Consider the system shown in Fig. Determine the value of 'a' such that the damping ratio is 0.5. Also obtain the values of the rise time 't_r' and maximum overshoot 'M_p' in its step response.

The characteristic equation of a feedback control system is $$$s^4+20s^3+15s^2+2s+K\;=\;0$$$
(i) Determine the range of K for the system to be stable.
(ii) Can the system be marginally stable? If so, find the required value of K and the frequency of sustained oscillation.

In the Bode-plot of a unity feedback control system, the value of phase of G($$j\omega $$) at the gain cross over frequency is $$ - 125^\circ $$. The phase margin of the system is

The Nyquist plot of a loop transfer function G$$(j\omega )$$ H$$(j\omega )$$, of a system encloses the (-1, j0) point. The gain margin of the system is