Question

In the signal flow diagram given in the figure, $u_1$ and $u_2$ are possible inputs whereas $y_1$ and $y_2$ are possible outputs. When would the SISO system derived from this diagram, be controllable and observable?

• A. When $u_1$ is the only input and $y_1$ is the only output.
• B. When $u_2$ is the only input and $y_1$ is the only output.
• C. When $u_1$ is the only input and $y_2$ is the only output.
• D. When $u_2$ is the only input and $y_2$ is the only output.

Answer 1

The correct option is B When $u_2$ is the only input and $y_1$ is the only output.

Equations from flow diagram,

${\dot{x}}_1=5x_1-2x_2+u_1;y_1=x_1$

${\dot{x}}_2=2x_1+x_2+u_1+u_2;y_2=x_1-x_2$

$\dot{x}=\frac{\left[\begin{array}{cc}5& -2\\ 2& 1\end{array}\right]}{A}x+\frac{\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]}{B}u;y=\frac{\left[\begin{array}{cc}1& 0\\ 1& -1\end{array}\right]}{C}x$

Considering the SISO cases:

$1.u_1\left(input\right)and{y}_1\left(output\right):$

$Adel=\left[\begin{array}{cc}5& -2\\ 2& 1\end{array}\right]$; $Bdel=\left[\begin{array}{c}1\\ 1\end{array}\right]$ and C = [1 0]

$Odel\left[\begin{array}{c}C\\ CA\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 5& -2\end{array}\right]$

$\Rightarrow Rank\left(O\right)\ne 2\Rightarrow non-controllable$

2. $u_2\left(input\right)and{y}_1\left(output\right):$

$Adel\left[\begin{array}{cc}5& -2\\ 2& 1\end{array}\right]$; $Bdel=\left[\begin{array}{c}0\\ 1\end{array}\right]andCdel=[1-1]$

$Odel=\left[\begin{array}{cc}1& 0\\ 5& -2\end{array}\right]$ $\Rightarrow$ Rank (O) = 2
$\Rightarrow$ Observable

$Cdel=\left[\begin{array}{cc}0& -2\\ 1& 1\end{array}\right]$ $\Rightarrow$ Rank(C) = 2
$\Rightarrow$ controllable

3. $u_2\left(input\right)and{y}_2\left(output\right):$

$Rank\left(O\right)\ne 2\Rightarrow non-observable$

$Rank\left(C\right)=2|\Rightarrow controllable$

4. $u_1\left(input\right)and{y}_2\left(output\right):$

Neither observable and non-controllable