P, PI and PID: what each term does

In short. P reacts to the present error, I to the accumulated error, D to its rate of change. Most process loops need only PI; D is added when the process is slow and the signal is clean. Useful to know which knob to turn and what to expect.

Section conventions: e = SP − PV, ideal form, time in seconds (details).

Equation

u(t) = u0 + Kc · [ e(t) + (1/Ti)·∫e dt + Td · de/dt ]
         └ P ┘   └──── I ────┘   └── D ──┘

Proportional action (P)

The output is proportional to the error: with Kc = 2 %/°C an error of 3 °C adds 6 % to the output.

• Larger Kc — faster response, smaller offset, but more oscillation. With dead time, oscillation starts already at moderate Kc.
• Offset. A P controller needs a non-zero error to keep the output away from u0. For a self-regulating process, after a set-point step ΔSP an error of ΔSP/(1 + Kc·K) remains.

Example. Process K = 1.8 °C/%, τ = 45 s, θ = 12 s; P controller Kc = 1.04 %/°C. Set point +1 °C → PV stops at 0.652 °C; error 1/(1 + 1.04·1.8) = 0.348 °C. The simulator with an empty Ti gives the same.

Integral action (I)

The integrator accumulates the error and moves the output until the error is zero. This is why PI removes the offset.

• Ti is the time in which, at constant error, the I term adds as much to the output as the P term gives ("one repeat"). Smaller Ti — stronger integral action.
• Ti too small — slow oscillation and overshoot: the integrator "overpumps" the output.
• Ti too large — PV creeps to the set point for a long time.
• When the output hits a limit the integrator keeps growing — windup protection is needed.

Derivative action (D)

D reacts to the rate of change and brakes the output in advance, reducing overshoot on slow processes.

• Td is how many seconds D "looks ahead" when the error grows linearly.
• D amplifies noise: without a filter a 0.1 % step of the measurement in one 0.1 s sample becomes a rate of 1 %/s. Filtering and "D on measurement" are covered in D, noise and derivative kick.

Comparison on one process

Here PID rises faster (11.1 s against 22.0 s) but does not settle faster: the setting is aggressive and D adds noise sensitivity. This is why PID is not "always better".

When D is needed and when not

What changes with the parameters (PI/PID, all else equal)

Typical mistakes

• Mixing up Ti and Ki. Ti is a time (s or min), Ki = Kc/Ti is a gain (1/s). Increasing Ti weakens, increasing Ki strengthens the integral action. See Kp, Ki, Kd, Ti, Td and PB.
• Adding Td to a noisy signal without a filter — the output jitters and the valve wears out.
• Copying settings from another loop without considering K, τ, θ and units.

Limits

The description refers to the ideal form. A particular PLC block may compute the terms differently (parallel form, P and D on measurement, set-point weighting) — check the block documentation. NI states plainly that there is no industry-wide standard PID algorithm.