Decoupling of multivariable control systems over unique factorization domains

Decoupling of multivariable control systems over unique

factorization domains

Citation for published version (APA):

Datta, K. B., & Hautus, M. L. J. (1984). Decoupling of multivariable control systems over unique factorization domains. SIAM Journal on Control and Optimization, 22(1), 28-39. https://doi.org/10.1137/0322003

DOI:

10.1137/0322003

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Vol.22,No.1,January1984 0363-0129/84/2201-0003$01.25/0

DECOUPLING

OF

MULTIVARIABLE CONTROL

SYSTEMS

OVER

UNIQUE FACTORIZATION

DOMAINS*

K. B. DATTAS" AND M. L. J. HAUTUS

Abstract. Necessary and sufficient conditions are established for the existence of a state variable

feedback decouplingof anm-input, m-outputtime invariant linearcontrol system over auniquefactorization domain. An explicit computation is provided forthe feedback and the feedforward gain matrix. Also

necessary andsufficient conditionsforthe existence of astability-preservingstatefeedback decouplingare

given.Theresults areillustratedbysomeexamples.

Keywords, decoupling, delay sygtems,systems overrings,multivariablesystems,statefeedback

1. Introduction. Thedesignandsynthesis of noninteractingcontrol in multivari-able controlsystemsby state-variable feedbackwereinitiated by

Morgan

(1964)and definitive results in this direction by establishing necessary and sufficient conditions

for the existence of a

decoupling

feedback, as well as an explicit construction, were first given by Falb andWolovich (1967). Their resultswere formulated for systems

withreal coefficientsbuttheyareeasilyseen tobeextendibletosystems overarbitrary

fields. The extension of these resultsto systemsover rings, however,is lessobvious.

Onthe otherhand, systemsoverrings have showntopossessawiderange of potential applications such asdelay systems,2-D systems, parametrized systems,discrete time distributed systems, systems with integer coefficients, etc. We refer to the survey papers E. D.

Sontag

(1976), (1981), E. W.

Kamen

(1978),and thereferences therein. Thisabundance ofcontrolsystemswhich canconveniently be modelledassystems over rings is a motivation for a systematic investigation of systems over rings. This

investigationwas startedwiththe thesis Rouchaleau

(1972)

and thepaper Rouchaleau,

Wyman

and

Kalman (1972)

andithas receivedmuch attention recently.

The purpose ofthispaperistoformulate necessaryand sufficient conditions for the existence of adecouplingstate feedback for alinear time-invariant system over aunique factorizationdomain.This particular class of ring is wideenoughtoencompass

almost all themodels arising from the applicationsmentionedbefore andontheother hand, it allows a complete solution of the problem. The conditions which will be

obtained reduce to the Falb-Wolovich conditions when applied to systems over a field. The method ofproof, however, is completely different fromthe proof in Falb andWolovich (1967). Itcan beregarded ageneralizationto systemsoverrings of the type of proof given in Hautus and Heymann (1980), (1983) and it is based on a

characterization of feedback transformations given inHautus and

Heymann

(1978). Itis possibletoaxiomatize the concept of stability for systemsoverrings in such

awaythatineach particular specification and application(delaysystems,2-Dsystems)

the notion ofstabilitycustomary in thatfieldaccommodates conveniently in the general

framework.

An

examplewillbe given in 2. The treatmentisbasedonwhatwehave called "denominator set". This concept was introduced for systems over a field in

*Receivedby theeditorsMay26, 1981, andinrevisedform March8, 1982.Thisresearchwaspartially supported bythe National Science Foundation undergrant ECS-7908673 andby theNational Science

Council,TaiwanundergrantVE80003.

tDepartmentof ElectricalEngineering,liT,Kharagpur-721302, India.Formerlyof theDepartment

ofApplied Physics,CalcuttaUniversity.

tDepartmentof Mathematics, University ofTechnology, Eindhoven,The Netherlands. Thispaper was completedwhile thisauthor was on leave asTRWVisitingLecturerattheDepartmentofElectrical

EngineeringSystems,UniversityofSouthernCalifornia,LosAngeles,California90007.

and Khargonekar and Sontag

(1981)

(where the name Hurwitz set is used).

In

the general framework of stability thusprovidedwegivenecessaryand sufficient conditions for the existence ofastability preserving decouplingstate feedback.

The problem formulation and the main results are given in 3.

ExampIes,

illustrating theresults,aregiven in 4,and 5isdevotedtotheproofof our main result. 2. Stability ofsystems overrings. Throughout the paper

g

willdenoteaunique factorizationdomain (= UFD) orfactorialring

(see

Samuel (1963),Barshay

(1969)).

We

usethe notationsg/[z and

(z)

todenote the rings ofpolynomialsand rational

functions over

,

respectively.

A

polynomialq iscalledmonic ifitsleading coefficient equals 1.

A

rational functionis

_call.ed

causal (or proper) if .it has a

representation

of the formp/q,where q is a monicpolynomial anddegp_-<_{deg q.}

A

denominator set is asubset @ of

[z

satisfying the following conditions’

(i) ismultiplicative,i.e. 1 andifp,q

(ii) Eachpolynomialp @ is monic (inparticular 0

:).

(iii) 5 issaturated, i.e.ifp and q is monic anddividesp then q

(iv) There existsa g such thatz-a

.

Since a denominator setis multiplicative, itis possibletoassociate with it a ring

of

fractions

tobe denotedby

[z] (see

Barshay (1969,Chap.

3)).

Specifically

[z]

isthe set ofrational functionshavingarepresentation ofthe form

p/q,

where p and q are polynomials and .q

.

It is well known and easily seen that

[z]

is a ring,

even a

UFD

(see Samuel (1963, Thm.4, p. 29)).

In

addition,weintroducethesetof causal fractions in

[z],

i.e. elements of

[z]

that are causal rational

functions.

This set isdenoted by

[z

],

or, if thedenominator setdoesnothave tobe specified, by

.

LEMMA2.1. isa

UFD.

For a proof, see 5. The set of all monic polynomials, which is denoted 0,

is an example of a denominator set. The corresponding set of causal fractions is denoted

0.

A

(free) linearsystem is identified by aquadruple (A,B, C, D) ofmatricesover ofsuch dimensions thatthe

following

equationsarewell defined.

(2.2) x,+

Ax

+

But,

y,

Cxt

+

Du,,

where

x,:=",

utq/:=

,

Equations (2.2) give a discrete time interpretation of the system

,v.,

:= (A,

B,

C, D).

The system is calledreachableifthe columns of the matrix

[B, AB,

,

A"-IB

span

the total state space

".

(See Sontag

(1976)

fordetails.)

To

the system

X

a

_transfer

function

(2.3) W(z) :=

Wx(z):=C(zI-A)-IB

+D

isassociated. This is a matrixwhoseentries are causalrationalfunctions.

For

agiven transferfunction W(z),

E

(A,B, C, D)iscalledarealization if(2.3) holds.

Other interpretations of

E

canbe given.

Systems

over ringscanbe

used

tomodel systemswithparameters,systemswithdelays, 2-D systems, neutralsystems

(see

Eising

(1980),

Hautus

andSontag (1981), E. W.

Kamen

(1978), Rouchaleau (1972),

Sontag

(1976),

(1981)). We

will give an example below.

By

a suitable choice of (and

sometimes of

g,

see Eising (1980,

4.3))

one can accommodate various stability

DATTA AND

denominator set have been chosen, we call a rational function stable if it is in

Y[z]. A

(single variable) stable transferfunction is an element of

[z].

An

nxn

matrix

A

is called astability matrix if det

(z!-A)

.

Obviously,

W(z)

isstable if

A

isastabilitymatrix.Theconverse isnotalwaystrue,however,foranystable ti’ansfer function matrix, there exists a

(free)

realization

X

which is stable, i.e., for which

A

is

astabilitymatrix

(see

Sontag (1976)).

Letus give someexamplesof interpretations of systems over rings and particular choicesofdenominator sets.

Example 2.4.

In

the case that

Y

(the fieldof realnumbers) stabilityoftenis

formulated in terms ofpole location. Specifically, a set C-__.Cis given and amonic denominator q(z) is in ittithas nozeros outside C-.

It

is easilyseenthat

,

thus defined,is adenominatorsetprovidedC-f’]

.

[3

Example 2.5.

One

canmodel adelay system withdelays allmultiple of agiven positive real numberz byasystem over the ring

g

[cr]

of polynomials inor, where

o-stands for thedelay operator

crx(t)

x(t-r).

The system then will be of the form

(2.6)

A(cr)x+B(cr)u, y

C(cr)x

+D(r)u

where

A, B, C, D

arepolynomial matrices. The systemic significance of the transfer function

(2.7)

W(s,

r)=D(cr)+C(cr)(sI-A(cr))-XB(cr)

isdescribedin

Sontag (1976). In

particular,applyingaLaplacetransformto

(2.6)

yields

(2.8)

(s)

W(s,

e-)a(s).

It

is well known (see Hale (1977,

7.4))

that X=(A,B, C,D) is (externally) stable iff W(s,e

-)

has nopolein

Re

s ->0.Thus,herewedefine

(2.9)

:=

(p

[s,

r]lp

is monic with respect to s and p(s,e

-)

0 for

Re

s

=>

0}.

When saying p ismonic withrespectto s we mean that p is of the form

p(s,

r)

s"

+p(r)s

"-

+" +p,(r)

where p,..., p,

R[o,]. It

is easily seen that is a denominator set.

In

order that

thesystem beinternally stableone mustrequire that det

(sI-A(cr))

be in

.

[3

Furtherexamplesdemonstrating thegeneralityof the stability concept described here can be given.

Compare Datta

and

Hautus

(1981), Eising (1980),

Hautus

and

Sontag

(1981),

Kamen (1980).

3. Problem formulation and statement of themainresults. First,wegiveageneral

formulation ofthe decouplingproblem.

We

introducethe i/s-map correspondingto

system (i.e., system

(2.2))

by

(see

Fig.

3.1)

(3.1)

W(z):=(zI-A)-B,

so that W

_CW

+

D. Let

F

and G be dynamical systems withdimensionssuch that

theformula

(3.2)

u

-F(z

)x

+

G(z

)y

+ u

Ws

x

FIG. 3.1

into asystem

E,6

withtransfermatrix

(3.3)

Wl,O(z)

W(z)L,o(z),

where

(3.4)

Lv,(z)

(I

+F(z)

Ws(z))-tG(z).

We

noticethat thesametransfermatrixis obtained if onereplaces(F, G)by(0,

Lv,).

A

compensatorin which

F

0 iscalledaprecompensator. IfG isstatic

(no

dynamics

in the precompensatorpart) thenwe saythat (F,

G)

is pure (dynamic)

_feedback

and if, in addition,

F

is staticthen (F,

G)

is called a static state

_feedback.

Our

objective

istofind acompensator ofaspecified class(precompensator,pure dynamicfeedback,

staticfeedback)such that the resultingtransfermatrix

_W,

is diagonal, in which case

we call the resulting system decoupled.

In

order to guarantee that each output can effectively be controlled, werequire in addition that the diagonal elements of

WF,

be nonzero.This is equivalenttorequiring

G

tobenonsingular.Sometimes onewants toimpose stronger conditions, such as theinvertibility (over

)

of

G

(compare

Datta

and

Hautus (1981)).

Finally, assuming that theoriginalsystem is internallystable,we

trytofind(F, G)such that the resulting system is internally stable. Such acompensator

will be called stabilitypreserving. Notice that we do not attempt to stabilize and to

decouple the system simultaneously. Rather, wetrytodecouple itwhilemaintaining

itsstability. If the system isnot stable atthe outset, it has to be stabilized first and afterwards one has to design the decoupling compensator.

It

will follow from the results of this paper, that one cannot destroy the existence of such a decoupling

compensator whenapplying thestabilizing feedback.

We

assume that q/=

,

i.e., the numberofinput and output variables areequal.

It

turns out that the problem of decoupling by precompensation or combined compensation (i.e., no restrictions on

F, G)

is very simple even if

Y

is an arbitrary integral domain.

TI-IEOREM 3.5.

In

the situation described above, the following statements are

equivalent.

(i) Thereexists a (stability preserving) decoupling combined compensator (F, G).

(ii) Thereexists a (stability preserving)decouplingprecompensator(0, G).

(iii) Wisnonsingular, i.e.,det Wis notidentieallyzero.

Proof.

(ii)

:ff

(i) is trivial.

(i) :::), (iii). Accordingto(3.3)wehave

det

W

detLv,6 det

WF,

a 0,

(iii)

=:>

(ii). Let adj Wdenote the adjointof W (occurringinCram6r’s

rule).

Then

Wadj W (det W)L

Choose a

Y

such that z-a @. Then, for sufficiently high k,

(z-a)-k

adj W G

is causaland stable since theentriesof

W

arestable.

Hence,

by

W. G (z

a)-k

(det W)./,

G is a stable decoupling compensator, which has an internally stable realization. If

G isinternallystable, then thetotal realization isinternallystable. [3

The condition for the existence of pure feedback decoupling compensators is moreinvolved. Toformulateit weneedsome notation.

We

write

(3.6)

W(z)

where wi(z) denotesthe ith rowof

W.

Let di(z) denoteaGCD over of the entries of wi(z). Such a GCD existsbecause is a

UFD.

An

explicit construction of sucha

GCD isgiven in Lemma 3.11. We can write

_w

_dw*

for suitable

w*

withentries in

.

Hence

(3.7) W(z A(z)

W*(z

),

where A(z)=diag(dl,...,d,,) and

_,

W*

is the matrix consisting of the rows

W ,’’’,Wm.

Now..

we areinthe positiontoformulatethemain result ofthispaper.

THEOREM 3.8. Let

,

be a reachable, internally stablesystem with respectto the denominatorset@. Then the followingstatementsareequivalent"

(i) can be decoupled by a stability preserving static state

_feedback

with G

invertible

over

.

(ii) Y_, can be decoupled by a stability preserving, stable dynamic state

_feedback

withGinvertibleover (andF(z) stable).

(iii) Y.,

_can

be decoupledby astableprecompensatorL which is invertible over

.

(iv)

W*,

as given in (3.7), isinvertibleover

.

Theproofofthisresultwillbe given in 5.

in

the theorem it is assumedthat the gain matrix G isinvertible, although this

is not necessary in the original problem formulation: G nonsingular would do.

It

is

possible to generalize the theorem to this more general case, but the formulation

becomes more involved. There are two remarkable consequences of Theorem 3.8,

already noted for systems over fields in Hautus and

Heymann

(1980), (1983). In

the firstplace, if decouplingis possible by dynamicstate feedback, itis also possible bystaticfeedback.

In

thesecondplace,the conditionfor theexistenceofadecoupling

statefeedbackdoesnotdependonthe realization, provided the realizationisreachable

(forsystems over fieldsthislatter restriction isnotnecessary).

The GCD’sused indefining

W*

are not unique andconsequentlyso is not

W*

itself.

However,

the invertibility of

W*

isindependent of the particularchoice of the

GCD’s,

as easily can be seen. The condition on

W*

can be checked by computing

w(z)

=det

W*

and checking whether

(w(z))-l

.

Whether this condition can be

verified effectively depends on

.

In

the particular case that @ 0, the set of all

extent,expandw(z) inpowersofz

-x,

W

(Z)

WO q-WIZ-1-[-_W2Z

-2’’[-"

and

(w(z))

-1

o

iffw0is invertible over

Y.

Itisevennotnecessarytocompute

w(z).

COROLLARY3.9.

_If

0

in Theorem 3.8 and

_if

W*

isexpandedas

W*(z)

W*o

+

W’z

-

+...

then the condition (iv)

of

Theorem 3.8 may bereplaced by"

W*o

isinvertibleovergt.

When restricted to the case that

Y

is a field, thisis exactly the condition given inFalb and Wolovich

(1967).

Althoughthe particular choice of the GCD’s used in the definition of

W*

is of

no relevance for condition (iv) of Theorem 3.8, it will turn out that for the actual

construction of a decoupling feedback it is imperative that the

GCD’s

satisfy an

additional condition. To express this condition we use the notation (p,q

l[z])

for the

GCD’s

of elements in

Y[z].

DEFINITION 3.10. Let wl,’.’,w,,. A GCD d of Wl,"’,w,, is called

admissible ifthere exist polynomials q and p such thatpd andqwi/pd, 1,...,m

arepolynomials, and(p,

qlY[z])

1.

The existence and the construction of an admissible GCD is guaranteed by the following result.

LEMMA 3.11. Let wx,’’’,Wm and letq be a leastcommon denominator

_of

Wl,..’,w,,.

Define

fii:=qwi and let v:=

(,61,.."

,/,,lY[z]).

Finally, write pi:=ffi/V.

Choose a

e

such that

z-ae

and

_define

lz

:=min{degq-degpili=l,...,m}.

Then

d:=v

.(z-a)-*"

is anadmissible GCD

of

Wx, ",w,,.

For a proof see 5. The lemma provides us with an actual construction of an

admissible

GCD,

atleast ifwehaveanalgorithmfor computing

GCD’s

in

Y[z].

This is for instancethe case when

Y

Rio,

I,

seeBose

(1976).

Now

wecanspecify Theorem 3.8"

PROPOSITXON3.12.

_If

theelements

_di

usedintheconstruction

_of

W*

areadmissible

GCD’s

_of

withen thereexistsa static

feedback

(F,

G)

suchthat

Lv.

W*-I.

Foraproofsee 5.

Onceithasbeenproved thatagiven precompensatorL canbe implemented by

a(static) feedback (F, G), the actual computation ofF and G isstraightforward. Let

usstart from

L

-1,

ratherthan

L

(recallthatL-1

W*).

The relation

L

-1

Lv,-l

reads

G-I(I

+FWs)=L

-1

=Mo+MlZ

-1

+...,

wherewehave expandedL-1

intopowers of z

-1.

Invertibility ofL-1over implies that

M0

isinvertible. Since

_Ws

is strictly causalwe musthave

(3.13) G

_=M1.

Then it follows that

GL-I=I=:

V(z)

is strictly causal and the map

F

has to be computed from

FW, (z V(z ).

Onsubstitution of

W

(zI-A)-IB

into this equation and expanding both sides as

aseriesin z

-x,

one obtains

DATTA AND

wherewehave used theexpansion

V(z)

Vlz

-1

+

V2z

-2+.

..

Because

of

reachabil-ity, the map

[B,

AB,...,A"-IB]

has a right inverse and hence

F

can be solved

uniquelyfrom

(3.14).

Notice thatthemaps

F

and

G

are uniquelydeterminedby

L. In

particular,the

stability-preservationproperty of

(F, G)

isautomatically guaranteed bytheinvertibility

of

W*

over

.

4. Examples.

Example 4.1. Consider the system

_’=

(A,

B, C, D)

over

[cr],

where

B=

C

-or

2+cr+1

cr D=0.

Suppose

that

o.

It

is easily seen that

E

isreachable. The transfermatrixhas the following expansion

W(z

CBz

-1

+

CABz

-2 +.

--[ltr

_tr+ll

]

_1

[

lz

+

2tr

]

-2 cr+l

20.2+0._

1 z 2o"

+

_{2o.2+o._1}

2o’2-o

"+1]

-3 20

"3+2

z +" ’.

Thesystemcanbe decoupled bystatefeedback since

_W0*

[

+1]

isinvertible. Accordingto(3.13), wehave

Furthermore,

F

hastobe computed from

(3.14),

where

_V

isthe coefficient of z in theexpansionof

_W*o

-

W(z).

(Noticethat

W*(z)=

z

-1W(z).)

It

follows that

F[0

1 1

tr-1]

[01

tr+l tr+l

o’2-1]

1 tr tr 0

"2+

1 o’-1 o’-1 0

"2-O"+2

-i-1

Equating thefirst twocolumns we obtain

The transfer function of theresultingsystem is

z-lL

Example 4.2. The purpose of this example is to show that the reachability

condition inTheorem 3.8 is essential.Let

N[o-],

N

No

The systemis notreachable,since

2

0

and hence allreachablevectorshavetobedivisiblebytr.Thetransfer functionis

Following theproceduredescribedatthe beginning ofthissectionweobtain

W*

toO" Z--

Wo*

+...

0 1 where

W’

=I

is invertible. We show that, nevertheless, decoupling by dynamic state feedback is impossible. Suppose that (F,G) is a dynamic state feedback decoupling the system. Then

W(I+FW)-IG=A

for some diagonal matrix A (recall that

W=

Ws).

Equivalently,

(4.3) W

AG-I(I

+FW).

Expanding thematrices

D, W, F

inpowersof z-1wehave

W

Wlz

-1

+ W:zz

-z+.

.,

A

_{=Do+DlZ-l+D:z-2+...,}

F

Fo

+Flz

-1+"

,

where

2

Substitution into

(4.3)

yields

Wlz

-

+

W2z-2

+

(Do+Dlz

-

+"

")G-I(I

+

(Fo+’’

")(Wlz

-

+"

")).

The coefficients of Zo yields: D0=0, and of z

-"

W1

=D:G

-

hence

trG

=D.

In

particular,

G

isdiagonal. Finally, equating the coefficients of z

-

weobtain

Wz

DG-1Fo

W

+ D2G

-

o-2Fo

+ D2G

-1.

Thematrix

D2G

-1 isdiagonal and hence r2

F0,2

tr. whichhasno solution in

[r].

Furtherexamplescanbefound inDattaand Hautus

(1981).

5. Proofs.

Proof of

Lemma 2.1. Itiswell known that

[z]

isa

UFD

if is amultiplicative

set

(see

Samuel (1969, Thm. 4, p.

29)). We

define an isomorphism which maps

onto

_l[z]

for some

@1.

Theisomorphismis

q"

r(z)->(z):=r

a

+

definedon (z),where a is chosensuch that z-a

,

and

1

:={z(z)lp

e

,

n

degp}.

@1

is easily seen to be a multiplicative set in

9[z].

The map q is invertible, and

q r(z)

r((z- a)-

The homomorphism propertiesarereadilyverified.

It

remains

-1

36 K. 13. DATTA AND M. L. J. HAUTUS

Letr

p/q

.

Then

z"(z)

where n degq. Notice that because of the causality of r, the numerator

z"ff(z)

isa

polynomial. Conversely,letr

_l[z],

say p(z)

r(z)

z 4(z)

for some q withdeg q n.Then

_,

_{(z._-__a.)"p(}

1

)

o

r(z)=

q(z) \z-a

since(z -a)-1 and hencep((z

-a)-),

and q and hence

(z-a)"/q(z).

Since

l[z

is a

UFD

and isomorphicto

,

itfollows that

Remark. need not be adenominator set, in particular, the elements of neednotbemonic.

Remark. The isomorphism q is a standard device for transferring properties

known about quotient ringstorings of causalquotients (Eising(1980, 4.2),

Hautus

and

Sontag (1980)).

Proof

of

Lemma 3.11. Becauseof the assumption that

g

isaunique factorization domain, the ring

g[z]

is alsoaunique factorizationdomain

(see

Barshay

(1969,

Thm.

4.7)). Let

usdenote by theset of polynomials v of the form v uw, where u is a

unit and w

.

If v is any polynomial in

[z],

we can factorize it into primes v p _Pro.

As

aconsequence,wecandecomposev into v

v/v

where v isthe

productof theprimefactorsof v whichare in and v/

consists of theother factors.

Except

for unit factorsthisdecomposition is unique. Ifweinsiston aunique decomposi-tion we can achieve this by requiring that

v-

be monic.

We

call

v-

the

-part

and

+

v the

non-

part ofv.The following results follow easily from this definition.

PROPOSITION5.1.

(i) (ab

)/

a/b

/,

(ab

)-

a-b-(ii)

p[q

(in

[z ])

iff

p/[q/

and

p-[q-.

+

(iii) (GCD (v,..., v,))

/=GCD

(v,...,

v,). (iv)

If

p

,

q

lP

thenq

.

In

view of thesaturation conditionimposedon

,

property (iv)follows from the fact that divisors of elements of are monic upto aunit factor.

Afterthispreparation we are inthe positiontoprove that d

:= v(z-a)-

is an admissible

GCD

ofw1,.

.,

w,, over

.

In

the first place d is a divisor of w,..., w,

because(recallthat wi=piv/q)

Wi

_pit)(Z

a)U

pi(z

-a)",

d qv q

since q

and/z

+degpi=<degq.

Now

let

dt

be also adivisor of wt,...,Win.

We

havetoshowthat

d/d1

,

Let

dl

a/, wi/d =:

a

b/ci,withfl,

c

We

noticethat

bi

Wi

aidl

,

andhence

qob

i,Bci.

+b?

Taking the

non-

partweobtaina

/?.

Thisshows that

a/l/

ing[z].Observing that v is a GCD of

the/i’s

and hence v+ is a GCD of

the/’s,

we conclude that

a

+Iv

+,

sayv+ ya+with y

g [z ]. It

follows that

d

flv

/33,v

e[z]

=a(z-a)

ot-(z-a)

since the denominator is in

.

It

remainsto beshown that

d/d1

is causal. Choose

such thatdeg q degpi

+

_Ix (recall the definition of

Ix).

Thenwehave that

d d wi

dl

Wi

dl

v q

(z-a)

iscausal,since ai is.

It

follows that dis aGCDofwl,., ’, w,,. Finally,weshow that

d is admissible.

We

choose p

(z- a)

",

and q as already defined. Thenpd v is a

polynomial and pd and q are coprime.

In

addition, qwi/pd-pi is alsoapolynomial.

Thiscompletestheproof.

Proof of

Theorem 3.8 (andProposition 3.12).

(i) =),(ii)is evident.

(ii)

:=>

(iii).If

(F, G)

is astability preserving,stable dynamicstatefeedback and if

G is invertible, then, according to(3.3), W isdecoupledalsobytheprecompensator

LF,

defined by

(3.4). It

remains to be shown that the entries of

_LF.

are in and that

LF,

is invertible over

.

It

is easily seen that

_Lv.

is invertible as a rational

--1

matrix and that

Lr,

G-I(I

+FWs)

has entries in

.

So,

only the stability of itself has tobe shown.

By

assumption, the resulting system is internally stable. This implies that thematrix

V

WsLF,

is stable.Since

itfollows that

(I

+

FWs)-I

I

_-FWs

(I

+

FWs)-1

LF.C

G

-FV

isstable.

(iii)::> (iv).

Suppose

that for some nonsingular diagonal matrix

E

diag(el,’’’,e,,) we have W

EL,

where L is amatrix invertible over

.

Thefirst row ofthismatrixequation reads

[w11,

wlm]=el[111,

11,,,],

which implies thatel is a divisor ofwl

=[w11,

",

wl,,].

Since

di

isa

GCD

of wl it

follows that el divides dl, i.e.,

dl

hie1

for some

hi

s

.

Similar results hold for the

otherrows,sothat we canwrite

A

EH

where

H

diag(h1, ",h,).

It

follows that

EL

W

A

W*

EHW*

andhenceL

HW*.

Therefore

W

*-1

_HL

-1

iscausal and stable.(The nonsingularity

ofthe matricesinvolvedisobvious).

(iv)

::>

(i).If

W*

isinvertiblewedefine

L

W

*-1,

andformula

(3.7)

reads

AND M. L. J. HAUTUS

Hence,

the system is decoupled bythe precompensatorL which is amatrix over

,

hencecausal and stable.

We

havetoshow that

F

andG exist such that

L

LF,

(see

(3.4)). To

this extent, we formulate a generalization to systems over of a result given for systemsover a fieldinHautusand

Heymann (1978).

The result inquestion

is:

THEOREM 5.2.

Let

(A,

B, C, D)

denote a reachable system andL be abicausal isomorphism (i.e., causal and with a causal inverse). Then there exists a static state

feedback

compensator (F,G), with Ginvertible, andL

LI,

iff

for

each polynomial

u

"[z]

wehave:

_If

Wsu

ispolynomialthen

L-lu

ispolynomial.

Theproof of thisresultis completely analogoustotheproofin the fieldcase,so it will not be repeated here.

Contrary

tothe field case, however, the reachability of the system is essential. (A counterexample in the nonreachable case canbe deduced from Example

4.2.) In

order to apply the theorem to our

L,

wehave toprove that

L-lu

W*u

is polynomialwhenever u and

_Wsu

arepolynomial.

We

show that the following stronger statement: u and

Wu

are polynomial implies

W*u

ispolynomial holds,provided

W*

isconstructed via admissible

GCD’s.

(Thiswill alsoprove

Proposi-tion

3.12.)

Letus assume that

Wu

ispolynomial.Thefirst entry of thisvector iswlu,

whichis apolynomial. Let

dl

bean admissibleGCD of w, and let

w’

d-(lw.

We

have to show that

_W*lU

is polynomial. According to Definition 3.10, there exist

polynomialsp and q such thata := pdand v

/a,

where v :=qw arepolynomials and

(a,

ql[z])

1. Since wlu

=q-Xvxu

ispolynomialwe have

qlv u.

Also,

alvxu,

since

u and v

_x/a

are polynomial.

Hence

qalvu,

q and a being coprime. Butthis means that

w*u=d-Xwxu

(qa)-Xvxup

is polynomial. The same argument applies to each

row. This shows thatLcanbe realized bystatefeedback.

It

remainstobe shown that the resulting systemisinternallystable. Since

W,v.,

the i/s-mapwhich results after feedback is applied, isequalto

_WLF,

oand hencestable,the desired resultfollowsfrom:

LEMMA

5.3. Let

W,

(zI-A)-IB

beareachable i/s-map. Then

W

isstable

_iff

det

(zI-A).

Proof.

Since

(zI-A)-l=adj

(zI-A)/det

(zI-A), the "if" part is obvious. To

prove the "only-if" partwenotethat because of the reachability of(A, B),there exist polynomial matrices

P(z)

andQ(z)such that

(zI

-A )P(z

+

BO(z

I

(see

Khargonekarand

Sontag

(1981, Lemma

3.2)). It

follows that

(zI-A)-=P(z)+

W(z)Q(z)

is stable whenever

Ws(z)

is.

But

then also det

(zI-A)-a=

1/det

(zI-A)

is stable,

and hence det

(zI-A)

(since is

saturated).

Acknowledgment.

K.

B.

Datta

isgratefultotheEindhovenUniversity of Tech-nology,

Department

of Mathematics for offeringhim a Research Fellowship during

the tenure of which he workedontheproblemreported here.

REFERENCES

J.BARSHAY(1969),TopicsinRingTheory,W. A.Benjamin,NewYork.

N. K. BOSE (1976),An algorithmfor GCFextractionfrom two multivariable polynomials,Proc. IEEE, pp.185-186.

K. I. DATTA ANDM. L. J. HAUTUS (1981),Decoupllngofsystemsoveraunique_{factorization}domain,

inProc.International Symposiumon MathematicalTheory otNetworks andSystems, N. Levan, ed., Santa Monica,CA,pp. 35-39.

F.EISING(1980),2-D systems, analgebraicapproach, Ph.D. Dissertation,Dept.ofMathematics,Univ. ofTechnology,Eindhoven.

P. L.FALBANDW.A. WOLOVICH (1967),Decouplinginthe design and synthesisofmultivariable control

systems,IEEETrans. Aut. Contr., AC-12,pp. 651-659.

J.HALE (1977),TheoryofFunctionalDifferentialEquations, Springer,NewYork.

M. L. J.HAUTUSANDM. HEYMANN (1978),Linear_feedback--analgebraic approach,thisJournal,.16, pp.83-105.

(1980), New results in linear _feedback decoupling, in Analysis and Optimization of Systems, A. Bensoussan andJ. L. Lions, eds., Lecture Notes inControl and Information Sciences 28, Springer,NewYork, pp. 562-577.

(1983),Linearfeedbackdecoupling-transferfunctionanalysis,IEEE Trans. Aut.Control,toappear. M. L. J. HAUTUSANDE. D.SONTAG (1981),Anapproach todetectability and observers, in Algebraic and Geometric Methods i.n Linear System Theory, C. Byrnes and C. Martin, eds., Reidel, Dordrecht.

E. W. KAMEN(1978),LecturesonAlgebraicSystemTheory:Linear systems over rings,NASAContractor Report3016.

(1980), Onthe relationship betweenzero criteriafortwo-variable polynomials and asymptotic stability ofdelaydifferentialequations,IEEETrans. Aut.Control,AC-25, pp. 983-984.

P.P.KHARGONEKARANDE. D.SONTAG(1981),On therelationbetween stablematrix_fraction factoriz-ationsand regulable realizationsoflinear systems over rings,IEEE Trans. Aut.Control,AC-27, pp. 627-638.

B. S. MORGAN,JR. (1964), The synthesis_oflinearmultivariable systemsbystate-variablefeedback,IEEE Trans. Aut.Control, AC-9, pp.405-411.

A.S. MORSE(1976), Systeminvariantsunder_feedbackandcascadecontrol,inMathematicalSystemsTheory, G.MarchesiniandS.K.Mitter,eds.,Lecture Notesin Economics and MathematicalSystems,131, Springer,Berlin,pp. 61-74.

Y. ROUCHALEAU(1972),Linear,discrete-time,_{finite-dimensional}dynamicalsystems over someclassesof

commutative rings,Ph.D. Dissertation, Stanford Univ., Stanford,CA.

Y.ROUCHALEAU, B. F. WYMANANDR. E. KALMAN(1972), Algebraicstructureoflineardynamical

systems,III.Realization theoryover a commutative ring,Proc. Nat.Acad.Sci.,69, pp. 3404-3406. P. SAMUEL (1963), Anneaux factoriels, Redaction de ArtibanoMicali,SociedadedeMatemiticadeSo

Paulo.

E. D.SONTAG (1976),Linear systems over commutative rings: a survey, Ricerche diAutomatica,7, pp. 1-34. (1981),Linear systems over commutativerings:a(partial)updated survey,Proc.8th WorldCongress