Dynamics and Stability of Phase Controlled Oscillators

This paper systematically analyzes linear oscillators, e.g., spring-mass-damper systems or RLC-circuits that are controlled by an extension of a phase-locked loop (PLL). These systems are often used in measurement applications where the stability and dynamics directly influence the measurement quality. Therefore, a description of the control loop in terms of phase signals is sought. However, the classical oscillator turns into a highly nonlinear system when it is formulated in amplitude/phase-variables of its input and output signals. Up to now, there were made either ab-initio assumptions of slowly varying parameters or trial-and-error designs. The novel approach proposed in this paper derives a universally valid description in state space form which enables the use of standard methods of nonlinear system theory. Using this description, the stability of phase controlled oscillators is analyzed by means of Lyapunov functions. A linearization is applied in order to effectively design the controller and optimize the closed-loop dynamics. Simulations with the original nonlinear systems are conducted to justify the linear approach. Thereby, two application scenarios are under consideration: Tracking of the desired target value (target phase shift) and resonance tracking (changes of the system parameters). It is found that including the phase dynamics of the oscillator significantly improves the description of the closed-loop behavior. Finally, the results are validated experimentally for an application measuring the viscosity of fluids.