Editing Step response (section)

==== System Identification using the Step Response: System with two real poles ====
[[File:PT2 System Step-Response Diagram with required Measurements (2018).png|thumb|340x340px|Step response of the system with <math>x(t)=1</math>. Measure the significant point <math>k</math>, <math>t_{25}</math>and <math>t_{75}</math>.]]
This method uses significant points of the step response. There is no need to guess tangents to the measured Signal. The equations are derived using numerical simulations, determining some significant ratios and fitting parameters of nonlinear equations. See also.<ref>{{Cite web|url=https://hackaday.io/page/4829-identification-of-a-damped-pt2-system|title=Identification of a damped PT2 system {{!}} Hackaday.io|website=hackaday.io|language=en|access-date=2018-08-06}}</ref>

Here the steps:

* Measure the system step-response <math>y(t)</math>of the system with an input step signal <math>x(t)</math>.
* Determine the time-spans <math>t_{25}</math>and <math>t_{75}</math>where the step response reaches 25% and 75% of the steady state output value.
* Determine the system steady-state gain <math>k=A_0</math>with <math>k=\lim_{t\to\infty} \dfrac{y(t)}{x(t)}</math>
* Calculate <math display="block">r=\dfrac{t_{25}}{t_{75}}</math> <math display="block">P=-18.56075\,r+\dfrac{0.57311}{r-0.20747}+4.16423</math> <math display="block">X=14.2797\,r^3-9.3891\,r^2+0.25437\,r+1.32148</math>
* Determine the two time constants <math display="block">\tau_2=T_2=\dfrac{t_{75}-t_{25}}{X\,(1+1/P)}</math> <math display="block">\tau_1=T_1=\dfrac{T_2}{P}</math>
* Calculate the transfer function of the identified system within the Laplace-domain <math display="block">G(s) = \dfrac{k}{(1+s\,T_1)\cdot(1+s\,T_2)}</math>