While control loops are in the focus of engineers, physicists under normal conditions don't touch this issue. So what follows may look a little strange.
To start with, I wish to look for the most simple case: have a P control.
What is a controller good for? If something goes wrong, the idea is, to make it better. As this is very general, I want to be more destinct and introduce a simple example.
Imagine somethings (an object) exists, then, if not a ghost, it is in a place. This place now might be the place, I want the object to be. But, as there are many places, the probability that the object is located else is huge. We introduce the distance between the wanted place (set-value) and the actual place (is-value) as the "error" we want to minimize.
To correct the error we need an action. Like: apply to the object a directed force. But: how will the force change the place? Changing a place we call movement and movement is characterized by velocity. So how is the application of force related to that velocity? Our gut feeling says: the more force, the more speed we will gain, the faster the object reaches the set value of place. But from experience we know, an object has a property of mass. So acceleration and speed is not only related to force, but also to mass. An object without mass can not be controlled, as the minimum force will result in infinite speed.
For this reason, an object that has to be controlled MUST have a mass and so inertia.
Now the term "the faster the object reaches the set value of place" shows ambiguity. Two meanings: reach set position in short time AND at high velocity. So we do not reach the set value, but pass that value and introduce a new error.
A p-controller by definition creates a force, that is proportional to the error. If the initial condition is: no error, no force is created and nothing happens. But it there is an initial error (in a positive direction), there is a negative force, the object starts to move and gains kinetic energy. As long as the positive error exists, there is force and kinetic energy increases. That means: the moment the set position is reached, the speed is at maximum and the next moment we will have a negative error and generate a positive force which now decelerates the object. The moment the object is at stand still, the error will be the initial error but in negative direction. Now the game starts from the beginning.
What we see here is: a P controller in the case of a mass with inertia is nothing but a harmonic oscillator and without damping will oscillate forever if once excited.
That is, where the "D" steps in.
If there is no error at start-up, the system is at rest in peace. Now we establish an error by creating a displacement. That means, the error changes so the derivative is large. To correct the error we apply a force propertional to the error (the "P"component) and a second force proportional to the change rate of the error ( the "D" component). That means: with a D, the back driving force is higher and the error gets smaller. But now the derivative becomes negative and the backdriving force is reduced. When the compents P and D are properly adjusted, the D part over compensates the P part, that means, when returning to the set position, the object is decelerated and comes to stand still at the set position.
But what, if the D compensates oscillation, but the error doesn't come to zero? In this case P creates a back driving force, but there is no movement. That only can be the case, if there is another force of value -P, that has an unknown (external) origine. So, if a PD controller comes to stand still at an error position it actually is a weigh that measures an external force. To compensate this force we have to apply a third current, we call the "I" component.
Now we have a PID controller completed: The D components dampends the movement by controlled extraction of kinetic energy from the object. The P component brings back the object to the set position, but if there is still an error, this means the existance of an external force and this force is proportional to the error and compensated by the I component.