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Offset error happens when there is a constant force on the system. Say you have a robot arm. If you want the arm to point strait up or strait down A P controller would do a decent job because there is no constant force pulling on the arm. However if you want the arm to be parallel to the ground you have gravity that will pull the arm down. A simple P controller can't overcome this sort of constant force because when the arm is in position sideways e = 0 so the controller is applying no force to oppose gravity: so the arm sinks. The higher the gain the closer the arm will be to the set-point but without using infinite gain (which is impossible in real life) the arm will never be able to stay at the setpoint.
When the arm is close to the setpoint the I tem builds up until it matches the constant force (gravity in the robot case). When I matches the constant force the controller will reach the setpoint.
From the above you should see why a PI controller is useful. The D term is a damping term. It works like the shock absorbers in your car: it keeps the system from oscillating a lot. If you look at a car with bad shocks it will keep bouncing for a while after it goes over a bump. With good shocks the car bounces once and then comes back to the neutral position.
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