Global Analysis of Limit Cycles in Networks of Oscillators ...stan/SlidesNolcos2004_Beamer.pdf ·...

Goals and motivationThe dissipative oscillator

Global analysis of PLSPL Fitzhugh-Nagumo oscillator

Conclusions

Global Analysis of Limit Cycles in Networks ofOscillators

Guy-Bart STAN, Rodolphe SEPULCHRE

Guy-Bart Stan

University of Liege

Nolcos, September, 3, 2004

−0.8−0.6

−0.4−0.2

00.2

0.40.6

−1

−0.5

0

0.5

1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

X

ξ

G.-B. Stan- ULg Nolcos 2004 [1/14]



Conclusions

Outline of the talk

1 Goals and motivation2 The dissipative oscillator3 Global analysis of PLS4 PL Fitzhugh-Nagumo oscillator5 Conclusions




Conclusions

Goals and motivation

Main theme: Global analysis and control of sustainedoscillations (limit cycles) in networks of oscillators.Based upon:

high-dimensional generalizations of the Van der Pol andFitzHugh-Nagumo equations,passivity and related concepts,application of input-output stability theory to the analysis oflimit cycles.

Goal: Definition of a class of oscillators whose orbital stabilityproperties extend directly to networks of oscillators.Motivation:

Bridge the gap between equilibrium points and limit cyclesanalysis tools in nonlinear systems,Control of amplitude, frequency and phase of oscillations innetworks of oscillators.

Applications:

robotics: rhythmic tasks robots (i.e. walking robots, jugglingrobots, dexterous robots),biology: fundamental oscillation mechanism.




Conclusions

CharacterizationAnalytical results for k & k∗

The dissipative oscillator - Characterization

u yO

Internal description: ’global’ limit cycleExternal description: a dissipation inequality, suitable foranalysis of interconnectionsIncludes prototype oscillators as simplest examples (Van derPol, Fitzhugh-Nagumo) but not restricted to low-dimensionalmodels




Conclusions


Analytical results for k & k∗

−k

−

passiveu y

Σ

φk(y)

A parameter (k) controls the negative slope of the staticnonlinearity φk(y) = −ky + φ(y), with φ(·) a stiffeningnonlinearity, satisfying yφ(y) > 0, ∀y 6= 0 (e.g. cubic)Absolute stability for k = 0; equilibrium loses stability atk = k∗ ≥ 0Bifurcation codimension (one or two) dictates one of twoscenarios for global oscillations for k & k∗




Conclusions



−k

−

passiveu y

Σ

φk(y)

Bif. codim. = 2 → Hopf bifurcation : GAS\Es(0) limit cyclefor k & k∗.Bif. codim. = 1 → pitchfork bifurcation, global bistability




Conclusions



y

−−

Σ

R

adaptation

1τs+1

φk(y)

u

global bistability + feedback adaptation loop : GAS\Es(0)limit cycle for k & k∗.




Conclusions



−k

−

passiveu y

Σ

φk(y)

y

−−

Σ

R

adaptation

1τs+1

φk(y)

u

What about a particular value of k > k∗?




Conclusions

Definition of a PL dissipative oscillatorNumerical results for a particular value of k > k∗, n = 2Upper bounds on the switching times, n = 2

Global analysis of PLS

RESULT : global stability analysis of limit cycles of PLS canbe proven using quadratic stability of Poincare maps(Goncalves (2000))HOW : express Poincare maps induced by an LTI flowbetween 2 switching surfaces as linear transformationsanalytically parameterized by a scalar function of the state.IDEA : compute quadratic Lyapunov functions forPoincare maps.The search for Lyapunov functions is done by solving a setof LMIs.




Conclusions


Poincare maps for PLS

T1

limit cycle

T2

S0

3

eq. point

eq. point

v

v

v

vx∗

1

x∗

2

vx∗

0

S1




Conclusions


Individual Poincare maps (Impact maps)

S0

V0(·)

x∗0

∆0

x∗0 + ∆0

x = Ax − By = Cx

T1S1

V1(·)

x∗1 + ∆1

∆1

x∗1

S1 = {x ∈ Rn : Cx + d = 0}

T1 : S0 → S1

x∗

1 = T1(x∗

0 )

or

∆1 = T1(x∗

0 + ∆0) − x∗

1

= H(t)∆0

Quadratic stability of impact maps

r(t) = ∆T0 P(t)∆0

= V0(∆0) − V1(H(t)∆0) > 0, ∀t ∈ [tmin, tmax ]LMIs−−−→ [ t1

︸︷︷︸

=tmin

, t2, . . . , tN︸︷︷︸

=tmax

]




Conclusions


Definition of a PL dissipative oscillator

Σ is a linear systemFeedback static nonlinearity : φk(y) = −ky + y3 → fpls(y)

−1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

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1

1.5

2

y

y3 −k py

Cubic nonlinearity (blue); piecewise linear approximation (red)

−m

m

y

fpls(y)

py − d

py + d

−m

m

km

m =√

k3

d = m(p + k)

NLS: Analytical results for k & k∗ PLS: Numerical results for k > k∗




Conclusions


PL dissipative oscillator numerical analysis

S0 S1

m−m

x∗0

−x∗0

x∗1

y = Cx

−x∗1

(R3)(R1)

∆0

x = A1x

(R2)

x = A2x − dB x = A2x + dB

V0(·)V1(·)

∆1 = H(t)∆0

A1 is not Hurwitz, A2 is Hurwitz, m > −CA−12 dB

1 Existence of limit cycles

f1(t∗

1 , t∗2 ) = Cx∗

0 (t∗1 , t∗2 ) + m = 0

f2(t∗

1 , t∗2 ) = Cx∗

1 (t∗1 , t∗2 ) − m = 0

2 Quadratic stability of impact maps

r(t) = ∆T0 P(t)∆0

= V0(∆0) − V1(H(t)∆0) > 0, ∀t ∈ [tmin, tmax ]LMIs−−−→ [ t1

︸︷︷︸

=tmin

, t2, . . . , tN︸︷︷︸

=tmax

]




Conclusions


Upper bounds on the switching times, n = 2

Generic assumption: A1 has no real unstable eigenvalue.

1 (R2), A1 is not Hurwitz

Generically, A1 has stable and unstable eigenspaces: switching time is unboundedn = 2 : A1 is anti-Hurwitz (⇔ no stable eigenspace): tmax is given by the worst switching scenario(Cx0 = −m, CAx0 = 0)

2 (R1) and (R3): A2 is Hurwitz

guarantee contraction in (0, timax)

prove that contraction holds ∀t > timax




Conclusions

Simulation results

PL Fitzhugh-Nagumo oscillator

−−

G

1τs+1

y

fpls(·)

u

0 2 4 6 8 10 12 14 16 180.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Oscillator with ki=1, k

p=1.200000e+000 and p=5

Min

(eig

(Ri(t

)))

with

impl

emen

tatio

n of

the

coni

c re

latio

n

Time (s)

R1

R2

G (s) =1

s + 1, τ = 20, k∗ = 1 +

1

τ= 1.05 < 1.2 = k

t∗1 = 8.88, t∗2 = 8.4

tmax(R2)= 9




Conclusions

Simulation results

PL Fitzhugh-Nagumo oscillator

−−

G

1τs+1

y

fpls(·)

u

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−0.6

−0.4

−0.2

0

0.2

0.4

0.6

y

ξ

State space for one oscillator with ki=1, k

p=1.200000e+000 and p=5

G (s) =1

s + 1, τ = 20, k∗ = 1 +

1

τ= 1.05 < 1.2 = k

t∗1 = 8.88, t∗2 = 8.4

tmax(R2)= 9




Conclusions

Conclusions

Global analytical results for k & k∗ and n > 2 : dissipativeoscillator;

Global numerical results for a particular value of k > k∗ andfor n = 2 : PL dissipative oscillator;

Other results (not presented): interconnection,synchronization of dissipative oscillators.




Conclusions

That’s all !

Questions...


Global Analysis of Limit Cycles in Networks of Oscillators ...stan/SlidesNolcos2004_Beamer.pdf ·...

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