A polynomial characterization of (A,B)-invariant and reachability subspaces

A polynomial characterization of (A,B)-invariant and

reachability subspaces

Citation for published version (APA):

Emre, E., & Hautus, M. L. J. (1980). A polynomial characterization of (A,B)-invariant and reachability subspaces. SIAM Journal on Control and Optimization, 18(4), 420-436. https://doi.org/10.1137/0318031

DOI:

10.1137/0318031

Document status and date: Published: 01/01/1980

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SIAMJ. CONTROLANDOPTIMIZATION

Vol.18, No.4, July 1980

980 Society for Industrial andAppliedMathematics 0363-0129/80/1804-0007$01.00/0

A POLYNOMIAL

CHARACTERIZATION OF (/, M)-INVARIANT

AND

REACHABILITY

SUBSPACES*

E. EMRE-? AND M. L. J. HAUTUS{

Abstract.BasedonthestatespacemodelofP.Fuhrmann,a link is laid between thegeometric approach

tolinear system theory, as developed byW. M.WonhamandA. S. Morse, and the approach basedon polynomial matrices. In particular polynomial characterizations of (A,B)-invariant and reachability subspacesaregiven. Thesecharacterizationsare usedtoprovetheequivalence of thedisturbancedecoupling problemand the exactmodel matching problem and alsoto connect thepolynomialmatrixand the geometric approachtotheconstruction of observers.

Finally,constructiveprocedures andconditions aregiven for computing the supremal(A,B)-invariant subspaceandreachability spaceand forcheckingthesolvability ofthe exact modelmatching problem.

1. Introduction.The geometricapproachtolinear systemtheory hasprovedvery successfulinsolvingavarietyofproblems

(see

17]

foradetailed account of thistheory).

The principal conceptsin thistheory,which are instrumental inthe description of many results,are(A, B)-invariantsubspaces and reachability(controllability) subspaces.

An

alternativeapproachtolinearsystemdesignhas beendevelopedin

[13],

[14],

[16].

This

theory dependsto alargeextent onpolynomial matrix techniques. Itis evidentthat a

method for translating results ofonetheorytoanotherisvery desirable,becausesucha

method would yielda better understanding of the relations between the twodifferent

approaches.Thiswould be veryuseful,inparticular since the geometric methodmaybe

viewed as exponent of the so-called "modern control theory" and the polynomial matrix method may be considered ageneralization of the classicalfrequencydomain

methods.

A

number of papers with the objective of translating the results of geometric

controltheoryintopolynomialmatrix terms haveappeared (e.g.,

[1], [3],

[10], [12]).

It is the purpose of thispaper to show that a very useful link between the two

approachescanbebasedonthe work ofP.Fuhrmann

([7], [8], [9]).

Specifically,it will

be shown that usingthe statespace modelassociatedwithasystemmatrix, introduced by Fuhrmann, one can give characterizations of the concepts of (A, B)-invariant subspaces and teachabilitysubspaces intermsof polynomialmatrices.Thiswillbe the subject of 3 and 6. One application of the polynomial characterization of (A,

B)-invariant subspaces will be given in 4,where it will be shown that the disturbance decouplingproblem

(see

[

17,Chap.

4])

andthe exactmodel matchingproblem

(see

[ 16],

[

14], [5])

areequivalentproblems.Another applicationisgivenin 5,where it is shown

that the equivalence of the polynomial matrix and the geometric formulation of

observers can bederived fromthe results of 3.In 7,theconcept ofrowproperness

defined in

[14],

[16]

is used to formulate anecessary and sufficientcondition for the existence of a solution of the exact model matching problem and, hence, of the

disturbance decouplingproblemin termsof degrees of polynomial matrices. Also in 7

a constructive characterization of the supremal (A,B)-invariant subspace and reachability space contained in kerC is given. Finally, in 8, the results of 3 are

extendedtothe situationwhere the system is describedbyRosenbrock’s systemmatrix.

The preliminary 2contains ashort description of Fuhrmann’sstatespacemodel.

* Receivedbytheeditors October18, 1978,and in revised formOctober5, 1979.

5" Centerfor MathematicalSystemTheory, UniversityofFlorida, Gainesville,Florida32611. The work of this authorwassupportedinpart bytheUnitedStatesArmyResearchOfficeunder Grant DAA29-77-G-0225,andbythe UnitedStatesAirForce underGrant AFOSR76-3034Mod.B throughtheCenterfor MathematicalSystemsTheory, Universityof Florida, Gainesville, Florida.

Departmentof Mathematics,EindhovenUniversityofTechnology, Eindhoven,theNetherlands. 420

2. The state spacemodelassociatedwith a matrix fractionrepresentation. Let

K

be a field. Wedenote by

K[s]

the set ofpolynomials, andby

K(s)

thesetof rational functionsover

K.

If 6isanysetand p,q

N,

wedenote by6ep_{the setof p-vectors}with

componentsin6e and by5epqthe setof pxq matriceswith entriesin5

.

If

A

isapxq matrix, we denote by

{A}

the K-linear space generated by the columns of

A.

If

U(s)eKqr[s]

and

:Kq[s]K"[s]

is a linearmap, then

U(s)

denotes the result

obtained by applying toeachof the columns of

U(s).

Let

x(s)6K’(s). We

denote by

(x(s))_,

the strictly proper part of

x(s)

and by

(x(s))_l,the coefficient ofs-1in theexpansionof

x(s)

inpowers ofs

-1.

DEFINITION 2.1. Let

T(s)

Keq[s].

Then

_Xr

denotes the set

_of

x(s)

K[s]

_for

whichthereexists astrictly proper

u(s)

Ka(s)

suchthat

T(s)u(s)

x(s).

In

whatfollows,

_XT-

playsafundamentalrole

(compare

theclosely related concept

ofrightrational annihilator

[4]).

In

particular,if p=q and

T(s)

isnonsingular, then

XT-

{x(s)

K"[s]

T-X(s)x(s)

is strictly

proper}.

In

this particular situationwedefine the map

zrT-:KP[s]X7

x

(s)

--

T(s)(T-l(s)x(s))_.

(Compare

[7]

and

[8]

wherefurtherpropertiesof this map aregiven.)

In

the following

weconsider

XT-

aK-linearspace.Weconsideralinear system whose transfermatrixis givenbythe leftmatrix fraction representation

(2.2)

G(s)

T-(s)U(s).

Weassume that

G(s)

isstrictlyproper,

T(s)

K[s],

U(s)

Kr[s].

Definethelinearmaps

:X-

---,

_X-

_x(s)

_(sx(s)),

(2.3)

Yd

K

--

_XT-

u

U(s)u,

:XT-

K

q

_{:x(s)--(T-(s)x(s))-.}

By

definition, for

x(s)XT,

we have

4x(s)= sx(s)-T(s)c(s)

for some

c(s)6Kq[s].

Since

T-l(s)x(s)

and

T-l4,x(s)

arestrictlyproperitfollowsthat

c(s)

mustbe constant.

Hence

(2.4)

x(s)

sx(s)- T(s)c

for some c

K

q,

dependingon

x(s).

Thefollowingresult isprovedin

[7].

THEOrEM 2.5. The system

,

:= ((,

.,

)

with state space

_XT-

is an observable

realization

_of

G(s).

The realization isreachable

_iff

T(s)

and

U(s)

are

_left

coprime.

We

will call this realization

Z

the T-realization of

G(s).

Conversely, ifwearegivenanobservable system

Z

(C,

A,

B),thenwe construct a

leftmatrixfractionrepresentation of the transfer matrix of

Z

in thefollowing way.Let

(2.6)

C(sI-a)

-=

T-(s)S(s),

whereS and T areleft coprime. Define

(2.7)

U(s):=S(s)B.

422 E. EMRE AND M. L. J. HAUTUS

THEOREM2.8. The T-realization

_of

G(s)

T-I(s)U(s),

whereTand

Uare

_defined

by

(2.6)

and

(2.7)

isisomorphictothesystem

,.

Proof.

Using the dual of

[11,

Cor.

4.11],

we see that $(s) is a basis matrix of

XT.

Hence

thelinearmap

:K

_XT

:x--S(s)x

is anisomorphism. Usingtheequation

T(s)C=S(s)(sI-A),

whichfollows from

(2.6),

onederives easily therelationsg6e _{6eA, 6eB, 6e C.}

In

particular, g6ex=g(S(s)x)=

zrT.(S(s)(sI-A)x)+zrT-(S(s)Ax)= zrT-(T(s)Cx)+

zrT-(S(s)Ax)

$(s)Ax

YAx.

It

follows that(C,

A, B)

and (c,

g,

)

areisomorphic. [3

Using Theorem2.8,wemaytransform results obtained for the particular realiza-tion

(’,

,

)

toanyobservable system.

3. (,)-invariant

_sulslaees.

Wegiveacharacterization ofthe

(’,

)-invariant

subspaces of the T-realization ofatransfer matrix

G(s)

T-l(s)U(s),

asdefinedin the previous section.

For

the definition of(4, )-invariantsubspaceswereferto

17].

THEOREM 3.1. Let

(s)

be a q

m

polynomical matrix. Then

{(s)}

is an

(, )-invariantsubspace

_ofXT-iff

thereexist

_Ca

g

qm,

F1 Kr’and

A

K""such

that

(3.2)

T(s)C1

+

U(s)F1

(s)(sI-A1).

Proof.

Suppose

that

{(s)}

is an(’, )-invariant subspace,i.e.,

(3.3)

{(s)}

c_

_{(s)}

₊

Im

.

Applying

(2.4)

toeachcolumn of

(s),

wefine that

4(s)=

l(S), where

(3.4)

a(s)

:=

_s(s)-T(s)C1

for some

C1

K

q". Onthe otherhand,

(3.3)

implies

(3.5)

l(S)

(s)A1

+

U(s)F1

for some

A

K

""

and

_F1

K

r".

Combining

(3.4)

and

(3.5)

yields

(3.2).

Conversely,

ifweassume

(3.2),

then

(3.6)

T-l(s)(s)

(C1

+

T-I(s)U(s)F1)(sI-A1)

-1

isstrictlyproperand hence

{(s)}

Xr.

Furthermore,if wedefine

l(s)

by(3.4),then

(3.5)

followsfrom

(3.2)

and, hence,

{l(S)}

XT-.

Itfollowsthat

’(xI)’(S))

"rt’T(SXIt(s))

7rT(XI(S)

+

T(s)C1)

xItl(s).

Thus,

(3.5)

implies

(3.3).

El

If the matrix

(s)

occurring in Theorem 3.1 has full columnrank, itispossibleto

give an interpretationtothematrices

A

1,F1,

C1.

Forin that case there existsa K-linear

map

:XT

K

satisfying

(3.7)

(s)

F.

Then

(3.2)

implies

It

followsthat

{(s)}

is an

(M-

-)-invariantsubspace,andthat

A

isthe matrixofthe

restriction of s4- to

{(s)}

withrespecttothe basismatrix

(s). In

addition,

F1

is

the matrix(withrespecttothebasismatrix

(s)

of

{(s)}

andthe naturalbasisin

K r)

of

.

Finally,wehave

(3.9)

(s)--

C1

so that

_C1

isthematrixof the restriction of to

{(s)}

withrespecttothe basismatrix

(s)

of

{(s)}

andthenaturalbasisofK

q.

The last result givesacharacterizationof(d, )-invariantsubspacescontained in

ker.

COROLLARY 3.10. Let

(s)Kq"[s].

Then

{(s)}

is an (M,)-invariant

subspacecontained in ker

_iff

thereexist matrices

F, A

such that

(3.11)

U(s)F

(s)(sI-Ax).

Proof.

Accordingto(3.9), wemusthave

C1

0 informula

(3.2).

[3

COROLLARY3.12.

Xu

isthe largest (M, )-invariantsubspaceo]XTcontainedin

ker

’.

Proof.

According to (3.10), we have for an arbitrary

(M,

)-invariant subspace

{(s)}

contained in ker

:

W(s)= U(s)F(sI-A)

-.

Hence

{(s)}c__Xu (see

Definition

2.1).

It remains to be shown that

_Xu

itself is an

(s4, )-invariant subspace. If

(s)

is a basis matrix of

_Xu

then there exists astrictly propermatrixQ(s)such that U(s)Q(s)

(s).

Let (F1,A1,

B1)

be a reachable realiza-tionof Q(s),so that

(I)(s)

U(s)FI(sI-A1)-IB1.

Itfollows fromLemma3.13 that

W(s)

:=

_U(s)F(sI-AI)

-1

isapolynomialmatrix.

By

Corollary 3.10,

{(s)}

isan(s4, )-invariant subspace. Hence

{(s)}_ {(s)}.

Onthe other hand,since

(s)

(s)B,

itfollows that

{(s)}

c__

_{(s)}.

Consequently,

Xu

{(s)}

{(s)}

is an(M, )-invariantsubspace.

I-I

LEMMA3.13.

Let O(s)

gln[s],

A

K

nn,

B

K

nr,

(A,

B)

reachable.

_If

O(s)

(sI-A)-aB

is apolynomialmatrix then

O(s)(sI-A)

-

isapolynomialmatrix.

Proof.

Wedecompose therational matrix

O(s)(sI-A)

-

into itspolynomialand its

strictly proper part"

O(s)(sI -A)

-

P(s)

+

R

(s);

then

Ro

:=

_{R(s)(sI-A)= O(s)-P(s)(sI-A)}

is apolynomial of degreezeroandhenceconstant. Itfollowsthat

Ro(sI-A)-XB

O(s)(sI-A)-B-P(s)B

is astrictlyproper polynomial and hencezero. Since (A,

B)

is reachable,thisimplies

Ro

0 andhence

424 .E.EMRE AND M.L.J. HAUTUS

The foregoing implies that the set of

(M,

)-invariant subspaces in ker is uniquely determined by the numerator polynomial matrix of the matrix fraction

representation of the transferfunction matrix:

COROLLARY3.14.Let

U(s)

gqXr[s],

Ti(s)

gqXq[s],

1,2, such that

G,(s) :=

T

.1

_(s)U(s)

isstrictly proper

_for

1,2.Let %,Mi, i)bethe

Ti-realization

of

Gi(s)

fori

1,2. Then

M

_Xcr

is an (M1, 31)-invariant subspace

o]

_Xrl

contained in ker

₁

_iff

M is an

(2,

32)-invariantsubspace

o]’Xr2

containedin ker

2.

REMARK

3.15. Theorem 3.1 may be specializedtothecase

U(s)

0,that is, 0.

In

thiscasewehavea realizationof

G(s)

0 with the samestatespace

_Xr

andthe same

map as before.

An

(M, )-invariant subspace of

_Xr

then is just an M-invariant subspace. Thusweobtainthe following characterization of M-invariant subspaces.

PROPOSITION. Let

(s)

be a qxm polynomial matrix. Then,

{(s)}

is an

sC-invariantsubspace

_{of Xr}

_iff

thereexist

_C

K

q", A1

Km"

suchthat

T(s)C1

(s)(sI

A 1).

4. Exact modelmatchinganddisturbance decoupling. Ifwe have anobservable system

(C,

A, B)

with statespace

X

thenwemayconsidertheproblem of characterizing

the (A, B)-invariant subspaces contained in kerC. Using the isomorphism given in

Theorem2.8,wetransform theproblemtothecase ofa suitable T-realization.

For

this

case wemayappealtoCorollary3.10, by whichacomplete characterization is given.It

is important that,asalready notedinCorollary 3.14,thischaracterizationdepends only

onthe numerator

poly_nomial

U(s). Consequently,wehavethefollowingresult.

THEOREM. Let

,

(C,

A,

B) be a realization with state space X

_of

a

_transfer

matrix

G(s)

T-l(s)U(s),

andlet (q,

sg,

3)

bethe T-realization

_of

G(s).

_If

,

and

Z

are isomorphic by the isomorphism

."

X-,

Xr,

then

M

_

X

is an

(A,

B)-invariant subspacecontained in ker

c

_iff

thereexist constantmatricesF1,

A

satisfying

U(s)F

W(s)(sI-A

), where q(s) isa basismatrix

_of

(M).

Thuswe seehowcharacterizationsfor

(,

)-invariant subspacesof the particular

state spacemodelY., can be generalizedto arbitrary

(observable)

state space models.

In

this section we usethe theorydevelopedthusfartoshow the equivalenceofthe

exactmodel matching problem and the disturbance decoupling problem.

PROBLEM 4.1. (Disturbancedecoupling problem

(DDP)).

Given thesystem

(4.2)

2(t)

Ax(t)

+

Bu(t)

+

Eq(t), y(t) Cx(t),

where

(C, A)

isobservable,determine a constant matrix

F

such that

_if

u(t)

Fx(t),

>-_

O,

the outputy(t) doesnotdependonq(t),

>=

0.

The following result has been given in

[17,

Thm.

4.2]

in aslightly different but equivalent formulation"

THEOREM4.3. Problem 4.1 hasasolution

_iff

thereexists asubspace

M

_of

thestate

spacesuchthat

AM

_

M

+ {B},

{E}

c_

M

c_kerC. l-1

PROBLEM 4.4.(Modifieddisturbance decouplingproblem

(MDDP)).

Givensystem

(4.2),

determine constant matrices

F

andDsuch that

_if

u(t)

=Fx(t)+Dq(t),

theoutputdoesnotdependonq(t).

In

the modified problem, one assumes that not only the state but also the disturbance is directly available for measurement. Similarly to

(4.3)

we have the following result.

THEOREM4.5.Problem 4.4hasasolution

_iff

thereexists asubspace

M

suchthat

AMcM+{B},

{E}cM+{B},

MckerC.

The exactmodel matchingproblemis defined as follows.

PROBLEM 4.6. Given

_transfer

_function

matrices

Gx(s)

and

G2(s)

determine a (i)

strictly properor (ii)properrational matrix

O(s)

such that

Gl(s)O(s)=G2(s).

Problem 4.6(i) will be called the exact model matching problem

(EMMP),

and

Problem4.6(ii)willbecalledthe

_modified

exactmodel matchingproblem

(MEMMP).

It

is thepurpose of thissection to show that theexistence ofasolution of Problem 4.1 is

equivalenttotheexistenceofasolution of Problem4.6(i).Similarly" Problem 4.4 hasa

solution itt Problem 4.6(ii) has a solution.

We

will concentrate on the modified

problems.The originalproblemscanbe dealtwith similarly.

Firstwehavetoindicate which

MEMMP

correspondstoagiven

MDDP

andvice versa. Letusstart withsystem

(4.2).

The data

Gx(s)

and

G2(s)

of

MEMMP

arethen defined by

GI(S)

:-

C(sI-A)-IB,

G2(s)

:=

C(sI-A)-XE.

Conversely,if we are given

Gx(s)

and

G2(s)

in

MEMMP,

we construct anobservable realization

(C, A,

[B, El)

of thetransfer matrix

[Gx(s), G2(s)].

Then

C,

A,

B,

E

arethe data for

MDDP.

Thus,wehavea one to onecorrespondencebetween

MEMMP’s

and

MDDP’s.

Following Theorem 2.8,weassume that

C(sI-A)-X= T-l(s)S(s)

with

T(s)

and $(s) relatively prime, and

U(s)=S(s)B;

and we consider the

T-realization

(,

,

N)

of Gl(S)=

T-l(s)U(s).

According to Theorem 2.8, the map

5:x

S(s)x :K

Xr

is anisomorphism. Consequently,weintroduce thepolynomial

matrix

R (s)

:= $(s)E asrepresentativeofEin

Xr.

Thenwehave

G2(s)=

T-I(s)R

(s)

andwe canstatethe following result.

THEOREM4.7. Let

{(s)}

bean (sg, )-invariantsubspacein ker

,

sothatthere

existconstant matrices

Fx

and

A

satisfying

(4.8)

U(s)F1

(s)(sI

A 1).

In

addition,assumethat

{R (s)}_

{(s)}+{U(s)},

sothatthereexist matrices

B1

and

D1

such that

(4.9)

R (S)= xIt(s)B1

+

U(s)D1.

ThenQ(s) :=

Fl(sI-Ax)-XB1 +Dx

is asolution

_of

MEMMP.

Conversely, let Q(s) bea

solution

_of

MEMMP

and let

(F1, A

x,

Bx, Dx)

bea reachablerealization

of

Q(s). Then

426 E.EMRE ANDM. L. J.HAUTUS

Proof.

If

(s)

satisfies

(4.8)

and

(4.9)

then

U(s)Q(s)-

(s)B1

+

U(s)D1

R (s),

which implies Gl(s)Q(s)

G2(s).

Conversely thelatterequation implies

U(s)Q(s)-R (s).

Hence,

(4.10)

U(s)FI(sI-A1)-IB1

R(s)-

U(s)D1.

Since

(A1, B1)

isreachable itfollowsfrom

Lemma

3.13 that

(4.11)

(s)

:=

_U(s)F(sI-A1)

-1

is apolynomial.Now

(4.10)

and

(4.11)

imply

(4.9)

and

(4.8).

COROLLARY 4.12.

MEMMP

has a solution

_iff

the corresponding

MMDP

has a

solution.

Similarly oneproves

PROPOSITION4.13.

EMMP

hasasolution

iff

thecorresponding

DDP

hasasolution. Thus, ifwe want to solve

(M)EMMP,

we may construct the data

A, B, C, E

of

(M)DDP

andsolve the latter problem. Thenwedo notonlyobtain asolution Q(s)of

(M)EMMP

but alsoarealizationOf this solution.

In

thisrespect,it is importantto note

that thesolutionof

(M)EMMP

onlydependsonthenumeratorpolynomials

U(s)

and

R (s).

Consequently, byasuitable choice of

T(s) (not

necessarily equaltotheoriginal

denominatorpolynomial)wemaytrytoobtainasimple

(M)DDP;

compare

[3]. We

will

formulate thisideamoreexplicitlyin 6.Alsoin 6,wewillgive existence conditions

for asolution of

(M)EMMP

and,hence, of

(M)DDP

in termsof

U(s)

and

R (s).

The following result states that if disturbance decoupling is at all possible by a

(dynamic)controldependingcausally uponq(t),thenitis possible byafeedbackcontrol

of the form u

Fx

+

D

lq.

COROLLARY 4.14. Let there existaproper rational matrix

H(s)

such that,

if

the controlu

H(s)q

isusedin

(4.2),

theoutput doesnotdependonq. Then

MDDP

hasa

solution.

_If

thereexists a strictly propermatrix

H(s)

with this property, then

DDP

has a

solution.

Proof.

If thecontrol u

H(s)q

isused in

(4.2),

then the transferfunction matrix

from q to y is

Gl(s)H(s)+ G(s).

If y does notdependon q, then thistransfermatrix mustbezero,hence

GI(s)H(s)

-a2(s),

that is,

-H(s)

is a solutionof

MEMMP.

Consequently, byCorollary 4.12,

MDDP

hasa solution.

5. Observers.

We

considerseveral formulationsof the observerproblem,which is

awell-knownprobleminlinear system theory. Further referencesonthe subjectcanbe found in

[14],

[15],

[16],

[19], [20],

[2]

and

[6].

Thus fartwotypes of formulation of this problem have appeared in the literature: the geometric formulation

(see

[19], [20],

and

[2])

and the polynomialmatrix

formula-tion

(see

[14],

[15],

and

16]).

Here,

our purpose is

(based

on the results on the connections of the geometric theory oflinearsystemsandpolynomialmatrixapproaches developedin

3)

to show explicitly the algebraic equivalence of the geometric and the polynomial matrix

formulations of this problem, including the case where some of the inputs may be

Let 5; (C,

A, B)

be a given system over R. Let C- be a subset of C satisfying

C-f’lR#

.

We

call a rationalfunction

u(s)

stable (withrespectto

C-)

if

u(s)

hasno

poles in C\C-.

In

the continuous time interpretation of

E,

one might choose C-=

{s

e

_CIRe

s

< 0}

and in discrete time C-

{s

e

_CI

_Isl

<

1},

but also different choices ofC- arepossible.

We

assume that C

Nqn,

Be

Nnr

and that in addition to

,

we are given a

feedthroughmatrix

D

x Nqr.

In

continuous time, the interpretation(Y_,,

D)

reads:

(5.1)

2

Ax

+

Bu,

y

Cx

+

Du,

and the transfer function of

(, D)

is

G(s)

G.D(S)

C(sI-A)-IB

+

D.

DEFINITION 5.2.LetL

".

A

system

(,, )

(,

:,

,

:)

is anL-observer

_of

(,

D)

_iffor

every initial valueXo

of

E,

Yo

of

X

and every control

function

u, theoutput

_of

(5.3)

=+iy,

=+Dy

satisfies"

-

Lx

isstable (inparticularrational).

The observerusesonlythe output of(Y_,,

D).

If onewants toconsider the situation in whichpartialor totalknowledge of theinputof5;_{isavailable, onecan incorporate}

this intheproblembyasuitable choice ofD.

In

particular,if the input iscompletely known,oneintroduces newmatrices

,/

and a newoutput

₃₇

of

E

accordingto

sothat

₇

x

+/u

represents thetotal data availablefortheestimation of

Lx.

Letususe thefollowingnotation"

G(s)

:=

_G.,(s)=

C(sI-A)-B

+D,

(S)

:=

_O,,l(S)

(sI-

/)-IB

-1-/,

GL(S)

:=

L(sI-A)-IB.

Thenwehavethe following result.

THEOREM 5.4.

Let

the system be reachable and let be observable. Then the followingstatements areequivalent:

(i) (Y.,,

D)

isanL-observer

_of

(:E,

D).

(ii)

GL(s)= G(s)G(s)

andtr(a)_ C-. (iii) Thereexists arealmatrix

M

suchthat

DD

O,

MA

AM +BC,

L

CM+DC,

MB

BD

and

o’(A

c_C-.

Proof.

(i)=), (ii). IfXo 0,

o

0, thenwehave

-L

=H(s)a,

428 E.EMREAND M.L.J. HAUTUS

particular, u

e^’uo

withA

C,

wefind

:9-L=

y.(s-A)-"-H(s)Uo.

Since

)3-L

hastobe stable for all n

N,

h C,

Uo

,

itfollowsthat

H(s)

O. Furthermore,ifXo

O,

u

O,

then

:-L,f

(sI-)-$o.

Since

(C,

A)

is observable,thestabilityof

:

-L

implies that

r(A)

_

C-. (ii)

=>

(iii).

We

use the matrix fraction representation

O’o(S)

:=

_{B’(sI-A’)-IC}

’=

T-I(s)U(s)

defined

by B’(sI-A’)

-=

T-(s)S(s),

U(s)= S(s)C’,

andweconsiderthe

T-realiza-tion

(,

M,

)

of

G’o(S) (see

Theorem

2.8).

Then the equation

G(s)G(s)= GL(s)

may

berewritten as

Hence,

ifwewrite

(5.5)

then

(5.6)

U(s)

+

T(s)D’)’(s)

S(s)L’.

W(s)

:=

(U(s)+

T(s)D’):’(sI-’)

-,

W(s)C’

SL’

U

+

TD’)D’.

Since

(C, A)

isobservable,itfollows fromLemma3.13 that

W(s)

isapolynomial matrix.Equation

(5.5)

implies

U(s)B’

+

T(s)D’B’=

W(s)(si-A’).

Hence, {W(s)}

is an

(M,

)-invariantsubspace, and

(5.7)

M(s)

sg(s)- T(s)D’B’ = (s)A’

+

U(s)B’

(see

(3.4)). We

consider again the

map

defined in the proof of Theorem 2.8"

x

S(s)x.

Thenwedefine

M’

b-lW(s),

so that

S(s)M’

W(s).

Itfollows from

(5.7)

that

A’M’

MSfM’

dW(s)

W(s)A’

+

U(s)B’

M’A’

x

C’B’.

Hence,

A’M’

M’A’ +

C’B’.

Furthermore,

(5.6)

implies"

T-l(s)W(s)

’-

T-(s)S(s)L’

+ T

-1UD’=

Since the left-hand Side is strictly

proper,

itfollowsthat

D’D

0 and

5M’

C’ L’ +

C’D

O,

Hence,

M’C’-L’+C’D’

=0.

Finally,itfollows from

(5.7)

that

T-l(s)U(s)’

+

T-(s)(s)fix’

sT-XW(s)-D’:

’.

Hence,

since

T-l(s)U(s)

and

T-l(s)(s)

arestrictlyproper. (iii)

::>

(i).

A

short calculationyields

-Lx

C(-Mx),

d

d--

Mx

A

Mx

),

and the result follows fromr(A)_ C- [3

The equivalence (ii):> (i) isgiven in

15],

andthe equivalence (i):> (iii), withthe a priori assumption(intheproofof(i)::), (ii))that

-

Mx

isstable, isgivenin

[2].

Notice that here the stability of

-

Mx

is aconsequence,rather

than

anassumption

(see

the proof of (iii)::),(i)).

For

the situation of availability of the whole input, thiswas also shown in

[6].

REMARK

5.8. The results of thissectioncaneasilybeextendedtosystems overan arbitraryfield

K,

providedanappropriate definitionofstable rational function has been defined. Suchadefinitioncanbe given as follows:

Let

:t/be amultiplicativesubsetof

K[s]

(i.e.,

p(s)

l,

g(s)

ell::>p(s)g(s)

;

1

).

Then we say that a rational function

r(s)e K(s)

is stable if

r(s)

has the representation

r(s)=

p(s)/q(s) with

p(s)

K[s], q(s)

l. Then the stable functions formaring.

In

the situation describedabove

wehave

{p(s)

K[s]lp(s)

0=>

s

C-}.

In

the general situationTheorem5.4 remains validifone replacesthe condition

r(A)

C-with

"(sI-

fi)-i

isstable".

A

particularexample,which is relevantfor discrete time systems, over arbitrary fields,is

/:=

{s"

_In

=0,

1,...}.

An

observer constructed according this multiplicative set is called a deadbeat

observer.

{i. Reaehalility

_snlslaees.

Let

(s)

be a full column rank basis matrix of an

(,

)-invariantsubspace.

Recall

theinterpretation of the matrices

A

1,F1,

C1

givenin

(3.7),

(3.8)

and

(3.9).

Let

_B1

beanyconstantmxp matrixsuchthat

{(s)B1}

{U(s)},

say

(s)B1

U(s)L1.

Then

B1

isthematrix ofthe (codomain) restrictionof

_Y3L1

to

{(s)}. It

followsthat (g-

3)’Ll

v

(s)A

B

v

for everyv e

K p.

Consequently,

(6.1)

(4-IL)={(s)[B,...,

An-IB1]}.

Thisformula immediatelyimpliesthe following result.

THEOREM6.2.

Let

(s)

be

a (full

column

rank)

basis matrix

_of

an

(4, )-invariant

subspace. Then

(i)

{(s)}

isa teachabilitysubspace

_iff

thereexistsa constant

matrix

B

such that

{T(s)B1}

_

{U(s)},

and

(A1,

Ul) isreachable

(here

A1

isgivenby

(3.2)).

(ii)

_If

B

isa constantmatrix such that

430 E.EMRE AND M. L. J. HAUTUS

then

{q(s)[B1,’’’,

A’-IB1]}

isthe supremalreachabilitysubspace contained

in

{(s)}.

Letus nowconsiderreachabilitysubspacescontainedinker

c.

Let

(s)

be a basis matrixofsuch aspace. AccordingtoCorollary3.1 0, thereexist matrices

_F1

andA such that

(6.4)

(s)

U(s)FI(sI-A1)

-1.

Itfollows from Theorem6.2 that there exists

B1

suchthat (A1,

B1)

is reachable and

{(s)B1}

_

{U(s)},

say

(s)B1

U(s)L1. Hence

(6.5)

U(s)Q(s) U(S)tl,

where Q(s):=

FI(sl-A1)-IB1.

Also, since

(s)

has full column rank, (F1,

A1)

is

observable,asfollows from

(6.4).

Hence

(F1, A

1,

B1)

is a minimalrealization ofQ(s).

COROLLARY6.6. Thereexists anontrivial reachabilitysubspacecontained in ker

iff

{ U(s)}

0

xt,

{o}.

Proof.

If

(s)

is a basis matrix of the

(,

)-invariant subspace

Xtr

and

(s)=

U(s)FI(sI-A1)

-1,

then the supremal reachability subspace contained in

_Xt

(or,

equivalently,inker

)

is nontrivial iff

_B1

0,whereB1is a matrixsatisfying

(6.3).

[3

According to

(6.5), O(s)-L1

is a nontrivial right zero matrix of

U(s).

Consequently, if the supremal reachability subspace contained in is nonzero, then

U(s)

is not left invertible. The converse, however, is not true.

For

example, if

U(s)

Ul(S),

0]

where

Ul(s)

isleft invertible,then it iseasilyseen that

U(s)

is notleft invertible, and

{U(s)}

71Xtr

{0}.

In

orderto giveanecessary and sufficient condition forthe existenceofamaximalreachability subspace containedinker

c,

weconsider the

K[s]-module

(6.7)

A:=

_{v(s)

_Kr[s]l

U(s)v(s)

0}.

Thismoduleisgenerated by the columns ofamatrix

M(s)

(see 15,

Thm.

3.1]).

COROLLARY6.8. Thereexists anontrivial reachability subspace containedin ker

c

iff

themodule A

_defined

in

(6.7)

is notgeneratedbya constant matrix.

Proof.

Let

M(s)

be a generator matrix of minimal degree, say

M(s)=

Mos

k

+"

"+Mk.

Then

s-kM(s)=

O(s)-L1,

where

O(s)=Mls

-1+.

_+Mks

-k

and

L1

=-M0.

Wehave

U(s)O(s)=U(s)L1

and

U(s)L1

O,

since, otherwise,

[M(s)-skMo,

M0]

would be agenerator matrixof

lowerdegree than k.Itfollows that

{U(s)L1} {U(s)}f’]Xu,

sothat{U(s)}fqXcr#

{0}.

Conversely, suppose that A is generated by a constant matrix, say

D,

and that

v{U(s)}f’lXtz,

say v

U(s)c

U(s)r(s),

wherec is a constant vector and

r(s)

is a

strictlyproperrational vector.

It

follows that there existsa rationalvector

q(s)

such that

c-r(s)=Dq(s).

Decomposing q(s) into a polynomial and a strictly proper part

q(s)

=ql(s)+q2(s), we conclude that c=Dql(s), so that v

U(s)c

0. Hence,

{ U(s)}

o

x

{0}.

Nowwe have a procedure for constructing reachability subspaces contained in

ker

.

Choosing anymatrix

_L1

suchthat

{U(s)L1}_Xu,

wehave

U(s)O(s)= U(s)L1

for some strictly proper

O(s).

If(F1,A1,

B1)

is aminimalrealizationof

O(s),

itfollows

thatq(x) :=

U(s)FI(sI-A 1)

-1isabasis matrix ofareachabilitysubspace, provided the columns ofq(s)areindependent.

In

general,it seems difficult toformulateconditions

upon

L1

andQ(s)thatguarantee that

(s)

hasfullcolumnrank.

A

sufficientcondition

for this isthat Q(s)be astrictlyproperrational matrix with minimal McMillandegree

satisfying the equation U(s)Q(s)=

U(s)LI.

Indeed, ifin thiscase

(s)

doesnothave full columnrank, thereexists

(s)

withlesscolumns than such that

{(s)}

{(s)}.

Since

{(s)}

is an (sg, )-invariant subspace, there exist F2,

A2

such that

(s)=

U(s)F2(sI-A2)

-a.

Also,there exists

D1

suchthat

(s)= (s)D1. Hence,

U(s)O(s)--

xIt(s)B1

((s)D1B1

U(s)Q2(s)

U(s)L1,

where Q2(s):=

F2(sI-A2)-aDIB1

has lower McMillandegreethan

O(s).

THEOREM6.9.

Let

L1

bea constant matrixsuchthat {U(s)LI}={U(s)}fqXu.

Let

Q(s) be a strictly proper rational matrix

_of

minimalMcMillan degree, satisfying the

equation U(s)Q(s)

U(s)LI.

Let

(F1, AI, B1)

bea minimal realization

_of

Q(s). Then

xlz(s)

:=

_{U(s)FI(sI-AI)}

-1

isabasis matrix

_of

thesupremalreachability spacecontained in ker

.

Proof.

The supremal reachability subspace contained in ker

c

is the (unique)

minimal (, )-invariantsubspace satisfying

(Im )

(3 o/_ 7/"

_

o/g, where o/# isthe

supremal (s

g,

)-invariant subspacegontained in ker

.

To see this, observe that an

(4, )-invariantsubspace7/"satisfying

(Im )

f3/d/"

_

7/’

_

is

(

--)-invariantfor every such that is

(1

-’)-invariant. Indeed,

(

")

7/"

_

(4

-)

//V

W

and

(4- Y3’)

7/’

_

7/’

+

Im

imply

(1-

)c_ Wf3

(+Im )=

o//.+o/g.f-)im

_

.

Since

{U(s)}f]Xu={U(s)L1}={xIt(s)B}_{xIt(s)}Xu,

and because of the minimal

McMillandegree ofQ(s), the resultfollows. 71

In

thenextsection,it willbe shownhow Theorem 6.9canbe usedfor the explicit construction of the supremal reachability subspace.

7. Constructive characterizations. Conditions for solvability and the charac-terization of solutions of variousproblemscanbe made explicit by theuseofrowand column propermatrices

(see

[16]).

Thiswill bethe subject ofthissection.

If

R

KPq[s]

hasrows

rx(s),

",

rp(s),

thendegri(s)iscalledthe ithrowdegree

of

R (s).

The coefficientvectorofs

’

inri(s),whereui degri(s)iscalled theithleading

coefficient

rowvector, and is denoted by

[ri]r. We

denote by

[R]r

thematrixof leading coefficient row vectors, that is, theconstantmatrix with rows

[rx]r,

",

[rp]r.

Similarly,

[R ]c

denotes the matrix ofleadingcoefficient column vectors,thatis,

[R

]c

([R’])’. A

matrix iscalledrow

(column)

proper if

[R

]([R

]c)

isnonsingular.

A

rowpropermatrixis

easilyseen tobe right invertible.Conversely, wehave

(see [16,

Thm,

2.5.7])

LEMMA 7.1.

_If

L(s)

KPq[s]

is right invertible thereexists a unimodular matrix

M(s)

KPP[s]

such that

M(s)L(s)

is rowproper withrowdegreesu,

,

upsatisfying

,

<-...

_<=

_up.

_If

_{L(s) KPq[s]}

_{is not right invertible, them exists a unimodularmatrix}

M(s)

such that

M(s)L(s)=[

Ll(s)]

0

where

L(s)

is rowproper withrowdegrees

,

<-.

_<=

_’l. Thenumber

_of

rows

_of

Ll(S)

equals the rank

_of

L(s).

The row _{degrees u} are independent of

M(s)

(which is not unique) andwill be

calledthe row indices of

L(s).

The following result

(see

[14, Property

2.2])

states a simple criterion for the

propernessofarational matrix

T-(s)

U(s)

if the denominator polynomialmatrixisrow

432 E. EMRE AND M. L.J.HAUTUS

LEMMA

7.2.Let

T(s)

berowproper withrowdegrees1)1, 1)q.

If

therowdegrees

of

U(s)

arehi,"

,

hq then

T-(s)U(s)

isproper

_iff

h <-_vi(i 1,.

,

q)and strictly proper

iff

Ai

<

1}i (i i, ",q).

Observethat if

T

is not rowproper,there exists a unimodular matrixM(s) such that

T(s):=

M(s)T(s)

is row proper. If we define Ua(s):= M(s)U(s), we have

T-(s)U(s)

T-

(s)U(s),

andwemayapply Lemma7.2.

Letus now consider

(M)EMMP

asdefinedinProblem 4.6.

Assume

thatwehavea

matrixfraction representation

T-(s)[

U(s), R

(s)]

of

[G(s), G_(s)].

Thenthe equation for

O(s)

reads

(7.3)

U(s)Q(s)=R(s).

In

order that this equation has a

(not

necessarily proper) rational solution, it is

necessary and sufficient that rank U(s)=rank

[U(s),R(s)].

For the existence of a

proper solution additional conditionshave to beimposed. Writingdown the ith rowof

(7.3)

Ui(S)((S)- ri(s),

wenotethat anecessaryconditionfor the existence ofapropersolutionisdeg ui(s)>=

degri(s).Thefollowingresult shows that this is also sufficient provided that

U(s)

hasthe form

with

U(s)

row proper. According to Lemma 7.1, this can always be obtained by premultiplying

(7.3)

with a suitableunimodular matrix

M(s).

THEOREM7.4.Let

M(s)

bea unimodularmatrixsuch that

U(s)

M(s)R(s)

I_R(s)J

M(s)U(s)

0

where

U(s)

isrowproper Let the row degrees

_of

U(s)

be v, 1)l and let the row

degrees

_of

R(s)

beA,

.,

At.

Then

(7.3)

hasaproper solution

_iff

R(s)=0

andAi<-_

(i 1,...,

l).

Equation

(7.3)

has a strictly proper solution

_iff

R(s)=0

and _Ai

<

(i=1,...,l).

Proof.

The conditions are necessary accordingto theforegoing discussions. Now

assumethatthe conditionshold. Then there existsL grl

such that

U(s)L

isa row

proper matrix with rowdegrees u,

,

Ul. Define

O(s)

:=

_{L(UI(s)L)-IRI(S).}

Then Q(s) satisfies

(7.3). It

follows from

(7.2)

that Q(s)is proper. Theproofforthe

strictlypropersolution is similar.

We

can express the result of Theorem 7.4 in a way not involving explicitlythe

matrix

M(s):

COROLLARY 7.5. Equation

(7.3)

hasaproper solution

_iff

U(s)

and

[U(s), R(s)]

have thesamerankandthesame row indices.

In [14],

noexplicitconditionfor thesolvabilityisgiven.

In

[5],

acondition is given intermsof the kernel ofthematrix

U(s), R

(s)].

The conditions given in Theorem 7.4 andCorollary7.5 aredirectlyexpressed in termsof the matrices

U(s)

and

R (s).

The set

Xu

is the largest

(M,

)-invariant subspace contained in ker

c.

By

definition

x(s)

Xu

iff the equation

has a strictly proper solution

v(s).

Therefore, using Theorem 7.4, we can give a

constructive characterizationof

_Su.

COROLLARY 7.6. Let

M(s)

be as in Theorem 7.4. Then

x(s)Xu

_iff

y(s) :=

M(s)x(s)

_satisfies

the conditions

degyi(s)

<

vi (i i, l),

yi(s) 0 (i=/+1,...

,q).

Here

yi(s)denotes the ithcomponent

o] y(s). In

particular,

_if

weintroducetherow vector

w(s)

:=

[s

-,

,1],

then

M-(s)

W(s)

is abasismatrix

_of

Xu,

where

W(s)

:=

[

W(s)

with

Wx(s)

:= diag

(wl-a(S),

Wl-X(s)).

Oneway of solving(7.3),alreadymentioned in 4,isthereformulation of

(7.3)

as a

(M)DDP. In

doing so,itisnotnecessaryto use theoriginal denominatormatrix

T(s).

Onemight tryto find a newdenominatormatrix

Tx(s)

suchthat

_T-

(s)U(s)

isstrictly

properand

Tl(s)

isassimpleaspossible.Ifwechoose

Tl(s)

rowproper,thenaccording

toLemma 7.2,it sufficesfor the strictpropernessof

T

-1

U,

thattherowdegreesof

_Tx

arelargerthan therowdegreesof

U.

Ifwedenote the latterbyAx,

,

A1,the simplest

choiceof

Tx(s)

is

Tl(s)=

diag

(sX,+x,

.,

sXl+l).

For

this computation,itisnotnecessarythat

U(s)

be inrowproperform.Butifwe

transform

U(s)

such thatithastheform given in Theorem7.4,then the dimension of the statespacewill be minimal.Theseideas areworkedoutin more detail in

[3].

We

conclude thissection with a constructionof the supremal reachability subspace

contained in ker

.

Tothisend,weconsiderthespace

A

:-

_{v(s)K(s)l

U(s)v(s)-O},

and wechoose aminimalbasis for

A (see [5]),

thatis, abasis forA

_(see

_(6.7))

which is

column proper. We define

_La

:-[M]c.

Furthermore we choose any

D(s)Kl[s]

whichhas thesamecolumndegrees

M(s)

and suchthat

[D]c

L

Thenweobserve

(by

Lemma

7.2)

that,if

N(s):=LaD(s)-M(s),

then

O(s)

:=

_N(s)D-X(s)

is strictlyproper.Nowwehave

THEOREM7.7.(i)

{ U(s)L1}

Xu

(

{ U(s)},

(ii)

O(s)

is astrictlyproperrational matrix

_of

minimalMcMillandegreesatisfying

(7.8)

U(s)O(s)=U(s)La.

Hence,

_if

(F1,

A1,

B1)

is aminimal realization

_of

O(s),

then

(s)

:=

_{U(s)Fx(sI-Ax)}

-1

is a basis

_of

thesupremalreachability subspace containedin ker

.

Proof.

(i)Since

U(s)M(s)

0,itiseasily seen that

(7.8)

issatisfied. Thisimpliesthat

{ U(s)L 1}

c__

_Xu

71

{ U(s)}. Suppose

thatthere existsa matrix

L

Offull column rank such

that

{U(s)L1}c_

{U(s)/Sx},

and

U(s)/S1

U(s)O(s)

for some strictlyproper

O(s).

Let

),/

beright coprimepolynomialmatricessuch that

((s)- l(s)l-a(s), and/(s)

is

columnproperwith

[D]

I.Then

U(s)(N(s)- LID(S))

O.

434 E. EMRE AND M.L.J. HAUTUS

over

K(s).

Butthen

L1

cannothave more columns than

L1.

Consequently,

{U(s)L}

{U(s)L}.

(ii)

Suppose

that

O(s)-

lql’(s)-(s)

hasalower McMillandegree than

O(s)

and

that

N(s)

and

D(s)

arerelatively primeandthat

D(s)

iscolumnproperwith

[D

(s)]

I.

Thenwehave

U(s)(N(s)-LD(s))

O,

andhence,

N(s)-LID(S)= M(s)R(s). By

the "predictable degree

property" (see [5,

3,

Remark]),

this implies that the sum of the column degreesof

D(s),

and hence deg det

D(s)

isnotlessthandegdet

D(s),

which contradictsourassumption.

8. Generalization tosystemsrepresented by Rosenbrock’ssystemmatrix.

In

this section,weindicatehow the result of 3canbe generalizedtothe casewhere the system isrepresented byasystem matrix

T(s)

_U(s)]

(8.1)

P(s)

V(s)

W(s)J’

where

T(s)

K’’[s]

is nonsingular and

P(s) K

(q+l)(q+r)

Is].

We

assume that the transfer function matrix

G(s)

:=

V(s)T-I(s)U(s)+

W(s)

and thematrix

T-l(s)U(s)

are strictlyproper. Ifthe latter conditionisnotsatisfied,we

can obtain this by strict system equivalence

(see

[13,

3.1]).

Indeed, ifwedefine

U(s)

:=

rr(U(s)),

then

O(s)

:=

_T-l(s)(U(s)

_Ul(S))

isapolynomial matrix.Therefore,

[

T(s)

P(s)

:=

I.-V(s)

U(s)

W(s)

+

V(s)O(s)

isapolynomial system matrixwiththe same transfermatrix

G(s).

In

[9],

it isshown that the maps

sC

_Xr

--,

_Xr

_x

_(s

_{zrr(sx (s )),}

3

K

-->

XT

U

U

s u,

"XT

">

Kl’x(s)

’’>

(V(s)

T-l(s)x(s))_l

yield a realization

(,

M,

)

of

G(s)

which is reachable iff

T(s)

and

U(s)

are left coprime, and observable iff

T(s)

and

V(s)

areright coprime.

It

is easily seen that Theorem 3.1 is equally valid in this situation. Instead of

Corollary3.10weget

THEOREM8.2.

Let W(s)

beaq

x

mpolynomialmatrix. Then

{(s)}

isan

(M,

)-invariantsubspace in ker

_iff

there exists

CI

K

",

F

K

"*,

A

K

""

and an

xmpolynomialmatrix

(s)

such that

(8.3)

P(s)

IV1]

[*(s)](si_Ax)"

F

!_

_(s).l

Cx,

El,

A1

we have

(3.2)

and hence

(3.6).

Butthen

xlt(s)

V(s)

T-(s)W(s))_l

=((V(s)C +(G(s)-

W(s)F))(sI-A)-)-((V(s)C-

W(s)F)(sI-A)-)_

since

G(s)

and

(sI-A)

-

arebothstrictlyproper.

Now

itfollows fromLemma

(8.5)

that

(8.4)

(s)

:=

_(-V(s)CI

₊

_{W(s)Fx)(sI-Ax)}

-is apolynomialiff

xlt(s)=

0. Combining

(3.2)

and

(8.4)

yieldsthe desiredresult.

LEMMA

8.5.

Let

Q(s)

Ktn[s],

A

K

.

_If

(O(s)(sI-A-1))_l

=0, then

O(s)(sI-A)

-x is apolynomialmatrix.

TheproofisanalogoustotheproofofLemma3.13 and will beomitted. ThegeneralizationofCorollary3.12 can beexpressedintermsof the map

Kq[s]"

r

/x(s)/x(s).

Kq+l[s].->

y(s)J

L

COROLLARY

8.6. The largest (sg,Yd)-invariant subspace

of

XT-contained

in ker

is(Xp).

Theproofis similarto theproofofCorollary3.12 andwillbe omitted.

Acknowledgment. Oneof the authors

(E.

Emre)

wouldlike tothank the

Depart-mentof Mathematics of theEindhovenUniversity ofTechnology forfinancialsupport and hospitalitywhilethisresearch wasbeing done.

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