01.1 Basics of Open Loop
Derivation of P(s) - Second-Order System
$P(s)$ comes from the standard mass-spring-damper system:
\[m\ddot{x} + c\dot{x} + kx = F\]Taking the Laplace transform (zero initial conditions):
\[ms^2X(s) + csX(s) + kX(s) = F(s)\] \[\frac{X(s)}{F(s)} = \frac{1}{ms^2 + cs + k} = \frac{1/m}{s^2 + \frac{c}{m}s + \frac{k}{m}}\]Standard form substitution:
- Natural frequency: $\omega_n = \sqrt{k/m}$
- Damping ratio: $\zeta = \frac{c}{2\sqrt{km}}$
This gives: \(P(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}\)
The $\omega^2$ in the numerator ensures unity DC gain (when $s=0$, $P(0)=1$), which is the standard normalized form for second-order systems. For details, refer to 01.2 DC Gain Explained
Then from open loop, we will continue to 02.1 Basics of Closed Loop