Open Loop vs Closed Loop

An open-loop system's control action does not depend on the output. A toaster runs for the time you set; a washing machine fills for a fixed period; a traffic light cycles regardless of whether anyone is there.

It works when the relationship between input and output is known and stable. Its entire behaviour rests on a calibration: *this much input produces that much output*. Where that holds, open loop is cheap, simple and — a point worth making early — unconditionally stable.

Change the plant or the environment and the calibration is wrong. A heater set to 3 kW for 22 °C will settle at 17 °C once a window is opened, and stay there.

The controller is not badly designed. It has no channel through which the error could reach it. Anything an open-loop controller does not measure, it cannot respond to — and it measures nothing.

• Disturbances — anything affecting the output that is not the control input.
• Parameter drift — components age, motors wear, insulation degrades.
• Modelling error — the calibration was an approximation to begin with.
• Component failure — a half-dead heating element looks exactly like a working one from the controller's side.

A closed-loop controller measures the output, subtracts it from the reference, and drives the plant with the resulting error signal. e = r − y.

The shift is more profound than it looks. The controller no longer needs to know that 22 °C requires 3 kW; it only needs to know which way to push when the error is positive. That is a far weaker requirement, and unlike a calibration it does not go stale.

1. 1Error is 5 °C, so the controller raises the power.
2. 2The room warms, so the error falls, so the controller eases off.
3. 3The process continues until the error is small and the power has settled wherever it needed to be.
4. 4The controller never learned that a window was open. It converged on 4.5 kW as a *consequence*, not a calculation.

That last point is the whole value of feedback. One controller handles an open window, worse insulation, a colder day and a partly failed heater, without being modified for any of them.

The reduction in gain is not simply a loss. The same factor 1/(1+GH) that shrinks the gain also shrinks the *sensitivity* to changes in the plant — which is the trade at the heart of feedback amplifier design: gain given away in exchange for predictability.

Every real loop has delay: sensors lag, actuators have inertia, controllers take time. If a correction arrives late enough to be *in phase* with the error rather than against it, the feedback has effectively become positive, and each correction is larger than the last.

• Low gain — slow correction, and a steady-state error that never fully disappears.
• Moderate gain — faster and more accurate, with some overshoot.
• High gain — overshoot, correction, overshoot: the loop rings.
• Too high — the oscillation grows and the loop destroys itself or saturates.

This is what Routh-Hurwitz, root locus, Nyquist, and gain and phase margins are all for. Every one of them is a way of answering "how much gain is safe given the delay in this loop" — a question that only exists once the loop is closed.

A toaster is open loop and correctly so: the plant is well characterised, disturbances are small, and a browning sensor is not worth its cost. An autopilot is closed loop and correctly so. The engineering question is whether disturbances matter enough to pay for a sensor and a stability analysis.