Consider the plant defined by
$$
\begin{aligned}
\mathbf{x}(k+1) & =\mathbf{G} \mathbf{x}(k)+\mathbf{H} u(k) \\
y(k) & =\mathbf{C x}(k)
\end{aligned}
$$
where
$$
\mathbf{G}=\left[\begin{array}{ccc}
0 & 0 & -0.25 \\
1 & 0 & 0 \\
0 & 1 & 0.5
\end{array}\right], \quad \mathbf{H}=\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right], \quad \mathbf{C}=\left[\begin{array}{lll}
1 & 0 & 0
\end{array}\right]
$$
Design a control system for the plant. For the pole placement part, we want to have three closed-loop poles at the origin, or
$$
H(z)=z^3
$$
and for the characteristic equation for the minimum-order observer, we want to have
$$
F(z)=z^2
$$
Use the polynomial equations approach to the design