8/7/2019 Block Diagram Algebra
1/16
ME2142/TM3142 Feedback Control Systems1
Block Diagram Algebra
Block Diagram Algebra
ME2142/ ME2142E Feedback Control SystemsME2142/ ME2142E Feedback Control Systems
8/7/2019 Block Diagram Algebra
2/16
ME2142/TM3142 Feedback Control Systems2
The Transfer Function Block
Block Diagram RepresentationBlock Diagram Representation
A block diagramis agraphicaltoolcan help us to visualize themodelof a system and evaluate the mathematical relationshipsbetween their elements, using their transfer functions.
A block diagramis agraphicaltoolcan help us to visualize themodelof a system and evaluate the mathematical relationshipsbetween their elements, using their transfer functions.
G(s)
System
Input OutputR(s) C(s)
)(
)()(
sR
sCsG =
The transfer function G(s) is
defined only for a linear time-invariant system and notfor nonlinear systems.
Is a property of the system and is independent of the
input to the system.
Commutative1221
GGGG = Associative
1221GGGG +=+
8/7/2019 Block Diagram Algebra
3/16
ME2142/TM3142 Feedback Control Systems3
The Summing Point
Block Diagram ElementsBlock Diagram Elements
Any number of inputs. Only one output
Signed inputs
X
-
Z
Y
++ +X+ Y-Z
output
8/7/2019 Block Diagram Algebra
4/16
ME2142/TM3142 Feedback Control Systems4
Blocks in series or cascaded blocks
Block Diagram AlgebraBlock Diagram Algebra
When manipulating block diagrams, theoriginal relationships, or
equations, relating the various variablesmust remain the same.
When manipulating block diagrams, theoriginal relationships, or
equations, relating the various variablesmust remain the same.
When blocks are connected in series, there must be noloading effect.
G1 G2XY Z
G1G
2ZX
8/7/2019 Block Diagram Algebra
5/16ME2142/TM3142 Feedback Control Systems5
Blocks in parallel
Block Diagram AlgebraBlock Diagram Algebra
G1
G2
++ YX
G1 + G2YX
Y=G1X+G2X Y=G1X+G2X
8/7/2019 Block Diagram Algebra
6/16ME2142/TM3142 Feedback Control Systems6
G+
+ ZX
Y
G+
+ ZX
Y
G
X Y
X
G
X Y
X
XG
Y
Z
+
+XG
Y
Z
+
+
Block Diagram AlgebraBlock Diagram Algebra
G
G
+
+ ZX
Y
GX Y
1/GX
+
+ Z
Y
X
G
1/G
Z=G(X+Y)
Z=GX+GY
Z=GX+Y
Z=G(X+Y/G)
8/7/2019 Block Diagram Algebra
7/16ME2142/TM3142 Feedback Control Systems7
Closed-Loop Feedback SystemClosed-Loop Feedback System
is called the open-loop transfer functionGHE
B =
is called the feedforward transfer functionGE
C=
R is called the reference inputC is the output or controlled variableB is the feedbackE = (R B) is the error
RG
C
-
+ E
HB
8/7/2019 Block Diagram Algebra
8/16ME2142/TM3142 Feedback Control Systems
8
Closed-Loop Feedback SystemClosed-Loop Feedback System
C = GE
= G(R B)= G(R HC)
C(1 + GH) = GR
GH
G
R
C
+=
1
is the closed-loop transfer functionR
C
Also andG
CE=
GHR
C
GR
E
+==
1
11
is called the error transfer functionR
E
R G C
-
+ E
HB
8/7/2019 Block Diagram Algebra
9/16ME2142/TM3142 Feedback Control Systems
9
Closed-Loop Control Feedback SystemClosed-Loop Control Feedback System
is the open-loop transfer functionHGGE
Bpc
=
is the feedforward transfer functionpcGGE
C=
Gc is the controller transfer function
Gp is the plant transfer functionM is the manipulated variableD is the external disturbance
R Gp
C
-
+ E
HB
Gc
+ +
D
M
8/7/2019 Block Diagram Algebra
10/16ME2142/TM3142 Feedback Control Systems
10
HGG
GG
GH
G
R
C
pc
pc
+=
+=
11
Assuming R = 0, we can re-draw
HGGG
GHG
DC
cp
p
+=+= 11
Closed-Loop Control Feedback SystemClosed-Loop Control Feedback System
RG
p
C
-
+ E
HB
Gc
+ +
D
M
DG
p
C
-
+
GcH
8/7/2019 Block Diagram Algebra
11/16
ME2142/TM3142 Feedback Control Systems11
Block Diagram ManipulationBlock Diagram Manipulation
Example: Determine C(s)/R(s)Example: Determine C(s)/R(s)
R G-
+
H
F+
+
-
+
I
++
ED
C
a Da
When manipulating blocks,must ensure C(s) does not
change, so that C(s)/R(s)
remains same.
8/7/2019 Block Diagram Algebra
12/16
ME2142/TM3142 Feedback Control Systems12
Block Diagram ManipulationBlock Diagram Manipulation
Example: Determine C(s)/R(s)Example: Determine C(s)/R(s)
b = Fa-Fc+Da
Assume names of
signals as shown
We wish to move this
signal to before block F
8/7/2019 Block Diagram Algebra
13/16
ME2142/TM3142 Feedback Control Systems13
Block Diagram ManipulationBlock Diagram Manipulation
Example: Determine C(s)/R(s)Example: Determine C(s)/R(s)
b = Fa-Fc+Da
Assume names of
signals as shown
b = Fa-Fc+(D/F)aF
To move signal b to
after Block G
8/7/2019 Block Diagram Algebra
14/16
ME2142/TM3142 Feedback Control Systems14
Block Diagram ManipulationBlock Diagram Manipulation
Example: Determine C(s)/R(s)Example: Determine C(s)/R(s)
R G-
+
H
F+
+
-
+
I
++
ED
C
aDa
bEb
R G-
+
H
F-
+
I
++
ED/F
C+
a(D/F)a
Gb
R
-
+
H
FG-
+
I
++
E/G
C1+D/F
Gb
Eb
R
-
+
H
FG-
+
I
1+E/GC
1+D/F
C = Gb + Eb
C = Gb + (E/G)Gb
8/7/2019 Block Diagram Algebra
15/16
ME2142/TM3142 Feedback Control Systems15
Block Diagram ManipulationBlock Diagram Manipulation
ExampleExample
R
-
+
H
FG-
+
I
1+E/GC
1+D/F R
-
+
I
1+E/GC
1+D/FFGH1
FG
+
R
-
+
I
C
+
+
+
G
E
FGH
FG
F
D1
11
CR
IG
E
FGH
FG
F
D
G
E
FGH
FG
F
D
+
+
++
+
+
+
1111
11
1
GH
G
+1
8/7/2019 Block Diagram Algebra
16/16
ME2142/TM3142 Feedback Control Systems16
End