Control algorithm

Figure $1$ shows a position control scheme of robotic arm link, which has three feedback loops: two inner loops and one outer loop, and a feedforward compensator transfer function $s^2.$

Suppose that the estimates of the mass, damping, and stiffness are accurate so that $m_e=m=3$;$c_e=c=4$;$k_e=k=5$ and $I=2.$

a$.$ Derive the transfer functions: $\frac{x(s)}{x_r(S)}$ and $\frac{x(s)}{F_d(S)}.$

b$.$ How well does this control scheme work?

$(6$ Marks$)$

$(3$ Marks$)$

c$.$ Obtain the expressions for the PID gains to achieve a damping ratio of $=1$ and a closed$-$loop time constant of specified value ${}_d=1$ second.

$(8$ Marks$)$

Discuss any practical limitations to this scheme.

$(3$ Marks$)$

$[$Total: $20$ Marks$]$

Figure $1$: Position control scheme of a robotic arm link.