Second-Order Response
One parameter reshapes the whole response — overdamped, critical, underdamped, undamped — on the same plot.
Skip to the animationA second-order system is described by a natural frequency that sets how fast it responds and a damping ratio that sets how it settles — and because overshoot depends on damping alone while settling time depends on their product, a specification translates directly into a region of the s-plane.
The standard form
G(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²). A first-order system can only decay; two poles are the fewest that can oscillate, which is why this form is the workhorse of the subject.
| ζ | Poles | Response |
|---|---|---|
| > 1 | Two real, distinct | Overdamped — slow, no overshoot |
| = 1 | Real, repeated | Critically damped — fastest with no overshoot |
| 0 < ζ < 1 | Complex conjugate pair | Underdamped — overshoot, then ringing |
| = 0 | On the imaginary axis | Undamped — oscillates forever |
| < 0 | Right half-plane | Unstable — growing oscillation |
Most designs sit near ζ = 0.7: about 5% overshoot and a far faster rise than critical damping. A little overshoot buys a lot of speed.
Which parameter owns which specification
- Rise time
t_r ≈ 1.8/ωₙ— owned byωₙ. - Peak overshoot
M_p = e^(−πζ/√(1−ζ²))— owned by `ζ` alone;ωₙdoes not appear. - Settling time
t_s ≈ 4/(ζωₙ)— owned by the product, which is the real part of the poles. - Damped frequency
ω_d = ωₙ√(1−ζ²)— the ringing you actually observe.
Since overshoot depends only on ζ, an overshoot specification *is* a damping specification. That is the single most useful fact in the topic.
Design as pole placement
Each specification becomes a region of the s-plane: a wedge from the origin for overshoot (constant ζ is a constant angle), a vertical boundary for settling time (constant ζωₙ), and a circle for rise time (constant ωₙ). The design target is where the regions overlap, and root locus is the tool for steering the poles into it.
Real systems have additional poles and zeros that distort these relations, so the second-order approximation is a starting point. It is a good one when one pole pair is much closer to the imaginary axis than everything else — the dominant pole assumption.
The numbers you will be asked for
- Standard form
ωₙ² / (s² + 2ζωₙs + ωₙ²)
Two parameters, and they mean different things.
- Peak overshoot
M_p = e^(−πζ/√(1−ζ²))
Depends on ζ alone.
- Settling time
t_s ≈ 4 / (ζωₙ)
To within 2%. Set by the real part of the poles.
- Damped frequency
ω_d = ωₙ·√(1 − ζ²)
The observed ringing frequency.
- Pole locations
s = −ζωₙ ± jωₙ√(1−ζ²)
ζ is the cosine of the angle from the negative real axis.
Advantages and disadvantages
Advantages
- Two parameters describe the whole step response.
- Each specification maps to one parameter or one product.
- Specifications become regions on the s-plane, so design becomes geometry.
- A good approximation for higher-order systems with a dominant pole pair.
Disadvantages
- Real systems have extra poles and zeros that distort every relation.
- The formulas assume a step input and a unity-gain standard form.
- Says nothing about disturbance rejection or noise.
- Critical damping is a knife edge — slightly less and it overshoots.
Watch it work
Check yourself
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