How to use the PID Controller Tuning Calculator
- 1. Choose a methodUltimate gain (Ku, Tu), plant G(s), or reaction curve (K, L, T).
- 2. Enter the dataFor example G(s) = 1/(s(s+1)(s+5)).
- 3. Get the gainsKp, Ti and Td in ideal and parallel forms for every rule.
Ziegler-Nichols tuning methods
Second method (ultimate gain): with only proportional control, raise K until the loop oscillates steadily. That gain is Ku and the period is Tu. The classic PID settings are Kp = 0.6Ku, Ti = 0.5Tu and Td = 0.125Tu. From a transfer function, Ku and ωu come from putting s = jω in D(s) + K·N(s) = 0, which gives the same answer as the Routh array. First method (reaction curve): an open-loop step test gives the gain K, dead time L and time constant T, and the PID settings are Kp = 1.2T/(KL), Ti = 2L and Td = 0.5L.
Worked example (Ogata): G(s) = 1/(s(s+1)(s+5)). The characteristic equation is s³ + 6s² + 5s + K = 0. The imaginary part 5ω − ω³ = 0 gives ωu = √5 = 2.236 rad/s, and the real part gives Ku = 6ω² = 30. Then Tu = 2π/√5 = 2.81 s, Kp = 18, Ti = 1.405 s and Td = 0.351 s, so Gc(s) = 6.32(s + 1.4235)²/s.
Exam tips
- The ideal form is Kp(1 + 1/(Ti·s) + Td·s). The parallel form is Kp + Ki/s + Kd·s, with Ki = Kp/Ti and Kd = Kp·Td.
- A second-order plant without delay never oscillates under P control, so the ultimate gain method does not apply.
- Z-N settings usually give around 25% overshoot. The 'some overshoot' and 'no overshoot' rules are gentler.
- Always state which form your gains are in.
Frequently asked questions
How does the tool find Ku from G(s)?
It puts s = jω in 1 + K·G(s) = 0 and solves the imaginary part for ω, then the real part for K. The smallest positive K is Ku, and Tu = 2π/ω.
Which Ziegler-Nichols table is used?
The classic tables from Ziegler and Nichols (1942), plus the commonly used Pessen, 'some overshoot' and 'no overshoot' variants for the ultimate gain method.
Are these the final controller settings?
Treat them as a starting point. Fine-tune on the real plant or in a simulation to get the response you need.
Is this tool free?
Yes. You fill in a short form once to see your result, and the other Uncodemy tools then open without it. All the maths runs in your browser.