Closing the Loop

Open loop vs closed loop

A toaster is open-loop: it runs its heater for a fixed time and hopes. Change the bread, the room, the mains voltage — and hope is all it has. A closed-loop controller instead measures the result, compares it with the goal, and corrects continuously:

You have built this twice without the vocabulary: the op-amp with negative feedback (6.2) closes a loop a million times a second to hold V₋ equal to V₊; your night-light (12.1) closes one twenty times a second. Control theory is the study of what happens in between the corrections — because in between is where the trouble lives.

Proportional control and its two failures

The obvious rule: push proportionally to the error — drive = Kp × error. It works! And it fails twice:

• Steady-state offset. Holding a heater above room temperature needs nonzero drive — but P-control's drive is zero when the error is zero. So it settles where the leftover error, times Kp, exactly sustains the temperature: always a little short. Raise Kp and the offset shrinks… but:
• Oscillation. Real systems answer late — heat takes time to travel from element to sensor (a deadtime). A high-gain controller keeps pushing on stale information, overshoots, slams the other way, overshoots again: the loop rings like your LC tank (9.1), and past a critical gain the ringing grows instead of dying. Feedback + delay + too much gain = an oscillator. (Sometimes on purpose — that's precisely how oscillators are built. In a heater, it's a defect.)