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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

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