Manual PID Tuner and Simulator

Free-form closed-loop simulation for any process model. Pick FOLPD, SOPDT, or an arbitrary transfer function, dial in PID gains manually, and watch the step response and Bode plot update in real time. No auto-tuning — just explore.

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.

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

Open-Loop Bode Plot

Bandwidth ωgc

—

rad/s

Phase Margin

—

degrees

Gain Margin

—

Process Model

Gain K: 2.0

Time Constant T (s): 5.0

Dead Time L (s): 1.0

PID Parameters

Kp: 2.50

Ki: 0.50

Ti (s): 5.00

Kd: 1.25

Td (s): 0.50

Derivative Filter τd (s): 0.10

Plant Transfer Function

General form (FOLPD):

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

K = process gain

T = time constant (s)

L = dead time (s)

Current model:

DC gain = 2.00

Rise time (≈ 2.2T) ≈ 11.0 s

Controllability ratio L/T = 0.20

DC gain = 2.00

Dominant time constant ≈ 4.0 s

Controllability ratio L/(T1+T2) = 0.17

Order = 2

DC gain = 0.50

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