2025-01-17 20:15:01 +01:00
|
|
|
# Smith Predictor
|
|
|
|
|
Once you have your system with `pure delays`, desing $G_c$
|
|
|
|
|
as if $G_p$ ha no `delay` at all. The resulting system
|
|
|
|
|
should be something similar to: $\frac{G_cG_p}{1 + G_cG_p}$.
|
|
|
|
|
|
|
|
|
|
Now consider the real delay of the system and desing $\hat{G}_c$
|
|
|
|
|
so that the new $\hat{G}(s) = G(s)e^{-sT}$:
|
|
|
|
|
$$
|
|
|
|
|
\begin{align*}
|
|
|
|
|
\frac{\hat{G}_cG_pe^{-sT}}{1 + \hat{G}_cG_pe^{-sT}}
|
|
|
|
|
&= \frac{G_cG_p}{1 + G_cG_p}e^{-sT} \rightarrow \\
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\rightarrow
|
|
|
|
|
\frac{\hat{G}_c}{1 + \hat{G}_cG_pe^{-sT}}
|
|
|
|
|
&= \frac{G_c}{1 + G_cG_p} \rightarrow \\
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\rightarrow
|
|
|
|
|
\hat{G}_c + \hat{G}_c G_cG_p
|
|
|
|
|
&= G_c + G_c\hat{G}_cG_pe^{-sT} \rightarrow \\
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\rightarrow
|
|
|
|
|
\hat{G}_c \left[
|
|
|
|
|
1 + G_cG_p(1 - e^{-sT})
|
|
|
|
|
\right]
|
|
|
|
|
&= G_c \rightarrow \\
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\rightarrow
|
|
|
|
|
\hat{G}_c
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
&= \frac{ G_c}{ 1 + G_cG_p(1 - e^{-sT})}
|
|
|
|
|
\end{align*}
|
|
|
|
|
|
|
|
|
|
$$
|
