damping_from_fractional_overshoot Function

public pure function damping_from_fractional_overshoot(x) result(rst)

Employs the method of fractional overshoot to estimate the damping ratio from the response of a system to a step input. This method is useful for cases where the damping ratio is between approximately 0.5 to 0.8. In such range, the logarithmic decrement approach becomes less precise.

The fractional overshoot method locates the amplitude of the first peak of oscillation ($x_p$) and the settling amplitude ($x_f$), and the estimates the damping ratio as follows.

$$s = \frac{x_p - x_f}{x_f}$$

$$\zeta = \frac{1}{\sqrt{1 + \left( \frac{\pi}{\ln{s}} \right)^2}}$$

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:)	::	x	The step response of the system.

Return Value real(kind=real64)

The estimated damping ratio.

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