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Been brushing up on my control systems knowledge and I keep seeing that a P controller WILL NOT have 0 error and that doesn't make sense to me.

Imagine a tank problem, the setpoint is 50 cm, you open the drain valve and the level goes down to 49. Controller output goes up and you start putting water into the tank. Controller output does not go to 0 until the level is back to 50. So while this may take a while if your Kp is low, it will get there.

I understand that steady-state error is not measured at infinite time but the idea that a P controller will not reach the SP is odd to me.

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If you open your drain valve, and then close it again, then you're correct, the tank will refill back up to 50, leaving you with no offset. If, however, the valve is left open, then the water may not come into the tank fast enough to compensate for the water that's leaving. This means that the level will continue to fall, up to the point where the proportional gain means that the rate of water in = water out.

In a P controller, the rate of water in will only be non-zero when there is an error, therefore, if the outlet valve is open, then there must be an error, or no water would come in at all. Integral action corrects for this offset.

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