The signal flow graph of Fig. shown below, has ___________ forward paths / and ________feed - back loops.

A first order system and its response to a unit step input are shown in Fig. below. The system parameters are
$$a=$$ ____________
$$K=$$ ___________

A unity feedback system has an open-loop transfer function of the form $$KG\left( s \right) = {{K\left( {s + a} \right)} \over {{s^2}\left( {s + b} \right)}};\,\,\,\,b > a$$ Which of the loci shown in Fig. can be valid root-loci for the system?

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

(a) Determine the range of $$K$$ for stable operation

(b) Determine the imaginary axis crossover points

(c) Without calculating the real axis break - away points, sketch the form of root loci for the system.