Let's solve the problem step-by-step: --- ## 1. **Given Transfer Function** \[ G(s) = \frac{6}{s^2 + s + 8} \] --- ## 2. **Standard Second-Order System Form** The standard form is: \[ G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \] Comparing: - \( \omega_n^2 = 8 \implies \omega_n = \sqrt{8} = 2.828 \) - \( 2\zeta\omega_n = 1 \implies \zeta = \frac{1}{2\omega_n} = \frac{1}{2 \times 2.828} = .177 \) But numerator is 6, not 8. For step response, we need the closed-loop transfer function for a unit step input: \[ C(s) = G(s) \cdot \frac{1}{s} = \frac{6}{s(s^2 + s + 8)} \] But the final value depends on the DC gain: - \( G() = \frac{6}{8} = .75 \) --- ## 3. **Step Response Characteristics** ### **(a) Steady-State Value, \(C_{SS}\):** By Final Value Theorem: \[ C_{ss} = \lim_{s \to } s \cdot \frac{6}{s(s^2 + s + 8)} = \frac{6}{8} = .75 \] ### **(b) Damping Ratio (\(\zeta\)) and Natural Frequency (\(\omega_n\)):** Already found: - \( \omega_n = 2.828 \) - \( \zeta = .177 \) ### **(c) Peak Time (\(T_P\)):** \[ T_P = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}} \] Calculate: \[ \sqrt{1-\zeta^2} = \sqrt{1-(.177)^2} = .984 \] \[ T_P = \frac{\pi}{2.828 \times .984} = \frac{3.142}{2.783} = 1.13 \, \text{sec} \] ### **(d) Settling Time (\(T_S\)):** (For 2% criterion) \[ T_S = \frac{4}{\zeta \omega_n} \] \[ T_S = \frac{4}{.177 \times 2.828} = \frac{4}{.5} = 8 \, \text{sec} \] ### **(e) Percent Overshoot (\%OS):** \[ \%OS = e^{-\frac{\zeta \pi}{\sqrt{1-\zeta^2}}} \times 100\% \] \[ \frac{\zeta \pi}{\sqrt{1-\zeta^2}} = \frac{.177 \times 3.142}{.984} = .565 \] \[ \%OS = e^{-.565} \times 100\% = .568 \times 100\% = 56.8\% \] ### **(f) Maximum Value (\(C_{MAX}\)):** \[ C_{MAX} = C_{SS} \times \left(1 + \frac{\%OS}{100}\right) = .75 \times (1 + .568) = .75 \times 1.568 = 1.176 \] --- ## 4. **Summary Table** | Quantity | Value | |---------------------|-----------| | \(T_P\) | 1.13 sec | | \(T_S\) | 8 sec | | \(C_{SS}\) | .75 | | \(C_{MAX}\) | 1.176 | | Percent Overshoot | 56.8% | --- ## 5. **Sketch** (Description) - **Starts at **, rises quickly - **Overshoots** the final value (.75) to a peak (\(\approx 1.18\)) - **Oscillates** and settles at .75 after about 8 sec --- ### **Final Answers** \[ \boxed{ \begin{aligned} &\text{Peak Time, } T_P = 1.13\,\text{sec} \\ &\text{Settling Time, } T_S = 8\,\text{sec} \\ &\text{Steady-State Value, } C_{SS} = .75 \\ &\text{Maximum Value, } C_{MAX} = 1.176 \\ &\text{Percent Overshoot} = 56.8\% \end{aligned} } \] --- **Let me know if you want a plot of the step response!**