Consider a plant defined by
$$
\begin{aligned}
\mathbf{x}(k+1) & =\mathrm{Gx}(k)+\mathrm{H} u(k) \\
y(k) & =\mathrm{Cx}(k)
\end{aligned}
$$
where The pulse transfer function for the plant can be written as
$$
\frac{Y(z)}{U(z)}=\frac{B(z)}{A(z)}
$$
Determine polynomials $A(z)$ and $B(z)$.
Using the polynomial equations approach, design a control system for the plant.
It is desired that the block diagram configuration of the designed system is the same as that of Figure 7-4. In solving the Diophantine equation
$$
\alpha(z) A(z)+\beta(z) B(z)=F(z) H(z)
$$
assume that $H(z)$ and $F(z)$ are, respectively, as follows:
$$
H(z)=z^3, \quad F(z)=z^2
$$
Obtain the unit-step response and unit-ramp response of the designed control system. The sampling period $T$ is $1 \mathrm{sec}$.