How can I reduce the Short Period frequency of a Pitch Hold Auto-Pilot while maintaining stability?
You seem to have too high an alpha feedback. For intuitive understanding, alpha feedback mainly increases the pitch stiffness, analogous to increasing the static margin; pitch rate feedback mainly increases the pitch damping. For your purpose, you would need to significantly reduce the alpha feedback.

For example, with your provided state space model,

$$\dot{x}=Ax+Bu$$

using a linear feedback,

$$u=Kx$$

where

$$K=\begin{bmatrix}0 &amp; 0.5 &amp; 0.25 &amp; 0 \end{bmatrix}$$

satisfies your damping ratio requirements. SP frequency is 3.9 rad/s, SP damping is 0.53, Phugoid frequency is 0.023 rad/s, Phugoid damping is 0.14.