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I have been assigned a control systems problem, where I am supposed to stabilize a system using a controller technique from the frequency domain. When I implemented the system on Matlab and calculated the phase margin of the system it turned out to be infinite. I am not exactly sure what that means.

Can someone tell me what exactly does it mean that the phase margin comes out infinite, and how can such a system be stabilised?

This is my code:

clc 
clear all
close all
num=[1];
den=[1 0 -9];
sys=tf(num,den);
bode(sys);
[Gm,Pm,wcp,wcg]=margin(num,den)

Here is the phase margin displayed on Matlab

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  • \$\begingroup\$ What is the step response of the open loop system? Is the open loop system unstable? Does it have right hand plane poles? If yes, you have to use the full Nyquist criterion. Don't just look at the phase margin. It is misleading. \$\endgroup\$ – AJN Dec 31 '20 at 1:21
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Your system \$\frac{1}{s^2 - 9}\$ has gain less than 0 dB at all frequencies. The bode plot you have plotted shows that clearly. Since the gain doesn't cross 0 dB anywhere, the phase margin is not well defined.

Apart from the above, this system is open loop unstable. The open loop system sys has a pole at \$s = +3\$. For systems which are open loop unstable like this system, use the full Nyquist criterion (counting the encirclements) to determine stability.

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  • \$\begingroup\$ The system needs to be stabilized by using a controller technique from the frequency domain. Usually, in the frequency domain, we proceed by adding a compensator that would increase the phase margin when it is too small. But I have no idea how to proceed here with the design when the phase margin is coming out to be infinite. Can you guide me about this? Thank you so much for your previous answer! \$\endgroup\$ – Areen Dec 31 '20 at 18:36
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    \$\begingroup\$ Since this is a second order system with a pole on the RHS, I think that you need a a derivative feedback apart from proportional feedback. This will introduce a zero on the left hand side. Also you need to increase the gain of the system so that the Nyquist diagram will have exactly one encirclement. \$\endgroup\$ – AJN Dec 31 '20 at 23:47
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To answer yout question:

What is the phase margin? It is the additional phase shift which must be inserted into the feedback loop in order to bring the closed-loop system to its stability limit (at the frequency where the loop gain is unity in magnitude). For a pure resistive negative feedback, this margin apparently is 180 deg. However, that is pure theory because each real amplifier will cause phase shifts for rising frequencies. I really cannot see how the phase margin could be "infinite".

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