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I have a question regarding the poles of a transfer function given a state space diagram. There is a formula for it. This is my answer:

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Unfortunately, the answer is apparently long. The answer says that the pole of the transfer function G(s) is located at -3. So it doesn't mention the -2.

Anyone know where I went wrong?

Translation of question :

Given is the step response h(t) of a dynamic system whose dynamics are described by the state space model. Which poles does the transfer function G(s) = Y(s) / U(s) have?

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  • $\begingroup$ Could you translate the question in the screenshot to English? There could be some property of $\alpha$ and $\beta$ which "cancel" the pole at -2 with a zero at -2. What is $h(t)$? if it is something like impulse or step response and it contains only $e^{-3t}$, I strongly suspect that there is a zero involved in the transfer function. $\endgroup$
    – AJN
    Commented Nov 25 at 15:34
  • $\begingroup$ is $G(s) = \Lambda / \det {(sI-A)}$? What is $\Lambda$? $\endgroup$
    – AJN
    Commented Nov 25 at 15:37
  • $\begingroup$ So this is the translation: Given is the step response h(t) of a dynamic system whose dynamics are described by the state space model. Which poles does the transfer function G(s) = Y(s) / U(s) have? $\endgroup$
    – CheesyBeqa
    Commented Nov 25 at 19:13

1 Answer 1

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Let the system be $H(s)$. Laplace transform of the step response is $$ \begin{align} \frac{1}{s}\cdot H(s) &= \mathscr{L}\left(2-2e^{-3t}\right)\\ &= \frac{2}{s} - \frac{2}{s+3}\\ \implies H(s) &= \frac{6}{s+3} \end{align} $$ The above system has a pole at $s=3$.

Moreover, if you find the expression for the transfer function from the formula $C\cdot (sI-A)^{-1}\cdot B$. You will be able to note that for certain values of $\alpha$ and $\beta$, the pole at $s=2$ is cancelled by a zero at $s=2$. I am leaving that unsolved since this looks like a homework question.

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  • $\begingroup$ Thank you so much for this answer. I realize as well that for some values of alpha, one of the poles ( in this case 2 ) will cancel out. $\endgroup$
    – CheesyBeqa
    Commented Nov 26 at 15:03

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