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