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.
- 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)
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.
- 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
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-ζ2))
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.
- 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%.
 Step Response
 The title image shown below is by author Krishnavedala