Signal Flow Graph, Mason's Gain RUle

1. AE 313
AE Systems & Control
Semester 222
Dr. Syed Saad Azhar Ali
75-110
syed.ali@kfupm.edu.sa

2. Reduction of Complex Systems
• For a complex system, the block diagram reduction technique is cumbersome.
• An alternative method for determining the relationship between system variables has
been developed by Mason and is based on a signal flow graph.
• A signal flow graph is a diagram that consists of nodes that are
• connected by branches.
• A node is assigned to each variable of interest in the system, and branches are used
to relate the different variables.
• The main advantage for using SFG is that a straight forward procedure is available for
finding the transfer function in which it is not necessary to move pickoff point around or
to redraw the system several times as with block diagram manipulations.

3. Basic Elements
System
Gain
Signal
Node
Connection

4. Converting a Block Diagram to SFG
4
• Denote the signals after every summing point
• Denote the signals after every block
• Keep the nodes
• Connect the nodes
• Add the gains and signs

5. Equivalent SFG
5

6. Equivalent SFG
6
Combine the series e.g.
G1(s) and 1
H1(s) and -1
H2(s) and -1
H3(s) and -1

7. Simplify using Mason’s Gain Rule
𝑃1 = 𝐺1 𝑠 𝐺2 𝑠 𝐺3 𝑠
𝑃2 = 𝐺1 𝑠 1 𝐺3 𝑠
𝑃2 = 𝐺1 𝑠 𝐺3 𝑠
𝐿1 = −𝐺1 𝑠 𝐺2 𝑠 𝐻1 𝑠
𝐿2 = −𝐺2 𝑠 𝐻2 𝑠
𝐿3 = −𝐺3 𝑠 𝐻3 𝑠
𝐿1 𝐿3 = 𝐺1 𝑠 𝐺2 𝑠 𝐻1 𝑠 𝐺3 𝑠 𝐻3 𝑠
𝐿2 𝐿3 = 𝐺2 𝑠 𝐻2 𝑠 𝐺3 𝑠 𝐻3 𝑠

8. 3/8/2024 Syed Saad Azhar Ali 8

9. 3/8/2024 Syed Saad Azhar Ali 9