Improving frequency stability using slowly modulated adaptive feedback

Abstract

We present a new method for improving frequency stability in a self-sustained oscillator using a secondary feedback loop acting outside the main sustaining loop with adaptively-controlled amplitude and phase shift. We show that quite simple adaptive control laws for the control states of the secondary feedback signal can affect the way that noise circulates in the oscillator so as to reduce or even eliminate (in theory) phase diffusion, i.e., the rate of linear growth with time of the variance of the oscillator output phase. Our rigorous treatment of this effect is based on a linearized analysis of the noisy slow-flow amplitude and phase equations for a relevant general class of systems, from which we derive an explicit expression for the corresponding asymptotic rate of phase diffusion. Using this result, we consider different choices for tailoring the actuation from the secondary feedback loop. This includes tuning a phase coupling constant that ensures that the noise driving the output phase also drives one of the adaptive control states, which represents a generalization of a desirable behavior observed in internally resonant coupled mode operation. We show that such feedback design may be used to eliminate effects that commonly arise from the conversion of amplitude fluctuations to phase diffusion, analogous to operation at zero dispersion points, and to improve upon the phase cleaning effect associated with internal resonance to even achieve the ideal situation of zero phase diffusion. We validate our theoretical findings with numerical simulations that agree remarkably well with the theory. The presented results establish a framework for achieving extraordinary frequency stability using slowly varying states of the resonator and feedback.