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On Adaptive Control of Distributed Parameter Systems
Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto, Japan , 1981
ON ADAPTIVE CONTROL OF DISTRIBUTED PARAMETER SYSTEMS G. Hulko, B. Rohai-I'lkiv and
i. Sapak
Department of Automatic Control and Measurement, Slovak Technical University, Bratislava, CSSR
b!?~1!:~£1.!
A X-representation of the distributed parameter S;]'Stems /further DPS/ is introduced on the input/output quantities measured in their finite number of points. The nonlinear DPS /interconnected set of nonlinear DPS/ adaptive control problem is solved by means of this representation. Some results of hierarchical control of technological /glass/ furnace physical model are presented. ~~~2!:£~.! X-representation of distributed parameter systems ; nonlinear distributed parameter systems interconnected set of nonlinear distributed parameter systems; adaptive control; hierarchical control ; process control •
INTRODUCTION necessary for control under the uncer tainties must be obtained directly in the control process by the measurements in finite number of DPS points.
In the recent years the increasing attention has been devoted to control problems of the plants with unneglected spatial distribution. This fact is a consequence of their remarcable social-economic importance. In many situations the theory of the DPS deals witn these problems. Many fundamental results of this theory are summarized in the publications of A.G.Butkovskij, A.I.Egorov, E.D. Gilles, J.L.Lions, K.A.Lurie, Y.Sawaragi-T.Soeda-S.Omatu, T.K.Sirazetdinov, P.K.C.Wang, in the collection of works edited by W.H.Ray and D.G.Lainiotis and in the contributions of the IFAC/IFIP meetings devoted to the DPS control problems. For a present only a few works deal with the DPS control problems when: * the controlled plant consists of a set of interconnected nonlinear DPS • the distributed parameter control system as a subsystem of the large sea le system must adapt to varying cont-rol conditions, controlled plant variations and/or variations of control system environment-the distributed pa rameter control system thus operate under controlled plant and its environment uncertainties * the crucial part of information,
887
The submitted paperdiscusses the DPS control problems just in these conditions, which are frequently present at the control of the continuous tech nological processes. On the input/output quantities measured in the finite number of DPS points the paper introduces the X-representa tion of DPS. By means of this repre-sentation are solved the nonlinear DPS adaptive control problems for various amount of a priori information. At the same time there are employed some results of A.V.Balakrishnan and H.C.Hsieh related to multivariable lumped systems. The paper indicates a possibility of a priori information transformation, expressed in the linear partial differential equations form, to the X-representation structu re and calls attention to of utilizing the extensive A.G.Butkovskij's/1979/ summary of DPS charakteristics. It solves the decomposition of controlled plant, which can be comprehended as interconnected set of nonlinear DPS. Finally, there is demonstrated the applicability of suggested approach on some results of hierarchical cont-
888
C. Hulk6, B. Rohal-Ilkiv a nd
rol 0 f t:'1e technologi cal /glass/ fur nace physical model. I.
X-REPRESEN!ATION OF THE DIST~IBUTED ?AR~METER SYSTEMS
E
x ... y..
IIkn.LII~O 100
I I 1
I
I Xax( m,c)
z
Fig. ,. Input/output quantities of DPS.
Sapak
In these points let us determine the following curves x(l',t) , ••• , x(Li,t) , ••• ,y(kj,t) , ••• ,y(kn,t) on the input/outp~t functions from X and Y designate them as vectors of functions ~ (m, t)::
For simplicity of further considerations let us assume that the control led physical DPS is distributed on interval of E, space. For the abstract DPS definition purposes on physical DPS-at its certain state and values of ou~put quantities on system boundary-let us realize with the system multiple experiments E-Moore /'955/. In addition to obvious assumptions at this method of abstract system defining, let us ass~ me that distributed input/output qua~ tities can be measured-without incorrectness-in every point of interval. Let us denote the set of distributed input/output quantities on <0, L>x interval we can locate only the finite numberm/n, ~3, n~m - of measured points, e.g. with accordance to Fig. ,. 110, k11i- 0
L.
LX (L,
, t) , ••• , x (lm, t) ]
y (n, t):: fy(k1, t), ••• ,y (kn, t)] where Li,kj are coordinates of points i,j on interval; further, for simplicity, x(li,t)::x(i,t), y(kj,t)= ::y(j,t) - ~ee Fig. ,. Let us assign to vector x(m,t) a polynomial of tyP(; xm (z , t):: X ( , , t) L, (z) ......... x (m, t) ~ (z ) /1.'/ To the polynomial /1.'/ let us assign subdomain - Xax(m,t) - of all elements X(Z,t)EX which contain vector x(m,t) and for a norm holds: 11 xm ( z , t) - x ( z , t) 11 X ~ a / I • 2/ To the polynomial x (z,t) - at bounda ry condition oO,oL ~nd plO - by R-reE resentation let us assign in the selected "n" points vector ~(n,t). To this vector let us assign the followbg polynomial of the type : y n (z , t) =y ( , , t) B1 (z) + ......y (n, t) 3 (z) n/ L 3/ By R-representation let us map the subdomain Xax(m,t) to the Y and the image denote Xayx(m,t). To the polynomial y (z,t) let us assign subdomain X~~(n,t)~ which contains XaYx(m,t) and all elements from Y for which in certain norm holds : 11 y (z, t) - y (z, t) IIX ~ ~ /1.4/ Wit~ all possible pa1rs of the subdomains of type [Xax(m,t);X~i(n,t)J - at which the image of Xax(m,t) in the sense of R-representation is contained in the X~~(n,t) - on the set nwe define the system of subsets : X={[njRC.(X ~ Y);[Xax(m,t);XBy(n,t)]]} Let us fu~~her call this system of sub sets X ,X-representation of investiga-ted DPS fat this certain state and va lues of output quantities on system boundary/. For a D?S the analytical dependence exists among the input functions xm(z,t), boundary conditions 00, oL, p10 and the output in point "j" y(j,~)=FNj[oO, x(m,t), oL, p10J /1.5/ Hulko/1977/; the existence of this re ~ation ~as showed by elementary means 1n Hulko/l979/; see relation /111.4/. ~(n,t)={FN1[00, x(m,t), oL, p10], ••• , ••• l-FNn[20, x(m,t), oL, p10]}= =FNLoO, x(m,t), oL, p'O] /1.6/ Then we can relate to the system of subsets X a mathematical structure X[xm(z,t);II. II Xa;FN[.]; Yn(z,t); d • Ily~J, which to the subset
889
On Adaptive Control of Distribut e d Parameter Systems
X~x(m,t)
of input quantities, con:aining the vector i(m,t), by relation FN[.] land vector y(n,t)1 relates sub set ~~y{n,t) of output quantieties from Y, so tha~ the image X~Yi(m,t) of X~i(m,t) is'nX~y(n,t) subset. In the frame·.... ork of X -represents tiO!1 let us define the prob:ems of control synthesis and identification. Unde~ the DPS control synthesis we understand to find todthe given desired output function y (z,t)£Y lat cer tain DPS state/ and for specificed "control quali ty v " the pair [X~x(m,t)j X~y(n,t~ fro~ the suita2 le system of subsets from X so that yd(zlt)E.X~Y(n,t) and max IXBy {n,t)1I y~V The a~~Grmination of the suitable subsets system from X-structure of type X [xm(z,t); iI • Ilx~jFN[ • ] j Yn(z,t) j 11 • IlyBJ- which suits the control synthesis problem we will call the identification problem. I I. ADAPT r{E CONTROL
Let us solve the control synthesis problem for the nonlinea~ DPS at the gi ven desired function y (z, t) and v value. In the X and Y let us consi6eJ.· Tchebycheff norms. During the solution we will employ various assumptions-sometimes only for illustrative purposes-which cadprise from a priori information l i f there are any availab lel or from current information about the controlled process. Let us consider that the X and Y belong to the certain classes of functions-Collatz-Krabs/'97J/j Dzisdyk 1'977/; Laurent/'972/j Nikolskij/19771 etc. Let us consider further, that to V/k valuel "k" will be determined laterl -for certain DPS state-it is possible to determine ~(v/k) value in X so that when 11 x' (z , t ) x 2 ( z , t) 11 ~ ~ ( v Ik ) /1 1. , I then 11 y' (z,t) - y2(z,t) 11 ~ V/k /11.21 when {[x' (z,t) j y' (z,t)], [x 2 (z,t); y2(z,t)]tR and at the same time the superposition principle holds in the small sourrouding of [x(z,t), y(z,t)J pair from R. In the X let us choose the x (z,t) polynomial/the degree and mWasurement points location "m"/ so that it aproximates with the error s~aller than ~(v/k) every functions from X, !hich contains the polynomial vector x(m,t). Similarly in Y let us choose Yn(z,t)
polynomial Ithe degree and the ~easu points locations "n"/so that it appr oxima tea ',d th the error' smaller than v/k every fu~ctions from Y£ which contains the polynomial vector y(n,t), see relation IT. J/. Supposing that xm(z,t), Yn(z,t) polynomials also belcng to the certain classes of functions Ige::erally different from above mentioned/. 1 Let be x' {z,t)=x (z,t)"then the image of x (z,t) in~Y is y (z,t). In accordan~e yith previous consider~ tions to y (z,t) let us assign y (z,t) so, that n !I y' (z, t) - y ~ (z ,t) 1I <: v Ik III. JI Because the ~ ( v Ik) surrou!1ding of xm(z,t) images to the v/k surronding +\ +h" .&.'X-( 0_of' y 1( Z,MI, ~ en 1mage o~ ~x m, t ) " 19 situated in the 2(v/k) surronding of
re~ent
y~(z,t). Ba3ed on the Frechet generalized Weierstra3se theorem the relation/I.51 for t»Ts we approximate by the N deg ree functional power series : a m~' N Ts Ts Y (j, t)= r r 5 ) i= 0 h= 1 0 0 r (1 '+ T1) ••• X\l, " t -T h,'J'"c;T, ••• d T = Wh ji LX ,"h
N(=P WN j !!!) X!!! ,
/II.4/
where xm = [00, x (m, t) , oLJ:; [x (0, t) , x (, , t) , : •• , x (m , t) , x ( m ,t) J ,!!!= m" 2 , - , -- LwN r.: ' , , ••• ,wN ", , ••• ,wN, J 1, wN J!!! J J1 J!!! wN ' , = [w (T, ) .. , ••• ,w ( T, , ••• ,T ) .. ] N J1 J1 J1 and Ts is a finite setting time. Let U3 introduce the functional t L
+'
I:.~
~ {B~(z)""1LPN(WN1m)xm-Y(1,t)J2't'
t-T 0 J2 + ••• +B22() z .... 2 [PN(wN )x y(2,t) 2m m - )x -- - y(n,t) J2 }dz dt ... B2() z .... [N P (wN n n nID m / II. 5I Let be
/II.6/
Q =
o
..
Ill. 7I
,0, ••• il'nQn
....i - weighting funetions, Q is an n x n positive definite symetric
G. Hulk6, B. Rohai-likiv and
890
I ::
S
tT~t
eT Qed t
I l l . 81
t-T
be given le.g. as the result o~ thW succesive determining of optimal input ~uantity during adaEtive control by Ht" time/.Instead o~ y(n,t) we sub~ti " lte into the func tional /II.SI the ~E(n,t)-vector of measured output quantities. It is the true output vector corrupted by additive noise E (t). E(t) includes the measurements errors in observation the cutput and the effects of random disturbances entering into the DPS. Then let us denote the arising functional as I n ...." Mihimizing the functional InA relating to Vol~erra kernels-Hsieh/1965/-so, that Ily~(z,t) - ~';:z,t) 11 ~vlk III.91 where
Let the vector x
y~ (z, t) =yE { , , t)8, (z) ... y E (2 , t} 8 ( z) + 2
111.'01 n (Z) y~(z,t)::ya(, ,t)B, (z)+ya(2,t)B 2 (z)'" + ••• +yE(n,t)3
+ ••• ~ya(n,t)Bn(z)
./ e
matrix
Sapak
of IIl.51 type
matrix.Then t
L.
111.1'1
obtain the Volterra kernels
vm, O,vlN, 1""
,wN, ,m+1
wN 20' wN 2' , ••• ,wN 2 ,mT
.
~,Q,
~=
,
0 , ••• , 0
.o
Let us minimize the functional InAD relating the Xm - Hsieh/196S1 -so,that I1 y~(z,t) - y~(z,t) 11 ~ vlk III.161 whe::e y~(~,t) is determined by the optlmal xm vector. According to the /11.161 and so 11 y~(z,t) - yd(z,t) 11 ~ vlk IIl.,7/ and a 11 y(z,t) - Yn(z,t) 11 ~J v/k IIl.'SI where y(z,t)~Xa(V/k)Yx(m,t) then
1~~_!~~g~~_Q!_!h~_Qi~~£i2~~~Q_iDE~!
S~~~1i~i~~_Q~~~£~i~~£_~y
---
11 • 11
J ( V Ik )]
III.121
IIl.DI
i~-~~~-~i~~~l~~~~!2~-~!-!UE¥~-g~~~~i= ~les
- measureo ln m pOln.s from interval with coordinates {ti} - £~!~~~~ - by means of Volterra kernels matrix, the values of ya(n,t) vector in the points "n" with coordinates {kj}, the polynomials B.(z) and the value of the no~m 11 • 11 3evlk) ~~~_~~E!~~_~1~~~¥ly-i~~~11-!hi£h-£2~:
tne-
.
~re~nulnt·
Y~£!Q£ -
of the-subset their images y(z,t)EX3(vlk)y (n,t)cy --- ~~1i~fY 11 y (z, t) - y (z , t) 11 ~ 5v Ik 1I1. , 9I and when k=S 11 yd(z,t) - y(z,t) 11 !:.v /11.20/ So under the assumption, that relation of structure X[xm(z,t)j 11. lIa(vIS)j pN[WNn!E{')];
y~ (z, t) ; 11 • 11 3 ( viS) ]
y ~ ( z , t) ;
im
Yi~~S2 elements Xa(Vlk)x(ffi,t)E~aand
from
-----------------------
IIl.1SI
iu~~~~1i~y
'
. WNno,WNnl,···,WNn,m+'
InAC:: 5 e,T ~ e' dt III. 1 41 t where N " d e'/ [p (wNj~)X~ - ~l (j,t)] and
~
~~!U!L~h~_~·~.f~~!:_~L~~HSie~2!_12~~!.
Distributed output quantity yd(z,t) in the point~ "n" with coordinates {kj} gives y (n,t) vector. Supposing that relations of the /11.131 structure holds on the (t, t+~t) interval too. Let us consider the functional
Ill. 2 , I
(t-T,t) interval holds on (t,t+~t)interval too, in the sence of relation /11.201 the output quantity y(z,t) with error smaller then v will a8proximate desired output quantity y (z,t). With regard tG~~umptiorr, which will be mostly used for illustrative purposes it should be noticed, that e.g. assumption about inputloutput quantities-namely that they belong to certain classes of functions-is naturally not necessary, see part I and V. In usual industrial conditions mostly a simplier a priori information can be also sufficient. For example from the manner of introducing the input quantities lin som~~r.S~~~~·sl is often possible to de~~rmln~~f x (z,t) in which practically the distrib~ted input quantities are found. For deter mination of the relation between Vlkand above mentioned x (z,t) surrounding may be often sufficient also sta tic measurements le.g. during opera-tionl realized directly on the plant.
On Adaptive Control of Distributed Parameter Systems
But in the real situ8ticns we eften have so a priori information that we can use such assumptio~s which considerably decrease require~ents relating to mln numbers,E.g. if in the di rection of NZ" axis in.lout. quant.hi ve character of polynomial of 2,3,4.:. degree,at the stagger distr.of mln,the m=n=3,4,5 ••• provides high accuracy of approximation. It is similar to the Tchebycherr norm which we used mainly to illustrate our aims. Naturally, every consideration ceuld be dene e.g. for ~p norm etc. By shiftir.g the intervals (t-T,t), (t,t~~t) along the time axis and by succesiv~ minimizing functionals of type :!:n; , Il".AC so, that at the same time the presence of the appropriate structures of type 111.211 is secured-when relation 111.201 holds-I at the assumption of complete identificationability and controllabilityl we realize the adaptive control of DPS. At the assumpticn, that in the course of adaptive control appropriate struc tures of type 111.21/ are secured - satisfaing the g{~aH~~~~t~~l of DPSit "reduces" to adaptive control of multivariatle lumped parameter system. It "reduces" to succesive minimization of functionals of type InA, InAO at the shifting of intervals (t-T,t), (t,t~~t). At these conditions we can use fer solution of the adaptive cont rol problem of nonlinear DPS e.g. A.V.Balakrishnan's and H.C.Hsieh's re suIts refering to the adaptive control of multivariable nonlinear lumped parameter systems by function space met hods. The minimization functionals or type InA, InAO was solved by H.C.Hsieh 119651 by steepest descent method in corresponding adjoint Hilbert space 2f inputloutput vectors o~ type x(m,t) y(n,t). Beside the analysis computational requirements of optimization problems H.C.Hsieh/19651 presents also a special computationally very effective method to matrix of Volterra kernels on-line determination. I
Ill. TRANSFORMATION OF THE A PRIORI INFORMATION At the concrete control realization there is often at our disposal some a priori information about structure andlor coefficients of linear partial differential equation, which roughly approximates the dynamics of controlled distributed parameter plant lin the surrounding of certain operation regime etc.l. The solution of this linear partial differential equation at the zero initial and boundary conditions let have the form :
L
t
y(z,t)= )
)
o
891
G(Z,~,t,T)x(~,T) d~dT
IIII.11 is the Green function. ~hen Y(J,t} the output corresponding to xm(z,t) in "j" is t L y(j,t)= 5 S G(j,~,t,T) [x(1 ,T)L 1 (~h
:here
0
G~Z,~~,T)
o
0
..... +x(m, T )L,J~)]
d~dT=
L
t
= ) { x ( 1 , T) [S G ( j , ~ , t , T ) L, ( ~) ~ ~ ] •
.: o
0
L x(m,T) [S G(j,~,t,T)L (~) d~! dT o m lIIl.2/ when L S G( j , ~, t , T ) Li (~) d ~ =g ( j , i , t ,T ) o IIII.3/ then t y(j,t)= S [x{1,T)g(j,1,t,T)+x(2,T). +
o
.g(j,2,t,T)+ ••• +x(m,T)g(j,m,t,T)] dT <=FNj[oO=O,x(m,t), oL=O, p10=0]> /III.41 The functions g(j,i,t,T) are elements of . Vol~erra kernel matrix for DPS, whlch lS represented by the given linear partial differential equation. /'
W, nm
=
~(1'1,t'T), ••• ,g(1,m,t'T)J • • .g(n,1,t,T), . • • ••• ,g(n,m,t,T)
IIII.51 these relations.a~d by utlllzatlon e.g. A.G.Butkovsklj s 11979/ summary of DPS charaeteristics, which containes Green functions to mo re then 500 boundary value problems,we can transform a priori information, expressed in the ~orm of lin.partial differential equations, into the struc ture of the introduced X-representa- tion. St~r~ing.from
IV. MULTILEVEL DECOMPOSITION OF DPS To the controlled distributed parameter plant it is many times possible to relate interconnected set o~ nonlinear DPS DISTURBANCES
tl~t.cr~· ~~f.U~l~~OU~~Yl l.l Output ~J~~ trans-
D P S
~
form.
3 YJ T
P
U T
Fig. 2. Interconnected distributed parameter systems.
892
G. Hulk6, B. Rohai-likiv and
XG={Xl, ••• ,XI, ••• ,XH}, XI is the set of I-th admissible distributed input quantities XI={xI(z,t)}. YG={Yl, ••• ,YJ, ••• ,YK}, YJ is the set of J-th corresponding distributed output quantities YJ={yJ(z,t)}. UG={Ul, ••• ,UJ, ••• ,UK}, UJ is the set of J-th state quantities of the int erconnecteC set of DPS,UJ={uJ(z,t)} E'}={El, ••• ,£I, ••• ,EH}, El is the set of I-th distributed disturbances. For simplicity let us assume, that UG=YG and E}=O, respectively. Now to single DPS let us determine polynomials of type xm(z,t) and y (z,t) so, that together with relat£ons : ---yyn=FNG(Xx m) when /IV.l/ YY;'= [ Y1n' : •• , yJ n ' • • • , yK n ] /IV • 2/ /IV.3/ xXm= [X'!!)"" ,XI!!), ••• ,XH!!) ] FNG(xx m) =
FNK1(x1 m),
possibilities of proposed approach to solution of distri~uted parameter sy~ tems adaptive control problems. The dynamics of the technological /glass/ furnace was investigated on the physical model see Fig. ~, Kacha nak et al./1975-78,1975-80/, Sapak fin preparation/. Experimental apparatus and its mathematical model in the form of three nonlinear partial differential equations was described in Hulko, Sapsk /1980/.Starting from the choosen ope ration regime the model of the appa: ratus can be consider as an intercon nected set of nonlinear DPS Fig. 3.Power input l.DPS on outside surface of heating cylinder
FNJI(Xl m)
I.DPS
2.DPS ,FNKH(XH m)
/IV.4/ where yJ and xlm are vectors related to surfances: yJ(z,t) and xI(z,t) from the sets XI and YJ b~ m/n points of interval <0, 1>. FNJI(xIm) = =FNJI [oOJ, xl m , oLJ J, where oOJ and oLJ are corresponding boundary conditions. --- Sometimes the interconnected set of DPS structure may require a finer decomposition. For example between the J-th output and I-th input there is a "block" of n serially connected distributed parameter subsystems.Then at the fullfilment of the citated assumptions the further stages of deco!!) position arise: - =••• FNJIn••• 0FNJI T-xlm ] ••• / IV.5 / yJ n where T FNJI [.], ••• ,FNJln [.l correspond to single elements of serially connected "block"-at the given state and bound~ ry conditions.--arise structure of type /11.13/ in which-at the fullfilment identificationability and controllability conditions-similarily as in part 11, it is possible to realize control by given desired output quantities and at the required "quality". V.
Sapak
Temper~ture
FNl 1 (Xl m) , • ••
i.
ILWSTRATIVE
EXAMPLE
Temperature on inside surface of heating cylinder
NOi=i,;S; I 4.DPS
).DPS
] 2.DPS
Temperature of glass Fig. 3. Decomposition of the physical model On the basis of modelling studies and experiments on the physical model the structure in Fig. 3. can be approximated by system Power input xl I.DPS yl Temperature on inside surface of heating cylinder x2 d ~.DPS
Yrr Temperature of glass
Fig. 4. Approximation of the decomposed model. By some results of technological/glass/ furnace physical model hierarchical At the constitution of the control control we suggest the applicability
89 3
On Adaptive Control o f Di s trib u t e d Parameter Systems
system let us start from the structu re in Fig. 4. Let us relate to it tne relation of type /IV.5/ YII(n,t)=FN~[FNT (x1 m)] IV.3/ where FN~ [.] corresponds to ~ .DPS IV .4/ T FN (.) corresponds to T.DPS IV.5/ We investigated the possibilities of control partly in a small area near an operation point-linear approximation, LP-partly at the transition be~ ween the operation regimes-nonlinear approximation, NP. In the LP case Vol terra kernels matrix contains 1th or der kernels and in the NP case con-tains also 2nd order kernels. The identification was realised in se veral stages. At the structures choice to t and ~.DPS of type /11.21/ we 3tart with the requirement that on ~.DPS v=3% from mean value of the de ,ired output quantity. We exploited;.: artly a priori information,conta~ned in models Fig. 3., 4. , what 'Ne Slm~ lated in an oriented language SIKOS by digital computer SIEMENS 4004 and partly by measurements on physical model. - For polynomials L, (z) and B, (z) were choosen Lagrange1interpolaJtion polynomials. - We choose measuring poin t nu~e~ for ~P m=n=5 and for NP m=n=6 by~t~6~r distribution along the apparatus. - At the Volterra kernels starting va lues determination we exploite A.G. Butko"Skij's /1979/ summary of DPS charakteristics. - Volterra kernels determination was realized on two levels: * On the first level by minimizing the functional of type InA from the Volterra kernels initial values we determined the T.DPS Volterra kernel matrix. * On the second level we proceed similarly as on the first level, but wi th a differeT"ce, that model in put in InA for ~.DPS was computed from model output of the T.DPS < e.g. for j-th component of ~. DPS in theLP c~ se ~ T , { ••• +w 1 ..\ [ ••• +w 1 .\ [x ( 1 , t) ] + ••• ] + ••• }>J1
11
In the Fig. 7,8. are presented some elements of Volterra kernel matrixes. On the first and second level of Volterra kernel matrixes determination we used the time courses of input/out put quantities, obtained from experi: mental measurements on the physical model. The experiments were realized with Automatic Data Acquisition System HP 3050 B in Czechoslovak Institute of Metrology in Bratislava, connected with digital computer SIEMENS 4004 for the realization of identification and computing of optimal control ,Fig. 5, (:"C:. T?
... .
The aim of control was to take, at the given state, the output quantity from value Yr to YIr and from YJII to y /LP and NP case/, Iwe al'Nays as sumeT{complete iden:ificationability and controllability/, at the minimum energy. In our case-for LP- it means the minimization o~ functional: min fj,t I= 1 ('), t) [ ( ) e~ ~ d + 11 x1 (5, t) 11 ~ t: (O,fj,t) 0
x
e
+{i~e{~~t) [fj,h} ~e~+ 1I (O,fj,t ,
0
x2 d (5,t)
~
lI~tJ }]'" /V.6/
The energy requirements denote values \I x1(5,t) ilK and 11 x2 d (5,t) ilK ,when e.g. 11 X1(5,t) IIK=[x1(1,t)L1(z), ••• ,x1(5,t~. .L,,(zL,.
"~_1(t) 91
, ••• K2 (t), 0·
J
6) . , ,";K~{t)
~'(1,t)L1(Z~
, x 1{ 2,t)L2 (z)
~1(5,t)L5(z)
-In 1 ~ T)d(')l ed'l: (w1 J ' 5_x15 J _ - xZ J,t~ /;1
ef'=~ J
where
{~ ,. )-2d
( '
)il
w1j5_x_5 - YII J,t~
are kernels of
IV -- matrix/si positive definit/ of weighding functl.ons. x1(0,t)=y1(O,t), ••• ,x2 (5,t) =YTI(L,t) are boundary conditions on the transition interval (O,fj,t). When determining the optimal control of these serially connected systems we start from the M.I}.Singh's and J.F.Coales's/1975/ ideas. On the second level of control there is minimised the functional I?,as the problem of final value and minimum energy. The result is the function X2~(z,t). To this function-as to the desired output quantity-on the first level of control then there is deter mined optimal input x1 (z,t) for i.DPS by minimization Of 5 I 1 functional. Functionals of identification and syn thesis of control were minimized by method of the steepest descent in cor responding adjoint Hilbert spaces ofinput/output vectors. The method of optimization was taken over from H.C. HSieh 's/1965/ monographs. If conditions of adaptive control from part 11. hold, the described offline hierarchical control of model T, and
metri~
~.DPS subsystems.
894
G. Hulk6, B. Rohai-likiv and
furnace on inte~vals (t-T, t), (t,t+~t) car. be consider as k-th step of adaE tive control. The arisen adrtive control system ha ve only u3ual computational re~uire= ments, that is why its exploitation to adaptive centrol of slowe~ techno logical processes than DPS-3 shouldnot m~et with greater p~actical obstacles not even under current engineering conditions. CONCWSIONS Naturally, in the paper there were studied only fundamental properties and possibilities of the proposed aE proach to solution of the DPS adapt~ ve contrel problems. In future it will be necessary to de vete the attention mainly to DPS i-dentificatienability and controllabi lity problems in the fraffiework of tne X-representation with the aim to de termine principles of constituing optimal adaptive control systems in dependence on controlled distributed parameter plant and its environments uncertainties /the amount and kind of a priori information, the possibi lities of obtaining the current in-formation etc./. REFERENCES Butkovskij, A.G./1979/.Characteristics of distrinuted 2arameter-sistems:7in-russian7~auKa~-Mo3cOw:2~j-pp:
Collatz, L., Krabs, W./197J/. AE2ro~1~~~~2~~1h~2r1;_!~£b~2~~£5~!r:Scne AE2roxlmat~on m~t Anwenuun-
g~5~-~·G:Taubner~-Stuttgaro:27Tpp.
Dziadyk, V.K./1977/. Introduction to Theorv of Uniform-Annroi1mat1onrunct!ons-Si-~oIinoi!als:71n-ru: ssian7-~auKa;-~oscow:-;TO pp. Hsieh! H.C./1965/. §~~~h!!1! 2f_~~aE tlve Control Svstems bv functlon ~Ea£e:~~sfi§~s:~in-Aava~ces-in-~on1 roI ~ystems ~fieory and Applications /Ed./ C.T.Leondes. Academic Press, N.Y. and London. Vol.2. 117-268 pp. Hulko, G./1977/. Contribution to The orv of Distributea-~arameter-Sys te~s:7generaI1zea-moaeIs~-aaapt1 ve-control/. Ph.D.Thesis./in slo vak/ Slovak Technical University, Bratislava. 147 pp. Hulko, G./1979/. On Nonparametric ReE resentation of Distributed Parameter Systems. Preprints of 5th IFAC Svmnosium on Identification
~5~:~~£~~~~~~:~~!I§~!I25~~armstadt
Hulko, G., Sapak ~./'980/. On Multile vel Decomposition of DistributedParameter Systems. Preprints of ~n2_JE~Q_§l~E2!i~~_Qn_~~g~_§£~!~
L.
Sapak
SV$tems:Theo~~
and
~nElications.
T~uIouge:----~------~
----------
Kachahak, A., Belans~y, J., Hu lko,G., Rohal-Ilkiv, B., Sapak, f./1975i980/. Models of Glass Furnace I., JJ./in s!ovaK7-~esearch-report-- unaer the contract with Slovak Works of Technical Glass, Bratislava. Kachanak, A., Belanskj, J., Hulk6, G., Rohsl-Ilkiv, B., Sapak, ~./'9751980/. Identifi catio n Pr oblems of Adantive-~ontroI-o~-~istribu
~~g:r~r~~e1~£:Si§!~~s~-nesearch-
report uncer tne contract with Insti tute of Informatior, Theory end Automation-Czechoslovak Academ;;- of Sciences Frague. Bratislava. Kornej~uk, N.P./1975 / . On Extremel Att ributes of Splines. Froceedings of !n1i~~~11gD~1_£Q~gr~~§ 9D ~EErCXlma.lon o~ runct~ons. Rau~B, Moscow. 440 Pt>. Kornej6uk, N.P./1976/. Extremal Prob 1~~~.2f ~2Er~~i~ali§5:~5eo£i:/in russlar.7 Nau~a, 9.oscow. pp. Laurent, F.J./1972/. AEErcximation et
"0
QE1i~i~!1129. Ferien~-Pirri;19;p~.
Moore, E.F./1956 / . §~!C~l!r~!!E~!i;_ ments on SequentlaI Macr.lr.es.7~n russIan7-in-Avtorna{y-7ro:7-~hS
nnon, C.E., M~ Certhy,J. IL. Moscow. Nikolskij, S.M./1977/. AEircximetion Functions of Several vari5fes:7In russran7-~auka~-~osccw~-!~)PP;
Saplh, L./in preparation/. AdaEtive Subontimal Con t rol of DIstrIbuted ~araieter-~isterns:-~h:~:Tnesrs:- ~IovaK-TecnnrcaI-University.
Rratislava. Singh, M.G., Coales, J.F./1975/. A he uristic approach to the h~erarchi cel control of multivariab:"e se-rially interccnnected dynamical systems. Int.J.Control,Vel.21 , No.4. 575=;S;-pp:-----ACKNOW".uEDGEMENT
The authors wisches to express his thanks and appreciation to PROF.J. S k k a 1 a - scientific director of Czechoslovak Institute of Metrology in Bratislava - for kindly enab ling the realization ef experimen~s in Institute a~d the first fro~ authors to Ph.D.L. S u t e k for SUE porting through many years his research in the theory of automatic control.
a
895
On Adaptive Control of Distributed Parameter Systems r
1\ i
tf"ater
InsulHlon
'1
r;~~t:-Da~ I:CqUl:-;.:J I System HP 3050 B I
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.
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1.
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2.
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Thermocouples
Fig. 6. Axial cross section of the experimental apparatus. 5. The flow of the information during the experiments.
F~g.
Digital computer SIEMENS
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:ig. 7. First order kernels.
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500
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450 Initial 0
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)(
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t
11
state ~
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150
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250
Initial state
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300
,;tate
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X
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6
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400
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Healinl: prO):rarn [A] - LP approximation O,09-a power lnpu - a power In u 0,
0
62 124 186 248 310 372 434 496 558 620 68:2 744 8061W1
t 2
3
Fig. 9. Measured and computed profiles.
t
4
Measuring' poinu for LP approxi m~tion
0
Finite state Initial state
930~2
t 5
·H m,n
896
C. HulkS, B. Rohal-I1Kiv and
1.
Sapak
Discussion to Paper 31.2 K.J. Astrom (Sweden): Were the experiments mentioned in your paper made on a laboratory process or on an industrial plant? G. Hulko (Czechoslovakia): The experiments mentioned in the paper were made on a laboratory process.
ON ADAPTIVE CONTROL OF DISTRIBUTED PARAMETER SYSTEMS G. Hulko, B. Rohai-I'lkiv and
i. Sapak
Department of Automatic Control and Measurement, Slovak Technical University, Bratislava, CSSR
b!?~1!:~£1.!
A X-representation of the distributed parameter S;]'Stems /further DPS/ is introduced on the input/output quantities measured in their finite number of points. The nonlinear DPS /interconnected set of nonlinear DPS/ adaptive control problem is solved by means of this representation. Some results of hierarchical control of technological /glass/ furnace physical model are presented. ~~~2!:£~.! X-representation of distributed parameter systems ; nonlinear distributed parameter systems interconnected set of nonlinear distributed parameter systems; adaptive control; hierarchical control ; process control •
INTRODUCTION necessary for control under the uncer tainties must be obtained directly in the control process by the measurements in finite number of DPS points.
In the recent years the increasing attention has been devoted to control problems of the plants with unneglected spatial distribution. This fact is a consequence of their remarcable social-economic importance. In many situations the theory of the DPS deals witn these problems. Many fundamental results of this theory are summarized in the publications of A.G.Butkovskij, A.I.Egorov, E.D. Gilles, J.L.Lions, K.A.Lurie, Y.Sawaragi-T.Soeda-S.Omatu, T.K.Sirazetdinov, P.K.C.Wang, in the collection of works edited by W.H.Ray and D.G.Lainiotis and in the contributions of the IFAC/IFIP meetings devoted to the DPS control problems. For a present only a few works deal with the DPS control problems when: * the controlled plant consists of a set of interconnected nonlinear DPS • the distributed parameter control system as a subsystem of the large sea le system must adapt to varying cont-rol conditions, controlled plant variations and/or variations of control system environment-the distributed pa rameter control system thus operate under controlled plant and its environment uncertainties * the crucial part of information,
887
The submitted paperdiscusses the DPS control problems just in these conditions, which are frequently present at the control of the continuous tech nological processes. On the input/output quantities measured in the finite number of DPS points the paper introduces the X-representa tion of DPS. By means of this repre-sentation are solved the nonlinear DPS adaptive control problems for various amount of a priori information. At the same time there are employed some results of A.V.Balakrishnan and H.C.Hsieh related to multivariable lumped systems. The paper indicates a possibility of a priori information transformation, expressed in the linear partial differential equations form, to the X-representation structu re and calls attention to of utilizing the extensive A.G.Butkovskij's/1979/ summary of DPS charakteristics. It solves the decomposition of controlled plant, which can be comprehended as interconnected set of nonlinear DPS. Finally, there is demonstrated the applicability of suggested approach on some results of hierarchical cont-
888
C. Hulk6, B. Rohal-Ilkiv a nd
rol 0 f t:'1e technologi cal /glass/ fur nace physical model. I.
X-REPRESEN!ATION OF THE DIST~IBUTED ?AR~METER SYSTEMS
E
x ... y..
IIkn.LII~O 100
I I 1
I
I Xax( m,c)
z
Fig. ,. Input/output quantities of DPS.
Sapak
In these points let us determine the following curves x(l',t) , ••• , x(Li,t) , ••• ,y(kj,t) , ••• ,y(kn,t) on the input/outp~t functions from X and Y designate them as vectors of functions ~ (m, t)::
For simplicity of further considerations let us assume that the control led physical DPS is distributed on
L.
LX (L,
, t) , ••• , x (lm, t) ]
y (n, t):: fy(k1, t), ••• ,y (kn, t)] where Li,kj are coordinates of points i,j on
889
On Adaptive Control of Distribut e d Parameter Systems
X~x(m,t)
of input quantities, con:aining the vector i(m,t), by relation FN[.] land vector y(n,t)1 relates sub set ~~y{n,t) of output quantieties from Y, so tha~ the image X~Yi(m,t) of X~i(m,t) is'nX~y(n,t) subset. In the frame·.... ork of X -represents tiO!1 let us define the prob:ems of control synthesis and identification. Unde~ the DPS control synthesis we understand to find todthe given desired output function y (z,t)£Y lat cer tain DPS state/ and for specificed "control quali ty v " the pair [X~x(m,t)j X~y(n,t~ fro~ the suita2 le system of subsets from X so that yd(zlt)E.X~Y(n,t) and max IXBy {n,t)1I y~V The a~~Grmination of the suitable subsets system from X-structure of type X [xm(z,t); iI • Ilx~jFN[ • ] j Yn(z,t) j 11 • IlyBJ- which suits the control synthesis problem we will call the identification problem. I I. ADAPT r{E CONTROL
Let us solve the control synthesis problem for the nonlinea~ DPS at the gi ven desired function y (z, t) and v value. In the X and Y let us consi6eJ.· Tchebycheff norms. During the solution we will employ various assumptions-sometimes only for illustrative purposes-which cadprise from a priori information l i f there are any availab lel or from current information about the controlled process. Let us consider that the X and Y belong to the certain classes of functions-Collatz-Krabs/'97J/j Dzisdyk 1'977/; Laurent/'972/j Nikolskij/19771 etc. Let us consider further, that to V/k valuel "k" will be determined laterl -for certain DPS state-it is possible to determine ~(v/k) value in X so that when 11 x' (z , t ) x 2 ( z , t) 11 ~ ~ ( v Ik ) /1 1. , I then 11 y' (z,t) - y2(z,t) 11 ~ V/k /11.21 when {[x' (z,t) j y' (z,t)], [x 2 (z,t); y2(z,t)]tR and at the same time the superposition principle holds in the small sourrouding of [x(z,t), y(z,t)J pair from R. In the X let us choose the x (z,t) polynomial/the degree and mWasurement points location "m"/ so that it aproximates with the error s~aller than ~(v/k) every functions from X, !hich contains the polynomial vector x(m,t). Similarly in Y let us choose Yn(z,t)
polynomial Ithe degree and the ~easu points locations "n"/so that it appr oxima tea ',d th the error' smaller than v/k every fu~ctions from Y£ which contains the polynomial vector y(n,t), see relation IT. J/. Supposing that xm(z,t), Yn(z,t) polynomials also belcng to the certain classes of functions Ige::erally different from above mentioned/. 1 Let be x' {z,t)=x (z,t)"then the image of x (z,t) in~Y is y (z,t). In accordan~e yith previous consider~ tions to y (z,t) let us assign y (z,t) so, that n !I y' (z, t) - y ~ (z ,t) 1I <: v Ik III. JI Because the ~ ( v Ik) surrou!1ding of xm(z,t) images to the v/k surronding +\ +h" .&.'X-( 0_of' y 1( Z,MI, ~ en 1mage o~ ~x m, t ) " 19 situated in the 2(v/k) surronding of
re~ent
y~(z,t). Ba3ed on the Frechet generalized Weierstra3se theorem the relation/I.51 for t»Ts we approximate by the N deg ree functional power series : a m~' N Ts Ts Y (j, t)= r r 5 ) i= 0 h= 1 0 0 r (1 '+ T1) ••• X\l, " t -T h,'J'"c;T, ••• d T = Wh ji LX ,"h
N(=P WN j !!!) X!!! ,
/II.4/
where xm = [00, x (m, t) , oLJ:; [x (0, t) , x (, , t) , : •• , x (m , t) , x ( m ,t) J ,!!!= m" 2 , - , -- LwN r.: ' , , ••• ,wN ", , ••• ,wN, J 1, wN J!!! J J1 J!!! wN ' , = [w (T, ) .. , ••• ,w ( T, , ••• ,T ) .. ] N J1 J1 J1 and Ts is a finite setting time. Let U3 introduce the functional t L
+'
I:.~
~ {B~(z)""1LPN(WN1m)xm-Y(1,t)J2't'
t-T 0 J2 + ••• +B22() z .... 2 [PN(wN )x y(2,t) 2m m - )x -- - y(n,t) J2 }dz dt ... B2() z .... [N P (wN n n nID m / II. 5I Let be
/II.6/
Q =
o
..
Ill. 7I
,0, ••• il'nQn
....i - weighting funetions, Q is an n x n positive definite symetric
G. Hulk6, B. Rohai-likiv and
890
I ::
S
tT~t
eT Qed t
I l l . 81
t-T
be given le.g. as the result o~ thW succesive determining of optimal input ~uantity during adaEtive control by Ht" time/.Instead o~ y(n,t) we sub~ti " lte into the func tional /II.SI the ~E(n,t)-vector of measured output quantities. It is the true output vector corrupted by additive noise E (t). E(t) includes the measurements errors in observation the cutput and the effects of random disturbances entering into the DPS. Then let us denote the arising functional as I n ...." Mihimizing the functional InA relating to Vol~erra kernels-Hsieh/1965/-so, that Ily~(z,t) - ~';:z,t) 11 ~vlk III.91 where
Let the vector x
y~ (z, t) =yE { , , t)8, (z) ... y E (2 , t} 8 ( z) + 2
111.'01 n (Z) y~(z,t)::ya(, ,t)B, (z)+ya(2,t)B 2 (z)'" + ••• +yE(n,t)3
+ ••• ~ya(n,t)Bn(z)
./ e
matrix
Sapak
of IIl.51 type
matrix.Then t
L.
111.1'1
obtain the Volterra kernels
vm, O,vlN, 1""
,wN, ,m+1
wN 20' wN 2' , ••• ,wN 2 ,mT
.
~,Q,
~=
,
0 , ••• , 0
.o
Let us minimize the functional InAD relating the Xm - Hsieh/196S1 -so,that I1 y~(z,t) - y~(z,t) 11 ~ vlk III.161 whe::e y~(~,t) is determined by the optlmal xm vector. According to the /11.161 and so 11 y~(z,t) - yd(z,t) 11 ~ vlk IIl.,7/ and a 11 y(z,t) - Yn(z,t) 11 ~J v/k IIl.'SI where y(z,t)~Xa(V/k)Yx(m,t) then
1~~_!~~g~~_Q!_!h~_Qi~~£i2~~~Q_iDE~!
S~~~1i~i~~_Q~~~£~i~~£_~y
---
11 • 11
J ( V Ik )]
III.121
IIl.DI
i~-~~~-~i~~~l~~~~!2~-~!-!UE¥~-g~~~~i= ~les
- measureo ln m pOln.s from interval with coordinates {ti} - £~!~~~~ - by means of Volterra kernels matrix, the values of ya(n,t) vector in the points "n" with coordinates {kj}, the polynomials B.(z) and the value of the no~m 11 • 11 3evlk) ~~~_~~E!~~_~1~~~¥ly-i~~~11-!hi£h-£2~:
tne-
.
~re~nulnt·
Y~£!Q£ -
of the-subset their images y(z,t)EX3(vlk)y (n,t)cy --- ~~1i~fY 11 y (z, t) - y (z , t) 11 ~ 5v Ik 1I1. , 9I and when k=S 11 yd(z,t) - y(z,t) 11 !:.v /11.20/ So under the assumption, that relation of structure X[xm(z,t)j 11. lIa(vIS)j pN[WNn!E{')];
y~ (z, t) ; 11 • 11 3 ( viS) ]
y ~ ( z , t) ;
im
Yi~~S2 elements Xa(Vlk)x(ffi,t)E~aand
from
-----------------------
IIl.1SI
iu~~~~1i~y
'
. WNno,WNnl,···,WNn,m+'
InAC:: 5 e,T ~ e' dt III. 1 41 t where N " d e'/ [p (wNj~)X~ - ~l (j,t)] and
~
~~!U!L~h~_~·~.f~~!:_~L~~HSie~2!_12~~!.
Distributed output quantity yd(z,t) in the point~ "n" with coordinates {kj} gives y (n,t) vector. Supposing that relations of the /11.131 structure holds on the (t, t+~t) interval too. Let us consider the functional
Ill. 2 , I
(t-T,t) interval holds on (t,t+~t)interval too, in the sence of relation /11.201 the output quantity y(z,t) with error smaller then v will a8proximate desired output quantity y (z,t). With regard tG~~umptiorr, which will be mostly used for illustrative purposes it should be noticed, that e.g. assumption about inputloutput quantities-namely that they belong to certain classes of functions-is naturally not necessary, see part I and V. In usual industrial conditions mostly a simplier a priori information can be also sufficient. For example from the manner of introducing the input quantities lin som~~r.S~~~~·sl is often possible to de~~rmln~~f x (z,t) in which practically the distrib~ted input quantities are found. For deter mination of the relation between Vlkand above mentioned x (z,t) surrounding may be often sufficient also sta tic measurements le.g. during opera-tionl realized directly on the plant.
On Adaptive Control of Distributed Parameter Systems
But in the real situ8ticns we eften have so a priori information that we can use such assumptio~s which considerably decrease require~ents relating to mln numbers,E.g. if in the di rection of NZ" axis in.lout. quant.hi ve character of polynomial of 2,3,4.:. degree,at the stagger distr.of mln,the m=n=3,4,5 ••• provides high accuracy of approximation. It is similar to the Tchebycherr norm which we used mainly to illustrate our aims. Naturally, every consideration ceuld be dene e.g. for ~p norm etc. By shiftir.g the intervals (t-T,t), (t,t~~t) along the time axis and by succesiv~ minimizing functionals of type :!:n; , Il".AC so, that at the same time the presence of the appropriate structures of type 111.211 is secured-when relation 111.201 holds-I at the assumption of complete identificationability and controllabilityl we realize the adaptive control of DPS. At the assumpticn, that in the course of adaptive control appropriate struc tures of type 111.21/ are secured - satisfaing the g{~aH~~~~t~~l of DPSit "reduces" to adaptive control of multivariatle lumped parameter system. It "reduces" to succesive minimization of functionals of type InA, InAO at the shifting of intervals (t-T,t), (t,t~~t). At these conditions we can use fer solution of the adaptive cont rol problem of nonlinear DPS e.g. A.V.Balakrishnan's and H.C.Hsieh's re suIts refering to the adaptive control of multivariable nonlinear lumped parameter systems by function space met hods. The minimization functionals or type InA, InAO was solved by H.C.Hsieh 119651 by steepest descent method in corresponding adjoint Hilbert space 2f inputloutput vectors o~ type x(m,t) y(n,t). Beside the analysis computational requirements of optimization problems H.C.Hsieh/19651 presents also a special computationally very effective method to matrix of Volterra kernels on-line determination. I
Ill. TRANSFORMATION OF THE A PRIORI INFORMATION At the concrete control realization there is often at our disposal some a priori information about structure andlor coefficients of linear partial differential equation, which roughly approximates the dynamics of controlled distributed parameter plant lin the surrounding of certain operation regime etc.l. The solution of this linear partial differential equation at the zero initial and boundary conditions let have the form :
L
t
y(z,t)= )
)
o
891
G(Z,~,t,T)x(~,T) d~dT
IIII.11 is the Green function. ~hen Y(J,t} the output corresponding to xm(z,t) in "j" is t L y(j,t)= 5 S G(j,~,t,T) [x(1 ,T)L 1 (~h
:here
0
G~Z,~~,T)
o
0
..... +x(m, T )L,J~)]
d~dT=
L
t
= ) { x ( 1 , T) [S G ( j , ~ , t , T ) L, ( ~) ~ ~ ] •
.: o
0
L x(m,T) [S G(j,~,t,T)L (~) d~! dT o m lIIl.2/ when L S G( j , ~, t , T ) Li (~) d ~ =g ( j , i , t ,T ) o IIII.3/ then t y(j,t)= S [x{1,T)g(j,1,t,T)+x(2,T). +
o
.g(j,2,t,T)+ ••• +x(m,T)g(j,m,t,T)] dT <=FNj[oO=O,x(m,t), oL=O, p10=0]> /III.41 The functions g(j,i,t,T) are elements of . Vol~erra kernel matrix for DPS, whlch lS represented by the given linear partial differential equation. /'
W, nm
=
~(1'1,t'T), ••• ,g(1,m,t'T)J • • .g(n,1,t,T), . • • ••• ,g(n,m,t,T)
IIII.51 these relations.a~d by utlllzatlon e.g. A.G.Butkovsklj s 11979/ summary of DPS charaeteristics, which containes Green functions to mo re then 500 boundary value problems,we can transform a priori information, expressed in the ~orm of lin.partial differential equations, into the struc ture of the introduced X-representa- tion. St~r~ing.from
IV. MULTILEVEL DECOMPOSITION OF DPS To the controlled distributed parameter plant it is many times possible to relate interconnected set o~ nonlinear DPS DISTURBANCES
tl~t.cr~· ~~f.U~l~~OU~~Yl l.l Output ~J~~ trans-
D P S
~
form.
3 YJ T
P
U T
Fig. 2. Interconnected distributed parameter systems.
892
G. Hulk6, B. Rohai-likiv and
XG={Xl, ••• ,XI, ••• ,XH}, XI is the set of I-th admissible distributed input quantities XI={xI(z,t)}. YG={Yl, ••• ,YJ, ••• ,YK}, YJ is the set of J-th corresponding distributed output quantities YJ={yJ(z,t)}. UG={Ul, ••• ,UJ, ••• ,UK}, UJ is the set of J-th state quantities of the int erconnecteC set of DPS,UJ={uJ(z,t)} E'}={El, ••• ,£I, ••• ,EH}, El is the set of I-th distributed disturbances. For simplicity let us assume, that UG=YG and E}=O, respectively. Now to single DPS let us determine polynomials of type xm(z,t) and y (z,t) so, that together with relat£ons : ---yyn=FNG(Xx m) when /IV.l/ YY;'= [ Y1n' : •• , yJ n ' • • • , yK n ] /IV • 2/ /IV.3/ xXm= [X'!!)"" ,XI!!), ••• ,XH!!) ] FNG(xx m) =
FNK1(x1 m),
possibilities of proposed approach to solution of distri~uted parameter sy~ tems adaptive control problems. The dynamics of the technological /glass/ furnace was investigated on the physical model see Fig. ~, Kacha nak et al./1975-78,1975-80/, Sapak fin preparation/. Experimental apparatus and its mathematical model in the form of three nonlinear partial differential equations was described in Hulko, Sapsk /1980/.Starting from the choosen ope ration regime the model of the appa: ratus can be consider as an intercon nected set of nonlinear DPS Fig. 3.Power input l.DPS on outside surface of heating cylinder
FNJI(Xl m)
I.DPS
2.DPS ,FNKH(XH m)
/IV.4/ where yJ and xlm are vectors related to surfances: yJ(z,t) and xI(z,t) from the sets XI and YJ b~ m/n points of interval <0, 1>. FNJI(xIm) = =FNJI [oOJ, xl m , oLJ J, where oOJ and oLJ are corresponding boundary conditions. --- Sometimes the interconnected set of DPS structure may require a finer decomposition. For example between the J-th output and I-th input there is a "block" of n serially connected distributed parameter subsystems.Then at the fullfilment of the citated assumptions the further stages of deco!!) position arise: - =••• FNJIn••• 0FNJI T-xlm ] ••• / IV.5 / yJ n where T FNJI [.], ••• ,FNJln [.l correspond to single elements of serially connected "block"-at the given state and bound~ ry conditions.--arise structure of type /11.13/ in which-at the fullfilment identificationability and controllability conditions-similarily as in part 11, it is possible to realize control by given desired output quantities and at the required "quality". V.
Sapak
Temper~ture
FNl 1 (Xl m) , • ••
i.
ILWSTRATIVE
EXAMPLE
Temperature on inside surface of heating cylinder
NOi=i,;S; I 4.DPS
).DPS
] 2.DPS
Temperature of glass Fig. 3. Decomposition of the physical model On the basis of modelling studies and experiments on the physical model the structure in Fig. 3. can be approximated by system Power input xl I.DPS yl Temperature on inside surface of heating cylinder x2 d ~.DPS
Yrr Temperature of glass
Fig. 4. Approximation of the decomposed model. By some results of technological/glass/ furnace physical model hierarchical At the constitution of the control control we suggest the applicability
89 3
On Adaptive Control o f Di s trib u t e d Parameter Systems
system let us start from the structu re in Fig. 4. Let us relate to it tne relation of type /IV.5/ YII(n,t)=FN~[FNT (x1 m)] IV.3/ where FN~ [.] corresponds to ~ .DPS IV .4/ T FN (.) corresponds to T.DPS IV.5/ We investigated the possibilities of control partly in a small area near an operation point-linear approximation, LP-partly at the transition be~ ween the operation regimes-nonlinear approximation, NP. In the LP case Vol terra kernels matrix contains 1th or der kernels and in the NP case con-tains also 2nd order kernels. The identification was realised in se veral stages. At the structures choice to t and ~.DPS of type /11.21/ we 3tart with the requirement that on ~.DPS v=3% from mean value of the de ,ired output quantity. We exploited;.: artly a priori information,conta~ned in models Fig. 3., 4. , what 'Ne Slm~ lated in an oriented language SIKOS by digital computer SIEMENS 4004 and partly by measurements on physical model. - For polynomials L, (z) and B, (z) were choosen Lagrange1interpolaJtion polynomials. - We choose measuring poin t nu~e~ for ~P m=n=5 and for NP m=n=6 by~t~6~r distribution along the apparatus. - At the Volterra kernels starting va lues determination we exploite A.G. Butko"Skij's /1979/ summary of DPS charakteristics. - Volterra kernels determination was realized on two levels: * On the first level by minimizing the functional of type InA from the Volterra kernels initial values we determined the T.DPS Volterra kernel matrix. * On the second level we proceed similarly as on the first level, but wi th a differeT"ce, that model in put in InA for ~.DPS was computed from model output of the T.DPS < e.g. for j-th component of ~. DPS in theLP c~ se ~ T , { ••• +w 1 ..\ [ ••• +w 1 .\ [x ( 1 , t) ] + ••• ] + ••• }>J1
11
In the Fig. 7,8. are presented some elements of Volterra kernel matrixes. On the first and second level of Volterra kernel matrixes determination we used the time courses of input/out put quantities, obtained from experi: mental measurements on the physical model. The experiments were realized with Automatic Data Acquisition System HP 3050 B in Czechoslovak Institute of Metrology in Bratislava, connected with digital computer SIEMENS 4004 for the realization of identification and computing of optimal control ,Fig. 5, (:"C:. T?
... .
The aim of control was to take, at the given state, the output quantity from value Yr to YIr and from YJII to y /LP and NP case/, Iwe al'Nays as sumeT{complete iden:ificationability and controllability/, at the minimum energy. In our case-for LP- it means the minimization o~ functional: min fj,t I= 1 ('), t) [ ( ) e~ ~ d + 11 x1 (5, t) 11 ~ t: (O,fj,t) 0
x
e
+{i~e{~~t) [fj,h} ~e~+ 1I (O,fj,t ,
0
x2 d (5,t)
~
lI~tJ }]'" /V.6/
The energy requirements denote values \I x1(5,t) ilK and 11 x2 d (5,t) ilK ,when e.g. 11 X1(5,t) IIK=[x1(1,t)L1(z), ••• ,x1(5,t~. .L,,(zL,.
"~_1(t) 91
, ••• K2 (t), 0·
J
6) . , ,";K~{t)
~'(1,t)L1(Z~
, x 1{ 2,t)L2 (z)
~1(5,t)L5(z)
-In 1 ~ T)d(')l ed'l: (w1 J ' 5_x15 J _ - xZ J,t~ /;1
ef'=~ J
where
{~ ,. )-2d
( '
)il
w1j5_x_5 - YII J,t~
are kernels of
IV -- matrix/si positive definit/ of weighding functl.ons. x1(0,t)=y1(O,t), ••• ,x2 (5,t) =YTI(L,t) are boundary conditions on the transition interval (O,fj,t). When determining the optimal control of these serially connected systems we start from the M.I}.Singh's and J.F.Coales's/1975/ ideas. On the second level of control there is minimised the functional I?,as the problem of final value and minimum energy. The result is the function X2~(z,t). To this function-as to the desired output quantity-on the first level of control then there is deter mined optimal input x1 (z,t) for i.DPS by minimization Of 5 I 1 functional. Functionals of identification and syn thesis of control were minimized by method of the steepest descent in cor responding adjoint Hilbert spaces ofinput/output vectors. The method of optimization was taken over from H.C. HSieh 's/1965/ monographs. If conditions of adaptive control from part 11. hold, the described offline hierarchical control of model T, and
metri~
~.DPS subsystems.
894
G. Hulk6, B. Rohai-likiv and
furnace on inte~vals (t-T, t), (t,t+~t) car. be consider as k-th step of adaE tive control. The arisen adrtive control system ha ve only u3ual computational re~uire= ments, that is why its exploitation to adaptive centrol of slowe~ techno logical processes than DPS-3 shouldnot m~et with greater p~actical obstacles not even under current engineering conditions. CONCWSIONS Naturally, in the paper there were studied only fundamental properties and possibilities of the proposed aE proach to solution of the DPS adapt~ ve contrel problems. In future it will be necessary to de vete the attention mainly to DPS i-dentificatienability and controllabi lity problems in the fraffiework of tne X-representation with the aim to de termine principles of constituing optimal adaptive control systems in dependence on controlled distributed parameter plant and its environments uncertainties /the amount and kind of a priori information, the possibi lities of obtaining the current in-formation etc./. REFERENCES Butkovskij, A.G./1979/.Characteristics of distrinuted 2arameter-sistems:7in-russian7~auKa~-Mo3cOw:2~j-pp:
Collatz, L., Krabs, W./197J/. AE2ro~1~~~~2~~1h~2r1;_!~£b~2~~£5~!r:Scne AE2roxlmat~on m~t Anwenuun-
g~5~-~·G:Taubner~-Stuttgaro:27Tpp.
Dziadyk, V.K./1977/. Introduction to Theorv of Uniform-Annroi1mat1onrunct!ons-Si-~oIinoi!als:71n-ru: ssian7-~auKa;-~oscow:-;TO pp. Hsieh! H.C./1965/. §~~~h!!1! 2f_~~aE tlve Control Svstems bv functlon ~Ea£e:~~sfi§~s:~in-Aava~ces-in-~on1 roI ~ystems ~fieory and Applications /Ed./ C.T.Leondes. Academic Press, N.Y. and London. Vol.2. 117-268 pp. Hulko, G./1977/. Contribution to The orv of Distributea-~arameter-Sys te~s:7generaI1zea-moaeIs~-aaapt1 ve-control/. Ph.D.Thesis./in slo vak/ Slovak Technical University, Bratislava. 147 pp. Hulko, G./1979/. On Nonparametric ReE resentation of Distributed Parameter Systems. Preprints of 5th IFAC Svmnosium on Identification
~5~:~~£~~~~~~:~~!I§~!I25~~armstadt
Hulko, G., Sapak ~./'980/. On Multile vel Decomposition of DistributedParameter Systems. Preprints of ~n2_JE~Q_§l~E2!i~~_Qn_~~g~_§£~!~
L.
Sapak
SV$tems:Theo~~
and
~nElications.
T~uIouge:----~------~
----------
Kachahak, A., Belans~y, J., Hu lko,G., Rohal-Ilkiv, B., Sapak, f./1975i980/. Models of Glass Furnace I., JJ./in s!ovaK7-~esearch-report-- unaer the contract with Slovak Works of Technical Glass, Bratislava. Kachanak, A., Belanskj, J., Hulk6, G., Rohsl-Ilkiv, B., Sapak, ~./'9751980/. Identifi catio n Pr oblems of Adantive-~ontroI-o~-~istribu
~~g:r~r~~e1~£:Si§!~~s~-nesearch-
report uncer tne contract with Insti tute of Informatior, Theory end Automation-Czechoslovak Academ;;- of Sciences Frague. Bratislava. Kornej~uk, N.P./1975 / . On Extremel Att ributes of Splines. Froceedings of !n1i~~~11gD~1_£Q~gr~~§ 9D ~EErCXlma.lon o~ runct~ons. Rau~B, Moscow. 440 Pt>. Kornej6uk, N.P./1976/. Extremal Prob 1~~~.2f ~2Er~~i~ali§5:~5eo£i:/in russlar.7 Nau~a, 9.oscow. pp. Laurent, F.J./1972/. AEErcximation et
"0
QE1i~i~!1129. Ferien~-Pirri;19;p~.
Moore, E.F./1956 / . §~!C~l!r~!!E~!i;_ ments on SequentlaI Macr.lr.es.7~n russIan7-in-Avtorna{y-7ro:7-~hS
nnon, C.E., M~ Certhy,J. IL. Moscow. Nikolskij, S.M./1977/. AEircximetion Functions of Several vari5fes:7In russran7-~auka~-~osccw~-!~)PP;
Saplh, L./in preparation/. AdaEtive Subontimal Con t rol of DIstrIbuted ~araieter-~isterns:-~h:~:Tnesrs:- ~IovaK-TecnnrcaI-University.
Rratislava. Singh, M.G., Coales, J.F./1975/. A he uristic approach to the h~erarchi cel control of multivariab:"e se-rially interccnnected dynamical systems. Int.J.Control,Vel.21 , No.4. 575=;S;-pp:-----ACKNOW".uEDGEMENT
The authors wisches to express his thanks and appreciation to PROF.J. S k k a 1 a - scientific director of Czechoslovak Institute of Metrology in Bratislava - for kindly enab ling the realization ef experimen~s in Institute a~d the first fro~ authors to Ph.D.L. S u t e k for SUE porting through many years his research in the theory of automatic control.
a
895
On Adaptive Control of Distributed Parameter Systems r
1\ i
tf"ater
InsulHlon
'1
r;~~t:-Da~ I:CqUl:-;.:J I System HP 3050 B I
q Cool ing air
I r----, I '---l<:--'
In
_ ._ ._
.
H".tinx . Cooiing
Heatmg Wires
.
t-
.r.?V
I
1
Glass bar
I
1.
I
16.
1S .
1.
2.
16 .
17.
Thermocouples
Fig. 6. Axial cross section of the experimental apparatus. 5. The flow of the information during the experiments.
F~g.
Digital computer SIEMENS
1 -1 23 . w - f rom exrCrtment _ ._
4
WIT - f mm mat. mod,,1 ZJ inirial value - 11 . w 2.3 - f rom experiment ",11
_ from mat.modd
initial value
2J
OJ-____
~
______
L __ _ _ _- L_ _ _ _ _ _L __ _ _ __ L_ _ _ __ _
10
)0
20
40
~
TEni~60
50
:ig. 7. First order kernels.
Measuring points for NP approximation 2
4
6
1
l
1
Heating program [A] . NI> approximation Q25
Q24
T CC] 019
19 )9
9
6S0
600
~
~
202 021 021 l5 a5 25 :24 4023 Q23 6 l'l ~9 C\I9 19 16 6 16 6016 ,16 Finite ~tate o Measured I'rofil" ~ ~ I;'NP
5l
0
550
~
X
X
500
~
'g
><
450 Initial 0
0
350
0
0
0
0
state 0 0
0
0
X
Ym
)(
0 0
200
Finite
100
'g
~
0
0
t
11
state ~
0
'0
LP 'a
..
Y"![ 'is 'g
0
Initial
0
50
-0
0
150
Fig. 8. Second order kernel.
~
X
0
250
Initial state
'g
NP
300
,;tate
YiJ.. X
X
Finite
6
>< Desired
400
m ,n
3
0
0
0
0
Itate LP 0
0
0
0
0
Y1 o
0 0
Healinl: prO):rarn [A] - LP approximation O,09-a power lnpu - a power In u 0,
0
62 124 186 248 310 372 434 496 558 620 68:2 744 8061W1
t 2
3
Fig. 9. Measured and computed profiles.
t
4
Measuring' poinu for LP approxi m~tion
0
Finite state Initial state
930~2
t 5
·H m,n
896
C. HulkS, B. Rohal-I1Kiv and
1.
Sapak
Discussion to Paper 31.2 K.J. Astrom (Sweden): Were the experiments mentioned in your paper made on a laboratory process or on an industrial plant? G. Hulko (Czechoslovakia): The experiments mentioned in the paper were made on a laboratory process.