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An Experimental Study of the Multivariable Control of a Distillation Column
Copyright © IFAC Control Science a nd T ec hnology (8th T rienni al Worl d Congress) Kyoto. J apa n . 198 1
AN EXPERIMENTAL STUDY OF THE MULTIVARIABLE CONTROL OF A DISTILLATION COLUMN T. Takamatsu*, I. Hashimoto* , M. Iwasaki* and M. Nakaiwa** *Department of Ch emical Engineering, Kyoto University, Kyoto, japan **National Ch emical Laboratory for Industry, Tsukuba, Ibaragi Pref ecture, japan Abstract . I n th i s pape r, a mu lti va ri ab l e contro l system design problem of a b i nar y disti llat i on co l umn is de a lt wi th by app l y i ng a frequency - doma i n method cal l ed "the GG- pseudo- band me t hod " wh i ch was proposed by Araki as an extension of the INA method . By usi n g the ethanol- water system , the dynamic performance of the control system obta i ned by t his method i s experimental ly evaluated in a sma l l scal e disti l lation co l umn . Th i s design method seems to be eas i ly hand l ed by process contro l eng inee r s , a nd i t provides practical advantages . Keywords . Process contro l; d i s t illat i on co l umn ; mult i var iab l e control ; f r equency domain approach .
other interesting methods, such as the Inverse Nyquist Arr ay method (Rosenbrock , 1974) , and Characte r istic Locus method (McFarlane , 1973), to a distil l ation column have been reported . Tyr~us ' paper (1979) is the first report on the experimental study of the application of the INA method to an industrial scale distil lation co l umn . The control system obtained achieved considerab l e improvement in dynamic behaviour of the co l umn . The application of the Character i stic Locus method to a pi l ot scale co l umn is also reported by Schwanke and co- autho r s (1977) . These case studies are very he l pfu l not only to clarify some advan tages and disadvantages of each design method , but also to evaluate the performance of the control system obtained from a more practical viewpoint .
INTRODUCTION In recen t years , the rapid i ncrease i n raw material and energy costs , espec i a ll y the enormous increase in energy costs s i nce t he o i l embar go of 1973 , has caused a gr eat change in the philosophy of process sys t em design and control , and has forced p r ocess des i gners to uti l ize energy and other natura l resources mo r e eff i cientl y. Di sti l lation is widely used for the separation of products i n refinaries or in many chemical processes . I t i s a typical energy- con suming process , and it i s a prime candidat e fo r the improvement of energy efficiency (Kenney , 1979) . Even in a binary dist i llation column , effective contro l greatly contr i butes to mi nimiz ing e n e r gy usage . Because ope r a t ion c l ose to the desired product set po i nt can dec r ease off- specifi cati on products which mus t be rep r ocessed before sale . Hence , significant efforts have been devoted to the appl i cation of modern multivar i able control techniques to a b i nary d i stillation co l umn control , that is to control the composit i ons of both t he top and bottom streams s i mu ltaneous l y (Rijnsdorp , 1976; Edgar , 1 977) . Mos t of t h i s research has been based on simulat i on stud i es applying state space methods such as decoup l ing contro l (Dav i son , 1 970; Wa I ler , 1974) , opt i mal feedback con t ro l (Schwanke , 1 976) , d i s turb a n ce rejection control (Takamatsu , 1979 ). F r equency domain methods fo r a mu l t i va ri able system are p r esently of great inte r est , but mos t reported d i sti l lat i on co l umn app licati ons have been limited to the well known decoup l ing appr oach (Luben , 1970 ; Wood , 1973 ; We i schede l, 1 979) . To da t e , very few appl i cations of
In this study , the Gereralized Gershgorin Psuedo- Band method which was developed by Araki (1 98 1 ) as an extension of the INA method , is emp l oyed i n t h e design of a multivariable con trol system for a binary experimental distil l a tion column . The aim of this paper is to ana lyse the advantageous features of this newly developed design method and at the same time to c l arify many practical problems incidental to the implementation of the control system for an experimental column .
GG- PSEUDO - BAND METHOD GG- pseudo- band method was proposed by Ar aki (198 1 ) as an e x tension of the INA method . Here , only a short description is given in order to c l ar i fy the main features of the design procedure . Cons i der an n - input n - output system as shown in F i g .l. He r e , r , u and y are n - vectors representing the reference , manipulating and
2 77 9
2780
T. Takamatsu
et aL .
output variables, respective l y . G (s) is the transfer matr i x of the p l ant to bePcontroll ed . Gc(s ) is the precompensator to make G(s) ~ G (s)G (s) = {g .. (s)} nearly diagonal. K(s) = {d~ag. (~. (s»} i§J the diagonal main contro ller.
G(S)
r--------- - ---- - -----, I
,
~
~-- - - - --- - - - -- ---- - - -~
For an arbitrary square matr~x Q {qi~} ' here the ~nte raction matr~x c(sIQ) = {c .(s IQ)} is introduced as follows ; ~J
(~
.. (s)
~J
/
i i
q .. (s) JJ
t
Fi g . l
A controlled system
j
Then, the maximum eigenvalue of C(s IQ), A (sl Q) , is called the i nteraction index of Q. When n=2, c(sIQ) and A(sIQ) are given by 0 ( q21 (s)/qll (s)
5.400
q12(S)/q22(S»)
7.40
0
and
1
It is clear from the definition of A (sIQ) that A (sIGK) = A (sIG) , that is , the interaction index only depends on the non- diagonal matrix G(s) and is not influenced by the diagona l controller K(s).
Fig.2
It is also verified that the relative change of the open- loop transfer function of the i - th loop of the system shown in Fig . l when the other loops are closed, is bounded by A (s i G). The disk with center k. (jW)g .. (jw) and radius A(j w IG) Ik (j w)g . . (j~) I ohJthe complex plane and th~ band~§wept out by the disk when w changes from 0 to + co , are called " the GGdisk (Generalized Gershgorin Disk)" and "the GG-band (Generalized Gershgorin Band)", re spectively (see Fig . 2) . A theorem has been derived which implies that the polar plot of the i - th loop lies inside the i - th GG- band if the GG- bands of the other loops do not include the point (- 1 , 0) . The disk and the band obtained by mapping the GG-disk and the GG- band on the gain-phase plane are called "the GG- pseudo- disk" and " GGpseudo - band", respectively (see Fig.3) . Here it should be noticed that the GG- pseudo- disk is determined completely by A(j w IG), and the change of k. (j w) only causes the parallel displacement of the GG- pseudo- disk of the i - th loop . Thus, the theorem mentioned above can be restated as follows: the log- modulus plot of the open- loop transfer function of the i - th loop lies inside the i - th GG- pseudo- band if the GG- pseudo-bands of the other loops do not include the point (OdB , - 180°) . As for the stability condition, the following theorem has also been derived : When G(s) ~ G (s)G (s) has no unstable po l es , every closed ~oop In the system of Fig . l is stable if each GG- band neither includes nor encircles the point ( - 1 , 0) ( equivalently, none of the GG- pseudo-bands include the point (OdB , -1 8 0°» . Based on the theorems derived , Araki proposed the following design procedure:
G.G . Band in the complex plane
El ", en o
Z aD
o~
",
4--rI~~~~~~~T~~~~~~~~ ~
-IBO .
Fig.3
. 0.0
G.G . P . Band in the gain - ph a s e plane
step i) Find suitable pairing between the input and output variables by taking into account the strength of their physical connection . step ii) Determine the precompensator G (s) so as to make G(s) (=G (s)G (s» diagona~ dominant by checking tRe interaction index A (s I G) •
step iii) Draw the GG- band on the complex plane or equivalently, the GG-pseud o-band on the gain-phase plane. By regarding the boundaries of these bands as the true response of the controlled object , evaluate the stability characteristics of the controlled process and adjust the main controller by applying some classical design technique. When the GG - pseudo- band on the gain- phase plane is used , the Nichols chart technique can be very easily applied to adjust gain constants of the controller because the radius of the GG- pseudo- disk is not changed , and the GGpseudo- band moves only up and down when the gain parameter of the controller is changed. step iv) Evaluate the over- all performance of the controller obtained in the above steps by performing simulation calculations . When a
278 1
Multiv ari a bl e Co ntr o l of a Di s till at i o n Co lumn
ind irect way in this experimenta l study. Th e compos i tions are ca l cu l ated from measu r emen t s of the li quid t empe r ature at the overhead and the 7-th plate . In orde r to measur e the tempe r atur e , a thermocoup l e cons i st i ng of al umel- ch r ome l was used.
des ir ab l e re s p o n se i s not ob t a i ned fo r some disturban ces , r eturn to step i i i ) to redes i gn t he ma in co ntro ller . I f t he i nte r act i on between loops i s s t rong , re t urn to s t ep ii ) t o r edes i gn the precompen sato r. Repeat i ng these s t eps , a r at i onal mu l tivari ab l e contro l sys t e m can be des i gned .
The a i m of contro l i s to maintain as much as poss i ble two output variables at their des i red set points , by manipulating two input var i ables.
EXPERIMENTAL DI STILLATION COLUMN SYSTEM AND TESTING PROCEDURE
Dynami c mode l of the system I n o r de r to design a control system for the d i st illation co l umn by app l ying the GG- pseudoband method , a transfer function model G(s) which represents the dynamics of the column , has to be obtained first .
Expe rimenta l System The exper i mental system used in th i s work is schemat i cal l y shown in F i g . 4 . A binary mixtu r e of ethanol - water is p rocessed in the perforated dist il lation co l umn. The dimens i ons of the co l umn are as fo l lows: the diamete r i s l20mm , there are 1 0 p l a t es , the spacing between stages is 292mm , there are 72 ho l es on each plate, the diameter of which i s 4mm , and t he he i ght of the ove r flow we i r is 35mm . The ove r head vapour is total l y condensed in the condenser. The liquid feed , con t a i ning 25 . 6 we i gh t % ethanol and r egul ated at its boiling point , is supp l ied to the middle plate (the 5- th plate) of the column . The reflux f l ow is regulated at its boil i ng po i nt , and the range of the ref l ux flow ratio is from 2 . 4 t o 5.0 . The upper range of the heat i ng power supplied to the reboiler is 20KW .
I n th i s study , the so- c alled " step response test " was used i n order to obta i n G(s) . The dist il lation column is a two- inputs two outputs system , and each input affects both outputs . Therefore , the dynamic model of the column can b e expressed by
G G 2 1 22
where 6 is the deviation from the nomi nal steady state condition . The manipulating var iables l'L and 6V are Lapalace transforms of the reflu~ flow rate and the heating power , re spective l y. The output variables 6X and 6X7 a r e a l so Laplace transforms of the l~quid compositions of the overhead and the 7- th plate.
In order t o implement the con t rol system obtained , a mi ni - computer MELCOM- 70 , CPU capacity of which i s 20kwords , i s connected to t h e co l umn. The manipulating variables are the reflux f l ow rate and the heating power supp li ed to the reboi l er . The output var i ables to be contro ll ed are t he compos i t i on of the d i sti ll ate a nd the li quid composition at the 7- th plate . Since the on-l ine measurement of the l iquid composition is extreml y d i ff i cult to make , the t wo output compositions were measured in an
[G ll G12 ] [:vT~ R]
e 6X ] 6X = 7
In step response testing, 6V was held constant at its steady state , while a step input was
:!:1.0 1---------,;0-0-0-0-0-----0-o-o-un--~o'o-,o,---o 0----0"-°'::"°_ °-,,-°-"-_ 0>-1
~
0
UJ
0c:oo
0
:J
0000
0
0
0
00
0° (/)
..
°
~
0°
~
0
fX)0
CONDENSER
0.0 L - - - - - - - -"'0C - - - - -- - -----c:'=0- - - -3 6
-..J
TlME ( mln)
Fig.5(a) Res p ons e of t he o p e n - l oo p system t o the step c hange of ~ LR' Res ponse of 6 X D DISTILLATION COLUMN
V
~ HERM ' .. . ~
COUPL '''~ MINI 'COMPUTER
o FE ED TANK
Fig . 4
eST 5 _ K
: a-
EED PU
0
0
0
0
~BOICEft
"
'
,,'
Schematic diagram of the ex",erir,',ental d i still ation co l umn
,1.0 l< o UJ
.
N
::;
~
~
°
od'
0,O L\l..-------3 ;!;0,.--------:;6~ 0 ---------'
TlIoolE(min)
F ig.5(b) Response of the open - loop system to the step change of 6L , Re sponse of 6 X 7 R
2782
T . Takama t s u
°0°0 0
0 0
o dJCX)% O OO O OO o
0
00 00
°0
0
0o-oo°~oOc
00
et al.
freque n cy. Besi de the above reqis i te s , the simp l er it is , t he better the structure of G(s) wil l be . In order to determine Gc(s) , several methods have already been proposed by Hawkins ( 1972) and othe r author s .
0
0
In this study , Gc(s) was determi ned as fo l lows;
o
-1 Let g ' .. b e t he (i - j) e l ement of G (s) , a nd introgi3.~e £,~. (s) defined by p
~
~J
g~ij (s)
o.oL-----------:3o;;---------:6~0--------.J
g~jj(s)
TIIo4E(mln)
Fig . 6(a) Response of the open - loop system. to the step change of 6V , Response of 6X D
Then G (s) = {gcij (s)} c g
,
1.0
0 00
00°00 000000
K
0
0 0 "
UJ N
.
:J
00000
0
..
00 ' 000000 0
00
0
a: 0
z
0 .0
0 00
~J
(s)h .. (s)
JJ
where h .. is chosen arbitrarily. JJ
~J
60
30 TlME ( mi n)
Fig . 6(b) Response of the ope n-loop system to the step change of 6V, Response of 6 X 7 loaded to A L . Then, G (s) and G (s) were 21 ll R obta~ned from the responses of 6X and 6X . Similarly, G (s) and G (s) wereDobtaineJ 22 21 from the responses of 6X and 6X when a step 7 input was loaded at 6V b? holding 6L at its steady state . Some r es ults of the s~ep response test are shown in Fig.5 and 6 . Judging from the experimental results , it appears that each transfer function in the mode l could be approximated by the first - order model. By fitting the first- order model to the experimental data , a matrix of process transfer func tions, G(s) , was obtained as fol l ows: G
£'?
When h .. (s) i s chosen as shown above , G(s)= G (s)GJ~s) becomes a diagona l matrix . If hP.(S)£ l, the po l es in every diagonal e l emen t olJ G (s) are preserved in G(s) . I f £, ~.(s) be corneR too comp l ex, we can use a simp l §t e l ement which can sufficiently approximate £,~. (s) in the Bode diagram , as a substitute fofJ£,~.(s) .
00"'"
00
~
. . (s) =
c~J
is determined such that
Consequently , the followin g structure was chosen for G (s) : c Kl (l+Ti S ) 1 l +T s l G (s) K2 (l+T S) c 1 1+T (S) J 2
1
2
Every parameter in G (s) was adjusted by us i ng the interaction inde~ a nd was finally deter mined as follows; Precompensator 1 G (s)
c
1
1
0 . 883 10 . 4s+1 '
- 2 . 07] 14.5s+1
2.44 23 . 7s+1 '
-1. 87 10 . 6s+1
where time constants are measured in (minJ . Design p rocedure The first step in the des ign procedure of the GG- pseudo- band method is to perform variables pairing , that is , to choose the most c l osely coup l ed input- output pairs . Judging from the static gain of G(O) , the pairing of ( 6 X : 6 V) and (t X : 6 L) is preferable to that of : 6L) 7 and (6X : 6V). Thus , in this study the foPmer 7 pairing was chosen .
(Rx
The next step in the design procedure is to achieve diagonal dominance similar to that in the INA method. The precompensator G (s) is determined so as to make G(s) (=G (s)Gc(s)) di ago nal - dominated . G (s) shoulg be §hosen so that the interaction igdex of G(s) becomes sufficient l y small over the desired range of
- 0 . 42(5s+1) 10 . 36s +1
- 0 . 77 (5s+ 1 ) 10 . 6s+s
1
The las t step in the design procedure is to determine the diagonal main con troller K(s) . In this study, each main control ler element is presumed to be a proportional- integral (P.I.) controller . Af t er setting the reset time of each contro l ler at an empirical l y recommended va l ue , the GG - pseudo- band of G(s)K(s) i n the gain - phase plane is drawn . Even if the gain constant ~n the P .I . contro l ler is changed , the shape of the GG- pseudo-band is unchanged and moves up and down in a vertical line . The refore , by regarding the upper boundary of the GG - pse udo- band as the true response of the controlled object, we can evaluate the stability characteristics, and at the same time we can adjust the gain constant so as to satisfy the des i gn criteria proper ly chose n in the Nichols chart . (Here , a criteria such that M =1.2 ~ 1 . 5 for w =0 . 2rad/min is used.) After tRe above adjust~ng procedure , the main con trol l er was determined as follows:
Multivariable Control of a Distillation Column Main controller
K(s)
= [
1 O.35(lU""s)
o
o
1 O.40(lU""s)
The GG-bank and the GG-pseudo-band corresponding to the above controller are shown in Fig.7. The block diagram of the multivariable control system for the experimental distillation column is shown in Fig.S.
RESULT AND DISCUSSION
2783
But in reality, the actual feasible ranges of manipulating variables are usually restricted in some intervals due to the plant characteristics. For example, 6L is only movable from o to the total reflux, a~d 6 V is also restricted in some range so as not to cause both flooding and weeping in the column. Therefore, parameters of the controller are actually restricted in some ranges. But this fact is not taken into account in the design procedure mentioned in the previous section. Thus, simulation calculations are necessary t o fully check whether or not the obtained controller K(s) forces the actual manipulating variables to be operated in the non-feasible regions.
41eSET
4x.,
Fig. 9 shows the result of the experiment where the step change is loaded in the feed composition. The two output variables seem to be maintained closer to their set points by the controller obtained in the previous section. In the simulation calculation, it may be possible to i mprove the performance of the control system much more by freely adjusting the gain constant Ki and the reset time T . i
0.730
dX7 dX7SET -
Fig.S
Block diagram of the mul tivariable control system for the experimental distillation column
1. 73
RE-RXIS
0.05
No Control
Fig.7(a) The G.G.-Band and the G.G.P.-Band corresponding to the finally obtained controller kl (s), G.G.- Band
Fig.9(a) The results of the experiment, Respons e of 6 X D
0.05
,
Control
l-----Simulation
0.0
Fig.7(b) The G. G.- Band and the G.G. P .-Band corresponding to the finally obtained controller kl (s), G. G.P. -Band
Fig.9(b) The results of the experiment, Response of 6 X 7
T. Takamatsu e t al.
2784
In this study, the o utput variables, that is , the composition of the distil l ate and the liquid composition at the 7- th p l ate, are calculated from the measured value of temperature. In the ethanol-water system, the relationship between the temperature and compositon i s very sensitive around the steady state composition of the distillate . In other words, a small measurement error of the temperature of the distillate causes very large inaccuracy in the calculated value of the composition . This means that the signals feeding back to the controller are easily contaminated by the noise relating to the temperature measurement . In actual process contr o l , this kind of noise problem has to be carefully handled otherwise degradation in the performance of the contro l system easily occurs. In the GG-pseudo- band method applied here, the ordinary (not inverse) transfer function of the plant , G (s), is utilized to make the GGpseudo-band ~n the gain- phase plane . The design and evaluation of the controller, K(s), can be eas ily pe rformed in the gain- phase plane by applying some classical design technique. This point provides thi s method with a g reat advantage when compared with the I.N.A method. In this study , the dynamics of the experimental distillation co lumn was approximated by a fairly simple transfer function matrix G (s). But very strong interaction exists iR G (s). The interaction index ~ (jwIG ) becomes greRter than 1 for w = 0 . 473 r ad/min . p In such a case where G (s) has very strong interaction, it is sti~l a very difficult problem to design a precompensator G (s) which is not on ly simple , but also has en08g h performance to be able to decouple the strong interaction . Though several methods are proposed , it i s still necessary to design a precompensator depending on the designer ' s intuition or some tri a l - and-error approach. Since G(s) 6 G (s)G (s), based on which the design proc~du~e ofCthe controller K(s) is deve l oped , always has uncertainty bounded by A(S IG(s», the dynamic performance of the control system designed by this method has, in the end, to be checked in the simulation. In order to apply this method to some practical mu l tivariabl e control problem, a computeraided design pac kage should be developed first and the design p ro cedure has to be pe rformed repeatedly by using a computer- aided graphic disp l ay.
CONCLUSION In this study , the problem of the multivariable control system design of a binary distil lation column was dealt with by applying a frequency - domain method called "the GG - pseudoband method" proposed by Araki. The performance of the control sys tem ob tained was experimentally investigated by using a small scale experimental disti ll at i on co l umn . The fact that the GG - pseudo- band method is advantageous in that it can be easily handled by process control engineers was shown. But
~ ':lme difficulty still remains in designing a precompensator which has necessary and sufficient performance for a practical use.
More case studies, such as those discussed here will be necessary to r escrut inize many multivariable control techniques from the practical viewpoint and at the same time to design a more powerful control system for many complex chemical processes.
REFERENCES Araki, M., K. Yamamoto and B. Kondo (1981) . "GG-Pseudo- Band Method for the Design of Multivariable Control Sys tems" submitted to 8-th IFAC Congress, Kyoto, Japan . Davison, E . J. (1970). "The Interaction of Control Systems in a Binary Distillation Column", A)]tomatica , vo1.6 pp.447 - 461. Edgar, T.F. and C.O. Schwanke (1977). "A Review of the App li ca ti on of Modern Control Theory to Distillation Columns", J.A . C.C . , pp. 1370- 1376 . Hawkins, D. J. (1972). "ps eudo- Diagona li zat ion and the Inverse Nyquist Array Method " Proc . I. E . E . Vol.119. Kenney , W. F . (1979). "Reducing the Energy Demand of Separati on Processes", ~ Engng. Progr. , Vol . 7S , No . 3 pp . 68- 71. Lube n , W.L . (1970). " Dis till at i on Decoupling", A. I . Ch.E.J . Vol . 16 , p . 198. McFarlane, A.G . J. and J . J . Belletrutti (1973) . " Th e Characteristic Locus Method", Automatica , Vol.9 pp . S7S-S 88 . Rijnsdorp, J.E. and D.E . Seborg (1976). "A Survey of Experimen t al App li cations on Multivariable Control to Process Control Problems", A.I.Ch.E. Syrn . Ser. No.lS9, Vol.72, pp . 112 - 123. Rosenbrock, H. H. (1974) . "Computer Aided Con trOl System Design" Academi c Press, London. Schwanke, C . O., T.F. Edga r and J.O. Hougen (1976). " Prob lems in the Application of Optima l Multivariable Control Theory to Distillation Columns ", A.I.Ch.E. Annual Meet ing, Chicago , November. Schwanke , C.O . , T.F. Edgar and J.O . Hougen (1977). "Development of Multivariable Control Strategies for Distillation Columns", I , S.A . Trans, Vol.16 , No . 4, pp . 69 - 81. Takamatsu, T . , I. Hashimoto and Y. Nakai (1979). "A Geometric Approach to Multivariable Con trol System Design of a Distill ation Co lumn" , Automatica, Vol . lS, pp . 387 - 402 . Tyr~us , B.D. (1979) . "Multivariable Control System Design for an Industrial Distillation Column" , Ind . Eng. Ch em . Process Des. Dev ., Vol.18 , No .l , pp . 177- 182. Waller , K.V . T. (1974) . "De coupling in Distill ation", A.LCh.E . J . , Vo 1.2 0 , pp . S92- S94 . weischedel, K. and T.J. McAvoy (1979). "Feasibility of Deco up lin g Distillation Composition Loops " , A. LCh . E. 72nd Annual Meeting , San Francisco, Nov.2S - 28. Wood , R.K. and M.W . Berry (1973) . "Terminal Composition Control of a Binary Distillation Column", ~ Vo1.28 , pp . 1707- 1727 .
For Discussion see page 2792
AN EXPERIMENTAL STUDY OF THE MULTIVARIABLE CONTROL OF A DISTILLATION COLUMN T. Takamatsu*, I. Hashimoto* , M. Iwasaki* and M. Nakaiwa** *Department of Ch emical Engineering, Kyoto University, Kyoto, japan **National Ch emical Laboratory for Industry, Tsukuba, Ibaragi Pref ecture, japan Abstract . I n th i s pape r, a mu lti va ri ab l e contro l system design problem of a b i nar y disti llat i on co l umn is de a lt wi th by app l y i ng a frequency - doma i n method cal l ed "the GG- pseudo- band me t hod " wh i ch was proposed by Araki as an extension of the INA method . By usi n g the ethanol- water system , the dynamic performance of the control system obta i ned by t his method i s experimental ly evaluated in a sma l l scal e disti l lation co l umn . Th i s design method seems to be eas i ly hand l ed by process contro l eng inee r s , a nd i t provides practical advantages . Keywords . Process contro l; d i s t illat i on co l umn ; mult i var iab l e control ; f r equency domain approach .
other interesting methods, such as the Inverse Nyquist Arr ay method (Rosenbrock , 1974) , and Characte r istic Locus method (McFarlane , 1973), to a distil l ation column have been reported . Tyr~us ' paper (1979) is the first report on the experimental study of the application of the INA method to an industrial scale distil lation co l umn . The control system obtained achieved considerab l e improvement in dynamic behaviour of the co l umn . The application of the Character i stic Locus method to a pi l ot scale co l umn is also reported by Schwanke and co- autho r s (1977) . These case studies are very he l pfu l not only to clarify some advan tages and disadvantages of each design method , but also to evaluate the performance of the control system obtained from a more practical viewpoint .
INTRODUCTION In recen t years , the rapid i ncrease i n raw material and energy costs , espec i a ll y the enormous increase in energy costs s i nce t he o i l embar go of 1973 , has caused a gr eat change in the philosophy of process sys t em design and control , and has forced p r ocess des i gners to uti l ize energy and other natura l resources mo r e eff i cientl y. Di sti l lation is widely used for the separation of products i n refinaries or in many chemical processes . I t i s a typical energy- con suming process , and it i s a prime candidat e fo r the improvement of energy efficiency (Kenney , 1979) . Even in a binary dist i llation column , effective contro l greatly contr i butes to mi nimiz ing e n e r gy usage . Because ope r a t ion c l ose to the desired product set po i nt can dec r ease off- specifi cati on products which mus t be rep r ocessed before sale . Hence , significant efforts have been devoted to the appl i cation of modern multivar i able control techniques to a b i nary d i stillation co l umn control , that is to control the composit i ons of both t he top and bottom streams s i mu ltaneous l y (Rijnsdorp , 1976; Edgar , 1 977) . Mos t of t h i s research has been based on simulat i on stud i es applying state space methods such as decoup l ing contro l (Dav i son , 1 970; Wa I ler , 1974) , opt i mal feedback con t ro l (Schwanke , 1 976) , d i s turb a n ce rejection control (Takamatsu , 1979 ). F r equency domain methods fo r a mu l t i va ri able system are p r esently of great inte r est , but mos t reported d i sti l lat i on co l umn app licati ons have been limited to the well known decoup l ing appr oach (Luben , 1970 ; Wood , 1973 ; We i schede l, 1 979) . To da t e , very few appl i cations of
In this study , the Gereralized Gershgorin Psuedo- Band method which was developed by Araki (1 98 1 ) as an extension of the INA method , is emp l oyed i n t h e design of a multivariable con trol system for a binary experimental distil l a tion column . The aim of this paper is to ana lyse the advantageous features of this newly developed design method and at the same time to c l arify many practical problems incidental to the implementation of the control system for an experimental column .
GG- PSEUDO - BAND METHOD GG- pseudo- band method was proposed by Ar aki (198 1 ) as an e x tension of the INA method . Here , only a short description is given in order to c l ar i fy the main features of the design procedure . Cons i der an n - input n - output system as shown in F i g .l. He r e , r , u and y are n - vectors representing the reference , manipulating and
2 77 9
2780
T. Takamatsu
et aL .
output variables, respective l y . G (s) is the transfer matr i x of the p l ant to bePcontroll ed . Gc(s ) is the precompensator to make G(s) ~ G (s)G (s) = {g .. (s)} nearly diagonal. K(s) = {d~ag. (~. (s»} i§J the diagonal main contro ller.
G(S)
r--------- - ---- - -----, I
,
~
~-- - - - --- - - - -- ---- - - -~
For an arbitrary square matr~x Q {qi~} ' here the ~nte raction matr~x c(sIQ) = {c .(s IQ)} is introduced as follows ; ~J
(~
.. (s)
~J
/
i i
q .. (s) JJ
t
Fi g . l
A controlled system
j
Then, the maximum eigenvalue of C(s IQ), A (sl Q) , is called the i nteraction index of Q. When n=2, c(sIQ) and A(sIQ) are given by 0 ( q21 (s)/qll (s)
5.400
q12(S)/q22(S»)
7.40
0
and
1
It is clear from the definition of A (sIQ) that A (sIGK) = A (sIG) , that is , the interaction index only depends on the non- diagonal matrix G(s) and is not influenced by the diagona l controller K(s).
Fig.2
It is also verified that the relative change of the open- loop transfer function of the i - th loop of the system shown in Fig . l when the other loops are closed, is bounded by A (s i G). The disk with center k. (jW)g .. (jw) and radius A(j w IG) Ik (j w)g . . (j~) I ohJthe complex plane and th~ band~§wept out by the disk when w changes from 0 to + co , are called " the GGdisk (Generalized Gershgorin Disk)" and "the GG-band (Generalized Gershgorin Band)", re spectively (see Fig . 2) . A theorem has been derived which implies that the polar plot of the i - th loop lies inside the i - th GG- band if the GG- bands of the other loops do not include the point (- 1 , 0) . The disk and the band obtained by mapping the GG-disk and the GG- band on the gain-phase plane are called "the GG- pseudo- disk" and " GGpseudo - band", respectively (see Fig.3) . Here it should be noticed that the GG- pseudo- disk is determined completely by A(j w IG), and the change of k. (j w) only causes the parallel displacement of the GG- pseudo- disk of the i - th loop . Thus, the theorem mentioned above can be restated as follows: the log- modulus plot of the open- loop transfer function of the i - th loop lies inside the i - th GG- pseudo- band if the GG- pseudo-bands of the other loops do not include the point (OdB , - 180°) . As for the stability condition, the following theorem has also been derived : When G(s) ~ G (s)G (s) has no unstable po l es , every closed ~oop In the system of Fig . l is stable if each GG- band neither includes nor encircles the point ( - 1 , 0) ( equivalently, none of the GG- pseudo-bands include the point (OdB , -1 8 0°» . Based on the theorems derived , Araki proposed the following design procedure:
G.G . Band in the complex plane
El ", en o
Z aD
o~
",
4--rI~~~~~~~T~~~~~~~~ ~
-IBO .
Fig.3
. 0.0
G.G . P . Band in the gain - ph a s e plane
step i) Find suitable pairing between the input and output variables by taking into account the strength of their physical connection . step ii) Determine the precompensator G (s) so as to make G(s) (=G (s)G (s» diagona~ dominant by checking tRe interaction index A (s I G) •
step iii) Draw the GG- band on the complex plane or equivalently, the GG-pseud o-band on the gain-phase plane. By regarding the boundaries of these bands as the true response of the controlled object , evaluate the stability characteristics of the controlled process and adjust the main controller by applying some classical design technique. When the GG - pseudo- band on the gain- phase plane is used , the Nichols chart technique can be very easily applied to adjust gain constants of the controller because the radius of the GG- pseudo- disk is not changed , and the GGpseudo- band moves only up and down when the gain parameter of the controller is changed. step iv) Evaluate the over- all performance of the controller obtained in the above steps by performing simulation calculations . When a
278 1
Multiv ari a bl e Co ntr o l of a Di s till at i o n Co lumn
ind irect way in this experimenta l study. Th e compos i tions are ca l cu l ated from measu r emen t s of the li quid t empe r ature at the overhead and the 7-th plate . In orde r to measur e the tempe r atur e , a thermocoup l e cons i st i ng of al umel- ch r ome l was used.
des ir ab l e re s p o n se i s not ob t a i ned fo r some disturban ces , r eturn to step i i i ) to redes i gn t he ma in co ntro ller . I f t he i nte r act i on between loops i s s t rong , re t urn to s t ep ii ) t o r edes i gn the precompen sato r. Repeat i ng these s t eps , a r at i onal mu l tivari ab l e contro l sys t e m can be des i gned .
The a i m of contro l i s to maintain as much as poss i ble two output variables at their des i red set points , by manipulating two input var i ables.
EXPERIMENTAL DI STILLATION COLUMN SYSTEM AND TESTING PROCEDURE
Dynami c mode l of the system I n o r de r to design a control system for the d i st illation co l umn by app l ying the GG- pseudoband method , a transfer function model G(s) which represents the dynamics of the column , has to be obtained first .
Expe rimenta l System The exper i mental system used in th i s work is schemat i cal l y shown in F i g . 4 . A binary mixtu r e of ethanol - water is p rocessed in the perforated dist il lation co l umn. The dimens i ons of the co l umn are as fo l lows: the diamete r i s l20mm , there are 1 0 p l a t es , the spacing between stages is 292mm , there are 72 ho l es on each plate, the diameter of which i s 4mm , and t he he i ght of the ove r flow we i r is 35mm . The ove r head vapour is total l y condensed in the condenser. The liquid feed , con t a i ning 25 . 6 we i gh t % ethanol and r egul ated at its boiling point , is supp l ied to the middle plate (the 5- th plate) of the column . The reflux f l ow is regulated at its boil i ng po i nt , and the range of the ref l ux flow ratio is from 2 . 4 t o 5.0 . The upper range of the heat i ng power supplied to the reboiler is 20KW .
I n th i s study , the so- c alled " step response test " was used i n order to obta i n G(s) . The dist il lation column is a two- inputs two outputs system , and each input affects both outputs . Therefore , the dynamic model of the column can b e expressed by
G G 2 1 22
where 6 is the deviation from the nomi nal steady state condition . The manipulating var iables l'L and 6V are Lapalace transforms of the reflu~ flow rate and the heating power , re spective l y. The output variables 6X and 6X7 a r e a l so Laplace transforms of the l~quid compositions of the overhead and the 7- th plate.
In order t o implement the con t rol system obtained , a mi ni - computer MELCOM- 70 , CPU capacity of which i s 20kwords , i s connected to t h e co l umn. The manipulating variables are the reflux f l ow rate and the heating power supp li ed to the reboi l er . The output var i ables to be contro ll ed are t he compos i t i on of the d i sti ll ate a nd the li quid composition at the 7- th plate . Since the on-l ine measurement of the l iquid composition is extreml y d i ff i cult to make , the t wo output compositions were measured in an
[G ll G12 ] [:vT~ R]
e 6X ] 6X = 7
In step response testing, 6V was held constant at its steady state , while a step input was
:!:1.0 1---------,;0-0-0-0-0-----0-o-o-un--~o'o-,o,---o 0----0"-°'::"°_ °-,,-°-"-_ 0>-1
~
0
UJ
0c:oo
0
:J
0000
0
0
0
00
0° (/)
..
°
~
0°
~
0
fX)0
CONDENSER
0.0 L - - - - - - - -"'0C - - - - -- - -----c:'=0- - - -3 6
-..J
TlME ( mln)
Fig.5(a) Res p ons e of t he o p e n - l oo p system t o the step c hange of ~ LR' Res ponse of 6 X D DISTILLATION COLUMN
V
~ HERM ' .. . ~
COUPL '''~ MINI 'COMPUTER
o FE ED TANK
Fig . 4
eST 5 _ K
: a-
EED PU
0
0
0
0
~BOICEft
"
'
,,'
Schematic diagram of the ex",erir,',ental d i still ation co l umn
,1.0 l< o UJ
.
N
::;
~
~
°
od'
0,O L\l..-------3 ;!;0,.--------:;6~ 0 ---------'
TlIoolE(min)
F ig.5(b) Response of the open - loop system to the step change of 6L , Re sponse of 6 X 7 R
2782
T . Takama t s u
°0°0 0
0 0
o dJCX)% O OO O OO o
0
00 00
°0
0
0o-oo°~oOc
00
et al.
freque n cy. Besi de the above reqis i te s , the simp l er it is , t he better the structure of G(s) wil l be . In order to determine Gc(s) , several methods have already been proposed by Hawkins ( 1972) and othe r author s .
0
0
In this study , Gc(s) was determi ned as fo l lows;
o
-1 Let g ' .. b e t he (i - j) e l ement of G (s) , a nd introgi3.~e £,~. (s) defined by p
~
~J
g~ij (s)
o.oL-----------:3o;;---------:6~0--------.J
g~jj(s)
TIIo4E(mln)
Fig . 6(a) Response of the open - loop system. to the step change of 6V , Response of 6X D
Then G (s) = {gcij (s)} c g
,
1.0
0 00
00°00 000000
K
0
0 0 "
UJ N
.
:J
00000
0
..
00 ' 000000 0
00
0
a: 0
z
0 .0
0 00
~J
(s)h .. (s)
JJ
where h .. is chosen arbitrarily. JJ
~J
60
30 TlME ( mi n)
Fig . 6(b) Response of the ope n-loop system to the step change of 6V, Response of 6 X 7 loaded to A L . Then, G (s) and G (s) were 21 ll R obta~ned from the responses of 6X and 6X . Similarly, G (s) and G (s) wereDobtaineJ 22 21 from the responses of 6X and 6X when a step 7 input was loaded at 6V b? holding 6L at its steady state . Some r es ults of the s~ep response test are shown in Fig.5 and 6 . Judging from the experimental results , it appears that each transfer function in the mode l could be approximated by the first - order model. By fitting the first- order model to the experimental data , a matrix of process transfer func tions, G(s) , was obtained as fol l ows: G
£'?
When h .. (s) i s chosen as shown above , G(s)= G (s)GJ~s) becomes a diagona l matrix . If hP.(S)£ l, the po l es in every diagonal e l emen t olJ G (s) are preserved in G(s) . I f £, ~.(s) be corneR too comp l ex, we can use a simp l §t e l ement which can sufficiently approximate £,~. (s) in the Bode diagram , as a substitute fofJ£,~.(s) .
00"'"
00
~
. . (s) =
c~J
is determined such that
Consequently , the followin g structure was chosen for G (s) : c Kl (l+Ti S ) 1 l +T s l G (s) K2 (l+T S) c 1 1+T (S) J 2
1
2
Every parameter in G (s) was adjusted by us i ng the interaction inde~ a nd was finally deter mined as follows; Precompensator 1 G (s)
c
1
1
0 . 883 10 . 4s+1 '
- 2 . 07] 14.5s+1
2.44 23 . 7s+1 '
-1. 87 10 . 6s+1
where time constants are measured in (minJ . Design p rocedure The first step in the des ign procedure of the GG- pseudo- band method is to perform variables pairing , that is , to choose the most c l osely coup l ed input- output pairs . Judging from the static gain of G(O) , the pairing of ( 6 X : 6 V) and (t X : 6 L) is preferable to that of : 6L) 7 and (6X : 6V). Thus , in this study the foPmer 7 pairing was chosen .
(Rx
The next step in the design procedure is to achieve diagonal dominance similar to that in the INA method. The precompensator G (s) is determined so as to make G(s) (=G (s)Gc(s)) di ago nal - dominated . G (s) shoulg be §hosen so that the interaction igdex of G(s) becomes sufficient l y small over the desired range of
- 0 . 42(5s+1) 10 . 36s +1
- 0 . 77 (5s+ 1 ) 10 . 6s+s
1
The las t step in the design procedure is to determine the diagonal main con troller K(s) . In this study, each main control ler element is presumed to be a proportional- integral (P.I.) controller . Af t er setting the reset time of each contro l ler at an empirical l y recommended va l ue , the GG - pseudo- band of G(s)K(s) i n the gain - phase plane is drawn . Even if the gain constant ~n the P .I . contro l ler is changed , the shape of the GG- pseudo-band is unchanged and moves up and down in a vertical line . The refore , by regarding the upper boundary of the GG - pse udo- band as the true response of the controlled object, we can evaluate the stability characteristics, and at the same time we can adjust the gain constant so as to satisfy the des i gn criteria proper ly chose n in the Nichols chart . (Here , a criteria such that M =1.2 ~ 1 . 5 for w =0 . 2rad/min is used.) After tRe above adjust~ng procedure , the main con trol l er was determined as follows:
Multivariable Control of a Distillation Column Main controller
K(s)
= [
1 O.35(lU""s)
o
o
1 O.40(lU""s)
The GG-bank and the GG-pseudo-band corresponding to the above controller are shown in Fig.7. The block diagram of the multivariable control system for the experimental distillation column is shown in Fig.S.
RESULT AND DISCUSSION
2783
But in reality, the actual feasible ranges of manipulating variables are usually restricted in some intervals due to the plant characteristics. For example, 6L is only movable from o to the total reflux, a~d 6 V is also restricted in some range so as not to cause both flooding and weeping in the column. Therefore, parameters of the controller are actually restricted in some ranges. But this fact is not taken into account in the design procedure mentioned in the previous section. Thus, simulation calculations are necessary t o fully check whether or not the obtained controller K(s) forces the actual manipulating variables to be operated in the non-feasible regions.
41eSET
4x.,
Fig. 9 shows the result of the experiment where the step change is loaded in the feed composition. The two output variables seem to be maintained closer to their set points by the controller obtained in the previous section. In the simulation calculation, it may be possible to i mprove the performance of the control system much more by freely adjusting the gain constant Ki and the reset time T . i
0.730
dX7 dX7SET -
Fig.S
Block diagram of the mul tivariable control system for the experimental distillation column
1. 73
RE-RXIS
0.05
No Control
Fig.7(a) The G.G.-Band and the G.G.P.-Band corresponding to the finally obtained controller kl (s), G.G.- Band
Fig.9(a) The results of the experiment, Respons e of 6 X D
0.05
,
Control
l-----Simulation
0.0
Fig.7(b) The G. G.- Band and the G.G. P .-Band corresponding to the finally obtained controller kl (s), G. G.P. -Band
Fig.9(b) The results of the experiment, Response of 6 X 7
T. Takamatsu e t al.
2784
In this study, the o utput variables, that is , the composition of the distil l ate and the liquid composition at the 7- th p l ate, are calculated from the measured value of temperature. In the ethanol-water system, the relationship between the temperature and compositon i s very sensitive around the steady state composition of the distillate . In other words, a small measurement error of the temperature of the distillate causes very large inaccuracy in the calculated value of the composition . This means that the signals feeding back to the controller are easily contaminated by the noise relating to the temperature measurement . In actual process contr o l , this kind of noise problem has to be carefully handled otherwise degradation in the performance of the contro l system easily occurs. In the GG-pseudo- band method applied here, the ordinary (not inverse) transfer function of the plant , G (s), is utilized to make the GGpseudo-band ~n the gain- phase plane . The design and evaluation of the controller, K(s), can be eas ily pe rformed in the gain- phase plane by applying some classical design technique. This point provides thi s method with a g reat advantage when compared with the I.N.A method. In this study , the dynamics of the experimental distillation co lumn was approximated by a fairly simple transfer function matrix G (s). But very strong interaction exists iR G (s). The interaction index ~ (jwIG ) becomes greRter than 1 for w = 0 . 473 r ad/min . p In such a case where G (s) has very strong interaction, it is sti~l a very difficult problem to design a precompensator G (s) which is not on ly simple , but also has en08g h performance to be able to decouple the strong interaction . Though several methods are proposed , it i s still necessary to design a precompensator depending on the designer ' s intuition or some tri a l - and-error approach. Since G(s) 6 G (s)G (s), based on which the design proc~du~e ofCthe controller K(s) is deve l oped , always has uncertainty bounded by A(S IG(s», the dynamic performance of the control system designed by this method has, in the end, to be checked in the simulation. In order to apply this method to some practical mu l tivariabl e control problem, a computeraided design pac kage should be developed first and the design p ro cedure has to be pe rformed repeatedly by using a computer- aided graphic disp l ay.
CONCLUSION In this study , the problem of the multivariable control system design of a binary distil lation column was dealt with by applying a frequency - domain method called "the GG - pseudoband method" proposed by Araki. The performance of the control sys tem ob tained was experimentally investigated by using a small scale experimental disti ll at i on co l umn . The fact that the GG - pseudo- band method is advantageous in that it can be easily handled by process control engineers was shown. But
~ ':lme difficulty still remains in designing a precompensator which has necessary and sufficient performance for a practical use.
More case studies, such as those discussed here will be necessary to r escrut inize many multivariable control techniques from the practical viewpoint and at the same time to design a more powerful control system for many complex chemical processes.
REFERENCES Araki, M., K. Yamamoto and B. Kondo (1981) . "GG-Pseudo- Band Method for the Design of Multivariable Control Sys tems" submitted to 8-th IFAC Congress, Kyoto, Japan . Davison, E . J. (1970). "The Interaction of Control Systems in a Binary Distillation Column", A)]tomatica , vo1.6 pp.447 - 461. Edgar, T.F. and C.O. Schwanke (1977). "A Review of the App li ca ti on of Modern Control Theory to Distillation Columns", J.A . C.C . , pp. 1370- 1376 . Hawkins, D. J. (1972). "ps eudo- Diagona li zat ion and the Inverse Nyquist Array Method " Proc . I. E . E . Vol.119. Kenney , W. F . (1979). "Reducing the Energy Demand of Separati on Processes", ~ Engng. Progr. , Vol . 7S , No . 3 pp . 68- 71. Lube n , W.L . (1970). " Dis till at i on Decoupling", A. I . Ch.E.J . Vol . 16 , p . 198. McFarlane, A.G . J. and J . J . Belletrutti (1973) . " Th e Characteristic Locus Method", Automatica , Vol.9 pp . S7S-S 88 . Rijnsdorp, J.E. and D.E . Seborg (1976). "A Survey of Experimen t al App li cations on Multivariable Control to Process Control Problems", A.I.Ch.E. Syrn . Ser. No.lS9, Vol.72, pp . 112 - 123. Rosenbrock, H. H. (1974) . "Computer Aided Con trOl System Design" Academi c Press, London. Schwanke, C . O., T.F. Edga r and J.O. Hougen (1976). " Prob lems in the Application of Optima l Multivariable Control Theory to Distillation Columns ", A.I.Ch.E. Annual Meet ing, Chicago , November. Schwanke , C.O . , T.F. Edgar and J.O . Hougen (1977). "Development of Multivariable Control Strategies for Distillation Columns", I , S.A . Trans, Vol.16 , No . 4, pp . 69 - 81. Takamatsu, T . , I. Hashimoto and Y. Nakai (1979). "A Geometric Approach to Multivariable Con trol System Design of a Distill ation Co lumn" , Automatica, Vol . lS, pp . 387 - 402 . Tyr~us , B.D. (1979) . "Multivariable Control System Design for an Industrial Distillation Column" , Ind . Eng. Ch em . Process Des. Dev ., Vol.18 , No .l , pp . 177- 182. Waller , K.V . T. (1974) . "De coupling in Distill ation", A.LCh.E . J . , Vo 1.2 0 , pp . S92- S94 . weischedel, K. and T.J. McAvoy (1979). "Feasibility of Deco up lin g Distillation Composition Loops " , A. LCh . E. 72nd Annual Meeting , San Francisco, Nov.2S - 28. Wood , R.K. and M.W . Berry (1973) . "Terminal Composition Control of a Binary Distillation Column", ~ Vo1.28 , pp . 1707- 1727 .
For Discussion see page 2792