Relay Feedback PID Auto-Tuner

Relay Feedback PID Auto-Tuning Under Construction

Interactive Åström–Hägglund relay feedback auto-tuner. Pick any process model, run relay experiment to measure ultimate gain and period, then switch to automatically-tuned PID control.

Plant: G(s) = K·e^−Ls / (Ts + 1) | Relay: u = d·sgn(e) | PID: u = Kp·e + Ki∫e + Kd·de/dt

Setpoint y(t) u(t) PID Control

Auto Y-axis

Measured / Computed Values

Limit Amp a

—

Ultimate Period Tu

—

Ultimate Gain Ku

—

κ (Ku/K)

—

Error e

0.00

Simulation Controls

Process Model

K 2

T (s) 10.0

L (s) 1.0

Relay

Amplitude d 1.0

Hysteresis ε 0.05

Tuning Method

ZN Variant

PID Gains (Manual)

Kp 0.44

Ki 0.23

Kd 0.21

Reference r(t)

Type

Period (s) 20

Setpoint 1.0

📚 Read the tutorial — relay feedback theory and ZN frequency rules

Tuning Formulas & References

All rules below are derived from a single relay-feedback experiment that produces the ultimate gain K_u and ultimate period T_u via describing-function analysis:

K_u = 4d / (π·a), ω_u = 2π / T_u

where d is the relay amplitude and a is the resulting limit-cycle amplitude.

Ziegler–Nichols (1942)

The classic frequency-domain rules. Aggressive — expect ~25% overshoot on step setpoints. For aggressive setpoint tracking (e.g. square waves), ZN Classic is the default choice.

Variant	K_p	T_i	T_d	Notes
Classic	0.60 · K_u	0.50 · T_u	0.125 · T_u	Quarter-decay, ~25% overshoot
Some Overshoot	(1/3) · K_u	0.50 · T_u	(1/3) · T_u	Less overshoot, slower
No Overshoot	0.20 · K_u	0.50 · T_u	(1/3) · T_u	Conservative, monotonic step

AMIGO — Åström & Hägglund (2006), Eq. 7.7

Model-based AMIGO. Uses the FOLPD parameters L, T together with the measured K_u, T_u to produce a controller with target overshoot around 10% — less aggressive than ZN, but slower setpoint tracking. For SOPDT, the effective time constant is taken as max(T₁, T₂).

Parameter	Formula
K_p	(0.2 + 0.45 · T_u/T) / K_u
T_i	0.4 · T_u / (1 + 0.5 · L/T)
T_d	0.5 · T_u · L / T

Convert to standard gains: K_i = K_p / T_i and K_d = K_p · T_d. The process model gain K does not appear directly — only the ratio T_u / T matters, which is what makes AMIGO tolerant of static-gain errors in the FOLPD model.

Process model used for the relay experiment

The relay feedback rules above are model-free (they only use K_u, T_u). The simulator additionally supports an FOLPD model G(s) = K · e^−L·s / (T·s + 1) for users who want to compare with model-based rules. The κ = K_u / K gain ratio reported in the live values panel is useful for diagnostic purposes — a very large κ (≥ 10) usually indicates either a poorly-conditioned plant or a relay amplitude that is too small to overcome hysteresis.

References

Åström, K. J. & Hägglund, T. (1984). "Automatic tuning of simple regulators with specifications on phase and amplitude margins." Automatica, 20(5), 645–651.
Ziegler, J. G. & Nichols, N. B. (1942). "Optimum settings for automatic controllers." Trans. ASME, 64(8), 759–768.
Åström, K. J. & Hägglund, T. (2006). Advanced PID Control. ISA — The Instrumentation, Systems, and Automation Society. Chapter 7 (PID Design), §7.4 (AMIGO).
Hornsey, S. (2010). "A piecewise linear extension of the Ziegler–Nichols method." IEEE Control Systems Magazine, 30(6), 124–127. (cited for variant definitions; not the rule set used here.)