How can I reduce the Short Period frequency of a Pitch Hold Auto-Pilot while maintaining stability?

Question

I am working on a Pitch Hold Auto-pilot for F-104, which has the following damping characteristics;

which clearly shows that it has degenerated and unstable phugoid mode. well, I made a SAS which consists of 2 loops / Feedbacks. 1) The angle of attack feedback to increase the Cm alpha or static margin of the aircraft 2 Pitch rate feedback to dampen the short period mode. but it has eventually increased the natural frequency to almost 365 rad/s as shown below,

but this frequency is not according to the handling qualities as it should be less than 3 rad /s. what feedbacks should I try so that all the states are stable as well as desired handling qualities can be achieved? states are velocity, angle of attack, pitch, pitch rate. Simulink Model of SAS:

Root Locus of the outer Loop:

How can i decrease the frequency to the desired range keeping it stable and keeping in mind that it was unstable first. Also, with not much decrease in damping as it should be greater than 0.5.(short period mode) Phugoid mode >0.04

Answer 1

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

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

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.