Lec 4 . Graphical System Representations and Simplifications
• Block Diagrams
• Signal Flow Graphs and Mason’s Formula
• Reading: 3.9-3.10
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
Basic Components of Block Diagrams
(Functional) block
Summing point+
Branch point
Signal flow
Cascaded/Parallel Connected Systems
Cascaded systems:
Parallel connected systems:
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(Negative) Feedback Connected Systems
Feedback connected systems:
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Feedforward transfer function (FTF):
Open-loop transfer function (OTF):
Closed-loop transfer function (CTF):
Positive Feedback Connected Systems
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Closed-loop transfer function:
Unity Feedback System
Unit feedback connected systems: H(s)=1
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Closed-loop transfer function:
Feedback Control System
Closed-loop transfer function:
Remark: by adjusting the controller C(s), one can change the close-loop transfer function to achieve desired properties.
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plantcontroller
Block Diagram Reduction
Often times the block diagram under study is complicated
Use previous basic steps to reduce the complexity of block diagram
Example:
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Another Example
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Operations for Simplifying Block Diagrams
“Slide a branch point past a functional block (forward)”
Application to Previous Example
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Another Operation for Simplifying Block Diagrams
“Slide a summation point past a functional block (backward)”
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Application to Previous Example
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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
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
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Type of Nodes
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
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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 W:
At node U:
At node W:
A More Complicated Example
Signal flow graph:
Block diagram:
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Simplifying Signal Flow Graphs
Cascaded systems:
Parallel connected systems:
Feedback connected systems:
However, In General
Transfer function from U to Y?
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
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
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
Application to the Previous Example
Forward path Forward path gain Pi i
Another Example
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:
Example I• One input: F• Two outputs: x and y
• Transfer matrix H(s)=[H1(s), H2(s)]
Example II
• Two inputs: u1, u2
• Two outputs: y1, y2
• Transfer matrix H(s)?
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