Second order model from step response (peak 1.25)

The response has an overshoot, so model it as an underdamped second order system with unity DC gain (final value 1):

$$T(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$$

Step 1: Read the response

• Final value $= 1$, so the DC gain is 1.
• Peak value $= 1.25$, so $M_p = \dfrac{1.25 - 1}{1} = 0.25$ (25%).
• Peak time $t_p = 0.01$ s.

Step 2: Damping ratio

$$M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}} \;\Rightarrow\; \zeta = \frac{-\ln M_p}{\sqrt{\pi^2 + (\ln M_p)^2}} = \frac{1.3863}{\sqrt{9.8696 + 1.9218}} = \frac{1.3863}{3.4339} = 0.4037$$

Step 3: Natural frequency

$$t_p = \frac{\pi}{\omega_n\sqrt{1-\zeta^2}} \;\Rightarrow\; \omega_n = \frac{\pi}{0.01\sqrt{1 - 0.4037^2}} = \frac{3.1416}{0.01(0.9149)} = 343.4\ \text{rad/s}$$

Step 4: Coefficients

$$\begin{aligned} \omega_n^2 &= 343.4^2 = 1.179\times 10^5 \\ 2\zeta\omega_n &= 2(0.4037)(343.4) = 277.3 \end{aligned}$$

Answer:

$$T(s) = \frac{C(s)}{R(s)} = \frac{117914}{s^2 + 277.3s + 117914}$$

with $\zeta = 0.404$ and $\omega_n = 343.4$ rad/s. If the system is taken as unity feedback, the equivalent open-loop model is $G(s) = \dfrac{117914}{s(s + 277.3)}$.