## PREFACE

Automatic control plays an important role in the advance of engineering and science.

It is of extreme importance in most of the engineering fields; such as the aerospace

engineering, chemical engineering, robotic systems, automotive and mobile

equipment engineering as well as manufacturing and industrial processes. Automatic

control provides the means of understanding the problems of stability and precision

of dynamic systems. Actually most engineers must have good understanding of this

field.

The majority of textbooks on automatic control are most appropriated for electrical

engineers. The main problem in designing and analyzing a control loop for nonelectrical

systems normally arises when deducing adequate mathematical model for

the system. Generally, the components cannot readily be represented by simple

discrete ideal elements. The classical approach based on the transfer function and

associated techniques of analysis is more easily comprehended and related to

practice by the beginners than is the modern control theory. Therefore this text is

prepared for the mechanical engineering students. It deals with the basics of linear

control theory. The text includes simple examples enabling applicants to understand

the problems of dynamic systems accuracy and stability. The text includes examples

and exercises that facilitate the comprehension of the control theory, especially for

the mechanical engineering students. The text is arranged in ten chapters dealing

with the following topics:

1. An introduction giving the basic definitions and methods of system

representation, Chapter 1.

2. A revision of selected topics from mathematics, Chapter 2.

3. Deduction of the transfer functions using mathematical models, block

diagrams and signal flow graphs, Chapters 3, 4 & 5.

4. Analysis of the transient and frequency responses of the system and how

does the response vary with the form of transfer function and the input

excitation. The text discusses also how a transfer function can be determined

by practical testing of a system, Chapters 6 & 7.

5. Analysis of the accuracy and stability of the feedback system, Chapter 8.

6. Root locus analysis, Chapter 9.

7. Improvement of the system stability by introducing and designing different

types of compensators, mainly the P, PI & PID controllers, Chapter 10.

I am indebted to my colleague Prof. Dr. Gamal Ahmed El-Sheikh, for his objective

comments on the whole text.

## CONTENTS

CHAPTER 1: INTRODUCTION TO AUTOMATIC CONTROL 1

1.1 INTRODUCTION 1

1.2 SYSTEM DEFINITION 1

1.3 SYSTEM CONTROL 3

1.3.1 Open Loop Control 3

1.3.2 Closed Loop (Feedback) Control 5

1.4 SYSTEM REPRESENTATION 8

1.4.1 Schematic Diagrams 8

1.4.2 Mathematical Model 8

1.4.3 Transfer Function 9

1.4.4 Block Diagram 10

1.4.5 Signal Flow Graph 10

1.4.6 State Space Representation 10

1.4.7 Bond Graph 11

1.5 SYSTEM ANALYSIS 12

1.6 EXERCISE 12

CHAPTER 2: MATHEMATICAL TOPICS 13

2.1 INTRODUCTION 13

2.2 DIFFERENTIAL EQUATIONS 13

2.3 LAPLACE TRANSFORM 17

2.3.1 Direct Laplace Transform 17

2.3.2 Inverse Laplace Transform 18

2.3.3 Properties of Laplace Transform 18

2.3.4 Partial Fraction Expansion 20

2.3.5 Solving Differential Equations Using Laplace Transform 22

2.4 COMPLEX VARIABLES 24

2.5 LAPLACE TRANSFORM TABLES 25

2.6 EXERCISE 25

CHAPTER 3: TRANSFER FUNCTIONS 29

3.1 BASIC DEFINITIONS 29

3.2 TRANSFER FUNCTION OF SOME BASIC ELEMENTS 31

3.2.1 Proportional Element 31

3.2.2 Integrating Elements 32

3.2.2.1 Ideal hydraulic cylinder 32

3.2.2.2 Valve controlled actuator 33

3.2.3 First Order Element 35

3.2.3.1 Hydraulic servo actuator 35

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3.2.3.2 Resistance-capacitance network 36

3.2.4 Second Order Element 37

3.3 EXERCISE 38

CHAPTER 4: BLOCK DIAGRAM 43

4.1 INTRODUCTION 43

4.2 CONVENTIONS FOR BLOCK DIAGRAMS 43

4.3 DEDUCING SYSTEM TRANSFER FUNCTION 44

4.4 BLOCK DIAGRAM ALGEBRA 46

4.5 EXERCISE 50

CHAPTER 5: SIGNAL FLOW GRAPH 53

5.1 INTRODUCTION 53

5.2 CONVENTIONS FOR SIGNAL FLOW GRAPHS 53

5.3 MASON’S FORMULA 56

5.4 EXERCISE 60

CHAPTER 6: TIME DOMAIN ANALYSIS 63

6.1 INTRODUCTION 63

6.2 TIME RESPONSE OF BASIC ELEMENTS 65

6.2.1 Integrating Member 65

6.2.1.1 Response to step input 65

6.2.1.2 Response to ramp input 66

6.2.1.3 Response to input impulse 66

6.2.2 First Order Element 66

6.2.2.1 Step response 66

6.2.2.2 Response to ramp input 67

6.2.2.3 Response to input impulse 68

6.2.3 Second Order Element 69

6.2.3.1 Step response of second order element 69

6.2.3.1.1 Step response of over-damped 2nd order element 70

6.2.3.1.2 Step response of critically-damped 2nd order element 71

6.2.3.1.3 Step response of under-damped 2nd order element 71

6.2.3.1.4 Step response of un-damped 2nd order element 75

6.2.3.2 Response of second order element to ramp input 76

6.2.3.3 Response of second order element input impulse 77

6.2.4 Third and Higher Order Systems 78

6.2.5 Effect of Root Location 79

6.3 TRANSIENT RESPONSE CHARACTERISTICS 81

6.4 STEP RESPONSE TESTING OF PRACTICAL SYSTEMS 83

6.4.1 Response Apparently of First Order 83

6.4.2 Response Apparently of Under Damped Second Order 84

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6.5 EXERCISE 85

CHAPTER 7: FREQUENCY RESPONSE 91

7.1 INTRODUCTION 91

7.2 CALCULATION OF THE FREQUENCY RESPONSE 93

7.3 POLAR PLOT (NYQUIST DIAGRAM) 95

7.3.1 Polar Plot for First Order Element 95

7.3.2 Polar Plot of Second Order Element 96

7.3.3 Polar Plot of Integrating Member 98

7.3.4 Polar Plot of Higher Order Elements 98

7.4 BODE DIAGRAM 99

7.4.1 Introduction 99

7.4.2 Bode Plot of Basic Elements 101

7.4.2.1 Proportional Element 101

7.4.2.2 Integrating Element 101

7.4.2.3 First order element (simple lag) 102

7.4.2.4 Simple lead element 104

7.4.2.5 Second order element 104

7.4.2.6 Quadratic lead element 106

7.5 NICHOL’S CHART 108

7.6 EXERCISE 109

CHAPTER 8: FEEDBACK SYSTEM ACCURACY AND STABILITY 113

8.1 INTRODUCTION 113

8.2 STEADY STATE ERROR 113

8.2.1 Steady State Error with Step Input 114

8.2.2 Steady State Error with Ramp Input 116

8.3 STABILITY OF FEEDBACK SYSTEMS 117

8.3.1 Routh-Hurwitz Stability Criterion 117

8.3.2 Nyquist Stability Criterion 120

8.3.2.1 Stability analysis using Nyquist plot 120

8.3.2.2 Stability analysis using Bode plot 129

8.3.2.3 Stability analysis using Nichol’s chart 130

8.4 EXERCISE 131

CHAPTER 9: ROOT LOCUS ANALYSIS 137

9.1 INTRODUCTION 137

9.2 INTERPRETATION OF ROOT LOCUS 138

9.3 EXERCISE 144

CHAPTER 10: COMPENSATION OF CONTROL SYSTEMS 147

10.1 INTRODUCTION 147

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10.2 PHASE LEAD COMPENSATOR 150

10.3 PHASE LAG COMPENSATOR 153

10.4 LAG-LEAD COMPENSATOR 155

10.5 P, PI AND PID CONTROLLERS 157

10.6 EXERCISE 162

REFERENCES 163

INDEX 165

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