On the fundamental matrix of the inverse of a polynomial matrix and applications N. P. Karampetakis...

On the fundamental matrix of the inverse of a polynomial matrix and applicationsN. P. KarampetakisS. VologiannidisDepartment of MathematicsAristotle University of ThessalonikiThessaloniki 54006, Greecehttp://anadrasis.math.auth.gr

Contents

The fundamental matrix of the inverse of a matrix pencil and applications

Problem statement Computation of the fundamental matrix of the

inverse of a polynomial matrix and its properties Applications to the solution of ARMA models Conclusions

Forward fundamental matrix of regular matrix pencils

nE I= , ( ) 1nzI A -- is called the resolvent of A .

( ) 1zE A -- is called generalized resolvent [Rose, 1978].

Definition. [Mertzios, 1989] If , n nE A ´ä C and zE A- is invertible for some

z ä C , then for some 1 0R > and 1z R> , we have the Laurent expansion at infinity of

( ) 1 1 1 2 10 1

ii

i

zE A z z z z zmm

m

¥- - - - - -

-=-

- = F + + F + F + = FåL L

where the coefficient matrices n ni

´F ä C are independent of z and are uniquely determined by ,E A . , , 1,..i i m mF = - - + are known as the (forward)

fundamental matrix sequence.

Backward fundamental matrix of regular matrix pencils

Definition. [Langenhop, 1971] If , n nE A ´ä C and zE A- is invertible for some

z ä C , then for some 2 0R > and 20 z R< < , we have the Laurent expansion at zero of

( ) 1 1 20 1 2

p ip i

i p

zE A V z V V z V z V z¥

- -- - -

=-

- = + + + + + = åL L

where the coefficient matrices n niV

´ä C are independent of z and are uniquely determined by ,E A . , , 1,..iV i p p= - are known as the backward fundamental

matrix sequence.

Computation of the fundamental matrix of a regular matrix pencil

[Langenhop, 1971]. Computation of iV in terms of 0V and 1V-

[Rose, 1978], [Campbell, 1980]. Computation of iV using the Drazin inverse [Yimin Wei, 2000] of E, A.

[Mertzios, 1989]. Leverrier-Fadeev method.

Properties of the fundamental matrix of regular matrix pencils.

Theorem. [Langenhop, 1971], [Mertzios 1989] With ( )zE A- regular and iF the forward fundamental matrix sequence the following equations hold: 1. 1i i iE A I d-F - F = 2. 1i i iE A I d-F - F =

3. ( )

( )

0 0

11 1

0

0

i

i i

A i

E i- -- -

ì üF F ³ï ïï ïï ïF = í ýï ï- F F <ï ïï ïî þ

4. i j j iE EF F = F F

5.

0, 0

0, 0

0 otherwise

i j

i j i j

i j

E i j

+

+

ì üï ï- F < <ï ïï ïï ïï ïF F = F ³ ³í ýï ïï ïï ïï ïï ïî þ

6.

1

1

0, 0

0, 0

0 otherwise

i j

i j i j

i j

A i j

+ +

+ +


where id is the Kronecker delta.

An application of the forward fundamental matrix sequence.

Consider the singular dynamical system of equations 1 0,1,..., 1k k kEx Ax Bu k N+ = + = -

with ,n mk kx uä R ä R . Let the interval of interest of the index k be [0, ]N and

the forward fundamental matrix sequence sequence of ( ) 1zE A -- be the following

( ) 1 1 ii

i

zE A z zm

¥- - -

=-

- = Få

[Lewis, 1990]. Forward solution. Consider that the initial condition 0x is given and that it is desired to determine the state kx in a forward fashion from the input sequence and the previous values of the semistate.

10 10

kk k k i iix Ex Bu

m+ -- -=

= F + Få

Another application of the forward fundamental matrix sequence.



the forward fundamental matrix sequence sequence of ( ) 1zE A -- be the following

( ) 1 1 ii

i

zE A z zm

¥- - -

=-

- = Få

[Lewis, 1990]. Symmetric solution. Another interpretation, arising in economics (where k might not be the time variable) and elsewhere, is to determine the semistate kx for intermediate values of k , given the sequence { }ku and admissible 0x and Nx .

10 10

N

k k N k N k i iix Ex Ex Bu

-- + - -=

= F - F + Få

An application of the backward fundamental matrix sequence.



the backward fundamental matrix sequence sequence of ( ) 1zE A -- be the following

( ) 1 ii

i p

zE A V z¥

--

=-

- = å

[Lewis, 1990]. Backward solution. Consider that the final condition Nx is given and then determine kx in a backward fashion from the input and future values of the semistate.

11

Nk k N N k i ii k px V Ex V Bu

-- - -= -

= - + å

Fundamental matrix of the inverse of a polynomial matrix.

Definition. Consider the polynomial matrix 0 1( ) [ ]q r r

qA z A A z A z z ´= + + +K ä R a) By assuming that ( )A z is regular i.e. ( )det 0A z ¹ , then for some 1 0R > and 1z R> , we have the Laurent expansion at infinity of

ˆ ˆ1 1ˆ ˆ 1

ˆ( ) r r

r r

r

q q iq q i

i q

A z H z H z H z¥

- - --

=-

= + + = åK

where the coefficient matrices n niH

´ä C are independent of z and are uniquely determined by ( )A z . ˆ ˆ, , 1,..i r rH i q q= - - + are known as the (forward)

fundamental matrix sequence.

b) By assuming that ( )A z is regular i.e. ( )det 0A z ¹ , then for some 2 0R > and 20 z R< < , we have the Laurent expansion at zero of

1 11( ) i

ii

A z N z N z N zn nn n

n

¥- - - +

- -=-

= + + = åL

where the coefficient matrices n niN

´ä C are independent of z and are uniquely determined by ( )A z . , , 1,...iN i n n= - are known as the backward fundamental

matrix sequence

First Problem

Computation of the forward and backward fundamental matrix sequence of a the inverse of a polynomial matrix [Fragulis, 1991]. Recursive solution of the forward fundamental

matrix without reference to its properties.

Properties of the forward and backward fundamental matrix sequence. No properties like in the matrix pencil case have been found.

AutoRegressive Moving Average representations (ARMA)

Definition. A linear, time invariant discrete time system, described by the difference equation:

0 1 1 0 , 0,1,...,k k q k q k q k qA y A y A y B u B u k N q+ + ++ + + = + + = -L L

or ( ) ( )k kA y B us s=

is called an ARMA representation of the system, where s denotes the shift-forward operator, [ ]: 0, r

ky N ® R is the output of the system, [ ]: 0, mku N ® R is a

known input of the system, and ( ) [ ] ( )

( ) [ ]

0 1

0 1

, 0q r rq

q r mq

A A A A A

B B B B

s s s s s

s s s s

´

´

= + + + ¹

= + + +

L

L

ä R

ä R

Second problem Solutions of ARMA Systems [Karampetakis et al, 2001]

The solutions have already been found in a closed form, as a result of cumbersome calculations.

First thoughts

0 1 1 0, 0,1,...,k k q k qA y A y A y k N q+ ++ + + = = -L

or ( )

( )

0 1 0, 0,1,...,qq k

A

A A A y ks

s s+ + + = = NL1444444444442444444444443

In case where 1 1TT T T

k k k k qx y y y+ + -é ù= ê úë ûL then we define as

1 0k kEx Ax+ + = or ( ) 0kE A xs + = with 0,1,...,k = N

0 1 2 1

0 0 0 0 0 0

0 0 0 00 0 0

,

0 0 00 0 0

0 0 0

r r

r

rq rq rq rq

rr

q qq

I I

I

E A

II

A A A AA

´ ´

- -

-é ù é ùê ú ê úê ú ê úê ú ê úê ú ê úê ú= = ê úê ú ê úê ú ê ú-ê ú ê úê ú ê ú- - - -ê ú ê úë ûë û

K K

KK

M M O M MM M O M M

KK

KK

ä R ä R

First thoughts

Why matrix pencils ? A number of recent algorithms are trying to reduce the

polynomial matrix problems to matrix pencil problems since we can use more robust and reliable algorithms for their solutions.

Why not this matrix pencil ? However, the connection between the fundamental

matrix sequence of A(z)^(-1) and the one of (zE+A)^(-1) is complicated.

Rewriting the system matrix equationsWe may rewrite the system equations for 0,1,.., 1k q= - as follows

1 1 0

2 1 0

1 2 0

11 1 0

2 1 0

0

1 2 0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

q q

q

q q q N

Nq q

q

q q q

q

q

A A A A

A A A A

A A A A y

yA A A A

A A A Ay

A A A A

B B

B B

-

- -

--

- -

é ùê úê úê úê úê úê úê úé ùê úê úê úê úê úê úê ú =ê úê úê úê úê úê úê úë ûê úê úê úê úê úê úê úê úê úë û

=

L L

L L

M M O M M M O M

L L

L L

ML L

M M O M M M O M

L L

M O O O

L L

L O 1

01 0

0

0

N

N

q q

u

u

uB B B

-

-

é ùé ùê úê úê úê úê úê úê úê úê úê úê úê úê úê úë ûê úë û

MM O O O O M

L L

E

A

B

Rewriting the system matrix equations

1 0,1,..., 1k k kN

Ex Ax Bv kq+

é ù+ = = -ê ú

ê úë û%% %

where 1 1 0

2 1 0

1 2 0

0 0

0 0, ,

0 0

q q

qqr qr qr qr

q q q

A A A A

A A A AE A

A A A A

-

´ ´

- -

é ù é ùê ú ê úê ú ê úê ú ê ú

= =ê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úë û ë û

L L

L L%%

M M O M M M O M

L L

ä R ä R

0

02

1 0

0 0

0 0

0

q

qqr qr

q q

B B

B BB

B B B

´

-

é ùê úê úê ú

= ê úê úê úê úê úë û

L L

L O%

M O O O O M

L L

ä R

and 1 2 1

2 2 2

0 0

;

kq q kq q

kq q kq q

k k

kq kq

y u

y ux v

y u

+ - + -

+ - + -

+ +

é ù é ùê ú ê úê ú ê úê ú ê ú= =ê ú ê úê ú ê úê ú ê úê ú ê úë û ë û

M M

Relation between ( )A z and ( )zE A+ %%

Same infinite elementary divisors

If l is a finite zero of ( )A z , ql is a finite zero of ( )zE A+ %%

The above properties ensure that stability/controllability/observability is preserved.

The Laurent expansion of the 1( )A z - can be easily computed through the

Laurent expansion of 1( )zE A -+ .

Computation of the fundamental matrix sequence of 1( )A z -

1

ˆ( )

r

i ii i

i q i

A z H z N zn

¥ ¥- -

-=- =-

= =å å

( ) 1 1 i ii i

i i p

zE A z z V zm

¥ ¥- - -

-=- =-

+ = F =å å%%

Theorem The coefficients iH (resp. iN ) of the Laurent series expansion at infinity

(zero) of 1( )A z - and those iF ( iV ) of ( ) 1zE A

-+ %% are connected by:

1 2 1

1 2 2,

1 2

q qi q qi q qi

q qi q qi q qiqq

i q qi

qi qi qi q

H H H

H H HH

H H H

- - - - - - - +

- - + - - - - +

- -

- - - - - -

é ùê úê úê úF = = ê úê úê úê úê úë û

L

L

M M O M

L

and 1 1

1 2,

1 2

qi qi qi q

qi qi qi qqq

i qi

qi q qi q qi

N N N

N N NV N

N N N

- - - - - +

- + - - - +

-

- + - - + - -

é ùê úê úê ú= = ê úê úê úê úê úë û

L

L

M M O M

L

Computation of the fundamental matrix sequence of 1( )A z -

Proof 1 1 1 2 1

1 2 22

1

1 2

0

1 0

1 2 0

0

0 0

0 0

0

q q q qi q qi q qi

q qi q qi q qiq

i i

qi qi qi qq

qi

q q

A A A H H H

H H HA AE A

H H HA

A H H

A A

A A A

- - - - - - - - +

- - + - - - - +

-

- - - - - -

- -

- -

é ùé ùê úê úê úê úê úê úF + F = +ê úê úê úê úê úê úê úê úê úê úë ûë û

é ùê úê úê úê úê úê úê úê úë û

L L

LL%%

M M O MM M O M

LL

L

L

M M O M

L

1 1

1 2

1 2

qi q qi

qi qi q qi

qr i

qi q qi q qi

H

H H HI

H H H

d

- - - +

- + - - - +

- + - - + - -

é ùê úê úê ú=ê úê úê úê úê úë û

L

L

M M O M

L

where

( ) ( ) 1

ˆ0

0 0

or

r

qi i

r i i ri i q

q q

i i k k r i k i k ri i

A z A z I A z H z I

AH I H A Id d

¥--

= =-

- -= =

æ öæ ö ÷ç÷ç ÷= Û = Ûç÷ç ÷÷ç÷ç ÷÷çè øè ø

æ ö÷ç= = ÷ç ÷÷çè ø

å å

å å

Step 1. Construct the compound matrices ,E A%% defined previously.

2 1 0

2 1 0

1 1 0 00 0 1 1

0 1 0 0 00 1 0 0,

0 0 0 0 1 1 1 10

0 0 0 1 0 0 1 0

A A AE A

A A A

-é ùé ùê úê úê úê úé ù é ù ê úê úê ú ê ú= = = = ê úê úê ú ê ú -ê úê úê ú ê úë û ë û ê úê úê úê úê ú ê úë û ë û

%%

Algorithm for the computation of the fundamental matrix

Consider the inversion of the polynomial matrix

( )

1 20

20 01 1 1 1

0 0 0 11 0

A AA

A z z z-é ù é ù é ù

ê ú ê ú ê ú= + +ê ú ê ú ê úê ú ê úê ú ë û ë ûë û 1442443 1442443144424443

Algorithm for the computation of the fundamental matrixStep 2. Determine the matrices 0 1, -F F of the resolvent ( ) 1

zE A-

+ %% using one of

the known computing techniques described in [Campbell], [Dziurla], [Mertzios], [Rose] and [Schweitzer], and therefore compute the coefficients 2 1 1, ,q qH H- + -K

from the following relations 1 2 1 0 1 1

1 2 2 1 0 2

0 1

1 2 1 2 0

,

q q q q

q q q q

q q q

H H H H H H

H H H H H H

H H H H H H

- - - - + - - +

- + - - + - +

-

- - - - -

é ù é ùê ú ê úê ú ê úê ú ê úF = F =ê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úë û ë û

L L

L L

M M O M M M O M

L L

2 3 0 10 1

1 2 1 0

1 1 2 2 0 0 1 0

0 1 1 0 0 0 0 0,

0 0 0 01 0 1 1

0 0 0 00 0 0 1

H H H H

H H H H

- - --

- -

- -é ù é ùê ú ê úê ú ê ú-é ù é ùê ú ê úê ú ê úF = = F = =ê ú ê úê ú ê ú- -ê ú ê úê ú ê úë û ë ûê ú ê úê ú ê úê úê ú ë ûë û

Therefore

3 2 1 0 1

2 2 1 1 0 0 0 01 0, , , ,

0 0 0 00 01 0 0 1H H H H H- - -

- -é ù é ù é ù é ù é ùê ú ê ú ê ú ê ú ê ú= = = = =ê ú ê ú ê ú ê ú ê ú- ë û ë ûë ûë û ë û

Step 3. The rest terms can be determined by using the following equations

( )

( )

1 2 1

1 2 20 0

11 1

1 2

0

0

q qi q qi q qii

q qi q qi q qi

i i

qi qi qi q

H H H

H H HA i

E iH H H

- - - - - - - +

- - + - - - - +

- -- -

- - - - - -

é ùê ú

ì ü ê úï ï- F F ³ï ï ê úï ïF = =í ý ê úï ï ê úï ï- F F <ï ï ê úî þê úê úë û

L% L

M M O M%

L

Algorithm for the computation of the fundamental matrix

Properties of the forward fundamental matrix sequence

1 2 1

1 2 2,

1 2

q qi q qi q qi

q qi q qi q qiqq

i q qi

qi qi qi q

H H H

H H HH

H H H

- - - - - - - +

- - + - - - - +

- -

- - - - - -

é ùê úê úê úF = = ê úê úê úê úê úë û

L

L

M M O M

L

,

( ) 1 1 ii

i

zE A z zm

¥- - -

=-

+ = Få%%

1 1 0

2 1 0

1 2 0

0 0

0 0, ,

0 0

q q

qqr qr qr qr

q q q

A A A A

A A A AE A

A A A A

-

´ ´

- -

é ù é ùê ú ê úê ú ê úê ú ê ú

= =ê ú ê úê ú ê úê ú ê úê ú ê úê ú ê úë û ë û

L L

L L%%

M M O M M M O M

L L

ä R ä R

Theorem. With ( )A z regular and iF defined earlier we can prove the following equations:

1. 1i i iE A I d-F + F =%%

2. 1i i iE A I d-F + F =%%

3. ( )

( )

0 0

11 1

0

0

i

i i

A i

E i- -

- -

ì üï ï- F F ³ï ïï ïF = í ýï ïï ï- F F <ï ïî þ

%

%

4. i j j iE EF F = F F% %

5.

0, 0

0, 0

0 otherwise

i j

i j i j

i j

E i j

+

+


%

6.

1

1

0, 0

0, 0

0 otherwise

i j

i j i j

i j

A i j

+ +

+ +


%

The above properties will be crucial to the analysis of ARMA representations.

Properties of the forward fundamental matrix sequence

Applications to difference equations - forward solutionTheorem The forward solution of ARMA is the following:

0

11

1 1

11 2

0 1 0

10 1ˆ1

ˆ0 1

0 0

0

0 0

0 0

0 0

r

r

q

q q

k k q k q k

qq

q

qk k q

k q qq

A y

yA Ay H H H

yA A A

B B B u

uB B BH H H

uB B B

-

- - - - + - -

-

- - +

+ +

é ùé ùê úê úê úê úê úé ù ê ú= +ê úê ú ê úë ûê úê úê úê úê úê úë ûê úë ûé ùé ùê úê úê úêê úé ù êê úê ú êë ûê úêê úêê úêë ûê úë û

L

LL

MM M O M

L

L L

L OL

MM O O O O O M

L L

úúúúúú

Proof Solution of the corresponding descriptor system

10 10

kk k k i iix Ex Bu

m+ -- -=

= F + Få% %

Using the connections between the fundamental matrices and their properties, we arrive at the above equations.

1 2 1 1 2 1

2 2 21 2 2

0 01 2

; ;

q qk q qk q qk kq q iq q

kq q iq qq qk q qk q qk

k k i

kq iqqk qk qk q

H H H y u

y uH H Hx u

y uH H H

- - - - - - - + + - + -

+ - + -- - + - - - - +

+ +- - - - - -

é ù é ù é ùê ú ê ú ê úê ú ê ú ê úê ú ê ú ê úF = = =ê ú ê ú ê úê ú ê ú ê úê ú ê ú ê úê ú ê ú ê úë û ë ûë û

L

L

M MM M O M

L

Theorem The backward solution of ARMA is the following: 0

11 0

1 1

11 2 0

0

101

1 0

0 0

0

0 0

0 0

0

N

N

k N k N k N k q

N qq q

q N

NqN k q N k q p

k pq q

A y

yA Ay N N N

yA A A

B B u

uB BN N N

uB B B

-

- - - - - +

- +- -

-

- - - - - -

--

é ùé ùê úê úê úê úê úé ù ê ú= +ê úê ú ê úë ûê úê úê úê úê úê úë ûê úë ûé ùéê úêê úêê úé ù êê úê ú êë ûê úê úê úëê úë û

L

LL

MM M O M

L

L L

L OL

MM O O O O M

L L

ùúúúú

ê úê úê úû

Applications to difference equations - backward solution

Applications to difference equations - symmetric solution

Theorem The symmetric solution of ARMA is the following: 1 1 1

22

1 2

0

0

11 0

1 1

11 2 0

0

0 0

0 0

0

q q q

qq

k k k k q

q

N

N

N k N k N q k

N qq q

A A A y

yA Ay H H H

yA

A y

yA AH H H

yA A A

- -

-

- - - - - -

-

- - - - - +

- +- -

é ùé ùê úê úê úê úê úé ù ê ú= +ê úê ú ê úë ûê úê úê úê úê úê úë ûê úë ûé ùé ùê úê úê úê úê úé ù ê úê úê ú ê úë ûê úê úê úê úê úêë ûê úë û

L

LL

MM M O M

L

L

LL

MM M O M

L

0

101

01 0

0 0

0 0

0

q N

Nqq N k q N k k

q q

B B u

uB BH H H

uB B B

-

- + - - + - - -

-

+

ú

é ùé ùê úê úê úê úê úé ù ê úê úê ú ê úë ûê úê úê úê úê úê úë ûê úë û

L L

L OL

MM O O O O M

L L

Conclusions.

Computation of the fundamental matrix Advantages

The problem has been reduced to a matrix pencil problem that can be solved by robust and reliable algorithms

DisadvantagesMatrices of larger dimensions are used.

Several new properties for the fundamental matrices of the inverse of a polynomial matrix have been found similar to [Langenhop, 1971]

The forward, backward and symmetric solutions of ARMA equations have been found through this reduction technique.

Further research Definition of controllability, observability and reachability for ARMA

representations. Tests for controllability, observability and reachability of ARMA

representations.

Further research – symmetric reachability

Theorem The descriptor symmetric reachability matrix over the interval 0,N is defined as

( ) ( ) ( )0 1 1 1 1( )S N N NR N B B B- - + - -é ù= F + F F + F F + Fê úë ûL

Then the descriptor system is reachable iff ( )( )Srank R N r=

On the fundamental matrix of the inverse of a polynomial matrix and applications N. P. Karampetakis...

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