Overshoot refers to the output of a control system exceeding its final, steady-state value.

In this post we present two ways to calculate the Percentage Overshoot.

From

- Maximum and Step Values
- The damping ratio in a second order system

**From Max and Step Values**

To calculate the percentage (%) overshoot, enter:

- Maximum Value (Max_val)
- Step Value (Step_val)

**Formula**

**PO = ((Max_val – Step_val)/(Step_val))*100**

*Note: Both the maximum and the step values must have the same units. *

For a step input, the *percentage overshoot* (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step (value=1), the *overshoot* is just the maximum value of the step response minus one.

The picture above shows an *ideal* step input where the signal rises instantaneously to a fixed value and the output that rises or (overshoots) the value to which it finally settles.

**Example Calculation**

For

- Max value of 2 and a step value of 1, the percentage overshoot is 100%
- Max value = 10 and step value = 1, the overshoot is 900%

**Second Order Systems**

**Formula**

For second-order systems, the percentage overshoot is a function of the damping ratio **ζ** and is given by

**PO = 100*exp((-ζ*π)/(√(1-ζ ^{2}))**

Enter *ζ* to calculate the percentage overshoot

The damping ratio *ζ* is a system parameter, that can be

- undamped (
*ζ*= 0) - underdamped (
*ζ*< 1) - critically damped (
*ζ*= 1) - overdamped (
*ζ*> 1)

However, the formula above works only for the undamped or underdamped case or *ζ* < 1.

The picture below shows a graph of the time response of a second order system with various damping ratios. The horizontal axis is in radians, and represents the time multiplied by the natural frequency of the system. A range of damping ratios are depicted between 0 and 2.

Note that damping ratios of *ζ*=1 and *ζ*=2 are not associated with any overshoot.

**Example Calculations**

For

*ζ*=0, the overshoot is 100%*ζ*=0.5, the overshoot is 25%

These values can be seen in the plot as well. For instance with *ζ*=0, the max value is 2 and the step value is 1. These values can be used in the first calculator on this page to give the same value of 100%.

**Related Calculators**

**Wikipedia References**

[1] Overshoot

[2] Damping

[3] Step Response

[4] The title image shown below is by author Krishnavedala