In a robotic arm's control system, the angle \(\theta(t)\) represents the robotic arm orientation, \(\omega(t)\) is the robot's angular speed, \(u(t)\) is the input force, and the output \(y(t)\) is the angle \(\theta(t)\). The robotic arm control model can be represented by the following equations:

\[
\begin{aligned}
\dot{\theta}(t) & =\omega(t) \\
\dot{\omega}(t) & =2 u(t)
\end{aligned}
\]

(a) Find the plant State-Space model \((A, B, C, D)\) for the system.

(b) Use two different methods to show that the plant Transfer Function model of the system is given by

\[
G(s)=\frac{2}{s^{2}}
\]

(c) From continuous-time State-Space model \((A, B, C, D)\) obtained in (a) derive the digital StateSpace model \((F, G, H, J)\).

(d) From the results in (c), find the discrete plant Transfer Function model \(G(z)\).

(e) Using the direct discrete design method, from the continuous plant transfer in (b), develop the discrete plant Transfer Function model \(G(z)\). Is the \(G(z)\) in (e) the same as that obtained in \((d)\) ? Why is this the case?

(f) Explain the significance of the poles of the continuous and digital Transfer Function models of the robot arm. What is the impact of digitisation on the stability of a dynamic system?

(g) Show that the digital system is controllable and observable for all sample times \((T>0)\).

(h) Use two different values of \(T\) to demonstrate the correctness of ( \(g\) ).