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Why does increasing derivative gain $K_d$ in a PID controller loop lead to a reduction in overshoot and an increase in rise time?

This is not a homework question, it is a question to improve my understanding. I know that the output of the derivative path of the controller is $K_d \times \frac{de}{dt}$, but I don't understand why increasing $K_d$ would lead to the rise time increasing and less of an overshoot as when $\ K_d $ is greater, the controller output would be greater, thus leading to an increased overshoot?

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It is basically the same as a damper on a car. It will counteract on any change in "velocity". Hence, it will decrease the overshot, but it will also try to resist any initial change of velocity, thus increase the rise time.

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  • $\begingroup$ Please don't encourage homework questions with no attempt at a solution. We'll be swamped! $\endgroup$ – Transistor Feb 24 '20 at 18:54
  • $\begingroup$ I didn't think this was a homework questions, but more a question out of curiosity. $\endgroup$ – useless-machine Feb 24 '20 at 20:38
  • $\begingroup$ Fair enough. Let's see if the OP comes back to clarify. $\endgroup$ – Transistor Feb 24 '20 at 21:01
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The derivative path of a PID controller "predicts" the future error regarding the variable which needs to be controlled and reach a certain reference point. More specifically, the outcome of the derivative path is analogous to the rate of change of the error:

$$ error = reference - measurement $$

For example, if the error is quickly decreasing then the output of the derivative would produce a negative value which will then be added to the summation of the outputs of the proportional and integral paths resulting in a reduction of the total control signal. The reduction of the control signal generally means a reduction on the rate of change of the controlled variable (for example the velocity, if position needs to be controlled).

The gain $\ K_d $ corresponds to the weight the control engineer adjusts to this behavior regarding the derivative path. Increasing the derivative gain $\ K_d $ means that it is important for the system to react more aggressively to the rate of change of the error. Increasing $\ K_d $ means the output of the derivative path is larger which, as mentioned above, will reduce the total amount of the control signal, the value of rate of change of the controlled variable (velocity) will decrease and as a result the system will approach more slowly towards the desired reference point $\ \rightarrow $ $\ t_r $ increases (lower velocity) & $\ \%OS $ decreases (controller slows down the system before it reaches the reference point).

In conclusion, the derivative path of the PID controller determines how the system approaches the reference point, whether this happens fast or slowly, and correspondingly reduces or increases the total control signal following the below pattern:

  • error decreases $\ \rightarrow $ output is negative
  • error increases $\ \rightarrow $ output is positive

and the amplitude of the output depends on how fast the error changes.

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