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I know that that the poles of this system are: $s_1 = -1 + 2j$ and $s_2 = -1 - 2j$. I think to solve this problem I would need to find the laplace of this plot, with the formula: $$ W(s) = \frac{\rm zeros}{\rm poles} $$ Then I would need to get the inverse Laplace. I am stuck on how to get the zero when it is not provided in the plot.


1 Answer 1


You don't need the zeros, you just need to understand the relation between poles and the corresponding time domain function. A real-valued pole $s_{\infty}=\sigma$ corresponds to an exponential function

$$\frac{k}{s-\sigma}\Longleftrightarrow ke^{\sigma t},\qquad t>0$$

A double real-valued pole corresponds to an exponential multiplied by $t$:

$$\frac{k}{(s-\sigma)^2}\Longleftrightarrow kte^{\sigma t},\qquad t>0$$

[You can look up how this generalizes to poles of higher order.]

A pair of complex conjugate poles $s_{\infty}=\sigma\pm j\omega$ corresponds to a sinusoid with frequency $\omega$ multiplied by an exponential:

$$ke^{\sigma t}\cos(\omega t+\phi),\qquad t>0$$

The scaling $k$ and the phase $\phi$ depend on the numerator of the Laplace transform, but here they are irrelevant.

So for the given pair of complex conjugate poles, we know that the corresponding time domain function must be an exponentially damped ($\sigma=-1<0$) sinusoid with frequency $\omega=2$.


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