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A Disturbance Decoupling Control Law with Output Dynamic Matching for Nonlinear Systems. Application to a Binary Distillation Column
Copyright © IFAC Simulation of Control Systems, Vienna, Austria, 1986
A DISTURBANCE DECOUPLING CONTROL LAW WITH OUTPUT DYNAMIC MATCHING FOR NONLINEAR SYSTEMS. APPLICATION TO A BINARY DISTILLATION COLUMN R. Castro*,
J.
Alvarez* and G. Bornard**
*Department oJ Electrical Engineering, CIEA-IPN, Apdo. Postal 14-740, 07000 Mexico 14, D.F. , AUxico ** Laboratoire d'Automalique de Grenoble, ENSIEG-INPG, B.P. 46 - 38402 St-Martin-d'Heres, Grenoble, France
Abstract . The prob l em of usin0 state-feedback and feedforwa rd control in order to solve the d istur bance decoupl in~ prob l e~ IrDP) of a nonlinear system and to ohtain, at the same ti me, an outp ut matchino of this nonl inear system wi th a 1 inear reference mode l is discussed. The resu lts obtained in simulation Ylhen the nonl inear system is a binary distillation column are s hown . Keywords. Nonl in ear control systems; disturbance decour l in!]; output ma tchin']; feedbacl:; feedforward; ~eome tric arproach; chemicul industr y; disti l lation column. that the disturbance w has no influence on the out DUt :t. This control law has the fom
I tHROOUCT ION In the last years, the nonl inear dis turbance decoup I ing problem has been wi de l y studied Ilsidor i , 193 1a; Hi rscho rn, l ~J
i (2 )
j:
=
'L
= ~(2)
+ 'J (2 ).:!. + p (2)::::
m U.
1
a . lx) + l: S .. lx)v .. , l i=l , .. . ,m). 1 i=l IJ '.1
(:)
In vie\'} of some earl ie r resu l ts (Isidori, 1935h) the ?rob lem consists in fin di n" a f eedb ac~ pair (~, S ) and a distribution 6 which is in variant under
( 1a) (1 bj
m
1n the other hand, there has bee n a cons iderab l e interest in the use of feedback wi th the rurpose of obtui nin~ a 1 inear behavio.r out out of a 'Jiven nonl inear sys t em (:iun t , 1903 ; Isidori, 1985a) . 110t i vated by these achievments \'/e study in t:,is purer t he problem of using state-feedback a n feedforward control in order to so l ve the disturbance decouf'l in!] pro bl em (iJCP) of the nonl inear system (1) and t~ . achieve, at the same ti me, an output matchin~ of this system wi th a 1 inear one . For this purpose, we ma l:e ex tensive use of the differential ~eometr i c concepts now widely used in the study of nonl inear systems. Th e paper is orga ni zed in the fo ll owi n'} way. \le fi r st review the DDP for nonl inear s ystems of the form (1) ( Isido ri, 1931a, 1901 b) . li e re, we just state the re sults that show the ex istance of a feed back which so l ves the problem. Then we propose a control stra:e~y that permits to match the output of a nonl inear syste~ ( 1) to t hat of a I inear mode l and to solve the ~"P 3t the sa me time. We a l so state the sufficient conditions fo un d in order to solve both prob l ems. Finall ,!,~:e s ho'lI the resu lt s obtai ned in simu l ation whe n the nonl inear system corresponds to the model of a b in ary disti 1 lation column tOCJether Ylith some conc lu din~ remurl:s. THE 0 I STUR3ANCE DECOUPLl IIG PR.O[lLEI1 (DDP) IN NONLINEAR SYSTEMS
1'1x) = -f (,,) + l: C\. (x) <1. (x) I - -I i=l
(3a)
-
.9. ,. (x)
m l: S . . (x)O. ( )' IJ - -J 2
j =l
(i =l , ... ,m),
conta i ns the vecto r fie l ds £ . , (i=l , . . . ,r), and is contained in I(er(dh) . This iL con tro l lee i nvariant. d~ is the differential of ~ , i. e ., d~ = aya ~. Th e noti on of l ocul co ntr o l led invariance is sometimes eas i e r to dea l with than (J l oha l) con trol l ed invariance and we wi 1 1 just consider the l ocal treatment. Theorem 1 (o n l ocal flDP). I f we define a constant r a nk involutive distribut ion 6 upon an open U of o I\n such th a t ( I s idor i, 19(11 1)) l i ) 6 is
( .!.-~)-in va ri ant
( i i) .piS 6 C: Ker(d~) Then for a ll x E U there ex i sts an Doen U of x in U where the --oDD P °ad~'its loca ll y a s~lution. c3n v2rsely, if the nrp has a l oca l so l ution upon an o~en U of Rn, it is rossib l e to define a constant rank invo luti ve distributi on on U thut fulfi Is co nditions (i) 2nd (i i).
We cons ider nonl inea r systems of the form ( 1) with m sta te x E ~n ,input u E R , pe rt urb atio n w E ~r and p outpu t 'L E 1\ . i , the m co l umns .21" " , ~ of the r.1a trix 9 and the r co lumns £1"'" £ of the matrix pa r e comp l et smooth vecto r fields on rRn The ad d itio na l input w represents an undesired sional which in f lu ences the behavio r of t he s ys t em thr o u ~h p.
wa~ to dea l with the DCr is to examine f ir st Ylhethe r or no t the fami l y of a ll control l ed in var iant distributions contained i n :;er(dh) has a "max ir.1a l" e leme nt (un element which contains a ll other r.1embe rs of the f am i 1,, ). I f this i s true, then the problem is solved if ~~d on l y if this maximal e l eme nt contains the vector fields £i ( i=l, . . . , r ) .
It is desired to modify the system ( 1) via state-feed back control on t he in puts u , ... , u in such a wu y 1 m
Hhen usin'] "locall y" contro ll ed i nvariant distributions, the existance of such a maximal elemen t may
193
n systematic
R. Castro, J. Alvarez and C. Bornard
194
be shown under rather mild conditions. It can also be calculated by means of local representations for the vector fields i, ~, , .. . , 2m as fol lows. We introduce the functions k
(ii) 6 SKer(db,) (iii) PS6 where
k
~
is a modified matrix
~iven
by
k
(~)
(L Lfh.), (L Lfh.), ... , (L Lf·h.), .2, ' .9.2 - ' .9m - ' i=I, ... ,p with Lz~ (2) denoting the Li e der ivative of a smooth function ~(2) alon:] the vecto r field :: '. These functions are obtained when actina upon the function hi along the vector fields i, .21"'" ~ k ti~es . Let P. be the smallest value of k for which at least one'of the functions (I,) is not identically cero on J.n; if all the functions are cero for an~1 k, one sets p .= "' .
,
Then for all x £ U , there exis~s an open of x , l!~1I where theo~Qr'loadm its a 1oca 1 so I ut ion. C~ver sel?, if the D"P'1 adrlits locully a solution in an open I! of 'Rn, then one can define a re'lular distribution on 'J that fulfi Is conditions (i) throu"h (i i i). The calculation of y (2) can be easi l y done once we have found the rlaximal locall y (f,'l)-invariant distribution contained in Ker(dh), this is 6". Some nice examplzs fer th e orerative part of the algorithms described above are re~orted by Isidori (1')81b) and Castro(I~C5).
Suppose p .~ "' . Let n(~) be the matrix with e lements a . . (x) (i,!,', ... , m; j=I, ... , m), and b(:d be the col u~n vMtor with comronents b. (x) (i=1 ,.~"-;-m) such that
DY~!Alll
,-
C OIJTPUT t1ATCH I tiC; /\'1(1 THE nnp
Besides a fixed non 1 i near system described by (1) , we assume no", a 1 i near reference model 'liven by (5b)
\Ie then have the foIl owi n] theorem.
-
,
( G)
-
and the feedback functions a (x) and S(x} w:,ich leave this distribution i nvariant-are any solution s of '~( 2)~(2)
A(2) S(2)
1\~2p. + Iln,~R
1.~
r.ft'?:R
(const~nt
matrix)
Ob)
When obtaining a and B one seeks for an invertible solution to (7bT s o that one has fu ll control. In some real cases (e.~., retrochemical processes), some of the disturbances correspond to phys ical variables which a r e measurable. This partia l measurement consideration pe rmits to solve the ~DP in appl ications where the non- meas urement condi t ion does not allo\-I a complete disturbance decoupl in]. cor the disturbance decoupl in~ problem .dth rartial measurement (DDPI1) one considers the control law (11009, 1932)
(C) Where a (x) and S (x) are defined as above and Vex ) is a m x r matrix. The- 'ith column of y( x) is associated to the ith perturbation, .' . , It is also assumed that only the first r , perturbations are measu rable so that the r-r columns of y (2) are cero. l There already exists local and DOpn t0get her with interestin,) These results are an extension the OOP and we wi 11 just state treat ment of the problem.
state 2~ £ 1\ ", input ~o £ l ., and output 1.R £ J. . Th~ cour1e (An,iC ) is contro ll ahle, '~' q is asym£ totically stable ~nd R q~ has maxium r~nk m . It is n evident that (10) is the most natural reference model, since the c l ass of 1 inear and contro llable systems is the best understood.
The prOblem of interest is to find cond i tions for the existance of smooth feedhack and feedfon-lard functionsn(~' S , Y ) defined on an open dense subset of U of ~ , such that ( i) 1 i m ~ ( t) = 1 i m
(i~ (tl
-
i (tl)
= Q wit h
t --+o:l
t~
sui table ini t ial condi t ions
(~(0)
~ 1.(0))
(i i) the ~CP:, is so 1ved for the non 1 i near s,/s tem ( 1) \·Ihen the flOPi; was discussed, we stated the necessary and sufficient conditions for th e existance of a control la", (13) that solves t~e proble", as wel l as procedure for its construction. So, in order to find the conditions that sa tisfv (i), consider the output derivat ive of (I) .
.,e
v
.L
where and
=
~ = L h(x) f- -
dt
Lf~(2)is
+ L h(x)u + L h(x)w
the Lie
'l- -
re- - -
-
~er i vative
( 11)
as defined before
r
L h(x) p--
i L h (x) i=Il!j-
Fr om (10) we also have d1.p
F
=
CR~~
( 12)
nlobal results for the . ('1,00:), 19~2) On the other hand, if there exists a contro l law app l'Icat,ons · u '(n) of the ones found in u that solves the Drp n then, us i n1 (7), (11) can be .Jr it ten uS and use the local o·
Theorem 3 (on local ODP/;). If we define a rellular dis tribution 6 upon an open U of ~n such that ' o (i) 6 is (~.- ~) -in variunt
( lOb) Iilo
w~th
(7a) C
( lOa )
np
TheoreM 2 (construction of a maximal element). Su~po se P . < '" for every i anci the rank of A(x) is equal to the 'dimension of y = h(x) for every x ~ tn, the maxi mal locally (f,g)-invariant distribution contained in Ker(db,) is ( Isidori, 19C1 a)
() k
~~
i
Lf~(2)
-
Ln~(2)A-l(2)£(2)
+
l
+ L h(x)A - (x) C v + 'l"- -
-
-
( 13 )
A Disturba nce Decouplin g Control Law and the fulfi lment of the conditions
Lf~(~)
=
L~(~)A-l(~)£(~)
L h(x) y(x) g--
= -
L h(x)
( 14<))
( 14c ) to~ether
with YR(O) = y(O) assures that
APPL I C/IT 1011 TO A B 1:lARY D1ST I LL!\T I O~ COLUMN The resu 1ts a 1 ready descr i bed above v'ere app 1 i e d, in si,."u l
~(t)
1 im
In the next section, we app l y thes e resu lts when the nonl inear s ysten is that of a hinary distil lation colu mn.
( l'lb)
p-
)95
li m (lR(t) -
t-+oo
i(tl) = Q
t -+oo
Th e condit ion s given by (1~) are suff ici ent condi tions that permit the output dynamic matchin ~ of a nonl inear system ( 1) to a 1 inear reference mode l (10). ~emark
1
One can notice that the control law used
( 15) preserves the same fo rm of (D). SO, once the ~DP~ bas been so l ved one can obtain, a t the same time, an output dynamic match in g if conditions ( 1 ~) a r e accom pI ished. cig. 1 shows the co rr espo ndin:) bloc!, d i
L h( x) g--
£(~) y(~)
C y = C ~F\ R
(lGa)
- L h (x)
( 1Gb)
iR
( 1Gc)
r>- -
lost p lote
Reboiler
which are easier to use in some practica l
Bottom Product
Fi g. 2. 5chem
d i
of a binary dist ill atio n
The feed is a binary mixture (wa t e r- a lcohol) and the column has n e l e"1ents, inc l udin '1 the renoi l e r and the condense r . Th e mode l is formed by the mate rial - halance eo uat ions assoc i ated to the process and the ther"1odynamic node l of the bin3ry mi xture. Th e nost inportant assu,.,~t i o n s introduced are : - The hinary system has constant re lative vo l ati 1 ity throu~ht the co l umn. - A
fe ed stream i s fed as a sa t urated 1 i(at it s ~ubble roint) onto the feed tra y.
s in ~ l e
~uid
- The mo lal f l m, of the va~o r and 1 i(1uid ti1rou'lh the str ir p inn and r ec tif yi n" sect ions are constant, resnectivel" .
The rest of t he assu m ~t i ons are the ones usua ll y done for this kind of ~rocesses (90 rnar d, 1~81). Th e symho l s used are the Pi g. 1. Block dia~ram for the control l aw t ha t solves th e ODPM to~eth er with the out put m at ch in ~ p rob lem.
fol l owin~
ones:
_i: index o f the cu rr en t eleme nt f : index of the feed e l ement n: nUOl he r of elenents Of the column 1 inuid and '1a~()II' co"'rosition on the jth e lement , res~ective l y (mol fraction]
R. Castro, J. Alvarez and G. Bornard
196
It.: vapour-I iquid equi I ibrium coefficient on the Jth J element L: I iquid flow rate (enrichin~ section) [mol/se']1 V: vapour flow rate throuaht the column t~ol/se~l L : I iquid flow rate of the feed tmol/seal x F: I iquid co~position of the feed (MOl fraction1 F H, H , Ha: tray, cone:enser ane: uottoo I i~uid holdup, D respectively [nolI a: relative volatil ity The model in a state-space representation is aiven by ( 17a) ( 17b) where x is formed by the 1 iquid composition on each element of the column and
(X _ - x.) Ulj+1Xj+l- M.x.) j l J J J
From (17) and (13) cne can notice that one of the system outruts is x?' the 1 iquid composition on the first plate, insteaa of Xl which is the 1 iquid composition of the distillate. This output choice was mainly done in order to avoid an undefined steady state control, this is that u(t) ... 00 when t ... "'. In Castro (1~n5) this case is wIdely discussed.However, from a practical point of view, this choice is not com~letly inade~uate since Xo depends directly on Xl' .50, we can expect a nood~re'1ulation on Xl when trylnn to net a ~ood one on x . 2
If we try to solve t~e ~DP for the binary disti llation column describe~ by (17) assuminn that it is not rossi~le to measure the distur~ances, one finds that the output system can not be decoupled from the disturbance ~Jl=LF/f: (r-authier, 19!)2;Castro, 19(5). On the other hand, when usin'1 theorem 3 one sees that there exists n solution of the nnp IL .'!.nd .,hen one supposes that w is measurable, a control law l u(x,w) which el iminates the effect of ~ upon y can be-eas i 1,' calculated . The
~aximal
element 6
..
is
(x _ - x ) (M + x + - Ilfx ) f l f f l f l f
'J(~)=
(x _ - x ) (tlj+ 1x + 1- IljX ) j j l j j
",here T In is the tannent s"ace of ln at x al1l n is x
equal to the number of elements of the column. H/Ha(xn_ixn) fl/HB (x n- l - x n )
o
o
0
0
-x p(~)=
1
f
(x f - x f + l )
0
(x. 1- x.)
0
H/H[)(xn_l-x n )
(1
r
The invariance of 6 as related to the modified dynamics of (17) can be obtained with (r.astro, 1~~5)
J
(21 ) where the c .. (i~l, ... ,2; j=l, ... ,Z) are the elements of th~Jmatrix C.
The input and di sturbance vectors are u =
T
LlH V/H1 , w = C LF/H
(LFxF)/H J
T
( 19)
"Ii th
H. = a / (1 + (a - 1)x. ) J
J
V is directly related to the power consumed by the reboi ler and the feed stream LF comes usually from another production sta'Je in a petrochemical complex. In practice, it is not possible to have on 1 ine measurement of x ; however, LF can be measured by an F adequate flux sensor.
The second column of y (x) is cero hecause of the measurement condition assumed on w and l
f1emark 1
In this ap~l icaticn. it is possible to o~tain at the same ti me an outQut matchinl of (17) to a 1 inear reference model; the conditions that must be checked out are the ones niven ry (16) since r.(x)= L h(x) (Bornard, IJCI; ; Castro, 19'35). ~Condition (l(a) is easi 1)' satisfied since f(x)=O (and therefore ~(x)=O) and some strai<)ht forward calculations (Castro:l~.'J5) nermit to verify (16b). Then, in order to accompl ish the output matchin9, we just modify the control law found above in such a way that condition (lCc) is fulfilled bein,] no",
The physical state space is the open cube C=(0,1)n. It can be shown (Sornard, 1902) that the system (17) is always inside this invariant domain for positive inputs (which are the only on~ physically sinnificative). One can also show that > x
j
for
j=2, ... , f-l, f+I, ... ,n
and that every point of the accessible domain satisfies these relations.
-x) /;i(x)=(x -, ~x ) (x -x,) - ('1,x.-/1 x ) (x n n n 1 .: ) j ? 2 n-l n
(23)
From (21), (::'2) and (~3) one can \-/rite directly the control law which solves the rnpH.
Remark 2 or
197
A Disturbance Decoupling Control Law H
TA~Lr:
(x n -tl nX n )YRl - tf(t\3x3-1:2x2)Yru + (l1 x - 11 x ) (x _ -x )w
3 3
1
u2(~)= 6 [ - (xn-l- xn)Ynl
2 2
n 1
He.
+
~( xl - x2)Y ~~
n
1
1 Values for siMulati on of the hinarv distillation co lumn
Tota l number of elements
n:9
Feed element number
f=5
Holdup of the condenser
Ho= 10 000 mol
Liquid holdup on each plate
H=I 000 mol
Holdup of the reboiler
H8 =2 0 000 mol
Relative volatility
a = 3.212
USa)
(25b) ~Ihen substituting expressions USa) and (25b) in (17) we easi ly see that
(2 Cia) (Z Gb )
X
n
So that the outp ut y is tota ll y decoupled frOM ~ and it is dynamicall y matched to the output y~ of a " 1 inear reference model . SIIIUUIT10l1 A digital simu l at i on was carried out with the mode l of the bina r y distillation co lumn (17), a 1 in ear reference mode l (10) and the contro l l aw (25). The properties of the feed a nd the initial ope rati on condi tion are given in Table 1. The ini tia l steady state of the disti llati on column is sho>m in Tab l e 2 . The 1 inear reference mode l co nsidered was a second order sys t em whose outputs correspond dynamical l y to the outputs of two input decoup l ed first o rd er systems. Then
Feed flow rote at the Initial steady stote
LF =1. 8 mol/seo
Composition of the feed stream at the Imial steady state
iF = 0.5
Product at the flow rate and the compoSition feed atream
I. F iF = 0.9
flow rote of the vapor stream at the initial .teady sta te
V= 1.36
Flow rote of the liquid · .tream at the initial steady state
I= 0 .52 mol/seg
TA n L~
~
Liquid Cam posit i on
Element number I
In the control l aw (25), four state variables (x?' x ' 3 x _ and x ) are fed back. This eas y-to-implement n 1 controlle r n can comp l etel y reject the effect of the dis turbances w and w? on th e both outruts Yl and yz and~ 1 at t he same time, - it matches them to the outputs y ~ 1 and Y07 of the linear reference mode l 'liven by (11)) ' and " (z71 . To check the perfomance of the control l~w, a set of simu la tions was carried out for step changes Lx" LL , F LUr.l and Lu oZ as shown in Table 3. These simu la~i ons show that tHe differences between the output trajecto ries of the linear reference model a nd the correspo n~ ding ones for the nonl inear system are negl i~ible. Even more, the influence of the disturbances x F and LF is complete l y rejected from the output va ri ab l es. nG REtlAf\KS
In this artic le, we have presented an extension of the OOPII for nonl inear systems: the output ma tchin g of a class of nonl in ear systems to a 1 in ea r referen ce mode l . We have found conditions under which this matchino is poss ibl e t oge ther with the so l vabi 1 it y of the OnPH. These condi tions may be s imply exr r essed in terms of Lie derivatives associated wit h the system and with th e structure of the model. Based on the n-dimensional nonl inear mode l of a bina ry d i sti llation column, the DDPtI and the output match in ~ problem were investigated. Fo r the model of the di5ti llation col umn, i t was sho./n that the disturbance re jection and output matchin~ is accompl ished by feedfor ward and state feedback.
0 .884 0.719
3
0.615
4
0.552
5 (Feed Element)
0.514
6
0.493
7
0 .458
8
0.346
9
(Reboiler)
TABLE 3
0.144
Step cha n'l es of ~r.2
xF~F2.....!:!.q 1
and
used for numerical cal cu -
lat i on Values of the previous
CO~ICLUDI
xi
(Condenser)
2
(27c)
mol/seg
Composition dist ribution at the initial steady state
(27a) (Z7b)
mol/se;
Absolute Value 161
steady state
Relative Chanoe 0/0
0 .1
20 .0
LF = 1.8 mOI/.. o
0 . 3 mol/lIg
16 .6
a/a
'Kf'I f = 0 . 9 mol/a.g
0 .06 mol/s.g
6 .6
%
uRI = 0 .713
0 .00
0 .0
a/a
uR2= 0.11 (see fiO.3)
0 . 00
0.0
a/a
i F =0.4 LF: 2 . 1 mol/eeo
0.1
25.5
%
0 .35 mol/seo
16 .6
%
iF LF = 0 . 84 mol/sell
0 .035 mol/.eg
4 .2
a/a
xF
= 0.5
uRI = 0 . 79
0 .08
10.1
uR2 = 0 .1I
0 .02
18.2
(••• RQ. 4)
a/a
:0
R. Castt·o, J. Alvarez and G. Bornard
198
~I;:seg] ,fmol
hact ~"!)J
These results were confirmed by a di~ital simulation of a distillation column composed of nine elements, so they show the solvabi I ity of both problems for this kind of processes.
REFERENCES Bornard, G., J.P. Gauthier (1~8J). ~lodelisation dynamique des colonnes a disti ller. In C:!RS (Ed.), Outi Is et tlod~les r'lathematiques pour l',l\utomatique;I 'Analyse de Systemes et le Traitement du Signal, TRCP No. 1, ~, Pari s. Bornard, G., J.P. Gauthier (198 1:). Etude experimentale des problemes poses par la comande d'un systeme non-I ineaire simple . .1\TP-Syst~mes Complexes, CNRS, Grenoble, "rance.
3.2
2.'
Castro, R., J. Alvarez (19G5). Singularities and aproxi mated matching. In CI~IVESTAV-IE (Ed.), The Distur-bance Decoupl ing Proble,n, Sing~lari ties and .l\proximated Hatching in a Distillation Column, Technical Report No. 17, rlexico,D.F., pp. 79-,)Lf. Gauthier, J.P., G. Bornard, S. Bacha, and ,'1. Idir (1982) Rejet de perturbations pour un modele non-I ineaire de collone distiller. Gel le-fie Con~ress, Jelle, lie, France.
3.
a
Hirschorn, R.II. (lgGl). (A,C)-invariant distributions and the disturbance decoupl in~ of nonl inear systems. SIAll J. on Control and Optimization, ~,~. Hunt, L.rt . , R. Su, and C. multi-input nonl inear Differential Geometry ~'Iathematics, 1.'01. 27,
Di~ital
160X 10
'20
simulation for the first experiment of Table 3 (Cont.).
as
~leyer (1983).
Desi~n for systems. In r<.VI. Brocket (Ed.) Control Theory, Pro~ress in pp. 2G8-293.
Isidori, A., A.J. :
200
'60
05
240XIO'S
200
'60
rm'i!l.~eg\.fmol
Isidori, A. (198Sb). Disturbance decoupl in'l and noninteracting control. In 11. Thoma (Ed . ), ilonl inear Con trol Systems: An Introduction, Lectures Notes and Information Sciences, \LQL..B, Springer-Verla~, Berl in, pp. 121:-177 .
fractionl
" -" 1
I
I I I
1
Moog, C.H., A. Glumineau (1982). Le probleme du rejet de perturbations measurable dans les systemes nonI in~ai res: Appl ication a I 'amarra~e en un seul point des grands petrol iers. Bel le-fie Congress, Belle-ile, France. 200
'60
!Jl~!/5~gl ,
.., ' I
., I I
2.14
0.5
I
( '60
Fig. 3. Digital simulation for the first of Table 3.
e~periment
Fi~.
200
4. Digital simulation for the second experiment of Table 3.
A DISTURBANCE DECOUPLING CONTROL LAW WITH OUTPUT DYNAMIC MATCHING FOR NONLINEAR SYSTEMS. APPLICATION TO A BINARY DISTILLATION COLUMN R. Castro*,
J.
Alvarez* and G. Bornard**
*Department oJ Electrical Engineering, CIEA-IPN, Apdo. Postal 14-740, 07000 Mexico 14, D.F. , AUxico ** Laboratoire d'Automalique de Grenoble, ENSIEG-INPG, B.P. 46 - 38402 St-Martin-d'Heres, Grenoble, France
Abstract . The prob l em of usin0 state-feedback and feedforwa rd control in order to solve the d istur bance decoupl in~ prob l e~ IrDP) of a nonlinear system and to ohtain, at the same ti me, an outp ut matchino of this nonl inear system wi th a 1 inear reference mode l is discussed. The resu lts obtained in simulation Ylhen the nonl inear system is a binary distillation column are s hown . Keywords. Nonl in ear control systems; disturbance decour l in!]; output ma tchin']; feedbacl:; feedforward; ~eome tric arproach; chemicul industr y; disti l lation column. that the disturbance w has no influence on the out DUt :t. This control law has the fom
I tHROOUCT ION In the last years, the nonl inear dis turbance decoup I ing problem has been wi de l y studied Ilsidor i , 193 1a; Hi rscho rn, l ~J
i (2 )
j:
=
'L
= ~(2)
+ 'J (2 ).:!. + p (2)::::
m U.
1
a . lx) + l: S .. lx)v .. , l i=l , .. . ,m). 1 i=l IJ '.1
(:)
In vie\'} of some earl ie r resu l ts (Isidori, 1935h) the ?rob lem consists in fin di n" a f eedb ac~ pair (~, S ) and a distribution 6 which is in variant under
( 1a) (1 bj
m
1n the other hand, there has bee n a cons iderab l e interest in the use of feedback wi th the rurpose of obtui nin~ a 1 inear behavio.r out out of a 'Jiven nonl inear sys t em (:iun t , 1903 ; Isidori, 1985a) . 110t i vated by these achievments \'/e study in t:,is purer t he problem of using state-feedback a n feedforward control in order to so l ve the disturbance decouf'l in!] pro bl em (iJCP) of the nonl inear system (1) and t~ . achieve, at the same ti me, an output matchin~ of this system wi th a 1 inear one . For this purpose, we ma l:e ex tensive use of the differential ~eometr i c concepts now widely used in the study of nonl inear systems. Th e paper is orga ni zed in the fo ll owi n'} way. \le fi r st review the DDP for nonl inear s ystems of the form (1) ( Isido ri, 1931a, 1901 b) . li e re, we just state the re sults that show the ex istance of a feed back which so l ves the problem. Then we propose a control stra:e~y that permits to match the output of a nonl inear syste~ ( 1) to t hat of a I inear mode l and to solve the ~"P 3t the sa me time. We a l so state the sufficient conditions fo un d in order to solve both prob l ems. Finall ,!,~:e s ho'lI the resu lt s obtai ned in simu l ation whe n the nonl inear system corresponds to the model of a b in ary disti 1 lation column tOCJether Ylith some conc lu din~ remurl:s. THE 0 I STUR3ANCE DECOUPLl IIG PR.O[lLEI1 (DDP) IN NONLINEAR SYSTEMS
1'1x) = -f (,,) + l: C\. (x) <1. (x) I - -I i=l
(3a)
-
.9. ,. (x)
m l: S . . (x)O. ( )' IJ - -J 2
j =l
(i =l , ... ,m),
conta i ns the vecto r fie l ds £ . , (i=l , . . . ,r), and is contained in I(er(dh) . This iL con tro l lee i nvariant. d~ is the differential of ~ , i. e ., d~ = aya ~. Th e noti on of l ocul co ntr o l led invariance is sometimes eas i e r to dea l with than (J l oha l) con trol l ed invariance and we wi 1 1 just consider the l ocal treatment. Theorem 1 (o n l ocal flDP). I f we define a constant r a nk involutive distribut ion 6 upon an open U of o I\n such th a t ( I s idor i, 19(11 1)) l i ) 6 is
( .!.-~)-in va ri ant
( i i) .piS 6 C: Ker(d~) Then for a ll x E U there ex i sts an Doen U of x in U where the --oDD P °ad~'its loca ll y a s~lution. c3n v2rsely, if the nrp has a l oca l so l ution upon an o~en U of Rn, it is rossib l e to define a constant rank invo luti ve distributi on on U thut fulfi Is co nditions (i) 2nd (i i).
We cons ider nonl inea r systems of the form ( 1) with m sta te x E ~n ,input u E R , pe rt urb atio n w E ~r and p outpu t 'L E 1\ . i , the m co l umns .21" " , ~ of the r.1a trix 9 and the r co lumns £1"'" £ of the matrix pa r e comp l et smooth vecto r fields on rRn The ad d itio na l input w represents an undesired sional which in f lu ences the behavio r of t he s ys t em thr o u ~h p.
wa~ to dea l with the DCr is to examine f ir st Ylhethe r or no t the fami l y of a ll control l ed in var iant distributions contained i n :;er(dh) has a "max ir.1a l" e leme nt (un element which contains a ll other r.1embe rs of the f am i 1,, ). I f this i s true, then the problem is solved if ~~d on l y if this maximal e l eme nt contains the vector fields £i ( i=l, . . . , r ) .
It is desired to modify the system ( 1) via state-feed back control on t he in puts u , ... , u in such a wu y 1 m
Hhen usin'] "locall y" contro ll ed i nvariant distributions, the existance of such a maximal elemen t may
193
n systematic
R. Castro, J. Alvarez and C. Bornard
194
be shown under rather mild conditions. It can also be calculated by means of local representations for the vector fields i, ~, , .. . , 2m as fol lows. We introduce the functions k
(ii) 6 SKer(db,) (iii) PS6 where
k
~
is a modified matrix
~iven
by
k
(~)
(L Lfh.), (L Lfh.), ... , (L Lf·h.), .2, ' .9.2 - ' .9m - ' i=I, ... ,p with Lz~ (2) denoting the Li e der ivative of a smooth function ~(2) alon:] the vecto r field :: '. These functions are obtained when actina upon the function hi along the vector fields i, .21"'" ~ k ti~es . Let P. be the smallest value of k for which at least one'of the functions (I,) is not identically cero on J.n; if all the functions are cero for an~1 k, one sets p .= "' .
,
Then for all x £ U , there exis~s an open of x , l!~1I where theo~Qr'loadm its a 1oca 1 so I ut ion. C~ver sel?, if the D"P'1 adrlits locully a solution in an open I! of 'Rn, then one can define a re'lular distribution on 'J that fulfi Is conditions (i) throu"h (i i i). The calculation of y (2) can be easi l y done once we have found the rlaximal locall y (f,'l)-invariant distribution contained in Ker(dh), this is 6". Some nice examplzs fer th e orerative part of the algorithms described above are re~orted by Isidori (1')81b) and Castro(I~C5).
Suppose p .~ "' . Let n(~) be the matrix with e lements a . . (x) (i,!,', ... , m; j=I, ... , m), and b(:d be the col u~n vMtor with comronents b. (x) (i=1 ,.~"-;-m) such that
DY~!Alll
,-
C OIJTPUT t1ATCH I tiC; /\'1(1 THE nnp
Besides a fixed non 1 i near system described by (1) , we assume no", a 1 i near reference model 'liven by (5b)
\Ie then have the foIl owi n] theorem.
-
,
( G)
-
and the feedback functions a (x) and S(x} w:,ich leave this distribution i nvariant-are any solution s of '~( 2)~(2)
A(2) S(2)
1\~2p. + Iln,~R
1.~
r.ft'?:R
(const~nt
matrix)
Ob)
When obtaining a and B one seeks for an invertible solution to (7bT s o that one has fu ll control. In some real cases (e.~., retrochemical processes), some of the disturbances correspond to phys ical variables which a r e measurable. This partia l measurement consideration pe rmits to solve the ~DP in appl ications where the non- meas urement condi t ion does not allo\-I a complete disturbance decoupl in]. cor the disturbance decoupl in~ problem .dth rartial measurement (DDPI1) one considers the control law (11009, 1932)
(C) Where a (x) and S (x) are defined as above and Vex ) is a m x r matrix. The- 'ith column of y( x) is associated to the ith perturbation, .' . , It is also assumed that only the first r , perturbations are measu rable so that the r-r columns of y (2) are cero. l There already exists local and DOpn t0get her with interestin,) These results are an extension the OOP and we wi 11 just state treat ment of the problem.
state 2~ £ 1\ ", input ~o £ l ., and output 1.R £ J. . Th~ cour1e (An,iC ) is contro ll ahle, '~' q is asym£ totically stable ~nd R q~ has maxium r~nk m . It is n evident that (10) is the most natural reference model, since the c l ass of 1 inear and contro llable systems is the best understood.
The prOblem of interest is to find cond i tions for the existance of smooth feedhack and feedfon-lard functionsn(~' S , Y ) defined on an open dense subset of U of ~ , such that ( i) 1 i m ~ ( t) = 1 i m
(i~ (tl
-
i (tl)
= Q wit h
t --+o:l
t~
sui table ini t ial condi t ions
(~(0)
~ 1.(0))
(i i) the ~CP:, is so 1ved for the non 1 i near s,/s tem ( 1) \·Ihen the flOPi; was discussed, we stated the necessary and sufficient conditions for th e existance of a control la", (13) that solves t~e proble", as wel l as procedure for its construction. So, in order to find the conditions that sa tisfv (i), consider the output derivat ive of (I) .
.,e
v
.L
where and
=
~ = L h(x) f- -
dt
Lf~(2)is
+ L h(x)u + L h(x)w
the Lie
'l- -
re- - -
-
~er i vative
( 11)
as defined before
r
L h(x) p--
i L h (x) i=Il!j-
Fr om (10) we also have d1.p
F
=
CR~~
( 12)
nlobal results for the . ('1,00:), 19~2) On the other hand, if there exists a contro l law app l'Icat,ons · u '(n) of the ones found in u that solves the Drp n then, us i n1 (7), (11) can be .Jr it ten uS and use the local o·
Theorem 3 (on local ODP/;). If we define a rellular dis tribution 6 upon an open U of ~n such that ' o (i) 6 is (~.- ~) -in variunt
( lOb) Iilo
w~th
(7a) C
( lOa )
np
TheoreM 2 (construction of a maximal element). Su~po se P . < '" for every i anci the rank of A(x) is equal to the 'dimension of y = h(x) for every x ~ tn, the maxi mal locally (f,g)-invariant distribution contained in Ker(db,) is ( Isidori, 19C1 a)
() k
~~
i
Lf~(2)
-
Ln~(2)A-l(2)£(2)
+
l
+ L h(x)A - (x) C v + 'l"- -
-
-
( 13 )
A Disturba nce Decouplin g Control Law and the fulfi lment of the conditions
Lf~(~)
=
L~(~)A-l(~)£(~)
L h(x) y(x) g--
= -
L h(x)
( 14<))
( 14c ) to~ether
with YR(O) = y(O) assures that
APPL I C/IT 1011 TO A B 1:lARY D1ST I LL!\T I O~ COLUMN The resu 1ts a 1 ready descr i bed above v'ere app 1 i e d, in si,."u l
~(t)
1 im
In the next section, we app l y thes e resu lts when the nonl inear s ysten is that of a hinary distil lation colu mn.
( l'lb)
p-
)95
li m (lR(t) -
t-+oo
i(tl) = Q
t -+oo
Th e condit ion s given by (1~) are suff ici ent condi tions that permit the output dynamic matchin ~ of a nonl inear system ( 1) to a 1 inear reference mode l (10). ~emark
1
One can notice that the control law used
( 15) preserves the same fo rm of (D). SO, once the ~DP~ bas been so l ved one can obtain, a t the same time, an output dynamic match in g if conditions ( 1 ~) a r e accom pI ished. cig. 1 shows the co rr espo ndin:) bloc!, d i
L h( x) g--
£(~) y(~)
C y = C ~F\ R
(lGa)
- L h (x)
( 1Gb)
iR
( 1Gc)
r>- -
lost p lote
Reboiler
which are easier to use in some practica l
Bottom Product
Fi g. 2. 5chem
d i
of a binary dist ill atio n
The feed is a binary mixture (wa t e r- a lcohol) and the column has n e l e"1ents, inc l udin '1 the renoi l e r and the condense r . Th e mode l is formed by the mate rial - halance eo uat ions assoc i ated to the process and the ther"1odynamic node l of the bin3ry mi xture. Th e nost inportant assu,.,~t i o n s introduced are : - The hinary system has constant re lative vo l ati 1 ity throu~ht the co l umn. - A
fe ed stream i s fed as a sa t urated 1 i(at it s ~ubble roint) onto the feed tra y.
s in ~ l e
~uid
- The mo lal f l m, of the va~o r and 1 i(1uid ti1rou'lh the str ir p inn and r ec tif yi n" sect ions are constant, resnectivel" .
The rest of t he assu m ~t i ons are the ones usua ll y done for this kind of ~rocesses (90 rnar d, 1~81). Th e symho l s used are the Pi g. 1. Block dia~ram for the control l aw t ha t solves th e ODPM to~eth er with the out put m at ch in ~ p rob lem.
fol l owin~
ones:
_i: index o f the cu rr en t eleme nt f : index of the feed e l ement n: nUOl he r of elenents Of the column 1 inuid and '1a~()II' co"'rosition on the jth e lement , res~ective l y (mol fraction]
R. Castro, J. Alvarez and G. Bornard
196
It.: vapour-I iquid equi I ibrium coefficient on the Jth J element L: I iquid flow rate (enrichin~ section) [mol/se']1 V: vapour flow rate throuaht the column t~ol/se~l L : I iquid flow rate of the feed tmol/seal x F: I iquid co~position of the feed (MOl fraction1 F H, H , Ha: tray, cone:enser ane: uottoo I i~uid holdup, D respectively [nolI a: relative volatil ity The model in a state-space representation is aiven by ( 17a) ( 17b) where x is formed by the 1 iquid composition on each element of the column and
(X _ - x.) Ulj+1Xj+l- M.x.) j l J J J
From (17) and (13) cne can notice that one of the system outruts is x?' the 1 iquid composition on the first plate, insteaa of Xl which is the 1 iquid composition of the distillate. This output choice was mainly done in order to avoid an undefined steady state control, this is that u(t) ... 00 when t ... "'. In Castro (1~n5) this case is wIdely discussed.However, from a practical point of view, this choice is not com~letly inade~uate since Xo depends directly on Xl' .50, we can expect a nood~re'1ulation on Xl when trylnn to net a ~ood one on x . 2
If we try to solve t~e ~DP for the binary disti llation column describe~ by (17) assuminn that it is not rossi~le to measure the distur~ances, one finds that the output system can not be decoupled from the disturbance ~Jl=LF/f: (r-authier, 19!)2;Castro, 19(5). On the other hand, when usin'1 theorem 3 one sees that there exists n solution of the nnp IL .'!.nd .,hen one supposes that w is measurable, a control law l u(x,w) which el iminates the effect of ~ upon y can be-eas i 1,' calculated . The
~aximal
element 6
..
is
(x _ - x ) (M + x + - Ilfx ) f l f f l f l f
'J(~)=
(x _ - x ) (tlj+ 1x + 1- IljX ) j j l j j
",here T In is the tannent s"ace of ln at x al1l n is x
equal to the number of elements of the column. H/Ha(xn_ixn) fl/HB (x n- l - x n )
o
o
0
0
-x p(~)=
1
f
(x f - x f + l )
0
(x. 1- x.)
0
H/H[)(xn_l-x n )
(1
r
The invariance of 6 as related to the modified dynamics of (17) can be obtained with (r.astro, 1~~5)
J
(21 ) where the c .. (i~l, ... ,2; j=l, ... ,Z) are the elements of th~Jmatrix C.
The input and di sturbance vectors are u =
T
LlH V/H1 , w = C LF/H
(LFxF)/H J
T
( 19)
"Ii th
H. = a / (1 + (a - 1)x. ) J
J
V is directly related to the power consumed by the reboi ler and the feed stream LF comes usually from another production sta'Je in a petrochemical complex. In practice, it is not possible to have on 1 ine measurement of x ; however, LF can be measured by an F adequate flux sensor.
The second column of y (x) is cero hecause of the measurement condition assumed on w and l
f1emark 1
In this ap~l icaticn. it is possible to o~tain at the same ti me an outQut matchinl of (17) to a 1 inear reference model; the conditions that must be checked out are the ones niven ry (16) since r.(x)= L h(x) (Bornard, IJCI; ; Castro, 19'35). ~Condition (l(a) is easi 1)' satisfied since f(x)=O (and therefore ~(x)=O) and some strai<)ht forward calculations (Castro:l~.'J5) nermit to verify (16b). Then, in order to accompl ish the output matchin9, we just modify the control law found above in such a way that condition (lCc) is fulfilled bein,] no",
The physical state space is the open cube C=(0,1)n. It can be shown (Sornard, 1902) that the system (17) is always inside this invariant domain for positive inputs (which are the only on~ physically sinnificative). One can also show that > x
j
for
j=2, ... , f-l, f+I, ... ,n
and that every point of the accessible domain satisfies these relations.
-x) /;i(x)=(x -, ~x ) (x -x,) - ('1,x.-/1 x ) (x n n n 1 .: ) j ? 2 n-l n
(23)
From (21), (::'2) and (~3) one can \-/rite directly the control law which solves the rnpH.
Remark 2 or
197
A Disturbance Decoupling Control Law H
TA~Lr:
(x n -tl nX n )YRl - tf(t\3x3-1:2x2)Yru + (l1 x - 11 x ) (x _ -x )w
3 3
1
u2(~)= 6 [ - (xn-l- xn)Ynl
2 2
n 1
He.
+
~( xl - x2)Y ~~
n
1
1 Values for siMulati on of the hinarv distillation co lumn
Tota l number of elements
n:9
Feed element number
f=5
Holdup of the condenser
Ho= 10 000 mol
Liquid holdup on each plate
H=I 000 mol
Holdup of the reboiler
H8 =2 0 000 mol
Relative volatility
a = 3.212
USa)
(25b) ~Ihen substituting expressions USa) and (25b) in (17) we easi ly see that
(2 Cia) (Z Gb )
X
n
So that the outp ut y is tota ll y decoupled frOM ~ and it is dynamicall y matched to the output y~ of a " 1 inear reference model . SIIIUUIT10l1 A digital simu l at i on was carried out with the mode l of the bina r y distillation co lumn (17), a 1 in ear reference mode l (10) and the contro l l aw (25). The properties of the feed a nd the initial ope rati on condi tion are given in Table 1. The ini tia l steady state of the disti llati on column is sho>m in Tab l e 2 . The 1 inear reference mode l co nsidered was a second order sys t em whose outputs correspond dynamical l y to the outputs of two input decoup l ed first o rd er systems. Then
Feed flow rote at the Initial steady stote
LF =1. 8 mol/seo
Composition of the feed stream at the Imial steady state
iF = 0.5
Product at the flow rate and the compoSition feed atream
I. F iF = 0.9
flow rote of the vapor stream at the initial .teady sta te
V= 1.36
Flow rote of the liquid · .tream at the initial steady state
I= 0 .52 mol/seg
TA n L~
~
Liquid Cam posit i on
Element number I
In the control l aw (25), four state variables (x?' x ' 3 x _ and x ) are fed back. This eas y-to-implement n 1 controlle r n can comp l etel y reject the effect of the dis turbances w and w? on th e both outruts Yl and yz and~ 1 at t he same time, - it matches them to the outputs y ~ 1 and Y07 of the linear reference mode l 'liven by (11)) ' and " (z71 . To check the perfomance of the control l~w, a set of simu la tions was carried out for step changes Lx" LL , F LUr.l and Lu oZ as shown in Table 3. These simu la~i ons show that tHe differences between the output trajecto ries of the linear reference model a nd the correspo n~ ding ones for the nonl inear system are negl i~ible. Even more, the influence of the disturbances x F and LF is complete l y rejected from the output va ri ab l es. nG REtlAf\KS
In this artic le, we have presented an extension of the OOPII for nonl inear systems: the output ma tchin g of a class of nonl in ear systems to a 1 in ea r referen ce mode l . We have found conditions under which this matchino is poss ibl e t oge ther with the so l vabi 1 it y of the OnPH. These condi tions may be s imply exr r essed in terms of Lie derivatives associated wit h the system and with th e structure of the model. Based on the n-dimensional nonl inear mode l of a bina ry d i sti llation column, the DDPtI and the output match in ~ problem were investigated. Fo r the model of the di5ti llation col umn, i t was sho./n that the disturbance re jection and output matchin~ is accompl ished by feedfor ward and state feedback.
0 .884 0.719
3
0.615
4
0.552
5 (Feed Element)
0.514
6
0.493
7
0 .458
8
0.346
9
(Reboiler)
TABLE 3
0.144
Step cha n'l es of ~r.2
xF~F2.....!:!.q 1
and
used for numerical cal cu -
lat i on Values of the previous
CO~ICLUDI
xi
(Condenser)
2
(27c)
mol/seg
Composition dist ribution at the initial steady state
(27a) (Z7b)
mol/se;
Absolute Value 161
steady state
Relative Chanoe 0/0
0 .1
20 .0
LF = 1.8 mOI/.. o
0 . 3 mol/lIg
16 .6
a/a
'Kf'I f = 0 . 9 mol/a.g
0 .06 mol/s.g
6 .6
%
uRI = 0 .713
0 .00
0 .0
a/a
uR2= 0.11 (see fiO.3)
0 . 00
0.0
a/a
i F =0.4 LF: 2 . 1 mol/eeo
0.1
25.5
%
0 .35 mol/seo
16 .6
%
iF LF = 0 . 84 mol/sell
0 .035 mol/.eg
4 .2
a/a
xF
= 0.5
uRI = 0 . 79
0 .08
10.1
uR2 = 0 .1I
0 .02
18.2
(••• RQ. 4)
a/a
:0
R. Castt·o, J. Alvarez and G. Bornard
198
~I;:seg] ,fmol
hact ~"!)J
These results were confirmed by a di~ital simulation of a distillation column composed of nine elements, so they show the solvabi I ity of both problems for this kind of processes.
REFERENCES Bornard, G., J.P. Gauthier (1~8J). ~lodelisation dynamique des colonnes a disti ller. In C:!RS (Ed.), Outi Is et tlod~les r'lathematiques pour l',l\utomatique;I 'Analyse de Systemes et le Traitement du Signal, TRCP No. 1, ~, Pari s. Bornard, G., J.P. Gauthier (198 1:). Etude experimentale des problemes poses par la comande d'un systeme non-I ineaire simple . .1\TP-Syst~mes Complexes, CNRS, Grenoble, "rance.
3.2
2.'
Castro, R., J. Alvarez (19G5). Singularities and aproxi mated matching. In CI~IVESTAV-IE (Ed.), The Distur-bance Decoupl ing Proble,n, Sing~lari ties and .l\proximated Hatching in a Distillation Column, Technical Report No. 17, rlexico,D.F., pp. 79-,)Lf. Gauthier, J.P., G. Bornard, S. Bacha, and ,'1. Idir (1982) Rejet de perturbations pour un modele non-I ineaire de collone distiller. Gel le-fie Con~ress, Jelle, lie, France.
3.
a
Hirschorn, R.II. (lgGl). (A,C)-invariant distributions and the disturbance decoupl in~ of nonl inear systems. SIAll J. on Control and Optimization, ~,~. Hunt, L.rt . , R. Su, and C. multi-input nonl inear Differential Geometry ~'Iathematics, 1.'01. 27,
Di~ital
160X 10
'20
simulation for the first experiment of Table 3 (Cont.).
as
~leyer (1983).
Desi~n for systems. In r<.VI. Brocket (Ed.) Control Theory, Pro~ress in pp. 2G8-293.
Isidori, A., A.J. :
200
'60
05
240XIO'S
200
'60
rm'i!l.~eg\.fmol
Isidori, A. (198Sb). Disturbance decoupl in'l and noninteracting control. In 11. Thoma (Ed . ), ilonl inear Con trol Systems: An Introduction, Lectures Notes and Information Sciences, \LQL..B, Springer-Verla~, Berl in, pp. 121:-177 .
fractionl
" -" 1
I
I I I
1
Moog, C.H., A. Glumineau (1982). Le probleme du rejet de perturbations measurable dans les systemes nonI in~ai res: Appl ication a I 'amarra~e en un seul point des grands petrol iers. Bel le-fie Congress, Belle-ile, France. 200
'60
!Jl~!/5~gl ,
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., I I
2.14
0.5
I
( '60
Fig. 3. Digital simulation for the first of Table 3.
e~periment
Fi~.
200
4. Digital simulation for the second experiment of Table 3.