PARA~ETRIC
METHOD OF COORDINATION
USING FEEDBACK FROM TBE REAL PROCESS
Adam Woiniak Institute of Automatic Control Technical University of warsaw 00-665 Warszawa, Poland
ABSTRACT The inaccuracy of models used by local optimizers is a major problem when we want to use a multi-level system to the on-line control of the complex process. In the paper a new two-level control structure, based on parametric method of coordination is presented. In this structure a feedback from the real system is used to reduce the influence of above mentioned inaccuracy on control's performance. Applicability conditions of the structure and the coordinator strategy are discussed. The method is illustrated by a computational example. 1.INTRODUCTION The decision problem of a control system is usually an optimization problem. If the process subject to the control action of the control system is large, complex system fan industrial production process is a typical example/ the finding of the solution of this problem meets difficulties. In order to avoid these difficulties the original problem may be replaced by a set of interrelated simpler decision subproblems for the control subsystems which are then arranged in two levels
(4),[8],[9). The decomFosition of decision problem may be carried out in a two-step sequence. First, the process is conceptually parti tioned into separate subprocesses. Second, for each sub-process its decision problem Iso-called local problem/ is defined, depending on coordination variables, and in addition a second level problem Iso-called coordinator problem/ is d2 fined. The set of local problems and the coordinator problem have to be defined in such a way that the solution obtained in this two -level system in fact corresponds to the solution of the original problem. The described method of replacing the original problem by the set of the simpler problems is termed decomposition on the basis of structure (4),[9). In most industrial processes the above partition is such that the dynamics of each individual subprocess does not influence other subprocesses. This follows from the fact that, mostly, each subprocess has a direct controller which objective is to suppress the dynamics
571
of the subprocess, i.e., to hold the subprocess variables at particular desired values /which may change in time because of disturbances, tut at a slow rate compared with the frequency of direct control intervention/. The union of subprocess and direct controller then represents a subsystem which is in the near-steady state. Assuming that the disturbances are step time functions with long intervals of constancy, ~nd the direct controllers are ideal, the original decision problem can be considered as a sequence of static optimization problems "tracking" the disturbances, i.e., each measured or predicted value of disturbance define a "new" static optimization problem. Even this is not exactly the case, most inaustrial processes are built so that above approximation is appropriate [4],[9]. Models used by local optimizers are mostly not accurate. This implies that the calculated and realized outputs and interactions are not the same and consequently the real system operation is nonoptimal and constraints may be violated. Because the use of feedback from the real system is the only way of increasing the available information about it, the closed loop structures of control, most often the structure with adaptation, have been proposed. In this paper another closed loop structure, proposed particulary for two-level control system by W.lindeisen (4),[5), will be considered. As Fig.1.2 shows, the feedback from the real system in this structure is used to improve directly on the algorithm of the coordinator. As is known (8),[9) all ways of transforming a given static optimization problem into a decomposed two-level one are combinations of two different approaches WI.ich may be termed: constraint or model coordination mode, and the goal coordination mode. The parametric /primal, feasible/ method, e.g. [5),[6),[91, is a typical representative of the first mode of coordination and interaction balance /price adjustment/ method, e.g. [5],(8),(9], of the second. 'l'he structure proposed by W.Findeisen is applicable to both methods of coordination. The interaction balance method with feedback is described in (7] and the closed loop structure using parametric method of coordination will be presented below.
2.DESChIPTION OF THE
ME~HOD
basing upon model, at first solves the upper local problem:
Let us assume that the model of controlled system is featured by the following mappings which describe: /il model of subsystems P : U~ ~ C~ ~ Y k
k,
/ULPkl find a point u~(v) in UVk~ ](kU(UCkn {Cu,c): v k= Pk(u,c)}> such that the distance between the point Hk(v) and
_ If) k € , ,n
where the set Ck is the set of local manipulated /control/ variables, the set Uk is the set of interactions through which the k-th subsystem is coupled with the other and is the set of subsystems outputs; /ii/ interconnections between subprocesses
Y;
H : Y;~ •••• Y~ k
--+
U~, kE~,
which are known exactly. Because the static optimization will be considered, in the sequel we shall assume that the sets are finite-dimensional Hilbert spaces. Suppose that the mapping
U:, C;, Y:
p.: C1' x ••• x
k Ck'
s. U x and they are known exactly. The only overall constraint has a form
= 'l'0(
fl.,
R, k e
1\
=
y(v)
p.(c(v»
is measured and transmitted to the coordinator. The controls ak(v) are sent upwards as well. The task of the coordinator is minimization of performance functional Q(c (.),p.(c(·)))
find a coordination variable
Q(V) ••• )(
f;Ii
:=
where
y~,
the above constraints that the overall of the subsystem
v such
D
g
Vl'
min Q(')' D
{v: p .. (c (v) e Y
11
(Yk €.
(Hk(P.(C(v »),Ck(v
f;Ii) »E UCk} n Vp '
P,(UC,)x ••• x Pn(UC ). n
The introduction of the upper local problems and the additional constraint v E Vp needs explanation. When these problems are not introduced the mapping c(·) is defined only on a set
••• ,Qn) ,
--+
A
as produced are applied to the real system and after decay of the transient process the output
ICP/
where ~: C~ ~ Y'
1\
(c,(v), ••• ,cn(v» = c(v)
that
where C ,~ C ,' x ••• xC'n' y,~ y', )l is as small as possible and are not violated. We assume cost functional is composed performance functionals
= Pk(u~(v),c)}.
based on real values c(v), y(v) = p.(c(v» subject to the real constraints:
Suppose that is desirable to control the system in such a way that the overall performance functional ~
k
vk
12.4/ Q ~
(Y1' ••• 'Yn) = yE Yf Y1'x ••• xY~.
Q: C')I y'
/LLF / find a point ~k(v) such that k ~(ck(v),v) = m i n ~(. ,v), Wk(v) where /2.3/ Wk(v) ~ [c: (u~(v),c)E: UC 11
I'
represents collectively the entire real system. As we mentioned earlier this mapping is not known analytically, however we assume that for each value of manipulated variables (c 1 ' ••• 'c n ) we can measure the output y = p .. (c 1 , ••• ,c n ). Next let us assume that the constraints associated with each subsystem are the following
12.2/
C:
ITI'kuis a projection mapping from U~ ~ onto Uk/. Next they solve the lower local problems:
The values of manipulated variables
C~ ~ Y1' x ••• x y~
/2.'/ (uk,c k ) € UC k
the set Uvk is equal to the distance between these points
and
"I":Rn~1R
V o£
is a strict order preserving function. The structure of two-level control system with feedback in which decisions problems are specified on the ground of parametric method of coordination is presented in Fig.2.'. :r'or a given coordination variable v € y' set by the coordinator each local controller,
[v: (Yk E f;'ii)(3Ck) ~(ck'v) = m i n ~(.,v)}, Ck(v)
where Ck(v) ~
(Hk(v),c)€. UC k 11 v k = Pk(Hk(v),c)}, and as a result of this the constraint
.) The disturbances are neglected because in each step of "tracking" they are constant.
572
{Cl
V€V
p
in above coordinator problem /CP/ would have to be replaced by the constraint v E: V o ·)' Except syecial cases, methods of determining the set V o are not known. Thus the introduction of upper local problems enables an extension of mapping cC,) onto the set Vp which is easier determined. Moreover, it is easy to see that when for all k ~ the sets U are closed, the upper local problems have vk a solution for all v in Vp ' Before passing to the investigation of applicability conditions of the parametric coordination using feedback from the real process and to a method of solving the coordinator problem, we shall pay some attention to what one-level problem is related to the two-level problem just described·"~ Let for all k€ 1,Ii: Dk be a mapping from C' in U:. C: defined as
;-;n
c ~ Dk(c) = (Hk(P.. (c »,c ). k Now, the definition of the feasible set D of the coordinator may be expressed as follows A-1(-1()n" -1(_ 4 D = c p.. Y n D lIC k » n VP = k /2.5/ { t-\G*)nVp~'1 where
G
1 ~ P- (Y) n
n" D - 1 (UC
k k It can be easily proved that so
It
1I
k; 1
.-1(,,( c c Vp »
f the feasible sets C and F. If we know exactly the mapping p .. /i.e. the moaels Pk , k = 1, ... ,n, are accurate/ the constraint c E c(Vp ) is unessential lin this case it is fulfilled for each c which is the problem's /ecP/ solution/ and problems /OCP/ and fOri are equivalent and consequently the problems «(ULPk,LLPk}k~l'CP) and IOCPI are related. Because that is not the case, the value of performance functional Q Gorresponding to the solution of the two-level parametric problem with feedback is not greater than the value associated with the solution of the o~timizing controller problem. In principle the more inaccurate model of system is used, the greater loss of performance will be obtained. 3.APPLICABILITY CONDITIONS OF THE PARAHl,TRIC COORDINATION WITH F~ELBACK when formulating the two-level problem with feedback we have assumed tacitly that the mappings u~ (.) and k ( . ) are well defined and the feasible set D is not empty. First we briefly consider the non-emptiness conditions. From the expression /2.5/ we have
c
(Vv)[vE D (=,..,> c(v)€ G,,] ,
).
hence the necessary and sufficient condition of the non-emptiness of the coordinator feasible set D is the following
= Vp '
/,*/{G*'~
() D ,. C...-1 ( Gll n c" V p) •
(3 v)
This immedietly suggests the following proposi tion. /2.6/PROPOSITION: The two-level problem
C(v) E
G If
•
The set G" is not empty if the system of constraints /2.1-2/ is not contradictory for the ({ULPk,LLPk}k~l'CP) is related to the following real process /the constraints are not stiff/.On the other hand the above v exists when the difproblem ference between model and real system is not f very large. /OP/ find in C such that When these requirements are not fulfilled, Q(c,p. (c » min the system model is too inaccurate, or the feaf sible sets are wrongly defined and both must be C improved. Therefore, from this point we restrict where f our attention to cases in which conditions / C ~ {c: Pit (c ) € Y 1\ (Vk € ~) are fulfilled. (Hk(F,,(c)),Ck)E UCk}n c(Vp ), Now, we pass on to the applicability conditions of the considered method /applicability and cC) is the relating mapping. [J in the sense that the solutions of upper and It is easy to see that the problem /OP/ is lower local problems and of the coordinator approximate to the ideal optimizing controller problem exist/. problem /3.1/PROPOSITION: If the controlled s~stem and /OCP/ find c in F such that its model are such that
c
'* /
Q(c,P,,(C»
= min
/3.2/ the mapping P" representing overall real system is continuous,
Q(·,p_C»,
F
where
F~
{c:
p.. (c)€YI\(Vk€~)
/3.3/ the set Y is closed, for all k e
(Ek(P .. (c» ,c ) E UC }. k k The only difference consists in definition of
<; Vp ' o ··)Problem 2 is related to problem 1 if: /i/ s~ lution w of the problem 2 exists iff solution of the problem 1 exists, and /ii/ there exists a mapping ~ such that ~(w). 11) It is easy to verify that V
x
r,n :
/3.4/ the set UC
x=
573
is compact, k /3.5/ the mapping P k is continuous on Uk • C: , open /i.e. converts open sets on open sets/ on UC , k /3.6/ the mapping H is continuous on Vp ' k
13.71 the mapping Pk and the set UC k are such that for all v in Vp the upper local
Hence
Q C4
problem IULPkl has at most unique solution,
13.81 the functional ~k and the set UCk are such that for all v in Vp the lower local problem ILLPkl has at most unique solution,
-J:,
4 + ~).
~1
(4,4)
-(7)2
11m (4,4) Q1(v) ~
v _
then for all k e 1;'Ii and all v ~ Vp solution of IULP and ILLPkl exists and there exists the solution of coordinator problem Icp/, i.e. the parametric method with feedback is applicable.
I
We shall now briefly discuss the foreeoing applicability conditions. The assumptions that the mappings p., P k , Hk , ~k and ¥ are continuous, the sets TICk are compact and the set Y is closed are natural and fulfilled in most practical cases. The first crucial assumption is the requirement that the mappings Pk are open on UC k • Unfortunatelly, an example can be constructed which shows that when this assumption is dropped the mapping Q Idefined by is not even lower-semicontinuous. Consequently the coordinator problem may not have a solution. 13.1.1/EXA}lPLE: Let U; = C; = Y; = Y;' = R+. Consider a system where the first subsystem model is given by the equation 2 1R+ 3CU1'C 1) ~ P 1 (u 1 ,c 1 ) = + c for u + c U 1 1 1 EIR { 41 otherwise
12.411
41
-
~~ < - ~g
Q1 C4,4)
which means that Q (.) is not lower-semiconti1 nuous.O After L.Schwartz [10] we can state the following sufficient condition for openess of a mapping. 12/PHOfOSITION: A continuos differentiable mapping P : U ~ Ok ~ Y is open on UC <; Uk~Ck k k if for all (u,c)€: TIC
13.
Proof is given in Appendix A.[J
~
~ _..!L
TheI'efore
13.91 the functional ~ is continuous, 13.101 the function y is continuous,
+
and coupled according to the equation
=
+
1 J J-+I)O 4 but the value of the functional ~1(') in the point (4,4) equals
1
k
rank [Pk(u,c)]
= dim
Yk ,
where (P;C·)] denotes the Jacobi matrix of mapping Pk .0 The second restrictive assumption is requirement that for all v in Vp the upper and lower local problems have a unique solution. This assumption is necessary because otherwise u~(·) or k (') will only be relations but not mappings. I f for all kE::r:n and for all v E: Pk(UC ) k k the set Uvk=JlkU(UCk'" {Cu,c): v k = Pk(u,c)})
e
is convex, then the theorem about existence of a unique element with minimal norm in a convex set in Hilbert space implies uniqueness of the upper local problem solution. In general the following proposition has been proved. 13.13/PROPOSITION: Let US U' be compact. 'rhe set J(U) ~ ij', {u: (3! uOe U) d(u,U) =
lIu _ uOIl}
(Y1'Y2) ~ H 1 (Y1'Y2) ;2 Y2' 2 The local feasible set is given in IR+ as 1 1 UC = {Cc 1 ,u 1 ): c 1 ~4 + ;2u 1 11u 1 ~ 10} 1 and local performance functional as 2 IR + 3 c ~ Q Cc ) = -c E IR. 1 1 1 1 It is easy to Vrove that only the assumption is not fulfilled because ~ is non-open
11)
of these points in which the uniqueness assumption is not satisfied has an empty interior. Proof is given in Appendix
B.O
Therefore, in numerical search of a solution of the coordinator problem the lack of fulfilment 01 assUJlll'tion 13.71 should not prevent finding the solution. The uniqueness property of lower local problem is a necessary requirement in most 01' the multilevel methods, and as is known, except for the strictly convex problems*~~ we must investigate the fulfilment of this property for all problems in particular.
ICPI
13.5(
mapp~ng.
Let Q C·) be defined as follows 1 (vv~ [4,oo[ ~ IR+) Q1(v) = m i n Q1(')' W (v) A
1
where W (·). is defined by 12.3/. 1 Let {(y~, y~ )} j':1 be a sequence defined as follows yj ~ 4 + ~ 2 J It is obvious that this sequence converges to (4,4). F~r ap j ~ 1 it is easy to compute the value ~1(Y)' It equals A( 1 1 1(1 12 Q,1 4 + j 4 + j) = ;2 + ( 4 + (y j
~ 1)
j Y1
9"
ll)
d(u,U) denotes a distance between the point u and the se tU.
.*)
T» .
574
That is for such problems, where for each v in Vp and for all k E ~ sets Wk (v) are convex and functionals Qk(' ,v) are strictly convex.
4.THE COORDINATOR STRATEGY We are now in position to approach the problem of choosing the method of solving the coordinator problem Icp!, or the so called coordinator strategy. As we recall /CP/ is formulated as an optimization problem in which we do not ~now analytical form of minimized functional Q and of the constraining mappings. That is because these functions are defined in terms of mappings c(·) and y(.) for which we can only calculate or measure their values. In these circumstances the penalty function method is the only one which enables the finding of /CP/ solution. Therefore we shall have to replace the coordinator problem by a sequence of appropriate defined problems. The set D was defined in such a way that for all v in D the values S(v) and y(v) are such that real constraines are fulfilled. Therefore as a penalty function we should choose the interior penalty function of the set D. As is well known however, the interior penalty funct10n can be used only for such a set D that
/**/
D = D.
cl int
Since we do not know the analytical description of this set, it seems that statement of requirements which would assure the holding of /**/ 1S impossible. Let us suppose instead that we fo~nd a partition of the set D on the subsets D and DO such that /al int DW ~ and DW ~ Vp ' /4.1/ Ibl int DWn DO:! ~,
!
{ /c/
W
W
D = D I"l DO = cl(int D I"l nO).
Let PD: { r € lR: r> o} x Y' ~ IR U {oo} be a mixed W . penalty function of the set D n DO ,1.e., a mapping such that li/ for each sequence ~j}j~l C int DW convergent
f
to v
h j } J'~1- . convergent .
reals
. J /11/
DO, and for each sequence of positive
Urn (f:J' v J ) = - pD
(X)
-tooo
(VC> O)(V
VE
to zero
, W
fr D
)
PD(£"v) =
W
/11i/ (V vE int D 11 1'°) Um
00,
0,
PD(f. ,v)
£ -to 0+ and {PCP j}.oo £.
J=1
be a sequence of problems defined j
W
~ind v € int D j (f:j v ) Q,(V J ) + PD ' int
{£j}.oo J=1
",-1 (
D = c
=
GIl)nV p '
If we suppose that
Cv k E f;li)
in tUCk
~
nW DO
Vp
and
c- 1(G .. )
satisfies /4.1a/ and the assumptions of rroposition /4.2/, whenever the assumptions in Proposition /3.1/ hold. On the other hand, from the fact that the exterior "part" of penalty function Po is responsible for holding the constraint v€c- 1 (G .. ) it follows that above vartition does not require checking the requirement /4.1b/ before solving the probl~m /PCPf;j /. It is an essential advantage of th1s vartition, besides its simplicity. A drawback of it is a possibility of violation of constraints in course of searching the solution. Some of these violations can be suppressed when the subsystems feasible sets are defined by individual constraints of interactions u k and manipulated variables c k ' and also some joint ones. So, let us assume that
f;li)
UC =c UC~ ,,(U x C ), k k k UkCU andCkCC
k•
Hence, the coordinator feasible s~t equals O ~-1 -1 0 0) "'-1( D = c (p. (Y ) " C 11 VP 11 C k= 1 D -1( UC k k where o A -1( U ) 1 1 · · · " Hn -1(U n ), Y =YI)H 1 1 CO ~ C x ••• x C , n 1 and we obtain the following partition 1 o /403/ D'" ~ (y ) I) Co) n V and o ~ ~-1(D -1(UC )f) I1D -1 (UCo». Do c 1 1 ••• n n
n
.
DW
is appropriate sequence of reals. {
:I
and for all k in f;li: P k is continuous open mapping, then the set Vp has a non-void interior. In the proof of Proposition /3.1/ we showed that the set Vp is compact and the 1 set 2- (G.)is closed, whenever the assumptions of this proposi tion are fulfilled. From this follows that a natural partition of the set D on
k
such that
m i n [Q(.) + PD(£J,·)] where
It results from the proof of Proposition /3.1/ that its assumptions are sufficient conditions of the above required continUity. nence we have to consider only the question how to find the requested partition of the coordinator feasible set. The statement of sufficient conditions for fulfilment of the requirement /4.1c/ is rather impossible, but fortunately, in /4.1/ this last requirement has a rather theoretical meaning. Therefore we shall restrict our attention to investigating satisfaction of the other conditions in /4.1/. Fir"st we recall that the set D equals
(V k €
as follows /PCP£j /
Proof is omitted since it is a rather simple application of Theorem 1 in [3], for detail s see [12J. 0
j}
00
/4. 2/PROPOSITION: I f the sequence £. j =1 of positive reals converges to zero and the functioW nal is continuous, the set D is compa~t, the set DO is closed, then the sequence {V J }i=1 , where for all j ~ 1: v J is a solution of the problem /PCP£i/, has an accumulation poi~t, and any accumulation point of this sequence 1S the solution of coordinator problem /CP/.
Q
575
c-\p:
p
The fulfih,ent by this partition of the requirement /4.1a/
»
int nW 1 int 8-\p: (yo)" CO) (', int VI- ~ ~ can be verified onl;y ex!-·erimentally using measurements in the real system, because we do not know the analytical form of the mapping F•. This testing is faGilitated by the following lemma. 14.4/LE¥~: If a mapping f: A'~ B' is continuou s and BC;; B', then
/il
f-\ int B) ~ int f-\B),
/ii/ CV
aEA')[f(a)Eint B
=>
aEint C\B)].
Proof is obvious.CJ Now, it can be easily seen that tbe following chain of implications is true, whenever the mappings p. and c(·) are continuous. v "int Vp A C(V)E int CO /I p.(c(v»€int y O
.u. Le~a
A
-1.
It is known !hat, even for a simple problem, the functional Q may not be differentiable, see e.g. [4]. Since also the statement of differentiability conditions is rather a formidable problem, we have to apply the direct search methods for solving the coordinator problem /}CPj~ In example described below the Coml'lex method £ of M.J.Box was used. ) .AN EXAJ·;PLE USING ThE DESChIbBD LEThOD
In this section we will demonstrate a simple example of parametric coordination. Consider a system illustrated in Fig.5.1. Let the real subprocesses and their models be defined by the equations Yk = ~kuk + 'kck + T k , k where for real subsystems
0
vEint Vp"c(vlE. int C n int P lI (Y ) vt
C-\ int
CO n
A_1(
vtintc
0
in~ p~\Yo» V._fe~a)
('I
0(
=
2
int Vp
CnPll(Y
k
~ '/J "int U
k
~ I/J
1\
int Y ~ ~ ,
0<1
=
[:J,
[:
~1
0(
-8
[~
[2
2] , 0<3
~] , ~2
[3
2
=
CJ
1]
2]
-1 -1
-1
T1 = .Q , t 2 = -4, r 3 = .Q. • Let the minimized performance functional be a sum of functions:
r;n)
lUCk = Ukx Ck " int Uk of W then the set D , defined in the set D, and consequently coordinator problem IPCPEi/
1] ,
while for models
Notice, that if £
[:J '
-3.8 ,
int C of '/J ] k
then the satisfying of requirement /4.1a/ can be ascertained by searching such v in int Vp for which c(v) lies in int CO and value y(v) = Pll (c (v» , measured in real system, is in intY. For above mentioned search we can use, e.g~ the Fiacco and McCormick method of finding the interior point [2]. If the point VO in int DW is found, then we can proceed to numerical solution of the coordinator problem /PCP£i / with V O as a starting point. (Y k
=
-0.8
(V k E ~)
[int UC
2 ],
~J ' 1.6]
)nintVp Hence, when th~ assum~tions in Proposition 13.1/ hold and, in addition, we assume the following
and
[2
= 1,2,3,
'/Jllint Ck i ~], /4.3/, equals to in solving the the real con-
Q1(c 1 'Y1)
straints will not be violated. Since holding to the constraint v€: DO is ensured by the exterior "part" of penalty function, then /similarly as for natural partition/ we need not verify the requirement /4.1b/ before solving the problem. A remark about the choice of numerical method of solving the local and coordinator problem is appropriate. For fixed values of coordination variable both local problems are standard nonlinear programming problems. Hence the choice depends on their properties, like differentiability of performance functionals, linearity of models and the like.
576
=
c 11 c 12 +
1
:2c 13
2
+
20IY11- 4.131 - 3Y11' 2 2 10 c c 21 22 + c 22 + (c 21 - 4) + 2 8Y 2 '
2c
31
20(y
222 + 3c + (c + 1) + 32 31 2
- 4) 31 The constraints associated with each subsystem have the following forms UC 1==
{(U 'C
1
1): -3 ~u1.s 20" Ic 11 / ~ 8 "lc121 ~ 4/1 Ic 13 1 ~ 10 /I 2c 11 + 5c12~ O},
UC 2=
{Cu 2 ,C 2 ): 0~u21+u22~20
1\
1 ~ C21 ~ 10 1\ IC221~5}, UC = {CU ,C ): IU31~ 5 A IC311 ~ 10 1\ 3 3 3 -7~ C 32 ~100 1\ IC331 ~ 91\ c + c ~-8} . 31 33 It was an easy task to formulate both local problems and the coordinator problem. The upper and lower local problems are standard nonlinear programming problems which were resol~ed by ~e nalty function shifting method [11J wlth cO~Ju gate gradient as a subroutine for unconstralned minimization. As a coordinator strategy the Complex method [1J was chosen. The obtained results are shown in a table and in Fig.5.2, where on x-axis is laid off, for g-raph 1, the number of centroid compu t~ tions, and for graph 2, the number of exper1ments on the real system. The table shows that the loss of performance amounts to 0.415 per cent of the optimal performance, that is not large. We can say from the graphs that after 11 iterations /what corresponds to 33 experiments on the real system/ the minimal value of performance was equal to 128.021309, that is it decreased about 33.2 per cent of the initial value, and was invariable for next 12 iterations /27 experiments/. The next 58 iterations /147 additional experiments/ improved the performance only by 1.3 per cent. This ~eans ~ha~ ~ essential improvement was ach1eved 1n 1n1tial iterations. Hence, although the numerical experience in parametric coordination with feedback is not large now, it seems that the Complex method is appropriate as the coordinator strategy whenever a substantial improvement of performance /rather than obtaining ultimate minimum/ is an objective of the control system. 6.CONCLUSIONS The presented parametric method of coordination with feedback brings new qualities: it partly eliminates the difficulties of determining the set Vo of solvability of local pro blems and it is less sensitive to the model inaC~uracy than the classical parametric method (4), (9]. Consequently it is possible to apply the l'arametric method of coordinatoon as a method of on-line control of complex processes with a result which is better than the open -loop optimization. 7.
ACtNO~LEDG}~NTS
The author wishes to thank Prof. W.Findeisen for his valuable suggestions and comments. The author also expresses his thanks to Dr. M.Brdys, Mr. T.Kr~glewski, Dr. K.Malin~wski and Mr. P.Tatjewski for continuing discuss10ns on the subject. 8.REFERENCES
[2J l"iacco, A., G. McCormick, "Nonlinear Programming: Sequential, Unconstrained I';inimization Techniques", J.Wiley, New York 1968. (3) Fiacco, A., A.P.Jones,"Generalized Penalty
Methods in Topological Spaces", SIAM J. APPL. MATH., Vol.17(1969J, no.5. (4) Findeisen, W., "Wielopoziomowe ukiady sterowania /Mul tilevel Control Systems/", PWN, Warszawa 1974. (5) Findeisen, W., "Control and Cooruination in Multilevel Systems", Proceedings of 2nd Polish-Italian Conference on Application of System ~heory to Economy, Managem~nt anu Technology, Pugnochiuso 1974. [6J Findei sen, W., "Parametric Optimi zation by Primal Method in Multilevel Systems", IEEE TRANS. ON SYSTEMS SCIENC~S AND CYBERNETICS, Vol.SSC-4(1968), no.2. [7J Malinowski, K., A.RuszczyUski, "Application
of Interaction Balance Method to Real Process Coordination", CONTROL AND CYB~kN~TICS, Vol.4(1·975J, no.2. [8] Nesarovic ,M.D., D.Macko, Y.Takahara, "Theory of Hierarchical, Multilevel, Systems, Academic Press, New York 1970. C9J Schoeffler ,J .D., "Static Mul tilevel Systems; On-line Multilevel Systems" in: "Optimization Methods for Large-scale Systems",ed. D.A.Wismer, McGraw-Hill, New York 1971.
[10J Schwartz, L., "Analyse Hermann, Paris 1967.
[11] Wierzbicki, A.P.,"A Penalty Function Shifting Method in Constrained Static Optimization and its Convergence Properties" ,ARCH. AUT. I TELEr~CH, Tom XVI(1971), z.4. [12] Woiniak, A., ·Sterowanie ziozonymi systemami i parametryczna metoda koordynacji /Control of Complex Systems and Parametric Coordination/", Ph.D.Dissertation, Institute of Automatic Control, Tecrmical University of Warsaw, 1975. APPENDIX A Proof of Proposition /3.1/ The proof will be based on the well known Weierstrass's theorem: "a continuous functional defined on a compact set achieves its maximum and minimum". Now, it is easy to see that the assumptions /3.4-5/ and /3.9/ imply the existence of upper and lower local problems' solutions. Similarly the coordinator problem /Cp/ has a solution whenever the functional
Q=
[1J Box, M.J.,"A New Constrained Optimization Method and a Comparison with Other Methods", COJ..FUTER J., Vo1.8(1965), no.1.
577
U1ath~matique",
Q(C'(')'p.(c(.»)
is continuous and the feasible set D of the coordinator is compact.
Ye recall that the set D equals
r(2
-1 -1 D = c,,-1( P -1 (Y)nD lI 1 (uc 1 )n ••• nDn (UCn»nVp ' where (V
k
€
r;n)
D : C ~ c k
k
-.+
Dk(c) = (Hk(P.(c»,c )€ k
k
U x Ck' The mappings D are continuous because the mapk pings H and p. are continuous. The s~ts UC k k -1( Y)n and Y are closed, so the set p. D -1 (UC ) k k=1 k
n
other hand the definition and assumptions /3.2/
"
A
"
(
.»
;:n
and complicated it requires a few steps. First we shall prove the following lemmas. For a given k ( 1,n let u 0 (·) k /A.1/ u~(.): V p 3 v 1--+ u~(v) =
u;.
Uvk= rrkU(UCkn
nkU(UC) = ftkU(UC)
Let in the sets r(2
UCk
U
) and /::(2 .)
E t(2
Uk ).
the ex-
ponential /Vietoris/ topology [A1), and in the U set Ukx t(2 .) the product topology be introduced. As is well known composition is continuous if composing mappings are continuous. It is easy to prove that projections Jt and TT are yk kU continuous. Because of the assumption /3.5/ that the mapping G is continuous. The mapping k H is continuous due to the assumption 13.6/. k So,only the proof of continuity of mapping Sk
that
is sufficient for proving Proposition
u~ €
1--+
Hence in proof of Sk continuity we can assume
/3.1/. Because the proof of this is rather long
where
UC
the topology induced by Hausdorff distance [All.
Therefore the continuity of 0k(-) for all k in
3
remains. Since U ~ ~(2U,) the topology induced in this H set by exponential topology is equivalent to
Qis
continuous whenever the mapping c(·)= (c 1C') , ••• ,c n is continuous.
)
from Theorem 2 of §'7,III in [A1) it follows
is closed. From /3.4-5/ we obtain compactness of the set Vp ' Then the set D is compact whenever the mapping c(·) is continuous. On the and /3.9-10/ imply that the functional
UCk
p~\{vkl»
is a metric space.
j j Let {Cu ,u )}j':1 C Hk(Vp)XU H be a convergent sequence with limit (ug,U )€ Hk(Vp )X U • g H From definition of the fami~y U it follows H
be the mapping €
UH
that there is a sequence {uJ}j:',CU
k
such that
Uk
for all j ~ ,: d (u j ,U j ) = 11 u j - ujll, and there
is such that
iSUgEU
d(Hk(v),U vk ) = IIHk(v) - u~1I • /A.2/LEMMA: The assumptions /3.4-7/ imply that
k
such that d(u ,U) =lIu g
g
g
'"'u 11. It g
has been shown in [A2] that the functional
for all k in 1;0 the mapping u~(.) is continuous. Proof: Let us introduce the following notation:
U'
.
U ,t:(2 ·)~(u,U) _ .1)(u,U) = d(u,U)€ IR+ k is continuous,hence j j j lIu - iijll c J)(u ,U ) _ 3>Cu ,U ) = lIu - ii: 11. .1):
g
g
g
g
The set U is compact as a projection ~f comp~t k set. This implies that the sequence {uJ}j:' j has a subsequence {u '} which converges to iiEU • k Norm is a continuous mapping, therefore lIu j '- u j 'lI _ lIu - ulI
U ~ ll (UC ) and kU k k Hk(V p ), U H 3 (u,U) ~ ~k(u,U) = uOe Uk' where
UH~ {U vk } VkE
Uk Pk(UC ) £ t'(2 ) ·)and k u O E: U is such that d (u,U) = I/u _ uOIl.
g
j
From definition /A.1'; i t follows that ~(-) is
j where {uj'}c{u } corresponds to {u '} • On the
the following composition
other hand j
u~(.): 5ko(H k , TTkUo GkOJLyk!Vp) ,
j
lIu '- u 'lI - l I u
g
- U 11 , e;
then
where
ull = lIu g - ii g 11 = Mu g ,U g ). It is easy to prove that u E U • Hence, due to
1*/ lIu g -
g
uniqueness assumption .) For any topological space X: ~(2X) denotes the family of all compact non-void subsets of the set X.
I~tfl
ug =u.
/3.71 from
/~/
we have
The point u was any accumulation point of
578
j:, ,
{uji
/* */
therefore
means that {ii
j}
X' X', X sets :D(2 ) ' :D(~ ), where X is open in X',
has
unique accumulation point. Hence ii j _ ii . g , because {u J } is a subset of the com~act set Uk.
5k
This latter result i~plies
is a subbase of exponential topology, the mapping B is continuous • Continuing, we shall calculate the inverse
is continucus.[]
we recall that for given k in ~ the mapping
images 01 sets in subbase under mapping B : k -, X 0 Bk (.1)(2 » = {v: {uk(v)Sx C~c X!
W (·) is defined as follows k Vp ) v ~ Yi Cv) =' k J(kC C(\ Cv) ()
{
(fu~ Cv)}
k
x C » e: t(2
where -
G
A
k
'" G
k
~
0
/A.3/LE}~~:
Jl
Ck
The assumptions /3.4-7/ imply that
.1)(2U~'
[
is continuous.
Ck') and :l)(2
Ck
the product topology be introduced
X' 2 )) : X,n X
='
2
f.
~}
X, n X e: J) (2 2 oX' {u (v)}xC E:c1J(2), k
~nkC(X)
X'
='
)
k=
X '" UXC
XI
X',
k
is
closed/, and due to Lemma /A.2/ the
(
.)
= (c,(.) , ••• "
A ,C
(·».
1;n :
k
='
min
Q
C
:iLkC(X)EC(~C;).
V p .,(J'3(V,C)
~ 1k(v,C)
c
"k
(.
,v) € lR
o ECk' k
where C g Jl (UC ) , kC k k C {Cl (3ve:Vpl C '" W (V)}<;C(2 ,) k and c~ is such that o (V ve: Vpl qk (v,C) =' ~(ck'v).
tV~
and kC are con tinuou s. Then it remain s to p~·ove the
k continuity 01 mappings B and B • k l'irst we do this for mapping B. For any open
From definition of the mapping lows that for all k in
set X ~ X' we have
1-;n
ek (·)
it fol-
(·»·
ckrl = "lko(id,W k Because of Lemma /A.3/ the mapl'ing WkC)
X B-\ d) (2 » '" X\{(X"X )e:X: X, 1"1 X 1"1(X', X) 'f~} = 2 2 X, {X,: X, 1"1 (X', X) f ~} x {X : X~ (l (X', X) 2 <-
X', X
»
~}
f.
=
sets~(01
is
continuous. Therefore we need only to show
{X,: X, n X -I ~} x {X : X 1"1 X f. ~} • 2 2 A set {X : X n A f ~} is open /closed/ iff when o 0 a set A is open /closed/ [A']. Therefore, since
the family of all
U is
pings defined for all k in
n
), .D( 2
/U'C
k) X"
A starting point for n proving it is provided by the following map-
=' 0 BO(Gk,B ). kC k k It can be easily shown that mappings n
X'
.hen (jU£lJ
k= X
" C
,
Now, the mapping W (·) is the following comk position
B- (:D (2
u}
oH,erwi se.
imaees are open. This is sufficient to prove
k'
{(X, ,X ) E: (.1)(2 2
Vp.)v~Bk(v)
and,
Vp\{v: u~(v)e:
='
tl~e B continui ty. 0 k Now, we can prove continuity of the mapping
. X ) (X, ,X 2 ) ~ :B(X, ,X 2 '
G
' , X)) '"
mapping u~(.) is continuous, then both inverse
x,gu'xc'
Yi (·)
k
{u~(v)lXCkCX'}\ {v: {u~(v)}~C~C X', X]
VP
set
.)
For convenience we employ the following no-
L(2 ') .) X
X
UXC
X
open /closed/iff when the set U is open /the
tations
X
J:'l(2
when(3U~Uk)
Since the set U • C )
U'x C'2 the exponential topology, and in (.;oC2 ~ ~»
k
otherwise,
{v: k.
Proof: Let in the sets
g
u~(vH: U}
~
B~\ II (2 X ),
) ,
Y
for all k in ~ the mapping Yi (·) k
X
£v:
continuity of mapping
~k.
First we shall prove the continuity of the C auxiliary mapping qk. Since the set Vp ' 'r( 2 k') is a domain of this mapping, therefore, similarly as in proof of Lemma / A. 2/, we can assume that this domain is a metric space. Eecause of functional
and family of all
( v ' ,c 1) E V x p
.) For any topological space X: ,1)\0) denotes the family of all closed. subsets of the set X.
579
c'k
we b.ave
~
continuity for any
2
2
> 0)(3S v ' 0c)(V(y ,c 1€
/A.4/ CH
1 (lIv _ v 2 11
V
1
A
1 IIc _ c 2 11< [)
1
2
C~)
R1FEBbNCES
'>
[A1]
Vp '
c
I'\lc ,v) - ~cc ,v )1<£]. 1
2
C'
1
2
c-")
Let C ,C €. t(2 k) Le such that dist CC ,C )< 0c • This means that 20 2 OC E:C )(y c1EC1)lIc1 _ c 2 °11<&
/A.S/
C~E:C2)lIc10
{ (3c 10E:C 1)(y
Functional
_ c 2 11<
is continuous, the sets
~
,c C
min ~(. ,v
j
),
j
2
ard C
1Qk(c
2*
1
20
~ '~(cjO,vj) ,
h 1
1
2
2
=
'I'he inequality IIvll~ r = lIu'lI
= 1,2
j
IIv - u'
11
)1. It is obvious
and f or all € > 0 1 2 1 Cllv _ v 2 11d 1\ distCc ,C >< S) = )
Let u" lie in frX(8,r)
2
/qk(v ,C) - qk(Y ,C )1<£
Therefore, from
It is easy to see that now we can prove con~k
f'\
frX(v,r - livID,
= IIv
hence lIu"lI = lIu'l/ and /Iv - u"lI
what means that the mapping qk is continuous. tinuity of mapping
that the point u' lies in
/*/lIv-u'II+IIvIl=lIu'lI.
c
2
S1"vll implies
11 - 011'lIvll = C01- 1)lIvll lIu'lI- I/vll = r - IIvl/.
fr X(e,r) n fr XCv,r - IIvll) and
v 1
+ S1v},
equals
1 1 2 2 Therefore we showed that for any (v ,C ), (v ,C )
1
{u
that 51 ~ 1. 'l'he norm of difference v - u'
1 2* 2 ,v) - ~(c ,v)1
Iqk(v ,C ) - qkCv ,C
ve:1(CiI,r) then
(3S1~O)u'=cS"1v.
we obtain £) ',\(c
f.
u'e: R} {u: (:3'l~ 0) u :. 'lv} be such that lIu'lI =- r. Since u'E:R ' then v
Hence, in the presence of an obvious fact that )
the following lemma.
only the case when u = 8. Let
2
,v)1 < £
2 10 1 ,v) - ~Cc ,v)1 < f.
,\:(cj",v j
APP1NDIX li Froof of Proposition 13.13/
where 1 ~ 51 = II~II • Proof: For convenience we shall consider
C ,v) - ~(c
ci t.
fr J(iI,r) n fr X(iI + v,r - liviD =
1,2.
/A.S/ we have
1~
Ope
LEMMA: I f UkE: D~, r >0, e
j
1'\ (c
[A2} Yioir.iak, A.,
Before we prove the proposition we state
SC
are compact, therefore
Now, from
Kuratowski, K., "Topology", Academic Press, New York, vol.1 -1966, vol.2 - 1966.
2
in the same manner as
/*/
- u'lI.
we have
lIu"- vII + IIvll = lIu"lI = lIu"- v + vII. Because U
k is
a Hilbert space the above' equali-
continui ty of nal'ping ~k /the mapping qk plays
ty implies that
the same part as a mapping ~ and uniqueness
/**/
(302 ~ 0) u " - v = 02v •
assumption /3.8/ as /3.7//. Consequently we
I t is easy to see that lIu"- vII = IIv - u'lI
omit details.
(01 - 1)lIvll. Therefore from
As we mentioned in the
be~inning,
continu-
/IB/ we have
ity of cC·) is sufficient for the existence of
S2 = 51 -1. It means that u"= 51 v = u', what terminate s
coordinator problem's solution. Hence the pro-
the proof.
position is proven.[]
0
The proof of Proposition /3.13/ is very
/A.6/REMARK: From the above proof it is easy to
easy now. Let ue.JY(U) and d(u,U) = lIu - uo" = r.
see that Proposition /3.1/ is valid when the
Let us assuIl;e that
sets only.
U~, C~, Y~,
k = 1, ••• ,n, are metric spaces
VE.Ju,uo[ ~ {wE-D': (30
0
Becuse of above Lemma we have .).
d~stCC
0
(1 -A)u +llu } •
1
2
,C ) denotes the Hausdorff distance 1
between the sets C
2
and C •
580
fr1(u,r) () fr K(v,r that is v
t
J'{(U).
= {uOl ,
IIv - ull)
Because v (; ]u,uo[
this im-
plies that a point which does not lie in $(U) exists in any non-void open set. Hence the set
$(U) has a void interior.[]
Coord
/",,,..~ v(lrtables
LocaL
LocCl. L
controll~r
controtler
MeClsurlments
Set po1(lt
or pred 'ch 0"
or predietLon
r - -
f-
- l
-
O'r.ct contreller of subs~lte",1
I
I I
I
I Sub"ste ," 1 "n a I -stead~ n~ar
1
I
r
I
stl1te I
Sa b pro_eH 1
I"ter,).cticns
Disturban'es I
I L
Mea sure ,nent"
v(ll ... ~s
-
-
- -
.J
,- -
- - -
I
t cc Dcr
I I Su b"
1 r
~"
I I I
-
Il
et
I
Su "process 2 L -
-
- -
sie",
l
i1ea.r - ,tead~
I
Q
state
I
j
I
D~st.(bance5
I .J
Outputs
Fig.1.1. Converting a dynamic process to a static system in order to use a sequence of static optimization problems, and decomposition of static problem. Coordinator
/"".'"
va.r"LabliS
L oc 0. L
L cc a L
CC nt (elLer
controller
Set
r
- -
pOlnt v/ltues
- - - - - - - -
- - ..,
I I I
I S"bs~st~m
1
I nte ractLo";
Subs~ltem
2 I
___
L
..J
Real system output. or ~,.,terotti.o.,~
Fig.1.2. Two-level system with feedback.
581
9C v)
Coordinator problem
Setc,Hi lev.L
c,( v)
-~
-
~
c/v)
~._-
Uppu local
proi>lem
L o.,e ('
Lower lCCQt
l eCc1l prcbLem2
proolem 1
-
.
.- ' - . -
G1(v)
r - -
2
l!~ (v)
l!~(v )
-
'-'-'
Upper local
problem 1
Fi.nt Level
_. _.
-- -
I I
I L
-- -
~1
-
- u- -
.
-
c\cv )
_
.
-.
.
-
'_0-
- ---,
-
2
I
P.. 2
P.. 1
I
'"
- -
--
;(v)
= P,,(C(v»
PlO ~
Fig.2.1. Structure of two-level system with parametric coordination with feedback.
c'1
o
Fig.3.1. The sets UC , P1-1qv1P 1 in the example /3.11/.
Fig.5.1. The structure of system in the example.
and ""1 (v)
582
Val .. e of pertcr",a.nce
HO 146
Coord~na t~o.
var'l1bles 11,2
~.O850
~11
~.522336
"12 2·1010
~l
2..]256]-2
"z.
O. C, 5'60
111.
V3i
l.j, 15 ~O
V>l
-0.4160
vii
Ini.tLa.l pOL.t (eq,u.a.ls "'oc!.el opti. .. al)
156
2,1..14566
~~L
-0.%2606
3.G~H30
l311
4.n01b4
Vu 2t&3~05r
~"2.
2.g128Ilj
':lz
0. 51 1023
~,1
~.02.21S5
"11
0.5
"2
paro."'etr,c
150
COOydi.no..ti.o..
V~1
0. 53 t wo G,O-114g8
V~l -O.O22~~5 146
Optif\1i.2irli Gn ...~a.l s~,h",)
1'j1. 625
125",
~1t
5O.~~8g
~)2 -O.~ 1650fi
V 11
4.1B384
~11
4.12338lj
V1L
~.01Ug~
~IL
3.01l8g~
V7.
O.lt~H61
~l
O.lj~26G1
V,1
3·~161U
':l~1
3.<;f61U
"3L -O.5l5'6~3
~)2
-0.515&43
pOl.t
ValILe of pH for ",,,,nee
O.4G222.lj
~>1
154
Po,. t obta.'ned. a resl4.lt of
Real s~ste... ou.tpu.ts
/l4.g84g~~
HO
Wumous 10
20
30 B
~o
50
60
/0
(f0
90
100
of
iterotlons, or up"rLIY1tnts
Fig.5.2. Convergence behaviour of performance junctional in the example.
583