JournaloftheFranklinInstitute354(2017)2712–2724
www.elsevier.com/locate/jfranklin
Dwell-time-based
stabilization
of
switched
linear
singular
systems
with
all
unstable-mode
subsystems
Jinghan
Li
a,
Ruicheng
Ma
a ,∗,
Georgi
M.
Dimirovski
b ,c,
Jun
Fu
da SchoolofMathematics,LiaoningUniversity,Shenyang110036,China bFacultyofEngineering,DogusUniversityofIstanbul,Istanbul,Turkey cStCyril&StMethodiusUniversity,SchoolFEEIT,Skopje,Macedonia
dStateKeyLaboratoryofSyntheticalAutomationforProcessIndustries,NortheasternUniversity,110189,China
Received26February2016;receivedinrevisedform14November2016;accepted12January2017 Availableonline1February2017
Abstract
Theglobalstabilizationdesignofaclassofswitchedlinearsingular systemsviaanoveldwell-time switching is investigated and solved in this work. Thedistinguishing feature of the proposed method is that stability of all subsystems ofthe switchedsystemsis not necessarilyrequired. Atime-varying coordinatetransformation isintroducedfirst in orderto convertthe probleminto anequivalent oneof reduced-order switchedconventionallinearsystemwithstatejumps.Then,byconstructing certainnew multipletime-varying Lyapunovfunctions,computable sufficientconditionsforthe globalstabilization taskareproposedwithintheframeworkofdwell-timeswitching.Giventheassumed instabilityof indi-vidualsubsystems,thestabilizationoftheswitchedsystemisachievedundertheconditionofconfining the dwell time by a certainpair of upperand lower bounds, which restrict the growth of Lyapunov functionfortheactivelyoperatingsubsystem,thusdecreasethe energyofthe Lyapunovfunctionofthe overallswitched system atswitchingtimes.In addition, the multipletime-varyingLyapunovfunctions method is also used to analyze the stability analysis of a class of switched linear singular systems withstable subsystems.Two illustrativeexamplesarepresentedtodemonstratetheeffectiveness ofthe proposed method.
© 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
∗Correspondingauthor.
E-mailaddresses:[email protected](J.Li),[email protected](R.Ma),[email protected](G.M. Dimirovski),[email protected](J.Fu).
http://dx.doi.org/10.1016/j.jfranklin.2017.01.015
1. Introduction
In recent years, an increasing number of researchers have investigated switched systems duetotheirnumerousapplications invariousfieldsandsystemengineeringproblemssuchas powerelectronics,flightcontrol,networkcontrolsystems,see,e.g.,[14–16] andthereferences therein. Major effortsare devotedtoproblemsof stability,stabilization, H∞ control, optimal control, and so on [1,4,9,11] . So far it has been well established that switching between unstable subsystems may yield stability of the overall system. As far as the stability of switched systems is concerned, most of the reported results appear confined to cases of switched systems where at least one stable subsystem within switched system exists. Thus, switching strategy is viewed as powerful control inputeven to stabilize a switched systems withallunstablesubsystems[18] andisemployedtoachievetheobserversdesignofswitched linear systems with unknown inputswithoutposing any strong detectability requirements on subsystems of switched systems [10] . How todesign such an appropriate switching became one of the most challenging problems inthe study of switched systems. It is therefore that this paper is focused on the design of stabilizing switching signal for the class of switched systems with allunstable subsystems.
On the other hand, since linear singular system models not only describe the system dynamics, but also cancapture algebraicconstrains [3] , switchedsingular systems have also drawn considerable attention in the recent past [2,6–8,12,17,19–21] . As stated in [5] , it is necessary toallowfor solutionsinswitched singularsystems withinstantaneous state jumps, which are unavoidable even if all subsystems are regular and impulse-free. This is one of the major distinctionsbetweenswitched singular systems andswitchedconventional systems
[21] . Thus, in turn, considerable research attention has been devotedto the crucialproperty of stability under arbitrary switchings and under someconstrained switching laws. With the help of consistency projectors, [5] proposed a common Lyapunov function for the stability analysis and designof switched singular systems underarbitrary switchingpolicies.
Work[13] developedstate-dependentswitchingstrategiesforstabilizationof switched sys-tems. On the dwell-time-based approach, the stability analysis is studied in [[2,6–8,12,17, 19–21] .Thesestudiesexploit thefeatures ofthe dwelltimetechnique:adwell timeof active subsystemcansubsideforpotentiallypossiblelargestatetransients.Inaddition,itappearedto be agoodalternativefor studyingelectricalcircuitswithphysicalswitches orcaseswith sud-den component faults in electricaland mechanical systems. Thesegave the main motivation forchoosingdwelltimetechniqueinthepresentexplorationstudy.However,dwell-time-based methods in [2,6–8,12,17,19–21] require that there must be at least one stable subsystem of the switched systemtobe switchedon for thestability analysistobe successful.Thus, natu-rally questions arise: Isit possible, intheframework of the dwelltimetechnique, toachieve the stabilization design of switched linear singular systems without posing any stability re-quirements on subsystems ofthe switched systems? If possible,then under which conditions and how can we come up witha switching policy to achieve this goal? To the best of our knowledge, in the literature there are no results which provide answers to these questions. This preciselyis the motivationof thepresent paper.
To study the problemsproposed above,a time-varyingcoordinate transformation is intro-ducedfirstinordertoconverttheproblemintoanequivalentoneofforreduced-orderswitched conventional linearsystemwithstate jumps.Then,byconstructingcertainnewmultiple time-varying Lyapunovfunctions,computablesufficient conditions forthe globalstabilization task are proposed within the framework of adwell-time switching.Given the assumed instability
of individual subsystems,the stabilization of theswitched systemis achieved underthe con-ditionof confiningthedwell timebyacertainpairofupperandlowerbounds,whichrestrict the growth of Lyapunov function for the actively operating subsystem, thus decrease energy of theoverallswitchedsystematswitchingtimes.Inaddition,themultipletime-varying Lya-punov functions method is also used to analyzethe stability analysis of a class of switched linear singular systems with stable subsystems. Two illustrative examples are presented to demonstrate the effectivenessof theproposed method.
Further thispaper isorganizedas follows. In Section 2 ,systemdescription andnecessary preliminaries are given. The main results are presented in Section 3 . An illustrative exam-ple along with numerical and simulation results is provided for in Section 4 . Finally, some conclusionsare drawn inSection 5 .
Notations: The notations used in this paper are fairly standard. Rn is the n dimensional
Euclidean space, Rn× m denotes the set of n × m real matrices, R+=(0,+∞), and N =
{0,1,2,· · ·}. xT denotes the transpose of the vector x. The symbol “∗ represents arbitrary blockmatrixofappropriatedimensions.P>(≥,<,≤)0isusedtodenoteapositive-definite (positive-semidefinite, negative, negative-semidefinite)matrix.Denote the maximal (minimal) eigenvalues of a matrix P by λmax(P) (λmin(P)). I and 0 represent identity matrix and zero
matrix withproper dimension,respectively.
2. Systemdescription and preliminaries
Consider the followingswitched linearsingular systems:
Eσ (t )x˙(t )=Aσ (t )x(t ),
x(0)=x0, (1)
where x(t) ∈ Rn is the system state, x
0 ∈ Rn is a vector-valued initial state, σ (t ):R+→
M ={1,2,...,m}isthe switchinglaw,whichisassumedtobeapiecewisecontinuous(from the right) function of time andm >0 is the number of modes of the switched system (i.e., subsystems). Throughoutthispaper, we assumethat σ (t )=σ (tk)=ik,ik∈M,t ∈
tk,tk+1),
where tk is the switching instant, this means that the ikth subsystem is activated when t ∈
tk,tk+1).Foreveryi∈M,Ei andAi areconstantmatrices,anditisassumed thatrank(Ei)=
r≤ n.Forsimplicity,we use (Ei,Ai)todenote the ithsubsystem. The set{tk}generated by
σ (t) ∈ T[τ1,τ2] denotesthe switching sequenceswithτ1≤ tk− tk−1≤ τ2, k∈ N.
Definition 1. [3] . Foreveryi ∈ M,the singular system(Ei,Ai)is said tobe
(i)regular if det(sEi− Ai)isnot identically zero;
(ii)impulse-freeif deg(det(sEi− Ai))=rank(Ei).
Assumption 1. Forevery i∈ M,the singular system(Ei,Ai) is regularandimpulse-free.
Due to the fact rank(Ei)=r≤ n, we can find nonsingular matrices Hi and Ni(i ∈ M),
such that HiEiNi= Ir 0 0 0 =: ¯E, HiAiNi= A11(i) A12(i) A21(i) A22(i) =: ¯Ai. (2)
By introducing the state transformation: ¯x(t )= ¯x1(t ) ¯x2(t ) =Ni−1x(t ),t∈ tk,tk+1), (3)
switched system(1) takes the followingform inthe newcoordinates: ˙
¯x1(t )=A11(σ (t ))¯x1(t )+A12(σ (t ))¯x2(t ), (4)
0=A21(σ (t ))¯x1(t )+A22(σ (t ))¯x2(t ). (5)
Note that at the switching instant tk,the systemswitches from (Ej, Aj) to (Ei,Ai). Then,
considering the switching law dependent feature of the state transformation(3) ,we have
xtk−=Nj¯x
tk−,xtk+=Ni¯x
tk+. (6)
According tothe analysis presented in[21] , we havethat ¯xtk+=i j¯x tk−,i,j∈M, (7) with i j= I 0 −A−1 22(i)A21(i) 0 Ni−1Nj. (8)
In addition,we canobtain that
¯x1 tk+=I 0i j¯x tk−=I 0i j I −A−1 22( j)A21( j) ¯x1 tk−=i j¯x1 tk−, (9) with i j= I 0Ni−1Nj I −A−1 22( j)A21( j) . (10)
Under Assumption 1 , we know that A22 is nonsingular. Thus, by (5) , we can obtain a
reduced-order switched conventional linearsystem withstate jumps(9) : ˙¯x1(t )=Aˆ(σ (t ))¯x1(t ), ¯x1 tk+=i j¯x1 tk−, (11) where Aˆ(σ (t ))=A11(σ (t ))− A12(σ (t ))A−122(σ (t )) A21(σ (t)).
Theobjectiveofthisworkistoderiveandestablishsufficientconditionsfortheexistenceof theswitchingsignaltogloballyasymptoticallystabilizetheswitchedsingularsystem(1) under the switching law design σ (t). If one of the subsystems is asymptotically stable, then this problem is no longer a challenge. Therefore, herein none of the individual subsystems is assumed tobe asymptoticallystable.
3. Main results
In this section, we consider two classes of switched singular systems. First we consider the stabilization problem for a switched singular system with all unstable subsystems. And thereafter we study the stability analysis problem for a switched singular system with all stable subsystems for the sake of establishing aparallel.
3.1. Case a: allsubsystems are unstable
Theorem 1. Consider switched singular system (1) satisfying Assumption 1 . If there exist constants τ2 ≥ τ1 > 0, λ > 0, 1 > μ > 0, μi > 1, i ∈ M, and positive definite matrices
Pi1 >0, Pi2 >0, and anyappropriatedimensional matrix Pi3 and Pi4,such that
θilq− λPil ≤ 0, ∀i∈M, l,q=1,2, (12) −μPj1 Ti jPi2 ∗ −μiPi2 <0,∀i,j∈M,i= j, (13) lnμ +λτ2<0, (14) where
θilq=ϑiPil+AˆT(i)Pil+PilAˆ(i)+
1 τq (Pi1− Pi2),∀i∈M, l,q=1,2, (15) i j= I 0Ni−1Nj I −A−1 22( j)A21( j) (16) withϑi=lnτ(μi) 1 ,Aˆ(i)=A11(i)− A12(i)A −1
22(i)A21(i),switchedsingularsystem(1) isglobally
uniformly asymptotically stableunder switchinglaw σ (t)∈ T[τ1,τ2].
Proof. For{tk} generated byσ (t) ∈ T[τ1,τ2] andt ∈
tk,tk+1), we define: ρ(t )= t− tk tk+1− tk ,ρ(t )˜ =1− ρ(t ),ρ1(t )= 1 tk+1− tk ,φ(t )=μρ(t )i −1. (17)
Whent ∈tk,tk+1),we consider thefollowing Lyapunovfunction:
Vi(t,x(t))=φ(t )¯xT(t )¯E¯Pi(t )¯x(t ), (18) where ¯Pi(t )= Pi(t ) 0 Pi3 Pi4 withPi(t )=ρ(t)Pi1+ρ(t˜ )Pi2, i ∈ M.
From (18) ,we can obtain that
Vi(t,x(t))=φ(t )¯xT(t )¯E¯Pi(t )¯x(t ) =φ(t )¯xT 1(t ) ¯x T 2(t ) Ir 0 0 0 Pi(t ) 0 Pi3 Pi4 ¯x1(t ) ¯x2(t ) =φ(t )¯xT 1(t ) 0 Pi(t ) 0 Pi3 Pi4 ¯x1(t ) ¯x2(t ) =φ(t )¯xT 1(t )Pi(t ) 0 ¯x¯x1(t ) 2(t ) =φ(t )¯xT 1(t )Pi(t )¯x1(t ) =Vi(t,¯x1). (19)
It is obviousthat Vi(t,¯x1)satisfies
α1( ¯x1 )=
λ2
where ν =max{μi,i∈M}, λ1=max{λmax(Pil),i∈ M,l =1,2}, and λ2=
min{λmin(Pil),i∈M,l=1,2}.
Whent ∈tk,tk+1), thetime derivative ofVi(t,¯x1) is
˙ Vi(t,x1(t))=ρ(t )˙ ln(μi)φ(t )¯x1T(t )Pi(t )¯x1(t )+φ(t )¯x1T(t ) ˙ ρ(t )Pi1+ρ(t )˙˜ Pi2 ¯x1(t ) +2φ(t )¯xT 1(t )Pi(t )˙¯x1(t ) ≤ φ(t )¯xT 1(t ) ln(μi) τ1 Pi(t )+ρ1(t )(Pi1− Pi2) ¯x1(t)+2φ(t )¯x1T(t )Pi(t )Aˆ(i)¯x1(t ) =φ(t )¯xT 1(t ) ln(μi) τ1 Pi(t )+Pi(t )Aˆ(i)+AˆT(i)Pi(t )+ρ1(t )(Pi1− Pi2) ¯x1(t). (21)
We choose afunction ρ2(t)∈ [0, 1] and ρ2(t )=1− ˜ρ2(t ),such that
ρ1(t )= 1 τ1ρ2(t )+ 1 τ2ρ˜2(t ). (22)
Apparently, such a function ρ2(t) can be fairly easyobtained. Forexample, when τ2 > τ1,
we choose ρ2(t )=(ρ1(t )−τ12)/(τ11 −τ12). In addition, if τ1=τ2,then ρ1(t )=τ1=τ2, we
can easilyto choose, for example,ρ2(t )=ρ˜2(t )= 12,whichsatisfies (22) .
From (21) and(22) , onehasthat ˙ Vi(t,¯x1)≤ φ(t )¯xT1(t ) ϑiPi(t )+Pi(t )Aˆ(i)+AˆT(i)Pi(t )+ρ1(t )(Pi1− Pi2) ¯x1(t) =φ(t )¯xT 1(t )[ϑi(ρ(t )Pi1+ρ(t )˜ Pi2)+(ρ(t )Pi1+ρ(t )˜ Pi2)Aˆ(i) +AˆT(i)(ρ(t )Pi1+ρ(t )˜ Pi2)+ρ1(t )(Pi1− Pi2)]¯x1(t ) =φ(t )¯xT 1(t ) ρ(t )ϑiPi1+AˆT(i)Pi1+Pi1Aˆ(i) ¯x1(t) +φ(t )¯xT 1(t ) ˜ ρ(t )ϑiPi2+AˆT(i)Pi2+Pi2Aˆ(i) ¯x1(t) +φ(t )¯xT 1(t ) (τ1 1 ρ2(t )+ 1 τ2 ˜ ρ2(t ))(Pi1− Pi2) ¯x1(t ). (23)
With the help of (12) and (23) , we canobtain ˙ Vi(t,¯x1)≤ φ(t )¯xT1(t ) ρ(t )ρ2(t )θi11+ρ˜2(t )θi12 ¯x1(t ) +φ(t )¯xT 1(t ) ˜ ρ(t )ρ2(t )θi21+ρ˜2(t )θi22 ¯x1(t ) ≤ φ(t )¯xT 1(t ) ρ(t )ρ2(t )λPi1+ρ˜2(t )λPi1 ¯x1(t ) +φ(t )¯xT 1(t ) ˜ ρ(t )ρ2(t )λPi2+ρ˜2(t )λPi2 ¯x1(t ) =λφ(t )¯xT 1(t ) ρ(t )Pi1+ρ(t )˜ Pi2 ¯x1(t ) =λVi(t,¯x1). (24)
On the other hand,according to(13) ,onecan find that
Vi tk+,¯x1 =φtk+¯x1Ttk+Pi tk+¯x1 tk+ =μρ(tk+)−1 i ¯x T 1 tk+ρtk+Pi1+ρ˜ tk+Pi2 ¯x1 tk+ =μ−1 i ¯x T 1 tk+ Pi2¯x1 tk+
=μ−1 i ¯x T 1 tk−Ti jPi2i j¯x1 tk− < ¯xT 1 tk−μPj1¯x1 tk− =μμρ(tk−)−1 i ¯x T 1 tk− Pj1¯x1 tk− =μ ¯xT 1 tk− ρtk− Pj1+ρ˜ tk− Pj2 ¯x1 tk− =μφtk−¯xT1tk−Pj tk−¯x1 tk− =μVj tk−,¯x1 . (25)
For simplicity, we define V(t,¯x1)=
m
i=1αi(t )Vi(t,¯x1), where αi(t )=
1, ifσ (t )=i,
0, otherwise, ∀i ∈ M.Assumingαi(t )=1 and αj(t )=0, whent ∈[tk,tk+1).
From (24) ,we can deriveVi(t,¯x1)≤ eλ(t−tk)Vi(tk,¯x1),t ∈[tk,tk+1).
Then, by supposing system (11) switches from subsystem j to i at switching instant tk,
since σ (t)is continuousfrom the right,we have
Vi(tk,¯x1)≤ μeλ(tk−tk−1)Vj(tk−1,¯x1). (26)
Since tk− tk−1≤ τ2, k=1,2,· · ·, whichtogether with(14) ,we can obtain that
μeλ(tk−tk−1)<1. (27)
Thus, we can see Vi(tk,¯x1)<Vj(tk−1,¯x1) . Then, for any ε > 0, we can choose
¯x1(t0) <δ(ε)=α−12
e−λτ2α
1(ε)
. Thus, this yields V(t0,¯x1)≤ α2( ¯x1(t0) )<e−λτ2α1(ε).
Since Vi(tk,¯x1) is strictly decreasing, we have Vi(tk,¯x1)≤ e−λτ2α1(ε). Then, we have
V(t,¯x1)≤ α1(ε).Furthermore,from(20) ,we canconclude ¯x1(t ) <ε.Obviously,for ∀δ >
0, wehave ¯x1(t ) <ε.Duetothe factthat thesequenceVi(tk,¯x1),k=0,1,2,...isstrictly
decreasing, we obtained that lim
t→∞ ¯x1(t ) =0. Therefore, we can conclude switched system
(11) isglobally asymptotically stabilizedunder switching law σ (t) ∈ T[τ1, τ2].
Itfollows from(5) that ¯x2(t )=−A−122(σ (t ))A21(σ (t ))¯x1(t),thus, ¯x2(t)alsoglobal
asymp-toticalstable.Thisindicatesthat system(4) ,orequivalently,thesystem(1) isglobally asymp-totically stabilizeunder switching law σ(t)∈ T[τ1,τ2].Thus,the proof is completed.
Remark 1. Theorem 1 provides asufficient condition toachieve the stabilizationfor a class of switched singular systems via dwell-time-based switchings.Our method does not require stability of each subsystem, which nontrivially generalizes the result of [21] obtained under the assumption that allor part of subsystemsare stable.On the otherhand, when Ei=I,∀i
∈ M,theswitchedsingularsystemdeducestotheswitchednormalsystemin[18] .Therefore, ourobtainedresultextendsthatoftheswitchednormalsystemtotheswitchedsingularsystem case.
Remark 2. In our results, the common λ and μ are used to confine accordingly the upper bound of the dwell time τ2. Since these two parameters may be dependent on individual
subsystems, values of these two parameters can be different for different subsystems. It is thereforethathereproposedmethodsappearslessconservativeandthusyieldabetteroutcome design result.
3.2. Case b: allsubsystems are stable
It is wellknown that switching among stablesystems mayyield instabilityof the overall switchedsystem. Itisthereforethatisnecessary alsotoinvestigatethislineof design
deriva-tion for thecase where all subsystemsof switchedsingular systems are stable.In particular, we have found that if the subsystems of switched singular system (1) are stable, then the condition on the upper bound of the dwell-time (i.e., τ2 in Theorem 1 ) can be removed.
Thus, we havethe followingtheorem.
Theorem 2. Consider switched singular system (1) satisfying Assumption 1 . If there exist constants τ1 > 0, λ > 0, 1 > μ > 0, μi >1, i ∈ M,and positive definite matrices Pi1 >
0, Pi2 >0, such that θilq+λPil ≤ 0, ∀i∈M, l,q=1,2, (28) −μPj1 Ti jPi2 ∗ −μiPi2 <0,∀i,j∈M,i= j, (29) Pi1− Pi2>0, (30) where
θilq=ϑiPil+AˆT(i)Pil+PilAˆ(i)+
1 τ1 (Pi1− Pi2),∀i∈M, l,q=1,2, (31) i j= I 0Ni−1Nj I −A−1 22( j)A21( j) , (32) with ϑi=lnτμi
1 , and Aˆ(i)=A11(i)− A12(i)A
−1
22(i)A21(i), thenswitched singularsystem(1) is
globally uniformly asymptotically stableunder switchinglaw σ (t)∈ T[τ1,∞).
Proof. The proof isvery similar tothe proof of Theorem 1 , andit can be easilyderived by the methodology as above. Therefore,the proof of Theorem 2 isomitted here.
4. Illustrative examples
In this section, we present illustrative examples along with the respective numerical and simulation results todemonstratethe effectiveness of the proposedswitching designmethod.
4.1. Example 1
Consider the following switched singular system (1) with two subsystems of the fourth order: E1= ⎡ ⎢ ⎢ ⎣ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 ⎤ ⎥ ⎥ ⎦,A1= ⎡ ⎢ ⎢ ⎣ −1 1 0 −1 1 −2 −1 1 1 −3 −0.2 −0.2 2 1 −2 −1 ⎤ ⎥ ⎥ ⎦, E2= ⎡ ⎢ ⎢ ⎣ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎦, A2 = ⎡ ⎢ ⎢ ⎣ 1 −2 −1 2 −2 −2 −1 2 −1 1 −1 −2 1 1 −1 1 ⎤ ⎥ ⎥ ⎦.
Time(s) 0 0.5 1 1.5 2 x(t) -200 -150 -100 -50 0 50 100 150 x1 x 2 x 3 x 4
Fig.1. Thetrajectoriesofsubsystem1.
By H1=N1=N2= ⎡ ⎢ ⎢ ⎣ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ⎤ ⎥ ⎥ ⎦andH2= ⎡ ⎢ ⎢ ⎣ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ⎤ ⎥ ⎥
⎦, we canobtain (2) with
¯A1=H1A1N1 = ⎡ ⎢ ⎢ ⎣ −1 1 0 −1 1 −2 −1 1 1 −3 −0.2 −0.2 2 1 −2 −1 ⎤ ⎥ ⎥ ⎦, ¯A2=H2A2N2= ⎡ ⎢ ⎢ ⎣ 1 −2 −1 2 −1 1 −1 −2 −2 −2 −1 2 1 1 −1 1 ⎤ ⎥ ⎥ ⎦, and(11) withAˆ1= ⎡ ⎣−33 −10 −32 0.6 −3.2 0.2 ⎤ ⎦, Aˆ2= ⎡ ⎣−11 −43 −31 −4 −4 1 ⎤ ⎦,12=21= ⎡ ⎣10 01 00 0 0 1 ⎤ ⎦. From Figs. 1 and 2 , we see that both subsystem 1 and 2 are unstable with x(t0)=
(3,4,−2,14)T andx(t0)=(3,4,−2,9)T,respectively.Therefore,themethodin[21] cannot
beappliedtothestudiedswitchedsystem.However,sincethe hereproposedmethoddoesnot require stability of the subsystems of switched singular system (1) , we can apply it to this example inorder toasymptotically stabilizethe switched system, as shownfurtherbelow.
Let λ =1, μ =0.9, μ1=2.1, μ2=2.2, τ1=0.08s, τ2 =0.1s. Then by means of
Theorem 1 , we obtain the followingmatrices:
P11= ⎡ ⎣ 5755..94022065 6055..20650027 −18−21..15908876 −21.1590 −18.8876 28.4523 ⎤ ⎦, P12 = ⎡ ⎣5658..67591763 5867..17638901 −32.−25.89769528 −25.8976 −32.9528 38.5350 ⎤ ⎦,
Time(s) 0 0.5 1 1.5 2 x(t) -500 0 500 1000 x1 x 2 x 3 x4
Fig.2. Thetrajectoriesofsubsystem2.
0 2 4 6 8 10 12 −15 −10 −5 0 5 10 15 x(t) Time(s) x 1 x 2 x 3 x 4
Fig.3. Stateresponsesoftheswitchedsingularsystem.
P21= ⎡ ⎣4446..86495320 4653..53201657 −12.−9.44281667 −9.4428 −12.1667 22.8090 ⎤ ⎦,P22 = ⎡ ⎣ 8376..21911833 8076..18336093 −46−47..73943264 −47.7394 −46.3264 52.6676 ⎤ ⎦. Let x(t0)=(3,4,−2,14)T.Fig. 3 shows the state trajectories of the switched systemunder
thedwelltimeswitchingsignalσ (t)∈T[τ1,τ2)showninFig. 4 .ApparentlyFig. 3 showsthat
0 2 4 6 8 10 12 1
2
The i−th subsystem
Time(s)
Fig.4. Thedesignedswitchingsignalσ(t).
0 2 4 6 8 10 0 1 2 3 4 5 6 x(t) Time(s) x 1 x 2
Fig.5. Stateresponseoftheswitchedsingularsystem.
simulation resultsdemonstrate prettywellthe effectiveness of theproposed switchingdesign method.
4.2. Example 2
Consider the sameswitched singular systemas in[21] ,where
E1= 0 1 0 0 , A1= 0 −1 1 a , E2= 1 1 0 0 , A2= −1 −1 1 0 . Asin[21] ,we setN1= 0 1 1 0 ,N2= 0 1 1 −1
,thenget ¯E =E1N1=E2N2 =
1 0 0 0 , ¯A1= A1N1= −1 0 a 1 , ¯A2=A2N2 = −1 0 0 1 .
0 2 4 6 8 10 1
2
The i−th subsystem
Time(s)
Fig.6. Thecorrespondingswitchingsignal.
When a=−2, it is reported in [21] that when the average dwell time is greater than
1
2ln(|1− a|)=0.5493, thesystem isasymptotical stability.
Now, weapplyourTheorem 2 tothestudiedsystem,andithasbeenfoundthat,whena=
−2, thereexist τ1=0.05s,λ =10, μ1=μ1=1.002, andP11 =[11.0254],P12=[7.0712],
P21 =[3.6752],P22 =[2.3893],suchthat(28), (29) and(30) hold.Thus,thestudiedswitched
singular systemis asymptotically stable under switching law σ (t) ∈ T[0.05, ∞). When a=
−2,thestateresponseoftheswitchedsingularsystemandthecorrespondingswitchingsignal are illustrated inFig. 5 and Fig. 6 ,respectively.
Remark 3. It canbe seenthat the dwell time0.05 obtained by Theorem 2 is much smaller than the one 12ln(|1− a|)=0.5493 in [21] . Thus, our obtained stability results are less conservative thanthe ones givenin [21] .
5. Conclusions
This paper has investigated the global stabilization problem of a class of switched linear singularsystemsviadwell-timeswitchingmechanismwithoutrequiringstabilityofsubsystems of the switched systems. In spiteof theinstability of individual subsystems,the stabilization of the switchedsystemisachieved underthe conditionof confining thedwell timewithin an upper and lowerbounds representing arelevant pair of bounds, whichrestrict the growthof Lyapunov function of the active subsystem. At the same time such switching law decreases the energy of the Lyapunovfunction of the overallswitched systemat switchinginstants. In addition, the multiple time-varying Lyapunov functions method is also used to analyze the stability analysis of a class of switched linear singular systems with stable subsystems. A future research along the lines of the method proposed in this manuscript is envisaged in solving the finite-time on stabilization problemfor the considered class of switched singular systems in ordertoimprove the overall performance.
Acknowledgement
This work was partially supported by the National Natural Science Foundation of China
(61673198 ,61304055 ,61473063 ,61590924 ),Provincial Natural Science Foundation of Liaon- ing Province (2015020088 ), the Fundamental Research Funds for the Central Universities
(N150802001), as well as supported by the Fund for Science of Dogus University and by the grant TUBITAK-RFBR-113E595 for the Turkish–Russian scientific project inAerospace Sciences. Last but not the least, we would thank the three anonymous reviewers for their constructive comments whichgreatly improvedthe qualityandpresentation of thispaper.
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