Understanding of the stability criterion for a double-feedback loop system

Recently, low-frequency instability has been observed in single-phase AC railway systems with modern railway vehicles, equiped with asynchronous motor and PWM line converter. Multiple approaches to study such a phenomenon have been introduced. One of the proposed impedance-based methods makes use of dq-decomposition of signals - a technique that is commonly used for three-phase systems. Such a representation results in a multiple-input multiple-output control system. In the present article, this complex control loop is studied using well-established control-theory techniques, such as the Nyquist criterion. A stability criterion is proposed and tested on a sample system.