Question

The state diagram of a system is shown below is described by the state -variable equations:
$\dot{x}=AX+Bu;y=CX+Du$

The state -variable equations of the system in the figure above are

• A. $\dot{X}=\left[\begin{array}{cc}-1& 0\\ 1& -1\end{array}\right]X+\left[\begin{array}{c}-1\\ 1\end{array}\right]uy=\left[\begin{array}{cc}1& -1\end{array}\right]X+u$
• B. $\dot{X}=\left[\begin{array}{cc}-1& 0\\ -1& -1\end{array}\right]X+\left[\begin{array}{c}-1\\ 1\end{array}\right]uy=\left[\begin{array}{cc}-1& -1\end{array}\right]X+u$
• C. $\dot{X}=\left[\begin{array}{cc}-1& 0\\ 1& -1\end{array}\right]X+\left[\begin{array}{c}-1\\ 1\end{array}\right]uy=\left[\begin{array}{cc}-1& -1\end{array}\right]X-u$
• D. $X=\left[\begin{array}{cc}-1& -1\\ 0& -1\end{array}\right]X+\left[\begin{array}{c}-1\\ 1\end{array}\right]uy=\left[\begin{array}{cc}1& -1\end{array}\right]X-u$

Answer 1

The correct option is A $\dot{X}=\left[\begin{array}{cc}-1& 0\\ 1& -1\end{array}\right]X+\left[\begin{array}{c}-1\\ 1\end{array}\right]uy=\left[\begin{array}{cc}1& -1\end{array}\right]X+u$


So, ${\dot{x}}_1=-x_1-u$
${\dot{x}}_2=-(x_2+x_1)=-(x_2-x_1-u)$
${\dot{x}}_2=x_1-x_2+u$
$y=\dot{x_2}$
$y=x_1-x_2+u$
$\left[\begin{array}{c}\dot{x_1}\\ \dot{x_2}\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 1& -1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}-1\\ 1\end{array}\right]u$
$\dot{x}=\left[\begin{array}{cc}-1& 0\\ 1& -1\end{array}\right]X+\left[\begin{array}{c}-1\\ 1\end{array}\right]u$