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