Control-Network-Systems/docs/Chapters/8-SMITH-PREDICTOR.md

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*}