Control Systems
Lecture (3)
Constantinos Siettos
Typical Flow Diagram of a Feedback Control System
control G c (s)
system G p (s)
feedback H(s) pre-filter
G 1 (s)
post-filter G 2 (s) reference input, R(s)
error, E(s)
system inputs, U(s)
output, Y(s)
feedback, H(s)Y(s)
BLOCKS
Rules of Reduction
G 1 G 2 G 1 G 2
U Y U Y
G 1 G 2
U + Y
+ U G 1 + G 2 Y
G 1 G 2
U + Y
- U G 1 /(1+G 1 G 2 ) Y
In Series
In parallel
Feedback-Normal Form
Rules of Reduction
Transfer of the point of summation in front of the block
G 1
U + Y
+
Ζ
G 1
U Y
+ +
1/G 1 Ζ
Transfer of the point of summation behind the block
G 1
U + Y
+ Ζ
U +
+ G 1
Υ G 1
Ζ
H2
Reduction of Diagrams with 1 input
+ -
+ -
+ +
H2
G1 G2 G3 G4
H1 H3
R(s)
Y(s)
H2
+ -
+ -
+ +
H2/G4
G1 G2 G3 G4
H1 H3
R(s)
Y(s)
Reduction of Diagrams with 1 Input
H2
+ -
+ -
+ +
H2/G4
G1 G2 G3 G4
H1 H3
R(s)
Y(s)
H2
+ -
+ -
H2/G4
G1 G2 G3G4/(1-G3G4H1)
H3 R(s)
Y(s)
+ -
G1 G2G3G4/(1-G3G4H1+G2G3H2)
H3 R(s)
Y(s)
Matlab for Reduction of Diagrams with 1 Input
H2
+ -
+ - +
+
H2
G1 G2 G3 G4
H1 H3
R(s)
Y(s)
Let
G1(s)=1/(s+10) G2(s)=1/(s+1)
G3(s)=(s^2+1)/(s^2+4s+4) G4(s)=(s+1)/(s+6)
H1(s)=(s+1)/(s+2) H2(s)=2
H3(s)=1 ng1=[1];dg1=[1 10];G1=tf(ng1,dg1);
ng2=[1];dg2=[1 1];G2=tf(ng2,dg2);
ng3=[1 0 1];dg3=[1 4 4];G3=tf(ng3,dg3);
ng4=[1 1];dg4=[1 6];G4=tf(ng4,dg4);
nh1=[1 1];dh1=[1 2];H1=tf(nh1,dh1);
nh2=[2];dh2=[1];H2=tf(nh2,dh2);
nh3=[1];dh3=[1];H3=tf(nh3,dh3);
sys1=H2/G4;
sys2=series(G3,G4);
sys3=feedback(sys2,H1,+1);
sys4=series(G2,sys3);
sys5=feedback(sys4,sys1);
sys6=series(G1,sys5);
sys=feedback(sys6,H3)
Rules of Reduction with Multiple Inputs
G1 G2
R(s)
U(s)
Y(s) +
-
+ +
1. Set all inputs zero except from one
2. Transform the diagram in the normal form making use of the rules of reduction 3. Compute the response from a chosen input
4. Repeat Steps 1 to 4 for each one of the inputs
5. Add algebraically all the responses from steps 1-4. This sum is the total response
ofa mutliple input system
Rules of Reduction for Multiple Inputs
1. Set U=0 G1G2/(1+G1G2)
R(s)
G1 G2
R(s)
U(s) + Y(s)
-
+ +
Y(s)
2. Set R=0
G2/(1+G1G2)
U(s) Y(s)
3. Sum Responses Υ(s)=[ G2/(1+G1G2)] [G1 R(s) +U(s)]
Matlab for the computation of the response of a loop with multiple inputs
G1 G2
R(s)
U(s) + Y(s)
-
+ + LET
G1(s)=1/(s+10) G2(s)=1/(s+1)
ng1=[1];dg1=[1 10];G1=tf(ng1,dg1);
ng2=[1];dg2=[1 1];G2=tf(ng2,dg2);
ng3=[1];dg3=[1];G3=tf(ng3,dg3);
sys1=series(G1,G2);
sysYR=feedback(sys1,G3);
sysYU=feedback(G2,G1);
sys=[sysYR sysYU];
step(sys)
0 2 4 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 From: In(1)
0 2 4 6
From: In(2) Step Response
Time (sec)
Amplitude
