Control Systems

CONTROL SYSTEMS
Miss. Poonam A. Desai,
Assistant Professor
Electrical Engineering Department
1

CONTENTS
INTRODUCTION TO CONTROL SYSTEMS &
ITS MODELLING
2

Control Systems
● Control means to regulate or to direct. Thus, a control system is
the interconnection of various physical elements connected in a way to
regulate or direct itself or the other system.
● A Control System is a system in which the output is controlled by varying
the input.
● The first control system device was James Watt's Flyball governor, which
was invented in 1767. The aim of inventing Flyball governor was to keep
the speed of the engine constant by regulating the supply of the steam to
the engine. 3

● A system can be defined as a mathematical relationship between the input, output and the
states of a system.
● In control theory, a system is represented a a rectangle with an input and output.
● In a control system, the behavior of the system is described by the differential
equations. The differential equations can be either ordinary differential equations
or the difference equations.
4

System
A system is an arrangement of different components that act together as a collective unit to perform a
certain task. The physical system is defined as the collective action of physical objects in a system to
perform an objective, for example a classroom. The classroom with benches, tables, fans, lights, blackboard,
etc., collectively forms a physical system.
A kite is made up of sticks and paper, and the water in a lake is also a type of physical system.
5

Basic Elements of Control System
● Plant/Process
● Input
● Output
● Controller
● Disturbances
6

Plant
The portion of the system that is to be controlled or regulated is known as plant of the process. In a control
system, it is often referred to a transfer function, which specifies the relationship between the input and output of
the system without feedback. It means that plant can be anything that has an input and provides the output.
The plant can have one or more output and inputs. The sensors are used to measure the plant's output, while
actuators drive the plants inputs.
The diagram of the plant is shown below
The input variable in a system is generally called reference input and the output is called as the controller output.
7

Controller
The controller is the element of the system. It can also lie external to the system. The function of the
controller is to control the plant or process. Every system accepts an input and defines the output after
analyzing the nature of the input. The controller in the control system is a mechanism that reduces the
difference between the actual value and desired value of the system. Here, the actual value signifies the real
value, while desired value is the set-point or target value.
Input
It is a signal from the external energy source applied to the control system to produce the desired output.
Or
It is the desired action that is capable of producing any response in a system.
8

Output
It is an actual response of the applied input signal from the control system. The inputs are excited into the
system and the outputs are the processed results of that inputs. The outputs are the results of either a small
part of the process or the entire process.
Disturbances
Disturbances are a type of signal which adversely affects the output value of the control system. The
disturbances can be internal or external. The internal disturbances that arise in the system itself and the
external disturbances are generated outside the system. Such disturbances act as an extra input to the system
and the normal input and further affect the system's output.
9

Requirements of a control system
1. Stability
2. Accuracy
3. Response
The basic concepts of a control system are:
1. To minimize the error
2. To minimize the time-response
The lesser the error between the actual value and the desired value, the better the system's response will be. It is because no system
wants any error in between. The minimum time response of the system helps to load changes in the system.
10

Terminologies of the control system
The terminologies of the control system are categorized as:
● Automatic control system
● Manual control system
● Linear control system
● Nonlinear control system
● Time-variant control system
● Time-invariant control system
● Open loop control system
● Closed loop control system
11

Other Two types of control system in detail:
1. Open loop control system.
2. Closed loop control system.
12

Open loop control system
An open-loop control system is a system in which the control action is independent of the desired output
signal. In this system, the output signal is not compared with the input signal which means there is no
feedback signal in this system. The open-loop control system is also known as a non-feedback control system
or control system without feedback.
13

Examples
1. Automatic washing machine- In this system, the operating time is set manually. After the completion
of the set time, the machine stops whether the desired cleanness is obtained or not because there is no
feedback signal which the machine can sense.
2. Immersion rod- In this system, the rod heats up the water but how much hot water is required that
cannot be sensed by the rod.
14

Advantages of an open-loop control system
1. Open loop systems are simple.
2. These are economical.
3. Less maintenance is required and is not difficult.
Disadvantages of open loop control system
1. Open loop systems are inaccurate.
2. These systems are not reliable.
3. These are slow.
4. Optimization is not possible.
15

Closed loop control system
A closed loop control system is a system in which control action is dependent on the desired output. In this
system, the output signal is compared with the reference input signal, and an error signal is produced then
this error signal is fed to the controller to reduce the error to obtain the desired output.
16

Examples
1. Automatic electric iron.
2. Servo voltage stabilizer.
3. An air conditioner.
17

Advantages of closed-loop systems
1. These systems are more reliable.
2. Closed-loop systems are faster.
3. Many variables can be handled simultaneously.
4. Optimization is possible.
Disadvantages of closed-loop systems
1. Closed-loop systems are expensive.
2. Maintenance is difficult.
3. Installation is difficult for these systems.
18

Open Loop system Closed Loop system
These are easier to build. These are difficult to build.
These systems are not reliable. These systems are reliable.
These systems are slow. These systems are faster.
These systems are generally more stable. These systems are less stable.
Optimization is not possible. Optimization is possible.
Examples - Hand dryer, washing machine. Servo voltage stabilizer, air conditioner.
19

Revision: Laplace Transform of Derivatives
Let’s recall Laplace transform for a function, first order derivative and second order derivative. For
a given continuous and differentiable function f(t), the following Laplace transforms properties
applies:
Finding the transfer function of a systems basically means to apply the Laplace transform to the set
of differential equations defining the system and to solve the algebraic equation for Y(s)/U(s).
20

Mathematical Modelling of Control System
The process of drawing the block diagram for mechanical and electrical systems to ﬁnd the performance
and the transfer functions is called the mathematical modeling of the control system.
The control systems can be represented with a set of mathematical equations known as mathematical
model. These models are useful for analysis and design of control systems.
Analysis of control system means finding the output when we know the input and mathematical model.
Design of control system means finding the mathematical model when we know the input and the output.
The following mathematical models are mostly used.
● Differential equation model
● Transfer function model
● State space model
21

Differential Equation Model
Differential equation model is a time domain mathematical model of control systems. Follow these
steps for differential equation model.
● Apply basic laws to the given control system.
● Get the differential equation in terms of input and output by eliminating the intermediate
variable(s).
22

Example 1: Electrical System
Consider the following electrical system as shown in the following figure. This circuit consists of
resistor, inductor and capacitor. All these electrical elements are connected in series. The input
voltage applied to this circuit is vi and the voltage across the capacitor is the output voltage Vo
23

The above equation is a second order differential equation.
24

Differential Equation Modeling of Mechanical Systems:
There are two types of mechanical systems based on the type of motion.
● Translational (Linear) mechanical systems
● Rotational mechanical systems
Translational or Linear system: The motion that takes place along a straight line is called a
translational motion. There are three different types of forces that we have to study.
Rotational System:When the motion of a body takes place about a ﬁxed axis, this type of motion is
known as rotational motion. There are three types of torques that resist the rotational motion.
25

Modeling of Translational Mechanical Systems
Translational mechanical systems move along a straight line. These systems mainly consist of
three basic elements. Those are mass, spring and dashpot or damper.
If a force is applied to a translational mechanical system, then it is opposed by opposing forces
due to mass, elasticity and friction of the system. Since the applied force and the opposing forces
are in opposite directions, the algebraic sum of the forces acting on the system is zero.
Let us now see the force opposed by these three elements individually.
1. Mass (Inertia Force)
2. Spring (Spring Force)
3. Dashpot (Damping Force)
26

Mass (Inertia Force)
Mass is the property of a body, which stores kinetic energy. If a force is applied on a body having mass M, then it is
opposed by an opposing force due to mass. This opposing force is proportional to the acceleration of the body.
Assume elasticity and friction are negligible.
According to Newton’s Second law of motion
Where,
● F is the applied force
● Fm is the opposing force due to
mass
● M is mass
● a is acceleration
● x is displacement
27

Spring (Spring Force)
Spring is an element, which stores potential energy. If a force is applied on spring K, then it is opposed by an opposing
force due to elasticity of spring. This opposing force is proportional to the displacement of the spring. Assume mass and
friction are negligible.The restoring force of a spring is proportional to the displacement.
Where,
● F is the applied force
● Fk is the opposing force due to
elasticity of spring
● K is spring constant of stiffness
Unit of K = N/m
28

Dashpot (Damping Force)
If a force is applied on dashpot B, then it is opposed by an opposing force due to friction of the dashpot. This opposing force is proportional to the
velocity of the body. Assume mass and elasticity are negligible.
For viscous friction, we assume that the damping force is proportional to the velocity.
Where,
● Fb is the opposing force due to friction of dashpot
● B is the frictional coefficient (Damping Coeﬃcient)
Unit of B = N/m/sec.
● v is velocity
29

Modeling of Rotational Mechanical Systems
Rotational mechanical systems move about a fixed axis. These systems mainly consist of three
basic elements. Those are moment of inertia, torsional spring and dashpot.
If a torque is applied to a rotational mechanical system, then it is opposed by opposing torques
due to moment of inertia, elasticity and friction of the system. Since the applied torque and the
opposing torques are in opposite directions, the algebraic sum of torques acting on the system is
zero.
Let us now see the torque opposed by these three elements individually.
1. Moment of Inertia (Inertia Torque)
2. Torsional Spring (Spring Torque)
3. Dashpot (Damping Torque)
30

Moment of Inertia (Inertia Torque)
In translational mechanical system, mass stores kinetic energy. Similarly, in rotational mechanical system, moment of
inertia stores kinetic energy.
The property of an element that stores the kinetic energy of rotational motion is called inertia (J). The inertia Torque T1
is
the product of the moment of inertia J and angular acceleration α (t).
If a torque is applied on a body having moment of inertia J, then it is opposed by an opposing torque due to the moment of
inertia. This opposing torque is proportional to angular acceleration of the body.
Assume elasticity and friction are negligible.
Where,
● T is the applied torque
Unit of T = N-m
● Tj is the opposing torque due to
moment of inertia
● J is moment of inertia
● α is angular acceleration
● θ is angular displacement
31

Where,
● T is the applied torque
● Tk is the opposing torque due to elasticity of torsional
spring
● K is the torsional spring constant
Unit of K = N-m/rad.
● θ is angular displacement
Torsional Spring (Spring Torque)
In translational mechanical system, spring stores potential energy. Similarly, in rotational mechanical system, torsional
spring stores potential energy. Spring Torque T is the product of torsional stiffness and angular displacement.
If a torque is applied on torsional spring K, then it is opposed by an opposing torque due to the elasticity of torsional spring.
This opposing torque is proportional to the angular displacement of the torsional spring. Assume that the moment of inertia
and friction are negligible.
32

Dashpot (Damping Torque)
If a torque is applied on dashpot B, then it is opposed by an opposing torque due to the rotational friction of the dashpot.
This opposing torque is proportional to the angular velocity of the body.
Assume the moment of inertia and elasticity are negligible.
The product of angular velocity ω and damping coeﬃcient B is known as Damping Torque
Where,
● Tb is the opposing torque due to the
rotational friction of the dashpot
● B is the rotational friction coefficient
● ω is the angular velocity
● θ is the angular displacement
33

The simplest representation of a system is through Ordinary Differential Equation (ODE). When
dealing with ordinary differential equations, the dependent variables are function of a positive real
variable t (often time). By applying Laplace’s transform we switch from a function of time to a
function of a complex variable s (frequency) and the differential equation becomes an algebraic
equation.
The transfer function defines the relation between the output and the input of a dynamic system,
written in complex form (s variable).
For a dynamic system with an input u(t) and an output y(t), the transfer function H(s) is the ratio
between the complex representation (s variable) of the output Y(s) and input U(s).
34

TRANSFER FUNCTION
To find the transfer function, first take the Laplace Transform of the differential equation (with zero
initial conditions). Recall that differentiation in the time domain is equivalent to multiplication by "s"
in the Laplace domain.
The transfer function of a system is deﬁned as the ratio of Laplace transform of output to the Laplace
transform of input where all the initial conditions are zero is often called H(s).
35

Where,
1. T(S) = Transfer function of the system.
2. C(S) = output.
3. R(S) = Reference output.
4. G(S) = Gain.
36

Steps to obtain transfer function -
Step-1 Write the differential equation.
Step-2 Find out Laplace transform of the equation assuming 'zero' as an initial condition.
Step-3 Take the ratio of output to input.
Step-4 Write down the equation of G(S) as follows - C(S)/R(s)
37

Transfer Function Model
Transfer function model is an s-domain mathematical model of control systems.
The Transfer function of a Linear Time Invariant (LTI) system is defined as the ratio of Laplace
transform of output and Laplace transform of input by assuming all the initial conditions are zero.
38

The transfer function model of an LTI system is shown in the following figure.
Here, we represented an LTI system with a block having transfer function inside it. And this block
has an input X(s) & output Y(s)
39

Example 2
Previously, we got the differential equation of an electrical system as
40

Where,
● Vi(s) is the Laplace transform of the input voltage vi
● Vo(s) is the Laplace transform of the output voltage vo
The above equation is a transfer function of the second order electrical system. The transfer function
model of this system is shown below.
Here, we show a second order electrical system with a block having the transfer function inside it. And
this block has an input Vi(s) & an output Vo(s)
41

SOLVED NUMERICALS ON TRANSFER FUNCTION
42

Transfer function example for a mechanical system
Example 1. Find the transfer function for a single translational mass system with spring and damper.
The methodology for finding the equation of motion for this is Mechanical systems modeling using
Newton’s and D’Alembert equations.
43

The ordinary differential equation describing the dynamics of the system is:
where:
m [kg] – mass
k [N/m] – spring constant (stiffness)
c [Ns/m] – damping coefficient
F [N] – external force acting on the body (input)
x [m] – displacement of the body (output)
44

The input of the system is the external force F(t) and the output is the displacement x(t). First we’ll
apply the Laplace transform to each of the terms of the equation (1):
The initial conditions of the mass position and speed are:
45

Replacing the Laplace transforms and initial conditions in the equation (1) gives:
We have now found the transfer function of the translational mass system with spring and
damper:
46

The response given by the transfer function is identical with the response obtained by integrating
the ordinary differential equation of the system.
We have now defined the same mechanical system as a differential equation and as a transfer
function. Both representations are correct and equivalent.
47

Transfer function example for a electrical system
Example 2. Find the transfer function of a series RL circuit connected to a continuous current voltage
source.
The methodology for finding the electrical current equation for the system is RL circuit – detailed
mathematical analysis.
48

The ordinary differential equation describing the dynamics of the RL circuit is:
where:
R [Ω] – resistance
L [H] – inductance
u [V] – voltage drop across the circuit
i [A] – electrical current through the circuit
49

The input of the system is the voltage u(t) and the output is the electrical current i(t). First we’ll
apply the Laplace transform to each of the terms of the equation (2):
The initial condition of the electrical current is:
50

Replacing the Laplace transforms and initial conditions in the equation (2) gives:
We have now found the transfer function of the series RL circuit:
51

The response given by the transfer function is identical with the response obtained by integrating
the ordinary differential equation of the system.
We have now defined the same electrical system as a differential equation and as a transfer
function. Both representations are correct and equivalent.
52

By comparing equations, we get an analogous system:
S.No. Translational Rotational
1 Force, F Torque, T
2 Acceleration, a angular acceleration, α
3 Velocity, v angular velocity, ω
4 Displacement, x angular displacement, θ
5 Mass, M Moment of inertia, J
6 Damping Coeﬃcient, B Rotational damping Coeﬃcient, B
7 Stiffness torsional stiffness
53

Example 3: Find the transfer function of the given network.
Step 1
54

Step 2: By taking the Laplace transform of eq (1) and eq (2) and assuming all initial condition to be zero.
Step 3: Calculation of transfer function
Eq (5) is the transfer function
55

Single block diagram representation
56

Components of Linear Time Invariant Systems (LTIS)
57

Block diagram of a closed-loop system with a feedback
element
60

Block Diagram Reduction Rules
Follow these rules for simplifying (reducing) the block diagram, which is having
many blocks, summing points and take-off points.
● Rule 1 − Check for the blocks connected in series and simplify.
● Rule 2 − Check for the blocks connected in parallel and simplify.
● Rule 3 − Check for the blocks connected in feedback loop and simplify.
● Rule 4 − If there is difficulty with take-off point while simplifying, shift it towards
right.
● Rule 5 − If there is difficulty with summing point while simplifying, shift it towards
left.
● Rule 6 − Repeat the above steps till you get the simplified form, i.e., single
block.
Note − The transfer function present in this single block is the transfer function of
the overall block diagram.
61

BLOCK DIAGRAM SIMPLIFICATIONS
Cascade (Series) Connections
62

Shifting of take-off point ahead of the block
66

Shifting of take-off point behind the block
67

Block Diagram Algebra for Branch Points
68

Shifting of Summing point ahead of the block
69

Shifting of summing point behind the block
70

Block Diagram Algebra for Summing Junctions
71

Eliminating the feedback loop
the gain of a closed-loop system with
positive feedback is defined as:
So to remove the feedback loop, feedback
gain must be used.
72

Interchange of the summing point
We can use associative
property and can
interchange these directly
connected summing points
without altering the output.
73

Splitting/ combining the summing point
A summing point having 3 inputs can
be split into a configuration having 2
summing points with separated inputs
without disturbing the output. Or three
summing points can be combined to
form a single summing point with the
consideration of each given input.
74

Block Diagram Reduction Rules
75

Basic rules with block diagram transformation
76

Example: Consider the block diagram shown in the following figure.
77

Step 1 − Use Rule 1 for blocks G1 and G2. Use Rule 2 for blocks G3
and G4. The modified block diagram is shown in the following figure.
78

Step 2 − Use Rule 3 for blocks G1G2 and H1. Use Rule 4 for shifting
take-off point after the block G5. The modified block diagram is shown in
the following figure.
79

Step 3 − Use Rule 1 for blocks (G3 + G4) and G5. The modified block
diagram is shown in the following figure.
80

Step 4 − Use Rule 3 for blocks (G3 + G4) G5 and H3. The modified
block diagram is shown in the following figure.
81

Step 5 − Use Rule 1 for blocks connected in series. The modified block
diagram is shown in the following figure.
82

Step 6 − Use Rule 3 for blocks connected in feedback loop. The modified
block diagram is shown in the following figure. This is the simplified block
diagram.
Therefore, the transfer function of the system is
83

Note − Follow these steps in order to calculate the transfer function of the
block diagram having multiple inputs.
● Step 1 − Find the transfer function of block diagram by considering
one input at a time and make the remaining inputs as zero.
● Step 2 − Repeat step 1 for remaining inputs.
● Step 3 − Get the overall transfer function by adding all those transfer
functions.
84

Reducing the 3 directly connected blocks in series into a single block, we will have:
86

Further on simplifying the internal closed-loop system, the overall internal gain will be
87

Further, we can see 3 blocks are present that are connected parallely. Thus on reducing blocks in parallel, we will have:
88

Further on simplifying the internal closed-loop system, the overall internal gain will be
89

We know gain of the closed-loop system is given as:
91

On simplifying the equation
This is the overall transfer function of the given control system.
92

Example 3: Consider the block diagram shown in the figure below. Lets simplify it step-by-step using block
diagram reduction rules.
93

Step-1 :- Use rule 4 for shifting take-off point after block G3
.
94

Step-2 :- Use rule 1 for blocks G2
and G3
.
95

Step-3 :- Use rule 6 for loop containing blocks G2
G3
and H2
.
96

Step-4 :- Use rule 1 for blocks G1
and G2
G3
/(1+ G2
G3
H2
) .
97

Step-5 :- Finally, use rule 6 to get simplified block diagram.
Therefore, the transfer function of the system is
98

Signal flow graph is a graphical representation of algebraic equations. In this chapter, let us discuss the basic concepts
related signal flow graph and also learn how to draw signal flow graphs.
A graphical method of representing the control system using the linear algebraic equations is known as the signal
flow graph. It is abbreviated as SFG. This graph basically signifies how the signal flows in a system.
The equation representing the system holds multiple variables that perform a crucial role in forming the graph.
99

Basic Elements of Signal Flow Graph
Nodes and branches are the basic elements of signal flow graph.
Node
Node is a point which represents either a variable or a signal. There are three types of nodes — input node, output node
and mixed node.
● Input Node − It is a node, which has only outgoing branches.
● Output Node − It is a node, which has only incoming branches.
● Mixed Node − It is a node, which has both incoming and outgoing branches.
Branch
Branch is a line segment which joins two nodes. It has both gain and direction.
100

Terms used in Signal Flow Graph
1. Node: The small points or circles in the signal flow graph are called nodes and these are used to show the
variables of the system. Each node corresponds to an individual variable. In the figure given below x1 to x8 are
nodes.
2. Input node or source: The one having branches with only outgoing signal is known as the input node or
source. For the below shown SFG, x1 is the source.
3. Output node or sink: A node that only contains branches with entering signal is known as a sink or output
node. Both input and output nodes act as dependent variables. Here, x8 is the sink.
102

4. Mixed node: It is also known as chain node and consists of branches having both entering as well as leaving
signals. For this SFG, x2 to x7 are mixed nodes.
5. Branch: The lines in the SFG that join one node to the other is known as a branch. It shows the relationship
existing between various nodes.
6. Transmittance: The transfer function or gain between two nodes is said to be the transmittance of the SFG. Its
value can be either real or complex. These are from ‘a’ to ‘j’ here.
7. Forward path: A path proceeding from input to output is known as forward path. Here one of the paths is x6 –
x7 – x8.
103

8. Feedback loop: A loop which is formed through a path starting from a specific node and ending on the same
node after travelling to at least one node of the graph is known as a feedback loop. It is to be noted that in the
feedback loop, no node should be traced twice. Here the loop is x2 to x6 then back to x2.
9. Self-loop: A self-loop is the one which has only a single node. The paths in such loops are never defined by a
forward path or feedback loop as these never trace any other node of the graph. Here it is formed at node x5
having gain f.
10. Path gain: When the signal flows in a forward path then the product of gains of each branch gives the overall
path gain. For above-given graph, i*j is path gain for forward path, x6 to x8.
11. Loop gain: The product of individual gains of the branches that are in the form of loop is said to be the overall
loop gain. For the loop forming between x3 and x4 the gain is c*d.
12. Non-touching loops: When two or more loops do not share common nodes then such loops are called
non-touching loops. For this SFG, the two non-touching loops are x2-x6-x2 and x3-x4-x3.
104

Mason’s Gain Formula
t is a technique used for finding the transfer function of a control system. Basically, a formula that determines the transfer
function of a linear system by making use of the signal flow graph is known as Mason’s Gain Formula.
It shows its significance in determining the relationship between input and output.
To determine the transfer function of the system using signal flow graph, the formula given below is used.
105

K denotes the number of forward paths,
TK shows the gain of the kth forward path,
Δ is the determinant of the system calculated as:
Δ = 1 – (sum of loop gains of total individual loops in the SFG) + (sum of the product of gains of all pairs
of two non-touching loop in SFG) + (sum of the product of all pairs of three non-touching loops in the
SFG) + ——
ΔK denotes the Δ for the region of SFG which do not touches the Kth forward path
106

SFG from system equations
The steps are as follows:
● First represent each variable of the equation by an individual node.
● Further, we know the value of the variable at a node is obtained by summing up all the signals that are
coming to that node. So, draw the branches to that node accordingly.
● The coefficient of the variables present in an equation will be the gain of the branches that are joining
the nodes.
● Furthermore, locate the input and output variables of the equations in the signal flow graph.
107

SFG from Block Diagram
The following are the steps to draw a signal flow graph:
● Here the summing and take-off points are represented by the nodes. Thus, all the summing and
take-off points of the block diagram should be named and must indicate a specific node.
● Now, interconnect the various nodes using branches which was earlier connected through blocks in
the block diagram representation.
Also, the block transfer function will now act as branch gain of the corresponding branch.
● Further, show the input and output nodes individually in the signal flow graph to complete it.
108

Consider an example. The figure below shows a block diagram which is to be converted into the signal flow graph.
109

Example
Let us consider the following signal flow graph to identify these nodes.
● The nodes present in this signal flow graph are y1, y2, y3 and y4.
● y1 and y4 are the input node and output node respectively.
● y2 and y3 are mixed nodes.
For example, there are four branches in the above signal flow graph. These branches have gains of a, b, c and -d.
111

Construction of Signal Flow Graph
Let us construct a signal flow graph by considering the following algebraic equations −
There will be six nodes (y1, y2, y3, y4, y5 and y6) and eight branches in this signal flow graph. The gains of the branches are
a12, a23, a34, a45, a56, a42, a53 and a35.
112

To get the overall signal flow graph, draw the signal flow graph for each equation, then combine all these signal flow graphs and then
follow the steps given below −
Step 1 − Signal flow graph for is shown in the following figure.
113

114

115

116

117

Step 6 − Signal flow graph of overall system is shown in the following figure.
118

Here we will apply mason’s gain formula but first, determine each component of the formula step by step.
Here the total number of forward paths are two.
T1 = G1G2G3G4G5
T2 = G1G2G6
Also, the above signal flow graph contains, 2 individual feedback loops
L1 = – G1H1
L2 = – G4H2
These two loops of the SFG are also the two non-touching loops.
120

Therefore, substituting the values in the formula to calculate Δ, we will get,
Δ = 1 – (L1 + L2) + (L1L2)
Δ = 1 – (- G1H1 – G4H2) + [(- G1H1) (- G4H2)]
So,
Δ = 1 + G1H1 + G4H2 + [(G1G4 H1H2)]
Further, we will now calculate ΔK
So,
Δ1 = 1 – (loops that are not touching first forward path)
121

And here there is no such loop which is not touching the first forward path, hence,
Δ1 = 1 – (0)
Δ1 = 1
Now,
Δ2 = 1 – (loops not touching the second forward path)
And here L2 is not touching the second forward path, therefore,
Δ2 = 1 – (L2)
Δ2 = 1 – (- G4H2)
122

On simplifying
Δ2 = 1 + G4H2
So, now substituting the values in the mason’s gain formula:
This is the gain of the system with the above given SFG.
123

Here we have only a single forward path.
Thus, K = 1
So,
T1 = G1G2G3G4G5
The 4 individual feedback loops of the above shown signal flow graph are:
L1 = – G1H1
L2 = – G2H2
L3 = – G4H3
L4 = – G4G5H4
125

The various combinations of 2 non-touching loop of the SFG are:
L1L3 = (- G1H1)(- G4H3) = G1H1G4H3
L1L4 = (-G1H1)(-G4G5H4) = G1H1G4G5H4
L2L3= (-G2H2)(-G4H3) = G2H2G4H3
L2L4= (-G2H2)(-G4G5H4) = G2H2G4G5H4
126

n this SFG, there are no 3 non-touching loops, thus we will stop right here.
So, Δ will be given as:
Δ = 1 – (-G1H1 – G2H2 – G4H3– G4G5H4) + [(G1H1G4H3) + (G1H1G4G5H4) + (G2H2G4H3) + (G2H2G4G5H4)]
So, we will have,
Δ = 1 + G1H1 + G2H2 + G4H3 + G4G5H4 + G1H1G4H3 + G1H1G4G5H4 + G2H2G4H3 + G2H2G4G5H4
Now, as we have a single forward path,
Therefore, Δ1 = 1 – (0)
This is so because we have no such loops that are not touching the first forward path.
So, Δ1 = 1
127

Hence, the transfer function of the above signal flow graph will be:
On substituting the values, we will get,
This is the transfer function of the system with the above-given signal flow graph.
128

Conversion of Block Diagrams into Signal Flow Graphs
Follow these steps for converting a block diagram into its equivalent signal flow graph.
● Represent all the signals, variables, summing points and take-off points of block diagram as nodes in signal flow
graph.
● Represent the blocks of block diagram as branches in signal flow graph.
● Represent the transfer functions inside the blocks of block diagram as gains of the branches in signal flow graph.
● Connect the nodes as per the block diagram. If there is connection between two nodes (but there is no block in
between), then represent the gain of the branch as one. For example, between summing points, between summing
point and takeoff point, between input and summing point, between take-off point and output.
129

With the help of Mason’s gain formula (discussed in the next chapter), you can calculate the transfer function of this signal flow graph.
This is the advantage of signal flow graphs. Here, we no need to simplify (reduce) the signal flow graphs for calculating the transfer
function.
131

Control Systems

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Control Systems

Similar to Control Systems (20)

Recently uploaded

Recently uploaded (20)

Control Systems