Assume that we have a SISO transfer function model $G(s)$ and we want to look how the output looks like when we have a input signal $u(t)$ that looks like this:

$$u_1(t) = \left\{\begin{matrix} 50 + 5\sin(2\pi10 t) & u(t) > 0 \\ 0 & u(t) \leq 0 \end{matrix}\right.$$

$$u_2(t) = \left\{\begin{matrix} A_2\sin(2\pi\omega_2 t) & u(t) > 0 \\ 0 & u(t) \leq 0 \end{matrix}\right.$$

Is it possible that I can prove that $G(s)U_1(s) = G(s)U_2(s)$ or $G(s)U_1(s) \sim G(s)U_2(s)$ by changing $\omega_2$ and $A_2$?

$U_i(s), i = 1, 2$ is the frequency domain signal on the view of $u_i(t), i = 1, 2$.

  • $\begingroup$ um, what's $u(t)$ here, then. Is it deterministic? Is it linked to either of $u_1$ or $u_2$? Without very specific restrictions on $u(t)$, your desired statements are plain wrong; without any restricitions, it's trivially true (set $u(t) = -1$ for all $t$). $\endgroup$ – Marcus Müller Jun 10 '20 at 11:12
  • $\begingroup$ @MarcusMüller $u$ is the input signal to the system $G(s)$. Control engineering. $\endgroup$ – Daniel Mårtensson Jun 10 '20 at 11:14
  • $\begingroup$ well, point is that if you can't define what $u(t)$ is, your statements are plain wrong $\endgroup$ – Marcus Müller Jun 10 '20 at 11:15
  • $\begingroup$ @MarcusMüller Input signal? $\endgroup$ – Daniel Mårtensson Jun 10 '20 at 11:16
  • $\begingroup$ Daniel, sorry misunderstanding with "you can't define $u(t)$" I didn't mean "you can't explain $u(t)$" but "you can't set $u(t)$ so that you end up in a corner case where there might be solutions". $\endgroup$ – Marcus Müller Jun 10 '20 at 12:39

No, at least not the way it's written.

$U_1(\omega)$ contains two frequency peaks (10Hz and 0Hz) and $U_2(\omega)$ only one at $\omega _2$. $G(\omega)$ can change the relative height of the peaks but the overall but not create new ones. $A_2$ and $U_2(\omega)$ can vary the height and position of the peak but not the number of peaks.

  • $\begingroup$ So there is no way I can get a similar output from $G(s)$ if I use both the output signals where $u_2$ is tuned with $\omega_2$? $\endgroup$ – Daniel Mårtensson Jun 10 '20 at 12:14
  • $\begingroup$ @DanielMårtensson exactly what the answer says. (and what I've been saying in the comments). ("similar" is a problematic term: it's not really well-defined. Your $\sim$ means "proportional", and yes, we can definitely preclude that). $\endgroup$ – Marcus Müller Jun 10 '20 at 12:38
  • $\begingroup$ @MarcusMüller I think you need more practical experience how to understand different questions. Some questions are based on clean theory, some other on real train and error. $\endgroup$ – Daniel Mårtensson Jun 10 '20 at 14:01
  • $\begingroup$ :D practical experience is key to successful development; tell me about it, I lead software projects! You, however, state provably incorrect equations, so that doesn't really apply here. $\endgroup$ – Marcus Müller Jun 10 '20 at 14:35

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