# Negative gain in Ziegler-Nicholas Oscillation Method?

For a plant G(s), the transfer function is given by: $$G(s) = \frac{(10s+50)}{s(s^2+20s+2500)}$$

Using Ziegler-Nicholas method to get the parameter: $k_p = \dfrac{-1000}{3}$ and $w =-28.868j$.

Is it correct? what should I do in case of negative $k_p$ and how do I calculate the PID parameters?

The Ziegler-Nichols cycling method is not appropriate for this system. Because of the zero located in -5, the closed loop system is always stable and will not oscillate for proportional compensator (poles will never be conjugated on top of the $j\omega$ axis), as can be evidenced by the root locus plot: