83
1 INTRODUCTION
Adynamicalsystemiscalledpositiveifitstrajectory
starting from any nonnegative initial condition state
remains forever in the positive orthant for all
nonnegativeinputs.Anoverview of stateofthe art
inpositivesystemtheoryisgiveninthemonographs
Farina & Rinaldi 2000, Kaczorek 2001 and in the
pa
pers Kaczorek 1997, 1998a, 2011, 2015. Models
havingpositivebehaviorcanbefoundinengineering,
economics,socialsciences,biologyandmedicine,etc.
The Laypunov, Bohl and Perron exponents and
stabilityof time‐varying discrete‐time linear systems
havebeeninvestigatedinCzorniket.all2012,2013a,
2013b,2013c,2013d,2014.Thepositivestandardand
descriptor systems and their st
ability have been
analyzed in Kaczorek 1998a, 2001, 2011, 2015.The
positivelinearsystemswithdifferentfractionalorders
havebeenaddressedinKaczorek2011,2012andthe
singular discrete‐time linear systems in Kaczorek
1998a.The switched discrete‐time systems have been
considered in Zha
ng et. all 2014a, 2014b and the
extremal norms for positive linear inclusions in
Zhonget.all2013.
Inthispaperthepositivityandasymptoticstability
of the descriptor time‐varying discrete‐time linear
systemswithregularpencilswillbeinvestigated.
Thepaperisorganizedasfollows.Insection2the
Weierstrass‐Kronecker decomposit
ion of the regular
pencilisextendedtodescriptortime‐varyingdiscrete‐
time linear systems and thesolution of the state‐
equation describing the time‐varying discrete‐time
linear system is derived. Necessary and sufficient
conditionsforthepositivityofthedescriptorsystems
are established in section 3. The st
ability of the
positivedescriptorsystemsisaddressedinsection4.
Concludingremarksaregiveninsection5.
Thefollowingnotationwillbeused:
‐theset
of real numbers,
mn
‐ the set of mn
real
matrices,
mn
‐the set of mn matrices with
nonnegativeentriesand
1
nn
,
n
I
‐the
nn
i
dentitymatrix.
Positive Descriptor Time-varying Discrete-time Linear
Systems and Their Asymptotic Stability
T.Kaczorek
BialystokUniversityofTechnology,Bialystok,Poland
ABSTRACT:Thepositivityandasymptoticstabilityofthedescriptortime‐varyingdiscrete‐timelinearsystems
areaddressed.TheWeierstrass‐Kroneckertheoremonthedecompositionoftheregularpencilisextendedto
the time‐varying discrete‐time descriptor linear systems. Using the extension necessary and sufficient
condit
ionsforthepositivityofthesystemsareestablished.Sufficientconditionsforasymptoticstabilityofthe
positivesystemsarepresented.Theeffectivenessofthetestsisdemonstratedontheexample.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 9
Number 1
March 2015
DOI:10.12716/1001.09.01.10
84
2 POSITIVETIME‐VARYINGDISCRETE‐TIME
LINEARSYSTEMS
Consider the descriptor time‐varying discrete‐time
linearsystem
iii
uiBxiAxiE )()()(
1
, ,...}1,0{
Zi (2.1a)
ii
xiCy )( (2.1b)
where
n
i
x
,
m
i
u
,
p
i
y
are the state,
input and output vectors and
,)(
nn
iA
mn
iB
)( ,
np
iC
)( are matrices with entries
dependingon
Zi .
Itisassumedthat
0)(det
iE ,
Zi and
det[ () ()] 0Ei Ai

(2.2)
forsome
C
(thefieldofcomplexnumbers)and
Zi
.
Itiswell‐known(Kaczorek2015)thatif(2.2)holds
then there exists a pair of nonsingular matrices
nn
iQiP
)(),( suchthat
2
1
0
0)(
0
0
)()]()()[(
1
n
n
I
iA
N
I
iQiAiEiP
, (2.3)
where
Zi , )]()(det[deg
1
iAiEn
,
,)(
11
1
nn
iA
22
nn
N
is the nilpotent matrix
withtheindexμ(i.e.
0
N
and
0
1
N
).
The matrices
)(),(),(
1
iAiQiP can be found by
for example the use of elementary row and column
operations(Kaczorek1998b).
Premultiplying (2.1a) by the matrix
)(iP ,
introducingthenewstatevector
in
i
i
i
in
i
i
i
i
i
ii
x
x
x
x
x
x
x
x
x
x
xiQx
,2
,22
,21
2
,1
,12
,11
1
,2
,1
1
21
,,)(
(2.4)
andusing(2.3)weobtain
1, 1 1 1, 1
() ()
iii
x
Aix Biu
 (2.5a)
2, 1 2, 2
()
ii i
Nx x B i u

(2.5b)
where
)(
)(
)()(
2
1
iB
iB
iBiP
,
,)(
1
1
mn
iB
mn
iB
2
)(
2
. (2.5c)
Theorem 2.1. The solution of equation (2.5a) for
known initial condition
1
10
n
x and input
m
i
u ,
Zi isgivenby
1
0
110,11,1
)()1,()0,(
i
j
ji
ujBjixix
,
Zi (2.6a)
where
0
0
for
for
)()...2()1(
),(
111
1
jk
jk
jAkAkA
I
jk
n
. (2.6b)
Proofisgivenin(Kaczorek2015).
To simplify the notation it is assumed that the
matrixNin(2.5b)hastheform
22
0...000
1...000
0...100
0...010
nn
N
. (2.7)
From(2.5b)and(2.7)wehave
i
n
in
i
i
in
i
i
u
iB
iB
x
x
x
x
x
x
)(
)(
0...000
1...000
0...100
0...010
2
22
2
21
,2
,22
,21
1,2
1,22
1,21
(2.8a)
for
Zi and
iii
ininin
inin
uiBxx
uiBxx
uiBx
)(
,)(
,)(0
21,211,22
12,121,2
2,2
222
22
,
Zi . (2.8b)
Solving the equations (2.8b) with respect to the
componentsofthevector
i
x
,2
weobtain
.)(...)1(
,)()1(
,)(
21122,21
1212,12
2,2
22
222
22
inini
ininin
inin
uiBuniBx
uiBuiBx
uiBx
(2.9)
The considerations can be easily extended to the
casewhenthematrixNin(2.5b)hastheform
1],,...,[blockdiag
1
qNNN
q
(2.10)
andN
kfork=1,2,…,qhastheform(2.7).
Example 2.1. Consider the descriptor time‐varying
system described by the equation (2.1a) with the
matrices
85
,
0000
000
1
2
2)cos(
)1(
0
1
)1))(sin(2(
2)cos(
000
)(
2
2
i
i
i
ee
e
i
ii
i
e
iE
ii
i
i
1
)2(2
)(
1
)1))(cos(2(2
)sin()(
)1))(sin(sin(
1
)1))(sin(1))(cos(2(2
)(
2)cos(
1
)(
2)cos(
1
)(
,
)(0
)(0
)()(
0)(
)(
5
4
3
2
1
5
4
32
1
i
ii
ib
i
iii
iib
ii
i
iiii
ib
i
e
eib
i
ib
ib
ib
ibib
ib
iB
i
i
(2.11)
,
)(000
)(00)(
)()()()(
0)(00
)(
44
3431
24232221
13
ia
iaia
iaiaiaia
ia
iA
where
,
2)cos(
1
)(
13
i
ia
,
)2))(sin(1(
)3)sin()cos()sin()sin(2)cos(2)(2(
)(
21
ii
iiiiiiii
ia
,2)(
2
22
ii
eeia
,
2)cos(
1
)(
23
i
e
ia
i
,
1
)1))(sin(1))(cos(2(
)(
2
24
i
iiie
ia
i
,
2)sin(
2
)(
31
i
i
ia
,
1
)1))(cos(2(
)(
2
34
i
iie
ia
i
.
1
)2(
)(
2
44
i
ie
ia
i
Thecondition(2.2)issatisfiedsince
0
)2))(sin(2)(cos()1(
)1)sin(2)(12()2(
)]()(det[
2
2
iii
iieei
iAiE
ii
(2.12)
Inthiscase
,
2
1
000
000)cos(2
)cos(1100
0)sin(111
)(
i
i
i
i
ie
iP
i

i
i
e
e
i
i
iQ
2
000
0100
000
00
2
1
0
)(
(2.13)
and
,
1000
0100
00
)sin(2
1
0
00)cos(12
)()()(
0
0)(
,
0000
1000
0010
0001
)()()(
0
0
2
1
1
i
i
ie
iQiAiP
I
iA
iQiEiP
N
I
i
n
n
)2(
,
20
01
)sin(0
0
)()(
)(
)(
21
2
1
nn
i
i
e
iBiP
iB
iB
i
(2.14)
Theequation(2.5)havetheform
i
i
i
i
i
i
i
i
u
u
i
e
x
x
i
i
ie
x
x
2
1
,12
,11
1,12
1,11
)sin(0
0
)sin(2
1
0
)cos(12
(2.15a)
and
i
i
i
i
i
i
u
u
i
x
x
x
x
2
1
,22
,21
1,22
1,21
20
01
00
10
(2.15b)
The solution of (2.15a) is given by (2.6) with the
matrices
)(
1
iA and )(
1
iB definedby(2.14).
From(2.15b)wehave
.,)1(2
,2
1211,211,21
2,22
Ziuiuxux
iux
iiiii
ii
(2.16)
The solution of the equation (2.1a) with (2.11) is
givenby
86
Zi
x
x
x
x
iQ
ix
ix
ix
ix
ix
i
i
i
i
,)(
)(
)(
)(
)(
)(
,22
,21
,12
,11
4
3
2
1
(2.17)
where
)(iQ isdefinedby(2.13)andthecomponents
of the state vector
)(ix by (2.6) with )(
1
iA and
)(
1
iB definedby(2.14)and(2.16).
3 POSITIVESYSTEMS
Definition3.1.The descriptortime‐varying discrete‐
time linear system (2.1) is called the (internally)
positive if and only if
n
i
x
and
p
i
y
,
Zi foranyadmissibleinitialconditions
n
x
0
andallinputs
,
m
i
u
Zi .
The matrix
nn
iQ
)( ,
Zi is called
monomialifineachrowandcolumnonlyoneentry
ispositive and theremaining entries arezero forall
Zi .
It is well‐known (Kaczorek 1998a) that
nn
iQ
)(
1
,
Zi if and only if the matrix is
monomial.
Itisassumedthatforthepositivesystem(2.1)the
decomposition (2.3) is positive for the monomial
matrix
)(iQ .Inthiscase
n
ii
xiQx
)(
ifandonlyif
n
i
x
,
Zi .(3.1)
Itisalsowell‐knownthatpremultiplicationofthe
equation(2.1a)bythematrix
)(iP doesnotchange
itssolution
i
x ,
Zi .
From (2.9) it follows that
2
,2
n
i
x
,
Zi
for
,
m
i
u
Zi ifandonlyif
mn
iB
2
)(
2
for
Zi . (3.2)
In(Kaczorek2015)hasbeenshownthatthetime‐
varyingdiscrete‐timesystem (2.5a)ispositive ifand
onlyif
mnnn
iBiA
111
)(,)(
11
,
Zi
. (3.3)
From(2.1b)and(2.4)wehave
iii
xiCxiQiQiCy )()()()(
1
,
Zi . (3.4a)
where
)()()( iQiCiC . (3.4b)
For monomial matrix
nn
iQ
)( from (3.4) we
have
np
iC
)( ,
Zi if and only if
np
iC
)( ,
Zi and
p
i
y
,
Zi .
Therefore,thefollowingtheoremhasbeenproved.
Theorem 3.1. The descriptor time‐varying discrete‐
timelinearsystem(2.1)ispositiveifandonlyif
1 thereexiststhedecomposition(2.3)formonomial
matrix
nn
iQ
)( ,
Zi ;
2 theconditions(3.2)and(3.3)aresatisfied;
3
np
iC
)( for
Zi .
Example3.1.Considerthedescriptortime‐varying
system described by the equation (2.1) with the
matrices
,
0000
0001
4)sin(2
2
0
2)cos(
1
1)cos(
4)sin(2
1
000
)(
i
e
i
i
i
iE
i
,
10
2)sin(0
)2)sin()(1)(cos(
2)sin(
2
0
2)sin(
1
)(
ie
iei
i
e
e
i
iB
i
i
i
i
,
0
1
0
1
2
5.00
2)cos(
1
0
)(
i
i
e
e
i
i
i
iC
(3.5)
,
)(000
)(00)(
)()()()(
0)(00
)(
44
3431
24232221
13
ia
iaia
iaiaiaia
ia
iA
where
,
)1)(2)(sin(
1
)(
13
i
ei
ia
,1.0)cos(3.0)sin(2.0)cos(3.03.0)(
21
ieiieia
ii
,
)2))(cos(2(
)1(1.0
)(
22
ii
i
ia
,
)1)(2)(sin(
2
)(
23
i
i
ei
e
ia
,
)1(2
)1)(1))(cos(2(
)(
24
i
eii
ia
i
),1(3.0)(
31
i
eia
,
)1(2
)1)(2(
)(
34
i
ei
ia
i
.
)1(2
2
)(
44
i
i
ia
Thecondition(2.2)issatisfiedsince
87
22
det[ () ()]
(3 60 30 3) 3 70 100 200 40 3
100( 1)( 1)(sin(2 ) 4cos( ) 4sin( ) 8)
0
i
i
Ei Ai
ei i i i i
ie i i i


 

(3.6)
Inthiscase
,
2
1
000
000)sin(2
1100
0)cos(112
i
i
i
e
ie
P
i
i
2000
0100
000)cos(2
0010
i
e
i
Q
(3.7)
and
,
1000
0100
00)1(3.00
00))sin(1(2.0
2
1
1.0
)()()(
0
0)(
,
0000
1000
0010
0001
)()()(
0
0
2
1
1
i
n
n
e
i
i
i
iQiAiP
I
iA
iQiEiP
N
I
(3.8)
.
0
1
2
0
1001
)()()(
,
2
1
0
01
)sin(10
0
)()(
)(
)(
2
1
i
i
e
i
i
iQiCiC
i
i
i
e
iBiP
iB
iB
(3.8)
The descriptor system is positive since the three
conditions of Theorem 3.1 are satisfied. The matrix
)(iQ defined by (3.7) is monomial, the conditions
(3.2)and(3.3)aremetsince
,
2
1
0
01
)(
22
2
R
i
i
iB
22
1
)1(3.00
))sin(1(2.0
2
1
1.0
)(
R
e
i
i
i
iA
i
, (3.9)
,
)sin(10
0
)(
22
1
R
i
e
iB
i
Zi
and
42
0
1
0
1
2
5.00
2)cos(
1
0
)(
R
e
e
i
i
i
iC
i
i
for
Z
.
Thesolutiontotheequation(2.1)withthematrices
),(iE ),(iA )(iB given by (3.5)can be found ina
similarwayasinExample2.1.
4 STABILITYOFTHEPOSITIVEDESCRIPTOR
LINEARSYSTEMS
From (2.1a) and (2.6a) for
n
IiE )( , 0)(
i
uiB ,
Zi itfollowsthat
0,11,1
)(
ˆ
xix
i
,
Zi (4.1a)
where
)0()...2()1()0,()(
11111
AiAiAii
(4.1b)
isthesolutionoftheequation
ii
xiAx
,111,1
)(
,
Zi . (4.2)
From(4.1b)wehave
)()()1(
111
iiAi
,
Zi . (4.3)
Definition 4.1. The positive system (4.2) is called
asymptotically stable if the norm
i
x
,1
of the state
vector
1
,1
n
i
x
,
Zi satisfiesthecondition
0lim
,1
i
i
x foranyfinite
1
10
n
x
. (4.4)
Theorem4.1.Thepositivesystem(4.2)is
asymptotically stable if the norm
)(
1
iA of the
matrix
)(iA
,
Zi satisfiesthecondition
1
1
A for
Zi (4.5a)
where
)(max
1
0
1
iAA
i
for
Zi . (4.5b)
Proofisgivenin(Kaczorek2015).
Theorem4.2.Thepositivesystem(4.2)is
asymptotically stable if its system matrix
nn
jk
iaiA
)]([)(
)1(
1
satisfiesthecondition
88
1
1
1
)1(
0
1)(max
n
k
jk
nj
ia
for
Zi
(4.6a)
or
1
1
1
)1(
0
1)(max
n
j
jk
nk
ia for
Zi . (4.6b)
Proofisgivenin(Kaczorek2015).
Theorem4.3.Thepositivesystem(4.2)is
asymptoticallystableifitssystemmatrix
11
)(...)()()(
1...000
0...100
0...010
)(
)1(
1
)1(
2
)1(
1
)1(
0
1
nn
n
iaiaiaia
iA
(4.7)
satisfiesthecondition
1
0
)1(
1)(
n
k
k
ia for
Zi . (4.8)
Proofisgivenin(Kaczorek2015).
Considerthepositivedescriptorsystemdescribed
by(2.1a)for
0)(
i
uiB ,
Zi
ii
xiAxiE )()(
1
. (4.9)
If the assumption (2.2) is satisfied then the
characteristic polynomial of the system (4.9) and of
thesystem
i
n
i
n
x
I
iA
x
N
I
2
1
0
0)(
0
0
1
1
(4.10)
arerelatedby
),()(
)(
0
0)(
0
0
)(det
)]()(det[),(
1
1
1
2
1
izpik
iQ
I
iA
z
N
I
iP
iAziEizp
n
n
(4.11a)
where
).()(det)(
)],(det[
0
0)(
0
0
det),(
11
1
1
1
2
1
iQiPik
iAzI
I
iA
z
N
I
izp
n
n
n
(4.11b)
From(4.11)wehavethefollowinglemma.
Lemma 4.1. The positive descriptor time‐varying
discrete‐time linear system (4.9) is asymptotically
stable if and only if the positive time‐varying linear
system
1, 1 1 1,
()
ii
x
Aix
(4.12)
isasymptoticallystable.
From Theorem 4.1 and Lemma 4.1 we have the
followingtheorem.
Theorem 4.4. The positive descriptor time‐varying
discrete‐time linear system (4.9) is asymptotically
stableifthecondition
1)(max
1
0
1
iAA
i
(4.13)
issatisfied.
Similarly, from Theorem 4.2, 4.3 and Lemma 4.1
wehavethefollowingtheorems.
Theorem 4.5. The positive descriptor time‐varying
discrete‐time linear system (4.9) is asymptotically
stable if the matrix
11
)]([)(
)1(
1
nn
jk
iaiA
,
Zi
satisfiesthecondition
n
k
jk
nj
ia
1
)1(
0
1)(max for
Zi (4.14a)
or
n
j
jk
nk
ia
1
)1(
0
1)(max for
Zi . (4.14b)
Theorem 4.6. The positive descriptor time‐varying
discrete‐time linear system (4.9) is asymptotically
stable if the matrix
)(
1
iA has the canonical
Frobeniusform
11
)(...)()()(
1...000
0...100
0...010
)(
)1(
1
)1(
2
)1(
1
)1(
0
1
nn
n
iaiaiaia
iA
(4.15)
anditsatisfiesthecondition
1
0
)1(
1)(
n
k
k
ia
for
Zi . (4.16)
Example 4.1. (continuation of Example 3.1). By
Theorem 4.4 the positive descriptor time‐varying
discrete‐time linear system(2.1)with thematrixA (i)
givenby(3.5)isasymptoticallystablesince
89
1)1(3.0)],sin(1[2.0
2
1
1.0max
)(max
0
1
0
1
i
i
i
ei
i
i
iAA
(4.17)
forall
Zi .
5 CONCLUDINGREMARKS
The positivity and asymptotic stability of the
descriptor time‐varying discrete‐time linear systems
with regular pencils have been addressed. The
Weierstrass‐Kroneckertheoremonthedecomposition
of the regular pencils has been extended to the
descriptor time‐varying discrete‐time linear systems.
Solutions to the decomposed
systems have been
derived (Theorem 2.1). Necessary and sufficient
conditionsforthepositivityofthesystemshavebeen
established (Theorem 3.1). Using the norms of the
vectors and matrices sufficient conditions for
asymptoticstabilityofthepositivesystemshavebeen
derived(Theorems4.1–4.6).Theeffectivenessofthe
test
are demonstrated on examples. The proposed
method can be applied in analysis of marine
navigationandsafetyofsea transportationproblems.
The considerationscan beextended to the fractional
descriptortime‐varyingdiscrete‐timelinearsystems.
ACKNOWLEDGMENT
ThisworkwassupportedbyNationalScienceCentre
inPolandunderworkNo.2014/13/B/ST7/03467.
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