Formula Sheets Block Diagrams & Signal Flow Graphs - Control Systems - Electrical

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Control S ystems F ormula Sheet: Block
Diagr ams & Signal Flow Gr aphs
1. Block Diagr am Reduction
• Series Connection : Blocks in series combine b y multiplying their tr ansfer
functions:
G
eq
(s)=G
1
(s)·G
2
(s)·...·G
n
(s)
• Par allel Connection : Blocks in par allel combine b y adding their tr ansfer func-
tions:
G
eq
(s)=G
1
(s)+G
2
(s)+...+G
n
(s)
• Closed-Loop S ystem (negative feedback):
T(s)=
G(s)
1+G(s)H(s)
where G(s) is the forward path tr ansfer function, H(s) is the feedback path
tr ansfer function.
• Closed-Loop S ystem (positive feedbac k):
T(s)=
G(s)
1-G(s)H(s)
• Moving a Summing Point Before a BlockG(s) :
Input to summ ing point becomes
R(s)
G(s)
• Moving a Summing Point After a BlockG(s) :
Input to summing point becomesR(s)·G(s)
• Moving a T ak e-off Point Before a Block G(s) :
Output becomes
Y(s)
G(s)
• Moving a T ak e-off Point After a Block G(s) :
Output becomesY(s)·G(s)
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Page 2


Control S ystems F ormula Sheet: Block
Diagr ams & Signal Flow Gr aphs
1. Block Diagr am Reduction
• Series Connection : Blocks in series combine b y multiplying their tr ansfer
functions:
G
eq
(s)=G
1
(s)·G
2
(s)·...·G
n
(s)
• Par allel Connection : Blocks in par allel combine b y adding their tr ansfer func-
tions:
G
eq
(s)=G
1
(s)+G
2
(s)+...+G
n
(s)
• Closed-Loop S ystem (negative feedback):
T(s)=
G(s)
1+G(s)H(s)
where G(s) is the forward path tr ansfer function, H(s) is the feedback path
tr ansfer function.
• Closed-Loop S ystem (positive feedbac k):
T(s)=
G(s)
1-G(s)H(s)
• Moving a Summing Point Before a BlockG(s) :
Input to summ ing point becomes
R(s)
G(s)
• Moving a Summing Point After a BlockG(s) :
Input to summing point becomesR(s)·G(s)
• Moving a T ak e-off Point Before a Block G(s) :
Output becomes
Y(s)
G(s)
• Moving a T ak e-off Point After a Block G(s) :
Output becomesY(s)·G(s)
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2. Signal Flow Gr aphs
• Basic Elements :
– Node : Represents a variable (input, output, or intermediate).
– Br anch : Directed path with gain (tr ansfer function) between nodes.
– F orward Path : Path from input to output without touching an y node more
than once.
– Loop : Closed path starting and ending at the same node without repeating
nodes (except start/end).
• Mason’ s Gain F ormula (Over all Tr ansfer Function):
T(s)=
?
P
k
?
k
?
where:
– P
k
: Gain of thek -th forward path.
– ? : Determinant of the gr aph:
?=1-
?
L
i
+
?
L
i
L
j
-
?
L
i
L
j
L
k
+···
(L
i
: Gain of the i -th loop,
?
L
i
L
j
: Sum of products of gains of all pairs of
non-touching loops, etc.)
– ?
k
: Determin ant of the gr aph excluding loops touching the k -th forward
path.
• Non-T ouching Loops : Loops that do not share an y common nodes.
• Loop Gain : Product of br anch gains in a loop.
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