vol1_pp44-58. 160KB Jun 04 2011 12:06:07 AM

(1)

The Electronic Journal of Linear Algebra.

A publication of the International Linear Algebra Society. Volume 1, pp. 44-58, November 1996.

ISSN 1081-3810. http://gauss.technion.ac.il/iic/ela

ELA

PERIODIC COPRIME MATRIXFRACTION

DECOMPOSITIONS RAFAEL BRUyx, CARMEN COLLyx

AND JOSEP GELONCH zx

Abstract. A study is presented of right (left) coprime decompositions of a collection of N-periodic rational matrices, with some ordered structure. From a block-ordered right coprime decomposition of a rational matrix of the given periodic collection, the corresponding block-ordered right coprime decompositions of the remaining matrices of the collection are constructed. In addition, those decompositions areN-periodic.

Keywords. Periodic rational matrices, coprime decompositions, Smith canonical form

AMS(MOS) subjectclassication. 15A23, 93C50

1. Intro duction. It is well-known that in control theory there are

dif-ferent approaches for studying linear multivariable systems. One of the most important advantages of the fraction matrix approach is that it permits the use of polynomial matrices which form the decomposition of the transfer ma-trix. Those polynomial matrices have all information about the system. In the study of rational matrices the right and left coprime polynomial fraction decompositions play an important role in control theory, because they are di-rectly related to the concept of controllability and observability; see [4] and [6]. Further, those kind of decompositions apply to a lot of problems of mul-tivariable systems as the minimal realization problem, the regulator problem with internal stability, the output feedback compensator problem, etc. Dier-ent studies of fraction decompositions with applications in control invariant systems are [3], [5], [7]. and [8]

A discrete-time linear periodic system in thez-domain can be dened by

a periodic collection of rational matrices of the following form (see [2]),

fHs(z);s2Zg; Hs +N(

z) =Hs(z)2RpN mN

[z];

(1) such that

Hs +1(

z) =SpN;p(z)Hs(z)S ,1

mN;m(z); s2Z;

(2)

where SpN;p(z) and SmN;m(z) are given in (3), and one of the matrix of the

collection, say H 0(

z), is proper with lower block triangular polynomial part.

It seems that the knowledge of coprime decompositions of the collection (1){(2) can help to solve problems of discrete-time linear periodic systems

Received by the editors on 2 June 1996. Final manuscript accepted on 6 November 1996. Handling editor: Daniel Hershkowitz.

yDept. Matematica Aplicada, Univ. Politecnica de Valencia, 46071 Valencia, Spain zDept. Matematica, Univ. de Lleida, 25198 Lleida, Spain

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ELA

Periodic Coprime Matrix Fraction Decompositions 45

as those mentioned above in the invariant case. For instance, one can use those decompositions to obtain, with an output feedback, a stable closed-loop periodic system of minimal dimension. The main aim of this work is to obtain a right (left) coprime decomposition of a collection of periodic rational matrices and study the relationship among them. This paper is structured as follows: In Section 2 we state our denitions and give some notation to simplify the paper. Then, in Section 3 we study the block-ordered right coprime decomposition of a rational matrix. Next, in Section 4, we study the Smith canonical form of the natural decomposition of Hs

+1, from a given decomposition of

Hs. That

canonical form is basic in the last section, where we shall construct a block-ordered right coprime decomposition ofHs

+1 from one of Hs.

2. Preliminaries and notation.

Given two nonnegative integers ,

with , we dene the polynomial matrices:

S;(z) =

O I

,

zI O

:

(3)

Some properties of those matrices can be found in [1]. Here, we explicitly state the following properties which are complementaries to those given in [1].

Lemma 2.1. The matrices given in (3) satisfy the following properties: (i) S;

1(

z)S; 2(

z) =

S; 1

+ 2(

z) if

1+

2

zS; 1

+ 2

,(

z) if 1+

2

>

(ii) det S;(z) = (,1) (,)

z

(iii) S ,1

;(z) =z ,1

S; ,(

z):

In the rest of the paper we shall denote, in general, rational matrices byHs,

because most of the results will be applied in the context of rational matrices dened in (1) satisfying property (2). We recall that a pair of polynomial matrices

Ds(z);Ns(z)

, with Ds(z) 2RmN mN[

z] and Ns(z) 2RpN mN[

z],

is a right decomposition of Hs(z) if

Hs(z) =Ns(z)D ,1

s (z) :

(4)

In what follows we shall work with right coprime decompositions. A pair of polynomial matrices

D(z);N(z)

is said to be right coprime if the Smith canonical form of the matrix

F

D(z);N(z)

= D(z) N(z)

2R

(m+p)NmN

[z]

is the matrix

ImN

O

:

(3)

ELA

46 Rafael Bru, Carmen Coll, and Josep Gelonch

From the right decomposition (4) of

H

s(

z

) and the relation (2) we have

H

s+1(

z

) =

S

pN;p(

z

)

H

s(

z

)

S

,1

mN;m(

z

) =

S

pN;p(

z

)

N

s(

z

)

D

,1

s (

z

)

S

,1

mN;m(

z

) =

S

pN;p(

z

)

N

s(

z

)

S

mN;m(

z

)

D

s(

z

)

,1

;

(6)

which is a right decomposition of

H

s+1(

z

). In general, the right

decom-position (6) is not right coprime even in the case the right decomdecom-position

D

s(

z

)

;N

s(

z

)

is right coprime, as the following example shows.

Example 2.2. Let us consider the matrices

H

0(

z

) = 2

6 6 6 6 4

1

z

0

1

z

1

,1 3

7 7 7 7 5

and

H

1(

z

) = 2

6 6 6 6 4

1

z

,1

1

z

0

z

1

3

7 7 7 7 5

:

The periodic set f

H

0(

z

)

;H

1(

z

)

;H

s+2(

z

) =

H

s(

z

)

;s

2 Zg is a periodic

collec-tion of racollec-tional matrices with period

N

= 2 and, according to the notation given in (1),

p

= 1 and

m

= 1. The pair of matrices

D

0(

z

)

;N

0(

z

)

=

z

2

,

z

2

,

z z

,1

;

z

,1 1

z

2

z

+ 1

is a right coprime decomposition of

H

0(

z

). Then, compute the matrices

S

2;1(

z

)

D

0(

z

) =

0 1

z

0

z

2

,

z

2

,

z z

,1

=

z

2

,

z z

,1

z

3 ,

z

2

z

2

;

S

2;1(

z

)

N

0(

z

) =

0 1

z

0

z

,1 1

z

2

z

+ 1

=

z

2

z

+ 1

z

2

,

z z

:

The pair

S

2;1(

z

)

D

0(

z

)

;S

2;1(

z

)

N

0(

z

)

(7)

is a right decomposition of

H

1(

z

). Since the Smith canonical form of the

matrix F

S

2;1(

z

)

D

0(

z

)

;S

2;1(

z

)

N

0(

z

)

is

2

6 6 4

1 0 0

z

0 0 0 0

3

7 7 5

;

(4)

ELA

For simplicity, we will omit the variable

z

in rational and polynomial matrices, however, when

z

= 0, we will always make it explicit.

Let us consider the following row-block partition of the matrices

D

s and

N

s,

D

s=

2

6 6 6 6 6 6 6 4

e

D

1 e

D

2

...

e

D

N

3

7 7 7 7 7 7 7 5

; N

s=

2

6 6 6 6 6 6 6 4

e

N

1 e

N

2

...

e

N

3

7 7 7 7 7 7 7 5

;

with

D

e

i 2 Rm

mN[

z

] and e

N

i 2 Rp

mN[

z

], for

i

= 1

;

2

;:::;N

. We construct

the block-polynomial matrices

Fk(

D

s

;N

s) = 2

6 6 6 6 6 6 6 6 6 6 6 6 6 4

e

D

k ...

e

D

N

e

N

k ...

e

N

3

7 7 7 7 7 7 7 7 7 7 7 7 7 5

2R

(m+p)(N,k+1)mN

[

z

]

;

for

k

= 1

;

2

;:::;N

. Note thatF dened in the rst section is justF

1. Denote

r

sk = rankFk

D

s(0)

;N

s(0)

; k

= 1

;

2

;:::;N:

(8)

Definition 2.3. Let(

D

s

;N

s) denote a right decomposition of a rational

matrix. We shall call the sequence (

r

s1

;r

s2

;:::;r

sN) the characteristic rank

of the pair (

D

s

;N

s). Clearly,

mN

r

s 1

r

s 2

r

sN 0

:

When denition 2.3 applies to a coprime decomposition then the rst inequal-ity of the above expression becomes equalinequal-ity, i.e.,

mN

=

r

s1.

Definition 2.4. A right decomposition of

H

s, (

D

s

;N

s), with

charac-teristic rank (

r

s1

;r

s2

;:::;r

sN) is said to be block-ordered if for each matrix Fk(

D

s

;N

s),

k

= 1

;

2

;:::;N

, there exists a partition in two blocks,

Fk(

D

s

;N

s) = [

G

k

G

k]

; G

k 2R

(5)

ELA

such that Gk(0) =O :

The next result gives a practical characterization of block-ordered decom-positions. This characterization will be useful throughout the paper.

Lemma 2.5. Let (Ds;Ns) be a right coprime decomposition of Hs with

characteristic rank (rs 1

;rs 2

;:::;rsN). Then (Ds;Ns) is block-ordered if and

only if there exists a partition of the matrices Ds and Ns,

Ds= 2

6 6 6 6 6 6 6 6 6 6 4

D 11

D 12

D 13

::: D 1;N,1

D 1N zD

21

D 22

D 23

::: D 2;N,1

D 2N zD

31

zD 32

D 33

::: D 3;N,1

D 3N

... ... ... ... ...

zDN ,1;1

zDN ,1;2

zDN ,1;3

::: DN ,1;N,1

DN ,1;N zDN

1

zDN 2

zDN 3

::: zDN;N ,1

DNN 3

7 7 7 7 7 7 7 7 7 7 5 ;

Ns= 2

6 6 6 6 6 6 6 6 6 6 4

N 11

N 12

N 13

::: N 1;N,1

N 1N zN

21

N 22

N 23

::: N 2;N,1

N 2N zN

31

zN 32

N 33

::: N 3;N,1

N 3N

... ... ... ... ...

zNN ,1;1

zNN ,1;2

zNN ,1;3

::: NN ,1;N,1

NN ,1;N zNN

1

zNN 2

zNN 3

::: zNN;N ,1

NNN 3

7 7 7 7 7 7 7 7 7 7 5 ;

(9)

with Dii2Rm (r

si ,r

s;i+1 )[

z]; Nii2Rp (r

si ,r

s;i+1 )[

z] and

rank Dii(0) Nii(0)

=rsi,rs;i +1

;

(10)

for i= 1;2;:::;N, where rs;N

+1= 0 .

Proof. Assume that (Ds;Ns) is block-ordered. Consider a block partition

of matrices Ds and Ns with the same block sizes as partitions (9), where we

denote the blocks of the lower block triangular part by ^Dij and ^Nij; i > j.

We have to prove that each of those blocks are multiples of zand the equation

(10) holds.

Since (Ds;Ns) is block-ordered the matrix

FN(Ds;Ns) = "

^

DN 1

::: D^N;N ,1

DNN

^

NN 1

::: N^N;N ,1

NNN #

evaluated at z = 0 is FN(Ds(0);Ns(0)) = [O GN(0)]. Then, we can write

^

(6)

ELA

rank

DNN(0) NNN(0)

=rsN:Next, similarly,

FN ,1(

Ds(0);Ns(0))

=

2

6 6 4

^

DN ,1;1(0)

::: D^N

,1;N,2(0) DN

,1;N,1(0) DN

,1;N(0)

O ::: O O DNN(0)

^

NN ,1;1(0)

::: N^N

,1;N,2(0) NN

,1;N,1(0) NN

,1;N(0)

O ::: O O NNN(0)

3

7 7 5

= [O GN ,1(0)]

;

implying that ^DN ,1;j=

zDN

,1;j and ^ NN

,1;j = zNN

,1;j, for

j= 1;2;:::;N,

2:By the expression (8) and the characteristic rank of the decomposition, we

can conclude that rank DN

,1;N,1(0) NN

,1;N,1(0)

= rs;N ,1

,rsN: Reasoning in this

way we complete the proof. The converse is straightforward.

As we said in the introduction, the aim of this paper is to nd a block-ordered right coprime decomposition of matrix Hs

+1, from a block-ordered

right coprime decomposition of Hs;s2Z.

3. Block ordered right coprime decomposition of a rational

ma-trix.

In this section, we prove the existence of a block-ordered right coprime

decomposition of an arbitrary rational matrix Hs, which will be used in the

rest of the paper. Let A2Rp

q[

z] be a polynomial matrix, with pq. Let S be its Smith

canonical form

S =

Iq O

:

It is easily seen that there exists an unimodular polynomial matrix, V 2 Rp

p[

z], such that

VA=

Iq O

:

(11)

Theorem 3.1. Let Hs;s2Zbe a rational matrix. Then, there exists a

block-ordered right coprime decomposition of Hs.

Proof. Let Hs = NsD ,1

s be a right coprime decomposition, with the following sizes Ds 2 RmN

mN[

z] and Ns 2 RpN mN[

z]. We know that the

Smith canonical form of the matrix

F 1(

Ds;Ns) =

(7)

ELA

is given by (5). There exists, (see (11)), an unimodular polynomial matrix,V,

such that

VF 1(

Ds;Ns) =

ImN O

:

Let (rs 1

;rs 2

;:::;rsN) be the characteristic rank of the pair of matrices (Ds;Ns).

Then, there exists a nonsingular constant matrixW

1of adequate size such that F

2

Ds(0);Ns(0)

W 1= [

O G

2(0)]

where G 2

2R

(m+p)(N,1)r s2[

z]. Then,

DsW 1= 2 6 6 6 6 6 6 6 6 6 6 4 D 11 e D 12 zD 21 e D 22 zD 31 e D 32 ... ... zDN 1 e DN 2 3 7 7 7 7 7 7 7 7 7 7 5

; NsW 1 = 2 6 6 6 6 6 6 6 6 6 6 4 N 11 e N 12 zN 21 e N 22 zN 31 e N 32 ... ... zNN 1 e NN 2 3 7 7 7 7 7 7 7 7 7 7 5 with D 11

2Rm

(mN,r s2

)[

z] and N 11

2Rp

(mN,r s2

)[ z].

Obviously, the characteristic rank of the pair (DsW 1

;NsW

1) coincides with

that of (Ds;Ns). Then, there exists a nonsingular constant matrix,

W 2 = Ir s2 O O W 2

of adequate size such that

F 3

Ds(0)W 1

;Ns(0)W 1

W

2 = [

O G

3(0)]

with G 3

2R

(m+p)(N,2)r s3[

z]:Therefore,

DsW 1 W 2 = 2 6 6 6 6 6 6 6 6 6 6 4 D 11 D 12 e D 13 zD 21 D 22 e D 23 zD 31 zD 32 e D 33 ... ... ... zDN 1 zDN 2 e DN 3 3 7 7 7 7 7 7 7 7 7 7 5

; NsW 1 W 2 = 2 6 6 6 6 6 6 6 6 6 6 4 N 11 N 12 e N 13 zN 21 N 22 e N 23 zN 31 zN 32 e N 33 ... ... ... zNN 1 zNN 2 e NN 3 3 7 7 7 7 7 7 7 7 7 7 5 with D 22

2 Rm (r

s2 ,r

s3 )[

z] and N 22

2 Rp (r

s2 ,r

s3 )[

z]. The process can be

repeated until we have constructed a pair of matrices (Ds;Ns),

(8)

ELA

which admits a partition of the form (9) with the conditions (10), where

W

=

W

1

W

2

W

N

,1 .

The matrices

D

s and

N

s form a right coprime decomposition for

H

s, since

N

s(

D

s),1 = (

N

s

W

)(

D

s

W

),1= (

N

s

W

)(

W

,1

D

,1

s ) =

N

s

D

,1

s

;

and

V

D

s

N

s

W

,1 =

V

D

s

W

,1

N

s

W

,1

=

V

D

s

N

s

=

I

mN

O

:

Clearly, the pair (

D

s

;N

s) is a block-ordered right coprime decomposition of

H

s.

4. The Smith canonical form of the matrix

F(

S

mN;m

D

s

;S

pN;p

N

s)

.

Let (

D

s

;N

s) be a block-ordered right coprime decomposition of a rational matrix

H

s

;s

2 Z, and let (

r

s

1

;r

s2

;:::;r

sN) be its characteristic rank. For

simplicity, we rewrite the partition (9) in a more compact form as

D

s=

D

11

D

12

zD

21

D

22

; N

s=

N

11

N

12

zN

21

N

22

;

(12) where

D

11=

D

11 2Rm

(mN,r s2

)[

z

]

; D

12

2Rm r

s2[

z

]

;

D

21 2Rm

(N,1)(mN,r s2

)[

z

]

; D

22

2Rm

(N,1)r s2[

z

]

;

N

11=

N

11 2Rp

(mN,r s2

)[

z

]

; N

12

2Rp r

s2[

z

]

;

N

21 2Rp

(N,1)(mN,r s2

)

[

z

]

;

N

22 2Rp

(N,1)r s2[

z

]

;

We shall construct a polynomial matrix

P

, with a precise structure, such that

P

F(

D

s

;N

s) =

I

mN

O

. To that end, we dene the matrices

F = 2

6 6 4

D

11

D

12

N

11

N

12

zD

21

D

22

zN

21

N

22 3

7 7

5 and

C

=

2

6 6 4

I

mN,r

s2

O

I

rs2

O

3

7 7 5

:

From (10), the matrices

D

11(0)

N

11(0)

and

D

22(0)

N

22(0)

are full column rank. Then, there exists a nonsingular constant matrix

V

of order (

m

+

p

)

N

such that

V

F(0) =

C

, that is,

V

F =

C

+

zA

.

The structure ofF(0) allows us to choose the matrix

V

so that its lower

left block, of size (

m

+

p

)(

N

,1)(

m

+

p

), is null; that is,

V

=

O

(9)

ELA

The Smith canonical form of the matrix VF is still (5). Therefore, by

equation (11), with an adequate row permutation, there exists an unimodular polynomial matrix X such that

XVF =C ;

(13)

that is, X(C+zA) = C. The matrix X satisfying this equality has a very

special structure. It admits a partition such that

X=

2

6 6 4

ImN ,r

s2+ zX

11 X

12

zX 13

X 14 zX

21

X 22

zX 23

X 24 zX

31

X 32

Ir s2+

zX 33

X 34 zX

41

X 42

zX 43

X 44

3

7 7 5 :

Since this matrix is unimodular, the matrix

X(0) = 2

6 6 4

ImN ,r

s2 X

12(0)

O X

14(0)

O X

22(0)

O X

24(0)

O X

32(0) Ir

s2 X

34(0)

O X

42(0)

O X

44(0) 3

7 7 5

is a nonsingular constant matrix and the submatrix

X

22(0) X

24(0) X

42(0) X

44(0)

is also nonsingular. Then, we can nd a nonsingular constant matrix,

Y = 2

6 6 4

ImN ,r

s2

O O O

O Y

22

O Y

24

O Y

32 Ir

s2 Y

34

O Y

42

O Y

44 3

7 7 5 ;

such that

YX = 2

6 6 4

ImN ,r

s2+ zX

11 X

12

zX 13

X 14 zX

0 21

X 0 22

zX 0 23

X 0 24 zX

0 31

zX 0 32

Ir s2+

zX 0 33

X 0 34 zX

0 41

zX 0 42

zX 0 43

X 0 44

3

7 7 5 :

Moreover, it is easy to see that

YC=C :

(14)

Let us dene the matrixP =YXV. Equations (13) and (14) imply that PF =C. By construction, the matrixP has the structure,

P =

z

(10)

ELA

where the lower left block has size (

m

+

p

)(

N

,1)(

m

+

p

). We can construct

a partition of the matrix

P

, in blocks with adequate sizes, such that

P

F = 2

6 6 4

P

11

P

12

P

13

P

14

P

21

P

22

P

23

P

24

zP

31

zP

32

P

33

P

34

zP

41

zP

42

P

43

P

44 3

7 7 5

2

6 6 4

D

11

D

12

N

11

N

12

zD

21

D

22

zN

21

N

22 3

7 7 5=

C:

Hence, the matrix

P

=

2

6 6 4

P

11

P

13

P

12

P

14

zP

31

P

33

zP

32

P

34

P

21

P

23

P

22

P

24

zP

41

P

43

zP

42

P

44 3

7 7 5

satises

P

F(

D

s

;N

s) =

I

mN

O

:

This whole discussion can be summarized in the following resulr.

Proposition 4.1. Let (

D

s

;N

s) be a block-ordered right coprime

decom-position of

H

s with characteristic rank (

r

s1

;r

s2

;:::;r

sN). There exists an

unimodular polynomial matrix

V

, such that

V

F(

D

s

;N

s) =

I

mN

O

with the following structure

V

=

2

6 6 4

V

11

V

12

V

13

V

14

zV

21

V

22

zV

23

V

24

V

31

V

32

V

33

V

34

zV

41

V

42

zV

43

V

44 3

7 7 5

;

where

V

21 2Rr

s2

m[

z

]

; V

23

2Rr s2

p[

z

],

V

41

2R

[(m+p)(N,1),r s2

]m[

z

]

; V

43

2 R

[(m+p)(N,1),r s2

]p[

z

]

:

Next, dene the matrix

V

0

=

S

mN;rs2

O

S

pN;(p+m)(N,1),r s2

,1

V

S

mN;m(N,1)

O

S

pN;p(N,1)

:

(15)

It is easy to check that the matrix

V

0is unimodular.

(11)

ELA

Theorem 4.2. Let(

D

s

;N

s) be a block-ordered right coprime decomposi-tion of

H

s. Then, the Smith canonical form of the matrixF(

S

mN;m

D

s

;S

pN;p

N

s)

is

2

4

I

rs2

O

O zI

mN,r s2

O

3

5

:

Proof. Let

V

0 be the matrix dened in (15). Then,

V

0

F(

S

mN;m

D

s

;S

pN;p

N

s) =

V

0

S

mN;m

O

S

pN;p

D

s

N

s

=

z

,1

S

mN;mN,r

s2

O

S

pN;m(1,N)+p+r s2

V z

D

s

N

s

=

S

mN;mN,r s2

O

(16)

Multiplying this equation on the left by (the permutation matrix)

S

mN;rs2(1),

we have

V

0

F(

S

mN;m

D

s

;S

pN;p

N

s)

S

mN;r s2(1) =

2

6 6 4

I

rs2

O

O zI

mN,r s2

O

3

7 7 5

:

5. A block-ordered right coprime decomposition of the matrix

H

s+1

.

Consider the periodic collection of rational matrices

f

H

s

; s

2Zggiven

in (1) satisfying (2). Let (

D

s

;N

s) be a block-ordered coprime decomposition of

H

s dened in (9). Then, we dene the matrices

D

s+1=

S

mN;m

D

s

S

,1

mN;mN,r

s2 and

N

s

+1=

S

pN;p

N

s

S

,1

mN;mN,r s2

:

(17)

Note that the block structures of

D

s+1 and

N

s+1 are parallel to those of the

D

s and

N

s, but now the rst row block and the rst column block are the second row block and second column block of (9). In general, for 1

j

N

,

D

s+j =

S

mN;jm

D

s

S

,1

mN;mN,r

s;j +1 and

N

s

+j =

S

pN;jp

N

s

S

,1

mN;mN,r s;j +1

:

(18)

This last expression has been obtained using Lemma 2.1 and Lemma 5.1. In this section we shall prove that the pair (

D

s+1

;N

s+1) written in (17)

(more in general, (

D

s+j

;N

s+j)

; j

1, as dened in (18)) is a block-ordered

right coprime decomposition of

H

s+1, (

H

s+j). First, we prove the following

lemma.

Lemma 5.1. Let (

r

s+1;1

;r

s+1;2

;:::;r

s+1;N) be the characteristic rank of

the right coprime pair (

D

s+1

;N

s+1), where the matrices

D

s+1 and

N

s+1 are

given in (17). Then,

r

s+1;k =

mN

+

r

s;k+1 ,

r

s

(12)

ELA

Proof. From the denition of

r

s+1;k (see (8) and (10)), we have

r

s+1;k = rank Fk

D

s+1(0)

;N

s+1(0)

= rank

D

k+1;k+1(0)

N

k+1;k+1(0)

++ rank

D

NN(0)

N

NN(0)

+ rank

D

11(0)

N

11(0)

=

r

s;k+1 ,

r

s;k

+2+

r

s;k+2 ,

r

s;k

+3+

+

r

sN +

r

s 1

,

r

s 2

=

r

s;k+1+

mN

,

r

s

2

:

The result of Lemma 5.1 can be generalized to the right coprime pair (

D

s+j

;N

s+j), 1

j

N

. Indeed, in this case one obtains the relations

r

s+j;k =

mN

+

r

s;k+j ,

r

s;

1+j, with

r

s;k+j =

r

s;k+j,N

,

mN

if

k

+

j > N

. Theorem 5.2. Consider the periodic collection of rational matrices de-ned in (1) satisfying (2), and consider the block-ordered right coprime decom-position (12) of

H

s for some

s

2Z. Then the pair of matrices (

D

s

+1

;N

s+1),

dened in (17), is a block-ordered right coprime decomposition of

H

s+1.

Proof. First, we shall see that the pair of matrices (

D

s+1

;N

s+1) is a right

decomposition of

H

s+1. By equation (6), we have

H

s+1 = (

S

pN;p

N

s)(

S

mN;m

D

s) ,1

= (

N

s+1

S

mN;mN,r s2)(

D

s

+1

S

mN;mN,r s2)

,1

=

N

s+1

D

,1

s+1

;

then, the pair of matrices (

D

s+1

;N

s+1) is a right decomposition of

H

s+1. On

the other hand, by Theorem 4.2, more precisely by the expression (16), we have

V

0 F(

D

s

+1

;N

s+1) =

V

0

F(

S

mN;m

D

s

;S

pN;p

N

s)

S

,1

mN;mN,r s2 =

I

mN

O

;

that is, the decomposition (

D

s+1

;N

s+1) is right coprime (recall that the matrix

V

0is unimodular). Moreover,

rank

D

ii(0)

N

ii(0)

=

r

s+1;i,1 ,

r

s

+1;i

;

(19)

by Lemma 5.1. From the equalities (17) and (19), the pair of matrices (

D

s+1

;N

s+1)

is block-ordered (see Lemma 2.5).

Hence, it is clear that (

D

s+1

;N

s+1) is a block-ordered right coprime

decom-position of

H

s+1 .

Notice that Theorem 5.2 applies to decompositions of

H

s+j

;

1

j

N

,

(13)

ELA

We present now some nal remarks.

1. All results obtained (with the corresponding denitions) for right

decom-positions can be directly translated to left decomdecom-positions.

2. The denition of the matricesD

s+1and N

s+1given in (17) provides a simple

algorithm for the construction of a block-ordered right coprime decomposition of the rational matrixH

s+1 from one of H

s.

3. Repeating this algorithm, we construct a block-ordered right coprime

de-composition of the rational matrices H

s+j dened in (18), for 1

j N.

In fact, we can say more than Remark 3, that is, the decompositions of

H s+j, (

D s+j

;N s+j)

; j2Z, areN-periodicas can be seen using the expressions

(18). Then, we have the next theorem.

Theorem 5.3. Consider the periodic collection of rational matrices de-ned in (1) satisfying (2). Let (D

s ;N

s) be a block-ordered right coprime

de-composition of the rational matrix H

s for some

s 2 Z. If we construct the

pairs of matrices (D s+j

;N s+j)

; j 2Z

+, using (18), the equalities

D s+N =

D s and

N s+N =

N s

are satised.

Finally, we illustrate the above results with the following example.

Example 5.4. The matrices

H 0=

2

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

1

,2,z+z

2 0 0

1

,2,z+z 2

,3,2z,z 2

,2z,z 2+

z 3

1

z

,

1

z

,5,3z ,2z,z

2+ z

3

,3,6z,2z 2

,z 3

,2z,z 2+

z 3

1 + 2z z

,1,z z

,5,9z,2z 2

,2z,z 2+

z 3

(,1 +z)z ,2,z+z

2 0 0

2

,2,z+z 2

,12,7z,10z 2

,2z,z 2+

z 3

4 +z+z 2

z

,4,z z

,20,11z,6z 2

,2z,z 2+

z 3

,9,3z,8z 2

,z 3

,2z,z 2+

z 3

3 +z 2

z

,3,z z

,15,8z,6z 2

,2z,z 2+

z 3

3

(14)

ELA

H 1= 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 2

,2,z+z 2

,1 +z ,2,z+z

2 0

,4,z z

,20,11z,6z 2

,2z,z 2+

z 3

,12,7z,10z 2

z 2(

,2,z+z 2)

4 +z+z 2

z 2

,3,z z

,15,8z,6z 2

,2z,z 2+

z 3

,9,3z,8z 2

,z 3

z 2(

,2,z+z 2)

3 +z 2

z 2

0 z

,2,z+z 2

1

,2,z+z

2 0

,1

,5,3z ,2,z+z

2

,3,2z,z 2

,2z,z 2+

z 3

1

z

,1,z

,5,9z,2z 2

,2,z+z 2

,3,6z,2z 2

,z 3

,2z,z 2+

z 3

1 + 2z z 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 ;

form a collection of periodic rational matricesH swith

N = 2, sinceS 6;3 H 0 S ,1 4;2= H 1 and S 6;3 H 1 S ,1 4;2= H

0. We have the block-ordered right coprime

decompo-sition H 0= N 0 D ,1

0 , where

N 0= 2 6 6 6 6 6 6 4

1 0 0 0

,1 ,1 0 0

1 ,z 1 0

,z 0 0 1

,2z ,4 z 1

,z ,4 ,1 +z 0 3 7 7 7 7 7 7 5 ; D 0 = 2 6 6 4

,2,z 0 0 1

3 1 1 0

,z 1 +z 1 ,1

z

2 0 0

,1 3 7 7 5 :

Further, the characteristic rank of the pair (D 0

;N

0) is (4

;3). Then, the

ma-trices N 1= S 6;3 N 0 S ,1 4;1= 2 6 6 6 6 6 6 4

0 0 0 1

,4 z 1 ,2

,4 ,1 +z 0 ,1

0 0 0 1

,z 0 0 ,1

,z 2

z 0 1

3 7 7 7 7 7 7 5 ; D 1= S 4;2 D 0 S ,1 4;1 = 2 6 6 4

1 +z 1 ,1 ,1

0 0 ,1 z

0 0 z ,2,z

z z 0 3

3

7 7 5 ;

are a block-ordered right coprime decomposition ofH

1, according with

Theo-rem 5.2. By Lemma 5.1 the characteristic rank of the pair (D 1

;N

(15)

ELA

In addition, itis straightforwardtoseethat

S 4;2

D 1

S ,1 4;3

=D 2

=D 0

and S 6;3

N 1

S ,1 4;3

=N 2

=N 0

:

REFERENCES

[1] C. Coll, R. Bru, and V. Hernandez. Generalized Bezoutian for a Periodic Collection of Rational Matrices. Kibernetica, 30(4):393{402, 1994.

[2] C. Coll, R. Bru, V. Hernandez, and E. Sanchez. Discrete-Time Periodic Realizations in the Frequency Domain.Linear Algebra and its Applications, 203-204:301{326, 1994. [3] E. Emre, H. M. Tai, and J. H. Seo. Transfer Matrices, Realizations and Control of Continuous-Time Linear Time-Varying Sistem via Polynomial Fractional Represen-tations. Linear Algebra and its Applications, 141:79{104, 1990.

[4] T. Kailath. Linear Systems. Prentice Hall, Englewood Clis, N.J., 1980.

[5] N. P. Karampenakis, A. C. Pugh, and I. Vardulakis. Equivalence Transformations of Rational Matrices and Applications. Int. J. of Control, vol. 59(4):1001{1020, 1994. [6] V. Kucera. Analysis and Design of Discrete Linear Control Systems. Prentice Hall,

London, 1991.

[7] V. Kucera and P. Zagalak. Fundamental theorem of state feedback for singular systems.

Automatica, pages 653{658, 1988.