Effect of three PID gains on each other?
They are independent variables with known frequency or slew rate effects for controlling the desired settling time, TS and peak over-shoot %POS to a step input. However that is just for a time or frequency normalized response.

In a true 2nd order system, the parameter ratios are very dependent on each value for the overall total error measured by some method, such as integrated average squared error (IASE) or noise gain peak error (NGPE) for example.

Just imagine your system is driving on the freeway, and for now your yourshoot error to avoid a collision, 10% overshoot, 20% speed error to avoid a traffic ticket and 25% IASE to avoid most close calls, and 25% error to react to a potential accident and swerve to miss.

So it turns out that not only the gain but it is the ratios that become more important to tune  the optimum response due to variable interference levels and different rules for dynamic response. 

Yet (no pun by accident ) every “specs” are very specific to the system needs. 

It may be for allowed latency, noise , or jerk  gain, impulse attenuation response or peak  excursion response to an emergency disturbance or smooth minimal motion error.


If you were drunk and always overshooting with slower response, which ratio or gains would expect was the cause?
If you had too much coffee and kept over-reacting to small noise bursts , which gains could be the cause?


The answer to your question, again, it is that these ratios somewhat standardized to desired curve shapes, so the effects of cascaded transfer functions are dependent on each other.

When the system is tested for nominal and worst case inputs, it is then tuned depending on time , inputs and error criteria to either meet new specs like peak jerk or ramp latency , or quadratic input overshoot and then retune the original specs.


This is especially critical when an output is limited or “saturates” like an Op Amp because now your gain drops to ZERO. If you lock your front brakes, you can no longer steer.  If the steering wheel direction is maxed out you cannot respond by turning more. That’s zero gain to a particular condition and can cause an accidental error.


These ratios are fairly common for second order systems. However most things in life and to be more complex and so the goal is to have a compensator work towards making it look like a second order system with a partial filter gain or k over a limited frequency range.

Most of the time from a cascade of stage effects in order to maximize the desired and minimize the unstable errors like a signal to noise ratio. Thus adding compensation to reduce the harmful errors of higher than 2nd order effects in some spectrum while reducing the unstable errors at other ranges of noise input spectrum to make it look more like a well behaved 2nd order system.

Of course if it were possible to make more like a 1st order system , all we would need was mostly kP gain control.

Now go to test drive to a disturbance of pot hole, curved road and tight traffic and change your response time to a step inputting speed,  a pulsing driver who cuts you off or see how you collision avoidance gains can be improved. The. For giggles compare your IASE for a minute with a good truck driver compared to a car driver and measure the error between lane and each side of the vehicle, sum it and square it and accumulate it over time.  You will find good drivers with larger vehicles must have much better specs and lower IASE errors.

Yes and No they are dependent but control different aspects which are different so independent outputs from an input, impulse, ramp and step input effects. but the starting ratios are often fixed. for typical design. So the response to a glitch is only affected by Kd gain is independent, but that affects jitter and proportional gain Kp  may need to be adjusted at same time as Kd. to minimize overall system errors.  

Specific freq. spectrum range of inputs = independent  ( f = DC to ?)

Overall freq. spectrum range of inputs,  Dependent on ratios for Kp,Ki,Kd

This is my nutshell answer

T.S. EE75