Both drbitboy and OldChemEng seemed to favor a rate of change mode but that isn't close to optimal. Basically your gain is an integrator gain because the output is integrated.
The problem is that this method results in a slow response. In the case of drbitboy's example the closed loop time constant is 2 times the open loop time constant. That isn't good. Secondly, by adding, what is effectively an integrator gain, you have introduced two poles. This means it takes 6.64 time constants to get with 1% of the set point instead of 5 time constants. So in drbitboy's example it takes 13.28 seconds for the response to settle within 1%.
Increase the gain does not result in a faster settling time. It does result on more overshoot.
The is a general rules for choosing gains or if to choose a gain. There should be one controller gain for every open loop pole. The integrator does not count because it has its own pole. So in this example a true proportional gain is required.
I used a standard PI controller. My closed loop time constant is 0.1. There are two poles but my method of placing poles puts a zero on top of one of the poles so effectively there is only one closed loop pole with a time constant of 0.1 second so it takes only 0.5 seconds to settle within 1% of the set point.
A controller should make the response faster not slower.
On page 5/13 of
https://deltamotion.com/peter/Mathcad/Mathcad - t0p1 pi tc direct synthesis drbitboy.pdf
You can see that the pole closest to the origin is cancelled out by a zero. This is one reason response of the PI controller much faster.
On page 6/13 I implement a controller like drbitboy's. I get the same response using the same numbers.
I also calculate the integrator gain Ki, what drbitboy calls Kc, that results in a crtically damped response. It is simple finding the roots of a quadratic equation.
On page 7/13 I calculate the symbolic formulas for the integrator gain and closed loop time constant that results in a critically damped response. I then use the invlaplace function to find the response in the time domain just to check the results. You can see it takes about 13.28 seconds for the response to get within 1% of the setpoint.
On page 8/13 I calculate the symbolic formulas for the underdamped case where the integrator gain is greater than 2.5. I also compute the response as a function of time using the invlaplace function. Unfortunately the equation is long and continued on page 12/13. Notice that the exponential term is exp(-1/(2*Tp)). This shows that the decay is independent of the integrator gain. Increasing the integrator gain only increases the amplitude of the overshoot.
On page 9/13 I do a similar calculation for the over damped response when the integrator gain is below 2.5.
On page 10/13 I plot the response using formulas from previous pages as a function of time and the integrator gain. As drbitboy rightly noticed, the setting time is contant. Now you know why and have seen the proof.
On page 11/13 I plot the close loop pole locations as a function of the integrator gain. At first the pole locations start at -0.1 and 0 and move toward 0.05 as the integrator gain is increased from 0 to 2.5. Integrating the integrator gain above 2.5 only moves the poles in the plus and minus imaginary directions resulting in high frequencies and magnitudes of oscillation.
The short story is there is a right way and wrong way to do closed loop control.
I can understand cases where one may want to integrate the output but in drbitboy's example there should be a derivative gain that gets integrated into an effect proportional gain.
