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docs/Formularies/CONTROL-FORMULARY.md

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+# Control Formulary
+## Settling time
+$
+T_s = \frac{\ln(a_{\%})}{\zeta \omega_{n}} 
+$
+
+- $\zeta$ := Damping ratio
+- $\omega_{n}$ := Natural frequency
+
+## Overshoot
+$
+\mu_{p}^{\%} = 100 e^{
+    \left(
+        \frac{- \zeta \pi}{\sqrt{1 - \zeta^{2}}}
+    \right)
+}
+$
+
+## Reachable Space
+$X_r = Span(K_c)$
+
+$X = X_r \bigoplus X_{r}^{\perp} = X_r \bigoplus X_{nr}$
+
+> [!TIP]
+> Since $X_{nr} = X_r^{\perp}$ we can find a set of perpendicular
+> vectors by finding $Ker(X_r^{T})$
+
+## Non Observable Space
+$X_no = Kern(K_o)$
+
+$X = X_no \bigoplus X_{no}^{\perp} = X_no \bigoplus X_{o}$
+
+> [!TIP]
+> Since $X_{o} = X_no^{\perp}$ we can find a set of perpendicular
+> vectors by finding $Ker(X_{no}^{T})$

docs/Formularies/GEOMETRY-FORMULARY.md

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+# Geometry Formulary
+
+## Inverse of a Matrix
+$A^{-1} = \frac{1}{det(A)} Adj(A)$
+
+## Adjugate Matrix
+The adjugate of a matrix $A$ is the `transpose` of the `cofactor matrix`:\
+$Adj(A) = C^{T}$
+
+### $(i-j)$-minor (AKA $M_{ij}$)
+$M_{ij}$ := Determinant of the matrix $B$ got by removing the 
+***$i$-row*** and the ***$j$-column*** from matrix $A$
+
+### Cofactors
+$C$ is the matrix of `cofactors` of a matrix $A$ where all the elements $c_{ij}$
+are defined like this:\
+$
+c_{ij} = \left( -1\right)^{i + j}M_{ij}
+$
+
+## Eigenvalues
+By starting from the definition of `eigenvectors`:\
+$A\vec{v} = \lambda\vec{v}$
+
+As we can see, the vector $\vec{v}$ was unaffected by this matrix 
+multiplication appart from a scaling factor $\lambda$, called `eigenvalue`
+
+By rewriting this formula we get:\
+$
+\left(A - \lambda I\right)\vec{v} = 0 
+$ 
+
+This is solved for:\
+$\det(A- \lambda I) = 0$
+
+> [!NOTE]
+> If the determinant is 0, then $(A - \lambda I )$ is not invertible, so
+> we can't solve the previous equation by using the trivial solution (which
+> can't be taken into account since $\vec{v}$ is not $0$  by definition)
+
+## Caley-Hamilton
+Each square matrix over a `commutative ring` satisfies its own 
+characteristic equation $det(\lambda I - A)$
+
+> [!TIP]
+> In other words, once found the characteristic equation, we can 
+> substitute the ***unknown*** variable $\lambda$ with the matrix itself
+> (***known***),
+> powered to the correspondent power

docs/Formularies/PHYSICS-FORMULARY.md

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+# Physics Formulary
+
+> [!TIP]
+>
+> You'll often see $\vec{v}$ and $\vec{\dot x}$, and $\vec{a}$ and 
+> $\vec{\ddot{x}}$.
+>
+> These are equal, but the latter forms, express better the relation
+> between state space variables
+
+## Hooke's Law (AKA Spring Formula)
+$\vec{F} = -k\vec{x}$
+- $k$: Spring Constant
+- $\vec{x}$: vector of spring stretch from rest position
+  
+## Fluid drag
+$\vec{F_D} = \frac{1}{2}b\vec{v}^2C_{D}A$
+- $b$: density of fluid
+- $v$: speed of object ***relative*** to the fluid
+- $C_D$: drag coefficient
+- $A$: cross section area
+
+### Stokes Drag
+$\vec{F_D} = -6 \pi R\mu \vec{v}$
+- $\mu$: dynamic viscosity
+- $R$: radius (in meters) of the sphere
+- $\vec{v}$: flow velocity ***relative*** to the fluid
+
+### Simplified Fluid Drag (Simplified Stokes Equation)
+$\vec{F_D} = -b \vec{v}$
+- $b$: simplified coefficient that has everything else
+- $\vec{v}$: flow velocity ***relative*** to the fluid
+##
+
+## Newton force
+
+