What a pole does to the loop. Adding a pole to G(s) inserts an extra lag: each pole contributes up to −90° of phase and an extra −20 dB/decade of roll-off. Both effects work against stability.

The frequency-domain reading. The extra phase lag pulls the open-loop phase curve down, so the phase crosses −180° at a lower frequency and the phase margin shrinks. Since phase margin and damping are tied together by the familiar rule of thumb

\(PM \approx 100\,\xi\ \text{degrees}\)

a smaller phase margin means a smaller damping ratio ξ. And overshoot rises steeply as damping falls:

\(M_p=e^{-\pi\xi/\sqrt{1-\xi^{2}}}\times100\%\)

so the maximum overshoot increases.

The root-locus reading, which shows the same thing. Adding a pole pushes the asymptotes of the root locus to the right: the centroid moves right and the asymptote angles widen, so the complex branches bend towards — and eventually into — the right half plane. The closed-loop poles therefore sit closer to the jω axis at any given gain, i.e. less damped, more oscillatory, larger overshoot, and the system becomes unstable at a lower gain.

The mirror-image result is worth learning alongside it: adding a zero to the forward path contributes phase lead, pulls the locus to the left, raises the phase margin and reduces the overshoot. That is precisely why a derivative or lead compensator (a zero) is used to tame an oscillatory system, while integral action (a pole at the origin) improves steady-state accuracy at the cost of transient behaviour.

Hence, adding a pole generally has the effect of increasing the maximum overshoot.