engineering recuitment GATE IN Test Series 2023-24 Mock Test Control Systems State Space Analysis State Transition Matrix
A state variable system
\(\dot X (t) = \begin{bmatrix} 0 & 1 \\\ 0 & -3 \end{bmatrix} X(t) + \begin{bmatrix} 1 \\\ 0 \end{bmatrix} U(t)\) with the initial condition X(0) = [-1 3]T and the unit step input U(t) so the state transition matrix will be
1
\(\begin{bmatrix} 1 & \dfrac {1}{3} (1 - e^{-3t}) \\\ 0 & e^{-3t} \end{bmatrix}\)
2
\(\begin{bmatrix} 1 & (1 - e^{-t}) \\\ 0 & e^{-3t} \end{bmatrix}\)
3
\(\begin{bmatrix} 1 & \dfrac {1}{3} (e^{-t} - e^{-3t}) \\\ 0 & e^{-t} \end{bmatrix}\)
4
\(\begin{bmatrix} 1 & \dfrac {1}{3} (e^{-t} - e^{-3t}) \\\ 0 & e^{-3t} \end{bmatrix}\)