CONTROL SYSTEMS for
EC / EE / IN By
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Syllabus
Control Systems
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 ProportionalIntegral-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%
th
th
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Contents
Control Systems
CONTENTS Chapter #1.
#2.
Basics of Control System
1 - 30
1- 4 4-5 5-8 8 - 10 11 - 17 18 - 22 22 - 24 25 25 - 30
#4.
Classification of Control Systems Effect of Feedback Transfer Functions Signal Flow Graphs Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Time Domain Analysis
31 - 58
31 - 32 32 - 33 33 - 35 35 - 39
#3.
Page No.
Introduction Standard Test Signals Time Response of First Order Control System Time Response of Second Order Control System Time Response of the Higher Order Control System & Error Constants Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
39 - 41 42 - 46 47 - 51 51 - 53 54 54 - 58
Stability &Routh Hurwitz Criterion
59 - 79
59 - 61 61 - 62 62 - 63 64 - 67 68 - 70 71 - 72 73 73 - 79
Introduction Relative Stability Analysis Routh – Hurwitz Criterion Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Root Locus Technique
80 - 99
80 80 - 81 81 - 83 84 - 88
Introduction Rules for the Construction of Root Locus Complementary Root Locus Solved Examples th
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Contents
#5.
#6.
#7.
#8.
Control Systems
89 - 92 93 - 94 95 95 - 99
Assignment 1 Assignment 2 Answer Keys Explanations
Frequency Response Analysis Using Nyquist Plot
100 -131
100 - 101 101 - 103 103 - 107 107 - 108 109 - 113 114 - 119 119 - 125 126 126-131
Frequency Domain Specifications Polar Plot Nyquist Plot and Nyquist Stability Criteria Gain Margin Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Frequency Response Analysis Using Bode Plot
132 -156
132 - 134 134 - 137 137 - 140 141 - 145 146 - 150 151 - 152 153 153 - 156
Bode Plots Bode Magnitude Plots for Typical Transfer Function M & N Circles Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
Compensators & Controllers
157 - 175
157 157 - 158 159 - 160 161 - 163 163 - 164 165 - 168 169 - 170 170 - 172 173 173 - 175
Introduction Phase Lag Compensator Phase Lead Compensator Phase Lag – Lead Compensator Controllers Solved Examples Assignment 1 Assignment 2 Answer Keys Explanations
State Variable Analysis
176 - 198
176 176 - 179 179 - 180 180 - 181 182 - 186
Introduction State Space Representation State Transition Matrix Characteristic Equation & Eigen Values Solved Examples th
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Contents
Control Systems
187 - 190 190 - 192 193 193 - 198
Assignment 1 Assignment 2 Answer Keys Explanations
Module Test
199 - 216 199 - 210 211 211 - 216
Test Questions Answer Keys Explanations
Reference Books
217
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Chapter 1
Control Systems
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. Control System
Reference input r(t)
Controlled output c(t)
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 = T(s) =
, ( )= , ( )-
|initial conditions = 0
( ) |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
Where L and L operator.
(T (s) . R (s))
L
(T(s)) * r(t)
are forward and inverse Laplace transform operators and * is convolution
Classification of Control Systems Control systems can be classified based on presence of feedback as below, 1. 2.
Open loop control systems Closed loop control systems.
Open-loop Control System Reference input
Controller
Actuating Signal
Process
Output
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. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 1
Chapter 1
Control Systems
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. Reference input r(t)
Controller
Process
Output c(t)
Feedback signal (t)
Feedback Network
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) r(t)
e(t)
G(s)
c(t)
f(t)
H(s) 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. THE GATE ACADEMY PVT.LTD. H.O.: #74, KeshavaKrupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, info@thegateacademy.com © Copyright reserved. Web: www.thegateacademy.com Page 2
Chapter 1
Positive feedback Control Systems Unity F/B (H(s) = 1)
:
:
Non Unity F/B (H(s)
1)
T(s) =
T(s) =
T(s) =
Non Unity F/B
:
( ) ( ) ( ) ( ) ( )
T(s) =
Negative feedback Control Systems Unity F/B :
Control Systems
( ) ( ) ( ) ( ) ( )
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. R(s)
G(s)
C(S)
F(s)
H(s) 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 1.
Open-loop C.S.
Closed-loop C.S.
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 closeloop system works more accurately.
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Chapter 1
Control Systems
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 nonlinearities in its elements.
In terms of the performance, the closeloop 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.
3.
Effect of feedback on overall gain
Negative feedback decreases the gain of the system and Positive feedback increase the gain of the system.
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. 5. 6.
Negative feedback improves the dynamic response of the system Negative feedback reduces the effect of disturbance signal or noise. Negative feedback improves the Bandwidth of the system. Let ∝ β ∝
A variable that changes its value
A parameter that changes the value of ∝ change in ∝ change in β
∝ ∝
β ∝
∝ β
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Chapter 1
Control Systems
Open Loop Control System R(s)
C(s) G(s)
∝
M(s),open loop control system-
( ) ( )
G(s)
( ) ( ) ( )
( ) ( )
( ) ( )
x
, ∝
β-
Closed Loop control system R(s)
+–
G(s)
C(s)
H (s) ∝ β
M(s),closed loop control systemG(s) ( ) ( ) ( )
( ) ( )
( ) ( )
( )
(
0 )
( ) ( ) ( )
( ) –––––––––––––––––1 ( )
( ) ( ) ( )
( )
( ) ( )
G(s)H(s) ( ) 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.
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