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