A unity feedback system has open-loop transfer function $$G(s) = {1 \over {s\left( {2s + 1} \right)\left( {s + 1} \right)}}$$

Sketch Nyquist plot for the system and there from obtain the gain margin and the phase margin.

A process with open-loop model $$G(s) = {{K{e^{ - s{\tau _d}}}} \over {\tau s + 1}},$$ is controlled by a PID controller. For this process

A linear time-invariant system is described by the state variable model $$$\left[ {\matrix{ {{{\mathop x\limits^ \bullet }_1}} \cr {{{\mathop x\limits^ \bullet }_2}} \cr } } \right] = \left[ {\matrix{ { - 1} & 0 \cr 0 & { - 2} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 0 \cr 1 \cr } } \right]u.$$$ $$$Y = \left[ {\matrix{ 1 & 2 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$$

The logic realized by the circuit shown in figure is