CONTROL SYSTEM MODELING

6. INTRODUCTION TO
CONTROL SYSTEM

7. INPUT
 The stimulus or excitation applied to a control system
from an external source in order to produce the
output is called input
Input

8. OUTPUT
 The actual response obtained from a system is
called output.
Output
Input

9. “SYSTEM”
 A system is an arrangement of or a combination of
different physical components connected or related in
such a manner so as to form an entire unit to attain a
certain objective.
SYSTEM
Input Output

10. CONTROL
 It means to regulate, direct or command a system so
that the desired objective is attained

11. Combining above definitions
System + Control = Control System

12. CONTROL SYSTEM
 It is an arrangement of different physical elements
connected in such a manner so as to regulate, direct
or command itself to achieve a certain objective.
CONTROL
SYSTEM
Input Output

13. DIFFERENCE BETWEEN SYSTEM AND CONTROL
SYSTEM
System
Input Control
System
Input Desired
Output
Proper
Output
(May or may not
be desired)
13

14. DIFFERENCE BETWEEN SYSTEM AND CONTROL
SYSTEM
An example : Fan
Fan
(System)
230V/50Hz
AC Supply
Air Flow
Input Output
14

15. A FAN:CAN BE A SYSTEM
230V/50Hz
AC Supply
Airflow
(Proper Output)
 A Fan without regulator can be a “SYSTEM” Because it can
provide a proper output i.e. airflow
 But it cannot be a “Control System” Because it cannot
provide desired output i.e. controlled airflow
Input Output
15

16. A FAN:CAN BE A CONTROL SYSTEM
 A Fan with blades and with regulator can be a “CONTROL
SYSTEM” Because it can provide a Desired output.
i.e. Controlled airflow
230V/50Hz
AC Supply
Controlled Airflow
(Desired Output)
Input Output
Control
Element
16

17. UNIT 1-CONTROL SYSTEM MODELLING
Open loop, Closed loop,
Feedback, Feed forward control theory
Electrical and Mechanical Transfer function
models
Block diagram models
Signal Flow Graph models.

18. Classification of Control System
(Depending on control action)
Open Loop Control
System
Closed Loop Control
System
CLASSIFICATION OF CONTROL SYSTEM
18

19. OPEN LOOP CONTROL SYSTEM
Definition:
“A system in which the control action is totally
independent of the output of the system is called as Open
loop system”
Fig. Block Diagram of Open loop Control System
Controller Process
Reference I/p
r(t) u(t)
Controlled
o/p
c(t)
19

20. OLCSEXAMPLES
 Electric hand drier – Hot
air (output) comes out as
long as you keep your
hand under the machine,
irrespective of how much
your hand is dried.
20

21.  Light switch – lamps glow whenever light switch is on
irrespective of light is required or not.
adjusted
 Volume on stereo system – Volume is
manually irrespective of output volume level.
OLCS: EXAMPLES
21

22. ADVANTAGES OF OLCS
 Simple in construction and design.
 Economical.
 Easy to maintain.
 Generally stable.
 Convenient to use as output is difficult to measure.
22

23. DISADVANTAGES OF OLCS
 They are inaccurate
output cannot be corrected
 They are unreliable
 Any change in
automatically.
23

24. CLOSED LOOP SYSTEM
Definition:
“A system in which the control action is somehow
dependent on the output is called as "Closed loop
system”
24

25. BLOCK DIAGRAM OF CLCS
Reference
Transducer
Controller Plant
Feedback
Transducer
Command
I/p
Reference
I/p
Feedback
Signal
Manipulated
Signal
Error
Signal
Controlled
O/p
r(t)
e(t)
b(t) c(t)
c(t)
m(t)
Forward Path
Feedback Path
25

26. CLCS EXAMPLES
 Automatic Electric Iron- Heating elements are
controlled by output temperature of the iron.
26

27. ADVANTAGES OF CLCS
 Closed loop control systems are more accurate even in the
presence of non-linearity.
 Highly accurate as any error arising is corrected due to
presence of feedback signal.
 Bandwidth range is large.
 Facilitates automation.
 The sensitivity of system may be made small to make
system more stable.
 This system is less affected by noise.
27

28. DISADVANTAGES OF CLCS
 They are costlier.
 They are complicated to design.
 Required more maintenance.
 Feedback leads to oscillatory response.
 Overall gain is reduced due to presence of feedback.
 Stability is the major problem and more care is needed
to design a stable closed loop system.
28

29. DIFFERENCE BETWEEN OLCS & CLCS
2. They consume less 2. They consume more
power. power.
systems are
3. The OL
easier to construct
because of less number
of components required.
4. The open loop systems
are inaccurate &
unreliable
Open Loop Control System Closed Loop Control System
1. The open loop systems 1. The
are simple & economical.
closed loop systems
are complex and costlier
3. The CL systems are not
easy to construct because
of more number of
components required.
4. The closed loop
are accurate &
systems
more
reliable.
29

30. is not a major
in OL control
5. Stability
problem
systems. Generally OL
systems are stable.
6. Small bandwidth.
7.Feedback element is
absent.
8.Output measurement is
not necessary.
5. Stability is a major problem
in closed loop systems & more
care is needed to design a
stable closed loop system.
6. Large bandwidth.
7.Feedback element is
present.
8.Output measurement is
necessary.
Open Loop Control System Closed Loop Control System
30
DIFFERENCE BETWEEN OLCS & CLCS

31. 9. The changes in the output due
to external disturbances are not
corrected automatically. So they
are more sensitive to noise and
other disturbances.
10. Examples:
Coffee Maker,
Automatic Toaster,
Hand Drier.
9.The changes in the output
due to external disturbances
are corrected automatically. So
they are less sensitive to noise
and other disturbances.
10. Examples:
Guided Missile,
Temp control of oven,
Servo voltage stabilizer.
DIFFERENCE BETWEEN OLCS & CLCS
31
Open Loop Control System Closed Loop Control System

40. ELECTRICAL AND MECHANICAL
TRANSFER FUNCTION MODELS

41. MECHANICAL SYSTEM-TRANSFER
FUNCTION
1.TRANSLATIONAL SYSTEM
2. ROTATIONAL SYSTEM

86.  If the system is simple & has limited parameters then it
is easy to analyze such systems using the methods
discussed earlier i.e. transfer function, if the system is
complicated and also have number of parameters then
it is very difficult to analyze it.
NEED OF BLOCK DIAGRAM ALGEBRA

87. block diagram
 It is a simple way to represent any practically
complicated system. In this each component of the
system is represented by a separate block known as
functional block.
 These blocks are interconnected in a proper sequence.
Need of Block Diagram Algebra
 To overcome this problem
representation method is used.

88.  Block Diagram: It is shorthand, pictorial
representation
 of the cause and effect relationship between input
and output of a physical system.
BLOCK DIAGRAM FUNDAMENTALS
BLOCK
Input Output

89.  Output: The value of the input is multiplied to the
value of block gain to get the output.
3s
X(s) Y(s)
Output Y(s)= 3s. X(s)
BLOCK DIAGRAM FUNDAMENTALS

90. Output =x+y-z
+
x
-
z
 Summing Point: Two or more signals can be added/
substracted at summing point.
y
+
output
BLOCK DIAGRAM FUNDAMENTALS

91.  Take off Point: The output signal can be applied to two
or more points from a take off point.
Z
Z
Z
Z
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Take off point
Amit Nevase 103
BLOCK DIAGRAM
FUNDAMENTALS

92. G1 G2
H1
-
R(s) + C(s)
 Forward Path: The direction of flow of signal is from input
to output
Forward Path
Feedback Path
Feedback Path: The direction of flow of signal is from
output to input
BLOCK DIAGRAM
FUNDAMENTALS

93. Rule 1: For blocks in cascade
Gain of blocks connected in cascade gets
multiplied with each other.
BLOCK DIAGRAM REDUCTION
TECHNIQUES
G1
R(s) R1(s) G2 C(s)
R1(s)=G1R(s)
C(s) =G2R1(s)
=G1G2R(s)
C(s)= G1G2R(s)
G1G2
R(s) C(s)

94. G1 G2 G3
R(s) C(s)
Find Equivalent
G1G2G3
R(s) C(s)

95. G1 G2 G3
R(s) C(s)
R1(s)
Find Equivalent
G1G2G3
R(s) C(s) G1G2
R(s) C(s)
G3
R1(s)
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96. Rule 2: For blocks in Parallel
Gain of blocks connected in parallel gets added
algebraically.
BLOCK DIAGRAM REDUCTION
TECHNIQUES
C(s)= (G1-G2+G3) R(s)
G1-G2+G3
R(s) C(s)
G1
G2
R(s) C(s)
G3
R1(s)
R3(s)
+
+
R2(s) -
C(s)= R1(s)-R2(s)+R3(s)
= G1R(s)-G2R(s)+G3R(s)
C(s)=(G1-G2+G3) R(s)

97. Rule 3: Eliminate Feedback Loop
BLOCK DIAGRAM REDUCTION
TECHNIQUES
C(s)
G
H
R(s)
+
+
- R(s) C(s)
G
1  G H
B(s)
E(s)
C(s) G
R(s)

1 GH
In General

98. R(s)
C(s)
G
H
B(s)
E(s)
+-
From Shown Figure,
and
E(s)  R(s) B(s)
C(s) G.E(s)
 G[R(s)  B(s)]
 GR(s)GB(s)
B(s) H.C(s)
C(s) G.R(s)G.H.C(s)
C(s)  G.H  GR(s)
C(s){1G.H}G.R(s)

C(s)

G
R(s) 1 GH
But
For Negative Feedback

99. R(s)
C(s)
G
H
B(s)
E(s)
+
+
and
But
For Positive Feedback
From Shown Figure,
E(s)  R(s) B(s)
C(s) G.E(s)
 G[R(s)  B(s)]
 GR(s) GB(s)
B(s) H.C(s)
C(s) G.R(s)G.H.C(s)
C(s)G.H  GR(s)
C(s){1G.H}G.R(s)

C(s)

G
R(s) 1GH
111

100. Rule 4: Associative Law for Summing Points
The order of summing points can be changed if two or more
summing points are in series
BLOCK DIAGRAM REDUCTION
TECHNIQUES
C(s)
B2
R(s) + X +
-
C(s)
B1
R(s) + X +
-
B1
X=R(s)-B1
C(s)=X-B2
C(s)=R(s)-B1-B2
B2
X=R(s)-B2
C(s)=X-B1
C(s)=R(s)-B2-B1

101. Rule 5: Shift summing point before block
BLOCK DIAGRAM REDUCTION
TECHNIQUES
R(s) C(s)
X
+
G
C(s)=R(s)G+X
C(s)=G{R(s)+X/G}
=GR(s)+X
+
C(s)
R(s) +
G
1/G
X
+

102. Rule 6: Shift summing point after block
BLOCK DIAGRAM REDUCTION
TECHNIQUES
C(s)
R(s) +
G
X
C(s)=G{R(s)+X}
=GR(s)+GX
C(s)=GR(s)+XG
=GR(s)+XG
+
R(s) C(s)
X
+
G
G
+

103. Block Diagram Reduction Techniques
Rule 7: Shift a take off point before block
R(s) C(s)
G
C(s)=GR(s)
and
X=C(s)=GR(s)
C(s)=GR(s)
and
X=GR(s)
X
R(s) C(s)
X
G
G

104. Rule 8: Shift a take off point after block
BLOCK DIAGRAM REDUCTION
TECHNIQUES
R(s) C(s)
X
C(s)=GR(s)
and
X=C(s).{1/G}
=GR(s).{1/G}
= R(s)
G R(s) C(s)
G
X
1/G
C(s)=GR(s)
and
X=R(s)
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105.  While solving block diagram for getting single block
equivalent, the said rules need to be applied. After
each simplification a decision needs to be taken. For
each decision we suggest preferences as
BLOCK DIAGRAM REDUCTION
TECHNIQUES

106. BLOCK DIAGRAM REDUCTION
TECHNIQUES
First Choice
First Preference: Rule 1 (For series)
Second Preference: Rule 2 (For parallel)
Third Preference: Rule 3 (For FB loop)

107. Block Diagram Reduction Techniques
Second Choice
(Equal Preference)
Rule 4 Adjusting summing order
Rule 5/6 Shifting summing point before/after block
Rule7/8 Shifting take off point before/after block

108. EXAMPLE
1
-
+
-
+
+
G1
H1
G2 G3
G4
G5
H2
G6
+
R(s) + C(s)

109. Rule 1 cannot be used as there are no
immediate series blocks.
Hence Rule 2 can be applied to G4, G3, G5 in
parallel to get an equivalent of G3+G4+G5

110. -
+
-
+
+
G1
H1
G2 G3
G4
G5
H2
G6
+
R(s) + C(s)
Apply Rule 2
Blocks in Parallel
EXAMPLE 1 cont….

111. -
+
-
G1
H1
G2
H2
G6
R(s) + C(s)
G3+G4+G5
Apply Rule 1 Blocks in series
EXAMPLE
1 cont….

112. +
-
+
-
G1
H1
H2
G6
R(s) C(s)
G2(G3+G4+G5)
Apply Rule 3 Elimination of feedback loop
EXAMPLE
1 cont….

113. -
H2
G6
R(s) + C(s)
G2(G3+G4+G5)
Apply Rule 1 Blocks in series
G1
1G1H1
EXAMPLE
1 cont….

114. -
H2
G6
R(s) + C(s)
Apply Rule 3 Elimination of feedback loop
G1G2(G3G4G5)
1 G1H1
EXAMPLE
1 cont….

115. G6
R(s)
C(s)
Apply Rule 1 Blocks in series
G1G2(G3G4G5)
1G1H1G1G2H2(G3G4G5)
EXAMPLE
1 cont….

116. R(s) C(s)
G1G2G6(G3G4G5)
1G1H1G1G2H2(G3G4G5)
EXAMPLE
1 cont….

117. G1G2G6(G3G4G5)
1G1H1G1G2H2(G3G4G5)
R (s)
C (s)

EXAMPLE
1 cont….

118. SIGNAL FLOW GRAPH

119. 1
1
OUTLINE
• Introduction to Signal Flow Graphs
 Definitions
 Terminologies
• Signal-Flow Graph Models
• BD to SFG
 Example
• Mason’s Gain Formula
Example

120. 1
2
INTRODUCTION
• Definition:-
“ A signal flow graph is a graphical representation of the
relationship between variables of a set of linear algebraic
equation.”
• A signal-flow graph consists of a network in which nodes
are connected by directed branches.
• It depicts the flow of signals from one point of a system to
another and gives the relationships among the signals.

121. FUNDAMENTALS OF SIGNAL FLOW GRAPHS
• Consider a simple equation below and draw its signal flow graph:
y  ax
• The signal flow graph of the equation is shown below;
x a y
• Every variable in a signal flow graph is designed by a Node.
• Every transmission function in a signal flow graph is designed by a
Branch.
• Branches are always unidirectional.
• The arrow in the branch denotes the direction of the signal flow.
1
2

122. TERMINOLOGIES
• An input node or source contain only the outgoing branches. i.e., X1
• An output node or sink contain only the incoming branches. i.e., X4
• A path is a continuous, unidirectional succession of branches along which no
node is passed more than ones. i.e.,
• A forward path is a path from the input node to the output node. i.e.,
X1 to X2 to X3 to X4 , and X1 to X2 to X4 , are forward paths.
• A feedback path or feedback loop is a path which originates and terminates on
the same node. i.e.; X2 to X3 and back to X2 is a feedback path.
X2 to X3 to X4
X1 to X2 to X3to X4 X1 to X2 to X4
1
2

123. TERMINOLOGIES
1
2
• A self-loop is a feedback loop consisting of a single branch. i.e.; A33 is a self
loop.
• The gain of a branch is the transmission function of that branch.
• The path gain is the product of branch gains encountered in traversing a path.
i.e. the gain of forwards path X1 to X2 to X3 to X4is A21A32A43
• The loop gain is the product of the branch gains of the loop. i.e., the loop gain
of the feedback loop from X2 to X3 and back to X2 is A32A23.
• Two loops, paths, or loop and a path are said to be non-touching if they have
no nodes in common.

124. SFG TERMS REPRESENTATION
input node (source)
b
a
x4
d
1
x3
x3
mixed node
mixed node
forward path x2
path
lo
cop
branch
node
x1
transmittance
input node (source)
1
2

125. SIGNAL-FLOW GRAPH MODELS
2 1 3
x3  fx0 gx2
x4  hx3
x  dx ex
x1ax0 bx1 cx2
b
x4
x3
x1
x0
h
f
x2 g
e
d
c
a
xo is input and x4 is output

126. CONSTRUCT THE SIGNAL FLOW GRAPH FOR THE FOLLOWING SET
OF SIMULTANEOUS EQUATIONS.
• There are four variables in the equations (i.e., x1,x2,x3,and x4) therefore four nodes are
required to construct the signal flow graph.
• Arrange these four nodes from left to right and connect them with the associated
branches.
• Another way to arrange this
graph is shown in the figure.

127. BD TO SFG
Block Diagram Signal Flow Graph

128. BD TO SFG
Block Diagram
Block
Diagram
Signal Flow Graph

129. EXAMPLE:-

130. MASON’S RULE (MASON, 1953)
• The block diagram reduction technique requires successive
application of fundamental relationships in order to arrive at the
system transfer function.
• On the other hand, Mason’s rule for reducing a signal-flow graph
to a single transfer function requires the application of one
formula.
• The formula was derived by S. J. Mason when he related the
signal-flow graph to the simultaneous equations that can be
written from the graph.

131. MASON’S RULE:
Where,
n = number of forward paths.
Pi = the i th forward-path gain.
∆ = Determinant of the system
∆i = Determinant of the ith forwardpath
• ∆ is called the signal flow graph determinant or characteristic function. Since
∆=0 is the system characteristic equation.

• The transfer function T, of a system represented by a signal-flow graph is;
n
Pii
 i1
R(s)
T  C(s)

132. MASON’S RULE

133. SYSTEMATIC APPROACH
1. Calculate forward path gain Pi for each forward path i.
2. Calculate all loop transfer functions.
3. Consider non-touching loops 2 at a time.
4. Consider non-touching loops 3 at a time.
5. etc
6. Calculate Δ from steps 2,3,4 and 5
7. Calculate Δi as portion of Δ not touching forward path i

134. EXAMPLE 1:
APPLY MASON’S RULE TO CALCULATE THE TRANSFER
FUNCTION OF THE SYSTEM REPRESENTED BY FOLLOWING
SIGNAL FLOW GRAPH

135. CONTINUED…..
• In this system there is only one forward path between the input R(s) and the
output C(s). The forward path gain is
𝑃1 = 𝐺1𝐺2𝐺3
• we see that there are three individual loops. The gains of these loops are
𝐿1 = 𝐺1𝐺2𝐻1
𝐿2 = −𝐺2𝐺3𝐻2
𝐿3 = −𝐺1𝐺2𝐺3
• Note that since all three loops have a common branch, there are no non-touching
loops. Hence, the determinant ∆ is given by
∆= 1 − (𝐿1+ 𝐿2 + 𝐿3)
= 1 − 𝐺1𝐺2𝐻1 +𝐺2 𝐺3𝐻2 +𝐺1 𝐺2𝐺3
• There is no any non touching loop so we get,
∆𝑙= 1
• Therefore, the overall gain between the input 𝑅𝑠 and the output 𝐶𝑠 or the closed
loop transfer function, is given by
𝐶𝑠
=
𝐺1𝐺2𝐺3
𝑅𝑠 1 − 𝐺1𝐺2𝐻1 +𝐺2 𝐺3𝐻2 +𝐺1 𝐺2𝐺3

136. T 
C(s)

P11P22
R(s) 
Therefore,
L3  G1G3G4 H2
L2  G1G2G4H2 ,
L1  G1G4 H1,
There are three feedbackloops
Example 2 :
Apply Mason’s Rule to calculate the transfer function of the system
represented by following Signal Flow Graph

137. There are no non-touching loops, therefore
∆ = 1- (sum of all individual loop gains)
  1 L1  L2  L3 
  1 G1G4H1  G1G2G4H2  G1G3G4H2 
Example2:
Apply Mason’s Rule to calculate the transfer function of the system
represented by following Signal Flow Graph

138. Eliminate forward path-1
∆1 = 1- (sum of all individual loopgains)+...
∆1 = 1
Eliminate forward path-2
∆2 = 1- (sum of all individual loopgains)+...
∆2 = 1
EXAMPLE2 :
APPLY MASON’S RULE TO CALCULATE THE TRANSFER
FUNCTION OF THE SYSTEM
represented by following Signal Flow Graph

139. EXAMPLE 2: CONTINUE