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Completely Decentralized Output Control Based on an Approximate Mathematical Model
COMPLETELY DECENTRALIZED OUTPUT CONTROL BASED ON AN APPROXIMATE MATHEMATICAL MODEL P. Tatjewski and M. Cygler Institute of Automatic Control, Technical University of Warsaw, 00-665 Warsaw, Poland
Abstract. The aim of this paper is to present an algorithm of adjusting the real system outputs to given desired values. The controlled system is assumed to be described by an approximate mathematical model, i.e. differences between models (used by control units) and the real system characteristics are taken into account. The algorithm is designed for the optimizing control layer of the hierarchical (multilayer) control structure ot the system. The goal of the control is to obtain such stead3-state of the system"that given values of real system outputs are achieved, the constraints are satisfied and optimality of performance is taken into account. To achieve these goals feedback information from the real system is used on-line in the algorithm, along with the mathematioal models mentioned before. The system is assumed to be largescale, oonsisting of interconneoted SUbsystems. Por such system the algorithm is oompletely decentralized, consisting of a set of its parts, each working with one subsystem. The algorithm is thoroughly presented and analysed in the paper, a number of numerical results is also given. Keywords. Large-scale systems; hierarchical systems; control theo~; optimizing control; decentralized control; nonlinear systems.
INTRODUCTION The most widespread concept in hierarchical control is perhaps that of splitting the task of determining the control m of the controlled system into two mutually related oontrol functions: the follow-up control where a direot controller sets m so as to Cause the (appropriately chosen) regulated variables 0 to possibly follow their desired values (set-points) Cd' and the optimizing oontrol, where the 157
optimal trajectory of Cd is determined. Pigure 1 presents this kind of structure, with reference to oontrol of a large-scale system - that is a system composed of N subsystems interconneoted via an interconnection matrix H. Every SUbsystem is influenced by disturbances zi and f i , and by other subsystems thro~h interaction input ui • It influences other subsystems ~ its interconnection output Yi. The subsystem control inputs are denoted by ~, i=1, ••• ,N.
158
P. Tatjewski and M. Cygler
oprlMIZINS
CONTIlDL
FOl.LOH-UP CONTROL
riq 1. Hierarchical cont~ol. of a complex SI/stem layers
J4ith folloH-up and opllmLzLng controL
The concept of splitting the task of determining the control into some mutually related control functions, introduced by Lefkowitz (1965), is generally known 88 miltilayer control hierarch3. The follow-up and optimizing control layers constitute the most important part of such hierarcb3. General aspects of multilayer control structures were discussed in the literature, see e.g. (~indeisen, 1974; Pindeisen and others, 1978; Sandell and others, 1978), along with presenting several control algorithms for both follow-up and optimizing control l~ers. In this paper we are concerned with optimizing control layer and, moreover, with the case when its algorithms can treat the system (together with the follow-up layer) as non-dynamic, i.e. operating in steady state (or, more precisely, in a sequenoe of steady states). It is the most important case which clearly shows the appealing feature of splitting the control into two layers: the task of optimizing control may be considerably simplified in comparison to dete~1ng m directly from a
dynamioal optimal control solution. Conditions under which steady-state optimizing control acting on a follow-up layer is a reasonable approximation to the true dynamic problem are exposed in(Pindeisen, 1974; Pindeisen and others, 1980). Intuitively, this approach is valid for slow disturbances acting on 8 fast system. In a continuous chemical prooess, for example, the prooess parameters c (see ~ig.1) such as temperatures, pressures, concentrations, etc. may have to be varied from time to time in adjustment to some slow disturbances z, say changes in raw material composition. The new set-points cd for these process parameters may be determinable by steady-state approach (e.g. optimization). On the other hand, the process control inputs m such BS valve positions have to be adjusted much more frequently, in response to fast disturbances f such as heating gas pressure, for example. This adjustment can successfully be done by the follow-up control systems, trying to keep the process temperatures, pressures, etc. to their desired and constant values cd as prescribed by the optimizing control. If the model of the system and the measurements of disturbances are sufficiently accurate the optimizing control l~er can operate open-loop, i.e. for every new value of disturbances z the optimal model-based value of set-points Sd(z), where i is the estimate of z, is evaluated and transmitted to the follow-up control layer. Unfortunately, it is not so often in real systems that the measurements of disturbances (or their estimates) and, first of all, the system model are accurate in the above sense - the optimal
Completely Decentralized Output Control
model-based set-points cd(Z) may fail to be real-optimal, and, what is even worse, they mB3 violate the system constraints. However, in on-line control one is not left with his model of the system alone, there exists the possibility to observe the actual values of the system variables and to use this information as a feedback to the control algorithm. Extensive studies have been made of iterative control algorithms for that purpose, see, e.g., (Pindeisen and others, 1978; Brd1 s, 1978). Thus the optimizing control layer consists in fact of two sublQ3ers: the model-based optimization layer (acting open-loop) and the correction l~er (acting iteratively and using feedbaok information) as Fig. 2 is showing.
HOJJEL-BASED
OPTIMIZATION
CO~RECTION
(SUB)LAYER
Fig 2. Optimizl'ny control uSl'ng (eed.hack
and iterative correction.
The task of model optimization is to oalculate the optimal model-based set-points cd(z), when new value of z occurs (is measured or estimated on the basis of present and past measurements). Then, if needed, the Correction layer improves iterative13 these set-points on the basis of measuring the interaction variables
159
in the system, i.e., the inputs ui or outputs Yi of the Bubsystems, see Fig. 2. These measurements are feedback information. Let us note that for the complex system each of the layers should be structured as a multilevel hierarchy consisting of local units and of a coordinator. :Por many reasons (Findeisen,1974, 1978; Pindeisen and others, 1980; Sandell and others, 1978; Singh, 1977)it is desirable even for model-based optimization (although it is not shown in Fig.2) , and it is necessary for the oorrection layer. Moreover, the most desirable situation occurs when the correotion layer is designed in such a way as to work satisfactorily without its coordinator unit, in a fully decentralized way. An idea of such algorithm, basing on price approach (with Lagrange function: was presented ~ Findeisen (1976) and thoroughly investigated by Brdys and co-workers (Brdys , 1978; Brdys and Michalak,1978; Brdys and Ulanicki, 1978). This algorithm oan be used, however, only when the system satisfies some "weak coupling oondition", what significantly limits its applicability. It has also some features typioal for price-type methods. The aim of this paper is to present a new fully decentralized correction layer algorithm, basing on direct approach. The main advantage of this algorithm is that the system does not have to be "weakly ooupled", and that the constraints more general then in the price approach can be handled. The algorithm has some other advantages and drawbacks, rather typical for direct-type methods. DESCRIPTION OF THE SYSTEM It is assumed that the controlled system inclusive of its follow-up controllers (see Pig.2) is desoribed Qy a set of subsystem input-output
P. Tatjewski and M. Cygler
160
vector mappings :p. i:
ei x U i x
~i --..
whioh are only apprOXimations of reality
":1 1 ' i=1, ••• ,N,
Yi ::
where N denotes the number of subsystems and ~i' U i , 2,i' ~i are fini te-
y :: K(o,z) ,
i=1, ••• ,N (1 )
where the variables c i E. ~i' u i E U i , Yi E 1J i , z1€ Zi are the i-th subsystem set-points (called "controls" below, since the set-points are controls for t'he optimizing control layer), inputs, outputs and disturbances, respectively. The sUbsystems are interconnected, the structure equations are assumed to be in the form N
ui
= Hi' = ~
Hij Yj' i=1, ••• ,N, j=1 (2) where Hi =[Hi1 ••• HiN ] are matrices
composed of zeros and ones. Introducing the global desoriptions c
T
6
T
(c1, •••
T
~
~
,oN)E~1x••• xJi
T.4 TT." Y = (Y 1 ' • • • , YN) E. '1j 1x. • •x IN
4
::
Y7
~
4.....,
,
= .J,
et
u
::z
c.,
=K*(c,
where
(6 )
Gij : ~i)( U i )( 1ji...... R' j , J i ,
and J i denotes a set of integer indexes, i =1, ••• ,N. Due to the, rather very general, form of the constraint sets (6) they oan be assumed to be exaotly known (Brdys, 1978). With each subsystem a known local performance function ~i'>
--..... R
is associated,
and we assume that it is to be minimized. The overal performance funotion of the system is assumed to have the form 1
,QN(cN,uW'YN»'
()
Of course y
,j~J1J,
1
= Hy, 4 (FT , ••• ,F*N)' T =
y :: K*(c, z).
CUY1 ~ {CC 1 ,u i 'Yi): Gij CC 1 ,ui 'Y1 )flO,
Q't' (c ,u,y) = \f'(Q1 (0 ,u 'Y1 ), ••• ,
F H (c, u, z),
T] HT 4[ :: HT a1 1 •• .HN, and superscript T denotes transposition. We will assume that in the system as a whole for each 0 E ~ and z £ ~ there exists exaotly one output, that is that there exists a mapping K*: e" 2,--...y
T where It'*
where Pi' F and K are models of It'*i' F". and K 1I , respectively. Next we assume there exist local constraint sets
Qi:
the system as a whole can be described ae follows y
i=1, ••• ,N (5)
y = F(c,u,z)
dimensional spaces, as follows Yi = F*1(c i ,ui ,Zi)'
~i(ci,ui,Zi)'
z)~y =It'~(c,Hy,z)
(4) It is assumed that the system mappings m~ be not known exactly and, consequently, we have to use models
where 't': ~N 1R is some stricty order preserving function (for example, 't' =
:i!: ). i=1
THE
DIRECT-TYPE CORRECTION
ALGORITHM WITH WeAL It'EEDBACKS Let us assume that for given value i of the disturbanoe estimate the model-based optimization has been performed resulting in the values 0d(Z) and y(i) of set-points and interconnection outputs. Then the optimal model-based set-points Cd(Z)
Completely Decentralized Output Control
161
have been applied in the follow-up control la3er and after the transient process in the system has decayed Z some value y. = K*(c d (z) ,z) of the
output equations shift, the following local decision problems are solved
interconnection outputs has resulted. Of course, if our lmowledge of the system would be perfect ( that is,
find ci(si'Y) such that
A-
li-P.i' z 1.
i=1, ••• ,N, and
Itt-
and
y(z)
:I
€CUYi ,
z
,..
i
-
z = z) Z
z
(ci(z),u.i'Yai)
= 1, ••• ,N,
ci(si,y)=arg min Qi(ci,HiY,y i ) ci
then
,.
Bubject to €
j
where
y~ ~ A
y(z),
Z
ce (c i (z), Uti' Y.i) ,. CUY i can occur, at least for some i. The correction layer action is now necessary, first of all to satisfy the constraints in the real system. Observe that this goal will be satisfied if we modify the controls Sd(Z) to a value o~(z) • Bd(z) + dO in such a way as to force the equality K~(C~(Z) ,z)
:I
y(i) despite
J., 1.
i · 1 , ••• ,N.
generally, and hen-
Z
-
£,
Yi+si=Fi(ci,Hiy),
z
uai • HiY~ • But where it is not the case
(7)
A
Gij(ci,HiY'Yi)ECUYi'
of
It should be noted that the local problems (7) are precisely local optimization problems of the direct method of hierarchical optimization A for y = y, (see,e.g.,Tatjewski, 1978) if 8i=0 is set, i =1, ••• ,N. However, the model-based optimization was accomplished previously (with the result Od) and we try to improve its result using nonzero values of Si,
the model-reality differences, and
i.e. shifting the values of Pi(Ci,HiY)
demanding (c~i (z) ,ui
to equal them with the values of
,..
_
(z)
A
where ui(z) = Hiy(z), i
,y i (z») €
CUYi ,
= 1, ••• ,N.
The optimality of the performance must be of course also taken into account, hence optimization basing on models of the SUbsystems will be used when generating c~(Z). Since the disturbances z and their estimate are assumed to be constant dUring the correction process, in the sequel they will be dropped in all mappings, to simplify the notation. The value c~ will be generated iteratively, U8ing using feedback from the real process. Por each value of some aUXiliary parameter
z
F*i(ci,HiK~(c)),
i
= 1, ••• ,N.
Of course, the way how to choose the values of output shifts Si cannot be learned from the models· only, hence it will be obtained using feedback information from the real process in form of the measured interactions K*(c(s, y)). Our goal is, therefore, to find such value s of 8 that K*(C(B,y)) = Y or, in other words, to find the solution 8 of the operator equation R~(s)
= 0 ,
where
aT. T T =- (s1'··· (8 1 , ••• ,sN) £ '.11)( .... x'.i N' having the interpretation of subsystem LS.s.--G
The equation (8) will be solved
(8 )
P. Tatjewski and M. Cygler
162
iteratively, using the following Newton-like sheme sn+1 = an -
£L(sn)R*(sn)
,
A
I
[
4
]
K(o)= I-P* u (c ,~. )H (10)
where L > 0 is some step coefficient and L (. ): ~ - '1j. If £ = 1 and L(.) = [R~(.)] -1, then (10) would be precisely the Newton scheme. However, even if all the models and system mappings (Qi' Fi , F*i' etc.) are differantiable the operator R* may be nondifferentiable, since may be nondifferentiable (the known feature of optimal point mapping of parametrized optimization problems). Even being differentiable, R~(.) would be not possible for evaluation due to the fact that the operator K* appearing in its definition is not known precisely. Moreover, the I A evaluation of c s (.' y) would be
c(., y)
extremely difficult. Therefore the Newton-type scheme (10) 1s used where L(.) is a reasonable approximation of [R~(.)] -1, basing on the models only. To evaluate L(.) the following methodology will be used in the sequel: assuming R*(.) is differentiable we evaluate the precise formula for R~(.) and then, making some neoessary but reasonable approximations, we simplify this formula to obtain L(.) in a form possible for using also in nondifferentiable case, and basing on models only. The simplification will be justified by pricise mathematical oonsiderations. Let us assume therefore, for the time be ini , t hat K ~ and y) are differentiable, hence from (9)
"
•
A
-1
I
"
F. c (c ,H3.) (12 ) 4,.
A
where c = o(s,y) and y.= K.(o), for notation simplicity (these abbreviations will be also used in the sequel). The assumption of differentiability of C(., is almost equivalent to the assumption that, for the oonsidered variations of s, only the same set of inequality oonstraints G ij is active. Let us assume therefore, without loss of generality, that all inequality oonstraints are active. Then the Kuhn-Tuoker conditions for all local problems (7) have the form
y)
'(A A A T I ~ ~ T~ 'ft ~ A T Qc c,HY,y) -F c (c,H3) / l +Gc(c,HY,y) =
=0 G(o,HY,y) • A
0
(1) iI\
,..
Y + s - F(c,BY) = 0,
N :z: i::1
A
where Q(c,u,y)= A
G
(G 1J .. )(.J
Qi(oi'u. ,y.), ~
J i; J.= . 1,
Eo
~
••• , N)'
and A ',If are Kuhn-Tucker multipliers for equality and inequality constraints, respectively. Using the implicit function theorem for the set of equations (13) we obtain " '" I
A
A cs(s,y)
A
Ps (s ,y) =_ I
A
I
A
o
B
As(S'y)
o
o
o
I
(14 )
where A : Qcc 11 (,. u~ A )_p 11 (A U~) C ,J.J.J ,y cc C ,J.J.J
cc
C 'J.J.J
,y
cc
C 'J.J.J
A+
c(.,
B AI " R*(s) = - K.(c(s,y».cs(s,y). ,
'"
A 1:1
'A
A
A
Gc(C'HY,y).
A
(11 )
Using the implicit function theorem and taking into account the identity (4) we have
After oomputing the inversion (assuming it exists) and operator compositions we find
Completely Decentralized Output Control
where D = A-1_A-1BT(BA-1BT)-1BA-1. (16) Putting now (12) and (15) into (11) we obtain
163
minIlYi+si-Fi(Ci,Hiy)D ci
(20 )
subject to Gij CCi'HiY 'Yi)
R/Cs) .. -[r - P~Cc,H§)HJ-1
(18)
Taking F instead of F~ is necessary since F is just the known model of :P~. On the other hand, taking y instead of YI- ~ K.( (s ,y » is re as onable since K#(c(s,y» is just the value of real system outputs deviating from due to model-reality differences, and forced to be equal to y by our algorithm. It should be pointed out that the above simplifioations, being natural and reasonable, lead at the same time to very simple form of R. In all our considerations we have tacitly assumed that for all Si from some appropriate neighbourhood of the origin the local decision problems (7) have feasible solutions. However, it may not be true due to the constraints. Then the following modification of (7) can be made. Assuming Si is a given value, the projection ~i(8i) of Si on the set
a
y
)
•
The described modification does not alter our previous oonsiderations, only (18) becomes a bit more crude approximation, since now
R~Cs)
= -[r-p~CC,H3*)
H]-1p~cCC,Hy*).
'DP~CC,HY)T[p~CC,HY)DP~CC,BY)TJ-1W~ (21 )
where
11 ~ (1r1 ,··.,1fN ).
=[R1-
1 we get the following Taking L iteration formula (10)
+f[I-F~ (0 (sn ,y
),H3 )H] (y-Kt(8(s~ y»))
(22) Let us show. finally. that the algorithm (22) is decentralized, resulting in the control structure with local feedbacks only. It can be easily obtained, ~ straightforward calculation, that (22) is equivalent to a set of N equations
n+1 n [~ n Si =si +t (Yi-Y:JJi) + I
A
nA
I'
N
A
+ (Fi)Ui(ci(s ,y),HY)ZH .. (y.- Y~J.)] j=1 J.J
J
(23 )
~{si£ ":1 i :
3c i
A
Yi+si
12
~ ~i
(
H" ) ci ' i Y ,
(19 )
i=1, ••• ,N, is first performed, and then 'ffi (si) instead of 8 i is used in (7).
Ji ,
(17 )
Si (y) ,
Si Cy
'i
• 1, ••• ,.N.
i
R;CS) .. -[r - P: u cO,H3.) HJ-1 P;cCC,H32C) DP~CC,HY)T[p~CC,HY)DP~Cc,Hi)TJ-1 A rather very reasonable approximation R of RJcan be now obtained by putting into (17) P and y instead of F~ and y¥ , respectively. Thus
O,j
6
i=-1 , ••• ,N , where y~ : KJt(c(sn,y». Taking into account the structure relations (2) the above equations can be written in the form
164
P. Tatjewski and M. Cygler
(24)
n 6 where u,* =
~
HK~(c(s
n
i=1, ••• ,N, ~
A
A
,y)) and ui=Hiy.
The variables U~i are interconnection inputs to the i-th real subsystem oorresponding to to the oontrols " n" c(s ,y) , and they can be measured locally, like the outputs Y~i .Hence, the algorithm (24) is decentralized, the control decisions are taken locally in each local decision unit, using local input and output measurements and subsystem models. Only the value of parameter f should be equal for each subsystem, and the next iteration should be made in all local decision units simultaneously, after the transient process in the system resulting from the previous iteration is finished. The structure of the local decision unit is shown in Pig.).
The iterations atop whenJYi-y~iD are below the prescribed treshold for each i =1, ••• ,N. In all previous considerations the assumption was made that the real system outputs y~ should be foreced to be equal to Y~ the optimal model-based outputs y. It is quite reasonable when the model-based optimization can be performed. However, the algorithm can also work when insted of y another, quite arbitra~ value y8 of the outputs is required. It should be only not too far from the actual system outputs to assure the convergence in the nonlinear case (see next section). CONVERGENCE AND OPTIMALITY
CONDITIONS
0°
set
n=O,
s/=o
Let come back to the more compact form of the iteration equation (22) I -1 sn+1 =sn + £ [R(Sn)] R (an) (25)
and t}o to
n n 1 mea-Jure lJ,i' u.i
*'
0
"
nf1
n
fi "
n
n"
2 Si::l Si +EL(lJi -Ij"i)-~' (Uj -ull'iJ), n~( )1 ('" n If. " I' i= ~. ni Ci (Si' Ij)) If'/) j
The following theorem formulates sufficient conditions for the convergence and gives the convergence rate estimate of the algorithm (25), with starting point so.
nolI" d' 3" eva lua te c,."" (Si' IJ) accor 1ng to (7) and appllj to the sljstem 4" HQit until the transient proce'1'J is f£nished and go to t" Hith n=n+1
~.
i-rH SUBSYSTEM Hith 'OLLOH-UP CONTROLLER
lj.i
Fig 3. The i-th local decision unit
Theorem 1. If on some set S c1j the following conditions are satisfied: (i ) DC R,. (8 ) R(8 1 )) - (R:le( s 2) 1
f
-t
R(8 2 )) U~ k·l[ 8 1-s 2 11
(it) 11 Rts 1 )- R'(s2)H
The first step (n=O) is explicitly shown in Pig.3,consisting of the application of the optimal model-based control 0di= ci(O,Y) to the system.
(iii) UR'(8)-1U ~
f
~
£kf
U8 1 - 8 2 "
(iv) IlR(so)U~~, h = 1 -
So
+ k~ + ~kl,2rz. ~ 1
Completely Decentralized Output Control
Cv)
Bca o,r) ~{a ~ ":1: lIa_aoll~ for r =
~
r} ~ S
outputs y*= K~(C(8,y)) are equal to then, since at each iteration step
y)
,
(ci(Si,y),HiY'Yi)~CUYi i=1, ••• ,N,
1-h -then there exists a solution
s
,r) of the equation (8) and the sequence {sk} generated by (19) converges to this solution, moreover
n 2
,
if
2 h n ,.. s -all~-·-n ,if k~ 1_h2
1 -£+ kA~ 0 1
-~+
k~
= 0
•
The above theorem is a rather straightforward application of the powerful theorem 1 from (Zinchenko, 1973) to our problem, hence the proof is omitted. Let us comment the assumptions. The first one is on quality of the approximation - in ideal case, when R = R s (and Newton step-legnth Eo = 1 is taken) it is always satisfied with k=O. (ii) is a standard requirement of Lipschitz type, and (iii) requires that variations of R,(.)-1 on S are not too great • The assumptions (iv) and (v) , typical for nonlinear problems, require appropriate quality of the starting point so, combining it with previous requirements - observe that if the approximation is good (k small) and o s is appropriately close to the solution (~small) then (iv) and (v) are satisfied. In fact, £ = 1 could be set in all our considerations, like Zinchenko (1973) has done, but then we would loose an additional degree of freedom. Choosing £ -F 1 we have the possibility to change the convergence rate of the algorithm. This can happen to be successful, hat some computation results have shown (Cygler, 1978). en the convergence is achieved for some s = (that is the real system
s
have (ci(si,y),HiY,*,Y.-i)E CUYi , i = 1, ••• ,N.The constraints are satisfied in the real system. It is the most desired feature of our algorithm. However, immediately after that one asks about the optimality of the control C(B,y). It is obvious that this control cannot be strictly optimal for the real system, since we do not know the real mappings F~i and y is, in general, not realsystem optimal. However, the following property is an appealing feature of the algorithm.
we
~ -B(s 0
lIan-ell ~ rhn
165
Theorem 2. If the differences between real and model subsystem output mappings are additive, i.e. F#i(ci,Ui ) = Fi(ci,ui ) +~i' (26) i:::1, ••• ,N, where
are some elements from ".1 i , i = 1, ••• ,N, -t'hen J. =-ti.., i = 1, ••• ,N and the ]. control 0(8,y) is the best possible for the real system with y*= y, i.e. it is the solution of the problem 0(
i
s.
~" min Q't' (c,HY,y)
c
sub.to
(27)
Yi=F~i(ci,HiY)'
1=1, ••• ,N. ~.
A
A
A
The controls ci(s,y) are
solutions of the local decision problems (7) for 8 i =8 i , hence they are solution of the problem min Q 't' (c ,Hy ,y ) c sub.to Yi+si=F.i(ci,HiY)-D(,i (c.
J.
,H.Y,Y·)E CUY., ]. J. ].
(28)
i=1, ••• ,N,
166
P. Tatjewski and M. Cygler A
A
and Yi :: P*i(ci,Hiy), i = 1, ••• ,N
(c
».
since y • K ~ (s,y Comparison of (27) and (28) constitutes the proof.
CUY2 = {c 2 : 4c21+2c21u21+0.4U21+ +
•
Por more complex model-reality differences it was not possible, as yet, to achieve constructive results. It can be only shown (by reasonable assumptions) that a certain "continuity property" is satisfied: when the model-reality differences tend to zero then s ~ 0 and c (s,y) tends to the real optimal value. Howover, the computation results show that the control c(a,y) is, as a rule, superior to the model-based Od = = o(O,y) not only in terms of the contraints satisfaction, but also as far as performance deterioration is considered (Cygler, 1978). ,.
"
...
c21c23+0.5C~1+u~1::: 4,
0.5c21+c22+2C23 ~ 1, IC2jl~10, j
= 1,2,3 }
Subsystem 3: 2+2.5032 2 Q3(c 3 ,u3 )=(c 31 +1) 2 +(u J -1) P3(cJ,uJ)=c31+2.5cJ2-4U3
A
CUY3
= {c J :
c 21 +u 3 ~ -0.5,
0 ~c32~1}.
The output relations of the real system differ from its models and are as follows: P*1(c 1 ,u 1 )=a11c11-c12+2u1+812c11u1 2
PS21 (c2,u2)=c21-c22+a21u21-Ju22+a22c22 P*22(c2,u2)=2c22-a2Jc23-u21+u22+
EXAMPLE RESULTS
Let us consider the system structure shown in Fig.4.
+ a24c22c23+a25u21u22 P~3(c3,u3)=a31c31+2.5c32-a32u3'
and the values of parameters
Fi9 4. The
example
s!lstem structure
8
i j
are
given in Table 1, together with the corresponding values in the system model. Notice that the model output equations Pi are approximate linear parts of the real output equations F#i' which are nonlinear, 1 1,2,3. The optimal model-based solution AT A y -(Y1'Y21'Y22'Y3) is aa follows 3
The subsystem models are as follows: Subsystem 1: Q1(c 1 ,u 1 }1ll(U1-1}4+5(C11+c12-2}2
P1 (c 1 ,u1 )=c11-c12+2u1 CUY1 lll{c 1 :
c~1+0~2 ~
1
J
Subsystem 2: 22222 Q2(c 2 ,u2 )-2(c 21 -2) +c22+3023+4u21+u22 P21(c2,u2)=c21-c22+u21+3u22 P22(c2,u2)=2C22-023-u21+U22' 4
P'2=(P 21 ,P 22 )
It>
"
"
A
Y1= 0.028 AT Y2= (0.179, 0.005) Y3= 0.331 When the optimal model-based controls cd=c(O,y) are applied to the real system the constraints are violated, and the application of the correction algorithm seems indispensable. Applying our algorithm, the local decision problem (7) corespond1ng to the se-
Completely Decentralized Output Control
oond Subsystem was solved numericaly using conjugate gradients and penalt, shifting technique to handle the constraints (first and third looal decision problem could be solved analytically). Also the projections ~i on the sets (19) were solved using the same numerical technique. The iterations were stopped when the Euclidean norm of the difference ,.
A
n
A
A
n
K*(c(s ,y)= R~(s ) was below the given accuracy parameter ~=10-6 The re8ults, for a few values of ateplength parameter £, are given in Table 2. The obtained optimal shift parameters
y -
"T =(s1,S21,s22'S3) ". ,. " " were
8
81 =-0.142 AT
8
2
=(-0.017,-0.051 )
S :a-0.30) To evaluate the sUboptimality of the A " obtained control It.c(s,y) the real optimal control 0 (unknown in practical applications) was also computed, as well as the real optimal A ". value Q = Q(c,H3) of the performance function. It is shown below, together with the value Qs=Q(C(s,y),HY) obtained in the real system after the application of the controls C(B,y) resulting from the correction algorithm " :z 10.83453 Q A
Qs = 10.97342 Hence, ~he suboptimality index is as follows A Qs - Q • 100% = 1.28%. 6 = A Q
It should be realized that comparison of the performance function value obtained for the model-based control A Cd with the value Qs is unreasonable since the control Cd violates the system constraints. Next, the correction algorithm was tested using nother than y and
167
quite arbitrarily chosen desired values of the outputs. The results are shown in Table 3. The casea 5,6 and 7 are chosen in such a way that the optimal shifts lie on the boundry of their feasible sets (19). In all cases the steplength La 0.9 and the accuracy 9= 10-6 were used. Pinally, the correction algorithm, with desired outputs Y,l-O.9 and 9- 10-6 ,was tested several times using the real system parameters different than in Table 1 (and keeping the model fixed). The results are shown in Table 4. In summarizing, it can be concluded that for the given interconnected nonlinear system example with the linear model the correotion algorithm occured to be quick convergent and this property was rather insensitive to values of desired real system outputs and model-reality differences.
s
CONCLUSIONS An important class of hierarchical
oontrol systems are those where the task of the optimizing control layer oan be reduced to steady-state optimization. In these cases the iterative correction schemes can be applied, that can bring about improvement over model-based control by skillful use of feedback from the real system. In this paper the direct-type correotion algorithm with decentralized structure was presented. The main task of the algorithm is to improve the pure model-based control cd(Z) so as to satisfy the real system constraints, at the same time taking into account the system performance. As opposed to price-type correction mechanism (Brdy s, 1978) which can work with the local system constraints on controls and inputs only (Gij (c i ,ui ) ~ 0, j € J , i:z1, ••• ,N) a wider class of i
P. Tatjewski and M. Cygler
168
TABLE 1 Values of System and Model Parameters
a 11
8
model
1
system
1.3
8
ij
a 21
8
0
1
0
1
0
0
1
4
0.15
1.2
0.1
1.25
0.25
0.1
0.9
4.2
12
8
22
8
23
TABLE 2 Convergence for Various Values of iteration number n 1
t =
0.6
.020017 .001161 .000043 .000012 .000008 .000003 .000001 .000000
2
3 4 5 6 7 8
t.
= 0.8
25
8
8
31
32
£-
0.9
L= 1.0
E..= 1.1
.020017 .000466 .000041 .000002 .000000
.020011 .001520 .000031 .000002 .000000
.020011 .003216 .000172 .000012 .000001 .000000
(=
.020017 .000054 .000051 .000009 .000001 .000000
8
24
TABLE 3 Convergence for Various Desired Values of Real System Outputs desired values of outputs 1 2 3 4
5
6 7
0.028 0.000 0.100 1.000 -0.400 0.000 -1.626 TABLE 4
8
11
0.0 1.1 1.5
8
12
0.15 -0.15 0.2
number of iterations
0.0005 0.000 0.100 0.000 -0.186 0.000 0.000
0.179 0.000 0.100 0.000 0.020 0.000 0.000
0.331 0.000 0.100 0.000 0.500 -0.450 0.000
5 5 5 6 6 5 5
Convergence for Various Model-Reality Differences
8
21
1.0 0.8 1.3
8
22
-0.1 -0.1 0.1
8
23
0.8 1.2 1.3
constraints (Gij(ci,ui'Yi) ~O, j 6 J , i = 1, ••• ,N) can be i handled with the direct-type algorithm. The difference between the two mentioned classes of constraints is signifioant, sinoe when a constraint
8
24
-0.25 0.25 -0.25
a
25
0.20 -0.1 -0.1
a
31
1.30 1.20 1.3
8
32
4.50 3.90 3.8
num r of iterati..o.ns 10 6 7
Gij(Ci,Ui'Yi) in the system is pre-
sent we can transform it into a form that does not explicitly depend on the outputs by using the model output equations
Gij(CitUitZi)=Gij(CitUitFi(CitUi·i~) (29 )
Completely Decentralized Output Control
but then (29) is only a model of the real system constraint
169
other values reasonably chosen) should be feasible in the real system (i.e., it should exist such G ij(Ci,Ui,Zi)=Gij(Ci,Ui,F.i(Ci,Ui,Zi)), control c that y(z)=F~(c,H§(z),z) ()O ) and the constraints are satisfied). and G#ij ~ Gij , and is not exactly It is rather difficult to formulate general condition assuring the above kIlo when F1I'i ~ F i • feasibility property, hence it must The applicability conditions of the be rather checked individually when algorithm do not include any "weak applying the algorithm. ooupling condition", as it is the Stating the problems which are still case ith the price correction algoopen one should think about the rithm, significantly reducing its possibility to find a system output application possibilities. However, that would be closer to the real this feature is obtained at a cost optimal one than the model-based of measuring both interaction inputs output y(z) is, especially in the and outputs of each subsystem (in cases when the model-reality differenthe price method case only the inces are significant. This,however, puts are measured). Since each implies the application of the coorsubsystem input is at the same time dinator over the local units of our an output of another subsystem it algorithm, and one step of this may not necessarily mean more meacoordinator procedure would mean the surements, but a transmission of whole action of our algorithm. This measurements (which may also be expendoes not seem to be unreasonable sive) • It should be, however, poinsince, how the computation results ted out that the algorithm can also show, our algorithm is rather effiork using the following, more simcient, terminating its action in a ple than (24), correction formula few iterations. n+1 n An. Si = si + L (Yi - y~i)' 1=1, ••• ,N. The authors of this paper would like ()1 ) to express their gratitude to prof. In this case only the subsystem inteW.Findeisen and to the colleagues raction outputs should be measured, from his Hierarchical Control Group but at the same time the system must in Warsaw for valuable discussions satisfy some tlweak coupling condiconcerning the considered subject. tion", to assure the convergence; we do not describe it here in any detail. REFERENCES The correction algorithm presented in Brdys, • (1978). Hierarchical conthis paper is direct-type, hence it trol of steady-state systems. uffers also some drawbacks typical In 'i. Findeisen (Ed.), Second for direct approach. First, each suborkshop on Hierarchical Conystem must have at least as many ~.Institute of Automatic controls as outputs (dim c 1. ., dim y.1 , Control, Technical University , ••• , ), since otherwise the local of Varsaw, Part 1, pp.19-68. oblems (7) have, in general, no Brdys,M., and P. ichalak (1978). easible points. Next, the values of On-line coordination with local tputs hich have to be obtained in feedback for steady-state sye real system using the algorithm stems. Arch.Autom. Telemech., optimal model-based y(z), or
·e., -G
£2,
40)-422.
170
P. Tatjewski and M. Cyglrt
Brdys, BrdYs, M., and B.Ulanicki (1978). On the completely decentralized control with local feedback in large-scale system. ~. Aut om. Telemech.,gJ, 23-36. Cygler, M. (1918). An Algorithm of Local Control Units by Hierarchical Control of the System Using Direct Method (in Polish) M.Se.Thesis, Institute of Automatic Control, Technical University of Warsaw. Pindeisen, w. (1974). Multilevel Control Systems (in Polish). PWN, Warszawa. (German translation: Hierarchische Stenerungssysteme. Verlag Technik, Berlin 1977) • Findeisen, w. (1976). Coordination by price methods. In Proc.3-rd Polish-Italian Conf. Applications of Systems Theory. Bialow1eza, Poland. Findeisen, w. (1978). Introduction. In W.Findeisen (Ed.), Second Workshop on Hierarchical Control. Institute of Automatic Control, Technical University of Warsaw. Pp. 7-18. Findeisen, W., M.Brdys, M.BrdYs, K. Malinowski, P.Tatjewski, and A. Wozniak (1978). On-line hierarchical control for steady-state systems. IEEE Trans. Aptom.Control, Z2" 189-209. Findeisen, W., F.N. Bailey, M.Brdys K.Malinowski, P.Tatjeweki, and A. Wozniak (1980). Control and Coordination in Hierarchical Systems. J. Wiley, London, to be published. Lefkowitz, I. (1966). Multilevel approach applied to control system design. Trans. ASME, !illSandell H.R., P.Varaiya, M. Athans, and M.G. Safonov (1978). Survey of decentralized control methods
for large scale systems. IEEE Trans. Aut om. Control, ~, 108-128. Singh, M.G. (1977). DYnamical Hierarchical Control. North Holland, Amsterdam. Tatjewski,P.(1978). Multilevel optimization methods. In W.Findeisen (Ed.), Second Workshop on Hierarchical Control. Institute of Automatic Control, Technical University of Warsaw. Part.4, pp. 241-266. Zinchenko, A.I. (1973). About approximate solution of functional equations with nondifferentiable operators (in Russian). Matematich. 'fizika, ll, 55-58.
Abstract. The aim of this paper is to present an algorithm of adjusting the real system outputs to given desired values. The controlled system is assumed to be described by an approximate mathematical model, i.e. differences between models (used by control units) and the real system characteristics are taken into account. The algorithm is designed for the optimizing control layer of the hierarchical (multilayer) control structure ot the system. The goal of the control is to obtain such stead3-state of the system"that given values of real system outputs are achieved, the constraints are satisfied and optimality of performance is taken into account. To achieve these goals feedback information from the real system is used on-line in the algorithm, along with the mathematioal models mentioned before. The system is assumed to be largescale, oonsisting of interconneoted SUbsystems. Por such system the algorithm is oompletely decentralized, consisting of a set of its parts, each working with one subsystem. The algorithm is thoroughly presented and analysed in the paper, a number of numerical results is also given. Keywords. Large-scale systems; hierarchical systems; control theo~; optimizing control; decentralized control; nonlinear systems.
INTRODUCTION The most widespread concept in hierarchical control is perhaps that of splitting the task of determining the control m of the controlled system into two mutually related oontrol functions: the follow-up control where a direot controller sets m so as to Cause the (appropriately chosen) regulated variables 0 to possibly follow their desired values (set-points) Cd' and the optimizing oontrol, where the 157
optimal trajectory of Cd is determined. Pigure 1 presents this kind of structure, with reference to oontrol of a large-scale system - that is a system composed of N subsystems interconneoted via an interconnection matrix H. Every SUbsystem is influenced by disturbances zi and f i , and by other subsystems thro~h interaction input ui • It influences other subsystems ~ its interconnection output Yi. The subsystem control inputs are denoted by ~, i=1, ••• ,N.
158
P. Tatjewski and M. Cygler
oprlMIZINS
CONTIlDL
FOl.LOH-UP CONTROL
riq 1. Hierarchical cont~ol. of a complex SI/stem layers
J4ith folloH-up and opllmLzLng controL
The concept of splitting the task of determining the control into some mutually related control functions, introduced by Lefkowitz (1965), is generally known 88 miltilayer control hierarch3. The follow-up and optimizing control layers constitute the most important part of such hierarcb3. General aspects of multilayer control structures were discussed in the literature, see e.g. (~indeisen, 1974; Pindeisen and others, 1978; Sandell and others, 1978), along with presenting several control algorithms for both follow-up and optimizing control l~ers. In this paper we are concerned with optimizing control layer and, moreover, with the case when its algorithms can treat the system (together with the follow-up layer) as non-dynamic, i.e. operating in steady state (or, more precisely, in a sequenoe of steady states). It is the most important case which clearly shows the appealing feature of splitting the control into two layers: the task of optimizing control may be considerably simplified in comparison to dete~1ng m directly from a
dynamioal optimal control solution. Conditions under which steady-state optimizing control acting on a follow-up layer is a reasonable approximation to the true dynamic problem are exposed in(Pindeisen, 1974; Pindeisen and others, 1980). Intuitively, this approach is valid for slow disturbances acting on 8 fast system. In a continuous chemical prooess, for example, the prooess parameters c (see ~ig.1) such as temperatures, pressures, concentrations, etc. may have to be varied from time to time in adjustment to some slow disturbances z, say changes in raw material composition. The new set-points cd for these process parameters may be determinable by steady-state approach (e.g. optimization). On the other hand, the process control inputs m such BS valve positions have to be adjusted much more frequently, in response to fast disturbances f such as heating gas pressure, for example. This adjustment can successfully be done by the follow-up control systems, trying to keep the process temperatures, pressures, etc. to their desired and constant values cd as prescribed by the optimizing control. If the model of the system and the measurements of disturbances are sufficiently accurate the optimizing control l~er can operate open-loop, i.e. for every new value of disturbances z the optimal model-based value of set-points Sd(z), where i is the estimate of z, is evaluated and transmitted to the follow-up control layer. Unfortunately, it is not so often in real systems that the measurements of disturbances (or their estimates) and, first of all, the system model are accurate in the above sense - the optimal
Completely Decentralized Output Control
model-based set-points cd(Z) may fail to be real-optimal, and, what is even worse, they mB3 violate the system constraints. However, in on-line control one is not left with his model of the system alone, there exists the possibility to observe the actual values of the system variables and to use this information as a feedback to the control algorithm. Extensive studies have been made of iterative control algorithms for that purpose, see, e.g., (Pindeisen and others, 1978; Brd1 s, 1978). Thus the optimizing control layer consists in fact of two sublQ3ers: the model-based optimization layer (acting open-loop) and the correction l~er (acting iteratively and using feedbaok information) as Fig. 2 is showing.
HOJJEL-BASED
OPTIMIZATION
CO~RECTION
(SUB)LAYER
Fig 2. Optimizl'ny control uSl'ng (eed.hack
and iterative correction.
The task of model optimization is to oalculate the optimal model-based set-points cd(z), when new value of z occurs (is measured or estimated on the basis of present and past measurements). Then, if needed, the Correction layer improves iterative13 these set-points on the basis of measuring the interaction variables
159
in the system, i.e., the inputs ui or outputs Yi of the Bubsystems, see Fig. 2. These measurements are feedback information. Let us note that for the complex system each of the layers should be structured as a multilevel hierarchy consisting of local units and of a coordinator. :Por many reasons (Findeisen,1974, 1978; Pindeisen and others, 1980; Sandell and others, 1978; Singh, 1977)it is desirable even for model-based optimization (although it is not shown in Fig.2) , and it is necessary for the oorrection layer. Moreover, the most desirable situation occurs when the correotion layer is designed in such a way as to work satisfactorily without its coordinator unit, in a fully decentralized way. An idea of such algorithm, basing on price approach (with Lagrange function: was presented ~ Findeisen (1976) and thoroughly investigated by Brdys and co-workers (Brdys , 1978; Brdys and Michalak,1978; Brdys and Ulanicki, 1978). This algorithm oan be used, however, only when the system satisfies some "weak coupling oondition", what significantly limits its applicability. It has also some features typioal for price-type methods. The aim of this paper is to present a new fully decentralized correction layer algorithm, basing on direct approach. The main advantage of this algorithm is that the system does not have to be "weakly ooupled", and that the constraints more general then in the price approach can be handled. The algorithm has some other advantages and drawbacks, rather typical for direct-type methods. DESCRIPTION OF THE SYSTEM It is assumed that the controlled system inclusive of its follow-up controllers (see Pig.2) is desoribed Qy a set of subsystem input-output
P. Tatjewski and M. Cygler
160
vector mappings :p. i:
ei x U i x
~i --..
whioh are only apprOXimations of reality
":1 1 ' i=1, ••• ,N,
Yi ::
where N denotes the number of subsystems and ~i' U i , 2,i' ~i are fini te-
y :: K(o,z) ,
i=1, ••• ,N (1 )
where the variables c i E. ~i' u i E U i , Yi E 1J i , z1€ Zi are the i-th subsystem set-points (called "controls" below, since the set-points are controls for t'he optimizing control layer), inputs, outputs and disturbances, respectively. The sUbsystems are interconnected, the structure equations are assumed to be in the form N
ui
= Hi' = ~
Hij Yj' i=1, ••• ,N, j=1 (2) where Hi =[Hi1 ••• HiN ] are matrices
composed of zeros and ones. Introducing the global desoriptions c
T
6
T
(c1, •••
T
~
~
,oN)E~1x••• xJi
T.4 TT." Y = (Y 1 ' • • • , YN) E. '1j 1x. • •x IN
4
::
Y7
~
4.....,
,
= .J,
et
u
::z
c.,
=K*(c,
where
(6 )
Gij : ~i)( U i )( 1ji...... R' j , J i ,
and J i denotes a set of integer indexes, i =1, ••• ,N. Due to the, rather very general, form of the constraint sets (6) they oan be assumed to be exaotly known (Brdys, 1978). With each subsystem a known local performance function ~i'>
--..... R
is associated,
and we assume that it is to be minimized. The overal performance funotion of the system is assumed to have the form 1
,QN(cN,uW'YN»'
()
Of course y
,j~J1J,
1
= Hy, 4 (FT , ••• ,F*N)' T =
y :: K*(c, z).
CUY1 ~ {CC 1 ,u i 'Yi): Gij CC 1 ,ui 'Y1 )flO,
Q't' (c ,u,y) = \f'(Q1 (0 ,u 'Y1 ), ••• ,
F H (c, u, z),
T] HT 4[ :: HT a1 1 •• .HN, and superscript T denotes transposition. We will assume that in the system as a whole for each 0 E ~ and z £ ~ there exists exaotly one output, that is that there exists a mapping K*: e" 2,--...y
T where It'*
where Pi' F and K are models of It'*i' F". and K 1I , respectively. Next we assume there exist local constraint sets
Qi:
the system as a whole can be described ae follows y
i=1, ••• ,N (5)
y = F(c,u,z)
dimensional spaces, as follows Yi = F*1(c i ,ui ,Zi)'
~i(ci,ui,Zi)'
z)~y =It'~(c,Hy,z)
(4) It is assumed that the system mappings m~ be not known exactly and, consequently, we have to use models
where 't': ~N 1R is some stricty order preserving function (for example, 't' =
:i!: ). i=1
THE
DIRECT-TYPE CORRECTION
ALGORITHM WITH WeAL It'EEDBACKS Let us assume that for given value i of the disturbanoe estimate the model-based optimization has been performed resulting in the values 0d(Z) and y(i) of set-points and interconnection outputs. Then the optimal model-based set-points Cd(Z)
Completely Decentralized Output Control
161
have been applied in the follow-up control la3er and after the transient process in the system has decayed Z some value y. = K*(c d (z) ,z) of the
output equations shift, the following local decision problems are solved
interconnection outputs has resulted. Of course, if our lmowledge of the system would be perfect ( that is,
find ci(si'Y) such that
A-
li-P.i' z 1.
i=1, ••• ,N, and
Itt-
and
y(z)
:I
€CUYi ,
z
,..
i
-
z = z) Z
z
(ci(z),u.i'Yai)
= 1, ••• ,N,
ci(si,y)=arg min Qi(ci,HiY,y i ) ci
then
,.
Bubject to €
j
where
y~ ~ A
y(z),
Z
ce (c i (z), Uti' Y.i) ,. CUY i can occur, at least for some i. The correction layer action is now necessary, first of all to satisfy the constraints in the real system. Observe that this goal will be satisfied if we modify the controls Sd(Z) to a value o~(z) • Bd(z) + dO in such a way as to force the equality K~(C~(Z) ,z)
:I
y(i) despite
J., 1.
i · 1 , ••• ,N.
generally, and hen-
Z
-
£,
Yi+si=Fi(ci,Hiy),
z
uai • HiY~ • But where it is not the case
(7)
A
Gij(ci,HiY'Yi)ECUYi'
of
It should be noted that the local problems (7) are precisely local optimization problems of the direct method of hierarchical optimization A for y = y, (see,e.g.,Tatjewski, 1978) if 8i=0 is set, i =1, ••• ,N. However, the model-based optimization was accomplished previously (with the result Od) and we try to improve its result using nonzero values of Si,
the model-reality differences, and
i.e. shifting the values of Pi(Ci,HiY)
demanding (c~i (z) ,ui
to equal them with the values of
,..
_
(z)
A
where ui(z) = Hiy(z), i
,y i (z») €
CUYi ,
= 1, ••• ,N.
The optimality of the performance must be of course also taken into account, hence optimization basing on models of the SUbsystems will be used when generating c~(Z). Since the disturbances z and their estimate are assumed to be constant dUring the correction process, in the sequel they will be dropped in all mappings, to simplify the notation. The value c~ will be generated iteratively, U8ing using feedback from the real process. Por each value of some aUXiliary parameter
z
F*i(ci,HiK~(c)),
i
= 1, ••• ,N.
Of course, the way how to choose the values of output shifts Si cannot be learned from the models· only, hence it will be obtained using feedback information from the real process in form of the measured interactions K*(c(s, y)). Our goal is, therefore, to find such value s of 8 that K*(C(B,y)) = Y or, in other words, to find the solution 8 of the operator equation R~(s)
= 0 ,
where
aT. T T =- (s1'··· (8 1 , ••• ,sN) £ '.11)( .... x'.i N' having the interpretation of subsystem LS.s.--G
The equation (8) will be solved
(8 )
P. Tatjewski and M. Cygler
162
iteratively, using the following Newton-like sheme sn+1 = an -
£L(sn)R*(sn)
,
A
I
[
4
]
K(o)= I-P* u (c ,~. )H (10)
where L > 0 is some step coefficient and L (. ): ~ - '1j. If £ = 1 and L(.) = [R~(.)] -1, then (10) would be precisely the Newton scheme. However, even if all the models and system mappings (Qi' Fi , F*i' etc.) are differantiable the operator R* may be nondifferentiable, since may be nondifferentiable (the known feature of optimal point mapping of parametrized optimization problems). Even being differentiable, R~(.) would be not possible for evaluation due to the fact that the operator K* appearing in its definition is not known precisely. Moreover, the I A evaluation of c s (.' y) would be
c(., y)
extremely difficult. Therefore the Newton-type scheme (10) 1s used where L(.) is a reasonable approximation of [R~(.)] -1, basing on the models only. To evaluate L(.) the following methodology will be used in the sequel: assuming R*(.) is differentiable we evaluate the precise formula for R~(.) and then, making some neoessary but reasonable approximations, we simplify this formula to obtain L(.) in a form possible for using also in nondifferentiable case, and basing on models only. The simplification will be justified by pricise mathematical oonsiderations. Let us assume therefore, for the time be ini , t hat K ~ and y) are differentiable, hence from (9)
"
•
A
-1
I
"
F. c (c ,H3.) (12 ) 4,.
A
where c = o(s,y) and y.= K.(o), for notation simplicity (these abbreviations will be also used in the sequel). The assumption of differentiability of C(., is almost equivalent to the assumption that, for the oonsidered variations of s, only the same set of inequality oonstraints G ij is active. Let us assume therefore, without loss of generality, that all inequality oonstraints are active. Then the Kuhn-Tuoker conditions for all local problems (7) have the form
y)
'(A A A T I ~ ~ T~ 'ft ~ A T Qc c,HY,y) -F c (c,H3) / l +Gc(c,HY,y) =
=0 G(o,HY,y) • A
0
(1) iI\
,..
Y + s - F(c,BY) = 0,
N :z: i::1
A
where Q(c,u,y)= A
G
(G 1J .. )(.J
Qi(oi'u. ,y.), ~
J i; J.= . 1,
Eo
~
••• , N)'
and A ',If are Kuhn-Tucker multipliers for equality and inequality constraints, respectively. Using the implicit function theorem for the set of equations (13) we obtain " '" I
A
A cs(s,y)
A
Ps (s ,y) =_ I
A
I
A
o
B
As(S'y)
o
o
o
I
(14 )
where A : Qcc 11 (,. u~ A )_p 11 (A U~) C ,J.J.J ,y cc C ,J.J.J
cc
C 'J.J.J
,y
cc
C 'J.J.J
A+
c(.,
B AI " R*(s) = - K.(c(s,y».cs(s,y). ,
'"
A 1:1
'A
A
A
Gc(C'HY,y).
A
(11 )
Using the implicit function theorem and taking into account the identity (4) we have
After oomputing the inversion (assuming it exists) and operator compositions we find
Completely Decentralized Output Control
where D = A-1_A-1BT(BA-1BT)-1BA-1. (16) Putting now (12) and (15) into (11) we obtain
163
minIlYi+si-Fi(Ci,Hiy)D ci
(20 )
subject to Gij CCi'HiY 'Yi)
R/Cs) .. -[r - P~Cc,H§)HJ-1
(18)
Taking F instead of F~ is necessary since F is just the known model of :P~. On the other hand, taking y instead of YI- ~ K.( (s ,y » is re as onable since K#(c(s,y» is just the value of real system outputs deviating from due to model-reality differences, and forced to be equal to y by our algorithm. It should be pointed out that the above simplifioations, being natural and reasonable, lead at the same time to very simple form of R. In all our considerations we have tacitly assumed that for all Si from some appropriate neighbourhood of the origin the local decision problems (7) have feasible solutions. However, it may not be true due to the constraints. Then the following modification of (7) can be made. Assuming Si is a given value, the projection ~i(8i) of Si on the set
a
y
)
•
The described modification does not alter our previous oonsiderations, only (18) becomes a bit more crude approximation, since now
R~Cs)
= -[r-p~CC,H3*)
H]-1p~cCC,Hy*).
'DP~CC,HY)T[p~CC,HY)DP~CC,BY)TJ-1W~ (21 )
where
11 ~ (1r1 ,··.,1fN ).
=[R1-
1 we get the following Taking L iteration formula (10)
+f[I-F~ (0 (sn ,y
),H3 )H] (y-Kt(8(s~ y»))
(22) Let us show. finally. that the algorithm (22) is decentralized, resulting in the control structure with local feedbacks only. It can be easily obtained, ~ straightforward calculation, that (22) is equivalent to a set of N equations
n+1 n [~ n Si =si +t (Yi-Y:JJi) + I
A
nA
I'
N
A
+ (Fi)Ui(ci(s ,y),HY)ZH .. (y.- Y~J.)] j=1 J.J
J
(23 )
~{si£ ":1 i :
3c i
A
Yi+si
12
~ ~i
(
H" ) ci ' i Y ,
(19 )
i=1, ••• ,N, is first performed, and then 'ffi (si) instead of 8 i is used in (7).
Ji ,
(17 )
Si (y) ,
Si Cy
'i
• 1, ••• ,.N.
i
R;CS) .. -[r - P: u cO,H3.) HJ-1 P;cCC,H32C) DP~CC,HY)T[p~CC,HY)DP~Cc,Hi)TJ-1 A rather very reasonable approximation R of RJcan be now obtained by putting into (17) P and y instead of F~ and y¥ , respectively. Thus
O,j
6
i=-1 , ••• ,N , where y~ : KJt(c(sn,y». Taking into account the structure relations (2) the above equations can be written in the form
164
P. Tatjewski and M. Cygler
(24)
n 6 where u,* =
~
HK~(c(s
n
i=1, ••• ,N, ~
A
A
,y)) and ui=Hiy.
The variables U~i are interconnection inputs to the i-th real subsystem oorresponding to to the oontrols " n" c(s ,y) , and they can be measured locally, like the outputs Y~i .Hence, the algorithm (24) is decentralized, the control decisions are taken locally in each local decision unit, using local input and output measurements and subsystem models. Only the value of parameter f should be equal for each subsystem, and the next iteration should be made in all local decision units simultaneously, after the transient process in the system resulting from the previous iteration is finished. The structure of the local decision unit is shown in Pig.).
The iterations atop whenJYi-y~iD are below the prescribed treshold for each i =1, ••• ,N. In all previous considerations the assumption was made that the real system outputs y~ should be foreced to be equal to Y~ the optimal model-based outputs y. It is quite reasonable when the model-based optimization can be performed. However, the algorithm can also work when insted of y another, quite arbitra~ value y8 of the outputs is required. It should be only not too far from the actual system outputs to assure the convergence in the nonlinear case (see next section). CONVERGENCE AND OPTIMALITY
CONDITIONS
0°
set
n=O,
s/=o
Let come back to the more compact form of the iteration equation (22) I -1 sn+1 =sn + £ [R(Sn)] R (an) (25)
and t}o to
n n 1 mea-Jure lJ,i' u.i
*'
0
"
nf1
n
fi "
n
n"
2 Si::l Si +EL(lJi -Ij"i)-~' (Uj -ull'iJ), n~( )1 ('" n If. " I' i= ~. ni Ci (Si' Ij)) If'/) j
The following theorem formulates sufficient conditions for the convergence and gives the convergence rate estimate of the algorithm (25), with starting point so.
nolI" d' 3" eva lua te c,."" (Si' IJ) accor 1ng to (7) and appllj to the sljstem 4" HQit until the transient proce'1'J is f£nished and go to t" Hith n=n+1
~.
i-rH SUBSYSTEM Hith 'OLLOH-UP CONTROLLER
lj.i
Fig 3. The i-th local decision unit
Theorem 1. If on some set S c1j the following conditions are satisfied: (i ) DC R,. (8 ) R(8 1 )) - (R:le( s 2) 1
f
-t
R(8 2 )) U~ k·l[ 8 1-s 2 11
(it) 11 Rts 1 )- R'(s2)H
The first step (n=O) is explicitly shown in Pig.3,consisting of the application of the optimal model-based control 0di= ci(O,Y) to the system.
(iii) UR'(8)-1U ~
f
~
£kf
U8 1 - 8 2 "
(iv) IlR(so)U~~, h = 1 -
So
+ k~ + ~kl,2rz. ~ 1
Completely Decentralized Output Control
Cv)
Bca o,r) ~{a ~ ":1: lIa_aoll~ for r =
~
r} ~ S
outputs y*= K~(C(8,y)) are equal to then, since at each iteration step
y)
,
(ci(Si,y),HiY'Yi)~CUYi i=1, ••• ,N,
1-h -then there exists a solution
s
,r) of the equation (8) and the sequence {sk} generated by (19) converges to this solution, moreover
n 2
,
if
2 h n ,.. s -all~-·-n ,if k~ 1_h2
1 -£+ kA~ 0 1
-~+
k~
= 0
•
The above theorem is a rather straightforward application of the powerful theorem 1 from (Zinchenko, 1973) to our problem, hence the proof is omitted. Let us comment the assumptions. The first one is on quality of the approximation - in ideal case, when R = R s (and Newton step-legnth Eo = 1 is taken) it is always satisfied with k=O. (ii) is a standard requirement of Lipschitz type, and (iii) requires that variations of R,(.)-1 on S are not too great • The assumptions (iv) and (v) , typical for nonlinear problems, require appropriate quality of the starting point so, combining it with previous requirements - observe that if the approximation is good (k small) and o s is appropriately close to the solution (~small) then (iv) and (v) are satisfied. In fact, £ = 1 could be set in all our considerations, like Zinchenko (1973) has done, but then we would loose an additional degree of freedom. Choosing £ -F 1 we have the possibility to change the convergence rate of the algorithm. This can happen to be successful, hat some computation results have shown (Cygler, 1978). en the convergence is achieved for some s = (that is the real system
s
have (ci(si,y),HiY,*,Y.-i)E CUYi , i = 1, ••• ,N.The constraints are satisfied in the real system. It is the most desired feature of our algorithm. However, immediately after that one asks about the optimality of the control C(B,y). It is obvious that this control cannot be strictly optimal for the real system, since we do not know the real mappings F~i and y is, in general, not realsystem optimal. However, the following property is an appealing feature of the algorithm.
we
~ -B(s 0
lIan-ell ~ rhn
165
Theorem 2. If the differences between real and model subsystem output mappings are additive, i.e. F#i(ci,Ui ) = Fi(ci,ui ) +~i' (26) i:::1, ••• ,N, where
are some elements from ".1 i , i = 1, ••• ,N, -t'hen J. =-ti.., i = 1, ••• ,N and the ]. control 0(8,y) is the best possible for the real system with y*= y, i.e. it is the solution of the problem 0(
i
s.
~" min Q't' (c,HY,y)
c
sub.to
(27)
Yi=F~i(ci,HiY)'
1=1, ••• ,N. ~.
A
A
A
The controls ci(s,y) are
solutions of the local decision problems (7) for 8 i =8 i , hence they are solution of the problem min Q 't' (c ,Hy ,y ) c sub.to Yi+si=F.i(ci,HiY)-D(,i (c.
J.
,H.Y,Y·)E CUY., ]. J. ].
(28)
i=1, ••• ,N,
166
P. Tatjewski and M. Cygler A
A
and Yi :: P*i(ci,Hiy), i = 1, ••• ,N
(c
».
since y • K ~ (s,y Comparison of (27) and (28) constitutes the proof.
CUY2 = {c 2 : 4c21+2c21u21+0.4U21+ +
•
Por more complex model-reality differences it was not possible, as yet, to achieve constructive results. It can be only shown (by reasonable assumptions) that a certain "continuity property" is satisfied: when the model-reality differences tend to zero then s ~ 0 and c (s,y) tends to the real optimal value. Howover, the computation results show that the control c(a,y) is, as a rule, superior to the model-based Od = = o(O,y) not only in terms of the contraints satisfaction, but also as far as performance deterioration is considered (Cygler, 1978). ,.
"
...
c21c23+0.5C~1+u~1::: 4,
0.5c21+c22+2C23 ~ 1, IC2jl~10, j
= 1,2,3 }
Subsystem 3: 2+2.5032 2 Q3(c 3 ,u3 )=(c 31 +1) 2 +(u J -1) P3(cJ,uJ)=c31+2.5cJ2-4U3
A
CUY3
= {c J :
c 21 +u 3 ~ -0.5,
0 ~c32~1}.
The output relations of the real system differ from its models and are as follows: P*1(c 1 ,u 1 )=a11c11-c12+2u1+812c11u1 2
PS21 (c2,u2)=c21-c22+a21u21-Ju22+a22c22 P*22(c2,u2)=2c22-a2Jc23-u21+u22+
EXAMPLE RESULTS
Let us consider the system structure shown in Fig.4.
+ a24c22c23+a25u21u22 P~3(c3,u3)=a31c31+2.5c32-a32u3'
and the values of parameters
Fi9 4. The
example
s!lstem structure
8
i j
are
given in Table 1, together with the corresponding values in the system model. Notice that the model output equations Pi are approximate linear parts of the real output equations F#i' which are nonlinear, 1 1,2,3. The optimal model-based solution AT A y -(Y1'Y21'Y22'Y3) is aa follows 3
The subsystem models are as follows: Subsystem 1: Q1(c 1 ,u 1 }1ll(U1-1}4+5(C11+c12-2}2
P1 (c 1 ,u1 )=c11-c12+2u1 CUY1 lll{c 1 :
c~1+0~2 ~
1
J
Subsystem 2: 22222 Q2(c 2 ,u2 )-2(c 21 -2) +c22+3023+4u21+u22 P21(c2,u2)=c21-c22+u21+3u22 P22(c2,u2)=2C22-023-u21+U22' 4
P'2=(P 21 ,P 22 )
It>
"
"
A
Y1= 0.028 AT Y2= (0.179, 0.005) Y3= 0.331 When the optimal model-based controls cd=c(O,y) are applied to the real system the constraints are violated, and the application of the correction algorithm seems indispensable. Applying our algorithm, the local decision problem (7) corespond1ng to the se-
Completely Decentralized Output Control
oond Subsystem was solved numericaly using conjugate gradients and penalt, shifting technique to handle the constraints (first and third looal decision problem could be solved analytically). Also the projections ~i on the sets (19) were solved using the same numerical technique. The iterations were stopped when the Euclidean norm of the difference ,.
A
n
A
A
n
K*(c(s ,y)= R~(s ) was below the given accuracy parameter ~=10-6 The re8ults, for a few values of ateplength parameter £, are given in Table 2. The obtained optimal shift parameters
y -
"T =(s1,S21,s22'S3) ". ,. " " were
8
81 =-0.142 AT
8
2
=(-0.017,-0.051 )
S :a-0.30) To evaluate the sUboptimality of the A " obtained control It.c(s,y) the real optimal control 0 (unknown in practical applications) was also computed, as well as the real optimal A ". value Q = Q(c,H3) of the performance function. It is shown below, together with the value Qs=Q(C(s,y),HY) obtained in the real system after the application of the controls C(B,y) resulting from the correction algorithm " :z 10.83453 Q A
Qs = 10.97342 Hence, ~he suboptimality index is as follows A Qs - Q • 100% = 1.28%. 6 = A Q
It should be realized that comparison of the performance function value obtained for the model-based control A Cd with the value Qs is unreasonable since the control Cd violates the system constraints. Next, the correction algorithm was tested using nother than y and
167
quite arbitrarily chosen desired values of the outputs. The results are shown in Table 3. The casea 5,6 and 7 are chosen in such a way that the optimal shifts lie on the boundry of their feasible sets (19). In all cases the steplength La 0.9 and the accuracy 9= 10-6 were used. Pinally, the correction algorithm, with desired outputs Y,l-O.9 and 9- 10-6 ,was tested several times using the real system parameters different than in Table 1 (and keeping the model fixed). The results are shown in Table 4. In summarizing, it can be concluded that for the given interconnected nonlinear system example with the linear model the correotion algorithm occured to be quick convergent and this property was rather insensitive to values of desired real system outputs and model-reality differences.
s
CONCLUSIONS An important class of hierarchical
oontrol systems are those where the task of the optimizing control layer oan be reduced to steady-state optimization. In these cases the iterative correction schemes can be applied, that can bring about improvement over model-based control by skillful use of feedback from the real system. In this paper the direct-type correotion algorithm with decentralized structure was presented. The main task of the algorithm is to improve the pure model-based control cd(Z) so as to satisfy the real system constraints, at the same time taking into account the system performance. As opposed to price-type correction mechanism (Brdy s, 1978) which can work with the local system constraints on controls and inputs only (Gij (c i ,ui ) ~ 0, j € J , i:z1, ••• ,N) a wider class of i
P. Tatjewski and M. Cygler
168
TABLE 1 Values of System and Model Parameters
a 11
8
model
1
system
1.3
8
ij
a 21
8
0
1
0
1
0
0
1
4
0.15
1.2
0.1
1.25
0.25
0.1
0.9
4.2
12
8
22
8
23
TABLE 2 Convergence for Various Values of iteration number n 1
t =
0.6
.020017 .001161 .000043 .000012 .000008 .000003 .000001 .000000
2
3 4 5 6 7 8
t.
= 0.8
25
8
8
31
32
£-
0.9
L= 1.0
E..= 1.1
.020017 .000466 .000041 .000002 .000000
.020011 .001520 .000031 .000002 .000000
.020011 .003216 .000172 .000012 .000001 .000000
(=
.020017 .000054 .000051 .000009 .000001 .000000
8
24
TABLE 3 Convergence for Various Desired Values of Real System Outputs desired values of outputs 1 2 3 4
5
6 7
0.028 0.000 0.100 1.000 -0.400 0.000 -1.626 TABLE 4
8
11
0.0 1.1 1.5
8
12
0.15 -0.15 0.2
number of iterations
0.0005 0.000 0.100 0.000 -0.186 0.000 0.000
0.179 0.000 0.100 0.000 0.020 0.000 0.000
0.331 0.000 0.100 0.000 0.500 -0.450 0.000
5 5 5 6 6 5 5
Convergence for Various Model-Reality Differences
8
21
1.0 0.8 1.3
8
22
-0.1 -0.1 0.1
8
23
0.8 1.2 1.3
constraints (Gij(ci,ui'Yi) ~O, j 6 J , i = 1, ••• ,N) can be i handled with the direct-type algorithm. The difference between the two mentioned classes of constraints is signifioant, sinoe when a constraint
8
24
-0.25 0.25 -0.25
a
25
0.20 -0.1 -0.1
a
31
1.30 1.20 1.3
8
32
4.50 3.90 3.8
num r of iterati..o.ns 10 6 7
Gij(Ci,Ui'Yi) in the system is pre-
sent we can transform it into a form that does not explicitly depend on the outputs by using the model output equations
Gij(CitUitZi)=Gij(CitUitFi(CitUi·i~) (29 )
Completely Decentralized Output Control
but then (29) is only a model of the real system constraint
169
other values reasonably chosen) should be feasible in the real system (i.e., it should exist such G ij(Ci,Ui,Zi)=Gij(Ci,Ui,F.i(Ci,Ui,Zi)), control c that y(z)=F~(c,H§(z),z) ()O ) and the constraints are satisfied). and G#ij ~ Gij , and is not exactly It is rather difficult to formulate general condition assuring the above kIlo when F1I'i ~ F i • feasibility property, hence it must The applicability conditions of the be rather checked individually when algorithm do not include any "weak applying the algorithm. ooupling condition", as it is the Stating the problems which are still case ith the price correction algoopen one should think about the rithm, significantly reducing its possibility to find a system output application possibilities. However, that would be closer to the real this feature is obtained at a cost optimal one than the model-based of measuring both interaction inputs output y(z) is, especially in the and outputs of each subsystem (in cases when the model-reality differenthe price method case only the inces are significant. This,however, puts are measured). Since each implies the application of the coorsubsystem input is at the same time dinator over the local units of our an output of another subsystem it algorithm, and one step of this may not necessarily mean more meacoordinator procedure would mean the surements, but a transmission of whole action of our algorithm. This measurements (which may also be expendoes not seem to be unreasonable sive) • It should be, however, poinsince, how the computation results ted out that the algorithm can also show, our algorithm is rather effiork using the following, more simcient, terminating its action in a ple than (24), correction formula few iterations. n+1 n An. Si = si + L (Yi - y~i)' 1=1, ••• ,N. The authors of this paper would like ()1 ) to express their gratitude to prof. In this case only the subsystem inteW.Findeisen and to the colleagues raction outputs should be measured, from his Hierarchical Control Group but at the same time the system must in Warsaw for valuable discussions satisfy some tlweak coupling condiconcerning the considered subject. tion", to assure the convergence; we do not describe it here in any detail. REFERENCES The correction algorithm presented in Brdys, • (1978). Hierarchical conthis paper is direct-type, hence it trol of steady-state systems. uffers also some drawbacks typical In 'i. Findeisen (Ed.), Second for direct approach. First, each suborkshop on Hierarchical Conystem must have at least as many ~.Institute of Automatic controls as outputs (dim c 1. ., dim y.1 , Control, Technical University , ••• , ), since otherwise the local of Varsaw, Part 1, pp.19-68. oblems (7) have, in general, no Brdys,M., and P. ichalak (1978). easible points. Next, the values of On-line coordination with local tputs hich have to be obtained in feedback for steady-state sye real system using the algorithm stems. Arch.Autom. Telemech., optimal model-based y(z), or
·e., -G
£2,
40)-422.
170
P. Tatjewski and M. Cyglrt
Brdys, BrdYs, M., and B.Ulanicki (1978). On the completely decentralized control with local feedback in large-scale system. ~. Aut om. Telemech.,gJ, 23-36. Cygler, M. (1918). An Algorithm of Local Control Units by Hierarchical Control of the System Using Direct Method (in Polish) M.Se.Thesis, Institute of Automatic Control, Technical University of Warsaw. Pindeisen, w. (1974). Multilevel Control Systems (in Polish). PWN, Warszawa. (German translation: Hierarchische Stenerungssysteme. Verlag Technik, Berlin 1977) • Findeisen, w. (1976). Coordination by price methods. In Proc.3-rd Polish-Italian Conf. Applications of Systems Theory. Bialow1eza, Poland. Findeisen, w. (1978). Introduction. In W.Findeisen (Ed.), Second Workshop on Hierarchical Control. Institute of Automatic Control, Technical University of Warsaw. Pp. 7-18. Findeisen, W., M.Brdys, M.BrdYs, K. Malinowski, P.Tatjewski, and A. Wozniak (1978). On-line hierarchical control for steady-state systems. IEEE Trans. Aptom.Control, Z2" 189-209. Findeisen, W., F.N. Bailey, M.Brdys K.Malinowski, P.Tatjeweki, and A. Wozniak (1980). Control and Coordination in Hierarchical Systems. J. Wiley, London, to be published. Lefkowitz, I. (1966). Multilevel approach applied to control system design. Trans. ASME, !illSandell H.R., P.Varaiya, M. Athans, and M.G. Safonov (1978). Survey of decentralized control methods
for large scale systems. IEEE Trans. Aut om. Control, ~, 108-128. Singh, M.G. (1977). DYnamical Hierarchical Control. North Holland, Amsterdam. Tatjewski,P.(1978). Multilevel optimization methods. In W.Findeisen (Ed.), Second Workshop on Hierarchical Control. Institute of Automatic Control, Technical University of Warsaw. Part.4, pp. 241-266. Zinchenko, A.I. (1973). About approximate solution of functional equations with nondifferentiable operators (in Russian). Matematich. 'fizika, ll, 55-58.