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GATE ECE 2008 | Question-40

A signal flow graph of a system is given below.


The set of equations that correspond to this signal flow graph is

  1. $\frac{d}{d t}\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)=\left[\begin{array}{ccc}\beta & -\gamma & 0 \\ \gamma & \alpha & 0 \\ -\alpha & -\beta & 0\end{array}\right]\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)+\left[\begin{array}{ll}1 & 0 \\ 0 & 0 \\ 0 & 1\end{array}\right]\left(\begin{array}{l}u_{1} \\ u_{2}\end{array}\right)$
  2. $\frac{d}{d t}\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)=\left[\begin{array}{ccc}0 & \alpha & \gamma \\ 0 & -\alpha & -\gamma \\ 0 & \beta & -\beta\end{array}\right]\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)+\left[\begin{array}{ll}0 & 0 \\ 0 & 1 \\ 1 & 0\end{array}\right]\left(\begin{array}{l}u_{1} \\ u_{2}\end{array}\right)$
  3. $\frac{d}{d t}\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)=\left[\begin{array}{ccc}-\alpha & \beta & 0 \\ -\beta & -\gamma & 0 \\ \alpha & \gamma & 0\end{array}\right]\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)+\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 0\end{array}\right]\left(\begin{array}{l}u_{1} \\ u_{2}\end{array}\right)$
  4. $\frac{d}{d t}\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)=\left[\begin{array}{ccc}-\gamma & 0 & \beta \\ \gamma & 0 & \alpha \\ -\beta & 0 & -\alpha\end{array}\right]\left(\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right)+\left[\begin{array}{ll}0 & 1 \\ 0 & 0 \\ 1 & 0\end{array}\right]\left(\begin{array}{l}u_{1} \\ u_{2}\end{array}\right)$
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