Show that each of the following problems can be viewed as a

Show that each of the following problems can be viewed as a root locus problem, and find the root-locus function L(s) in each case: A plot of the roots of s^3 + cs^2 + cs + 1, as c varies from 0 to infinity. A plot of the roots of cs^3 + s^2 + s + 1, as c varies from 0 to infinity. A plot of the roots of s^3 + cs^2 + s + 1 as c varies from 1 to infinity. For the plant G(s) = 1/s(s+3) and a proportional controller (in the unity-feedback-gain configuration), a plot of the closed-loop system\'s poles for all gains that make the closed loop system stable.

Solution

The root locus analysis is a graphical method for exmning how the roots of the system change with variation of certain system parameter commonly a gain with the feedback system in the above equation the stability of the system can be determined by RH criterion.But in c case the value of of the value of c with s parameter effect the stability.

The characterstic equation is guven as 1 + G(S)H(S)=0

here since it is unity feedback system .Therefore, the characterstic equation is S² + 3S + 1=0

Here the system is stable since tehre is no sign change in the s domain characterstic equation.