Question

The block diagram of a system with one input $u$ and two output $y_1$ and $y_2$ is given below.
A state space model ofthe above system in terms ofthe state vector $x$ and the output vector $y={\left[\begin{array}{cc}{y}_1& y_2\end{array}\right]}^T$ is

• A. $\dot{x}=\left[1\right]x+\left[1\right]u;y\left[\begin{array}{cc}1& 2\end{array}\right]x$
• B. $\dot{x}=[-2]x+\left[1\right]u;y\left[\begin{array}{c}1\\ 2\end{array}\right]x$
• C. $\dot{x}=\left[\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right]x+\left[\begin{array}{c}1\\ 1\end{array}\right]u;y\left[\begin{array}{cc}1& 2\end{array}\right]x$
• D. $x=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]x+\left[\begin{array}{c}1\\ 2\end{array}\right]u;y\left[\begin{array}{cc}1& 2\end{array}\right]x$

Answer 1

The correct option is C $\dot{x}=\left[\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right]x+\left[\begin{array}{c}1\\ 1\end{array}\right]u;y\left[\begin{array}{cc}1& 2\end{array}\right]x$

$\frac{Y_1\left(s\right)}{U\left(s\right)}=\frac{1}{s+2}$

$\frac{Y_2\left(s\right)}{U\left(s\right)}=\frac{2}{s+2}$

$\frac{Y_1\left(s\right)X_1\left(s\right)}{X_1\left(s\right)U\left(s\right)}=\frac{1}{s+2}$

$\frac{Y_2\left(s\right)X_2\left(s\right)}{X_2\left(s\right)U\left(s\right)}=\frac{2}{s+2}$

$\frac{X_1\left(s\right)}{U\left(s\right)}=\frac{1}{s+2}\text{and}\frac{Y_1\left(s\right)}{U\left(s\right)}=1$

$\frac{X_2\left(s\right)}{U\left(s\right)}=\frac{1}{s+2}\text{and}\frac{Y_2\left(s\right)}{U\left(s\right)}=2$

$s{X}_1\left(s\right)+2X_1\left(s\right)=U\left(s\right)and{Y}_1\left(s\right)=X_1\left(s\right)$

$s{X}_2\left(x\right)+2X_2\left(s\right)=U\left(s\right)and{Y}_2\left(s\right)=2X_2\left(s\right)$

${\dot{x}}_1\left(t\right)+2x_1\left(t\right)=U\left(t\right)$

$and{y}_1\left(t\right)=x_1\left(t\right)$

${\dot{x}}_2\left(t\right)+2x_2\left(t\right)=U\left(t\right)$

$and{y}_2\left(t\right)=2x_2\left(t\right)$

From question,

$y={\left[\begin{array}{cc}{y}_1& y_2\end{array}\right]}^T={\left[\begin{array}{cc}1& 2\end{array}\right]}^T=\left[\begin{array}{c}1\\ 2\end{array}\right]$
${\dot{x}}_1\left(t\right)=-2x_1\left(t\right)+u\left(t\right)$
${\dot{x}}_2\left(t\right)=-2x_2\left(t\right)+u\left(t\right)$
$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}-2& 0\\ -2& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]u\left(t\right)$
$or\dot{x}=[-2]x+\left[1\right]u$
only option (b) is satisfied.