How is phase lag/lead provided using a phase-lag lead filter?

[amsdesign]

Hello,

I have a negative feedback control system (a PLL)
The loop filter is a phase lead-lag filter. Can you tell me how such a filter can compensate for leading and lagging actions of the input signals ?

This is part of the text that says so, how exactly does does it compensate?

A note on terminology: a lead-lag[also called lag-lead] filter combines a phase-leading with a phase-lagging net-work. The phase-leading action comes from the numerator [in other words, from the zero] in the transfer function in Eq.(2.29a), whereas the denominator [that is, the pole] produces the phase lag. All filters used as loop filters are lead-lagfilters)

Here is the filter :

 

[FvM]

Can you tell me how such a filter can compensate for leading and lagging actions of the input signals ?

No, because the statement doesn't make sense.

A "lag-lead" loop filter (lag-lead because the pole comes before the zero) acts as a low-pass filter without reducing the loop phase margin too much. It's necessary because the VCO acts as an intergrator, so a simple first order low-pass filter would reduce the phase margin to zero and result in an unstable loop. By placing the lead zero before or near to the loop crossover frequency, you can achieve sufficient phase margin.

The RC circuit does only act as a filter in combination with the source impedance or a series resistor.