(Solved): Consider the torsional mechanical system in Figure 1(a). The torque due to the twisting of the shaf ...

Consider the torsional mechanical system in Figure 1(a). The torque due to the twisting of the shaft is $?k?$; the damping torque due to the braking device is $?b?$; the disturbance torque is $t_d(t)$; the input torque is $\mathrm{r}(\mathrm{t})$; and the moment of inertia of the mechanical system is $J$. The transfer function of the torsional mechanical system is $G(s)=\frac{\frac{1}{J}}{s^2+\frac{b}{J}s+\frac{k}{J}}$ A closed-loop control system for the system is shown in Figure 1(b). Suppose the desired angle ${?}_d(t)=$ $0^{?},k=5,b=0.9$, and $J=1$. (a) Determine the open-loop response $?(t)$ of the system for a unit step disturbance (set $r(t)=0$ ). (b) With the controller gain $K0=50$, determine the closed-loop response, $?(\mathrm{t})$ to a unit step disturbance. (c) Using MATLAB, plot the open-loop versus the closed-loop response to the disturbance input. Discuss your results and make an argument for using closed-loop feedback control to improve the disturbance rejection properties of the system.