This is the enhanced version of the single phase PLL, based on the basic version of the single phase PLL, avoiding the mentioned double-frequency ripple on the estimated frequency.
The output y[1] is a cosine signal in phase with
the input u, the output y[2] is a sine
signal 90° phase shifted to y[1]. The output
y[3] is a cosine signal in phase with the input
u and the estimated amplitude, i.e. the fundamental
harmonic.