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A second-order system represented by state variables has 

\(A = \left[ {\begin{array}{*{20}{c}} -2&-4\\ 1&0 \end{array}} \right]\)

The values of natural frequency and damping ratio are respectively

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  1. 2 and 0.5
  2. 2 and 1
  3. 1 and 2
  4. 0.5 and 2

Answer (Detailed Solution Below)

Option 1 : 2 and 0.5
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\(A = \left[ {\begin{array}{*{20}{c}} -2&-4\\ { 1}&0 \end{array}} \right]\)

\(sI - A = \left[ {\begin{array}{*{20}{c}} s&0\\ 0&s \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} -2&-4\\ {1}&0 \end{array}} \right] \)

\(\\ {\left( {sI - A} \right)^{ - 1}} = \frac{1}{s(s+2)+4}\left[ {\begin{array}{*{20}{c}} {s +2}&4\\ {-1}&{s} \end{array}} \right]\)

The characteristics equation is:

s(s + 2) + 4 = 0

s2 + 2s + 4 = 0

Comparing with: 

s2 + 2ζ ωn + ωn2 = 0

ωn = 2 rad/s

ζ = 0.5 

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