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Nonlinear Analysis and Design
of
Phase-locked loops (PLL) & Costas loop
Nikolay V. Kuznetsov Gennady A. Leonov
Department of Applied CyberneticsFaculty of Mathematics and Mechanics
Saint-Petersburg State Universityhttp://www.math.spbu.ru/user/nk/
http://www.math.spbu.ru/user/nk/Nonlinear_analysis_of_PLL.htm
tutorial last version:
http://www.math.spbu.ru/user/nk/PDF/Nonlinear-analysis-Phase-locked-loop-PLL.pdf
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 1 / 25
Phase-locked loops (PLL): history
I Radio and TVI de Bellescize H., La reception synchrone, L'Onde Electrique, 11, 1932I Wendt, K. & Fredentall, G. Automatic frequency and phase control of
synchronization in TV receivers, Proceedings IRE, 31(1), 1943
I Computer architectures (frequency multiplication)Ian Young, PLL in a microprocessor i486DX2-50 (1992)
(in Turbo regime stable operation was not guaranteed)
I Theory and TechnologyX F.M.Gardner, Phase-Lock Techniques, 1966X A.J. Viterbi, Principles of Coherent Comm., 1966X W.C.Lindsey, Synch. Syst. in Comm. and C., 1972X W.F.Egan, Freq. Synthesis by Phase Lock, 2000X B. Razavi, Phase-Locking in High-Perf. Syst., 2003X R.Best, PLL: Design, Simulation and Appl., 2003
X V.Kroupa, Phase Lock Loops and Freq. Synthesis, 2003
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 2 / 25
PLL in Computer architecturesClock Skew Elimination:multiprocessor systems
Frequencies synthesis: multicoreprocessors, motherboards
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 3 / 25
PLL analysis & design
PLL operation: generation of electrical signal (voltage), phase of whichis automatically tuned to phase of input reference signal (after transientprocess the signal, controlling the frequency of tunable osc., is constant)
0.5(sin(ωt+θ(t)))+sin(ωt-θ(t)))
cos(θ(t))Filter
VCO
11 2
2
REF
21
g(t)
0.5sin(ωt+ωt))
cos(ωt)
1 1 1
const1
REF
VCO
Filter
Design: Signals class (sinusoidal,impulse...), PLL type (PLL,ADPLL,DPLL...)
Analysis: Choose PLL parameters (VCO, PD, Filter etc.) to achievestable operation for the desired range of frequencies and transient time
Analysis methods: simulation, linear analysis, nonlinear analysisof mathematical models in signal space and phase-frequency space
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 4 / 25
PLL analysis & design: simulation
D.Abramovitch, ACC-2008 plenary lecture:Full simulation of PLL in space of signals isvery dicult since one has to observesimultaneously very fast time scale of inputsignals and slow time scale of signal's phases.How to construct model of signal's phases?
Simulation in space of phases: Could stableoperation be guaranteed for all possibleinputs, internal blocks states by simulation?N.Gubar' (1961), hidden oscillation in PLL:global stability in simulation, but onlybounded region of stability in reality
Control signal in signals space
Control signal in phase-freq. space
η
σσ2
σ − 2π 2
σ − 4π 2
σ − 6π 2
Hidden oscillaton in PLL model
Survey: Leonov G.A., Kuznetsov N.V., Hidden attractors in dynamical systems. From hiddenoscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor inChua circuits, International Journal of Bifurcation and Chaos, 23(1), 2013, art. no. 1330002
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 5 / 25
PLL analysis & design: analytical linear methods0.5(sin(θ (t)-θ (t))+sin(θ (t)+θ (t)))
cos(θ(t))
1 2
2
21
g(t)
Filter
VCO
PDsin(θ (t)
1
θ2
F(s) g
θ1
+0.5 θ=(θ -θ )
1 2
-0.5
VCO
Linearization: while this is useful for studying loops that are near lock, it does
not help for analyzing the loop when misphasing is large.⊗−PD : sin(θ1) cos(θ2) = 0.5 sin(θ1+θ2)+sin(θ1+θ2) ≈ 0.5(θ1+θ2)
VCO : θ2(t) = ωfree + Lg(t)
Linearization justication problems: Harmonic linearization (describingfunction method), Aizerman & Kalman conjectures,
Time-varying linearization & Perron eects of Lyapunov exponent sign inversions
Survey: V.O. Bragin, V.I. Vagaitsev, N.V. Kuznetsov, G.A. Leonov, Algorithms for nding hiddenoscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits, J. ofComputer and Systems Sciences Int., 50(4), 2011, 511-544 (doi:10.1134/S106423071104006X)Survey: G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron eects, Int. J.of Bifurcation and Chaos, Vol. 17, No. 4, 2007, 1079-1107 (doi:10.1142/S0218127407017732)
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 6 / 25
Classical PLL:models in time&phase-freq. domains
?) Control signals in signals and phase-frequency spaces are equal.?) Qualitative behaviors of signals and phase-freq. models are the same.
REF Filter
VCO
free
⇔
REF PDFilter
VCO
free
θj≥ωmin, |θ1−θ2|≤∆ω, |θj(t)−θj(τ)|≤∆Ω, |τ−t|≤δ, ∀τ, t∈ [0, T ] (∗)waveforms: fp(θ) =
∑∞i=1
(api sin(iθ) + bpi cos(iθ)
), p = 1, 2
bpi = 1π
∫ π−π f
p(x) sin(ix)dx, api = 1π
∫ π−π f
p(x) cos(ix)dx,
Thm. If (*), ϕ(θ)= 12
∑∞l=1
((a1l a
2l +b
1l b
2l ) cos(lθ)+(a1l b
2l −b1l a2l ) sin(lθ)
)then |G(t)− g(t)| = O(δ), ∀ t ∈ [0, T ]
Leonov G.A., Kuznetsov N.V., Yuldahsev M.V., Yuldashev R.V., Analytical method forcomputation of phase-detector characteristic, IEEE Transactions on Circuits and Systems Part II,vol. 59, num. 10, 2012 (doi:10.1109/TCSII.2012.2213362)
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 7 / 25
PD characteristics examples
f 1,2(θ) ϕ(θ)
-1-0.5
0 0.5
1 1.5
2
0 2 4 6 8 10 12 14
-1-0.5
0 0.5
1 1.5
2
0 2 4 6 8 10 12 14
f 1,2(θ) ϕ(θ)
-1-0.5
0 0.5
1 1.5
2
0 2 4 6 8 10 12 14
-1-0.5
0 0.5
1 1.5
2
0 2 4 6 8 10 12 14
f 1,2(θ) ϕ(θ)
-1-0.5
0 0.5
1 1.5
2
0 2 4 6 8 10 12 14
-1-0.5
0 0.5
1 1.5
2
0 2 4 6 8 10 12 14
REF Filter
VCO
free
I f 1,2 waveforms
REF PDFilter
VCO
free
I ϕ PD characteristic
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 8 / 25
Dierential equations of PLL
REF Filter
VCO
free
⇔
REF PDFilter
VCO
free
I master oscillator phase: θ1(t) ≡ ω1
I phase & freq. dierence: θd(t) = θ1(t)− θ2(t), ω2 − ω1 = ∆ω
Nonautonomous equationsx = Ax+ bf1(ω1t)f2(θd + ω1t),
θd = (ω2 − ω1) + Lc∗x,x(0) = x0, θd(0) = θ0
⇔
Autonomous equationsx = Ax+ bϕ(θd),
θd = ∆ω + Lc∗x,x(0) = x0, θd(0) = θ0
?) Qualitative behaviors of signals and phase-freq. models are the same.
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 9 / 25
Autonomous dierential equations of PLL
?) Qualitative behaviors of signals and phase-freq. models are the same.
z = Az + bf 1(ω1t)f 2(ηd + ω1t),ηd = ωd + Lc∗z
⇔x = Ax+ bϕ(θd)
θd = ωd + Lc∗x
If ω1(t) ≡ ω1 then averaging method [Bogolubov, Krylov] allows oneto justify passing to autonomous di. equations of PLL.
τ = ω1t, ω1 dudτ
= Au+ bf 1(τ)f 2(ηd + τ),
u(τ) = z(τω1
)= z(t), ω1 dηd
dτ= ωd + Lc∗u.
Aver. method requires to prove 1T
T∫0
f 1(τ)f 2(ηd + τ)− ϕ(ηd)dt ⇒T→∞
0
(it is satised for phase detector characteristic ϕ computed above).Solutions (z(t), ηd(t)) & (x(t), θd(t)) (with the same initial data)) ofnonautonomous and autonomous models are close to each other on acertain time interval as ω1 →∞.
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 10 / 25
Nonlinear analysis of PLL
Continuousx = Ax+ bϕ(θd),
θd = ∆ω + Lc∗x,x(0) = x0, θd(0) = θ0
Discretex(t+ 1) = Ax(t) + bϕ(θd(t)),
θd(t+ 1) = θd(t) + ∆ω + Lc∗x(t),
x(0) = x0, θd(0) = θ0
Here it is possible to apply various well developed methods ofmathematical theory of phase synchronization
•Nonlinear Analysis and Design of Phase-Locked Loops, G.A. Leonov, N.V. Kuznetsov, S.M.Seledzhi, (chapter in "Automation control - Theory and Practice", In-Tech, 2009), pp. 89-114• V. Yakubovich, G. Leonov, A. Gelig, Stability of Systems with Discontinuous Nonlinearities,(Singapore: World Scientic), 2004• G. Leonov, D. Ponomarenko, V. Smirnova, Frequency-Domain Methods for Nonlinear Analysis.Theory and Applications (Singapore: World Scientic, 1996).• G. Leonov, V. Reitmann, V. Smirnova, Nonlocal Methods for Pendulum-Like Feedback Systems(Stuttgart; Leipzig: Teubner Verlagsgesselschaft, 1992).
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 11 / 25
PLL simulation: Matlab Simulink
Filter
1
s+1
PD
In 1
In 2Out1
VCO
In 1Out1
ScopeIntegrator
1
s
REF
frequency
100
0.2 seconds-1
-0.5
0
0.5
1
1.5
2
0 2 4 6 8 10 12 14
Filter
1
s+1
In 1Out1
ScopeMultiplication
Interpreted
MATLAB FcnIntegrator
1
s
REF
frequency
100
VCO
20 seconds-1
-0.5
0
0.5
1
1.5
2
0 2 4 6 8 10 12 14
Kuznetsov,Leonov et al., Patent 2449463, 2011Kuznetsov,Leonov et al., Patent 112555, 2011
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 12 / 25
Costas loop: digital signal demodulation
John P. Costas. General Electric, 1950s.I signal demodulation in digital communication
1
2(
1
2(
)
)
VCO Filter
=
m(t) = ±1 data,m(t) sin(2ωt) and m(t) cos(2ωt) can be ltered out,m(t) cos(0) = m(t),m(t) sin(0) = 0
I wireless receivers
I Global Positioning System(GPS)
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 13 / 25
Costas Loop:PD characteristic computationAnalysis of Costas Loop can reducedto the analysis of PLL:u1(θ1(t))=f 1(θ1(t))f 1(θ1(t)),u2(θ2(t))=f 2(θ2(t))f 2(θ2(t)− π
2),
Costas Loop PD characteristic:Akl , B
kl Fourier coecients of uk(θ)
1
2(
1
2(
)
)
VCO Filter
=
Filter
⇔PD Filter
Theorem. If (1)(2) and
ϕ(θ)=A0
1A02
4+ 1
2
∞∑l=1
((Al1A
l2+Bl
1Bl2) cos(lθ)+(Al1B
l2−Bl
1Al2) sin(lθ)
)then |G(t)− g(t)| ≤ Cδ, ∀ t ∈ [0, T ]
G.A. Leonov, N.V. Kuznetsov, M.V. Yuldashev, R.V. Yuldashev, Dierential equations of CostasLoop, Doklady Mathematics, 2012, 86(2), 149-154 (doi:10.1134/S1064562412050080)
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 14 / 25
Costas loop:PD characteristic example
f 1,2(θ)
-1-0.5
0 0.5
1 1.5
2
0 2 4 6 8 10 12 14
ϕ(θ)
-0.15-0.1
-0.05 0
0.05 0.1
0.15
0 2 4 6 8 10 12 14
f 1,2(θ) = − 2π
∞∑n=1
1n
sin(nθ)
ϕ(θ)=− 1
72+
1
2
∞∑l=1
16π4 l4
cos(lθ), l = 4p, p ∈ N0
− 8π3 l3
sin(lθ), l = 4p+ 2, p ∈ N0
−4(π l−2)π4 l4
cos(lθ)− 4(π l−2)π4 l4
sin(lθ), l = 4p+ 1, p ∈ N04(π l+2)π4 l4
cos(lθ)− 4(π l+2)π4 l4
sin(lθ), l = 4p+ 3, p ∈ N0
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 15 / 25
Costas loop simulation: Matlab Simulink
30 c
0.3 c-1
-0.5
0
0.5
1
1.5
2
0 2 4 6 8 10 12 14
REF
frequency
−C−
NCOIntegrator
1
s
T
1 - exp(T)z-1
90 о
Filter
Triangular
waveform
Multiplication
MultiplicationMultiplication
1
s+1
In1
In2Out1
In1Out1
Scope
1
s
REF
frequency
-c-
Integrator
Phase Detector Filter
VCO
-1
-0.5
0
0.5
1
1.5
2
0 2 4 6 8 10 12 14
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 16 / 25
Squaring loop
PLL-based carrier recovery circuitGolomb et al., 1963; Stier, 1964; Lindsey, 1966
I wireless recievers
I demodulation
Squarer s s
2
m(t)sin(ωt) sin(ωt) 0.5cos(2ωt)
sin(θ(t))
I m(t) = ±1 data,
I (m(t)sin(ω1t))2 = 1−cos(2ωt)2
squaring allows to remove data
I 0.5 cos(2ωt) lter removes the constant
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 17 / 25
Classical DPLL: nonlinear analysis
sin(ωint+θ0)
Gain
tk =tk-1+(t0-yk)
sin(ωintk+θ0)
yk=G sin(ωintk+θ0)
φk+1−(φk+2π)==2π(ωin
ω−1)− ωin
ωωG sin(φk)
ωin = ω, φk=ωtk+θ0,φk + 2π → σk ∈ [π, π]σt+1 =σt− r sinσt, r=ωG
r1 = 2: Global asymptotic stability disappears andappears unique and globally stable on (−π, 0)∪(0,−π) cycle with period 2
r2 = π: Bifurcation of splitting: unique cycle loses stabilityand two locally stable asymmetrical cycles with period 2 appear
r3 = Two locally stable asymmetrical cycles with period 2 lose stability√π2+2 : and two locally stable asymmetrical cycles with period 4 appear
G.A. Leonov, N.V. Kuznetsov, S.M. Seledzhi, Nonlinear Analysis and Design of Phase-LockedLoops, in 'Automation control - Theory and Practice, In-Tech, 2009, 89-114E.V. Kudryashova, N.V. Kuznetsov, G.A. Leonov, P. Neittaanmaki, S.M. Seledzhi, Analysis andsynthesis of clock generator, in From Physics to Control Through an Emergent View, WorldScientic, 2010, 131-136 (doi:10.1142/9789814313155_0019)
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 18 / 25
Classical DPLL: simulation of transition to chaos
j Bifurcation value Feigenbaum const
of parameter, rj δj =rj−rj−1
rj+1−rj
1 22 π 3.75973373263 3.445229223301312 4.48746758424 3.512892457411257 4.62404520675 3.527525366711579 4.66014783206 3.530665376391086 4.66717650897 3.531338162105000 4.66876798838 3.531482265584890 4.66907465829 3.531513128976555 4.669111696510 3.531519739097210 4.669025736511 3.531521154835959
E.V. Kudryashova, N.V. Kuznetsov, G.A. Leonov,P. Neittaanmaki, S.M. Seledzhi, Analysis andsynthesis of clock generator, in From Physics toControl Through an Emergent View, World Sci.,2010, 131-136 (doi:10.1142/9789814313155_0019)
Feigenbaum const for unimodal maps:= limn→+∞
δn=4.669..
σ(t+1)=σ(t)−r sinσ(t)
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
x
y
y=xy=f(x)y=f(f(x))
± xm, f(± xm)
f(xm), f(f(xm))
f(f(xm)),f(f(f(xm)))
f(x) = x− 2.2 sin(x)nonunimodal map
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 19 / 25
Publications: PLL and Costas loop, 2012-2011• Leonov G.A., Kuznetsov N.V., M.V. Yuldashev, R.V. Yuldashev, Dierential equations of CostasLoop, Doklady Mathematics, 86(2), 2012, 723-728 (doi: 10.1134/S1064562412050080)• Kuznetsov N.V., Leonov G.A., Neittaanmaki P., Seledzhi S.M., Yuldashev M.V., Yuldashev R.V.,Simulation of phase-locked loops in phase-frequency domain, International Congress on UltraModern Telecommunications and Control Systems and Workshops, 2012, IEEE art. no. 6459692,351-356 (doi:10.1109/ICUMT.2012.6459692)• Kuznetsov N.V., Leonov G.A., Neittaanmaki P., Seledzhi S.M., Yuldashev M.V., Yuldashev R.V.,Nonlinear mathematical models of Costas Loop for general waveform of input signal, IEEE 4thInternational Conference on Nonlinear Science and Complexity, NSC 2012 - Proceedings, art. no.6304729, 2012, 75-80 (doi:10.1109/NSC.2012.6304729)• Leonov G.A., Kuznetsov N.V., Yuldashev M.V., Yuldashev R.V., Analytical method forcomputation of phase-detector characteristic, IEEE Transactions on Circuits and Systems II,59(10), 2012, 633-637 (doi:10.1109/TCSII.2012.2213362)• G.A. Leonov, N.V. Kuznetsov, M.V. Yuldashev, R.V. Yuldashev, Dierential equations of CostasLoop, Doklady Mathematics, 2012, 86(2), 149-154 (doi:10.1134/S1064562412050080)• Kuznetsov N.V., Leonov G.A., Yuldashev M.V., Yuldashev R.V., Nonlinear analysis of Costas loopcircuit, 9th International Conference on Informatics in Control, Automation and Robotics, Vol. 1,2012, 557-560, (doi:10.5220/0003976705570560)• Kuznetsov N.V., Leonov G.A., Neittaanmaki P., Seledzhi S.M., Yuldashev M.V., Yuldashev R.V.,Simulation of Phase-Locked Loops in Phase-Frequency Domain, IEEE IV International Congress onUltra Modern Telecommunications and Control Systems, 2012, 364-368• G.A. Leonov, N.V. Kuznetsov, M.V. Yuldashev, R.V. Yuldashev, Computation of Phase DetectorCharacteristics in Synchronization Systems, Doklady Mathematics, 84(1), 2011, 586-590,(doi:10.1134/S1064562411040223)•N. Kuznetsov, G. Leonov, P. Neittaanmaki, S.Seledzhi, M. Yuldashev, and R. Yuldashev,High-frequency analysis of phase-locked loop and phase detector characteristic computation, 8th Int.Conf. on Informatics in Control, Automation and Robotics, 2011, 272-278,(doi:10.5220/0003522502720278).
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 20 / 25
Publications: PLL and Costas loop, 2011-...• N.V. Kuznetsov, G.A. Leonov, M.V. Yuldashev, R.V. Yuldashev, Analytical methods forcomputation of phase-detector characteristics and PLL design, ISSCS 2011-IEEE InternationalSymposium on Signals, Circuits and Systems, Proceedings, 2011, 7-10,(doi:10.1109/ISSCS.2011.5978639)• G.A. Leonov, S.M. Seledzhi, N.V. Kuznetsov, P. Neittaanmaki, Nonlinear Analysis of Phase-lockedloop, IFAC Proceedings Volumes (IFACPapersOnline), 4(1), 2010(doi:10.3182/20100826-3-TR-4016.00010)• G.A. Leonov, S.M. Seledzhi, N.V. Kuznetsov, P. Neittaanmaki, Asymptotic analysis of phasecontrol system for clocks in multiprocessor arrays, ICINCO 2010 - Proceedings of the 7thInternational Conference on Informatics in Control, Automation and Robotics, 3, 2010, 99-102(doi:10.5220/0002938200990102)• N.V. Kuznetsov, G.A. Leonov, S.M. Seledzhi, Nonlinear analysis of the Costas loop andphase-locked loop with squarer, IASTED Int. conference on Signal and Image Processing, SIP 2009,2009, 1-7• N.V. Kuznetsov, G.A. Leonov, S.M. Seledzhi, P. Neittaanmaki, Analysis and design of computerarchitecture circuits with controllable delay line. ICINCO 2009 - 6th International Conference onInformatics in Control, Automation and Robotics, Proceedings, 3 SPSMC, 2009, 221-224(doi:10.5220/0002205002210224).• N.V. Kuznetsov, G.A. Leonov, S.M. Seledzhi, Phase locked loops design and analysis. ICINCO2008 - 5th International Conference on Informatics in Control, Automation and Robotics,Proceedings, SPSMC, 2008, 114-118 (doi:10.5220/0001485401140118)• N.V. Kuznetsov, G.A. Leonov, S.M. Seledzhi, Analysis of phase-locked systems with discontinuouscharacteristics of the phase detectors, IFAC Proceedings Volumes (IFACPapersOnline), 1 (PART 1),2006, 107-112 (doi:10.3182/20060628-3-FR-3903.00021)•Nonlinear Analysis and Design of Phase-Locked Loops, G.A. Leonov, N.V. Kuznetsov, S.M.Seledzhi, Automation control - Theory and Practice, In-Tech, 2009), 89-114
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 21 / 25
Lyapunov exponent: sign inversions, Perron eects, chaos, linearizationx = F (x), x ∈ Rn, F (x0) = 0
x(t) ≡ x0, A = dF (x)dx
∣∣x=x0
y = Ay + o(y)y(t) ≡ 0, (y=x−x0)
z=Azz(t)≡0
Xstationary: z(t) = 0 is exp. stable ⇒ y(t) = 0 is asympt. stablex = F (x), x(t) = F (x(t)) 6≡ 0
x(t) 6≡ x0, A(t) =dF (x)
dx
∣∣∣∣x=x(t)
y=A(t)y + o(y)y(t) ≡ 0, (y=x−x(t))
z=A(t)zz(t)≡0
? nonstationary: z(t) = 0 is exp. stable ⇒? y(t) = 0 is asympt. stable
! Perron eects:z(t)=0 is exp. stable(unst), y(t)=0 is exp. unstable(st)
Positive largest Lyapunov exponentdoesn't, in general, indicate chaos
Survey: G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron eects,Int. Journal of Bifurcation and Chaos, 17(4), 2007, 1079-1107 (doi:10.1142/S0218127407017732)N.V. Kuznetsov, G.A. Leonov, On stability by the rst approximation for discrete systems, 2005Int. Conf. on Physics and Control, PhysCon 2005, Proc. Vol. 2005, 2005, 596-599
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 22 / 25
Hidden oscillations in control: Aizerman & Kalman problem:Harmonic linearization without justication may lead to wrong results
Harmonic balance & Describing function method in Absolute Stability Theoryx = Px+ qψ(r∗x), ψ(0) = 0 (1) x = P0x+ qϕ(r∗x)
W (p) = r∗(P − pI)−1qImW (iω0) = 0, k = −(ReW (iω0))
−1P0=P + kqr∗, ϕ(σ)=ψ(σ)− kσP0 : λ1,2 =±iω0, Reλj>2<0
DFM: exists periodic solution σ(t) = r∗x(t) ≈ a cosω0t
a :∫ 2π/ω0
0ψ(a cosω0t) cosω0tdt = ka
∫ 2π/ω0
0(cosω0t)
2dtAizerman problem: If (1) is stable for any linear ψ(σ)=µσ, µ∈(µ1, µ2)then (1) is stable for any nonlinear ψ(σ) : µ1σ < ψ(σ) < µ2σ, ∀σ 6= 0DFM: (1) is stable ⇒ k : k<µ1, µ2<k ⇒ kσ2<ψ(σ)σ, ψ(σ)σ<kσ2
⇒ ∀a 6= 0 :∫ 2π/ω0
0(ψ(a cosω0t) a cosω0t− k(a cosω0t)
2)dt 6= 0⇒ no periodic solutions by harmonic linearization and DFM, but
Survey: V.O. Bragin, V.I. Vagaitsev, N.V. Kuznetsov, G.A. Leonov (2011) Algorithms for ndinghidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits,J. of Computer and Systems Sciences Int., V.50, N4, 511-544 (doi:10.1134/S106423071104006X)
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 23 / 25
Hidden attractor in classical Chua's systemIn 2010 the notion of hidden attractor was introduced
hidden chaotic attractor was found in Chua circuits for the rst time by the authors
[Leonov G.A., Kuznetsov N.V., Vagaitsev V.I., Physics Letters A, 375(23), 2011]
x=α(y−x−m1x−ψ(x))y=x−y+z, z=−(βy+γz)ψ(x)=(m0−m1)sat(x)
α=8.4562, β=12.0732γ=0.0052m0 =−0.1768,m1 =−1.1468
equilibria: stable zeroF0 &2 saddlesS1,2
trajectories: 'from'S1,2 tend (black)to zero F0 or tend (red) to innity;
Hidden chaotic attractor (in green)
with positive Lyapunov exponent
−15
−10
−5
0
5
10
15
−5
−4
−3
−2
−1
0
1
2
3
4
5
−15
−10
−5
0
5
10
15
x
y
z
M2unst
M1
unst
Ahidden
S2
S1
M2
stM1
st
F0
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 24 / 25
Kuznetsov N.V., Leonov G.A. Nonlinear analysis and design of phase-locked loop (PLL) & Costas loop 25 / 25
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