38 lines
840 B
Markdown
38 lines
840 B
Markdown
# Smith Predictor
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Once you have your system with `pure delays`, desing $G_c$
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as if $G_p$ ha no `delay` at all. The resulting system
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should be something similar to: $\frac{G_cG_p}{1 + G_cG_p}$.
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Now consider the real delay of the system and desing $\hat{G}_c$
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so that the new $\hat{G}(s) = G(s)e^{-sT}$:
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$$
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\begin{align*}
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\frac{\hat{G}_cG_pe^{-sT}}{1 + \hat{G}_cG_pe^{-sT}}
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&= \frac{G_cG_p}{1 + G_cG_p}e^{-sT} \rightarrow \\
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\rightarrow
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\frac{\hat{G}_c}{1 + \hat{G}_cG_pe^{-sT}}
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&= \frac{G_c}{1 + G_cG_p} \rightarrow \\
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\rightarrow
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\hat{G}_c + \hat{G}_c G_cG_p
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&= G_c + G_c\hat{G}_cG_pe^{-sT} \rightarrow \\
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\rightarrow
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\hat{G}_c \left[
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1 + G_cG_p(1 - e^{-sT})
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\right]
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&= G_c \rightarrow \\
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\rightarrow
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\hat{G}_c
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&= \frac{ G_c}{ 1 + G_cG_p(1 - e^{-sT})}
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\end{align*}
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$$ |