Exercise 6.1#
Analysis ∙ Linear dynamical systems
A state-space representation of \(G(s) = \frac{1}{s+1}\) is given by
(25)#\[\begin{split}
\begin{dcases}
\dot{x} = \begin{bmatrix} -1 & 0 \\ 0 & 2 \end{bmatrix} x + \begin{bmatrix} 1 \\ 1 \end{bmatrix} u \\
y = \begin{bmatrix} 1 & 0 \end{bmatrix} x
\end{dcases}
\end{split}\]
Compute the poles of (25) and compare them with those of \(G(s)\).
Is \(G(s)\) input-output stable? Is (25) asymptotically stable?
Examine the controlalbility and observability of the system.
Explain why (25) is not a good realization of \(G(s)\), and provide a better state-space realization.