Control engineering module 3 part-A

CONTROL ENGINEERING
Course Code 18ME71 CIE Marks 40
Teaching Hours / Week (L:T:P) 3:0:0 SEE Marks 60
Credits 03 Exam Hours 03
[AS PER CHOICE BASED CREDIT SYSTEM (CBCS) SCHEME]
SEMESTER – VII
Dr. Mohammed Imran
B. E. IN MECHANICAL ENGINEERING

Course Objectives
 To develop comprehensive knowledge and understanding of
modern control theory, industrial automation, and systems
analysis.
 To model mechanical, hydraulic, pneumatic and electrical
systems.
 To represent system elements by blocks and its reduction
 To represent system elements by blocks and its reduction
techniques.
 To understand transient and steady state response analysis
of a system.
 To carry out frequency response analysis using polar plot,
Bode plot.
 To analyse a system using root locus plots.
 To study different system compensators and characteristics
of linear systems.
Dr. Mohammed Imran

Course outcomes
On completion of the course the student will be able to
CO1: Identify the type of control and control actions.
CO2: Develop the mathematical model of the physical systems.
CO3: Estimate the response and error in response of first and
second order systems subjected standard input signals.
second order systems subjected standard input signals.
CO4: Represent the complex physical system using block diagram
and signal flow graph and obtain transfer function.
CO5: Analyse a linear feedback control system for stability using
Hurwitz criterion, Routh‟s criterion an root Locus technique in
complex domain.
CO6: Analyse the stability of linear feedback control systems in
frequency domain using polar plots, Nyquist and Bode plots.
Dr. Mohammed Imran

Module-3
 Block diagram algebra,
 Reduction of block diagram,
 Signal flow graphs,
 Signal flow graphs,
 Gain formula for signal flow graphs,
 State diagram from differential
equations.
10 Hours
Dr. Mohammed Imran

Text Books:
 Automatic Control
Systems, Farid G., Kuo
B. C, McGraw Hill
Education, 10th
Edition,2018
Dr. Mohammed Imran
Edition,2018
 Control systems, Manik
D. N, Cengage, 2017

Reference Books:
 Modern control Engineering K. Ogeta Pearson 5th
Edition, 2010
 Control Systems Engineering Norman S Nice Fourth
Edition, 2007
Modern control Systems Richard C Dorf Pearson
Dr. Mohammed Imran
 Modern control Systems Richard C Dorf Pearson
2017
 Control Systems Engineering IjNagrath, M Gopal
New Age International (P) Ltd 2018
 Control Systems Engineering S Palani Tata McGraw
Hill Publishing Co Ltd ISBN-13 9780070671935

MODULE -3
PART-A
PART-A
Dr. Mohammed Imran CONTROL ENGINEERING

Introduction
TRANSFER FUNCTION
 Any physical system is mathematically described by a set of linear differential
equations which are known as system equations.
 These equations are simplified by using Laplace transformations and taking the
system initial conditions to be zero.
 Initial conditions are taken to be zero, as in actual analysis or design of control
system, the system is assumed to be initially at rest i.e., initially they have no
energy stored in their elements.
energy stored in their elements.
 These simplified system equations are manipulated to determine the ratio of
Laplace transform of system output to the Laplace transform of the system input
to investigate the system properties. This ratio is known as transfer function of the
system.
 The complete transfer function of the system is obtained by the determining the
transfer function of various components in the system and then combining them
according to their connections.
 Definition : The transfer function of a linear time invariant system is defined as the
ratio of the Laplace transform of the system output to the Laplace transform of
the system input with all initial condition assumed to be zero.
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Introduction
TRANSFER FUNCTION
 Consider a control system represented in time - domain as shown in figure 1.
 In Laplace domain as shown in figure 2 (s-plane)
Fig.1.
Fig.2.
 If G(s) be the transfer function of system, mathematically
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Fig.2.
is the transfer function of the system.

Introduction
BLOCK DIAGRAM
 A Control system can be simple or complicated, but it consists
of number of components.
 Each components performs some specific functions.
 One of the easy and convenient way of representing the
function of each component is by the unidirectional blocks.
 All the blocks are interconnected by lines with arrows
indicating flow of signals from output of one block to the
input of another.
 Such block diagram gives an overall idea of the inter
relationship that exist among various components.
 Thus, block diagram is a pictorial representation of the given
control system.
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Generally, any block diagram has following five basic elements
associated with it:
Introduction
Basic Elements of Block Diagram
 Block - It is a rectangular box, or a symbol
that explain the mathematical operation on
the input to produce the corresponding
output. The flow of information is
unidirectional.
 Transfer function - The mathematical
function of each block is shown by inserting
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function of each block is shown by inserting
corresponding transfer function of the
element inside the block.
 Summing points - It is a symbol that shows
the algebraic summation of two or more
signals. The plus or minus sign at each arrow
head indicates whether that signal is to be
added or subtracted.
 Take off point - It is a point from which the
signal is taken for the feedback purpose or
distribution to the other blocks.
 Arrows - These are the symbols that
indicates the direction of flow of signal from
one block to the another
Fig. 3. Basic elements of Block diagram

1. Block Diagram Algebra
1.1 Block Diagram Of A Closed Loop System
 A block diagram in which forward path contain only one block,
feedback path contains only one block, one summing point and one take
off point is known as closed loop system or simple or canonical form of
a closed loop system.
 For example, consider the block diagram of a closed loop system in
Laplace domain as shown in figure 4.
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Fig. 4 : Block diagram of a closed loop system
Where,
R(s) = Reference Input
C(s) = Controlled variable (output signal)
B(s) = Feedback signal
E(s) = Actuating signal (error signal)
G(s) = Forward path transfer function
H(s) = Feedback path transfer function

1. Block Diagram Algebra
1.2 Transfer Function Of Closed Loop System
 The mathematical function relating C(s) to R(s) is called the transfer function of
closed loop system.
 In figure (4), C(s) - controlled variable (output signal) and R(s) - reference input
are related as follows
--(1)
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is the Transfer function of closed loop system
Where
Positive (+) sign is for negative feedback
Negative (-) sign is for positive feedback.
The closed loop T.F. can be represented as in figure 5
Fig. 5
Eq(1)
Fig. 4

2. Reduction of block diagram
 The transfer function of a block diagram can be obtained easily when it is
represented in its simple closed loop form in which forward path consists of
one block, feedback path consists of one block, one summing point and one
take off point.
 But, block diagram of a control system, generally consists number of blocks,
interconnected to each other.
 These are brought to the simple closed loop form by the reduction of block
 These are brought to the simple closed loop form by the reduction of block
diagram, this is achieved by the use of the block diagram reduction rules are
as follows .
Rule 1: Combining blocks in cascade (series)
Rule 2: Combining blocks in parallel
Rule 3: Moving a summing point after a block
Rule 4: Moving a summing point before a block
Rule 6: Moving take off point before a block
Rule 5: Moving take off point after a block
Rule 7: Eliminating a feedback loop
Rule 8: Rearranging summing points
Rule 9: Splitting a summing point
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Rule 1: Combining blocks in cascade (series)
The transfer function of the blocks which are in cascade get multiplied with each
other
Hint: If there is a summing point or take off point in between the blocks, the
blocks cannot be said to be in cascade (series).
Rule 2: Combining blocks in parallel
The transfer function of blocks which are in parallel get added algebraicall
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Rule 3: Moving a summing point after a block
Hint: The output must remain same, while moving the summing point
Rule 4: Moving a summing point before a block
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Rule 5: Moving take off point after a block
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Hint: After moving take off, value of the signal taken off must remain same

Rule 7: Eliminating a feedback loop
Rule 8: Rearranging summing points
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Rule 9: Splitting a summing point
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General Procedure
The general steps involved in the reduction of block
diagram are
1. Combine all the blocks in cascade
2. Combine all the blocks in parallel
3. Eliminate all minor feedback loops
4. Shift Summing points and take off points if necessary
5. Repeat steps 1to 4 until canonical form has obtained
6. Using standard transfer function of simple closed loop
system, obtain the closed loop T.F C(s)/R(s) of overall system.
Hint: As for as possible try to shift take off points towards right
and summing points to the left.
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Problems
1. Reduce the block diagram shown is figure Q(1) to its
simplest possible form and find its closed loop transfer
function.
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Problem-1
Solution:
Define the names to start the reduction process, to its simplest possible form and
find its closed loop transfer function =?.
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Step 1: Move the take off point (4) after the block G2

2. Reduction of block diagram: Problem-1
Step 2: Move the summing point (3) before the block GI
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Step 3: Inter change the summing point 2 and 3.
Combine the
parallel blocks
G4 /G2 and G3
combine the
cascade blocks
1/G1 and H2
Also G1 and G3
Separate the
path along the
dotted line

Step 4: Eliminate minor feedback and
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Combine the parallel blocks of 1 and

Combine the parallel blocks of 1 and
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Step 5 : Combine the cascade blocks

Step 5 : Combine the cascade blocks
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2. Reduce the block diagram shown is figure Q(1) to its
simplest possible form and find its closed loop transfer
function.
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Solution
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Solution
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Step 6: Eliminate the minor feedback loop
Thus,

Solution
Overall Transfer functions
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4 . Reduce given block diagram into canonical from and
determine closed loop transfer function. Also represent in open
loop from
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MODULE -3
PART-B
PART-B
Dr. Mohammed Imran CONTROL ENGINEERING

3. SIGNAL FLOW GRAPHS (SFG)
 A signal flow graph is an another way of representing a linear control systems, when
any control system is described by a set of linear algebraic equations having the
form
 The signal flow graph (SFG) was developed by S.J. Mason for the control
system described by an algebraic equations to represent the cause and
system described by an algebraic equations to represent the cause and
effect relationship among the various variables of the system, Thus signal flow
graph can be defined as :
 The diagramatic or graphical representation of the variables of a set of linear
algebraic equations representing the linear control system is called signal flow
graph (SFG
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3.1 BASIC Elements of a SFG
 In SFG, variable plays a vital role, these are represented first. Thus variables of the
set of equations representing the control system are considered as the first basic
element and are represented by the small circles called nodes in SFG.
 The control system consists of number of variables, which are both dependent and
independent in nature.
 Each (dependent and independent) variables are presented by separated node.
 The nodes are connected by directed line segments called branches, according
to the cause and effect equations.
 The branch is associated with the transfer function and an arrow.
 The branch is associated with the transfer function and an arrow.
 The transfer function represents (branch gain) mathematical operation on one
variable to produce the other variable.
 The direction of the arrow indicates the flow of signal.
 For instance, consider that a linear system is represented by the simple algebraic
equation.
 where X1, is the input, X2 is the output and a12 is the gain between the two
variables.
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 In addition to the nodes and branches defined earlier, the following terms are useful
for the analysis of SEG.
 Consider a signal flow graph shown in the figure
 Node : Nodes are the variables of the system represented by small circles.
 Input Node : The node that has only outgoing branches is known, as Input or source
node. Example : x1 is Input node.
 Output Node: The node that has only incoming branches is known as output or sink
node. Example : x7 in output node.
 Mixed Node : The node that having both incoming and outgoing branches is known
as mixed or chain node. Example x2, x3, x4, x5 and x6.
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 Branch : Directed line segment joining two nodes is known as branch.
 Path : It is a traversal from one node to another node in the direction of the branch
arrow, such that no node traverse more than once.
 Branch gain : The gain between nodes is known as branch gain or transmittance. such
gains are expressed in terms of transfer functions.
Cont……
gains are expressed in terms of transfer functions.
 Forward path : The path that starts from an input node and ends at an output node
and along which no node is traversed more than once is known as forward, path.
 Path Gain : The product of the branch gains encountered while going through the a
forward path is known as path gain or forward path gain.
Example : Consider a forward path
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 Path Gain : The product of the branch gains encountered while going through the a forward
path is known as path gain or forward path gain.
Example : Consider a forward path its path gain
is
Cont……
 Feedback loop : A path which starts from a particular node and ends at the same node.
travelling through at least one other node, and along which no node is traversed more than
once is known as feedback loop or closed loop
 Self loop : A path which starts from a particular node and ends at the same node.
Example : x3 – x3
Hint : A self loop should not be considered while defining the forward path.
 Non - touching loop : If there is no node common between the two or more loops, such loops
are said to be non - touching loops.
 Loop Gain : The product of all the gains of the branches Conning loop is known as loop gain.
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1. The SFG can be applied only to linear systems.
2. The equations for which SFG is drawn must be a linear algebraic
equations in the form of cause and effect.
3. Nodes are used to represent variables and line connecting
between nodes in represented by branches.
4. The signals along the branches represented by arrow.
5. The signal traveling along branch is multiplied by the gain of the
branch.
3.1.1 PROPERTES OF SFG
Cont……
branch.
6. The branch indicates the functional dependence of one signal on
another.
7. At each node all the signals of in coming branches added and
transmitted to the out going branches.
Hint : The value of variable at each node is an algebraic sum of all
the signals of incoming branches.
8. A mixed node may be treated as an output node by adding an
outgoing branch with unit branch gain.
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3.1.1 PROCEDURE or DRAW SFG
Case I : From block diagram Cont……
1. Replace all the summing points by nodes.
2. Replace all the take off points by nodes.
3. Replace all the blocks by branches, indicating block.
4. If the branch connecting a summing point and take off point has unity gain, then
summing and take off point can be combined and represented by a single node
5. If there are wan take off points from the same signal then all take off points can be
combined and represented by a single node
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3.1.1 PROCEDURE or DRAW SFG
Cont……
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Mason’s Gain Formula
Mason's gain formula is used for the determination of overall transfer function (Input
- output relationship) of a system.
It is a convenient and easy way of finding the relation between the input and output
variables of a system.
The number of steps involved in the block diagram reduction are more and it is a
time consuming procedure and also the task of solving for the input-output
relationship by algebraic manipulation could be quite tedious.
An advantage of Mason's gain formula is that the system transfer functions are
readily obtained without manipulation of the graph.
readily obtained without manipulation of the graph.
Mason's gain formula is given by
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4. Steps involved in SFG
Mason’s formula for T.F
Step 1: Identify the number of forward path and their gain (Forward path gains
are P1, P2, P3, P4 ,P5 , P6 , ……) (K=nth Ex: 6 Forward path P6 , then K=6)
Step 2: Identify the individual loops and their loop gain (loop gains are L1, L2, L3,
L4 ,L5 , L6 , …… Lm)
Step 3: find the combination of non touching loops
Ex: May be L1L2 or L1L6 or L2L3 or L4L5 or any of them identified
Step 4: Find the value of determinant 
 = Determinant of the graph
 = 1- (sum of individual loop gain) + (sum of gain products of all
 = 1- (sum of individual loop gain) + (sum of gain products of all
combinations of two non touching loops) - (sum of gain products of
all Combinations of three non touching loops) +…….
 = 1-(L1 + L2 + L3+L4 + L5 + L6 +…)+( L1L2 + L1L6 +..)
Step 5: Find the value of K
K = 1 , 2 , 3 , 4 , …… k
1 = 1+ non touching loops, =1+Lm ; 2 = =1+Lm …….
Step 6: From Mason’s Gain Formula T
Step 7: Overall transfer function C/R = T
…….
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Problems on Signal flow graphs
Problem -1
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Problem -2 Using signal flow graph and Mason's gain formula, obtain the overall
transfer function of the system depicted in figure
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Problem -3 Using signal flow graph and Mason's gain formula, obtain the overall
transfer function of the system depicted in figure
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Problem -4
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Control engineering module 3 part-A

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