545
1 INTRODUCTION
Adynamicalsystemiscalledpositiveifitstrajectory
starting from any nonnegative initial state remains
forever in the positive orthant for all nonnegative
inputs. An overview of state of the art in positive
systems theory is given in the monographs [2, 14].
Variety of models having positive
behavior can be
found in engineering, economics, social sciences,
biologyandmedicine,etc.[2,14].
ThedeterminationofthematricesA,B,C,Dofthe
state equations of linear systems for given their
transfermatricesiscalledtherealizationproblem.The
realizationproblemisaclassicalproblemof
analysis
of linear systems and has been considered in many
booksandpapers[4‐6,12,13,23,25].Atutorialonthe
positive realization problem has been given in the
paper [1] and in the books [2, 14]. The positive
minimal realization problem for linear systems
withoutandwith
delayshasbeenanalyzedin[3,7,9,
10,14‐18,21,22,24].Theexistenceanddetermination
of the set of Metzler matrices for given stable
polynomials have been considered in [11]. The
realization problem for positive 2D hybrid systems
has been addressed in [20]. For fractional linear
systemsthe
realizationproblemhasbeenconsidered
in [4, 19, 23, 25]. A method for computation of
positive realizations of descriptor continuous‐time
linearsystemshasbeenproposedin[8].
Inthis paper a new method for determination of
positiverealizationsofdescriptorlineardiscrete‐time
systemsisproposed.
The paper is
organized as follows. In section 2
some definitions and theorems concerning the
positive discrete‐time linear systems are recalled. A
newmethodforcomputationofpositiverealizations
for single‐input single‐output linear systems is
proposed in section 3 and for multi‐input multi‐
outputsystemsinsection4.Concludingremarks
are
giveninsection5.
Thefollowingnotationwillbeused:
‐theset
of real numbers,
mn
‐ the set of
mn
real
matrices,
mn
‐ the set of
mn
real matrices
with nonnegative entries and
1
nn
,
Z
‐the
setof nonnegativeintegers,
n
I ‐the nn
identity
matrix.
Computation of Positive Realizations for Descriptor
Linear Discrete-time Systems
T.Kaczorek
BiałystokUniversityofTechnology,Białystok,Poland
ABSTRACT:Anewmethod forcomputationof positiverealizations ofgiven transfermatrices ofdescriptor
lineardiscrete‐timelinearsystemsisproposed.Necessaryandsufficientconditionsfortheexistenceofpositive
realizations of transfer matrices are given. A procedure for computation of the
positive realizations for
descriptordiscrete‐timelinearsystemsisproposedandillustratedbyexamples.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 13
Number 3
September 2019
DOI:10.12716/1001.13.03.08
546
2 PRELIMINARIES
Considerthediscrete‐timelinearsystem
iii
BuAxx
1
,(2.1a)
iii
DuCxy ,(2.1b)
where
n
i
x
,
m
i
u
,
p
i
y
are the state,
inputand output vectors and
nn
A
,
mn
B
,
np
C
,
mp
D
.
Definition 2.1. [2, 14] The system (2.1) is called
(internally) positive if
n
i
x
and
p
i
y
,
Zi
for any initial conditions
n
x
0
and all
inputs
m
i
u
,
Zi
.
Theorem2.1.[2,14]Thesystem(2.1)ispositiveif
andonlyif
nn
A
,
mn
B
,
np
C
,
mp
D
. (2.2)
Thetransfermatrixofthesystem(2.1)isgivenby
DBAzICzT
n
1
][)(
. (2.3)
Thetransfermatrixiscalledproperif
mp
z
DzT

)(lim
(2.4)
anditiscalledstrictlyproperif
0D
.
Definition2.2.[1,25]Thematrices(2.2)arecalled
a positive realization of
)(zT
if they satisfy the
equality(2.3).
Definition2.3.[1,25]Thematrices(2.2)arecalled
asymptoticallystablerealizationof(2.3)ifthematrix
nn
A
is an asymptotically stable matrix (Schur
matrix).
Theorem2.2.[1,25]Thepositiverealization(2.2)is
asymptoticallystableifandonlyifallcoefficientsof
thepolynomial
01
1
1
...])1(det[)( azazazAzIzp
n
n
n
nA
(2.5)
arepositive,i.e.
0
i
a for 1,...,1,0
ni .
Thepositive realization problemforthe standard
system can be stated as follows. Given a proper
transfer matrix
)(zT
find its positive realization
(2.2).
Theorem2.3.[25]If(2.2)isapositiverealizationof
(2.3)thenthematrices
1
PA
P
A , PB
B
,
1
CPC
, DD (2.6)
arealsoapositiverealizationof(2.3)ifandonlyifthe
matrix
nn
P
isamonomialmatrix(ineachrow
andineachcolumnonlyoneentryispositiveandthe
remainingentriesarezero).
Proof. Proof follows immediately from the fact
that
nn
P
1
if and only if P is a monomial
matrix.□
Theorem 2.4. The polynomial
)(zp
n
with zeros
k
z , 0Re
k
z ,
nk ,...,1
hastheform
0
3
3
2
2
1
1
)1(...)( azazazazzp
nn
n
n
n
n
n
n
n
(2.7)
anditsrealcoefficients
k
a satisfythecondition
0
k
a for
1,...,1,0
nk
. (2.8)
Proof. Proof will be accomplished by induction.
Thehypothesisistruefor
1n
and
2n
since
01
)( azzp
,
0
0
a
and
222
212
2))(())(()(
zzjzjzzzzzzp
.
Assumingthatthehypothesisistrueforkweshall
showthatitisalsovalidfor
1k
:
1
12
12 0
111
12 0
() ()( )
( ... ( 1) )( )
( ) ( ) ... ( 1) .
kk
kk k k
kk
kkkk
kk
pz pzz
zaz az az
za za z a








Therefore, the hypothesis is true for any k. The
proofforapairofcomplexconjugatezerosissimilar.
□
3 COMPUTATIONOFPOSITIVEREALIZATIONS
OFDESCRIPTORSINGLE‐INPUTSINGLE‐
OUTPUTSYSTEMS
Considerthedescriptordescriptor‐timelinearsystem
iii
BuAxEx
1
,(3.1a)
ii
Cxy
,(3.1b)
where
n
i
x
,
m
i
u
,
p
i
y
are the state,
input and output vectors and
nn
AE
, ,
mn
B
,
np
C
,
mp
D
.
It is assumed that
0det E
and the pencil of
),( AE
isregular,i.e.
0]det[
AEz
forsome
Cz
(thefieldof
complexnumbers). (3.2)
Definition 3.1. The descriptor system (3.1) is
called (internally) positive if
n
i
x
,
p
i
y
,
Zi
foranyconsistentinitialconditions
n
x
0
andallinputs
m
i
u
, qi ,...,1,0
.
Thetransfermatrixofthesystem(3.1)
547
)(][)(
1
zBAEzCzT
mp
(3.3)
can be decomposed in the polynomial part
)(zP
andstrictlyproperpart
)(zT
sp
,i.e.
)()()( zTzPzT
sp
, (3.4a)
where
][...)(
10
zzPzPPzP
mpq
q
(3.4b)
and
BAzICzT
nsp
1
][)(
. (3.4c)
Firstthenewmethodforcomputationofapositive
realization of given transfer function will be
presented.
Theorem3.1.Thereexiststhepositiverealization
n
n
z
z
z
z
A
1000
0000
0001
0000
1
2
1
,
n
b
b
b
B
2
1
,
]100[ C (3.5)
ofthetransferfunction
01
1
1
01
1
1
...
...
)(
dzdzdz
mzmzm
sT
n
n
n
n
n
sp
 (3.6)
ifandonlyif
n
n
n
n
n
m
m
m
zzzzz
zzzzzz
b
b
b
B
1
1
0
1
12121
121211
2
1
1000
...10
...1
,(3.7)
where
k
z ,
nk ,...,1
are the zeros of the
denominator
))...()((...)(
2101
1
1 n
n
n
n
zzzzzzdzdzdzzd
(3.8)
whicharenonnegative,i.e.
0
k
z ,
nk ,...,1
.
Proof.Theproofisgivenin[6].
Therealizationisasymptoticallystableifandonly
if
10
k
z for
nk ,...,1
.
Remark 3.1. The positive realization (3.5) is
asymptoticallystableifandonlyifallcoefficientsof
thepolynomial
11
110110
(1)
( 1) ( 1) ... ( 1) ...
nn nn
nn
dz
zdz dzdzdz dzd




arepositive,i.e.
0
k
d
,
1,...,1,0 nk
[6].
Theorem 3.1 and Remark 3.1 can be easily
extended to the multi‐input multi‐output linear
systems[6].
Theorem 3.2. There always exists the positive
asymptotically stable realization (3.5) of the transfer
function
01
1
1
0
...
)(
dzdzdz
m
sT
n
n
n
sp
. (3.9)
if and only if
0
0
m and the zeros of (3.8) satisfy
thecondition
10
k
z ,
nk ,...,1
.
Proof. From (3.7) it follows that if
0
k
m for
nk ,...,1
then
01
mb
, 0
k
b , nk ,...,2 and
nT
mB
]00[
0
. The positive
realization is asymptotically stable if and only if
10
k
z for
nk ,...,1
.□
Remark 3.2. The Theorems 3.1 and 3.2 are also
validifthematrix
A hasmultipleeigenvalues.
Example 3.1. Compute the positive realization
(3.5)ofthetransferfunction
006.011.06.0
12
)(
23
2
01
2
2
3
01
2
2
zzz
zz
dzdzdz
mzmzm
sT
sp
.(3.10)
Thedenominator
)3.0)(2.0)(1.0(006.011.06.0)(
23
zzzzzzzd
(3.11)
has the real positive zeros
1.0
1
z
,
2.0
2
z
,
3.0
3
z and the matrix A is the Schur matrix of
theform
3.010
02.01
001.0
10
01
00
3
2
1
z
z
z
A
. (3.12)
Note that the polynomial (3.11) satisfies the
conditionsofTheorem2.4.
Using(3.7)and(3.11)weobtain
1
7.1
81.0
1
2
1
100
3.010
02.01.01
100
10
1
1
2
1
0
1
21
211
m
m
m
zz
zzz
B
.(3.13)
Inthiscasethematrix
C
hastheform
]100[C . (3.14)
The positive asymptotically stable realization of
(3.10)isgivenby(3.12)–(3.14).
Itiseasytocheckthatthematrices
3.000
12.00
011.0
ˆ
A
,
1
0
0
ˆ
B
,
]17.181.0[
ˆ
C
(3.15)
548
arealsothepositiveasymptoticallystablerealization
ofthetransferfunction(3.10).
Remark 3.3. If the matrices (3.5) are positive
realizationof(3.6)thenthematrices
n
n
z
z
z
z
A
0000
1000
0010
0001
ˆ
1
2
1
,
1
0
0
0
ˆ
B
,
][
ˆ
21 n
bbbC
(3.16)
arealsothepositiverealizationof(3.6).
Theorem 3.3. Let the matrices (3.5) be a positive
realizationofthestrictlypropertransferfunction(3.6)
thenthematrices
1 ,][
,
0
0
1
0
,
10000
00100
00010
000
0100
0010
0001
0000
,
0
0
1
10
1
)1()1(
qnnPPPCC
B
BA
A
N
N
I
E
n
q
nnn
qqnn
n
(3.17)
areapositiverealizationofthetransferfunction(3.3)
ifandonlyif
k
P for
qk ,...,1,0
and 0
k
z for
nk ,...,1
(3.18)
Proof.Using(3.17)itiseasytoverifythat
....][
1
][
][
0
0
1
0
1000
0010
00010
000
][][
10
1
1
10
1
10
1
q
qn
q
n
q
n
q
zPzPPBAzIC
z
z
BAzI
PPPC
z
z
BAzI
PPPCBAEzC
(3.19)
Therefore, the matrices (3.17) are the positive
realizationofthetransferfunction(3.3).□
Remark 3.4. Note that the positive realization
(3.17)forthedescriptorlinearsystemshasthematrix
n
B
.
Remark 3.5. The positive realization (3.17) is
asymptotically stable if and only if the matrix
nn
A
isSchurmatrixwithonlynonnegativereal
partsofeigenvalues.
Example 3.2. Compute the positive realization
(3.17)ofthetransferfunction
006.011.06.0
88.016.209.04.1
)(
23
234
zzz
zzzz
zT
. (3.20)
Thetransf erfunction(3.20)canbedecomposedas
follows
)()()( zTzPzT
sp
,(3.21)
where
zzPPzP
2)(
10
,
006.011.06.0
12
)(
23
2
zzz
zz
zT
sp
. (3.22)
Thepositiverealizationof
)(zT
sp
givenby(3.22)
has been found in Example 3.1 and it is given by
(3.12)–(3.14).
TheconditionsofTheorem3.3fortheexistenceof
thepositiverealizationof(3.20)aresatisfiedsincein
thiscase
2
0
P and
1
1
P
.
Therefore, by Theorem 3.3 the desired positive
realizationofthetransferfunction(3.20)hastheform
].21100[ ,
0
1
0
0
0
10000
01000
013.010
07.102.01
081.0001.0
,
01000
00000
00100
00010
00001
CB
AE
(3.20)
4 COMPUTATIONOFPOSITIVEREALIZATIONS
OFDESCRIPTORMIMOSYSTEMS
Inthissectionthemethodpresentedinsection3will
be extended to multi‐input multi‐output (MIMO)
lineardiscrete‐timesystems.
The strictly proper transfer matrix (3.4c) can be
written in the form with common least row
denominator
mkpidzdzdzzd
mzmzmzm
zd
zm
zd
zm
zd
zm
zd
zm
sT
ii
n
in
n
i
ikik
n
iknik
p
pm
p
p
m
sp
,...,1 ;,...,1 ,...)(
,...)( ,
)(
)(
)(
)(
)(
)(
)(
)(
)(
01
1
1
01
1
1
1
1
1
1
11
(4.1)
orwithcommonleastcolumndenominator
549
.,...,1 ;,...,1 ,
ˆˆ
...
ˆ
)(
ˆ
,
ˆˆ
...
ˆ
)(
ˆ
,
)(
ˆ
)(
ˆ
)(
ˆ
)(
ˆ
)(
)(
ˆ
)(
)(
ˆ
)(
01
1
1
01
1
1
1
1
1
1
11
mkpidzdzdzzd
mzmzmzm
zd
zm
zd
zm
zd
zm
zd
zm
zT
kk
n
kn
n
k
ikik
n
iknik
m
pmp
m
m
sp
(4.2)
Furtherwe shallconsider in detailsonly thefirst
case(4.1)sincetheconsiderationsfor(4.2)aresimilar
(dual).
The matrices
A and
B
of the desired
realizationhavetheforms
]blockdiag[
1 p
AAA
, (4.3a)
where
nn
in
in
i
i
i
z
z
z
z
A
1000
0000
0001
0000
1
2
1
, 0
ik
z 
for
pi ,...,1
,
nk ,...,1
(4.3b)
and
mnp
pmp
m
BB
BB
B
1
111
,
i
ikn
ik
ik
ik
b
b
b
B
2
1
,
pi ,...,1
,
mk ,...,1
. (4.4)
The entries of the matrices
ik
B
are computed in
thesamewayasofthematrix
B
insection3using
theequation
i
n
iii
MZB
1
,
pi ,...,1
, (4.5a)
where
1000
...10
...1
12121
121211
iniiii
iniiiii
i
zzzzz
zzzzzz
Z
,
pi ,...,1
, (4.5b)
1
1
0
i
ikn
ik
ik
i
m
m
m
M
,
pi ,...,1
,
mk ,...,1
. (4.5c)
Thematrix
C
isgivenby
]blockdiag[
1 p
CCC
,
i
n
i
C
1
]100[
. (4.6)
Theorem 4.1. If the matrices (4.3), (4.4) and (4.6)
areapositiverealizationofthestrictlypropertransfer
matrix(4.1)thenthematrices
mqnnPPPCC
I
B
I
I
I
BA
A
I
I
I
E
np
q
mn
m
nn
m
m
m
nn
m
m
n
)1( ,][
~
,
0
0
0
~
,
0000
0000
0000
000
~
,
0000
0000
00000
0000
~
10
(4.7)
areapositiverealizationofthetransfermatrix(3.3)if
andonlyif
mp
k
P
for
qk ,...,1,0
and 0
ik
z ,
pi ,...,1
,
nk ,...,1
(4.8)
Proof.TheproofissimilartotheproofofTheorem
3.2.
From the above considerations we have the
following procedure for computation of the positive
realization(4.7)ofthegiventransfermatrix
)(zT
.
Procedure4.1.
Step 1. Decompose the given matrix
)(zT
in the
polynomialpart(3.4b)andstrictlyproperpart(3.4c).
Step 2. Compute the zeros
ij
z
,
pi ,...,1
,
j
nj ,...,1
of the denominator )(zd
i
,
pi ,...,1
andfindthematrices(4.3b),(4.3a).
Step 3.Using (4.5b) and (4.5c) compute the matrices
i
Z ,
i
M and check the conditions (4.5a). If the
conditions (4.5a) are satisfied then there exists
mnp
B
andthepositive realizationof
)(zT
.
Thedesiredpositiverealizationisgivenby(4.7).
Example 4.1. Compute the positive realization
(4.7)ofthetransfermatrix
06.05.0
12.118.05.03
03.04.0
23.066.02.02
)(
2
23
2
23
zz
zzz
zz
zzz
zT
. (4.9)
UsingProcedure4.1weobtainthefollowing.
Step 1. The matrix (4.9) can be decomposed in the
polynomialpart
550
2
1
3
2
)( zsP (4.10)
andstrictlyproperpart
06.05.0
1
03.04.0
2.0
)(
2
2
zz
z
zz
z
zT
sp
. (4.11)
Step2.Thezerosofthefirstdenominator
03.04.0)(
2
1
zzzd (4.12)
are:
1.0
11
z
,
3.0
12
z
and of the second
denominator
06.05.0)(
2
2
zzzd (4.13)
are:
2.0
21
z
,
3.0
22
z
.
Therefore,thematrix
A hastheform
3.0100
02.000
003.01
0001.0
0
0
2
1
A
A
A
. (4.14)
Step3.Inthiscase
2
1
B
B
B
,
12
11
1
b
b
B
,
22
21
2
b
b
B
(4.15a)
andusing(4.5a)weobtain
1
1.0
1
2.0
10
1.01
10
1
11
10
1
11
1
m
m
z
B
(4.15b)
and
1
8.0
1
1
10
2.01
10
1
21
20
1
21
2
m
m
z
B
. (4.15c)
Therefore,thematrix
1
2
0.1
1
0.8
1
B
B
B










(4.16)
andthematrix
1000
0010
0
0
2
1
C
C
C
. (4.17)
Thedesiredpositiverealizationof(4.9)isgivenby
.
231000
120010
~
,
0
1
0
0
0
~
100000
010000
013.0100
08.002.000
01003.01
01.00001.0
~
,
010000
000000
001000
000100
0000
10
000001
~
CB
AE
 (4.18)
Now let us consider the strictly proper transfer
matrix(4.11)asthematrixwithleastcommoncolumn
denominator
1.09.0
02.03.0
006.011.06.0
1
)(
2
2
23
zz
zz
zzz
sT
sp
,(4.19)
where
)3.0)(2.0)(1.0(006.011.06.0)(
23
 zzzzzzzd
(4.20)
hasthezeros:
1.0
1
z
,
2.0
2
z
, 3.0
3
z .
Therefore,thematrix
A hastheform
3.000
12.00
011.0
A
. (4.21)
Inthiscasethematrix
B
isgivenby
1
0
0
B
. (4.22)
Using the dual method to the method for
computationofthematrix
B
weobtain
16.002.0
100
C . (4.23)
Therefore,thedesired positive realizationof (4.9)
hastheform
551
.
2316.002.0
12100
ˆ
,
0
1
0
0
0
ˆ
10000
01000
013.000
0012.00
00011.0
ˆ
,
01000
00000
00100
00010
00001
ˆ
CB
AE
(4.24)
5 CONCLUDINGREMARKS
A new method for determination of positive
realizations of transfer matrices of descriptor linear
discrete‐time systems has been proposed. Necessary
and sufficient conditions for the existence of the
positiverealizationshavebeenestablished(Theorems
3.1,3.2and4.1).Aprocedureforcomputationofthe
positive
realizations has been proposed and
illustrated by an example (Example 4.1). The
presented method can be extended to fractional
descriptor linear continuous‐time discrete‐time
systems.
The presented method can be considered as an
extension of the method presented in [8] for
continuous‐timesystemstothediscrete‐timesystems.
Between
themethodswehavethefollowingessential
differences:
1 Themethodpresentedinthispapercanbeapplied
only to discrete‐time linear systems with zeros
satisfyingthecondition(3.18).
2 For discrete‐time systems the matrix B may have
negativeentries(seeRemark3.4).
ACKNOWLEDGMENT
Thestudieshavebeencarriedoutintheframeworkofwork
No.S/WE/1/2016andfinancedfromthefundsforscienceby
thePolishMinistryofScienceandHigherEducation.
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