# Bode plot and stability margins

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## 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.

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

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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.

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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.

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

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