For the control system shown in the Figure, the transfer function of a plant, $G(s)=\frac{1}{(s+1)(s+2)}$ is connected in cascade with a compensator $C(s)=K(s+\alpha)$, where $K$ and $\alpha$ are positive real valued constants.

Which of the following pairs ( $K, \alpha$ ) represent the correct values for the closed loop system to have poles at $(-3 \pm j \sqrt{5})$ ?

Let $G(s)=\frac{1}{10 s^2}$ be the transfer function of a second-order system. A controller $M(s)$ is connected to the system $G(s)$ in the configuration shown below. Consider the following statements.

(i) There exists no controller of the form $M(s)=\frac{K_I}{s}$, where $K_I$ is a positive real number, such that the closed loop system is stable.

(ii) There exists at least one controller of the form $M(s)=K_P+s K_D$, where $K_P$ and $K_D$ are positive real numbers, such that the closed loop system is stable.

Which one of the following options is correct?

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 ____.