From 7a5e7782f7a80af3becc70de96739a70a98568c9 Mon Sep 17 00:00:00 2001 From: Christian Risi <75698846+CnF-Gris@users.noreply.github.com> Date: Fri, 17 Jan 2025 19:04:40 +0100 Subject: [PATCH] V0.8.9.9 --- docs/Chapters/5-STATE-FEEDBACK.md | 87 +- .../reference-input-1.excalidraw.json | 1903 +++++++++++++++++ .../reference-input-2.excalidraw.json | 721 +++++++ .../State-Feedback/reference-input-1.png | Bin 0 -> 168808 bytes .../State-Feedback/reference-input-2.png | Bin 0 -> 50122 bytes 5 files changed, 2710 insertions(+), 1 deletion(-) create mode 100644 docs/Images/.excalidraw/State-Feedback/reference-input-1.excalidraw.json create mode 100644 docs/Images/.excalidraw/State-Feedback/reference-input-2.excalidraw.json create mode 100644 docs/Images/State-Feedback/reference-input-1.png create mode 100644 docs/Images/State-Feedback/reference-input-2.png diff --git a/docs/Chapters/5-STATE-FEEDBACK.md b/docs/Chapters/5-STATE-FEEDBACK.md index e346f8c..f5d2620 100644 --- a/docs/Chapters/5-STATE-FEEDBACK.md +++ b/docs/Chapters/5-STATE-FEEDBACK.md @@ -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 \ No newline at end of file +$G_p(s)$ 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` diff --git a/docs/Images/.excalidraw/State-Feedback/reference-input-1.excalidraw.json b/docs/Images/.excalidraw/State-Feedback/reference-input-1.excalidraw.json new file mode 100644 index 0000000..af23697 --- /dev/null +++ b/docs/Images/.excalidraw/State-Feedback/reference-input-1.excalidraw.json @@ -0,0 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