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Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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 Electrical Engineering and Computer ScienceUniversity of Liège
March, 7, 2005
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Oscillations: why is it important?
Oscillation is ubiquitous in nature:
I breathing, walking, heart beating, sleeping cycles,seasons, etc.
I SARCOMAN
Currently, no general theory for the global analysis orsynthesis of oscillators!
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Goal of our research
Open avenues towards the development of a generalsystem theory for oscillators:
I global stability resultsI dimension independent resultsI interconnection results (complex oscillatory systems
≡ interconnections of simpler oscillatory 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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Outline
Introduction
Global oscillations for the passive oscillator
Global oscillations for networks of passive oscillators
Synthesis of oscillations in stabilizable systems
Conclusions
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Outline
Introduction
Global oscillations for the passive oscillator
Global oscillations for networks of passive oscillators
Synthesis of oscillations in stabilizable systems
Conclusions
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Thesis
Results:I global analysis of oscillations
I synthesis of oscillations
Approach:Dissipativity theory ≡ “efficient tool for global analysisand synthesis of oscillations”
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Thesis
Results:I global analysis of oscillations
I high-dimensional (global) oscillatorsI networks of oscillatorsI synchronization in networks of identical oscillators
I synthesis of oscillationsI simple method for generating oscillations in systems
Approach:Dissipativity theory ≡ “efficient tool for global analysisand synthesis of oscillations”
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Why a dissipativity approach? (1)
Dissipativity theory ≡ Stability theory for open systems(WILLEMS, 1972)
Syst.
u y
System is dissipative (resp. passive) if there exists astorage fcn S(x) ≥ 0:
S ≤ w(u, y)
(resp. S ≤ uT y )
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Why a dissipativity approach? (2)
Dissipativity has increasingly proved useful as a nonlineartool for
I analysis of stability of eq. points of open systemsI stabilization of open systems
Advantages:
I dimension independent resultsI interconnection theory (complex systems ≡
interconnections of simpler systems)
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Outline
Introduction
Global oscillations for the passive oscillator
Global oscillations for networks of passive oscillators
Synthesis of oscillations in stabilizable systems
Conclusions
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
The passive oscillator
φk (y)
yPassive
static nonlinearity
−k
−
u
+φ(y)= −ky
x1
LimitCycle
x2
I Includes two well-known low-dimensional oscillators:VAN DER POL and FITZHUGH-NAGUMO
I Characterization by a specific dissipation inequality:
S︸︷︷︸
storage variation
≤(k − k∗
passive)
y2
︸ ︷︷ ︸
local activation
−
>0︷ ︸︸ ︷
yφ(y)︸ ︷︷ ︸
global dissipation
+ uy︸︷︷︸
ext. supply
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Our results on this class of systems
φk (y)
yPassive
static nonlinearity
−k
−
Stable Unstable
Bifurcation
0 k∗
kGAS
Generically two types of bifurcation (HOPF or pitchfork)
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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∗
Stable
Limit cycle
Unstable
GAS GloballyAttractive
at k = k∗
kk∗
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
First scenario: HOPF bifurcation (2)
A ’basic’ global oscillation mechanism inelectro-mechanical systemsSimplest example: VAN DER POL oscillator:
i = φk (v)L C
Passive
−−
1s
1s
φk (·)
Global oscillation mechanism:
I Continuous lossless exchange of energy betweenthe storage elements
I Static nonlinear element regulates the sign of thedissipation
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
HOPF scenario: example
−−
φk (·)
1s
H(s)y
Passive
I H(s) = τs+ω2n
s2+2ζωns+ω2n
I 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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
HOPF scenario: example
−−
φk (·)
1s
H(s)y
Passive
I H(s) = τs+ω2n
s2+2ζωns+ω2n
I φk (y) = y3 − ky
State-space (k∗ = 1)
−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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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
kk∗
at k = k∗
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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∗
k & k∗, without adaptation k & k∗, with adaptation
unstablestable stable
x2
Relaxation Oscillationx1
x2
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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∗
(Slow) “adaptation” converts the bistable system into aglobal oscillator
φk(·)
− −
1τs+1
Passive
τ À 0
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Second scenario: pitchfork bifurcation (2)
A ’basic’ global oscillation mechanism in biologySimplest example: FITZHUGH-NAGUMO oscillator:
insidethe cell the cell
outsideall ions all ions
VE+ E−
Adaptation
Passive
φk (·)
− −
1τs+1
1s
τ À 0
Global oscillation mechanism:
I Continuous switch between 2 quasi stable eq. points
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Pitchfork scenario: example
Passive
H(s) y
−
φk(·)
I H(s) = τs+ω2n
s2+2ζωns+ω2n
I φk (y) = y3 − ky
State-space (k∗ = 1)
−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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Pitchfork scenario: example
Passive
H(s) y
−−
φk (·)
1τs+1
τ À 0
I H(s) = τs+ω2n
s2+2ζωns+ω2n
I φk (y) = y3 − kyI Adaptation
State-space (k∗ = 1)
−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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Outline
Introduction
Global oscillations for the passive oscillator
Global oscillations for networks of passive oscillators
Synthesis of oscillations in stabilizable systems
Conclusions
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Networks of oscillators
In nature, oscillation is the result of interconnectedoscillators!
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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 oscillations for networks (1)
Question: “What are the topologies that lead to globaloscillations in the network?”
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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 oscillations for networks (1)
Question: “What are the topologies that lead to globaloscillations in the network?”Answer: Passive coupling (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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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 oscillations for networks (1)
Question: “What are the topologies that lead to globaloscillations in the network?”Answer: Passive coupling (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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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 oscillations for networks (2)
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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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 oscillations for networks (3)
Network of identical passive oscillatorsIf 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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Examples
We consider 6= networks of identical passive oscillators
−−
φk (·)
1s
H(s)y
Passiveu
I H(s) = τs+ω2n
s2+2ζωns+ω2n
I φk (y) = y3 − ky
O1
+1
+1
O2
I Each oscillator is represented as a circleI The arrows denote the linear input-output
interconnection between the oscillators
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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)
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Synchronization and incremental passivityQuestion: Under which conditions do all oscillatorssynchronize?
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Synchronization and incremental passivityQuestion: Under which conditions do all oscillatorssynchronize?
ApproachIncremental dissipativity ≡ dissipativity expressed interms of the difference between solutions of systems
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Synchronization and incremental passivityQuestion: Under which conditions do all oscillatorssynchronize?
ApproachIncremental dissipativity ≡ dissipativity expressed interms of the difference between solutions of systems
A B
C=A-B
Do A and B synchronize?
I Study stability of C through dissipativity theory: if Cis stable then A and B synchronize
I Stability of C generally depends on the topology ofthe network
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Synchronization result
Theorem (3rd result)If
I network of identical, incrementally passive oscillatorscharacterized by a global limit cycle oscillation
I linear, passive (Γ ≥ 0) coupling, andker (Γ) = ker
(ΓT ) = range
(
(1, . . . , 1)T)
I strong enough coupling (λmin6=0 (Γs) > k − k∗passive)
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))
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
ExamplesThis result is useful to prove global synchronization innetworks with specific topologies such as:
O1 O2
O3O4
O1 O2
O3O4
All-to-all Bidirectional ringO1 O2
O3O4 O1 O2 · · · ON
Unidirectional ring Open chain
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
ExamplesThis result is useful to prove global synchronization innetworks with specific topologies such as:
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)
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Outline
Introduction
Global oscillations for the passive oscillator
Global oscillations for networks of passive oscillators
Synthesis of oscillations in stabilizable systems
Conclusions
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Synthesis of oscillations
The structure of the passive oscillator suggests a methodfor generating oscillations in passive systems
−
Φk (·) + KI1s
Passive y
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
The cart-pendulum
θ
mc
x
+
+F
m
−
Φk (·) + KI1s
Passive y
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Cart-pendulum simulation: k = −1
Stabilization
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Cart-pendulum simulation: k = 1
Oscillation
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
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
dotθθ
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
dotθθ
dot
k = −1 k = 1
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Outline
Introduction
Global oscillations for the passive oscillator
Global oscillations for networks of passive oscillators
Synthesis of oscillations in stabilizable systems
Conclusions
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Conclusions
Dissipativity allows us to
I Uncover 2 ’basic’ global oscillations mechanisms inhigh-dimensional systems
I Generalize these results for networks of oscillatorsI Obtain global synchronization resultsI Propose a simple controller for the synthesis of
oscillations in stabilizable systems
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
Future works
I Study other classes of oscillators with input-outputtools
I Extend our results to the study of synchronization ofnon-identical oscillators
I Apply the synthesis method to generate oscillationsin complex mecatronic systems (RABBIT,SARCOMAN, COG)
Global Analysisand Synthesis of
Oscillations
Guy-Bart STAN
IntroductionThesis
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
That’s all
Thank you for your attention !!!