Consider the following three State-Space models. Define the terms A, B, C, D, x and u. Find the observability and controllability matrices for each of the three systems and determine whether each system is controllable or observable.

(a)

\[\begin{array}{ll}A=\left[\begin{array}{ccc}0 & 1 & 0 \\0 & 0 & 1 \\-4 & -3 & -2\end{array}ight] & B=\left[\begin{array}{l}0 \\0 \\1\end{array}ight] \\C=\left[\begin{array}{lll}0 & 5 & 1\end{array}ight] & D=[0]\end{array}\]

(b)

\[\begin{array}{ll}A=\left[\begin{array}{cc}0 & 1 \\-5 & -\frac{21}{4}\end{array}ight] & B=\left[\begin{array}{l}0 \\1\end{array}ight] \\C=\left[\begin{array}{ll}5 & 4\end{array}ight] & D=[0]\end{array}\]

(c)

\[\begin{array}{ll}A=\left[\begin{array}{ccc}-2 & -1 & -3 \\0 & -2 & 1 \\-7 & -8 & -9\end{array}ight] & B=\left[\begin{array}{l}2 \\1 \\2\end{array}ight] \\C=\left[\begin{array}{lll}4 & 6 & 8\end{array}ight] & D=[0]\end{array}\]