# Why do high-frequency dynamics quickly go away in a step response?

As we know, a step input hits all the frequencies of a dynamical system. However, my professor told me today that the high-frequency response is only present for a short time at the very start, and then only the low-frequency parts remain for the rest of the step response. Here's an example:

The first image shows two identified systems - $P_1(j\omega)$ and $P_2(j\omega)$ - of the actual system $P(j\omega)$. As you see, $P_1(j\omega)$ matches closely at low frequencies while $P_2(j\omega)$ matches closely at high frequencies.

When we apply a step response, you can see that $P_2(j\omega)$ responds almost like the real system $P(j\omega)$ at the very start (what my professor says is the part where high frequencies are "activated") while $P_1(j\omega)$ is quite accurate for the rest "slow dynamics" part of the response - where my professor says the "low frequencies" remain.

Question : what is it that makes the high frequencies be present only at the start and dissipate away quickly as time goes by, while the low frequencies persist throughout the step response, decaying more slowly?

Thanks a lot for your help!

• First of all, your identified system number 2 seems to have poor estimation accuracy of the static gain. I don't know how you judge the identification accuracy in this regard. Sep 17 '15 at 10:30

## 1 Answer

High-frequency dynamics are fast. If they are stable, their effect goes away after a few time constants, which are short, by definition.

The thing to keep in mind is that the time constant of a mode is the inverse of its frequency. The higher the frequency, the shorter the time a stable mode takes to decay to a specified fraction if its initial value.

To make this all more precise, just look at the expansion of a system's time-response in complex-exponential terms of the type $e^{(a+jb)t}$.

• You're on the right track I think... Care to develop this more formally? Sep 17 '15 at 15:32
• See last paragraph above.
– Pait
Jan 14 '16 at 21:05