Which one of the following is the correct state-space model of the circuit shown below?
1
\(\left[ {\begin{array}{*{20}{c}} {{{\dot v}_c}}\\ {{{\dot i}_L}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {0.25}&1\\ { - 0.5}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{v_c}}\\ {{i_L}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0\\ {0.5} \end{array}} \right]{v_s}\)
2
\(\left[ {\begin{array}{*{20}{c}} {{{\dot v}_c}}\\ {{{\dot i}_L}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 0.125}&1\\ { - 0.5}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{v_c}}\\ {{i_L}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} {0.5}\\ 0 \end{array}} \right]{v_s}\)
3
\(\left[ {\begin{array}{*{20}{c}} {{{\dot v}_c}}\\ {{{\dot i}_L}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 0.5}&1\\ { - 0.25}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{v_c}}\\ {{i_L}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 1\\ {0.5} \end{array}} \right]{v_s}\)
4
\(\left[ {\begin{array}{*{20}{c}} {{{\dot v}_c}}\\ {{{\dot i}_L}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - 0.125}&0.5\\ { - 0.5}&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{v_c}}\\ {{i_L}} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0\\ {0.5} \end{array}} \right]{v_s}\)