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Research ArticleStabilization of a Class of Switched Positive Nonlinear Systems
Xing Xing12 Zhichun Jia3 Yunfei Yin1 and Tingting Wu1
1College of Engineering Bohai University Jinzhou Liaoning 121013 China2Space Control and Inertial Technology Research Center Harbin Institute of Technology Harbin Heilongjiang 150001 China3College of Information Science and Technology Bohai University Jinzhou Liaoning 121013 China
Correspondence should be addressed to Xing Xing superfatherxgmailcom
Received 22 September 2014 Accepted 5 December 2014
Academic Editor Guangming Xie
Copyright copy 2015 Xing Xing et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The problem of switching stabilization for a class of switched positive nonlinear systems (switched positive homogeneouscooperative system (SPHCS) in the continuous-time context and switched positive homogeneous order-preserving system (SPHOS)in the discrete-time context) is studied by using average dwell time (ADT) approach where the positive subsystems are possibly allunstable To tackle this problem a new class of ADT switching is first defined which is different from the previous defined ADTswitching in the literature Then the proposed ADT is designed via analyzing the weighted 119897
infinnorm of the considered systemrsquos
state A sufficient condition of stabilization for SPHCSs with unstable positive subsystems is derived in continuous-time contextFurthermore a sufficient condition for SPHOSs under the assumption that all modes are possibly unstable is also obtained Finallya numerical example is given to demonstrate the advantages and effectiveness of our developed results
1 Introduction
Switched system is a class of hybrid systems that involves acoupling between continuous dynamics and discrete eventswhich has wide application areas such as traffic control pro-cess control network control systems automotive industryandmechanical systems Due to their numerous applicationsswitched systems have been a hot research topic in thepast decades [1ndash5] Most efforts in researches of switchedsystems are mainly devoted to the dynamic behavior analysisand property characterization for example controllabilityobservability realizability optimized performance and espe-cially the stability The authors in [6] applied a class ofLyapunov-like function to study the problem of stability forswitched systems comprising unstable subsystems
On the other hand many physical systems encounteredin practice involve state variables that are always confinedto be nonnegative For instance density of a object absolutetemperatures and concentration of substances in chemicalprocesses are always positive Such systems are generallytermed as positive systems whose states and output arepositive whenever the initial conditions and input are non-negative [7] Numerous models having positive behavior
can be found in many areas such as biology sociology andcommunication networks Given their practical importancerecently positive system has been paid much attention bya large number of researchers [8ndash14] As it is well knownpositivity of the system state can yield some interestingproperties and bring about some efficiently tackling tech-niques to positive linear systems [7 15] For example theso-called copositive linear Lyapunov function with positivevector parameters can be particularly chosen as a systemenergy function for switched positive linear systems In themost recent few years positive nonlinear systems have alsobeen noticed due to their important applications [16ndash18]A valuable issue naturally arises whether those propertiesresults and methods for positive linear systems can beextended to positive nonlinear systems In [19] the authorsproved that same to positive linear systems with constantdelay the stability of homogeneous cooperative system isalso independent of delays Afterwards the authors of [20]extended this fact to continuous-time homogeneous coop-erative systems and discrete-time homogeneous monotonesystems with bounded time-varying delays
Furthermore switched positive system that possesses theessences of both switched systems and positive systems is
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 152410 7 pageshttpdxdoiorg1011552015152410
2 Mathematical Problems in Engineering
recently studied due to its wide applications in practice As aspecial class of switched systems and positive systems somerelevant methods applicable to positive systems or switchedsystems are still suitable for switched positive systems [21 22]But anyway it should be pointed out that those methodsmay not be as efficient as they are for general switchedsystems or positive systems Therefore many researchershave recently focused on exploring efficient approaches andresults for switched positive systems The authors of [23]introduced a multiple linear copositive Lyapunov function(MLCLF) to investigate the problem of stability for switchedpositive systems Then the authors of [24] presented ageneralization of copositive types of Lyapunov function forstability analysis of switched positive systems Note that thoseworks all consider switched positive linear systems So farthere are few results reported for switched positive nonlinearsystems which are however of both theoretical and practicalimportances
As far as the stabilization of switched positive systems isconcerned how to find an appropriate switching signals toguarantee system stability is an important issue Switchingsignals that may be either autonomous or controlled gener-ally include arbitrary switching signals stochastic switchingsignals and constraint switching signals Time constrainedswitching signal that is viewed as an important class ofconstraint switching signals naturally has been extensivelyinvestigated during the past several decades because of itsstrong applicability Time constrained switching signal canbe also called slow switching and it is classified into threetypes dwell time (DT) average dwell time (ADT) andmode-dependent average dwell time switching (MDADT)
In the recent few years time constrained switching hasbeen successfully used for switching stabilization of bothgeneral switched systems and switched positive systems Tolist a few when all the subsystems of a switched linearsystem are stable the author of [25] had shown that thesystem is exponentially stable if the DT is sufficiently largeIt was proved in [26ndash28] that ADT turns out to be veryuseful for analysis and synthesis of switched systems TheADT was extended to MDADT in [29] for stabilization ofgeneral switched systems In [23] the ADT switching wasutilized for switched positive linear systems It is noted thatmost recently the time constrained switching stabilizationfor switched (positive) linear systems with partlyall unstablesubsystems have been paid some attentions [6 30 31] How-ever such an attempt is just a start and the slow switchingstabilization for switched (positive) systems with possibly allunstable subsystems has not been fully solved especially forswitched positive nonlinear systems
In summary the existing time constrained switchingstabilization results for switched positive systems are mainlyfocused on the linear case and those promising ideas thereinare not applicable for switched positive nonlinear systemsSuch a problem becomes more complicated once all the non-linear subsystems become unstable To the best of the authorsrsquoknowledge up to now there is no literature reporting theswitching stabilization problem for switched positive nonlin-ear systems with possibly all unstable subsystems even in thelinear case
All the above observations motivate us to carry outthe present work Compared with some existing resultsour results can solve the switching stabilization problem ofswitched positive nonlinear systems though all the subsys-tems are unstable The most important of the paper liesin that a new class of ADT is proposed and the weighted119897infin
norm of the considered systemrsquos state is introduced totackle the considered problem The layout of the paperis organized as follows Section 2 reviews some necessarydefinitions of switched positive nonlinear systems and definesa new concept of ADT which characterizes a different setof switching signals from traditional ADT In Section 3stabilization criteria for switched positive nonlinear systemswith possibly all unstable subsystems via proposed ADTswitching are derived in continuous-time and discrete-timecase Section 4 provides a numerical example to demonstratethe feasibility and effectiveness of the proposed techniquesand Section 5 concludes the paper
Notations In this paper the notations used are standard RR119899 and R119899
+denote the field of real numbers 119899-dimensional
Euclidean space and the nonnegative orthant of R119899 respec-tively and R119899
0stands for R119899
+ 0119899
and 119872119890119905119911119897119890119903 is a matrixwhose off diagonal entries are nonnegative The notation sdot refers to the Euclidean norm For x isin R119899 119909
119894denotes the 119894th
component of x In addition x ⪰ y (or x ≻ y) mean that allentries of vector 119909
119894ge 119910119894(or 119909119894gt 119910119894)
2 Problem Formulation and Preliminaries
This section presents some definitions and preliminaryresults which will be used throughout the paper We firstrecall some preliminaries about positive nonlinear systemsfor the introduction of switched positive nonlinear systemshereafter Consider the following nonlinear system
120599x (119905) = f (119909 (119905)) (1)
where x(119905) isin R119899 is the state vector f R119899 rarr R119899 is asmooth functions vector and is homogeneous cooperative inthe continuous-time case (homogeneous order-preserving inthe discrete-time case) the symbol 120599 denotes the derivativeoperator in the continuous-time context 120599x(119905) = (119889119889119905)x(119905)and the shift forward operator in the discrete-time case120599x(119905) = x(119905 + 1)
Definition 1 System (1) is said to be positive if and only if forevery nonnegative initial state its state is nonnegative
Definition 2 A continuous vector field f R119899 rarr R119899 is saidto be homogeneous if for all x(119905) isin R119899 and all real 120582 gt 0f(120582x) = 120582f(x)
Definition 3 A continuous vector field f R119899 rarr R119899 whichis C1 on R119899 0 is said to be cooperative if the Jacobian(120597f120597x)(a) is Metzler for all a isin R119899
0
Definition 4 A continuous vector field f R119899 rarr R119899 is saidto be order-preserving onR119899
+ if f(x) ≻ f(y) for any x y isin R119899
+
such that x ⪰ y
Mathematical Problems in Engineering 3
Lemma 5 If f R119899 rarr R119899 is cooperative then for any twovectors x y isin R119899
0satisfying x ⪰ y and 119909
119894= 119910119894 119891119894(x) ge 119891
119894(y)
Remark 6 It should be pointed out that if f is defined byhomogeneous and cooperative vector field in the continuous-time context (or f is defined by homogeneous and order-preserving vector field in the discrete-time case) the non-linear system (1) is positive which means that for everynonnegative initial condition x(0) isin R119899
+ the corresponding
state trajectory x(119905) isin R119899+for all 119905 ge 0
In this paper we consider the following SPHCS (SPHOSin the discrete-time case) consisting of a family of subsystems(1)
120599x (119905) = f120590(119905)
(x (119905)) (2)
where x(119905) isin R119899 is the state vector 120590(119905) is a switching signalwhich is a piecewise constant function from the right of timeand takes its values in the finite set 119878 = 1 119898 where119898 gt 1 is the number of subsystems f
119901 R119899 rarr R119899
are smooth functions for any 120590(119905) = 119901 isin 119878 Moreover allthe subsystems in system (1) may be unstable Also for aswitching sequence 0 lt 119905
1lt sdot sdot sdot lt 119905
119901lt 119905119901+1
lt sdot sdot sdot 120590(119905)may be either autonomous or controlled When 119905 isin [119905
119901 119905119901+1
)we say 120590(119905
119901)th mode is active With respect to switching law
120590(119905) the following exponential stability definition of system(2) is given andwe denote time by 119896 in the discrete-time case
Definition 7 Switched system (2) with switching 120590(119905) is saidto be globally uniformly exponentially stable (GUES) if thereexist constants 120572 gt 0 120573 gt 0 (resp 0 lt 120582 lt 1) such thatthe solution of the system satisfies x(119905) le 120572119890
minus120573(119905minus1199050)x(1199050)
forall119905 ge 1199050(resp x(119896) le 120572120582
minus(119896minus1198960)x(1199050) forall119896 ge 119896
0 with any
initial conditions x(1199050) (or x(119896
0)))
Due to the fact that all the positive nonlinear subsystemsof system (2) may be unstable we need to design timeconstrained switching sequences such that the switchednonlinear system (2) is GUES In order to achieve this goallet us first define the following ADT switching
Definition 8 For a switching signal 120590(119905) and each 1199052ge 1199051ge 0
let 119873120590(1199052 1199051) denote the number of discontinuities of 120590(119905) in
the interval (1199051 1199052) One says that 120590(119905) has an average dwell
time 120591119886if there exist two positive numbers 119873
0and 120591119886such
that
119873120590(1199052 1199051) ge 119873
0+1199052minus 1199051
120591119886
forall1199052ge 1199051ge 0 (3)
3 Main Results
In this section we consider the problem of stabilization forswitched positive nonlinear system (2) with our proposedADT switching
First we are in a position to provide the continuous-timeversion of stabilization condition in the case that the f
119901is
defined in homogeneous and cooperative vector field
Theorem 9 (continuous-time version) Consider SPHCS (2)possibly composed of all unstable modes If there exist a set ofvectors ^
119901≻ 0 119901 isin 119878 and two positive numbers 120578 gt 0 and
0 lt 120583 lt 1 such that forall119894 = 1 2 119899 forall119901 isin 119878
f119901(^119901) ≻ 0 (4)
119891119901119894(]119901119894)
]119901119894
minus 120578 lt 0 (5)
10038171003817100381710038171003817x (119905+119901)10038171003817100381710038171003817
infin
k119901minus 120583
10038171003817100381710038171003817x (119905minus119901)10038171003817100381710038171003817
infin
k119902le 0 (6)
where the 119909(119905119901)infin
]119901 is the weight 119897infin norm defined as
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901= max1le119894le119899
10038161003816100381610038161003816119909 (119905119901)119894
10038161003816100381610038161003816
]119894
(7)
then SPHCS (2) is globally uniformly exponentially stable forevery switching signal 120590(119905) with average dwell time
120591119886le 120591lowast
119886=minus ln 120583120578
(8)
Proof For any 119879 gt 0 1199050
= 0 we denote 1199051 1199052 119905
119901
119905119901+1
119905119873119873120590(1198790)
as the switching times on time interval [0 119879]Then we consider the function
119882(119905) = 119890minus120578(119905minus119905120590(119905)) x(119905)infin^120590(119905) (9)
It is immediately clear that 119882(119905) is a continuously differen-tiable function When 119905 isin [119905
119901 119905119901+1
) we get from (9) that
119882(119905) = 119890minus120578(119905minus119905119901) x(119905)infin^119901 (10)
It will show that 119882(119905) is nonincreasing when 119905 isin [119905119901 119905119901+1
)Next we suppose that there exists a point-in-time isin
[119905119901 119905119901+1
] such that
119882() =10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901 (11)
Then one can obtain from (7) and (11) that
119882() = 119890minus120578(119905minus119905120590(119905))max
1le119894le119899
1003816100381610038161003816119909119894 ()1003816100381610038161003816
]119901119894
= 119890minus120578(119905minus119905119901)
119909119896()
]119901119896
(12)
This together with (11) yields
119909119896() =
1003817100381710038171003817x()1003817100381710038171003817
infin
^119901]119901119896
= 119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901]119901119896
x () ⪯ 1003817100381710038171003817x()1003817100381710038171003817
infin
^119901^119901
= 119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901^119901
(13)
4 Mathematical Problems in Engineering
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is in homogeneous
and cooperative vector field based on Lemma 5 it can bederived from (5) (12) and (13) that
(119905) = 119890minus120578(minus119905119901)
119891119901119896(x ())]119901119896
minus 120578119890minus120578(minus119905119901)
119909119896()
]119901119896
le 119890minus120578(minus119905119901)
119891119901119896(119890120578(minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901^119901)
]119901119896
minus 120578119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901]119901119896
]119901119896
= 119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901119891119901119896(]119901)
]119901119896
minus 1205781003817100381710038171003817x ()
1003817100381710038171003817
infin
^119901
=10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901(119891119901119896(]119901)
]119901119896
minus 120578) lt 0
(14)
Note that if (11) holds we can get () lt 0 which impliesthat the upper-right Dini derivative 119863+(119882()) lt 0 Thus itcan be seen that 119882(119905) is nonincreasing when 119905 isin [119905
119901 119905119901+1
)This together with (10) gives that for forall119905 isin [119905
119901 119905119901+1
)
119882 (119905) = 119890minus120578(119905minus119905119901) x(119905)infin^119901
le 119882(119905119901)
=10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901
(15)
According to (6) and (15) one can obtain that10038171003817100381710038171003817x(119905+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119905minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583119890120578(119905119901+1minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901
(16)
By integrating this for 119905 isin [119905119901 119905119901+1
) one gets that
1003817100381710038171003817x(119879minus
)1003817100381710038171003817
infin
^119873120590le 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590
= 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905+119873120590
)10038171003817100381710038171003817
infin
^119873120590
le 120583119890120578(119879minus119905119873120590minus1
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590minus1
le 120583119873120575119890120578119879
x(0)infin^0
(17)
Moreover it can be derived from (3) and (17) that
x(119879)infin^119873120590 le 120583119873120575119890120578119879
x(0)infin^0
le 119890120578119879
119890(1198730+119879120591119886) ln 120583 x(0)infin^0
= 1198901198730 ln 120583119890(120578+ln 120583120591119886)119879 x(0)infin^0
(18)
Therefore by denoting 120572 = 1198901198730 ln 120583 120573 = minus(120578 + 120583120591
119886) one can
get from (18) that
119909 (119879) le 120572119890minus120573119879
119909 (0) forall119879 ge 0 (19)
By Definition 7 we conclude that SPHCS (2) is GUES byour proposed ADT switching signals (3) satisfying (8) if theconditions (4)ndash(6) hold This completes the proof
Remark 10 It has been shown in [19] that if there exists avector ^
119901≻ 0 satisfying f(^) ≻ 0 the corresponding homoge-
nous cooperative system is not globally asymptotically stableThus all the homogenous cooperative subsystems of SPHCS(2) may be unstable if (4) holds Under the ADT designedin Theorem 9 the stabilization can be achieved even if allthe subsystems are unstable On the other hand it should bepointed out that different from the existing results where thelower bound of the ADT should be specified a priori we needto set the upper bound in advance in our results
Next the following theorem is provided for the discrete-time version of stabilization condition for SPHOS (2) wherethe f119901is defined in homogeneous and order-preserving vector
field
Theorem 11 (discrete-time version) Consider SPHOS (2)possibly composed of all unstable modes If there exist a set ofvectors ^
119901≻ 0 119901 isin 119878 and two positive numbers 0 lt 120578 lt 1 and
0 lt 120583 lt 1 such that forall119894 = 1 2 119899 forall119901 isin 119878
f119901(^119901) ≻ ^119901 (20)
119891119901119894(]119901119894)
]119901119894
minus1
120578le 0 (21)
10038171003817100381710038171003817x(119896+119901)10038171003817100381710038171003817
infin
k119901minus 120583
10038171003817100381710038171003817x(119896minus119901)10038171003817100381710038171003817
infin
k119902le 0 (22)
where the 119909(119905119901)infin
]119901 is the weight 119897infin norm defined as
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901= max1le119894le119899
10038161003816100381610038161003816119909 (119905119901)119894
10038161003816100381610038161003816
]119894
(23)
then SPHOS (2) is globally uniformly exponentially stable forevery switching signal 120590 with average dwell time
120591119886le 120591lowast
119886=ln 120583ln 120578
(24)
Proof For any 119870 gt 0 1198960= 0 we denote 119896
1 1198962 119896
119901
119896119901+1
119896119873119873120590(1198700)
as the switching times on time interval[0 119879] Then we consider the function
119882(119905) = 120578119896minus119896120590(119896) x(119896)infin^120590(119896) (25)
When 119896 isin [119896119901 119896119901+1
) we get from (25) that
119882(119896) = 120578119896minus119896119901 x(119896)infin^119901 (26)
Mathematical Problems in Engineering 5
Next it will be shown that 119882(119896) le x(119896119901)infin
^119901 when 119896 isin
[119896119901 119896119901+1
) First when 119896 = 119896119901 the above inequality obviously
holdsThen we suppose that 119896 gt 119896119901 for a given 119896 isin [119896
119901 119896119901+1
)
the above inequality is satisfied and it proves that it is alsocorrect for 119896 + 1 isin (119896
119901 119896119901+1
) Due to the fact that for 119896 isin
[119896119901 119896119901+1
)119882(119896) le x(119896119901)infin
^119901 is true one can obtain that
119882(119896) = 120578119896minus119896119901max1le119894le119899
1003816100381610038161003816119909119894 (119896)1003816100381610038161003816
]119901119894
le10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901 (27)
which implies that
x (119896) ⪯ 120578119896119901minus119896
10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901^119901 (28)
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is defined in
homogeneous and order-preserving vector field it can beseen from (21) (26) and (28) that
119882(119896 + 1) = 120578119896+1minus119896119901
10038171003817100381710038171003817x((119896 + 1)
119901)10038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901
10038171003817100381710038171003817fp(x(119896))
10038171003817100381710038171003817
infin
^119901
le 120578119896+1minus119896119901
1003817100381710038171003817100381710038171003817fp (120578119896119901minus119896
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901^119901)
1003817100381710038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901120578
119896119901minus11989610038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 12057810038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 120578max1le119894le119899
10038161003816100381610038161003816119891119901119894(^119901)10038161003816100381610038161003816
]119901119894
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
le10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(29)
Thus we can obtain that (27) is right for forall119896 isin [119905119901 119905119901+1
)Combining this together with (22) one can get
10038171003817100381710038171003817x(119896+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119896minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583120578119896119901+1minus119896119901
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(30)
Similar to (17) and (18) we have for forall119870 gt 0
1003817100381710038171003817x(119870minus
)1003817100381710038171003817
infin
^119873120590le 120583119873120575120578minus119870
x(0)infin^0
le exp(119870120591119886
+ 1198730) ln 120583 exp minus119870 ln 120578 x(0)infin^0
= exp 1198730ln 120583 exp(
ln 120583120591119886
minus ln 120578)119870 x(0)infin^0
(31)
Therefore by denoting 120572 = exp1198730ln 120583 120582 =
exp(ln 120583120591119886minus ln 120578) one can get from (31) that
119909 (119870) le 120572120582119870
119909 (0) forall119870 ge 0 (32)
We conclude that SPHOS (2) is GUES by our proposed ADTswitching signals (3) satisfying (24) if the conditions (20)ndash(22) hold This completes the proof
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Stat
e res
pons
es
40
30
20
10
0
Figure 1 State response of the subsystem Σ1
Remark 12 InTheorems 9 and 11 based on a novel proposedswitching signals sufficient conditions of stabilization for aclass of switched positive nonlinear systems are obtainedwhere the positive subsystems are possibly all unstable
4 A Numerical Example
We provide the following numerical example in this sectionto verify our main results developed in this paper
Example 1 Consider a switched nonlinear system consistingof two homogeneous cooperative subsystems described by
Σ1=
1(119905) = 119891
11(1199091(119905) 1199092(119905))
= minus21199091(119905) + 4119909
2(119905) minus 3radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
12(1199091(119905) 1199092(119905))
= minus021199091(119905) + 01119909
2(119905)
+001radic11990921(119905) + 119909
2
2(119905)
Σ2=
1(119905) = 119891
21(1199091(119905) 1199092(119905))
= minus21199091(119905) + 3119909
2(119905) + 12radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
22(1199091(119905) 1199092(119905))
= 121199091(119905) minus 119909
2(119905) + 06radic119909
2
1(119905) + 119909
2
2(119905)
(33)
It is seen in (33) that f1 f2are defined in homogeneous
cooperative vector field which means that the switchednonlinear system is a switched positive system There exist avector ^
1= [3 78] ≻ 0 with f(^
1) = [01289 02636] ≻ 0 and
a vector ^2= [3 12] ≻ 0 with f(^
2) = [18754 04613] ≻ 0
Thus both of them are unstable and the state trajectories areshown in Figures 1 and 2
Next we are interested in designing a kind of switchingsignal 120590(119905) with property (3) to asymptotically stabilize thesystem Furthermore we generate two possible switchingsequences by 120591
119886= 05 and 120591
119886= 2 respectively the
corresponding state responses of the systemunder initial state
6 Mathematical Problems in EngineeringSt
ate r
espo
nses
1200
1000
800
600
400
200
0
Sample time (s)0 2 4 6 8 10
x1(t)
x2(t)
Figure 2 State response of the subsystem Σ2
Stat
e res
pons
es
15
10
5
0
Sample time (s)0 10 20 30 40 50
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Figure 3 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 05
condition 119909(0) = [10 15]119879 are shown in Figures 3 and 4 from
which we can see that the switched nonlinear system is stableunder 120591
119886= 05 but unstable under 120591
119886= 2
5 Conclusions
The stabilization problem for switched positive nonlinearsystems composed of possibly all unstable subsystems arestudied in both continuous-time and discrete-time domainsby using ADT switching As a first attempt a new class ofADT switching signal is proposed and then it is designedfor the system via the analysis of the weight 119897
infinnorm of
the system Two sufficient stabilization conditions for theunderlying systems are derived The highlight of the paperlies in that it is the first time the stabilization is solved forour considered switched positive nonlinear system wherethe system is possibly composed of all unstable subsystemsA numerical example is provided to show the effectivenessof our proposed approach In our future work we aim at
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50St
ate r
espo
nses
350
300
250
200
150
100
50
0
Figure 4 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 2
developing numerially easily checked conditions for the con-sidered systems and the problem of stabilization for switchedpositive nonlinear system with delays also needs to bestudied
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Kao CWang F Zha andH Cao ldquoStability inmean of partialvariables for stochastic reaction-diffusion systems with Marko-vian switchingrdquo Journal of the Franklin Institute vol 351 no 1pp 500ndash512 2014
[2] X-M Sun W Wang G-P Liu and J Zhao ldquoStability analysisfor linear switched systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 38 no 2 pp 528ndash533 2008
[3] L Wu and W X Zheng ldquoWeighted 119867infin
model reduction forlinear switched systems with time-varying delayrdquo Automaticavol 45 no 1 pp 186ndash193 2009
[4] L Zhang E-K Boukas and P Shi ldquoExponential 119867infin
filteringfor uncertain discrete-time switched linear systems with aver-age dwell time a 120583-dependent approachrdquo International Journalof Robust and Nonlinear Control vol 18 no 11 pp 1188ndash12072008
[5] X Zhao L Zhang and P Shi ldquoStability of a class of switchedpositive linear time-delay systemsrdquo International Journal ofRobust and Nonlinear Control vol 23 no 5 pp 578ndash589 2013
[6] L Zhang and H Gao ldquoAsynchronously switched control ofswitched linear systems with average dwell timerdquo Automaticavol 46 no 5 pp 953ndash958 2010
[7] L Farina and S Rinaldi Positive Linear SystemsTheory and Ap-plications vol 50 John Wiley amp Sons 2011
Mathematical Problems in Engineering 7
[8] X Chen J Lam P Li and Z Shu ldquoL1-induced norm andcontroller synthesis of positive systemsrdquoAutomatica vol 49 no5 pp 1377ndash1385 2013
[9] J-E Feng J Lam P Li and Z Shu ldquoDecay rate constrainedstabilization of positive systems using static output feedbackrdquoInternational Journal of Robust and Nonlinear Control vol 21no 1 pp 44ndash54 2011
[10] P Li J Lam and Z Shu ldquo119867infin
positive filtering for positivelinear discrete-time systems an augmentation approachrdquo IEEETransactions on Automatic Control vol 55 no 10 pp 2337ndash2342 2010
[11] P Li J Lam Z Wang and P Date ldquoPositivity-preserving 119867infin
model reduction for positive systemsrdquo Automatica vol 47 no7 pp 1504ndash1511 2011
[12] O Mason and R Shorten ldquoOn linear copositive Lyapunovfunctions and the stability of switched positive linear systemsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1346ndash1349 2007
[13] M A Rami and F Tadeo ldquoController synthesis for positivelinear systems with bounded controlsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 54 no 2 pp 151ndash1552007
[14] X Zhao P Shi and L Zhang ldquoAsynchronously switched controlof a class of slowly switched linear systemsrdquo Systems amp ControlLetters vol 61 no 12 pp 1151ndash1156 2012
[15] X Liu and C Dang ldquoStability analysis of positive switchedlinear systems with delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1684ndash1690 2011
[16] S Fadali and S Jafarzadeh ldquoStability analysis of positive intervaltype-2 TSK systems with application to energy marketsrdquo IEEETransactions on Fuzzy Systems vol 22 no 4 pp 1031ndash1038 2013
[17] L Imsland G O Eikrem and B A Foss ldquoA state feedbackcontroller for a class of nonlinear positive systems applied tostabilization of gas-lifted oil wellsrdquo Control Engineering PracticeN vol 3 pp 7ndash15 2006
[18] S K Nguang and P Shi ldquoFuzzy119867infinoutput feedback control of
nonlinear systems under sampled measurementsrdquo Automaticavol 39 no 12 pp 2169ndash2174 2003
[19] O Mason and M Verwoerd ldquoObservations on the stabilityproperties of cooperative systemsrdquo Systems amp Control Lettersvol 58 no 6 pp 461ndash467 2009
[20] H R Feyzmahdavian T Charalambous and M JohanssonldquoExponential stability of homogeneous positive systems ofdegree one with time-varying delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 6 pp 1594ndash1599 2014
[21] X Ding L Shu and X Liu ldquoOn linear copositive Lyapunovfunctions for switched positive systemsrdquo Journal of the FranklinInstitute vol 348 no 8 pp 2099ndash2107 2011
[22] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003
[23] X Zhao L Zhang P Shi and M Liu ldquoStability of switchedpositive linear systems with average dwell time switchingrdquoAutomatica vol 48 no 6 pp 1132ndash1137 2012
[24] X Zhao X Liu S Yin and H Li ldquoImproved results on stabilityof continuous-time switched positive linear systemsrdquo Automat-ica vol 50 no 2 pp 614ndash621 2014
[25] A S Morse ldquoSupervisory control of families of linear set-point controllersmdashpart 1 exactmatchingrdquo IEEETransactions onAutomatic Control vol 41 no 10 pp 1413ndash1431 1996
[26] D Liberzon Switching in Systems and Control Springer 2003[27] H Liu Y Shen and X Zhao ldquoAsynchronous finite-time
Hinfin
control for switched linear systems via mode-dependentdynamic state-feedbackrdquo Nonlinear Analysis Hybrid Systemsvol 8 no 1 pp 109ndash120 2013
[28] J Zhang Z Han H Wu and J Huang ldquoRobust stabilizationof discrete-time positive switched systems with uncertaintiesand average dwell time switchingrdquo Circuits Systems and SignalProcessing vol 33 no 1 pp 71ndash95 2014
[29] X Zhao L Zhang P Shi andM Liu ldquoStability and stabilizationof switched linear systems with mode-dependent average dwelltimerdquo IEEE Transactions on Automatic Control vol 57 no 7 pp1809ndash1815 2012
[30] J Lian and J Liu ldquoNew results on stability of switched positivesystems an average dwell-time approachrdquo IET Control Theoryand Applications vol 7 no 12 pp 1651ndash1658 2013
[31] X Zhao S Yin H Li and B Niu ldquoSwitching stabilization for aclass of slowly switched systemsrdquo IEEE Transactions on Auto-matic Control 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
recently studied due to its wide applications in practice As aspecial class of switched systems and positive systems somerelevant methods applicable to positive systems or switchedsystems are still suitable for switched positive systems [21 22]But anyway it should be pointed out that those methodsmay not be as efficient as they are for general switchedsystems or positive systems Therefore many researchershave recently focused on exploring efficient approaches andresults for switched positive systems The authors of [23]introduced a multiple linear copositive Lyapunov function(MLCLF) to investigate the problem of stability for switchedpositive systems Then the authors of [24] presented ageneralization of copositive types of Lyapunov function forstability analysis of switched positive systems Note that thoseworks all consider switched positive linear systems So farthere are few results reported for switched positive nonlinearsystems which are however of both theoretical and practicalimportances
As far as the stabilization of switched positive systems isconcerned how to find an appropriate switching signals toguarantee system stability is an important issue Switchingsignals that may be either autonomous or controlled gener-ally include arbitrary switching signals stochastic switchingsignals and constraint switching signals Time constrainedswitching signal that is viewed as an important class ofconstraint switching signals naturally has been extensivelyinvestigated during the past several decades because of itsstrong applicability Time constrained switching signal canbe also called slow switching and it is classified into threetypes dwell time (DT) average dwell time (ADT) andmode-dependent average dwell time switching (MDADT)
In the recent few years time constrained switching hasbeen successfully used for switching stabilization of bothgeneral switched systems and switched positive systems Tolist a few when all the subsystems of a switched linearsystem are stable the author of [25] had shown that thesystem is exponentially stable if the DT is sufficiently largeIt was proved in [26ndash28] that ADT turns out to be veryuseful for analysis and synthesis of switched systems TheADT was extended to MDADT in [29] for stabilization ofgeneral switched systems In [23] the ADT switching wasutilized for switched positive linear systems It is noted thatmost recently the time constrained switching stabilizationfor switched (positive) linear systems with partlyall unstablesubsystems have been paid some attentions [6 30 31] How-ever such an attempt is just a start and the slow switchingstabilization for switched (positive) systems with possibly allunstable subsystems has not been fully solved especially forswitched positive nonlinear systems
In summary the existing time constrained switchingstabilization results for switched positive systems are mainlyfocused on the linear case and those promising ideas thereinare not applicable for switched positive nonlinear systemsSuch a problem becomes more complicated once all the non-linear subsystems become unstable To the best of the authorsrsquoknowledge up to now there is no literature reporting theswitching stabilization problem for switched positive nonlin-ear systems with possibly all unstable subsystems even in thelinear case
All the above observations motivate us to carry outthe present work Compared with some existing resultsour results can solve the switching stabilization problem ofswitched positive nonlinear systems though all the subsys-tems are unstable The most important of the paper liesin that a new class of ADT is proposed and the weighted119897infin
norm of the considered systemrsquos state is introduced totackle the considered problem The layout of the paperis organized as follows Section 2 reviews some necessarydefinitions of switched positive nonlinear systems and definesa new concept of ADT which characterizes a different setof switching signals from traditional ADT In Section 3stabilization criteria for switched positive nonlinear systemswith possibly all unstable subsystems via proposed ADTswitching are derived in continuous-time and discrete-timecase Section 4 provides a numerical example to demonstratethe feasibility and effectiveness of the proposed techniquesand Section 5 concludes the paper
Notations In this paper the notations used are standard RR119899 and R119899
+denote the field of real numbers 119899-dimensional
Euclidean space and the nonnegative orthant of R119899 respec-tively and R119899
0stands for R119899
+ 0119899
and 119872119890119905119911119897119890119903 is a matrixwhose off diagonal entries are nonnegative The notation sdot refers to the Euclidean norm For x isin R119899 119909
119894denotes the 119894th
component of x In addition x ⪰ y (or x ≻ y) mean that allentries of vector 119909
119894ge 119910119894(or 119909119894gt 119910119894)
2 Problem Formulation and Preliminaries
This section presents some definitions and preliminaryresults which will be used throughout the paper We firstrecall some preliminaries about positive nonlinear systemsfor the introduction of switched positive nonlinear systemshereafter Consider the following nonlinear system
120599x (119905) = f (119909 (119905)) (1)
where x(119905) isin R119899 is the state vector f R119899 rarr R119899 is asmooth functions vector and is homogeneous cooperative inthe continuous-time case (homogeneous order-preserving inthe discrete-time case) the symbol 120599 denotes the derivativeoperator in the continuous-time context 120599x(119905) = (119889119889119905)x(119905)and the shift forward operator in the discrete-time case120599x(119905) = x(119905 + 1)
Definition 1 System (1) is said to be positive if and only if forevery nonnegative initial state its state is nonnegative
Definition 2 A continuous vector field f R119899 rarr R119899 is saidto be homogeneous if for all x(119905) isin R119899 and all real 120582 gt 0f(120582x) = 120582f(x)
Definition 3 A continuous vector field f R119899 rarr R119899 whichis C1 on R119899 0 is said to be cooperative if the Jacobian(120597f120597x)(a) is Metzler for all a isin R119899
0
Definition 4 A continuous vector field f R119899 rarr R119899 is saidto be order-preserving onR119899
+ if f(x) ≻ f(y) for any x y isin R119899
+
such that x ⪰ y
Mathematical Problems in Engineering 3
Lemma 5 If f R119899 rarr R119899 is cooperative then for any twovectors x y isin R119899
0satisfying x ⪰ y and 119909
119894= 119910119894 119891119894(x) ge 119891
119894(y)
Remark 6 It should be pointed out that if f is defined byhomogeneous and cooperative vector field in the continuous-time context (or f is defined by homogeneous and order-preserving vector field in the discrete-time case) the non-linear system (1) is positive which means that for everynonnegative initial condition x(0) isin R119899
+ the corresponding
state trajectory x(119905) isin R119899+for all 119905 ge 0
In this paper we consider the following SPHCS (SPHOSin the discrete-time case) consisting of a family of subsystems(1)
120599x (119905) = f120590(119905)
(x (119905)) (2)
where x(119905) isin R119899 is the state vector 120590(119905) is a switching signalwhich is a piecewise constant function from the right of timeand takes its values in the finite set 119878 = 1 119898 where119898 gt 1 is the number of subsystems f
119901 R119899 rarr R119899
are smooth functions for any 120590(119905) = 119901 isin 119878 Moreover allthe subsystems in system (1) may be unstable Also for aswitching sequence 0 lt 119905
1lt sdot sdot sdot lt 119905
119901lt 119905119901+1
lt sdot sdot sdot 120590(119905)may be either autonomous or controlled When 119905 isin [119905
119901 119905119901+1
)we say 120590(119905
119901)th mode is active With respect to switching law
120590(119905) the following exponential stability definition of system(2) is given andwe denote time by 119896 in the discrete-time case
Definition 7 Switched system (2) with switching 120590(119905) is saidto be globally uniformly exponentially stable (GUES) if thereexist constants 120572 gt 0 120573 gt 0 (resp 0 lt 120582 lt 1) such thatthe solution of the system satisfies x(119905) le 120572119890
minus120573(119905minus1199050)x(1199050)
forall119905 ge 1199050(resp x(119896) le 120572120582
minus(119896minus1198960)x(1199050) forall119896 ge 119896
0 with any
initial conditions x(1199050) (or x(119896
0)))
Due to the fact that all the positive nonlinear subsystemsof system (2) may be unstable we need to design timeconstrained switching sequences such that the switchednonlinear system (2) is GUES In order to achieve this goallet us first define the following ADT switching
Definition 8 For a switching signal 120590(119905) and each 1199052ge 1199051ge 0
let 119873120590(1199052 1199051) denote the number of discontinuities of 120590(119905) in
the interval (1199051 1199052) One says that 120590(119905) has an average dwell
time 120591119886if there exist two positive numbers 119873
0and 120591119886such
that
119873120590(1199052 1199051) ge 119873
0+1199052minus 1199051
120591119886
forall1199052ge 1199051ge 0 (3)
3 Main Results
In this section we consider the problem of stabilization forswitched positive nonlinear system (2) with our proposedADT switching
First we are in a position to provide the continuous-timeversion of stabilization condition in the case that the f
119901is
defined in homogeneous and cooperative vector field
Theorem 9 (continuous-time version) Consider SPHCS (2)possibly composed of all unstable modes If there exist a set ofvectors ^
119901≻ 0 119901 isin 119878 and two positive numbers 120578 gt 0 and
0 lt 120583 lt 1 such that forall119894 = 1 2 119899 forall119901 isin 119878
f119901(^119901) ≻ 0 (4)
119891119901119894(]119901119894)
]119901119894
minus 120578 lt 0 (5)
10038171003817100381710038171003817x (119905+119901)10038171003817100381710038171003817
infin
k119901minus 120583
10038171003817100381710038171003817x (119905minus119901)10038171003817100381710038171003817
infin
k119902le 0 (6)
where the 119909(119905119901)infin
]119901 is the weight 119897infin norm defined as
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901= max1le119894le119899
10038161003816100381610038161003816119909 (119905119901)119894
10038161003816100381610038161003816
]119894
(7)
then SPHCS (2) is globally uniformly exponentially stable forevery switching signal 120590(119905) with average dwell time
120591119886le 120591lowast
119886=minus ln 120583120578
(8)
Proof For any 119879 gt 0 1199050
= 0 we denote 1199051 1199052 119905
119901
119905119901+1
119905119873119873120590(1198790)
as the switching times on time interval [0 119879]Then we consider the function
119882(119905) = 119890minus120578(119905minus119905120590(119905)) x(119905)infin^120590(119905) (9)
It is immediately clear that 119882(119905) is a continuously differen-tiable function When 119905 isin [119905
119901 119905119901+1
) we get from (9) that
119882(119905) = 119890minus120578(119905minus119905119901) x(119905)infin^119901 (10)
It will show that 119882(119905) is nonincreasing when 119905 isin [119905119901 119905119901+1
)Next we suppose that there exists a point-in-time isin
[119905119901 119905119901+1
] such that
119882() =10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901 (11)
Then one can obtain from (7) and (11) that
119882() = 119890minus120578(119905minus119905120590(119905))max
1le119894le119899
1003816100381610038161003816119909119894 ()1003816100381610038161003816
]119901119894
= 119890minus120578(119905minus119905119901)
119909119896()
]119901119896
(12)
This together with (11) yields
119909119896() =
1003817100381710038171003817x()1003817100381710038171003817
infin
^119901]119901119896
= 119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901]119901119896
x () ⪯ 1003817100381710038171003817x()1003817100381710038171003817
infin
^119901^119901
= 119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901^119901
(13)
4 Mathematical Problems in Engineering
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is in homogeneous
and cooperative vector field based on Lemma 5 it can bederived from (5) (12) and (13) that
(119905) = 119890minus120578(minus119905119901)
119891119901119896(x ())]119901119896
minus 120578119890minus120578(minus119905119901)
119909119896()
]119901119896
le 119890minus120578(minus119905119901)
119891119901119896(119890120578(minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901^119901)
]119901119896
minus 120578119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901]119901119896
]119901119896
= 119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901119891119901119896(]119901)
]119901119896
minus 1205781003817100381710038171003817x ()
1003817100381710038171003817
infin
^119901
=10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901(119891119901119896(]119901)
]119901119896
minus 120578) lt 0
(14)
Note that if (11) holds we can get () lt 0 which impliesthat the upper-right Dini derivative 119863+(119882()) lt 0 Thus itcan be seen that 119882(119905) is nonincreasing when 119905 isin [119905
119901 119905119901+1
)This together with (10) gives that for forall119905 isin [119905
119901 119905119901+1
)
119882 (119905) = 119890minus120578(119905minus119905119901) x(119905)infin^119901
le 119882(119905119901)
=10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901
(15)
According to (6) and (15) one can obtain that10038171003817100381710038171003817x(119905+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119905minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583119890120578(119905119901+1minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901
(16)
By integrating this for 119905 isin [119905119901 119905119901+1
) one gets that
1003817100381710038171003817x(119879minus
)1003817100381710038171003817
infin
^119873120590le 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590
= 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905+119873120590
)10038171003817100381710038171003817
infin
^119873120590
le 120583119890120578(119879minus119905119873120590minus1
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590minus1
le 120583119873120575119890120578119879
x(0)infin^0
(17)
Moreover it can be derived from (3) and (17) that
x(119879)infin^119873120590 le 120583119873120575119890120578119879
x(0)infin^0
le 119890120578119879
119890(1198730+119879120591119886) ln 120583 x(0)infin^0
= 1198901198730 ln 120583119890(120578+ln 120583120591119886)119879 x(0)infin^0
(18)
Therefore by denoting 120572 = 1198901198730 ln 120583 120573 = minus(120578 + 120583120591
119886) one can
get from (18) that
119909 (119879) le 120572119890minus120573119879
119909 (0) forall119879 ge 0 (19)
By Definition 7 we conclude that SPHCS (2) is GUES byour proposed ADT switching signals (3) satisfying (8) if theconditions (4)ndash(6) hold This completes the proof
Remark 10 It has been shown in [19] that if there exists avector ^
119901≻ 0 satisfying f(^) ≻ 0 the corresponding homoge-
nous cooperative system is not globally asymptotically stableThus all the homogenous cooperative subsystems of SPHCS(2) may be unstable if (4) holds Under the ADT designedin Theorem 9 the stabilization can be achieved even if allthe subsystems are unstable On the other hand it should bepointed out that different from the existing results where thelower bound of the ADT should be specified a priori we needto set the upper bound in advance in our results
Next the following theorem is provided for the discrete-time version of stabilization condition for SPHOS (2) wherethe f119901is defined in homogeneous and order-preserving vector
field
Theorem 11 (discrete-time version) Consider SPHOS (2)possibly composed of all unstable modes If there exist a set ofvectors ^
119901≻ 0 119901 isin 119878 and two positive numbers 0 lt 120578 lt 1 and
0 lt 120583 lt 1 such that forall119894 = 1 2 119899 forall119901 isin 119878
f119901(^119901) ≻ ^119901 (20)
119891119901119894(]119901119894)
]119901119894
minus1
120578le 0 (21)
10038171003817100381710038171003817x(119896+119901)10038171003817100381710038171003817
infin
k119901minus 120583
10038171003817100381710038171003817x(119896minus119901)10038171003817100381710038171003817
infin
k119902le 0 (22)
where the 119909(119905119901)infin
]119901 is the weight 119897infin norm defined as
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901= max1le119894le119899
10038161003816100381610038161003816119909 (119905119901)119894
10038161003816100381610038161003816
]119894
(23)
then SPHOS (2) is globally uniformly exponentially stable forevery switching signal 120590 with average dwell time
120591119886le 120591lowast
119886=ln 120583ln 120578
(24)
Proof For any 119870 gt 0 1198960= 0 we denote 119896
1 1198962 119896
119901
119896119901+1
119896119873119873120590(1198700)
as the switching times on time interval[0 119879] Then we consider the function
119882(119905) = 120578119896minus119896120590(119896) x(119896)infin^120590(119896) (25)
When 119896 isin [119896119901 119896119901+1
) we get from (25) that
119882(119896) = 120578119896minus119896119901 x(119896)infin^119901 (26)
Mathematical Problems in Engineering 5
Next it will be shown that 119882(119896) le x(119896119901)infin
^119901 when 119896 isin
[119896119901 119896119901+1
) First when 119896 = 119896119901 the above inequality obviously
holdsThen we suppose that 119896 gt 119896119901 for a given 119896 isin [119896
119901 119896119901+1
)
the above inequality is satisfied and it proves that it is alsocorrect for 119896 + 1 isin (119896
119901 119896119901+1
) Due to the fact that for 119896 isin
[119896119901 119896119901+1
)119882(119896) le x(119896119901)infin
^119901 is true one can obtain that
119882(119896) = 120578119896minus119896119901max1le119894le119899
1003816100381610038161003816119909119894 (119896)1003816100381610038161003816
]119901119894
le10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901 (27)
which implies that
x (119896) ⪯ 120578119896119901minus119896
10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901^119901 (28)
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is defined in
homogeneous and order-preserving vector field it can beseen from (21) (26) and (28) that
119882(119896 + 1) = 120578119896+1minus119896119901
10038171003817100381710038171003817x((119896 + 1)
119901)10038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901
10038171003817100381710038171003817fp(x(119896))
10038171003817100381710038171003817
infin
^119901
le 120578119896+1minus119896119901
1003817100381710038171003817100381710038171003817fp (120578119896119901minus119896
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901^119901)
1003817100381710038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901120578
119896119901minus11989610038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 12057810038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 120578max1le119894le119899
10038161003816100381610038161003816119891119901119894(^119901)10038161003816100381610038161003816
]119901119894
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
le10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(29)
Thus we can obtain that (27) is right for forall119896 isin [119905119901 119905119901+1
)Combining this together with (22) one can get
10038171003817100381710038171003817x(119896+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119896minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583120578119896119901+1minus119896119901
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(30)
Similar to (17) and (18) we have for forall119870 gt 0
1003817100381710038171003817x(119870minus
)1003817100381710038171003817
infin
^119873120590le 120583119873120575120578minus119870
x(0)infin^0
le exp(119870120591119886
+ 1198730) ln 120583 exp minus119870 ln 120578 x(0)infin^0
= exp 1198730ln 120583 exp(
ln 120583120591119886
minus ln 120578)119870 x(0)infin^0
(31)
Therefore by denoting 120572 = exp1198730ln 120583 120582 =
exp(ln 120583120591119886minus ln 120578) one can get from (31) that
119909 (119870) le 120572120582119870
119909 (0) forall119870 ge 0 (32)
We conclude that SPHOS (2) is GUES by our proposed ADTswitching signals (3) satisfying (24) if the conditions (20)ndash(22) hold This completes the proof
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Stat
e res
pons
es
40
30
20
10
0
Figure 1 State response of the subsystem Σ1
Remark 12 InTheorems 9 and 11 based on a novel proposedswitching signals sufficient conditions of stabilization for aclass of switched positive nonlinear systems are obtainedwhere the positive subsystems are possibly all unstable
4 A Numerical Example
We provide the following numerical example in this sectionto verify our main results developed in this paper
Example 1 Consider a switched nonlinear system consistingof two homogeneous cooperative subsystems described by
Σ1=
1(119905) = 119891
11(1199091(119905) 1199092(119905))
= minus21199091(119905) + 4119909
2(119905) minus 3radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
12(1199091(119905) 1199092(119905))
= minus021199091(119905) + 01119909
2(119905)
+001radic11990921(119905) + 119909
2
2(119905)
Σ2=
1(119905) = 119891
21(1199091(119905) 1199092(119905))
= minus21199091(119905) + 3119909
2(119905) + 12radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
22(1199091(119905) 1199092(119905))
= 121199091(119905) minus 119909
2(119905) + 06radic119909
2
1(119905) + 119909
2
2(119905)
(33)
It is seen in (33) that f1 f2are defined in homogeneous
cooperative vector field which means that the switchednonlinear system is a switched positive system There exist avector ^
1= [3 78] ≻ 0 with f(^
1) = [01289 02636] ≻ 0 and
a vector ^2= [3 12] ≻ 0 with f(^
2) = [18754 04613] ≻ 0
Thus both of them are unstable and the state trajectories areshown in Figures 1 and 2
Next we are interested in designing a kind of switchingsignal 120590(119905) with property (3) to asymptotically stabilize thesystem Furthermore we generate two possible switchingsequences by 120591
119886= 05 and 120591
119886= 2 respectively the
corresponding state responses of the systemunder initial state
6 Mathematical Problems in EngineeringSt
ate r
espo
nses
1200
1000
800
600
400
200
0
Sample time (s)0 2 4 6 8 10
x1(t)
x2(t)
Figure 2 State response of the subsystem Σ2
Stat
e res
pons
es
15
10
5
0
Sample time (s)0 10 20 30 40 50
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Figure 3 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 05
condition 119909(0) = [10 15]119879 are shown in Figures 3 and 4 from
which we can see that the switched nonlinear system is stableunder 120591
119886= 05 but unstable under 120591
119886= 2
5 Conclusions
The stabilization problem for switched positive nonlinearsystems composed of possibly all unstable subsystems arestudied in both continuous-time and discrete-time domainsby using ADT switching As a first attempt a new class ofADT switching signal is proposed and then it is designedfor the system via the analysis of the weight 119897
infinnorm of
the system Two sufficient stabilization conditions for theunderlying systems are derived The highlight of the paperlies in that it is the first time the stabilization is solved forour considered switched positive nonlinear system wherethe system is possibly composed of all unstable subsystemsA numerical example is provided to show the effectivenessof our proposed approach In our future work we aim at
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50St
ate r
espo
nses
350
300
250
200
150
100
50
0
Figure 4 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 2
developing numerially easily checked conditions for the con-sidered systems and the problem of stabilization for switchedpositive nonlinear system with delays also needs to bestudied
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Kao CWang F Zha andH Cao ldquoStability inmean of partialvariables for stochastic reaction-diffusion systems with Marko-vian switchingrdquo Journal of the Franklin Institute vol 351 no 1pp 500ndash512 2014
[2] X-M Sun W Wang G-P Liu and J Zhao ldquoStability analysisfor linear switched systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 38 no 2 pp 528ndash533 2008
[3] L Wu and W X Zheng ldquoWeighted 119867infin
model reduction forlinear switched systems with time-varying delayrdquo Automaticavol 45 no 1 pp 186ndash193 2009
[4] L Zhang E-K Boukas and P Shi ldquoExponential 119867infin
filteringfor uncertain discrete-time switched linear systems with aver-age dwell time a 120583-dependent approachrdquo International Journalof Robust and Nonlinear Control vol 18 no 11 pp 1188ndash12072008
[5] X Zhao L Zhang and P Shi ldquoStability of a class of switchedpositive linear time-delay systemsrdquo International Journal ofRobust and Nonlinear Control vol 23 no 5 pp 578ndash589 2013
[6] L Zhang and H Gao ldquoAsynchronously switched control ofswitched linear systems with average dwell timerdquo Automaticavol 46 no 5 pp 953ndash958 2010
[7] L Farina and S Rinaldi Positive Linear SystemsTheory and Ap-plications vol 50 John Wiley amp Sons 2011
Mathematical Problems in Engineering 7
[8] X Chen J Lam P Li and Z Shu ldquoL1-induced norm andcontroller synthesis of positive systemsrdquoAutomatica vol 49 no5 pp 1377ndash1385 2013
[9] J-E Feng J Lam P Li and Z Shu ldquoDecay rate constrainedstabilization of positive systems using static output feedbackrdquoInternational Journal of Robust and Nonlinear Control vol 21no 1 pp 44ndash54 2011
[10] P Li J Lam and Z Shu ldquo119867infin
positive filtering for positivelinear discrete-time systems an augmentation approachrdquo IEEETransactions on Automatic Control vol 55 no 10 pp 2337ndash2342 2010
[11] P Li J Lam Z Wang and P Date ldquoPositivity-preserving 119867infin
model reduction for positive systemsrdquo Automatica vol 47 no7 pp 1504ndash1511 2011
[12] O Mason and R Shorten ldquoOn linear copositive Lyapunovfunctions and the stability of switched positive linear systemsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1346ndash1349 2007
[13] M A Rami and F Tadeo ldquoController synthesis for positivelinear systems with bounded controlsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 54 no 2 pp 151ndash1552007
[14] X Zhao P Shi and L Zhang ldquoAsynchronously switched controlof a class of slowly switched linear systemsrdquo Systems amp ControlLetters vol 61 no 12 pp 1151ndash1156 2012
[15] X Liu and C Dang ldquoStability analysis of positive switchedlinear systems with delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1684ndash1690 2011
[16] S Fadali and S Jafarzadeh ldquoStability analysis of positive intervaltype-2 TSK systems with application to energy marketsrdquo IEEETransactions on Fuzzy Systems vol 22 no 4 pp 1031ndash1038 2013
[17] L Imsland G O Eikrem and B A Foss ldquoA state feedbackcontroller for a class of nonlinear positive systems applied tostabilization of gas-lifted oil wellsrdquo Control Engineering PracticeN vol 3 pp 7ndash15 2006
[18] S K Nguang and P Shi ldquoFuzzy119867infinoutput feedback control of
nonlinear systems under sampled measurementsrdquo Automaticavol 39 no 12 pp 2169ndash2174 2003
[19] O Mason and M Verwoerd ldquoObservations on the stabilityproperties of cooperative systemsrdquo Systems amp Control Lettersvol 58 no 6 pp 461ndash467 2009
[20] H R Feyzmahdavian T Charalambous and M JohanssonldquoExponential stability of homogeneous positive systems ofdegree one with time-varying delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 6 pp 1594ndash1599 2014
[21] X Ding L Shu and X Liu ldquoOn linear copositive Lyapunovfunctions for switched positive systemsrdquo Journal of the FranklinInstitute vol 348 no 8 pp 2099ndash2107 2011
[22] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003
[23] X Zhao L Zhang P Shi and M Liu ldquoStability of switchedpositive linear systems with average dwell time switchingrdquoAutomatica vol 48 no 6 pp 1132ndash1137 2012
[24] X Zhao X Liu S Yin and H Li ldquoImproved results on stabilityof continuous-time switched positive linear systemsrdquo Automat-ica vol 50 no 2 pp 614ndash621 2014
[25] A S Morse ldquoSupervisory control of families of linear set-point controllersmdashpart 1 exactmatchingrdquo IEEETransactions onAutomatic Control vol 41 no 10 pp 1413ndash1431 1996
[26] D Liberzon Switching in Systems and Control Springer 2003[27] H Liu Y Shen and X Zhao ldquoAsynchronous finite-time
Hinfin
control for switched linear systems via mode-dependentdynamic state-feedbackrdquo Nonlinear Analysis Hybrid Systemsvol 8 no 1 pp 109ndash120 2013
[28] J Zhang Z Han H Wu and J Huang ldquoRobust stabilizationof discrete-time positive switched systems with uncertaintiesand average dwell time switchingrdquo Circuits Systems and SignalProcessing vol 33 no 1 pp 71ndash95 2014
[29] X Zhao L Zhang P Shi andM Liu ldquoStability and stabilizationof switched linear systems with mode-dependent average dwelltimerdquo IEEE Transactions on Automatic Control vol 57 no 7 pp1809ndash1815 2012
[30] J Lian and J Liu ldquoNew results on stability of switched positivesystems an average dwell-time approachrdquo IET Control Theoryand Applications vol 7 no 12 pp 1651ndash1658 2013
[31] X Zhao S Yin H Li and B Niu ldquoSwitching stabilization for aclass of slowly switched systemsrdquo IEEE Transactions on Auto-matic Control 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Lemma 5 If f R119899 rarr R119899 is cooperative then for any twovectors x y isin R119899
0satisfying x ⪰ y and 119909
119894= 119910119894 119891119894(x) ge 119891
119894(y)
Remark 6 It should be pointed out that if f is defined byhomogeneous and cooperative vector field in the continuous-time context (or f is defined by homogeneous and order-preserving vector field in the discrete-time case) the non-linear system (1) is positive which means that for everynonnegative initial condition x(0) isin R119899
+ the corresponding
state trajectory x(119905) isin R119899+for all 119905 ge 0
In this paper we consider the following SPHCS (SPHOSin the discrete-time case) consisting of a family of subsystems(1)
120599x (119905) = f120590(119905)
(x (119905)) (2)
where x(119905) isin R119899 is the state vector 120590(119905) is a switching signalwhich is a piecewise constant function from the right of timeand takes its values in the finite set 119878 = 1 119898 where119898 gt 1 is the number of subsystems f
119901 R119899 rarr R119899
are smooth functions for any 120590(119905) = 119901 isin 119878 Moreover allthe subsystems in system (1) may be unstable Also for aswitching sequence 0 lt 119905
1lt sdot sdot sdot lt 119905
119901lt 119905119901+1
lt sdot sdot sdot 120590(119905)may be either autonomous or controlled When 119905 isin [119905
119901 119905119901+1
)we say 120590(119905
119901)th mode is active With respect to switching law
120590(119905) the following exponential stability definition of system(2) is given andwe denote time by 119896 in the discrete-time case
Definition 7 Switched system (2) with switching 120590(119905) is saidto be globally uniformly exponentially stable (GUES) if thereexist constants 120572 gt 0 120573 gt 0 (resp 0 lt 120582 lt 1) such thatthe solution of the system satisfies x(119905) le 120572119890
minus120573(119905minus1199050)x(1199050)
forall119905 ge 1199050(resp x(119896) le 120572120582
minus(119896minus1198960)x(1199050) forall119896 ge 119896
0 with any
initial conditions x(1199050) (or x(119896
0)))
Due to the fact that all the positive nonlinear subsystemsof system (2) may be unstable we need to design timeconstrained switching sequences such that the switchednonlinear system (2) is GUES In order to achieve this goallet us first define the following ADT switching
Definition 8 For a switching signal 120590(119905) and each 1199052ge 1199051ge 0
let 119873120590(1199052 1199051) denote the number of discontinuities of 120590(119905) in
the interval (1199051 1199052) One says that 120590(119905) has an average dwell
time 120591119886if there exist two positive numbers 119873
0and 120591119886such
that
119873120590(1199052 1199051) ge 119873
0+1199052minus 1199051
120591119886
forall1199052ge 1199051ge 0 (3)
3 Main Results
In this section we consider the problem of stabilization forswitched positive nonlinear system (2) with our proposedADT switching
First we are in a position to provide the continuous-timeversion of stabilization condition in the case that the f
119901is
defined in homogeneous and cooperative vector field
Theorem 9 (continuous-time version) Consider SPHCS (2)possibly composed of all unstable modes If there exist a set ofvectors ^
119901≻ 0 119901 isin 119878 and two positive numbers 120578 gt 0 and
0 lt 120583 lt 1 such that forall119894 = 1 2 119899 forall119901 isin 119878
f119901(^119901) ≻ 0 (4)
119891119901119894(]119901119894)
]119901119894
minus 120578 lt 0 (5)
10038171003817100381710038171003817x (119905+119901)10038171003817100381710038171003817
infin
k119901minus 120583
10038171003817100381710038171003817x (119905minus119901)10038171003817100381710038171003817
infin
k119902le 0 (6)
where the 119909(119905119901)infin
]119901 is the weight 119897infin norm defined as
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901= max1le119894le119899
10038161003816100381610038161003816119909 (119905119901)119894
10038161003816100381610038161003816
]119894
(7)
then SPHCS (2) is globally uniformly exponentially stable forevery switching signal 120590(119905) with average dwell time
120591119886le 120591lowast
119886=minus ln 120583120578
(8)
Proof For any 119879 gt 0 1199050
= 0 we denote 1199051 1199052 119905
119901
119905119901+1
119905119873119873120590(1198790)
as the switching times on time interval [0 119879]Then we consider the function
119882(119905) = 119890minus120578(119905minus119905120590(119905)) x(119905)infin^120590(119905) (9)
It is immediately clear that 119882(119905) is a continuously differen-tiable function When 119905 isin [119905
119901 119905119901+1
) we get from (9) that
119882(119905) = 119890minus120578(119905minus119905119901) x(119905)infin^119901 (10)
It will show that 119882(119905) is nonincreasing when 119905 isin [119905119901 119905119901+1
)Next we suppose that there exists a point-in-time isin
[119905119901 119905119901+1
] such that
119882() =10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901 (11)
Then one can obtain from (7) and (11) that
119882() = 119890minus120578(119905minus119905120590(119905))max
1le119894le119899
1003816100381610038161003816119909119894 ()1003816100381610038161003816
]119901119894
= 119890minus120578(119905minus119905119901)
119909119896()
]119901119896
(12)
This together with (11) yields
119909119896() =
1003817100381710038171003817x()1003817100381710038171003817
infin
^119901]119901119896
= 119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901]119901119896
x () ⪯ 1003817100381710038171003817x()1003817100381710038171003817
infin
^119901^119901
= 119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901^119901
(13)
4 Mathematical Problems in Engineering
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is in homogeneous
and cooperative vector field based on Lemma 5 it can bederived from (5) (12) and (13) that
(119905) = 119890minus120578(minus119905119901)
119891119901119896(x ())]119901119896
minus 120578119890minus120578(minus119905119901)
119909119896()
]119901119896
le 119890minus120578(minus119905119901)
119891119901119896(119890120578(minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901^119901)
]119901119896
minus 120578119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901]119901119896
]119901119896
= 119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901119891119901119896(]119901)
]119901119896
minus 1205781003817100381710038171003817x ()
1003817100381710038171003817
infin
^119901
=10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901(119891119901119896(]119901)
]119901119896
minus 120578) lt 0
(14)
Note that if (11) holds we can get () lt 0 which impliesthat the upper-right Dini derivative 119863+(119882()) lt 0 Thus itcan be seen that 119882(119905) is nonincreasing when 119905 isin [119905
119901 119905119901+1
)This together with (10) gives that for forall119905 isin [119905
119901 119905119901+1
)
119882 (119905) = 119890minus120578(119905minus119905119901) x(119905)infin^119901
le 119882(119905119901)
=10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901
(15)
According to (6) and (15) one can obtain that10038171003817100381710038171003817x(119905+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119905minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583119890120578(119905119901+1minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901
(16)
By integrating this for 119905 isin [119905119901 119905119901+1
) one gets that
1003817100381710038171003817x(119879minus
)1003817100381710038171003817
infin
^119873120590le 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590
= 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905+119873120590
)10038171003817100381710038171003817
infin
^119873120590
le 120583119890120578(119879minus119905119873120590minus1
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590minus1
le 120583119873120575119890120578119879
x(0)infin^0
(17)
Moreover it can be derived from (3) and (17) that
x(119879)infin^119873120590 le 120583119873120575119890120578119879
x(0)infin^0
le 119890120578119879
119890(1198730+119879120591119886) ln 120583 x(0)infin^0
= 1198901198730 ln 120583119890(120578+ln 120583120591119886)119879 x(0)infin^0
(18)
Therefore by denoting 120572 = 1198901198730 ln 120583 120573 = minus(120578 + 120583120591
119886) one can
get from (18) that
119909 (119879) le 120572119890minus120573119879
119909 (0) forall119879 ge 0 (19)
By Definition 7 we conclude that SPHCS (2) is GUES byour proposed ADT switching signals (3) satisfying (8) if theconditions (4)ndash(6) hold This completes the proof
Remark 10 It has been shown in [19] that if there exists avector ^
119901≻ 0 satisfying f(^) ≻ 0 the corresponding homoge-
nous cooperative system is not globally asymptotically stableThus all the homogenous cooperative subsystems of SPHCS(2) may be unstable if (4) holds Under the ADT designedin Theorem 9 the stabilization can be achieved even if allthe subsystems are unstable On the other hand it should bepointed out that different from the existing results where thelower bound of the ADT should be specified a priori we needto set the upper bound in advance in our results
Next the following theorem is provided for the discrete-time version of stabilization condition for SPHOS (2) wherethe f119901is defined in homogeneous and order-preserving vector
field
Theorem 11 (discrete-time version) Consider SPHOS (2)possibly composed of all unstable modes If there exist a set ofvectors ^
119901≻ 0 119901 isin 119878 and two positive numbers 0 lt 120578 lt 1 and
0 lt 120583 lt 1 such that forall119894 = 1 2 119899 forall119901 isin 119878
f119901(^119901) ≻ ^119901 (20)
119891119901119894(]119901119894)
]119901119894
minus1
120578le 0 (21)
10038171003817100381710038171003817x(119896+119901)10038171003817100381710038171003817
infin
k119901minus 120583
10038171003817100381710038171003817x(119896minus119901)10038171003817100381710038171003817
infin
k119902le 0 (22)
where the 119909(119905119901)infin
]119901 is the weight 119897infin norm defined as
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901= max1le119894le119899
10038161003816100381610038161003816119909 (119905119901)119894
10038161003816100381610038161003816
]119894
(23)
then SPHOS (2) is globally uniformly exponentially stable forevery switching signal 120590 with average dwell time
120591119886le 120591lowast
119886=ln 120583ln 120578
(24)
Proof For any 119870 gt 0 1198960= 0 we denote 119896
1 1198962 119896
119901
119896119901+1
119896119873119873120590(1198700)
as the switching times on time interval[0 119879] Then we consider the function
119882(119905) = 120578119896minus119896120590(119896) x(119896)infin^120590(119896) (25)
When 119896 isin [119896119901 119896119901+1
) we get from (25) that
119882(119896) = 120578119896minus119896119901 x(119896)infin^119901 (26)
Mathematical Problems in Engineering 5
Next it will be shown that 119882(119896) le x(119896119901)infin
^119901 when 119896 isin
[119896119901 119896119901+1
) First when 119896 = 119896119901 the above inequality obviously
holdsThen we suppose that 119896 gt 119896119901 for a given 119896 isin [119896
119901 119896119901+1
)
the above inequality is satisfied and it proves that it is alsocorrect for 119896 + 1 isin (119896
119901 119896119901+1
) Due to the fact that for 119896 isin
[119896119901 119896119901+1
)119882(119896) le x(119896119901)infin
^119901 is true one can obtain that
119882(119896) = 120578119896minus119896119901max1le119894le119899
1003816100381610038161003816119909119894 (119896)1003816100381610038161003816
]119901119894
le10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901 (27)
which implies that
x (119896) ⪯ 120578119896119901minus119896
10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901^119901 (28)
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is defined in
homogeneous and order-preserving vector field it can beseen from (21) (26) and (28) that
119882(119896 + 1) = 120578119896+1minus119896119901
10038171003817100381710038171003817x((119896 + 1)
119901)10038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901
10038171003817100381710038171003817fp(x(119896))
10038171003817100381710038171003817
infin
^119901
le 120578119896+1minus119896119901
1003817100381710038171003817100381710038171003817fp (120578119896119901minus119896
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901^119901)
1003817100381710038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901120578
119896119901minus11989610038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 12057810038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 120578max1le119894le119899
10038161003816100381610038161003816119891119901119894(^119901)10038161003816100381610038161003816
]119901119894
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
le10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(29)
Thus we can obtain that (27) is right for forall119896 isin [119905119901 119905119901+1
)Combining this together with (22) one can get
10038171003817100381710038171003817x(119896+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119896minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583120578119896119901+1minus119896119901
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(30)
Similar to (17) and (18) we have for forall119870 gt 0
1003817100381710038171003817x(119870minus
)1003817100381710038171003817
infin
^119873120590le 120583119873120575120578minus119870
x(0)infin^0
le exp(119870120591119886
+ 1198730) ln 120583 exp minus119870 ln 120578 x(0)infin^0
= exp 1198730ln 120583 exp(
ln 120583120591119886
minus ln 120578)119870 x(0)infin^0
(31)
Therefore by denoting 120572 = exp1198730ln 120583 120582 =
exp(ln 120583120591119886minus ln 120578) one can get from (31) that
119909 (119870) le 120572120582119870
119909 (0) forall119870 ge 0 (32)
We conclude that SPHOS (2) is GUES by our proposed ADTswitching signals (3) satisfying (24) if the conditions (20)ndash(22) hold This completes the proof
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Stat
e res
pons
es
40
30
20
10
0
Figure 1 State response of the subsystem Σ1
Remark 12 InTheorems 9 and 11 based on a novel proposedswitching signals sufficient conditions of stabilization for aclass of switched positive nonlinear systems are obtainedwhere the positive subsystems are possibly all unstable
4 A Numerical Example
We provide the following numerical example in this sectionto verify our main results developed in this paper
Example 1 Consider a switched nonlinear system consistingof two homogeneous cooperative subsystems described by
Σ1=
1(119905) = 119891
11(1199091(119905) 1199092(119905))
= minus21199091(119905) + 4119909
2(119905) minus 3radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
12(1199091(119905) 1199092(119905))
= minus021199091(119905) + 01119909
2(119905)
+001radic11990921(119905) + 119909
2
2(119905)
Σ2=
1(119905) = 119891
21(1199091(119905) 1199092(119905))
= minus21199091(119905) + 3119909
2(119905) + 12radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
22(1199091(119905) 1199092(119905))
= 121199091(119905) minus 119909
2(119905) + 06radic119909
2
1(119905) + 119909
2
2(119905)
(33)
It is seen in (33) that f1 f2are defined in homogeneous
cooperative vector field which means that the switchednonlinear system is a switched positive system There exist avector ^
1= [3 78] ≻ 0 with f(^
1) = [01289 02636] ≻ 0 and
a vector ^2= [3 12] ≻ 0 with f(^
2) = [18754 04613] ≻ 0
Thus both of them are unstable and the state trajectories areshown in Figures 1 and 2
Next we are interested in designing a kind of switchingsignal 120590(119905) with property (3) to asymptotically stabilize thesystem Furthermore we generate two possible switchingsequences by 120591
119886= 05 and 120591
119886= 2 respectively the
corresponding state responses of the systemunder initial state
6 Mathematical Problems in EngineeringSt
ate r
espo
nses
1200
1000
800
600
400
200
0
Sample time (s)0 2 4 6 8 10
x1(t)
x2(t)
Figure 2 State response of the subsystem Σ2
Stat
e res
pons
es
15
10
5
0
Sample time (s)0 10 20 30 40 50
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Figure 3 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 05
condition 119909(0) = [10 15]119879 are shown in Figures 3 and 4 from
which we can see that the switched nonlinear system is stableunder 120591
119886= 05 but unstable under 120591
119886= 2
5 Conclusions
The stabilization problem for switched positive nonlinearsystems composed of possibly all unstable subsystems arestudied in both continuous-time and discrete-time domainsby using ADT switching As a first attempt a new class ofADT switching signal is proposed and then it is designedfor the system via the analysis of the weight 119897
infinnorm of
the system Two sufficient stabilization conditions for theunderlying systems are derived The highlight of the paperlies in that it is the first time the stabilization is solved forour considered switched positive nonlinear system wherethe system is possibly composed of all unstable subsystemsA numerical example is provided to show the effectivenessof our proposed approach In our future work we aim at
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50St
ate r
espo
nses
350
300
250
200
150
100
50
0
Figure 4 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 2
developing numerially easily checked conditions for the con-sidered systems and the problem of stabilization for switchedpositive nonlinear system with delays also needs to bestudied
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Kao CWang F Zha andH Cao ldquoStability inmean of partialvariables for stochastic reaction-diffusion systems with Marko-vian switchingrdquo Journal of the Franklin Institute vol 351 no 1pp 500ndash512 2014
[2] X-M Sun W Wang G-P Liu and J Zhao ldquoStability analysisfor linear switched systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 38 no 2 pp 528ndash533 2008
[3] L Wu and W X Zheng ldquoWeighted 119867infin
model reduction forlinear switched systems with time-varying delayrdquo Automaticavol 45 no 1 pp 186ndash193 2009
[4] L Zhang E-K Boukas and P Shi ldquoExponential 119867infin
filteringfor uncertain discrete-time switched linear systems with aver-age dwell time a 120583-dependent approachrdquo International Journalof Robust and Nonlinear Control vol 18 no 11 pp 1188ndash12072008
[5] X Zhao L Zhang and P Shi ldquoStability of a class of switchedpositive linear time-delay systemsrdquo International Journal ofRobust and Nonlinear Control vol 23 no 5 pp 578ndash589 2013
[6] L Zhang and H Gao ldquoAsynchronously switched control ofswitched linear systems with average dwell timerdquo Automaticavol 46 no 5 pp 953ndash958 2010
[7] L Farina and S Rinaldi Positive Linear SystemsTheory and Ap-plications vol 50 John Wiley amp Sons 2011
Mathematical Problems in Engineering 7
[8] X Chen J Lam P Li and Z Shu ldquoL1-induced norm andcontroller synthesis of positive systemsrdquoAutomatica vol 49 no5 pp 1377ndash1385 2013
[9] J-E Feng J Lam P Li and Z Shu ldquoDecay rate constrainedstabilization of positive systems using static output feedbackrdquoInternational Journal of Robust and Nonlinear Control vol 21no 1 pp 44ndash54 2011
[10] P Li J Lam and Z Shu ldquo119867infin
positive filtering for positivelinear discrete-time systems an augmentation approachrdquo IEEETransactions on Automatic Control vol 55 no 10 pp 2337ndash2342 2010
[11] P Li J Lam Z Wang and P Date ldquoPositivity-preserving 119867infin
model reduction for positive systemsrdquo Automatica vol 47 no7 pp 1504ndash1511 2011
[12] O Mason and R Shorten ldquoOn linear copositive Lyapunovfunctions and the stability of switched positive linear systemsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1346ndash1349 2007
[13] M A Rami and F Tadeo ldquoController synthesis for positivelinear systems with bounded controlsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 54 no 2 pp 151ndash1552007
[14] X Zhao P Shi and L Zhang ldquoAsynchronously switched controlof a class of slowly switched linear systemsrdquo Systems amp ControlLetters vol 61 no 12 pp 1151ndash1156 2012
[15] X Liu and C Dang ldquoStability analysis of positive switchedlinear systems with delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1684ndash1690 2011
[16] S Fadali and S Jafarzadeh ldquoStability analysis of positive intervaltype-2 TSK systems with application to energy marketsrdquo IEEETransactions on Fuzzy Systems vol 22 no 4 pp 1031ndash1038 2013
[17] L Imsland G O Eikrem and B A Foss ldquoA state feedbackcontroller for a class of nonlinear positive systems applied tostabilization of gas-lifted oil wellsrdquo Control Engineering PracticeN vol 3 pp 7ndash15 2006
[18] S K Nguang and P Shi ldquoFuzzy119867infinoutput feedback control of
nonlinear systems under sampled measurementsrdquo Automaticavol 39 no 12 pp 2169ndash2174 2003
[19] O Mason and M Verwoerd ldquoObservations on the stabilityproperties of cooperative systemsrdquo Systems amp Control Lettersvol 58 no 6 pp 461ndash467 2009
[20] H R Feyzmahdavian T Charalambous and M JohanssonldquoExponential stability of homogeneous positive systems ofdegree one with time-varying delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 6 pp 1594ndash1599 2014
[21] X Ding L Shu and X Liu ldquoOn linear copositive Lyapunovfunctions for switched positive systemsrdquo Journal of the FranklinInstitute vol 348 no 8 pp 2099ndash2107 2011
[22] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003
[23] X Zhao L Zhang P Shi and M Liu ldquoStability of switchedpositive linear systems with average dwell time switchingrdquoAutomatica vol 48 no 6 pp 1132ndash1137 2012
[24] X Zhao X Liu S Yin and H Li ldquoImproved results on stabilityof continuous-time switched positive linear systemsrdquo Automat-ica vol 50 no 2 pp 614ndash621 2014
[25] A S Morse ldquoSupervisory control of families of linear set-point controllersmdashpart 1 exactmatchingrdquo IEEETransactions onAutomatic Control vol 41 no 10 pp 1413ndash1431 1996
[26] D Liberzon Switching in Systems and Control Springer 2003[27] H Liu Y Shen and X Zhao ldquoAsynchronous finite-time
Hinfin
control for switched linear systems via mode-dependentdynamic state-feedbackrdquo Nonlinear Analysis Hybrid Systemsvol 8 no 1 pp 109ndash120 2013
[28] J Zhang Z Han H Wu and J Huang ldquoRobust stabilizationof discrete-time positive switched systems with uncertaintiesand average dwell time switchingrdquo Circuits Systems and SignalProcessing vol 33 no 1 pp 71ndash95 2014
[29] X Zhao L Zhang P Shi andM Liu ldquoStability and stabilizationof switched linear systems with mode-dependent average dwelltimerdquo IEEE Transactions on Automatic Control vol 57 no 7 pp1809ndash1815 2012
[30] J Lian and J Liu ldquoNew results on stability of switched positivesystems an average dwell-time approachrdquo IET Control Theoryand Applications vol 7 no 12 pp 1651ndash1658 2013
[31] X Zhao S Yin H Li and B Niu ldquoSwitching stabilization for aclass of slowly switched systemsrdquo IEEE Transactions on Auto-matic Control 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is in homogeneous
and cooperative vector field based on Lemma 5 it can bederived from (5) (12) and (13) that
(119905) = 119890minus120578(minus119905119901)
119891119901119896(x ())]119901119896
minus 120578119890minus120578(minus119905119901)
119909119896()
]119901119896
le 119890minus120578(minus119905119901)
119891119901119896(119890120578(minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901^119901)
]119901119896
minus 120578119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901]119901119896
]119901119896
= 119890minus120578(minus119905119901)
119890120578(minus119905119901)
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901119891119901119896(]119901)
]119901119896
minus 1205781003817100381710038171003817x ()
1003817100381710038171003817
infin
^119901
=10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901(119891119901119896(]119901)
]119901119896
minus 120578) lt 0
(14)
Note that if (11) holds we can get () lt 0 which impliesthat the upper-right Dini derivative 119863+(119882()) lt 0 Thus itcan be seen that 119882(119905) is nonincreasing when 119905 isin [119905
119901 119905119901+1
)This together with (10) gives that for forall119905 isin [119905
119901 119905119901+1
)
119882 (119905) = 119890minus120578(119905minus119905119901) x(119905)infin^119901
le 119882(119905119901)
=10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901
(15)
According to (6) and (15) one can obtain that10038171003817100381710038171003817x(119905+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119905minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583119890120578(119905119901+1minus119905119901)
10038171003817100381710038171003817x (119905119901)10038171003817100381710038171003817
infin
^119901
(16)
By integrating this for 119905 isin [119905119901 119905119901+1
) one gets that
1003817100381710038171003817x(119879minus
)1003817100381710038171003817
infin
^119873120590le 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590
= 119890120578(119879minus119905119873120590
)10038171003817100381710038171003817x(119905+119873120590
)10038171003817100381710038171003817
infin
^119873120590
le 120583119890120578(119879minus119905119873120590minus1
)10038171003817100381710038171003817x(119905119873120590)10038171003817100381710038171003817
infin
^119873120590minus1
le 120583119873120575119890120578119879
x(0)infin^0
(17)
Moreover it can be derived from (3) and (17) that
x(119879)infin^119873120590 le 120583119873120575119890120578119879
x(0)infin^0
le 119890120578119879
119890(1198730+119879120591119886) ln 120583 x(0)infin^0
= 1198901198730 ln 120583119890(120578+ln 120583120591119886)119879 x(0)infin^0
(18)
Therefore by denoting 120572 = 1198901198730 ln 120583 120573 = minus(120578 + 120583120591
119886) one can
get from (18) that
119909 (119879) le 120572119890minus120573119879
119909 (0) forall119879 ge 0 (19)
By Definition 7 we conclude that SPHCS (2) is GUES byour proposed ADT switching signals (3) satisfying (8) if theconditions (4)ndash(6) hold This completes the proof
Remark 10 It has been shown in [19] that if there exists avector ^
119901≻ 0 satisfying f(^) ≻ 0 the corresponding homoge-
nous cooperative system is not globally asymptotically stableThus all the homogenous cooperative subsystems of SPHCS(2) may be unstable if (4) holds Under the ADT designedin Theorem 9 the stabilization can be achieved even if allthe subsystems are unstable On the other hand it should bepointed out that different from the existing results where thelower bound of the ADT should be specified a priori we needto set the upper bound in advance in our results
Next the following theorem is provided for the discrete-time version of stabilization condition for SPHOS (2) wherethe f119901is defined in homogeneous and order-preserving vector
field
Theorem 11 (discrete-time version) Consider SPHOS (2)possibly composed of all unstable modes If there exist a set ofvectors ^
119901≻ 0 119901 isin 119878 and two positive numbers 0 lt 120578 lt 1 and
0 lt 120583 lt 1 such that forall119894 = 1 2 119899 forall119901 isin 119878
f119901(^119901) ≻ ^119901 (20)
119891119901119894(]119901119894)
]119901119894
minus1
120578le 0 (21)
10038171003817100381710038171003817x(119896+119901)10038171003817100381710038171003817
infin
k119901minus 120583
10038171003817100381710038171003817x(119896minus119901)10038171003817100381710038171003817
infin
k119902le 0 (22)
where the 119909(119905119901)infin
]119901 is the weight 119897infin norm defined as
10038171003817100381710038171003817x(119905119901)10038171003817100381710038171003817
infin
^119901= max1le119894le119899
10038161003816100381610038161003816119909 (119905119901)119894
10038161003816100381610038161003816
]119894
(23)
then SPHOS (2) is globally uniformly exponentially stable forevery switching signal 120590 with average dwell time
120591119886le 120591lowast
119886=ln 120583ln 120578
(24)
Proof For any 119870 gt 0 1198960= 0 we denote 119896
1 1198962 119896
119901
119896119901+1
119896119873119873120590(1198700)
as the switching times on time interval[0 119879] Then we consider the function
119882(119905) = 120578119896minus119896120590(119896) x(119896)infin^120590(119896) (25)
When 119896 isin [119896119901 119896119901+1
) we get from (25) that
119882(119896) = 120578119896minus119896119901 x(119896)infin^119901 (26)
Mathematical Problems in Engineering 5
Next it will be shown that 119882(119896) le x(119896119901)infin
^119901 when 119896 isin
[119896119901 119896119901+1
) First when 119896 = 119896119901 the above inequality obviously
holdsThen we suppose that 119896 gt 119896119901 for a given 119896 isin [119896
119901 119896119901+1
)
the above inequality is satisfied and it proves that it is alsocorrect for 119896 + 1 isin (119896
119901 119896119901+1
) Due to the fact that for 119896 isin
[119896119901 119896119901+1
)119882(119896) le x(119896119901)infin
^119901 is true one can obtain that
119882(119896) = 120578119896minus119896119901max1le119894le119899
1003816100381610038161003816119909119894 (119896)1003816100381610038161003816
]119901119894
le10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901 (27)
which implies that
x (119896) ⪯ 120578119896119901minus119896
10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901^119901 (28)
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is defined in
homogeneous and order-preserving vector field it can beseen from (21) (26) and (28) that
119882(119896 + 1) = 120578119896+1minus119896119901
10038171003817100381710038171003817x((119896 + 1)
119901)10038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901
10038171003817100381710038171003817fp(x(119896))
10038171003817100381710038171003817
infin
^119901
le 120578119896+1minus119896119901
1003817100381710038171003817100381710038171003817fp (120578119896119901minus119896
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901^119901)
1003817100381710038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901120578
119896119901minus11989610038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 12057810038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 120578max1le119894le119899
10038161003816100381610038161003816119891119901119894(^119901)10038161003816100381610038161003816
]119901119894
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
le10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(29)
Thus we can obtain that (27) is right for forall119896 isin [119905119901 119905119901+1
)Combining this together with (22) one can get
10038171003817100381710038171003817x(119896+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119896minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583120578119896119901+1minus119896119901
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(30)
Similar to (17) and (18) we have for forall119870 gt 0
1003817100381710038171003817x(119870minus
)1003817100381710038171003817
infin
^119873120590le 120583119873120575120578minus119870
x(0)infin^0
le exp(119870120591119886
+ 1198730) ln 120583 exp minus119870 ln 120578 x(0)infin^0
= exp 1198730ln 120583 exp(
ln 120583120591119886
minus ln 120578)119870 x(0)infin^0
(31)
Therefore by denoting 120572 = exp1198730ln 120583 120582 =
exp(ln 120583120591119886minus ln 120578) one can get from (31) that
119909 (119870) le 120572120582119870
119909 (0) forall119870 ge 0 (32)
We conclude that SPHOS (2) is GUES by our proposed ADTswitching signals (3) satisfying (24) if the conditions (20)ndash(22) hold This completes the proof
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Stat
e res
pons
es
40
30
20
10
0
Figure 1 State response of the subsystem Σ1
Remark 12 InTheorems 9 and 11 based on a novel proposedswitching signals sufficient conditions of stabilization for aclass of switched positive nonlinear systems are obtainedwhere the positive subsystems are possibly all unstable
4 A Numerical Example
We provide the following numerical example in this sectionto verify our main results developed in this paper
Example 1 Consider a switched nonlinear system consistingof two homogeneous cooperative subsystems described by
Σ1=
1(119905) = 119891
11(1199091(119905) 1199092(119905))
= minus21199091(119905) + 4119909
2(119905) minus 3radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
12(1199091(119905) 1199092(119905))
= minus021199091(119905) + 01119909
2(119905)
+001radic11990921(119905) + 119909
2
2(119905)
Σ2=
1(119905) = 119891
21(1199091(119905) 1199092(119905))
= minus21199091(119905) + 3119909
2(119905) + 12radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
22(1199091(119905) 1199092(119905))
= 121199091(119905) minus 119909
2(119905) + 06radic119909
2
1(119905) + 119909
2
2(119905)
(33)
It is seen in (33) that f1 f2are defined in homogeneous
cooperative vector field which means that the switchednonlinear system is a switched positive system There exist avector ^
1= [3 78] ≻ 0 with f(^
1) = [01289 02636] ≻ 0 and
a vector ^2= [3 12] ≻ 0 with f(^
2) = [18754 04613] ≻ 0
Thus both of them are unstable and the state trajectories areshown in Figures 1 and 2
Next we are interested in designing a kind of switchingsignal 120590(119905) with property (3) to asymptotically stabilize thesystem Furthermore we generate two possible switchingsequences by 120591
119886= 05 and 120591
119886= 2 respectively the
corresponding state responses of the systemunder initial state
6 Mathematical Problems in EngineeringSt
ate r
espo
nses
1200
1000
800
600
400
200
0
Sample time (s)0 2 4 6 8 10
x1(t)
x2(t)
Figure 2 State response of the subsystem Σ2
Stat
e res
pons
es
15
10
5
0
Sample time (s)0 10 20 30 40 50
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Figure 3 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 05
condition 119909(0) = [10 15]119879 are shown in Figures 3 and 4 from
which we can see that the switched nonlinear system is stableunder 120591
119886= 05 but unstable under 120591
119886= 2
5 Conclusions
The stabilization problem for switched positive nonlinearsystems composed of possibly all unstable subsystems arestudied in both continuous-time and discrete-time domainsby using ADT switching As a first attempt a new class ofADT switching signal is proposed and then it is designedfor the system via the analysis of the weight 119897
infinnorm of
the system Two sufficient stabilization conditions for theunderlying systems are derived The highlight of the paperlies in that it is the first time the stabilization is solved forour considered switched positive nonlinear system wherethe system is possibly composed of all unstable subsystemsA numerical example is provided to show the effectivenessof our proposed approach In our future work we aim at
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50St
ate r
espo
nses
350
300
250
200
150
100
50
0
Figure 4 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 2
developing numerially easily checked conditions for the con-sidered systems and the problem of stabilization for switchedpositive nonlinear system with delays also needs to bestudied
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Kao CWang F Zha andH Cao ldquoStability inmean of partialvariables for stochastic reaction-diffusion systems with Marko-vian switchingrdquo Journal of the Franklin Institute vol 351 no 1pp 500ndash512 2014
[2] X-M Sun W Wang G-P Liu and J Zhao ldquoStability analysisfor linear switched systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 38 no 2 pp 528ndash533 2008
[3] L Wu and W X Zheng ldquoWeighted 119867infin
model reduction forlinear switched systems with time-varying delayrdquo Automaticavol 45 no 1 pp 186ndash193 2009
[4] L Zhang E-K Boukas and P Shi ldquoExponential 119867infin
filteringfor uncertain discrete-time switched linear systems with aver-age dwell time a 120583-dependent approachrdquo International Journalof Robust and Nonlinear Control vol 18 no 11 pp 1188ndash12072008
[5] X Zhao L Zhang and P Shi ldquoStability of a class of switchedpositive linear time-delay systemsrdquo International Journal ofRobust and Nonlinear Control vol 23 no 5 pp 578ndash589 2013
[6] L Zhang and H Gao ldquoAsynchronously switched control ofswitched linear systems with average dwell timerdquo Automaticavol 46 no 5 pp 953ndash958 2010
[7] L Farina and S Rinaldi Positive Linear SystemsTheory and Ap-plications vol 50 John Wiley amp Sons 2011
Mathematical Problems in Engineering 7
[8] X Chen J Lam P Li and Z Shu ldquoL1-induced norm andcontroller synthesis of positive systemsrdquoAutomatica vol 49 no5 pp 1377ndash1385 2013
[9] J-E Feng J Lam P Li and Z Shu ldquoDecay rate constrainedstabilization of positive systems using static output feedbackrdquoInternational Journal of Robust and Nonlinear Control vol 21no 1 pp 44ndash54 2011
[10] P Li J Lam and Z Shu ldquo119867infin
positive filtering for positivelinear discrete-time systems an augmentation approachrdquo IEEETransactions on Automatic Control vol 55 no 10 pp 2337ndash2342 2010
[11] P Li J Lam Z Wang and P Date ldquoPositivity-preserving 119867infin
model reduction for positive systemsrdquo Automatica vol 47 no7 pp 1504ndash1511 2011
[12] O Mason and R Shorten ldquoOn linear copositive Lyapunovfunctions and the stability of switched positive linear systemsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1346ndash1349 2007
[13] M A Rami and F Tadeo ldquoController synthesis for positivelinear systems with bounded controlsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 54 no 2 pp 151ndash1552007
[14] X Zhao P Shi and L Zhang ldquoAsynchronously switched controlof a class of slowly switched linear systemsrdquo Systems amp ControlLetters vol 61 no 12 pp 1151ndash1156 2012
[15] X Liu and C Dang ldquoStability analysis of positive switchedlinear systems with delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1684ndash1690 2011
[16] S Fadali and S Jafarzadeh ldquoStability analysis of positive intervaltype-2 TSK systems with application to energy marketsrdquo IEEETransactions on Fuzzy Systems vol 22 no 4 pp 1031ndash1038 2013
[17] L Imsland G O Eikrem and B A Foss ldquoA state feedbackcontroller for a class of nonlinear positive systems applied tostabilization of gas-lifted oil wellsrdquo Control Engineering PracticeN vol 3 pp 7ndash15 2006
[18] S K Nguang and P Shi ldquoFuzzy119867infinoutput feedback control of
nonlinear systems under sampled measurementsrdquo Automaticavol 39 no 12 pp 2169ndash2174 2003
[19] O Mason and M Verwoerd ldquoObservations on the stabilityproperties of cooperative systemsrdquo Systems amp Control Lettersvol 58 no 6 pp 461ndash467 2009
[20] H R Feyzmahdavian T Charalambous and M JohanssonldquoExponential stability of homogeneous positive systems ofdegree one with time-varying delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 6 pp 1594ndash1599 2014
[21] X Ding L Shu and X Liu ldquoOn linear copositive Lyapunovfunctions for switched positive systemsrdquo Journal of the FranklinInstitute vol 348 no 8 pp 2099ndash2107 2011
[22] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003
[23] X Zhao L Zhang P Shi and M Liu ldquoStability of switchedpositive linear systems with average dwell time switchingrdquoAutomatica vol 48 no 6 pp 1132ndash1137 2012
[24] X Zhao X Liu S Yin and H Li ldquoImproved results on stabilityof continuous-time switched positive linear systemsrdquo Automat-ica vol 50 no 2 pp 614ndash621 2014
[25] A S Morse ldquoSupervisory control of families of linear set-point controllersmdashpart 1 exactmatchingrdquo IEEETransactions onAutomatic Control vol 41 no 10 pp 1413ndash1431 1996
[26] D Liberzon Switching in Systems and Control Springer 2003[27] H Liu Y Shen and X Zhao ldquoAsynchronous finite-time
Hinfin
control for switched linear systems via mode-dependentdynamic state-feedbackrdquo Nonlinear Analysis Hybrid Systemsvol 8 no 1 pp 109ndash120 2013
[28] J Zhang Z Han H Wu and J Huang ldquoRobust stabilizationof discrete-time positive switched systems with uncertaintiesand average dwell time switchingrdquo Circuits Systems and SignalProcessing vol 33 no 1 pp 71ndash95 2014
[29] X Zhao L Zhang P Shi andM Liu ldquoStability and stabilizationof switched linear systems with mode-dependent average dwelltimerdquo IEEE Transactions on Automatic Control vol 57 no 7 pp1809ndash1815 2012
[30] J Lian and J Liu ldquoNew results on stability of switched positivesystems an average dwell-time approachrdquo IET Control Theoryand Applications vol 7 no 12 pp 1651ndash1658 2013
[31] X Zhao S Yin H Li and B Niu ldquoSwitching stabilization for aclass of slowly switched systemsrdquo IEEE Transactions on Auto-matic Control 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Next it will be shown that 119882(119896) le x(119896119901)infin
^119901 when 119896 isin
[119896119901 119896119901+1
) First when 119896 = 119896119901 the above inequality obviously
holdsThen we suppose that 119896 gt 119896119901 for a given 119896 isin [119896
119901 119896119901+1
)
the above inequality is satisfied and it proves that it is alsocorrect for 119896 + 1 isin (119896
119901 119896119901+1
) Due to the fact that for 119896 isin
[119896119901 119896119901+1
)119882(119896) le x(119896119901)infin
^119901 is true one can obtain that
119882(119896) = 120578119896minus119896119901max1le119894le119899
1003816100381610038161003816119909119894 (119896)1003816100381610038161003816
]119901119894
le10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901 (27)
which implies that
x (119896) ⪯ 120578119896119901minus119896
10038171003817100381710038171003817x(119896119901)10038171003817100381710038171003817
infin
^119901^119901 (28)
Moreover since f119901 R119899 rarr R119899 and 119901 isin 119878 is defined in
homogeneous and order-preserving vector field it can beseen from (21) (26) and (28) that
119882(119896 + 1) = 120578119896+1minus119896119901
10038171003817100381710038171003817x((119896 + 1)
119901)10038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901
10038171003817100381710038171003817fp(x(119896))
10038171003817100381710038171003817
infin
^119901
le 120578119896+1minus119896119901
1003817100381710038171003817100381710038171003817fp (120578119896119901minus119896
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901^119901)
1003817100381710038171003817100381710038171003817
infin
^119901
= 120578119896+1minus119896119901120578
119896119901minus11989610038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 12057810038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
10038171003817100381710038171003817fp (^119901)
10038171003817100381710038171003817
infin
^119901
= 120578max1le119894le119899
10038161003816100381610038161003816119891119901119894(^119901)10038161003816100381610038161003816
]119901119894
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
le10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(29)
Thus we can obtain that (27) is right for forall119896 isin [119905119901 119905119901+1
)Combining this together with (22) one can get
10038171003817100381710038171003817x(119896+119901+1
)10038171003817100381710038171003817
infin
^119901le 120583
10038171003817100381710038171003817x(119896minus119901+1
)10038171003817100381710038171003817
infin
^119901
le 120583120578119896119901+1minus119896119901
10038171003817100381710038171003817x (119896119901)10038171003817100381710038171003817
infin
^119901
(30)
Similar to (17) and (18) we have for forall119870 gt 0
1003817100381710038171003817x(119870minus
)1003817100381710038171003817
infin
^119873120590le 120583119873120575120578minus119870
x(0)infin^0
le exp(119870120591119886
+ 1198730) ln 120583 exp minus119870 ln 120578 x(0)infin^0
= exp 1198730ln 120583 exp(
ln 120583120591119886
minus ln 120578)119870 x(0)infin^0
(31)
Therefore by denoting 120572 = exp1198730ln 120583 120582 =
exp(ln 120583120591119886minus ln 120578) one can get from (31) that
119909 (119870) le 120572120582119870
119909 (0) forall119870 ge 0 (32)
We conclude that SPHOS (2) is GUES by our proposed ADTswitching signals (3) satisfying (24) if the conditions (20)ndash(22) hold This completes the proof
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Stat
e res
pons
es
40
30
20
10
0
Figure 1 State response of the subsystem Σ1
Remark 12 InTheorems 9 and 11 based on a novel proposedswitching signals sufficient conditions of stabilization for aclass of switched positive nonlinear systems are obtainedwhere the positive subsystems are possibly all unstable
4 A Numerical Example
We provide the following numerical example in this sectionto verify our main results developed in this paper
Example 1 Consider a switched nonlinear system consistingof two homogeneous cooperative subsystems described by
Σ1=
1(119905) = 119891
11(1199091(119905) 1199092(119905))
= minus21199091(119905) + 4119909
2(119905) minus 3radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
12(1199091(119905) 1199092(119905))
= minus021199091(119905) + 01119909
2(119905)
+001radic11990921(119905) + 119909
2
2(119905)
Σ2=
1(119905) = 119891
21(1199091(119905) 1199092(119905))
= minus21199091(119905) + 3119909
2(119905) + 12radic119909
2
1(119905) + 119909
2
2(119905)
2(119905) = 119891
22(1199091(119905) 1199092(119905))
= 121199091(119905) minus 119909
2(119905) + 06radic119909
2
1(119905) + 119909
2
2(119905)
(33)
It is seen in (33) that f1 f2are defined in homogeneous
cooperative vector field which means that the switchednonlinear system is a switched positive system There exist avector ^
1= [3 78] ≻ 0 with f(^
1) = [01289 02636] ≻ 0 and
a vector ^2= [3 12] ≻ 0 with f(^
2) = [18754 04613] ≻ 0
Thus both of them are unstable and the state trajectories areshown in Figures 1 and 2
Next we are interested in designing a kind of switchingsignal 120590(119905) with property (3) to asymptotically stabilize thesystem Furthermore we generate two possible switchingsequences by 120591
119886= 05 and 120591
119886= 2 respectively the
corresponding state responses of the systemunder initial state
6 Mathematical Problems in EngineeringSt
ate r
espo
nses
1200
1000
800
600
400
200
0
Sample time (s)0 2 4 6 8 10
x1(t)
x2(t)
Figure 2 State response of the subsystem Σ2
Stat
e res
pons
es
15
10
5
0
Sample time (s)0 10 20 30 40 50
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Figure 3 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 05
condition 119909(0) = [10 15]119879 are shown in Figures 3 and 4 from
which we can see that the switched nonlinear system is stableunder 120591
119886= 05 but unstable under 120591
119886= 2
5 Conclusions
The stabilization problem for switched positive nonlinearsystems composed of possibly all unstable subsystems arestudied in both continuous-time and discrete-time domainsby using ADT switching As a first attempt a new class ofADT switching signal is proposed and then it is designedfor the system via the analysis of the weight 119897
infinnorm of
the system Two sufficient stabilization conditions for theunderlying systems are derived The highlight of the paperlies in that it is the first time the stabilization is solved forour considered switched positive nonlinear system wherethe system is possibly composed of all unstable subsystemsA numerical example is provided to show the effectivenessof our proposed approach In our future work we aim at
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50St
ate r
espo
nses
350
300
250
200
150
100
50
0
Figure 4 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 2
developing numerially easily checked conditions for the con-sidered systems and the problem of stabilization for switchedpositive nonlinear system with delays also needs to bestudied
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Kao CWang F Zha andH Cao ldquoStability inmean of partialvariables for stochastic reaction-diffusion systems with Marko-vian switchingrdquo Journal of the Franklin Institute vol 351 no 1pp 500ndash512 2014
[2] X-M Sun W Wang G-P Liu and J Zhao ldquoStability analysisfor linear switched systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 38 no 2 pp 528ndash533 2008
[3] L Wu and W X Zheng ldquoWeighted 119867infin
model reduction forlinear switched systems with time-varying delayrdquo Automaticavol 45 no 1 pp 186ndash193 2009
[4] L Zhang E-K Boukas and P Shi ldquoExponential 119867infin
filteringfor uncertain discrete-time switched linear systems with aver-age dwell time a 120583-dependent approachrdquo International Journalof Robust and Nonlinear Control vol 18 no 11 pp 1188ndash12072008
[5] X Zhao L Zhang and P Shi ldquoStability of a class of switchedpositive linear time-delay systemsrdquo International Journal ofRobust and Nonlinear Control vol 23 no 5 pp 578ndash589 2013
[6] L Zhang and H Gao ldquoAsynchronously switched control ofswitched linear systems with average dwell timerdquo Automaticavol 46 no 5 pp 953ndash958 2010
[7] L Farina and S Rinaldi Positive Linear SystemsTheory and Ap-plications vol 50 John Wiley amp Sons 2011
Mathematical Problems in Engineering 7
[8] X Chen J Lam P Li and Z Shu ldquoL1-induced norm andcontroller synthesis of positive systemsrdquoAutomatica vol 49 no5 pp 1377ndash1385 2013
[9] J-E Feng J Lam P Li and Z Shu ldquoDecay rate constrainedstabilization of positive systems using static output feedbackrdquoInternational Journal of Robust and Nonlinear Control vol 21no 1 pp 44ndash54 2011
[10] P Li J Lam and Z Shu ldquo119867infin
positive filtering for positivelinear discrete-time systems an augmentation approachrdquo IEEETransactions on Automatic Control vol 55 no 10 pp 2337ndash2342 2010
[11] P Li J Lam Z Wang and P Date ldquoPositivity-preserving 119867infin
model reduction for positive systemsrdquo Automatica vol 47 no7 pp 1504ndash1511 2011
[12] O Mason and R Shorten ldquoOn linear copositive Lyapunovfunctions and the stability of switched positive linear systemsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1346ndash1349 2007
[13] M A Rami and F Tadeo ldquoController synthesis for positivelinear systems with bounded controlsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 54 no 2 pp 151ndash1552007
[14] X Zhao P Shi and L Zhang ldquoAsynchronously switched controlof a class of slowly switched linear systemsrdquo Systems amp ControlLetters vol 61 no 12 pp 1151ndash1156 2012
[15] X Liu and C Dang ldquoStability analysis of positive switchedlinear systems with delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1684ndash1690 2011
[16] S Fadali and S Jafarzadeh ldquoStability analysis of positive intervaltype-2 TSK systems with application to energy marketsrdquo IEEETransactions on Fuzzy Systems vol 22 no 4 pp 1031ndash1038 2013
[17] L Imsland G O Eikrem and B A Foss ldquoA state feedbackcontroller for a class of nonlinear positive systems applied tostabilization of gas-lifted oil wellsrdquo Control Engineering PracticeN vol 3 pp 7ndash15 2006
[18] S K Nguang and P Shi ldquoFuzzy119867infinoutput feedback control of
nonlinear systems under sampled measurementsrdquo Automaticavol 39 no 12 pp 2169ndash2174 2003
[19] O Mason and M Verwoerd ldquoObservations on the stabilityproperties of cooperative systemsrdquo Systems amp Control Lettersvol 58 no 6 pp 461ndash467 2009
[20] H R Feyzmahdavian T Charalambous and M JohanssonldquoExponential stability of homogeneous positive systems ofdegree one with time-varying delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 6 pp 1594ndash1599 2014
[21] X Ding L Shu and X Liu ldquoOn linear copositive Lyapunovfunctions for switched positive systemsrdquo Journal of the FranklinInstitute vol 348 no 8 pp 2099ndash2107 2011
[22] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003
[23] X Zhao L Zhang P Shi and M Liu ldquoStability of switchedpositive linear systems with average dwell time switchingrdquoAutomatica vol 48 no 6 pp 1132ndash1137 2012
[24] X Zhao X Liu S Yin and H Li ldquoImproved results on stabilityof continuous-time switched positive linear systemsrdquo Automat-ica vol 50 no 2 pp 614ndash621 2014
[25] A S Morse ldquoSupervisory control of families of linear set-point controllersmdashpart 1 exactmatchingrdquo IEEETransactions onAutomatic Control vol 41 no 10 pp 1413ndash1431 1996
[26] D Liberzon Switching in Systems and Control Springer 2003[27] H Liu Y Shen and X Zhao ldquoAsynchronous finite-time
Hinfin
control for switched linear systems via mode-dependentdynamic state-feedbackrdquo Nonlinear Analysis Hybrid Systemsvol 8 no 1 pp 109ndash120 2013
[28] J Zhang Z Han H Wu and J Huang ldquoRobust stabilizationof discrete-time positive switched systems with uncertaintiesand average dwell time switchingrdquo Circuits Systems and SignalProcessing vol 33 no 1 pp 71ndash95 2014
[29] X Zhao L Zhang P Shi andM Liu ldquoStability and stabilizationof switched linear systems with mode-dependent average dwelltimerdquo IEEE Transactions on Automatic Control vol 57 no 7 pp1809ndash1815 2012
[30] J Lian and J Liu ldquoNew results on stability of switched positivesystems an average dwell-time approachrdquo IET Control Theoryand Applications vol 7 no 12 pp 1651ndash1658 2013
[31] X Zhao S Yin H Li and B Niu ldquoSwitching stabilization for aclass of slowly switched systemsrdquo IEEE Transactions on Auto-matic Control 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in EngineeringSt
ate r
espo
nses
1200
1000
800
600
400
200
0
Sample time (s)0 2 4 6 8 10
x1(t)
x2(t)
Figure 2 State response of the subsystem Σ2
Stat
e res
pons
es
15
10
5
0
Sample time (s)0 10 20 30 40 50
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Figure 3 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 05
condition 119909(0) = [10 15]119879 are shown in Figures 3 and 4 from
which we can see that the switched nonlinear system is stableunder 120591
119886= 05 but unstable under 120591
119886= 2
5 Conclusions
The stabilization problem for switched positive nonlinearsystems composed of possibly all unstable subsystems arestudied in both continuous-time and discrete-time domainsby using ADT switching As a first attempt a new class ofADT switching signal is proposed and then it is designedfor the system via the analysis of the weight 119897
infinnorm of
the system Two sufficient stabilization conditions for theunderlying systems are derived The highlight of the paperlies in that it is the first time the stabilization is solved forour considered switched positive nonlinear system wherethe system is possibly composed of all unstable subsystemsA numerical example is provided to show the effectivenessof our proposed approach In our future work we aim at
Sample time (s)0 10 20 30 40 50
x1(t)
x2(t)
Syste
m m
ode 3
2
1
0
Sample time (s)0 10 20 30 40 50St
ate r
espo
nses
350
300
250
200
150
100
50
0
Figure 4 State responses of switched nonlinear system (33) underswitching signal 120590(119905) with 120591
119886= 2
developing numerially easily checked conditions for the con-sidered systems and the problem of stabilization for switchedpositive nonlinear system with delays also needs to bestudied
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] Y Kao CWang F Zha andH Cao ldquoStability inmean of partialvariables for stochastic reaction-diffusion systems with Marko-vian switchingrdquo Journal of the Franklin Institute vol 351 no 1pp 500ndash512 2014
[2] X-M Sun W Wang G-P Liu and J Zhao ldquoStability analysisfor linear switched systems with time-varying delayrdquo IEEETransactions on Systems Man and Cybernetics Part B Cyber-netics vol 38 no 2 pp 528ndash533 2008
[3] L Wu and W X Zheng ldquoWeighted 119867infin
model reduction forlinear switched systems with time-varying delayrdquo Automaticavol 45 no 1 pp 186ndash193 2009
[4] L Zhang E-K Boukas and P Shi ldquoExponential 119867infin
filteringfor uncertain discrete-time switched linear systems with aver-age dwell time a 120583-dependent approachrdquo International Journalof Robust and Nonlinear Control vol 18 no 11 pp 1188ndash12072008
[5] X Zhao L Zhang and P Shi ldquoStability of a class of switchedpositive linear time-delay systemsrdquo International Journal ofRobust and Nonlinear Control vol 23 no 5 pp 578ndash589 2013
[6] L Zhang and H Gao ldquoAsynchronously switched control ofswitched linear systems with average dwell timerdquo Automaticavol 46 no 5 pp 953ndash958 2010
[7] L Farina and S Rinaldi Positive Linear SystemsTheory and Ap-plications vol 50 John Wiley amp Sons 2011
Mathematical Problems in Engineering 7
[8] X Chen J Lam P Li and Z Shu ldquoL1-induced norm andcontroller synthesis of positive systemsrdquoAutomatica vol 49 no5 pp 1377ndash1385 2013
[9] J-E Feng J Lam P Li and Z Shu ldquoDecay rate constrainedstabilization of positive systems using static output feedbackrdquoInternational Journal of Robust and Nonlinear Control vol 21no 1 pp 44ndash54 2011
[10] P Li J Lam and Z Shu ldquo119867infin
positive filtering for positivelinear discrete-time systems an augmentation approachrdquo IEEETransactions on Automatic Control vol 55 no 10 pp 2337ndash2342 2010
[11] P Li J Lam Z Wang and P Date ldquoPositivity-preserving 119867infin
model reduction for positive systemsrdquo Automatica vol 47 no7 pp 1504ndash1511 2011
[12] O Mason and R Shorten ldquoOn linear copositive Lyapunovfunctions and the stability of switched positive linear systemsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1346ndash1349 2007
[13] M A Rami and F Tadeo ldquoController synthesis for positivelinear systems with bounded controlsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 54 no 2 pp 151ndash1552007
[14] X Zhao P Shi and L Zhang ldquoAsynchronously switched controlof a class of slowly switched linear systemsrdquo Systems amp ControlLetters vol 61 no 12 pp 1151ndash1156 2012
[15] X Liu and C Dang ldquoStability analysis of positive switchedlinear systems with delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1684ndash1690 2011
[16] S Fadali and S Jafarzadeh ldquoStability analysis of positive intervaltype-2 TSK systems with application to energy marketsrdquo IEEETransactions on Fuzzy Systems vol 22 no 4 pp 1031ndash1038 2013
[17] L Imsland G O Eikrem and B A Foss ldquoA state feedbackcontroller for a class of nonlinear positive systems applied tostabilization of gas-lifted oil wellsrdquo Control Engineering PracticeN vol 3 pp 7ndash15 2006
[18] S K Nguang and P Shi ldquoFuzzy119867infinoutput feedback control of
nonlinear systems under sampled measurementsrdquo Automaticavol 39 no 12 pp 2169ndash2174 2003
[19] O Mason and M Verwoerd ldquoObservations on the stabilityproperties of cooperative systemsrdquo Systems amp Control Lettersvol 58 no 6 pp 461ndash467 2009
[20] H R Feyzmahdavian T Charalambous and M JohanssonldquoExponential stability of homogeneous positive systems ofdegree one with time-varying delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 6 pp 1594ndash1599 2014
[21] X Ding L Shu and X Liu ldquoOn linear copositive Lyapunovfunctions for switched positive systemsrdquo Journal of the FranklinInstitute vol 348 no 8 pp 2099ndash2107 2011
[22] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003
[23] X Zhao L Zhang P Shi and M Liu ldquoStability of switchedpositive linear systems with average dwell time switchingrdquoAutomatica vol 48 no 6 pp 1132ndash1137 2012
[24] X Zhao X Liu S Yin and H Li ldquoImproved results on stabilityof continuous-time switched positive linear systemsrdquo Automat-ica vol 50 no 2 pp 614ndash621 2014
[25] A S Morse ldquoSupervisory control of families of linear set-point controllersmdashpart 1 exactmatchingrdquo IEEETransactions onAutomatic Control vol 41 no 10 pp 1413ndash1431 1996
[26] D Liberzon Switching in Systems and Control Springer 2003[27] H Liu Y Shen and X Zhao ldquoAsynchronous finite-time
Hinfin
control for switched linear systems via mode-dependentdynamic state-feedbackrdquo Nonlinear Analysis Hybrid Systemsvol 8 no 1 pp 109ndash120 2013
[28] J Zhang Z Han H Wu and J Huang ldquoRobust stabilizationof discrete-time positive switched systems with uncertaintiesand average dwell time switchingrdquo Circuits Systems and SignalProcessing vol 33 no 1 pp 71ndash95 2014
[29] X Zhao L Zhang P Shi andM Liu ldquoStability and stabilizationof switched linear systems with mode-dependent average dwelltimerdquo IEEE Transactions on Automatic Control vol 57 no 7 pp1809ndash1815 2012
[30] J Lian and J Liu ldquoNew results on stability of switched positivesystems an average dwell-time approachrdquo IET Control Theoryand Applications vol 7 no 12 pp 1651ndash1658 2013
[31] X Zhao S Yin H Li and B Niu ldquoSwitching stabilization for aclass of slowly switched systemsrdquo IEEE Transactions on Auto-matic Control 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
[8] X Chen J Lam P Li and Z Shu ldquoL1-induced norm andcontroller synthesis of positive systemsrdquoAutomatica vol 49 no5 pp 1377ndash1385 2013
[9] J-E Feng J Lam P Li and Z Shu ldquoDecay rate constrainedstabilization of positive systems using static output feedbackrdquoInternational Journal of Robust and Nonlinear Control vol 21no 1 pp 44ndash54 2011
[10] P Li J Lam and Z Shu ldquo119867infin
positive filtering for positivelinear discrete-time systems an augmentation approachrdquo IEEETransactions on Automatic Control vol 55 no 10 pp 2337ndash2342 2010
[11] P Li J Lam Z Wang and P Date ldquoPositivity-preserving 119867infin
model reduction for positive systemsrdquo Automatica vol 47 no7 pp 1504ndash1511 2011
[12] O Mason and R Shorten ldquoOn linear copositive Lyapunovfunctions and the stability of switched positive linear systemsrdquoIEEE Transactions on Automatic Control vol 52 no 7 pp 1346ndash1349 2007
[13] M A Rami and F Tadeo ldquoController synthesis for positivelinear systems with bounded controlsrdquo IEEE Transactions onCircuits and Systems II Express Briefs vol 54 no 2 pp 151ndash1552007
[14] X Zhao P Shi and L Zhang ldquoAsynchronously switched controlof a class of slowly switched linear systemsrdquo Systems amp ControlLetters vol 61 no 12 pp 1151ndash1156 2012
[15] X Liu and C Dang ldquoStability analysis of positive switchedlinear systems with delaysrdquo IEEE Transactions on AutomaticControl vol 56 no 7 pp 1684ndash1690 2011
[16] S Fadali and S Jafarzadeh ldquoStability analysis of positive intervaltype-2 TSK systems with application to energy marketsrdquo IEEETransactions on Fuzzy Systems vol 22 no 4 pp 1031ndash1038 2013
[17] L Imsland G O Eikrem and B A Foss ldquoA state feedbackcontroller for a class of nonlinear positive systems applied tostabilization of gas-lifted oil wellsrdquo Control Engineering PracticeN vol 3 pp 7ndash15 2006
[18] S K Nguang and P Shi ldquoFuzzy119867infinoutput feedback control of
nonlinear systems under sampled measurementsrdquo Automaticavol 39 no 12 pp 2169ndash2174 2003
[19] O Mason and M Verwoerd ldquoObservations on the stabilityproperties of cooperative systemsrdquo Systems amp Control Lettersvol 58 no 6 pp 461ndash467 2009
[20] H R Feyzmahdavian T Charalambous and M JohanssonldquoExponential stability of homogeneous positive systems ofdegree one with time-varying delaysrdquo IEEE Transactions onAutomatic Control vol 59 no 6 pp 1594ndash1599 2014
[21] X Ding L Shu and X Liu ldquoOn linear copositive Lyapunovfunctions for switched positive systemsrdquo Journal of the FranklinInstitute vol 348 no 8 pp 2099ndash2107 2011
[22] A Jadbabaie J Lin andA SMorse ldquoCoordination of groups ofmobile autonomous agents using nearest neighbor rulesrdquo IEEETransactions on Automatic Control vol 48 no 6 pp 988ndash10012003
[23] X Zhao L Zhang P Shi and M Liu ldquoStability of switchedpositive linear systems with average dwell time switchingrdquoAutomatica vol 48 no 6 pp 1132ndash1137 2012
[24] X Zhao X Liu S Yin and H Li ldquoImproved results on stabilityof continuous-time switched positive linear systemsrdquo Automat-ica vol 50 no 2 pp 614ndash621 2014
[25] A S Morse ldquoSupervisory control of families of linear set-point controllersmdashpart 1 exactmatchingrdquo IEEETransactions onAutomatic Control vol 41 no 10 pp 1413ndash1431 1996
[26] D Liberzon Switching in Systems and Control Springer 2003[27] H Liu Y Shen and X Zhao ldquoAsynchronous finite-time
Hinfin
control for switched linear systems via mode-dependentdynamic state-feedbackrdquo Nonlinear Analysis Hybrid Systemsvol 8 no 1 pp 109ndash120 2013
[28] J Zhang Z Han H Wu and J Huang ldquoRobust stabilizationof discrete-time positive switched systems with uncertaintiesand average dwell time switchingrdquo Circuits Systems and SignalProcessing vol 33 no 1 pp 71ndash95 2014
[29] X Zhao L Zhang P Shi andM Liu ldquoStability and stabilizationof switched linear systems with mode-dependent average dwelltimerdquo IEEE Transactions on Automatic Control vol 57 no 7 pp1809ndash1815 2012
[30] J Lian and J Liu ldquoNew results on stability of switched positivesystems an average dwell-time approachrdquo IET Control Theoryand Applications vol 7 no 12 pp 1651ndash1658 2013
[31] X Zhao S Yin H Li and B Niu ldquoSwitching stabilization for aclass of slowly switched systemsrdquo IEEE Transactions on Auto-matic Control 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
