# PD (proportional derivative) controller

1. Sep 12, 2011

### phiby

All the books I have read say that when a proportional derivative controller is used the natural frequency remains the same.

However, this is true only when the proportional part of the PD is unity.

Otherwise the natural frequency is multiplied by Kp

i.e. if the original Characteristeric equation is

s2 + 2Zwns + w2n

(I am using Z instead of Zeta for damping ratio &
w for frequency instead of omega - it's difficult to type those out)

With a PD controller (P & D connected additively)
the new charac eqn becomes

s2 + 2Zwns + Kdw2n + Kpw2n

So now natural frequency here is Kp multiplied by the original frequency.

So why do all textbooks say that natural frequency remains unchanged by PD controller?

Also, above, I have TF of the Controller to be
Gc = Kp + Kds

However, in one textbook, I noticed that they have the TF of the Controller to be
Gc = Kp(1 + Kds)

I tried to figure out why they have it this way
I feel the above will be true only if they have the connection in the following way.

After the proportional gain, the line is split (with a takeoff point). The Takeoff point does
a positive feed forward before it's connected to the plant/process.
There is the derivative controller in one path of the split & a unity gain on the other path.
Is this a standard way of connecting a PD controller?

2. Sep 12, 2011

### jim hardy

""Is this a standard way of connecting a PD controller? ""

well, in my day it depended on who built the controller.
Some manufacturers let the K value apply to the derivative, others split it as you describe.

By now there are doutless industry standards and they may be different for US and Europe i dont know.

So you are at the mercy of what your textbook's author grew up with.

Be aware that "Modern Control Theory" was largely developed after WW2.
Descartes stumbled across the behavior of feedback systems but in his day there were no automatic machines to apply it . So it remained just a curiosity for nearly three centuries. When the Germans built their rockets they revived his math and their textbooks on the subject were among the War Prizes brought back with Von Braun et al.

So it's a relatively young field whose teaching methods are still being shaken out, imho.

old jim