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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?

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  • \$\begingroup\$ You want it to be an oscillator then? \$\endgroup\$
    – Andy aka
    Commented Jan 28, 2020 at 12:26
  • \$\begingroup\$ @Andyaka No, I want to know if I can design a PD controller in order to have the entire system (with controller) with 35º phase margin. \$\endgroup\$
    – Carmen González
    Commented Jan 28, 2020 at 12:29
  • \$\begingroup\$ I would resort to classical pole-zero placement then convert them into \$k_p\$ and \$k_d\$ parameters. However, before you do this, you have to select a crossover frequency \$f_c\$ and extract the magnitude and phase at this point from your Bode plot. Then you'll know how much gain (or attenuation) and phase boost are needed at \$f_c\$ to satisfy your phase margin design. You can have a look at a seminar I taught at APEC some time ago, I showed how to match PID parameters with poles and zeroes: cbasso.pagesperso-orange.fr/Downloads/PPTs/… \$\endgroup\$
    – Verbal Kint
    Commented Jan 28, 2020 at 12:35
  • \$\begingroup\$ @CarmenGonzález why are you saying you want a phase margin of -10 if in fact you want a phase margin of +35? You currently have a PM of about 20 deg so what are you really trying to achieve? \$\endgroup\$
    – Andy aka
    Commented Jan 28, 2020 at 12:49
  • \$\begingroup\$ @Andyaka I want a phase margin of 35º with the controller. Since the controller increases 45 degrees in the phase for high frequencies, so I want to obtain a "phase margin" for the system without controller of -10º (35º-45º). In summary, I want a phase margin of 35º for the system with the controller. I need to find Kp and Kd for the controller. \$\endgroup\$
    – Carmen González
    Commented Jan 28, 2020 at 13:16

1 Answer 1

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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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