The document discusses signal flow graphs (SFG), which provide a pictorial representation of systems of equations. SFGs show variables as nodes, relationships between variables as branches connecting nodes, and coefficients as gains of branches. Key concepts discussed include input and output nodes, forward paths and their gains, loops and loop gains, and the Mason Gain Formula for calculating the overall system gain based on these elements. Two examples demonstrate applying the Mason Gain Formula to SFG representations of feedback control systems.

1. Topic: Signal Flow Graph
week: 2
Lecture: 1
Level: 7th Semester
program: BE Mechanical Engineering
Course Teacher: Engr. Muhammad Akmal Qaisar
Lecturer, Mechanical Engineering Department
SUIT Peshawar.
ME 352 Control
Engineering
1

2. Signal Flow Graph- S. J. Mason
1953
• Though the block diagram approach is commonly used for
simple systems, it quickly gets complicated when there are
multiple loops and subsystems or in MIMO cases. Thus, we need
a more streamlined and systematic approach for such systems.
• Signal Flow Graph (SFG): Pictorial representation of a system
of equations, in which:
• variables → nodesof SFG
• relationship between variables →
• branches of SFG coefficients →
• gains of branches in SFG
• Example: F = Ma

3. Signal Flow Graph-Cont.
• Input Node: Node with only outgoing branches;
• Output Node: Node with incomingbranches.
• Note: Any non-input node can be made an output node by
adding a branch with gain= 1.
• Path: Collection of branches linked together in same direction.
• Forward Path: Path from input node to output node where
node is visited more than once.
.
Key Definitions:

4. Signal Flow Graph-Cont.
• Gain of Forward Path: Product of all gains of branches in the
forward path
• Loop: Path that originates and terminates at the same node.
No other node is visited more than once.
• Loop Gain: Product of branch gains in a loop.
• Non-Touching: Two parts of a SFG are non-touching if they do
not share at least one node.

5. Signal Flow Graph-Cont.
• Example:

6. Signal Flow Graph-Cont.
• Input Node: x1
• Output Node: All nodes besides x1
.
• Forward Path: Assume x5 as output node,
then
Path 1 = x1, x2, x3, x4, x5;
Path 2 = x1, x2, x4, x5

7. Signal Flow Graph-Cont.
• Gain of Forward Path:
• Path 1: M1 = abcd;
• Path 2: M2 = afd
• Loop: x3 → x4 → x5
• Loop Gain: P1 = −ce

8. Signal Flow Graph-Cont.
• Algebra of SFG:
1. Output variable of a node = weighted sum (by the gains of
branches) of all incoming branches. For previous example
x2 = ax1, x3 = bx2 − ex4, x4 = cx3 + fx2, x5 = dx4

9. Signal Flow Graph-Cont.
2. Parallel branches
Note: all branches must be in same direction (otherwise they form
a loop).

10. Signal Flow Graph-Cont.
3. Series branches
Note: No intermediate incoming or outgoing branches
between x1 and x4.

11. Mason Gain Formula
• Mason Gain Formula:
Let yin : Input variable (s-domain)
yout : Output variable (s-domain)
Then the gain, M , is:
M = = 𝑘=1
𝑁
𝑀𝑘Δ𝑘
Δ
• Mk: Gain of kth forward path between yout andyin.
• N : Number of forward paths
𝑦𝑜𝑢𝑡
𝑦𝑖𝑛

12. Mason Gain Formula-Cont.
Δ = 1 − 𝑃𝑚1 + 𝑃𝑚2 − 𝑃𝑚3 … .
Pm1: Loop gain of mth loop
Pm2: Product of mth combination of pairs of non-touching loop
Gains
Pm3: Product of mth combination of triplets of non-touching
loop gains
∆k: value of ∆ for part of SFG which is non-touching with kth
forward path.

13. Mason Gain Formula-Cont.
• Steps:
• Arrange SFG (from Block Diagram) and identify input and
output nodes.
• List all forward paths and gains Mks.
• List all loops and gains, Pm1, Pm2, . . . and form ∆.
• Determine state of path k with loops and form ∆ks.
• Apply Mason Gain Formula.

14. Example 1.
• Example 1:
Consider a standard closed-loop system as shown.

15. Example 1
R - input
C – output
M1 = G
P11 = −GH
∆ = 1 − P11 = 1 + GH
∆1 = 1, since path 1 is touching with the loop.
𝑀 =
𝑀1 ∆1
∆
=
𝐺(𝑠)
1+𝐺(𝑠)𝐻(𝑠)

16. Example 2.
Example 2:
Step 1:
Input: R1, output: C1
G (s) =
𝐶1(𝑠)
𝑅1(𝑠)

17. Example 2
Step 2:
M1 = G1
Step 3:
Loop gains: P11 = −G1G2G3G4
Thus ∆ = 1 − P11 = 1 + G1G2G3G4
Step 4:
∆1 = 1 since forward path is touching with the loop.

18. Example 2
• 𝐺 𝑠 =
𝑀1∆1
∆
=
𝐺1
1+𝐺1𝐺2𝐺3𝐺4

19. End