A satellite attitude control system, as shown below, has a plant with transfer function $G(s) = \frac{1}{s^2}$ cascaded with a compensator $C(s) = \frac{K(s +\alpha)}{s + 4}$, where $K$ and $\alpha$ are positive real constants.

In order for the closed-loop system to have poles at $-1 \pm j \sqrt{3}$, the value of $\alpha$ must be ______.

The position control of a DC servo-motor is given in the figure. The values of the parameters are
K_t=1 N-m/A, R_a=$$1\Omega ,$$ L_a=0.1H,
J=5kg-m², B=1 N-m/(rad/sec) and K_b=1V/(rad/sec).
The steady-state position response (in radians) due to unit impulse disturbance torque T_d is ____.

A lead compensator network includes a parallel combination of 'R' and 'C' in the feed-forward path. If the transfer function of the compensator is $${G_C}(s) = {{s + 2} \over {s + 4}}.$$ The value of RC is ____

The transfer function of a compensator is given as $${G_C}(s) = {{s + a} \over {s + b}}.$$

The phase of the above lead compensator is maximum at