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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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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