(Solved): The figure below shows a DC motor speed control diagram where y(t) is the motor speed, va is the ...

The figure below shows a DC motor speed control diagram where $y(t)$ is the motor speed, $v_a$ is the armature voltage and $w$ is the load torque. Assume the following PI control law is being used $D_c(s)=\left(k_P+\frac{k_I}{s}\right)$. a) Compute the transfer function from $W$ to $Y$ as a function of $k_P$ and $k_I$. b) Compute values for $k_P$ and $k_I$ so the characteristic equation of the closed-loop system will have roots at $?60\pm 60j$. c) Considering the values computed in b), compute the following steady-state errors: 1. to a unit-step reference input; 2. to a unit-ramp reference input; d) Verify your answers to c.1) and c.2) using MATLAB and/or Simulink and show the plots of $e(t)$ for both cases. If you use just MATLAB, note that a ramp response can be generated as a step response of a system modified by adding an integrator at the reference input. If you use Simulink, there are both step and ramp input blocks available. Also, if you use Simulink, make sure to change the solver configuration to ode $23\mathrm{t}$ to avoid inconsistent results due to solver issues.