Docstoc

Dissipativity theory for oscillator analysis

Document Sample
Dissipativity theory for oscillator analysis Powered By Docstoc
					                                                               Dissipativity theory
                                                                  for oscillator
                                                                    analysis

                                                                Guy-Bart STAN


                                                               Introduction
Dissipativity theory for oscillator analysis                   Global oscillations
                                                               for the passive
                                                               oscillator
                                                               The passive oscillator
                                                               Global osc. for the passive

                     Guy-Bart STAN                             oscillator


                                                               Global oscillations
                                                               for networks of
   Department of Electrical Engineering and Computer Science   passive oscillators
                                                               Extension of the results for
                       University of Liège                     one passive oscillator
                                                               Sync. in networks of
                                                               identical passive oscillators


                     March, 24, 2005                           Conclusions




                                                                                     1/24
Outline                                                       Dissipativity theory
                                                                 for oscillator
                                                                   analysis

                                                               Guy-Bart STAN


                                                              Introduction
Introduction                                                  Global oscillations
                                                              for the passive
                                                              oscillator
                                                              The passive oscillator

Global oscillations for the passive oscillator                Global osc. for the passive
                                                              oscillator

   The passive oscillator                                     Global oscillations
                                                              for networks of
   Global oscillation mechanisms for the passive oscillator   passive oscillators
                                                              Extension of the results for
                                                              one passive oscillator
                                                              Sync. in networks of

Global oscillations for networks of passive oscillators
                                                              identical passive oscillators


                                                              Conclusions
   Extension of the results for one passive oscillator
   Sync. in networks of identical passive oscillators

Conclusions



                                                                                    2/24
Outline                                                       Dissipativity theory
                                                                 for oscillator
                                                                   analysis

                                                               Guy-Bart STAN


                                                              Introduction
Introduction                                                  Global oscillations
                                                              for the passive
                                                              oscillator
                                                              The passive oscillator

Global oscillations for the passive oscillator                Global osc. for the passive
                                                              oscillator

   The passive oscillator                                     Global oscillations
                                                              for networks of
   Global oscillation mechanisms for the passive oscillator   passive oscillators
                                                              Extension of the results for
                                                              one passive oscillator
                                                              Sync. in networks of

Global oscillations for networks of passive oscillators
                                                              identical passive oscillators


                                                              Conclusions
   Extension of the results for one passive oscillator
   Sync. in networks of identical passive oscillators

Conclusions



                                                                                    3/24
Dissipativity and oscillations                                    Dissipativity theory
                                                                     for oscillator
                                                                       analysis

                                                                   Guy-Bart STAN


                                                                  Introduction
Results:                                                          Global oscillations
                                                                  for the passive
    global analysis of oscillations                               oscillator
                                                                  The passive oscillator
                                                                  Global osc. for the passive
           high-dimensional (global) oscillators                  oscillator


           networks of oscillators                                Global oscillations
                                                                  for networks of
           synchronization in networks of identical oscillators   passive oscillators
                                                                  Extension of the results for

    synthesis of oscillations
                                                                  one passive oscillator
                                                                  Sync. in networks of
                                                                  identical passive oscillators

           simple method for generating oscillations in systems   Conclusions



Approach:
Dissipativity theory ≡ “efficient tool for global analysis
and synthesis of oscillations”



                                                                                        4/24
Dissipativity and oscillations                                    Dissipativity theory
                                                                     for oscillator
                                                                       analysis

                                                                   Guy-Bart STAN


                                                                  Introduction
Results:                                                          Global oscillations
                                                                  for the passive
    global analysis of oscillations                               oscillator
                                                                  The passive oscillator
                                                                  Global osc. for the passive
           high-dimensional (global) oscillators                  oscillator


           networks of oscillators                                Global oscillations
                                                                  for networks of
           synchronization in networks of identical oscillators   passive oscillators
                                                                  Extension of the results for

    synthesis of oscillations
                                                                  one passive oscillator
                                                                  Sync. in networks of
                                                                  identical passive oscillators

           simple method for generating oscillations in systems   Conclusions



Approach:
Dissipativity theory ≡ “efficient tool for global analysis
and synthesis of oscillations”



                                                                                        4/24
Outline                                                       Dissipativity theory
                                                                 for oscillator
                                                                   analysis

                                                               Guy-Bart STAN


                                                              Introduction
Introduction                                                  Global oscillations
                                                              for the passive
                                                              oscillator
                                                              The passive oscillator

Global oscillations for the passive oscillator                Global osc. for the passive
                                                              oscillator

   The passive oscillator                                     Global oscillations
                                                              for networks of
   Global oscillation mechanisms for the passive oscillator   passive oscillators
                                                              Extension of the results for
                                                              one passive oscillator
                                                              Sync. in networks of

Global oscillations for networks of passive oscillators
                                                              identical passive oscillators


                                                              Conclusions
   Extension of the results for one passive oscillator
   Sync. in networks of identical passive oscillators

Conclusions



                                                                                    5/24
The passive oscillator                                                                            Dissipativity theory
                                                                                                     for oscillator
                                                                                                       analysis

                                                                                                   Guy-Bart STAN

                                                                         Limit                    Introduction
                                                                         Cycle                    Global oscillations
u                            Passive                            y                                 for the passive
                                                                                                  oscillator
       −                                                                               x2         The passive oscillator

                     static nonlinearity                                                          Global osc. for the passive
                                                                                                  oscillator


                                                                                                  Global oscillations
             φk (y )                                                                         x1   for networks of
                                                                                                  passive oscillators
            = −ky                                                                                 Extension of the results for
                                                                                                  one passive oscillator
            +φ(y )              −k                                                                Sync. in networks of
                                                                                                  identical passive oscillators


                                                                                                  Conclusions
    Includes two well-known low-dimensional oscillators:
    VAN DER P OL and F ITZHUGH -N AGUMO
    Characterization by a specific dissipation inequality:
                                                                 >0
            ˙                                       2
            S            ≤     k−    kpassive
                                      ∗
                                                    y −       y φ(y )          +      uy
     storage variation                                    global dissipation       ext. supply
                                 local activation

                                                                                                                        6/24
Our results on this class of systems                        Dissipativity theory
                                                               for oscillator
                                                                 analysis

                                                             Guy-Bart STAN


                                                            Introduction

                                                            Global oscillations
                           Passive               y          for the passive
                                                            oscillator
          −                                                 The passive oscillator

                    static nonlinearity                     Global osc. for the passive
                                                            oscillator


                                                            Global oscillations
              φk (y )                                       for networks of
                                                            passive oscillators
                                                            Extension of the results for
                                                            one passive oscillator
                                 −k                         Sync. in networks of
                                                            identical passive oscillators


                                                            Conclusions
              0         Stable        k∗   Unstable
              GAS                Bifurcation          k

Generically two types of bifurcation (H OPF or pitchfork)



                                                                                  7/24
First scenario: H OPF bifurcation (1)                            Dissipativity theory
                                                                    for oscillator
                                                                      analysis

                                                                  Guy-Bart STAN

             st
Theorem (1 result)                                               Introduction

Passivity for k ≤ k∗ and two eigenvalues on the                  Global oscillations
                                                                 for the passive
imaginary axis at k = k ∗ implies global oscillation through     oscillator
                                                                 The passive oscillator

H OPF bifurcation for k k ∗                                      Global osc. for the passive
                                                                 oscillator


                                                                 Global oscillations
    at k = k ∗                                Limit cycle
                                                                 for networks of
                                                                 passive oscillators
                                                                 Extension of the results for
                                                                 one passive oscillator
                                                                 Sync. in networks of
                                                                 identical passive oscillators


                       Stable                     Unstable       Conclusions

                                k∗                           k


                       GAS           Globally
                                     Attractive


                                                                                       8/24
First scenario: H OPF bifurcation (2)                    Dissipativity theory
                                                            for oscillator
                                                              analysis

                                                          Guy-Bart STAN
A ’basic’ global oscillation mechanism in
electro-mechanical systems                               Introduction

Simplest example: VAN DER P OL oscillator                Global oscillations
                                                         for the passive
                                                         oscillator
                                                         The passive oscillator
                                         Passive
                                                         Global osc. for the passive
                                                         oscillator
                                             1
                                             s           Global oscillations
                   i = φk (v )   −   −
                                                         for networks of
                                             1
      L        C                             s           passive oscillators
                                                         Extension of the results for
                                                         one passive oscillator
                                                         Sync. in networks of
                                          φk (·)         identical passive oscillators


                                                         Conclusions

Global oscillation mechanism:
    Continuous lossless exchange of energy between
    the storage elements
    Static nonlinear element regulates the sign of the
    dissipation

                                                                               9/24
H OPF scenario: example                                                                                                                                                                         Dissipativity theory
                                                                                                                                                                                                   for oscillator
                                                                                                                                                                                                     analysis

                                                                                                                                                                                                 Guy-Bart STAN
                                 Passive
                                                                                                                            τ s+ωn2
                                                                           y
                                  H(s)                                                                         H(s) = s2 +2ζω s+ω2                                                              Introduction
−              −                                                                                                                n   n
                                                                                                                                                                                                Global oscillations
                                       1
                                       s                                                                       φk (y ) = y 3 − ky                                                               for the passive
                                                                                                                                                                                                oscillator
                                   φk (·)                                                                                                                                                       The passive oscillator
                                                                                                                                                                                                Global osc. for the passive
                                                                                                                                                                                                oscillator

                                                                        State-space (k ∗ = 1)                                                                                                   Global oscillations
                                                                                                                                                                                                for networks of
                                                                                                                                                                                                passive oscillators
                                                                                                                                                                                                Extension of the results for
                                 State−space of a SINGLE oscillator for kp=9.000000e−01                                      State−space of a SINGLE oscillator for kp=1.100000e+00             one passive oscillator
                                                                                                                                                                                                Sync. in networks of
                                                                                                                                                                                                identical passive oscillators

         1.4                                                                                             1.5

         1.2
                                                                                                                                                                                                Conclusions
          1
                                                                                                          1
         0.8

         0.6
                                                                                                         0.5
    ξ




                                                                                                    ξ




         0.4

         0.2
                                                                                                          0
          0

        −0.2

        −0.4                                                                                            −0.5
         1.5                                                                                             1.5

                   1                                                                            1              1                                                                            1
                                                                                          0.5                                                                                         0.5
                       0.5                                                                                         0.5
                                                                                   0                                                                                           0
                                        0                                                                                           0
                                                                    −0.5                                                                                        −0.5
                        X                      −0.5    −1                                                           X                      −0.5    −1
                             2                                                X                                          2                                                X
                                                                               1                                                                                           1




                                         k = 0.9                                                                                     k = 1.1
                                                                                                                                                                                                                      10/24
Second scenario: pitchfork bifurcation (1)                     Dissipativity theory
                                                                  for oscillator
                                                                    analysis

Theorem (2nd result)                                            Guy-Bart STAN


Passivity for k ≤k∗   and one eigenvalue on the imaginary      Introduction

axis at k = k ∗ implies global bistability through pitchfork   Global oscillations
                                                               for the passive
bifurcation for k k ∗                                          oscillator

(Slow) “adaptation” converts the bistable system into a
                                                               The passive oscillator
                                                               Global osc. for the passive
                                                               oscillator

global oscillator                                              Global oscillations
     at k = k ∗                            Eq. point
                                                               for networks of
                                                               passive oscillators
                                                               Extension of the results for
                                                               one passive oscillator
                                                               Sync. in networks of
                                                               identical passive oscillators


                      Stable                   Unstable        Conclusions


                               k∗                         k


                       GAS          Globally
                                    Bistable


                                                                                     11/24
Second scenario: pitchfork bifurcation (1)                                      Dissipativity theory
                                                                                   for oscillator
                                                                                     analysis

Theorem (2nd result)                                                             Guy-Bart STAN


Passivity for k ≤     and one eigenvalue on the imaginary
                     k∗                                                         Introduction

axis at k = k ∗ implies global bistability through pitchfork                    Global oscillations
                                                                                for the passive
bifurcation for k k ∗                                                           oscillator

(Slow) “adaptation” converts the bistable system into a
                                                                                The passive oscillator
                                                                                Global osc. for the passive
                                                                                oscillator

global oscillator                                                               Global oscillations
                                                                                for networks of
                                                                                passive oscillators
                                                                                Extension of the results for
                                                                                one passive oscillator
  k   k ∗ , without adaptation            k   k ∗ , with adaptation             Sync. in networks of
                                                                                identical passive oscillators
                                                            x1
                                                  Relaxation Oscillation        Conclusions


        stable   unstable   stable
                                     x2                                    x2




                                                                                                      11/24
Second scenario: pitchfork bifurcation (1)                     Dissipativity theory
                                                                  for oscillator
                                                                    analysis

Theorem (2nd result)                                            Guy-Bart STAN


Passivity for k ≤k∗   and one eigenvalue on the imaginary      Introduction

axis at k = k ∗ implies global bistability through pitchfork   Global oscillations
                                                               for the passive
bifurcation for k k ∗                                          oscillator

(Slow) “adaptation” converts the bistable system into a
                                                               The passive oscillator
                                                               Global osc. for the passive
                                                               oscillator

global oscillator                                              Global oscillations
                                                               for networks of
                                                               passive oscillators
                                                               Extension of the results for
                                                               one passive oscillator


                          Passive
                                                               Sync. in networks of
                                                               identical passive oscillators


        −        −                                             Conclusions




                           φk (·)

                               1
                            τ s+1
                                    τ     0
                                                                                     11/24
Second scenario: pitchfork bifurcation (2)                        Dissipativity theory
                                                                     for oscillator
                                                                       analysis

                                                                   Guy-Bart STAN


                                                                  Introduction
A ’basic’ global oscillation mechanism in biology                 Global oscillations
Simplest example: F ITZHUGH -N AGUMO oscillator                   for the passive
                                                                  oscillator
                                                                  The passive oscillator

                                                Passive           Global osc. for the passive
                                                                  oscillator

                                                  1               Global oscillations
                                                  s               for networks of
                                        −   −
                                                                  passive oscillators
                                                                  Extension of the results for
    all ions             all ions               φk (·)            one passive oscillator
    inside E+         E− outside    V                             Sync. in networks of
    the cell Adaptation the cell                                  identical passive oscillators
                                                   1
                                                τ s+1             Conclusions
                                                          τ   0


Global oscillation mechanism:
    Continuous switch between 2 quasi stable eq. points



                                                                                        12/24
Pitchfork scenario: example                                                                                                                                 Dissipativity theory
                                                                                                                                                               for oscillator
                                                                                                                                                                 analysis

                                                                                                                                                             Guy-Bart STAN
                  Passive                                                                               τ s+ωn2
                                                                                           H(s) = s2 +2ζω s+ω2                                              Introduction
                                                 y                                                          n   n
                  H(s)                                                                                 3 − ky
                                                                                                                                                            Global oscillations

−                                                                                          φk (y ) = y                                                      for the passive
                                                                                                                                                            oscillator
                                                                                           Adaptation                                                       The passive oscillator

                  φk (·)                                                                                                                                    Global osc. for the passive
                                                                                                                                                            oscillator


                                                                                                                                                            Global oscillations
                                                                                                                                                            for networks of
                                                      State-space (k ∗ = 1)                                                                                 passive oscillators
                        State−space for k =1 and k =9.000000e−01
                                        i        p
                                                                                                          State−space for ki=1 and kp=2                     Extension of the results for
          0.5                                                                      1.5                                                                      one passive oscillator

          0.4
                                                                                                                                                            Sync. in networks of
                                                                                                                                                            identical passive oscillators
                                                                                    1
          0.3


          0.2
                                                                                                                                                            Conclusions
                                                                                   0.5

          0.1


           0                                                                        0
                                                                             X2
     2
    X




         −0.1

                                                                                  −0.5
         −0.2


         −0.3
                                                                                   −1

         −0.4


         −0.5                                                                     −1.5
           −1.5    −1    −0.5               0            0.5       1   1.5          −0.4    −0.3   −0.2   −0.1         0           0.1    0.2   0.3   0.4
                                            X1                                                                         X1




    k = 0.9, without adaptation                                              k = 2, without adaptation
                                                                                                                                                                                  13/24
Pitchfork scenario: example                                                                                                  Dissipativity theory
                                                                                                                                for oscillator
                                                                                                                                  analysis

                                                                                                                              Guy-Bart STAN
        Passive
                                                                                                              τ s+ωn2

        H(s)          y                                                                          H(s) = s2 +2ζω s+ω2         Introduction
                                                                                                                  n   n
−   −                                                                                                                        Global oscillations

        φk (·)                                                                                   φk (y ) = y 3 − ky          for the passive
                                                                                                                             oscillator
                                                                                                 Adaptation                  The passive oscillator
                                                                                                                             Global osc. for the passive
           1                                                                                                                 oscillator
        τ s+1
                  τ             0                                                                                            Global oscillations
                                                                                                                             for networks of
                                                                                                                             passive oscillators
                                      State-space (k ∗ = 1)                                                                  Extension of the results for
                                               State−space of a SINGLE relaxation oscillator for ki=1 and kp=2               one passive oscillator
                                1.5
                                                                                                                             Sync. in networks of
                                                                                                                             identical passive oscillators
                                 1
                                                                                                                             Conclusions
                                0.5




                                 0
                          X2




                               −0.5




                                −1




                               −1.5
                                 −0.3   −0.2          −0.1            0            0.1            0.2            0.3   0.4
                                                                             X1




                                  k = 2, with adaptation
                                                                                                                                                   13/24
Outline                                                       Dissipativity theory
                                                                 for oscillator
                                                                   analysis

                                                               Guy-Bart STAN


                                                              Introduction
Introduction                                                  Global oscillations
                                                              for the passive
                                                              oscillator
                                                              The passive oscillator

Global oscillations for the passive oscillator                Global osc. for the passive
                                                              oscillator

   The passive oscillator                                     Global oscillations
                                                              for networks of
   Global oscillation mechanisms for the passive oscillator   passive oscillators
                                                              Extension of the results for
                                                              one passive oscillator
                                                              Sync. in networks of

Global oscillations for networks of passive oscillators
                                                              identical passive oscillators


                                                              Conclusions
   Extension of the results for one passive oscillator
   Sync. in networks of identical passive oscillators

Conclusions



                                                                                    14/24
Networks of oscillators                                    Dissipativity theory
                                                              for oscillator
                                                                analysis

                                                            Guy-Bart STAN


                                                           Introduction

                                                           Global oscillations
                                                           for the passive
                                                           oscillator
                                                           The passive oscillator
                                                           Global osc. for the passive
                                                           oscillator


                                                           Global oscillations
                                                           for networks of
                                                           passive oscillators
                                                           Extension of the results for
                                                           one passive oscillator
                                                           Sync. in networks of
                                                           identical passive oscillators


                                                           Conclusions




In nature, oscillations are the result of interconnected
oscillators!


                                                                                 15/24
MIMO representation of a network of passive                                        Dissipativity theory
                                                                                      for oscillator
                                                                                        analysis
oscillators                                                                         Guy-Bart STAN

                                      Passive
                                                   y1                              Introduction
     W         U                        P1                       Y                 Global oscillations
                                                   yN                              for the passive
         −         −                    PN                                         oscillator
                                                                                   The passive oscillator
                                                                                   Global osc. for the passive
                                                                                   oscillator
                                      φk (y1 )
                        Φk (Y )                                                    Global oscillations
                                                                                   for networks of
                                      φk (yN )                                     passive oscillators
                                                                                   Extension of the results for
                                                                                   one passive oscillator
                                                                                   Sync. in networks of
                                    COUPLING                                       identical passive oscillators

                                        (Γ)                                        Conclusions



Characterization through dissipativity theory

                                              ≥0

   ˙
   S≤ k−      kpassive
               ∗
                           Y Y − Y Φ(Y ) −Y T ΓY + W T Y
                                T         T

                                     global dissipation   coupling   ext. supply
             local activation
                                                                                                         16/24
Global oscillations for networks (1)                                Dissipativity theory
                                                                       for oscillator
                                                                         analysis

                                                                     Guy-Bart STAN


Question: What are the topologies that lead to global               Introduction

oscillations in the network?                                        Global oscillations
                                                                    for the passive
Answer: Passive coupling (Γ ≥ 0)                                    oscillator
                                                                    The passive oscillator


Characterization (analogue to that for 1 oscillator!)               Global osc. for the passive
                                                                    oscillator


                                                                    Global oscillations
                                                                    for networks of
    S ≤ k − kpassive Y T Y − Y T Φ(Y ) + W T Y
    ˙        ∗                                                      passive oscillators
                                                                    Extension of the results for
                                                                    one passive oscillator

                                 global dissipation   ext. supply   Sync. in networks of
              local activation                                      identical passive oscillators


                                                                    Conclusions

Consequence:   1st (H OPF) and   2nd(pitchfork +
adaptation) results generalize to networks of passive
oscillators
Identical osc.: behaviour of the network may be deduced
from that of its constituting oscillators


                                                                                          17/24
Global oscillations for networks (1)                                Dissipativity theory
                                                                       for oscillator
                                                                         analysis

                                                                     Guy-Bart STAN


Question: What are the topologies that lead to global               Introduction

oscillations in the network?                                        Global oscillations
                                                                    for the passive
Answer: Passive coupling (Γ ≥ 0)                                    oscillator
                                                                    The passive oscillator


Characterization (analogue to that for 1 oscillator!)               Global osc. for the passive
                                                                    oscillator


                                                                    Global oscillations
                                                                    for networks of
    S ≤ k − kpassive Y T Y − Y T Φ(Y ) + W T Y
    ˙        ∗                                                      passive oscillators
                                                                    Extension of the results for
                                                                    one passive oscillator

                                 global dissipation   ext. supply   Sync. in networks of
              local activation                                      identical passive oscillators


                                                                    Conclusions

Consequence:   1st (H OPF) and   2nd(pitchfork +
adaptation) results generalize to networks of passive
oscillators
Identical osc.: behaviour of the network may be deduced
from that of its constituting oscillators


                                                                                          17/24
Global oscillations for networks (1)                                Dissipativity theory
                                                                       for oscillator
                                                                         analysis

                                                                     Guy-Bart STAN


Question: What are the topologies that lead to global               Introduction

oscillations in the network?                                        Global oscillations
                                                                    for the passive
Answer: Passive coupling (Γ ≥ 0)                                    oscillator
                                                                    The passive oscillator


Characterization (analogue to that for 1 oscillator!)               Global osc. for the passive
                                                                    oscillator


                                                                    Global oscillations
                                                                    for networks of
    S ≤ k − kpassive Y T Y − Y T Φ(Y ) + W T Y
    ˙        ∗                                                      passive oscillators
                                                                    Extension of the results for
                                                                    one passive oscillator

                                 global dissipation   ext. supply   Sync. in networks of
              local activation                                      identical passive oscillators


                                                                    Conclusions

Consequence:   1st (H OPF) and   2nd(pitchfork +
adaptation) results generalize to networks of passive
oscillators
Identical osc.: behaviour of the network may be deduced
from that of its constituting oscillators


                                                                                          17/24
Global oscillations for networks (1)                                Dissipativity theory
                                                                       for oscillator
                                                                         analysis

                                                                     Guy-Bart STAN


Question: What are the topologies that lead to global               Introduction

oscillations in the network?                                        Global oscillations
                                                                    for the passive
Answer: Passive coupling (Γ ≥ 0)                                    oscillator
                                                                    The passive oscillator


Characterization (analogue to that for 1 oscillator!)               Global osc. for the passive
                                                                    oscillator


                                                                    Global oscillations
                                                                    for networks of
    S ≤ k − kpassive Y T Y − Y T Φ(Y ) + W T Y
    ˙        ∗                                                      passive oscillators
                                                                    Extension of the results for
                                                                    one passive oscillator

                                 global dissipation   ext. supply   Sync. in networks of
              local activation                                      identical passive oscillators


                                                                    Conclusions

Consequence:   1st (H OPF) and   2nd(pitchfork +
adaptation) results generalize to networks of passive
oscillators
Identical osc.: behaviour of the network may be deduced
from that of its constituting oscillators


                                                                                          17/24
Global oscillations for networks (2)                       Dissipativity theory
                                                              for oscillator
                                                                analysis

                                                            Guy-Bart STAN


                                                           Introduction

                                                           Global oscillations
                                                           for the passive
                                                           oscillator
                                                           The passive oscillator
                                                           Global osc. for the passive
                                                           oscillator

Dissipativity is useful for proving global limit cycle     Global oscillations
oscillations in networks composed of a large number of     for networks of
                                                           passive oscillators
oscillators with various topologies including all-to-all   Extension of the results for
                                                           one passive oscillator

coupling, bidirectional ring coupling, etc.                Sync. in networks of
                                                           identical passive oscillators


                                                           Conclusions




                                                                                 18/24
Synchronization and incremental dissipativity            Dissipativity theory
                                                            for oscillator
                                                              analysis

Question: Under which conditions do all oscillators       Guy-Bart STAN

synchronize? (global synchronization)
                                                         Introduction
Approach                                                 Global oscillations
                                                         for the passive
Incremental dissipativity ≡ dissipativity expressed in   oscillator
                                                         The passive oscillator
terms of the difference between solutions of systems     Global osc. for the passive
                                                         oscillator


                                                         Global oscillations
                                                         for networks of
                                                         passive oscillators
                                                         Extension of the results for
                                                         one passive oscillator
                                                         Sync. in networks of
                                                         identical passive oscillators


                                                         Conclusions




                                                                               19/24
Synchronization and incremental dissipativity            Dissipativity theory
                                                            for oscillator
                                                              analysis

Question: Under which conditions do all oscillators       Guy-Bart STAN

synchronize? (global synchronization)
                                                         Introduction
Approach                                                 Global oscillations
                                                         for the passive
Incremental dissipativity ≡ dissipativity expressed in   oscillator
                                                         The passive oscillator
terms of the difference between solutions of systems     Global osc. for the passive
                                                         oscillator


                                                         Global oscillations
                                                         for networks of
                                                         passive oscillators
                                                         Extension of the results for
                                                         one passive oscillator
                                                         Sync. in networks of
                                                         identical passive oscillators


                                                         Conclusions




                                                                               19/24
Synchronization and incremental dissipativity                 Dissipativity theory
                                                                 for oscillator
                                                                   analysis

Question: Under which conditions do all oscillators            Guy-Bart STAN

synchronize? (global synchronization)
                                                              Introduction
Approach                                                      Global oscillations
                                                              for the passive
Incremental dissipativity ≡ dissipativity expressed in        oscillator
                                                              The passive oscillator
terms of the difference between solutions of systems          Global osc. for the passive
                                                              oscillator


                        A           B                         Global oscillations
                                                              for networks of
                                                              passive oscillators
                                                              Extension of the results for
                                                              one passive oscillator
                                                              Sync. in networks of
                            C=A-B                             identical passive oscillators


                                                              Conclusions



Do A and B synchronize?
    Study stability of C through dissipativity theory: if C
    is stable then A and B synchronize
    Stability of C generally depends on the topology of
    the network
                                                                                    19/24
Synchronization and incremental dissipativity                     Dissipativity theory
                                                                     for oscillator
                                                                       analysis

Incremental dissipativity is useful to prove global                Guy-Bart STAN

synchrone oscillations in networks with specific
                                                                  Introduction
topologies including:                                             Global oscillations
                                                                  for the passive
                                                                  oscillator
             O1        O2                  O1       O2
                                                                  The passive oscillator
                                                                  Global osc. for the passive
                                                                  oscillator


                                                                  Global oscillations
             O4        O3                  O4       O3
                                                                  for networks of
                                                                  passive oscillators
              All-to-all                Bidirectional ring        Extension of the results for
                                                                  one passive oscillator
             O1        O2                                         Sync. in networks of
                                                                  identical passive oscillators


                                                                  Conclusions

             O4        O3          O1      O2     ···        ON


      Unidirectional ring                 Open chain

    Γ ≥ 0,
                            “               ”
    ker (Γ) = ker ΓT = range (1, . . . , 1)T ,
                 ` ´

    λmin,=0 (Γs ) > k − kpassive
                         ∗

                                                                                        20/24
Synchronization and incremental dissipativity                                                     Dissipativity theory
                                                                                                     for oscillator
                                                                                                       analysis

                                                                                                   Guy-Bart STAN


                                                                                                  Introduction
Global synchrone oscillation
                                                                                                  Global oscillations
                                                                                                  for the passive
                             Time evolution of the five outputs for kp=2                          oscillator
            3                                                                                     The passive oscillator
                                                                                     y (t)
                                                                                      1
                                                                                     y (t)
                                                                                      2
                                                                                                  Global osc. for the passive
           2.5                                                                       y3(t)        oscillator
                                                                                     y4(t)
            2                                                                        y5(t)
                                                                                                  Global oscillations
                                                                                                  for networks of
           1.5
                                                                                                  passive oscillators
                                                                                                  Extension of the results for
            1
                                                                                                  one passive oscillator
                                                                                                  Sync. in networks of
           0.5                                                                                    identical passive oscillators

            0                                                                                     Conclusions
          −0.5


           −1


          −1.5


           −2
                 0   2   4   6         8         10        12         14   16   18           20




                                                                                                                        21/24
Outline                                                       Dissipativity theory
                                                                 for oscillator
                                                                   analysis

                                                               Guy-Bart STAN


                                                              Introduction
Introduction                                                  Global oscillations
                                                              for the passive
                                                              oscillator
                                                              The passive oscillator

Global oscillations for the passive oscillator                Global osc. for the passive
                                                              oscillator

   The passive oscillator                                     Global oscillations
                                                              for networks of
   Global oscillation mechanisms for the passive oscillator   passive oscillators
                                                              Extension of the results for
                                                              one passive oscillator
                                                              Sync. in networks of

Global oscillations for networks of passive oscillators
                                                              identical passive oscillators


                                                              Conclusions
   Extension of the results for one passive oscillator
   Sync. in networks of identical passive oscillators

Conclusions



                                                                                    22/24
Conclusions                                                Dissipativity theory
                                                              for oscillator
                                                                analysis

                                                            Guy-Bart STAN


                                                           Introduction

                                                           Global oscillations
                                                           for the passive
                                                           oscillator
                                                           The passive oscillator
Dissipativity allows us to                                 Global osc. for the passive
                                                           oscillator


    Uncover 2 ’basic’ global oscillations mechanisms in    Global oscillations
                                                           for networks of
    high-dimensional systems                               passive oscillators
                                                           Extension of the results for

    Generalize these results for networks of oscillators
                                                           one passive oscillator
                                                           Sync. in networks of
                                                           identical passive oscillators

    Obtain global synchrone oscillation results            Conclusions




                                                                                 23/24
Thank you for your attention                   Dissipativity theory
                                                  for oscillator
                                                    analysis

                                                Guy-Bart STAN


                                               Introduction

                                               Global oscillations
                                               for the passive


               Questions?
                                               oscillator
                                               The passive oscillator
                                               Global osc. for the passive
                                               oscillator


                                               Global oscillations
                                               for networks of
                                               passive oscillators
                                               Extension of the results for
                                               one passive oscillator
                                               Sync. in networks of
                                               identical passive oscillators


                                               Conclusions
           (Ph.D. thesis available online at
          www.montefiore.ulg.ac.be/~stan/)




                                                                     24/24

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:15
posted:8/1/2011
language:English
pages:33