Added modern control and material

docs/Chapters/MODERN-CONTROL.md

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+# Modern Control
+
+Normally speaking, we know much about classical control, in the form
+of:
+
+$$
+\dot{x}(t) = ax(t) + bu(t) \longleftrightarrow sX(s) - x(0) = aX(S) + bU(s)
+$$
+
+With the left part being a derivative equation in continuous time, while the 
+right being its tranformation in the complex domain field.
+
+> [!NOTE]
+>
+> $$
+> \dot{x}(t) = ax(t) + bu(t) \longleftrightarrow x(k+1) = ax(k) + bu(k)
+> $$
+>
+> These are equivalent, but the latter one is in discrete time.
+>
+
+## A brief recap over Classical Control
+
+Be $Y(s)$ our `output variable` in `classical control` and $U(s)$ our 
+`input variable`. The associated `transfer function` $G(s)$ is:
+
+$$
+G(s) = \frac{Y(s)}{U(s)}
+$$
+
+### Root Locus
+<!-- TODO: write about Root Locus -->
+
+### Bode Diagram
+
+### Nyquist Diagram
+
+## State Space Representation
+
+### State Matrices
+A state space representation has 4 Matrices: $A, B, C, D$ with coefficients in 
+$\R$:
+- $A$: State Matrix `[x_rows, x_columns]`
+- $B$: Input Matrix `[x_rows, u_columns]`
+- $C$: Output Matrix `[y_rows, x_columns]`
+- $D$: Direct Coupling Matrix `[y_rows, u_columns]`
+
+$$
+\begin{cases}
+\dot{x}(t) = Ax(t) + Bu(t) \;\;\;\; \text{Dynamic of the system}\\
+      y(t) = C{x}(t) + Du(t) \;\;\;\; \text{Static of the outputs}
+\end{cases}
+$$
+
+This can be represented with the following diagrams:
+
+Continuous Time:
+![continuous state space diagram](..\Images\Modern-Control\state-space-time.png)
+
+---
+Discrete time:
+
+![discrete state space diagram](..\Images\Modern-Control\state-space-discrete.png)
+
+### State Vector
+This is a state vector `[x_rows, 1]`:
+$$
+x(t) =  \begin{bmatrix}
+            x_1(t)\\
+            \dots\\
+            x_x(t)
+        \end{bmatrix}
+
+\text{or} \:
+
+x(k) =  \begin{bmatrix}
+            x_1(k)\\
+            \dots\\
+            x_x(k)
+        \end{bmatrix}
+$$
+
+Basically, from this we can know each next step of the state vector, represented 
+as:
+
+$$
+x(k + 1) = f\left( 
+    x(k), u(k)
+\right) = Ax(k) + Bu(k)
+$$
+
+
+### Case Studies
+<!-- TODO: Complete case studies -->
+- PAGERANK 
+- Congestion Control
+- Video Player Control
+- Deep Learning