Problem 2
Given:
Der(s) = K
H(s) = 1
a) Assume W(s) = 0. Draw the root locus of the system assuming the compensator consists only of the adjustable gain parameter K, i.e. D(s) = K.
b) Determine the approximate range of values of K for which the rise time requirement is satisfied (maximum damping factor) you can achieve with the P-type control? Choose K to satisfy the rise-time requirement and plot the time-domain response of the closed-loop system.
c) Add a lead-compensator Da(s) to the system, so that the magnitude of zero (z) = 10. Pick a suitable value of p, so that the rise-time requirement for the closed-loop system is satisfied, and so that the damping factor is at least 0.5. What would be the settling time for this case? Choose an appropriate value for p, and the gain K to achieve the rise-time requirement as well as damping factor requirement (or possibly even better than the requirement). Plot the time domain response of the system to a unit-step input. Compare to the unit-step response in part b).
NOTE: there is more than 1 correct answer - provide appropriate plots, explanations/comments and reasoning behind your choices!!!!!!
d) What is the steady-state error for the compensated closed-loop system in part c)? Derive from the Closed-loop transfer function and compare the value to the output from the step-response.
Design a lag-type compensator that will reduce the steady-state error by a factor of 5. Use design recommendations we discussed in class.
Plot the root-locus / step response with both the lead and lag compensator and make sure that all parameters remain within requirements after the introduction of the lag-compensation. Compare the steady-state error with the lead+lag-compensator with the one with the lead compensator.
