Formula for PI-regulation Proportional Integral algorithm - c++

Question

I've been reading this website: http://www.csimn.com/CSI_pages/PIDforDummies.html and I'm confused about the proportional integral part. Here's what it says.
Proportional control
Here’s a diagram of the controller when we have enabled only P control:
In Proportional Only mode, the controller simply multiplies the Error by the Proportional Gain (Kp) to get the controller output.
The Proportional Gain is the setting that we tune to get our desired performance from a “P only” controller.
A match made in heaven: The P + I Controller
If we put Proportional and Integral Action together, we get the humble PI controller. The Diagram below shows how the algorithm in a PI controller is calculated.
The tricky thing about Integral Action is that it will really screw up your process unless you know exactly how much Integral action to apply.
A good PID Tuning technique will calculate exactly how much Integral to apply for your specific process - but how is the Integral Action adjusted in the first place?
As you can see, the proportional part is easy to understand it says that you multiply error by tuning variable. The part that I don't get is where you get the P and I from on the second part, and what mathematical operation you do with them. I don't have a degree in mathematics or advanced calculus knowledge, so I would appreciate it if you would try to keep it algebra level.

Answer 1

There is a big part missing from the text, the actual physical system that turns the control into a process and the actual physical variable.
Think of the integral as some kind of averaging operation that filters out small oscillations in the PV input. It also represents some kind of memory of the immediate past of the process.
A moving exponential average, for instance, can be thought of being a mix of integral and proportional action.
Staying with the car driving example, if you come to a curb where you need the steering wheel in a certain position to go in a circle, you don't just yank the wheel to that position, you move it gradually (most of the time). Exactly such ramp-up and -down actions are effects of using the integral action part.

Answer 2

I integral part is just summation also multiplied by some constant.
Analogue integration is done by nonlinear gain and amplifier.
Digital integration of first order is just:
output += input*dt;
second order is:
temp += input*dt;
output += temp*dt;
dt is the duration time of iteration loop (timer or what ever)
do not forget that PI regulator can have more complicated response
i1 += input*dt;
i2 += i1*dt;
i3 += i2*dt;
output = a0*input + a1*i1 + a2*i2 +a3*i3 ...;
where a0 is the P part
Now the I regulator adds more and more amount of control value
until the controlled value is the same as the preset value
the longer it takes to match it the faster it controls
this creates fast oscillations around preset value
in comparison to P with the same gain
but in average the control time is smaller then in just P regulators
therefore the I gain is usually much much smaller which creates the memory and smooth effect LutzL mentioned. (while the regulation time is similar or smaller then just for P regulation)
The controlled device has its own response
this can be represented as differential function
there is a lot of theory in cybernetics about obtaining the right regulator response
to match your process needs as:
quality of control
reaction times
max oscillations amplitude
stability
but for all you need differential math like solving system of differential equations of any order
strongly recommend use of Laplace transform
but many people also use Z transform instead
So I-regulator add speed to regulation
but it also create bigger oscillations
and when not matching the regulated system properly also creates instability
Integration adds overflow risks to regulation (Analog integration is very sensitive to it)
Also take in mind you can also substracting the I part from control value
which will make the exact opposite
sometimes the combination of more I parts are used to match desired regulation response shape