A second order low pass filter is an input-output block that computes
and
where \(r\) is a positive real number. This causes the input signal \(u(\cdot)\) to be converted to an output signal \(y(\cdot)\) that is differentiable in time. See the plot of a filtered step response.
For a control valve, the valve authority \(N\) is defined as
where \(\Delta p_v(1)\) is the pressure drop across the fully open valve, and \(\Delta p_v(1) + \Delta p_0\) is the pressure drop across the whole flow leg whose mass flow rate is controlled by the valve. Valves should be designed such that \(N\) is around 0.5, but not higher.