lead high pass filter and lag low pass filter
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&gt;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&lt;1\$. Its phase is smaller (i.e., less positive) than the phase of the lead high-pass filter (\$\Rightarrow\$ "lag").