In this video we are going to discuss about the block reduction technique in control systems.

1. RULES OF BLOCK DIAGRAM ALGEBRA
 COMBINING BLOCKS IN CASCADE (SERIES)
 COMBINING BLOCKS IN CASCADE (PARALLEL)
G1 G2X1
X1 * G1 X1 * G1*G2
G1*G2X1
X1 * G1*G2
G1 + G2X1
X1 (G1 + G2)
X1 * G2
G1
G2
X1
X1 * G1 X1 (G1 - G2)
-
G1 - G2X1
X1 (G1 - G2)
G1
G2
X1
X1 * G1 X1 (G1 + G2)
+
X1 * G2

2.  MOVING SUMMING POINT AFTER A BLOCK
GX1
X1 + X2 G (X1 + X2)
+
X2
+
X2
X2 * G
GX1 X1 * G G (X1 + X2)
+
G
+
GX1
X1 - X2 G (X1 - X2)
-
X2
+
X2 * G
GX1 X1 * G G (X1 - X2)
-
G
+
X2

3.  MOVING SUMMING POINT BEFORE A BLOCK
G
X1 X1*G ( X1*G ) + X2
+
X2
+
X2
X2/G
GX1
X1 + (X2/G) (G*X1) + X2
+
1/G
+
G
X1 X1*G ( X1*G ) - X2
-
X2
+
X2
X2/G
GX1
X1 - (X2/G) (G*X1) - X2
-
1/G
+

4.  MOVING A TAKE OFF POINT AFTER A BLOCK
 MOVING A TAKE OFF POINT BEFORE A BLOCK
GX1
G * X1
X1
X2
GX1
X1 * G
1/G
X1 * G
GX1
G * X1
G*X1
X1
GX1 X1 * G
G
X1 * G

5.  SWAP WITH TWO NEIGHBOURING SUMMING POINT
 WHEN TWO OR MORE SIGNALS ENTERING A SUMMING POINT
-
X3
+X1 X1 – X2
-
X2
+
X1 – X2 - X3
A B
X1 X1 – X3
-
X2
+
X1 – X2 - X3
+
-
AB
X3
+
-
+X1
X2
X3
X1 – X2 + X3

6.  ELIMINATING A FEEDBACK LOOP
G+
+
H
R(s) C(s)
𝐺
1 − 𝐺𝐻
R(s)
C(s)
G-
+
H
R(s) C(s)
𝐺
1 + 𝐺𝐻
R(s)
C(s)

7.  SHIFTING TAKE OFF POINT AFTER SUMMING POINT
 SHIFTING TAKE OFF POINT AFTER SUMMING POINT
X1
+
-
X1
X2
X1 – X2
-
+
-
X1
X2
X1 – X2
+
X1
+
-
X1
X2
X1 – X2
X1 – X2
-
+
-
X1
X2
X1 – X2
+
X1 – X2

8.  Example
 Find the transfer function for the following block diagram.
2G 3G1G
4G
1H
2H
)(sY)(sR

9.  Step 1: Moving take off point “A” before the block
2G
A
2G 3G1G
4G
1H
2H
)(sY)(sR
2G 3G1G
4G
1H
2H
)(sY)(sR
2G
B
B

10.  Step 2: Combine series blocks and
2G1G
2G 3G1G
4G
1H
2H
)(sY)(sR
B
2G
G2 * G31G
4G
1H
2H
)(sY)(sR
B
2G

11.  Step 3: Eliminate the Loop 1 (parallel blocks)
( it’s a forward loop not feedback loop, so simply add the blocks according to summing point signs)
Loop 1
G2 * G31G
4G
1H
2H
)(sY)(sR
B
2G
G4+(G2 * G3)1G
1H
2H
)(sY)(sR
B
2G

12.  Step 4: Combine series blocks and1H 2G
G4+(G2 * G3)1G
1H
2H
)(sY)(sR
B
2G
G4+(G2 * G3)
H1 * G2
1G
2H
)(sY)(sR
B

13.  Step 5: Moving take off point “B” after block G4+(G2 * G3)
G4+(G2 * G3)
H1 * G2
1G
2H
)(sY)(sR
B
G4+(G2 * G3)
H1 * G2
1G
2H
)(sY)(sR
1/(G4+G2*G3)

14.  Step 6: Eliminate Loop 2 (feedback loop)
G4+(G2 * G3)
H1 * G2
1G
2H
)(sY)(sR
1/(G4+G2*G3)
Loop 2
H1 * G2
1G
)(sY)(sR
1/(G4+G2*G3)
)(1 3242
324
GGGH
GGG



15.  Step 7: Combine series blocks , and ,1G )(1 3242
324
GGGH
GGG


H1 * G2
1G
)(sY)(sR
1/(G4+G2*G3)
)(1 3242
324
GGGH
GGG


H1 * G2 1/(G4+G2*G3)
)(sY)(sR
)(1
)(*
3242
3241
GGGH
GGGG


324
12
GGG
HG


16.  Step 8: Eliminate Loop 3 (Feedback loop)
)(sY)(sR
)(1
)(*
3242
3241
GGGH
GGGG


324
12
GGG
HG

)(sY)(sR
)(1
)(
3242121
3241
GGGHHGG
GGGG


Loop 3

17.  Step 9: Eliminate Loop 4 (Unity feedback loop, i.e., the value of feedback loop is 1 because
there is no feedback block value)
)(sY
)(sR
)(1
)(
3242121
3241
GGGHHGG
GGGG


)(sY)(sR
)()(1
)(
32413242121
3241
GGGGGGGHHGG
GGGG


Loop 4

18.  The given block diagram:
 Transfer function for the given block diagram
(from previous slide)
2G 3G1G
4G
1H
2H
)(sY)(sR
)()(1
)(
)(
)(
)(
32413242121
3241
GGGGGGGHHGG
GGGG
sR
sY
sT


