Question

A signal flow graph of a system is given below

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

• A. $\frac{d}{dt}\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\left[\begin{array}{ccc}\beta & -\gamma & 0\\ \gamma & \alpha & 0\\ -\alpha & -\beta & 0\end{array}\right]\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left[\begin{array}{cc}0& 0\\ 0& 1\\ 0& 0\end{array}\right]\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$
• B. $\frac{d}{dt}\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\left[\begin{array}{ccc}0& \alpha & \gamma \\ 0& -\alpha & -\gamma \\ 0& \beta & -\beta \end{array}\right]\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left[\begin{array}{cc}0& 0\\ 0& 1\\ 1& 0\end{array}\right]\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$
• C. $\frac{d}{dt}\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\left[\begin{array}{ccc}-\alpha & \beta & 0\\ -\beta & -\gamma & 0\\ \alpha & \gamma & 0\end{array}\right]\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$
• D. $\frac{d}{dt}\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\left[\begin{array}{ccc}-\beta & 0& \beta \\ \gamma & 0& \alpha \\ -\gamma & 0& -\alpha \end{array}\right]\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left[\begin{array}{cc}0& 1\\ 0& 0\\ 1& 0\end{array}\right]\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$

Answer 1

The correct option is D $\frac{d}{dt}\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\left[\begin{array}{ccc}-\beta & 0& \beta \\ \gamma & 0& \alpha \\ -\gamma & 0& -\alpha \end{array}\right]\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left[\begin{array}{cc}0& 1\\ 0& 0\\ 1& 0\end{array}\right]\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$


By solving this we get
$\frac{d}{dt}\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)=\left[\begin{array}{ccc}-\beta & 0& \beta \\ \gamma & 0& \alpha \\ -\gamma & 0& -\alpha \end{array}\right]\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)+\left[\begin{array}{cc}0& 1\\ 0& 0\\ 1& 0\end{array}\right]\left(\begin{array}{c}{u}_1\\ u_2\end{array}\right)$