1. Lec 4 . Graphical System Representations and Simplifications
Block Diagrams
Signal Flow Graphs and Mason’s Formula
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2. Block Diagrams Graphical representation of interconnected systems
A system may consist of multiple subsystems: the output of one may be the input to another
Each subsystem is represented by a functional block, labeled with the corresponding transfer function
Blocks are connected by arrows to indicate signal flow directions
Advantage
Easy for visualization purpose
Can represent a class of similar systems

3. Basic Components of Block Diagrams
Signal flow
(Functional) block
Summing point +
Branch point

4. Cascaded/Parallel Connected Systems
Cascaded systems:
Parallel connected systems: +

5. (Negative) Feedback Connected Systems+
Feedforward transfer function (FTF):
Open-loop transfer function (OTF):
Closed-loop transfer function (CTF):

6. Positive Feedback Connected Systems+
Closed-loop transfer function:
Remark: can be thought of as negative feedback connected system with feedback element –H(s)

7. Unity Feedback System Unit feedback connected systems: H(s)=1+
Closed-loop transfer function:

8. Feedback Control System
controller
plant +
Closed-loop transfer function:
Remark: by adjusting the controller C(s), one can change the close-loop transfer function to achieve desired properties.

9. Block Diagram Reduction
Often times the block diagram under study is complicated
Use previous basic steps to reduce the complexity of block diagram
Example: + +

10. Another Example + +

11. Operations for Simplifying Block Diagrams
“Slide a branch point past a functional block (forward)”

12. Application to Previous Example+ +

13. Application to Previous Example+ + + + +

14. Another Operation for Simplifying Block Diagrams
“Slide a summation point past a functional block (backward)” + +

15. Application to Previous Example+ +

16. Application to Previous Example+ + + + + +

17. Yet Another Simplifying Operation
“Slide a summation point past a functional block (forward)” + +
Compare with:
“Slide a branch point past a functional block (forward)”

18. Signal Flow Graphs
An alternative graphical representation of interconnections of subsystems
Advantage compared with block diagrams
A systematic way to compute the transfer function from any input to any output

19. A Simple Example Block diagram: Signal flow graph:+
Basic component of a signal flow graph:
Node: represents a signal
Each node is labeled with the corresponding signal
Branch: directed line segment connecting two nodes
Signal can only flow along the specified direction
Each branch is associated with a transmittance or gain

20. Type of Nodes Input nodes: nodes with only outgoing branches
Block diagram:
Signal flow graph: +
Input nodes: nodes with only outgoing branches
Output nodes: nodes with only incoming branches
Mixed nodes: both incoming and outgoing branches
An output node can be made from an arbitrary node by adding an outgoing branch of unit gain

21. What Happen At a Mixed Node?
At a mixed node, signals of all incoming branches are added and the result is transmitted to all outgoing branches
At node Z:
At node U:
At node W:
At node W:

22. A More Complicated Example
Block diagram: + +
Signal flow graph:

23. Simplifying Signal Flow Graphs
Cascaded systems:
Parallel connected systems:
Feedback connected systems:

24. However, In General
Transfer function from U to Y?

25. Mason’s Formula: A Direct Approach
Path: a sequence of connected branches (following arrow directions)
Forward path: start from an input node and end at an output node
Forward path gain: product of all branch gains along a forward path
Loop: a closed path (starts and ends at the same node)
Loop gain: product of all branch gains along a loop
Notouching loops: loops that do not have shared nodes

26. Determinant of A Graph =1- (sum of all individual loop gains)
+ (sum of gain products of all two nontouching loops)
- (sum of gain products of all three nontouching loops)
+ …
Determinant of a graph without any loop is 1
Example:

27. Mason’s Formula Transfer function from an input node to an output node
Compute the determinant  of the signal flow graph
Find all forward paths with path gains P1,…,Pk
For each forward path Pi, find its cofactor i , i.e., the determinant of the sub-graph with all the loops touching Pi removed
Transfer function from input node to the output node is given by

28. Application to the Previous Example
Forward path
Forward path gain Pi i

29. Another Example

30. Systems with Multiple Inputs and Outputs
MIMO system
m inputs u1,…,um
n outputs y1,…,yn
Laplace transform of the k-th output is
where is the transfer function from ui to yk
Transfer matrix:

31. Example I One input: F Two outputs: x and y
Transfer matrix H(s)=[H1(s), H2(s)]

32. Example II Two inputs: u1, u2 Two outputs: y1, y2
Transfer matrix H(s)? + +