Parameter Independent Evaluation of Decentralized Stabilizability

Copyright © I F.·\C Large Scale S, stelll" Theory and .-\pplicati"Il> I'/H6. Zurich . S,,·itl eriand. IlJ86

PARAMETER INDEPENDENT EVALUATION OF DECENTRALIZED ST ABILIZABILITY M. Vim and H. D. Wend i.:lli"l'l.\ity vj

[)/w/illlg.

Iwlill/II' of Cu//lro!

t:lIglIIl'l'rllIg.

Lulllar.,lr. 1-21.

D- -IIOO [)1/i.,!I/IIK I. FR(;

Abstract. A major problem due to the use of decentralized feedback for large scale or complex technical plants is the question of stabilizability, i. e. the existence of local controllers stabilizing the overall system. The paper describes a concept for investigating whether a plant, which consists of state-coupled SISO-subsystems, can be stabilized by local dynamic output feedbacks of fixed dynamic order. The investigation is carried out in a parameter independent way based on graph theoretic results such that stabilizability is guaranteed irrespective of the numerical values of the non-vanishing system parameters. The approach permits the analysis of systems with more general structures than known methods . The significance of the proposed algorithm is demonstrated by means of a hydroelectric power system. Keywords. Decentraliz ed control system; stabilit y; linear systems; graph-theoretic methods. I.

INTRODUCTION

and outputs. The results of Sezer and Siljak require the exclusion of any invariant zero of the coupled plant. Therefore, the system classes taken into account are quite conservatively chosen. In order to state system classes within these restrictions,which exhibit a maximum of structurally permissable subsystem couplings, Sezer and Siljak assign strong negative real parts to all poles of the isolated subsystems. For practicaY-[mplementation this leads to serious difficulties for instance due to bounded a c tuator output signals.

The application of decentralized control for large scale technical systems has found an increased acceptance through the intens i fied use and availability of microcomputers. A gl obal problem arising due to the use of a constrained feedback pattern is the one of decentralized stabilizability, i. e. the exisLence of a set of local, non-communi cating feedback controllers stabilizing the overall system. In a pioneering work the problem of decentralized stabilizability of linear, time-invariant systems and the related one of decentralized pole assignment was investigated by Wang and Davison (1973). By introduction of the concept of fixed modes Wang and Davison state an upper bound for the existence of local linear, time-invariant stabilizing controllers. Fixed modes prove to be those poles o f the open and the closed loop plant which are invariant for any arbitrary feedback law with identical structural constraints. The proposed stabilization scheme generally leads to local controllers of high dynamic order which presents a major drawback for practical purp os es . Cor f mat and Mo r s e (1976) suggest to apply static output feedback to all but one subsystem and a dynamic controller of sufficient order to the remaining subsystem. In this way they ensure both stability and even pole assignability for the overall system, but have to put up with a distinct asymmetry of the controllers . Another well-known concept f o r the verification of decentralized stabilizability is based on the application of vector Ljapunov functions and leads to the investigation of a so-called aggregation matrix. By means of this concept connective stability guarantees that an arbitrary subset of subsystem couplings may vanish without losing stability of the overall system. The entries of the aggregation matrix depend on numerical values of subsystem and coupling parameters.

The aim of this paper is to present a new criterion for the parameter independent characterization of decentrally stabilizable systems. In order to increase its potential applicability the new method requires that only an actually necessary number of subsystem poles take values with strong negative real parts. The knowledge of this minimal number is a valuable hint for the decentralized synthesis of the local control laws. In addition the new method permits the occurence of invariant zeros within the isolated subsystem plants. The approach presented differs clearly from those already discussed since it bases on the graph-theoretic interpretation of the closed loop system's characteristic polynomial. In contrast to system aggregation any coupling path is explicitly taken into account. In th e following it is assumed that the multivariable plant is described by a set of linear, timeinvariant state equations which represent state-coupled SISo-subsystems (i=I, .. • ,N): N ~i (t)

A. ~i (t) +

-1

I

A.. x. (t) + -1 b . u i (t) j=1 -lJ -J i*j

T y i (t) = -1 c· ~i (t)

(I )

~i

(t o )= ~i, O (x . ElR n u. E IR

-,

In contrast to the methods mentioned up to now Sezer and Siljak (1981) describe decentrally stabilizable plants in a parameter independent, structural way using the concept of system classes. A system class is defined by similar binary interaction properties of the plant's states, inputs

1

N Yi E IR

I i=1

n. 1

n).

Each of these subsystems is controlled by a local linear dynamic output feedback of order L.1

219

220

M. Urn and H. D. Wend §i'~i (t) + 1i' Yi (t)

(2)

r~ . z. (t) + k. 'y . (t)

-1

1.

-1

cycle family CF is defined to be the product of the weights of all cycles Ci being member of CF:

i=I ..... N.

1.

d(CF)

The vector h (t)E IRLi ([ Li = L) represents the additional states. which are introduced by the dynamic controller. In order to enable complete pole placement for the decoupled SISO subsystems the dimensions of these controllers are chosen to be Li - ni - 1. The paper is organized as follows. Section 2 contains a brief graph-theoretic discussion of the closed-loop characteristic polynomial if decentralized control is used. The new method of investigating decentralized stabilizability is presented in section 3. In section 4 the stabilizability criterion is transformed into an integer optimal matching problem which can be solved by a stated algorithm. The application of this algorithm is demonstrated by means of a practical example in section 5. The conclusion in sec tion 6 summarizes the main results.

(3)

and an edge set E. The elements of Vs are called state nodes and co rrespond to entries of ~i and ~i whereas the elements of VI/O (I/O-nodes) correspond to entries of ui and Yi. Each element e of the edge set E denotes one non-vanishing entry of the matrices ~i. ~ij. ~i. £i. §i. qi. Ei or ki' If the numerical value of a non-vanishing matrix entry is assigned t o the edge e it is called the weight f(e) of edge e. Using these definitions a system's interconnection structure may be easily visualized. for example see Fig. 1 representing the digraph of an open loop system.

f(Ci).

where d(CF) denotes the number of cycles Ci taking part in the cycle family CF. The characteristic polynomial p(s) of the closed-loop system of plant (I) and feedback (2) may be obtained by p(s)=s

n+L

+PI's

n+L-I

+ ..• + Pn+L ' s

0

(5)

A powerful result which has been derived by Reinschke (1984) is stated in Lemma I: Any coefficient Pi(i=I •...• n+L) of the characteristic polynomial p(s) in (5) may be computed by means of 1

In this section a graph-theoretic interpretation of a characteristi c polynomial is given. The structure of the plant (I) with the feedback law (2) applied to it's outputs and inputs may be represented by a weighted or an unweighted digraph in the usual fashion. Such a digraph G(V.E) con sists of a node set V

n i=l

p.

2. GRAPH-THEORETIC FUNDAMENTALS

V = Vs v VI / O

f(CF)=

~ (_I)d(CF) .f(CF). w(CF)=i

(6 )

where the summation is carried out for all cycle families CF with prescribed width i within the graph G(V.E) of the closed-loop system. In the sequel the notion of isolated subsystem i denotes the isolated plant i closed by a local dynamic output feedback. The name of any sub graph which contains at least one coupling edge (c-edge) representing anon-vanishing entry of a coupling matrix ~ij will be marked by the prefix nc_ n •

Definition I: A cycle C within the graph G(V.E) is called a c-cycle iff it contains at least one c-edge. A -cycle family CF within the graph G(V.E) is called a c-cycle family iff it contains at least one c-cycle. Using Definition it is obvious that the set of all cycle families within G(V.E) may be divided into two disjoint subsets. namely the set of all c-cycle families and the set of all non-c-cycle families. the latter describing the characteristic polynomial of the decoupled system: N

n p(i.s) •

(7)

i=l

Fig. 1.

Digraph of a dynamic system

where p(i.s) denotes the characteristic polynomial of the isolated subsystem i. On the contrary all c-cycle families lead to the occurence of a so called coupling polynomial Pc(s). Therefore.the closed-loop polynomial for decentralized control can be described by (8)

A sequence of nodes within the graph G(V.E) vi EV

(i=I .... ,m)

with vi*Vj if '*J and the sequence of existing edges e reaching from node vi to vi+ 1 (i= I •...• m; vm+l:=vl) is called a cycle C. A set of node disjoint cycles C within the graph G(V.E) is called a cycle family CF. If a cycle C resp. a cycle famil y CF reaches exactly w state-nodes this number is said to be the width of the cycle C resp. the cycle family CF: w = w(C)

resp.

The coupling polynomial pc(s) is anon-monic polynomial of maximum order n+L-2 because any c-cycle family has to reach at least one state node of at least two subsystems. Note that the coefficients of Pc(s) are in general simultaniously dependent on values of weighted c-edges. on plant parameters and on feedback parameters according to their occurence in c-cycle families . 3. INVESTIGATION OF DECENTRALIZED STAB IL IZAB IL ITY

w = w(CF).

The weight f(C) of a cycle C is defined to be the product of the weights of all edges being member of the cycle C. Analogously the weight f(CF) of a

In the following it is assumed that each decoupled subsystem is controllable and observable such that all poles of each of the decoupled subsystems may be placed arbitrarily by means of an appropiate

221

Decentr alized Stabilizability dynamic output feedbac k. The minimal dynamic order of a general dynamic output feedback , which establishes complet e pole placeme nt for the SISO case, points out to be (9)

In contras t to other approac hes for the paramet erindepen dent evaluat ion of decentr alized stabiliz ability (Sezer and Siljak, 1981; Bachmann and Konik, 1984) transmi ssion zeros within the isolated plants will not be exclude d. Assume that the isolated plant is describe d by the followin g scalar, strictly proper transfe r function zPL(i,s ) . . ) FpL (~,s) = PpL ( ~, s

( 10)

contain ing mZ i transmi ssion zeros. Let

~~L (i):=

[0 ... 0 zPL, I'" zPL,mzi+ l )

(I I)

I x (0.+1);

dim:

1

(ni-mzi)

(i):- [zC omp. 0' " zT -Comp

~~omp(i):= contai~

[I

( 18)

Zc omp. L) i

( 19)

Pc amp, I"'PC amp, L) i

the coeffic ients of zComp( i.s) and

PComp(~·s).

The well-kno wn relation ship (i,s)+zP L(i.s).z c p(i,s) = PpL(i.s )·PC omp

leads to p (i)

omp

~)l' [~PL(i); ~PL(i»)' [~comp( ~Comp

(i,s) (20) (21)

(1)

where the matrix of the right hand side of Eq. (21) is called Sylvest er resulta nt (Bitmead and co-workers. 1978) and depends only on PPL(i) and ~PL(i). Due to the special structu re of tfie Sylvest er resultant and Eq. (16) the entries of the feedback coeffic ient vector on the right hand side of Eq.(21) are determin ed by polynom ials in y with graduat ed maximal degrees as follows :

leading zero element s (12)

contain all coeffic ients of the polynom ials zPL(i,s ) and PpL(i,s ). Moreove r, in contras t to the known concept s our approac h does not require that all but only an actually necessa ry number m· (0 $ mi $ ni + Li) of subsyste m poles are shif~ed towards minus infinity in order to assure stabilit y. Therefo re, the values of mi subsyste m poles are chosen according to

~Comp (~l

The notation p+(i,y, j) denotes polynom ials in y with maximal degree j. Case 2 (m i > n i - mz i ): +

I

P (i.y.l)

(I3)

where 0i,j denotes a positive arbitra ry constan t (Oi.j > 0) and for the scope of this paper y is cons~dered to tend towards infinity

(22)

(~ z [ -Comp

EComp(i)

(23)

~comp(i)

( 14) In a graph-t heoreti c way it is possible to characterize the effect of a subsyste m dynamic feedback

The remainin g poles of the isolated closed- loop subsyste m may take any constan t but stable value (Oi.j>O ): s . .J 1.,

=-

•...• n~+L~). . . ) (j=m~+I 0 . . (± jw 1.,) ... ... ... 1.,J

Due to Vieta's Theorem the charact eristic polynomial of the isolated closed-l oop subsyste m i mi+Li mi n [s+o . . (±jw . . ») [s+O . .. y). p(i.s) = n ~,J ~,J jEJll.+I ~,J j=1 ~ ( 15)

[I

l

possess es the coeffic ients given by the vector p*(i,y. l)

£(i)

f(i)

R

p*(i; y .m i )

( 16)

p*(i.y.m i ) The abbrevi ations p*(i.y.k ) (I $ k S mi) denote monic polynom ials in y of maximal degree k. As mention ed above local dynamic output feedback describe d by the scalar proper transfe r function FComp( i.s) •

zComp (i.s) (. ) PComp ~.s

(17)

is applied to each subsyste m. Analogo usly to Eqs. (11) and (12) let

I. by cycle-f amilies of width I •...• L. within the graph of the feedback dynamic s accofdin g to ~ . (this is due to PComp( i.s) and Lemma I appli~d to an isolated subsyste ms i) and 2. a combina tion of both compens ator input/o utput paths reaching through the feedback and accompanying disjoin t cycle-f amilies . These combinations have a sum width of O•...• Li. This result follows from a theorem derived by Reinsch ke ( 1985) for a plant's zero polynom ial but wh ich in this case is applied to the feedback zero polynomial. Accordi ng to Eqs.(22 ) and (23) there exists a direct relation ship between the width and the y-weigh t for all cycle families within an isolated subsyste m. This depende ncy will be reflecte d by the fact that the coeffic ients of the closed- loop charact eristic polynom ial Pd (s) of the overall decoupl ed system and the cou~ting polynom ial Pc(s) are again polynom ials in y. Especia lly for pc(s) the maximal exponen t in y of these polynom ials is a function of the" specifi c configu ration of c-edges . The followin g theorem which is the fundame ntal result of this paper states a sufficie nt condition for decentr alized stabiliz ability . Theorem The plant (I) may always be decentr ally stabiliz ed

;\\. elm and H. D. \rend

222

by local dynamic output feedback meeting the requirements of Eq. (16) and Eq. (18) if for each prescribed width i=2, •.• ,n+L all cycle families of width i and maximal weight in y, i. e. maximal y exponent, are not c-cycle families, i. e. contain no c-edge. Remarks

~proof of this Theorem is given in Ulm (1985).

The proof leads to the result that for y ~ 00 one part of the roots of p(s) tends to the roots of the constant polynomial ni+Li

N

n

n j=m.+l

i=1

[s + 0i j(! jw . . )] ' 1,J

1

which is up to the choice of the feedback designer whereas the remaining roots of p(s) tend to minus infinity. By this method an approximate partial decentralized pole placement is enabled. 2. The Theorem provides even the proof of connective stability which is a reassuring fact if subsystem coupling links are endangered of being disconnected. The weight of each feedback edge is actually described by a polynomial in y such that the width and weight requirements of all subsystem cycle families and I/O-paths determined by Eqs. (22) resp. (23) are met. According to the Theorem all cycle families of the overall closed-loop system leading to maximal y-weights have to be taken into account. These maximal weights are characterized by parameters y with highest exponent in each edge weight polynomial, because the computation of cycle family weights is based on the mUltiplication of the edge weight polynomials. Therefore, it suffices to chose those parameters y with highest exponent as weights for the feedback edges. This is shown for an example in Fig. 2.

yO ~- - - , "

Fig. 2.

__yO__ ~ Subsystem 2 7eJ

yO

~

... ____ ",,/ yl

Artificial weights for a decentrally closed-loop system

In this example the cycles marked by dashed edges represent a cycle family of width 4 and maximal weight in y

Corollary Assume that non-negative integers are assigned to the edges of the closed-loop coupled system graph according to the rules stated above. If for each prescribed width i=2, ... ,n+L all cycle families of width i and maximal sum weight contain no c-edge then system (I) may always be decentrally stabilized by the local controllers chosen. By introduction of integer weights the computation of the maximal cycle family weights has been simplified. But the example stated above additionally shows that the investigation of all cycle families with maximal sum weight by visual inspection is not an easy task and prone to errors. This is especially true if the system graph contains a large number of nodes and edges. Therefore, in the following section an algorithm for the evaluation of cycle families with maximal weight is derived which can easily be implemented on a digital computer.

4. ALGORITHM BASED ON INTEGER OPTIMAL MATCHING Let G(V,E) be a digraph consisting of the node set V and edge set E. A subset Mp c E is called a perfect matching if every node of V is incident with exactly one incoming and exactly one outgoing edge. Assume that integer weights fi,j are assigned to all edges ei,j of the set E (ei,j denotes the edge reaching from node j to node i). The sum weight of a perfect matching Mp is defined by f. 1,

. J

e . . EM 1,J P The problem of obtaining special perfect matchings Mp max ,which result in a maximal sum weight fs,is well known and may be solved by several methods, for instance the Hungarian Method . These algorithms are readily available by means of already programmed procedures. For a detailed description of a reliable and fast algorithm see Burkard and Derigs (J 980) .

One way of evaluating cycle families with optimal integer weight is given by reduction on maximal perfect matchings. The notions of cycle family and of perfect matching are not directly interchangable,because the latter one does not distinguish between state and input/output nodes and may not be directly applied to subsets of V containing a prescribed number of nodes. Therefore, all input/output nodes have to be eliminated from the closed loop system graph which is described in detail in Ulm (1985). Moreover, if evaluating the Corollary for the optimal cycle families of width i it is necessary to select the optimal perfect matchings within subsets of exactly i nodes. This optimization task will be called the computation of an "optimal perfect matching with prescribed cardinali ty i". Algorithm

The experience of this example shows that the assignment of simply the integer y-exponents to the individual plant and feedback edges suffices in order to compute a so called sum weight of these integer values. So for example the sum weight fs of the cycle family in Fig. 2 yields f

s

=

0 + 0 + 0 + 0 + 3 + 0 + 0 + 1 = 4.

The sum weight equally well characterizes the cycle families' maximal y-weight. So for such an integer assignment the Theorem leads immediately to

("Matching procedure for stabilizability test") I. Construct the artificially integer weighted digraph of the locally closed loop system. A han~­ on approach will be stated after this algorithm.

2. Determine the total number of coupling edges. Let nc be this number. 3. Modify the integer weights within the graph of the locally closed loop system as follows: - Assign a new weight to each edge by multiplying the old weight by (nc+I). - Assign a unit weight (=1) to each coupling edge.

223

Decentr alized Stabiliza bility 4. DO for cardina lities i ranging from 2 to n+L :

Integer Weights Assigne d to the Local Subsyste m Feedbac k

TABLE 1

- Compute a maximal perfect matchin g with prescribed cardina lity i within the closed loop system graph taking into account the weighti ng scheme of 3.

m.~ 1

e

- Let fs be the integer sum weight of this maximal perfect matchin g. If

e e

m. > n . -mz.1

n.-mz.

1

1

1

1

l

2

2

2 3

n.-mz.- I 1

1

then

m.

m.

1

1

m.

m.

resume 4, with an increme nted cardina lity, e 2L + 1

else the conditio ns of the Corolla ry are violated . STOP. END DO 5. The conditio ns of the Corolla ry are fulfille d, so the decentr alized (connec tive) and paramet er indepen dent stabiliz ability holds. STOP.

1

1

3. The algorith m tries to compute a maximal perfect matchin g of prescrib ed cardina lity which violates the sufficie nt ~onditions of the Corollary. In step 3 of the algorith m the edge weights are modifie d such that the number of couplin g edges contain ed in the maximal perfect matchin g

equals Remarks

~algorithm may be complet ely impleme nted

in a compute r program using integer weighte d adjacen cy matrice s. 2. The followin g steps lead to the constru ction of the artific ially weighte d digraph required in step 1 of the algorith m. - Compute the structu ral number of transmi ssion zeros (mz. ; i=I, .•• ,N) for each decoupl ed subsyste mlby means of

5. EXAMPLE In this section the applica tion of the algorith m is demons trated by means of a practic al example , namely a hydroel ectric power station equiped with a synchro nous generat or. The model and it's parameters are taken from Korn and Wilfert (1982) and a rough schemat ic diagram is shown in Fig. 4.

mz. = n. - min 1

1

where min equals the number of state nodes within the decoupl ed subsyste m's input/o utput path of minimal width. - Decide on the number mi of subsyste m poles which are assigne d strong negativ e real parts. In order to keep feedback gains within practical bounds for a first approac h chose mi equal to the number of subsyste m plant poles already exhibit ing strong negativ e real parts. - If local dynamic feedback of order L is investiga ted close the subsyste m, for instance by the control ler graph of Fig. 3.

e

Hydroe lectric power station

Fig. 4.

th order linear The plant is describe d by an 8 state equatio n (I) with:

0

0

-0 . b83 -0 .0 34b-0 . 08Ib

0

0

0

0

0

0

0

0.882

0

I. 32

0

-0.0832

0

-0.0254 0.137

0.130

0

-0.107

0

0

0

0

-
0

-0 . 02b5

0

0

0

0

0

0

0

0

0

0

0

A •

0

0.0531

-0.137 -1 . 37

0

0

0

0

0

0

0

0

0

-
o.

0

0

-13.7

0

123

2.74

-
Couplin gs Fig. 3.

Graph of a subsyste m control ler

Assign integer numbers accordin g to Table 1 to the feedback edges marked with the letter e. Assign the value '0' to all other edges. - Elimina te all input/o utput nodes from the closed-l oop system graph.

!T. [

0

0

0

0

0

0

0

0

0 0

0

0

~]

The state variabl es Xl through x8 represe nt torque angle, angular velocit y, flux linkage , excitat ion voltage , interna l governo r voltage , power offered by falling water, interna l signal on turbine inlet and depth of water. The input variabl es u l and u are the actuato r signals for the generat or 2 excitati on and for the turbine inlet. The torque angle is chosen to represe nt an output signal.

224

M. elm and H. D. Wend

Computing the eigenvalues of the plant proves the plant's stability but shows that the dominating poles lead to slowly declining and weakly damped characteristic motions. It will be shown that independent of the exact system parameters an improvement of this behaviour is always possible by a structurely constrained dynamic feedback reaching from the plant's output to it's input u • In this l case the states xI to x5 and the states x6 to x8 belong to two subsystems, the first being closed by dynamic output feedback of order LI - 4. The plant of the isolated subsystem I contains no transmission zeros (mz l = 0) •. The . graph of the closed- loop coupled system shown in Fig. 5 includes nc = 4 c-edges.

state and output variables but not the exact system parameters. The new procedure relies on a graphtheoretic background and may be evaluated by a proposed algorithm which bases on an integer optimal matching computation. The new method takes into account general local dynamic output feedback and tolerates the existence of transmission zeros within the isolated subsystem plants. The procedure supports a decentralized synthesis of the local control laws because the stabilizabilit y analysis already provides some insight on the requir ed pole placement within the isolated subsystem in order to guarantee stability of the overa ll coupled system. 7. ACKNOWLEDGEMENT The work presented in this paper has been supported by the German Society for the Encouragement of Scientific Research (DFG), grant number We 977/1. The authors are grateful to the DFG . 8. REFERENCES

Fig. 5.

Graph of th e closed-loop coupled system

If ml=5 poles of subsystem I are shifted towards strong negative real parts then the modified integer weights of the plant and feedback vertices shown in Fig. 5 result. Remember that the stabi lization scheme of the The orem permits a partial pole assignment for the closed-loop coupled plant. If the conditions of the Corollary are satisfied then this property can be used to improve the system's characteristic motion by the structurally constrained feedback. The application of the proposed algorithm points out that none of the optimal perfect matchingswith cardinalities ranging from 2 to 12 contains a c-edge. Therefore, by virtue of the Corollary an improvement of the system's performance by the proposed constrained feedback is possible. This improvement is demonstrated by simulation results in Fig. 6 obtained using a de centralized synthesis of the local controller. 1.1111 ,,2

• 1!J1I1I

Open-loop case

Closed-loop case

-.811

11111.

Fig. 6.

Transient response for the angular velocity in the open - loop and the constraint closed-loop case 6. CONCLUSION

In this paper a new method for the stabilizability analysis of decentrally closed-loop systems has been proposed. The method requires the knowledge of a qualitative interaction scheme of all input,

Bachmann, W. and Konik, D. (1984). On stabilization of decentr a lized dynamic output feedback systems. Systems and Control Letters,S, 89 - 95 . Bi tmead, R.R., Kung, S. Y. and Anderson, B.D.O. (1978). Greatest common divisors via generalized Sylvester and Bezo ut matrices. IEEE Tr.-AC., 23, 1043 - 1047. -Burkard, E.B.-and Derigs, U. (1980). Assigment and matching prob lems: Solution Methods with FORTRAN. Springer, Berlin , 2 11. Corfmat, J.P. and Morse, A. S. (1976). Decentralized control of linear multivariable systems. Automati ca , 12, 479 - 495. Korn, U. and Wilfert, H.H. (1982). MehrgroBenregelungen (in German). VEB Verlag Technik, Berlin GDR, 124 - 127. Reinschke, K. (1984). Graph-theoretic characterization of fixed modes in centralized and decentralized control. Int. J. Control, 39, 7 I 5 - 729.

Reinschke, K. (1985). Structural properties of linear control systems should be investigated by means of graphs. Proc . 2nd Int. Symp . on Systems Analysis and Simulation, Berlin (GDR), 280 - 283. Sezer, M.E . and Siljak, D.D. (1981). On decentralized stabilization and structure of linear lar ge -scale systems. Automatica, 17, 641 - 644. Ulm, M. (1985). Evaluating decentraliZed stabilizability by integer optimal programming. Report No. 9/85, University of Duisbur g , Institute of Control Engineering, FB 7. Wang, S.H. and Davison, E.J. (1973). On the stabilization of decentralized control systems . IEEE Tr.-AC, ~, 473 - 478.