Under pure P control, increasing Kp eventually produces sustained oscillation. That critical gain is Ku and the oscillation period is Pu. Ziegler–Nichols maps (Ku, Pu) into suggested PID gains.

Why does Z–N sometimes look aggressive?

The classic rules aim for a quarter-amplitude decay response, which often has noticeable overshoot. Many modern tunings detune from Z–N for smoother behavior.

What does the disturbance step test?

A sudden load d shifts the plant input. Integral action should remove the resulting offset; derivative and proportional terms shape how quickly the output recovers.

How is this different from the cart PID page?

The cart demo is a live 1D mechanical toy with random kicks. This sandbox focuses on quantitative step-response metrics, plant parameters, and Ziegler–Nichols suggestions on a standard second-order plant.

PID Tuning Sandbox

Second-order plant step response: overshoot, rise/settling time, Ziegler–Nichols Ku/Pu hints, actuator saturation, and disturbance rejection.

Live graphs

About this model

This PID tuning sandbox closes a classical feedback loop around a second-order plant τ²ÿ + 2ζτẏ + y = K(u + d). A unit step in the setpoint r drives the error e = r − y into a saturated PID law u = Kp e + Ki ∫e dt + Kd ė. The page reports the usual step-response metrics: percent overshoot, 10–90% rise time, ±2% settling time, and steady-state error. A load disturbance step at a chosen time shows how well the tuned gains reject external forcing. The Ziegler–Nichols button estimates the ultimate gain Ku and ultimate period Pu under pure proportional control (sustained oscillation), then applies the classic PID rules Kp = 0.6 Ku, Ti = 0.5 Pu, Td = Pu/8. The model is pedagogical: it omits sensor noise, anti-windup beyond simple integral clamping, higher-order plant dynamics, and formal stability margins.

Who it's for: Students in classical control, mechatronics, and process control learning gain tuning, step-response metrics, and Ziegler–Nichols heuristics.

Key terms

PID controller
Step response
Overshoot
Settling time
Rise time
Ziegler–Nichols
Ultimate gain
Disturbance rejection
Actuator saturation

How it works

PID tuning sandbox: unit step response with overshoot, rise time, settling time, Ziegler–Nichols gain suggestions, and load-disturbance rejection on a second-order plant.

Key equations

u = Kp e + Ki ∫e dt + Kd ė, e = r − y

τ²ÿ + 2ζτẏ + y = K(u+d); Z–N: Kp=0.6Ku, Ti=0.5Pu, Td=Pu/8

Frequently asked questions

What are Ku and Pu?: Under pure P control, increasing Kp eventually produces sustained oscillation. That critical gain is Ku and the oscillation period is Pu. Ziegler–Nichols maps (Ku, Pu) into suggested PID gains.
Why does Z–N sometimes look aggressive?: The classic rules aim for a quarter-amplitude decay response, which often has noticeable overshoot. Many modern tunings detune from Z–N for smoother behavior.
What does the disturbance step test?: A sudden load d shifts the plant input. Integral action should remove the resulting offset; derivative and proportional terms shape how quickly the output recovers.
How is this different from the cart PID page?: The cart demo is a live 1D mechanical toy with random kicks. This sandbox focuses on quantitative step-response metrics, plant parameters, and Ziegler–Nichols suggestions on a standard second-order plant.

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Frequently asked questions