The Nyquist plot of loop transfer function $$G(s) H(s)$$ of a closed loop control system passes through the point $$(-1,j0)$$ in the $$G(s) H(s)$$ plane. The phase margin of the system is

In the system shown in figure, the input $$x(t)$$ $$=$$ $$sin$$ $$t.$$ In the steady-state, the response $$y(t)$$ will be

The open loop transfer function of a unity feedback control system is given as $$G\left( s \right) = {{as + 1} \over {{s^2}}}.$$. The value of $$‘a’$$ to give a phase margin of $${45^0}$$ is equal to

The state variable description of a linear autonomous system is, $$\mathop X\limits^ \bullet = AX,\,\,$$ where $$X$$ is the two dimensional state vector and $$A$$ is the system matrix given by $$A = \left[ {\matrix{ 0 & 2 \cr 2 & 0 \cr } } \right].$$ The roots of the characteristic equation are