Adaptive phasor control of a Duffing oscillator with unknown parameters

Abstract

Weakly damped electrical and mechanical oscillators that contain a cubic nonlinearity are described mathematically using Duffing's equation. In particular microelectromechanical systems (MEMS) exhibit mechanical structures that are characterized by nonlinear elasticity due to a hardening or softening spring constant. Driving these nonlinear oscillators by sinusoidal excitation with varying frequency and amplitude we observe typical characteristics of nonlinear oscillators in the frequency response as jump phenomena at bifurcation points. Especially in applications with varying physical parameters due to changes in environmental conditions or external disturbances a stable operation in resonance cannot be guaranteed by applying conventional Phase Locked Loops (PLLs). In this work we propose an adaptive phasor control approach that is based on a phasor representation of the linearized Duffing oscillator in order to control the amplitude and phase of the oscillation separately. The Duffing oscillator is linearized adaptively by using a dynamic parameter estimator for the unknown nonlinear spring constant.