FOLPD PID Tuner

Interactive closed-loop simulation with auto-tuning. Adjust PID gains and visualize step response in real time.

FOLPD PID Tuner

FOLPD = First-Order Lag Plus Dead Time

Interactive PID tuning and process control simulations. Adjust parameters in real time and visualize step responses and tuning rules.

PID Tuner — Closed-Loop Setpoint Step Response

Process Output Setpoint Control Output

Closed-Loop Disturbance Step Response

Unit step disturbance d = 1 at process input, setpoint = 0. Shows how well the controller rejects load disturbances.

Process Output y(t) Control Output u(t)

Open-Loop Bode Plot

Bandwidth ωgc

—

rad/s

Phase Margin

—

degrees

Gain Margin

—

dB

Process (FOLPD)

PID Parameters

Plant Transfer Function

General form (FOLPD):

$$G(s) = \frac{K \, e^{-L s}}{T \, s + 1}$$

K = process gain (steady-state output change per unit input)

T = time constant (s) — speed of first-order lag response

L = dead time (s) — pure transport delay

Current model:

DC gain = 2.00

Rise time (≈ 2.2T) ≈ 11.0 s

Controllability ratio L/T = 0.20

PID Form:

PID Controller Equations — Standard (Parallel) Form

Transfer function:

$$ G_c(s) = K_p + \frac{K_i}{s} + \frac{K_d \, s}{\tau_d \, s + 1} $$

Ki = integral gain  (Ki = Kp / Ti)

Kd = derivative gain  (Kd = Kp · Td)

τd = derivative filter time constant

Time-domain:

$$ u(t) = K_p\,e(t) + K_i\!\int_0^t e(\tau)\,d\tau + K_d\,\frac{de}{dt}\bigg|_{\text{filtered}} $$

P, I, and D terms add in parallel

Changing Ki or Kd affects only that term

Standard form: Each term is independent. Adjusting Ki or Kd changes only one zero. See the relay-feedback tutorial → for derivation.

New to PID tuning? Read the Getting Started Tutorial →

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