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77-92 SQU  Journal For  Science, 10 (2005) 
 © 2005 Sultan Qaboos University 

77 

Realization in Generalized State 
Space form for 2-D Polynomial 
System Matrices 

*M.S. Boudellioua and **B. Chentouf 

Department of Mathematics and Statistics, College of Science, Sultan 
Qaboos University, P.O. Box 36, Al-Khod 123, Sultanate of Oman,  
Email: *boudell@squ.edu.om, **chentouf@squ.edu.om.   

 تحقيق مصفوفات كثيرات الحدود الممثلة للنظم ثنائية البعد في شكل مجال الحاالت العام   

  محمد الصالح بودليوة  و بومدين شنتوف 

 في هذه الورقة يتم تقديم طريقة مباشرة لتحويل مصفوفة نظام عامة عناصرها كثيرات حدود ثنائية البعد               :خالصة  
التحويل يحافظ على الخاصيات الهامة بما فيها البنية الصفرية لمصفوفة النظام           هذا  . إلى شكل مجال الحاالت العام    

 مع تحديد طبيعـة  (zero coprime system equivalence)وتبين الورقة أن هذا التحويل هو تكافؤ . األصلية
 .المصفوفة الناتجة عن هذا التحويل وطبيعة مصفوفات التحويل نفسه

 
ABSTRACT: In this paper, a direct realization procedure is presented that brings a 
general 2-D polynomial system matrix to generalized state space (GSS) form, such that 
all the relevant properties including the zero structure of the system matrix are retained. It 
is shown that the transformation linking the original 2-D polynomial system matrix with 
its associated GSS form is zero coprime system equivalence. The exact nature of the 
resulting system matrix in GSS form and the transformation involved are established.   
 
KEYWORDS: 2-D Systems, system matrix, generalized state space form, zero coprime 
system equivalence, invariant polynomials, invariant zeros, grobner bases. 

1.  Introduction 

tate space models play an important role in the theory of 1-D finite-dimensional linear systems. In 
recent years attempts have been made toward extending the state space representation to more 

general systems, e.g. time-delay systems or systems described by partial differential equations. 
Another extension from 1-D to 2-D is the discrete linear state space model which has a number of 
variants as given by  Givone and Roesser (1972), Attasi (1973) or Fornasini and Marchesini (1976). 
One of the limitations of these models is that they can only be used to describe 2-D proper transfer 
functions. In other words, they are suitable only for the representation of northeast quarter plane 2-D 
systems. Several authors have suggested a generalized state space description for 2-D systems. Zak 

S 



M.S. BOUDELLIOUA AND B. CHENTOUF  

 78

(1984) suggested a generalized model based on Roesser’s model while Kaczorek (1988) proposed a 
model based on that of Fornasini-Marchesini.  The natural description of a system is not necessarily in 
a state space form, and it is often desirable to reduce such a description into a simpler but equivalent 
form. The reduction of an arbitrary 2-D polynomial system matrix to 2-D generalized state space 
(GSS) form was first studied by Pugh et al. (1998). Their algorithm involves the application of a two 
stage reduction procedure which includes the removal of factors from certain matrices to ensure that 
the transformations linking the original system matrix with the final GSS form are polynomial. The 
method does not give a priori the form of neither the resulting 2-D GSS system matrix nor the 
transformation linking it to the original 2-D polynomial system matrix. In the present work, we 
present a direct and simple procedure for the realization of a 2-D polynomial system matrix by an 
equivalent 2-D polynomial system matrix in GSS form. The exact nature of the resulting GSS system 
matrix and the transformation linking it with the original system matrix will be given. The 
transformation linking the original system matrix to its corresponding GSS form is shown to be zero 
coprime system equivalence. This type of equivalence has been studied by Levy (1981), Johnson 
(1993) and Pugh et al. (1996) and has been shown by Pugh et al. (1998) to provide the connection 
between all least order polynomial realizations of a given  2-D transfer function matrix. 

2.    2-D System Matrices and System Equivalence 

Consider the 2-D system matrix in the general form: 

         ( ) ( ) ( )( ) ( )
, ,

,
, ,

T s z U s z
P s z

V s z W s z
=

−

⎡ ⎤
⎢ ⎥
⎣ ⎦

                        (2.1) 

 
where T(s,z), U(s,z), V(s,z) and W(s,z) are respectively r×r, r×n, m×r and m×n polynomial matrices 
with T(s,z) invertible, in which case the system matrix in (2.1) is said to be regular. The transfer 
function matrix of the system matrix in (2.1) is given by 
 

G(s,z) = V (s,z)T-1(s,z)U(s,z) +W(s,z)                      (2.2) 
 
A special case of (2.1) is obtained from the system described by the following 2-D generalized state 
space discrete equations (Kaczorek 1988), 
                                               Ex(i + 1, j + 1) = ( ) ( )1 21 1, ,A x i j A i jx+ + +  

 + ( ),A x i j + B1u(i + 1, j) 
          + B2 u(i, j + 1) + B0 u(i, j),                                (2.3a)           

   y(i, j) =Cx(i, j) + Du(i, j)                                          (2.3b) 
 
where x(i,j) is the state vector, u(i,j) is the input vector, y(i,j) is the output vector, 

1 2 1 2, , , , , ,E A A A B B B  , C and  D are constant real matrices of appropriate dimensions and E may 
be singular.  Then, taking the 2-D z-transform of (2.3a) and (2.3b) and assuming zero boundary 
conditions yields 
 

    
( )
( ) ( )

1 2 0 1 2 0 0,
,,

B B B x s z

D y s zu s z

szE sA zA A s z
C

− − − + +
=

− −−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦⎣ ⎦
            (2.4) 

The polynomial matrix over [ ],s zR , 



REALIZATION IN GENERALIZED STATE SPACE FORM  

 79

      ( ) 1 2 0 1 2 0,
szE sA zA A sB zB B

P s z
C D

− − − + +
=

−

⎡ ⎤
⎢ ⎥
⎣ ⎦

         (2.5) 

 
in (2.4), is a system matrix in GSS form and the corresponding transfer function is given by 
 

G(s,z) = C(szE - sA1 - zA2 - A0)
-1(sB1 + zB2 + B0) + D              (2.6) 

Definition 2.1  Two polynomial matrices P1(s,z) and S1(s,z) of appropriate dimensions, are said to be 
zero left coprime if the matrix 

 
  [P1(s,z)     S1(s,z)]                          (2.7) 

has full rank for all (s,z) ∈ C 2. 
 
Similarly, P2(s,z) and S2(s,z), of appropriate dimensions, are said to be zero right coprime if the matrix 

           ( ) ( )2 2, ,   
TT TP s z S s z⎡ ⎤⎣ ⎦                                                    (2.8) 

has full rank for all (s,z) ∈ C 2. 
 
Following the results of Youla and Gnavi (1979), it follows that the polynomial matrices P1(s,z) and 
S1(s,z) are zero left coprime if and only if there exist zero right coprime polynomial matrices X(s,z) 
and Y (s,z) of appropriate dimensions satisfying the Bezout’s relation 
 

P1(s,z) X(s,z) + S1(s,z)Y (s,z) = I                                   (2.9) 
 
One immediate result given by Sontag (1980) is that a necessary and sufficient condition for the 
matrices P1(s,z) and S1(s,z) to be zero left coprime is that the matrix in (2.7) is unimodular equivalent 
to the matrix  [ I  0 ]. Similar results can be stated for zero right coprime matrices. 

Definition  2.2  Given a  p×q polynomial matrix P(s,z), the ith order invariant polynomial φi(s,z) of 
P(s,z) is defined by : 

( )
( )
( )

( )1
,

1
,,

1 min ,
0

i

ii

d s z
if i t

d s zs z
if i p q

φ −
≤ ≤

≤ ≤

⎧
⎪

= ⎨
⎪
⎩

            (2.10) 

 

where t is the normal rank of ( ) ( )P , d , 1,s z s z = , d0(s,z) =1 and di(s,z) is the greatest common 
divisor of all the ith order minors of the given matrix  P(s,z). 

As in 1-D case, the zero structure of 2-D systems is a crucial indicator of the system behavior. 
Zerz (1996) has shown that the controllability and observability of a system is connected to the zero 
structure of the associated polynomial matrix. However, unlike the 1-D case, the zero structure of a 
multivariate polynomial matrix is not completely captured by the invariant polynomials.  Therefore 
the following concept of invariant zeros as given by Pugh and El-Nabrawy (2003) is introduced. 

Definition 2.3  Given a p×q polynomial matrix P(s,z), the ith order invariant zeros of P(s,z) are the 
elements of the variety V R  (I i 

[P]) defined by the ideal [ ]PiI  generated by the ith order minors of 



M.S. BOUDELLIOUA AND B. CHENTOUF  

 80

P(s,z). 

Definition 2.4 Let P (m,n) denote the class of (r+m)×(r+n) polynomial matrices where m, n are fixed 
positive integers and r>- min(m, n).  Two polynomial system matrices P1(s,z) and P2(s,z) are said to be 
zero coprime equivalent if there exist polynomial matrices S1(s,z), S2(s,z) of appropriate dimensions 
such that  

( ) ( ) ( ) ( )2 1 2 1, , , ,S s z P s z P s z S s z=            (2.11) 
 
where P1(s,z), S1(s,z) are zero left coprime and P2(s,z), S2(s,z) are zero right coprime. Pugh et al. 
(1996) and Pugh and El-Nabrawy (2003) have shown that zero coprime equivalence exhibits 
fundamental algebraic properties amongst its invariants: 

Lemma 2.1   (Pugh et al. 1996) Suppose that two polynomial matrices P(s,z) and Q(s,z)∈ P (m, n), are 

related by zero coprime equivalence and let [ ] [ ] [ ]1 2, , ,
P P P

hφ φ φ… ,where h min=
[ ] [ ]( ),P Pr m r n+ + , 

denote the invariant polynomials of ( ),P s z  and [ ] [ ] [ ]1 2, , ,
Q Q Q

kφ φ φ… , where 
[ ] [ ]( ),Q Qk min r m r n= + + , denote the invariant polynomials of Q(s,z), then 

[ ] [ ] ( )for 0,1, , 1, 1P Qh i i k ic i max k hφφ − − = − −= …                             (2.12) 
where [ ] [ ] { }1, 1 for any 1, \ 0P Qj j ij c Rφ φ < ∈= =  
Lemma 2.2 (Pugh and El-Nabrawy 2003) Suppose that two polynomial matrices P(s,z) and Q(s,z) 
∈ P (m,n),are related by zero coprime equivalence and let [ ] for 1, , PjI j h= …  

[ ] [ ]( )min ,P Pr m r n= + + denote the ideal generated by the j×j minors of P(s,z) and 
[ ],QiI for

[ ]( )1, , min Qi k r n= = +… denote the ideal generated by the i×i minors of Q(s,z). Then 
                                            [ ] [ ] 0, , ,P Qh i k iI I i h− −= = …                       (2.13) 

 
where ( ) [ ]min 1, 1 and for any 1 in case or ., Ph ih h k i h I i h i k−= − − > = < <  
A basic transformation proposed for the study of 2-D systems is zero coprime system equivalence 
given by Levy (1981) and Johnson (1993).  This transformation, based on zero coprime equivalence is 
characterized by the following definition. 

Definition 2.5 Two polynomial system matrices P1(s,z) and P2(s,z) ∈ P (m,n), are said to be zero 
coprime system equivalent if they are related by the following 

                  
( )
( )

( )1

, 0

,

,
m

M s z

X s z I

S s z

⎡ ⎤
⎢ ⎥
⎣ ⎦

( ) ( )
( ) ( )

( )2

1 1

1 1

, ,

, ,

,

T s z U s z

V s z W s z

P s z

−

⎡ ⎤
⎢ ⎥
⎣ ⎦

 



REALIZATION IN GENERALIZED STATE SPACE FORM  

 81

( ) ( )
( ) ( )

( )1

2 2

2 2

, ,

, ,

,

T s z U s z

V s z W s z

P s z

=
−

⎡ ⎤
⎢ ⎥
⎣ ⎦

( ) ( )

( )2

, ,

0

,
n

N s z Y s z

I

S s z

⎡ ⎤
⎢ ⎥
⎣ ⎦

               (2.14) 

 
where P1(s,z), S1(s,z) are zero left coprime and P2(s,z), S2(s,z) are zero right coprime and M(s,z), 
N(s,z), X(s,z) and Y(s,z) are polynomial matrices of appropriate dimensions. 
The transformation of zero coprime system equivalence is an extension of Fuhrmann’s strict system 
equivalence from the 1-D to the 2-D setting and has been shown by Levy (1981), Johnson (1993) and 
Pugh et al. (1996, 1998) to preserve important properties of the system matrix P(s,z) and plays a key 
role in certain aspects of 2-D systems theory. 

Lemma 2.3   (Johnson 1993) the transformation of zero coprime system equivalence preserves the 
transfer function and in the sense described in Lemma 2.1, the invariant polynomials of the matrices: 

 
• Ti(s,z), i = 1, 2. 
• Pi(s,z), i = 1, 2. 
• [Ti(s,z)   Ui(s,z)], i = 1, 2. 

• ( ) ( ), , 1, 2, .
TT T

i iT s z V s z i− =⎡ ⎤⎣ ⎦  
 
The following lemma is a direct consequence of Lemmas 2.2 and 2.3. 

Lemma 2.4 The transformation of zero coprime system equivalence preserves, in the sense 
described in Lemma 2.2, the invariant zeros of the matrices given in Lemma 2.3. 

3.   Realization in 2-D Generalized State Space Form 

Let ( ) ( ) ( ) [ ], ,r m r nP s z s z+ × +∈ R  be a 2-D polynomial system matrix given by (2.1). First write 
P(s,z) as 

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

,
p q

i j p q
i j p q

i j
P s z P s z P s z P s z P s z P s z

= =

= = + + + +∑∑ …        (3.1) 
where  Pi,j , i = 0, 1, ..., p and  j = 0, 1, ..., q are (r + m)× (r + n) real constant matrices. Now construct 
the block matrices 

( )( ) ( )1 ,

1 1

0 r n pq r n pq

q q

E
E E E

+ − +

−

=
⎡ ⎤
⎢ ⎥
⎢ ⎥⎣ ⎦…

                   (3.2a) 

where 

, 1, 1, 1, 2, ,, .j p j p j jE P P P j q−= =⎡ ⎤⎣ ⎦ …                      (3.2b) 

( )( )( )0 0,01 ,r n pqA Diag I P+ −= − − ,                               (3.3) 



M.S. BOUDELLIOUA AND B. CHENTOUF  

 82

( ) ( ) ( )

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

( ) ( ) ( )

1 ,

1 1 , 1 1

,0 1,0 1,0, 1

0

0

0

r n p q r n pq

r n p r n pq p r n p

p pr m r n p q

A I

P P P

+ − +

+ − + − + + −

−+ + −

=

− − −

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

,                (3.4) 

and 

( ) ( ) ( ) ( ) ( )

( )( ) ( )

1 , 1

2 1 ,

2, 2, 1 2,1

0

0

r n p q r n p r n p q

r n p r n pq

q q

I

A

A A A

+ − + + −

+ − +

−

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

          (3.5a) 

where 

( ) ( )( )2, o,, 10 1, 2, , .,j jr m r n pA P j q+ + −= − =⎡ ⎤⎣ ⎦ …            (3.5b) 
 
Theorem  3.1  Let the matrices E, A0, A1 and A2 be as constructed in (3.2a), (3.3), (3.4) and (3.5a), 
respectively. Then the [(r +n)pq +2m]×[(r +n)pq +m+n] polynomial system matrix is in GSS form 
(2.5):] 

( )
1 2 0 0

, 0

0 0

m

n n

m

szE sA zA A Z

Q s z Z I

I

− − − −

−

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦

             (3.6) 

 

where ( ),0n nn r n pq r m nZ I⎡ + + + − ⎤⎣ ⎦
= ⎡ ⎤
⎣ ⎦

 and ( ),0
T
m mm r n pq m

Z I
⎡ + − ⎤⎣ ⎦

= ⎡ ⎤
⎣ ⎦

 is related to the original 

system matrix P(s,z) by the following :  
 

     ( ) ( ) ( ) ( )1 2, , , ,S s z P s z Q s z S s z=                                               (3.7) 
where 

( )

( )( ) ( )( )

( ) ( )
( )

1

1 , 1 , 2

,
1 2

, ,

,

,

0 0

0
, ,

0 0

0
0

,

r n pq r r n pq m

r r m

qm n r m n m

m r m

n r n

Y

Y

I
S s z S s z

Y

V WI
I

+ − + −

+ +

= =

−

⎡ ⎤
⎢ ⎥⎡ ⎤
⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥

⎣ ⎦ ⎢ ⎥
⎢ ⎥⎣ ⎦

                              (3.8) 

1 2 ,
Tp q j p q j q j

j r nY s z s z z I
− − − − −

+⎡ ⎤= ⊗⎣ ⎦            (3.9) 
 
where    j = 1, 2, . . . , q and ⊗ denotes the matrix Kronecker product. 
Proof.  The matrix Q(s,z) in (3.6) can be represented in the form 
 
 



REALIZATION IN GENERALIZED STATE SPACE FORM  

 83

( )

( ) ( )

( )

( ) ( )

1 2 1

0 0 0 0

0 0 0 0 0

, 0 0 0 0

0

0 0 0 0

0 0 0 0 0

p r n p r n

p r n

p r n p r n

q q m

n n

m

I zI

I

Q s z I zI

Q Q Q Q Z

Z I

I

+ +

+

+ +

−

−

= −

−

−

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦

           (3.10) 

where 
 

1

,1 ,0 1,1 10 1,1 1,0 0,1 0,0

0

0 0

0 0

,

r n r n

r n

r n

p p p p

I sI

I

Q

sI

szP sP szP sP szP sP zP P

+ +

+

+

− −

−

=

−

+ + + + +

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦

         (3.11) 

 
 

, 1, 2, 1, 0,

0 0 0 0

0 0 0 0

2, 3, ,

0 0 0 0

,j

p j p j j j j

Q j q

szP szP szP szP zP−

= =

+

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦

…         (3.12) 

 
and 
 

( ) ( )( ), , 10 0,
T

n n m mn p r n n m p r n rZ I Z I+ − − + += =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦             (3.13) 
 
and the matrix S2(s,z) can be written as 



M.S. BOUDELLIOUA AND B. CHENTOUF  

 84

( )

1 1

2 1

1

1 2

2 2

2
2

,

0

p q
r n

p q
r n

q
r n

p q
r n

p q
r n

q
r n

r n

n

s z I

s z I

z I

s z I

s z I

S s z

z I

I

V W

I

− −
+

− −
+

−
+

− −
+

− −
+

−
+

+

=

−

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

              (3.14) 

From which it can be easily verified that 

( ) ( ) ( ) ( )

( )( ) ( )( )1 , 1 ,

1 2
, ,

0 0

, , , ,
0 0

r n pq r r n pq n

m n r m n n

T U
S s z P s z Q s z S s z

V W

+ − + −

+ +

≡

−

⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

           (3.15) 

 
Lemma 3.1  The matrices in (3.7), Q(s,z) and S1(s,z) are zero left coprime and P(s,z) and S2(s,z) are 
zero right coprime. 
 
Proof.  The matrix [ Q(s,z)  S1(s,z) ] is given by 

( ) ( )

( )

( ) ( )

1 2 1

0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0

0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

p r n p r n

p r n

rp r n p r n

q q m

n n

m m

I zI

I

I zI I

Q Q Q Q Z

Z I

I I

+ +

+

+ +

−

−

−

−

−

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

      (3.16) 

 
It can be easily seen that the minor obtained by deleting the columns( )( )1r n pq+ − 1,+ ( ), r n+…  
pq  from the matrix in (3.16) is equal to ±1. 



REALIZATION IN GENERALIZED STATE SPACE FORM  

 85

Similarly, the matrix 
( )
( )2

,

,

P s z

S s z

⎡ ⎤
⎢ ⎥
⎣ ⎦

 is given by 

 

( )
1

2

1

2

,

,

0

p
r n

p
r n

r n

n r n

P s z

Y

Y

s I

s I

I

V W

I

−
+

−
+

+

−

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

                                     (3.17) 

 
It is clear that the matrix in (3.17) contains a block identity matrix Ir+n, and therefore it has the highest 
order minor which is equal to 1. 
Theorem  3.2  If P(s,z) is an arbitrary (r+m)×(r+n) polynomial system matrix over [ ]s,zR  given by 
(2.1) and Q(s,z) is the corresponding [(r+n) pq+2m]×[(r+n) pq + m + n] 2-D system matrix in GSS 
form (3.6), then P(s,z) and Q(s,z) are zero coprime system equivalent. 
 
Proof.  The result follows immediately from Theorem 3.1 and Lemma 3.1. 
 
4.   Example 
Consider the system matrix ( ) [ ]2 2, ,P s z s z×∈ R  given by 
 

( )
( ) ( )
( ) ( )

, ,
,

, ,

t s z u s z
P s z

v s z w s z
≡

−

⎡ ⎤
⎢ ⎥
⎣ ⎦

                                       (4.1) 

where 
t(s,z) =(z2 + 1)s2 - (2z2 - z - 3)s + z2 - 4z + 1,                          (4.2a) 
  u(s,z) =(z2 - z)s2 - (z2 - 2)s + z2 – z,              (4.2b) 
v(s,z) = - (z + 2)s2 + (z2 - z)s + 4z + 1,             (4.2c) 
w(s,z) =(2z2 - z)s2 + 5zs + z2 - z + 3             (4.2d) 

 
Here  r = m = n = 1 and  p = q = 2. 
 
Using Maple, the transfer function of the system matrix P(s,z) is given by : 
 



M.S. BOUDELLIOUA AND B. CHENTOUF  

 86

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

( ) ( )

1
2 2 2 2

3 2 4 4 3 2 3

4 3 2 4 3 3 2

, 1 2 3 4 1

 2 3 2 4

4 2 6 13 2 4 2 2 3],

                [

                

P
G s z z s z s z z

z z z s z z z z s

z z z s z z z s z z z

z
−

= + − − − + − +

× − + − + − + − −

− − + + − + + + − − +

⎡ ⎤
⎣ ⎦

(4.3) 

 
the invariant polynomials of   P(s,z) are computed as : 
 

[ ]

[ ] ( ) ( )
( )
( )

1

4 3 2 4 4 3 2 3
2

4 3 2 2

4 3 2

4 3 2

1

2 2 3 8 8 4

2 16 13 12 3

2 24 13 11

5 14 3,

      

      

      

P

P
z z z z s z z z s

z z z z s

z z z z s

z z z z

φ

φ

=

= − + + + − + + −

+ − + + +

+ − + − + +

+ − + − +

         (4.4) 

 
The reduced Grobner bases of the ideals generated by the minors of the matrices in Lemma  2.2 
associated with  P(s,z) are given by : 
 

Matrix Order of 
i minors 

Ideal generated by the  i i× minors 

( ) ( ), ,t s z u s z⎡ ⎤⎣ ⎦  1 
8 7 6 5 4

3 2

7 6 5 4

3 2

2 11 32 39

2 19 4 4

16 3 6 35 98

137 50 27 10

,

z z z z z

z z z

s z z z z

z z z

− + − +

− − − +

+ − + −

+ − − +

 

( )
( )
,

,

t s z

v s z−

⎡ ⎤
⎢ ⎥
⎢ ⎥⎣ ⎦

 
1 8 6 5 4

3 2

7 6

5 4 3

2

17 35 22

103 29 114 9

179880 1879 1563

37934 93203 202249

68516 12719 102237

,

z z z z

z z z

s z z

z z z

z z

+ − +

+ − − −

+ −

+ − +

− − +

 

( ),P s z  1 1                        



REALIZATION IN GENERALIZED STATE SPACE FORM  

 87

        ( ),P s z  2 ( )
( )
( )
( )

4 3 2 4

4 3 2 3

4 3 2 2

4 3 2

4 3 2

2 2

3 8 8 4

2 16 13 12 3

2 24 13 11

5 14 3

z z z z s

z z s

z z z z s

z z z z s

z z z z

z

+

+

− + +

+ − + + −

+ − + + +

+ − + − +

+ − + −
 

Writing  P(s,z) in the form (3.1), 

 

( )

0 ,0 0 ,1 0 ,2

1,1 1,21,0

2 ,0

0 0 0 1 0 2

1 0 1 1 1 2

0 0

1 0 4 1 1 1
,

1 3 4 1 0 1

3 2 1 0 2 1

0 0 1 5 1 0

1 0 0 1

2 0

             

             

P P

P s z s z s z s z

P P P

s z s z s z

P

s z

P

−
= + +

− − −

− −
+ + +

−

+ +

⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤
⎢ ⎥
⎣ ⎦

2 ,1 2 ,2

2 1 2 21 1

1 1 0 2
s z s z

P P

+
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

                     (4.6) 

 
and constructing the 10×10 polynomial system matrix in GSS form Q(s,z) corresponding to (3.6) 
gives 

( ) ( ) ( )( ) ( )

4 4

2 1 1

1

0 0

0, ,
,

0 0 1, ,

0 0 1 0

T

I zI

Q Q ZT s z U s z
Q s z

ZV s z W s z
≡

−

−⎡ ⎤
⎢ ⎥−⎡ ⎤ ⎢ ⎥≡⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎢ ⎥
⎢ ⎥⎣ ⎦

              (4.7) 

where 

2 2
1

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

1 0 0

0 1 0

3 4 1

2 4 1 5 3

   
2

I sI
Q

szP sP szP sP zP P

s

s

s sz sz s z s z

sz s sz sz z sz z

−
≡

+ + + +

−

−
=

− + − + −

+ − − − − +

⎡ ⎤
⎢ ⎥
⎣ ⎦
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦

             (4.8) 

 

  (4.5)



M.S. BOUDELLIOUA AND B. CHENTOUF  

 88

2,2 2,2
2

2,2 1,2 0,2

0 0 0 0

0 0 0 0 0 0

2

0 2

Q
szP szP zP sz sz sz sz z

sz sz z

≡ =
+ − − +

−

⎡ ⎤
⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦
⎢ ⎥
⎣ ⎦

           (4.9) 

and  1 1Z Z=  is the fourth column of  I4. 
The matrices E, A0, A1 and A2 corresponding to (3.2a, 3.3, 3.4 and 3.5a) are given by 
 

6,2 6,2 6,2 6,26,4 6,4

2,2 1,2 2,1 1,12 1

0 0 0 00 0

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 1 2 1 0 1 1 0
0 2 1 0 1 1 1 5

   

E
P P P PE E

≡ ≡

=

− − −
− −

⎡ ⎤⎡ ⎤
⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

,          (4.10) 

6
0

0,0

1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 00
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 1 3

I
A

P

−
−

−
−

=
−−

−
−

−

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥

−⎡ ⎤ ⎢ ⎥≡ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎢ ⎥

⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

,        (4.11) 

4,4 4,2 4,2

1 2,4 2,2 2

2,4 2,0 1,0

0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 00 0 0
0 0 0 0 0 0 0 0

0 0
0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1
0 0 0 0 1 0 3 2
0 0 0 0 2 0 0 0

A I

P P

≡ =

− − −
−

⎡ ⎤
⎢ ⎥
⎢ ⎥

⎡ ⎤ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥− −⎣ ⎦

⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

,             (4.12) 

 



REALIZATION IN GENERALIZED STATE SPACE FORM  

 89

2,2 2,2 2 2,2
4,4 4

2,2 2,2 2,2 2
2 2,4 2,4

2,2 2,2 2,2 2,2
2,2 2,1

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

0 0 0
0

0 0 0
0 0

0 0 0 0

0 0

0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 1 1 0 0 4 1
0 0 0 1 0 0 4 1

    

I
I

I
A

A A
P P

≡ ≡

=

− −
−

⎡ ⎤
⎡ ⎤ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦ − −⎢ ⎥⎣ ⎦
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢
⎢
⎢
⎢
⎢
⎢⎣ ⎦

⎥
⎥
⎥
⎥
⎥
⎥

          (4.13) 

 
By virtue of Theorem 3.1 and Theorem 3.2, the polynomial matrix P(s,z) in (4.1) and the 
corresponding system matrix in GSS form Q(s,z) in (4.7) are related by the zero coprime system 
equivalence transformation  S1(s,z)P(s,z) = Q(s,z)S2(s,z), where 
 

   ( )

6,1 6,1

1
2,1 2,1

0 0

1 0
,

0 0

0 1

S s z =

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

,                                            (4.14) 

and 

( )

( ) ( ) ( )

2

2 2 2 2 2

0

0

0

0

0
, 0

1 0

0 1

2 4 1 2 5 3

0 1

sz

sz

z

z

s
S s z s

z s z z s z z z s sz z z

=

+ − + − − − + + − +

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦

,           (4.15) 

 
In fact it can be easily verified that 



M.S. BOUDELLIOUA AND B. CHENTOUF  

 90

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

( ) ( )

6,1 6,1

1 2
2,1 2,1

0 0

, ,
, , , ,

0 0

, ,

t s z u s z
S s z P s z Q s z S s z

v s z w s z

= ≡

⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
−⎣ ⎦

       (4.16) 

 
where t(s,z), u(s,z), v(s,z) and w(s,z) are given by (4.2a), (4.2b), (4.2c) and (4.2d),  respectively. 
 
The matrices Q(s,z), S1(s,z) are zero left coprime and the matrices P(s,z), S2 (s,z) are zero right 
coprime since the matrices 

             [ Q(s,z)   S1(s,z) ] 
and 

    
( )
( )2

,

,

P s z

S s z

⎡ ⎤
⎢ ⎥
⎣ ⎦

 

 
have respectively a 10×10 and a 2×2 minor which is equal to 1. The transfer function of the system 
matrix Q(s,z) is given by : 

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

( ) ( )
[ ] ( )

1
2 2 2 2

3 2 4 4 3 2 3

4 3 2 4 3 3 2

, 1 2 3 4 1

2 3 2 4

4 2 6 13 2 4 2 2 3]

,

                 [

                

               

Q

P

G s z z s z z s z z

z z z s z z z z s

z z z s z z z s z z z

G s z

−
= + − − − + − +

× − + − + − + − −

− − + + − + + + − − +

=

⎡ ⎤
⎣ ⎦

   (4.17) 

 
and the invariant polynomials of Q(s,z) are : 
 

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

[ ]

[ ] ( ) ( )
( )

[ ]

1 2 3 4 5 6 7 8 9

1

4 3 2 4 4 3 2 3
10

4 3 2 2

4 3 2

2

1

2 2 3 8 8 4

2 16 13 12 3

5 14 3

     ,

     

     

    

Q Q Q Q Q Q Q Q Q

P

Q

P

z z z z s z z z s

z z z z s

z z z z

φ φ φ φ φ φ φ φ φ

φ

φ

φ

= = = = = = = = =

=

= − + + + − + + −

+ − + + +

+ − + − +

=

           (4.18) 

 
 
 
 
 
 
 



REALIZATION IN GENERALIZED STATE SPACE FORM  

 91

The reduced Grobner bases of the ideals generated by the minors of the matrices in Lemma 2.2 
associated with Q(s,z) are given by: 
 

Matrix Order i 
of minors 

Ideal generated by 
the i × i minors 

1, , 8i = …  1  

[ T(s,z)   U(s,z) ] 
9 

8 7 6 4

3 2

7 6 5 4

3 2

2 11 32 39

2 19 4 4,

16 3 6 35 98

137 50 27 10

z z z z z

z z z

s z z z z

z z z

+

+

− + − +

− − −

− + −

+ − − +

 

1, , 8i = …  1  ( )
( )
,

,

T s z

V s z−

⎡ ⎤
⎢ ⎥
⎣ ⎦

 

9 

8 6 5 4

3 2

7 6

5 4 3

2

17 35 22

103 29 114 9,

179880 1879 1563

37934 93203 202249

68516 12719 102237

z z z z

z z z

s z z

z z z

z z

+

− −

+ −

+ − +

− − +

− +

+ −

 

( ),Q s z  1, , 9i = …  1  
 
 
 

( ),Q s z  
10 

( )
( )
( )
( )

4 3 2 4

4 3 2 3

4 3 2 2

4 3 2

4 3 2

2 2

3 8 8 4

2 16 13 12 3

2 24 13 11

5 14 3

z z z z s

z z z s

z z z z s

z z z z s

z z z z

+ +

− + −

− + + +

− + − + +

+

+ +

+

+

+ − + −

 

(4.19) 
which is in accord with Lemmas 2.3 and 2.2. 

5.    Conclusions 

In this paper, a new direct reduction procedure to obtain a realization in GSS form for an 
arbitrary 2-D polynomial system matrix has been presented. The numerical operations required to 
determine the GSS form are simple. The exact connection between the original system matrix with its 
corresponding GSS form has been set out and shown to be zero coprime system equivalence. The zero 
structure of the original polynomial system matrix is preserved making it possible to analyze the 
polynomial system matrix in terms of its associated GSS form. The resulting 2- D system matrix may 
be larger in size than the one obtained by the algorithm used by Pugh et al. (1998), however, the 
method presented in this paper has the advantage of providing a priori both the final 2-D system 



M.S. BOUDELLIOUA AND B. CHENTOUF  

 92

matrix in GSS form and the transformation relating it to the original polynomial system matrix.  To 
reduce the size of the resulting system matrix while preserving its GSS form, a constant zero coprime 
system equivalence transformation may be used. 

6.       Acknowledgment 

The authors wish to express their thanks to Sultan Qaboos University (Oman) for their support in 
carrying out this research work and to the anonimous reviewers for their helpful comments. 

7.     References 

ATTASI, S.  1973.  Systemes lineaires a deux indices, Technical Report 31, IRIA. FORNASINI,  
E. and MARCHESINI, G. 1976. State space realization theory of two-dimensional filters, IEEE 
Trans. Autom. Control AC-21, 4: 484-492. 

FUHRMANN, P.A. 1977.  On strict system equivalence and similarity, Int. J. Control 25(1): 5–
10. 

GIVONE, D.D. and ROESSER, R.P. 1972. Multidimensional linear iterative circuits- General 
properties, IEEE Trans. Computers AC-21, 10: 1067-1073. 

JOHNSON, D.S. 1993. Coprimeness in multidimensional system theory and symbolic 
computation, PhD thesis, Loughborough University of Technology, UK. 

KACZOREK, T. 1988. The singular general model of 2-D systems and its solution, IEEE Trans. 
Autom. Control 33, 11: 1060-1061. 

LEVY, B.C. 1981. 2-D polynomial and rational matrices and their applications for the modelling 
of 2-D dynamical systems, PhD thesis, Stanford University, USA. 

PUGH, A.C. and EL-NABRAWY, E.M.O. 2003. Zero structures of n-D systems, Proceedings 
Mediterranean Conference on Control and Automation, Rhodes (Greece). 

PUGH, A.C., MCINERNEY, S.J., M. HOU and HAYTON, G.E. 1996. A transformation for 2-D 
systems and its invariants, in Proceedings of the 35th IEEE conference on decision and 
control, Kobe (Japan). 

PUGH, A.C., MCINERNEY, S.J., BOUDELLIOUA, M.S.,  JOHNSON, D.S. and HAYTON, 
G.E. 1998. A transformation for 2-D linear systems and a generalization of a theorem of 
Rosenbrock, Int. J. Control, 71(3): 491-503. 

PUGH, A.C., MCINERNEY, S.J., BOUDELLIOUA, M.S. and HAYTON, G.E. 1998. Matrix 
pencil equivalents of a general 2-D polynomial matrix, Int. J. Control, 71(6): 1027-1050. 

SONTAG, E.D. 1980. On generalized inverses of polynomial and other matrices, IEEE Trans. 
Autom. Control AC-25(3): 514-517. 

YOULA, D.C. and GNAVI, G. 1979. Notes on n-dimensional system theory, IEEE Trans. Circuits 
and Systems CAS-26(2): 105-111. 

ZAK, S.H. 1984. On state-space models for systems described by partial differential equations, in 
Proceedings of the 33rd IEEE conference on decision and control, Las Vegas (USA). 

ZERZ, E. 1996. Primeness of multivariate polynomial matrices, Systems and Control Letters 29: 
139-145. 

 
 
 

Received   8 September 2004        
Accepted   20 July 2005