Finite Spectrum Assignment of Time-Delay Systems — A Simplified Design Procedure

Cop yright © IFAC 11th Trien nial Wo rld Congress. Talhnn . Estonia. USS R. 1990

FINITE SPECTRUM ASSIGNMENT OF TIME-DELAY SYSTEMS - A SIMPLIFIED DESIGN PROCEDURE K. Watanabe, T. Ouchi and M. Nakatuyama Department of Electronic Engineering, Faculty of Engineering, Yamagata Un iversity, J onan, Yonezawa, 992, Japan

Finite spectruIII asslgnnnt of tin delay systelll is to locate n

Abstract

poles at an arb i trarily preass i gned set of points in the cOlllple x plane in the

saae

lIIanner

as

delay-free

characteristic equation . paper, a si.pl ified Keywords

systellls ,

where

nUlllber

location

of

Finite

of

all

spectruII!

feedback

spectral

pol e s

poles

and

is

controll i ng

practically

the

In this

is presented .

is

such

the

that

for

presented

a

and

finite

general

also

design

asslgnlllent .

further

is

spectrua The

the

procedure

algor i thlll

develoPlllent was

sufficient

ass i gnnnt of

was

required.

and finite

cOlllple x.

Hyun,

Shin

and Okubo(1987) presented a lIIethod which transforllls

static

delay operator

controllabil ity

condition spe c truIII

the

feasible .

linear

asslgnlllent

control

the dilllension of

Tillle-delay systellls; finite spectruIII assignlllent; algorlthlll

a retarded differential systelll has an

Infinite

is

design procedure of the control law

I NTRODUCT ION In general,

n

The previous design lIIethod was cOMPle x.

directly the control law with rational functions of

Is

el illlinated froa the characterist ic funct ion of the

the delay operator to the one with polynolll i als

and

corresponding

the

control

law

transforllls

In

closed-loop

systelll and

n poles

are

finite

Laplace

located at an arbitrarily preassigned set of points in

cOlllparlson of '!Iatanabe's algorithlll.

cOlllplex

plane

in

the

sallle

lIIanner

as

e x tra

finite

.

contains

the

lIIany

transforllls

The

Laplace

delay-free systelllS, where n is the dlnns ion of the In this paper, a silllpllfled design procedure of the

d i fferential equation describing the sy s telll .

control lIIatrix is presented by fus ing the result by The Idea of finite spectruIII assignlllent can be found

Nanit i us

in

'!Iatanabe <1983b , 1984c) .

papers

by

Ka.en(1978) .

Naeda

and

Yalllada(1975)

If the systelll is reachable,

lIIethods suggested by Norse (1976), Lee

and

Zak(1982)

give

silllple

and

Is

not

Sontag (1976)

Reachability , however,

reachable,

the

control

perforllled closed-loop

functions

feedback

by

control

dynalllical sys telll ,

however,

of

the

zx(t)=x(t-h ) A(z) E

delay

cOlllpensator . contains

u(t) ER, which

for

Rn>
has

delay and

(1)

and z the

is

the right shift

property such

duration h>O .

b(z) E Rn ><' [zl.

that

Furthenore, where R ' >
in z . The control is given by

The extra

u (t)

dyna.ical lIIodes wh i ch are not controllable . This is not finite spectruIII asslgnlllent .

o

=f >< (z) x (t) + [Nh ( o + f !/J (~)u(t+ ~)d -Nh

To overcon the problelll,

by

denotes the Ixj lIIatrl x cOlllposed of real polyno.ials

usually

is

x ERn,

(delay)operator

lIIatrix over

lIIatrix

rational

X(0 =A (z) x (0 +b (z) u (0 where

If the

ring of polynolllials can be easily enlarged to The

lIIethod

Consider the follow i ng systelll :

is a

the

wi th

the

FINITE SPECTRUN ASSIGNNENT

and

the

operator .

and

feedback su c h that

quite restr i ctive c ontrollability condition . systelll

Olbrot(1979)

algebraic

the control lIIatri x is over the ring of polynollllals in the delay ope r ator.

and

~ ) x (t + ~ ) d ~ ~

(2)

where N is a pos It Ive integer, ( L2([-Nh,OI,R'>
Nanltius and Olbrot(1979)

Introduced the f In I te Lap I a c e trans forllls in contro I

E

)

lIIatrix and showed that the necessary condition for f in I te spectruIII ass ignnnt spectral

Taking the Laplace transfon of

Is for the systn to be

controllable.

Ito(1983,1984a,1984b , 1984c,1986)

'!Iatanabe proved

that

(1)

yields (3)

sx (s) =A (z) x (s) +b (z) u (s)

and

where

the

187

z=e - eh.

The

characteristic

funct i on of

the

open-loop syste. is given by

t::.

(s, z)

=

=s h + a • (z) s h- ' + .. + a h (z) In

general,

trllnsfor.ed

to

the syste. hllS lIn

(14)

T I (z) M(z) =M I (z)

a. (z) (i=l, . ' ,n) are rell! po!yno.lals in s .

where

M(z) is

By ele.entllry row operations,

I s I -A (z) I

where

infinite nu.ber of

[ ... (.l

po les.

(15)

MI (z) = .hl (z)

The control (2) is transfor.ed to u(s)=f. (z,s)x(s)+f2(z,s)u(s)

ar\+1

where

The .,,(z)

t

:J

.hh (z)

(z)

• n+ In

(z)

is the real polyno.ial

in z, .,,(z)*O

0=1,2, ", n) fro. (10 . 1) lInd the degree of .,,(z) is

o J

f,,(z,s)= The

-Nh

l/J('t")exp(s't")d't"

chllracteristic

function

of

(6 . 2) the

luger then

is

closed-loop

thllt of . " (z) (j=I+I, . ·,n+1).

T
uni.odular .atrix.

1I

[le .. lI 11

syste. is given by

t::. , (s, z)

I

-b (z)

= lSI-A
(16)

(7)

1-f2(z, s)

."

(z)

The proo f is given in Append ix 1. If there exist f.(z,s) and f 2 (z,s) real nu.bers /3., .. , /3 h such that t::. ,(S,Z)=Sh+ /3 .Sh- '+ .. + /3 h then the syste. (1) ass ignab I e . sys te. is

It

(1)

is said to be finite spectru. known

spectru. A

co.putation

Le . . a 1 i.p lies thllt r(i)=(the degree of .,-I'-,(Z»

(8)

that

if

and only

is spectrally controllable,

finite

1984c).

Is

for arbitrary

the systn

ass Ignable (lIatanabe

(17)

- (the degree of .,,(z»;;';O

if the et.

Suppose thllt

al

<1 B. 1)

r(i»1

si.plifled design procedure for of f.(z,s) and f,,(z,s) is presented

for

l=cl,c2. ",Cd

(Cl
and

r (i) =0

be low .

for i

*

<1 B. 2)

Le t r=r(c .)+r(c2)+·· +r(c.) Cl, C 2,

" . C
(19)

SIMPLIFIED DESIGN PROCEDURE It

Is

supposed that

the syste.

(1)

By ele.entuy row operations, that T2(z)M,(z)=M 2 (z)

is spectrally

controllable. The following equation holds. rank[sl-A(e - an ), b(e - an)]=n, VsEC.

(9)

T2(z)=

and E

(J

(A)

[

T'" :

n /\

[ M21 : [

characteristic

function

of

the

*0

(23)

closed-loop

*

zr(cl)-l

(Z)]

ad j (s I -A (z» b [ I s I -A (z) I - [f.(z,s),f,,(z,s»)M(z)v(s)

•

(z)

C IC I

z

•

0

(z) C le I

•

C le I

(z)

o (24)

(11)

lISterisks denote polyno.llIls in z and the degree of

where M(z)= [

:1

M2. (z)

t::. , (s, z) I sl-A(z) I I 1-f,,(z,s)-f,(z,s) (sl-A(z»-'b(z) I s I-A (z) I

= I sl-A(z) I

(22) z

(Z)]

expanded to

- [f , (z,s),f,,(z,s»)

zr(cl)-l

c,

M2(Z)=

syste. is

(21)

T21 (z)= :

1)


.(20)

(Z)]

* .. *

rank[b(z),A(z)b(z), that .eans ", Ah - '(z)b(z»)=n for all but finitely .any ZEC, (J (A)={s E C I I sl-A(e- an ) I =01 and /\ ={s E C I rahk [ b (e - an), A(e - an) b (e - an), ", A(e -an) h-I b (e - an») (10.

such

T24 (z)

(l0. 2) where

hllve M,,(z)

where

This Is equivalent to (Spong and Tran, 1981) (a)ranko[b(z),A(z)b(z), ",Ah-'(z)b(z»)=n (10.1)

vs

we

adj (s I-A
v(s)=[1,s, . ·,Sh)T

the (12)

asterisks

in

the

j-th

colu.n

s.aller than that of .JJ(z) in M,(z).

(13) M2(Z)V(S) can be trarnsfor.ed to

188

of

M21 (z)

Is

M,,(z) v (s) =p (s) v. (z)

(25)

Kz(R. (s) +Rb(S) z} T 2(Z) IT I (z).

(26)

Substituting (32)

where v.(z)=[zr, ",z,ll T

into (7) yields

L::" ,(s,z)

[I elllla21 P (s)

Is non-s Ingu I ar .

=

The proof is given by Appendix 2.

degree

eleunts

r (c.)

denollinator

in

Q,(S)

is

at

of by

[f I (z,s), f,,(z,s)1M(z)v(s)

.. ", a

I

(z) -

!3 ,)

I s I-A (z) I

rows

least

I s)

*[k,(z,s),k2(z,slM(z)v(s)

(27)

the

-

-[ a n(Z)- !3

rows

Q. (s) of

-b(z) I-fz(z,

sI-A(z) -f,(z,s)

I sI-A(z) I sI-A(z)

[Jellu 31 Denote P-' (s) by r (c ,) [Q,(S)] P -I (S)=Q(s)= : The

I

the

non-zero

r(cl)-l

larger

-

than that of the nUllerator .

[a n (z) - !3 n,

a ,(z) - !3 ,) [

~: 1 s n- \

The proof is given by Appendix 3.

(34) The

Let

sys tell

is

finite

spectruII

ass Ignab I e . The algorithll is sUllllarized as follows:

(28)

Step I : ColIPute M(z) . Step 2:Transforll M(z) to M,(z) as shown in

(4) .

Step 3:Construct M.(z) froll M, (z) according to

RI(S)] R.(s)= : [

(29)

(20).

R. (s)

Step 4:Transfon M2(z) to pes)

as shown in (25)

Step 5:Collpute Q(S)=p-I(S) and R.(s) given by (29) . Froll

the non-zero elellents

I elllla 3,

strictly proper . there

Froll

in R.(s)

Manitius and Olbrot

are

Construct the finite Laplace transforM

(979),

lIatrix

is a lIatrix Rb(S) with rational functions of

R(z,s)=R.(z,s)+Rb(Z,S)Z.

Step 6:ColIPute M.(z)

finite Laplace transforlls . [Jeua 41

The step 2,3 and 4 (30)

R(z,s)M.(z)v(s)=M.(z)v(s) where M.(z)

in the

i-th colulln

(30) froll R(z,s) .

are

refined on by fusing the

results by Manitius and Olbrot(1979) and the one by

is the lIatrix with polynollials in z and

the degree of the elellents

given by

Step 7 : ColIPute r, and r" which satisfy (31). Step 8:Construct f,(z,s) and f.(z,s)given by (33).

s such that R(z,s)=R.(s)+Rb(s)z is the lIatrix with

Watanabe(1983).

of

The

algorithll

is

sillPlified

in

cOllparison with the previous one .

M.(z) is not larger than that of IIII(Z) in M,(z). The proof is gien by Appendix 4. [Ieua there

If

51 exist

the

the

systell

constant

NUMERICAL EXAMPLE

sat isf ies lIatrices

00.2),

r

then

Consider the systell

r.

I and

A (z) = [

such that

r

[ r"

,,1 [

I(Z)J

M

= [I

M.(z)

nxn,

0

(1)

and

Zo

is given by I sl-A(z)

Define K(z,s) by

Step 1 :M(z) is given by

r

r

,+

[z.

(32)

K,,(z,s) 1

M(z) =

~.

n

I

(z, s)

=[

(35)

(36)

z

(37)

Step 2:The T I (z) and MI (z) are given by

a

n

(z) -

a

!3

I

(z) -

!3 ,) K I

(z, s)

(33 . 1)

T, (z) =

a I (z)- !3 11K,,(z, s)

f,,(z,s)=[ a n(Z)-!3 n,

[~

(33 . 2)

The f,(z,s)

]

~]

0 -z

Let f

[~

=s 2_ ZS

,,(R.(s)+Rb(s)z)T,,(z)lT, (z)

= [K,(Z,S),

b (z) =

The characteristic function of the open-loop systell

(31)

The proof is given by Appendix 5.

K(z,s)=(

with h=l.

belongs to the class where the control

MI (z) =

consists of polynolliais in z and the fin ite Laplace f,,(z,s)

transforlls.

The

transforlls

becaus~

is

the

f 2 (z, s)

is

fin ite derived

0 1 0

[ z· -z·

z

-z"

0

n

(38)

0

0

J

(39)

Lapiace froll

Step 3:The T,,(z) and M,,(z) are cOllputed as follows:

189

[~

T.. (z)=

1

0

0

]

1

Z2

[-~"

M2(Z) =

~

z

(40)

~]

Z

K, ,, (z , s)=I+ (41)

-Z2 0 Step 4:The M2(Z) yields P(s)=

[·J ...LJ -1

0

R. (s)

Suppose

(55) (56)

that

the

characteristic

function

closed-loop sy s tem is required to be

(43)

of

the

(s +1) (s+2).

The c ontrol matrices are given by f , (z, s )=(O-2, - z -3)K , (z ,s)

s"

=(-13-2z- -2

0

1-z 1- (s+1) z (7+z) s s"

1-(0.5s"+s+1)z

s" 1-(s+1) z

0

s"

(54)

z

s

is given by

0

(53)

K",, (z,s ) =- -

----------- ---- --- ------ -0

(52)

z

s"

\- z

o o

s"

(1 +z)

S2

1-(s+1)z l-z + S2 s \-z 1-(s+1)z K'4(z,s)=I+-- <1+z) + s s" \-(s+1)z k", (z, s)= s"

Step 5: It fol lows froa (42) that 0, (s) o (s)= [ 0 2 (S)

s"

1-(s+1)z

( 1+2z)+--~~-=-

K,,,(z,s)=2+ -

(42)

s2

o

1-z s

1- <0. 5s 2+S+1) Z

+

J

1-(0 . 5s 2+s+1)z + =-...:..::..:...=.:=----=-~-=s" (51)

l-z 1-(s+1)z k , , (z, s) =3 . 5+2- + s s"

s"

(5+z)

1-z -5-z - - -

(5+8z+z 2)

1-(0 . 5s"+s +1)z

(2+5z+z 2) - 2z

(57)

(44)

R. (s) =

f ,, (z , s) = (0-2, -z-3) K2 (Z, s)

0

s"

l-z

(58)

(3+ z )

CONCLUSION

is computed a s follows: l-z

o

0

1-(s+1)z

l-z

s" l-
R(z,s)=

1- (s+1) z S2

=-2

0

S"

The R (z, s)

S2

o

s" 1-(s+1)z S2

o

A simplified design algorithm of control matrix for f i nite

o

re s ult de lays

1-(s+1) z S2

s pe ct rUM

as s ignment

is appliable via

the

is

presented.

to lI1ultivariable

r e sult s

by

Manitius

(1985)

l-z REFERENCES Hyun Y. T. , S. Shin and S. Okubo (1987) . A Finite SpectrUM As s ignaent Procedure for the

Step 6:The M,, (z)

i s given by

z"-z"

0

Single-Input Linear

z-z "-z "

0

T i Me-Delays, Tran s. S ICE Japa n , vol. 23 ,

[ l-z-0.5 z"

.Step

7 : The

(46)

-z

- z"

r

1 and

r

2

whi c h satisfy

(1978).

An Operator Th e ory of Linear

J . Differential Equat ions , vol . 27, pp . 274-297 . Le e , E. B. and S. H. Zak (1982) . On SpectrUM PlaceMent

3. 5 1

with Com.ensurate

Funct ional Different ial Equat ions,

(32) are given by

r

S y s te~s

no. 4, pp. 386-393 . Ka.en, E. \I.

l-z

.atr i c es

and

\latanabe (1986) .

(45 )

M,,(z)=

This

systell1s with

for Linear TiMe Invariant Delay Systeas,

(47 )

2

o

Trans . on

Auto~ati c

IEEE

Control,

Maeda, H. , and H. Yallada (1975) . On the

r

2=

21 [ 1

o o

o o

Stabilizability of Linear Systeas with Delays,

~]

Trans . SICE Japan, vol.l!, no . 4, pp.

( 48)

Manitius,A . Z.and A. \I .O lbrot (1979 ).

Spectrua AssignMent Proble. for Systeas with

Step 8:Coaputing K, (z ,s) and K,, (z,s) yields

K,

(z, s)

=

[ k,,(z s ) K, ,,(z , s )

k, ,, (z,s)

i

K'4(Z,S)

j

444-450 .

Finite

Delays,

(49)

IEEE Trans . on Autoaatic Control,

vo I . AC-21 , no . 4 pp . 541-553 . Man i t Ius, A. Z. and V. Manous iouthak is ( 1985 ). On

K2(Z, s)=

(50)

Spectral Controllability of Mult i -input tiae delay systeas, Syst . Contr . Lett . , vol. 6,

where

190

no. 3, pp. 199-205 . Morse, A. S.

Delay-Differential

Systeu,

Autoutica,vol.12,

Autout ica, vol. 7 no .

sl-A(z)

n- I

1

It follows frolll (A . 5) that

Linear SysteMs over

COMMutat ive Rings, A Survey, Richerche Spong,M. W.

(A . 5) ann+1 (z)

(1976).

•

t I

(z) =a

I 1+1

(z) a

0

T. J. Tarn (1981) . On the Spectral

and

APPENDIX 2

Equations,

squar Matrix and is denoted by

IEEE

Trans .

on

AutoiMatic Control,

(1983a).Finite

P (s) =

SpectruM AssignMentPro Contolr,

: P. I

(

bleM for Froll

SysteMs with Delay in State Variables, IEEE on AutoMat ic .

The

(25 ) ,

Pi i(s)

vol. AC-28, no . 4,

Proble.

for

SysteMs

the

with

equal in

Control, vol. 38, no.5

Condition Sy s teMs,

ones

eleMents of

degree

triangular

of

left is

of

p" (s)

than

c,-1.

triangular sllaller

is

c,-I is

at

IOOSt

sui ler

and

than

that

Cl-I.

of

P,,(s),

These

illlPI ies

that detP,,(s)=sq+(polynolOial of degree q-I at MOSt)

J . Control, vol.39 , no.2,

(A . 8)

pp. 363-374. Watanabe, K.

the

p" (s)

of

the degree of the eleMents of P"+I(S),

",P" -I (S)

for Spectral Controlabil ity of Delay

Int.

the

is c ,-I at 1II0St. The degree of

diagonal

··,P,.(s)

pp . 913-926. Watanabe, K., M. I to (1984a) . A Necessary

and

right

lower

FurtherMor,

J.

the diagonal

c I-I

in

Variables,

Int.

to

upper

eleMents

e x cept

is the rxr

(A . 7) the degree of

Multiple COMMensurate Delays in State

P(s)

P •• (s)

except diagonal ones

Watanabe, K. M. Ito and M. Kaneko (1983b). Finite

le . . a 2:

PI:.(S)]

(s)

is

eleMents

pp. 506-508 . Assignlllent

proof of

PIt(S)

vol . AC-26, no.2, pp . 527-528. Wat anabe, K. , M. I to, M. Kaneko and T. Ouch i

Spectrunl

(A . S)

• . a nn+ 1 (z)

1+' I + 2(Z}

This iMPlies
1 pp . 1-34.

Controllabil ity of Delay-Differential

Trans .

I

1

[

pp . 529-531. Sontag, E. D.

1

al"(Z)a"3(Z) .. a nn + I (Z) 1 sl-A(z)

a~3(Z) .. ann+ I:Z)

(1976) . Ring Models for

whrer


q=r (c I) (c

Controllability of Systeu with Multiple COMMensurate Delays

in State Variables,

The

Int . J .

,-I)

(A . 9)

+ .. +r (c.) (c .-1)

... trix P2(s)

0

is non-singular .

Contro I, vo I. 39, no.3 pp. 497-505. APPENDIX 3

Watanabe,k . ,M . Ito and M. Kaneko
lelllllla

Multiple COIII.,ensurate Delays in State and

p" (s) is

Control,

Int.

J. Control, vol. 39, no.5,

pp . 1073-1082. Watanabe, K.

[

and T.

Ouchi

(1985).

The proof of le . . a 3: Frolll the proof of the

degree

equal to

of

the

c ,-I.

O;(S)]

radj[P(S)l]

O. (s)

I IP

the

elelllents

of

(A . 10) (s)

and

SUM of the degree of the diagonal elelllents in P(s) . These the

non-zero

COMlllensurate Delays,

The

degree

Control, vol. AC-3I, no.

The proof of le . . a 1 : Tl

APPENDIX

(z)

=

rT

I I

eleMents

6, pp. 543-550.

(z)

(z)

,,(z)

T

I I

(z)

0

APPENDIX

0

n I

(z)

The

proo f

f

I e...

(A . 2)

~ 11

R.(s)+R.(s)e - · n

+

(A . 3)

where

11

4:

The

non-zero

following

(A . I I)

11

11

11

M", (e - · n)v(s)

R.(s)M,,(e .. an)v(s) •

11

(A. 12)

is the utrix no ...

Since R.(s)+Rb(s)z

is the ,.atrix with the finite Laplace transforMs,

ann (z)

the following inequal ity holds: 11 R.(s)+Rb(s)e ·· · n 11 < 00

b (z) =

the

-R.(s)M,,(z)v(s)

an - tn(Z)

a

of

This Yields R.(s)zM,,(e-· n)v (s)

a 1,1: (z) a I "' (Z) (

is at MOSt q-c ,+1.

zR. (s) M,,(z) v (s) = [R. (s) +R. (s) zl M,, (z) v (s)

Frolll (14) and <15), A(z) and b(z) are denoted by

A(z) =

the

least by c,-I higher than

that of the denollinator .

(z)

b (z)

is

identity holds:

I)

A (z) =T I (z)A (z>T , - I (z) I

nUMerator

is at

Def ine A (z) and b (z) by

b (z) =T

in 0, (s)

the

in O,(s)

q wh ich

the degree of the nUlllerators of

eleMents of

is

is denotd by

(A .

IT,

P(s)

1

illlPI ies that

Observer for Multivariable SysteMs with IEEE Trans . on Autolllatic

of

1

Int.JH.Control, vol.41, no . 1, pp . 217-229 . (1986) . Finite SpectrulI AssignMent and

degree

diagonal

FurtherMore,

Systellls with Delays in State Variables . Watanabe, k .

TI

An Observer of

2,

(A. 4)

Since

M,, (z)

cons ists

of

(A. 13)

polynolllials

in z and v (s)

in s, we have

(A . 14)

<00

It

This yields

TI

(z)

M(z) v

follows

R.(s)M,,(z)v(s)

(s)

191

froll are

(25)

that

the

elelOents

the quasi-polynoMials

of

in s with

the polynoalals in z as coefficeints . Then, 11

R.(sn!,,(e-·h)v (s)

Eqs. (A. 12)-(A. 14)

<

11

The (A. 15)

00

iMPly that of

polynoMials

be

represented

degree of the coluan of M,,(z) the

corresponding

the

co luan

of

coluan

M3(Z)

of

does

APPENDIX

5

The

proof

eleaent of MI(z)

(z) =c

I J

•

t J

oZ

P+

Since

Since

is less than that of

dq(Z,S)

exceed

degree

that

of

of

c

I J 1Z P - 1

+ . . +c

the

(A . 25)

*

The

(i, j)-th

The

q(z,s)

contains

degree of z (A . 17)

I Jp

0,

q(Z a,S)=O.

IZ=Z a = 0

d ' ' ' '- lq(Z,S)! d r ( I ) - 1Z Z=Z 5:

polynoMial

(A . 26)

Continuing this procedure, we have

0

leMaa

the

(A. 24) with respect to z yields

dZ

the

is

the zerose of a 1-11 (z)=O be

sl-A(za)

Different iat ing

the

Let

1 sl-A(z a)

p (s)

is represented by

where p is the degree of and cijk

In the

not

of

(30).

MI(Z),

corresponding coluan of Ml
q(Z a ,S)

[R.(s)+Rb(S)Z]

by

r(D>!.

Froa (A . 22), we have

(A. 16)

with polynoaial coe ffic i ent of z . can

al-"-I(z)/all(z)=al-"(z) degree

z a.

Rb(s)zM2(e - Oh )v(s) 11 < 00 Rb(s)zM2(e-· h)v(s) consists 11

*M,,(z)v(s)

of

q(z,s)

j-th coluan of MI (z)

a" - I(z)

in q (z, s)

is given frOM

are possibly non-zero real nUMbers .

q (z, s) =k (s) a

where

(A . 18)

k (s)

Matr Ices

is

I -

the

(A . 27)

as

a

factor

The q(z,s) 11

and

dose not exceed r (D [Ra(s)+Rb(s)zlT2(z)

is of the fora (22) .

Let

o 0

the

because

and T2(z)

is represented by

(A . 28)

(z)

non-zero

polynoMial

in

s.

The

p (s) and k (s) are co-pr iae.

and

c CI=

[

II]

Dividing (22) by al - lI(z) yields

(A. 19)

:

k (s)

Cid

p (s)

The Matr Ix C3 is constructed frOM M3 (z) in the salRe Manner as Cl.

<--i-l

Let

a 12 (z) a 23 (z) .. a I - 2 I -I (z)

[~J

C=

(A. 20)

a "3 (Z) .... a 1-21 - 1 (z) 1 sl-A(z) 1 I

*:

The C is the squar Matr ic of size n+r (c I) (c 1-1> + .. +r(cd) (Cd-1). exist

If

the

C

Is

singular,

row vector k I and k2 with real

then

there

[

nUMbers such

I,

NI N3

k2 ]

It follows

J

(z) (z)

o

(A . 21>

k (s)

1-2

sl-A(z )

I- I

p (s)

that

~

0] rT II(Z) adj(SI-A(Z»b(Z)] L i s I-A (z) 1

This iMplies that there

=0

is a i E

+

(CI,C2, ",Cd)

0 . . 0]

s

11+1

Since k(so) I-A (z)

1

1 I

Since

1 1

denote

(z) ..

3 nn+l

sl-A(z) 1 I-I

(z)

a

(A . 23)

1 sl-A
sa

I-A

1 sal-A(z) 1 1-' 1

* 0, (z)

1

o

a I 2 (z) a 23 (z) .. a I - I I (z)

: I

(A. 31)

it follows froa (A.29) that

q(z,s) /p(s) =k(s)a l-,I (z)/p(s)

1-

, I

(e -s ah ) =0

(A. 32) is

the

finite

(A . 33)

rank[s a l-A(e -s ah), b(e- s ah )] < n

p (s)

a23(z) .... al-"(z) Is l-A(z)

(z)

the

(A. 34)

It follows that

<-- i-I

I I

and

<00

1- ' =0

This contradicts to

*

z

they have liaited values

rank[s a l-A(e- s ah ), b(e - sah )]< n

q(z,s)

*

in

Eqs . (A. 32) and (A . 33) yield that

It follows that ....

(A . 30)

polynoMials

Laplace transforM,

ann+1 (z) 1 sl-A(z) 1 n- I

l

asterisks

k (s a ) 1 -p (S a)

=0

I -

*a 1- 21 - ' (z) 1 sl-A(z) 1 1-3 1

for S=S a which satisfies P(S a)=O . Therefore,

* q p(z,(s)s)

a 1,,(z)a"3(z) .. ann+1 (z)

a

I

Since

<-- i-I

[*

*a 23 (z) .... al- " I -' (Z) IsI-"(z) I11

+ 1 * 1 s I -A (z) 1

such taht

a

sl-"(z) 1 I -I 1

finite Laplace transforMs ,

....

(A . 29)

1 *a I 2 (z) a 23 (z) .. a I -2 I - I (z) +

(A. 22)

*

I

1-

T I ,,(z) 1

[*

1- 3

It follows that

[k l+k 2 (R.(s)+R b(S)Z ) T,,(z )]N I (z)v (s) =[k l+k,,(Ro(s)+R b(s)z ) T,,(z) ] *[ I

sl -A(z)

1 sl-A(z)

that

[k

0

a 1-2 1- 1 (z)

1 s

I-A (z)

1

sl-A(z) 1 I-I

1- "

(A. 24)

192

(A. 35)

(10.2) and coapletes the proof.