Exercise 6.0#
Analysis ∙ Linear dynamical systems
Consider the following system:
(21)#\[\begin{split}
\begin{dcases}
\dot{x} = \begin{bmatrix} -2 & 4 \\ 0 & -3 \end{bmatrix} x + \begin{bmatrix} 0 \\ 2 \end{bmatrix} u \\
y = \begin{bmatrix} 0 & 3 \end{bmatrix} x
\end{dcases}
\end{split}\]
Compute the poles of the state-space system (21), and check whether the system is asymptotially stable or not.
Is the system controllable and observable?
Compute the transfer function \(G(s)\) of system (21).
Compute the poles of \(G(s)\), and discuss whether they match those of (21).
Inverting \(2 \times 2\) matrices
Consider a generic matrix \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\). Its inverse can be easily computed as
(22)#\[\begin{split}
A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}
\end{split}\]