Question

Consider the state space system expressed by the signal flow diagram shown in the figure.
The corresponding system is

• A. always controllable
• B. always observable
• C. always stable
• D. always unstable

Answer 1

The correct option is A always controllable

The state equation from the given state diagram is
${\dot{x}}_1=x_2$
${\dot{x}}_2=x_3$
${\dot{x}}_3=a_3{x}_3+a_2{x}_2+a_1{x}_1+u$
Also ,
$y=c_1{x}_1+c_2{x}_2+c_3{x}_3$

Thus,state matrix

$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ a_1& a_2& a_3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u$
$and$ $y=\left[\begin{array}{ccc}{c}_1& c_2& c_3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$

Check for controability:

The system is said to be controllable if, the rank of controllability matrix $Q_c$ is equal to the rank of the state matrix $A$. However , if the controllability matrix $Q_c$ is a square matrix then the condition for controllability is
$Q_c|\ne 0$

where,

$Q_c=\left[\begin{array}{ccc}B& AB& A^{2}B\end{array}\right]$

$\therefore Q_c=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& a_3\\ 1& a_3& a_2+a_3^2\end{array}\right]$

$\because |Q_c|\ne 0$ and Rank of $Q_c$ Rank of $A=1$

$\therefore$ The system is controllable.

Check for observability:
The system is said to be observable if , the rank of observability matrix $Q_0$ is equal to the rank of the state matrix A. However , if the observability matrix $Q_0$ is a square matrix then the condition for observability is
$|Q_0|\ne 0$

where,

$Q_0=\left[\begin{array}{ccc}{C}^T& A^T{C}^T:& (A^T{)}^2{C}^T\end{array}\right]$

$\therefore Q_0=\left[\begin{array}{ccc}{c}_1& c_2& c_3\\ c_3{a}_1& c_1+c_2{a}_2& c_2+c_2{a}_3\\ c_1(c_2+c_3{a}_3)& c_3{a}_1+(c_2+c_3{a}_3)a_2& c_1+c_2{a}_2+(c_2+c_3{a}_3)a_3\end{array}\right]$

$\because |Q_0|$ depends on value of unknown
Hence,not observable always.