PID Controllers

PID action

A parallel ideal PID controller has C(s) = Kp + Ki/s + Kd s and acts on error e = r − y. Proportional action responds to present error, integral action accumulates error and derivative action responds to its rate. Their effects depend on the plant and loop; increasing a gain does not guarantee better performance. A realizable derivative commonly uses Kd s/(1 + Tf s), Tf > 0. Derivative on measurement avoids derivative kick from an abrupt reference step, and filtering limits high-frequency noise amplification.

Integral action and saturation

Integral action can remove constant steady-state error when the resulting loop is stable and not constrained by saturation. During actuator saturation, a naive integrator may continue accumulating error. Anti-windup methods such as conditional integration or back-calculation reduce this problem.

Worked example

For plant G(s) = 1/(s + 1) with unity negative feedback and PI controller C(s) = 2 + 1/s, loop transfer is (2s + 1)/[s(s + 1)]. The characteristic polynomial is s² + 3s + 1. Poles are (−3 ± √5)/2 = −0.382 and −2.618, both stable. For a unit-step reference, E(s) = (s + 1)/(s² + 3s + 1). The final-value theorem gives ess = lim sE(s) = 0 because its required pole condition holds. This is a zero-initial-condition linear calculation without actuator saturation.

Tuning approach

Start from a plant model or safe identification data, choose a target response and robustness, and test disturbances, noise, limits and uncertainty. Empirical tuning rules are starting points and may create aggressive responses; they are not permission to drive equipment into unsafe oscillation.

Quick check

1. Which term removes constant offset in a suitable stable loop? Integral action.
2. Which term is especially noise sensitive? Derivative action.
3. Why use anti-windup? To handle accumulated integral action during actuator saturation.