This page describes two methods to identify system parameters for first order systems. The first method uses a step response and the second method uses frequency response.
The general form of the transfer function of a first order system is
where K is the DC gain and τ is the time constant. The problem is to find these two parameters given an input signal and its output signal.
Input: apply a step with magnitude A
Measure the output's steady state value, yss. Then the DC gain is given by,
Measure the time when the output is 63.2% of steady state value. That time is the time constant τ.
Input: apply a sinusoid Asin(ωt)
Output (steady state):
Measure the horizontal time shift, Δt. Then the time constant is given by,
Measure the output's amplitude, A'. Then the DC gain is given by,