All the books I have read say that when a proportional derivative controller is used the natural frequency remains the same.(adsbygoogle = window.adsbygoogle || []).push({});

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

Otherwise the natural frequency is multiplied by K_{p}

i.e. if the original Characteristeric equation is

s^{2}+ 2Zw_{n}s + w^{2}_{n}

(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

s^{2}+ 2Zw_{n}s + K_{d}w^{2}_{n}+ K_{p}w^{2}_{n}

So now natural frequency here is K_{p}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

G_{c}= K_{p}+ K_{d}s

However, in one textbook, I noticed that they have the TF of the Controller to be

G_{c}= K_{p}(1 + K_{d}s)

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?

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# PD (proportional derivative) controller

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