Cluster Synchronization in Networks of Oscillators

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Cluster Synchronization in Networks of Oscillators

A Geometric Approach

Lorenzo Tiberi

^{†, ]}

Advised by Chiara Favaretto^{¶, ]}, Prof. Mario Innocenti^† & Prof. Fabio Pasqualetti^] Supervised by Prof. Lucia Pallottino^†

† Department of Information Engineering University of Pisa, Italy

¶ Department of Information Engineering University of Padua, Italy

] Department of Mechanical Engineering University of California, Riverside

University of Pisa, Italy - May 2, 2017

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Overview

1

History & Motivation

2

Introduction

3

Problem Setup

4

Analysis

5

Control

6

Simulations

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History & Motivation

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Where it all began

Christiaan Huygens (1629 - 1695) physicist & mathematician engineer & horologist

observed ”an odd kind of simpathy”

between coupled & heterogeneous clocks...

...which is still fascinating today:

Sync of 32 metronomes at Ikeguchi Laboratory, Saitama University, 2012

https://www.youtube.com/watch?

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Applications

Not just for metronomes...

Fields of application

1

Sync in mathematical biology, chemistry, mechanics & neuroscience

2

Sync in population of fireflies, neural networks and complex networks

3

Sync in tech applications: vehicle coordination, flocking & schooling,

AC power transmission networks, microgrids

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Motivation

Higher layer with respect to the applications...

Motivational aim

Derive graph-theoretical methods to analyze and control synchronization properties, within phase oscillators framework

Global Synchronization (all nodes synchronize)

Cluster Synchronization (group of nodes synchronize)

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Literature review

Complete synchronization:

F. D¨orfler and F. Bullo, “Synchronization in complex networks of phase oscillators: A survey,” Automatica, vol. 50, no. 6, pp. 1539–1564, 2014

J. G´omez-Garde˜nes, Y. Moreno, and A. Arenas, “Synchronizability determined by coupling strengths and topology on complex networks,” vol. 75, p. 066106, Jun 2007

Cluster synchronization:

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nature communications, vol. 5, 2014

C. Favaretto, D. Bassett, A. Cenedese, and F. Pasqualetti, “Bode meets kuramoto: Synchronized clusters in oscillatory networks,” Seattle, Wa, May 2017, to appear

C. Favaretto, A. Cenedese, and F. Pasqualetti, “Cluster synchronization in networks of kuramoto oscillators,” Toulouse, Fr, Jul. 2017, to appear

Geometric methods for clusterization:

N. Monshizadeh and A. van der Schaft, “Structure-preserving model reduction of physical network systems by clustering,” in Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on. IEEE, 2014, pp. 4434–4440

M. T. Schaub, N. O’Clery, Y. N. Billeh, J.-C. Delvenne, R. Lambiotte, and M. Barahona, “Graph partitions and cluster synchronization in networks of oscillators,”

Chaos, vol. 26, no. 9, p. 094821, 2016

Synchronization & brain disorders:

C. Hammond, H. Bergman, and P. Brown, “Pathological synchronization in parkinson’s disease: networks, models and treatments,” Trends in Neurosciences, vol. 30, no. 7, pp. 357–364, 2007

L. L. Rubchinsky, C. Park, and R. M. Worth, “Intermittent neural synchronization in parkinson’s disease,” Nonlinear Dynamics, vol. 68, no. 3, pp. 329–346, 2012 K. Lehnertz, S. Bialonski, M.-T. Horstmann, D. Krug, A. Rothkegel, M. Staniek, and T. Wagner, “Synchronization phenomena in human epileptic brain networks,”

Journal of neuroscience methods, vol. 183, no. 1, pp. 42–48, 2009

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Introduction

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Tools

Key ingredients:

Graph Theory:

Directed graph G = (V, E , W) V = {1, . . . , n} nodes

E ⊆ V × V edges

W → A = [a

_ij

] adjacency matrix L = D − A graph Laplacian

Small Example:

D =





a12 0 0

0 a21+ a23 0

0 0 a31



 A =





0 a12 0 a21 0 a23

a31 0 0





L =





a12 −a12 0

−a21 a21+ a23 −a23

−a31 0 a31





1 2

3

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Tools

Key ingredients:

Geometric Control Theory (Linear Systems):

Subspace invariance: AJ ⊆ J → AV = VX (V is a basis of J ) Controlled invariance: AJ ⊆ J + imB → AV = VX + BU

X and U compatible matrices

0 5 10 15 20 25 30

x2

State Evolutions

0 5 10 15 20 25 30

x2

State Evolutions

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Which Model?

Phase Oscillators

Dynamics: one dimensional ODE describing the state of each node Same dynamics in isolation

The amount of phase change induced by another node depends only on the phase difference between the two nodes

θ ˙

_i

= f (θ

_i

) + X

j 6=i

g

_ij

(θ

_i

− θ

_j

)

f (·): isolated dynamics on the phase g

ij

(·): input from area j to area i

⇑ simple to simulate

⇑ biological interpretation

⇑ good for analysis/control

⇓ still qualitative

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Which Model?

Phase Oscillators

Dynamics: one dimensional ODE describing the state of each node Same dynamics in isolation

The amount of phase change induced by another node depends only on the phase difference between the two nodes

θ ˙

_i

= f (θ

_i

) + X

j 6=i

g

_ij

(θ

_i

− θ

_j

)

f (·): isolated dynamics on the phase g

ij

(·): input from area j to area i

⇑ simple to simulate

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Which Model?

Phase Oscillators

Dynamics: one dimensional ODE describing the state of each node Same dynamics in isolation

The amount of phase change induced by another node depends only on the phase difference between the two nodes

θ ˙

_i

= f (θ

_i

) + X

j 6=i

g

_ij

(θ

_i

− θ

_j

)

f (·): isolated dynamics on the phase g

ij

(·): input from area j to area i

⇑ simple to simulate

⇑ biological interpretation

⇑ good for analysis/control

⇓ still qualitative

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Kuramato Oscillators

θ ˙

_i

= ω

_i

+ K

N

X

j =1

sin(θ

_j

− θ

_i

), i = 1 . . . N (Classical)

θ ˙

i

= ω

i

+

N

X

j =1

a

ij

sin(θ

j

− θ

_i

), i = 1 . . . N (Modified)

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Behaviors

Phase Syncronization

0 0.05 0.1 0.15 0.2 0.25 0.3

t(s)

10 20 30 40 50 60 70 80 90 100

θ(rad)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t(s)

10 20 30 40 50 60 70 80 90 100

θ(rad)

0 0.05 0.1 0.15 0.2 0.25 0.3

t(s)

10 20 30 40 50 60 70 80 90 100

θ(rad)

1 2 3 4 5 6

Networks of 100 nodes Incoherence, Global Synch, Cluster Sync

Insight: The role of ω

_i

and A

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Problem Setup

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Preliminary & Definitions

Preliminary

Let P = {P1, . . . , Pm} be a partition of V: V = ∪^m_{i =1}Pi and Pi∩ Pj = ∅ for all i , j ∈ {1, . . . , m} with i 6= j

We restrict the attention to the case m > 1, ruling out the global synchronization case (widely studied in the literature)

Graphical example: P = {P₁, P₂, P₃} with P₁= {1, 2, 7, 8}, P₂= {3, 6}

and P3= {4, 5}

1

2 3 4 5 6

7

8

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Preliminary & Definitions

Preliminary

and P3= {4, 5}

1 3 4 5 6

7

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Preliminary & Definitions

Preliminary

and P3= {4, 5}

1

2 3 4 5 6

7

8

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Preliminary & Definitions

Definition 1

(Phase synchronization) For the network of oscillators G = (V, E ), the partition P = {P1, . . . , Pm} is phase synchronizable if, for some initial phases

θ₁(0), . . . , θ_n(0), it holds

θi(t) = θj(t),

for all times t ∈ R≥0 and i , j ∈ P_k, with k ∈ {1, . . . , m}

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Preliminary & Definitions

Definition 2

(Characteristic matrix) For the network of oscillators G = (V, E ) and the partition P = {P₁, . . . , P_m}, the characteristic matrix of P is VP ∈ R^n×m, where V_P =v1 v2 · · · vm, v_i^T= 0 0 · · · 0

| {z }

Pi −1 j =1|Pj|

1 1 · · · 1

| {z }

|P_i|

0 0 · · · 0

| {z }

Pn j =i +1|P_j|

Example 1 - Ex1

4

6

5 1

2

3 9

5 9

10 2

10

7

2 5 A =







0 0 0 0 0 10

0 0 0 5 0 5

0 0 0 0 10 0

9 0 0 0 0 0

0 9 0 0 0 0

0 7 2 2 0 0





 VP=





 1 0 1 0 1 0 0 1 0 1 0 1







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Analysis

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Conditions for Cluster Synchronization

Assumption

(A1) For the partition P = {P1, . . . , Pm} there exists an ordering of the clusters P_i and an interval of time [t₁, t₂], with t₂> t₁, such that for all times t ∈ [t1, t2]:

max

i ∈P1

θ˙i> max

i ∈P2

θ˙i> · · · > max

i ∈Pm

θ˙i

Ex1 simulations: ω

P2

= 10, ω

P1

= 30 (left), ω

P1

= 19 (right)

(A1) satisfied over the entire interval (A1) satisfied over closed intervals

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Main Result for Analysis

Theorem 1 [Tiberi et al. 2017]

(Cluster synchronization) For the network of oscillators G = (V, E ), the partition P = {P

1

, . . . , P

m

} is phase synchronizable if and only if the following conditions are simultaneously satisfied:

1

the network weights satisfy P

k∈P`

a

_ik

− a

_jk

= 0 for every i , j ∈ P

_z

and z, ` ∈ {1, . . . , m}, with z 6= `;

2

the natural frequencies satisfy ω

_i

= ω

_j

for every k ∈ {1, . . . , m} and i , j ∈ P

_k

.

L. Tiberi, C. Favaretto, M. Innocenti, D. Bassett, and F. Pasqualetti, “Pattern formation in network of kuramoto oscillators: a geometric approach for analysis and control,” in IEEE Conference on Decision and Control, 2017, submitted

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Side results

Let A B denote the Hadamard product between A and B, and Im(V

P

)

^⊥

the orthogonal subspace to Im(V

P

)

Corollary 1 [Tiberi et al. 2017]

(Matrix condition for synchronization) Condition 1 in Theorem 1 is equivalent to ¯ V

_P^T

AV ¯

P

= 0, where ¯ V

P

∈ R

^n×(n−m)

satisfies

Im( ¯ V

P

) = Im(V

P

)

^⊥

, and

A = A − A V ¯

P

V

_P^T

.

Proposition 1 [MSc Thesis]

(Linear invariance on ¯ A) Condition ¯ V

_P^T

AV ¯

P

= 0, given in Corollary 1, is equivalent to the ¯ A-invariance of Im(V

P

) for the linear friend system

˙

x = ¯ Ax of the Kuramoto nonlinear dynamics.

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Side results

Let A B denote the Hadamard product between A and B, and Im(V

P

)

^⊥

the orthogonal subspace to Im(V

P

)

Corollary 1 [Tiberi et al. 2017]

(Matrix condition for synchronization) Condition 1 in Theorem 1 is equivalent to ¯ V

_P^T

AV ¯

P

= 0, where ¯ V

P

∈ R

^n×(n−m)

satisfies

Im( ¯ V

P

) = Im(V

P

)

^⊥

, and

A = A − A V ¯

P

V

_P^T

.

Proposition 1 [MSc Thesis]

(Linear invariance on ¯ A) Condition ¯ V

_P^T

AV ¯

P

= 0, given in Corollary 1, is

equivalent to the ¯ A-invariance of Im(V ) for the linear friend system

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Generalized external equitable partitions

An EP splits the graph in partitions P = {P1, . . . , Pm} such that each node has the same out-degree with respect to every partition

For an EEP this condition has to hold for connection between different partitions Pi, Pj with i 6= j

For a GEEP the in-degree for each node has to be the same with respect to every other partition

Insight: V¯_P^TAV¯ P = 0 describes exactly the GEEP condition!

1 2

3 4

7 2 7 2

7 2

(a) Equitable (EP)

1 2

3 4

10 7 2 7 2

5

(b) External EP (EEP)

1 2

3 4

10 1

7 2 8

5

(c) Gen. EEP (GEEP)

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Generalized external equitable partitions

1 2

7 2 7 2 2

1 2

10 7 2 7 2

1 2

10 1

7 2 8

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Generalized external equitable partitions

1 2

3 4

7 2 7 2

7 2

(a) Equitable (EP)

1 2

3 4

10 7 2 7 2

5

(b) External EP (EEP)

1 2

3 4

10 1

7 2 8

5

(c) Gen. EEP (GEEP)

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Generalized external equitable partitions

1 2

7 2 7 2 2

1 2

10 7 2 7 2

1 2

10 1

7 2 8

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Control

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Control paradigm

Motivational question

Given V

P

, let ω ∈ Im(V

P

) and suppose that A does not allow phase clusterization. What is the optimal modification for A such that the partition P is synchronizable?

Optimality measure: Frobenius norm of the perturbation ||∆||

_F

Synchronizability iff Theorem 1 is satisfied (conditions on ¯ A) Sparsity implementation: when perturbing, some edges must stay fixed while some others may be changed

1 a

21

2 1 a

21

2

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Control paradigm

Motivational question

Given V

P

, let ω ∈ Im(V

P

) and suppose that A does not allow phase clusterization. What is the optimal modification for A such that the partition P is synchronizable?

Optimality measure: Frobenius norm of the perturbation ||∆||

_F

Synchronizability iff Theorem 1 is satisfied (conditions on ¯ A) Sparsity implementation: when perturbing, some edges must stay fixed while some others may be changed

1 a

21

2

(a) Perturbable edge

1 a

21

2

(b) Fixed edge

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Control paradigm

Motivational question

Given V

P

, let ω ∈ Im(V

P

) and suppose that A does not allow phase clusterization. What is the optimal modification for A such that the partition P is synchronizable?

Optimality measure: Frobenius norm of the perturbation ||∆||

_F

Synchronizability iff Theorem 1 is satisfied (conditions on ¯ A) Sparsity implementation: when perturbing, some edges must stay fixed while some others may be changed

1 a

21

2 1 a

21

2

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Control paradigm

Motivational question

Given V

P

, let ω ∈ Im(V

P

) and suppose that A does not allow phase clusterization. What is the optimal modification for A such that the partition P is synchronizable?

Optimality measure: Frobenius norm of the perturbation ||∆||

_F

Synchronizability iff Theorem 1 is satisfied (conditions on ¯ A) Sparsity implementation: when perturbing, some edges must stay fixed while some others may be changed

1 a

21

2

(a) Perturbable edge

1 a

21

2

(b) Fixed edge

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What to control?

A perturbed adjacency matrix

Sync conditions are structural on ¯A and proper on ω

We seek a perturbation matrix ∆ such that the new adjacency matrix A + ∆ satisfies Theorem 1

min

∆ k∆k²_F (1a)

s.t. V¯_P^TA + ∆ V¯ P = 0 (1b)

∆ ∈ H (1c)

=⇒ min

∆ k∆ Hk²_F (2a)

s.t. V¯_P^TA + ∆ V¯ P = 0 (2b)

The two optimization problems are equivalent

(1b) is the ¯A-invariance of Im(VP), (1c) is a sparsity constraint on ∆

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What to control?

A perturbed adjacency matrix

Sync conditions are structural on ¯A and proper on ω

We seek a perturbation matrix ∆ such that the new adjacency matrix A + ∆ satisfies Theorem 1

min

∆ k∆k²_F (1a)

s.t. V¯_P^TA + ∆ V¯ P = 0 (1b)

∆ ∈ H (1c)

=⇒ min

∆ k∆ Hk²_F (2a)

s.t. V¯_P^TA + ∆ V¯ P = 0 (2b)

The two optimization problems are equivalent

(1b) is the ¯A-invariance of Im(VP), (1c) is a sparsity constraint on ∆ is the element-wise matrix division

H s.t. hij= 0 if the element aij cannot be perturbed, hij = 1 otherwise

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Main result for Control

Theorem 2 [Tiberi et al. 2017]

(Synchronization via structured perturbation) Let T = [V_P V¯_P], and let

A˜11 A˜12

A˜21 A˜22

= T⁻¹AT .¯

The minimization problem (2) has a unique solution if and only if there exists a matrix Λ satisfying ˜A21= ¯V_P^TXV_P where X has been defined as

X := ( ¯V_PΛV_P^T) H. Moreover, if it exists, the solution ∆^∗ to (2) is

∆^∗= T

∆˜^∗₁₁ ∆˜^∗₁₂

∆˜^∗₂₁ ∆˜^∗₂₂

T⁻¹,

where ˜∆^∗ = −V^TXV , ˜∆^∗ = −V^TX ¯V , ˜∆^∗ = − ˜A , and ˜∆^∗ = − ¯V^TX ¯V .

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Side results

Corollary 2 [Tiberi et al. 2017]

(Unconstrained minimization problem) Let

H = {H | h

_ij

= 1 for all i and j }. The minimization problem (2) is always feasible, and its solution is

∆

^∗

= − ¯ V

P

V ¯

_P^T

AV ¯

P

V

_P^T

.

Proposition 2 [MSc Thesis]

(Linear invariance on ¯ A + ∆

^∗

) Condition ¯ V

_P^T

[ ¯ A + ∆

^∗

]V

P

= 0 of (2) is

equivalent to the ( ¯ A, B) - controlled invariance of Im(V

P

) for the linear

friend nonautonomous system ˙ x = ¯ Ax + Bu of the Kuramoto nonlinear

dynamics.

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Side results

Corollary 2 [Tiberi et al. 2017]

(Unconstrained minimization problem) Let

H = {H | h

_ij

= 1 for all i and j }. The minimization problem (2) is always feasible, and its solution is

∆

^∗

= − ¯ V

P

V ¯

_P^T

AV ¯

P

V

_P^T

.

Proposition 2 [MSc Thesis]

(Linear invariance on ¯ A + ∆

^∗

) Condition ¯ V

_P^T

[ ¯ A + ∆

^∗

]V

P

= 0 of (2) is

equivalent to the ( ¯ A, B) - controlled invariance of Im(V

P

) for the linear

friend nonautonomous system ˙ x = ¯ Ax + Bu of the Kuramoto nonlinear

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Simulations

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Testing synchronizability on Ex1

Consider the network of Example 1, with ω ∈ Im(VP)

Theorem 1 is satisfied −→ phase synch & retaining [iff θ(0) ∈ Im(V_P)]

4

6 5 1

2 3 9

5 9

10 2

10 2 5 7

A =¯







0 0 0 0 0 10

0 0 0 5 0 5

0 0 0 0 10 0

9 0 0 0 0 0

0 9 0 0 0 0

0 7 2 0 0 0





 ω = [60, 60, 60, 20, 20, 20]

V¯P=





1 0 0 0

−1 1 0 0

0 −1 0 0





V¯^TAV¯ P=





0 0 0 −5 0 5

0 0 0 5 −10 5









 1 0 1 0 1 0





=



 0 0 0 0





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Testing synchronizability (Cont’d)

0 0.2 0.4 0.6 0.8 1

t[s]

-1 -0.5 0 0.5 1

sin(θ)

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

0 0.2 0.4 0.6 0.8 1

t[s]

10 20 30 40 50 60 70

rad˙ θ[]sec

˙θ₁ ˙θ₂ ˙θ₃ ˙θ₄ ˙θ₅ ˙θ₆

0 0.2 0.4 0.6 0.8 1

t[s]

-1 -0.5 0 0.5 1

sin(θ)

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

0 0.2 0.4 0.6 0.8 1

t[s]

10 20 30 40 50 60 70

rad˙ θ[]sec

˙θ1 ˙θ

2 ˙θ

3 ˙θ

4 ˙θ

5 ˙θ

6

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Modified weight

Consider the network of Example 1, with ω ∈ Im(V

P

) Notice the edge (1, 6). Now, Theorem 1 is not satisfied

Even starting with θ(0) ∈ Im(V

P

), de-synchronization takes effect

4

6 5 1

2 3 9

5 9

10 12 2

2 5 7

^V^¯^P^T^AV^¯ ^P ⁼





 0 2 0 0 0 0 0 0





 6=





 0 0 0 0 0 0 0 0







0.5 1

n(θ)

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

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Uncostrained control

Theorem 1 satisfied with adjacency (A + ∆

^∗

) −→ phase synch Recall Corollary 2: ∆

^∗

= − ¯ V

P

V ¯

_P^T

AV ¯

P

V

_P^T

4

6 5 1

2 3 9 -0.44

5.22 9

0.22 10.22 2

11.55 0.22

2 7 5.22 -0.44

0.22

^∆^∗⁼







0 0 0 −0.44 −0.44 −0.44

0 0 0 0.22 0.22 0.22

0 0 0 0 0 0





 ω = [60, 60, 60, 20, 20, 20]

V¯P^T( ¯A + ∆^∗)VP=







0 0 0 −5 0 5

0 0 0 5 −10 5

9 −9 0 0 0 0

0 2 −2 0 0 0







| {z }

V¯_P^T( ¯A+∆^∗)





 1 0 1 0 1 0 0 1 0 1 0 1







=





 0 0 0 0 0 0 0 0







||∆^∗||²F = 0.88

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Uncostrained control (Cont’d)

0 0.2 0.4 0.6 0.8 1

t[s]

-1 -0.5 0 0.5 1

sin(θ)

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

0 0.2 0.4 0.6 0.8 1

t[s]

0 10 20 30 40 50 60

rad˙ θ[]sec

˙θ₁ ˙θ₂ ˙θ₃ ˙θ₄ ˙θ₅ ˙θ₆

0 0.5 1

sin(θ)

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

20 30 40 50 60

rad˙ θ[]sec

˙θ₁ ˙θ₂ ˙θ₃ ˙θ₄ ˙θ₅ ˙θ₆

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Constrained control

Some edges are constrained (dashed), some perturbable (solid) Closed form solution for the perturbation ∆

^∗

as in Theorem 2

4

6 5 1

2 3 9

5 9

10 12 2

2 5 7

4

6 5 1

2 3 9

5 9

11 2

12 2 5 7

2 1

H =







0 1 1 0 0 0

1 0 1 0 1 0

1 1 0 0 1 1

0 1 1 0 1 1

1 1 1 1 0 1

1 0 0 1 1 0







∆^∗= T

∆˜^∗₁₁ ∆˜^∗₁₂

∆˜^∗21 ∆˜^∗22

T⁻¹

=







0 0 0 0 0 0

0 0 0 0 2 0

0 0 0 0 1 1

0 0 0 0 0 0







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Constrained control (Cont’d)

0 0.2 0.4 0.6 0.8 1

t[s]

-1 -0.5 0 0.5 1

sin(θ)

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

0 0.2 0.4 0.6 0.8 1

t[s]

0 20 40 60 80

rad˙ θ[]sec

˙θ₁ ˙θ₂ ˙θ₃ ˙θ₄ ˙θ₅ ˙θ₆

0 0.5 1

sin(θ)

θ₁ θ₂ θ₃ θ₄ θ₅ θ₆

20 40 60 80

rad˙ θ[]sec

˙θ₁ ˙θ₂ ˙θ₃ ˙θ₄ ˙θ₅ ˙θ₆

(49)

Conclusion

(50)

Summary

1

New analytical conditions ensuring multi-consensus on networks of Kuramoto oscillators

2

Link with subspace invariance for linear systems

3

Control mechanism to force pattern formation: optimality and sparsity constraints

4

Further developments regard the attractiveness of such clusterized

evolutions...

(51)

Summary

1

New analytical conditions ensuring multi-consensus on networks of Kuramoto oscillators

2

Link with subspace invariance for linear systems

3

Control mechanism to force pattern formation: optimality and sparsity constraints

4

Further developments regard the attractiveness of such clusterized

evolutions...

(52)

Summary

1

New analytical conditions ensuring multi-consensus on networks of Kuramoto oscillators

2

Link with subspace invariance for linear systems

3

Control mechanism to force pattern formation: optimality and sparsity constraints

4

Further developments regard the attractiveness of such clusterized

evolutions...

(53)

Summary

1

New analytical conditions ensuring multi-consensus on networks of Kuramoto oscillators

2

Link with subspace invariance for linear systems

3

Control mechanism to force pattern formation: optimality and sparsity constraints

4

Further developments regard the attractiveness of such clusterized

evolutions...

(54)

Summary

1

New analytical conditions ensuring multi-consensus on networks of Kuramoto oscillators

2

Link with subspace invariance for linear systems

3

Control mechanism to force pattern formation: optimality and sparsity constraints

4

Further developments regard the attractiveness of such clusterized

evolutions...

(55)

Closure

Thank you

It is theory that decides what can be observed.

— Albert Einstein

(56)

Appendix

(57)

Necessity of A1

Remark 1

(Necessity of assumption A1) Consider a network of oscillators with adjacency matrix

A =







0 a12 0 0

a21 0 a23 0 0 a32 0 a34

0 0 a43 0







and natural frequencies ωi = ¯ω for all i ∈ {1, . . . , 4}.

Condition (i) of Theorem may not be satisfied (general weights aij) Let θ1(0) = θ2(0) and θ3(0) = θ4(0) = θ1(0) + π

Notice that ˙θi = ¯ω at all times and for all i ∈ {1, . . . , 4} (Assumption A1 is not satisfied)

The partition P = {P1, P2}, with P1= {1, 2} and P2= {3, 4} is phase synchronized, independently of the interconnection weights

(58)

Necessity of A1

Remark 1

A =







0 a12 0 0

a21 0 a23 0 0 a32 0 a34

0 0 a43 0







The partition P = {P1, P2}, with P1= {1, 2} and P2= {3, 4} is phase

(59)

Necessity of A1

Remark 1

A =







0 a12 0 0

a21 0 a23 0 0 a32 0 a34

0 0 a43 0







(60)

Necessity of A1

Remark 1

A =







0 a12 0 0

a21 0 a23 0 0 a32 0 a34

0 0 a43 0







The partition P = {P1, P2}, with P1= {1, 2} and P2= {3, 4} is phase

(61)

Necessity of A1

Remark 1

A =







0 a12 0 0

a21 0 a23 0 0 a32 0 a34

0 0 a43 0







(62)

Necessity of A1 - Simulation

1 2

3 4

a

21

a

₁₂

a

32

a

23

a

43

a

₃₄

(c) Graph of Remark 1.

0 0.2 0.4 0.6 0.8 1

t[s]

-1 -0.5 0 0.5 1

sin(θ)

θ₃, θ₄ θ₁, θ₂

(d) Oscillator’s phases evolution.

For each choice of the arc weights a , the partition P = {P , P }

(63)

Theorem 1 - Sketch Proof

Proof - Sufficency

(If) Let θi = θjfor all i , j ∈ Pk, k ∈ {1, . . . , m}. Let i , j ∈ P`, and notice that θ˙i− ˙θj=X

z6=`

X

k∈P_z

aiksin(θk− θi) − ajksin(θk− θj)

=X

z6=`

sz`

X

k∈Pz

aik− ajk = 0,

Conditions 1 and 2 are plugged in.

The term szl is a parameter that depends on the clusters z and `, but not on the indices i , j , k.

θ ∈ Im(VP) implies ˙θ ∈ Im(VP), hence the subspace Im(VP) is invariant and the network is phase synchronizable.

Select θ(0) ∈ Im(VP) to remain within the subspace.

G. Basile and G. Marro, Controlled and conditioned invariants in linear system theory. Prentice Hall Englewood Cliffs, 1992

(64)

Theorem 1 - Sketch Proof (Cont’d)

Proof - Necessity

(Only if) Assume that the network is phase synchronized, and notice that it is also frequency synchronized. Let i , j ∈ P`. At all times:

0 = ¨θi− ¨θj=X

z6=`

X

k∈P_z

aikcos(θk− θi)( ˙θk− ˙θi)

−X

z6=`

X

k∈P_z

ajkcos(θk− θj)( ˙θk− ˙θj)

=X

z6=`

cz`vz`

X

k∈Pz

aik− ajk

| {z }

dz

,

(3)

cz`and vz`depend on the clusters z and `, but not on the indices i , j , k.

(65)

Theorem 1 - Sketch Proof (Cont’d)

Proof - Necessity

(By contradiction) Assume that the functions czlvzl are linearly dependent at all times.

Then it must hold that

X

z6=`

dz

dⁿ

dtⁿcz`vz`= 0,

Not only the functions czlvzl must be linearly dependent, but also all their derivatives, at some times in [t1, t2].

Let d16= 0. Notice that, because of assumption (A1), there exists an integer n such that d1dⁿ

dtⁿc1`v1` dz dⁿ

dtⁿcz`vz`, for all z 6= 1.

The functions czlvzl cannot be linearly dependent. Condition 1 is necessary for phase synchronization.

Let the network be phase synchronized, and let i , j ∈ P`. We have 0 = ˙θi− ˙θj= ωi− ωj+X

z6=`

sz`

X

k∈P_z

aik− ajk

| {z }

=0

,

(66)

Theorem 2 - Sketch Proof

Proof

(By construction) The Lagrangian, L, can be constructed as

L(∆, Λ) =

n

X

i =1 n

X

j =1

δ_ij²h_ij⁻¹+

m

X

i =1

λ^T_i V¯_P^T( ¯A + ∆)vi.

Equate partial derivatives of L to zero

∂L

∂λ_i = 0 ⇒ ¯V_P( ¯A + ∆)vi = 0

∂L = 0 ⇒ 2δ h⁻¹+

m

Xλ^Tv¯^Tv = 0

⇒

V¯_P^T( ¯A + ∆)V_P = 0 (5a)

T

(67)

Theorem 2 - Sketch Proof (Cont’d)

Proof

Apply the change of coordinates T = [VP V¯P], ¯A = T ˜AT⁻¹, ∆ = T ˜∆T⁻¹,

V¯_P^TT ( ˜A + ˜∆)T⁻¹V_P =

0 In−m

A˜₁₁+ ˜∆₁₁ A˜₁₂+ ˜∆₁₂ A˜₂₁+ ˜∆₂₁ A˜₂₂+ ˜∆₂₂

Im

0

= 0,

which leads to

∆˜^∗₂₁= − ˜A21. (6)

Equation (5b) is equivalent to

∆ + ( ¯V_PΛV_P^T) H = 0, (7)

(68)

Theorem 2 - Sketch Proof (Cont’d)

Proof

By means of the same change of coordinate

VP V¯_P

| {z }

T

∆˜₁₁ ∆˜₁₂

∆˜21 ∆˜22

V_P^T V¯_P^T

| {z }

T⁻¹

+( ¯V_PΛV_P^T) H = 0,

from which we obtain

(V_P∆˜11V_P^T− ¯V_PA˜12V_P^T+ V_P∆˜12V¯_P^T+ ¯V_P∆˜22V¯_P^T)+

( ¯VPΛV_P^T) H = 0, (8)

(69)

Theorem 2 - Sketch Proof (Cont’d)

Proof

Pre-multiply equation (8) by ¯V_P^Tand post-multiply it by VP

− ˜A21+ ¯V_P^TXV_P = 0, (9a) (9a) is a system of linear equations → solved with respect to the unknown Λ.

Following the same reasoning, we can obtain the following other three equations that entirely determine the solution ˜∆11, ˜∆12, and ˜∆22

∆˜₁₁+ V_P^TXV_P = 0,

∆˜₁₂+ V_P^TX ¯V_P = 0,

∆˜₂₂+ ¯V_P^TX ¯V_P = 0.

(70)

Theorem 2 - Sketch Proof (Cont’d)

Proof

Finally, the optimal matrix ∆

^∗

, solution to the problem (2), is given in original coordinates as

∆

^∗

= T

∆ ˜

^∗₁₁

∆ ˜

^∗₁₂

− ˜ A

₂₁

∆ ˜

^∗₂₂

T

⁻¹

(71)

(72)