This document discusses signal flow graphs and Mason's Rule for analyzing control systems. Signal flow graphs represent systems as nodes connected by branches, with transfer functions on the branches. Mason's Rule is used to evaluate closed-loop transfer functions. It involves identifying open-loop gains along forward paths and closed-loop gains for individual loops and combinations of non-touching loops. An example applies Mason's Rule by finding relevant gains and evaluating the closed-loop transfer function according to the rule. Laboratory materials for applying these concepts can be accessed online.

1. ME 176
Control Systems Engineering
Reduction of Multiple Subsystems
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2. Signal Flow Graphs:
Elements:
Nodes
Branches
Branches = Systems
Nodes = Signals
Arrow = Direction of signal flow
Transfer function written adjacent to line.
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3. Signal Flow Graphs:
Each signal is sum of signals
flowing into that node.
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4. Signal Flow Graphs: Cascade Form
Represent Singals as Nodes Interconnect the nodes with
system branches
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5. Signal Flow Graphs: Parallel Form
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6. Signal Flow Graphs: Feeddback Form
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7. Signal Flow Graphs:
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8. Signal Flow Graphs:
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9. Signal Flow Graphs:
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10. Mason's Rule: Definition
Loop Gain:
The product of branch gains found by
traversing a path that starts at a node and
ends at the same node, following the direction
of the signal flow, without passing through the
node more than once.
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11. Mason's Rule: Definition
Forward Path Gain:
The product of gains found by traversing
a path from the input node to the output
node of the signal-flow graph in the
direction of signal flow.
.
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12. Mason's Rule: Definition
Nontouching Loops:
Loops that do not have any nodes in common
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13. Mason's Rule: Definition
Nontouching-loop Gain:
The product of loop gains from nontouching
loops taken two, three, four, or more at a
time.
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14. Mason's Rule:
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15. Mason's Rule: Example
Step 1 : Identify the forwardpath gains:
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16. Mason's Rule: Example
Step 2 : Identify the loop gains:
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17. Mason's Rule: Example
Step 3 : Identify the loop gains (2 @ a time):
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18. Mason's Rule: Example
Step 4 : Identify the loop gains (3 @ a time):
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19. Mason's Rule: Example
Step 5 : Using Mason's rule - evaluate for :
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20. Mason's Rule: Example
Step 5 : Using Mason's rule - evaluate for :
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21. Mason's Rule: Example
Step 5 : Using Mason's rule - evaluate for :
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22. Laboratory :
Material for the laboratory sessions that come after this lecture can be viewed from
the class site at:
http://sites.google.com/a/vinta.com/me-176/
Please make sure to read the directions at the bottom of the page for instructions on
how to submit the laboratory working sessions.
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