Percent Overshoot (2nd-Order System)

Calculates the percentage by which a second-order underdamped system's response overshoots its final steady-state value.

Written by the Vorzep Team

RoboticsControl TheoryCalculator

⟹ Solve⟷%OS, ζ

Damping ratio (ζ)

—

Percent overshoot (%OS)

Please fix the errors above.

Solve for: %OS ζ (damping ratio)

Presets:

Percent overshoot (%OS)—

—

ζ: —%OS: —

✓ Copied!

Overshoot gauge—

Low (< 5%) Medium (5–25%) High (> 25%)

%OS = exp(-ζπ / √(1 - ζ²)) · 100 · Valid for 0 < ζ < 1 (underdamped system)

Variables

Symbol	Quantity	Unit
%OS	Percent overshoot	%
zeta	Damping ratio of the system (0 < zeta < 1)

What it means

Overshoot is the amount by which the response exceeds the final steady‑state value. For a second‑order system, it is determined solely by the damping ratio ζ. A lower ζ gives higher overshoot. This metric is used to evaluate the system’s transient behaviour and is often specified in design requirements. It guides the choice of controller gains.

Worked example

Percent Overshoot (2nd‑Order) – Two Detailed Examples

Real‑World

Scenario: A second‑order system has damping ratio ζ = 0.3. The percent overshoot %OS = exp(-ζπ/√(1‑ζ²)) × 100 ≈ 37.2%. This means the system will exceed its final value by 37.2% before settling. The control engineer finds this overshoot too high for precision applications and adjusts the controller to increase damping, trading off responsiveness for reduced overshoot.

Parameter	Value
ζ (damping ratio)	0.3

1%OS = exp(-0.3π/√(1-0.09)) × 100 = exp(-0.942/0.954) × 100 = exp(-0.987) × 100 ≈ 37.2%

Result 37.2% ✓ Overshoot

Scenario: A system with ζ = 0.7 has %OS = exp(-0.7π/√(1‑0.49)) × 100 ≈ 4.6%. This low overshoot is desirable for systems where accuracy is more important than speed, such as in medical devices or precision manufacturing. The engineer aims for ζ ≈ 0.7 to achieve a good balance between speed and stability.

Parameter	Value
ζ	0.7

1%OS = exp(-0.7π/√(1-0.49)) × 100 = exp(-2.199/0.714) × 100 = exp(-3.081) × 100 ≈ 4.6%

Result 4.6% ✓ Low overshoot

Insight: Percent overshoot depends only on the damping ratio ζ. It decreases as ζ increases. For ζ > 0.7, overshoot is less than 5%, making the system well‑damped. The trade‑off is slower response time.

Common mistakes

Percent overshoot: %OS = exp(−ζ·π / √(1−ζ²)) × 100 – for a step input.
ζ: Damping ratio – must be between 0 and 1 (underdamped).
Overshoot: The maximum peak value above the steady‑state value.
As ζ → 0: %OS → 100% (oscillatory). As ζ → 1: %OS → 0% (critically damped).
Assumes: Unity feedback, no zeros, and a standard second‑order system.

Applications

Percent overshoot for a second‑order system, %OS = exp(−ζ·π/√(1−ζ²))×100, quantifies the maximum peak value relative to the final value. This is a critical measure of system stability and quality. Engineers use it to ensure that the system does not exceed limits, especially in precision applications. A larger damping ratio ζ reduces overshoot. By specifying a maximum allowable overshoot, they can determine the required damping and thus the controller gains. This formula is essential for designing safe and reliable control systems in robotics, aerospace, and automotive applications.

Setting performance requirements for motion control systems
Controller design to limit overshoot in precision positioning
Stability assessment and compliance with specifications
Optimisation of damping ratio for desired response
Educational understanding of time‑domain specifications