Networks of Passive Oscillators

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Presentation transcript:

Networks of Passive Oscillators Vishwesh V. Kulkarni, Marc Riedel, and Guy-Bart Stan

Outline Passive oscillators using passive systems and static nonlinearities Modified MIMO Lure’ system Supercritical Hopf/Pitchfork bifurcation Networks of such passive oscillators Identical or non-identical How to choose the interconnection ? We establish a class of suitable - - -

Passive Oscillator - Stable, LTI Static Stiffening Nonlinearity If the system possesses a globally stable limit cycle that attracts all solutions except those in the origin’s stable manifold.

Main Problem: Oscillator Network - - Given: ODE model specifying and Qn: How to choose the coupling so that is oscillatory?

Passivity and Dissipativity

Dissipativity w.r.t. a Special w(u,y) supply local activation global dissipation For example:

Loop-Shift Transformed System where + Find a positivity preserving multiplier & finite normed N  stability. … Zames-Falb (1968)

Zames-Falb Multipliers Im s-plane Re

Passive Oscillator - If the system possesses a stable global limit cycle that attracts all solutions except those in the stable origin’s manifold.

Bifurcation for Oscillations: Hopf Global oscillations through Hopf bifurcation for if there exists a ZF multiplier such that is strongly passive for ; and has two eigenvalues on the axis at s-plane X stable unstable X (Stan-Sepulchre, 2007)

Bifurcation for Oscillations: Pitchfork Global oscillations through pitchfork bifurcation for if there exists a ZF multiplier such that is strongly passive for ; and has an eigenvalue on the axis at Fitzhuh-Nagumo Oscillator: … slow adaption added to enforce the relaxation phase (Stan-Sepulchre, 2007)

Networks of Passive Oscillators - Forcing Input Oscillatory Output Network of oscillators i-th Oscillator If the system possesses a stable global limit cycle that attracts all solutions except those in the stable origin’s manifold.

Feedback System Representation - - Given: ODE model specifying and Qn: How to choose the coupling so that is oscillatory?

Redrawn Oscillator Network - - repeated static monotone L2-stability of this system using relevant multipliers Substitution of those conditions in the Stan-Sepulchre results

KS Multipliers is a positive operator (Kulkarni-Safonov, 2002) Impulse response of a KS multiplier is given by is a positive operator (Kulkarni-Safonov, 2002) Hence, is L2-stable if a KS multiplier s.t.

Networked Oscillators: Bifurcation

Networked Oscillations

Summary Passive oscillators using passive systems and static nonlinearities Modified MIMO Lure’ system Supercritical Hopf/Pitchfork bifurcation Networks of such passive oscillators How to choose the interconnection ? We establish a class of suitable Direct extension of Stan-Sepulchre Asymmetry & non-identical oscillators considered We hope to reduce the global results to local results - -

Questions?

Thank You! Research Supported by NSF CAREER Award 0845650