Global Analysis and Synthesis of Introduction Oscillations Why a … · 2006. 10. 10. · Overview...

Global Analysisand Synthesis of

Oscillations

Guy-Bart STAN

IntroductionResults overview

Why a dissipativityapproach?

Global oscillationsfor the passiveoscillatorThe passive oscillator

Global oscillationmechanisms for the passiveoscillator

Global oscillationsfor networks ofpassive oscillatorsExtension of the results forone passive oscillator

Synchronization in networksof identical passiveoscillators

Synthesis ofoscillationsThe cart-pendulum

Conclusions

Global Analysis and Synthesis ofOscillations

A Dissipativity Approach

Guy-Bart STAN

Department of Engineering (Control Group)University of Cambridge

11th of October 2006


Oscillations

Guy-Bart STAN








Conclusions

Oscillations: why is it important?

Oscillation is ubiquitous in nature:

breathing, walking, heart beating, sleeping cycles,seasons, etc.

SARCOMAN

Currently, no general theory for the global analysis orsynthesis of oscillators!


Oscillations

Guy-Bart STAN








Conclusions

Goal of this research

Open avenues towards the development of a generalsystem theory for oscillators

Desired properties:

global results (i.e., independent from initialconditions)dimension independent resultsinterconnection results (complex osc. systems ≡

interconnections of simpler osc. systems)

x1

LimitCycle

x2

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5State space

y

ξ

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Time evolution

Time

y


Oscillations

Guy-Bart STAN








Conclusions

Outline

1 Introduction

2 Global oscillations for the passive oscillator

3 Global oscillations for networks of passive oscillators

4 Synthesis of oscillations in stabilizable systems

5 Conclusions


Oscillations

Guy-Bart STAN








Conclusions

Outline

1 Introduction




5 Conclusions


Oscillations

Guy-Bart STAN








Conclusions

Overview of the results

Results:global analysis of limit cycle oscillations

synthesis of limit cycle oscillations

Approach: dissipativity theoryDissipativity theory ≡ “efficient tool for global analysisand synthesis of limit cycle oscillations”


Oscillations

Guy-Bart STAN








Conclusions

Overview of the results

Results:global analysis of limit cycle oscillations

high-dimensional models of oscillatorsnetworks of oscillatorssynchronization in networks of identical oscillators

synthesis of limit cycle oscillations

simple method for generating oscillations instabilizable systems

Approach: dissipativity theoryDissipativity theory ≡ “efficient tool for global analysisand synthesis of limit cycle oscillations”


Oscillations

Guy-Bart STAN








Conclusions

Why a dissipativity approach? (1)

Dissipativity theory ≡ Stability theory for open systems(WILLEMS, 1972)

Syst.

u y

System is dissipative if there exists a storage fcnS(x) ≥ 0 and a supply rate w(u, y) such that

S ≤ w(u, y)

(passivity: S ≤ uT y )


Oscillations

Guy-Bart STAN








Conclusions

Why a dissipativity approach? (2)

Dissipativity has increasingly proved useful as a nonlineartool for

stability analysis of equilibrium points of opensystems

stabilization of open systems

Its advantages:

global results (i.e., independent from initialconditions)

dimension independent results

interconnection theory (complex diss. systems ≡

interconnections of simpler diss. systems)


Oscillations

Guy-Bart STAN








Conclusions

Outline

1 Introduction




5 Conclusions


Oscillations

Guy-Bart STAN








Conclusions

The passive oscillator

φk(y)

yPassive

static nonlinearity

−k

−

u

+φ(y)= −ky

x1

LimitCycle

x2

Includes two well-known low-dimensional oscillators:VAN DER POL and FITZHUGH-NAGUMO

Characterization by a specific dissipation inequality:

S︸︷︷︸

storage variation

≤(k − k∗

passive

)y2

︸︷︷︸

local activation

−

>0︷︸︸︷

yφ(y)︸︷︷︸

global dissipation

+ uy︸︷︷︸

ext. supply


Oscillations

Guy-Bart STAN








Conclusions

Results on this class of systems

φk(y)

yPassive

static nonlinearity

−k

−

Stable Unstable

Bifurcation

0 k∗

GAS k

Generically two types of bifurcation (HOPF or pitchfork)


Oscillations

Guy-Bart STAN








Conclusions

First scenario: HOPF bifurcation (1)

Theorem (1st result)Passivity for k ≤ k∗ and two eigenvalues on theimaginary axis at k = k∗ implies global oscillation throughHOPF bifurcation for k & k∗

Limit cycle

GAS GloballyAttractive

Stable Unstable

at k = k∗

k∗

x1

x2

k


Oscillations

Guy-Bart STAN








Conclusions

First scenario: HOPF bifurcation (2)

A ’basic’ global oscillation mechanism inelectro-mechanical systems

Simplest example: VAN DER POL oscillator:

i = φk(v)

L C

Passive

−−

1s

1s

φk(·)

Global oscillation mechanism:

Continuous lossless exchange of energy betweenthe storage elements

Static nonlinear element regulates the sign of thedissipation


Oscillations

Guy-Bart STAN








Conclusions

HOPF scenario: example

−−

φk(·)

1s

H(s)y

Passive

H(s) = τs+ω2n

s2+2ζωns+ω2n

φk (y) = y3 − ky

State-space (k∗ = 1)

−1

−0.5

0

0.5

1

−0.5

0

0.5

1

1.5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

X1

State−space of a SINGLE oscillator for kp=9.000000e−01

X2

ξ

−1

−0.5

0

0.5

1

−0.5

0

0.5

1

1.5−0.5

0

0.5

1

1.5

X1

State−space of a SINGLE oscillator for kp=1.100000e+00

X2

ξ

k = 0.9 k = 1.1


Oscillations

Guy-Bart STAN








Conclusions

HOPF scenario: example

−−

φk(·)

1s

H(s)y

Passive

H(s) = τs+ω2n

s2+2ζωns+ω2n

φk (y) = y3 − ky


−1

−0.5

0

0.5

1

−0.5

0

0.5

1

1.5−1

−0.5

0

0.5

1

1.5

2

X1

State−space of a SINGLE oscillator for kp=2

X2

ξ

−3−2

−10

12

3

−3

−2

−1

0

1

2

3

−15

−10

−5

0

5

10

15

X1

State−space of a SINGLE oscillator for kp=10

X2

ξ

k = 2 k = 10


Oscillations

Guy-Bart STAN








Conclusions

Second scenario: pitchfork bifurcation (1)Theorem (2nd result)Passivity for k ≤ k∗ and one eigenvalue on the imaginaryaxis at k = k∗ implies global bistability through pitchforkbifurcation for k & k∗

Stable

Eq. point

Unstable

GAS GloballyBistable

at k = k∗

k∗

k


Oscillations

Guy-Bart STAN








Conclusions


k & k∗, without adaptation k & k∗, with adaptation

unstablestable stable

x2

Relaxation Oscillation

x1

x2


Oscillations

Guy-Bart STAN








Conclusions


(Slow) “adaptation” converts the bistable system into aglobal oscillator

φk(·)

− −

1τs+1

Passive

τ ≫ 0


Oscillations

Guy-Bart STAN








Conclusions

Second scenario: pitchfork bifurcation (2)

A ’basic’ global oscillation mechanism in biology

Simplest example: FITZHUGH-NAGUMO oscillator:

insidethe cell the cell

outsideall ions all ions

VE+ E−

Adaptation

Passive

φk(·)

− −

1τs+1

1s

τ ≫ 0

Global oscillation mechanism:

Continuous switch between 2 quasi stableequilibrium points


Oscillations

Guy-Bart STAN








Conclusions

Pitchfork scenario: example

Passive

H(s) y

−

φk(·)

H(s) = τs+ω2n

s2+2ζωns+ω2n

φk (y) = y3 − ky


−1.5 −1 −0.5 0 0.5 1 1.5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

X1

X2

State−space for ki=1 and k

p=9.000000e−01

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1.5

−1

−0.5

0

0.5

1

1.5

X1

X2

State−space for ki=1 and k

p=2

k = 0.9, without adaptation k = 2, without adaptation


Oscillations

Guy-Bart STAN








Conclusions

Pitchfork scenario: example

Passive

H(s) y

−−

φk(·)

1τs+1

τ ≫ 0

H(s) = τs+ω2n

s2+2ζωns+ω2n

φk (y) = y3 − ky

Adaptation


−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−1.5

−1

−0.5

0

0.5

1

1.5

X1

X2

State−space of a SINGLE relaxation oscillator for ki=1 and k

p=2

k = 2, with adaptation


Oscillations

Guy-Bart STAN








Conclusions

Outline

1 Introduction




5 Conclusions


Oscillations

Guy-Bart STAN








Conclusions

Networks of oscillators

In nature, oscillation is the result of interconnectedoscillators!


Oscillations

Guy-Bart STAN








Conclusions

MIMO representation of a network of passiveoscillators

− −

YUW

Passive

(Γ)

P1

PN

Φk (Y )

y1

yN

φk (y1)

φk (yN)

COUPLING

Characterization through dissipativity theory

S ≤(k − k∗

passive

)Y T Y

︸︷︷︸

local activation

−

≥0︷︸︸︷

Y T Φ(Y )︸︷︷︸

global dissipation

−Y T ΓY︸︷︷︸

coupling

+ W T Y︸︷︷︸

ext. supply


Oscillations

Guy-Bart STAN








Conclusions

Global oscillations for networks (1)

Question: “What are the coupling topologies that lead toglobal oscillations in the network?”


Oscillations

Guy-Bart STAN








Conclusions


Question: “What are the coupling topologies that lead toglobal oscillations in the network?”Answer: Passive coupling topologies (Γ ≥ 0)

Characterization (analogue to that for 1 oscillator!)

S ≤(

k − k∗

passive

)

Y T Y︸︷︷︸

local activation

− Y T Φ(Y )︸︷︷︸

global dissipation

+ W T Y︸︷︷︸

ext. supply


Oscillations

Guy-Bart STAN








Conclusions


Question: “What are the coupling topologies that lead toglobal oscillations in the network?”Answer: Passive coupling topologies (Γ ≥ 0)

Characterization (analogue to that for 1 oscillator!)

S ≤(

k − k∗

passive

)

Y T Y︸︷︷︸

local activation

− Y T Φ(Y )︸︷︷︸

global dissipation

+ W T Y︸︷︷︸

ext. supply

Consequence: 1st and 2nd results generalize to networksof passive oscillators


Oscillations

Guy-Bart STAN








Conclusions


Theorem (Extension of 1st result for networks)Passivity for k ≤ k∗ and two eigenvalues on theimaginary axis at k = k∗ implies global oscillation throughHOPF bifurcation for k & k∗

Theorem (Extension of 2nd result for networks)Passivity for k ≤ k∗ and one eigenvalue on the imaginaryaxis at k = k∗ implies global bistability through pitchforkbifurcation for k & k∗

(Slow) adaptation converts the bistable system into arelaxation oscillation


Oscillations

Guy-Bart STAN








Conclusions


Network of identical passive oscillators

If coupling is linear, symmetric (Γ = ΓT ), passive (Γ ≥ 0),and connects all oscillators (rank(Γ) = N − 1), then thebehaviour of the network may be deduced from that ofone of its oscillators


Oscillations

Guy-Bart STAN








Conclusions

Examples

We consider 6= networks of identical passive oscillators

−−

φk(·)

1s

H(s)y

Passiveu H(s) = τs+ω2

ns2+2ζωns+ω2

n

φk (y) = y3 − ky

O1

+1

+1

O2

Conventions:

Each oscillator is represented as a circle

The arrows denote the linear input-outputinterconnection between the oscillators


Oscillations

Guy-Bart STAN








Conclusions

Examples (2 oscillators)

O1

+1

+1

O2

−1−1

O1

−1

−1

O2

−1 −1

Γ =

(1 −1−1 1

)

≥ 0 Γ =

(1 11 1

)

≥ 0

State-space for 2 coupled oscillators

−1−0.5

00.5

11.5

−2

−1

0

1

2

3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

X1

State−space of 2 oscillators for ki=1, k

p=3.000000e−01

X2

ξ

−1.5−1

−0.50

0.51

1.5

−2

−1

0

1

2

3−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

X1

State−space of 2 oscillators for ki=1, k

p=3.000000e−01

X2

ξ

k = 1.3 k = 1.3


Oscillations

Guy-Bart STAN








Conclusions

Examples (2 oscillators)

O1

+1

+1

O2

−1−1

O1

−1

−1

O2

−1 −1

Γ =

(1 −1−1 1

)

≥ 0 Γ =

(1 11 1

)

≥ 0

Time evolution of the 2 outputs

0 5 10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time evolution of the two outputs for ki=1, k

p=3.000000e−01

y1(t)

y2(t)

0 5 10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time evolution of the two outputs for ki=1, k

p=3.000000e−01

y1(t)

y2(t)

k = 1.3 k = 1.3


Oscillations

Guy-Bart STAN








Conclusions

Examples (N oscillators)

Useful for proving global oscillations in networkscomposed of a large number of oscillators with varioustopologies including all-to-all coupling, bidirectional ringcoupling, etc.

What can be said about the relative behaviour of theoscillators? (synchronization?)

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time evolution of the five outputs for kp=2

y1(t)

y2(t)

y3(t)

y4(t)

y5(t)


Oscillations

Guy-Bart STAN








Conclusions

Synchronization and incremental passivityQuestion: Under which conditions do all oscillatorssynchronize?


Oscillations

Guy-Bart STAN








Conclusions


Approach: incremental dissipativityIncremental dissipativity ≡ dissipativity expressed interms of the difference between solutions of systems


Oscillations

Guy-Bart STAN








Conclusions


Approach: incremental dissipativityIncremental dissipativity ≡ dissipativity expressed interms of the difference between solutions of systems

A B

C=A-B

Do A and B synchronize?

Study asymptotic stability of C through dissipativity theory: if Cis asymptotically stable then A and B synchronize

Asymptotic stability of C generally depends on the topology ofthe network


Oscillations

Guy-Bart STAN








Conclusions

Synchronization result

Theorem (3rd result)If

network of identical, incrementally passive oscillatorscharacterized by a global limit cycle oscillation

linear, passive coupling (Γ ≥ 0) such thatΓ1 = ΓT 1 = 0, and rank(Γ) = N − 1 (non symmetricΓ is allowed)

strong coupling (λ2 (Γs) > k − k∗

passive > 0)

Then, the network is characterized by a global limit cycleoscillation, and all oscillators synchronize exponentiallyfast

(in accordance with other results of the literature (POGROMSKY &NIJMEIJER (1998), SLOTINE & WANG (2003))


Oscillations

Guy-Bart STAN








Conclusions

ExamplesThis result is useful to prove global synchronization innetworks of oscillators.

O1 O2

O3O4

O1 O2

O3O4

All-to-all Bidirectional ringO1 O2

O3O4 O1 O2 · · · ON

Unidirectional ring Open chain


Oscillations

Guy-Bart STAN








Conclusions

ExamplesThis result is useful to prove global synchronization innetworks of oscillators.

Time evolution of the outputs

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time evolution of the five outputs for kp=2

y1(t)

y2(t)

y3(t)

y4(t)

y5(t)


Oscillations

Guy-Bart STAN








Conclusions

Outline

1 Introduction




5 Conclusions


Oscillations

Guy-Bart STAN








Conclusions

Synthesis of oscillations

The structure of the passive oscillator suggests a methodfor generating oscillations in passive systems

−

Φk(·) + KI1s

Passivey


Oscillations

Guy-Bart STAN








Conclusions

The cart-pendulum

θ

mc

x

+

+F

m

−

Φk(·) + KI1s

Passivey


Oscillations

Guy-Bart STAN








Conclusions

Cart-pendulum simulation: k = −1

Stabilization


Oscillations

Guy-Bart STAN








Conclusions

Cart-pendulum simulation: k = 1

Oscillation


Oscillations

Guy-Bart STAN








Conclusions

Cart-pendulum simulation results

k∗ = 0

Time evolution Time evolutionof the state variables of the state variables

0 5 10 15 20 25−8

−6

−4

−2

0

2

4

6

8

10

12

time

stat

e va

riabl

e

Time evolution of the state variables of the cart−pole system

xx

dot

θθ

dot

0 10 20 30 40 50 60 70−8

−6

−4

−2

0

2

4

6

8

10

12

time

stat

e va

riabl

e

Time evolution of the state variables of the cart−pole system

xx

dot

θθ

dot

k = −1 k = 1


Oscillations

Guy-Bart STAN








Conclusions

Outline

1 Introduction




5 Conclusions


Oscillations

Guy-Bart STAN








Conclusions

Conclusions

Dissipativity allows us to

Uncover 2 ’basic’ global oscillations mechanisms inhigh-dimensional systems

Generalize these results for networks of oscillators

Obtain global synchronization results

Propose a simple controller for the synthesis ofoscillations in stabilizable systems


Oscillations

Guy-Bart STAN








Conclusions

Future works

Study other classes of oscillators with input-outputtools

Extend our results to the study of synchronization ofnon-identical oscillators

Apply the synthesis method to generate oscillationsin complex mechatronic systems (RABBIT,SARCOMAN, COG)


Oscillations

Guy-Bart STAN








Conclusions

That’s all

Thank you for your attention !!!

Questions?

(webpage: http://www.montefiore.ulg.ac.be/~stan)

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Global Analysis and Synthesis of Introduction Oscillations Why a … · 2006. 10. 10. · Overview...

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