CP4.6 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 closedloop system to the parameter a.

(a) When a = 1, show analytically that the steadystate value of Y1s2 is equal to 2 when R1s2 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.

FIGURE CP4.6 A closed-loop control system with uncertain parameter a. Controller Process + 1 R(s) K Y(s) s-a