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Problems and Solutions ofControl SystemsWith Essential TheorySixth Edition

Problems and Solutions ofControl SystemsWith Essential TheorySixth EditionAK JAIRATHME, FIE, FIETECBS Publishers & DistributorsPvt LtdNew Delhi Bengaluru Chennai Kochi Mumbai PuneHyderabad Kolkata Nagpur Patna Vijayawada

DisclaimerScience and technology are constantly changing fields. New research and experience broaden the scope of information andknowledge. The authors have tried their best in giving information available to them while preparing the material for thisbook. Although, all efforts have been made to ensure optimum accuracy of the material, yet it is quite possible some errorsmight have been left uncorrected. The publisher, the printer and the authors will not be held responsible for any inadvertenterrors, omissions or inaccuracies.eISBN: 978 81 239 xxxx xCopyright Authors and PublisherFirst eBook Edition: 2017All rights reserved. No part of this eBook may be reproduced or transmitted in any form or by any means, electronic ormechanical, including photocopying, recording, or any information storage and retrieval system without permission, inwriting, from the authors and the publisher.Published by Satish Kumar Jain and produced by Varun Jain forCBS Publishers & Distributors Pvt. Ltd.Corporate Office: 204 FIE, Industrial Area, Patparganj, New Delhi 110092Ph: 91 11 49344934; Fax: 91 11 49344935; Website:; www.eduport;E mail:; marketing@eduport global.comHead Office: CBS PLAZA, 4819/XI Prahlad Street, 24 Ansari Road, Daryaganj, New Delhi 110002, India.Ph: 91 11 23289259, 23266861, 23266867; Fax: 011 23243014; Website:;E mail:; Bengaluru: Seema House 2975, 17 th Cross, K.R. Road, Banasankari 2 nd Stage, Bengaluru 560070, KarnatakaPh: 91 80 26771678/79; Fax: 91 80 26771680; E mail: Chennai: No.7, Subbaraya Street Shenoy Nagar Chennai 600030, Tamil NaduPh: 91 44 26680620, 26681266; E mail: Kochi: 36/14 Kalluvilakam, Lissie Hospital Road, Kochi 682018, KeralaPh: 91 484 4059061 65; Fax: 91 484 4059065; E mail: Mumbai: 83 C, 1 st floor, Dr. E. Moses Road, Worli, Mumbai 400018, MaharashtraPh: 91 22 24902340 41; Fax: 91 22 24902342; E mail: Kolkata: No. 6/B, Ground Floor, Rameswar Shaw Road, Kolkata 700014Ph: 91 33 22891126 28; E mail: kolkata@cbspd.comRepresentatives Hyderabad Pune Nagpur Manipal Vijayawada Patna

tomy wife Geeta andson Varunwho are my source of encouragement and inspirationandmy dearest father late Shri Ram Prasad Jairath andmother Shrimati Prem Rani Jairathwho are responsible forwhat I am today.

Preface to the Sixth EditionIt gives me immense pleasure to bring out the sixth edition of my book whichover the last 22 years has become immensely popular with the faculty aswell as the students who are pursuing undergraduate programs and also thosestudents who are appearing in the professional examinations like GATE, IES,etc. The sixth edition has been thoroughly revised and restructured to ensurethat it is student-friendly. Problems which already existed have been suitablyclubbed under new chapters for easier understanding of the topics. New problems have been added in each chapter to make the concepts more clear and togive the students extra problem-solving practice. Chapter on feedback and itseffect on system performance has been added. Nyquist stability criterionchapter has been fully revised and the explanation for each problem in thischapter will ensure better assimilation. Chapter 1 has been revised and theconcept of drawing block diagrams for practical systems have been explainedin a very simple way. Students must know how to recognize controllers, processes/plants and feedback elements of various types of control systems. I amsure that the explanation of the basics will lay the foundation for the controlsystem subject in totality and make understanding of the other related topicseasier.It will be my endeavour to keep on improving the book in the coming yearsand the faculty and students will be seeing new editions of this book shortly. Iplan to include design problems with case studies in my future editions. My aimis to ensure that this book is error-free. My students in Amity University havehelped me a lot in pointing out typing/spelling errors and I owe a lot to them. Iwill be grateful to all the readers of this book to point out such mistakes infuture also. I will acknowledge their efforts suitably.I thank my publishers, CBS Publishers & Distributors, in supporting myefforts. Ours have been a very long association and hope this bond will lastforever.AK Jairath

Preface to the Fifth EditionIt gives me immense pleasure in bringing out the fifth edition of this book whichhas the patronage of the student community all over India. The main endeavourof this book is to make available a series of worked exercises to assist students toprevail over difficulties in understanding and correlating with the theory. Theobjective has been not to just present an assortment of problems with solutions,but by means of straightforward preparatory problems leading to intricate ones,introduce the students to the fundamentals of control engineering. In keepingwith the aim, this edition has been compiled in a fashion to ensure the following: Step-by-step mathematical explanation for better understanding. Basics are explained in simple language. All possible variations to the principles of control engineering have beenexplained and illustrated by solving different problems. The problems and their solutions are listed in the order of difficulty andcomplexity. Step-by-step explanation will help the students in saving time which can bedevoted elsewhere, fruitfully. Most of the problems have appeared in different examinations. This willhelp the students gain more confidence in tackling the examinations. Essential theory included under each chapter in the beginning is to ensurethat students do not refer to any textbook for cross reference. It will alsohelp students to immediately correlate to various graphical illustrations depictedin the theory portions. To keep the size and cost of the book within realistic limits, introductorymaterial in the form of indispensable theory to each chapter contains onlythe vital features of the topics.I once again thank CBS Publishers & Distributors and their publishing staff forproviding help and support. I sincerely hope that this edition of the book will beas popular as the previous ones.AK Jairath

Prefaces to the Previous EditionsPreface to the Fourth EditionAdecade has passed and with the extreme cooperation from the studentsand the publisher, I present the fourth revised edition. The book has beenthoroughly revised, especially the chapter on “Root-locus”. Basic essentialtheory on sensitivity has been added in the fourth chapter. In addition, important solved problems in each chapter have been incorporated.I have no doubt that the fourth revised edition, with minimum errors willhelp the readers to assimilate the subject better.AK JairathIPreface to the Third Editiont is a matter of happiness for me that within a short span of seven years thisbook has become immensely popular and the students have benefited afterreading it. It has been my endeavour to improve the book constantly. Keepingthis in view, I have revised the third edition by adding most essential theorywhich will help students to assimilate the subject better.I have no doubt that the present revised edition will be liked by all thereaders.AK JairathIPreface to the Second Editiont gives me immense pleasure and satisfaction in bringing out this book inyour hands. This book contains typical examples in control system engineering divided into various chapters. A brief giving out the basic theory hasbeen added at the begining of each chapter which I am sure will help thestudents in understanding the solved problems. Problems have been solvedwith detailed explanation for better assimilation.Effort has been made to keep printing mistakes to the minimum. Even thenif the students are able to find any technical or printing mistakes, please bringthe same to the notice of the author for rectification in the next edition. Also,any suggestions to improve this book are welcome. I thank CBS Publishers &Distributors for their cooperation and help.AK Jairath

ContentsPreface to the Sixth EditionPreface to the Fifth EditionPrefaces to the Previous Editions1. CONTROL SYSTEMS: BASICS1. is Control Systems 1Classification of Systems 1Classification Based on the Parameters 2Analysis of Control Systems 3General Classification: Open and Closed-Loop Systems 3Elements of Automatic or Feedback Control Systems 5Requirements of Automatic Control Systems 62. MODELLING OF CONTROL SYSTEMS: ELECTRICAL SYSTEMS202.1 Introduction 202.2 Transfer Function 203. MODELLING OF CONTROL SYSTEMS:BLOCK DIAGRAM METHOD463.1 Introduction 463.2 Block Diagram 463.3 Block Diagram of a Closed-Loop System 474. MODELLING OF CONTROL SYSTEMS: SIGNAL FLOW GRAPH4. 92Signal Flow Graph 92Signal-Flow Diagram Reduction 94Mason’s Gain Formula 955. MODELLING OF CONTROL SYSTEMS: PHYSICAL SYSTEMS5.1 Introduction 1815.2 Basic Elements 1815.3 Translatory Motion 181181

xiiProblems and Solutions of Control Systems5.4 Rotational Motion 1835.5 Analogous System1856. TIME DOMAIN ANALYSIS2276.1 Introduction 2276.2 Standard Test Signals 2276.3 Static Accuracy 2296.4 Computation of Steady State Errors 2316.5 Transient Response: First Order System 2336.6 Transient Response: Second Order System 2376.7 Transient Response Specifications 2426.8 Control Actions 2466.9 Impulse Signal and Response 2586.10 Sensitivity 2597. CONTROL SYSTEMS: FEEDBACK CHARACTERISTICS3257.1 Introduction 3257.2 Steady State Error/Accuracy 3257.3 Sensitivity 3277.4 Overall Gain 3307.5 Time Constant 3307.6 Stability 3327.7 Disturbance 3338. POLAR PLOTS3468.1 General 3468.2 Polar Plot Sketching Procedure3439. STABILITY9.1 Introduction 3589.2 Stability 3589.3 Stability Study Methods35835810. ROUTH’S STABILITY CRITERION10.1 Introduction 36110.2 Procedure for Application 361361

Contentsxiii11. NYQUIST STABILITY on 392Relationship 392Encirclement 393Enclosed 394Stability Condition 395Principle of Argument 396Nyquist Contour and Nyquist Plot39912. BODE PLOT12.112.2468Introduction 468Method of Plotting 46813. ROOT LOCUS13.1 Introduction 52713.2 Simple Control System13.3 General Rules 52852752814. STATE SPACE ANALYSIS14.114.214.314.414.514.614.714.8General 595State 595State Variables 595State Vector 595State Space 595Controllability and ObservabilityControllability 596Observability 597595596

Problems And Solutions Of ControlSystemsPublisher : CBS PublicationsISBN : 9788123925721Author : A. K. JairathType the URL : this eBook

1. CONTROL SYSTEMS: BASICS 1 1.1 What is Control Systems 1 1.2 Classification of Systems 1 1.3 Classification Based on the Parameters 2 1.4 Analysis of Control Systems 3 1.5 General Classification: Open and Closed-Loop Systems 3 1.6 Elements of Automatic or Feedback Control Systems 5 1.7 Requirements of Automatic Control Systems 6 2.

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