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87
docs/Chapters/5-STATE-FEEDBACK.md
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@ -101,4 +101,89 @@ $$
$$
and so, you just need to substitute these matrices to the equation of the
$G_p(s)$ and you get you controller
$G_p(s)$ and you get you controller
## Inserting the reeference
Until now we've worked without taking the reference into consideration, so
let's add it like we normally do:
![reference input unitary feedback](../Images/State-Feedback/reference-input-1.png)
Here we can compute the error and our $G(s)$ we have in our system:
$$
e = r(t) - y(t) \\
G(s) = \frac{G_c(s)G_p(s)}{1 + G_c(s)G_p(s)}
$$
Another way to deal with the reference is to put it between our blocks:
![reference input unitary feedback](../Images/State-Feedback/reference-input-2.png)
$$
e = r(t) - \mathcal{L}^{-1}\left\{Y(s)G_c(s)\right\} \\
G(s) = \frac{G_p(s)}{1 + G_c(s)G_p(s)}
$$
Now, let's see the difference in both $G_1(s)$ and $G_2(s)$:
$$
\begin{align*}
G_1(s) &= \frac{G_c(s)G_p(s)}{1 + G_c(s)G_p(s)}
\\
&= \frac{
\frac{N_c}{D_c}\frac{N_p}{D_p}
}{
1 + \frac{N_c}{D_c} \frac{N_p}{D_p}
} \\
&= \frac{
\frac{N_c N_p}{D_c D_p}
}{
\frac{D_cD_p +N_cN_p}{D_cD_p}
} \\
&= \frac{
N_c N_p
}{
D_cD_p +N_cN_p
} \\
\end{align*}
\;\;\;\;
\begin{align*}
G_2(s) &= \frac{G_p(s)}{1 + G_c(s)G_p(s)}
\\
&= \frac{
\frac{N_p}{D_p}
}{
1 + \frac{N_c}{D_c} \frac{N_p}{D_p}
} \\
&= \frac{
\frac{N_p}{D_p}
}{
\frac{D_cD_p +N_cN_p}{D_cD_p}
} \\
&= \frac{
N_c D_c
}{
D_cD_p +N_cN_p
} \\
\end{align*}
$$
While both have the same `poles`, they differ in their `zeroes`

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