R E P R I N T DA " P R O C . 6 T H I N T , C O N F , ON A N A L Y S I S A N D O P T . OF S Y S T E M S /

NI ZZA/ 1 9 3 4 , L E C T . N O T E S IN C O N T R O L A N D INF, S C I . , V O L . 5 2 "

INFINITI:; ZERO MODULE AND INFINITE POLE MODULE

G. Conte

I s t . Mat. U n i v . Genova v i a L . B . A l b e r t i 4

16132 Genova - ITALY

A. Perdon

I s t . Mat. A p p i . Univ. Padova v i a Belzon.i 7

35100 Padova - ITALY

SUMMARY

I n t h i s paper we i n t r o d u c e t h e n o t i o n o f i n f i n i t e z e r o module Z (G) and i n f i n i t e oo

p o l e module P (G) a s s o c i a t e d w i t h a t r a n s f e r f u n c t i o n G ( z ) . We show t h a t Z (G) and co co

P (G) d e s c r i b e t h e z e r o / p o l e s t r u c t u r e a t i n f i n i t y o f G(z) and we i n v e s t i g a t e t h e i r

d y n a m i c a l and system t h e o r e t i c p r o p e r t i e s . F i n a l l y , we a p p l y these concepts t o the

s t u d y o f the i n v e r s e s o f G ( z ) .

INTP.ODUCTION

Let G(z) denote a r a t i o n a l t r a n s f e r f u n c t i o n m a t r i x of dimensions p x m . I n t h i s

paper we i n t r o d u c e two a b s t r a c t a l g e b r a i c òbjects, c a l l c d r e s p e c t i v e l y " i n f i n i t e z e r o

module" and " i n f i n i t e p o l e module" and denoted by Z (G) and P (G), which d e s c r i b e t h e

z e r o / p o l e s t r u c t u r e a t i n f i n i t y o f G ( z ) . More b r e c i s e l y , Z (G) and P (G) ar e f i n i t e l y CU PO

g e n e r a t e d t o r s i o n K|f z |-modules (and hence f i n i t e d i m e n s i o n a i K-vector spaces) whose

d e f i n i t i o n i n tlerms o f G(z) i s based on a dynamical c h a r a c t e r i z a t i o n o f zeros and po-

le s a t i n f i n i t y . Moreover, when S(z) •- d i a g {z z r ) i s the non trìvial p a r t

o f t h e S m i t h - M a c M i l l a n form a t i n f i n i t y o f G ( z ) , the f o l l o w i n g r e p r e s e n t a t i o n s h o l d :

2 (G) ~ * K|lz - 1 U/ z^iKlIz™1! and P (G) - • K|| a" 1/ z""Vi d z~111. co V^<0 Vj L>0

Both Z (tì) and P (G) can be d e s c r i b e d u s i n g s p e c i a l r e p r e s e n t a t i o n s o f G(z) : Au -* AY co oo -1 o f t h e form G(z) = T ( z ) V ( z ) , where T ( z ) and V(z) are matricès w i t h e n t r i e s from -1 , -1„ K|z [|, coprirne i n t h e a p p r o p r i a t e sense. I n f a e t , d c n o t i n g by \i U and u Y th e Kg z [|-

-1 m p modules o f s e r i e s i n z w i t h c o e f f i c i e n t s from U - K and Y = K r e s p e c t i v e l y , we

have t h a t Z (G) i s i s o m o r p h i c t o the t o r s i o n submodule o f f< Y/V(z)fì U and t h a t P (G) co OO 00 00

i s i s o m o r p h i c t o Q Y/T(z)Q Y. QO CO

When G(z) i s s t r i c t l y p r o p e r , the c o n n e c t i o n between Z (G) and the g e o m e t r i e d e f i n i -

t i o n o f zero s t r u c t u r e a t i n f i n i t y i s g i v e n by t h e e x i s t e n c e o f a K|| z I-isomorphism

f : S*/R* > Z (G). Here, assuming t h a t G(z) has the c a n o n i c a l r e a l i z a t i o n (X,A,B,C),

S* denotes, as u s u a i , t h e minimum c o n d i t i o n a l l y i n v a r i a n t subspace o f X c o n t a i n i n g

Im B and R* denotes t h e maximum r e a c h a b i l i t y subspace o f X c o n t a i ned i n Ker C. The

q u o t i e n t S*/R* i s endowed w i t h a n a t u r a i module s t r u c t u r e i nduced from those o f AU

and AY.

Since t h e zeros a t i n f i n i t y p l a y a fundamental r o l e i n many c o n t r o l problems

(such as : i n v e r s i o n o f l i n e a r systems, c a u s a i f a c t o r i z a t i o n , feedback e q u i v a l e n c e ) ,

the main m o t i v a t i o n t o i n t r o d u c e Z (G) and P (G) i s the i n t e r e s t i n h a v i n g a n a t u r a i CO co

a l g e b r a i c t o o l i n o r d e r t o handle such z e r o s .

Here we a p p l y t h e n o t i o n s o f i n f i n i t e z e r o module and i n f i n i t e p o l e module t o

the s t u d y o f t h e i n v e r s e s o f G ( z ) . We o b t a i n , f i r s t o f a l i , t h a t g i v e n a ( r i g h t o r

l e f t ) i n v e r s e H{•) o f G(z) t h e i n f i n i t e z e r o module Z (G) i s contaìned, i n an appro-CO o r i a t e sense, i n t h e i n f i n i t e p o l e module P (H) o f H ( z ) . The analogous r e s u l t , COn-

CO

c e r n i n g t h e ( f i n i t e ) z e r o module Z(G) and t h e ( f i n i t e ) p o l e module, i . e . t h e c a n o n i -

c a i s t a t e space, X(H) o f H(z) was proved by B.Wyman and M.Sain i n [ 8 ] ,

Then, t h e f a c t t h a t M a c M i l l a n degree H(z) = dim X(H) + dim P (H), enables us t o s t a t e

K K « the f o l l o w i n g r e s u l t : f o r any i n v e r s e H(z) o f G ( z ) , M a c M i l l a n degree H(Z) > dim Z(G)

K + dim Z (Gii Moreover, r e c a l l i n q t h a t g e n e r a l i z e d o r d . H(z) = dim X(H) + dim X (H) K oo K K « (see [ 2 | , [ 6 J ) , t h e e x i s t e n c e o f a c a n o n i c a ! p r o j e c t i o n <J> : X (H) -> P (H) and i t a

CO «XJ s p e c i a l p r o p e r t i e s a l l o w us t o s t a t e the f o l l o w i n g r e s u l t : f o r any i n v e r s e H(z) o f G( z ) , a e n e r a l i z e d o r d . H(z) > dim Z (G) + dim '/. (G) + (number o f c y c l i c submodules i n

K K °° d i r e c t sum d e c o m p o s i t i o n o f Z ( G ) ) .

co

I t i s p o s s i b l e t o - f i n d exampìes i n which t h e lower bounds d e s c r i b e d above cannot be

reached. Howevep:, f u r t h e r i n v e s t i g a t i o n s on t h i s l i n e seem a b l e t o g i v e r e s u l t s con-

c e r n i n g a c o n s t r u c t i v e c h a r a c t e r i z a t i o n o f the m i n i m a l i n v e r s e o f a g i v e n G ( z ) .

l.PRELIMINARIES AND NOTATIONS

ven a f i e l d K and a f i n i t e d i m e n s i o n a i K-vector spao- S AS denotes the s e t -1

o f ( f o r m a i ) L a u r e n t s e r i e s i n z w i t h c o e f f i c i e n t s i n S, i . e . s e r i e s o f the forni co — t

s = E s z , s e S and t è Z- The o r d e r o f s i s d e f i n e d as t h e ind e x o f t h e f i r s t non-t= k t t zero c o e f f i c i e n t s . A s e r i e s s w i l l be s a i d o r o o e r i f f i t s o r d e r i s 0 and s t r i c t l y t p r o p e r i f f i t s o r d e r i s g r e a t e r than 0. ^

- t

fiS denoted t h e p o l y n o m i a l subset o f AS, i . e . t h e s e t o f elements o f the form E s z ,

and fi S denotes the power s e r i e s subset o f AS, i . e . the s e t o f elements o f the form 00

^J S Z .. 0 t

AK i s a f i e l d and AS i s a AK-vector space; (e . e ..., e } denotes b o t h t h e ca-1 2 r

n o n i c a l b a s i s o f S over K and t h a t o f AS over AK. The f i e l d AK c o n t a i n s t h e r i n g

o f p o l y n o m i a l s K[ Z] , t h e r i n g o f power s e r i e s K 11 z Ij and the f i e l d o f f r a c t i o n s

K(z) . As a consequence A S i s , i n p a r t i c u l a r , a KÌ zi -module and, a l s o , a K (l z \ -module.

W i t h r e s p e c t t o these s t r u c t u r e s fiS t u r n s o u t t o be a k| z]-submodule and fi S t u r n s , OD

o u t t o be a K 1 z il -submodule. I n the f o l l o w i n g we w i l l c o n s i d e r t h e q u o t i e n t modules:

PS := A S/fiS Kl zi-module

I'*S :=AS/fi S and F S:= AS/z *Ù S K || z | -modules. CO CO do CO

The elements o f TS cari be u n i q u e l y r e p r e s e n t e d as s t r i c t l y p r o p e r s e r i e s , t h e p r o d u c t

by z b e i n g the u s u a i p r o d u c t by z i n As f o l l o w e d by t r u n c a t i o n o f the p o l y n o m i a l p a r t .

The elements o f V S ( f * S ) can be uni q u e l v r e o r e s e n t o d as p o l y n o m i a l s ( p o l y n o m i a l s co co

w i t h o u t Constant t e r m ) . The p r o d u c t by z i s the visual p r o d u c t i n AS f o l l o w e d by

t r u n c a t i o n o f t h e s t r i c t l y p r o p e r p a r t ( r e s p . : the p r o p e r p a r t ) .

L e t u and V .be K-vector spaces o f dimension m and p r e s p e c t i v e l y . I n the f o l l o w i n g

no d i s t i n c t i o n w i l l be made b e t w e e n A K - l i n e a r maps between AU and AY and pxm m a t r i c e s

a s s o c i a t e d t o them w i t h r e s p e c t t o the c a n o n i c a l b a s i s . -1

Any p xm m a t r i x w i t h e n t r i e s i n AK can be seen as a L a u r e n t s e r i e s i n z w i t h m a t r i x oo - t

c o e f f i c i e n t s , i . e . G(z) = l G z , G p x m m a t r i x w i t h e n t r i e s i n K, G(z) w i l l be s a i d k t t

p o l y n o m i a l , s t r i c t l y p r o p e r , p r o p e r i f f a l i i t s e n t r i e s are such and i t w i l l be s a i d

r a t i o n a l i f f a l i i t s e n t r i e s b e l o n g t o K ( z ) ,

By a t r a n s f e r f u n c t i o n we mean a A K-linear r a t i o n a l . map G(z) between AU and AY.

Any t r a n s f e r f u n c t i o n G(z) g i v e s r i s e t o the f o l l o w i n g commutative diagrams.- 1.1,

where a l i modules are K[ z)-modules and a l i maps are Kj zj-homomorphisms, and 1.2, where -1 -1 a l i modules are K|| z [1-modules and a l i maps are K|| Z ||-homomorphisms.

AU • AY j- | TI c a n o n i c a l p r o j e c t i o n

G , 0 0 - t 1.1 fiU TY p i ( pfcz ) = y }

i , i c a n o n i c a l i n c l u s i o n U x Y

B • c

X =Im Ga~ fiU/Ker G** i s a f.g. t o r s i o n K( Z] -module. De f i n i n g A: X -*• X as Ax = zx,

(X, A, B, C) i s a minimal r e a l i z a t i o n of the s t r i c t l y proper p a r t of G(z) (see(71)

G AU • AY f i ' , j ' c a n o n i c a l i n c l u s i o n i ' + ir' 1 G 1. 2 iì U p ' • F Y ir ' c a n o n i c a l p r o j e c t i o n

9 - t . j ' ! ^ B ' ^ ^ C- i p Q p Q ( l y t z ") = - y Q

Y B ' » C'

X - Im G - fi U/Ker G i s a f.g. t o r s i o n K|Z 1-module; d e f i n i n g A': X -*• X as c x > p o o * p - " o o c o

A'x = z x, (X ,A', B',C) i s a minimal g e n e r a l i z e d s t a t e space r e a l i z a t i o n of the oo

polynomial p a r t of G(z) ( s e e [ 2 ] ) .

L e t H(co) denote the Smith MacMillan form of G(l/to) and l e t p (co) = to J q (co) .with i i

q, (0) ^ 0 f o r i = 1 , . . , r=rank G, be the non-zero(diagonal) elements i n H(co) . Then,

the Smith-MacMillan form a t i n f i n i t y of G(z) i s the p x m matrix

'S(z) i

0 where S(z) = diag {z .., z V ) r ) v < ...< v

1— — r

The r-uple { v, ,.., v } i s c a l l e d s t r u c t u r e at i n f i n i t y of G(z) (see t s l 6.5). I r — """""" ~* 1 —

The Smith-MacMillan form a t i n f i n i t y can be obtained a l s o by the f o l l o w i n g procedure.

Write G(z) - z G(z) where s i s the minimum int e g e r such t h a t G(z) i s proper and l e t

M(z) be the Smith form of G(z) with r e s p e c t to the r i n g KIZ 1 1 As KIZ 1 I i s a l o c a i -1

r i n g whose maximal i d e a l i s generated by z , the non-zero (diagonal) elements of -1 s

M(z) can be assumed to be powers of z . Then z M(z) i s the Smith-MacMillan form a t s

i n f i n i t y of G(z) . I n p a r t i c u l a r G(z) * h^(z)z M(z)B (z) where B ^ z ) , BgU) are b i c a u s a l

matrices ( i . e . i n v e r t i t i l e i n the r i n g of proper m a t r i c e s ) .

G(z) i s s a i d to have a pole a t as of order -y for any negative v i h i t s s t r u c t u r e i 1

at <*>, and, analogously, i t i s s a i d to have a zero a t w of order w f o r any p o s i t i v e v i n i t s s t r u c t u r e a t °°. The t o t a l number of poles (zeros) a t » i s then E „ (-v.) j • v, <0 i ( Z n v.) . v, >0 ì

Supnose now t h a t v. < 0 f o r i = l , . . , k and that v > 0 for i = k + l , . . , r . Denotine i — ì

by S = diag{ z 1 , z k } and by S = diag {z z r } , a coprirne f a e t o n z i -

t i o n of the Smith-MacMillan forra a t °° of G(z) by proper m a t r i c e s i s given by

S(7) 1 0

0 ] 0

-1 -1 e (z) \p (z) - 4> (z) e (z) R L where

0 1 \ 1 0

0 S~ 1

0 1 0 i

, + I

s 0 * («) « I 0 I , 1 R I r-k 1

1 I 1 m-r

'S 0

di (a). «1 0 : I r-k

p-r/

Coprirne f a c t o r i z a t i o n s o f G(z) by proper matrices are then given by

G(z) = [B, ( z ) e ( z ) ] [ B* (Z)>|> ( Z ) ] _ 1

1 ^ Z R

G(z) = U ( z ) B ~ ( z ) ] ~ [ e(z)B •(*)] 2 -1 -1

I n the fo l l o w i n g , f o r any coprirne f a c t o r i z a t i o n G(z) = V ( Z ) T r (z) or G(z) = ( z ) V ( z )

by proper m a t r i c e s , V(z) w i l l be c a l l e d proper numerator of G(z) and T ( z ) , T (z) w i l l R L

be c a l l e d proper denominatorsof G ( z ) . I t can be proved as i n [5] t h a t E ( Z ) i s the

Smith form,with r e s p e c t to Kfz | , of every proper numerator of G(z) and t h a t ^ (z)

(resp. ( 2 ) ) has the sanie n o n t r i v i a l i n v a r i a n t f a c t o r s of any proper denominator of Li

G(z) .

2. INFINITE ZERO MODULE

The aim of t h i s s e c t i o n i s to define the modula of i n f i n i t e zeros of a t r a n s f e r

f u n c t i o n G ( z ) . I t s r e l a t i o n s with the classìcal notion of zeros a t i n f i n i t y and i t s

system t h e o r e t i c i n t e r p r e t a t i o n are i n v e s t i g a t e d .

DEFINITION 2.1 Given a t r a n s f e r function G(z) i t s i n f i n i t e zero module Z (G) i s d e f i -

ned by : •1 (fi Y) + n U Z (G) =

Ker G + fi U co

To motivate the d e f i n i t i o n given above, l e t us consider the case ra = p = 1. L e t

u(z) be an element i n AK and l e t k be i t s order, then, i f k < 0, u(z) i s s a i d to have

-k modes a t i n f i n i t y . C e r t a i n of these modes raay f a i l to appear i n the response of

the system y ( z ) = G ( z ) u ( z ) , i . e . ord y(2) =^k > k , and t h i s f a c t i s i n t e r p r e t e d as

thè presence of zeros a t i n f i n i t y i n G ( z ) . So i n d e f i n i n g the a b s t r a c t module we con­

s i d e r e x c i t a t i o n s which produce response having no modes a t i n f i n i t y , and we ignore

both proper inputs (which have no modes a t i n f i n i t y whose absence can be detected i n

the output) and Ker G ( s i n c e i d e n t i c a l l y zero outputs are of l i t t l e i n t e r e s t ) .

-1 PROPOSITION 2.2 Z (G) i s a f i n i t e l y generated t o r s i o n KÌz Smodule

• - •-— — — CO

..V

1 Proof. G(z) i s Kflz ! - l i n e a r and 0 Y i s a f i n i t e l y generated K| z 1-module, then

co G * (Q Y ) / k e r G and Z (G) , which can be viewed as a quotient of the previous one,

co co are f i n i t e l y generated KJ.Z J-modules. Everv element of Z (G) i s the equivalence c l a s s [ u] modulo Ker G + il U o f some

co co

u e AU such t h a t G ( z ) u i s proper. Let k be the degree of the polynomial p a r t of u; -k -k -k then G ( z ) ( z u) i s proper and z u i s proper. Therefore z [ u] = 0 and Z (G) i s

co

t o r s i o n .

PROPOSITION 2.3 Z (G) i s isomorphic to the t o r s i o n submodule of fi Y/ e ( z ) f i U, where • co co co

e(z) i s the Smith form of any proper numerator of G ( z ) .

Proof. We prove the P r o p o s i t i o n showing t h a t Z (G) i s isomorphic to the t o r s i o n

submodule of fi Y/V(z)fi U where G(z) = T ( z ) V ( z ) i s a coprirne proper f a c t o r i z a t i o n . co co

At t h i s aim, we r e p r e s e n t the elements of Z (G) as f u] , for some u e AU such t h a t co

G ( z ) u i s proper. I n p a r t i c u l a r , V ( z ) u i s proper, i . e . V ( z ) u <= fi Y. Moreover, i f CO

fu] • [ u'1 , i . e . (u - u 1) e (Ker G + fi U), we have, s i n c e Ker G(z) • Ker V ( z ) , 1 co

V(z) (u - u 1 ) B V(z)fi ììi hs a consequenee, we can define a K [ [ Z 1-homotnorphism co

f i Z (G) -* fi Y/V(z)fi U by f ( [ u ] ) * [ V ( z ) u ] ( c l a s s of V ( z ) u i n fi Y/V(z)fi U) . 00 OO CO 00 oo

As Z (G) i s t o r r i o n , f ( Z (G)) i s contained i n the t o r s i o n submodule of Q Y/V(z)fl U, eo co co 00

Suppose t h a t f ( [ u ] ) « [ V ( z ) u ] = Os then V ( z ) u e V(«5)flJU and u <3 Ker V + QJJ =

* Ker G + fi B. Therefore ( u] * 0 and f i s i n j e c t i v e . Co

L e t f y ] be a t o r s i o n element i n fi Y / v ( z ) f i U. By coprimnessof T(z) and V ( z ) , there 1 * J OO CO

e x i s t proper m a t r i c e s A(z) and B(z) such t h a t y - T ( z ) A ( z ) y + V ( z ) B ( z ) y . Then [y] m **k

* [ T(as) A ( z ) y ] and there e x i s t s a p o s i t i v e i n t e g e r k such t h a t z T ( z ) A ( z ) y « V(z) u k k f o r a s u i t a b l e u i n fi U. L e t v = z u t B ( z ) y e AO? then V ( z ) v = V ( z ) z u + V ( z ) B ( z ) y

do k -1 k = T ( z ) A ( z ) y + V ( z ) B ( z ) y " y. Now, z u belongs to G (fi Y) because G ( z ) z u = 00

= T _ 1 ( z ) V ( z ) z k u = T " 1 ( z ) V ( z ) ( v - B ( z ) y ) = T _ 1 ( z ) (y - V ( z ) B ( z ) y ) - T~ ( Z ) ( T ( z ) A ( z ) y )

= A ( z ) y e fi Y. Then [ y ] = [ V ( z ) v ] = f ( [ u ] ) , and f i s onto the t o r s i o n submodule of co

fi Y/V(z)fi U. OO 00

The b a s i c property of Z (G), i n connection with the notion of zero a t i n f i n i t y we r e -

c a l l e d i n the previous s e c t i o n , i s pointed out by the following C o r o l l a r y .

COROLLARE 2.4 The i n v a r i a n t fàctors of Z (G) over Kfl2 1 | c o i n c i d e with the non t r i -— — — — — — ^ ^ — co

v i a l eleraents of S .

REMARK 2.5 The above C o r o l l a r y says t h a t Z (G) contains a l i the information about the 00

zero s t r u c t u r e a t i n f i n i t y of G ( z ) . More p r e c i s e l y , i f {v , ,v^} i s the s t r u c t u r e

a t i n f i n i t y of G ( z ) , the i n f i n i t e zero module has the following c a n o n i c a l decomposi-

t i c n i n t o a d i r e c t sum of c y c l i c submodules : Z (G) = © K|Z * J / z V ; IK|z J . co v <0

i REMARK 2.6 The K|z *]]-module G ^ (fi Y) which appears i n the d e f i n i t i o n of Z (G) i s the

" co oo

l a t e n c y k e r n e l of G(z) introduced i n [ 4 ] . We w i l l i n v e s t i g a t e more deeply i t s r e l a -

t i o n s with the s t r u c t u r e at i n f i n i t y i n the following.

3. GEOMETRIC CHARACTERIZATION

I n t h i s s e c t i o n we assume t h a t G(z) i s a p xm s t r i c t l y proper t r a n s f e r function

provided with the minimal r e a l i z a t i o n (X,A,B,C). Moreover, we assume, without l o s t of

g e n e r a l i t y , t h a t X = Im G** and, as a consequenee, t h a t C* i s the i n c l u s i o n (see 1.1). ,

I t i s known ([ 1 ] ) that the zero s t r u c t u r e a t i n f i n i t y of G(z) can be obtained

from the quotient S*/R*, where S* i s the minimum ( A , C ) - i n v a r i a n t subspace of X con-

t a i n i n g Im B, and R* i s the maximum c o n t r o l l a b i l i t y subspace of X contained i n Ker C.

I n the f o l l o w i n g wè w i l l c h a r a c t e r i z e S* and R* i n terms of the t r a n s f e r function G(z)

and then we w i l l / prove t h a t Z (G) i s Kj z ^ - i s o m o r p h i c to S*/R*. co

PROPOSITION 3.1 Define S = { s e AY, s i s s t r i c t l y proper and s = G ( z ) u f o r some

u e fiU } = z ^fi Y fi G(fiU) and R = { s e AY, there e x i s t u <= fiU and 'S s t r i c t l y proper co

such t h a t G(z).u = G ( z ) u * s } = G(z fi U) O G(fiU) . Then S = S* and R = R* «

Proof. We remark, f i r s t of a l i , that both S and R are contained i n Im G f l = X,

Since G(z) i s s t r i c t l y proper, we have Im B C S. To prove the (A,C)-invariance of

S we show t h a t A(S fi Ker C) C S. Any element of S, i n f a c t , i s of the form s =

= s ^ " 1 + = G(z)u, with u e fiU. As C(s) = p ^ i s ) = s ^ s e Ker C i f f s i = 0.

For such an element s, A(s) = zG(z)u = G ( z ) z u = ( s 2 z 1 + ) belongs c l e a r l y

to S.

The m'-.imality of S among the ( A , C ) - i n v a r i a n t subspaces containing Im B w i l l be

p r o v e d by c o n t r a d i c t i o n . Suppose t h a t V i s an ( A , C ) - i n v a r i a n t subspace of X con-

t a i n i n g Im B but not co n t a i n i n g S, i . e . G ( z ) u e V for every C o n s t a n t u and there

e x i s t polynomials u(z) such t h a t G ( z ) u ( z ) i s s t r i c t l y proper b u t G ( z ) u ( z ) does not

belong to V. L e t p( z ) be such a polynomial of minimum degree : deg p(z) > 1, as

G(z) u e V f o r every C o n s t a n t u. Therefore, we have p(z) = z q ( z ) + r , with r e U -1

and G ( z ) p ( z ) = zG(z)q(Z) + G ( z ) r . Now, z G ( z ) q ( z ) = G ( z ) p ( z ) - G ( z ) r = y Z + -2 -2 + y z + = z ( y ^ z + ) and, s i n c e deg q ( z ) < deg p(z) , G ( z ) q ( z ) = , -2

= y z + i s an element of V H Ker C. By the (A,C)-invariance of V,

AG(z)q(z) = z G ( z ) q ( z ) = v e V. Thus G ( z ) p ( z ) = v - G ( z ) r belongs to V a g a i n s t the

hypothesis.

R = R* i s proved i n [ 3 ] §'4.

REMARK 3.2 S*/R* has a n a t u r a i K[z ^-module s t r u c t u r e d e f i n i e d as fo l l o w s . L e t [ s] -1. ,

denote an element i n S*/R*, s = G ( z ) u ( z ) where u ( z ) = zu(z) + U q e fiU. Then z [ sj =

[ G ( z ) u ( z ) ] . D e f i n i t i o n i s c o n s i s t e n t , i n f a c t G ( z ) u ( z ) i s s t r i c t l y proper and hence

[ G ( z ) u ( z ) ] e S*/R», moreover i f [ s] • [ s 1 ] and s ! • G ( z ) v ( z ) , v(z) = z v ( z ) + V q , we

have G ( z ) u ( z ) - G ( z ) v ( z ) e R*, i . e . G ( z ) u ( z ) - G ( z ) v ( z ) • G(z)w(z) with w(z) s t r i c t l y -1 -1

proper. As a consequenee G ( z ) u ( z ) - G ( z ) v ( z ) = G ( z ) ( z w(z)) + G ( z ) ( z ( V q - u Q ) ) e

e R* and [ G ( z ) u ( z ) ] = [ G ( z ) v ( z ) ] .

PROPOSITION 3.3 Z (G) and S#/R* are isomorphic as K([Z ^-modules. oo

Proof. As G(z) i s s t r i c t l y p r o p e r , Z (G) = G (fi Y) / (Ker G + fi U). L e t [ s ] , s = oo co oo

• G ( z ) u ( z ) , be an element of S*/R*. Then s i s , i n p a r t i c u l a r , s t r i c t l y proper and

zu(z) e G _ 1 ( f i Y) . We define f : S»/R* * Z (G) as follows : f ({ s] ) m [ z u ( z ) ] ( c l a s s co co

of zu(z) i n Z ( G ) ) . D e f i n i t i o n i s c o n s i s t e n t , i n f a c t , i f [ s] = [ a ' \ s' = G ( z ) v ( z ) co

then G ( z ) ( u ( z ) - v ( z ) ) £ R*, i . e . G ( z ) ( u ( z ) - v ( z ) ) « G(z)w(z) with w(z) s t r i c t l y

proper. As a consequenee zu(z) - z v ( z ) = zw(z) + p ( z ) , with p(z) e Ker G, and

f z u ( z ) ] = [ z v ( z ) ] i n Z (G). f i s c l e a r l y K - l i n e a r and, moreover, f ( z [ s] ) -CO

- z _ 1 f ([ s] ) = 0. i n f a c t , i f s = G(z) (zu (z} + U q ) , f ( z [ s] ) = f ([ G (z) u(z) ] ) = -1 -1 ?

= [ z u ( z ) ] i n Z (G) . On the other hand, z f ([ s] ) = z [ z " u ( z ) + zu ] = co U

= [ z u ( z ) + u ] = ( z u ( z ) ] i n Z (G) s i n c e u G f i U. Hence f i s Kffz 1-lir.ear as i c v 0 CO 0 CO

S*/R* .and Z (G) a r e t o r s i o n .

To show t h a t f i s i n j e c t i v e , assume chat, f o r s = G(z ) u ( z ) , f ( [ s ] ) = 0. Then

zu(z) Ker G + fi U, i . e . zu(z) = v(z) + w ( z ) , v ( z ) e Ker G •? id w(z) proper. M u l t i -ÙO

-1 ' -1

p l y i n g by z and applying G(z) we have G(z)u(z) = G(z)(z w ( z ) ) , hence G(z)u(z) e

e R* and [ s] = 0 i n S* /R* . To show t h a t f i s s u r j e c t i v e , l e t us r e c a l i t h a t any element i n Z (G) i s the equi-

-1 valence c l a s s , modulo Ker G & Sì, U, of an element i n G (fi Y) . Then any element can

oo oo

be represented as [ z u ( z ) l , where u(z) i s a polynomial such t h a t G(z)u(z) i s strìctly

proper, and i t f o l l o w s t h a t [ z u ( z ) ] = f ( [ s ] ) w i t h s = G(z)u(z) e s* .

COROLLARY 3.4 The i n v a r i a n t f a c t o r s of S*/R* over K|z J describe the zero s t r u c t u r e

at i n f i n i t y o f G(z).

Proof. T r i v i a l by 3.3 and 2.4.

4. INFINITE POLE MODULE

To apply the no t i o n of i n f i n i t e zero rnodule t o the study of inverse t r a n s f e r

f u n c t i o n s , we need the dual notion of i n f i n i t e pole module. I t has been remarked i n

[ 8 ] t h a t the f i n i t e pole module of a r a t i o n a l G(z) in e s s e n t i a l l y the state space of-

a minimal r e a l i z a t i o n of the s t r i c t l y proper p a r t of 0 ( 2 ) . C l e a r l y , the generalized

state space of a minimal r e a l i z a t i o n of the polynomial p a r t of G(z) cannot be chosen

to represent the ' i n f i n i t e pole module we need since i t may contain a nondynamical

component (see [ 2 ] ) .

In the sarné way as i n section 2, where we considered the d e f i n i t i o n of Z (G), the co

case in = p = 1 suggests t o us the f o l l o w i n g abstract d e f i n i t i o n :

DEFINITION 4.1 Given a t r a n s f e r f u n c t i o n G(z) i t s i n f i n i t e pole module P (G) i s d e f i -. — co

ned by : , J G{fi U) + fi Y

P (G) = fi Y

,1 PR0P0SITI0N 4.2 P (G) i s a f i n i t e l v generated t o r s i o n KtL z 11-module whose n o n t r i v i a l —-—,———. & „ - 1 , + i n v a r i a n t f a c t o r s over Kil z U coincide w i t h the n o n t r i v i a l elements of S .

Proof. G(fi U) and fi Y are f i n i t e l y generated K[1 z 1-modules, then P (G) i s f i n i t e l y

- Ao-

generated. Any element i n P (G) i s the equivalence cl a s s , modulo Q Y, of some y = co co

= G(z)u, w i t h u e 9. U. I f y i s proper, [ y) = 0 . I f y has a polynomial p a r t of de­ce -k , -k. , aree k, z y i s proper and z [ yj = 0; hence P (G) i s t o r s i o n i

oo

To prove the second p a r t of the p r o p o s i t i o n , we show t h a t P (G) i s isomorphic t o

the t o r s i o n Kjz S-modul'e fi Y/T(z)fi Y where G(z) = T *(z)V(z) i s a coprirne proper 00 00

f a c t o r i z a t i o n . Remark, f i r s t of a l i , t h a t f o r any u e fi U, T(z)G(z)u = V(z)u belongs oo

t o fi Y. Therefore, T(G(fi U) + fi Y) C fi Y and there e x i s t s h : P (G) •> fi Y/T(z)fi Y c o co CO OO OO 00 00

such t h a t the f o l l o w i n g diagram, where the upper v e r t i c a l maps are canonical i n c l u -sions and the lower ones are canonical p r o j e c t i o n s , cornrautes :

T o, Y • T(z) f i Y

CO co

T 4.3 G(fi U) + fi Y fi Y — ' CO oo oo

p + 4- q P (G) • • fi Y/T(z)fi Y

CO CO co

Assume t h a t h(y) = 0 , w i t h y = p(G(z)u) and u e fi U. Then qTG(z)u = hpG(z)u = 0 and OO

TG(z)u G T( z ) f i Y. Let TG(z)u = Tv, v e fi Y, then G(z)u = v e fi Y and y = pv = 0 i n co oo oo

P (G); hence h i s i n j e c t i v e . 00

Let v be an element i n fi Y/T(z)fi Y, i . e . v = qy, y 6 fi Y. By coprìmness of T(z) and CO OO CO

V(z) there e x i s t proper matrices A(z) and B(z) such t h a t y = T(z)A(z)y + V(z)B(z)y

and qy = q ( V ( z ) B ( z ) y ) . Take u * B(z)y i n fi U, then hpG(z)u * qTG(z)B(z)y = oc

= qV(z)B(z)y = qy = v and h i s onto.

REMARK 4.4 A consequence of 4.2 i s t h a t P (G) contains a l i the infor m a t i o n about the co

pole s t r u c t u r e a t i n f i n i t y of G(z). More p r e c i s e l y , i l {v , ,v } i s the s t r u c t u r e I r

at i n f i n i t y of G(z), thè i n f i n i t e pole module has the f o l l o w i n g canonical decomposition ' l ' I

i n t o a d i r e c t sum of c y c l i c submodules : P (G) = $ K|[ Z } / z -""KIZ ] . co V^>0

I n conclusion, the decomposition i n t o d i r e c t sums of c y c l i c submodules of Z (G) and 00

P (G) determines the non zero indices o f the s t r u c t u r e at i n f i n i t y of G(z). Moreover, 00

the s t r u c t u r e at i n f i n i t y contains a fiumber of zeros equal t o the d i f f e r e n c e (rank G - (number of c y c l i c submodules i n d i r e c t sum decompositions of Z (Q) and P (G)).

REMARK 4.5 I t i s easy t o see t h a t P (G) i s isomorphic t o the q u o t i e n t module — • — — co

<

fi u co

. This a l t e r n a t i v e representation p o i n t s out the r e l a t i o n between G (fi Y) H u

co co -1 . , P (G) and the latency kernel G (fi Y) (see l 4 J ) . This, together w i t h 2.6, gives an

i n s i g h t i n t o the connection between the concept of latency and the s t r u c t u r e at i n f i ­

n i t y . In p a r t i c u l a r , i t appears t h a t the latency kernel contains i n f o r m a t i o n on both

the i n f i n i t e zeros and the i n f i n i t e poles of G(z). However, as G (fi Y) i s not f i n i -co

t e l y generated unless G(z5 i s i n j e c t i v e , ([ 4 ] 6.16), Z (G) and P (G) are more handable CO 00

algebraic o b j e c t s .

In case G(z) i s i n j e c t i v e and s t r i c t l y proper, the latency indice:; {X , ,X } are 1 m

defined i n [ 4 ] i n the f o l l o w i n g way : l e t { d , ,d } be an ordered proper basis 1 m -1 of G (fi Y); then ord d < -1 and A = -ord d, - 1. Remarking t h a t the polynomial p a r t °° i i i

of any d. generates a c y c l i c submodule of order equa! t o - ord d, i n Z (G), we haye

t h a t the latency indices coincide w i t h the order of the i n f i n i t e zeros decreased by

1. As a consequence, G(z) i s non l a t e n t i f f a l i i t s i n f i n i t e zeros have order 1.

When G(z) i s proper, obviously G(fi U) C fi Y and P (G) = 0. Let now G(z) be a px m 00 co 00

t r a n s f e r f u n c t i o n of order k < 0. To c l a r i f y the r e l a t i o n between P (G) and X (G), the 00 co

generalized s t a t e space of the minimal r e a l i z a t i o n of the polynomial p a r t of G(z), l e t us consider the f o l l o w i n q diagram (see also 1.2) :

fi U

4.6 i d C'

Q U

P (G) 00

n Where TT* : V Y -> T* Y i s the projectìon TT* ( U Z f

r Y = AY/z * f i Y

r*.Y = AY/fi Y

+• u z + u ) = u z + 1 0 n + u^z and è i s the r e s t r i c t i o n of TT* (remark t h a t $ i s w e l l defined since TT*G = G * ) .

p p

PROPOfelTION 4.7 The morphism $ : X (G) •> P (G) i s s u r j e c t i v e . The c y c l i c submodules • a ? 00

of order k+1 of X ( G ) are mapped o r t o c y c l i c submodules of order k of P ( G ) .

Proof. The s u r j e c t i v i t y of <j> follows by the commuta t i v i ' t y of 4.6.

Let {x} be a c y c l i c submodule of order k+1 of X (G) , i . e . z 1 x f 0 i n T Y f o r i < 00 00 -k-1 -k-1 -k < k ahd z x = 0 i n T Y. I n other words, z x i s s t r i c t l y proper, z x i s pro-

co — i * — i — i - i per and z x has negative order f o r i < k. Then % $(x) ~ z ?r*x = v* (z x) 7* 0 f o r —k **k

i < k-1, Z 0 (x) = 7;* (z x) =0. Hence {<}> (x) } i s a c y c l i c submodule of P (G) of

order k.

- 4 2 -

REMARK 4.8 Let X (G) = © K|z 1/ z ^ K f z ] be the canonical decomposition of X (G) , 0 0 i °° i n t o a d\ j-ct sum of c y c l i c submodules. Then P (G) = ® K[z 1/ z ^ Kfz ] and the

i i n dices p -1 coincide w i t h the ind i c e s v of the pole s t r u c t u r e at. i n f i n i t y o f G(z). i i Moreover, denotino by G (z) the polynomial p a r t of G(z), we have by 4.7 and [ 2 ] :

p o i qeneralized ord. G (z) = dim X (G) = £ u > 2 (v +1) and (number o f indipendent p o i K 0 0 i i v i>0 i impulsive motions of G(z)) = dim P (G) = E v .

K 0 0 v i>0 i Hence, the d i f f e r e n c e between dim X (G) and dim P (G) i s equal t o the number of c y c l i c

K 0 0 K 0 0

submodules i n the d i r e c t sum decomposition of X^(G) or, e q u i v a l e n t l y , t o the number of

c y c l i c submodules i n the d i r e c t sum decomposition of P (G) plus the number of (non co ,

dynamical) c y c l i c submodules of order 1 of X (G). co

5. INVERSE TRANSFER FUNCTIONS

In t h i s section we i n v e s t i g a t e the connection between the i n f i n i t e zero module of

G(z) and the i n f i n i t e pole module of a ( r i g h t or l e f t ) inverse H(z) of G(z). I n the

case m = p = 1 any G(z) has a unique inverse 8(1) whose number of poles at i n f i n i t y i s

equal t o the number of zeros at i n f i n i t y of G(z). I n the m u l t i v a r i a b l e case, i t w i l l

be proved t h a t Z (G) i s a s o r t of lower bound, i n an module t h e o r e t i c sense, f o r P (H). 00 00

More p r e c i s e l y , we have the f o l l o w i n g two pr o p o s i t i o n s .

PROPOSITION 5.1 Let G(z) : AU -> AY be an i n j e c t i v e t r a n s f e r f u n c t i o n and l e t

H(z) : AY •* AU be a l e f t inverse of G(z), i . e . H(z)G(z) = 1 ^ . Then there e x i s t s an

i n j e c t i v e Klz 1-morphism j : Z (G) •> P (H) . OO OO

I

Proof. For any u € fi U such t h a t G(z)u = y belongs t o fi Y, we have H(z)G(z)u = CO CO

= u = H(z)y, hence G (fi Y) C H(fi Y) . This assure the existence of j : Z (G) ->- P (G) 00 CO 00 oo

such t h a t the f o l l o w i n g diagram commutes : i d

o • n u fi u • o 4- 4--1 i n c l 0 <• G (fi Y) + fi U • H (fi Y) + fi U

co co co co

4- « 4-Z (G) - • P (H)

CO CO

Moreover, j i s uniquely determined by the above property and i t . i s easìly seen,

using the snake lemma, t o be i n j e c t i v e .

PROPOSITION 5.2 Let G(z) : AU •* AY be a s u r j e c t i v e t r a n s f e r f u n c t i o n and l e t

H(z) : AY •+ AU be a r i g h t inverse of G(z), i . e . G(z)H(z) = 1 . Then there e x i s t s a AY

s u r j e c t i v e KIZ 1-morphism p : P (H) -*• Z (G) .

Proof. Let u = H(z)y be an element of H(fi^Y}, then G(z)u = G(z)H(z)y = y belongs t o

fi Y and H(fi Y) C G 1 ( f i Y). This assure the existence of p : P (H) •* Z (G) such t h a t

the f o l l o w i n g diagram commutes :

i n c l fi u — - — • — • Ker G + fi U co oo + , 4-i n c l -1 H(fi Y) + fi U G (fi Y) + fi U

P l 4- 4- P 2 p ( H) • Z (G)

CO 00

p i s uniquely determined by the above property.

Let x be an element-of Z (G) , x = p u w i t h G(z)u = y e fi Y. We have y = G(z)H(z)y co 2 0 0

and t h e r e f o r e G(z)(u - H(z)y) =0. This implies t h a t (u - H(z)y) belongs t o Ker G C

C Ker p 2 and t h a t pp^H(z)y - x = p^fHKzJy - u) - 0. As a consequence, x = pj>^H(z)y

and p i s s u r j e c t i v e .

Now, as MacMillan degree H(z) = dim X(H) + dim P (H) and generalized ord. H(z) = K K «>

= dim X(H) + dim X (H) (see [ 2 ] , [ 5 ] , [ 6 i ) , we have the f o l l o w i n g c o r o l l a r y : K K »

COROLLARY 5.3 Let H(z) be a ( r i g h t or l e f t ) inverse of the t r a n s f e r f u n c t i o n G(z).

Then MacMillan degree H(z) = dim X(H) + dim P (H) > dim Z(G) + dim Z (G) and K K » K K »

qeneralized ord. H(z) = dim X(H) 4- dim X (H) = dim X (H) + dim P (H) + (number of cy-3 K K c o K K » c l i c submodules i n d i r e c t sum decomposition of X ( H ) ) > dim Z(G) + dim Z (G) + (number

• » K K of c y c l i c submodules i n d i r e c t sum decomposition of Z (G)).

Proof. By [ 2 ] and by 4.6, 5.1, 5.2.

REMARK 5.4 We remark t h a t using the same techniques, w i t h the obvious m o d i f i c a t i o n s ,

as i n [ 3 ] 3.6 and 3.9 i t i s possible t o construct r i g h t or l e f t inverses such t h a t

j o r , r e s p e c t i v e l y , p are isomorphism.

- 4 4 -

CONCLUSION Twc abstract algebraic objects associateci w i t h any t r a n s f e r f u n c t i o n G(z), namely

the i n f i n i t e zero module Z (G) and the i n f i n i t e pole module P (G), have been i n t r o d u ­co co

ced. I t has been shown t h a t they describe the zero/pole s t r u c t u r e a t i n f i n i t y o f G(z)

and t h a t there e x i s t s a canonical r e l a t i o n between Z (G) and P (H) where H(z) i s a CO oo

( r i g h t or l e f t ) inverse of G(z). More p r e c i s e l y , Z (G) i s contained, i n a s u i t a b l e

sense, i n P (H). oo

These r e s u l t s complete the algebraic theory of the ( f i n i t e ) zero and pole module i n

the sense of B.Wyman and M.Sain [ 8 ] .

Moreover, together w i t h the r e a l i z a t i o n theory f o r non proper r a t i o n a l t r a n s f e r

f u n c t i o n s developed i n [ 2 ] , they give a b e t t e r understanding of the problems involved

i n the c o n s t r u c t i o n of the minimal inverse of a given G(z), as shown i n 5.3. Further

i n v e s t i g a t i o n s on t h i s subject w i t h the aid of the algebraic t o o l s described here w i l l

be the argument of a forthcoming pàper.

REFERENCES [ 1 ] C.Commault and J.M.Dion - Structure a t i n f i n i t y of l i n e a r m u l t i v a r i a b l e systems :

a geometrie approach - 20th IEEE Conf. on Decision and Control (1981)

[ 2 ] G.Conte and A.Perdon - Generalized s t a t e space r e a l i z a t i o n of non proper r a t i o n a l t r a n s f e r functions - System & Control L e t t e r s 1 (1982)

f 3 ] G.Conte and A.Perdon - An algebraic notion of zeros f o r systems over r i n g s - MTNS 1983 Conf., Beer Sheva (1983)

[ 4 ] j.Bammer and M.Heymann - Causai f a e t o r l z u t i o n an l i n e a r feedback - SIAM J. Control Opt. 19 (1981)

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C i r c u i t and Systems CAS-28 (1901)