Let's insert a compensator for the open-loop system is tangent to circle M=3dB in w=3rad/seg. The open-loop system is the following:
G (s) is the phase constant at -180 and the gain varies depending on the frequency.
We will draw Nichols for M = 3 dB
mtlb_axis([-230 -130 -5 15])
As we can see that the system is tangent, has to move to a phase of -135.
degrees. Ie we have to introduce a lead compensator compensates us 45 degrees.
We know that the tangent frequency at 45 degrees is 3rad/seg as the frequency where the compensation is maximum (where the open loop system has to be tangent).
Now we have to calculate
knowing that the gain has to be 3dB. If we see an expansion of the Nichols chart above that the tangent point in the gain will be 3dB.
We will draw plots Nichols of results with Scilab
mtlb_axis([-230 -90 -30 30])