Cleveland State University
MCE441: Intr. Linear Control Systems
Lecture 8: Final Value Theorem
Block Diagram Algebra
Prof. Richter
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The Final Value Theorem
The Final Value
⊲ Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
Provides information about the value of f (t) at steady state
(t = ∞)
The steady-state may not exist: the poles of sF (s) must lie
on the open left half of C (details later)
If limt→∞ f (t) exists, then
limt→∞ f (t) = lims→0 sF (s)
1
Example: For F (s) = s(s+1)
, we see that the only pole of
sF (s) lies on the left half plane. So
lim f (t) = lim sF (s) = 1
t→∞
s→0
Check by inversion of F (s)
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Block Diagram Algebra
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
The previous result allows us to multiply TF and input to get
output in the Laplace domain
Block diagrams are convenient representations of control
system structure
The following basic operations are allowed in block diagram
algebra:
1. Addition between signals. Result: another signal.
2. Multiplication between a signal and a system. Result:
another signal.
Systems (TFs) are represented by blocks.
Signals are represented by arrows.
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Block Diagram Algebra
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
A Summing junction or summing point is the symbol that
represents signal addition
A branch point indicates that the output from a block is
applied to other blocks or summing junctions.
Summing point
R(s)
E(s) = R(s) − Y (s)
Y (s) = G(s)E(s)
E(s)
+
−
G(s)
Y (s)
Branch point
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Block Diagram Manipulations
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
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Example
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
R(s)
Y (s)
+
G1
+
−
+
G2
+
⊲
G3
G4
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Solution
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
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Solution
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
(G +1)G
2
Check that the required TF is (G +1)G 1G +1−G
1
2 4
2 G3
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Example
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
Y (s)
R(s)
G1
+
G3
+
−
−
G2
+
G5
+
G4
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Solution
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
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Solution
The Final Value
Theorem
Block Diagram
Algebra
Block Diagram
Algebra
Block Diagram
Manipulations
Example
Solution
Solution
Example
Solution
Solution
⊲
G H
G
3 .
Check that the required TF is 1+G (G 1H 1G +G ) , where H1 = 1+G
3
5
1 1 4
2
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