Time-domain specifications of 25/(s²+8s+25)

Compare with the standard second-order form $\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$:

• $\omega_n^2=25 \Rightarrow \omega_n = 5$ rad/s
• $2\zeta\omega_n = 8 \Rightarrow \zeta = 0.8$ (underdamped)
• $\omega_d = \omega_n\sqrt{1-\zeta^2} = 5\times0.6 = 3$ rad/s
• $\theta = \cos^{-1}(0.8) = 36.87^\circ = 0.6435$ rad

$$\begin{aligned} t_r &= \frac{\pi-\theta}{\omega_d} = \frac{3.1416-0.6435}{3} = 0.833\ \text{s} \\ t_p &= \frac{\pi}{\omega_d} = \frac{3.1416}{3} = 1.047\ \text{s} \\ t_s &= \frac{4}{\zeta\omega_n} = \frac{4}{0.8\times5} = 1\ \text{s}\quad(2\%\ \text{criterion}) \\ M_p &= e^{-\zeta\pi/\sqrt{1-\zeta^2}} = e^{-0.8\pi/0.6} = e^{-4.189} = 0.0152 \end{aligned}$$

With the 5% criterion, $t_s = \frac{3}{\zeta\omega_n} = 0.75$ s.

Answer: $t_r = 0.833$ s, $t_p = 1.047$ s, $t_s = 1$ s (2%), $M_p = 1.52\%$.