Automatic control of helicopters is necessary because, unlike fixed-wing aircraft which possess a fair degree of inherent stability, the helicopter is quite unstable. A helicopter control system that utilizes an automatic control loop plus a pilot stick control is shown in Figure P7.5. When the pilot is not using the control stick, the switch may be considered to be open. The dynamics of the helicopter are represented by the transfer function
$$
G(s)=\frac{25(s+0.03)}{(s+0.4)\left(s^2-0.36 s+0.16\right)} .
$$
(a) With the pilot control loop open (hands-off control), sketch the root locus for the automatic stabilization loop. Determine the gain $K_2$ that results in a damping for the complex roots equal to $\zeta=0.707$. (b) For the gain $K_2$ obtained in part (a), determine the steady-state error due to a wind gust $T_d(s)=1 / s$. (c) With the pilot loop added, draw the root locus as $K_1$ varies from zero to $\infty$ when $K_2$ is set at the value calculated in part (a). (d) Recalculate the steady-state error of part (b) when $K_1$ is equal to a suitable value based on the root locus.