7/24/2019 Minimal Solutions to Transfer Matrix Equations

1/3

NOTES

AND

CORRESPONDENCE

131

o( A d)

denotes spectrum of A,.

It is shown3

in [ I ]

(cf. [ I , theorem 11) that

if

Z is jointly controllable

if (I)

is

true), then there exists a P E

9

such that ( 3 ) s a controllable

if and only

if

for each sE k* - k ) , he system

Z, s E k* k ) ) s complete (cf.

[

1, sec. 31)

if

and only if the

of Z, s E k*

k ) )

s nonzero and

[

1, lemma 31, the requirement that the transfer matrices of the X,,

k* -

k )

all be nonzero is equivalent to the following condition.

For each iE k ,

G ,

possesses a path from node 0 to node i. (111)

(I), (11), and (111) are necessary and sufficient

( A p , B p )

tobe structurally controllable. These conditions can

be

First note that Remark 2 implies that (111)

is

equivalent to

1) of Theorem 1. Next note that conditions

(I)

and (11) can be

n as

f l

ths is obviously equivalent to condition

2)

of Theorem

1.

Thus, he

0

IV.

CONCLUDING

EMARKS

For the special class of linear parameterizations treated in [ 2 ] and 131,

elements of ( A p , Q p ) ,except for certain fixed 0s and Is, are taken to

this case A,, B , C B,, D,,

, (i E k ) are binary matrices (i.e., matrices of 0s and 1s . If, in addition,

A , are assumed to arise from definitional relations [e.g.,

hen A , will almost certainly be a nilpotent matrix. In this

2)

of Theorem 1 canbe replaced with the simpler

A 0 Bo >

n,

sEk* .

ck-s D k -r

of

this

REFERENCES

J. P.Corfmat and

A .

S. Morse, uDecentralized

control

of linearmultivariable

C. T. in, Structural controllability,

E E E

Trans. Auromat. Conrr. vol. AC-19, pp.

systems,

Auromoticu,

to

be

published.

R. W Shields and J.

B.

Pearson,Structural controllability ofmulti-input Linear

201-208, June

1974.

J.P.Corfmat. Decentralized control of linear multivariable systems Yale Univ.

systems, Dep. Elec. Eng., Rice Univ., Houston, T X, Tech. Rep.

7502,

May 1975.

BectonCenter, New Haven, C T , Tech. Rep. CT-67, Oct. 1974; also Yale Univ.,

Ph.D. dissertation, Dec.

1974.

based

on

theassumption

[Ci,D,] O,

E k .

Remark

1

insures that hisassumption

is

of these conditions s based in part on [ I , proposition31, which, in turn

Minimal

Solutions

to

Transfer Matrix

Equations

A.

S.

MORSE

Abstract-It

is shown

that the problem of

finding

a solution

Te),f

least McMillan

degree, to the transfer matrix equation

Ti@ = ,(A)T@)

is in essenceequivalent to theproblem of finding an ( A ,B)-invariant

subspace of least dimension which

eontains

a given subspaca

The principal problem to be considered here

is

as

follows.

For

given

p

X

m and p X r transfer matrices TI@ nd

T2 X),

espectively, find if

possible) an

r X

m matrix

T* A),

of least McMillan degree, which

is

a

solution to the linear equation

TI A)

=

T , ( W ( Q .

(1)

Various algorithms for computing

P A)

re

known [ I ]

[ 2 ] .The purpose

of this note

is

to show that he problem of finding such a

minimal

solution is, in essence, equivalent to the problem

of

finding an

( A ,

B)-

invariant subspace of least dimension, which contains a given subspace.

The latter problem has been effectively solved in [3].

Since the set of all ransfer functions together with addition and

multiplication is known [ 4 ] to be a principal ideal domain

G ,

standard

existence and uniqueness tests

[ 5 ]

or solutions

to

linear matrix equations

over a principal ideal domain

are

applicable to (1). We are primarily

interested in the case when (1) has more than one solution, which we

henceforth assume. The set of

all

solutions to (1)

can

then be written as

S= { Q l ( A )+

Q2(X)Q

X): all transfer matrices Q X)} ( 2 )

where e l @ ) s any solution to

(1)

and Q 2 @ )

is

any transfer matrix with

columns spanning the free 9 -submodule generated by the set of r X 1

transfer matrices gQ satisfying T,(A)g(A)=O. Algorithms for calculating

PI@)

nd e,@) using (for example)

Smiths

canonical

form

for

Q -

matrices, can be found in [ 5 ] .

A state space system Z ( A , B , , B , ] , C , [ D l , D 2 ] )ith transfer matrix

[

Ql X), Q2(X)]s said to

generate

S, provided ( 2 ) holds with PI@)nd

Q 2 @ )

substituted for

Ql(X)

and

Q2(X),

respectively. Note that any real-

ization of

[ Q l ( X ) ,Q ,@) ]

must generate S.

Lemma

I : If

E = ( A , [ B , , B , ] , C , [ D , , D J )

generates S, then

so does

X , = ( A + B 2 F , [ B l , B 2 ] , C + D 2 F , [ D l , D 2 ] )or any m a t r i x F.

P roo j If

[ Q l ( A ) ,Q 2@) ]

nd

[ Q 1 @ ) ,

Q2@)] are transfer matrices of X

and

Z

respectively, then by direct computation

Ql@)=el@)+

2@)

. ( F ( A I - A B , F ) - B , )

and g2 X)= , ( X ) ( I -

F ( A I - A ) - B 3 - I .

Clearly

Ql X)ES.

In addition, since ( I - F ( h l - A ) - B & is an invert-

ible

9

-matrix (i.e., the reciprocal of its determinant is a transfer func-

tion), the columns of Q2 A) span he same 9-modu le spanned by the

columns of Q,(X).

Remark I : From Lemma

1

it follows that

if

Z generates S, then for

any

F

and

G ,

the transfer matrix of the system Z F , G = ( A

+

B 2 F , ( B I+

B 2 G ) , ( C + D 2 F ) , ( D , D , G ) ), namely G I @ ) +Q22(x)G,is a solution to

A

system X = ( A , [ B I , B 2 ] , C , [ D l , D ~ith state pace X

is

called

(1).

maxima

obsercable if the only subspace Y c X which satisfies

is the eroubspace.ts ossibleoeduce nyystem Z

= ~,[,[B,,~,],~,[D,,D,])hich generates

S

to a maximally observable

system Z which also genera tp S.

To

accomplish this let

3

denote $e

state space

of

and write

7

for the unique) largest subspace of

Office of Scientif ic Research under Grapt 72-221

I .

sity, New Haven, C T 6520.

Manuscript received August

14,

1975. This

work

aas supported by the US . Air Force

The author is with the Department of Engineering and Applied Science, Yale Univer-

7/24/2019 Minimal Solutions to Transfer Matrix Equations

2/3

IEEE

TRANSACTIONS

ON AUTOMATIC CONTROL, EBRU RY

1976

32

satisfying

The existence and uniqueness of can be deduced directly from the

results of [ 6 ] , as

can

the validity of the following algorithm for com-

puting 7 Set n=dim(%) and

T=

Tn,here To= and

Having calculated T, define

F

so that

[ ; I+[ ;]F scQ

Since

Tcq ,

this implies that ( i f + p z ) T c and that Tckernel

(c+2F ) .

Thus,

if ?X e / T nd P : X - 2 is the canonical projec-

tion, then theequations '+ D ,F =

CP

and A P = P ( x + E2F) have

unique solutions C and A , respectively. Set [ B , , B , ] = P [ ~ , , E , ] .

Lemma :

The system

Z ~ ( A , [ B I , B , ] , C . [ D I , D 2 ] )s

maximal(y obseru-

able and generates

S.

Proof: Observe that X is theystem induced by E F = ( A +

~ , ~ , [ ~ , , ~ ~ ] , ~ + D , ~ , [ D , . D , ] )n the quotient space /'T; since T i s

the unobservable space

of

E,, both Er and X must have the same

transfer matrix. But by Lemma 1, Z F generates S.Therefore, Z: generates

S.

Let Y be any subspace satisfying (3) . Set ?=

- Y;

then P ?+ T3

= T. It follows that

-

and thus, that

This implies that

and thus, that 4) is satisfied with + t replacing T.^Since is the

largest subspace of satisfying 4), itmust be hat Y +

q=

; but

Q=

kernel

P

and

P

3-+

73

= Y , o

7-

0. Thus,

X

is maximally observ-

able. 0

Remark

: From the preceding proof it is clear that if X is maximally

observable, then

Z

( A

+

B,F,

[

B , ,B,].

C

+D,F, [D D,] ) is observable

for all F

For

the remainder of this note, X r ( A , [ B , , B ] , C , [ D , , D l )s a maxim-

ally observable system which generates

s

is the state space of X, nd

f is the (nonempty) class of subspaces of defined by

f = { Y : % , + A Y c % +? f ,

where and 3, enote span B and span E , , respectively. Note that f

is also the class of subspaces for which there exist

F

and

G

such that

( A + B F ) T ~ + s p a n ( B I + B G ) c T

cf.

[6]).

Let

S*

denote the set of minimal solutions to ( I ) and write

f

* for the

set of subspaces in 5 of least dimension. Our main result is as follows.

Proposition: Let

TF,G

denote the transfermatrix of

Z, , ,= A +

BF,8, B G , C + D F , D ,+

DG . or each

Q*(A)ES*,

there exist

F , G

and

Y f

* such that

Q* A) =

TF, and

( A + B F ) T + S p a n ( B , + B G ) C ' 7 - . 5 )

Conuerseb,

i

Y E

f *

is arbitrary, then

T,,, E S*

for each pair ( F ,

G)

satisfving (5) .

Remark:

It

is

easy to show that if

M is

the set of

all

pairs

( F , G )

for

which the dimension of the controllable space of X , , is as small as

possible, then

S*

=

( TF,, F ,

G)

M } .

Thus, the problem of computing

a minimal solution to (1) is equivalent to the problem of finding a pair

( F , G )

for which the dimension of the controllable space of X , , is as

small as possible.

The preceding proposition implies that the set of all minimal solutions

to 1) coincides with the set of

all

transfer matrices TF,,, where F,G)

s

any pair for which

5 )

holds for some 5 - E

*.

Since a procedure for

computingasubspace 'E

*

is known (cf., Remark 3) , a

minimal

solution to

(1)

can be found by first computinga Y E f * and then

constructing a n y pair ( F ,G) for which

5 )

holds; the resulting transfer

matrix TF,, will then have the required properties.

Remark 3: To

make explicit the connection between the problem

of

findmg

T-

*, and the specific problem treated in

[ 3 ] ,

let

&

be any

completion from B to

X ,

write P

+

for he projection on &

along 9, nd define e ~ ( % : P A % c P A ~ + + , P 6 , c 3 ) . t is

easy to check that if

Y E 9 ,

then

PTEe;

onversely

if

and H

is any map such that

( P A + P ABH) '%

c

3

hen I +

B H ) %

E

9.

hus,

if Y E

9 ,

then there is a subspace

P

T) in C with dimension not

exceeding dim

Y ;

conversely if is a subspace of least dimension in e,

then there is a subspace ( ( I + B H )

)

in

f

of dimension not exceeding

dim

;

rom this it follows that (I+

B H ) %

E

f *.

Thus, to compute

an

element of

f *,

it is enough to use the index and decomposition algo-

rithms of [ 3 ] o find a subspace ' of least dimension in e ; f H is then

selected so that

( P A

+ P B H ) % c , the subspace ( I+ BH) ZL will be in

9 *.

ConcludingRemark:

It would be interesting to characterizehe

spectrum of

A

+

BF

for

( F , G )

in the set for which 5 ) holds for some

Y E

9

. Such a characterization would be useful in determining when

(say) a stable minimal solution to 1) exists. Results along these lines

would also be applicable to the problem of constructing a stable ob-

server of least dimension, capable of estimating a linear function of state

of a linear system (cf. [ 3 ] ) .

Proof of Proposition: Write [Q, ,Q ] or the transfer matrix

of

X and

let Q*

ESf

be fixed. Since X generates

S,

there must exist a transfer

matrix such that

Q* Q,

+

QG.

hus, if (x ,R ,C,D) realizes with

state space

x,

nd

if

we define

A BC

O K

then the system X * = ( A * , B * , [ C , D C ] , [ D , + D C ] ) , ith state space *

=X e , ealizes Q*. f V * is the unobservable space of X*, then

clearly

This implies that

But since X is maximally observable, the largest subspace of *

satisfying

(6)

must

be

. Hence

V C K . (7)

Let

Q

* denote the controllable space of Z* and define P *+X

so

that as a matrix P= I,O].Let & be any subspace such that

& @ * n = * .

8)

Since

&

n

3

=

0,

it is possible to find a map

F*

such that

7/24/2019 Minimal Solutions to Transfer Matrix Equations

3/3

NOTES AND

CORRESPONDENCE 133

A, = A + BE+ and define 2 55

*+X * so

that

(9) implies that

AF,Pr= PA*r,

r E E .

(10)

a n d P A * Q * c P % * ; h e

P i &

P % * so AE

Since

* + = & +

and A = O , it follows that

*+ 3) 51

*

+3.But span B* c * +3; ence the controllable

of the system Z ~ ( A , B * , [ C + D P , O ] , [ D , + D D * ] )ust satisfy

c Q *+

5 .

herefore,

% + $ c % + Q * . 1 1)

if q.is the unobservable space of Z, then clearly

R c4. (12)

Let T+ denote the transfer matrix of Xp,5 =(A

+

B P 2 B ,

+BD,

C +

DD).

learly T+ =

Q ,

the transfer matrix

of

X.

Thus, from

eory, the McMillan degree of P , written 6 P), ust equal

this and (12) imply that S(P) n * .

Let ?;E 5 be arbitrary and select F and G so that (5) holds. Since

the controllable space of (A+

B F , B ,

+ B G ) , is the smallest sub-

R satisfying (5), ,, c 7. It follows that 6(TF, , ) < S 51.,,)

d ( Y ) = n * ; thus, 6 ( T F , G ) 0. This together with the definitions

of

X* and

Z p , 5

show

systems have the same Markov parameters and hence the same

other words, Q*= TI which is the desired result. 0

REFERENCES

[I] M. K.

Sain,

A reemodular algorithm for

minimal

design of Linear multivariable

systems, in

Proc.

6th Triennial World Cong.

Inr.

Federcrion Automatic Conrrol, Aug.

121

S.

H. Wang and E. J. Davison, A minimization algorithm for the design of

Linear

1975.

multivariable system

IEEE

Tram. Auromr. Conlr., voL AG18, pp 22&225, June

1973.

W. M.

Wonham

and A. S.Morse,Feedback nvariants

of linear

multivariable

systems,Automarica, vol. 8, pp. 93-100, 1972.

Algebrak

System Theory,Udine, Italy, June 1975.

A. S. Morse, Sy st em invariants under feedback a n d cascade control,CISM S y q .

C . C.

MacDuffee,

The Theory of

Matrices. New York: Chelsea.

pp. 143-148, May 1971.

A.

S.

Morse, Output controllability and system synthesis,

SIAM

J.

Conrr.,

vol. 9,

Minimal Pol~momial rom the

Markov

Parameters of

a

System

QUANG C .THAM

Absrract-A procedure fordetermining the

minimal polynomial

of real-

ization

from the Markov parametersof a system is presented, The simple

method bases on introducing an inner product in the Hilbert space.

I. INTRODUCTION

This orrespondence concernsmainly thedetermination of the

minimal polynomial or lower bound of realization from theMarkov

parameters of a system. The idea follows closely that of Gupta and

Fairman [l]. The present method bases

on

introducing a simpler inner

product

in

the Hilbert space.

11.DEVELOPMENT

OF

THE METHOD

In this section, it is assumed that thedimension r of the minimal

polynomial is known. It is proved

[2]

hat if

r

is known to be the

dimension of the minimal polynomial orresponding to the given

Markov parameters

[

Yi),

i = 1,

;

. .

there always exists the set of real

constants

a ; ,

=O, 1,

2 ; - . , r - l

such that

Y,+,+,= C a i - I Y i + j , j = 0 , 1 , 2 , . . .

i =

1

and the minimal polynomial of the system can he expressed as

It is obvious at

t h i s

point that since

{

Yi},

i 1,

2

.

.

is given and

r

is

known, then the set [a i } = 0, 1, ;

.

r- can be obtained directly

through 1) as

long

as it is set up to be a problem of r equations and r

unknowns.

In

order to facilitate the calculation of a ; and to determine

the dimension

r

in the next section, it is desirable to have a simple inner

product defined for the purpose of this correspondence.

Definition: Inner product

of

and 5 is defined as

where tr is the trace operator and

(3 is

the transpose.

Obviously, theabove proposed inner product satisfies the required

axioms [3] of an innerproduct. Together with the act hat pace

spanned by finite dimensional real matrix is both Hilbert space and

closed subspace, the classical projection theorem [3] can be used to

Manuscript received August 14, 1975.

The author

is

with the Bendix Field Engineering

Corporation,

Columbia, MD 21045.