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F 31 Brasch. Jr.. and J B Pearson. “Pole placement using dynamic compsnsaton.” IEEE Tru,~r.A ~ r r o n m r .Cmrr.. w 1 . AC-15. pp. 34-43. Feh. 1970 S H. WangandE.J.Dalison.“Penetratingtransmlsslonzeros In the d e i g n of rohu*t remomechamrm s)stema.“ In Proc. I X r h A,r,lu. Allerron Con( CnnlllzutL Cnnrr. Conlprcr.. Oct 1980: also in I E E E Truum. .4111onlorConrr.. vol. AC-X. pp 784-787. June 19x1 R W Brockett. Fmrrr Dimensmrlol I J I W U SI~ rlenrr. Yeu York: Wilev. 1970. &I. .4ok1. “On feedbackstabiltzahllity of decentralizeddynamlc s!rtema.” 4uk> nturuu. ~ o 18. pp. 163-173. Mar. 1972. J. p CorfmatandA S . Morse.“Decentralized Lnntrol of linearmulrivanable sSstems.“ Aurrmmrrru. YOI. 12. pp. 479-495. Sept. 1976. W . S. Chan and C. A. Desoer. “Ekgenvalue assignment and atabilizatlon of Interconnected systems using local feedbacks” I E E E Trum. Auronwr. CoNr.. vol. AC-24. pp. 312-317. Apr 1979. R. Saeks. “ O n the decentralized control of interconnected dynarnlc systems.” I E E E Trum Auronfur. Cow-.. vol. AC-24. pp. 269-171, Apr. 1979. D D. Sillak and M. B. Vukcevic. “Decentralmtlon. stabilization. and estimation of large-scale linear systems.” I E E E Trum. Auronlur. Conrr.. vol. AC-21. pp 363-366. June 1976. S. H. Wang and E J. Davl.wn. “Minimization of transmission c o s In dmntralized control s>stems.” l n r . f. Cortrr.. vol. 28. no. 6. pp. 889-896. 1978. S. H. Wang. “An example in decentralized control systems.’‘ IEEE Trum. Aroonlur. Co,xrr. vol. AC-23. p. 93X. Ocl. 197X. I E E E Tuns. Auronmr. C o r m (Special Issue on Decentralized Control of Large-Scale Systems). vol. AC-23. Apr 197X B D 0 Anderson and J. B. Moore. “Tlme-baqing feedback laus for decentralmd control.” in PTM. 19rh I E E E Cod. Derrsron Conrr.. Dec. 1910.

VOL.

AC-27,NO. 3, JUNE 1982

developed here may be Liewed as an extension of Chan and Desoer’s work [3] to the discrete time case. Throughout the note. mill denote the unit circle of the complex plane. R” denotesn-dimensionalEuclideanvectorspace. For a given square and det A will denote the inverse and the determinant of A , matrix A . .4 respectively. For a given ~7 X nz matrix B with entries b,, (that is, B = [h,,],=,,, ,=,,), II B . x will denote the norm defined [7] as

-’

11. SYNTHESIS OF DECENTRALIZED STABILIZINGCOhTROLLER

Consider the large-scale system nith N subsystem Si. I = I , 2,. . . sh0v.m in Fig. I . where the dynamics of S, are given by

I

S,;~~(k+l)=A,x,(k)+b,v,(k),

W,

i=1,2:-.,N.

as

(1)

Here x , ( k ) E R“, is the state vector, e , ( ? )is scaler input to the ith subsystem. and A , and h, are the n , X n , and R , X 1 constant matrices, respectively. Let each subsystem S, be interconnected to other systems by the follodng relation:

x

.?i

c,(k)=u,(k)+

On Stabilization by Local State Feedback for Discrete-Time Large-Scale Systemswith Delays in Interconnections IL HONG SUH AND ZEUNGNAM BIEN Ahsrruct --By employing extended an Nyquist array technique, a sufficientcondition is obtained for decentralizedstabilization of a class of discrete-time largescale systems with delays in interconnections.Then

E,,x,(~-II,,).

(2)

i=1.2:..,~.

j = l 1‘1

Here u , ( k ) is the scaler control input to the ith subsystem S,. h , , a O is the delay-time in interconnections. and E,, is the I X 17., constant matrix of the form

E,, = [ eP, e ! , . . , e : / - ’ ] .

(3)

as

from ( I ) and (2). the composite system can berepresented x S , : . ~ , ( / i ? l ) = A , x , ( k ) + b , u , ( k ) + b,€,,.x,(k-k,,),

2

I. INTRODUCTION Centralizedcontroltechniqueshavebeen knonrn to beinefficient or even unsuccessful in some cases if applied for the control of large-scale control systems dvnamic [l],_To overcome the difficulties of centralized . methods.manyresearchershaveproposed as alternatives various decentralizedcontrolmethods invohing simplification of modeldescriptions.effectiveprocedures of testingthestability and/orherachical optimization [I]. However. most of the decentralized control techniques developed so far are derived to handle the continuous-time systems [2]. [3]. In particular. the assertion [2]. [3] that there al\va)-s exist local state feedback controllers stabilizing the large-scale systems in whch delayed and/or nondelayedinteractionsoccuronlythrough the input of each controllable subsystem is true for the continuous dynamic systems. but may not directly apply for the discrete-time systems. Inthisnoteit is shownthat if. inaddition to thecontrollability assumption on eachsubsystem as in thecontinuous-time case. certain restrictionsareimposed on the interactionsignal.thendiscrete-time large-scalesystems can bestabihzed by decentralized state feedback in [6] is controllers. For ths. an extendedNyquistarraytechnique employed. It is notedthat Chan andDesoer [3] utilized the Nyquistarray stabilizing technique by Rosenbrock [4] for the synthesis of decentralized controller of thecontinuous-timelarge-scalesystems.Thus.theresult ~ ~ ~ recetbed ~ ~lune n3. 1981: p treviwd June 17. 1981. December 1 1 . I9Pl.and J u n u a n 29. 1982 I H Suh is ,,i1h the h p a r t m e n t ofFJ~trical Engineering, Korea Advanced Inhrltute of Technology. SLxncs and Seoul 13I . Korea z. Blen ih u x h the Department of Electrical and Computer EngmR-ring. Unlverblt? of I ~ , , ~tows . C i p . IA 57241. on leave from the Korea Advanced Institute o f k i e n c e and Technology. Seoul I3 I . Korea.

J=l j#,

i=1,2;..

.!v.

(4)

For the continuous version of (4). it was shown in [2], [3] that under the assumption that each subsystem is controllable. there exist local decentralized state feedback controllers whch stabilize the continuous-time largescale systems. Hoxvever. in the case of the discrete-time large-scale system in (4). local stabilizing state feedback controllers may fail to exist under the local controllability assumption only as in the continuous-time case. This fact is easily shown by a simple example in the foIlo-ing. E.xunlp/e I: In (4).let X = 2. 11’ = n 2 = 1. A , = A , = 0. E,? = E,, = lo2, = 0. Let the local controller be u,( k ) = - u , x , ( k ) . for each and h I 2 = r = I . 2. Then the characteristic equation of the closed-loop system is given by

h,,

d(-) =(z

+al)(z

.,)104.

(5)

It easily follows from ( 5 ) that both of the zeros of d( 2 ) lie in the unit circle P of the z plane only if ‘ a l - a 2 J < 2 , and

la,~~-IO~l