Consider Pd control of this plant answer questions Problem 6

Problem 6 a. Consider proportional control of the plant G(6) (in the unity gain feedback con rar tion), and answer the following questions using the appropriate root locus: i) What is the smallest achievable rise time? iii) If we wish to limit the overshoot to 10%, what are the smallest achievable settling time and rise time? What gain achieves this rise time and settling time? (For finding the gain, please think about how you can do this by hand, in addition to using Matlab.) iv) If we wish to limit the steady state error due to a unit step reference to 5%, what is the smallest achievable overshoot? b. Consider PD control of this plant, using the controller D(s) K(s 10), where K 20 is a gain parameter. Answer the following questions (by drawing the appropriate root locus in Matlab): What is the smallest achievable settling time? ii) What is smallest achievable rise time? iii) If we wish to limit the overshoot to 10%, what are the smallest achievable settling time and rise time? iv) If we wish to limit the steady state error due to a unit step reference to 5%, what is the smallest achievable overshoot? c. Consider PD control of this plant, using the controller D(s) kr kas. Find kp and kd so that 8 m -2 4j is on the root locus. d. Consider control of this plant using the controller D(s) Ktikl), where K 20 is a gain parameter. (This controller is a lead compensator; it is a good approximation for a PD controller, but doesn\'t suffer from the noise issues that are typical of PD controllers lf we wish to limit the overshoot to 10%, what are the smallest achievable settling time and rise time?

Solution

MATLAB code for rootlocus

clear all
clc

syms s;
Gnum=sym2poly(s+4);
Gden=sym2poly((s+6)*(s+1)^2);

G=tf(Gnum,Gden);

k=20;

D=sym2poly(s+10);
Ds=k*tf(D,1);

figure
rlocus(G)

T=feedback(G,Ds);

figure
step(T)
stpinf=stepinfo(T)

For minimum settling time k=20

stpinf =

RiseTime: 0.1823
SettlingTime: 0.2776
SettlingMin: 0.0045
SettlingMax: 0.0051
Overshoot: 1.7825
Undershoot: 0
Peak: 0.0051
PeakTime: 0.4048

For limiting overshoot to 10% at k=7

stpinf =

RiseTime: 0.2271
SettlingTime: 0.7870
SettlingMin: 0.0127
SettlingMax: 0.0154
Overshoot: 9.9770
Undershoot: 0
Peak: 0.0154
PeakTime: 0.4885