engineering recuitment GATE ECE 2023-24 Test Series Control Systems State Space Analysis Solution To the State Equation
A linear time invariant system is characterized by the homogeneous state equation:
\(\left[ {\begin{array}{*{20}{c}} {{{\dot x}_1}}\\ {{{\dot x}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0\\ 1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}}\\ {{x_2}} \end{array}} \right]\)
The solution of homogeneous equation for the given initial state vector \({X_0} = \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]\) is
1
\(\left[ {\begin{array}{*{20}{c}} {{e^t}}\\ {{t^2}{e^t}} \end{array}} \right]\)
2
\(\left[ {\begin{array}{*{20}{c}} {{e^t}}\\ {t{e^t}} \end{array}} \right]\)
3
\(\left[ {\begin{array}{*{20}{c}} {{e^{2t}}}\\ {t{e^{2t}}} \end{array}} \right]\)
4
\(\left[ {\begin{array}{*{20}{c}} {{e^{2t}}}\\ { - t{e^{2t}}} \end{array}} \right]\)