(Solved):   The open-loop transfer function of a unity feedback control system is given by \[ G(s)=\fr ...

 

The open-loop transfer function of a unity feedback control system is given by \[ G(s)=\frac{k}{(s+2)(s+4)\left(s^{2}+6 s+25\right)} \] By applying Routh-Hurwitz criterion, determine (i) the range of \( k \) for which the closed-loop system will be stable and (ii) the values of \( k \) which will cause sustained oscillations in the closed-loop system. What are the corresponding oscillation frequencies? Sketch the root-locus plots and briefly explain qualitatively the improvements in system performance (stability and steady-state error) that are obtainable by introducing a compensating zero \( (s+3) \) to the unity feedback system whose open-loop transfer function is \[ G(s)=\frac{k}{s(s+2)(s+6)} \]