Feedback Fundamentals
Feedback compares output to reference and acts on the difference, reshaping a system's dynamics through the closed-loop gain.
The Feedback Loop
Consider a plant with transfer function G(s) driven by a controller C(s), with unity feedback of the measured output. The forward path is L(s) = C(s)G(s), called the loop gain. The closed-loop transfer function from reference to output is T(s) = L(s) / (1 + L(s)).
This single formula explains most of feedback's behavior. Where the loop gain is large, T approaches 1 and the output faithfully tracks the reference. Where the loop gain is small, T approaches L and the loop has little authority.
Sensitivity
Define the sensitivity S(s) = 1 / (1 + L(s)). It measures how the output responds to disturbances and to plant variation. Note that S + T = 1 always: you cannot make both small at the same frequency. This algebraic constraint is the heart of every feedback trade-off.
What high loop gain buys
- Small sensitivity S means good disturbance rejection.
- Closed-loop behavior becomes governed by the controller, not the plant.
- Steady-state tracking error shrinks as loop gain rises at DC.
What limits loop gain
You cannot raise loop gain everywhere. Every real plant has phase lag that grows with frequency; push the gain too high and the loop develops enough phase shift at the crossover frequency to oscillate. Stability, sensor noise, and actuator saturation all cap usable gain.
Positive versus negative feedback
Control uses negative feedback: the error drives the plant back toward the reference. Positive feedback reinforces deviation and is generally destabilizing, though it appears deliberately in oscillators and latches.
The engineering craft is shaping L(s) across frequency so that S is small where disturbances live, T is well behaved where the reference lives, and the loop stays stable through crossover. The Bode and Nyquist tools in this section make that shaping precise.