Notions of equivalence for Notions of equivalence for

linear multivariable systems

A.I.G. Vardulakis1 and N. Karampetakis1, E. Antoniou2 and S. Vologiannidis3

http://anadrasis.web.auth.gr

1. Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece, email : avardula@math.auth.gr, karampet@math.auth.gr

2. Department of Sciences, Alexander Technological Educational Institute of Thessaloniki, Thessaloniki, Greece, Email: eantonio@gen.teithe.gr

3. DataScouting Unlimited, Vakhou 30, 54629, Thessaloniki, Greece, Email: svol@datascouting.com

Contents

� Transformations that preserve :

� the finite zero structure of polynomial matrices,

� both the finite and infinite zero structure of

polynomial matricespolynomial matrices

and their extension to system matrices.

� Discrete time systems and elementary divisors.

� Conclusions

Гантмахер Феликс

Рувимович

Howard Harry

RosenbrockIsrael Gohberg

Jan Willems Paul Fuhrmann Wiliam Wolovich

SimilarityDefinition [Gantmacher 1959] Two constant matrices 1 2, ,p pA A ×∈ℝ are similar if there exists an invertible matrix ,M such that

11 2A MA M −=

11 2( ) ( )p psI A M sI A M −− = −

or equivalently

• It is an equivalence relation

• Invariants : eigenvalues with their algebraic and geometric multiplicities

2 1( ) ( )p pM sI A sI A M− = −

• Invariants : eigenvalues with their algebraic and geometric multiplicities

• Canonical form : By placing as columns in M the generalized

eigenvectors of A1 we get its Jordan Canonical form.

1, ,

1, ,

{ }

i

iji vj v

J diag J==

=……

1 0

0

1

0 0

ij ij

i

v viij

i

J

λλ

λ

×

= ∈

⋯

⋱ ⋮ℂ

⋮ ⋱

⋯

A real alternative of M, J is also possible.

Similarity – Geometrical interpretation

1 2( ) ( )x t Mx t=

We get a new description of the form

Consider the homogeneous system of the form

( )1 1 1 1 1 ,1( ) ( ) ( ) 0n nx t A x t I A x tρ ρ= ⇔ − =

where : /d dtρ = 1n nA ×∈ℝ .

We get a new description of the form

( )2 2 2 2 2( ) ( ) ( ) 0nx t A x t I A x tρ ρ= ⇔ − =

where 12 1A M A M−= .

System similarity

( ) ( )x t Mx t=

The classical time domain approach uses state space representations of the form

1 1 1 1

1 1

( ) ( ) ( )

( ) ( ) ( )

x t A x t B u t

y t C x t D u t

ρ = +

= +

where : /d dtρ = 1p pA ×∈ℝ , 1

p mB ×∈ℝ , 1l pC ×∈ℝ , 1

m lD ×∈ℝ ,

1( ) : px t →ℝ ℝ is the state vector and ( ) : lu t →ℝ ℝ , ( ) : my t →ℝ ℝ are

respectively the input and output vectors.

1 2( ) ( )x t Mx t=

The definition of the state of a system is nonunique !

It is easy to see that the above change of coordinates results in a new description

of the form

2 2 2 2

2 2 2

( ) ( ) ( )

( ) ( ) ( )

x t A x t B u t

y t C x t D u t

ρ = +

= +

where 12 1A M A M−= , 1

2 1B M B−= , 2 1C C M= and 2 1D D= .

System similarity

12 2 1 1

2 2 1 1

00

00

p p

ml

MsI A B sI A BM

IC D C DI

− − − = − −

5 or in system matrix terms

2 2 1 10 0p pM MsI A B sI A B− − =

2 2 1 1

2 2 1 1

0 0

0 0

p p

l m

M MsI A B sI A B

I IC D C D

− − = − −

( )( )

( )( )

1 20

0 m

Mx t x t

Iu t u t

= − −

( )( )

( )( )

1 2

2 1

00 0

0 m

Mx x

Iy t y t

=

ObservabilityControllability

1uX

2uX

uY

Commute

Unimodular equivalence

Definition [Gantmacher 1959, Rosenbrock 1970] Two polynomial matrices ( ) [ ] ,p q

iA s s ×∈ℝ 1,2i = are unimodular equivalent, if there exist unimodular

matrices (determinant constant) ( ) [ ]p pU s s ×∈ℝ , ( ) [ ]q qV s s ×∈ℝ , such that

1 2( ) ( ) ( ) ( ).A s U s A s V s=

Pre or post multiplication of a polynomial matrix by a unimodular one,

corresponds to performing row or column elementary operations

on the matrix.

• It is an equivalence relation

on the matrix.

a) Multiplication of any i-th row (column) by the number c<>0,

b) Addition to any i-th row (column) of the j-th row (column) multiplied

by any polynomial w(s),

c) Interchange of any two rows (columns).

( ) ( ) ( )

( )( ) ( )

2 2 31 13 314 23 8 310 5 10 5

22 23 3 3 9 3 1 161 2 12 10 2 10 8 3 310 5 10 5

1 01 14

4 0 2

CA s

U s V sA s S s

ss s s s

s s ss s s s s

− − − − + − = − − − − +− − − − + − ������� ���������������� �����

Unimodular equivalence – Canonical form

Theorem [Gantmacher 1959] Given a polynomial matrix ( )A s , there exist

unimodular matrices ( )U s , ( )V s such that

( )( ) ( ) ( ) ( )A sA s U s S s V s= ℂ

where

( ) 1( ) { ( ), , ( ), }A s rS s diag s sε ε= 0ℂ …

is the Smith form of ( )A s and 1

( ) ( )i

ijv

mi j

js sε λ

== ∏ − are monic polynomials

The roots of ( )i sε are the finite zeros of ( )A s , whereas the factors ( ) ijmjs λ− of

( )i sε , are the finite elementary divisors of ( )A s .

The finite elementary divisor structure and therefore the zeros and their

multiplicities form a complete set of invariants for unimodular equivalence.

( ) ( )( ) ( ) ( ) ( ) 11 2 1 2

p pM

p p p psI A U s sI A V s sI A M sI A M×∃ ∈

−− = − ⇔ − = −ℝ

1j

such that 1( ) | ( )i is sε ε + , 1, , 1i r= −… .

Finite Jordan pairsTheorem ([Gohberg et. al.], pp.184) Let ( )

0 0R , Rr n n n

s sC J× ×∈ ∈ be a matrix pair where

0sJ is a Jordan matrix with unique eigenvalue 0s . Then the following conditions are

necessary and sufficient in order that ( )0 0,s sC J be a Jordan pair of ( )A s

corresponding to 0s : (a) ( )det A s has a zero at 0s of multiplicity n , (b)

0s

s s

C

C Jrank n

=

Js0:=

s0 1 0 ⋯ 0

0 s0 1 ⋯ 0

⋮ ⋮ ⋱ ⋱ ⋮

0 0 0 ⋯ 1

0 0 0 ⋯ s0

∈ ℝn×n

0 0

1

0 0

n

s s

s s

C Jrank n

C J −

=

⋮

(c)

0 0 0 0 01 0 0q

q s s s s sA C J AC J A C+ + + =⋯

Taking a Jordan pair ( )R , Ri i i

i i

r n n ns sC J× ×∈ ∈ for every eigenvalue , 1,2,..,is i k= of

( )A s we define a finite Jordan pair ( )R , Rr n n nC J× ×∈ ∈ of ( )A s as

( ) ( )1 2 1 2

; k ks s s s s sC C C C J blockdiag J J J= =⋯ ⋯

where 1 2 kn n n n= + +⋯ is the total number of finite zeros (eigenvalues) of ( )A s (order

accounted for).

0 0 0 ⋯ s0

( ) 1 0

qqA s A s A s A= + + +⋯

Construction of the smooth solution spaceThe column span of JtCeΨ = defines a solution space CB of

( ) ( ) 0, : /A t d dtρ β ρ= = (Σ)

with dimension :

dim : total number of finite zeros of ( ) (order accounted for)CB n A s= = CB is known as the smooth solution space of (Σ) defined by (Gohberg et. al. 1982) and

(Vardulakis 1991).

( )( )

2 23 314 2110 5 10 5

04 tβρ ρ ρ ρ − + − =

( )

( )( )

2 2 61 2 1210 5 10 5

04

A t

t

ρ β

βρ ρ ρ ρ= − − − − + ����������������

( )( )

2 1 2 2

0 2

2 2

1 1 1

1 1 1

t ttJt

t t

e e tCe e

e e t

− − = = +

2 21 1

,1 1

C t t tB e e

t

− → = +

11 2

2 ,21 2

f k k kB

k− −

= +

The column span of kCJΨ = defines a solution space fB of

( ) ( ) ( ) ( )0, 1A k k kσ β σβ β= = +

with dimension :

dim : total number of finite zeros of ( ) (order accounted for)fB n A s= = fB is known as the forward solution space of (A(σ)) defined by (Gohberg et. al.

1982).

Unimodular equivalence – Geometrical interpretation

Theorem [Polderman and Willems 1998] Consider two homogeneous

systems of the form

( ) ( ) ( ) ( )1 1 2 20, 0A t A tρ β ρ β= =

where : /d dtρ = and ( ) ( ) [ ]1 2,p q

A Aρ ρ ρ ×∈ℝ . ( ) ( )1 2,A Aρ ρ define the

same smooth behavior B if and only if there exists a unimodular matrix

( ) [ ]q qV ρ ρ ×

∈ℝ such that ( ) ( ) ( )1 2V A Aρ ρ ρ= .

Consider two homogeneous systems of the form

( ) ( ) ( ) ( )1 1 2 20, 0A t A tρ β ρ β= =

where : /d dtρ = and ( ) ( ) [ ]1 2,p q

A Aρ ρ ρ ×∈ℝ . There exists a bijective map

of the form ( ) ( ) ( )2 1t V tβ ρ β= between the smooth behavior spaces

defined by ( ) ( )1 2,A Aρ ρ if and only if ( ) ( )1 2,A Aρ ρ are unimodular

equivalent.

Unimodular equivalence – Left MFDs

( )( ) ( )

( ) [ ] ( ) ( ) [ ],

: , ,0

p p p mL LL L L

p p m

A s B sL P s A s s rankA s p B s s

I× ×

= = ∈ = ∈ − ℝ ℝ

Definition [Kailath 1980, Smith 1981] Let ( ) ( )1 2,P s P s L L∈ ×

( ) ( )A s B s

( ) ( ) ( ) ( )( ) ( )

L L

p

A t B u t

y t I t

ρ β ρ

β

=

=

( )( ) ( )

,0

Li Lii

p p m

A s B sP s

I

= −

Then ( ) ( )1 2,P s P s are called unimodular equivalent if and only if there exist a

unimodular polynomial matrix ( ) [ ]p pLT s s

×∈ℝ such that

( ) ( ) ( ) ( ) ( )

2 2 1 1,

,, ,

0

00 0

L L L LL p p

p p pp p m p p m

A s B s A s B sT s

II I

= − −

•Same input decoupling zeros,

•Same transfer function

( ) ( )( ) ( ) ( ) ( )( )2 2 1 1L L L L LA s B s T s A s B s=

( ) ( ) ( ) ( )2 2 1 1

1 1

L L L LA s B s A s B s− −

=

Unimodular equivalence – Right MFDs

( )( )( ) ( ) [ ] ( ) ( ) [ ]

,

: , ,0

m m p mR mR R R

R p m

A s IR P s A s s rankA s p B s s

B s× ×

= = ∈ = ∈ − ℝ ℝ

Definition [Kailath 1980, Smith 1981] Let ( ) ( )1 2,P s P s R R∈ ×

( )( )( ) 0

iR p

i

A s IP s

B s

= −

( ) ( ) ( )( ) ( ) ( )

R m

R

A t I u t

y t B t

ρ β

ρ β

=

=

( )( ) ,0

i

iR p mB s

− Then ( ) ( )1 2,P s P s are called unimodular equivalent if and only if there exist a

unimodular polynomial matrix ( ) [ ]m mRT s s

×∈ℝ such that

( )( )

( )( )

( )2 1

2 1

,

,, ,

0

00 0

R p R p R m m

m m mR p m R p m

A s I A s I T s

IB s B s

= − −

•Same output decoupling zeros,

•Same transfer function

( )( )

( )( )

( )2 1

2 1

R R

RR R

A s A sT s

B s B s

= − −

( ) ( ) ( ) ( )2 2 1 1

1 1

R R R RB s A s B s A s− −=

Strict System Equivalence (PMDs)– Rosenbrock 1970otherwise called Unimodular Strict System Equivalent

Let

( ) ( )( ) , 1,2

( ) ( )

i ii

i i

A s B sP s i

C s D s

− = =

be the system matrices of two PMDs iΣ , where ( ) [ ]r riA s s ×∈ℝ , ( ) [ ]r l

iB s s ×∈ℝ

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

i i

i i

A t B u t

y t C t D u t

ρ β ρ

ρ β ρ

=

= +

i=1,2

be the system matrices of two PMDs iΣ , where ( ) [ ]iA s s∈ℝ , ( ) [ ]iB s s∈ℝ

, ( ) [ ]m riC s s ×∈ℝ , ( ) [ ]m l

iD s s ×∈ℝ .

Definition [Rosenbrock 1970] 1Σ and 2Σ are strictly system equivalent, if there

exist unimodular matrices ( ) ( ) [ ],r r

M s N s s×

∈ℝ and ( ) [ ] ,p r

X s s×

∈ℝ

( ) [ ]r mY s s

×∈ℝ such that

1 1 2 2

1 1 2 2

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )p m

M s N s Y sA s B s A s B s

X s I IC s D s C s D s

− = −

0

0

Strict System Equivalence – Invariants

• Input Decoupling Zeros

• Input-Output Decoupling Zeros

• Output Decoupling Zeros( )( )

1

1

A s

C s

−

( ) ( )( )1 1A s B s

• Order of Σ ( )1degn A s=

• Rosenbrock degree ( ){ }, ,...,i i i

• System Zeros

• System Poles

• Transfer function matrix

• Transmission Zeros-Poles

• Rosenbrock degree

( ){ } ( )1 2 1 2

1 2 1 2

, ,..., , ,...,

, ,..., , ,...,gcd , greatest value of q s.t. 0q q

q q

i i i i i ij j j j j jP s q q P s= = ≠ɶ ɶ

ɶ ɶɶ

( ) ( ) ( ) ( ) ( )1

1 1 1 1 1G s C s A s B s D s−

= +

( )1A s

( ){ }1 2

1 2

, ,...,

, ,...,max deg q

q

i i ir j j jd P s= ɶ

ɶ

Invariants – Geometric interpretation

• Karcanias, 1976. ‘Geometric Theory of zeros and its use in feedback analysis’,

Ph.D. Thesis, Electrical Engineering Department, UMIST, U.K.

• MacFarlane and Karcanias 1976, Poles and zeros of linear multivariable

systems: a survey of the algebraic, geometric and complex variable theory, Int. J.

Control, Vol. 24, 33-74.

• Desoer CA, Schulman JD (1974) Zeros and poles of matrix transfer functions

and their dynamical interpretations. IEEE Trans, vol CAS–21, No1, pp 3–8.

• Karcanias and Kouvaritakis, 1979 The output zeroing problem and its

relationship to the invariant zero structure: a matrix pencil approach", Int. J.relationship to the invariant zero structure: a matrix pencil approach", Int. J.

Control, Vol. 30, pp.395-415

• F.M. Callier, V.H.L. Cheng, and C.A. Desoer, Dynamic interpretation of poles

and zeros of transmission for distributed multivariable systems. IEEE Trans.

Circuits and Systems, CAS-28:300-306, April 1981

• C. B. Schrader and M. K. Sain, 1989, Research on System Zeros: A Survey,

International Journal of Control, Volume 50, Number 4, pp. 1407-1433

• Bourles H., Fliess M. (1997) "Finite poles and zeros of linear systems: an

intrinsic approach", Internal. J. Control, vol. 68, pp. 897-922

Special forms of Strict System Equivalence in different

system descriptions – Smith 1981

System similarity of state space systems coincides with strict system

equivalence [Rosenbrock 1970, pp.56]

Unimodular equivalence of MFDs coincides with strict system

equivalence [Smith 1981]

Unimodular equivalence

( )( )1

2 5 6

A s

s s+ +���

( )2

2 0

0 3

A s

s

s

+ + �����

Disadvantage : connect polynomial matrices of the same dimension!

1 1 2 0 1 301 s s− + + =

( ) ( ) ( ) ( )21

2

1 1 2 0 1 3

3 2 0 3 1 25 60

U s A s V sA s

s s

s s s ss s

− + + = + − − + ++ +

ɶ������������������

( ) ( ) ( ) ( )21

2

0

0

2 1 2 0 1 31

3 1 0 3 1 25 6

M s A s N sA s

s s s

s s ss s

+ + + = + + ++ +

ɶ��� �������������

Strict System Equivalence – extended (Rosenbrock 1970)

Let

( ) ( )( ) , 1,2

( ) ( )

i ii

i i

A s B sP s i

C s D s

− = =

be the system matrices of two PMDs iΣ , where ( ) [ ] i ir riA s s ×∈ℝ ,

( ) [ ] ir lB s s ×∈ℝ , ( ) [ ] im rC s s ×∈ℝ , ( ) [ ]m lD s s ×∈ℝ .

( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

1

1

2

2

iq r

i i i

i ii i

iI u t

A t B u t

y t C t D u t

t

t

ρ β ρ

ρ

β

β β ρ

− =

=

= + +

0

0

( ) [ ] ir liB s s ×∈ℝ , ( ) [ ] im r

iC s s ×∈ℝ , ( ) [ ]m liD s s ×∈ℝ .

Definition [Rosenbrock 1970] 1Σ and 2Σ are strictly system equivalent, if there

exist unimodular matrices ( ) ( ) [ ],r r

M s N s s×

∈ℝ and ( ) [ ] ,p r

X s s×

∈ℝ

( ) [ ]r mY s s

×∈ℝ such that

1 2

1 1 2 2

1 1 2 2

0 0 0 0( ) ( ) ( )

0 ( ) ( ) 0 ( ) ( )( )

0 ( ) ( ) 0 ( ) ( )

q r q r

p m

I IM s N s Y s

A s B s A s B sX s I I

C s D s C s D s

− −

= − −

0

0

where q is chosen to satisfy

( ){ } ( )1 2max deg ,degq A s A s≥

Extended Unimodular Equivalence – Pugh and Shelton 1978

A natural generalization of unimodular equivalence (Smith 1981)

Disadvantage of u.e. : connect polynomial matrices of the same dimensions.

Disadvantage of s.s.e. : extra pseudo-states

Let ( , )m lP be the class of polynomial matrices with dimensions ( ) ( )r m r l+ × + ,

for all 1,2,r = ….

Definition [Pugh and Shelton 1978] Two polynomial matrices 1( )A s , ( ) ( , )A s m l∈P are extended unimodular equivalent (e.u.e.) if there exist 2 ( ) ( , )A s m l∈P are extended unimodular equivalent (e.u.e.) if there exist

polynomial matrices of appropriate dimensions ( )M s , ( )N s such that

1 2( ) ( ) ( ) ( )M s A s A s N s= where the compound matrices

[ ] 1

2

( )( ) ( ) ,

( )

A sM s A s

N s

−

have full rank and no zeros in ℂ i.e. zero coprime.

Invariants : Same Smith form apart of a trivial expansion – same normal

rank defect.

Fuhrmann System Equivalence – Fuhrmann 1977

Let ( , )m lP be the class of polynomial matrices with dimensions ( ) ( )r m r l+ × + ,

for all 1,2,r = ….

Definition [Fuhrmann 1977] 1Σ and 2Σ are strictly system equivalent, if there

exist polynomial matrices ( ) ( ) [ ] 2 1,r r

M s N s s×

∈ℝ and ( ) [ ] 1 ,p r

X s s×

∈ℝ

( ) [ ] 2r mY s s

×∈ℝ such that

1 1 2 2( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

M s N s Y sA s B s A s B s

X s I IC s D s C s D s

− = −

0

0

Share the same invariants

1 1 2 2( ) ( ) ( ) ( ) ( )p mX s I IC s D s C s D s

= − 0

where the compound matrices

[ ] 12

( )( ) ( ) ,

( )

A sM s A s

N s

−

have full rank and no zeros in ℂ i.e. zero coprime.

Fuhrmann Strict System Equivalence and Rosenbrock Strict System

Equivalence give rise to the same equivalence relation (Levy et.al. 1977)

Geometric Interpretation of Strict System Equivalence

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

(*)i i

i i

A t B u t

y t C t D u t

ρ β ρ

ρ β ρ

=

= +i=1,2

Define , 1,2iuX i = as the sets of all (smooth) solutions of (*) corresponding to

the input u.

Definition [Pernebo 1977] 1Σ and 2Σ are strictly system equivalent, if and only if there exists a bijective mapping of the form

( ) ( ) ( ) ( ) ( )β ρ β ρ= + ( ) ( ) ( ) ( ) ( )2 1t N t Y u tβ ρ β ρ= +

between 1uX and 2

uX (where : /d dtρ = , ( ) ( ),N Yρ ρ are polynomial matrices).

The map is such that the diagram

2

1

u

u

u

X

Y

X

↑

ց

ր

commutes. uY is the set of outputs corresponding to the input u.

Alternative forms of the Definition of s.s.e.

• Wolovich W.A., 1974, Linear Multivariable Systems, New York,

Springer Verlag.

• M. Morf, 1975, Extended system matrices-Transfer functions and

system equivalence, IEEE Conference on Decision and Control

including the 14th Symposium on Adaptive Processes, Vol. 14, pp. 199

-206.

• B. Levy, S.-Y. Kung, M. Morf and T. Kailath, 1977, A unification of

system equivalence definitions, 1977 IEEE Conference on Decision and system equivalence definitions, 1977 IEEE Conference on Decision and

Control including the 16th Symposium on Adaptive Processes and A

Special Symposium on Fuzzy Set Theory and Applications, Vol.16, pp.

795 -800.

• Diederich Hinrichsen and Dieter Pratzel-Wolters, 1980, Solution

modules and system equivalence, International Journal of Control, 32,

pp. 777-802

Verghese, George

Tomas Kailath

Vardulakis, Antonis-Ioannis

• G.C. Verghese and T. Kailath, 1981, IEEE Transactions on Automatic Control,26, pp. 434-439

• A. I. G. Vardulakis, D. N. J. Limebeer, and N. Karcanias, “Structure and smith-MacMillan form of

a rational matrix at infinity,” International Journal of Control, vol. 35, no. 4, pp. 701–725, 1982.

Verghese, George

Nicos Karcanias

Clive Pugh

Biproper equivalence

Although unimodular equivalence operations preserve the finite zero

structure of polynomial matrices, they will in general destroy the

corresponding structure at infinity.

Definition [P. D. P. Van Dooren 1979, Vardulakis et. al. 1982] Two rational matrices ( ) ( ) ,p q

iA s s ×∈ℝ 1,2i = , are equivalent (at infinity) if there exist

biproper rational matrices ( ) ( ) , ( ) ( )p p q qpr prU s s V s s× ×∈ ∈ℝ ℝ , such that

( ) ( ) ( ) ( )A s U s A s V s=

Pre or post multiplication of a rational matrix by a biproper one,

corresponds to performing row or column elementary operations

on the matrix.

a) Multiplication of any i-th row (column) by the number c<>0,

b) Addition to any i-th row (column) of the j-th row (column) multiplied

by any proper rational function w(s),

c) Interchange of any two rows (columns).

1 2( ) ( ) ( ) ( )A s U s A s V s=

i.e. lim ( ) ,det 0p ps U s U U×→∞ = ∈ ≠ℝ (similar for V(s)).

Biproper equivalence – Canonical form

Theorem [Vardulakis et.al. 1982] Given a rational matrix ( )A s , there exist

biproper matrices ( )U s , ( )V s such that

( )( ) ( ) ( ) ( )A sA s U s S s V s∞=

where

1( ) ,ˆ ˆ1

1 1( ) , , , , , ,0k

k r

qqA s p r q rq q

S s diag s sss

∞− −+

= ⋯ ⋯

is the Smith-MacMillan form at infinity of ( )A s and is the Smith-MacMillan form at infinity of ( )A s and

1 2 0kn q q q= ≥ ≥ ≥ ≥⋯ are the orders of the poles at ∞ and

1 1ˆ ˆ ˆ 0r r kq q q− +≥ ≥ ≥ ≥⋯ are the orders of the respective zeros.

The indices qi, i=1,r form a complete set of invariants for biproper

equivalence.

Us

1 0

1

s3s2 − 1 1

As

s s2 + s + 1

−1 −s − 1

Vs

− 1s s + 1 − 1

s2s2 + s + 1

1 1s

=

SAs∞ s

s2 0

0 1

s2

Connection between the Smith form at infinity and the Smith form at C of polynomial matricesDefine the dual polynomial matrix of ( )A s

( ) 1

0 1

1q q qqA s s A A s A s A

s− = = + + +

ɶ ⋯

1 2

, 1,2,3,..,

, 1,2,..., with

j

j j k

s j r

q q j k q q q q

µ

µ=

= − = = ≥ ≥ ≥⋯

The infinite elementary divisors of ( )A s are given by (Hayton et.al.1988 & Vardulakis 1991):

The first kind of IED, i.e. the ones with degrees 0, 2,3,..,j jq q j kµ = − > = we call

infinite pole IEDs, since are connected with the orders of the poles , 1,2,...,jq j k= of

( )A s ats = ∞ .

1 2, 1,2,..., with j j kq q j k q q q q⋯

Ũs

1 0

s − s3 1

Ãs

s s2 + s + 1

−s2 −s2 − s

Vs

−1 − s −1 − s − s2

1 s=

SÃs0 s

s2−2 0

0 s2+2

where iq are the orders of the poles of ( )A s at s = ∞ defined above.

Connection between the Smith form at infinity and the Smith form at CDefine the dual polynomial matrix of ( )A s

( ) 1

0 1

1q q qqA s s A A s A s A

s− = = + + +

ɶ ⋯

The infinite elementary divisors of ( )A s are given by (Hayton et.al.1988 & Vardulakis 1991):

1 2

, 1, 2,..,

ˆ ˆ ˆ ˆ, 1, 2,..., with

j

j j v v r

q j k k v

q q j v r q q q

µ

µ τ + +

= = + +

= + = + + ≤ ≤ ≤⋯

The second kind of IED, i.e. the ones with degrees ˆ 0, 1, 2,...,j jq q j v v rµ = + > = + + we

call infinite zero IEDs, since are connected with the orders of the zeros

ˆ , 1, 2,...,jq j v v r= + + of ( )A s at s = ∞ .

where ˆiq are the orders of the zeros of ( )A s at s = ∞ defined above.

1 2ˆ ˆ ˆ ˆ, 1, 2,..., with j j v v rq q j v r q q qµ τ + += + = + + ≤ ≤ ≤⋯

Ũs

1 0

s − s3 1

Ãs

s s2 + s + 1

−s2 −s2 − s

Vs

−1 − s −1 − s − s2

1 s=

SÃs0 s

s2−2 0

0 s2+2

Infinite Jordan pairs

Theorem (Gohberg et. al.1982 ,pp.185) Let ( )R , RrC Jµ µ µ× ×∞ ∞∈ ∈ be a matrix pair

where J∞ is a Jordan matrix with unique eigenvalue 0 0s = . Then, the following

conditions are necessary and sufficient in order that ( ),C J∞ ∞ be an infinite Jordan pair

of ( )A s :

(a) ( )det A sɶ has a zero at 0 0s = of multiplicity µ , (b)

C

C Jµ

∞

∞ ∞

=

( ) 1

0 1

1q q qqA s s A A s A s A

s− = = + + +

ɶ ⋯

1

C Jrank

C J µ

µ∞ ∞

−∞ ∞

=

⋮

(c)

0 1 0qq qA C J A C J A C∞ ∞ − ∞ ∞ ∞+ + + =⋯

Taking a finite Jordan pair ( )R , Ri i iri iC Jµ µ µ× ×

∞ ∞∈ ∈ for different algebraic multiplicities

of the eigenvalue 0, 1,2,..,is i r= = of ( )A sɶ we define an infinite Jordan pair

( )R , RrC Jµ µ µ× ×∞ ∞∈ ∈ of ( )A s as

( ) ( )1 2 1 2R ; Rrr rC C C C J diag J J Jµ µ µ× ×

∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞= ∈ = ∈⋯ ⋯

where 1 2 rµ µ µ µ= + + +⋯ is the total number of infinite elementary divisors

(eigenvalues) of ( )A s (order accounted for).

Infinite Jordan pairs

Theorem (Gohberg et. al.1982 ,pp.185) Let ( )R , RrC Jµ µ µ× ×∞ ∞∈ ∈ be a matrix pair

where J∞ is a Jordan matrix with unique eigenvalue 0 0s = . Then, the following

conditions are necessary and sufficient in order that ( ),C J∞ ∞ be an infinite Jordan pair

of ( )A s :

(a) ( )det A sɶ has a zero at 0 0s = of multiplicity µ , (b)

C

C Jµ

∞

∞ ∞

=

1

C Jrank

C J µ

µ∞ ∞

−∞ ∞

=

⋮

(c)

0 1 0qq qA C J A C J A C∞ ∞ − ∞ ∞ ∞+ + + =⋯

It is easily seen from the above that ( ),C J∞ ∞ is an infinite Jordan pair of ( )A s iff it is

a finite Jordan pair of its dual polynomial matrix ( )A sɶ corresponding to its zero

(eigenvalue) at 0s = .

The column span of ( ) ( )

ˆ 1

, ,0

rqii

z zi

C J tδ−

∞ ∞ ∞=

Ψ = ∑ ɶ ɶ where

, , 1 , 2 ,

, , 1 , 2 ,

: , ,...,

: , ,...,

z

z z

r

z v v r

z v v r

C C C C R

J blockdiag J J J R

µ

µ µ

×

∞ ∞ + ∞ + ∞

×∞ ∞ + ∞ + ∞

= ∈

= ∈

ɶ ɶ ɶ ɶ

ɶ ɶ ɶ ɶ

and , lC∞ɶ (

, lJ∞ɶ ) , with 1, 2,...,l v v r= + + , consists of the first ˆ

lq columns

(the first ˆ ˆl lq q× block) of the matrix

, lC∞ ( , lJ∞ ), defines a solution space B∞

with 1ˆdim r

i vz iB qµ∞= += = ∑ .

Therefore to each infinite Jordan block of order ˆ , 1, 2,...,j jq q j v v rµ = + = + + Therefore to each infinite Jordan block of order ˆ , 1, 2,...,j jq q j v v rµ = + = + +

Geometric interpretation of finite and infinite zeros in

continuous time systems

The solution space B∞ of

( ) ( ) 0, : /A t d dtρ β ρ= = (Σ)

has dimension :

1

ˆdim : total number of infinite zeros of ( ) (order accounted for)r

z ii v

B q A sµ∞

= +

= = =∑

B∞ is known as the impulsive solution space of (Σ) defined by (Vardulakis 1991).

The whole solution space CB B B∞= + of (Σ) has dimension

( ) ( ) ( )( )1 1

ˆdim dim dim degdet degdetr k

Cz i i M

i v i

B B B n A s q A s q A sµ δ∞

= + =

= + = + = + = + =∑ ∑The whole solution space

CB B B∞= + of (Σ) has dimension

( ) ( ) ( )( )1 1

ˆdim dim dim degdet degdetr k

Cz i i M

i v i

B B B n A s q A s q A sµ δ∞

= + =

= + = + = + = + =∑ ∑

Therefore to each infinite Jordan block of order ˆ , 1, 2,...,j jq q j v v rµ = + = + +

corresponds ˆ jq linear independent impulsive solutions.

Therefore to each infinite Jordan block of order ˆ , 1, 2,...,j jq q j v v rµ = + = + +

corresponds ˆ jq linear independent impulsive solutions.

Strict equivalence – An extension of Matrix Similarity

H.J.S. Smith

1826-1883

C. Jordan

1838-1922

J. J. Sylvester

1814-1897K. Weierstrass

1815-1897

Leopold Kronecker

1823-1891

• H.J.S. Smith in 1861 (invariant factors of an integer matrix - Smith

normal form of these matrices).

• C. Jordan in 1870 (normal form for similarity classes).

• Sylvester in 1851 (study of the elementary divisors of sE-A, where E,A

are symmetric matrices)

• Weierstrass in 1868 (extend this methods by obtaining a canonical

form for a matrix pencil sE-A, with det(sE-A) not identically zero – define

the elementary divisors of a square matrix and proved that they

characterize the matrix up to similarity)

• Kronecker in 1890 (extend the results to nonregular matrix pencils)

Strict equivalence – Weierstrass canonical form

Weierstrass canonical form

Definition [Gantmacher 1959] Two regular matrix pencils ,i isE A− 1,2,i = with

, ,p pi iE A ×∈ℝ are strictly equivalent if there exist non singular matrices

, p pM N ×∈ℝ such that

1 1 2 2( ) .sE A M sE A N− = −

Every matrix pencil [ ]p psE A s ×− ∈ℝ , is strictly equivalent to a pencil of the

form

CnsI J− corresponds to the finite zeros (elementary divisors)

sJ Iµ∞ − corresponds to the infinite elementary divisors

{ }, 1, , , 1, ,C ij iJ diag J i v j v= = =… …

1 0

0

1

0 0

ij ij

i

v viij

i

J

λλ

λ

×

= ∈

⋯

⋱ ⋮ℂ

⋮ ⋱

⋯

{ }, 1, ,iJ diag J i µ∞∞ = = …

0 1 0

0 0

1

0 0 0

i iiJ µ µ×∞

= ∈

⋯

⋱ ⋮ℝ

⋮ ⋱

⋯

form

C( ) { , }nK s diag sI J sJ Iµ∞= − −

Strict equivalence – Kronecker canonical form

Definition [Gantmacher 1959] Two matrix pencils ,i isE A− 1,2,i = with

, ,p qi iE A ×∈ℝ are strictly equivalent if there exist non singular matrices

p pM ×∈ℝ , ,q qN ×∈ℝ such that

1 1 2 2( ) .sE A M sE A N− = −

Every matrix pencil [ ]p qsE A s ×− ∈ℝ , is strictly equivalent to a pencil of the

form

Kronecker canonical form

form

C( ) { , , ( ), ( )}nK s diag sI J sJ I L s L sµ ε η∞= − −

( 1) ( 1)( ) R[ ] , ( ) R[ ]T Ti i i i

i i i i i iL s sM N s L s sM N sε ε η ηε ε ε η η η

× + + ×= − ∈ = − ∈

( 1) ( 1)

1 0 0 0 1 0

R ; R .

0 1 0 0 0 1

i i i i

i iM Nε ε η η

ε η× + × +

= ∈ = ∈

⋯ ⋯

⋮ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋮

⋯ ⋯

iε ( iη ) are the right (left) Kronecker indices

{ }1 2

( ) , ,...,r

L s blockdiag L L Lε ε ε ε= { }1 2

( ) , ,...,l

L s blockdiag L L Lη η η η=

Restricted system equivalence (Rosenbrock 1974)

( ) ( )x t Nx t=

Let the descriptor state space representations of the form

1 1 1 1 1

1 1

( ) ( ) ( )

( ) ( ) ( )

E x t A x t B u t

y t C x t D u t

ρ = +

= +

where : /d dtρ = , 1 1, p pE A ×∈ℝ , 1p mB ×∈ℝ , 1

l pC ×∈ℝ , 1m lD ×∈ℝ ,

1( ) : px t →ℝ ℝ is the state vector and ( ) : lu t →ℝ ℝ , ( ) : my t →ℝ ℝ are

respectively the input and output vectors.

Premultiplying by M 1 2( ) ( )x t Nx t=Premultiplying by M

It is easy to see that the above change of coordinates results in a new description

of the form

2 2 2 2

2 2 2

( ) ( ) ( )

( ) ( ) ( )

Ex t A x t B u t

y t C x t D u t

ρ = +

= +

where 2 1 2 1,E ME N A MA N= = , 2 1B MB= , 2 1C C N= and 2 1D D= .

Restricted system equivalence (Rosenbrock 1974)

2 2 2 1 1 1

2 2 1 1

0 0

0 0l m

M NsE A B sE A B

I IC D C D

− − = − −

5 or in system matrix terms

12 2 2 1 1 1

2 2 1 1

00

00 m

NsE A B sE A BM

IC D C DI

− − − = − − 2 2 1 1 00 ml

IC D C DI − −

( )( )

( )( )

1 20

0 m

Nx t x t

Iu t u t

= − −

( )( )

( )( )

11 1 2 2

2 1

0 00

0 m

E x E xM

y t y tI

− =

ObservabilityControllability

1uX

2uX

uY

Commute

( ) ( ) ( ) ( ) ( )11 2 1 1 1 2 2 20 0 0 0 0x Nx E x E Nx M E x−= ⇒ = =

Strong equivalence – Verghese et.al. 1981

( ) 1 12 2

0

0

sE AsE A M N

I

− − =

Strong equivalence

2 2 2 1 1 1

2 2 1 1

0

0l m

M N RsE A B sE A B

Q I IC D C D

− − = − −

Operations of strong equivalence (can eliminate non-dynamic variables)

Disadvantage of strict equiv. : connect matrix pencils of same dimension

2 2 1 1 0l mQ I IC D C D− −

1uX

uY

Commute

Definition Two systems are termed strongly equivalent if one can

be obtained from the other by some sequence of allowed

transformations (i.e. operations of strong equivalence, and/or trivial

augmentations, and/or trivial deflations).

det 0,det 0M N≠ ≠ 1 10QE E R= =

2uX

Constant system equivalence – Anderson et.al. 1985

Disadvantage of str.eq. : lack of a closed-form expression as a system-

matrix transformation (the operation of trivial deflation (or inflation) has

not been formally embodied within the operations of strong equivalence)

Definition [Anderson et. al. 1985] Two descriptor systems are constant system equivalent if there exist invertible matrices ,M N and constant matrices ,X Y of appropriate dimensions such that

0 0 0 0I I

1 1 1 2 2 2

1 1 2 2

0 0 0 0

0 0

0 0m l

I IM N Y

sE A B sE A BX I I

C D C D

− − = − −

0

0

Complete equivalence – Pugh et.al. 1987

Definition [Pugh et. al. 1987] Two matrix pencils ,i isE A− 1,2,i = with , ( , ),i iE A m l∈P are completely equivalent if there exist constant matrices ,M N

of appropriate dimensions such that

Strict equivalence preserves both the finite and infinite elementary

divisors.

Identity block has no dynamic significance – Pugh et.al. 1987 – we are

interested only for the finite and infinite zeros.

of appropriate dimensions such that

1 1 2 2( ) ( )M sE A sE A N− = − where the compound matrices

[ ] 1 12 2 ,

sE AM sE A

N

− − −

have full rank and no zeros in { }∪ ∞ℂ .

Preserve both the finite and infinite zero structure.

If Σ1, Σ2 are completely equivalent then they are extended

unimodular equivalent and extended causal equivalent.

Complete system equivalence – Pugh et.al. 1987

Definition [Pugh et. al. 1987] Two descriptor systems are completely system equivalent if there exist constant matrices , , ,M N X Y of appropriate dimensions such that

1 1 1 2 2 2

1 1 2 2m l

M N YsE A B sE A B

X I IC D C D

− − − − =

0

0

where the compound matrices

[ ] 1 1,

sE AM sE A

− − [ ] 1 1

2 2 ,sE A

M sE AN

− − −

have full rank and no zeros in { }∪ ∞ℂ .

5 or the system matrices are complete equivalent

Complete System Equivalence – Invariants

• Finite and Infinite Input Decoupling Zeros

• Finite and Infinite Input-Output Decoupling Zeros

• Finite and Infinite Output Decoupling Zeros 1 1

1

sE A

C

− −

( )1 1 1sE A B−

• Generalized order of Σ ( )1

1

ˆdegr

ii

n A s qµ=

+ = +∑• Rosenbrock degree

• Finite and Infinite System Zeros

• Finite and Infinite System Poles

• Transfer function matrix

• Finite and Infinite Transmission Zeros-Poles

• Rosenbrock degree

( ) ( ) 1

1 1 1 1 1 1G s C sE A B D−

= − +

1 1sE A−

• Input and output minimal indices ( )1 1 1sE A B−1 1

1

sE A

C

− −

Invariants – Geometric interpretation

G. Verghese, Infinite frequency behavior in generalized dynamical systems, Ph.D.

dissertation. Stanford Univ.. Stanford. CA, 1978.

A. C. Pugh and V. Krishnaswamy. Algebraic and dynamic characterizations

of poles and zeros at infinity., Int. J. Contr., 1984.

N. P. Karampetakis, The output zeroing problem for general polynomial matrix

descriptions, International Journal of Control, vol. 71, no. 6, pp. 1069–1086, 1998.

Geometric interpretation of Complete System Equivalence

Definition [Hayton et. al. 1986] Let 1 2,P P be two descriptor systems. They are

said to be fundamentally equivalent if there exists :

a) a constant injective map

( )( )

( )( )

2 1

0

x s x sN Y

u s u sI

= − −

b) a constant surjective map

( ) ( )2 2 1 10 00E x E xM− − =

( )( )

( )( )

2 2 1 10 00E x E xM

y s y sX I

− − = − −

If P1, P2 are fundamentally equivalent then both maps are bijective.

Theorem P1, P2 are fundamentally equivalent if and only if they

are complete system equivalent.

Extended causal equivalence – Walker 1988

Definition [Walker 1988] Two polynomial matrices 1( )A s , 2 ( ) ( , )A s m l∈P are extended causal equivalent (e.u.e.) if there exist proper polynomial matrices of appropriate dimensions ( )M s , ( )N s such that

( ) ( ) ( ) ( )M s A s A s N s=

Disadvantage of bicausal equivalence : Connect polynomial matrices

with the same dimensions

Disadvantage of constant equivalence : Needs trivial inflation/deflations

1 2( ) ( ) ( ) ( )M s A s A s N s= where the compound matrices

[ ] 12

( )( ) ( ) ,

( )

A sM s A s

N s

−

have full rank and no zeros in ∞ i.e. zero coprime at infinity.

Preserves the infinite zero structure.

System Equivalence (PMDs) – Anderson et.al. 1984,

Coppel and Cullen 1985

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

i i

i i

A t B u t

y t C t D u t

ρ β ρ

ρ β ρ

=

= +

( ) ( )( ) ( )

( )( )( ) �

( )0 0

0

0 0

i i i

i i

A B t

C D I u t u t

I y t I

ρ ρ βρ ρ

− − =

− ������������

i=1,2

( ) ( ) 0 0

( ) ( ) 0 ( )( )

0 0 0

0 0 0

i i

i i m i ii

l l i

m

A s B s

C s D s I ss

I I

I

− − − = = −

T UP

V

Normalized form of the systems

( )

( )( )�

( ) ( ) ( )

0 0

0 0

ii i

i

t

i

I y t I

y t I t

ρ ξ

ξ

−

=

UT

V

������������

���

System equivalence – Coppel and Cullen 1985

Definition [Coppel and Cullen 1985] ( ) ( )1 2,P s P s are said to be system

equivalence at infinity if there exist causal rational matrices ( )M s , ( )N s , ( )X s, ( )Y s such that

Two separate transformations :

a) System equivalence (preserves the finite zero structure – it is the

extended strict system equivalence)

b) System equivalence at infinity (preserves the infinite zero structure)

, ( )Y s such that

1 1 2 2

1 2

( ) ( )( ) ( )( )

0 0( ) l

N s Y ss sM s

IX s I

− − =

0

0

T U T U

V V

where ( ) ( )2M s s T has a causal right inverse and ( )( )

1 s

N s

−

T has a causal

left inverse.

Full Equivalence – Ratcliffe 1982, Hayton et.al. 1988

Disadvantage of system equivalence : two transformations

Definition [Ratcliffe 1982, Walker 1988] Let 1 2( ), ( ) ( , ).A s A s m l∈P Then

1 2( ), ( )A s A s are fully equivalent (FE) if there exist polynomial matrices

( ), ( )M s N s of appropriate dimensions, such that

1 2( ) ( ) ( ) ( )M s A s A s N s= is satisfied and the compound matrices

[ ] 1( )( ) ( ) ,

A sM s A s

[ ] 1

2

( )( ) ( ) ,

( )

A sM s A s

N s

−

1) have full rank and no zeros in { }∪ ∞ℂ

2) [ ]2 2( ) ( ) ( )M MM s A s A sδ δ= , 1

1

( )( )

( )M M

A sA s

N sδ δ

= −

Mδ denotes the McMillan degree i.e. the total number of poles of the rational

matrix involved.

Preserves the finite and infinite zero structure (complete set of

invariants only for regular matrices !).

Geometric interpretation of Full Equivalence – Pugh et.al. 2007

Given two auto regressive (AR) representations ( ) 0i iA ρ ξ =

where ( ) [ ],i ip qiA s s×∈ℝ for 1,2,i = their behaviors are

{ : ( ) 0}iqi i imp i iAξ ρ ξ= ∈ =ℓB

where impℓ is the space of impulsive - smooth distributions.

Definition [Pugh et.al. 2007] The systems described above are fundamentally equivalent if there exists a bijective polynomial differential map

1 2( ) :N ρ →B B . 1 2( ) :N ρ →B B .

1 2( ), ( ) ( , )A s A s m l∈P

Full equiv.

Theorem [C. Pugh, E. Antoniou and N. Karampetakis 2007] The systems described by the above AR – representations are fundamentally equivalent iff there exists a polynomial differential operator 2 1( ) [ ]

q qN s s×∈ℝ satisfying the following conditions 1) 2 1

1 2( ) [ ] : ( ) ( ) ( ) ( ).p pM s s M s A s A s N s×∃ ∈ =ℝ

2) 11

( )( ( ))

( )M M

A sA s

N sδ δ

=

and [ ]2 2( ) ( ) ( ( )).M MM s A s A sδ δ=

3) [ ]1

2

( ), ( ) ( )

( )

A sM s A s

N s

have full rank and no zeros in { }.∪ ∞ℂ

1 1 2 2.q p q p− = −

Full system equivalence – Hayton et. al. 1990

Definition [Hayton et. al. 1990] The normalized system matrices ( )i sP 1,2i = are normal full system equivalent if there exist polynomial matrices ( )M s ,

( )N s , ( )X s , ( )Y s such that the relation

1 1 2 2

1 2

( ) ( )( ) ( )( )

0 0( ) l

N s Y ss sM s

IX s I

− − =

0

0

T U T U

V V

is a full equivalence relation.

Definition [Hayton et. al. 1990] The normalized system matrices ( )i sP 1,2i = are full system equivalent if there exist polynomial matrices ( )M s , ( )N s ,

( )X s , ( )Y s such that the relation

( ) ( )( ) ( )

( ) ( )( ) ( )

1 1 2 2

1 1 2 2

( ) ( )( )

( ) l

N s Y sA s B s A s B sM s

IC s D s C s D sX s I

= − −

0

0

is a full equivalence relation.

Coincides with strong system equivalence!

Full System Equivalence – Invariants

• Finite and Infinite Input Decoupling Zeros

• Finite and Infinite Input-Output Decoupling Zeros

• Finite and Infinite Output Decoupling Zeros( )

( ) ( )( ) ( )

1 1. .1

1 1

1 0

f eA s B s

sC s D s

I

= − −

T

V

( )( ) ( ) ( )( ) ( )

. .1 1

1 1

1 1

0f e A s B ss

C s D s I

= −

T U

• Generalized order of Σ ( )1

1

ˆdegr

ii

n A s qµ=

+ = +∑• Rosenbrock degree

• Finite and Infinite System Zeros

• Finite and Infinite System Poles

• Transfer function matrix

• Finite and Infinite Transmission Zeros-Poles

• Rosenbrock degree

( ) ( )

( ) ( ) ( ) ( )

1

1 1 1 1

1

1 1 1 1

G s s

C s A s B s D s

−

−

= =

= +

VT U

( )( ) ( )( ) ( )

1 1

1 1 1

0

0 0

A s B s

s C s D s I

I

= −

T

Geometrical Interpretation of full system equivalence

Two PMDs are fundamentally equivalent if and only if they are normal

Definition [C. Pugh, N. Karampetakis, A.I. Vardulakis and G.E. Hayton 1994] Let ( )i sP , 1,2i = be the normalized form of two PMDs. The PMDs are said to be fundamentally equivalent if there exists bijective polynomial differential map of the form

2 1( ) ( ) ( ) ( ) ( )t N t Y u tξ ρ ξ ρ= + and they have the same output ( )y t .

Two PMDs are fundamentally equivalent if and only if they are normal

full system equivalent.

Full Unimodular equivalence – Karampetakis and Vardulakis 1992

Definition [Karampetakis and Vardulakis 1992] Given two system matrices

( )( )( ) ,0

i

i

R p

iR p m

A s IP s

B s

= −

Then ( ) ( )1 2,P s P s are called fully unimodular equivalent if there exists a

unimodular polynomial matrix ( ) [ ]m mRT s s

×∈ℝ such that

( )( )

( )( )

( )2 1 ,

,, ,

0

00 0

R p R p R m m

m m mR p m R p m

A s I A s I T s

IB s B s

= − −

( ) ( )2 1 ,, ,

00 0 m m mR p m R p m IB s B s − −

where

( )( )

( )

2

2

R

R

R

A s

B s

T s

−

has no infinite zeros and

( )( )

( )

( )( )

2

2

2

2

RR

M R MR

R

A sA s

B sB s

T s

δ δ

− = −

.

Full Unimodular equivalence – Karampetakis and Vardulakis 1992

Definition [Karampetakis and Vardulakis 1992] Let ( ) ( )1 2,P s P s L L∈ ×

( )( ) ( )

,0

Li Lii

p p m

A s B sP s

I

= −

Then ( ) ( )1 2,P s P s are called fully unimodular equivalent if and only if there

exist a unimodular polynomial matrix ( ) [ ]p pLT s s

×∈ℝ such that

( ) ( ) ( ) ( ) ( )2 2 1 1,0

00 0

L L L LL p pA s B s A s B sT s

II I

= − −

,, ,00 0p p pp p m p p m

II I= − −

where ( ) ( ) ( )( )2 2L L LA s B s T s has no infinite zeros and

( ) ( ) ( )( ) ( ) ( )( )2 2 2 2M L L L M L LA s B s T s A s B sδ δ= .

Special cases of full system equivalence

Theorem [Antoniou et. al. 1998] If ( ),r n n nF fC J× ×∈ ∈ℝ ℝ is a finite (resp.

( ),rC Jµ µ µ× ×∞ ∞∈ ∈ℝ ℝ is an infinite) Jordan pair of [ ]( )

r rA s s

×∈ℝ , where

( )( )degdetn A s= (resp. ( ) 1

0

k

krankcol C J

µµ

−

∞ ∞ == ) then the columns of the matrix

Geometric interpretation of finite and infinite elementary

divisors in discrete time systems – Lewis 1984, Antoniou

et.al. 2004, Karampetakis 2004.

0k=

( ) , 0,1,...,kF F Fk C J k NΨ = =

( ) , 0,1,...,N kk C J k N−∞ ∞ ∞Ψ = =

are linearly independent solutions of ( ) ( ) 0A kσ β = for N n≥ ( )N µ≥ .

dim B n qrµ= + =

( ) [ ]0 1q

qr r

A s A A s A s s×

= + + + ∈⋯ ℝ

Divisor equivalence – Karampetakis et.al. 2004

Definition [N. Karampetakis, S. Vologiannidis and A.I. Vardulakis 2004] Two regular matrices ( ) [ ] ,i ir r

iA s s ×∈ℝ 1,2i = with 1 1 2 2deg ( ) deg ( )r A s r A s= , are said to be divisor equivalent if there exist polynomial matrices ( ), ( )M s N s of appropriate dimensions, such that

1 2( ) ( ) ( ) ( )M s A s A s N s= is satisfied and the compound matrices

( )A s

Preserves the finite and infinite elementary divisor structure (extends

the strict equivalence to regular polynomial matrices).

[ ] 12

( )( ) ( ) ,

( )

A sM s A s

N s

−

have no finite and infinite elementary divisors.

Geometric interpretation of divisor equivalence –

Vardulakis and Antoniou 2003

Definition [Vardulakis and Antoniou 2003] Two AR-representations ( ) 0,i iA ρ ξ =

where ( ) [ ] ,i ir riA s s ×∈ℝ 1,2i = with 1 1 2 2deg ( ) deg ( )r A s r A s= will be called

fundamentally equivalent over the finite time interval 0,1,2,..., ,k N= iff there

exists a bijective polynomial map between ( ) ( )1 2( ) : N N

A AN σ σσ →B B .

If 1 2( ), ( ) [ ] i ir rcA A Rσ σ σ ×∈ are divisor equivalent, then the corresponding AR-

representations are fundamentally equivalent.

Conclusions• A review of the main results concerning equivalence transformations

of polynomial matrices and their system theoretic counterparts has

been presented.

• Due to space limitations, we haven’t cover every aspect of the

subject i.e. linearization problem, n-D polynomial matrices and system

matrices.matrices.

• The present work will serve as a reference point for further studies

which will extend the existing theory and address open questions.