Electronic – Phase margin for a system (design of PD controller)

bode plotcontrolphasephase margin

I have a system with the following transfer function:

\$G(s)=\cfrac{20}{s^2+s+20K_f}\$

where \$K_f=1.75\$

I want to use a PD controller and I want obtain a phase margin of 35 degrees. I need to determine the parameters (\$K_p \text{ and } K_D\$) for the PD controller.

I made a Bode plot in Scilab for G(s):

enter image description here

As I want a phase margin of -10º, I would have to see the \$f_0\$ for which I have a phase of \$\phi=-10-180=-190º\$, but with -190º the system is unstable.

I can not determine \$f_0\$ from the Bode Plot, because the graph only reaches -180º. How can I design the PD controller for this system?

Best Answer

If I understood well, you want to compensate the plant whose transfer function is given so that once compensated you have a phase margin of 35°. The plant transfer function is \$H(s)=H_0\frac{1}{\tau_2s^2+\tau_1s+1}\$ with \$H_0=\frac{1}{k_f}\$, \$\tau_1=\frac{1}{20k_f}s\$ and \$\tau_2=\frac{1}{20k_f}s^2\$. The plot I got from Mathcad is this one and matches yours:

enter image description here

Then, you need to pick a crossover frequency \$f_c\$ at which you will measure the open-loop phase margin of 35°. Let's take \$f_c=10\;Hz\$. What is the attenuation of the plant at \$f_c\$ and what is its phase?

enter image description here

From these values, calculate the needed phase boost from the inverting compensator to meet the 35° design criteria and deduce the position of the zero:

enter image description here

Determine what the \$k_p\$ coefficient should be based on the zero position and plot the open-loop gain \$T(s)\$:

enter image description here

The crossover frequency is 10 Hz as arbitrarily selected and the phase margin is 35°. The compensator transfer function is \$G_1(s)\$ in the picture. Is this what you wanted to see?

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