Consider the system below. For this question, let C (s) = K,  in this question K  is a positive real number. a) Prove that if the root locus of the system is drawn, the points s1,2=−0.3214 ± 2.8951j will be on the curve. (Show it by doing mathematical calculation without drawing the curve.) b) Draw the root locus of the system. c) It is desired that the dominant (= near the imaginary axis) poles of the closed loop system be at the points s1,2= −0.3214 ± 2.8951j. What should K be for this? What can be said about the stability of the closed loop system for this K value? d) What are the unit step, ramp and parabolic reference tracking errors of the system for K in the option c?

Consider the system below. For this question, let C (s) = K,  in this question K  is a positive real number.

a) Prove that if the root locus of the system is drawn, the points s_1,2=−0.3214 ± 2.8951j will be on the curve. (Show it by doing mathematical calculation without drawing the curve.)

b) Draw the root locus of the system.

c) It is desired that the dominant (= near the imaginary axis) poles of the closed loop system be at the points s_1,2= −0.3214 ± 2.8951j. What should K be for this? What can be said about the stability of the closed loop system for this K value?

d) What are the unit step, ramp and parabolic reference tracking errors of the system for K in the option c?