Controller for damping ratio of 1

To obtain a damping ratio of 1 (critical damping: fastest response with no overshoot), a controller that adds damping is needed: a PD (proportional–derivative) controller, or equivalently derivative (tachometer) output feedback. Gain adjustment alone cannot usually do it without making the response very slow.

Why PD

For a typical underdamped plant $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$, PD control $G_c(s)=1+T_ds$ gives

$$\frac{C(s)}{R(s)}=\frac{\omega_n^2(1+T_ds)}{s^2+(2\zeta\omega_n+T_d\omega_n^2)s+\omega_n^2}$$ $$\zeta'=\zeta+\frac{T_d\omega_n}{2}$$

The derivative term raises ζ while $\omega_n$ stays the same. For $\zeta'=1$:

$$T_d=\frac{2(1-\zeta)}{\omega_n}$$

With derivative output feedback ($K_ts$ in the minor loop), $\zeta'=\zeta+\frac{K_t\omega_n}{2}$, so $K_t=\frac{2(1-\zeta)}{\omega_n}$; this has no added zero, so the response is truly non-overshooting.

Why not others

Example: $G(s)=\dfrac{16}{s(s+2)}$: $\omega_n=4$, $\zeta=0.25$. Rate feedback with $K_t=\dfrac{2(1-0.25)}{4}=0.375$ gives $s^2+(2+6)s+16=(s+4)^2$, i.e. $\zeta=1$, a critically damped response with $t_s\approx\frac{5.8}{\omega_n}=1.46$ s and no overshoot.