# Interdependence of Kp, Ki, and Kd in PID control

As shown in the attached photo, I have highlighted in yellow the purpose of a specific gain: Kp or Ki or Kd.

For example, Kp is especially useful for improving rise time.

But I am unable to understand and absorb the sentence that is underlined in red.

Are they really interdependent? If so, why?

• Think of it from the perspective of the total transfer function:$$K_p+\dfrac{K_i}{s}+sK_d=\dfrac{K_ds^2+K_ps+K_i}{s}$$The numerator is a 2nd order polynomial, so changing one of the parameters means changing the roots. The last phrase refers to the tuning part of the PID, where you first establish $K_p$, with the others seto to zero, then you add $K_i$, and only after that, if needed, $K_d$. Oct 12, 2021 at 11:36

It's not that they are dependant in themselves. It's just that the system final response is the sum of the three actions (it also depends on the exact PID topology, the most common are a dozen).

So if you first tune for P (typical) and you add some I most probably you'll have to slightly turn turn down P since I adds some control response in itself. Add to that that most of plants are actually everything but linear: the PID works in a 'small signal' way but the large signal transient can foil it (they invented setpoint ramping and similar tricks to compensate for it)

The underlined text means that adjusting one parameter, say Kp changes differently the total performance of the controlled system (=PID controller and process together) with different values of other controller parameters. You may for example find that increasing Kp a certain amount in a certain system decreases the rise time say 50%. Do not expect the effect is the same -50% if you change also integration or differentation. Your system may for ex. become unstable.

The thumb rules in elementary control theory texts are written for nicely behaving processes which resemble a 2nd order lowpass filter. The system is assumed to work properly in a certain operating point and your table gives the directions of the partial derivatives of rise time, overshoot, settling time and solid state error vs. controller parameters.

Often PID controllers are constructed by making Kp a commpon multiplier; the transfer function of the controller is

Kp(1 + 1/(sTi) + sTd) It increases the apparent independence of the parameters in many practical cases.

In pure math the next form would be as good, but it's not when one tries to tune the parameters in a practical system:

Kp + Ki/s + sKd

• " It increases the apparent independence of the parameters in many practical cases." here you intended/meant "interdependent" or dependent? Oct 17, 2022 at 5:17

PID is just a mathematical idea that one can implement in many ways.

IND method:

$$\G_{PID}(s) = K_p + K_i\frac{1}{s} + K_d s\$$

Every part is independent (hence the name).

ISA method:

$$\G_{PID}(s) = k_p \left[1 + \frac{1}{T_i s} + T_d s \right].\$$

We can see that Ki and Kd are dependant of kp:

$$\K_i = k_p \frac{1}{T_i}\$$

$$\K_d = k_p T_d\$$

There is a lot more methods, you need to know which one are you using. Maybe it is in your doc.

So in your implementation it could be ISA, because you can calculate kp to set desired Ki/Kd. That kp change will also affect Kp and Kd/Ki.

After theory, there is also real implementation (f.e. on op amps). Derivative part is not possible in reality, as you cannot predict y(t+dt) in time t. So there are different tricks to simulate it.