Let the state-space representation of an LTI system be \[\mathop {\rm{X}}\limits^ \cdot \](t) = AX(t) + Bu(t), y(t) = CX(t) + Du(t) where A, B, C are matrices, D is a scalar, u(t) is the input to the system, and y(t) is its output. Let B = [0 0 1]^T and D = 0. Which one of the following options for A and C will ensure that the transfer function of this LTI system is \[H\left( s \right) = \frac{1}{{{s^3} + 3{s^2} + 2s + 1}}\]

A) \[A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]{\rm{ and }} \,C = \left[ {0\,\,\,0\,\,\,1} \right]\] B) \[A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 1}&{ - 2}&{ - 3} \end{array}} \right]{\rm{ and }} \,C = \left[ {0\,\,\,0\,\,\,1} \right]\] C) \[A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 1}&{ - 2}&{ - 3} \end{array}} \right]{\rm{ and }} \,C = \left[ {1\,\,\,0\,\,\,0} \right]\] D) \[A = \left[ {\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ { - 3}&{ - 2}&{ - 1} \end{array}} \right]{\rm{ and }} \,C = \left[ {1\,\,\,0\,\,\,0} \right]\]

Answer & Solution

Correct Answer: Option C

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