I have a question regarding lead high pass filter and lag low pass filter.
I do not quite understand the reasoning behind these two special case (when z=0) ?
Lag low-pass and lead high-pass are in fact the "standard" low-pass and high-pass filters, in the sense that an ideal low-pass filter should have a gain of zero for \$\omega\rightarrow\infty\$, and an ideal high-pass filter should have a gain of zero at \$\omega=0\$. These conditions are satisfied by the lag low-pass filter (with a zero at \$s\rightarrow\infty\$), and by the lead high-pass filter (with a zero at \$s=0\$).
The phase of the lead low-pass filter is greater (i.e., less negative) than the phase of the lag low-pass filter (\$\Rightarrow\$ "lead"), but the magnitude is worse because its gain only decays from \$z_1/p_1>1\$ to \$1\$ for \$\omega\rightarrow \infty\$. A similar thing is true for the lag high-pass filter. Its gain is not zero at \$\omega=0\$ but it equals \$z_1/p_1<1\$. Its phase is smaller (i.e., less positive) than the phase of the lead high-pass filter (\$\Rightarrow\$ "lag").