Laboratory in Automatic Control Lab7
Transcript of Laboratory in Automatic Control Lab7
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Laboratory in Automatic Control
Lab 7
Root Locus
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Root Locus (1/12)
• MATLAB code
p=[1 1];q=[1 5 6 0];sys=tf(p,q);rlocus(sys);
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Root Locus (2/12)
• MATLAB code
p=[1 1];q=[1 5 6 0];sys=tf(p,q);[r,K]=rlocus(sys);
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Root Locus (3/12)
• MATLAB code
p=[1 1];q=[1 5 6 0];sys=tf(p,q);rlocus(sys);
rlocfind(sys)
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Root Locus (4/12)
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Root Locus (5/12)
• Consider the closed-loop control system as shown in the
following figure.
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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
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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
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Root Locus (8/12)
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Root Locus (9/12)
• The second-order transfer function
2
2 22
n
n ns s
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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.
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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
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Root Locus (12/12)
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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
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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.