control//step response

The step response is how a system answers a sudden change of its input, and it is the standard test of a control loop: change the setpoint in one jump, record the output, and read off four numbers that say whether the loop is fast enough, calm enough and accurate enough. Commissioning reports, tuning methods and specifications are written in these numbers.

The step response is how a system answers a sudden change of its input, and it is the standard test of a control loop: change the setpoint in one jump, record the output, and read off four numbers that say whether the loop is fast enough, calm enough and accurate enough. Commissioning reports, tuning methods and specifications are written in these numbers.

The rise time is how long the output takes to go from 10 % to 90 % of its final change. It measures speed.

The overshoot is how far the output passes its new value, as a percentage of the step. It measures how calm the loop is, and specifications usually put a ceiling on it.

The settling time is how long the output takes to enter a band around the final value, usually ±2 % or ±5 %, and stay there. It measures how long the loop is busy with the change.

The steady-state error is what remains at the end. It measures accuracy.

The four pull against each other. Raising a loop's gain shortens the rise time and raises the overshoot, and past some point the settling time grows again because the output rings around its target. A design picks a balance, and for a second-order loop the balance has a single knob, the damping ratio ζ\zetaζ, which fixes the overshoot as a fraction of the step:

Mp=e−πζ/1−ζ2,0<ζ<1.M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}},\qquad 0<\zeta<1.Mp​=e−πζ/1−ζ2​,0<ζ<1.

With ζ≈0.7\zeta\approx0.7ζ≈0.7 the overshoot is under 5 % and the settling close to the fastest; with ζ=0.5\zeta=0.5ζ=0.5 it is about 16 %. Real loops follow this formula when one pair of eigenvalues dominates their response (modes).

A first-order system never overshoots, and its time constant fixes its whole step response: 63 % of the change after one τ\tauτ, 95 % after three, 98 % after four (exponential decay).

The step is a convenient test, not the only one. A ramp tests tracking, a load step tests disturbance rejection (often the case that matters in process plants), and a sweep of sines gives the frequency response.

The numbers describe the tuning only for small steps. A step large enough to saturate the actuator measures the actuator's limit, and may show integral windup rather than the tuning.