Electronics and Communication Engineering : Control systems, THE GATE ACADEMY

CONTROL SYSTEMS
for
EC / EE / IN
By
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Syllabus Control Systems
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Syllabus for Control Systems
Basic control system components; block diagrammatic description, reduction of block diagrams.
Open loop and closed loop (feedback) systems and stability analysis of these systems. Signal
flow graphs and their use in determining transfer functions of systems; transient and steady
state analysis of LTI control systems and frequency response. Tools and techniques for LTI
control system analysis: root loci, Routh-Hurwitz criterion, Bode and Nyquist plots. Control
system compensators: elements of lead and lag compensation, elements of Proportional-
Integral-Derivative (PID) control. State variable representation and solution of state equation of
LTI control systems.
Analysis of GATE Papers
(Control Systems)
Year ECE EE IN
2013 11.00 10.00 10.00
2012 9.00 9.00 13.00
2011 8.00 8.00 12.00
2010 11.00 9.00 7.00
Over All
Percentage
9.75% 9.00% 10.5%

Contents Control Systems
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Cross, 10
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C O N T E N T S
Chapter Page No.
#1. Basics of Control System 1 - 30
 Classification of Control Systems 1- 4
 Effect of Feedback 4 - 5
 Transfer Functions 5 - 8
 Signal Flow Graphs 8 - 10
 Solved Examples 11 - 17
 Assignment 1 18 - 22
 Answer Keys 25
 Explanations 25 - 30
#2. Time Domain Analysis 31 - 58
 Introduction 31 - 32
 Standard Test Signals 32 - 33
 Time Response of First Order Control System 33 - 35
 Time Response of Second Order Control System 35 - 39
 Time Response of the Higher Order Control System &
Error Constants 39 - 41
 Answer Keys 54
#3. Stability &Routh Hurwitz Criterion 59 - 79
 Introduction 59 - 61
 Relative Stability Analysis 61 - 62
 Routh – Hurwitz Criterion 62 - 63
 Answer Keys 73
#4. Root Locus Technique 80 - 99
 Introduction 80
 Rules for the Construction of Root Locus 80 - 81
 Complementary Root Locus 81 - 83

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 Answer Keys 95
#5. Frequency Response Analysis Using Nyquist Plot 100 -131
 Frequency Domain Specifications 100 - 101
 Polar Plot 101 - 103
 Nyquist Plot and Nyquist Stability Criteria 103 - 107
 Gain Margin 107 - 108
 Answer Keys 126
 Explanations 126-131
#6. Frequency Response Analysis Using Bode Plot 132 -156
 Bode Plots 132 - 134
 Bode Magnitude Plots for Typical Transfer Function 134 - 137
 M & N Circles 137 - 140
 Answer Keys 153
#7. Compensators & Controllers 157 - 175
 Introduction 157
 Phase Lag Compensator 157 - 158
 Phase Lead Compensator 159 - 160
 Phase Lag – Lead Compensator 161 - 163
 Controllers 163 - 164
 Answer Keys 173
#8. State Variable Analysis 176 - 198
 Introduction 176
 State Space Representation 176 - 179
 State Transition Matrix 179 - 180
 Characteristic Equation & Eigen Values 180 - 181

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 Answer Keys 193
Module Test 199 - 216
 Test Questions 199 - 210
 Answer Keys 211
Reference Books 217

Chapter 1 Control Systems
THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th
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Main, Jayanagar 4th
Block, Bangalore-11
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CHAPTER 1
Basics of Control System
Introduction
It is a system by means of which any quantity of interest in a machine or mechanism is
controlled (maintained or altered) in accordance with the desired manner. Following diagram
depicts the block diagram representation of a control system.
Fig. 1.1 Block diagram of a control system
Any system can be characterized mathematically by Transfer function or State model.
Transfer function is defined as the ratio of Laplace Transform (L.T) of output to that of input
assuming initial conditions to be zero. Transfer function is also obtained as Laplace transform of
the impulse response of the system.
Transfer Function = |initial conditions = 0
T(s) =
, ( )-
, ( )-
=
( )
( )
|initial conditions = 0
For any arbitrary input r(t), output c(t) of control system can be obtained as below,
r(t) = L (R (s)) = L (T (s) . R (s)) L (T(s)) * r(t)
Where L and L are forward and inverse Laplace transform operators and * is convolution
operator.
Classification of Control Systems
Control systems can be classified based on presence of feedback as below,
1. Open loop control systems
2. Closed loop control systems.
Open-loop Control System
Fig 1.2. Block diagram of open-loop control system
Figure 1.2 depicts block diagram of a open loop control system. Also following are salient points
as referred to an open-loop control system.
 The reference input controls the output through a control action process. Here output has
no effect on the control action, as the output is not fed-back for comparison with the input.
 Accuracy of an open-loop control system depends on the accuracy of input calibration.
Reference input OutputController Process
Actuating
Signal
Control
System Controlled outputReference input
r(t) c(t)

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 The open –loop system is simple and cheap to construct.
 Due to the absence of feedback path, the systems are generally stable
 Examples of open loop control systems include Traffic lights, Fans, Washing machines etc,
which do not have a sensor.
 If R(s) is LT of input and C(s) is LT of output of a control system of transfer function G(s),
then
( )
( )
= G (s) C (s) = G (s) R (s)
Closed-loop Control System (Feedback Control Systems)
Figure shown below depicts the block diagram of a closed-loop control system. Closed-loop
control systems can be classified as positive and negative feedback (f/b) control systems. Also
following are the salient points related to closed-loop control systems.
 In a close-loop control system, the output has an effect on control action through a
feedback.
 The control action is actuated by an error signal ‘e (t)’ which is the difference between the
input signal ‘r(t)’and the feedback signal ‘f(t)’.
 The control systems can be manual or automatic control systems.
 Servomechanism is example of a close-loop (feedback) control system using a power
amplifying device prior to controller and the output of such a system is mechanical i.e.
position, velocity or acceleration.
Fig 1.3.Block diagram of closed loop control system
For Positive feedback, error signal e(t) = r(t) + f(t)
For Negative feedback, error signal e(t) = r(t) – f(t)
Fig. 1.4 Transfer function representation of a closed loop control system.
Generally, the purpose of feedback is to reduce the error between the reference input and the
system output.
Let G(s) be the forward path transfer function, H(s) be the feedback path transfer function and
T(s) be the overall transfer function of the closed-loop control system, then
T(s) =
( )
( ( ) ( ))
Here negative sign in denominator is considered for positive feedback and vice versa.
G(s)
H(s)
e(t)r(t)
f(t)
c(t)
Controller Process
Feedback Network
Feedback signal (t)
Reference input
r(t)
Output c(t)

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Positive feedback Control Systems
 Unity F/B (H(s) = 1) : T(s) =
( )
( )
 Non Unity F/B (H(s) 1) : T(s) =
( )
( ) ( )
Negative feedback Control Systems
 Unity F/B : T(s) =
( )
( )
 Non Unity F/B : T(s) =
( )
( ) ( )
Here, G(s) is T.F. without feedback (or) T. F of the forward path and H(s) is T.F. of the feedback
path.
Block diagram shown below corresponds to closed loop control system.
Fig. 1.5 Block diagram of a closed loop control system
The overall transfer function can be derived as below,
L{r(t)} = R(s) → Reference I/P
L{c(t)}= C(s) → Output (Controlled variable)
L*f(t)+ F (s) → Feedback signal
L{e(t)} = E(s) → Error or actuating signal
G( )H( )→ Open loop transfer function
E( )/R( )→ Error transfer function
G(s) → Forward path transfer function
H(s) → Feedback path transfer function
E(s) = R(s) F(s) ; F(s) = H(s) C(s)
C(s) = E(s) G(s) C(s) = { R(s) H(s)C(s)+ G(s)
( )
( )
( )
( ) ( )
Also,
( )
( ) ( ) ( )
Here negative sign is used for positive feedback and positive sign is used for negative feedback.
The transfer function of a system depends upon its elements assuming initial conditions as zero
and it is independent of input function
Comparison of Open-loop and Close-loop Control Systems
Table below summarizes the comparison between open and closed loop control systems.
S.No Open-loop C.S. Closed-loop C.S.
1. The accuracy of an open – loop system
depends on the calibration of the input.
Any departure from pre – determined
calibration affects the output.
As the error between the reference
input and the output is continuously
measured through feedback, the close-
loop system works more accurately.
G(s)
H(s)
R(s)
F(s)
C(S)

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2. The open-loop system is simple to
construct.
The close-loop system is complicated to
construct
3. Cheaper Costly.
4. The open-loop systems are generally
stable.
The close-loop systems can become
unstable under certain conditions.
5. The operation of open – loop system is
affected due to presence of non-
linearities in its elements.
In terms of the performance, the close-
loop systems adjusts to the effects of
non - linearities present in its elements.
Effect of Feedback
The feedback has effects on system performance characteristics such as stability, bandwidth,
overall gain, impedance and sensitivity.
1. Effect of feedback on Stability
 Stability is a notion that describes whether the system will be able to follow the
input command.
 A system is said to be unstable, if its output is out of control or increases without
bound.
 Negative feedback in a control system improves stability and vice versa.
2. Effect of feedback on overall gain
 Negative feedback decreases the gain of the system and Positive feedback increase
the gain of the system.
3. Effect of feedback on Sensitivity
Consider G as a parameter that can vary. The sensitivity of the gain of the overall system T
to the variation in G is defined as
=
/
/
=
Where denotes the incremental change in T due to the incremental change in G; /
and / denote the percentage change in T and G, respectively.
= =
Similarly, =
/
/
= -1
Negative feedback makes the system less sensitive to the parameter variation.
4. Negative feedback improves the dynamic response of the system
5. Negative feedback reduces the effect of disturbance signal or noise.
6. Negative feedback improves the Bandwidth of the system.
Let ∝ A variable that changes its value
β A parameter that changes the value of ∝
∝ change in ∝
change in β
∝
∝ β
∝
∝
β

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Open Loop Control System
∝ M(s),open loop control system-
( )
( )
G(s)
( )
( )
( ) ( )
( )
( )
( )
x , ∝ β-
Closed Loop control system
∝ M(s),closed loop control system-
( )
( )
( )
( ) ( )
β G(s)
( )
( ) ( )
( )
( )
( )
–––––––––––––––––1
( )
( )
( )
( )
( ) ( )
G(s)H(s)
( )
( ) ( )
0
( )
( ) ( )
1
( ) ( ) ( ) ( )
, ( ) ( )- , ( ) ( )-
From equation 1
( )
( )
, G(s) H(s)- x , ( ) ( )-
( ) ( )
1 + G(s) H(s) = Noise Reduction factor
= Return Difference
 Sensitivity of closed loop system is reduced by factor [1 + G(s) H(s)]
 Open Loop control systems are more sensitive to any external or interval disturbance.
 Complementary sensitivity function =
( ) ( )
( ) ( )
T(s)
(s) T(s) = 1
Transfer Functions
Transfer function of a generic control system can be found using block diagram approach or
signal flow graphs as described in the following sections.
G(s)
H (s)
R(s)
–+ C(s)
R(s)
G(s)
C(s)

Electronics and Communication Engineering : Control systems, THE GATE ACADEMY

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