Sunmix,
I don't mean to belabor the analogy any further but here goes. Phase lag in a system results from the simple fact reaction always follows an action; there is a time delay between the two. Get back in your car for a description of the PID algorithm:
This time your task is to stay even with a car in the freeway lane next to you. That car represents the servo command and and you will servo your car to it.
We start off with both cars stopped side by side. The car next to you accelerates from 0 MPH to 80 MPH in an instant (it can do that, you can't). It takes you a second to realize it is rapidly pulling away, putting distance between itself and you.
You gauge the size of this distance and decide to accelerate. This is the Proportional term in PID. It is the size of the distance between you and the target; the bigger it is the faster you accelerate.
You also notice the distance is increasing and accelerate even harder. This is the Derivative term in PID. It is the rate of change in the distance between you and the target. It will be very important in a little bit.
At some point during acceleration your speed will match the target's 80 MPH. That is the instant the distance between you and the target stops growing. You will have to go faster than 80 MPH to catch up so you keep accelerating but less quickly (there is still distance to be made up but it is now shrinking). The Derivative term goes strongly negative because the distance is rapidly shrinking and the P term, while still positive, is decreasing as well. You may actually start applying the brakes. The Derivative term will keep you from overshooting the target. If your judgment is good (P and D terms correct), you will be someplace near the target when your speeds match.
"Someplace near" the target when "your speeds match" also means the P and D terms (separation distance and rate of distance change) become too small to be usable. They are coarse tools and you now need a precision tool to be exactly even with the target. It is the Integral term, the 'I' in PID. If you are behind the target, accelerate very gently, if you are ahead, decelerate very gently. Your position is now exactly even with the target. The Integral term does result in a continuous "pull an inch ahead, fall an inch behind" motion but it is too small to see.
PID feedback gives the shortest time to target without overshoot. You are a PID servo-loop every time you drive your car even if you are unaware of it.
Where phase lag comes into play is the reaction time to the events described above. Your eyes are the "encoder" and there is a delay while your mind processes what you see and sends commands to your foot on the pedal. Say that takes a second while all of the above took 1 minute. That is not a problem.
Now imagine what would happen if it took 30 seconds to react. You would zag when you should have zigged, accelerated when you should have braked. That is what happens when phase lag is 180 degrees; your reaction is the opposite of what's needed because it's delayed by 1/2 a cycle.
Mariss