Fundamental of Control Systems

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EES32002

Fundamental of Control Systems

Reduction of Multiple Subsystems And Signal Flow Graphs

Lecturer:

Dr. Wahidin Wahab M.Sc.

Aries Subiantoro, ST. MSc.

Electrical Engineering Department

University of Indonesia

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Figure 5.1

The space shuttle consists of multiple subsystems. Can you identify those that are control systems, or parts of control systems?

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Control Systems Engineering, Fourth Edition by Norman S. Nise

Figure 5.2

Components of a block diagram for a linear, time-invariant system

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Figure 5.3 a. Cascaded subsystems;

b. equivalent transfer

function

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Figure 5.4

Loading in

cascaded

systems

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Figure 5.5 a. Parallel subsystems;

b. equivalent transfer function

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Figure 5.6

a. Feedback control system;

b. simplified model;

c. equivalent transfer

function

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Figure 5.7 Block diagram algebra for summing junctions—

equivalent forms for moving a block a. to the left past a summing junction;

b. to the right past a

summing junction

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Figure 5.8

Block diagram algebra for pickoff points - equivalent forms for

moving a block a. to the left past a pickoff point;

b. to the right

past a pickoff

point

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Figure 5.9

Block diagram for Example5.1

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Figure 5.10

Steps in solving Example 5.1:

a. collapse summing junctions;

b. form equivalent cascaded system in the forward path and equivalent

parallel system in the feedback path;

c. form equivalent

feedback system and

multiply by cascaded

G

₁

(s)

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Figure 5.11

Block diagram for

Example 5.2

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Figure 5.12

Steps in the

block diagram

reduction for

Example 5.2

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Figure 5.13 Block diagram for

Skill-

Assessment

Exercise 5.1

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Figure 5.17

Signal-flow graph components:

a. system;

b. signal;

c. interconnection of systems and signals

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Figure 5.18

Building signal-flow graphs:

a. cascaded system

nodes (from Figure 5.3(a));

b. cascaded system signal-flow graph;

c. parallel system

nodes (from Figure 5.5(a));

d. parallel system signal-flow graph;

e. feedback system nodes (from Figure 5.6(b));

f. feedback system

signal-flow graph

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Figure 5.19

Signal-flow graph development:

a. signal nodes;

b. signal-flow graph;

c. simplified signal-

flow graph

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Figure 5.19

Signal-flow graph development:

a. signal nodes;

b. signal-flow graph;

c. simplified signal-

flow graph

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Mason‘s Rule

Loop gain: the product of branch gains found by traversing a path that starts at a node and ends at the same node

Forward-path gain: the product of gains found by traversing a path from the input node to the output node of the signal-flow graph in the

direction of signal flow

Nontouching loops: loops that do not have any nodes in common

Nontouching loop gain: the product of loop

gains from nontouching loops taken two, three,

four, or more at a time

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Mason‘s Rule

Δ Δ

=

= ∑

k

T k

s R

s s C

G ( )

) ) (

(

k = number of forward paths T _k = the kth forward-path gain

Δ = 1 - Σloop gains + Σnontouching loop gains taken two at a time - Σnontouching loop gains taken three at a time +