Time response

first order.

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

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

For inspect of poles, form of system’s response

c (t) = K_{1} + K_{2} e^{- σ_{1} t} + K_{3} e^{- σ_{2} t}

System’s response:

c (t) = K_{1} + K_{2} e^{- σ_{1} t} + K_{3} t e^{- σ_{2} t}

where $- σ_{1} = - 3$ is our pole location.

Unit step response to the system:

C (s) = \frac{K _{1}}{s} + \frac{α + j β}{s + 1 + j 8} + \frac{α - j β}{s + 1 - j 8}

Thus the form of system’s response:

c (t) = K_{1} + e^{- σ_{d} t} [2 α cos ω_{d} t + 2 β sin ω_{d} t]

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 $K_4=\sqrt{(2\alpha)^2 + (2\beta)^2}$ ### general second-order systems - nature frequency $\omega_n$: frequency of oscillation of the system - damping ratio $\zeta = \frac{\text{exponential decay frequency}}{\text{natural frequency (rad/sec)}}$ _Deriving_ $\zeta$: - For _under-damped_ system, the poles are $\sigma = \frac{-a}{2}$ ### %OS (percent overshoot)

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