Redo Problem 19 using a lag-lead compensator and MATLAB in the following way:

a. MATLAB will generate the root locus for the uncompensated system along with the 0.5 damping-ratio line. You will interactively select the operating point. MATLAB will then proceed to inform you of the coordinates of the operating point, the gain at the operating point, as well as the estimated %O_S, T_s, T_p, ζ, ω_n, and K_p represented by a second-order approximation at the operating point.

b. MATLAB will display the step response of the uncompensated system.

c. Without further input, MATLAB will calculate the compensated design point and will then ask you to input a value for the lead compensator pole from the keyboard. MATLAB will respond with a plot of the root locus showing the compensated design point. MATLAB will then allow you to keep changing the lead compensator pole value from the keyboard until a root locus is plotted that goes through the design point.

d. For the compensated system, MATLAB will inform you of the coordinates of the operating point, the gain at the operating point, as well as the estimated %O_S, T_s, T_p, ζ, ω_n, and K_p represented by a second-order approximation at the operating point.

e. MATLAB will then display the step response of the compensated system.

f. Change the compensator’s zero location a few times and collect data on the compensated system to see if any other choices of the compensator zero yield advantages over the original design.

g. Using the steady-state error of the uncompensated system, add a lag compensator to yield an improvement of 30 times over the uncompensated system’s steady-state error, with minimal effect on the designed transient response. Have MATLAB plot the step response. Try several values for the lag compensator’s pole and see the effect on the step response.

Data From Problem 19:

Given the unity feedback system of Figure P9.1 with

Find the transfer function of a lag-lead compensator that will yield a settling time 0.5 second shorter than that of the uncompensated system. The compensated system also will have a damping ratio of 0.5, and improve the steady-state error by a factor of 30. The compensator zero is at -5. Also, find the compensated system’s gain. Justify any second-order approximations or verify the design through simulation.

Transcribed Image Text:

K G(s) = (s + 2)(s + 4)(s +6)(s+ 8) %3D