Control systems tutorial

Control Systems
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AbouttheTutorial
This tutorial is meant to provide the readers the knowhow of how to analyze the control
systems with the help of mathematical models. After completing this tutorial, you will be
able to learn various methods and techniques in order to improve the performance of the
control systems based on the requirements.
Audience
This tutorial is meant for all those readers who are aspiring to learn the fundamental
concepts of Control Systems.
Prerequisites
A learner who wants to go ahead with this tutorial needs to have a basic understanding of
Signals and Systems.
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Control Systems
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TableofContents
About the Tutorial ............................................................................................................................................i
Audience...........................................................................................................................................................i
Prerequisites.....................................................................................................................................................i
Copyright & Disclaimer.....................................................................................................................................i
Table of Contents ............................................................................................................................................ ii
1. Control Systems − Introduction.................................................................................................................1
Classification of Control Systems.....................................................................................................................1
2. Control Systems − Feedback .....................................................................................................................4
Types of Feedback ...........................................................................................................................................4
Effects of Feedback .........................................................................................................................................5
3. Control Systems − Mathematical Models..................................................................................................8
Differential Equation Model............................................................................................................................8
Transfer Function Model .................................................................................................................................9
4. Control Systems – Modeling of Mechanical Systems...............................................................................11
Modeling of Translational Mechanical Systems............................................................................................11
Modeling of Rotational Mechanical Systems ................................................................................................13
5. Control Systems – Electrical Analogies of Mechanical Systems ...............................................................16
Force Voltage Analogy...................................................................................................................................16
Force Current Analogy...................................................................................................................................18
6. Control Systems − Block Diagrams ..........................................................................................................20
Basic Elements of Block Diagram...................................................................................................................20
Block Diagram Representation of Electrical Systems ....................................................................................22
7. Control Systems − Block Diagram Algebra...............................................................................................25
Basic Connections for Blocks.........................................................................................................................25
Block Diagram Algebra for Summing Points..................................................................................................27
Block Diagram Algebra for Take-off Points....................................................................................................29
8. Control Systems – Block Diagram Reduction...........................................................................................32
Block Diagram Reduction Rules.....................................................................................................................32
9. Control Systems − Signal Flow Graphs.....................................................................................................36
Basic Elements of Signal Flow Graph.............................................................................................................36
Construction of Signal Flow Graph................................................................................................................37
Conversion of Block Diagrams into Signal Flow Graphs ................................................................................38
10. Control Systems – Mason’s Gain Formula ...............................................................................................41
Calculation of Transfer Function using Mason’s Gain Formula.....................................................................42
11. Control Systems − Time Response Analysis.............................................................................................44
What is Time Response?................................................................................................................................44
Standard Test Signals.....................................................................................................................................45
12. Control Systems – Response of the First Order System ...........................................................................48

Control Systems
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Impulse Response of First Order System.......................................................................................................49
Step Response of First Order System ............................................................................................................50
Ramp Response of First Order System ..........................................................................................................51
Parabolic Response of First Order System.....................................................................................................52
13. Control Systems – Response of Second Order System.............................................................................54
Step Response of Second Order System .......................................................................................................55
Impulse Response of Second Order System..................................................................................................58
14. Control Systems − Time Domain Specifications.......................................................................................60
Delay Time.....................................................................................................................................................60
Rise Time .......................................................................................................................................................61
Peak Time ......................................................................................................................................................61
Peak Overshoot .............................................................................................................................................62
Settling time ..................................................................................................................................................63
15. Control Systems − Steady State Errors ....................................................................................................65
Steady State Errors for Unity Feedback Systems...........................................................................................65
Steady State Errors for Non-Unity Feedback Systems...................................................................................67
16. Control Systems − Stability .....................................................................................................................69
What is Stability?...........................................................................................................................................69
Types of Systems based on Stability..............................................................................................................69
17. Control Systems − Stability Analysis........................................................................................................71
Routh-Hurwitz Stability Criterion ..................................................................................................................71
Routh Array Method......................................................................................................................................71
Special Cases of Routh Array .........................................................................................................................73
18. Control Systems − Root Locus .................................................................................................................76
Basics of Root Locus ......................................................................................................................................76
19. Control Systems – Construction of Root Locus ........................................................................................78
Rules for Construction of Root Locus ............................................................................................................78
20. Control Systems – Frequency Response Analysis ....................................................................................82
What is Frequency Response?.......................................................................................................................82
Frequency Domain Specifications .................................................................................................................83
21. Control Systems − Bode Plots..................................................................................................................86
Basics of Bode Plots.......................................................................................................................................86
22. Control Systems – Construction of Bode Plots ........................................................................................90
Rules for Construction of Bode Plots.............................................................................................................90
Stability Analysis using Bode Plots ................................................................................................................90
23. Control Systems − Polar Plots..................................................................................................................92
Rules for Drawing Polar Plots ........................................................................................................................93
24. Control Systems − Nyquist Plots..............................................................................................................95
Nyquist Stability Criterion..............................................................................................................................95
Rules for Drawing Nyquist Plots ....................................................................................................................96

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Stability Analysis using Nyquist Plots ............................................................................................................96
25. Control Systems − Compensators............................................................................................................98
Lag Compensator...........................................................................................................................................98
Lead Compensator.........................................................................................................................................99
Lag-Lead Compensator................................................................................................................................100
26. Control Systems − Controllers...............................................................................................................101
Proportional Controller ...............................................................................................................................101
Derivative Controller ...................................................................................................................................102
Integral Controller .......................................................................................................................................102
Proportional Derivative (PD) Controller ......................................................................................................103
Proportional Integral (PI) Controller............................................................................................................104
Proportional Integral Derivative (PID) Controller........................................................................................104
27. Control Systems − State Space Model...................................................................................................106
Basic Concepts of State Space Model..........................................................................................................106
State Space Model from Differential Equation............................................................................................107
State Space Model from Transfer Function.................................................................................................108
28. Control Systems − State Space Analysis ................................................................................................112
Transfer Function from State Space Model.................................................................................................112
State Transition Matrix and its Properties...................................................................................................113
Controllability and Observability.................................................................................................................114

Control Systems
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A control system is a system, which provides the desired response by controlling the output.
The following figure shows the simple block diagram of a control system.
Here, the control system is represented by a single block. Since, the output is controlled by
varying input, the control system got this name. We will vary this input with some mechanism.
In the next section on open loop and closed loop control systems, we will study in detail about
the blocks inside the control system and how to vary this input in order to get the desired
response.
Examples: Traffic lights control system, washing machine
Traffic lights control system is an example of control system. Here, a sequence of input
signal is applied to this control system and the output is one of the three lights that will be
on for some duration of time. During this time, the other two lights will be off. Based on the
traffic study at a particular junction, the on and off times of the lights can be determined.
Accordingly, the input signal controls the output. So, the traffic lights control system operates
on time basis.
ClassificationofControlSystems
Based on some parameters, we can classify the control systems into the following ways.
Continuous time and Discrete-time Control Systems
 Control Systems can be classified as continuous time control systems and discrete time
control systems based on the type of the signal used.
 In continuous time control systems, all the signals are continuous in time. But, in
discrete time control systems, there exists one or more discrete time signals.
SISO and MIMO Control Systems
 Control Systems can be classified as SISO control systems and MIMO control systems
based on the number of inputs and outputs present.
1. Control Systems − Introduction

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 SISO (Single Input and Single Output) control systems have one input and one output.
Whereas, MIMO (Multiple Inputs and Multiple Outputs) control systems have more
than one input and more than one output.
Open Loop and Closed Loop Control Systems
Control Systems can be classified as open loop control systems and closed loop control
systems based on the feedback path.
In open loop control systems, output is not fed-back to the input. So, the control action is
independent of the desired output.
The following figure shows the block diagram of the open loop control system.
Here, an input is applied to a controller and it produces an actuating signal or controlling
signal. This signal is given as an input to a plant or process which is to be controlled. So, the
plant produces an output, which is controlled. The traffic lights control system which we
discussed earlier is an example of an open loop control system.
In closed loop control systems, output is fed back to the input. So, the control action is
dependent on the desired output.
The following figure shows the block diagram of negative feedback closed loop control
system.

Control Systems
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The error detector produces an error signal, which is the difference between the input and
the feedback signal. This feedback signal is obtained from the block (feedback elements) by
considering the output of the overall system as an input to this block. Instead of the direct
input, the error signal is applied as an input to a controller.
So, the controller produces an actuating signal which controls the plant. In this combination,
the output of the control system is adjusted automatically till we get the desired response.
Hence, the closed loop control systems are also called the automatic control systems. Traffic
lights control system having sensor at the input is an example of a closed loop control system.
The differences between the open loop and the closed loop control systems are mentioned in
the following table.
Open Loop Control Systems Closed Loop Control Systems
Control action is independent of the
desired output.
Control action is dependent on the desired
output.
Feedback path is not present. Feedback path is present.
These are also called as non-feedback
control systems.
These are also called as feedback
control systems.
Easy to design. Difficult to design.
These are economical. These are costlier.
Inaccurate. Accurate.

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If either the output or some part of the output is returned to the input side and utilized as
part of the system input, then it is known as feedback. Feedback plays an important role in
order to improve the performance of the control systems. In this chapter, let us discuss the
types of feedback & effects of feedback.
TypesofFeedback
There are two types of feedback:
 Positive feedback
 Negative feedback
Positive Feedback
The positive feedback adds the reference input, 𝑅(𝑠) and feedback output. The following figure
shows the block diagram of positive feedback control system.
The concept of transfer function will be discussed in later chapters. For the time being,
consider the transfer function of positive feedback control system is,
𝑻 =
𝑮
𝟏−𝑮𝑯
(Equation1)
Where,
 T is the transfer function or overall gain of positive feedback control system.
 G is the open loop gain, which is function of frequency.
 H is the gain of feedback path, which is function of frequency.
Negative Feedback
2. Control Systems − Feedback

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Negative feedback reduces the error between the reference input, 𝑅(𝑠) and system output.
The following figure shows the block diagram of the negative feedback control system.
Transfer function of negative feedback control system is,
𝑻 =
𝑮
𝟏+𝑮𝑯
(Equation 2)
Where,
 T is the transfer function or overall gain of negative feedback control system
 G is the open loop gain, which is function of frequency
 H is the gain of feedback path, which is function of frequency
The derivation of the above transfer function is present in later chapters.
EffectsofFeedback
Let us now understand the effects of feedback.
Effect of Feedback on Overall Gain
 From Equation 2, we can say that the overall gain of negative feedback closed loop
control system is the ratio of ‘G’ and (1+GH). So, the overall gain may increase or
decrease depending on the value of (1+GH).
 If the value of (1+GH) is less than 1, then the overall gain increases. In this case, ‘GH’
value is negative because the gain of the feedback path is negative.
 If the value of (1+GH) is greater than 1, then the overall gain decreases. In this case,
‘GH’ value is positive because the gain of the feedback path is positive.
In general, ‘G’ and ‘H’ are functions of frequency. So, the feedback will increase the overall
gain of the system in one frequency range and decrease in the other frequency range.

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Effect of Feedback on Sensitivity
Sensitivity of the overall gain of negative feedback closed loop control system (T) to the
variation in open loop gain (G) is defined as
𝐒 𝐆
𝐓
=
𝛛𝐓
𝐓⁄
𝛛𝐆
𝐆⁄
=
𝐏𝐞𝐫𝐜𝐞𝐧𝐭𝐚𝐠𝐞 𝐜𝐡𝐚𝐧𝐠𝐞 𝐢𝐧 𝐓
𝐏𝐞𝐫𝐜𝐞𝐧𝐭𝐚𝐠𝐞 𝐜𝐡𝐚𝐧𝐠𝐞 𝐢𝐧 𝐆
(Equation 3)
Where, 𝞉T is the incremental change in T due to incremental change in G.
We can rewrite Equation 3 as
𝑺 𝑮
𝑻
=
𝝏𝑻
𝝏𝑮
𝑮
𝑻
(Equation 4)
Do partial differentiation with respect to G on both sides of Equation 2.
∂T
∂G
=
∂
∂G
(
G
1+GH
) =
(1+GH).1−𝐺(𝐻)
(1+GH)2 =
1
(1+GH)2 (Equation 5)
From Equation 2, you will get
𝐺
𝑇
= 1 + 𝐺𝐻 (Equation 6)
Substitute Equation 5 and Equation 6 in Equation 4.
𝑆 𝐺
𝑇
=
1
(1 + GH)2
(1 + 𝐺𝐻) =
1
1 + 𝐺𝐻
So, we got the sensitivity of the overall gain of closed loop control system as the reciprocal
of (1+GH). So, Sensitivity may increase or decrease depending on the value of (1+GH).
 If the value of (1+GH) is less than 1, then sensitivity increases. In this case, ‘GH’ value
is negative because the gain of feedback path is negative.
 If the value of (1+GH) is greater than 1, then sensitivity decreases. In this case, ‘GH’
value is positive because the gain of feedback path is positive.
In general, ‘G’ and ‘H’ are functions of frequency. So, feedback will increase the sensitivity of
the system gain in one frequency range and decrease in the other frequency range. Therefore,
we have to choose the values of ‘GH’ in such a way that the system is insensitive or less
sensitive to parameter variations.
Effect of Feedback on Stability
 A system is said to be stable, if its output is under control. Otherwise, it is said to be
unstable.
 In Equation 2, if the denominator value is zero (i.e., GH = -1), then the output of the
control system will be infinite. So, the control system becomes unstable.

Control Systems
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Therefore, we have to properly choose the feedback in order to make the control system
stable.
Effect of Feedback on Noise
To know the effect of feedback on noise, let us compare the transfer function relations with
and without feedback due to noise signal alone.
Consider an open loop control system with noise signal as shown below.
The open loop transfer function due to noise signal alone is
𝐶(𝑠)
𝑁(𝑠)
= 𝐺 𝑏 (Equation 7)
It is obtained by making the other input 𝑅(𝑠) equal to zero.
Consider a closed loop control system with noise signal as shown below.
The closed loop transfer function due to noise signal alone is
𝐶(𝑠)
𝑁(𝑠)
=
𝐺 𝑏
1+𝐺 𝑎 𝐺 𝑏 𝐻
(Equation 8)
It is obtained by making the other input 𝑅(𝑠) equal to zero.
Compare Equation 7 and Equation 8,
In the closed loop control system, the gain due to noise signal is decreased by a factor of
(1 + 𝐺 𝑎 𝐺 𝑏 𝐻) provided that the term (1 + 𝐺 𝑎 𝐺 𝑏 𝐻) is greater than one.

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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
Let us discuss the first two models in this chapter.
DifferentialEquationModel
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).
Example
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.
3. Control Systems − Mathematical Models

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Mesh equation for this circuit is
𝑣𝑖 = 𝑅𝑖 + 𝐿
𝑑𝑖
𝑑𝑡
+ 𝑣𝑜
Substitute, the current passing through capacitor i = C
d𝑣 𝑜
dt
in the above equation.
=> 𝑣𝑖 = 𝑅C
d𝑣𝑜
dt
+ 𝐿C
𝑑2
𝑣𝑜
d𝑡2
+ 𝑣𝑜
=>
𝑑2
𝑣𝑜
d𝑡2
+ (
𝑅
𝐿
)
d𝑣𝑜
dt
+ (
1
𝐿𝐶
) 𝑣𝑜 = (
1
𝐿𝐶
) 𝑣𝑖
The above equation is a second order differential equation.
TransferFunctionModel
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.
If 𝑥(𝑡) and 𝑦(𝑡) are the input and output of an LTI system, then the corresponding Laplace
transforms are 𝑋(𝑠) and 𝑌(𝑠).
Therefore, the transfer function of LTI system is equal to the ratio of 𝑌(𝑠) and 𝑋(𝑠).
i.e., Transfer function =
Y(s)
X(s)
The transfer function model of an LTI system is shown in the following figure.

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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).
Example
Previously, we got the differential equation of an electrical system as
𝑑2
𝑣𝑜
d𝑡2
+ (
𝑅
𝐿
)
d𝑣𝑜
dt
+ (
1
𝐿𝐶
) 𝑣𝑜 = (
1
𝐿𝐶
) 𝑣𝑖
Apply Laplace transform on both sides.
𝑠2
𝑉𝑜(𝑠) + (
𝑠𝑅
𝐿
) 𝑉𝑜(𝑠) + (
1
𝐿𝐶
) 𝑉𝑜(𝑠) = (
1
𝐿𝐶
) 𝑉𝑖(𝑠)
=> {𝑠2
+ (
𝑅
𝐿
) 𝑠 +
1
𝐿𝐶
} 𝑉𝑜(𝑠) = (
1
𝐿𝐶
) 𝑉𝑖(𝑠)
=>
𝑉𝑜(𝑠)
𝑉𝑖(𝑠)
=
1 𝐿𝐶⁄
𝑠2 + (
𝑅
𝐿
) 𝑠 +
1
𝐿𝐶
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).

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In this chapter, let us discuss the differential equation modeling of mechanical systems.
There are two types of mechanical systems based on the type of motion.
 Translational mechanical systems
 Rotational mechanical systems
ModelingofTranslationalMechanicalSystems
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.
Mass
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.
𝐹𝑚 ∝ 𝑎
=> 𝐹𝑚 = 𝑀𝑎 = 𝑀
𝑑2
𝑥
𝑑𝑡2
𝐹 = 𝐹𝑚 = 𝑀
𝑑2
𝑥
𝑑𝑡2
Where,
 𝑭 is the applied force
 𝑭 𝒎 is the opposing force due to mass
4. Control Systems – Modeling of Mechanical Systems

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 𝑴 is mass
 𝒂 is acceleration
 𝒙 is displacement
Spring
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.
𝐹𝑘 ∝ 𝑥
=> 𝐹𝑘 = 𝐾𝑥
𝐹 = 𝐹𝑘 = 𝐾𝑥
Where,
 𝑭 is the applied force
 𝑭 𝒌 is the opposing force due to elasticity of spring
 𝑲 is spring constant
Dashpot
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.

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𝐹𝑏 ∝ 𝑣
=> 𝐹𝑏 = 𝐵𝑣 = 𝐵
𝑑𝑥
𝑑𝑡
𝐹 = 𝐹𝑏 = 𝐵
𝑑𝑥
𝑑𝑡
Where,
 𝑭 𝒃 is the opposing force due to friction of dashpot
 𝑩 is the frictional coefficient
 𝒗 is velocity
ModelingofRotationalMechanicalSystems
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.
Moment of Inertia
In translational mechanical system, mass stores kinetic energy. Similarly, in rotational
mechanical system, moment of inertia stores kinetic energy.
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.

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𝑇𝑗 ∝ 𝛼
=> 𝑇𝑗 = 𝐽𝛼 = 𝐽
𝑑2
𝜃
𝑑𝑡2
𝑇 = 𝑇𝑗 = 𝐽
𝑑2
𝜃
𝑑𝑡2
Where,
 𝑻 is the applied torque
 𝑻𝒋 is the opposing torque due to moment of inertia
 𝑱 is moment of inertia
 𝛂 is angular acceleration
 𝛉 is angular displacement.
Torsional Spring
In translational mechanical system, spring stores potential energy. Similarly, in rotational
mechanical system, torsional spring stores potential energy.
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.

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𝑇𝑘 ∝ 𝜃
=> 𝑇𝑘 = 𝐾𝜃
𝑇 = 𝑇𝑘 = 𝐾𝜃
Where,
 𝑻 is the applied torque
 𝑻 𝒌 is the opposing torque due to elasticity of torsional spring
 𝑲 is the torsional spring constant
 𝛉 is angular displacement
Dashpot
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.
𝑇𝑏 ∝ 𝜔
=> 𝑇𝑏 = 𝐵𝜔 = 𝐵
𝑑𝜃
𝑑𝑡
𝑇 = 𝑇𝑏 = 𝐵
𝑑𝜃
𝑑𝑡

Control Systems
20
Where,
 𝑻 𝒃 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.

Control Systems
21
Two systems are said to be analogous to each other if the following two conditions are
satisfied.
 The two systems are physically different
 Differential equation modelling of these two systems are same
Electrical systems and mechanical systems are two physically different systems. There are
two types of electrical analogies of translational mechanical systems. Those are force voltage
analogy and force current analogy.
ForceVoltageAnalogy
In force voltage analogy, the mathematical equations of translational mechanical system
are compared with mesh equations of the electrical system.
Consider the following translational mechanical system as shown in the following figure.
The force balanced equation for this system is
𝐹 = 𝐹𝑚 + 𝐹𝑏+𝐹𝑘
=> 𝐹 = 𝑀
𝑑2 𝑥
𝑑𝑡2 + 𝐵
𝑑𝑥
𝑑𝑡
+ 𝑘𝑥 (Equation 1)
of a resistor, an inductor and a capacitor. All these electrical elements are connected in a
series. The input voltage applied to this circuit is 𝑉 volts and the current flowing through the
circuit is 𝑖 Amps.
5. Control Systems – Electrical Analogies of Mechanical
Systems

Control Systems
22
Mesh equation for this circuit is -
𝑉 = 𝑅𝑖 + 𝐿
𝑑𝑖
𝑑𝑡
+
1
𝐶
∫ 𝑖𝑑𝑡 (Equation 2)
Substitute, i =
dq
dt
in Equation 2.
𝑉 = 𝑅
dq
dt
+ 𝐿
𝑑2
𝑞
𝑑𝑡2
+
𝑞
𝐶
=> 𝑉 = 𝐿
𝑑2 𝑞
𝑑𝑡2 + 𝑅
dq
dt
+ (
1
𝐶
) 𝑞 (Equation 3)
By comparing Equation 1 and Equation 3, we will get the analogous quantities of the
translational mechanical system and electrical system. The following table shows these
analogous quantities.
Translational Mechanical System Electrical System
Force (F) Voltage (V)
Mass (M) Inductance (L)
Frictional coefficient (B) Resistance (R)
Spring constant (K) Reciprocal of Capacitance (
1
𝐶
)
Displacement (x) Charge (q)
Velocity (v) Current (i)
Similarly, there is torque voltage analogy for rotational mechanical systems. Let us now
discuss about this analogy.
Torque Voltage Analogy
In this analogy, the mathematical equations of rotational mechanical system are
compared with mesh equations of the electrical system.
Rotational mechanical system is shown in the following figure.

Control Systems
23
The torque balanced equation is -
𝑇 = 𝑇𝑗 + 𝑇𝑏+𝑇𝑘
=> 𝑇 = 𝐽
𝑑2 𝜃
𝑑𝑡2 + 𝐵
𝑑𝜃
𝑑𝑡
+ 𝑘𝜃 (Equation 4)
By comparing Equation 4 and Equation 3, we will get the analogous quantities of rotational
mechanical system and electrical system. The following table shows these analogous
quantities.
Rotational Mechanical System Electrical System
Torque (T) Voltage (V)
Moment of Inertia (J) Inductance (L)
Rotational friction coefficient (B) Resistance (R)
Torsional spring constant (K) Reciprocal of Capacitance (
1
𝐶
)
Angular Displacement (𝜃) Charge (q)
Angular Velocity (𝜔) Current (i)
ForceCurrentAnalogy
In force current analogy, the mathematical equations of the translational mechanical
system are compared with the nodal equations of the electrical system.
of current source, resistor, inductor and capacitor. All these electrical elements are connected
in parallel.

Control Systems
24
The nodal equation is -
i =
𝑉
𝑅
+
1
𝐿
∫ V dt + C
dV
dt
(Equation 5)
Substitute, 𝑉 =
𝑑𝛹
𝑑𝑡
in Equation 5.
i =
1
𝑅
𝑑𝛹
𝑑𝑡
+ (
1
𝐿
) 𝛹 + C
𝑑2
𝛹
𝑑𝑡2
=> i = C
𝑑2 𝛹
𝑑𝑡2 + (
1
𝑅
)
𝑑𝛹
𝑑𝑡
+ (
1
𝐿
) 𝛹 (Equation 6)
By comparing Equation 1 and Equation 6, we will get the analogous quantities of the
translational mechanical system and electrical system. The following table shows these
analogous quantities.
Translational Mechanical System Electrical System
Force (F) Current (i)
Mass (M) Capacitance (C)
Frictional coefficient (B) Reciprocal of Resistance (
1
𝑅
)
Spring constant (K) Reciprocal of Inductance (
1
𝐿
)
Displacement (x) Magnetic Flux (𝛹)
Velocity (v) Voltage (V)
Similarly, there is a torque current analogy for rotational mechanical systems. Let us now
discuss this analogy.
Torque Current Analogy
In this analogy, the mathematical equations of the rotational mechanical system are
compared with the nodal mesh equations of the electrical system.

Control Systems
25
By comparing Equation 4 and Equation 6, we will get the analogous quantities of rotational
mechanical system and electrical system. The following table shows these analogous
quantities.
Rotational Mechanical System Electrical System
Torque (T) Current (i)
Moment of Inertia (J) Capacitance (C)
Rotational friction coefficient (B) Reciprocal of Resistance (
1
𝑅
)
Torsional spring constant (K) Reciprocal of Inductance (
1
𝐿
)
Angular displacement (𝜃) Magnetic flux (𝛹)
Angular velocity (𝜔) Voltage (V)
In this chapter, we discussed the electrical analogies of the mechanical systems. These
analogies are helpful to study and analyze the non-electrical system like mechanical system
from analogous electrical system.

Control Systems
26
Block diagrams consist of a single block or a combination of blocks. These are used to
represent the control systems in pictorial form.
BasicElementsofBlockDiagram
The basic elements of a block diagram are a block, the summing point and the take-off point.
Let us consider the block diagram of a closed loop control system as shown in the following
figure to identify these elements.
The above block diagram consists of two blocks having transfer functions G(s) and H(s). It is
also having one summing point and one take-off point. Arrows indicate the direction of the
flow of signals. Let us now discuss these elements one by one.
Block
The transfer function of a component is represented by a block. Block has single input and
single output.
The following figure shows a block having input X(s), output Y(s) and the transfer function
G(s).
Transfer function, 𝐺(𝑠) =
𝑌(𝑠)
𝑋(𝑠)
=> 𝑌(𝑠) = 𝐺(𝑠)𝑋(𝑠)
6. Control Systems − Block Diagrams

Control Systems
27
Output of the block is obtained by multiplying transfer function of the block with input.
Summing Point
The summing point is represented with a circle having cross (X) inside it. It has two or more
inputs and single output. It produces the algebraic sum of the inputs. It also performs the
summation or subtraction or combination of summation and subtraction of the inputs based
on the polarity of the inputs. Let us see these three operations one by one.
The following figure shows the summing point with two inputs (A, B) and one output (Y).
Here, the inputs A and B have a positive sign. So, the summing point produces the output, Y
as sum of A and B.
i.e., 𝑌 = 𝐴 + 𝐵.
The following figure shows the summing point with two inputs (A, B) and one output (Y).
Here, the inputs A and B are having opposite signs, i.e., A is having positive sign and B is
having negative sign. So, the summing point produces the output Y as the difference of A
and B.
𝑌 = 𝐴 + (−𝐵) = 𝐴 − 𝐵.
The following figure shows the summing point with three inputs (A, B, C) and one output (Y).
Here, the inputs A and B are having positive signs and C is having a negative sign. So, the
summing point produces the output Y as
𝑌 = 𝐴 + 𝐵 + (−𝐶) = 𝐴 + 𝐵 − 𝐶.

Control Systems
28
Take-off Point
The take-off point is a point from which the same input signal can be passed through more
than one branch. That means with the help of take-off point, we can apply the same input to
one or more blocks, summing points.
In the following figure, the take-off point is used to connect the same input, R(s) to two more
blocks.
In the following figure, the take-off point is used to connect the output 𝐶(𝑠), as one of the
inputs to the summing point.

Control Systems
29
BlockDiagramRepresentationofElectricalSystems
In this section, let us represent an electrical system with a block diagram. Electrical systems
contain mainly three basic elements — resistor, inductor and capacitor.
Consider a series of RLC circuit as shown in the following figure. Where, 𝑉𝑖(t) and 𝑉𝑜(t) are the
input and output voltages. Let 𝑖(𝑡) be the current passing through the circuit. This circuit is in
time domain.
By applying the Laplace transform to this circuit, will get the circuit in s-domain. The circuit
is as shown in the following figure.
From the above circuit, we can write –
𝐼(𝑠) =
𝑉𝑖(𝑠) − 𝑉𝑜(𝑠)
𝑅 + 𝑠𝐿

Control Systems
30
=> 𝐼(𝑠) = {
1
𝑅+𝑠𝐿
} {𝑉𝑖(𝑠) − 𝑉𝑜(𝑠)} (Equation 1)
𝑉𝑜(𝑠) = (
1
𝑠𝐶
) 𝐼(𝑠) (Equation 2)
Let us now draw the block diagrams for these two equations individually. And then combine
those block diagrams properly in order to get the overall block diagram of series of RLC Circuit
(s-domain).
Equation 1 can be implemented with a block having the transfer function,
1
𝑅+𝑠𝐿
. The input and
output of this block are {𝑉𝑖(𝑠) − 𝑉𝑜(𝑠)} and 𝐼(𝑠). We require a summing point to get {𝑉𝑖(𝑠) − 𝑉𝑜(𝑠)}.
The block diagram of Equation 1 is shown in the following figure.
Equation 2 can be implemented with a block having transfer function,
1
𝑠𝐶
. The input and output
of this block are 𝐼(𝑠) and 𝑉𝑜(𝑠). The block diagram of Equation 2 is shown in the following
figure.
The overall block diagram of the series of RLC Circuit (s-domain) is shown in the following
figure.

Control Systems
31
Similarly, you can draw the block diagram of any electrical circuit or system just by following
this simple procedure.
 Convert the time domain electrical circuit into an s-domain electrical circuit by applying
Laplace transform.
 Write down the equations for the current passing through all series branch elements
and voltage across all shunt branches.
 Draw the block diagrams for all the above equations individually.
 Combine all these block diagrams properly in order to get the overall block diagram of
the electrical circuit (s-domain).

Control Systems
32
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