$H_{\infty}$ Robust Second-Order Generalized Integrator Phase-Locked Loop for Grid Synchronization of Single-Phase Inverters

This article proposes a new design framework based on $H_{\infty}$ control theory to synthesize the loop filter (LF) of phase-locked loops (PLLs). Taking the frequency-fixed and frequency-adaptive second-order generalized integrator (SOGI) PLLs as representative cases, the control problem is formulated in terms of the PLL's voltage-controlled oscillator while considering the influence of the SOGI block on the PLL dynamics. By specifying the weighting function as a function of PLL settling time in the $H_{\infty}$ control problem, optimal LFs are obtained to achieve robustness in the PLL's tracking performances. Furthermore, reduction of the LF's implementations and computational costs without loss in performance are achieved by means of Hankel singular value approximation. Performance-wise, the proposed SOGI PLLs provide robust estimations of a single-phase voltage's frequency and phase angle when compared with their existing pairs through experimental studies. Finally, employing the proposed PLLs for grid synchronization of a single-phase inverter demonstrates reduced power oscillation when compared to the conventional proportional-integral control-based PLLs.