I am given the Laplace transform of the output of a LTI system: $$Y(s) = \frac{1}{s((s+2)^2+1)}$$ Asked is what the steady state response $y_{ss}(t)$ would be. I think that $y_{ss}(t) = \lim_{t\to\infty} y(t)$, since after waiting infinit long, the system should be in steady state. (Right?)

I thought to use the final value theorem:

$$\lim_{t\to\infty}y(t)=\lim_{s\to 0}sY(s)$$ This gives $$\lim_{s\to 0} sY(s)=\lim_{s\to 0}\frac{1}{(s+2)^2+1} = \frac{1}{5}.$$

This is different from $\frac{1}{10}$. When I let a computer algebra system calculate $\mathscr{L}^{-1}[Y(s)] \bigg{|}_{t=\infty}$ I get $\frac{1}{10}$. (I'm using wxMaxima and used limit(ilt(1/(s*(s^2 + 2*s + 10)), s, t), t, inf);.)

What am I doing wrong? Thanks in advance.

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    $\begingroup$ Your code should be s^2+4*s+5. $\endgroup$ – msm Dec 3 '16 at 21:03
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    $\begingroup$ unlike transient response, which is with zero input but some states are non-zero at $t=0$, the steady state response needs a defined input. $\endgroup$ – robert bristow-johnson Dec 4 '16 at 2:37
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    $\begingroup$ $Y(s)$ is the output of the system, so its fine as written. $\endgroup$ – Batman Dec 4 '16 at 5:28

For these calculations, it is better to give the Wolfram Alpha answers:

inverse Laplace transform 1/(s*(s^2 + 4*s + 5))

Which gives the correct expression, consistent with the Final Value expression of 1/5:

$$\frac{1}{5} - \frac{1}{5} e^{-2 t} (2 sin(t) + cos(t))$$

This is very different from the complex (phasor) expression, which could inexplicably have a 1/10, but when evaluated it is still 1/5:

$$\frac{1}{10} i (e^{(-2 - i) t} ((2 + i) e^{2 i t} + (-2 + i)) - 2 i)$$


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