Control Systems

Resolution 20

Natural Frequency

\omega_n

1.000

Damping ratio

\zeta

0.700

\zeta = \sqrt[4]{\frac{tan^4{\phi_M}}{(4 + 2tan^2{\phi_M})^2 - 4tan^4{\phi_M} }}

Phase margin

\phi_M

21.000

\omega_{BW} = \omega_n \sqrt{(1-2\zeta^2) + \sqrt{4\zeta^4 - 4\zeta^2 + 2}}

9.000

\%overshoot = e^{-(\zeta\pi / \sqrt{1 - \zeta^2})} 100 = \

4.599

Tp = \frac{\pi}{\omega_n \sqrt{1 - \zeta^2}} = \

4.399

Ts = \frac{4}{\zeta \omega_n} = \

5.714

Tr = \frac{0.8 + 2.5 * \zeta} {\omega_n} = \

2.550

Frequency Domain

f(t) = M_icos(\omega t + \phi) = M_i(\omega)<\phi(\omega)

G(j\omega) = G(s)\Big|_{s->j\omega}

\mathcal{L}\{f(t)\} = \int_{0}^{\infty}f(t)e^{-st}dt

\mathcal{F}\{f(t)\} = \int_{0}^{\infty}f(t)e^{-j\omega}dt

Bode plot

20logM - \omega

\phi - log\omega

G(\omega) = \frac{2-j\omega}{\omega^2 + 4}

|G(\omega)| = \frac{1}{\omega^2 + 4} \sqrt{2^2 + \omega^2}

|\phi| = tan^{-1}{(-\frac{\omega}{2})}

G(s) = \frac{K(s + z_1)(s + z_2)...(s + z_k)}{s^m(s + p_1)(s + p_2)...(s + p_n)}

|G(j\omega)| = \frac{K|(s + z_1)||(s + z_2)|...|(s + z_k)|}{|s^m||(s + p_1)||(s + p_2)|...|(s + p_n)|}

20log|G(j\omega)| = 20logK + 20log|(s + z_1)| + 20log|(s + z_2)| + ... - 20log|s^m| - 20log|(s + p_1)| - 20log|(s + p_2)|...

<G(j\omega) = K + (s + z_1) + ... - (s + p_1) - ... -(s + p_n)

G(s) = (s + a)

20log|G(j\omega)| = 20log(a^2 + \omega^2)^{1/2}

At high frequency at w >> a

20log|G(j\omega)| = 20loga

At high frequency at w >> a

20log|G(j\omega)| = 20log\omega

Decade: 10 times the initial frequency, i.e. 1 decade = 10^0 ~ 10^1 Hz or rad/sec

Phase: 0 at w = 0.1a, 45 at w = a, 90 at w = 10a