AGGREGATION IN HIERARCHICAL SYSTEM: APPROXMATE DECOMPOSITIONS*

Douglas P. Looze and Bassen Sa lh i Coordinated Science Laboratory

Universi ty of I l l inois Urbana, I l l i n o i s 61801

Abstract

One problem facing hierarchical systems theory is that the allowed exact decompositions may result i n problems which a re too complex to be feas ib ly imple- mented. The approach of this paper is to allow aggre- gations of t he exact decomposition. The questions of convergence of the simplified i teration and the near- ness o f the resu l t ing so lu t ion to the t rue so lu t ion a re addressed.

1. Introduction

The topic of hierarchical control has-emerged as an important area i n the s tudy of large scale systems i n both theory and appl ica t ions . Among the var ious appl i - ca t ions which have been reported in t h e l i t e r a t u r e are app l i ca t ions t o steel roughing processes [l], urban

synchronous machines [4], and river pollution control t r a f f i c networks [2], water supply network [31,

c lass of problems which can be handled by current [5] . Despi te th i s d ivers i ty o f appl ica t ions , the

hierarchical techniques is l imi ted .

Several of the present l imitations of h i e ra rch ica l cont ro l theory a re d i rec t ly re la ted to the t echnica l development of the methodology which is in cu r ren t u se . The theory has developed as an extension of decomposi- t ion techniques in mathematical programming to the par t icular s t ructure of opt imizat ion problems considered by cont ro l theor i s t s . The theory insists on the decomposition of the original optimization problem with the i teration tending to the exact solution. This exactness requirement can lead to extremely complex subproblems i n many cases (e.g. , s tochas t i c problems or the completely decentralized l inear quadratic problem).

The purpose of t h i s paper is to re lax the exact- ness requirement and define approximate decompositions. The formulation of decomposition algorithms developed i n [ E l is used and is res ta ted in Sec t ion 2 . In this formulation, hierarchical decompositions are determined by one po in t i t e r a t ions . In Section 3 sub- problem s impl i f ica t ions a re descr ibed as operator theoret ic aggregat ions of t he i t e r a t ion ope ra to r . The approximate decomposition is then represented as a composition of an exact decomposition and an aggre- gation, and is defined in terms of the aggregated i te ra t ion equat ion .

Sec t ion 3a lso d i scussesre la t ionships between the original, exact decomposition and the resulting appro-

near ly the s implif ied problem represents the o r ig ina l ximate decomposition. F i r s t , the ques t ion of how

problem is addressed. Then, the convergence pro- pe r t i e s of the approximate decomposition is re la ted to those of the original decomposition.

2 . Exact Decompositions

For the purposes of this paper, the decomposition formulation of 181 w i l l be used. Assume the cont ro l problem to be solved can be expressed i n t h e form

f(x) = 0 (1)

where 5 is a Banach space and f : Z + % . The desired

*This work was supported i n p a r t by the Jo in t Services Electronics Program under Contract N00014-79- C-0424 and i n p a r t by the National Science Foundation under Grant ENG-79-08778.

solut ion of (1) will be denoted by x*. L e t f :%x* +A be a continuously Frechet differentiable funceion such t h a t a f (x*,x*) is invert ible . Define the nonl inear s p l i t &

f (x ) = f0(X,Y) + fl(X,Y) VX,Y€%. (2)

fo(%+l’%) + f l f \ , + - 0. ( 3)

The i te ra t ion def ined by t h i s s p l i t t i n g is given by the equation

Conditions for local convergence and the asymptotic convergence ra te of this i terat ion are given by the following theorem.

Theorem 1: Assume f , f o , and f l are def ined as above, and le t x* be a so lu t ion to (1) for which af(x*) and alf(x*,x*) are nonsingular. If

y = P~-alfo(X*,x*)-lalfl(x*,~*)}< 1 (4) then there exists an open neighborhood d Cs of x* such

which sa t i s f i e ; ( 3 ) . Moreover t h a t f o r any x € 4 there is a unique sequence I%};-o

lim “ k = x* (5) k+=

and for each E > 0 t h e r e e x i s t s an in teger k, such that

I\-x*l -< Vk_> ko. ( 6 )

Proof: See 181.

A complete discussion of this formulation and its appl icabi l i ty to the ana lys i s o f h ie rarch ica l cont ro l algorithms can be found i n (81 . It is s u f f i c i e n t t o no te here tha t mst such algorithms can be described and analyzed within this framework.

above is tha t t he ana lys i s of general hierarchical s t ruc tu res is equivalent to the analysis of the non- l inear d i scre te equa t ion ( 3 ) . In turn, propert ies of the i t e ra t ion equat ion can be re la ted to p roper t ies of the sp l i t t ing func t ion fo and the function f which descr ibes the o r ig ina l problem. This advantage includes the abi l i ty to s tudy the effects of problem s i m p l i f i c a t i o n s a t any l eve l of the hierarchy. In par t icu lar , the s impl i f ica t ion of these problems can be viewed as a nonlinear aggregation of ( 3 ) . To i l l u s - t r a t e t hese i deas mre c l ea r ly , on ly t he spec ia l ca se fo r which f and f o a r e l i n e a r i n x will be considered in this paper . The in s igh t s and results can be extended to the genera l nonl iaear problem i n two ways: f i r s t through the use of l inear aggrega t ions a t each i teration; or second, through the use of a nonlinear aggregation technique (c.f. is] 1. These cases w i l l be considered in the fu l l vers ion of th i s paper .

The advantage of the abstract formulation outl ined

Assume now tha t is a separable Hilber t space and that equation (1) is l i nea r in x

Ax - b. (7)

The l i n e a r s p l i t t i n g of the operator A is defined by a nonsingular operator A. : X + % and the equation

Ax = Aox + %x. (8)

The i te ra t ion equat ion can then be written

A0\+ l i -9% + ‘ (9) The convergence covdition (4) of Theorem 1 becows

p{AilA., 1 1 (10)

445 0191-2216/80/0000-0445$00.75 0 1980 IEEE

3. Approximate Decompositions

The d i f f icu l ty assoc ia ted wi th h ie rarch ica l decom- posit ion algorithms is t h a t t h e r e s u l t i n g i t e r a t i o n equation may be so complex that the implementation of the algorithm is not feasible . A natural approach to surmount t h i s d i f f i c u l t y is t o attempt to s impl i fy the i teration equation through an approximation to A,,. I n

on a subspace of the original space will be t h i s s e c t i o n , t h e problem of approximating an i t e r a t i o n

formulated.

L e t 3 be a sub-Iiilbert space o f 3 with the same topology +d l e t T :A+ 3 be a (possibly skew) projec- t ion onto&. The pseudo-inverse T i of T is defined uniquely by (91

T T T = T #

T'TT' = T

TT' = P (13)

T'T = I-Q (14)

where P and Q are the Or thogonal p ro jec t ions on to i and qT) r e spec t ive ly .

T onto X , the best approximation- of L on X defined by the p ro jec t ion T is given by

Giyen a l inear opera tor L :$+$, and-a pro jec t ion

L - T L T . # (15)

Thus, t h e i t e r a t i o n ( 9 ) r e s t r i c t e d t o 3 is given by

Two questions concerning the approximate i teration arise. F i r s t , what conditions can be placed on the subspzcez and the p ro jec t ion T such that the solut ion to (16) ( i f i t e x i s t s ) is nea r t he so lu t ion t o (7)? Secondly, what conditions can be placed o n 3 and T t o ensu re t ha t t he i t e r a t ion (16) converges?

Any s o l u t i o n t o (16) m u s t s a t i s f y i ; = i

where A = Ao+$ = TAT - b - #

GA= Tb. I f 3 is an invariant subspace of A (i.e., &) then the projecton onto 3 which minimizes the maximum e r r o r

e(T) = b sup IxCb)-T'i(b)l ( 2 0 ) I bl=1

is the orthogonal projection, and the value of (2O)_fs the spec t ra l rad ius of t h e o p e r a t o r r e s t r i c t e d t o 1. . I f b is f i x e d i n 4 t h e e r r o r ( 2 0 ) is zero for the orthogonal projection.

similar fashion. Assuming t h a t t h e o r i g i n a l i t e r a t i o n The convergence of (16) can be analyzed i n a

r e l a t e d t o t h a t of (9) through the concept of a (9) is convergent, the convergence rate of (16) can be

def la t ing pair of subspaces for (&,%) [lo]. A de f l a t ing pa i r of subspaces 74 and Y of J; are sub- spaces which are isomorphic and which s a t i s f y

AoZCClf A , U C t / . (21) This concept is re lated to the analysis of the conver- gence of (9) and (16) in the fo l lowing way.

The spec t r a l r ad ius of Ai1% is the magnitude of the largest generalized eigenvalue of (A,,,%). A gen- eral ized e igenvalue A of (Ao, and the corresponding generalized eigenvector x are 3' efined by the equation

AAox+qx = 0. (22)

Equation (22), %x, and A1x must l i e i n t h e same sub-

space i n which &x lies, a de f l a t ing pa i r by (211, gen- space. Thus, the subspace i n which x l i e s and the sub-

eral ize the concept of an invariant subspace to the generalized eigenvalue problem.

With th i s p re lude , ve can state the following theorem.

Theorem 21 L e t t 4 and 2/ be 5 def l a t ing pa i r of sub- spaces for (A,,,%) and le t y denote the magnitude of the largest associated generalized eigenvalue. Let T be a skew p r o j e c t i o n o n t o 4 a l o n g YL. Then the rate of convergence of (16) is c . Proof: L e t .

lJ = (U1,Uz) v = ( v V ) (23) 1' 2

be orthogonal operators with V1 :?/+x, v2 : f A + X , U1 : U + 3 , and U2 :2/'+%. Then, fo r i4,l

A, = V&.U*

us ing the par t i t ion def ined by (23). Thus, - Ai = TAiT #

But the propert ies of the skew pro jec t ion imply

m=(: 1) Hence (24) becomes

(25) . .

But (25) implies the generalized eigenvalues of (io,il) are those of (A&,<l) which i n t u r n are those of (&,A1) assoc ia ted wi th the def la t ing pa i r Y,V. .

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