How are Kp, L, and tau typically estimated from a step-response for PID tuning?

How are Kp, L, and tau typically estimated from a step-response for PID tuning?

• A. From the peak overshoot only.
• B. From the process reaction curve using the tangent method: L is the delay before the tangent, tau is the time from end of L to the tangent intersection with the axis, and Kp is the steady-state gain from input to output.
• C. From frequency response alone.
• D. From the final value of the output.

Answer: (B)

The step response of a real process is well approximated by a first-order system with dead time, so three parameters—gain, delay, and time constant—capture its behavior. The steady-state gain is the ratio of the final output to the input step, giving how much the output settles for a given input change. After the delay, the response behaves like a first-order process with a time constant, so tuning rests on L and tau being read from the reaction curve. To extract them, look at the steepest part of the curve after the delay and draw the tangent there. The time where this tangent crosses the time axis (the x-axis) is the delay, L. Then measure how long after the end of that delay the tangent takes to reach the final value level; that time interval is the time constant, tau. This tangent-based geometry ties directly to the FOPDT model: y(t) ≈ Kp [1 − e^{-(t−L)/τ}] for t ≥ L, with Kp from the final value. The other options don’t provide all three parameters from the time-domain step response in this way.

The step response of a real process is well approximated by a first-order system with dead time, so three parameters—gain, delay, and time constant—capture its behavior. The steady-state gain is the ratio of the final output to the input step, giving how much the output settles for a given input change. After the delay, the response behaves like a first-order process with a time constant, so tuning rests on L and tau being read from the reaction curve.

To extract them, look at the steepest part of the curve after the delay and draw the tangent there. The time where this tangent crosses the time axis (the x-axis) is the delay, L. Then measure how long after the end of that delay the tangent takes to reach the final value level; that time interval is the time constant, tau. This tangent-based geometry ties directly to the FOPDT model: y(t) ≈ Kp [1 − e^{-(t−L)/τ}] for t ≥ L, with Kp from the final value. The other options don’t provide all three parameters from the time-domain step response in this way.