A Frequency Domain Approach to the Decoupling Problem for Singular Systems

2a-17 4

Copyright © 1996 IFAC

13th Triennial World Congress. San Francisco. US.\

A FREQUENCY DOMAIN APPROACH TO THE DECOUPLING PROBLEM FOR SINGULAR SYSTEMS' D. Vafiadis and N. Karcanias Control Enginec1·ing Centre, City University, London ECl V OIlB, U./C e-mail: D.Vafiadisl!!city.ac.uk.N.Karcaniasillcity.ac . uk Ahstract: The input.--ollt.pnt decoupling problem for singular systems via state feedback and regular input transformation is considered. The treatment of the problem is in the frequency domain alat it is an extension of the state-space systems case. The polynomial matrix approach is ~hown to be considerably simpler than the existing methcds and the solviLbility condition~ are readily obtained from the "numera.tor" matrix of a. coprime and column reduced MFD I)f the given system. Pole placement, issues are. also consi4tered and a charaderisatioll of the fixed poles of the system is givell. Keywords: Singlllar syst.ems, decoupling, state feedback, minimal bases.

INTRODIJCTION This problem of input,~outPllt decoupling by state feedback and regular input transformation has been st.udied in detail for the da..;;;s of st.ate--spare systems during the last three decades (Falb and \Volovich, 1067), (Gilbert., 1969), (Wollham and Morse, 11170) etc.. Recent.ly the problem was solved and for tlte case of I:'iingula.r or generalised statf'~spa(':e systelTls in (Ailot!, 1991), (Minarnide et at., 1991), (Paraskl"!vopoulos and }\:oumboulis, 1991). These papers use generalised sta.te~spare tools for the solution of the problem. The aim of t.he. present papn is to give a solution to the row by row decoupling of singular systems by state feedback ill t.he fre(llIency domain and t.hus generalise t.he frequency domain results for st.ate-spac:e syst.ems (strictly proper systems) t.o tht' c:a.s(' of singlllar (nonproper) systems. The approa.c:h of 1.lw paper is along the linE'S of (Kollssiolll'is, lH70) and kads to a very simple necessary and suHicient ('ondit.ion for the solvability of t.he problem. The eondition involves ouly t.he llumerator ma.trix of a coprirne and ('olumn reduc:ed MFD of the transfer fUllct.ion of the given syst.t'l1l. This eondition is much simpler thall t.he exisLillg olles , sin('e the only transformation need III order to t.c:;;t. it. is t.he reduct.ion of a given MFD of th", SySt.Nfl t.o a (:cprime and column reduced form. Another generalisa.tion obtained in t.he present paper is t.he chararJerisation of the fixed poles of the decoupled system in terms of t.he IHllTleICI.t.or 11li:l.t.rix of an MFD of ·Work sllpported by

~ESDIP

the original system. It is shown tha.t the fixed poles of the system are eharac.terised in exactly the same way as in (Koussiouris, 1980) for the case of state-space systems. The proof of the above juggests a way of choosing feedbac:k for arbitrary placem:~Ht of the assignable poles.

2

STATEMENT OF THE PROBLEM AND PRELIMINARIES

Consider t.he singular syst.em

Ex

= Ax + Bl!, y = C'x,

det(sE - A)

t

0

(2.1)

with u states, t inputs and f outputs. The problem of decoupling is the problem of finding a control law of the t.ype 1L

= Fx + Cv detG oF 0

(2.2)

such that the t.ransfer functic,n (tJ.) of the closed loop system is diagonal. System (:'.1) has t.f. H(.) which is, in general, nonproper. If H( .. ) is written in MFD form i.e. H(s) ~ N(s)D-l(B), t'le problem of decoupling may be formulat.ed a.<:> follow~, (Kollssiouris, 1979): For a system described by a cop rime and column reduced MFD H(s) N(s)D- l (8) having a realisat.ion of the type (2. I) find a wnt,ral law "r the type (2.2) such that the U. of t.he closed loop sys'em H,(s) = N,(s)D;l(s) is diagonal which means

=

N,(s) = rl;ag{h;(s)}D(s)

(2.3)

where hi(s) are ra.t,ional scalsrs. The above yields that Hc(s) is der:oupled if and onl:1 if

ESPRIT III Project No. 8924

(2.4)

1560

where, !!~(S), d~(s) denote the i-th row of Ne(s) and D(:(s) respectively. Throughout the paper it will be assumed that the system (2.1) is minimal. The study nonminimal c.ase is similar a.nd may be found in (Vafiadis and Karcanias, 1995a). The following result gives the solvability conditions of the decoupling problem for strictly proper systems and it is fundamental for the development of the solution of the decoupling for singular systnm,.

Theorell12.1 (Kou8siouris, 1979) Let O"j denote the control/ability indices (col.) of thr .state-space system with t.f H(s) = N(,)D-1(s) and" = max(,,;}. Consider th, matT'lx N a (,,) = N(.,)
-PR] ()

=

[L(.,) sf\ - A

0]

-1

A notion which is important for the solution of the problem is t.hat of the pivot indices (p.i) of a minimal basis (Forney, 1975). The p.l. are invariants of the minimal bases of a rational vect.or space, i.e. all the minimal bases of a space have the sa.me p.i.. [n our case the minimal basis under consideration is the cop rime and column ",duced matrix [NT (s), DT (s)]T The following definition classifies the p.i. into two types:

Definition 2.1 Let ql," " q.' denote the pivot indices oJ[NT(s),DT(s)f. Then qi is called proper ifqi > £ and llonpropcr If qj :S f.

Let [NT(s), DT(s)f have T :JOnproper p.i. and define 'P = (Pl'''',PI-,j C {l, ... ,tj as the set of integers such that qp, is proper and P = {jj" ... ,PT j C (1,·· . ,f) its complement. Clearly 15 is the set of integers Pi such t.hat Cfp, is nonproper. The invarianee of N(s) under control law (2.2) yields t.he invariance of the nonproper p.i. under •. liese t.ransformations. This is an important observation which is fundamental for the solution of the problem. From the above definitions it is clear that O"p; are nonpropef and i1"pj are proper c.L (Malabre et al.,

1990).

(2.5 )

=

where L(.<) = block - dia!l{L,,(s)} and L,,(s) s[le;,O(;xd - [O(;XI, le,]. Then the input-state t.f. of t.he system is

=

where 8(s) =block-diag{[l, s, .. ·. se, 'f} and ri + 1 are the reachabilit.y indices of the t.riple (E, A, B). From (2 ..5) i~ is dear ~hai D(s) = [sf{ - A]8(s). Now, application of cont.rol law (2.2) to the system with system matrix as in (2,Fi), yield:;.; dosf'd loop syst.em wit.h dellominatnr

3 THE SOLUTION OF THE PROBLEM [n this sect.ion the necessary
)!'(s)

Ei

Note that the numerator matrix of the input-state tJ. after the transformation (2.5) is Se,,) = q-' Ni'(S). The numerator matrix of the input.- output. t.f. of the closed loop syst.em is

= [,,;(s).···, ":(8)],

.r(s) = [d'l(s),"" d;(s)]

The following propositions ;Lre given wit.hout proofs which may be found in (Vafia(lis and Karcanias, 1995a). Proposit.ion 3.1 Let [N T (,,), DT(s)]T be a coprime and column reduced C011l1/osite matrix of a MFD. Then the ZC7'OS at infinity of the corresponding transfer Junction an~ gilJen by the zero.s at infinity of ihe matrix N(s)diag{s-t1;} where. (Ti dCII.oic the column degrees of [NT(B), [JT(s)JT. Propositio" 3.2 Let Q.(s) = [fr1(8),"', fr'(s)] and J!.(s) = [f:Il(S)," ·/J.(s)] be polynomial veelors such that Q,(s) ~J!.(s). where p(s) and q(s) are relatIVely prim' polynomi(J.h. Thctl.:

=

which shows that. t.he numera.t.or is 110t. affected by control law (2.2). Note t.hat statt" ft'edbac.k and regular inpnt transformat.ion do not. affect. the rank of the composite ma.trix of thf' MFn of t.he 1..f. Throughout t.he paper it will be a.<;surned, without. losfo, of generalit.y, that. the system mat.rix jl:> as in (2.:)).

(i) '1(8) i" a COIIIlllon divisol of (3, (s),' .. , (3,(s). (ii) deg ads) - degf:li(s) = dog aj(s) - deg{3j(s) deg/,(,,) - degq(s), Vi,j E I,"·,k.

1561

Let.
1f!i

$" f a.nd

= 0 alld consider t he ma-

{'f'1, . .. ,'f'I -, } n (
trires N«( s ) and No of theorem 2 .1, where now t1j are the c.i. of I,he singular sysl,cm (2.1) . Let J; be I.he orders

of the zeros at infinity of the t .r. !!~ '(s)D-' (s) . From proposition 3.1 it follows that h are equal to th e orders of zeros at infini Ly of !!<;:'; (s )diag( s-" j), j = 1,·, ·f. Note that. h are the orders of the illfinite zeros of thl~ systems (E,A,B, cr,p;) where (', denotes t.he i-th row of C . Consider now the rows !!qp; (s), i = J,"., T and let

fj, i = 1, ·· " T be the degrees of the greatE"st common divisors of the entrieE- of thesp rows. The indices hand f j will be referred to as prolif~r and 1lonpt'oper dccoupling indices resp ec.t.ivdy. Denote by mat-rix

Hr}

t.h~

entr ies of No and hy N~ the square

,,"«. l ,"'_ T ,.0 "', _

T

N°(.) and DO(s) respectively. Clearly the t.r. GO(s) = N"(s)(D"(s»-' is strictly proper. Let "p be the maximal proper c.. i. of (2 .1). From tbe solvability condition of the de(,-Oupling problem for state- space systems it follows that the matrix N'"(s)dilJg(s(1p-tt,;. ) is row reduced and thus the matrix (~O'-O', )N'"(s)diag(s(1,-a,;) is row reduced . The nQnsingulari ty of D he yields that there exists k is such that deg d:; = "p". Then from proposition 3.2 it follows thai,

degn~7'(s) - degd~:;(sl and since deg

,I::' (s) S

deg ,,~:, (s)

+ (0' -

"P>

ITp, tbe following holds tnle:

ITp,) S deg ,,~:, (s)

+ (" - "P.)

aud t.hus deg {sq-·,,"~:,(s)}

]

= degn~:;(s) -

S deg{s·-a" "::'(sj}

The a bove means t,hat the r ow degrees of the matrif.es

,'~ t_,

The main result follows:

Theor e lll 3.1 Nccf;s.~ary and .' jujJicient condition for lit e solvability of the dccol11Jiing 1l1"OiJlem is that N~ tS

i7LH'. rtibir.. Proof:

Lel, (2.1) be dccouplable.

Then, t.here ex-

ist. state feedback F and regula.r input transforrnat.ion G su('.h that. !!i (s) = i[(s)hd1>), whr.r~ D e ( ..;;) = l ("jT, .. . , Note that. deg
[d

'(slV.

- () s - (Ieg 1l)"'" () ... = (CJ!, deg d"'" I P. 1.


which contradicts t.he asslImptioLl tha!. p.i .. Thus the mat.rix n~ll( s )

ne:-~{s~ d~: (s)

(I eg Up", '"

_ J ( s) >

q,.. is a l1o nproper

7l~Lr (8) ~,-, t

Hl;![

( ) Sh

d:/_~(s)

-

[ N:(Sl] D (s

(3.1 )

ar~ equal and t.hus NI: is the row high order coefficient of N*(s)diag(sO'p-O',) which I,roves th e l1Ccessity.

For the proof the sutfic.ienc.y a regul ar feedback pair yielding diagonal dosed loop tJ. is e.onst.meted next. As a first. st.ep ollly input t ransformation is used 0 1 sHe,h that. (lp, = !.pi + f, wh ere ijp ; are the proper pj.

of the matrix [NT(s) ,DT{slf and DI( s) Note that

Ot l

= G,'D(s),

is a permutat.ion matrix. The mat.rix

[(N'(s»)", (Di(s)?V forme" from [NT (s), Di(sW as in (3.1) has column high ord er coefficient matrix Di .. c as in (:1.2) which is invertibk. The next step is to use an input transformat.ion O 2 011 the syst em (N(s) , D.(s))

sllch that D,(s) = (;,'U,' D , s ) has the following structure: All t he entries wit.h coe·rciin at.es (q;;. , pj) have degrt'*' less than ",,) . This is a lwayr; possible since Di,\c

is invert.ihle. Note that D:;" ., = Di'\ ~. Consider now the matrix 0 3 = D2,,-JN~)-I. Clearly , this matrix is invertible since N~ is invertil, lc and (0;)-1 D;"c = N~. Denot.e the ent.ries of ((;;) - 1 by gijl i = 1,··· ,£- T, i I " .. , f' - rand defille tlw matrix

=

0;;1 =

has c.oluffin high order (',oeftic.lf'ut matrix

[~~] , .

q3

wh ere

a::

-Pi

= Q,I; , g3 = --,--,:...p,

~ i!:gi,· ~ j =l

··r )

"-f

(:I.2) wll ~re det. Diu. # o. Since th(' syst.em is cler:.ouplablc it follows t hat !!"'( s ) = 4"'(s)"" ,( s ) and thns !!i(s) = l(s)h~ ,(s), where !!:(s) and !!7 ( s) are the i- th rows of

and ~ are the vedors of the s tandard basis of ~l. Let now D3(fO) = G:; I D;,!(s). Sil': ce Ng is invertible , there exists k E (I , ... , f - T) ",cb t hat 11~"", # O. The following may he easily verifh-d:

1562

It is shown next show that the system (N(s), D3(s)) is decouplable by state feedback only, or equivalently the

following equation has a solution with respect to F3 S(s):

D3(s) - F"S'(s)

= P{s)N(s)

~ W(s)

(3.4)

where

~ ~} V;(~)I i E {q-p" . "'"1Pr P(s) = diag{p,(s)}, 1>'(s) = {

!.J.:!.l . { } q,($)' l.E If'l'--·Ir.p~-T

(3.5) are pairs of relatively prime polynomials. The appropriate selection of pes) will give the feedback matrix. Eqllation (3.4) has a solution with respect to F38(S) only if Vi(S) and qj(s) are common divisors of the entries of !J:..1Jr:, (s), i = 1"", T and rr'Pi(s), j = L· . If - T, respedively. Let and (I',(s), v,(s», (t;(s), qj(s)

then, (3.4) is solvable with r"speet to F 3 . Note that if (i)-(iii) above are not satisfied then (3.4) is not solvable. Since F3 is a solution of (3.4), the system (N(s), D3(s») is deeollplaLle by state feedback alone, namely F3 • The realisat.ion of this system is the quadruple (K, A, BG,G,Ga , C). This system is decoupled the system (E, A + by state feedback Fa i.e. BG 1 G 2 G 3 F 3 , BG 1 (;2(;3, C) IHIS diagonal LL, i.e. t.he pair (F, G) (G,G2GaF3,c; C;,(3 ) decouples the syst.em (E,.I1, B, C) which proves the sufficiency.

=

Reulark 3.1 Tlu: pair (F, C) which decoup/es the system (2.1) is deJerUlzned from the polynomials vq,_(s), It,,;(S), q,;(s), I.,(s). Thus, when we select these "polynomials (Lccording to (i), (ii) and (iii) above, it is equivalent to selecting fcedbacks which decouple tlte system. If qip.i(S), tipj(s) arc not monic then W,7c = R- 1D~C! where it is dclrrmined from tlte hl.qhesi order coefficients of tluse polynomials. Then (3.4) has to be modified to

R-'[Da(s) - FaS(s)] = P(s)N(s)

=

where ,"'j(s) are the entries of W(s) P(s)N(s) in (3.4). Then since 8,,8(s) I, ()oiD3(s) ~ 1', - 1 if (Ti is nonproper and odD3(S) = 1'j if fTi hi ~lfoper (ad denotes the i-th column degre('), it follows that W(s) must be such that VV;~1' = D h<," Indeed, ifti(S) and qj(s) are chosen to be mOIEe the equality of the above column high order coefficient matrices is obta.ined. The polynomials J'q-. (s), Vq_. (,<;) ill (:J.!J) mu::;1 sat.isfy the degree P, 1', condition

= ", -

If the above is not. satisfied t.llClI the polynomial (.f; )/vq" (s )rrqp; (s) has degree greater than (Jp; and thus, (3.4) dOf"R not. have CL solut.ion with respect to F3S(S) becallse of tilt' limitatiOlI on t.lw column degrees of o5'(s). On the ot.her hand, since I\r~ is invertible, there exists always an entry nr.:( . . ) of ll
where R is determine(l from

k.

and the decoupling pair

is (G 1 G 2 G 3 F 3 , G 1 G 2 G 3 R). 4

FIXED POLES

In this section it is shown that a number of poles of the closed loop system are fixed and unobservable when dccoupling cont.rol is applied. The characterisation of these polcn in given and it is a direct generalisation of t.he fixed poles of dccoupling for the st.rid.ly proper systems case (Kollssiollris, 1980). From (:l,}), (3.4) it follows that the rows of tile cienorninator of the closed loop (decoupled) system are

/lq'i

(4.1 ) where !l'(s) = (I!~,(s»!!.\s). Since q,,(s)I~.Js) it follows that dcgq~,(s) ~ degE~,(s) and therefore degt~,(s) ~ degE.,(')+k Similarly, since I'",(S) isa divisor of ~""1', (8) follows that. deg- 1',_1', (8) ~ ~q_ (s). The tJ. of the closed loop system in ~,

H,(s) = di"y{h,(s)} = P-'(s)

Summarizing we have that if P(.5) is chosen such that

and degl,,(s)-

The characterisation of the fixed poles the of the decoupled syst.em is given by t.he fc,llowing theorem.

V,_ (s)I~... (s) and q, (sll~, (s), where ~,(s) is (,he

Theurem 4.1 LrI N I (!l'(s)f .. , (!l'(s))T where !l'(s), i = 1"",£ an, defined in (1,.1). Let ifs) = dctN(s}. Then The rools of i(s) are (i) the

(i) 0 ~ degv\7,(s) - degl' q" degq~,(s}

(ii)

(4.2)

1',

= j,;

1',

H <: T,

J )

greateHt common divisor of t,11f' eielnent.n of the k-t.h row of N(s) a.nd (lIb means" ({ divides b"

(iii) t,,(s) and

q~,(8)

are monif

r

fixed pole.~ of thf de.coupled 8ystcm and (iz) un observable polrs of IILl' clO.5f:d loop SystC11L.

1563

Proof: From (4.1) it follows t.hat. for ;my dcc.oupling law

D,(s)

= diag{pi(s»N(s) = diagN'i(s)}N(s)

(4.3)

where ,pitS) = pi(S){i(S) . T hus de/(D(s» = detN(8)nf=,,pi(S) = i(s) n f=,V'i(S) which means t hat t.he zeros o f z(ot-') arc C' tlways poles of th e doSt.'d loop syst.em. Now, if wc choose 'I,,(s) = {~,(s) and t~;(s) any polynomial satisfying (3.7), then li'~,(S) = t~ ;(s) wit,h t ~;(.) arbit.rary polynomial of,kgrce f;+deg{~;(s). On the other hand ifwe,hoosc vq_ (s) =~q_ (s) and /Iq_Cs) any polynomial of rlegree: is satisfied and

'1',

71

= 3, 1;._ (s) = .3 + I, <~ (s) = I, {~,(s) = s' + I , = 1, 11 "=- 1, 12 = 1. T he Illatrices No a nd Nt are No = [

100] ?? ~ , N~ = [ 0I 011

and th us, the system is dccoup lable, The matrices G l G'2 1 , G3 I , G are

(;-, = [ 001] I 0 0 010

1

" degree'" condit.ion (3 " .6) fi t.lw

0;;'=

. C-'

"

l

,

= [1001 0 0-1 I 0 ]

[6010 n] ,0=(;,0,0

3

=

[n 6] 110

We choose P{s) = diag{p'(-, ), p'{s), p3(.)) where: which dearly shows t.hat, t h t.' poles of the d osed loop system which ca.nnot bf> s hifted , an· on ly t.he rool.s of ifs) whieh proves (i). From (4 .3) it. 1" lIows t. hat

N, (S) ] = [diU9{{i( S)) ,\i(S) ] [ D,.(.) diny{ V'i( S)) N(s)

Then , from (3.4) and F = I!F" we take the feedback matri x F = [F' , F' , F 3 ] whele

Thus the zeros of N(s) are OlltpUt. decoup li ng zeros of the c-.Iose.d Joop system whic-h proves {ii}.

-:la l - 4 4al -V2 - :la, - 4

-V3 -

R e mark 4.1 If we select v,.1' , (.,) = ~q_P, (s) and /',_r, (s) any polYllom ia l of dl'grcc fi (lull (J!.p,(s) = ~if;(8) (wd t ~;(s) ""Y I'o/yn"""af of dry"" Ji + deg~~,( s), then the number of asszgut1bic POi(,8 i.~ 11I(11.'Zm.1Wl and it i.s cq1Lal T '-1' lu Li=' fi + L j= , (J; + deg ( 'p, (s)j. Exautple 4.1 CO lIsidt'r I,hc syst.em wit.h system malrix

• -I.<

0 0 0 0 0 0 I

0 0

0 0 0

0

0 0 0 1 0 1 :] 1 0

°

0 0 - I 0 s - I 0 0

0 0 0

s

0

0 0

0 0 0 0 0

0 0 I 0 2 0 3

0

0

()

0

0

0

F"

A r.op rim c and column reduced MFD of the t.f. is

D

I -

1 0

-m3

0 -V2

V;i

= [-,1--.3 '" -2 - -

(I,

- '
) -VI

-3 - :3

-1}1

gl ]

-r ]

]

The closed loop t.f. is H, (.) = p - I (.) and the number of the assignable poles is 7 whi ch is equal to 71 + /1 + h + deg6(s) + deg~3(')' !i

.,

°°

I 11 0 () 0 0

0 I

()

0 0 0 0 0 0 0 0 0 - I 0 0 0 0 0 0 -' - 1 0 - I 0 0 0 () R - I 0 I 0 0 -2 100 0 0 0 0" ... - 1 010 I 0 0 () 8-2 () 0 I I II ~ 0 0 [) 0 I 0 I 0 I 0

0 0

F'=

- tnl

- ' n~

"'3 FI= [ 1- 4a l

CONCLUSIONS

The problem of input-output ct ecoupling for square singular systems via state feed ba '~k a nd regular input transformation has he('n so lve:rl 1.sing a frequenc-.y domain method. The proposed approach has been shown to be a direct generalisation of the fr equency domain approaehes for stat.e--spact" systems. An easi ly t.est.able necessary and sufficient condition for t.h e solvability of the problem has been obtained alld the complet.e characterisation of the fixed poles of -, he decoupled system has been given. REfERE NCES

=

I , PI = 1, 1-12 T his system has: T = I , PI (11'1 = ;" (1" 2 1, qp, =- 1, q,J I =- 4 ,
=

= :1 , u P1 = a, = 5 , IP I = 2,

Ailon , A. ( 199 1) . "Decoupling of square singular systems Vi;:l proporlioll ai ::;Late feedback", IEEE Tran.~. Aulomat. C01l1rol, Vol. ,\(~36, pp. 95- 102.

1564

Falb, P. L. and W. A. Wolovich (HI67). "Decoupling in the design and synth~~sis of multi variable ('.ontrot systems") IEEE Traus. Altiomai. Control, Val. AC-12, pp. 651-659.

W, M, Wonham and A. S. Morse (1970). "Decoupling a.nd pole assignment in linear multi variable systems: A geometric approach", SIAM J. Control, Vol. 8 , pp. 1-18.

Forne.y,G. D. (197.5). "Minimal hases of rational vector spaces, with a.pplications to mllltivariable linea.r systems", SIAM J. Conlr'ol, vol. 13, pp. 193-520.

Gilbert, E. G. (1969), "The decoupling of multivariable systems by state feedback", SIAM 1. Control, Vol. 7, pp. 50-63. Koussiouris, T. G. (1979). "A frequency domain approach to the de('.Qupling problem" 1nl. j, Control, Vol. 29,991 1010. Koussiouri~)

T. G. (1980). "Pole assignment while block decoupling a minimal sysl.em by state feedback and a nonsingular input transformation and the observability of t.he hlock decoupJed system", 1nl. J. (:onlrol, Vo!. 2fJ, pp. 44:3-464.

Loiseau, J . .I., K. Ozealdiran K., M. Malabre and N. Karc.allias N. (UH)l). "A feedback classification of singular systPrI1s:', Kybe.ntelika, Vol. 27, No. 4, pp. 289-30,5. Malabre, M., V, KlI"era V. and P. Zagalak (1990). "Reachability and controllability indices for linear descriptor s),st('ms", Syst. ('(mlr. Letters, vol. I, pp. 119· In. Minamide, M., P. N. Nikiforuk and M. M. Goupta (1991). '~necollpling of (ksnipt,or syst.('ms", lEE Proceedings, part D, Vo1. 138, No. 5, pp. 453-459. Paraskevopoulos, P. N. and F. N. KOlllllbolllis (1991). "Decollpling and pole assignment in generahsed state-·space syskms", lEE Proceedings , part D, Vo!. 1:38, No. 6, pp. ,517-560. Rosenbrock, H. H. (1974). '·St,rucf,un.l Linear Dynamical Sys\,ems", ltlt. 20, no. 2, pp. 191~202.

Properties of 1. Contr., vol.

Vafiadis, D. and N. Karcallias (199:'a). "Decoupling and pole assignment. of singular syst.ems: A frequency domain approach", Rc~. Rep. DV INK CEC~140, Control Eng. Ccnt.re, Cit.y Oniversit.y. "Canonical Vafiadis, D. and N. Karcanias (lS.W5b). forms for singular syst.ems with outputs under rest.ric.ted syst.em equiva.lence", }'roc. JFAC Conference on System St.rudure and Control, Nantes, France, .1-7 July, pp. :.124--:12D.

1565