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docs/Chapters/11-CRYPTOGRAPHY-WITH-STATE-OBSERVER.md

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+# Chryptography with Chaotic Systems
+
+## Deterministic Chaotic Systems
+These are `non-linear` differential equations that are 
+***very*** sensitive to initial conditions. One of the most 
+famous is the `Lorentz-Attractor`.
+
+Since we cannot get back to these initial conditions, these are 
+***unpredictable systems***.
+
+>[!NOTE]
+> $s(x)$ is a sync message added to the other `system` to get it 
+> produce the same `output` as the `first one`.
+>
+> Here it is represented its effect in `red`.
+>
+> In `yellow` you see a simple gain factor coming from the same 
+> $s(x)$
+
+Let's say we have 2 of these `systems`:
+$$
+\begin{align*}
+    x(t); z(t) &\triangleq \text{chaotic systems}\\
+    e(t) = x(t) - z(t) \rightarrow 0 
+    &\triangleq \text{synchronization} \\
+    s(x) = f(x) + Kx &\triangleq \text{synchronization signal}
+
+   
+\end{align*}\\
+
+\begin{cases}
+        \dot{x}(t) = Ax(t) + Bf(x) + c\\
+        \dot{z}(t) = Az(t) + Bf(z) + c 
+        \textcolor{red}{+ Bf(x) - Bf(z)}\\
+        \dot{e}(t) = \dot{x}(t) - \dot{z}(t)
+\end{cases}\\
+
+\begin{align*}
+    \dot{e}(t) &= \dot{x}(t) - \dot{z}(t) = \\
+   &=  Ax(t) + Bf(x) + c -  Az(t) - Bf(z) - c = \\
+   &=  A(x(t) - z(t))+ B(f(x) - f(z)) = \\
+   &=  Ae(t)+ B(f(x) - f(z)) = \\
+   &=  Ae(t)+ B(f(x) - f(z)\textcolor{red}{-f(x) + f(z)}) = \\
+   &=  Ae(t) \longrightarrow e(t) \rightarrow 0 
+   \text{ if } eig(A)\in \R^- \\
+
+   &= Ae(t) \textcolor{yellow}{-BKx + BKz} = (A - BK)e
+\end{align*}
+$$
+
+At the end of all, you can basically add a message into 
+$\hat{s}(x) = s(x) + m(t)$ and after getting the new value $z(t)$
+I can extract from here the message:
+$$
+\hat{s}(x) - f(z) -Kz = f(x) + Kx + m(t) - f(x) - Kx = m(t)
+$$

docs/Chapters/8-SMITH-PREDICTOR.md

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+# 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 $\hat{G}(s) = G(s)e^{-sT}$:
+$$
+\begin{align*}
+   \frac{\hat{G}_cG_pe^{-sT}}{1 + \hat{G}_cG_pe^{-sT}} 
+   &= \frac{G_cG_p}{1 + G_cG_p}e^{-sT} \rightarrow \\
+
+
+   \rightarrow
+   \frac{\hat{G}_c}{1 + \hat{G}_cG_pe^{-sT}}
+   &= \frac{G_c}{1 + G_cG_p} \rightarrow \\
+
+
+   \rightarrow
+   \hat{G}_c + \hat{G}_c G_cG_p
+   &=  G_c + G_c\hat{G}_cG_pe^{-sT} \rightarrow \\
+
+
+   \rightarrow
+   \hat{G}_c \left[
+    1 + G_cG_p(1 - e^{-sT})
+   \right]
+   &=  G_c \rightarrow \\
+
+
+   \rightarrow
+   \hat{G}_c 
+   
+   
+   &= \frac{ G_c}{ 1 + G_cG_p(1 - e^{-sT})} 
+\end{align*}
+
+$$

docs/Chapters/EXTRAS.md

@@ -47,4 +47,6 @@ If $| rank_k(p) - rank_{k-1}(p)| < \epsilon$ we will stop
 iterating.
 
 At time $k = 0$ all pages have the same importance that is 
-$rank_0(p) = \frac{1}{n}$
+$rank_0(p) = \frac{1}{n}$
+
+## Economic System