Control-Network-Systems/docs/Formularies/CONTROL-FORMULARY.md

# 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 $Ker(X_{no}^{T})$

## Sensitivity Function
This function tells us how much `disturbances` in our system
affects our $G(s)$ (here $T$):\
$
S = \frac{dG}{dT} \frac{G}{T}
$

Once found $S$ we do a `Bode Plot` of it to 
see how much we differ from our original
`system`