Time response

first order.

G(s) = \frac{b}{s^2 + as + b}

C(s) = \frac{9}{s(s^2+9s+9)}

For inspect of poles, form of system’s response

c(t) = K_1 + K_2e^{-\sigma_1 t} + K_3e^{-\sigma_2 t}

System’s response:

c(t) = K_1 + K_2e^{-\sigma_1 t} + K_3te^{-\sigma_2 t}

where $-\sigma_1=-3$ is our pole location.

Unit step response to the system:

C(s) = \frac{K_1}{s} + \frac{\alpha + j\beta}{s+1+j\sqrt{8}}+ \frac{\alpha - j\beta}{s+1-j\sqrt{8}}

Thus the form of system’s response:

c(t) = K_1 + e^{-\sigma_dt} \lbrack 2\alpha \cos \omega_d t+ 2\beta \sin \omega_d t \rbrack

e^{-\sigma_dt} \lbrack 2\alpha \cos \omega_d t+ 2\beta \sin \omega_d t \rbrack = K_4 e^{-\sigma_d t} \cos (\omega_dt - \phi)

where $\phi = \tan^{-1}(\frac{\beta}{\alpha})$ and $K_4=\sqrt{(2\alpha)^2 + (2\beta)^2}$

nature frequency $\omega_n$ : frequency of oscillation of the system
damping ratio $\zeta = \frac{\text{exponential decay frequency}}{\text{natural frequency (rad/sec)}}$

Deriving $\zeta$ :

\%OS = e^{\zeta \pi / \sqrt{1-\zeta^2}} \times 100 \%