Chapter 3
Block Diagrams &
Signal-Flow Graphs
Automatic Control Systems, 9th Edition
F. Golnaraghi & B. C. Kuo
Section 3- 0, p. 104
Main Objectives of This Chapter
1.
To study block diagrams, their components, and their
underlying mathematics.
2. To obtain transfer function of systems through block
diagram manipulation and reduction.
3. To introduce the signal-flow graphs.
4. To establish a parallel between block diagrams and
signal-flow graphs.
5. To use Marson’s gain formula for finding transfer
function of systems.
6. To introduce state diagrams.
7. To demonstrate the Matlab tools using case studies.
3-1
Section 3- 1, p. 104
3-1 Block Diagrams
• Block diagrams: the composition and interconnection of
the components of a system
 describe the cause-and-effect relationships throughout
the system.
Disturbance
Input
actuator
Output
Error
comparator
3-2
Section 3- 1, p. 105
DC-Motor Control System (Fig. 3.2)
system components
(a) Block diagram of a dc-motor control system.
linear range
(b) Block diagram with transfer function and amplifier characteristic.
3-3
Section 3- 1, p. 106
Typical Elements of Block Diagrams in
Control Systems
comparator
feedback loop
Typical block elements: plant, controller, actuator, and sensor
3-4
Section 3- 1, p. 107
Block-Diagram Elements of Comparators
(a) Addition
(b) Subtraction
(c) Addition and subtraction
Fig. 3-4 Block diagram elements
of typical sensing devices of
control systems.
3-5
Section 3- 1, p. 107
Time & Laplace Domain Block Diagrams
• Example 3-1-1:
3-6
Section 3- 1, p. 108
Example 3-1-2
3-7
Section 3- 1, p. 109
Linear Feedback Control System
reference input
(command)
actuating signal
output
(controlled variable)
feedback signal
Y (s)  G(s) R(s)  G(s) H (s)Y (s)
Negative feedback:
Positive feedback:
3-8
Section 3- 1, p. 110
Relation between Mathematical Equations &
Block Diagram
• The second-order prototype system:
Figure 3-9
Figure 3-10
3-9
Section 3- 1, p. 111
Relation between Mathematical Equations &
Block Diagram (cont.)
3-10
Section 3- 1, p. 112
Example 3-1-3
Find the transfer function
of the system in Fig. 3-12.
Sol:
3-11
Section 3- 1, p. 112
Example 3-1-3 (cont.)
3-12
Section 3- 1, p. 112
Example 3-1-4
Find V(s)/U(s) = ?
Sol:
1
X ( s)  V ( s)
s
3-13
Section 3- 1, p. 113
Block Diagram Reduction:
Moving a Branch Point
• Moving a branch point from P to Q:
3-14
Section 3- 1, p. 114
Block Diagram Reduction:
Moving a Comparator
3-15
Section 3- 1, p. 114
Example 3-1-5
3-16
Section 3- 1, p. 115
Example 3-1-5 (cont.)
G1G2 H1
Y (s)
R(s)
3-17
Section 3- 1, p. 116
Block Diagram of Multi-input Systems
Special Case: Systems with a Disturbance
Disturbance
Reference input
• Super Position:
•
3-18
Section 3- 1, p. 117
Super Position
3-19
Section 3- 1, p. 117
Multivariable Systems
Block diagram:
Feedback control system:
3-20
Section 3- 1, p. 118
Example 3-1-6
Sol:
3-21
Section 3- 2, p. 119
3-2 Signal-Flow Graphs (SFGs)
• Input-output (cause-and effect) relations:
branch gain
• Basic elements of an SFG:
output
input
A signal can transmit through a branch only in the direction of the arrow.


3-22
Section 3- 2, p. 120
Example 2-3-1
Construction of an SFG:
step-by-step
3-23
Section 3- 2, p. 121
Definition of SFG Terms (1/3)
Input node (Source): only outgoing branches, e.g.,  in Fig. 3-26
Output node (Sink): only incoming branches, e.g.,  in Fig. 3-26
• We can make any noninput node of an SFG an output.
• We cannot convert a noninput node into an input node.
3-24
Section 3- 2, p. 122
Definition of SFG Terms (2/3)
• Path: any connection of a continuous succession branches
traversed in the same direction, e.g., y1– y2– y3 or y2– y3–y2.
• Forward Path: a path that starts at an input node and ends at an
output node and along which no node is traversed more than once.
• Path Gain: the product of the branch gains encountered in
traversing a path,
e.g., path = y1– y2– y3 – y4  path gain = a12a23a34
• Forward-Path Gain: the path gain for a forward gain.
3-25
Section 3- 2, p. 122
Definition of SFG Terms (3/3)
• Loop: a path that originates and terminates on the same node and
along which no other node is encountered more than once.
• Loop Gain: the path gain of a loop.
• Nontouching Loops: they do not share a common node.
Nontouching Loops
3-26
Section 3- 2, p. 123
SFG Algebra
•
the sum of all signals entering the node
•
transmit through all branches
leaving the node
• Parallel branches:
3-27
Section 3- 2, p. 124
SFG Algebra & Feedback Control
• Series connection:
• SFG of a Feedback Control System:
U (s)
3-28
Section 3- 2, p. 125
Relation between Block Diagram & SFGs
3-29
Section 3- 2, p. 125
Gain Formula for SFG
• Mason’s gain formula:
Lmr= gain product of the mth possible combination of r nontouching loops
 = 1  (sum of the gains of all individual loops) + (sum of products of gains of
all possible combinations of two nontouching loops)  (sum of products of
gains of all possible combinations of three nontouching loops) +  +  …
k = the  for that part of the SFG that is nontouching with the kth forward path
3-30
Section 3- 2, p. 126
Example 3-2-2
• Y(s) / R(s) = ?
3-31
Section 3- 2, p. 126
Example 3-2-3
y5 / y1 = ?
• Three forward paths:
• Four loops:
• One pair of nontouching loops:
•
3-32
Section 3- 2, p. 126
Example 3-2-3 (cont.)
3-33
Section 3- 2, p. 127
Example 3-2-4
3-34
Section 3- 2, p. 128
Gain Formula between Output Node and
Noninput Nodes & Example 3-2-5
Example 3-2-5:
3-35
Section 3- 2, p. 128
Application of Gain Formula
to Block Diagram
3-36
Section 3- 2, p. 129
Example 3-2-6
3-37
Section 3- 2, p. 129
Simplified Gain Formula
• All loops and forward paths are touching:
• Example:
3-38