A linear system is described by

\[\begin{aligned}\dot{\mathbf{x}} & =\left[\begin{array}{ccc}-2 & 0 & 1 \\0 & -2 & 1 \\0 & -3 & -2\end{array}ight] \mathbf{x}+\left[\begin{array}{l}1 \\2 \\2\end{array}ight] \mathbf{u} \\\mathbf{y} & =\left[\begin{array}{lll}-1 & 1 & 0\end{array}ight] \mathbf{x} .\end{aligned}\]

Using standard notation used in this chapter

(a) Find the transfer function of this system and establish its controllability, observability and stability.

(b) Find a non-singular transformation \(T\) such that \(T^{-1} A T\) is diagonal.

(c) Determine the State-Transition Matrix \(\Phi(s)\).