211
1 INTRODUCTION
Adynamicalsystemiscalledpositiveifitstrajectory
starting from any nonnegative initial states remains
forever in the positive orthant for all nonnegative
inputs. An overview of state of the art in positive
systemstheoryisgiveninmonographs[10,16].The
problems of stability and control
of system with
delayshavebeenconsideredin[3,11,12,13,25].The
stabilityandthe robust stabilityof positive discrete‐
time linear systems without delays and with delays
havebeeninvestigatedin[1‐10,14‐26].
The stability of positive continuous‐time linear
systemswithdelayshavebeen
addressedin[17].and
the stability of positive fractional systems with one
delayin[22].
In this paper new necessary and sufficient
conditions for asymptotic stability of positive
fractionalcontinuous‐timelinearsystemswithdelays
willbepresented.Itwillbeshownthattheasymptotic
stabilityof positive fractional continuous‐time linear
systemsisindependentoftheirdelaysandchecking
ofasymptoticstabilityofthesystemwithdelayscan
be reduced to checking of the stability of positive
systemswithoutdelays.
Thepaperisorganizedasfollows.Insection2the
fractional continuous‐time linear systems and their
solutions are recalled. Necessary
and sufficient
conditionsforthepositivityofthisclassoffractional
systemswithdelaysaregiveninsection3.Themain
resultofthepaperispresentedinsection4,wherethe
necessaryandsufficientconditionsfortheasymptotic
stabilityofthepositivefractionallinearsystemswith
delaysareestablished.Concluding
remarksaregiven
insection5.
Thefollowingnotationwillbeused:
‐theset
of real numbers,
Z
‐ the set of nonnegative
integers,
mn
‐ the set of
mn
real matrices,
mn
‐thesetof
mn
matrices with nonnegative
entries and
1
nn
, .
n
I ‐the
nn
identity
matrix.Thestrictlypositivevectorxwithallpositive
componentswillbedenotedbyx>0.
Stability Tests of Positive Fractional Continuous-time
Linear Systems with Delays
T.Kaczorek
FacultyofElectricalEngineering,BialystokUniversityofTechnology,Poland
ABSTRACT:Necessaryandsufficientconditionsfortheasymptoticstabilityofpositivefractionalcontinuous‐
timelinearsystemswithmanydelaysareestablished.Itisshownthat:
1)theasymptoticstabilityofthepositivefractionalsystemisindependentoftheirdelays,
2) the checking
of the asymptotic stability of the positive fractional systems with delays can be reduced to
checkingoftheasymptoticstabilityofpositivestandardlinearsystemswithoutdelays.
http://www.transnav.eu
the International Journal
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Volume 7
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June 2013
DOI:10.12716/1001.07.02.08
212
2 FRACTIONALCONTINUOUS‐TIMELINEAR
SYSTEMSANDTHEIRSOLUTIONS
InthispapertheCaputodefinitionwillbeused
,...}2,1{1
,
)(
)(
1
)(
0
1
)(
Nnn
d
t
f
n
tf
dt
d
t
n
n
(2.1)
where
is the order of fractional derivative,
n
n
n
d
fd
f
)(
)(
)(
and
0
1
)( dttex
xt
is the
gammafunction.
Consider the continuous‐time fractional linear
systemwithdelays
10
),()()()(
1
0
tBudtxAtxAtx
dt
d
q
k
kk
(2.2)
where
n
tx )( is the state vector,
m
tu )( is
the input vector and
,
nn
k
A
mn
B
, dk is a
delay
qk ,...,1
.
Theinitialconditionsfor(2.2)havetheform
)()(
0
txtx for ]0,[ dt ,
k
k
dd max (2.3)
Thesolutiontotheequation(2.2)with(2.3)canbe
foundbytheuseofthestepmethod[17].
For
1q and
dt 0
the solution has the
form[17]
t
dBudxAtxttx
0
0100
)]()()[()0()()(
(2.4)
where
0
0
0
)1(
)(
k
k
k
k
tA
t
,
0
1)1(
0
])1[(
)(
k
k
k
k
tA
t
(2.5)
Knowing the state vector
)(tx for dt
0 in
a similar way we can find the state vector for
dtd 2 andnextfor dtd 32
,…
3 POSITIVEFRACTIONALCONTINUOUS‐TIME
SYSTEMSWITHDELAYS
Definition3.1. The fractional continuous‐time linear
systems with delays (2.2) is called positive if
n
tx
)( , 0t foranyinitialconditions
n
tx
)(
0
for
]0,[ dt
(3.1)
andallinputvectors
n
tu
)( , 0t .
A real matrix
nn
A
is called the Metzler
matrixifitsoff‐diagonalentriesarenonnegative.
Let
n
M bethesetof
nn
Metzlermatrices.
Theorem 3.1. The fractional continuous‐time linear
systems(2.2)for
10
ispositiveifandonlyif
n
MA
0
,
nn
k
A
,
qk ,...,1
;
mn
B
. (3.2)
Proof.Itiswell‐known[17]that
nn
)(
0
and
nn
)(
if and only if
n
MA
0
. From (2.4) it
follows that
n
tx
)( , 0t if
nn
k
A
,
qk ,...,1
;
mn
B
and
n
tu
)( for 0t .
Thenecessitycanbeshowninsimilarwayasin[17].□
4 ASYMPTOTICSTABILITYOFTHEPOSITIVE
FRACTIONALSYSTEMS
Consider the autonomous fractional positive linear
systemwithdelays
10,)()()(
1
0
q
k
kk
dtxAtxAtx
dt
d
(4.1)
where
n
MA
0
,
nn
k
A
and 0
k
d ,
qk ,...,1
.
Definition 4.1. The positive system (4.1) is called
asymptoticallystableif
0)(lim
tx
t
foranyinitialconditions(3.1) (4.2)
Definition 4.2. A vector
n
e
x
is called the
equilibriumpointofthepositiveasymptoticallystable
system (2.2) for
nT
n
tBu
]1...1[1)( if
thefollowingconditionissatisfied
ne
Ax 10
and
q
k
k
AA
0
(4.3)
From(4.3)wehave
ne
Ax 1)(
1
(4.4)
sinceforasymptoticallystablesystem(2.2)thematrix
Aisinvertibleandtheinversematrix
nn
A
1
)(
[16].
213
Theorem 4.1. The positive fractional system with
delay(4.1)isasymptoticallystableifandonlyifthere
existsastrictlypositivevector
n
suchthat
0
A (4.5)
Proof.Firstweshallshowthatifthepositivesystem
(4.1)isasymptoticallystablethenthereexistsastrictly
positivevector
0
satisfying(4.5).Ifthepositive
system (4.1) is a asymptotically stable then the
equilibriumpoint(4.4)isastrictlypositivevectorand
we can choose
ne
Ax 1)(
1
. This vector
satisfiesthecondition(4.5)since
nn
AAA 11
1
(4.6)
Nowweshallshowthatthepositivesystem(4.1)
isasymptoticallystableifthereexistsstrictlypositive
vector λ satisfying (4.5). It is well‐known that the
positive system (4.1) is asymptotically stable if and
onlyifthecorrespondingtransposesystem
q
k
k
T
k
T
dtxAtxAtx
dt
d
1
0
)()()(
(Tdenotesthetranspose) (4.7)
isasymptoticallystable.AscandidateforaLyapunov
function for the positive system (4.7) we chose the
function
q
k
k
t
dt
TT
AdxtxtxV
k
1
)()()]([
(4.8)
whichispositiveforanynonzero
n
tx
)( , 0t .
Using(4.8)and(4.7)weobtain
AtxAdtxAtx
dtxAtxA
Adx
dt
d
dt
txd
dt
txVd
T
q
k
kk
T
k
T
T
q
k
k
T
k
T
t
dt
T
T
)(])()([
])()([
)(
)()]([
1
1
0
1
(4.9)
If(4.5) holds then from(4.9) we have
0
)]([
dt
txVd
andthesystem(4.1)isasymptoticallystable.□
Theorem 4.2. The positive fractional system with
delay(4.1)isasymptoticallystableifandonlyoneof
thefollowingequivalentconditionsissatisfied:
Thepositivesystemwithoutdelay
1
),()( tAxtx
(4.10)
isasymptoticallystable,
The matrix A is asymptotically stable Metzler
matrix.
Proof. In [20] itwas shown that the positive system
(4.10) is asymptotically stable if and only if there
exists a strictly positive vector
n
such that
(4.5)holds.HencebyTheorem2thepositivesystem
(4.1)isasymptoticallystableifandonlyifthepositive
system (4.10) is asymptotically stable. It is well‐
known[16]thatthepositivesystem(4.10)(and(4.1))
isasymptoticallystableifandonlyifthematrixAis
asymptotically
stableMetzlermatrix.□
To check the asymptotic stability of the Metzler
matrixAthefollowingtheoremisrecommended[23,
24].
Theorem 4.3. The matrix
nn
A
is a
asymptoticallystableMetzlermatrixifandonlyifone
ofthefollowingequivalentconditionsissatisfied:
all coefficients
10
,...,
n
aa of the characteristic
polynomial
01
1
1
...]det[ asasasAsI
n
n
n
n
(4.11)
arepositive,i.e.
0
i
a ,i=0,1,…,n–1,
thediagonalentriesofthematrices
)(k
kn
A
fork=1,…,n–1 (4.12)
arenegative,where
]...[
,
,
...
...
...
]...[,
...
...
...
,
...
...
...
)(
1,
)(
1,
)(
1
)(
,1
)(
,1
)(
1
)(
,
)(
1
)(
1
)(
1
)(
,
)(
1,
)(
,1
)(
11
)1(
1,1
)1()1(
)1()(
)0(
1,
)0(
1,
)0(
1
)0(
,1
)0(
,1
)0(
1
)0(
1,1
)0
(
1,1
)0(
1,1
)0(
11
)0(
1
)0(
,
)0(
1
)0(
1
)0(
1
)0(
,
)0(
1,
)0(
,1
)0(
11
)0(
k
knkn
k
kn
k
kn
k
knkn
k
kn
k
kn
k
knkn
k
kn
k
kn
k
kn
k
knkn
k
kn
k
k
n
k
k
knkn
k
kn
k
kn
n
kn
k
kn
nnnn
nn
n
n
nnn
n
n
nnn
nn
nnn
n
n
aac
a
a
b
ac
bA
aa
aa
a
cb
AA
aac
a
a
b
aa
aa
A
ac
bA
aa
aa
AA
(4.13)
for k = 0,1,…,n – 1.
214
the diagonal entries
kk
a
~
of the lower triangular
matrix
nnnn
aaa
aa
a
A
,2,1,
2221
11
~
...
~~
0...
~~
0...0
~
~
are negative, i.e.
0
~
kk
a for nk ,...,1 ; where
A
~
is obtained from A by elementary column
operations[23].
From Theorem 4.2 we have the following
importantcorollary.
Corollary4.1.Theasymptoticstabilityofthepositive
fractional linear systems (4.1) is independent of its
delays.
Theorem 4.4. The positive fractional linear system
(4.1)is unstable if atleast one diagonal
entry of the
MetzlermatrixAisnonnegative.
Theprooffollows immediatelyfromTheorem4.2
andTheoremin[16].
Example4.1.Considerthepositivefractionalsystem
(4.1)withthematrices
2.005.0
1.02.0
,
3.005.0
1.02.0
,
25.0
1
2
10
A
A
a
A
(4.14)
andarbitrarydelay
0
k
d ,
2,1k
.Findthevalue
of the parameter a for which the system is
asymptoticallystable.
ByTheorem4.4thepositivefractionalsystem(4.1)
with(4.14)isunstableifthediagonalentry(1,1)ofthe
Metzlermatrix
5.16.0
2.14.0
210
a
AAAA
(4.15)
is nonnegative i.e.
04.0 a . Using the condition
i)ofTheorem4.2weobtain
)32.15.1()1.1(
5.16.0
2.14.0
]det[
2
asas
s
as
AIs
 (4.16)
and the positive fractional system is asymptotically
stableifandonlyifthecoefficientsofthepolynomial
(4.16)arepositive
01.1 a and 032.15.1
a (4.17)
Therefore,thepositivefractionalsystem(4.1)with
(4.14) is asymptotically stable for arbitrary delay
0d if 88.0
5.1
32.1
a .
Thesameresultcanbeobtainedbytheuseofthe
conditionii)andiii)ofTheorem4.3.
5 CONCLUDINGREMARKS
Necessary andsufficient conditionsfor the
asymptotic stability of continuous‐timelinear
systemswithdelayshavebeenestablished(Theorem
4.1and4.2).Ithasbeenshownthat:1)The
asymptotic
stability of the positive fractional system is
independent of their delays, 2) The checking of the
asymptoticstabilityofthepositivefractionalsystems
with delays can be reduced to checking of the
asymptotic stability of positive standard linear
systems without delays. The considerations can be
also extended for fractional
positive 2D continuous‐
discretelinearsystemswithdelays.
ACKNOWLEDGMENT
ThisworkwassupportedbyMinistryofScienceand
HigherEducationinPolandunderworkS/WE/1/11.
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