Bode plot and stability margins

[gfd43tg]

Homework Statement

Homework Equations

The Attempt at a Solution

Hello, In part (b), I found $k_{cu} = 0.5$. I found in part (d) a controller gain of $k_{c} = -0.3$ yielded a diverging output. Here are the bode plots for parts (a),(c), and (d). I don't understand how I should use the "stability margins" which are the dots on the plots for part (e) in order to determine $k_{cu}$ without ziegler-nichols tuning.

 

[LvW]

Maylis - at first, are you familiar with the definition of stabiliy margins (phase resp. gain margin) ?

 

[gfd43tg]

I know the gain margin is the inverse of the amplitude ratio at the crossover frequency (the frequency at which $\phi = -180^{o}$), $GM = 1/AR_{co}$ and the phase margin is the phase angle at which $g_{c}g_{p} = 1$. $PM = \phi_{pm} + 180^{o}$

Admittedly I hadn't read this section in my textbook prior to posting this question, so I see I can use those dots to identify my crossover frequency and gain margins.

I think it doesn't make any sense to use the bode plot of $g_{cu}*g_{p}$ to find $k_{cu}$, I should use the bode plot of $g_{p}$ and use those stability margins to determine $k_{cu}$? And compare with what I determined it to be by playing with simulink to be $k_{cu} = 0.5$

Here is a bode plot for just the transfer function $g_{p}$

The phase margin is -31.8 degrees at 0.777 rad/s, and the gain margin is -6.21 dB at 0.633 rad/s. With this information, I'm not sure how to determine what $k_{cu}$ should be.

 

[LvW]

If you reduce the gain the cross-over frequency will be shifted to smaller values (the phase response remains the same).
And - as a consequence - the phase margin will increase. I think, that`s what the green curve in the first diagram shows.

 

[gfd43tg]

How can I use this information to find $k_{cu}$?

 

[gfd43tg]

Never mind, you just find the value that will give zero phase and gain margin