qel

5 - Negligible heat losses from column V

6 - Equal molal overflow.

~

o

power U.•.. 6000 watts L.

L21 (Reflux) L21 + F 10 VI

1

L.

1

V. 1

for all

i=11.20

for all

i=I.IU

for all

i=1.20

heat of vaporisation

depends of physi-

cal property data). Condenser and reflux - Funnel :

The basic tray to tray model was obtained by making an unsteady light component material balnnce over

The vapor is condensed totally. then we have

the reboiler. the reflux drum and each tr ay. The column base and the condenser were both

c! X

assumed

21

dt

to have negligible holdups. Reflux Funnel Balance equatlons: Stage i=1 • . ..• 20 with reflux ratio 1.=1007. reflux 0.= 0 7. reflux. (1)

(5)

The vapor compositions are calculated from a correlation of the vapor liquid equilibrium relationship IDENTIFICATION OF THE PILOT PLANT

taking into account the efficiency of the trays : if mass transfer would be ideal • we would have: Yi=y i

y~1 : the equilibrium composition

I

1.

variable was considered.

=«./ (x.) 1.

1.

1.

It was found from same

preliminary experiments that these two variables

with Qi=efficiency

0<0(.<1.

are

1

suitable for the control purpose.

quiet

A simple representation of the simplified model can

The equation (1) can be written as

be written as :

dx .

~ A~i+1Xi+l

- Lix i +

- Vio(i /+(x i ) +

Vi-l~i-1 Y~-1 (Xi-I)

Fi~J

(2)

This equation holds for all stages i=l •...• 20 with Fi x . vanishing for all stages but not for i=lU F 1

model of the pilot

plant, where control variable is the reflux ratio and the top temperature is taken as the output

Yi=f(x ) i y.

A single-input single output

A(q

-1

)y(t)

with A(q-l) B(q -1)

q

l+a q 1

-le

-1

b +b q o 1

B(q

-1

+ ..• +a

-1

(6)

)u(t)+f
+ ... +b

q

-n

A

A nil

q

-n..

IS

b ;0 o

Identification and Control of a Distillation Column q-I

the backward shift operator,

1S

process time delay,

u(t)

and

ytt)

k

represents

are the

process input treflux ratio) and output (top tempe-

u(t)

is the control signal

y(t)

is the output variable

1(t)

is a sequence of random disturbances(assumed to be of zero mean and finite variance).

rature) respectively, and f(t) is a bounded disturbance. It is well known that this form of models suitable for the design of various adaptive is control systems.

1721

v(t)

measurable process disturbances included in feedforward manner in the control strategy is the time delay associated with the feed-

The sampling period T and the

forward measurements.

process time delay have been determined from an a priori caracterisation study of the process, while the process model order has been chosen to allow

2. Minimum variance control and related self-tuners

satisfactory performances of adaptive control Self-tuning controllers based on the minimization

systems.

of the variance of either the system output or some SELF-TUNING CONTROL DESIGN

more general quantity generated from the controlled system output, input and set-point, have showed

The fundamental idea in self-tuning is to estimate

their worth both in simulation and in practical

recursively the unknown parameters of the process

trials.

and to repeat the control design, based on the up-

modest computational requirement.

dated parameters.

objectives are in good agreement with those formu-

In other words" self-tuning

Moreover, the algorithms are of relatively As these control

control algorithms result from proper combination

lated previously for the distillation process, the

of a recursive estimator and a control design

method seems to be well suited to the considered

methods.

control problem.

Since various methods for recursive

To be specific, the controller

parameter estimation and control design are avai-

design employed is based upon the minimization of

lable, many different self-tuners are derived,

the general performance function.

analysed and applied with success in some industrial adaptive control applications / 1-5 /.

two different ways to organize the combination of identification and control design methods yielding the well known explicit and implicit self-tuners. Explicit self-tuners identify the system model Implicit

where

(9)

CP(t+k) = Py(t+k) - Rr(t) + Qu(t)

is an auxiliary output, a function of weighted process output y(t), input u(t) and set-point r(t). E. denotes the expectation operator.

directly, and perform some algorithmic computation to update the controller parameters.

(8)

J = E.
There are

sel~

tuners, require no intermediate algorithmic computations , as they identify the parameters of the

p(q-I),R(q-l) and ~(q-I) are weighting polynomials. In predictive control theory the control law for the minimum variance strategy is derived in two steps :

appropriate controller ratherthan those of the Step 1 - Estimate the k-step ahead prediction of

plant model.

the output variable. Step 2 - Set the current control u(t) to mullify

I. Plant model

the predicted output. Self-tuning control design for the pilot plant is based upon a simplified model of the complex dynamics of the physical process. sed

A loccally lineari-

single-input single-output model which includes

both measurable load disturbances and random disturbances was assumed.

In this model, the key

To obtain the optimal predictor of the general output
parameters to achieve the control objectives are manipulated.

The corresponding difference equation

in operator form

using the polynomial representa-

A(q

y

)y(t)=q

-k

B(q

-I

-I { -kl-I )u(t)+C(q ) {(t)+q D(q )v(t) (7)

where A,B,C and D are polynomials in backward shilt -I

and ~ respectively. C is the process time delay in integer number

operator q

of

(ID)

with remnant :

tpy (t+k)=A,o(t+k/t)+E(t+k) 'f'y

tion is :

-I

C+olt+k/t)=Fy(t)+EHu(t)+EDqk-kl (t)

where E(q

-I

) and F(q

-I

(11)

) are defined by the polyno-

mial identity : CP/A = E + q-k F/A

( 12)

,of order nA,nB,n

sam~le

intervals.

This implies for the general output

cpo(t+k/t)=.p~(t+k/t) + Qu(t) - Rr(t)

(13)

B. Dahhou et aL

1722

parameter estimator.

and CP(t+k) = ~o(t+k/t) + E

f (t)

(14)

The control law with mUltifies1>°(t+k/t) is then

~

(Rr(t)

e

-~~(t+k/t»

be the data and parameters vectors

defined as :

written as

u(t)

and~(t)

Let X(t)

(15)

[o(o,oCl.·'~o'fl,.·,do,.r;' .. ''7o'~I'·.

=

X(t)

Or in explicit feedback from as :

[yet) ,y\t-I), .• ,u(t) ,u(t-I), .. ,v(t), V(t-I)' ..

I

(10)

u(t) = EB+QC (Fy(t)-CRr(t)+EDv(t+k-k l »

f(22)

'4>~(t+k-l/t-I), ..

]T

Then the prediction equation is simply , (24)

3. Self-tuning control equations

As The remnant in equation (14) can be rewritten to give the current value of the function~(t+k) as

is not known, the data
the estimate of

~~(t+k/t)

1

+~.p0 (t+k/t) +e (t)

,i

each sample interval, from

= O
( 17)

0(, ( ' , "(

are obtained using

y

G at

(24) as follows

CP(t+k)=o(y(t)+~u(t)+ l"r(t)+6v(t+k-k ) +

where the polynomials

(23)

+ (?>ou(t) + ~Iu(t-I) +

and" are obtained

+ [vet) + £"v(t-I) + o

by comparison of equation (17) with equation (14),

1

.... "0

....

10

+.., A. (t+k-I / t-I )+"'l 'I' \t+k/t-2)

(16) and (10) as

loiy

1 y

(25) q= F (as in the identity (12» ~ =

The control law is then derived from :

EB + QC "'0

Y = - CR


[

=

ED

"1

=

1-C

""

A.""

"'"

J..o

Cf+Q)U(t)=Rr(t)-"y(t)-[V(t)-;'t'/t+k-1 /t-I)

If u(t) is chosen so to set ~o\t+k/t)=O, then cf(t+k) = CJl.y(t)+(!>u(t)+ 'fr(t)+6v(t) + e(t)

(20)

Or (27)

This approach has some practical advantages. For (19)

example, it does not assume the controller operating in closed loop to obtain optimal estimates.

which is an appropriate condidate for least-squares estimation of the parameters in

c( ,

~,

using equation (9) to evaluate
"(

and

J

from a prior

The estima-

ted para.leters are then employed in the control law defined by u(t) = -I

to

~ y( t)

f calculate the

The method can also be used with data obtained identification.

Thus during the

commissioning phase of a control law various condidate self-tuners with different choices of the weighting functions P and Q, can be tested by

+Yr(t) +[v(t»

(20)

current control signal when self-

inspecting the control signals that they would produce.

tuning. However as it appears clear from the definitions

4. Parameter estimation

(18) the weighting polynomials of the cost function could not be modified without affecting the controller polynomials under estimation.

Furthermore,the

es timator could not give the optimal parameters as the cond itions ~o(t+k/t)=O

may not be achieved

The basic extended least squares algorithm to update the parameters in the control law (27) is described by the following set of equations

...

G(t) =8(t-I)+K(t)

(28) y T K(t) = P(t-l)X(t-k)/(f(t)-X (t-k)P(t-I)X(t-k» (29)

because of the saturation on the control signal, and will not operate in open-loop such as during the commissioning phase.

Therefore, as suggested

in /5 / the control law given by equation (15) may

system output may be used to adapt the controller The selected model can be written as

CPy(t+k)~y(t)+ru(t)+1v(t)+?+~(t+k/t-l)+e(t) Where the parameters in

,..

0(

,..""

[I-K(t)XT(t-k)] / f(t)

(30)

f(t)

Fof(t-I) + (I-fc,)f(t-I)

(31 )

F(o)~' ,

The prediction model of the weighted

parameters.

P (t)

with,

be implemented to obtain a more efficient selftuner.

[A
(21)

J\

'f'[ and '1 are generated

recursively using an extended least squares based

;0(0) =

f>0~1

/7, ~ 0

X and Sas defined in (22) and (L3) The variants of the basic algorithm obtained by factorizing pet) and propagating the factors themselves are found numerically more efficient.

Identifi ca tion and Control of a Distillation Column ~o

appropriate methods of factorization are

1723

In ord er to estimate the process tlme constant, a

employed :

number of step texts were performea

ps eud o random

The square root method, where P(t)=S(t)ST(t) with

binary eXltation wa s also applled be tween

S(t) an upper triangular matrix.

100% to the retlux rati o fo r ldenti f i ca ti on of

- The U/D method, for which P(t)=U(t)D(t) UT(t) where

507, ana

discrete time linear model at above ope r ating

D(t) is a dia go nal matrix and Ott) an upper-trian-

conditions.

gular matrix with units on the diagonal.

Besldes expe rimental identlficatlo n, the basi c 'l.V selt-tuner control algorithm was run for t op t empe -

EXPERIMENTS AND RESULTS

rature control. Figure Flg 3. are

The distillatlon pilot plant was interfaced with a the 16 bit micro-

disc mass storage, a console terminal and a

teletype writer.

The DEC real time operating system

RTII was the software tool for program development and execution contr o l.

The alg orlthms employed were

written in standard Fortran VI .

The fUllest form

of the appllcation pro gram includes a dlalogue for a full lnitialisation procedure, date logging and full pilot operation management.

On line dialo gue

is also included for settlng up or alterlnp, crucial values.

It sbould be mentioned that a large propor-

tion of the software is includea for plant operation safety . The measurements of temperatures

are fed

to the

mi c ro computer via analog to digital co nverters ln BCD values.

!·!hi le th e following control variables

are set via digital t o

analo~

CONCLUSION

preliminary results from modeling exercise of the column

This approach should lead to t he implemen-

tation of more effective control strategies.

On

the other hand on-line computer control based on simple self-tuner are investigated. mance achievea with the

sin~le

controller shows that thls a'lOroac h loop behaviour of the system.

The perfor-

varian te self -t uning gave g,ood c l osed

Future wo rk wil l b e

focussed on the multivariable co ntr o l sc heme s t o fit the dynamic models of the plant. Acknowledgement

Special thanks to Professor E.D. Gilles and his group e s peciall y Dr . B. Retzba c h and J. Luke for

- Feed flow rat e

their active part i Cipation and co ntri buti on to the

- Reflux ratio

project.

2. Experlmental results

REFERENCES

To investigate the steady state and the dynamic behaviour of t he column, a number of experiments carrled out at dlfferent operating ca ndit ions

In table I. ar e numerized the column operating

info~

mat ion

Table I. steady state condltions of the column Feed rate

IS l/h

Reflux ra t io

737,

Top product temperature

65°C

w

Heat input - Preheater

1300

Heat input - Reboiler

2900 I.

From the point of view of the composltion control (which is a c hieved via tope product temperature control) the maln co ntrol actlons are the reflux rate for the top pr oduct and the heating power for product.

have been initiated. Here in are r e ported some

Germany for its financial support.

- Heat input to the preheater

the botton

Two aspects of the aistillation pilot plant study

The authors would like to thank the G.T.Z. - Wes t

interfaces :

- Heat input to the reboiler

were

shown th e tuned controller parame t e r s

of this experiment.

proceSS0r is paeKaged wlth 64 koctets of memory a flopy

presents the r esults from

a run of a single va rlante self-tune r. l
I. Implementation of algorithms

microcompu ter DEC POP 11/03.

~

Astrom, K. J.(Guest Editor), Optimal Control Applic . and Methods I Y82, 3, 457. Astrom, K.J. Borison, U-:- Ljung L. and Hi tt enma rk , B. (1977) "Theory and application of self-tuning regulators", Automatica 13,457-47b. Clarck, D.H. and Gowthrop P.J.,(1975) "Self-tuning controller", Pro c . lEE, 122,929-934. Clarck, D.lJ. and Gawthrop P.J. (1979) " Self-tuning control", Proc. lEE, 12b, 633-64U Clarck, D.H and Gawthrop P.J., (19/l1) "Implementation and application of micropr oces sor-bas ed self-tuners", Automatica, 17, 233-244. Gilles, £OD. and Retzbach Il., "Reduced models and control of distlllation co lumns with sharp temperature profiles. IEEE Irans. A.C,vol AC 28 nO S , 628-630, 1983. Kumar,S. Wright J.D., and Taylor P.A. "Modeling a nd dynamics of an extractlve distillation column" ACC 19/13, vol -I. 185-192.

B. Dahhou et a l.

1724

1.37

~I

0 ' ~----------~3~0------------~6~0----~T~i-me-(~mn)

~.

60

30

Time (1'110)

-0.3

f'>2

-0.827

I.~

..

0(0

O.~

Figure I. Distillation column pilot plant

30

60

TUDelmn)

2. 0.2 t.

0.1 O.

30

60

0(1 Time (mn)

O.

- I.

JO

Prediction erTcr

Figure 3. Tuned controller parameters

75% 1-1-_ _ _ _ _....:::_ _ _ _ _....-_ _ __ JO

60

Ti.me(n1O)

Re f lux ratio

( 'C)

68 67

66

65

TLme(mn)

- 0. I

100%

T20

60

1---------..I...-3~0------------6-0~-------T-i-;m~8 (mn)

Figure 2. Results from se lf-tuning contro l Experlment .