0 Nonlinear Analysis and Design of Phase-Locked Loops - InTech

28 Pages·2012·0.38 MB·English

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(cid:2)
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Nonlinear Analysis and Design of
Phase-Locked Loops
G.A.Leonov,N.V.KuznetsovandS.M.Seledzhi
Saint-PetersburgStateUniversity
Russia
1. Introduction
Phase-lockedloops(PLLs)arewidelyusedintelecommunicationandcomputerarchitectures.
They were invented in the 1930s-1940s (De Bellescize, 1932; Wendt & Fredentall, 1943) and
thenintensivestudiesofthetheoryandpracticeofPLLswerecarriedout(Viterbi,1966;Lind-
sey, 1972; Gardner, 1979; Lindsey and Chie, 1981; Leonov et al., 1992; Encinas, 1993; Nash,
1994;Brennan,1996;Stensby,1997).
One of the first applications of phase-locked loop (PLL) is related to the problems of data
transfer by radio signal. In radio engineering PLL is applied to a carrier synchronization,
carrierrecovery,demodulation,andfrequencysynthesis(see,e.g.,(Stephens,2002;Ugrumov,
2000)).
After the appearance of an architecture with chips, operating on different frequencies, the
phase-lockedloopsareusedtogenerateinternalfrequenciesofchipsandsynchronizationof
operationofdifferentdevicesanddatabuses(Youngetal., 1992; Egan, 2000; Kroupa, 2003;
Razavi, 2003; Shu&Sanchez-Sinencio, 2005; Manassewitsch, 2005). Forexample, themod-
erncomputermotherboardscontaindifferentdevicesanddatabusesoperatingondifferent
frequencies, which are often in the need for synchronization (Wainner & Richmond, 2003;
Buchanan&Wilson,2001).
Anotheractual applicationof PLLis theproblem ofsaving energy. One ofthe solutionsof
thisproblemforprocessorsisadecreasingofkernelfrequencywithprocessorload. Thein-
dependentphase-lockedloopspermitonetodistributemoreuniformlyakernelloadtosave
theenergyandtodiminishaheatgenerationonaccountofthateachkerneloperatesonits
ownfrequency.Nowthephase-lockedloopsarewidelyusedforthesolutionoftheproblems
of clock skew and synchronization for the sets of chips of computer architectures and chip
microarchitecture. For example, a clock skew is very important characteristic of processors
(see,e.g.,(Xanthopoulos,2001;Bindal,2003)).
Variousmethodsforanalysisofphase-lockedloopsarewelldevelopedbyengineersandare
consideredinmanypublications(see,e.g.,(Banerjee,2006;Best,2003;Kroupa,2003;Bianchi,
2005;Egan,2007)),buttheproblemsofconstructionofadequatenonlinearmodelsandnon-
linearanalysisofsuchmodelsarestillfarfrombeingresolvedandrequireusingspecialmeth-
odsofqualitativetheoryofdifferential,difference,integral,andintegro-differentialequations
(Gelig et al., 1978; Leonov et al., 1996a; Leonov et al., 1996b; Leonov & Smirnova, 2000;
Abramovitch, 2002; Suarez & Quere, 2003; Margaris, 2004; Kudrewicz & Wasowicz, 2007;
Kuznetsov,2008;Leonov,2006).Wecouldnotlisthereallreferencesintheareaofdesignand
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analysisofPLL,soreadersshouldseementionedpapersandbooksandthereferencescited
therein.
2. MathematicalmodelofPLL
InthisworkthreelevelsofPLLdescriptionaresuggested:
1)thelevelofelectronicrealizations,
2)thelevelofphaseandfrequencyrelationsbetweeninputsandoutputsinblockdiagrams,
3)thelevelofdifference,differentialandintegro-differentialequations.
Thesecondlevel, involvingtheasymptoticalanalysisofhigh-frequencyoscillations, isnec-
essary for the well-formed derivation of equations and for the passage to the third level of
description.
ConsideraPLLonthefirstlevel(Fig.1)
Fig.1.BlockdiagramofPLLonthelevelofelectronicrealizations.
HereOSCmaster isamasteroscillator,OSCslave isaslave(tunable)oscillator,whichgenerate
high-frequency"almostharmonicoscillations"
f (t)= A sin(ω (t)t+ψ) j=1,2, (1)
j j j j
whereA andψ aresomenumbers,ω (t)aredifferentiablefunctions.Block isamultiplier
j j j
ofoscillationsof f (t)and f (t)andthesignal f (t)f (t)isitsoutput. Therelationsbetween
1 2 1 2 (cid:2)
theinputξ(t)andtheoutputσ(t)oflinearfilterhavetheform
t
σ(t)=α (t)+ γ(t τ)ξ(τ)dτ. (2)
0 −
(cid:3)
0
Hereγ(t)isanimpulsetransientfunctionoffilter,α (t)isanexponentiallydampedfunction,
0
depending on the initial data of filter at the moment t = 0. The electronic realizations of
generators, multipliers, and filters can be found in (Wolaver, 1991; Best, 2003; Chen, 2003;
Giannini & Leuzzi, 2004; Goldman, 2007; Razavi, 2001; Aleksenko, 2004). In the simplest
caseitisassumedthatthefilterremovesfromtheinputtheuppersidebandwithfrequency
ω (t)+ω (t)butleavesthelowersidebandω (t) ω (t)withoutchange.
1 2 1 − 2
Nowwereformulatethehigh-frequencypropertyofoscillations f (t)andessentialassump-
j
tion that γ(t) and ω (t) are functions of "finite growth". For this purpose we consider the
j
great fixed time interval [0,T], which can be partitioned into small intervals of the form
[τ,τ+δ],(τ [0,T])suchthatthefollowingrelations
∈
γ(t) γ(τ) Cδ, ω (t) ω (τ) Cδ,
| − |≤ | j − j |≤ (3)
t [τ,τ+δ], τ [0,T],
∀ ∈ ∀ ∈
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ω (τ) ω (τ) C , τ [0,T], (4)
| 1 − 2 |≤ 1 ∀ ∈
ω (t) R, t [0,T] (5)
j ≥ ∀ ∈
aresatisfied.Hereweassumethatthequantityδissufficientlysmallwithrespecttothefixed
numbersT,C,C ,thenumberRissufficientlygreatwithrespecttothenumberδ. Thelatter
1
meansthatonthesmallintervals[τ,τ+δ]thefunctionsγ(t)andω (t)are"almostconstants"
j
andthefunctions f (t)rapidlyoscillateasharmonicfunctions.
j
ConsidertwoblockdiagramsshowninFig.2andFig.3.
Fig.2.Multiplierandfilter.
Fig.3.Phasedetectorandfilter.
Hereθ (t) = ω (t)t+ψ arephasesoftheoscillations f (t),PDisanonlinearblockwiththe
j j j j
characteristic ϕ(θ)(beingcalledaphasedetectorordiscriminator). Thephasesθ (t)arethe
j
inputs of PD block and the output is the function ϕ(θ (t) θ (t)). The shape of the phase
1 − 2
detectorcharacteristicisbasedontheshapeofinputsignals.
Thesignals f (t)f (t)andϕ(θ (t) θ (t))areinputsofthesamefilterswiththesameimpulse
1 2 1 − 2
transientfunctionγ(t).Thefilteroutputsarethefunctionsg(t)andG(t),respectively.
AclassicalPLLsynthesisisbasedonthefollowingresult:
Theorem1. (Viterbi,1966)Ifconditions(3)–(5)aresatisfiedandwehave
1
ϕ(θ)= A A cosθ,
2 1 2
thenforthesameinitialdataoffilter,thefollowingrelation
G(t) g(t) C δ, t [0,T]
| − |≤ 2 ∀ ∈
issatisfied.HereC isacertainnumberbeingindependentofδ.
2
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ProofofTheorem1(Leonov,2006)
Fort [0,T]weobviouslyhave
∈
g(t) G(t)=
−
t
= γ(t s) A A sin ω (s)s+ψ sin ω (s)s+ψ
− 1 2 1 1 2 2 −
(cid:3) (cid:4) (cid:5) (cid:8)
0 (cid:6) (cid:7) (cid:6) (cid:7)
ϕ ω (s)s ω (s)s+ψ ψ ds=
− 1 − 2 1− 2
(cid:5) (cid:8)(cid:9)
t
A A
= 1 2 γ(t s) cos ω (s)+ω (s) s+ψ +ψ ds.
− 2 − 1 2 1 2
(cid:3) (cid:4) (cid:5) (cid:8)(cid:9)
0 (cid:6) (cid:7)
Considertheintervals [kδ,(k+1)δ], where k = 0,...,m andthenumber m issuchthat t
∈
[mδ,(m+1)δ].Fromconditions(3)–(5)itfollowsthatforanys [kδ,(k+1)δ]therelations
∈
γ(t s)=γ(t kδ)+O(δ) (6)
− −
ω (s)+ω (s)=ω (kδ)+ω (kδ)+O(δ) (7)
1 2 1 2
arevalidoneachinterval[kδ,(k+1)δ].Thenby(7)foranys [kδ,(k+1)δ]theestimate
∈
cos ω (s)+ω (s) s+ψ +ψ =cos ω (kδ)+ω (kδ) s+ψ +ψ +O(δ) (8)
1 2 1 2 1 2 1 2
(cid:5) (cid:8) (cid:5) (cid:8)
(cid:6) (cid:7) (cid:6) (cid:7)
isvalid.Relations(6)and(8)implythat
t
γ(t s) cos ω (s)+ω (s) s+ψ +ψ ds=
− 1 2 1 2
(cid:3) (cid:4) (cid:5) (cid:8)(cid:9)
0 (cid:6) (cid:7) (9)
(k+1)δ
m
= ∑ γ(t kδ) cos ω (kδ)+ω (kδ) s+ψ +ψ ds+O(δ).
− 1 2 1 2
k=0 (cid:3) (cid:4) (cid:5) (cid:8)(cid:9)
kδ (cid:6) (cid:7)
From(5)wehavetheestimate
(k+1)δ
cos ω (kδ)+ω (kδ) s+ψ +ψ ds=O(δ2)
1 2 1 2
(cid:3) (cid:4) (cid:5) (cid:8)(cid:9)
kδ (cid:6) (cid:7)
andthefactthatRissufficientlygreatascomparedwithδ.Then
t
γ(t s) cos ω (s)+ω (s) s+ψ +ψ ds=O(δ).
− 1 2 1 2
(cid:3) (cid:4) (cid:5) (cid:8)(cid:9)
0 (cid:6) (cid:7)
Theorem1iscompletelyproved.(cid:2)
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Fig.4.BlockdiagramofPLLonthelevelofphaserelations
Thus,theoutputsg(t)andG(t)oftwoblockdiagramsinFig.2andFig.3,respectively,differ
littlefromeachotherandwecanpass(fromastandpointoftheasymptoticwithrespecttoδ)
tothefollowingdescriptionlevel,namelytothesecondlevelofphaserelations.
InthiscaseablockdiagraminFig.1becomesthefollowingblockdiagram(Fig.4).
Considernowthehigh-frequencyimpulseoscillators,connectedasindiagraminFig.1.Here
f (t)= Asign(sin(ω (t)t+ψ)). (10)
j j j j
Weassume,asbefore,thatconditions(3)–(5)aresatisfied.
Consider2π-periodicfunctionϕ(θ)oftheform
A A (1+2θ/π) for θ [ π,0],
ϕ(θ)= 1 2 ∈ − (11)
(cid:10)A1A2(1−2θ/π) for θ∈[0,π].
andblockdiagramsinFig.2andFig.3.
Theorem2. (Leonov,2006)
Ifconditions(3)–(5)aresatisfiedandthecharacteristicofphasedetector ϕ(θ)hastheform(11),then
forthesameinitialdataoffilterthefollowingrelation
G(t) g(t) C δ, t [0,T]
| − |≤ 3 ∀ ∈
issatisfied.HereC isacertainnumberbeingindependentofδ.
3
ProofofTheorem2
Inthiscasewehave
g(t) G(t)=
−
t
= γ(t s) A A sign sin ω (s)s+ψ sin ω (s)s+ψ
− 1 2 1 1 2 2 −
(cid:3) (cid:4) (cid:5) (cid:8)
0 (cid:6) (cid:7) (cid:6) (cid:7)
ϕ ω (s)s ω (s)s+ψ ψ ds.
− 1 − 2 1− 2
(cid:9)
(cid:6) (cid:7)
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Partitioningtheinterval[0,t]intotheintervals[kδ,(k+1)δ]andmakinguseofassumptions
(5)and(10),wereplacetheaboveintegralwiththefollowingsum
(k+1)δ
m
∑ γ(t kδ) A A sign cos ω (kδ)ω (kδ) kδ+ψ ψ
− 1 2 1 2 1− 2 −
k=0 (cid:4) (cid:3) (cid:4) (cid:5) (cid:8)
kδ (cid:6) (cid:7)
cos ω (kδ)+ω (kδ) s+ψ +ψ ds
− 1 2 1 2 −
(cid:5) (cid:8)(cid:9)
(cid:6) (cid:7)
ϕ ω (kδ) ω (kδ) kδ+ψ ψ δ .
− 1 − 2 1− 2
(cid:5) (cid:8) (cid:9)
(cid:6) (cid:7)
Thenumbermischoseninsuchawaythatt [mδ,(m+1)δ].Since(ω (kδ)+ω (kδ))δ 1,
∈ 1 2 ≫
therelation
(k+1)δ
A A sign cos ω (kδ) ω (kδ) kδ+ψ ψ
1 2 1 − 2 1− 2 −
(cid:3) (cid:4) (cid:5) (cid:8)
kδ (cid:6) (cid:7)
(12)
cos ω (kδ)+ω (kδ) s+ψ +ψ ds
− 1 2 1 2 ≈
(cid:5) (cid:8)(cid:9)
(cid:6) (cid:7)
ϕ ω (kδ) ω (kδ) kδ+ψ ψ δ,
≈ 1 − 2 1− 2
(cid:5) (cid:8)
(cid:6) (cid:7)
issatisfied.Hereweusetherelation
(k+1)δ
A A sign cosα cos ωs+ψ ds ϕ(α)δ
1 2 − 0 ≈
(cid:3)
kδ (cid:11) (cid:6) (cid:7)(cid:12)
forωδ 1,α [ π,π],ψ R1.
Thus,T≫heorem∈2−iscomple0te∈lyproved.(cid:2)
Theorem2isabaseforthesynthesisofPLLwithimpulseoscillators. Fortheimpulseclock
oscillatorsitpermitsonetoconsidertwoblockdiagramssimultaneously:onthelevelofelec-
tronicrealization(Fig.1)andonthelevelofphaserelations(Fig.4),wheregeneralprinciples
ofthetheoryofphasesynchronizationcanbeused(Leonov&Seledzhi,2005b;Kuznetsovet
al.,2006;Kuznetsovetal.,2007;Kuznetsovetal.,2008;Leonov,2008).
3. DifferentialequationsofPLL
LetusmakearemarknecessaryforderivationofdifferentialequationsofPLL.
Consideraquantity
θ˙ (t)=ω (t)+ω˙ (t)t.
j j j
Forthewell-synthesizedPLLsuchthatitpossessesthepropertyofglobalstability,wehave
exponentialdampingofthequantityω˙ (t):
j
|ω˙j(t)|≤Ce−αt.
Here C and α are certain positive numbers being independent of t. Therefore, the quantity
ω˙ (t)t is, asarule, sufficientlysmallwithrespecttothenumber R (seeconditions(3)–(5)).
j
From the above we can conclude that the following approximate relation θ˙ (t) ω (t) is
j ≈ j
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valid. InderivingthedifferentialequationsofthisPLL,wemakeuseofablockdiagramin
Fig.4andexactequality
θ˙ (t)=ω (t). (13)
j j
Notethat,byassumption,thecontrollawoftunableoscillatorsislinear:
ω (t)=ω (0)+LG(t). (14)
2 2
Hereω (0)istheinitialfrequencyoftunableoscillator, Lisacertainnumber,andG(t)isa
2
controlsignal,whichisafilteroutput(Fig.4).Thus,theequationofPLLisasfollows
t
θ˙ (t)=ω (0)+L α (t)+ γ(t τ)ϕ θ (τ) θ (τ) dτ .
2 2 0 − 1 − 2
(cid:5) (cid:3) (cid:8)
0 (cid:6) (cid:7)
Assumingthatthemasteroscillatorissuchthatω (t) ω (0),weobtainthefollowingrela-
1 ≡ 1
tionsforPLL
t
θ1(t)−θ2(t) ′+L α0(t)+ γ(t−τ)ϕ θ1(τ)−θ2(τ) dτ =ω1(0)−ω2(0). (15)
(cid:5) (cid:3) (cid:8)
(cid:6) (cid:7) 0 (cid:6) (cid:7)
ThisisanequationofstandardPLL.Note,thatifthefilter(2)isintegratedwiththetransfer
functionW(p)=(p+α)−1
σ˙ +ασ= ϕ(θ)
thenforφ(θ)=cos(θ)insteadofequation(15)from(13)and(14)wehave
θ¨˜+αθ˜˙+Lsinθ˜=α ω (0) ω (0) (16)
1 − 2
π (cid:6) (cid:7)
withθ˜=θ θ + . So,ifherephasesoftheinputandoutputsignalsmutuallyshiftedby
1− 2 2
π/2thenthecontrolsignalG(t)equalszero.
Arguingasabove,wecanconcludethatinPLLitcanbeusedthefilterswithtransferfunctions
ofmoregeneralform
K(p)=a+W(p),
whereaisacertainnumber,W(p)isaproperfractionalrationalfunction.Inthiscaseinplace
ofequation(15)wehave
t
θ1(t)−θ2(t) ′+L aϕ θ1(t)−θ2(t) +α0(t)+ γ(t−τ)ϕ θ1(τ)−θ2(τ) dτ = (17)
(cid:4) (cid:3) (cid:9)
(cid:6) (cid:7) (cid:6) (cid:7) 0 (cid:6) (cid:7)
=ω (0) ω (0).
1 − 2
Inthecasewhenthetransferfunctionofthefiltera+W(p)isnon-degenerate,i.e. itsnumer-
atoranddenominatordonothavecommonroots,equation(17)isequivalenttothefollowing
systemofdifferentialequations
z˙ = Az+bψ(σ)
(18)
σ˙ =c∗z+ρψ(σ).
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Hereσ=θ θ ,Aisaconstant(n n)-matrix,bandcareconstant(n)-vectors,ρisanumber,
1− 2 ×
andψ(σ)is2π-periodicfunction,satisfyingtherelations:
ρ= aL,
−
W(p)=L−1c∗(A pI)−1b,
−
ω (0) ω (0)
ψ(σ)= ϕ(σ) 1 − 2 .
− L(a+W(0))
Thediscretephase-lockedloopsobeysimilarequations
z(t+1)= Az(t)+bψ(σ(t))
(19)
σ(t+1)=σ(t)+c∗z(t)+ρψ(σ(t)),
wheret Z, Z isasetofintegers. Equations(18)and(19)describetheso-calledstandard
∈
PLLs (Shakhgil’dyan & Lyakhovkin, 1972; Leonov, 2001). Note that there exist many other
modificationsofPLLsandsomeofthemareconsideredbelow.
4. MathematicalanalysismethodsofPLL
Thetheoryofphasesynchronizationwasdevelopedinthesecondhalfofthelastcenturyon
thebasisofthreeappliedtheories:theoryofsynchronousandinductionelectricalmotors,the-
oryofauto-synchronizationoftheunbalancedrotors,theoryofphase-lockedloops. Itsmain
principleisinconsiderationoftheproblemofphasesynchronizationatthreelevels:(i)atthe
levelofmechanical,electromechanical,orelectronicmodels,(ii)atthelevelofphaserelations,
and(iii)atthelevelofdifferential,difference,integral,andintegro-differentialequations. In
thiscasethedifferenceofoscillationphasesistransformedintothecontrolaction, realizing
synchronization. Thesegeneralprinciplesgaveimpetustocreationofuniversalmethodsfor
studying the phase synchronization systems. Modification of the direct Lyapunov method
with the construction of periodic Lyapunov-like functions, the method of positively invari-
ant cone grids, and the method of nonlocal reduction turned out to be most effective. The
last method, which combines theelements of thedirect Lyapunov method and thebifurca-
tion theory, allows one to extend the classical results of F. Tricomi and his progenies to the
multidimensionaldynamicalsystems.
4.1 MethodofperiodicLyapunovfunctions
HereweformulatetheextensionoftheBarbashin–Krasovskiitheoremtodynamicalsystems
withacylindricalphasespace(Barbashin&Krasovskii,1952). Consideradifferentialinclu-
sion
x˙ f(x), x Rn, t R1, (20)
∈ ∈ ∈
where f(x)isasemicontinuousvectorfunctionwhosevaluesaretheboundedclosedconvex
set f(x) Rn. Here Rn isann-dimensionalEuclideanspace. Recallthebasicdefinitionsof
⊂
thetheoryofdifferentialinclusions.
Definition1. WesaythatUε(Ω)isanε-neighbourhoodofthesetΩif
Uε(Ω)= x inf x y <ε ,
{ | y Ω| − | }
∈
where isanEuclideannorminRn.
|·|
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(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:4)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:4)(cid:8)(cid:5)(cid:12)(cid:13)(cid:6)(cid:13)(cid:10)(cid:8)(cid:4)(cid:14)(cid:10)(cid:15)(cid:7)(cid:13)(cid:6)(cid:16)(cid:4)(cid:10)(cid:3)(cid:17)(cid:10)(cid:18)(cid:19)(cid:8)(cid:13)(cid:7)(cid:20)(cid:21)(cid:3)(cid:22)(cid:23)(cid:7)(cid:14)(cid:10)(cid:21)(cid:3)(cid:3)(cid:24)(cid:13) (cid:26)+
Definition2. Afunction f(x)iscalledsemicontinuousatapoint x ifforanyε > 0thereexistsa
numberδ(x,ε)>0suchthatthefollowingcontainmentholds:
f(y)∈Uε(f(x)), ∀y∈Uδ(x).
Definition 3. A vector function x(t) is called a solution of differential inclusion if it is absolutely
continuousandforthevaluesoft,atwhichthederivativex˙(t)exists,theinclusion
x˙(t) f(x(t))
∈
holds.
Under the above assumptions on the function f(x), the theorem on the existence and con-
tinuabilityofsolutionofdifferentialinclusion(20)isvalid(Yakubovichetal.,2004). Nowwe
assumethatthelinearlyindependentvectorsd1,...,dmsatisfythefollowingrelations:
f(x+d )= f(x), x Rn. (21)
j ∀ ∈
Usually, d∗jx is called the phase or angular coordinate of system (20). Since property (21)
allows us to introduce a cylindrical phase space (Yakubovich et al., 2004), system (20) with
property(21)isoftencalledasystemwithcylindricalphasespace.
Thefollowingtheoremisanextensionofthewell–knownBarbashin–Krasovskiitheoremto
differentialinclusionswithacylindricalphasespace.
Theorem3. SupposethatthereexistsacontinuousfunctionV(x):Rn R1suchthatthefollowing
→
conditionshold:
1)V(x+d )=V(x), x Rn, j=1,...,m;
j ∀ ∈ ∀
m
2)V(x)+ ∑(d∗jx)2 →∞as|x|→∞;
j=1
3)foranysolutionx(t)ofinclusion(20)thefunctionV(x(t))isnonincreasing;
4)ifV(x(t)) V(x(0)),thenx(t)isanequilibriumstate.
≡
Thenanysolutionofinclusion(20)tendstostationarysetast +∞.
→
RecallthatthetendencyofsolutiontothestationarysetΛastmeansthat
lim inf z x(t) =0.
t +∞z Λ| − |
→ ∈
AproofofTheorem3canbefoundin(Yakubovichetal.,2004).
4.2 Methodofpositivelyinvariantconegrids.Ananalogofcircularcriterion
Thismethodwasproposedindependentlyintheworks(Leonov,1974;Noldus,1977).Itissuf-
ficientlyuniversaland"fine"inthesensethathereonlytwopropertiesofsystemareusedsuch
astheavailabilityofpositivelyinvariantone-dimensionalquadraticconeandtheinvariance
offieldofsystem(20)undershiftsbythevectord (see(21)).
j
Hereweconsiderthismethodformoregeneralnonautonomouscase
x˙ =F(t,x), x Rn, t R1,
∈ ∈
wheretheidentitiesF(t,x+d ) = F(t,x)arevalid x Rn, t R1 forthelinearlyinde-
pendentvectorsd Rn (j =j1,...,m). Letx(t) = x∀(t,t∈,x )is∀a∈solutionofthesystemsuch
j ∈ 0 0
thatx(t ,t ,x )=x .
0 0 0 0
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(cid:26)(cid:25) (cid:11)(cid:27)(cid:28)(cid:29)(cid:30)(cid:11)(cid:28)(cid:31)(cid:29)(cid:2)(cid:10) (cid:10)!(cid:29)(cid:2)(cid:28)"(cid:29)(cid:21)(cid:10)(cid:20)(cid:10)(cid:28)(cid:19)(cid:7)(cid:3)(cid:9)(cid:12)(cid:10)(cid:8)(cid:4)(cid:14)(cid:10)(cid:18)(cid:9)(cid:8)(cid:22)#(cid:6)(cid:22)(cid:7)
WeassumethatsuchaconeoftheformΩ = x∗Hx 0 ,whereHisasymmetricalmatrix
{ ≤ }
suchthatoneeigenvalueisnegativeandalltherestarepositive,ispositivelyinvariant. The
lattermeansthatontheboundaryofcone∂Ω= xHx=0 therelation
{ }
V˙(x(t))<0
issatisfiedforallx(t)suchthat x(t)=0,x(t) ∂Ω (Fig.5).
{ (cid:13) ∈ }
Fig.5.Positivelyinvariantcone.
Bythesecondproperty,namelytheinvarianceofvectorfieldundershiftbythevectorskd,
j
k Z,wemultiplytheconeinthefollowingway
∈
Ω = (x kd )H(x kd ) 0 .
k { − j − j ≤ }
Since it is evident that for the cones Ω the property of positive invariance holds true, we
k
obtainapositivelyinvariantconegridshowninFig.6.Ascanbeseenfromthisfigure,allthe
Fig.6.Positivelyinvariantconegrid.
solutionsx(t,t ,x )ofsystem,havingthesetwoproperties,areboundedon[t ,+∞).
0 0 0
IftheconeΩhasonlyonepointofintersectionwiththehyperplane{d∗jx = 0}andallsolu-
tionsx(t),forwhichatthetimettheinequality
x(t)∗Hx(t) 0
≥
issatisfied,havepropertyV˙(x(t)) ε x(t)2(hereεisapositivenumber),thenfromFig.6it
≤− | |
isclearthatthesystemisLagrangestable(allsolutionsareboundedontheinterval[0,+∞)).
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