laboratory in automatic control lab7

8/2/2019 Laboratory in Automatic Control Lab7

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Laboratory in Automatic Control

Lab 7

Root Locus



Root Locus (1/12)

• MATLAB code

p=[1 1];q=[1 5 6 0];sys=tf(p,q);rlocus(sys);



Root Locus (2/12)

• MATLAB code

p=[1 1];q=[1 5 6 0];sys=tf(p,q);[r,K]=rlocus(sys);



Root Locus (3/12)

• MATLAB code

p=[1 1];q=[1 5 6 0];sys=tf(p,q);rlocus(sys);

rlocfind(sys)



Root Locus (4/12)



Root Locus (5/12)

• Consider the closed-loop control system as shown in the

following figure.



Root Locus (6/12)

• The closed-loop transfer function is

•The characteristic equation can be written as

1 3

2 3 1

Y s K s sT s

R s s s s K s

11 0

2 3

sK

s s s



Root Locus (7/12)

• MATLAB code

num=[1 1]; den1=[1 0]; den2=[1 2]; den3=[1 3];den=conv(den1,conv(den2,den3));sys=tf(num,den);rlocus(sys);

rlocfind(sys)

1 0

1

2 3

s

s sK s



Root Locus (8/12)



Root Locus (9/12)

• The second-order transfer function

2

2 22

n

n ns s



Root Locus (10/12)

• Consider the feedback control system as shown in the

following figure. The design specifications are (1) Ts<=10seconds and (2) P.O. <=10% for a unit step input. For the

proportional controller, sketch the root locus for

0<K <infinite, and determine the value of K so that the

design specifications are satisfied.



Root Locus (11/12)

• MATLAB code

zeta=((log(10)/pi)^2/(1+(log(10)/pi)^2))^0.5; % P.O<=10%theta=acosd(zeta);plot([-0.4 -0.4],[-6 6],'--',...

[0 6/tand(180-theta)],[0 6],'--',... [0 -6/tand(180+theta)],[0 -6],'--')

hold on numg=[1];deng=[1 5 6];sys1=tf(numg,deng);rlocus(sys1)[kp,poles]=rlocfind(sys1)figuret=0:0.1:15;

sys1_o=kp*sys1;sys1_cl=feedback(sys1_o,[1]);[y1,t]=step(sys1_cl,t);plot(t,y1),gridxlabel('time (sec)'),ylabel('y(t)')

4s

n

T

21

. . 100 %P O e



Root Locus (12/12)



Lab 7 Homework

• MP7.4. A unity negative feedback system has the open loop

transfer function

Using MATLAB, obtain the root locus as p varies: 0< p<∞.

For what values of p is the closed-loop stable?

2

1

3 6

p s pG s

s s



Lab 7 Homework

• MP7.6. (b) Consider the feedback control system in figure.

The design specifications are Ts<=10 seconds and P.O. <=10% for a unit step input. For the integral controller,

sketch the root locus for 0<K <infinite, and determine the

value of K so that the design specifications are satisfied.

laboratory in automatic control lab7

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