Control Systems · Worked example

PID Response Basics: Overshoot, Rise Time, and Settling Time from a Second-Order Model

A worked control-system example connecting damping ratio and natural frequency to overshoot, peak time, rise time, and settling time.

By Neutron Studios LLC9 minute readPublished 2026-08-11Reviewed 2026-11-03

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Why this calculation matters

PID gains are tuning parameters, but the performance language used to judge a tuning—overshoot, rise time, peak time, damping, and settling time—is often easiest to understand through a standard second-order model.

This guide starts with a target closed-loop damping ratio and natural frequency. It does not claim that every PID-controlled plant becomes exactly second order; instead it gives a reference response that can be compared against a simulated or measured loop.

What you will calculate

Calculate damped natural frequency from ζ and ωn.
Estimate percent overshoot and 2% settling time.
Relate transient specifications to a PID simulation.
Recognize when higher-order poles, zeros, saturation, delay, or sampling invalidate the simple model.

Given values

Closed-loop damping ratio ζ = 0.50
Natural frequency ωn = 4 rad/s
Standard underdamped second-order reference model
Unit-step command

Governing equations

Damped natural frequency

\omega_d = \omega_n \sqrt{1 - \zeta ^{2}}

Applies for 0 < ζ < 1.

Percent overshoot

Mp = 100\; \exp \left[\frac{- \pi \zeta }{\sqrt{1 - \zeta ^{2}}}\right]

Step-response overshoot for the standard second-order form.

Peak time

tp = \frac{\pi }{\omega_d}

Time of the first response peak.

2% settling estimate

ts \approx \frac{4}{\zeta \omega n}

Common engineering approximation for the 2% settling band.

Approximate rise time

tr = \frac{\pi - \arctan \left(\frac{\sqrt{1 - \zeta ^{2}}}{\zeta }\right)}{\omega_d}

A common 0–100% rise-time expression for an underdamped second-order step.

Worked solution

1. Find the oscillation frequency

The damped natural frequency is lower than the undamped natural frequency because damping slows the oscillatory mode.

\omega_d = 4 \sqrt{1 - 0.5^{2}} = \frac{3.464\; \mathrm{rad}}{s}

2. Estimate overshoot

With ζ = 0.5, the expected first peak is about 16.3% above the final value. A unit-step response should therefore peak near 1.163 if the closed loop is well represented by this model.

Mp \approx 100 e^{\frac{- \pi \times 0.5}{\sqrt{0.75}}} = 16.3 \%

3. Estimate timing

The first peak occurs at about 0.907 s. The common 2% settling estimate is 2.0 s. The corresponding 0–100% rise-time estimate is about 0.605 s.

\begin{gathered}tp \approx 0.907\; s\\ts \approx 2.00\; s\\tr \approx 0.605\; s\end{gathered}

4. Use the model as a PID target, not a guarantee

In the Control System Simulator, compare these target metrics with the actual closed-loop model. If the plant has a nonminimum-phase zero, transport delay, actuator saturation, significant extra poles, or a slow sample time, the response can deviate substantially even when a dominant-pole estimate looks similar.

Result

Engineering interpretation

For ζ = 0.5 and ωn = 4 rad/s, the reference response has approximately 16.3% overshoot, 0.907 s peak time, 0.605 s rise time, and 2.0 s 2%-settling time.

A PID tuning can be judged against those numbers, but the final assessment should come from the complete plant/controller simulation and, for real hardware, measured response.

Interactive

Try your own damping ratio and natural frequency

Standard underdamped second-order step response. Values start at the worked example (ζ = 0.5, ωn = 4 rad/s).

Simulate a full PID loop in the Control System Simulator →

Sanity checks

Increasing ζ while holding ωn fixed should reduce overshoot.
Increasing ωn while holding ζ fixed should reduce the characteristic response times.
A model with ζ ≥ 1 should not use the underdamped overshoot equation.
If actuator saturation is active, linear second-order formulas can badly underpredict settling time.

Common mistakes

Treating Ziegler–Nichols or any other tuning rule as a final answer.
Ignoring actuator limits and integral windup.
Comparing continuous-time formulas with a discrete controller whose sample rate is too low.
Tuning from the reference response without checking control effort or noise amplification.

References and model boundaries

Standard second-order transient-response relationships used in introductory feedback-control texts.
Final PID tuning should include the complete plant dynamics, actuator constraints, sampling, disturbances, and measurement noise.

For safety-critical, regulated, production, or otherwise consequential work, independently verify the result using the governing standard, current manufacturer data, and qualified engineering review. See the site methodology and engineering disclaimer.