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

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  1. \({\rm{\xi }} = \frac{{\rm{\beta }}}{{\sqrt \alpha }};{\omega _n} = \sqrt \alpha \)
  2. \({\rm{\xi }} = \sqrt \alpha ;{\omega _n} = \frac{\beta }{{\sqrt \alpha }}\)
  3. \({\rm{\xi }} = \frac{{\sqrt \alpha }}{\beta };{\omega _n} = \sqrt \beta\)
  4. \({\rm{\xi }} = \sqrt \beta ;{\omega _n} = \sqrt \alpha\)

Answer (Detailed Solution Below)

Option 1 : \({\rm{\xi }} = \frac{{\rm{\beta }}}{{\sqrt \alpha }};{\omega _n} = \sqrt \alpha \)
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Detailed Solution

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From the given state model

\(\begin{array}{*{20}{c}} {A = \left[ {\begin{array}{*{20}{c}} 0&1\\ { - \alpha }&{ - 2\beta } \end{array}} \right]}&{B = \left[ {\begin{array}{*{20}{c}} 0\\ \alpha \end{array}} \right]}&{C = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]} \end{array}\)

Transfer function = C [sI - A]-1 B + D

\(\left[ {SI - A} \right] = S\left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 0&1\\ { - \alpha }&{ - 2\beta } \end{array}} \right]\)

\(= \left[ {\begin{array}{*{20}{c}} s&{ - 1}\\ \alpha &{s + 2\beta } \end{array}} \right]\)

\(= {\left[ {sI - A} \right]^{ - 1}} = \frac{1}{{s\left( {s + 2\beta } \right)\alpha }}\left[ {\begin{array}{*{20}{c}} {s + 2\beta }&1\\ { - \alpha }&s \end{array}} \right]\)

\(\frac{T}{F} = \frac{{\left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {s + 2\beta }&1\\ { - \alpha }&s \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0\\ \alpha \end{array}} \right]}}{{\left( {s\left( {s + 2\beta } \right) + \alpha } \right)}}\)

\(= \frac{1}{{\left( {{s^2} + 2\beta s + \alpha } \right)}}\left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} \alpha \\ {s\alpha } \end{array}} \right]\)

\(= \frac{\alpha }{{\left( {{s^2} + 2\beta s + \alpha } \right)}}\)

By comparing the above transfer function with the standard second order transfer function \(\frac{{\omega _n^2}}{{{s^2} + 2\xi {\omega _n}s}}\)

\({\omega _n} = \sqrt \alpha\)

And 2ξωn = 2

\(\Rightarrow \xi = \frac{\beta }{{\sqrt \alpha }}\)

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