Step 1: Determine the damping ratio.
The standard second-order system denominator is given by \( s^2 + 2 \zeta \omega_n s + \omega_n^2 \). From this, we identify \( \omega_n = 2 \) and \( 2 \zeta \omega_n = 2 \). Solving for the damping ratio \( \zeta \) yields \( \zeta = 0.5 \).
Step 2: Calculate the percent overshoot.
The formula for percent overshoot (PO) is: \[PO = e^{-\zeta \pi / \sqrt{1 - \zeta^2}} \times 100%\]With \( \zeta = 0.5 \) substituted into the formula: \[PO = e^{-0.5 \pi / \sqrt{0.75}} \times 100 \approx 16.3%.\]
Final Answer: \[\boxed{\text{C) 16.3}}\]
Given an open-loop transfer function \(GH = \frac{100}{s}(s+100)\) for a unity feedback system with a unit step input \(r(t)=u(t)\), determine the rise time \(t_r\).
Consider a linear time-invariant system represented by the state-space equation: \[ \dot{x} = \begin{bmatrix} a & b -a & 0 \end{bmatrix} x + \begin{bmatrix} 1 0 \end{bmatrix} u \] The closed-loop poles of the system are located at \(-2 \pm j3\). The value of the parameter \(b\) is: