Percentage Overshoot Calculator

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

  1. Maximum and Step Values
  2. 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. 





 High accuracy settling time measurements figure 1 





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 Î¶ 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.





 2nd Order Damping Ratios 





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





 Sine integral