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Algorithm for Optimization of Large-Scale Control Systems in Conditions of Uncertainty and Applications
ALGORITHM FOR OPTIMIZATION OF LARGE-SCALE CONTROL SYSTEMS IN CONDITIONS OF UNCERTAINTY AND APPLICATIONS F. Stanciulescu Research Department, Central Institute for Informalics, Bucharest, Romania
Abstract. The optimization problem of large-scale systems, composed of a set of interconnected subsystems, is approached in a new manner, with algorithmic finality: the global model of the large-scale system is decomposed into interconnected submodels, with minimal interaction, which are sequentially aggregated. The aggregated global model, obtained by concatenation of aggregated submodels is solved, the resulted values being used as reference values for the local controller (at the level of the subsystems, for which the subproblems can be solved). Using an uncertainty principle, the uncertainty which proceeds from the decomposition of the large-scale model (unsolvable) in interconnected submodels and of the aggregation of them, is evaluated. As examples are given: the control of a large-scale technological process, and the management of the production in a large-scale industrial system. Keywords. Large-scale system; interaction; non-linear system; cannonical model; decomposition; linearization; aggregation; submodels concatenation; dezaggregation; reference value; uncertainty.
THE CONCEPT OF LARGE-SCALE SYSTEM
x(t) = Ax+Bu+f(x,u).tw(t), (2) proceed from the separation of linear and of non-linear parts, in the natural model, are large-scale matrices, the vectors x and u having large dimension too. A consistent and less ambiguous characterization of a large-scale system can be made using the following: Definition: "By large-scale system we understand 8 system S, complex as struct ure and objectives, composed of a set of interconnected sUbsystems Si :
The large-scale systems characterization can be made by means of the cannonical form of their model [12]. Indeed, let be : x(t) = F(x,u) + wet), (1) (where: x(t), denotes the n - dimensional state vector; u(t), the m dimensional control vector; wet), the' 1 - dimensional vector of the interaction system-exosystem), the natural model of a large-scale system, written in the language of nonlinear dynamic systems. From a formal point of view a large-scale system is characterized by the fact that the matrices A and B, from the non-linear cannonical model*): i) A direct way to obtain the cannoonical form of the model is that to consider: f(x,u)=F(x,u)-Ax-Bu, where A and B result from the Taylor's series development about the nominal point of operation (xo,uo) :
'OFI
0
A=~X,
Si~Ui,Xi,Vi,Wi'Yi'~i,gi,hi,T),
(3)
(i=1,2,3, ••• , n), and a co-ordinator sub-system" , where: Vi is the control (or input) subset, of the i subsystem; Xi' the state subset, of the i subsystem; Vi' Wi , the interaction subset of the i subsystem with the other en-I) subsystem, respectively with the exosystem; Yi , the output subset of
UFI
0 B=15Uu.
603
604
F . Stanciulescu
the i subsystem; ~i' the transition function of the i subsystem; gi' the interaction function of the i subsy~ tem with the other (n-l) subsystems; hi' the output function of the i subsystem; T, the time moment subset. The functions ~i' gi and hi are defined as following Ui x Xi x Vi -+ X·l. If'i gi a UX j ~ Vi' hi : Xi _Yi j;li 11 we take into account both characterizations and introduce the interaction Vi' between the subsystem i and the other (n-l) subsystems, under the form :
system optimization problem is formulated as following : min xi' ui
{J
=
t
1= 1
Ji (Xi'u1 )},
xi=Aixi+Biui+vi+fi(xi,ui)+wi' Xi (0) =
xf '
(6)
(7)
(8)
n
Vi ="')' gi'(x,) ,
Vi
J
J
(l0) (Xi' ui ) ~ 0 (i = 1,2,3, "" n) the optimization subproblem for the i-subsystem can be brought to t he form [111 :
ri
n
Vi = g. (x) = ) . gij(X j ) , l. j= 1, j/i (4) (i = 1,2,3, •.• , n) , the cannonical model of a largescale system, results as a reunion of cannonical submodels of the form: x.(t) l.
= A.l. x.l. +B.u. l. 1
+L giJ'(x ,) + j;li J
= 1,2,3, •.. ,
(5)
n).
The method which we present in the seq uent sections is based on the philosophical approach of largescale system, which recommends to get ove r the following stages : Stage 1: The large-scale system obj ectl.ves are set up. Stage 2: The large-scale system is decomposed in n interconnected sub systems. Sta~e 3: The qualitative and quantitatl.ve properties of the subsystems are characterized by means of adequate methods and techniques. Stage 41 The qualitative and quantitative properties of the global system are infered from those of the subsystems and from the interaction structure (by means of specific techniques) • DECOi,iPOSITION OF LARGE-SCALE SYSTE1;S IN HTTERCCN1\TE;CT ED 3UBSYSTE&;S The decomposition of a large-scale system in interconnected subsystems is equivalent with the decomposition of the optimization problem in n optimization subproblems (one for each subsystem). If the large-sca l e
= Aixi+Biui+Vi+fi(xi'ui)+wi
Xi (0) = X~ , n
Vi
n
+ fi(xi,u i ) + wi(t).
(i
.
Xi
=J. #i
gij (x j )
,
.
r i (Xi' ui ) .(. 0 Ano t her methods of decomposition of the large-scale systems are given in (1) - [5] • LINEARIZATION OF LARGE-SCALE SY3'fEt"J Come ba ck to the cannonical model xi:; Aixi+Biui+Vi+fi(Xi,ui)+vi (i = 1,2,3, ••• , n) where : n Vi
L j /i = j=l,
gij(X j )
.
(ll)
If we know the opera tion conditions of each subsystem these can be linearized and subst itued for subsystems Si. Finally, we, obtai~ a new, linearized syste ~ (S ), having the same topology as S, and which substitutes for the system S. Intuitively, S' is expected to be t he cannonical linearized repre sentation of S. The linearizati on of the non-linear cannonical mod el (5) is the equivalent of the lineariza tion of nonlinear function fi(xi, ui ) and of the interactions gij(X j ), by Taylor's series development ab out the nominal operating point (xo,uo), neglecting the higher-or der terms. The resulting
Algorithm for Optimization
linear cannonical model is xi (t) = Ai xi (t) + Bi ui (t) + vi (t) + + f i (t) + wi (t ) , (12 ) where the linearized interaction vi(t), can be expresed as :
r:
On the other hand from (21) it results, by means of relations (17) and (18) I
(13 )
AGGREGATION OF LARGE-SCALE SYSTEM We consider a large-scale system, composed by several interconnected subsystems, each subsystem i being represented, about the nominal operating point (xr, uf) by means of the linear cannonical model (14-) xi = Aixi+Bi ui+vi+fi+w i , (15)
n
V.
1.
=
I: j;ii
A.. x ' 1.J J
,,-
+ C~Difi+ciFiwi '
where: Aij (i,j = 1,2,3, ••• , n) are the interaction matrices. Observation. The linearization of the large-scale systems about the nominal operating point (XO,U O ), valid in certain conditions, is indispensable from technical point of view,because in the real operation the nominal values are the most used.
xi (0) = xf '
n
#
:#
xi = CiFiCiXi+CiGiui+~iCiAijCjXj+
n
v . ( t) = A. . x ' ( t ) 1. J;ii 1.J J
605
(16)
The aggregation of the system (14-) (16) is equivalent with the substitution of the state equation (14-) and of the interaction (16), with another state-equation and interaction of reduced dimension, but which described the behaviour of the system in the same manner as the model (14) - (16), (6) - [8J. We introduce the aggregation matrices Ci and Cj and the aggregation vectors zi and Zj (of dimensions li.( n i and Ij
where Ci represents the pseudoinverse of Ci • By comparing the equations (22) and (14), it results ... (23) Fi = CiAiC i , Gi = CiBi , to' Aij = CiAijC j , Di = Ei = Ci and finally the aggregated state equations of the i subsystem I •
t
to'
Zi = FiZi+GiUi+Vi+Dif i +Eiw i , (19) where : n (20) vi = L A1.' J' Z J' , j;ii or by introd ucing of (20) in (19) : n
Z.=F.Z.+G.U·+LA .. Z .+D.f . +E.w i , (21) 1. 1. 1. 1. 1. j;ii 1.J J 1. 1. 1.
#'
zi = CiAiCiZi+CiBiui+j;iiCiAijCjZj + + Cifi+Ciw i
(24)
THE AGGREGATED SYSTEM If the procedure which leads to the re~ation (23) ,i~ sequentially appl1.ed of the 1.(1.=1,2,3, ••• ,n) subsystems, the aggregated state equation of the large-scale system is obtained by concatenation of the similar terms, in the equations (24), in the following manner I • 0 Q 0 z(t)=Az(t)+Bu(t)+Cf(t)+ wet), (25) where we note
8
z =[zl' Z2'· •• ,Zn]
=[u l '
u
C*
Cl Al 0
A=
I
C2 A2l C# 1
·C·• n AnI
C#C I
·.. ·..
Cl Aln Cn C2 A2n Ctt: n
·..
Cn An
B=
0
C=
0
C2 B2 • ••
·0·
0
'It
0
0
Cl Bl 0
,
u 2 ' ••• , U m]
(18) The aggregated state equation is written I
(22)
iI
·..
0 Cn Bn
Cl 0
0 C2
·..
0
0
0
• ••
Cn
··
0
Ctt: n
606
F. Stimciulescu
In order to reconstitute the optimization aggregated model of the largescale system it is necessary to aggregate the performance index J i and the constraints, linearized at the level of each subsystem : , " J i = Ci xi + Ci u i ' (26) Pi xi + % ui + Ri = t'i (27) Therefore we shall rewrite the performance index aa follows : J i = Ci zi + Ci ui ' (26') and the constraints Li zi + Mi ui + Ni = Ir'i
(27')
Tacking into account the relation (17) it results by identification of the relations (26) with (26') and (27) with (27'), the aggregated forms : t # " J i = Ci Ci zi + Ci ui ' (28) and respectively :
%
ui + Ri = O"i (29) By concatenation the terms of the same kind in relations (2 8) and (29) we obtain the aggregated form of the global performance index and constraints : , " J C z + C u , (3 0) and respectively Px + QU + R = t', where :
=
,
, C=
it
[r
1
0
C2' C211-
o
PICl
Q=
Ql 0 0 Q 2 0
0
o o
·.. '*f
P=
·..
0
C"=
o
0
0
P C#2 2
0
0
·· ...
Finally it results the o?timization aggregated model of a large-scale system : , ""\ min {j = CZ+CuJ' (32) Z,u o
and respectively
P i C~ zi +
(~, the vector obtained by the concatenation of 0i)'
:1 '
o o
·.. ·..
R:
0
0
...
0
(33) (34) (35)
OF THE UNCERTAINTY IN T:IE AGREGATTED SYSTEi:.l In [10) and [ll) we have given an uncertainty principle in the largescale system theory. This principle relies on the observation that the solution of the optimization problem, relating to a large-scale system, depends on the influences of the exosystem in which our system is "plunged"*), i.e. in fact on the intera ct ion between the system and the exosystem. Particularly , this observation holds for the subsystems rezulted from the decomposition of a large-scale system too, due t o the fact that for these subsystems the exosystem includes the large-s cale system itself. The uncertainty erinciple in the large-sc ale s~stem theory. Come back to he state equation • n x.:;;;A.x.+B. u.+ L g .. (x . )+f. +w. ESTl ~~TIO N
1 1
1
1
jii lJ
J
1
1
0
and de fine the uncertainty on the state Xi as the distance ( usually defined by means of the norm) between the optimal (unknown) value x; and the value Xi (det ermined by an approximation procedure) ~xi = d(X;, Xi) = \\ x; - XiII (36)
~
*) 'Ne allways consider our system
p:cJ ...
0
Re mark . Due to reduction of the optimization problem dimension ~ under aggregated form (32) -(35), solving of this problem is now possible, for exemple using an algorithm based on dynamic programming principle . If the problem has a special form {6), we can also nope to obtain, by dezaggregation, the very subvectors Xi and ui • In any case, t he values obtained in this way, can be now used as reference values for the local controller (at the level of the subsystems resulted from decomposition).
1
RI 0 0 R2
0
z(t) = Az(t)+Bu(t)+Cf(t)+Cw(t), z(O) = Zo ' Pz(t) + Qu(t) + R = O.
0
finite dimensional.
Al gorithm for Optimization
The sources of uncertainty in the large-scale systems are variousn~, [111 , but the uncertainty in the state xi proceed objectively from : • the fact that the number of unknown variables in the sUbsystem i (din
mension of x· + L dimension of x..,) 1. jii J exceeds the number of respective state equations ; • the interactions between the considered subsystem and the other en-I) subsystems, including the exosystem (the ignorance of the interactions, the interactions non-linearity etc.). Analogically, starting from the relation which express the connection between the interaction variable vi and the state variables x·(jii), we J define the uncertainty on interaction as thendistance between the value v.*
1.
~ = J+1. . J
.
The uncertainties ~i and I::lv i show the limits to which the concepts of large-scale systems theory can be used. Evaluation of the uncertaint~. Tacking into account the definition of the uncertainty ~ Xi and the relation (17) it results: ..1x.1. =lIx~-x·Il=I\Ct.'(z*:-z 1. 1. 1. 1. 1.. ) 1\ 5--
~
J
11
~ 11
•
\I z~
- zi 11 ;
or,. if we take ir:to account that
~z,=
11 zi-zi ll , we obta1.n :
t1 x i .{ \\
:It-
Ci
11 .llzi.
(8)
An another inequality can be obtained if we start from6v i n
"#
*
/::,v i = 1\ <-Vi 11 =Il~i CiAijCjZj n
tt-
-I: C . A. . C . Jii 1. 1.J J
g. . (x *. ) a nd v., i. e • :
1.J
607
1.
Z •
J
1I
finally :
= I' v~1.
-
tjii
g . . (x . )
1.J
J
11·
n
.,
AVi~ ?l}Cill.\\Aijl\·\\Cjll·..1Zj.
(9)
(7)
Theref ore, if the system is large and consequently non-resolvable on the whole, there is a certain uncertainty, .b.x., on the state x .• If we attempt to1.solve the large-Scale system by decomposition, the uncertainty ..6xi' on the state xi' persists due to the dependence of the solution on the interaction v., 1. on which an uncertainty ~vi exists. As we decompose the large-scale system in ever more interactive subsystems, the uncertainty.6xi decreases (as a result of the decrease of the subsystem dimension), but the uncertainty bv.1. increases (as a result of the increase of the interaction - vector vi dimension), the uncertainty ~xi and.b vi finding in relation of inverse proportionality (eventually nonlinear). All these observations, made so far, intuitivelly give us the opportunity to conclude the valability of the following principle of uncertainty : "In a large-scale system, composed of several interconnected subsystems, the state Xi of the subsystem i, and its interaction Vi with the other subsystems, can be simultaneously determined only up to a certain degree of accuracy".
The inequalities (38) and (39) emphasize the fact that the uncertainty is determined by the agg regation procedure, the minimization of the uncertainties II Xi and llv i being tied to the finding of an aggregation matrix Ci ' ci and cj or a matrix Aij of mi ni ma l norm ( with ~i,AZj computed). THE COMPUTATION ALGORITHM The method of dimensional reduction of the optimization model of a largescale control system, using techniques of decomposition-aggregation, can be finalized alg orithmically in the following manner : 1 st step: Generating of the largescale system global model (of large dimension), as a reunion of interconnected submodels, of the form (5). 2 nd step: Adequately decomposition of the global model in interconnected submodels (s o that the interaction between submodels should be minimal), of the form (7) and (9). 3rd steal Linearization of submodels rezulte from decomposition and of the interaction between them (12), (13) •
4th step: Aggregation of the reszulted submodels and of the interactions
F. Stanciulescu
608
between them, by means of the equation (24) and of the relations (23).
by decomposition-aggregation of management system of a large-scale industrial system. The optimization by decompozition-aggregation of large-scale system occurs when it is not economically reasonable to take into account all state variables of the system, but it is recommended to construct a model after the following criteria: (1) they are retained the all significant state variables (we write c ij > 0 in the aggregation matrix); (2) they are eliminated the non-significant state variables (cij=O, in the aggregation matrix); (3) they are aggregated several state variables in a single variable (by means of the n. 1 l. transformation ~ ~ c ij x j ). iJ J=l In the figure it is represented the bloCK - di~gram of a large-scale system, as hierarchical system with several levels. At the lowest level it is found the controlled process, composed of n interconnected subprocesses - SP i (control loops), every subproces being described by a linear-dynamic model (Non-aggregated Subsystems - NSSi - level). At the higher decisional level (ASSi),
~th step: Restructuring of the glo-
al model (aggregated) by concatenation of aggregated local submodels, equations (32)-(35). 6 th step: Solvin8 of the global optimum control problem with the aggregated model (32)-(35); the obtained solution becomes reference value for the local co~trollers.
7th step: Esti~ation of the uncertal.nty on state and interaction variables at the level of subsystems, by tacking into consideration the aggregation. th 8 step: (Optional) Dezaggregation of the aggregated state and control vectors (z,u) and obtaining of the state and control subvectors (xi' ui )· APPLICATION ON CON'l'ROL OF INDUS'l'RlAL PROCESSES AND SYSTEMS We refer to two applications: (a) optimization by decomposition-aggregation of control system of a largescale process and (b) optimization
A C 5 I z,
~
z2. Zn
A 5 52
~
,
A 551")
N 5 5,
N 5 52
N 5 5n
A 5 5,
•
j
r-:t du,
U1
z;
u~
---
5 P, I ~----"'
R, Figure.
L
J
t,
~
lb-:
-=--;.t J~J2
....
Z2
S P x? ... 2 I
.
~
r
z~
u;-
5 Pn ~ 10..
!I ----.
dun
I R2
I
t2
Rn
En
Block-diagram of a large-scale control and management system, with hierarchical structure: ACS - Aggregated Co-ordination System; ASS i - Aggregated Subsystem i ; NSS i - Non-aggregated Subsystem i; SP i - Subprocess i; Ri - Local controller (regulator) i.
Algorithm for Optimization
these submodels are aggregated, and finally concatenated in a single aggregated model (Ag~regated co-ordinator system - ACS). This last mOdlel sfuPPli~S the*aggrle~at;d opt~mal va ue 0 z, ~.e z = zl,z2 ••• ,znl * 1, and by and u* = {* up u*2 ",., ~ dezaggregation we obtain the optimal values (x;, ~;), which becomes now reference values for the local controller Ri' The block-diagram represented in the figure can be viewed also as the management block-diagram of a largescale industrial system: at the lower level it is found the managed system, composed by several interconnected subsystems. The highest decisional level (ACS) concatenates in a single aggregated model, the submodels aggregated (in ASS i decisional blocks), and solves it, the obtained values (synthetical planning indicators) being furnished to co-ordinated subsystems as reference values for the local production subprocesses. We remark the existence of the control-correction loop (Ri of the deviations from the reference va lues). The method has been experimented in the management and control of a technological process in an enterpreise with discrete production (electrical apparata). REFERENCES 1.
A.N. and R.K. Miller (1977). Qualitative analysis of Large-scale DYnamical s~stems. Academic Press, New-Yor • 2. Singh, M.G. and A. Titli (1978). Systems: Decomposition and Control. Pergamon - Press, London. 3. ~ak, D.D. (1978). Large-scale DYnamic Systems. Stability and Structure. North Holland, New York and Shannon. Michel~
609
4. Findeisen, W. (1974). Multi1eve1 Control ssstems. PWN, Warszawa (in Polis ). 5. Sandell, N.R., P. Varaya, M.Athans and m.G. ~afonov (1978). Survey of Decentralized Control Methods for Large-scale Systems. IEEE Trans. Automat. Contr. vol.AC-23, No.2, April, pp. 10~-128. 6. Aoki, M. (1971). Aggregation. In: Optimization Methods for Largescale s~stems. Ac a demic - Press, New Yor • 7. Bertrand, P., G. Mic hailesco and J. M. Siret (1976). Sur la synth~se de mod~les r~duits 8ar aggregation. RAIRO, vol.l , n o .7, juillet, pp. 105-112. 8. Luenberger, D.G. ( 19 78 ). Aggregatio n and Dual A~g regation of Positive Dyna mic s*ste ms. In: Preprints of the 7t Trien nial World Congress of the IFAC, Helsinki, Finland. 9 . Guran, M. and F. Filip (19 79) . Hierarchic a 1 Systems in t he Chemical Industry. I n: Auto matics, Mana gement and Co mput ers in the Chemical Industry, Editura Tehnica, Bucure : ; ti ( in Homani an , in press) • IO.Stanciulescu, F. (19 77). An Uncertainty Pri nciple in the LargeScale Systems Theory a nd Applications in their Optimization. In: System Theory - Pr eprints of National Confere nce on System 'f heory, Bra$ov ( Homa nia ) . 11.Stanciulescu, F. (1~7 7 ). Largescale Disc r ete Systems 0Btimization in the Light of an ncertainty Principle. In: Preprints of IFAC Workshop on Computer Applications in Discrete Manufacturing, Prague (Czec h osl ova kia). 12.Stanciulescu, F. ( 19 78). ft namic Model and Alg orit hm for he Control of Large-s ca l e Systems with Hierarc hical Structure. I n: Digital Processing and Transmis s i on of Data and Con trol of Pr oc esses of Computers. Pr obl e ms of Automatic Control, n o .1 0 , Bditura Academiei, Bucharest (Romania), pp . 111-122.
Abstract. The optimization problem of large-scale systems, composed of a set of interconnected subsystems, is approached in a new manner, with algorithmic finality: the global model of the large-scale system is decomposed into interconnected submodels, with minimal interaction, which are sequentially aggregated. The aggregated global model, obtained by concatenation of aggregated submodels is solved, the resulted values being used as reference values for the local controller (at the level of the subsystems, for which the subproblems can be solved). Using an uncertainty principle, the uncertainty which proceeds from the decomposition of the large-scale model (unsolvable) in interconnected submodels and of the aggregation of them, is evaluated. As examples are given: the control of a large-scale technological process, and the management of the production in a large-scale industrial system. Keywords. Large-scale system; interaction; non-linear system; cannonical model; decomposition; linearization; aggregation; submodels concatenation; dezaggregation; reference value; uncertainty.
THE CONCEPT OF LARGE-SCALE SYSTEM
x(t) = Ax+Bu+f(x,u).tw(t), (2) proceed from the separation of linear and of non-linear parts, in the natural model, are large-scale matrices, the vectors x and u having large dimension too. A consistent and less ambiguous characterization of a large-scale system can be made using the following: Definition: "By large-scale system we understand 8 system S, complex as struct ure and objectives, composed of a set of interconnected sUbsystems Si :
The large-scale systems characterization can be made by means of the cannonical form of their model [12]. Indeed, let be : x(t) = F(x,u) + wet), (1) (where: x(t), denotes the n - dimensional state vector; u(t), the m dimensional control vector; wet), the' 1 - dimensional vector of the interaction system-exosystem), the natural model of a large-scale system, written in the language of nonlinear dynamic systems. From a formal point of view a large-scale system is characterized by the fact that the matrices A and B, from the non-linear cannonical model*): i) A direct way to obtain the cannoonical form of the model is that to consider: f(x,u)=F(x,u)-Ax-Bu, where A and B result from the Taylor's series development about the nominal point of operation (xo,uo) :
'OFI
0
A=~X,
Si~Ui,Xi,Vi,Wi'Yi'~i,gi,hi,T),
(3)
(i=1,2,3, ••• , n), and a co-ordinator sub-system" , where: Vi is the control (or input) subset, of the i subsystem; Xi' the state subset, of the i subsystem; Vi' Wi , the interaction subset of the i subsystem with the other en-I) subsystem, respectively with the exosystem; Yi , the output subset of
UFI
0 B=15Uu.
603
604
F . Stanciulescu
the i subsystem; ~i' the transition function of the i subsystem; gi' the interaction function of the i subsy~ tem with the other (n-l) subsystems; hi' the output function of the i subsystem; T, the time moment subset. The functions ~i' gi and hi are defined as following Ui x Xi x Vi -+ X·l. If'i gi a UX j ~ Vi' hi : Xi _Yi j;li 11 we take into account both characterizations and introduce the interaction Vi' between the subsystem i and the other (n-l) subsystems, under the form :
system optimization problem is formulated as following : min xi' ui
{J
=
t
1= 1
Ji (Xi'u1 )},
xi=Aixi+Biui+vi+fi(xi,ui)+wi' Xi (0) =
xf '
(6)
(7)
(8)
n
Vi ="')' gi'(x,) ,
Vi
J
J
(l0) (Xi' ui ) ~ 0 (i = 1,2,3, "" n) the optimization subproblem for the i-subsystem can be brought to t he form [111 :
ri
n
Vi = g. (x) = ) . gij(X j ) , l. j= 1, j/i (4) (i = 1,2,3, •.• , n) , the cannonical model of a largescale system, results as a reunion of cannonical submodels of the form: x.(t) l.
= A.l. x.l. +B.u. l. 1
+L giJ'(x ,) + j;li J
= 1,2,3, •.. ,
(5)
n).
The method which we present in the seq uent sections is based on the philosophical approach of largescale system, which recommends to get ove r the following stages : Stage 1: The large-scale system obj ectl.ves are set up. Stage 2: The large-scale system is decomposed in n interconnected sub systems. Sta~e 3: The qualitative and quantitatl.ve properties of the subsystems are characterized by means of adequate methods and techniques. Stage 41 The qualitative and quantitative properties of the global system are infered from those of the subsystems and from the interaction structure (by means of specific techniques) • DECOi,iPOSITION OF LARGE-SCALE SYSTE1;S IN HTTERCCN1\TE;CT ED 3UBSYSTE&;S The decomposition of a large-scale system in interconnected subsystems is equivalent with the decomposition of the optimization problem in n optimization subproblems (one for each subsystem). If the large-sca l e
= Aixi+Biui+Vi+fi(xi'ui)+wi
Xi (0) = X~ , n
Vi
n
+ fi(xi,u i ) + wi(t).
(i
.
Xi
=J. #i
gij (x j )
,
.
r i (Xi' ui ) .(. 0 Ano t her methods of decomposition of the large-scale systems are given in (1) - [5] • LINEARIZATION OF LARGE-SCALE SY3'fEt"J Come ba ck to the cannonical model xi:; Aixi+Biui+Vi+fi(Xi,ui)+vi (i = 1,2,3, ••• , n) where : n Vi
L j /i = j=l,
gij(X j )
.
(ll)
If we know the opera tion conditions of each subsystem these can be linearized and subst itued for subsystems Si. Finally, we, obtai~ a new, linearized syste ~ (S ), having the same topology as S, and which substitutes for the system S. Intuitively, S' is expected to be t he cannonical linearized repre sentation of S. The linearizati on of the non-linear cannonical mod el (5) is the equivalent of the lineariza tion of nonlinear function fi(xi, ui ) and of the interactions gij(X j ), by Taylor's series development ab out the nominal operating point (xo,uo), neglecting the higher-or der terms. The resulting
Algorithm for Optimization
linear cannonical model is xi (t) = Ai xi (t) + Bi ui (t) + vi (t) + + f i (t) + wi (t ) , (12 ) where the linearized interaction vi(t), can be expresed as :
r:
On the other hand from (21) it results, by means of relations (17) and (18) I
(13 )
AGGREGATION OF LARGE-SCALE SYSTEM We consider a large-scale system, composed by several interconnected subsystems, each subsystem i being represented, about the nominal operating point (xr, uf) by means of the linear cannonical model (14-) xi = Aixi+Bi ui+vi+fi+w i , (15)
n
V.
1.
=
I: j;ii
A.. x ' 1.J J
,,-
+ C~Difi+ciFiwi '
where: Aij (i,j = 1,2,3, ••• , n) are the interaction matrices. Observation. The linearization of the large-scale systems about the nominal operating point (XO,U O ), valid in certain conditions, is indispensable from technical point of view,because in the real operation the nominal values are the most used.
xi (0) = xf '
n
#
:#
xi = CiFiCiXi+CiGiui+~iCiAijCjXj+
n
v . ( t) = A. . x ' ( t ) 1. J;ii 1.J J
605
(16)
The aggregation of the system (14-) (16) is equivalent with the substitution of the state equation (14-) and of the interaction (16), with another state-equation and interaction of reduced dimension, but which described the behaviour of the system in the same manner as the model (14) - (16), (6) - [8J. We introduce the aggregation matrices Ci and Cj and the aggregation vectors zi and Zj (of dimensions li.( n i and Ij
where Ci represents the pseudoinverse of Ci • By comparing the equations (22) and (14), it results ... (23) Fi = CiAiC i , Gi = CiBi , to' Aij = CiAijC j , Di = Ei = Ci and finally the aggregated state equations of the i subsystem I •
t
to'
Zi = FiZi+GiUi+Vi+Dif i +Eiw i , (19) where : n (20) vi = L A1.' J' Z J' , j;ii or by introd ucing of (20) in (19) : n
Z.=F.Z.+G.U·+LA .. Z .+D.f . +E.w i , (21) 1. 1. 1. 1. 1. j;ii 1.J J 1. 1. 1.
#'
zi = CiAiCiZi+CiBiui+j;iiCiAijCjZj + + Cifi+Ciw i
(24)
THE AGGREGATED SYSTEM If the procedure which leads to the re~ation (23) ,i~ sequentially appl1.ed of the 1.(1.=1,2,3, ••• ,n) subsystems, the aggregated state equation of the large-scale system is obtained by concatenation of the similar terms, in the equations (24), in the following manner I • 0 Q 0 z(t)=Az(t)+Bu(t)+Cf(t)+ wet), (25) where we note
8
z =[zl' Z2'· •• ,Zn]
=[u l '
u
C*
Cl Al 0
A=
I
C2 A2l C# 1
·C·• n AnI
C#C I
·.. ·..
Cl Aln Cn C2 A2n Ctt: n
·..
Cn An
B=
0
C=
0
C2 B2 • ••
·0·
0
'It
0
0
Cl Bl 0
,
u 2 ' ••• , U m]
(18) The aggregated state equation is written I
(22)
iI
·..
0 Cn Bn
Cl 0
0 C2
·..
0
0
0
• ••
Cn
··
0
Ctt: n
606
F. Stimciulescu
In order to reconstitute the optimization aggregated model of the largescale system it is necessary to aggregate the performance index J i and the constraints, linearized at the level of each subsystem : , " J i = Ci xi + Ci u i ' (26) Pi xi + % ui + Ri = t'i (27) Therefore we shall rewrite the performance index aa follows : J i = Ci zi + Ci ui ' (26') and the constraints Li zi + Mi ui + Ni = Ir'i
(27')
Tacking into account the relation (17) it results by identification of the relations (26) with (26') and (27) with (27'), the aggregated forms : t # " J i = Ci Ci zi + Ci ui ' (28) and respectively :
%
ui + Ri = O"i (29) By concatenation the terms of the same kind in relations (2 8) and (29) we obtain the aggregated form of the global performance index and constraints : , " J C z + C u , (3 0) and respectively Px + QU + R = t', where :
=
,
, C=
it
[r
1
0
C2' C211-
o
PICl
Q=
Ql 0 0 Q 2 0
0
o o
·.. '*f
P=
·..
0
C"=
o
0
0
P C#2 2
0
0
·· ...
Finally it results the o?timization aggregated model of a large-scale system : , ""\ min {j = CZ+CuJ' (32) Z,u o
and respectively
P i C~ zi +
(~, the vector obtained by the concatenation of 0i)'
:1 '
o o
·.. ·..
R:
0
0
...
0
(33) (34) (35)
OF THE UNCERTAINTY IN T:IE AGREGATTED SYSTEi:.l In [10) and [ll) we have given an uncertainty principle in the largescale system theory. This principle relies on the observation that the solution of the optimization problem, relating to a large-scale system, depends on the influences of the exosystem in which our system is "plunged"*), i.e. in fact on the intera ct ion between the system and the exosystem. Particularly , this observation holds for the subsystems rezulted from the decomposition of a large-scale system too, due t o the fact that for these subsystems the exosystem includes the large-s cale system itself. The uncertainty erinciple in the large-sc ale s~stem theory. Come back to he state equation • n x.:;;;A.x.+B. u.+ L g .. (x . )+f. +w. ESTl ~~TIO N
1 1
1
1
jii lJ
J
1
1
0
and de fine the uncertainty on the state Xi as the distance ( usually defined by means of the norm) between the optimal (unknown) value x; and the value Xi (det ermined by an approximation procedure) ~xi = d(X;, Xi) = \\ x; - XiII (36)
~
*) 'Ne allways consider our system
p:cJ ...
0
Re mark . Due to reduction of the optimization problem dimension ~ under aggregated form (32) -(35), solving of this problem is now possible, for exemple using an algorithm based on dynamic programming principle . If the problem has a special form {6), we can also nope to obtain, by dezaggregation, the very subvectors Xi and ui • In any case, t he values obtained in this way, can be now used as reference values for the local controller (at the level of the subsystems resulted from decomposition).
1
RI 0 0 R2
0
z(t) = Az(t)+Bu(t)+Cf(t)+Cw(t), z(O) = Zo ' Pz(t) + Qu(t) + R = O.
0
finite dimensional.
Al gorithm for Optimization
The sources of uncertainty in the large-scale systems are variousn~, [111 , but the uncertainty in the state xi proceed objectively from : • the fact that the number of unknown variables in the sUbsystem i (din
mension of x· + L dimension of x..,) 1. jii J exceeds the number of respective state equations ; • the interactions between the considered subsystem and the other en-I) subsystems, including the exosystem (the ignorance of the interactions, the interactions non-linearity etc.). Analogically, starting from the relation which express the connection between the interaction variable vi and the state variables x·(jii), we J define the uncertainty on interaction as thendistance between the value v.*
1.
~ = J+1. . J
.
The uncertainties ~i and I::lv i show the limits to which the concepts of large-scale systems theory can be used. Evaluation of the uncertaint~. Tacking into account the definition of the uncertainty ~ Xi and the relation (17) it results: ..1x.1. =lIx~-x·Il=I\Ct.'(z*:-z 1. 1. 1. 1. 1.. ) 1\ 5--
~
J
11
~ 11
•
\I z~
- zi 11 ;
or,. if we take ir:to account that
~z,=
11 zi-zi ll , we obta1.n :
t1 x i .{ \\
:It-
Ci
11 .llzi.
(8)
An another inequality can be obtained if we start from6v i n
"#
*
/::,v i = 1\ <-Vi 11 =Il~i CiAijCjZj n
tt-
-I: C . A. . C . Jii 1. 1.J J
g. . (x *. ) a nd v., i. e • :
1.J
607
1.
Z •
J
1I
finally :
= I' v~1.
-
tjii
g . . (x . )
1.J
J
11·
n
.,
AVi~ ?l}Cill.\\Aijl\·\\Cjll·..1Zj.
(9)
(7)
Theref ore, if the system is large and consequently non-resolvable on the whole, there is a certain uncertainty, .b.x., on the state x .• If we attempt to1.solve the large-Scale system by decomposition, the uncertainty ..6xi' on the state xi' persists due to the dependence of the solution on the interaction v., 1. on which an uncertainty ~vi exists. As we decompose the large-scale system in ever more interactive subsystems, the uncertainty.6xi decreases (as a result of the decrease of the subsystem dimension), but the uncertainty bv.1. increases (as a result of the increase of the interaction - vector vi dimension), the uncertainty ~xi and.b vi finding in relation of inverse proportionality (eventually nonlinear). All these observations, made so far, intuitivelly give us the opportunity to conclude the valability of the following principle of uncertainty : "In a large-scale system, composed of several interconnected subsystems, the state Xi of the subsystem i, and its interaction Vi with the other subsystems, can be simultaneously determined only up to a certain degree of accuracy".
The inequalities (38) and (39) emphasize the fact that the uncertainty is determined by the agg regation procedure, the minimization of the uncertainties II Xi and llv i being tied to the finding of an aggregation matrix Ci ' ci and cj or a matrix Aij of mi ni ma l norm ( with ~i,AZj computed). THE COMPUTATION ALGORITHM The method of dimensional reduction of the optimization model of a largescale control system, using techniques of decomposition-aggregation, can be finalized alg orithmically in the following manner : 1 st step: Generating of the largescale system global model (of large dimension), as a reunion of interconnected submodels, of the form (5). 2 nd step: Adequately decomposition of the global model in interconnected submodels (s o that the interaction between submodels should be minimal), of the form (7) and (9). 3rd steal Linearization of submodels rezulte from decomposition and of the interaction between them (12), (13) •
4th step: Aggregation of the reszulted submodels and of the interactions
F. Stanciulescu
608
between them, by means of the equation (24) and of the relations (23).
by decomposition-aggregation of management system of a large-scale industrial system. The optimization by decompozition-aggregation of large-scale system occurs when it is not economically reasonable to take into account all state variables of the system, but it is recommended to construct a model after the following criteria: (1) they are retained the all significant state variables (we write c ij > 0 in the aggregation matrix); (2) they are eliminated the non-significant state variables (cij=O, in the aggregation matrix); (3) they are aggregated several state variables in a single variable (by means of the n. 1 l. transformation ~ ~ c ij x j ). iJ J=l In the figure it is represented the bloCK - di~gram of a large-scale system, as hierarchical system with several levels. At the lowest level it is found the controlled process, composed of n interconnected subprocesses - SP i (control loops), every subproces being described by a linear-dynamic model (Non-aggregated Subsystems - NSSi - level). At the higher decisional level (ASSi),
~th step: Restructuring of the glo-
al model (aggregated) by concatenation of aggregated local submodels, equations (32)-(35). 6 th step: Solvin8 of the global optimum control problem with the aggregated model (32)-(35); the obtained solution becomes reference value for the local co~trollers.
7th step: Esti~ation of the uncertal.nty on state and interaction variables at the level of subsystems, by tacking into consideration the aggregation. th 8 step: (Optional) Dezaggregation of the aggregated state and control vectors (z,u) and obtaining of the state and control subvectors (xi' ui )· APPLICATION ON CON'l'ROL OF INDUS'l'RlAL PROCESSES AND SYSTEMS We refer to two applications: (a) optimization by decomposition-aggregation of control system of a largescale process and (b) optimization
A C 5 I z,
~
z2. Zn
A 5 52
~
,
A 551")
N 5 5,
N 5 52
N 5 5n
A 5 5,
•
j
r-:t du,
U1
z;
u~
---
5 P, I ~----"'
R, Figure.
L
J
t,
~
lb-:
-=--;.t J~J2
....
Z2
S P x? ... 2 I
.
~
r
z~
u;-
5 Pn ~ 10..
!I ----.
dun
I R2
I
t2
Rn
En
Block-diagram of a large-scale control and management system, with hierarchical structure: ACS - Aggregated Co-ordination System; ASS i - Aggregated Subsystem i ; NSS i - Non-aggregated Subsystem i; SP i - Subprocess i; Ri - Local controller (regulator) i.
Algorithm for Optimization
these submodels are aggregated, and finally concatenated in a single aggregated model (Ag~regated co-ordinator system - ACS). This last mOdlel sfuPPli~S the*aggrle~at;d opt~mal va ue 0 z, ~.e z = zl,z2 ••• ,znl * 1, and by and u* = {* up u*2 ",., ~ dezaggregation we obtain the optimal values (x;, ~;), which becomes now reference values for the local controller Ri' The block-diagram represented in the figure can be viewed also as the management block-diagram of a largescale industrial system: at the lower level it is found the managed system, composed by several interconnected subsystems. The highest decisional level (ACS) concatenates in a single aggregated model, the submodels aggregated (in ASS i decisional blocks), and solves it, the obtained values (synthetical planning indicators) being furnished to co-ordinated subsystems as reference values for the local production subprocesses. We remark the existence of the control-correction loop (Ri of the deviations from the reference va lues). The method has been experimented in the management and control of a technological process in an enterpreise with discrete production (electrical apparata). REFERENCES 1.
A.N. and R.K. Miller (1977). Qualitative analysis of Large-scale DYnamical s~stems. Academic Press, New-Yor • 2. Singh, M.G. and A. Titli (1978). Systems: Decomposition and Control. Pergamon - Press, London. 3. ~ak, D.D. (1978). Large-scale DYnamic Systems. Stability and Structure. North Holland, New York and Shannon. Michel~
609
4. Findeisen, W. (1974). Multi1eve1 Control ssstems. PWN, Warszawa (in Polis ). 5. Sandell, N.R., P. Varaya, M.Athans and m.G. ~afonov (1978). Survey of Decentralized Control Methods for Large-scale Systems. IEEE Trans. Automat. Contr. vol.AC-23, No.2, April, pp. 10~-128. 6. Aoki, M. (1971). Aggregation. In: Optimization Methods for Largescale s~stems. Ac a demic - Press, New Yor • 7. Bertrand, P., G. Mic hailesco and J. M. Siret (1976). Sur la synth~se de mod~les r~duits 8ar aggregation. RAIRO, vol.l , n o .7, juillet, pp. 105-112. 8. Luenberger, D.G. ( 19 78 ). Aggregatio n and Dual A~g regation of Positive Dyna mic s*ste ms. In: Preprints of the 7t Trien nial World Congress of the IFAC, Helsinki, Finland. 9 . Guran, M. and F. Filip (19 79) . Hierarchic a 1 Systems in t he Chemical Industry. I n: Auto matics, Mana gement and Co mput ers in the Chemical Industry, Editura Tehnica, Bucure : ; ti ( in Homani an , in press) • IO.Stanciulescu, F. (19 77). An Uncertainty Pri nciple in the LargeScale Systems Theory a nd Applications in their Optimization. In: System Theory - Pr eprints of National Confere nce on System 'f heory, Bra$ov ( Homa nia ) . 11.Stanciulescu, F. (1~7 7 ). Largescale Disc r ete Systems 0Btimization in the Light of an ncertainty Principle. In: Preprints of IFAC Workshop on Computer Applications in Discrete Manufacturing, Prague (Czec h osl ova kia). 12.Stanciulescu, F. ( 19 78). ft namic Model and Alg orit hm for he Control of Large-s ca l e Systems with Hierarc hical Structure. I n: Digital Processing and Transmis s i on of Data and Con trol of Pr oc esses of Computers. Pr obl e ms of Automatic Control, n o .1 0 , Bditura Academiei, Bucharest (Romania), pp . 111-122.