Consider a state-variable model of a system\(\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&1\\ { - \alpha }&{ - 2\beta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0\\ \alpha \end{array}} \right]r\)\(y = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right]\)where y is the output, and r is the input. The damping ratio ξ and the undamped natural frequency ωn (rad/sec) of the system are given by

• A. \({\rm{\xi }} = \frac{{\rm{\beta }}}{{\sqrt \alpha }};{\omega _n} = \sqrt \alpha \)
• B. \({\rm{\xi }} = \sqrt \alpha ;{\omega _n} = \frac{\beta }{{\sqrt \alpha }}\)
• C. \({\rm{\xi }} = \frac{{\sqrt \alpha }}{\beta };{\omega _n} = \sqrt \beta\)
• D. \({\rm{\xi }} = \sqrt \beta ;{\omega _n} = \sqrt \alpha\)