Circuits and Systems, 2010, 1, 41-48
doi:10.4236/cs.2010.12007 Published Online October 2010 (http://www.SciRP.org/journal/cs)
Decoupling Zeros of Positive Discrete-Time Linear Systems*
Tadeusz Kaczorek
Faculty of Electrical Engineering, Bialystok University of Technology, Bialystok, Poland
Received July 13, 2010; revised August 16, 2010; accepted August 20, 2010
Abstract
The notions of decoupling zeros of positive discrete-time linear systems are introduced. The relationships
between the decoupling zeros of standard and positive discrete-time linear systems are analyzed. It is shown
that: 1) if the positive system has decoupling zeros then the corresponding standard system has also decoup-
ling zeros, 2) the positive system may not have decoupling zeros when the corresponding standard system
has decoupling zeros, 3) the positive and standard systems have the same decoupling zeros if the rank of
reachability (observability) matrix is equal to the number of linearly independent monomial columns (rows)
and some additional assumptions are satisfied.
Keywords: Input-Decoupling Zeros, Output-Decoupling Zeros, Input-Output Decoupling Zeros, Positive,
Discrete-Time, Linear, System
1. Introduction
In positive systems inputs, state variables and outputs
take only non-negative values. Examples of positive sys-
tems are industrial processes involving chemical reactors,
heat exchangers and distillation columns, storage systems,
compartmental systems, water and atmospheric pollution
models. A variety of models having positive linear be-
havior can be found in engineering, management science,
economics, social sciences, biology and medicine, etc. An
overview of state of the art in positive linear theory is
given in the monographs [1,2].
The notions of controllability and observability and the
decomposition of linear systems have been introduced by
Kalman [3,4]. Those notions are the basic concepts of the
modern control theory [5-9]. They have been also ex-
tended to positive linear systems [1,2].
The reachability and controllability to zero of standard
and positive fractional discrete-time linear systems have
been investigated in [10]. The decomposition of positive
discrete-time linear systems has been addressed in [11].
The notion of decoupling zeros of standard linear systems
have been introduced by Rosebrock [8,12].
In this paper the notions of decoupling zeros will be
extended for positive discrete-time linear systems.
The paper is organized as follows. In Section 2 the ba-
sic definitions and theorems concerning reachability and
observability of positive discrete-time linear systems are
recalled. The decomposition of the pair (A,B) and (A,C)
of positive linear system is addressed in Section 3. The
main result of the paper is given in Section 4 where the
definitions of the decoupling-zeros are proposed and the
relationships between decoupling zeros of standard and
positive discrete-time linear systems are discussed. Con-
cluding remarks are given in Section 5.
2. Preliminaries
The set of nm

real matrices will be denoted by nm


and 1 :.
nn

 The set of mn real matrices with
nonnegative entries will be denoted by mn


 and
1
:.
nn


 The set of nonnegative integers will be de-
noted by
Z

and the nn

identity matrix by .
n
I
Consider the linear discrete-time systems
1, iii
iii
x
AxBui Z
yCxDu


 
 (2.1)
where ,
n
i
x

 m
i
u

, p
i
y are the state, input
and output vectors and ,
nn
A  nm B  ,
p
n
C

 ,
p
m
D

 .
Definition 2.1. The system (2.1) is called (internally)
positive if and only if ,
n
i
x

 and ,
p
i
y

 

Zi
for every 0,
n
x


 and any input sequence ,
m
i
u



iZ


.
Theorem 2.1. [1,2] The system (2.1) is (internally)
positive if and only if
*This work was supported by Ministry of Science and Higher Educa-
tion in Poland under work No NN514 1939 33.

T. KACZOREK 42
,,,
nn nmpnpm
AB CD
 
  
 . (2.2)
Definition 2.2. The positive system (2.1) is called
reachable in q steps if there exists an input sequence
,
m
i
u  0,1,..., 1iq which steers the state of the
system from zero (00x) to any given final state
,
n
f
x  i.e., 0f
x
x.
Let ei, i = 1,…,n be the ith column of the identity ma-
trix In. A column aei for a > 0 is called the monomial
column.
Theorem 2.2. [1,2] The positive system (2.1) is reach-
able in q steps if and only if the reachability matrix
1
[...]
qnqm
q
RBAB AB


 (2.3)
contains n linearly independent monomial columns.
Theorem 2.3. [1,2] The positive system (2.1) is reach-
able in q steps only if the matrix
][ AB (2.4)
contains n linearly independent monomial columns.
Definition 2.3. The positive systems (2.1) is called
observable in q steps if it is possible to find unique initial
state 0
n
x  of the system knowing its input sequence
,
m
i
u  0,1,..., 1iq and its corresponding output
sequence ,
p
i
y  0,1,..., 1iq
.
Theorem 2.4. [1,2] The positive systems (2.1) is ob-
servable in q steps if and only if the observability matrix
1
qp n
q
q
C
CA
O
CA










 (2.5)
contains n linearly independent monomial rows.
Theorem 2.5. [1,2] The positive system (2.1) is ob-
servable in q steps only if the matrix
C
A



(2.6)
contains n linearly independent monomial rows.
3. Decomposition of Positive Pair (A,B) and
(A,C) of Positive Linear Systems
Let the reachability matrix
1
[...]
nnmn
n
RBAB AB


 (3.1)
of the positive system (2.1) has n1 < n linearly inde-
pendent monomial columns and let the columns
12
,,..., k
iii
BBB (km) (3.2)
of the matrix nm B

 be linearly independent mono-
mial columns. We choose from the sequence
11 1
221 1
,..., ,,...,,...,,...,
kk k
nn
iii iii
A
BABABABABAB

(3.3)
monomial columns which are linearly independent from
(3.2) and previously chosen monomial columns. From
those monomial columns we build the monomial matrix
1112 221
1
12
[............ ]
[... ]
kk
iidi ididnn
n
P
PPPP PPP
PP P



(3.4a)
where
1
11 111
2
22 222
1
1 1
,..., ,
,...,,..., k
kk k
d
ii idi
d d
ii idi idi
PB PAB
PB PABPAB

 

 
(3.4b)
and di (1,..., )ik

are some natural numbers.
Theorem 3.1. Let the positive system (2.1) be unrea-
chable, the reachability matrix (3.1) have n1 < n linearly
independent monomial columns and the assumption
11
0 for 1,...,;1,...,
T
kj
PAPknnjn (3.5)
be satisfied.
Then the pair (,)
A
B of the system can be reduced by
the use of the matrix (3.4) to the form
112 2
12 1
11 112 1
2
12 21
12 1
,,
00
,,()
,
nnnn
nn nm
AA B
APAPBPB
A
AA nnn
AB





 
 
 


 
 
(3.6)
where the pair 11 (,)
A
B is reachable and the pair
22
(, 0)AB

is unreachable.
Proof is given in [11].
Theorem 3.2. The transfer matrix
1
() []
n
TzCIz AB D


 (3.7)
of the positive system (2.1) is equal to the transfer matrix
1
1
11 11
() []
n
TzCIz ABD



(3.8)
of its reachable part 111 (,,)
A
BC , where 12 [],CPC C
1
1
p
n
C


 .
Proof is given in [11].
By duality principle [11] we can obtained similar (dual)
result for the pair (A,C) of the positive system (2.1).
Let the observability matrix
1
qn n
n
n
C
CA
O
CA










 (3.9)
has 1 nn

linearly independent monomial rows.
In a similar way as for the pair (A,B) by the choice of n1
linearly independent monomial row for the pair (A,C) we
may find the monomial matrix nn Q

 of the form [11]
12 1
112 21
[............ ]
ll
TT TTT TTT
jj nn
jdj djd
QQQ QQQ QQ

 (3.10a)

T. KACZOREK 43
where
1
11 1
11
2
22 2
22
1
1 1
,..., ,
,...,,..., l
l
ll
d
jj j
jd
d d
jj jj
jd jd
QC QCA
QC QCAQCA

 

 
(3.10b)
and (1,..., )
j
dj l are some natural numbers.
Theorem 3.3. Let the positive system (2.1) be unob-
servable, the matrix (3.9) has n1 < n linearly independent
monomial rows and the assumption
0
T
kj
QAQ  for 1 1,...,kn; 11,...,jn n (3.11)
be satisfied.
Then the pair (A,C) of the system can be reduced by
the use of the matrix (3.10) to the form
112 2
21 1
1
11
1
21 2
12 21
21 1
ˆ0
ˆˆˆ
,[0]
ˆˆ
ˆˆ
,,()
ˆˆ
,
nnnn
nn pn
A
AQAQCCQC
AA
AA nnn
AC






 



 
 
(3.12)
where the pair 11
ˆˆ
(, )
A
C is observable and the pair
22
ˆˆ
(, 0)AC is unobservable.
Proof is given in [11].
Theorem 3.4. The transfer matrix (3.7) of the positive
system (2.1) is equal to the transfer matrix
1
1
11 11
ˆˆ ˆ
() []
n
TzCIz ABD


(3.13)
where
12
1
12
2
ˆˆˆ
,,
ˆ
nmn m
B
QB BB
B







. (3.14)
Proof is given in [11].
Remark 3.1. From Theorem 3.1 and 3.3 it follows that
the conditions for decomposition of the pair (A,B) and
(A,C) of the positive system (2.1) are much stronger than
of the pairs of the standard system.
4. Decoupling Zeros of the Positive Systems
It is well-known [5-8] that the input-decupling zeros of
standard linear systems are the eigenvalues of the matrix
A2 of the unreachable (uncontrollable) part of the system.
Similarly, the output-decoupling zeros of standard linear
systems are the eigenvalues of the matrix of the un-
reachable and unobservable parts of the system. In a
similar way we will defined the decoupling zeros of the
positive linear discrete-time systems.
Definition 4.1. Let 22
2
nn
A

 be the matrix of un-
reachable part of the system (2.1). The zeros
2
12
, ,...,
ii in
zz z of the characteristic polynomial
22
22
1
2110
det[ ]...
nn
nn
I
zAza zaza


 
(4.1)
of the matrix 2 A are called the input-decoupling zero of
the positive system (2.1).
The list of the input-decoupling zeros will be denoted
by 2
12
{, ,...,}
iii in
Z
zz z.
Example 4.1. Consider the positive system (2.1) with
the matrices
102 1
020, 0
003 0
AB











(4.2)
Note that the pair (4.2) has already the form (3.6) with
112
2
1
12
102
020,
0003
1
0, (1,2)
00
AA
AA A
B
BBnn



 








 



(4.3)
In this case the characteristic polynomial of the matrix
2
20
03
A






 has the form
22
2
20
det[]03
(
2
)(
3
)
56
z
Iz Az
zz zz

 


, (4.4)
the input-decoupling zeros are equal to 12
2, 3
ii
zz


and {2, 3}
i
Z

.
Definition 4.2. Let 22
ˆˆ
2
ˆnn
A

 be the matrix of un-
observable part of the system (2.1). The zeros
2
12
, ,...,
oo on
zz z of the characteristic polynomial
22
22
ˆˆ
1
ˆˆ
2110
ˆˆˆˆ
det[ ]...
nn
nn
I
zAza zaza



  (4.5)
of the matrix 2
ˆ
A
are called the output-decoupling zero
of the positive system (2.1).
The list of the output-decoupling zeros will be denoted
by 2
ˆ 12
{ ,,...,}
oooon
Z
zz z.
Example 4.2. Consider the positive system (2.1) with
the matrices (4.2) and
[0 1 0],[0]CD

. (4.6)
The observability matrix
3
2
010
020
040
C
OCA
CA











(4.7)
has only one monomial row 1[010]Q. In this case
the monomial matrix (3.10) has the form
1
2
3
010
100
001
Q
QQ
Q














 
(4.8)

T. KACZOREK 44
and the assumption (3.11) is satisfied since
12 3
10210
[][010]02000 [00]
00301
TT
QAQ Q






(4.9)
Using (3.12) and (4.8) we obtain
1
1
12
21 2
1
1
01010 2010
ˆ100020100
001003001
200 ˆ0
012,( 1,2)
ˆˆ
003
ˆˆ
[0][100]
AQAQ
Ann
AA
CCQC











 






 
Characteristic polynomial of the matrix 2
12
ˆ
03
A







has the form
2
22
12
ˆ
det[]( 1)(3)43
03
z
Iz Azzzz
z

 

the output-decoupling zeros are equal to 12 1, 3
oo
zz
and {1, 3}
o
Z.
Definition 4.3. Zeros (1)(2)( )
00 0
,,..., k
ii i
zzz which are si-
multaneously the input-decoupling zeros and the out-
put-decoupling zeros of the positive system (2.1) are
called the input-output decoupling zeros of the positive
system, i.e.,
i
j
iZz 
)(
0 and ()
0
j
io
zZ for j = 1,…,k; 22
ˆ
min( ,) knn
(4.10)
The list of input-output decoupling zeros will be de-
noted by (1)(2)()
000 0
{ ,,...,}
k
iii i
Z
zz z.
Example 4.3. Consider the positive system (2.1) with
the matrices (4.2) and (4.6). The system has the input-
decoupling 12
2, 3
ii
zz and {2, 3}
i
Z (Example
4.1) and the output-decoupling zero 12
1, 3
oo
zz and
{1, 3}
o
Z (Example 4.2). Therefore, by Definition 4.3
the positive system has one input-output decoupling zero
3
io
z, {3}
io
Z. This zero is the eigenvalue of the
matrix 12 [3]A of the unreachable and unobservable
part of the system. Note that the transfer function of the
system is zero, i.e.,

1
1
() []
1021
0100200 [0][0]
00 30
n
TsCIzABD
z
z
z











(4.11)
since it represents the reachable and observable part of
the system.
Example 4.4. Consider the positive system (2.1) with
the matrices
102 1
021,0 ,[010],[0]
003 0
ABCD






.
(4.12)
Note that the matrices B, C, D are the same as in Ex-
ample 4.1 and 4.2 and the matrix A differs by only one
entry 1
23

a.
The pair (A,B) has already the form (3.6) since
112
2
1
12
102
021,
0003
1
0, (1,2)
00
AA
AA A
B
BBn n



 








 



. (4.13)
The observability matrix
3
2
010
021
045
C
OCA
CA











(4.14)
has only one monomial row 1[0 1 0]Q and
010
100
001
Q











(4.15)
is the same as in Example 4.2. The positive pair (A,C)
can not be decomposed because the assumption (3.11) is
not satisfied, i.e.,
123
10210
[ ][010]02100
00301
[0 1][00]
TT
QAQQ






(4.16)
Now let us consider the standard system (2.1) with
(4.12). In this case the matrix (4.14) has two linearly
independent rows and
010
021
100
Q











(4.17)
Using (3.12) and (4.17) we obtain
1
1
21 2
0101020 01
ˆ0210211 00
100003 210
010 ˆ0
650 ˆˆ
421
AQAQ
A
AA


 

 


 

 


 



 







(4.18a)

T. KACZOREK 45
and
1
1
ˆˆ
[0][100]
CCQC

 , 21 ()nnn (4.18b)
The matrix 2
ˆ[1]
A of the unobservable part of the
standard system has one eigenvalue which is equal to the
output-decoupling zero 11
o
z. Note that the standard
system has two input-decoupling zeros 12
i
z, 3
2

i
z
and has no input-output decoupling zeros.
The transfer function of the positive and standard sys-
tem is equal to zero.
Consider the positive pair
012 1
010... 00
001... 01
,0
000... 1
... 0
nn n
n
AB
aaa a



 
 


 

 


 




(4.19)
with 01 0aa.
The reachability matrix of the pair (4.19)
1
010...0
100...0
[...] 000...0
...
000...0
nnn
n
RBAB AB











 
(4.20)
has rank equal to two and two linearly independent mo-
nomial columns.
In this case the monomial matrix (3.4) has the form
1
010...0
100...0
[...] 001...0
000...1
nn
n
PP P






 





(4.21)
and the assumption (3.5) is satisfied since
2[0 ... 0]T
AP  and 20
T
k
PAP for k = 3,…,n.
Using (3.6) and (4.21) we obtain
1
012 1
112
2
010 ...0 0 1 0...0
001 ...0
1 00... 0
0 0 1...0
000 ...1
... 000...1
010...0
100...0
,
001...0 0
000 ...1
n
APAP
aaa a
AA
A



 
 
 
 


 


 














222( 2)
112
(2)(2)
2
2341
00 10...0
,,
10 00...0
010... 0
001... 0
000... 1
...
n
nn
n
AA
A
aaa a





 
 
 
 










(4.22a)
and
1 2 1
1
010...00
100...011
, 001...00 0
0
000...10
B
BPB B





 



 







(4.22b)
Theorem 4.1. If the rank of the reachability matrix
(4.20) is equal to the number of linearly independent
monomial columns then the input-decoupling zeros of
the standard and positive system with (4.19) are the same
and they are the eigenvalues of the matrix 2
A
. The state
vector xi of the system is independent of the input-de-
coupling zeros for any input ui and zero initial conditions
(x0 = 0).
Proof. By Definition 4.1 the input-decoupling zeros
are the eigenvalues of the matrix 2
A
and they are the
same for standard and positive system since the similar-
ity transformation matrix P has in both cases the same
form (4.21). If the initial conditions are zero then the zet
transformation of xi is given by
111
1
1112 1
2
1
1
111
()()[]()
()
00
1
[] () ()
0
0
0
XzPXzP IzA BUz
Iz AAB
PUs
Iz A
z
Iz AB
PUzUz







  

















(4.23)
where U(z) is the zet transform of ui.
Dual result we obtain for the positive pair
0
1
2
1
1
00...0
10...0
01...0 ,
00...1
[0 1 0...0]
nn
n
n
a
a
Aa
a
C















  (4.24)

T. KACZOREK 46
with 01 0aa.
The observability matrix of the pair (4.24)
1
010...0
100...0
000...0
...
000...0
nn
n
n
C
CA
O
CA









  







 
(4.25)
has rank equal to two and two linearly independent mo-
nomial rows.
In this case the monomial matrix (3.10) has the form
1
010...0
100...0
001...0
000...1
nn
n
Q
P
Q






 







(4.26)
and the assumption (3.11) is satisfied since
2[0 ... 0]QA and 20 k
QAQ  for k = 3,…,n.
Using (3.12) and (4.25) we obtain
1
0
1
2
1
1
21 2
1
ˆ
00...0 0 1 0...0
10...0 1 00... 0
01...0
0 0 1...0
00...1 000...1
010...0
100...0 ˆ0,
001...0 ˆˆ
000 ...1
01
ˆ
00
n
AQAQ
a
a
a
a
A
AA
A




 
 


 


 
 
 
















  

22( 2)2
21
2
3
(2)(2)
24
1
10
00
ˆ
,,
00
00...0
10...0
ˆ01...0
00...1
n
nn
n
A
a
a
Aa
a






















 
(4.27a)
and
1
12
11
010...0
100...0
ˆ[010...0] 001... 0
000...1
ˆˆ
[1 0 ...0][0],[10]
CCQ
CC







 





(4.27b)
Theorem 4.2. If the rank of the observability matrix
(4.25) is equal to the number of linearly independent
monomial rows then the output-decoupling zeros of the
standard and positive system with (4.24) are the same
and they are the eigenvalues of the matrix 2
ˆ
A
. The out-
put yi of the system is independent of the output-decoup-
ling zeros for any input '
ii
uBu and zero initial condi-
tions (x0 = 0).
Proof is similar (dual) to the proof of Theorem 4.1.
Example 4.5. For the positive pair
000
100,( 0),[010]
01
AaC
a






(4.28)
the matrix (4.26) has the form
010
100
001
Q











. (4.29)
Using (3.12) and (4.29) we obtain
1 1
21 2
1212
010 ˆ0
ˆ000 ,
ˆˆ
10
01
ˆˆˆ
,[10],[]
00
A
AQAQ
A
A
a
A
AAa




 










(4.30a)
and
1
11
ˆˆ
[1 00][0],[10]CCQCC

 (4.30b)
The pair 11
ˆˆ
(, )
A
C is observable since
1
11
ˆ10
ˆˆ 01
C
CA








 

and the positive system has one ouput-
decoupling zero 01 za

.
The zet transform of the output for 0

o
x and
)()(' zBUzU

is given by
1
11
()[] '()
ˆˆ
[]'() '()
n
n
TsCIzAU z
CIzAUzz Uz



 (4.31)
and it is independent of the output-decoupling zero.
The presented results can be extended to multi-input
multi-output discrete-time linear systems as follows.
Theorem 4.3. Let the reachability matrix (3.1) of the
positive system (2.1) have rank equal to its 1
nn

line-

T. KACZOREK 47
arly independent monomial columns and the assumption
(3.5) be satisfied. Then the input-decoupling zeros of the
standard and positive system are the same and they are
the eigenvalues of the matrix 2
A
. The state vector xi of
the system is independent of the input-decoupling zeros
for any input vector ui and zero initial conditions.
Proof. If the reachability matrix (3.1) of the system
(2.1) has rank equal to its n1 linearly independent mono-
mial columns and the assumption (3.5) is satisfied then
the similarity transformation matrix P has the same form
for standard and positive system. In this case the matrix
2
A
is the same for standard and positive system. There-
fore, the input-decoupling zero for the standard and posi-
tive system is the same. The second part of the Theorem
can be proved in a similar way as of Theorem 4.1.
Theorem 4.4. Let the observability matrix (3.9) of the
positive system (2.1) has rank equal to its 1 nn

line-
arly independent monomial rows and the assumption
(3.11) be satisfied. Then the output-decoupling zeros of
the standard and positive system are the same and they
are the eigenvalues of the matrix 2
ˆ
A
. The output vector
yi of the system is independent of the output-decoupling
zeros for any input vector '
ii
uBu and zero initial
conditions.
Remark 4.1. Note that if the positive pair (A,B) can be
decomposed then the corresponding standard pair (A,B)
can also be decomposed. Therefore, if the positive sys-
tem (2.1) has input-decoupling zeros then the standard
system (2.1) has also input-decoupling zeros.
Similar (dual) remark we have for the pair (A,C) and
the output-decoupling zeros.
The following example shows that the positive system
(2.1) may not have input-decoupling zeros but the stan-
dard system has input-decoupling zeros.
Example 4.6. The reachability matrix for the positive
pair
121 1
010,0
131 1
AB






(4.32)
has the form
2
124
[] 000
124
BABAB





. (4.33)
It has no monomial columns and it can not be decom-
posed (as the positive pair) but it can be decomposed as a
standard pair since the rank of the reachability matrix
(4.33) is equal to one. The similarity transformation ma-
trix has the form
100
010
101
P





(4.34)
and we obtain
























 
01
01
],12[],2[
,
0
010
010
122
2121
2
121
1
AAA
A
AA
APPA
(4.35a)
]1[,
0
0
0
1
1
1
1

















 B
B
BPB (4.35b)
The matrix 2
A
has the eigenvalues 11
i
z, 20
i
z

.
Therefore, the positive system with (4.32) has not in-
put-decoupling zeros but it has input-decoupling zeros
(11
i
z

, 20
i
z

) as a standard system.
5. Concluding Remarks
The notions of the input-decoupling zero, output-decoup-
ling zero and input-output decoupling zero for positive
discrete-time linear systems have been introduced. The
necessary and sufficient conditions for the reachability
(observability) of positive linear systems are much
stronger than the conditions for standard linear systems
(Theorem 2.2 and 2.4). The conditions for decomposition
of positive system are also much stronger than for the
standard systems. Therefore, the conditions for the exis-
tence of decoupling zeros of positive systems are more
restrictive. It has been shown that: 1) if the positive sys-
tem has decoupling zeros then the corresponding stan-
dard system has also decoupling zeros, 2) the positive
system may not have decoupling zeros when the corre-
sponding standard system has decoupling zeros (Exam-
ple 4.6), 3) the positive and standard system have the
same decoupling zeros if the rank of reachability (ob-
servability) matrix is equal to the number of linearly in-
dependent monomial columns (rows) and the assumption
(3.5) ((3.11)) is satisfied (Theorem 4.3 and 4.4).
The considerations have been illustrated by numerical
examples. Open problems are extension of these consid-
erations to positive continuous-time linear systems and to
positive 2D linear systems.
6. References
[1] L. Farina and S. Rinaldi, “Positive Linear Systems, The-
ory and Applications,” Wiley, New York, 2000.
[2] T. Kaczorek, “Positive 1D and 2D Systems,” Springer
Verlag, London, 2001.
[3] R. E. Kalman, “Mathematical Descriptions of Linear
Systems,” SIAM Journal on Control, Vol. 1, No. 2, 1963,
pp. 152-192.
[4] R. E. Kalman, “On the General Theory of Control Sys-

T. KACZOREK 48
tems,” Proceedings of the First International Congress
on Automatic Control, Butterworth, London, 1960, pp.
481-493.
[5] P. J. Antsaklis and A. N. Michel, “Linear Systems,” Birk-
hauser, Boston, 2006.
[6] T. Kaczorek, “Linear Control Systems,” Vol. 1, Wiley,
New York, 1993.
[7] T. Kailath, “Linear Systems,” Prentice-Hall, Englewood
Cliffs, New York, 1980.
[8] H. H. Rosenbrock, “State-Space and Multivariable The-
ory,” Wiley, New York, 1970.
[9] W. A. Wolovich, “Linear Multivariable Systems,” Sprin-
ger-Verlag, New York, 1974.
[10] T. Kaczorek, “Reachability and Controllability to Zero
Tests for Standard and Positive Fractional Discrete-Time
Systems,” Journal Européen des Systèmes Automatisés,
Vol. 42, No. 6-8, 2008, pp. 770-781.
[11] T. Kaczorek, “Decomposition of the Pairs (A,B) and
(A,C) of the Positive Discrete-Time Linear Systems,”
Proceedings of TRANSCOMP, Zakopane, 6-9 December
2010.
[12] H. H. Rosenbrock, “Comments on Poles and Zeros of
Linear Multivariable Systems: A Survey of the Algebraic
Geometric and Complex Variable Theory,” International
Journal on Control, Vol. 26, No. 1, 1977, pp. 157-161.