Multivariable Self-tuning Control of a Binary Distillation Column

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MULTIVARIABLE SELF-TUNING CONTROL OF A BINARY DISTILLATION COLUMN F. Vagi*, R. K. Wood ** , A. ···. ~ ri,'rlllo'r/ ('olllm/,

IJt"I.'"JII.

J.

Morris* * and M. Tham***

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Abstract. Control of distillation columns r enains a key conce rn uf iJldustr~' because of the high ener:;;y consumptioll associated wit11 distillation sejlarations. TIle rep0rt~d results, i r onl n co ntinuing progran of experimcnta~ studies ~ dire~teJ to improved 4istilL.1tion (:0~umn c~)ntr01, are con~c rncd \",itll the evalua t ion of mul tivarlable selt - tunlng cont rol algorlthms fo r the slr.1.ult<.lnc o us cuntr ol ot both top and bottom pro duct CO j~lpositions. EvaluJtion (' f the performance of multivari.:lbl!2 :::icl[ -tuning control algorithms has been performt.!d b~' e:-:perimt!ntal testing using a 8-tray, 30.5 cm dL.lmeter metllanul-h'ater pilot scale column . The column i s controlled using a DEe LSI 11 /03 microcomputer witll c o nti!lUOUS capacitaIlce analysis of top pro duct comp os iti on h'ith bot t um composition analyzed on a 3 min cycle , by gas c.hruma t og raph. Column con t rol is studied fo r 25 /~ s tep incr eases and decreases in feed f1 0 \.,I r a te. The perfl)rmanCe o f the multivar i able algorithms is compa red with that achieved using conven tional digital PI/PID cont r ol algorithms. The results have demonstrated that use of self-tulling co ntr ol will prl)vide better con tr o l performance for feed flow rate disturbances th an can be acilieved using well - tulle d convell ti o l1al multiloop PI/PID control.

Kc y\.,;ords .

Pr0CCSS

Cl1

n tr o l; dis tilla tion column ; mul ti variable cont r ol sys tel!lS ~ <.:omputcr control; chemi cal

variables <.:o!ltrol ; adilptivc con tr ol . pro vide an experimental evaluation of the pe r fo rmance of single rate and multi - rate samp l ing forms of the mult ivarinble algorithm introduced by Mor ri s , ~az e r, Wood (1982). Since this "o r k represe n ts t he first experimental evaluation of the two differen t forms of the algorithm , the underlying t heo r y , Wl1icl1 follu~s from an extension of the sing l e var iable development given by Clarke and Gaw t hrop (1975) , is presented in detail. The cont r ol pe rformanc e achieved using the a l go rithms fo r t h e simultane o us con trol of top a nd bottom compos it io ns is compared t o tllat acl1ieved using conventional multi-loop control . Results are presented fo r the use of control weightings calculated from s i mp l e dynamic informa t i on and for "eightings determi ned bv tuning of the inverse PI/PID control weigh t ing matrices.

r :\TRODUCno "

Di st illati on is the mos t commonly used separation operati on in t he chemical and re fin ing industry to achieve product purification. Unfortunately , the distillation process is very energy intensive, with the energy requireme nts Close l y l inked to the desired separation. For specified product compositions , the energy requ i rements are minimized if satisfactory control of the compositions can be achieved despite disturbances to the column. However, tile composition control of co l umns is difficult due to nonlinear dynamic behavior and the i nterac tion that is inl1erent in a multi variable system. As a result of these characteristics the control performance that can be achieved using single loop compos i tion control with conventional PI/PID controllers will generally not be satisfactory . Consequently. considerable effort is devoted to developing and evaluatinQ new COntr o l schemes and strategies as documented by the review of Tolliver and Waggoner (1980) which lists 195 papers including the minimum variance control study of the column used in this work (Sastry , Seborg , Wood , 1977) . Despite the numerous appli cations of adapt i ve control strategies cited by ~strom (1983), no distillation column contro l examples such as that of Dahlqvist ( 1 981) are menti oned. Furthermore , IHttenmark and Astrom (1984) in t he recent special issue of Automatica on adap t ive control , in addressing implementation considerat i ons for self - tuning contro l. onl y cons i dered single i nput - s i ngle output systems .

THEORY a)

Single Rate

~ultivariable

Self-Tuning Con tro l

'me

multivariable process to be controlled is considered to be described by a discrete model of t he form . -k. .

z

1

l J ~lz - )~(t) + z

+

~(z

-1

- d. . 1 l J Qlz - )~(t)

);,(t)

(1)

Hithout l oss of generality it may be ass umed t ha t the system outputs are independent of one ano t he r and since the noise terms are independent the -1 -1 g(z ) and ~(z ) are diagonal mat r ices. Furt he r-

The experimental results reported in this paper have been obtained us i ng a multivariab l e self This work represents t u ning control st r a t egy. par t of a con t inuing program of st u dy di r ected to adap t ive control of dist i llation columns. The o r ig i na l 1977 study, concerned wi t h the minimum va ri ance con t ro l of top compos iti on , was extended by Mo r r i s and co- worke r s (19 8 l a , 198lb ) t o inc lude self -t uning con tro l of bottom composi t ion as we ll as s i mul t aneous con tr o l of both p r oduct compos iti ons us i ng multi - loop and mu l tiva r iab l e selft u ning contro l. The res ul ts of t he presen t study

mo r e it may be assumed that ~(O ) and p r esen t ed b y the ident i ty maEr ix.

S(O) a r e r e -

The object i ve i s the design of a con tr o ll e r whi ch minimi zes the performance index. J

= E{ [ P(z - l)Y( t +k .. ) - R(z -l )W( t )]T [ p(z -I) Y(t+k . . ) = ~~ = = 11 - !3(Z- l)W ( t) ] -

20i

[Q'(z - l)U(t )] T [ Q ' (z - l)U( t ) ] ltl ( 2)

20H

F. \'agi 1'1 fll.

Minimization of this index to establish a control law is accomplished by dropping the expectation operator, which corresponds to minimization at each

sample instant.

The signals U(t) and h'(t) are

equation (7) can be written as ~ ~(t)

py (t+k .. )

=-

11

+ z

kii-k ij ~ ~ ~(t)

( 8)

known at time "t" so minimization of ]-with respect to ~(t) requires the prediction of ~(z-l)~(t+kii) at time "t", that is ~(z-l)~(t+kii I t) given all the information up to and including time lit". The

weighted prediction can be established by making use of the assumed model description. Factoring the diagonal elements of the delay from both sides of equation (1), dividing by the output polvnomial matrix 6(z-1) and premultiplying by the weighting polynomial matrix ~(z-l) gives (omitting the z-l arguments) the following expression

X

~[z

kii -k ij

+ z

~~ ( t) + z

k ii -d i j

(3)

~- ~ (t)

~,

~

f(z-l)~(Z-l)

~(z-l)

f(Z-l)~(z-l)

PY(t+k 11 .. )

z

=-

All inf ormation for making the prediction is known except for future values of the noise, =(t); (values from ~(t+l) to ~(t+kiJ), but bv making use of th e diaphantine equation (separation identitv) (XstrHm, 1970) this difficulty can be overco~e. By defining f(z-l) and ~(z-l) as diagonal polynomial matrices, since

~(Z-l)

allows equation (8) to be expressed as

~~ ( t) ]

k ii PC

Defining polynomial matrices ~(z-l) and ~(z-l) as

and

~

are diagonal

+ z

kii-k ij ~ ~(t)

kii-d ij L Vet)

=--=-~

+ E (t+k .. ) =-

9)

11

by defining a vector of predicted weighted outputs of the process as the difference between the actual weighted output and the noise, based on

~ow

the information up to and including time "tl' as

polynomial matrices, the stochastic disturbance term of equation (3), in terms of past and future disturbances can be expressed as = z

k ii

F f~(t)

+ P=A~(t) d=

where

~(z-l)

P (z-l) n -1 P (z ) d

Pno + Pnl z Pdo + Pdl z

-1 -1

(4)

+ Pn2 z + Pd2 z

-2

-2

kii-k ij P B U(t)

kii-dij

- --- + z

-

~

and assuming that the best expected value of the future noise is zero i.e., E{ =( t+k .. ) It}=o, allows the perfo r mance index, equation (2J~ to be expressed as J = E{[Py"(t+k .. lt) - RW(t)]T[PY"(t+k .. l t)-RW(t)]

+

=

=-

=-

11

=-

11

=-

+

(11)

Substitution of equation (4) into equation (3) allows the weighted predicted ou tput to be expres sed as z

(10)

D

with the vector of predicted weighted outputs, by combination of equations (9) and (10), g iven by

Q ~(t) A

F

(5)

+ z

then using equation (1), the stochastic noise dis-

Now bv defining ~(z-l) as

+ ~~(t+kii) + P=A d=

kii-d ij ~ ~(t)

(12)

turbance estimate at time "t" can be expressed as (6)

Substitution of equation (6) into equation (5) and collecting common terms yields ~~(t+kii)

~ ~(t)

= ---+ SP d +

k ..

[ kii-kij P C z A

z

[zkii-dij P C ~

z

1J

~ ~~\!(t) ~Pd

d ij

and kii-dij P C z ~

-k ij

-

~ ~Pd

z

kii-k ij

=-

11

d

~

~;;rrt) (7)

(2), at each sampling instant yields

~(O)T~(O)T

Since the identity of equation (4) can be manipulated to give k i i -k ij P C -A-

F yet) kii-k ij ~~(t) PY*(t+k .. It) = ; + z

Minimization of the performance index, equation

+ =E(t+k 11 .. )

z

a recursive fo rm of equation (12) expressed in terms of the weighted output , process outputs, manipulated inputs, disturbances and previous predictions can be stated as

~(O)

(14)

1;; By defining a new control weighting polynomial matrix as

-

z

-d ij

F --

~d

z

kii-d ij

~

~(O) Q'(O)T~'(z-l) ~(O)T~(O)T

(15)

Self-tuning Control of a Binary Distillation Column equation (14) becomes

209

Writing equation (22) on a loop by loop basis, yields (16)

Rather than solve equation (16) t o es tablish th e control action that minimizes the performance index at each sample instant, the algorithm employs an auxilliary outpu t func ti on. For the auxilliary output function vector ~ defined as

~y~(t+kii It) = 0- ~. (t)X .. (t) ~ -11 -

for i=l to m but from equation (10) the weighted predicted output for the i t h loop is Ey.(t+k E:;( t+k 11 .. ) ii I t) + =ii ) = E/(t+k - 1 1 ..L.L.

(17)

then the predicted auxilliary output function vector, based on the weighted predicted outpu t is (18)

(23)

.L..L.

-

(24)

..L..L

It then follows that expressing equation (13) on a loop by loop basis, subs titutin g the definition for the predicted output from equation (24). and expressing th e result at time t-k ii , yields

Pv~(t) 1

---r;;-

= i\Yi(t -k i i ) , , +CiiUi (t - kii)+GijUj (t-k i i )

=' with

,

so setting ~*(t+kiilt) = 0 solving equation (17).

,

+Liivi (t-dii)+LijV j (t-d ij )

(19)

+H 1=. Pv ~1 ( t - 1 I t - k 11 .. -1 )

is equivalent to

The restrictions that apply to implementation of control action based on equation (17) can readil v be apprec iat ed by substituting fo r the weighted predicted output given by equation (13) to yield

(25)

with

F Y( t)

Yi

Yi

Yi

[p(t -k .. ). p (t-k .. -l), ... ,p-(t-k .. -n ), d ~1 d 11 .d 11 f i ui(t-k It can be seen that the dela y associated with off diagonal control inputs must be l a rger than the delay associated with the diagonal cont r ol inputs, that is k ij ? kii' i f j. Furthe r mo r e the delav associated wi t h disturbanc es must be greater than the delay for the control input, or d ij ~ k ii for all j . These conditions if not satisf~ed would involve future terms in the prediction so would not be realizable . These restrictions are gene rally applicable since the manipulated variable will have been selec ted to affect its output before o t her control inputs or disturbances. For an unknOlm sys tern the £, Q. !! and !: polynomial matrices in equation (20) are-replaced-by es timates of t~e true polynomial matrices. that is and!: so the single rate multivariable self =tuning control law can be sta ted as

t, G, B

ii

), u.(t-k .. -l), ... ,u . (t-k .. -n 1

11

1

11

gii

),

uj(t -k i j ), u . (t-k .. -l), ...• u.(t - ki·-n ), J 1J J J gij vi(t -d

ii

), v.(t-d .. - l) .... ,v. (t-d .. -n 1

11

1

11

d ii

),

r/(t - ~ It-k .. - ~ ) ]

-

i

1

11

i

and 0 . (t) -1

[fi (O) , fill),

... , fi(n ) , f

1

gii(O). g . . (1), ... , g .. (n ), 11 11 gii

(21) The algorithm is rendered operational by employing an identification routine to estimate the parameters of the ~, and t matrices in order to predict a weighted output of the process, E~(t+kii l t). The identification port i on of the algorithm is developed by writing equation (13) as

t,

8

(22)

whe r e

~(t)

giJ'(O) , g .. (1), ... , g .. 1J

1J

(n

gij

),

lii(O), 1 .. (1), ...• 1 .. (n ), d ii 1J 11 li J'(O), 1 .. (1), ... ,1 . . (n d ), 1J 1J ij h.(O), h.(1), 1

1

.. . , h.(~ )] 1

T

i

On the bas i s of equations (23) to (25) it follows that

is a diagonal matrix of the form (26)

~(t)

and

Equation (26) is in regression form which allows for parameter identification to be implemented on a loop by loop basis. Furthermore, the form of equation (26), allows for the use of a recursive least squares estimation of B.( t). In this study upper diagonal factorization-lof the covariance matrix (Bierman, 1976, 1977) is employed with the algorithm expressed as

210 r

=

F. Vagi 1'1 Ill. p

is utilized.

+ X. . (t-k)P (t)X!.(t-k) - 11

K. (t+1)

=C

-11

be equivalen t t o those of a discrete PID controlle r, are specif i ed on th e bas i s of some knowledge of system dynamics.

P (t)X:. (t - k)/r =c

- 1

-11

(P (t) - K. (t+l)KT (t+l)r)/p =c

- 1

(27)

-

EQlJlP~IE:-;T

0 .(t) + K.(t+1)(-~'",
~ i( t+1)

b)

-1

P

-1

-1 1

d

~ultirate ~ultivariable

- 1

The di stillation col umn used for all tests was t he 22.) cm diameter pilot scale column that has been used for a numbe r of prev i ous cont rol studies .

Self - Tun i ng Control

HO\o.'ever , unlike the initial adapt i ve con tr ol study

The mult ir ate form of the self -tuning control algorithm i s based on the r estr i ction that all sampling inter va ls of the process a re integral multiples of the sho rtest san~ling interval . In addition , the algorithm is formulated ,)n th e basis that the manipulated variables (control s i gnals) for all i nteracting loops are meas ura ble . This means that from equation (25) all the control actions u.(t) , for j=I , 2 , ... n , ih for th" i th J

g. .

.

loop are accessible f or time llsequences of t (i) , 2t s (i ). 3t s (i) , etc . Consequenth' the output SOt the i th proces s loop y . (kt) is written as 1

S

n gij -k B.. u.(kt )+ ~ 2 1 J~ '1.uJ.(kts) =11 1 s .i=l

(28)

+

for ui' u ' Vj and - 1 assumed known at each cont rol j

interval. Development of the cuntrol la" then fo ll ows i n the same manner as for th e s in gle ra t e algori t hm given in the pre viuus sec tion. ~ith th e requirement, for realizability that k .. >k . . th . 11 1.1 and kii>d , the i loop control law , expressed ij o n a loo p hy loop

h~~ i s ,

can he state d RS

( 29)

performed with the column interfaced to an IBM 1800 process cont r ol computer (Sastrv , Seborg, Wood , lY77) or to the mo r e re cent s tudies in which a distributed HPIOOO compu ter svstem was utilized (~orris and co- workers , 19H1a . lY81b , 1982) , the present study has employed a LSI 11/03 microcomputer for con tr ol of the column . A deta i led description o f the bubble cap column which contains 8 trays at a 30.5 cm spacing hus been pr esented in the previous studies and is fu llv described bv I'agi (1985) so fu rtil" r de tails wi 11 not be pro- . vided here . As can be seen from the schematic diagram in Figure 1 , the methanol - water fee d st ream , con tai ning 50 mass percent methanol is introduced to the fou rth tray. ~ominal steady sta te operating cond itions for tIle column, ope ra ted to produce a top composition of 95 ma ss per -

cent me than ol and a bo t toms con t aining 5 mass per cent methanol , are given in Table 1 . . TABL[ 1 Feed : Rate Composition Temperature

Top: Rate Compos i t ion Tempera t ure

~ominal St eady State Operating Conditions

18 g/s 50 mass % 61°C

Bottom: Rate Composition Temperat ure

9 . 4 g/s 5 mass %

8.6 g/s 95 mass %

Reflux: Rate Temperature

8.4 g/s

Steam Ra te:

11 g/s

60 ° C

37°C

5~oC

Top composition is measured by a continuous in-

- LijVj(t+kii - dij) -

~i~y~(t+k ii-l i t-l)

Just as for the single rate controller , identifi cation can be performed on a l oop bv l oop basis (cf. equation (26), so wit h the parameters identi fied then the con trol action can be dete r mined from equation (29). For the special case of kii=k , equation (29) is ij modified as th e solution for th~ cu rrent control action r equires the solu ti on of a set of equations since the t e rm G.. u.(t), representing the cont r ol action from anotfiJrJloop would be unknown at time t. It is t o be noted that thi s mul tir a te algorithm allows for individual tuning of each loop with respe ct to the t, ~ , and ~ con tr oll e r polynomia l s and t he weighting functions- ~ , Q and B. As will be documented in the next sect i on , a c ont rol weigh ting ma tr ix , Q (cf. equa ti on (15» exp r essed in the form of an inv erse PID struct ure, th a t is

&

Q

(30)

line capaci t ance cell , with bo tt oms composit i on analyzed using an HP-5720A gas chroma tograph (GC) interfaced with an HP 1000 computer for analysis of the ch r omatogram . 11,e r es ult of the analysis is transmitted t o the LSl 11/03 microcomputer by a RS - 232 communication with a program resident in the LSI to interpret the GC report from the HP computer to extract th e composition va lue . Condi tions for th e chromatog r aph were adjus t ed to pro vide fo r a cyc l e time of 3 minutes . Control performance was evaluated fo r feed flow r ate changes of + 25% f r om the steady state ope rating value of l8-g/s. Al l tests, as described by Vagi (1985) , were conduc t ed as a sequence of four step changes ; + 25% increase from the steady sta t e value; - 25% decrease t o the s t eady sta te val ue ; - 25% dec r ease from the s t eady state value and +25 % increase t o the steady s t ate value . Due to space limitat i ons , on l y results for the + 25% changes from steady state wi ll be presented.

ALGORITIDI

D1PLE~E~TATION

The LSI 11 /03 microcomputer i s interfaced with the column thr ough 16 single ended analog inputs for various f l ow and pressure measurements , 8 a nalog outputs for co ntr ol s i gna l output s , 8 di gi t a l outputs for samp ling , purge and GC control and 16 thermocouple inputs fo r temper a ture measurements t o al l ow for energy balance calculations . The op e ratin g environment of the LST 11 /03 is a si ng l e

Self-tuning Control of a Binary Distillation Column user "foreground/background" operating system which allows for a privileged foreground job and a secondary background job. Therefore to ensure that the self-tuning algorithm has first priority it is programmed as a foreground job with other tasks such as material and energy balance calculations allocated as background jobs. The form of the control law, based directly on equation (29) can be implemented in a slightly modified form since the column is only subjected to a single disturbance, feed flow rate. This modification is possible since ~(z-l) in equation (1) can be considered as a diagonal matrix, that is the off-diagonal elements are zero and the delays, d ij ; 0 for i # j. Designating top composition as loop "1", and bottom composition as loop "2", the control law for loop "1" can be written as F () lYl t IQ l + gll (O»)u l (t) ; Rlw l (t) - ---P-----

211

Nichols, Cohen-Coon and two error integral methods (Miller and co-workers, 1967) and also for tuned qo' ql and q2 values. TABLE 2

Number 01 Polynomial Parameters

Top Composition Loop "I" Polynomial

Bottom Composition Loop "2"

Number of Parameters

Polynomial

Number of Parameters

Fl

3

F2

3

g11

4

g21

6

g12

4

g22

5

L11

3

L22

3

d

n

g11

- L

k;l

(31)

gll(k)u l (t-k)

u(t) ; u(t-l) + Kle(t) - K e( t-l) + K3e (t-2) 2 K t K TO where Kl K +~ +~ P ts LTI

-Ll1vl(t+kll-dll)

-Hl~y~(t+kll-llt-l) for k12

>

kll and d ll

>

kll.

However

In order to evaluate the performance of the multivariable algorithms, the column was also operated using two single composition loops with the velocity form of the discrete PlO controller, using trapezoidal integration, expressed as:

if k 12 ;k ll

as is the case for the pilot scale column under study, equation (31) must be modified as the interaction term, u2(t+k ll -k12) becomes u2(t). Under this condition the following equation

K2

K

P

Kpts 2TI

+ 2

(33)

~ t

s

KpTD K3

ts

RESULTS n

g11 L

g11 (k)ul (t-k)

(32)

k;l n

-

g12 L

k;l

g12(k)u 2 (t-k)

-Lllvl(t+kll-d1l)

-Hl~y~(t+kll-1It-l) is used, in conjunction with a similar equation for loop "2", to solve simultaneously for U1(t) and u (t). As can be appreciated from equaEion (32),2 in order to render the algorithm operational, it is necessary that the number of parameters for each of the polynomial matrices, ~, g, ti and k be specified. For this study the li polynomial was not used and on the basis of dynamic testing of the column (Vagi, 1985) the number of parameters were selected to be as given in Table 2. Final specification of the algorithm requires the choice of weighting matrices, g, Q and~. In this work, the g and B matrices are taken as identity matrices and the Q weighting expressed by the inverse PlO structure (cf equation (30». Because of the PlO form of the Q weighting, it is possible (Vagi, 1985) to calculate values for q , ql and q2 using methods developed to estimate igitial settIngs for PI/PlO controllers for single loop systems. Performance of both the single rate and multi-rate algorithms was studied (Vagi, 1985) for Q weightings calculated using the methods of Ziegler-

As state~ in the previous section, the control performance that can be achieved using the multivariable self-tuning algorithms is compared to the best performance that is possible using a single variable composition loop to control each of the product compositions. Although PlO control algorithms were programmed for both composition loops, it was found that derivative action was only advantabeous for control of bottom composition (Vagi, 1985). The control performance shown in Figures 2 and 3 is for use of a PI controller in the top composition loop and for PlO control of bottom composition, with a sample time of 3 minutes for both loops. The results in Figures 2 and 3 show the control performance for disturbances of a 25% increase in feed rate from its normal steady value and 25% decrease in feed rate from its normal steady state value respectively. As can be observed, the control of top composition for both types of disturbances is superior to that for bottom composition. This type of contrasting behavior is not unexpected due to an inherent time delay associated with the effect of a feed rate change on bottom composition. Unlike the GC analysis of bottom composition, which operates on a three minute cycle, the top composition capacitance analyzer provides a continuous signal that can be sampled at any rate. If the top composition loop sampling rate is set to one minute and the controller re tuned , the controlled composition responses for feed rate disturbances of !25% are as shown in Figures 4 and 5 respectively. These results show that the faster sampling rate has improved the control of top composition for the feed rate increase but not for the decrease. Bottom composition control seems to be unaffected by the change in sampling rate for the top com-

212

F. Vagi et al.

position. These results suggest that despite the inherent interaction between contro l loops when two or more compositions are cont r o lled, improved con-

trol behavior may be possible by a judicious choice of sampling rate. The responses of the product compositions using the single rate sampling form of the multivariable selftuning algorithm for ~25% step changes in feed flow rate, with Q-weighting constants calculated using the Cohen-Coon method, are displayed in Figures 6 and 7. Comparison of the control performance in these figures with that presented in Figures 2 to 5 shows that use of the multivariable algorithm has markedly improved the control of bottom compositio~ For example, the IAE value for 150 minutes of ope ratio n , for the +25% step increase in feed rate was reduced to 61.4 compared with values of 99.0 and 101.6 computed for the responses in Figures 2 and 4. These results also sh ow that use of the algorithm with Q-wei ghting constants calculated using the method of Cohen and Coon gives rise to top composition control behavior that is generally comparable to that observed using a well - tuned conventional multi-loop strategy.

CONCLUSION Experimental testing of a multivariable selftuning algorithm for product composition control of a distillation column has shown that generally superlor control performance, compared to conven-

tional control, can be achieved. Comoarable control performance using a multiloop cont rol strategy with conventional PI and ?ID controlle rs could only be achieve~ due to the nonlinear column dynamics, if the controller constants were tuned for the specific disturbance. Furthermore the performance, under multivariable control improved If the multirate sampling form of the algorithm versus the single rate form was employed. Results of t esting the performance of the multivariable a l gorithm have demonstrated that satisfactory Qweighting constants can be calculated from a limited knowledge of svstem dynamics. The results showed that suitable constants could be obtained USing. the.Cohen-C?on formulae developed for selectlng lnltlal settlngs for single variable controllers.

However, it must

be noted that this control performance which is satisfactory for top composition but not bottom composition , was only attained after an extensive

program of on-line adjustment of the controller. For example in the case of a step decrease in feed rate, use of controller constants calculated bv the Zie gler - Nichols (Z- N) method for the multi ~ loop scheme resulted in an IAE value for top composition of 49.0 compared to the value of 19.5 for the response in Figure 7 (Vagi, 1985). The corresponding lAE values for bottom composition control are 421.5 using the calculated Z- N settings versus 93.3 for the response in Figure 7. The control behavior that results ",hen the Qweighting constants are tuned is shown in Figures 8 and 9 for the +25 % and - 25% step changes in feed rate respectively. As can be observed, only a marginal improvement in control performance results by tuning of the Q-weighting constants calculated using the Cohen-Coon formulae . However, adjusting the co nstants from values computed using the Ziegler-Nichols formulae and other formulae did significantly improve control performance. Despite the limited improvement in control performance that results from tuning of the Q-weighting constants, it is important to realize from comparison

of the results in Figures 6-9 with those in Figures 2-5 that use of the multivariable self-tuning algorithm provides significantly better control, particularly for bottom composition , than is possible with conventional multi -loop control stra tegy. If the sampling rate for the top composi ti on analyzer is increased to a rate of one sample per minute, then control of the column with the multirate form of the self-tuning algorithm gives rise to the control behavior illustrated in Figures 10 and 11. These figures show that for both the increase and decrease in feed rate, that improved control of bottom composition is achieved compared to the performance shown in Figures 8 and 9 obtained using the single rate form of the algorithm. Control of top composition, for a decrease in feed rate, using the multi-rate form of the algorithm also improves as documented by a reduction in IAE value from 17.7 for the response in Figure 9 to 12.0 for the Figure 11 response. However, as can be seen by comparing the response of top composition in Figures 8 and 10, for a feed rate increase even though the Q-weighting constants for loop "1" were adjusted due to the change in sampling rate, the control performance was unaffected by the higher sample rate.

ACKNOWLEDGEHENT The support of this work by the Science and Engineer ing Research Council (UK) and the Nat ional Sciences and Engineering Research Council (Canada) is gratefully acknowledged.

REFERENCES Astrom, K.J. (1970). Introduction t o Stochastic Control Theo r y. Academic Press, New York. Astrom , K.J. (1983). Theory and applications of adaptive control - a survey. Automatica, 19,

471-486. -Bierman, G.J. (1976). Measurement updating using the U-D factorization. Automatica, 12, 375-382. Bierman, G.J. (1977). Factorization MethOds for Discrete Sequent ial Estimation, Academic Press,

New York. Clarke, D.W., P.J. Gawthrop. (1975). Self-tuning controller, Proc. lEE, 122, 929-934. Dahlqvist, S .A. (1981). In ~Isermann and H. Kaltenecker (Ed.), Digital Computer Applications to Process Control, Pergamon Press,

New York . pp. 167-174. Miller , J.A., A.M. Lopez, C.L. Smith, and P.W. Murrill (1967). A comparison of controller tunin g te chniques , Cont. Eng., 14, No. 12, 72-75. -Morris, A.J., Y. Nazer, R.K. Wood, and H. Lieuson (1981a). In R. Isermann and H. Kaltenecker (Ed.), Digital Computer Applications to Process Control, Pergamon ?ress, New York. pp. 345-354. Morris, A.J., Y. Nazer, and R.K. Wood (198lb). In C.J. Harris and S.A. Billings (Ed.), SelfTuning and Adaptive Control: Theory a~pli cations, Peter Peregrinus Ltd., New York, Chap. 11, pp. 249-281. Morris, A.J., Y. Naze r, and R.K. Wood (1982). Multivariate self-tuning process control, Opt. Control App. Meth., 3, 363-387. Sastry, V.A., D.E. Seborg, and R.K. Wood (1977). Self-tuning regulator applied to a binary distillation column, Automatica, 13, 417-424. Tolliver, T.L. and R.C. Waggoner (1980). Distillation column control: a review and perspective form the CPI, Advances in Instrumentation ~, Pt.l, 83-106. ' Vagi, F. (1985). M.Sc. Thesis, University of Alberta, Edmonton, Canada. Wittenmark, B., and K.J. Astrom (1984). Practical issues in the implementation of self-tuning control, Automatica, 20, 595-605.

Self-tuning Control of a Binary Distillation Colu mn

2 13

NOTATION ~(Z-l) , ~(z - l), ~(z-l), 9(z - 1) = mxm polynomial matrIces in the backward shift operator, as defined by equation (1) d ij = integer that expresses the time delay between the i to measurab l e disturbance and the jth out put as a multiple of the sampl-ing interval E{* l t} = expectation operator given information up t o and includin g time t [(z-l), Q(z - l), ~(z-l) and ~(z - l) = mxm diagonal conttoller p~lynomial m~trices as defined by equation (19) K = Kalman gain k ii = integer that expresses the time delay, for the effect of the i th control action on the i th

95.50 ~

~

95.25

(/) (/)

~ 95.00 Cl X

94.75 94.50

output, as a multiple of the sampling interval k ij = integer that expresses the time delay of the transfer function relating the i th manipulated input to the jth output as a mul tiple of the sampling interval nd. . order of the L po l ynomia l for the i th loop 11 on the diagonal order of the L polynomial for the i th loop and the jth e!ement of the polynomial matrix order of the f polynomials for the i th loop

n

o rder of th e G polynomial for the i th on the diagonal order of the G polynom i al for the i th and th e jth element of the polynomial order of the A polynomial for the ith

hi

function transfer function matrices as defined by e quati on (2) ~c(t) = covariance matrix, initially 1000r r = intermediate value as defined by equation (26) t s = shortest samp le time , for multira te algorithm U(t) mxl vector of manipulated inputs V(t) mxl vector of measurable disturbances We t ) mxl vector of desired set points ~ (t) rnxl vector of system outputs Greek ~(t)

~

02

ii

=

60

120

0:6

CD

X3

0

TIME(MINS) Fig . 2.

Multi-loop compos ition control for a +25% step increase in feed rat e at a single sample rate of 3 minutes: top composition controller constants (K p =5.00, Tr =175); bottom composition controller constants (K p =1.43, Tr =450, TO=llO)

95.50

95.25

(/) (/)

Cl X

WATER

0

9

~ 95.00

. nOlse variance COOLING

120

1::

~

= vector of zero mean (random) noise with

known covariance p = forgetting factor, 0 . 998 t+k

~

loop matr i x loop

-

60

(/) (/)

loop

~(z - l), 8(z-1), Q' (z-l) = mxm d i agona l weighting

0 12

I

94.75

I 94 . 50+---~----r----r---'--~

12

o

60

120

~~~~~__~-L____~P

~9

PRODuCT

,

(/) (/)

~-- - e

0:6 1::

FEED

~

CD

,

X3

----8 ,,-'T----'-----*---

04---~----~--~----~--~

STEAM

o

60

120

TIME(MINS) CODE CR

-

ANAlYZER R(CORO£R

noVo

FRC _ RECOROER CONTROLLER GC _ GAS CHROMATOGRAPH le _ LEVEL CONTROLLER

~ -- --- - -- - - -----~

i BOTTOM

PRODUCT

Fi g _ 1.

DCA-H

Schema t ic diagram of distilla t ion column

Fig . 3.

Multi-loop composition control for a -25 % step decrease in feed rate at a single sample rate of 3 minutes: top composition con tr oller constants (K =5.00, Tr =175); bottom composition con t ~oller constants (K =1 .43, Tr =450, TD=l l O) p

2 14

F. Vagi et at.

95.50 ~

95.50

i:t

95.25

(f) (f)

95.25

(f) (f)

~ 95.00

~ 95.00

0

0

X 94.75

X 94.75

94.50

94.50 60

0

120

12

12

~9

;i9

(f) (f)

(f) (f)

CI:S

CI:S I:::

m

m

5

0

60

120

0

60

120

~

x3

x3

.0

0

0

60

120

TIME(MINS)

TIME(MINS) Fig . 4.

Multir a te multi-l oop composition co ntrol for a +25% s t ep incre ase in feed r a te: top composition control l er constants (t s ~1 . O . Kn ~ 5.00 . TI ~IO O. TD~ 30) ; bo ttom comp6si t~ on controlle r constants (ts2 ~ 3 . 0 . Kp~ I .43.

T I~450,

Fig . b.

TD~IIO)

95.50 ~

Multlvariable self-t uning composition con tr ol for a +25% step increase in feed r ate us in g Cohen- Coon Q-wei gh ting at a single sample r a t e of 3 mi nutes : top composition Q- weigh ti ng constants (qo~4.53, ql~ - l. 27 ); bottom composition Q- weighting constants (qo~4.53 , ql~-l . OI)

95.50

i:t

95.25

(f) (f)

95.25

(f) (f)

~ 95.00

~ 95.00

0 X 94.75

0

X 94.75

94.50

94.50

0

60

120

0

~9

120

60

120

;i9

~ (f) (f)

(f) (f)

CI:S

CI:S I:::

m

m

5

~

X3

x3

0

0

0

60

0

120

TIME(MINS) Fig. 5.

60

12

12

Multirate multi -loop composi ti on cont r o l f or a - 25 % step decrease in feed rate: top composition controller cons tant s ( t s ~1 . O , Kp~5.00, TI~IOO, TD~30); bot tom com- 1 position con troll er cons tants (t s ~ 3.0 , Kp~I.43, TI ~ 4 50, TD~ IIO) 2

TIME(MINS) Fi g . 7.

Mu ltiva riabl e se l f -tuning composition con tr ol for a - 25% s t ep dec r ease in feed rate using Cohen-Coon Q-weighting at a s ingle sample rate of 3 minutes: t op compo s ition Q-weighting constants (qo~4.53, ql~-l. 27); bottom composition Q-wei ghting constants (qo~4.53, q l~-l.Ol)

Self-tu n ing Co ntro l o f a Bina rv Dist illa tio n Colum n

95.50 ~

95.50

95.25

~

95.25

(f) (f)

(f) (f)

~ 95.00

~ 95.00

0

Cl

X

94.75

X

94.50

94 . 75 94.50

0

60

120

0

60

120

0

60

120

12

12 ~9

~9

(f) (f)

(f) (f)

CI:s

CI:s

~

~

1:: CD X 3

1:: CD X 3

0

0

0

60

120

TIME(MINS) Fig. 8.

TIME(MINS)

self -tunin g composition control for a +25 % s tep increase in feed rate using tuned Q-wei ghting at a single sample rate of 3 minu t es : top composition Q- weighting constants (qo=14 . 2 , ql= . 833); bottom composition Q- I
~ulti variab l e

Fig. 10.

multivariable self - tuning composition control for a +25 % s tep in crease in feed rate using tuned Q-I
~ultirate

I

pos iti on Q-we i ghting constants (t s =3.0 , Qo=3.20, QI=- .800) 2

95.50

95.50 ~

~

95.25

95.25

(f) (f)

(f) (f)

~ 95 . 00 +-++-+--+--+---J~rl'-*-+l4+

~ 95.00

0

Cl

X

2 15

X

94.75

94 . 75 94.50

94 . 50

0

60

0

120

~9

~9

(f) (f)

(f) (f)

120

60

120

CI:S

CI:s

!;

!;

CD

CD

X3

X3

0

0

0

60

0

120

TIME(MINS)

TIME(MINS) Fig . 9.

60

12

12

Multivariable se l f - tuning composi t ion control fo r a -25 % s tep decrease in feed rate using tuned Q- we i gh ti ng a t a sing le sample rate of 3 minutes: top composition Q- we i gh ting constants (qo=14.2, QI=. 833 ); bottom composition Q- weighting constants (qo=3 . 20, -.800 )

Fig. 11.

Multirate mu lti va riable self-tuning compos iti on control for a -25 % s tep decrease in feed rate us in g tuned Q-wei gh ting: top composition Q-wei gh tin g constants (t s =1.0, qo=24.6, QI=- 13.9); bottom comI

pos1t1on Q-weighting constants (t s =3.0, Qo=3.20, QI =-.800) 2