Notions of equivalence for linear multivariable systems

Notions of equivalence forNotions 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
Rosenbrock
Israel Gohberg
Jan Willems Paul Fuhrmann Wiliam Wolovich

Similarity
Definition [Gantmacher 1959] Two constant matrices 1 2, ,p p
A A ×
∈ℝ are
similar if there exists an invertible matrix ,M such that
1
1 2A MA M −
=
1
1 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
ij
i v
j v
J diag J
=
=
=
…
…
1 0
0
1
0 0
ij ij
i
v vi
ij
i
J
λ
λ
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 = 1
n n
A ×
∈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 1
2 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 = 1
p p
A ×
∈ℝ , 1
p m
B ×
∈ℝ , 1
l p
C ×
∈ℝ , 1
m l
D ×
∈ℝ ,
1( ) : p
x t →ℝ ℝ is the state vector and ( ) : l
u t →ℝ ℝ , ( ) : m
y 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 1
2 1A M A M−
= , 1
2 1B M B−
= , 2 1C C M= and 2 1D D= .

System similarity
1
2 2 1 1
2 2 1 1
00
00
p p
ml
MsI A B sI A BM
IC D C DI
−
 − −     
=       − −     
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
1
uX
2
uX
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 p
U s s ×
∈ℝ , ( ) [ ]q q
V 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 2
3 8 310 5 10 5
22 23 3 3 9 3 1 161 2 1
2 10 2 10 8 3 310 5 10 5
1 01 14
4 0 2
C
A 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
ij
v
m
i j
j
s sε λ
=
= ∏ − are monic polynomials
The roots of ( )i sε are the finite zeros of ( )A s , whereas the factors ( ) ijm
js 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.
( ) ( )( ) ( ) ( ) ( ) 1
1 2 1 2
p p
M
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 pairs
Theorem ([Gohberg et. al.], pp.184) Let ( )0 0
R , 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 J
rank 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 J
rank 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 n
s sC J× ×
∈ ∈ for every eigenvalue , 1,2,..,is i k= of
( )A s we define a finite Jordan pair ( )R , Rr n n n
C 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
q
qA s A s A s A= + + +⋯

Construction of the smooth solution space
The column span of Jt
CeΨ = defines a solution space C
B of
( ) ( ) 0, : /A t d dtρ β ρ= = (Σ)
with dimension :
dim : total number of finite zeros of ( ) (order accounted for)C
B n A s= =
C
B is known as the smooth solution space of (Σ) defined by (Gohberg et. al. 1982) and
(Vardulakis 1991).
( )
( )
2 23 314 2
110 5 10 5
04 tβρ ρ ρ ρ   − + −  
=    
( )
( )
( )
2 2 61 2 1
210 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 tt
Jt
t t
e e t
Ce e
e e t
 
 
 
 − − 
= =    +   
2 21 1
,
1 1
C t t t
B e e
t
−   
→ =    +   
11 2
2 ,2
1 2
f k k k
B
k
− −   
=    +   
The column span of k
CJΨ = defines a solution space f
B of
( ) ( ) ( ) ( )0, 1A k k kσ β σβ β= = +
with dimension :
dim : total number of finite zeros of ( ) (order accounted for)f
B n A s= =
f
B 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 q
V 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 L
L L L
p p m
A s B s
L 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 Li
i
p p m
A s B s
P s
I
 
=  
− 
Then ( ) ( )1 2,P s P s are called unimodular equivalent if and only if there exist a
unimodular polynomial matrix ( ) [ ]p p
LT 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 m
R R R
R p m
A s I
R 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 I
P s
B s
 
=  
 − 
( ) ( ) ( )
( ) ( ) ( )
R m
R
A t I u t
y t B t
ρ β
ρ β
=
=
( )
( ) ,0i
i
R 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 m
RT 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
R
R R
A s A s
T 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 1970
otherwise called Unimodular Strict System Equivalent
Let
( ) ( )
( ) , 1,2
( ) ( )
i i
i
i i
A s B s
P s i
C s D s
− 
= = 
 
be the system matrices of two PMDs iΣ , where ( ) [ ]r r
iA 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 r
iC 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 m
Y 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 i
j j j j j jP s q q P s= = ≠ɶ ɶ
ɶ ɶ
ɶ
( ) ( ) ( ) ( ) ( )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 i
r 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 i
i
i i
A s B s
P s i
C s D s
− 
= = 
 
be the system matrices of two PMDs iΣ , where ( ) [ ] i ir r
iA s s ×
∈ℝ ,
( ) [ ] ir l
B s s ×
∈ℝ , ( ) [ ] im r
C s s ×
∈ℝ , ( ) [ ]m l
D 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 l
iB s s ×
∈ℝ , ( ) [ ] im r
iC 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 m
Y 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 I
M s N s Y s
A s B s A s B s
X 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 exist2 ( ) ( , )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 s
M 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 m
Y 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
[ ] 1
2
( )
( ) ( ) ,
( )
A s
M 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 , 1,2i
uX 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 1
uX 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 andsystem 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 q
pr 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 p
s 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
qq
A s p r q rq q
S s diag s s
ss
∞
− −+
 
=  
 
⋯ ⋯
is the Smith-MacMillan form at infinity of ( )A s andis 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.
Us
1 0
1
s3
s2
− 1 1
As
s s2
+ s + 1
−1 −s − 1
Vs
− 1
s s + 1 − 1
s2
s2
+ s + 1
1 1
s
=
SAs
∞ s
s2
0
0 1
s2

Connection between the Smith form at infinity and
the Smith form at C of polynomial matrices
Define the dual polynomial matrix of ( )A s
( ) 1
0 1
1q q q
qA 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,..., withj j kq q j k q q q q⋯
Ũs
1 0
s − s3
1
Ãs
s s2
+ s + 1
−s2
−s2
− s
Ṽs
−1 − s −1 − s − s2
1 s
=
SÃs
0 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 C
Define the dual polynomial matrix of ( )A s
( ) 1
0 1
1q q q
qA 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 2
ˆ ˆ ˆ ˆ, 1, 2,..., withj j v v rq q j v r q q qµ τ + += + = + + ≤ ≤ ≤⋯
Ũs
1 0
s − s3
1
Ãs
s s2
+ s + 1
−s2
−s2
− s
Ṽs
−1 − s −1 − s − s2
1 s
=
SÃs
0 s
s2−2
0
0 s2+2

Infinite Jordan pairs
Theorem (Gohberg et. al.1982 ,pp.185) Let ( )R , Rr
C 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 J
µ
∞
∞ ∞
 
 
  =
( ) 1
0 1
1q q q
qA s s A A s A s A
s
− 
= = + + + 
 
ɶ ⋯
1
C J
rank
C J µ
µ∞ ∞
−
∞ ∞
  =
 
 
 
⋮
(c)
0 1 0q
q qA C J A C J A C∞ ∞ − ∞ ∞ ∞+ + + =⋯
Taking a finite Jordan pair ( )R , Ri i ir
i 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 , Rr
C Jµ µ µ× ×
∞ ∞∈ ∈ of ( )A s as
( ) ( )1 2 1 2R ; Rr
r 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 , Rr
C 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 J
µ
∞
∞ ∞
 
 
  =
1
C J
rank
C J µ
µ∞ ∞
−
∞ ∞
  =
 
 
 
⋮
(c)
0 1 0q
q 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
rq
ii
z z
i
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 ,l
C∞
ɶ ( ,l
J∞
ɶ ) , with 1, 2,...,l v v r= + + , consists of the first ˆl
q columns
(the first ˆ ˆl l
q q× block) of the matrix ,l
C∞
( ,l
J∞
), defines a solution space B∞
with 1ˆdim r
i vz i
B 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 i
i v
B q A sµ∞
= +
= = =∑
B∞
is known as the impulsive solution space of (Σ) defined by (Vardulakis 1991).
The whole solution space
C
B B B
∞
= + of (Σ) has dimension
( ) ( ) ( )( )
1 1
ˆdim dim dim degdet degdet
r k
C
z i i M
i v i
B B B n A s q A s q A sµ δ∞
= + =
= + = + = + = + =∑ ∑
The whole solution space
C
B B B
∞
= + of (Σ) has dimension
( ) ( ) ( )( )
1 1
ˆdim dim dim degdet degdet
r k
C
z 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 ˆjq linear independent impulsive solutions.
Therefore to each infinite Jordan block of order ˆ , 1, 2,...,j jq q j v v rµ = + = + +
corresponds ˆjq linear independent impulsive solutions.

Strict equivalence – An extension of Matrix Similarity
H.J.S. Smith
1826-1883
C. Jordan
1838-1922
J. J. Sylvester
1814-1897
K. 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 p
i iE A ×
∈ℝ are strictly equivalent if there exist non singular matrices
, p p
M N ×
∈ℝ such that
1 1 2 2( ) .sE A M sE A N− = −
Every matrix pencil [ ]p p
sE 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 vi
ij
i
J
λ
λ
λ
×
 
 
 = ∈
 
 
 
⋯
⋱ ⋮
ℂ
⋮ ⋱
⋯
{ }, 1, ,iJ diag J i µ∞
∞ = = …
0 1 0
0 0
1
0 0 0
i i
iJ 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 q
i iE A ×
∈ℝ are strictly equivalent if there exist non singular matrices
p p
M ×
∈ℝ , ,q q
N ×
∈ℝ such that
1 1 2 2( ) .sE A M sE A N− = −
Every matrix pencil [ ]p q
sE 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 i
L s sM N s L s sM N sε ε η η
ε 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 i
M 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 p
E A ×
∈ℝ , 1
p m
B ×
∈ℝ , 1
l p
C ×
∈ℝ , 1
m l
D ×
∈ℝ ,
1( ) : p
x t →ℝ ℝ is the state vector and ( ) : l
u t →ℝ ℝ , ( ) : m
y 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
− −      
=       − −      
or in system matrix terms
1
2 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
    
=    − −    
( )
( )
( )
( )
1
1 1 2 2
2 1
0 00
0 m
E x E xM
y t y tI
−
    
=     
    
ObservabilityControllability
1
uX
2
uX
uY
Commute
( ) ( ) ( ) ( ) ( )1
1 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 1
2 2
0
0
sE A
sE 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− −      
1
uX
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= =
2
uX

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 I
M N Y
sE A B sE A B
X 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− = −
[ ] 1 1
2 2 ,
sE A
M 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
[ ] 1 1
,
sE A
M sE A
− 
−  [ ] 1 1
2 2 ,
sE A
M sE A
N
− 
−  − 
have full rank and no zeros in { }∪ ∞ℂ .
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
ˆdeg
r
i
i
n A s qµ
=
+ = + ∑
• Finite and Infinite System Zeros
• Finite and Infinite System Poles
• Finite and Infinite Transmission Zeros-Poles
( ) ( ) 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=
[ ] 1
2
( )
( ) ( ) ,
( )
A s
M 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 β
ρ ρ
    
    − − =    
   −    
i=1,2
( ) ( ) 0 0
( ) ( ) 0 ( )
( )
0 0 0
0 0 0
i i
i i m i i
i
l l i
m
A s B s
C s D s I s
s
I I
I
− 
 − −  = =  −   
 
 
T U
P
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 s
M s A s
 
[ ] 1
2
( )
( ) ( ) ,
( )
A s
M 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 s
A s
N s
δ 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 q
iA s s×
∈ℝ for 1,2,i = their behaviors are
{ : ( ) 0}iq
i 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 q
N s s×
∈ℝ satisfying the
following conditions
1) 2 1
1 2( ) [ ]: ( ) ( ) ( ) ( ).p p
M s s M s A s A s N s×
∃ ∈ =ℝ
2) 1
1
( )
( ( ))
( )
M M
A s
A s
N s
δ δ
 
= 
 
and [ ]2 2( ) ( ) ( ( )).M MM s A s A sδ δ=
3) [ ]1
2
( )
, ( ) ( )
( )
A s
M 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 e
A s B s
s
C s D s
I
 
   
= −   −   
 
T
V
( )( )
( ) ( )
( ) ( )
. .
1 1
1 1
1 1
0f e A s B s
s
C s D s I
 
=  
− 
T U
• Generalized order of Σ ( )1
1
ˆdeg
r
i
i
n A s qµ
=
+ = + ∑
• Finite and Infinite System Zeros
• Finite and Infinite System Poles
• Finite and Infinite Transmission Zeros-Poles
( ) ( )
( ) ( ) ( ) ( )
1
1 1 1 1
1
1 1 1 1
G s s
C s A s B s D s
−
−
= =
= +
V T 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
i
R p m
A s I
P s
B s
 
=  
 − 
Then ( ) ( )1 2,P s P s are called fully unimodular equivalent if there exists a
unimodular polynomial matrix ( ) [ ]m m
RT 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
R
R
M R M
R
R
A s
A s
B s
B 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 Li
i
p p m
A s B s
P 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 p
LT s s
×
∈ℝ such that
( ) ( ) ( ) ( ) ( )2 2 1 1,0
00 0
L L L LL p p
A 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 n
F fC J× ×
∈ ∈ℝ ℝ is a finite (resp.
( ),r
C Jµ µ µ× ×
∞ ∞∈ ∈ℝ ℝ is an infinite) Jordan pair of [ ]( )
r r
A s s
×
∈ℝ , where
( )( )degdetn A s= (resp. ( )
1
0
k
k
rankcol C J
µ
µ
−
∞ ∞
=
= ) 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,...,k
F F Fk C J k NΨ = =
( ) , 0,1,...,N k
k C J k N−
∞ ∞ ∞Ψ = =
are linearly independent solutions of ( ) ( ) 0A kσ β = for N n≥ ( )N µ≥ .
dim B n qrµ= + =
( ) [ ]0 1
q
q
r 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).
[ ] 1
2
( )
( ) ( ) ,
( )
A s
M 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 r
iA 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 r
cA 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.

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