Linear feedback : an algebraic approach

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Linear feedback : an algebraic approach

Citation for published version (APA):

Hautus, M. L. J., & Heymann, M. (1978). Linear feedback : an algebraic approach. SIAM Journal on Control and Optimization, 16(1), 83-105. https://doi.org/10.1137/0316007

DOI:

10.1137/0316007

Document status and date: Published: 01/01/1978

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LINEAR

FEEDBACK--AN ALGEBRAIC

APPROACH*

M. L. J. HAUTUS" AND MICHAELHEYMANN

Abstract. The algebraic theoryof linearinput-output mapsis reexamined withthe objectiveof

accomodatingtheconcept of(state)feedbackinthistheory. The conceptsof extended and restricted lineari/o maps (andlineari/smaps) areintroduced andinvestigated. It isshownhow "fraction

representations"oftransfer matrices arisenaturallyin this new theoretical framework.

Conditions aregivenforwhen the changecaused to a linearinput-outputmap byan(open loop)

"cascadecompensator"canalsobeaccomplishedbyutilization of(closedloop) statefeedback.In

particular,it is shownthat thechangecaused toalinearinput-output mapbycascading (composing)it with an input space isomorphism, can also be effected by feedback, provided the input space isomorphism in "bicausal", i.e. it does notchangethe causal structure of theinput-output map. Furtherdetailed characterizationsoffeedbackarealso given especiallyin connection withthenewly introducedconcepts ofdegreechainanddegreelist.

1. Introduction.Probablyoneof themostimportantcontributions to linear

systems theory since the introduction of the concepts of controllability and observability, has been thediscovery by

R.

E. Kalman

(1965) (see

also Kalman

(1968)

and Kalmanetal.

(1969,

Chap.

10))

thatthetheoryoflinearsystemscanbe

naturally accommodated in classical module theory. This observation led to a

completely satisfactory theoryof realization,i.e. thetheorythat links

(external)

input-output descriptionswith(internal)statespace descriptionsofsystems,the

most recentcompletediscussion of which canbefoundinEilenberg

(1974,

Chap.

16).

Yet,

despite the power of the module theoretic approach in attacking the realizationproblem,there seemedtobenoapparentcontactbetween thistheory

and even some of the most elementary control theoretic questions of linear

systemsespeciallyinsofar as theconceptof feedbackis concerned.

Two,

completelyunrelated, approacheswere usedto studyfeedback

prob-lems: One is the so called "geometric" approach, forwarded and

promoted

by

Wonham and Morse

(see

e.g. Wonham

(1974)),

which has been successfully applied tosolve suchproblemsasdecoupling, regulator design, designofmodel

following systems, and investigating feedback invariant structures,

(see.

e.g.

Wonham andMorse

(1970), (1972), Morse

andWonham

(1970), (1971),

Morse

(1973), (1975),

Wonham andPearson

(1974)

and Wonham

(1973)).

Thesecond

approach, which was widely used and was developed mainly by Rosenbrock

(1970)

and by Wolovich

(see,

e.g., Wolovich

(1974)),

used polynomial matrix

techniques forthe study ofa varietyof control theoretic questions. This latter

approach,whoseprimary powerderives from thesurprising usefulness of fraction

*Receivedbythe editorsMay21, 1976,and in revised formJanuary25, 1977. This work was

supportedinpartbytheU.S.ArmyunderResearchGrantDAHCO4-76-G-0011through theCenter for MathematicalSystemTheory, UniversityofFlorida, Gainesville,Florida.

tDepartment of Mathematics, Technological University,Eindhoven, The Netherlands.This work wascompletedwhilethe author wasonleave at theCenterfor MathematicalSystemTheory, UniversityofFlorida, Gainesville,Florida.

$Departmentof ElectricalEngineering,Technionmlsrael Institute ofTechnology, Haifa, Israel.

This work was completedwhile the author was on leave attheCenterfor Mathematical System

Theory, UniversityofFlorida, Gainesville,Florida. 83

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representationsoftransferfunction matrices, seemedtobeespeciallysuccessful in

providingconvenient andquitepowerful computational algorithmswitha capa-bilityofyieldingvarious abstractresults.Also,fractionrepresentationsoftransfer matrices provide a convenient vehicle for studying such problems as minimal

realization

(see

e.g.

Heymann

(1972),

Forney (1975)

and Fuhrmann

(1976))

and feedback invariants asin

Heymann (1972).

Probably the most strikingparadox in this state of affairs is the fact that, historically,themoduletheoreticapproachtolinearsystemsseemedtosupport

the prevailing viewpointthat transferfunction matrices in the form

H(z)/(z)

(if(z)

apolynomial)are thenatural

(and

theoretically

sound)

concrete

representa-tions of linear input-output maps

(see

Kalman et al.

(1969,

Chap.

10)).

The

representationoftransfer function matricesasmatrix fractions seemed therefore

to be nothing more than a useful technical trick. This discrepancy has been

recognizednotably by Eckberg

(1974)

andbyFuhrmann

(1976)

whoattemptedto

reconcile the two representationswithinthe module theoreticframework.

Yet,

both of these attempts provide a rather ad-hoc accommodation. Specifically,

whilebothEckberg and

Fuhrmann.

makeaverysound case forviewingtransfer matrices as matrix fractions, no successful contact is made with the theory of

input-output maps.

More

importantly, there is no satisfactory contact with

feedbacktheory.WhileinFuhrmann

(1976)

thereis noattemptinthisdirection,

inEckberg

(1974)

thetreatmentoffeedback isnotverysuccessfulinthatit fails in

exhibiting module theory as a powerful or even a convenient framework for

dealingwiththefeedback concept altogether.

Themainpurposeofthepresentpaperis to reexaminethe module theoretic

settingof linearinput-output mapswiththeexplicit objectiveofaccommodating

theconceptof

"state

feedback"withinthisframework.

In

thetheoryof realization, theconcept"canonical"(equivalentlyreachable and

observable)

playsaverycentral andfundamentalroleinthatitdefines whatis

essentiallyauniquestatespace.

Yet,

thepropertyofbeingacanonical realization

is not invariantunder feedback (i.e. acanonicalstatespacecan be modifiedby

state feedbacktobecomenoncanonical andvice

versa).

Sincethe input-output

mapdefinesuniquely

(or

essentiallyuniquely) onlyacanonicalstatespace, it is

clear whythe conceptof statefeedback somehow seems incompatible with the "classical"module theoretic setting oflinearinput-output maps. Itiseasilyseen

however that reachability is invariant under feedback. The importance of the

matrix fraction representation for theinput-output maps derives from the fact that the representation essentiallyfixesareachablestatespace. Specifically, the

representation determines uniquelya reachable realization.

As

aconsequence, the conceptoffeedback

(and

especiallyits

effects)

canbe studiedatthe level of

input-outputmapswithoutgoingthroughthe

process

ofconstructingstatespace descriptionsfirst.

Hence,

inthestudyoffeedbackfromaninput-output pointof view, theprocessof

(concrete)

realization canessentiallybebypassed.

It

hasbeentechnicallywell known forquitesome time that the modification caused to the input-output map of a linear system by state feedback can be

accomplished by cascadingthe system withalinear device (sometimes called a

compensator).

Conversely,the effecton alinearinput-outputmapbycascadingit

(4)

feedbackimplementation.

Yet

thisconverseproblemismuch less wellunderstood

even on apurely"technical"

(rather

thantheoretical) level.

In

this paper we give necessary and sufficient conditions (in a module theoretic

framework)

foracascade

"compensator"

tobe realizablebyafeedback

implementation. To formulate these conditions certain revisions in the way an

input-outputmapis viewed are

necessary

inordertoaccommodate feedback.

We

shall adopt here a point of view that was

already

taken previously in the

unpublishednotesof

Wyman

(1972) (see

also

Sontag

(1976)).

The

paper

is organized as follows: In 2 the concepts of extended and restricted linear

i/o

maps are introduced and theirrelation is investigated

(the

latter concept coinciding withthe standard linear input-outputmapdefined in

Kalman et al.

(Chap.

10)).

In 3 the concepts of abstract realization and semirealizationarediscussed and theconceptoflinear

i/s

mapassociatedwith a

(reachable)

semirealization is introducedandinvestigated.Theorems3.5 and 3.9

givecharacterizations of linear

i/s

maps.

In

4 the results of 3 arespecializedto

the casewhere theinputandoutput spacesare finitedimensionaland inparticular

forthe case inwhich the realizationorsemirealization (i.e. the associated state

space)

is finite dimensional. Specifically, it is shown that with every reachable realizationthere is associatedaconcreterepresentationofthe extended linear

i/o

mapin terms ofa (matrix)fraction. This establishes the naturalness of fraction

representationsandexplaitheirusefulness in thestudyof feedback.

In

asimilar

manner it is shown how linear

i/s

maps are concretely represented as matrix

fractions in association with reachablesemirealizations oflinear

_i/o

maps.

In

5

the concept of feedbackis abstractly introduced and itsrelation withlinear

_i/s

mapsisinvestigated.Thenotionofabicausalisomorphism (inthe extendedinput

space)

is defined andinvestigated. Itis seen thateveryfeedback transformation

canbeimplementedin

"open

loop"throughabicausalisomorphismof theinput space. Conversely, conditions

are

found for the implementability of

a

bicausal

isomorphismof theinputspaceas afeedback transformation

(Theorems

5.7and

5.10).

The sectionisconcludedwithTheorem 5.13,whichstatesessentially thatif

thelinear

_i/o

mapis rational("rational"beingappropriately defined)thenevery

bicausalsomorphismof the inputspacecanbeimplementedasfeedbackinsome

finite dimensional reachable

(although

not necessarily

observable)

state space.

Thisresult has the intuitiveinterpretationthat thechangecausedto a linear

i/o

map by

(externally)

modifying its inputstructure can also be accomplished by (internally) implementing feedbackprovided the externalinputchangedoes not

alter the causal structure of the linear

_i/o

map and is reversible. The paper is concluded in 6,where thestudyoffeedbackisfurtherexpanded.

In

particular,it

isnoted thatfeedbackcanbeinvestigatedby studyingthestructureof the kernel of a (restricted) linear

i/s

map. Since this kernel is a submodule of the input module, the study of feedback is generalized by studying the structure of an

essentially generalsubmodule.Inthisconnection, theconceptofthedegreechain ofasubmoduleis introducedandinvestigated.The main result of 6isTheorem

6.10, which gives a complete characterization of "feedback equivalent" sub-modules. This reestablishes from a new viewpoint the central role of certain feedback invariants which have been introduced previously.

In

addition, it is shown how certainpreviouslyknown

(but

notwell

understood)

factsfind anatural

(5)

accommodation within this new theoretical framework (in particular

Forney’s

conceptof minimal basis and Wolovich’sconceptof column

properness).

Throughoutthispaperitwill beassumed thatthereaderisfamiliar withthe

now classical module theory oflinear systems as can be found for example in Kalman

(1968),

and Kalman et al.

(1969,

Chap.

10).

2. Extended and restricted linear i/omaps.

We

shall begin by introducing

some notation. Throughout the paper

K

will denote a fieldand

U

and Ywill denote K-linearspaces.Thespace

U

willbe referredtoasthe input valuespace

and

Y

asthe outputvalue space

(of

anunderlyingdynamicalsystem

E). We

shall make finitedimensionality assumptionson

U

andY onlywhenexplicitlystated.

We

let7/denotethesetof integers. IfS is a K-linearspace (in particular

U

and

Y),

we consider the set of all sequences s

(st)t

("

",s-a,So,sl,

")

possessingthefollowing properties" (i)st Sfor all 7/,and(ii)thereexiststo67/ such that st 0 for all

_{< to.}

These sequences will be identified with

(formal)

S-Laurentseries inz-1 i.e series oftheform

(2.1)

s

s,z-’.

t=to

We

shalldenotethesetofS-Laurentseriesby

S((z-))

oralternativelyby

AS.

Itis thenwell known that theset

AK

K((z-1))

of

(scalar)

K-Laurentseriesisafield

withconvolution asscalarmultiplicationand the obvious

(coefficientwise)

addi-tion.Also,withconvolution asscalarmultiplicationand the usual addition,

AS

is a

AK-linear space. This,inparticular, impliesthatAS isalsoK-linear and also a

K[z

]-module

(where

K[z

]

istheringofpolynomialsin

z).

Foranelements

AS

givenby

(2.1),

the order ofs is definedby

min

{t

_ZIst

rs

0}

ifs#0,

(2.2)

oFds

!

ifs=0.

Furthermore, for an elements

AS,

multiplicationbyz results inashiftofthe

sequence

(s,)

to theleft,that is

(St),

_

Z

(St)t

Z

(St+l)t

We

now introduce the extended input space

A U,

and the extended output

space

A Y.

If a map

[" A U- A Y

is K-linear, we say that

_f

is time invariant

provideditsatisfies

f

(z

w)

_zf

(w)

for all w

A

U.

The following elementary but important result then follows

(see

also

Wyman

(1972))"

THZORM2.3.

A

K-linearmap

_f"

A

U-,

_A

Yis time invariant

_if

andonly

_if

it is

AK-linear.

A

K-linearmap

_f"

A U- A Y

iscalledcausalif ord

_f

(w)

->ord w andstrictly causal iford

_f

(w)

>

ord w forallw6

A U. We

nowintroducethefollowing

DZlINITION2.4.

A

map

_f"

A U

A Y

is calledanextended linear input-output

map

(or

extended linear

i/o map)

if it is K-linear, strictly causal, and time invariant.

(6)

Theorem2.3 provides an algebraic characterization oftime invariance.

In

ordertocompletethe characterization ofextendedlinear

_i/o

mapswe nowturn to

thequestionofcausality.

Let

usdenoteby

L

the K-linearspaceof all K-linear

maps

U

-

Y

and considerthe

space

AL

of all

L-Laurent

series. This

space

canbe identified with thespaceof AK-linearmaps

A

U

A Y

asfollows"

We

define the K-linearmaps

iu" U AU:

uu (canonical injection),

Pk"

A Y-> Y:

,

ytz-t--> Yk,

and foreveryAK-linearmap

_f"

A U-> A Y

andevery k 7:wedefine the K-linear

map

Ak"

U-> Y

by

(2.5)

Ak

:= Ag

(f):=/g

_f

i.

The

L-Laurent

series associatedwiththemap

f

is then givenby

(2.6)

Zf(z

-1)

:=

_{Y At(f-)z}

-t

and iscalled theimpulse responseorthe

_transfer

_function

of

f-

Now,

if

Z

Y

Atz-t

isanyelement of

AL

wedefinethe action of

Z

onw

Y’.

utz

-

A U

by

(2.7)

Zw

:=

Y,

(AkUt-k)Z-’.

k

If

_f"

A U

A Y

is definedasthemapwhoseactionisgivenby

(2.7),

it isthen easily verified that

z.

We

nowhave thefollowingimmediate characterization ofcausality: Themap

f"

A U-+ A Y

iscausal

_if

andonly

if

Ak (f)

0

_for

k

<

0 andisstrictly causal

_if

and

only

_if

Ak

(f)=

0

_for

k

<-O.

The followingisthenproved"

THEORE2.8.

A

map

f"

A

U-+

A Y

isanextended linear

i/o

map

_if

andonly

if

itisAK-linearand

_Ak

(f

0

_for

all k <- O.

Foralinearspace S wedenoteby

S[z

oralternativelybyfS thesetof all

N k

polynomials

_Yk--O

SkZ with coefficientsSk inS. Obviously, fS isasubset of

AS

andit iseasilyseenthatit isalsoaK-linearsubspaceandeven a

K[z ]-submodule

of

AS. It

is not,however, aAK-linearsubspace.

In

the

development

ofthemoduletheoretic treatment of linearsystems,one of Kalman’sprimary objectiveswas toprovidearigorousframework forrelating

linearinput-outputmapstostatespace descriptions, (i.e.theproblemof

realiza-tion).

In

his treatmentKalman considered the following "experimental" setup

(see

e.g.Kalman

(1968)

and Kalmanetal.

(1969,

Chap.

10))"

Allinputsterminate

(i.e.becomeidentically

zero)

at t 0 and theoutputsareobservedonlyfort

=>

1.

Thisspecial set ofinputs

IU

willhenceforth be called the restricted inputspace,

and elements w _{fU will be called restricted inputs. Since in this}settingoutputs

areobservedonlyfort

=>

1,itfollows thattwooutputsareindistinguishableif their difference isinl

Y.

Thus, ifweintroducethenotation

(7)

for

any

K-linearspace

S,

thenaccordingtothe foregoingwedefine

F

Y

to be the restrictedoutput space.

Let

A/

Y

denotethesubset of

A Y

consistingof all elements

t

oftheform

Y,__

ytz

It

isreadilyverifiedthatthere is abijective

correspondence

between

F

Y

and A/

Y

with theproperty thatwithevery y

F

Y

is associateda

unique / A/

Y

satisfying

_zr7

y where7r"

A

Y

F

Y

isthe canonicalprojection.

Furthermore,

F

Y

inducesaunique modulestructureonA/

Y

bytherequirement

that the above bijection become a

K[z]-module

homomorphism.

We

shall henceforthidentifyelements of

F

Y

withthoseofA/

Y

andrefertothelatter as

restricted outputs.

(It

ispreciselythe moduleA/

Y

whichwasdefined inKalmanet

al.

(1969,

Chap.

10)

as theoutputmodule of a linear

system.)

DEFINITION2.10.

A

map/:

fU

FY

iscalled a restricted linear input-output

map

(or

restricted linear

i/o

map)

if it isa

K[z]-module

homomorphism. Remark 2.11.

In

the above definition of restricted linear

i/o

maps

the

properties of causality and time invariance are automatically built in.

Wyman

(1972)

called the same aKalmanlinear

i/o

map and itis, infact, preciselythe

input-output mapderived inKalman’swork

(see

e.g.Kalman et al.

(1969,

Chap.

10)).

We

shall latermakeextensive usebothof extended and of restricted linear

i/o

maps.

In

additiontotheextended and restricted

i/o

maps,we introducethe concept

of linear

i/o

value map

(2.12)

f:

fU-

Y

asfollows:

_Iff

isanextendedlinear

i/o

map,wedefine

_f

(associatedwithit) by

(2.13)

f(w)

:=

_{p iT(w);}

w fU.

Alternatively,ifj isarestrictedlinear

_i/o

map,weconstructthe

i/o

value

map

_f

by

(2.14)

f:=p

_f

where (with

F

Y

identifiedwithA/

Y)

(2.15)

Pl"

FY

Y:

Y

ytz-t--ya.

t=l

Conversely,if

_f:

OU

Y

isanyK-linearmap,wecanregarditasalinear

i/o

map

by

associatingwith it themaps

)v

and which areconstructedfrom

_f

using the

conditionsof time invariance andcausality.

In

particular,wehave

(2.16)

](w)

_Z

f(ztw)z

-‘-t_>o

and

(2.17)

f-(w)

Z

f((ztw))z

-t-1

tZ

where thetruncationoperator

:

A

U-.

IIU

is definedby

(2.18)

(Z

u,z-’)

:=

_Y

_utz-’.

(8)

(Compare

also Kalman and

Hautus (1972,

formulas

(2)

and

(4)).)

It is easily

verifiedthatthe

maps/

and as defined in

(2.16)

and

(2.17)

are,respectively,a

restrictedandanextended linear

i/o

mapand that the formulas

(2.13)

and

(2.14)

hold. The relation between

f,

_f

and

_f

isindicated in thefollowingcommutative

diagram"

AU-

AY.

U

-.

y

u

where isthe identity map,

]

isthe canonical injection, and 7r isthe canonical projection. The maps

f,

p and

/1

are K-linear, the maps

],

zr and

_f

are

K[z

]-homomorphisms

and

_f

is aAK-linearmap.The

maps/r

and

f

arecalled the restrictedandextended

i/o

mapsassociated with

_f.

3. Linear i/s

_mas.

Let

[:

fU

FY

be a restricted linear

_i/o

map.

By

an

abstractrealization of

_f

we refertoatriple

(X, g, h)

where

X

is a

K[z ]-module

and

g"

IUX

and h’X-

FY

are

K[z]-module

homomorphisms such that the

diagram

fU

---

FY

X

commutes.

For

thesystemtheoreticinterpretationof anabstract realization, the reader is referred toKalman

(1968)

and Kalman et al.

(1969,

Chap.

10).

The module

X

iscalledthestatespace

(and

issometimesregarded onlyas a K-linear

space).

If

(X,

g,

h)

isagivenabstract realization ofa restrictedlinear

i/o

map

f,

one canconstructfrom it a concrete realization of

_f

(see

Kalmanetal.

(1969,

Chap.

10)).

Suchaconcrete realization isuniquelydetermined

by t[e

abstractrealization

(X,

g,

h)

andweshall henceforthcall

(X, g, h)

simplyarealizationof

_f.

In

keeping

with standard systems terminology we then call a realization reachable if g is

suriective

andobservable if

h

isinjective.

A

realization iscalled canonicalif it is

bothreachable and observable.

Consider now a restricted linear

i/o

map/:U-

FY.

Let X

be a

K[z]-module and letg"

IU

-

X

bea

K[z]-module

homomorphism.Thepair

(X, g)

will

becalledasemirealization

of/

ifitcanbeextendedtoarealization

(X,

g,

h)

with some

K[z

]-homomorphism h"

X

-

F

Y. (X, g)

willbe calledareachable semireali-zation ifg issurjective, andcanonicalif

every

extension realization

(X,

g,

h)

of

(X, g)

is canonical. Thefollowingcharacterization of reachable semirealizationsis

(9)

THEOREM 3.1.

Let

_f:fU-

FY

i/o

map, let

X

be a

K[z ]-module,

andletg"fU-

X

beasurjective

_K[z,

]-homomorphism. Then (i)

(X,

g)

is a

(reachable)

semirealization

_of

_f

_ff

andonly

_if

ker gc

_ker]

.

(ii)

_If

(X,

g)

isareachablesemirealization

_off

thereisa uniqueh such that

(X,

g,

h)

is arealization

_off.

(iii)

(X, g)

is acanonicalsemirealization

_of

_f

_if

andonly

_if

kerg ker

_f.

Considernow a

K[z

]-homomorphismg"

IU- X

where

X

is a

K[z ]-module.

Ifwereferonly totheK-linear structureof

X and

regardg as aK-linearmap then, as in

(2.16)

and

(2.17),

we can construct the restricted and extended linear

i/o

maps

(3.2)

,

IU

+

FX,

(3.3)

g"

A U

AX

associated with g.

In

view of the fact that

X

is a

K[z]-module

and notjust a

K-linearspaceand gis a

K[z ]-homomorphism

andnotjustaK-linearmap,the

maps and havepropertiesthatdistinguishthemfromordinary

i/o

maps. For

thisreasonwe will call and

,

respectively,arestricted andanextended linear

i/s

map

(where i/s

stands forinput-state).

More

generally,arestricted linear

_i/o

map

f:

l)U-

FY

andthecorrespondingextended

i/o

map

_f"

AU- A Y

arecalled

i/s

maps if

Y

can be endowed with a

K[z]-module

structure, compatible with its

K-vector spacestructure, such that the associated

i/o

valuemap

f

:=

pao

_f

oj=plof

is a

K[z

]-module

homomorphism. Itiseasilyverified thatif

)z

is arestrictedlinear

i/s

mapand

_f

the associated valuemapthen

(3.4)

kerf

ker

_f.

Indeed, for w

U,

]7(w)

0if and only if

f(ztw)-O

for all =>0

(see

(2.16)).

However,

in view of the fact that

_f

is a

K[z]-homomorphism,

the latter is

equivalentto

f(w)

0since

f(z’w)

z’f(w).

Assume

now that

_f’fUoFY

is a (restricted) linear

i/s

map and that its

associatedvaluemap

_f

issurjective.Then

(3.4)

andTheorem 3.1implythat

(Y,

f)

is areachable(infacteven acanonical) semirealizationof

j

We

shall henceforth callarestricted linear

i/s

map

freachable

wheneverthepair

(Y,

f)

is areachable semirealization of

jr

or, equivalently,whenever the associated

i/o

valuemap

_f

is

surjective.

(The

name

i/s

wasadopted preciselyfor the reason thatinthesespecial

i/o

mapstheoutputvaluespace qualifiesas state

space.)

Assume

nextthat

_f"

flU

FY

isalinear

_i/o

mapsuch that its associated

i/o

valuemap

f

is

_surj.ective

and that

(3.4)

holds. Then

Y

isisomorphicasaK-linear

spaceto

flU/ker

f

which is a

K[z ]-module.

Thisisomorphisminducesa compati-ble

K[z]-module

structure on

Y

and it is easily seen that

_f

is then a

K[z]-homomorphism.

We

summarizethe situationinthefollowing

THEOREM 3.5. Let

_f:

I)U-

FY

i/o

map such thatthe

associated

i/o

value mapf pl

:

IU-

Yissur]ective. Then

is

an

i/s

map

if

and

(10)

If

H: X

Yis a K-linearmap thenit induces in anaturalwayaAK-linear

map

(3.6)

H:

AX

A Y:

Y

xtz-t

(Hxt)z

-t

and also a

K[z ]-homomorphism

(3.7)

H:

FX

FY:

x,z-’

-

(Hxt)z-’

t>l

wherewehave identified

_A+

Y

with

FX

and

_A+

Y

with

F Y.

Themapsdefinedby

(3.6)

and

(3.7)

arecalledstaticmapsandit willbeconvenient todenote themby

the samesymbol H.

We

nowhave thefollowing

TI-IEORE3.8.Let

_f

IqU-FYbearestricted linear

i/o

map,letg"fU-Xbe a

K[z ]-homomorphism

andlet

,

be therestricted linear

i/s

mapassociatedwithg.

(i)

_If

(X, g)

is asemirealization

off,

then thereexistsaK-linearmapH:X-

Y

such that

_f

H.

g (correspondingly

_f

H.

,).

(ii) Conversely,

_if

gissurjectiveand thereexistsH:

X-

Y

such that

=

H.

,,

then

(X,

g)

is asemirealization

_off.

Proof.

(i)

Let

h"

X-

F

Y

be a

K[z]-homomorphism

such that

(X,

g,

h)

is a realization of

_fi

and define

H:X-,

_{Y:x-p h(x),}

where paisdefinedin

(2.15).

Then,if

_[

:= p

f,

wehave

f

=Pl

of

=ploh

og=Hog

andconsequentlyfor w6fluwehaveby

(2.16)

and

(3.7)

Z

f(z’w)z

Y. Hg(z’w)z

-t-’

t>=o t>o

=t-/o

E

t=>0

andtheproofof(i)iscomplete.

(ii) Theexistenceof

H

such

that/r=

H. impliesthatkerg ker

Hence,

sincegissurjective the result follows from Theorem 3.1.

The following result gives another characterization of the linear

i/s

maps

amongst alllinear

i/o

maps.

THEOREM 3.9. Let

(,"

fU-

FY

bea restricted linear

i/o

map. Then

,

is a

reachablerestricted linear

i/s

_{map if}

andonly

_if

the following

_condition,

holds"

_for

every restricted linear

i/o

map

_f:

IU- FS

satisfying ker

_ff

c

_kerf

(where

Sis a

K-linear

space),

thereexists aunique K-linearmap

H" Y

S such that

H

Proof.

Assume

that is a reachable restricted linear

i/s

map and that satisfies ker c

_kerf.

Then theexistence of

H

such that

_f

H. followsfrom

Theorems 3.1 and 3.8.Also,

_f

H.g where

_f

p

_o.1

rand g p

if,

and fromthe

surjectivityofg

(the

reachabilityof

)

itfollows thatHisunique.

Conversely, assume the condition holds. Define S lqU/ker and let

f:

fU Sbe thecanonical

_p.rojection.

Let

]r:

flU

_-.FS

bethe restrictedlinear

_i/o

(11)

thereexists auniqueK-linearmap

H" Y-

Ssuch that

₁

H. or,equivalently,

1

H.g.

Hence

ker gc_ker

_]’

_ker _and, _since _ker _ker_g _is_obvious, _it fol-lows that ker kerg. Fromtheuniquenessof

H

wehave thatgissurjectiveand the result follows from Theorem 3.5.

We

conclude this section withthefollowing result

(due

to

Wyman

(1972))

whichsummarizesin a

_tra.nsparent

wayourcurrentpointofview.

TI4EOREM 3.10. Let

[:IIU-. FY

be arestricted linear

i/o

map. Then every reachable realization

(X,

g,

h)

_of

inducesauniquecommutativediagram

AU

_.AX

.

_AY

IIU, _FX J,FY

4. Finite dimensionality and matrixrepresentations. Let

):

U-

FY

be a

restricted linear

i/o

map.

Two

reachablerealizations

(X1,

gl,h

1)

and

(X2,

g2,

h2)

of

1

arecalledisomorphic ifthere existsa

K[z

]-isomorphisma"

_X1

X.-suchthat thediagram

llU

.

FY

commutes.

Let

Acker

"

beasubmodule. Thereis associated with A(uniquely)a

reachable realization

(Xa,

ga,

ha)

of

]"

where

XA

:=

fU/A,

gA’fU-Xa is the

canonical projection and h

_a:Xa-

F Y

is defined via Theorem 3.1(ii).

Suppose

nowthat

(X,

g,

h)

is anyreachablerealization of

1

and let

A

kerg. Itis easily verifiedthat

(X,

g,

h)

isthenisomorphicto

(XA,

ga,

ha).

Itfollows that the class of reachablerealizationsof

1

is inone-onecorrespondence

(modulo

isomorphisms)

withtheclassof submodules of ker

]

Specifically,witheverysubmoduleA ker

]r

(12)

realizations

(X,

g,

h)

forwhich A ker gareessentially completelydeterminedby

A.

Thisfact will beparticularlyusefulin ourstudyoffeedback whereitwill turn out tobe easiertodealwith

(the submodule)

Athanwith

(the

quotient

module)

ag/a.

Sofarwehaveimposednofinitedimensionalityconditionsoneither

U, X

or

Y. In

this section we shallspecializeourresultstothefinitedimensionalcaseand shall henceforth assume that

U

and

Y

are finite dimensional linearspaces. More specifically,welet

U K

m,

Y

Kp

sothat fU=

K"[z]

and

FY=

KP((z-1))/KP[z].

Thus,inthiscase

IIU

is afree

K[z]-module

andhasabasis of m elements.

Since

K[z]

is aprincipalideal domain, everysubmoduleA _of

_IIU

hasatmost m

free generators and their numberd,whichwecall rank

A,

isindependentoftheir choice

(see

e.g. Hartleyand Hawkes

(1970,

Thm.

7.8)).

Our

maininterestisin therepresentationof finitedimensional linearsystems, andwe areconcernedwiththecasein which forasubmoduleA _fU,

_)U/A

is a

finite dimensionalK-linearspace,orequivalently,a torsionmodule

(see

e.g.

Lang

(1965,

p.

388)).

Thusweshall makeuseof thefollowingstandard butimportant

result

(the

proofofwhich canbe found in Fuhrmann

(1976)).

THEOREM4.1.LetAcfU beasubmodule. Then

fU/A

is a torsionmodule

(or

equivalentlya

_finite

dimensional K-linear

space)

_if

andonly

_if

rankA m.

LetAc

_D.U

_be_a_{submodule of rank}_m _{and let}_dl,

,

_d,, _be _{a basis}_for

_A.

Define the

K[z

]-homomorphism

D" U

-

A _{by Dei} _di, 1,

,

m, where e,

,

e,,denotesthenatural basis inK

m,

(as

well as in

fU). We

canview

D

also asan m

x

mpolynomialmatrix (i.e. amatrixwith entries in

K[z])

by regarding

di

K"

[z

asthe th column ofD. Conversely,ifD D

(z)

is an

x

mpolynomial

matrix, we can regard

D

as aK[z]-homomorphism

K"[z]-,K[z]:e-d,

i=

1,...,m, where

di

di(z)

Kl[z

is the th column of

D.

If inparticular m, then dl,"" ,d,,, are elements of

IIU=Km[z]

and are thus generators of a

submodule A fU definedby

A=DU

:=

_{Dwlw

ClearlyrankA rankD

(rank

D beingthe matrix rank of

D),

and rankA m if

andonlyif

D

isnonsingular (i.e.0 det

D

K[z ]).

Consider now the K[z]-homomorphism

IIUIIU

defined by an m

x

m

nonsingular

polyn_omial

matrix

D.

It iseasily verified that thereexists a unique

AK-linear

map D" A U

-*

A

U

such thatthe diagram

AU

D

fU ---*- fU

commutes, where], asusual, denotes thecanonicalinjection. Itis readily noted thatthe transferfunctionofD isgiven bythe(polynomial) matrixD.

[We

shall

(13)

refer to a AK-linear map whose transfer function is apolynomial matrix as a

polynomial

map.]

Since

D

isnonsingular,itis obviously invertibleover

AK,

and thusD is aninvertiblemap.

We

shall henceforth denote both themaps D and

D

and their associatedpolynomialmatrixbythe singlesymbol D.Themeaningwill

alwaysbe clear from the context.

An

m rn polynomialmatrixR iscalled unimodularif itsdeterminantis a

nonzero constant, i.e. if its inverse is also a polynomial matrix. The following

theorem whose elementary proof is omitted and the correspondingimmediate

corollarywillbe useful:

THEOREM 4.2. Let

A,

A*c_12U _be _submodules _given _by A DfU and A*

D*OU

whereDand

D*

are nonsingular. Then A*cA

_if

andonly

_if

there

exists apolynomialmatrixR such that

D*=

DR.

COROLLARY4.3. Under theconditions

_of

Theorem 4.2,A* A

_if

andonly

_if

thereexists aunimodularmatrixRsuch that

D*

DR.

We

nowturn to the study of representations of linear

i/o

maps

(and

their

realizations). Let

)z.

A U- A Y

be an extended linear

_i/o

map, let

)z

be the associated restricted linear

i/o

mapand let

(X,

g,

h)

beareachable realization of

withXafinite dimensionalK-vector space (i.e.rank ker g

rn).

Letker g A DfU anddefinethemap

(4.4)

N"

AU

AY:

w

-f(Dw).

SinceNis acompositeoftwo AK-linearmaps, it isclearlyalsoAK-linear.

Let

z,(z)

denote the transferfunctionofN.

We

observe that

_Zu

isapolynomialmatrix(i.e.

Nt

0for all

> 0).

Indeed,forw

7r

(Nw

7to

f

(Dw

f

(Dw

O,

withthelastequality holdingsinceDw ker gc_ker

_f.

_It_{follows that}_Nw

_{KP[7. ].}

Since w

IU

waschoosen arbitrarily,wesee

(compare

(2.7))

that

z,(z)= Y,

Ntz-’

t=<o

asclaimed.

We

shallnow(justas wehave done for

D)

identify (notationally)

Zu

withNsothatNwilldenote both themapand its associatedpolynomialmatrix.

In

view of the relation

Nw

f(Dw)

for every w

A U,

and the fact that

D

is invertibleweobtainthefollowing expression"

(w)

ND-’w

(w

AU).

Thus,the transferfunctionof

_f

has therepresentation

(4.5)

Z(z)=

Az-=N(z)D-’(z).

t=l

A

AK-linear map

_f"

A U- A Y

and its L-Laurent series

Z(z)

are called rational ifand onlyifthereexists a (nonsingular) polynomial matrix

D(z)

such

(14)

that

_ZD

is apolynomial matrix. Thuswehaveprovedthenecessity partof the

following

THEOREM 4.6.

A

restricted linear

i/o

map

_f:IU-

FY

has a reachable

realization with

_finite

dimensionalstatespace

_if

andonly

_if

_f

isrational.

Proo[(sufficiency). If

_f

is rational, there existpolynomialmatricesNand

D

such that

f

ND

-1.

Then

(XA,

gA,

hA),

where

A

DU,

isareachable realization

of

_f

withfinite dimensionalstatespace.

TwomatricesNandD arecalled rightcoprime ifevery m m polynomial

matrixR satisfying N

NR

and

D

_DR

forsomepolynomialmatrices

_N

and

D,

isnecessarilyunimodular.

THEOREM 4.7.Let

(X,

g,

h)

beareachable realization

_of

arestricted linear

i/o

map

f.

Let ker g

A

DI)Uand assume

D

isnonsingular. Let N

_ZD.

Then

(X,

g,

h)

isobservable (i.e.ker

]=

ker

g)

_if

andonly

_if

_N,

andDare right coprime.

Proof.

Let

D

be chosen such that

kerf=Dl)U.

Then

ker g

=

_kerf

D

ll)U, and

by

Theorem4.2 there exists apolynomialmatrix

R

such that

_{D =D1R.}

_D

is then nonsingular and if

_N

is defined such that

N1D

-1

Z,

itisreadilyseenthatN

NR.

IfweassumethatNand

D

are

_right,

coprime,then thematrixRmustbeunimodularandbyCorollary4.3ker g

kerf

and observability holds. Conversely, let

N-NaR

and D

D1R.

Then R is

nonsingularand

_Z

ND

-

ND-X.

Itfollows thatD

lieU

ker

)

andwehave

kerg Dl)UC

DllU

c_ker

_fi

If

(X,g,h)

is observable, then kerg=kerf so that

DfU=DIU

and by Corollary4.3R isunimodular, whenceNand

D

areright coprime. [3

Remark 4.8. Therepresentation

_Zf

ND-1

ofatransferfunction iswidely

known asa

_fraction

representation.

We

havejustseen howfractionrepresentations

arisenaturallyfromreachable

(abstract)

realizations

(X,

g,

h)

ofaninput-output map. In fact,thereisessentiallyaone-onecorrespondence

between

the class of all

finitedimensionalreachablerealizationsof

_f

and the class offraction

representa-tions. It is important to observe thatby writing

_Zf

ND

-

weessentially have

(modulo

isomorphism)aspecificreachable realizationinmind. Theobservability alongwiththe associatedright coprimenessisessentially ofnoconsequencein our

studyof feedback asweshallseelater.

Let

X

K

n.

Assume

X is also endowed with a module structure and let g"IIU-*X be a surjective

K[z]-homomorphism. Let

"

IIU- FX

be the restricted

(reachable)

linear

i/s

mapassociated withg.Then

(X,

g)

isacanonical semirealizationof andthereexistsh" X

-

FX

suchthat

(X,

g,

h)

is a(canonical)

realizationof

.

Accordingto theforegoingwe canwrite

(4.9)

_Z

SD

-,

where

_Ze

isthe transfer function of

,

withD being a(nonsingular) polynomial

matrixsuchthat ker

DIIU,

andS beingan nxm polynomialmap. Inviewof theobservabilityof

(X,

g,

h)

itfollows from Theorem 4.7 thatD andS areright

coprime, and moreover, degdetD n.

If

_Z

isthe transfer function ofanextendedlinear

i/o

map

f,

then thestrict

causalityof

_f

is equivalentto

_Z

beingastrictly properrationalmatrix; that is, each

(15)

than thedegreeofthe numerator.

We

nowspecializeTheorem 3.9to the finite dimensionalcase.

TUEOREM 4.10. Let S and Dbe right coprimepolynomial matrices with

D

nonsingular, andassumethat

Z

:= SD-aisstrictly proper

(or

equivalently strictly

causal).

Then

Z

isthe

_transfer

_function

_of

areachable extendedlinear

i/s

map

,

_if

andonly

_{if for}

everypolynomialmatrixN satisfying theconditionND-a isstrictly proper, thereexists auniqueconstant matrix

H

(withentries in

K)

suchthat N HS.

Proof.

Assume

that

Z

Zg

where isareachable extendedlinear

i/s

map.

By

thecoprimenessofSandDwehave ker DI2U and

ifjrN

is therestricted linear

i/o

map associated with

ZN=ND

-a

(ND

-a strictly

proper),

then

_ker=

Dl)Uckerfu.

By

Theorem 3.9 there exists a unique H:X-

Y

such that

jrn

H. so that if

H

also denotes the corresponding matrix we have

N-(ND-a)D

(HSD-a)D

HS. Conversely,let

jr

bearestricted linear

i/o

mapwith

ker cker

jr.

ThenDI)U ker

jr

and there existsapolynomialmatrixNsuch that

f=ND

-a

(

denoting the extended linear

i/o

map associated with

jr).

By

hypothesis there exists a unique constant matrix

H

such that

N--HS,

and

consequently

_f

H. andbyTheorem3.9 theproofis complete.

COROLLARY4.11.Let

D

beanonsingularrnxtnpolynomialmatrixandlet det

D

have degree n. The set

_of

m-row polynomial vectors

v(z)

such that

v(z)D-a(z)

is strictly proper, is an n dimensionalvector space over

K. _If

S is a

polynomial matrix such that SD-a is strictly proper, then SD-a is the

_transfer

function

of

areachable

lineari/s

map

if

andonly

_if

therows

_of

S

_form

abasis

_of

X.

Proof.

That is aK-vector

_space

is obvious.

Suppose

SD-1

isthe transfer function ofalinear

_i/s

map.If v

(z)

6

X

isanyvector, thenbyTheorem 4.10 there existsa

(row)

matrix

H

suchthat v HS andthus,the rows ofS generate

’.

The uniqueness condition of

H

_implies

that therows of S arelinearly

_independent

(over K)

andthus formabasis forX. Conversely,iftherowsofSformabasis for

X

then SD-a is a strictly proper matrix and for each matrix Nwhose rows are

elementsof

X

there exists auniqueK-matrix

H

such thatN

HS

whence, by

Theorem4.10, SD-aisthetransferfunctionofalinear

i/s

map.Sincethe number of rows inS isequalto n,itfollows that dim n.

5. Feedback.Considerarestrictedlinear

i/o

map

jr:

U

FY,

let

(X, g)

bea

reachablesemirealization of

_f

andlet

"

A

U

AX

be the extendedlinear

i/s

map

associatedwithg.

Suppose

wemodify inthe followingway.

For

each w6

AU

instead ofletting

g

act on w we let

g

act on q :=

V-

w-L s

c,

where

V: U- U

and

L’X- U

areK-linear (static) mapswith

V

invertible, and wheresc issuch that

:

(qg).

Then w,q, and

:

arerelatedbytheequations

(5.)

o

v.

w-L

;

=

(16)

In

linewiththeaboveblock diagramwecallthe pair

(L, V)

a

_feedback

pair. The

map

L

will be called a

_feedback

map and

V

will be called a

(static)

input

transformation.

From

(5.1)

wecaneliminate

so,

to obtain

(5.2)

(I

+z;

g).0

v.

w.

SinceL

g

isstrictly causal,itiseasilyverifiedthat ord

(I

+

Lg)0

=ord0foreach q

A U.

This implies that ker

(I +Lg)=

0, from which it follows that I

+Lg

is

invertible.Thus,

(5.2)

hasaunique solution for₀ givenby qg=l.w,

where

(5.3)

-

:=

_t\.

:=

_(I

_+L)

-

v.

Upon

substitutingfor

_o

inthe second formula of

(5.1),

weobtain theexpression

(5.4)

=fL, v(W),

where

(5.5)

f,,,

:=

g

t.

The schematicinterpretationof

(5.5)

is given

by

the blockdiagram

where l"

A

U

A U

isregarded as a dynamic input

_{transformation.}

It is readily

noted

(see

alsoLemma5.6

below)

that themap hasthefollowing properties: (i) is an invertible AK-linearmap.

(ii) Both

-

and

--1

arecausalmaps.

We

shall henceforth call a map

I’AU-AU

which satisfies both (i) and (ii) a

bicausalisomorphism on

AU.

Itis obviousthat

fL,

Vis anextended linear

i/o

map.Thisfollowsimmediately

from the fact that the compositeofAK-linearmaps isAK-linear, and that the

compositeofacausalmapwithastrictlycausaloneisstrictlycausal.

In

fact,it will

beseen laterthat

fL,

vis even areachablelinear

i/s

map.

We

have seen that ifwe can construct

_f,v

from

g

byfeedback

(as

in our first

interpretation) then we can also construct it by cascading with a bicausal

isomorphism

l,v

(whichis an

"open

loop"construction).

We

shallnow turn to the more difficult question" when can a AK-linear map

I’AU-AU

be

expressedas in

(5.3)

for someL and

V.

If this is the case we call a

_feedback

transformation

(correspondingto

(L,

V)).

LEMMA

5.6. Let

’AU-

AU

beacausal AK-linearmapandlet

Zr(z-1)

.k=O

Ak(

)Z

-k

beits

_transfer

_function.

Then hasacausalinverse

_if

andonly

_if

Ao(1-)"

U-

Uisinvertible, inwhichcase

Ao(--1)

(Ao(-))-1.

(17)

THEOREM 5.7.

Let

g’AU

:A U

-

A

U

isaAK-linearmap, then thereexists a

_feedback

pair

(L,

V)

such that

l,v (as

in

(5.3))

_if

and

only

_if

(i) isabicausalisomorphism,

(ii)

_for

w

IU,

g(w)

IX

implies

--l(w)

U

(equivalently,

--1

(ker

)

OU).

Proof.

Assume

firstthat

lt,

v.Thebicausalityof hasbeen noted beforeso that(i)holds. Also,since

--

V

-

+

V-1L,

it isreadilyseenthatif w6fU,then

,(w)6lX

impliesthat

{-(w)OU

sothat

(ii)

alsoholds.

Conversely, assume satisfies (i)and(ii). Let V:U--)Ubedefined by V :=

Ao(1).

By (i)

and Lemma 5.6, V is invertible and

V-a=Ao(/-a).

Hence,

the map

-a-

V-1

(where

V-ais regardedas a static

map)

isstrictly causal and

(5.8)

v-’

is an extendedlinear

_i/o

map

A

U-)

_{A U. We}

claimthat ker ker

]r

(where

1

is

therestricted linear

_i/o

map associated with

f).

Indeed

if(w)=

0 implies

(w)

YX

and by (ii)

--l(w)6fU.

By (5.8)

it then follows that

f(w)612U

so that

iV(w)

0.

Upon

employing,

Theorem3.9,we

_{conclude_}

theexistenceofaK-linear

map H:X--)Usuch that

_f

H.

(or

equivalently F H.

).

LettingL"X--)

U

be definedby L VHweobtainupon substitutinginto

(5.8)

f--l__

_V-1

..[_

V-1Lg

and theproofiscomplete.

COROLLARY

5.9. Let

," A

U-*

AX

be areachable extended linear

i/s

map.

Then

_for

every

_feedback

pair

(L,

V)

the map

fL,

V

defined

by

(5.5)

isareachable

extended linear

i/s

map.

Proof.

We

shall provethe corollary by showingthat

fL,

V satisfies Theorem

3.9._

Let

_f:

IU

FSbeanyrestricted linear

i/o

map

such that

_{ker_ jL,V}

C_ker

I

7.

_Let

f

be the extendedlinear

i/o

map associated with

_f

anddefine

_fI’AU

AS

by

I

A U

-

A

S w

-

f{-

w

If we can show that ker c

_kerfa,

then by Theorem 3.9 there exists a unique

H’X-_S

such

_{_that jl}

H.

(or

_{equi.valentl,}

_f-1

H- g),

_whence,

_f=

_fl

-=

H.,

l=H.

f,v (or

equivalently

f=H’f,v)

showing that

f,v

satisfies

Theorem 3.9.

Hence,

the proof will be complete upon showing that ker

ker]’a.

If

w61Uand

(w)

0 then

(w)

6lXandbyTheorem5.7(ii)

w

:=

-a(w)6U.

Itfollows that

fL,

v(W1)=o

f(W1)=

io

f-I(w)=(W)E’X

andconsequentlyfL,

v(Wl)

0 fromwhich

f(wl)

0byassumption.Thus,

f(wl)

S

and

L(w)= o

(18)

We

shall nowspecializeourresultstothe finite dimensional case and assume,

justas in 4, that

U

K"

and

Y

K

p. Moreover,

if

:

U

FY

is a restricted

linear

i/o

map,weshall assume that

]

is rational,i.e.thatithasarealization with

finitedimensionalstatespace.

Firstwehave thefollowing specializationof Theorem5.7.

THEOREM5.10. Let g"

A U- AX

beareachable extended

_finite

dimensional

(i.e.

X

K

)

linear

i/s

map,and let

DIU

ker

.

_If

l"

AU- AU

is a AK-linear

map, therd exists apair

(L, V)

such that

_{l;v (see (5.3))}

_if

andonly

if

is a

bicausal isomorphism and

Z;ID

is apolynomial matrix

(where

Zr

denotes the

transfer

function

of

).

Remark 5.11. Thecondition that

_ZflD

is a polynomial matrix is obvious

upon examination of

(5.3). In

the finite dimensional case we have

Zg

SD-1

(compare

Theorem

4.10),

andconsequently

(5.3)

impliesthat

Z-aD

V-I(I+LSD-1)D

V-I(D

+LS).

It is interestingtoobserve thatif is givenby

,

(where

l,v),

then

Zir

S(D +LS)

-

V

and consequently

(5.12)

ker

f’r

V-’(D

+LS)nU.

Alternatively,

(5.12)

canalso bewritten inthe form ker

f’t-

V-’(D

+

Q)IU,

where

O

is anarbitrary polynomialmatrixsuch that

QD-

isstrictlyproper

(the

matrixL beingofcourseuniquely specified throughTheorem

4.10).

The

interest-ingpointinthisalternate formulationisthe fact thatwehave eliminated

L

from

explicitconsideration.

It is also interesting to make explicit the possible

i/o

maps

(or

transfer

functions) obtainable byfeedback. If ND

-

where it is understood that the factorizationdefines a realizationof

f,

then the transferfunctionsobtainableusing

feedbackaregiven by

ft,v=N(D+O)-lv,

where Qisany polynomialmatrixsuch that QD-1isstrictlyproper

(L

being, as

before, specified throughTheorem

4.10).

The reader should noticethesimilarityofthesituation withthesingleinput

single

_out,

putcase.

Let

_f"

IU

FY

bea rationalrestrictedlinear

i/o

map, and let l"

AU-

AU

bearationalcausalAK-linearmap. Let

(X,

g,

h)

beareachable finitedimensional

(X

K")

realizationof

_f

andlet be the extended linear

i/s

mapassociatedwith

g. Theorem 5.10 then states that there exists a feedback pair

(L, V)

such that

-=

(I

+

L)

-a

V

provided

-

is abicausal isomorphismand

Z{ID

is apolynomial

matrix, where

Zr

is the transfer function of

[,

and D is a polynomial matrix

satisfying

DIU

ker

.

Whilethebicausalityof isclearlyanecessarycondition

for itsfeedbackimplementationin agivenrealization,it is not sufficient.

We

shall conclude this sectionby showingthat therealwaysexists areachablerealizationin

which canbe implemented byfeedbackif and onlyif is a rational bicausal

(19)

THEOREM5.13.

Let:

fU- FYbearational restricted linear

i/o

mapand let

l"

A U- A U

be a rational causal AK-linear map.

A

necessary and

_sufficient

condition

_for

theexistence

_of

a

_finite

dimensionalreachablerealization

(X,

g,

h)

suchthat

Zr

(I

+

LSD-1)

-

V

for

some

_feedback

pair

(L, V) (where

SD-1

isthe

_{transfer function}

_of

thelinear

i/s

mapassociatedwith

g)

isthat is abicausalisomorphism.

Proof.

The

_necessity

ofthecondition is obviousfromTheorem5.10.

We

will

prove sufficiency.

_Iff

is

_{rational,_}

then

Z[

ND-1

forpolynomialmatricesNand

Dand

DU

ker

_f.

Alsoif is a bicausalisomorphism,thenit canbeexpressed

as

[

(I

+

po-1)-i

V

(with

PQ-I

strictlyproper).

Let

R

:= det

O

I.Then

_Zf

NR

(DR)-1

andconsidertherealization

(XA,

gA,

ha)

where

A

DR

fU. Clearly

ZflDR

V-I(I

+

PQ-1)D

det

O

Iisapolynomialmatrixsatisfying the condi-tionof Theorem 5.10.

Remark 5.14. Theorem 5.13 gives a complete characterization of those

dynamical compensators which can be implemented by

"state

feedback" in a

(possibly

unobservable)

reachable state-space. This characterization problem

receivedsome attentionin the literature and the readeris referredtoWolovich

(1974).

6. Chains and invariants. Throughout this section we shall assume that

U

K"

and

Y

K

p.

LetAc

OU

beasubmoduleofrankrn andlet

(XA, ga)

be the (finite dimensional) reachable semirealization given by

XA

:=

12U/A

and

ga"fU-

XA

the canonicalprojection.

Let

a

be the

(restricted)

linear

i/s

map

associated withga.Thenker Aand,aswehaveseenatthe beginningof 4,all reachable realizations and hence also all reachablelinear

i/s

mapswhose kernelis

A areisomorphic. Thus, modulo state space isomorphism, is uniquely deter-mined by

A.

If feedback is applied to

,

then is transformed into a new reachable linear

i/s

map

_fr

(see

_{(5.5))_}

_and.

correspondingly the submodule A is

transformedinto asubmodule

A(1) :=

ker

_fr

(see (5.12)). Two

submodulesAand A’offUof rankrn willbe called

_feedback

equivalentifthere existsafeedback such thatifA ker

,

then A’= ker

_ft.

Clearlyfeedbackequivalenceis an

equival-ence relation and as we _{have just seen, each feedback equivalence class of}

submodulescharacterizes a class of feedback equivalent linear

i/s

maps. In the presentsection westudythis equivalencerelationof submodules.

For

w fU define the(polynomial) degreeof wbydegw

:=

-ord w.LetA and A’ be submodulesof2U.

A K[z ]-homomorphism

q"A- A’ is calleddegree

preserving if for each w6

A,

_{deg q}

(w)

degw.

LEMMA 6.i. Two

submodules

A

and A’

_of

O

U

_of

rank m are

_feedback

equivalent

_if

and

only

if

thereexists a

degree

preserving

K[z

]-isomorphism

q

:A

-

A’.

Proof.

If

A

and

A’

are feedback equivalent then

A

ker and

A’=

ker where

f--/-

-

forafeedbacktransformation

-.

Let q denotetherestrictionof

/--

to

A. We

see thatq maps

A

into

A’.

Indeed,if w6

A,

then

(w)

6

OX,

andby

Theorem 5.7(ii) we also have

v:=q(w)=-l(w)612U.

Thus

/-/-(v)=

-o/-(w)

,(w)12X,

whence)r(w)

0and

q(w)=

v

cA’.

From

the

bicaus-alityof

t-]

itisclear thatord

--a(w)

ord w forall w

A U

and consequently the

(20)

Conversely,assumethath_and_{A’ are}_submodules_{of fU of}_rank_m_{and that a}

degree preserving q"

A-

A’

exists. Let d1,’’’,d,, be abasis for A and denote

d :=q(di), i=l,...,m.

Let

D

and

D

be the polynomial matrices

D

:=

_[dl,.

,

d,,, and

D’

:=

[d ],.

,

d’,,,].

Thensurely Disnonsingularandfor

each w6Awehave

q(w)

D’D-lw.

By

formally regardingthematrix

D’D

-

as a transfer function,weseethatqextendsuniquelytoa AK-linear map

"

AU->

AU;

Zq

:=

D’D

-1.

We

willcompletetheproof byshowing that is a bicausalisomorphismand that themap

-:=

]-1

isthedesiredfeedbacktransformationsuch that

A’

A().

First

note that

(and

hence also

l)

isabicausalisomorphismif andonlyif

(6.2)

ordt]

(w)

ord w

forall w6

A U.

Recallthat

(6.2)

holds for w

A

by assumption.Since

A

isof rank m

(and

hence

IU/A

is a torsion

module)

there exists a polynomial

q

such that

w

A_for_{all w}6

U.

Thus ord

(w)

=ord

w

forall w6

IU

anditfollows that

(6.2)

also holdsfor all w6flU.

Next,

for w

A U

let k be aninteger such that

w

:=

(zkw)O.

Then

wa

61U and ord

t](wa)=

ord

w

=ordzkw.

Sincefork

sufficiently largeord

](w)

=ord

gl(zkw),

itfollows that

(6.2)

holds forallw

AU

and isabicausalisomorphism. Finally,toseethat

---

-

isthedesiredfeedback transformation we needtoshow

(see

Theorem

5.10)

that

Z-ID

is apolynomial

matrix. Indeedthisis true since

Z{D

_Zr-D

D’D-aD

D’

and theproofis

complete.

El

In

ordertoapply Lemma6.1 incheckingastowhethertwosubmodules A and

A’

arefeedback equivalentonehastoverifythe existence

(or

nonexistence)ofa

degree preserving homomorphismq"A->

A’.

To

thisendweintroducethe concept ofdegreechain.

DEFINITION6.3.

Let

A beasubmodule of12U and for each 0, 1,2,

,

let

A

A_{be the submodule}_{generated by the}elements of A whosepolynomialdegree

is less than or equal to i. Let

v

:= vi(A):=rank

_Ai.

The sequence

(Ai)o

of

submodules iscalled the

degree

chain of A and thesequent

(u)=o

is called the

degreelist.

It follows immediatelythat

Ao

c

_A1

cA₂c. c

_A,

and Uo_-<Ul-<uz_-<" _-<

u:= rank

A.

Since

IU

is Noetherian,itsatisfiestheascendingchainconditionand we have

_Ar

A

forsomefiniter.

Hence

AoCAc

"CAr=A,

We

also introduce the following notation. Let a 0 be a polynomial or a

polynomialvector.

We

denoteby the leading coefficient

(vector)

ofa.That is,if

a

k=

_o

az

andak 0 then ak.Also,if a 0wedefined := 0. Recall thatif

dl,

,

dk

areelements ofa

K[z ]-module

which generateasubmodule

M,

they

are called

_free

generators

_of

M,

orsimply

_free

ifthey are

K[z

]-linearly

indepen-dent.

We

now introducethe following

DEFINITION 6.4.

Let

dx,’’ ",

d

be elements ofa finitely generated

K[z]-module (specifically of

U).

Thendl,"

,

dk

arecalled

properly

_free

if

1,"

,

dk

areK-linearlyindependent.

(21)

LEMMA

6.5.

_ff

dl," ",

dk

areproperly

free

thentheyare

free.

Proof.

We

provethelemmabynegation. Ifdl,

,

dk

arenotfree,there exist polynomials a1,’’" ,ak, not all zero, such that

Eik=l

aidi=O.

Let r:=

maxdeg

(aidi)

anddefine

0 ifdeg(otidi

<

r,

Ei

:-’-1 ifdeg(otidi) r.

Addingthetermsofdegree rin

_ol.idi

yields

k

E

Ei

idi

O,

i=1

implyingthat

al,

,

k

are K-linearlydependentsince notall

6ii

are zero.

DEFINIaION 6.6.

Let

A be afree

K[z]-module.

A

basisd1,"’,

dk

of A is called properifdl,.

,

dk

areproperlyfree.

The followingis an instrumentalresult.

LEMMA

6.7.

Let

Abeasubmodule

_of

12U

_of

rank

u(

>

0).

Thenthereexists a

properbasisdl, ",

d

of

A such that

(6.8)

deg dj

_for

12i--1<j 12iand O, 1, 2,.

where

(12i)i%0

isthedegreelist

_of

A

and12-- :’"O.

Proof.

Firstobserve thatifk_->0is thefirstinteger such that 12k 0,we can choose 0

_{d16 Ak}

such that deg d k. Then d is clearly properly free and satisfies

(6.8). We

proceed stepwise and assume that for >0, dl,...,

dt

are

properly free elements of

A

satisfying

(6.8)

and let S denote the submodule

generated by dl,

,

dl.

If 12

(and

hence S

A),

then there existsaninteger

such that

At-1

S

At.

Thus, thereexists an element

dl+l

At

such that

dt+

:

S anddeg

dl+l

t.

Obvi-ously d,

,

dl+

satisfies

(6.8)

andwecomplete

_theproof

by showingthat this

setis

_also

_propeOy

free.

Assume

tothecontrarythat

dl+

is a K-linear combina-tion ofdl,...,d"

Consider the element

p ::

dl+l--

Aiz6’di

i=l

where 6i := t-degdi, 1,-..,

I.

Obviouslyp A and it is easilyverified that

degp=<t- 1. Thus p At_lcS andweconclude that

dl+

S,

acontradictionto ourassumption, fq

COROLLARY 6.9.

Let

A

OU

beasubmodule

_of

rank uand let

(Ai)i=o

ahd 12i)i=obe thedegreechain anddegree list

_of

A respectively. Thereexists aproper basis d,

,

d

of

A such that

_for

each satisfyingui 0 thesetda,

,

d,

isa

basis