Basics of Autonomous Nonlinear Oscillators: Limit Cycle, Orbital Stability, and Synchronization

Abstract

Periodic motions are often desired in engineering applications. The traditional method of trajectory planning followed by feedback tracking may work well for maintaining the planned motion. However, it is desired for certain applications to achieve the motion as autonomous rather than forced oscillations so as to adapt the motion profile in response to disturbance and changing environment. This motivates us to consider feedback control design to embed a limit cycle oscillation in the closed-loop dynamics with the notion of orbital stability. Here we provide a tutorial review of basic concepts and analysis tools for limit cycles arising from autonomous nonlinear dynamics. The Floquet theory and contraction analysis are reviewed, simple planar oscillators are illustrated, and synchronization and coordination of coupled oscillators are discussed.

Keywords

• oscillation
• limit cycle
• orbital stability
• synchronization
• coordination