Consider the closed-loop control system shown in Figure CP4.6. The controller gain is K = 2. The nominal value of the plant parameter is a = 1. The nominal value is used for design purposes only, since in reality the value is not precisely known. The objective of our analysis is to investigate the sensitivity of the closed-loop system to the parameter a.

(a) When a = 1, show analytically that the steady-state value of Y(s) is equal to 2 when R(s) is a unit step. Verify that the unit step response is within 2% of the final value after 4 seconds.

(b) The sensitivity of the system to changes in the parameter a can be investigated by studying the effects of parameter changes on the transient response. Plot the unit step response for a = 0.5, 2, and 5. Discuss the results.