ME2142/ME2142E Control Systems
Block Diagram Algebra
ME2142/TM3142 Control Systems
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Block Diagram Representation
A block diagram is a graphical tool can help us to visualize the model of a system and evaluate the mathematical relationships between their elements, using their transfer functions. The Transfer Function Block Input
R(s)
G(s)
C(s)
Output
System
G ( s) =
C (s ) R( s )
The transfer function G(s) is § defined only for a linear time-invariant system and not for nonlinear systems. § Is a property of the system and is independent of the input to the system. § Commutative § Associative
G1G 2 = G2 G1 G1 + G2 = G2 + G1
ME2142/TM3142 Control Systems
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Block Diagram Elements
The Summing Point
X Signed inputs Y
+
+
+X+Y- Z
-
output Z
§
Any number of inputs. Only one output
ME2142/TM3142 Control Systems
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Block Diagram Algebra
When manipulating block diagrams, the original relationships, or equations, relating the various variables must remain the same.
Blocks in series or cascaded blocks X
G1
§
Y
G2
Z
X
G1 G2
Z
When blocks are connected in series, there must be no loading effect.
ME2142/TM3142 Control Systems
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Block Diagram Algebra
Blocks in parallel Y=G1X+G2X
Y=G1X+G2X
G2
X X
G1
+
+
ME2142/TM3142 Control Systems
Y
G1 + G2
Y
5
Block Diagram Algebra
X
+
Z
G
X
G
+
Z
+ +
Y
X
Z=G(X+Y)
G
Y
G
X
Y
X
X
Z=GX+GY
Y
G 1/G
Z=G(X+Y/G) X
G
X
Z
+
+
G
Z
+
+ Y
1/G
Y
Z=GX+Y
ME2142/TM3142 Control Systems
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Closed-Loop System
R
E
+
G
C
B
H
R is called the reference input C is the output or controlled variable B is the E = (R – B) is the error C =G E B = GH E
is called the feedforward transfer function
is called the open-loop transfer function
ME2142/TM3142 Control Systems
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Closed-Loop System
R
E
+
C
G
B
C R
C = GE = G(R – B) = G(R – HC) C(1 + GH) = GR C G = R 1 + GH
H
is the closed-loop transfer function
Also E =
E R
C G
and
E 1C 1 = = R G R 1 + GH
is called the error transfer function
ME2142/TM3142 Control Systems
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Closed-Loop Control System D R
E
+
Gc
M +
+ Gp
C
B
H
Gc is the controller transfer function Gp is the plant transfer function M is the manipulated variable D is the external disturbance C = Gc G p is the feedforward transfer function E B = Gc G p H E
is the open-loop transfer function
ME2142/TM3142 Control Systems
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Closed-Loop Control System D R
E
+
Gc
M +
+ Gp
C
B
H
Gc G p C G = = R 1 + GH 1 + Gc G p H Assuming R = 0, we can re-draw D
+
Gp -
C
Gp C G = = D 1 + GH 1 + G p Gc H
GcH
ME2142/TM3142 Control Systems
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Block Diagram Manipulation Example: Determine C(s)/R(s) a R
+
+ -
Da
D
F
+
E +
G
+
+
C
-
H I
ME2142/TM3142 Control Systems
When manipulating blocks, must ensure C(s) does not change, so that C(s)/R(s) remains same.
11
Block Diagram Manipulation Example: Determine C(s)/R(s)
We wish to move this signal to before block F
Assume names of signals as shown
b = Fa-Fc+Da
ME2142/TM3142 Control Systems
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Block Diagram Manipulation Example: Determine C(s)/R(s)
To move signal b to after Block G
Assume names of signals as shown
b = Fa-Fc+Da
ME2142/TM3142 Control Systems
b = Fa-Fc+(D/F)aF
13
Block Diagram Manipulation Example: Determine C(s)/R(s) a R
+
E
(D/F)a
D/F
a +
-
Da
D
+
+
F
+
G
+
C
R
b
+
-
+
+ -
F
Eb
E G
+
+
C
-
H
H
I
I
Gb
C = Gb + Eb Eb
E/G R
+
1+D/F -
+
FG
Gb
+
+
C
R
+
1+D/F -
-
+
FG
1+E/G
-
H
H
I
I
C = Gb + (E/G)Gb
ME2142/TM3142 Control Systems
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C
Block Diagram Manipulation Example
R
+
1+D/F -
G 1 + GH +
FG
1+E/G
C
-
R
+
1+D/F -
FG 1 + FGH
1+E/G
H I
R
+ -
D FG E 1 + 1 + F 1 + FGH G
I
C
R
D FG E 1 + 1 + F 1 + FGH G D FG E 1 + 1 + 1 + I F 1 + FGH G
C
I
ME2142/TM3142 Control Systems
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C
End
ME2142/TM3142 Control Systems
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