Modern Control Engineering
Modern ControlEngineeringFifth EditionKatsuhiko OgataPrentice HallBoston Columbus Indianapolis New York San Francisco Upper Saddle RiverAmsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal TorontoDelhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
VP/Editorial Director, Engineering/Computer Science: Marcia J. HortonAssistant/Supervisor: Dolores MarsSenior Editor: Andrew GilfillanAssociate Editor: Alice DworkinEditorial Assistant: William OpaluchDirector of Marketing: Margaret WaplesSenior Marketing Manager: Tim GalliganMarketing Assistant: Mack PattersonSenior Managing Editor: Scott DisannoArt Editor: Greg DullesSenior Operations Supervisor: Alan FischerOperations Specialist: Lisa McDowellArt Director: Kenny BeckCover Designer: Carole AnsonMedia Editor: Daniel SandinCredits and acknowledgments borrowed from other sources and reproduced, with permission, in thistextbook appear on appropriate page within text.MATLAB is a registered trademark of The Mathworks, Inc., 3 Apple Hill Drive, Natick MA 01760-2098.Copyright 2010, 2002, 1997, 1990, 1970 Pearson Education, Inc., publishing as Prentice Hall, One LakeStreet, Upper Saddle River, New Jersey 07458. All rights reserved. Manufactured in the United States ofAmerica. This publication is protected by Copyright, and permission should be obtained from the publisherprior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by anymeans, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use materialfrom this work, please submit a written request to Pearson Education, Inc., Permissions Department, OneLake Street, Upper Saddle River, New Jersey 07458.Many of the designations by manufacturers and seller to distinguish their products are claimed astrademarks. Where those designations appear in this book, and the publisher was aware of a trademarkclaim, the designations have been printed in initial caps or all caps.Library of Congress Cataloging-in-Publication Data on File10 9 8 7 6 5 4 3 2 1ISBN 10: 0-13-615673-8ISBN 13: 978-0-13-615673-4
CContentsPrefaceChapter 11–11–21–31–41–52–6Introduction to Control Systems1Introduction1Examples of Control Systems4Closed-Loop Control Versus Open-Loop Control7Design and Compensation of Control Systems9Outline of the Book10Chapter 22–12–22–32–42–5ixMathematical Modeling of Control SystemsIntroduction13Transfer Function and Impulse-Response Function15Automatic Control Systems17Modeling in State Space29State-Space Representation of Scalar DifferentialEquation Systems35Transformation of Mathematical Models with MATLAB1339iii
2–7Linearization of Nonlinear Mathematical ModelsExample Problems and SolutionsProblemsChapter 360Mathematical Modeling of Mechanical Systemsand Electrical SystemsIntroduction3–2Mathematical Modeling of Mechanical Systems3–3Mathematical Modeling of Electrical SystemsProblems63728697Mathematical Modeling of Fluid Systemsand Thermal Systems4–1Introduction4–2Liquid-Level Systems4–3Pneumatic Systems1064–4Hydraulic Systems1234–5Thermal SystemsProblems101136140152Transient and Steady-State Response Analyses5–1Introduction5–2First-Order Systems5–3Second-Order Systems1645–4Higher-Order Systems1795–5Transient-Response Analysis with MATLAB5–6Routh’s Stability Criterion5–7Effects of Integral and Derivative Control Actionson System Performance2185–8Steady-State Errors in Unity-Feedback Control SystemsProblems159159161263183212Example Problems and SolutionsContents100100Example Problems and SolutionsChapter 56363Example Problems and Solutionsiv463–1Chapter 443231225
Chapter 6Control Systems Analysis and Designby the Root-Locus Method6–1Introduction6–2Root-Locus Plots6–3Plotting Root Loci with MATLAB6–4Root-Locus Plots of Positive Feedback Systems6–5Root-Locus Approach to Control-Systems Design6–6Lead Compensation6–7Lag Compensation6–8Lag–Lead Compensation6–9Parallel Compensation269270Chapter 7290303308311321330342Example Problems and SolutionsProblems269347394Control Systems Analysis and Design by theFrequency-Response Method7–1Introduction7–2Bode Diagrams7–3Polar Plots7–4Log-Magnitude-versus-Phase Plots7–5Nyquist Stability Criterion7–6Stability Analysis7–7Relative Stability Analysis7–8Closed-Loop Frequency Response of Unity-FeedbackSystems4777–9Experimental Determination of Transfer Functions3984034274434454544624867–10 Control Systems Design by Frequency-Response Approach7–11 Lead Compensation7–12 Lag Compensation493511Example Problems and SolutionsChapter 8521561PID Controllers and Modified PID Controllers8–1Introduction8–2Ziegler–Nichols Rules for Tuning PID ControllersContents4915027–13 Lag–Lead CompensationProblems398567567568v
8–38–48–58–68–7Design of PID Controllers with Frequency-ResponseApproach577Design of PID Controllers with Computational OptimizationApproach583Modifications of PID Control Schemes590Two-Degrees-of-Freedom Control592Zero-Placement Approach to Improve ResponseCharacteristics595Example Problems and Solutions614ProblemsChapter 99–19–29–39–49–59–69–7Control Systems Analysis in State SpaceChapter 10vi648Introduction648State-Space Representations of Transfer-FunctionSystems649Transformation of System Models with MATLAB656Solving the Time-Invariant State Equation660Some Useful Results in Vector-Matrix le Problems and 10–610–710–810–9641720Control Systems Design in State SpaceIntroduction722Pole Placement723Solving Pole-Placement Problems with MATLAB735Design of Servo Systems739State Observers751Design of Regulator Systems with Observers778Design of Control Systems with Observers786Quadratic Optimal Regulator Systems793Robust Control Systems806Example Problems and Solutions817Problems855Contents722
Appendix ALaplace Transform Tables859Appendix BPartial-Fraction Expansion867Appendix CVector-Matrix Algebra874References882Index886Contentsvii
This page intentionally left blank
PPrefaceThis book introduces important concepts in the analysis and design of control systems.Readers will find it to be a clear and understandable textbook for control system coursesat colleges and universities. It is written for senior electrical, mechanical, aerospace, orchemical engineering students. The reader is expected to have fulfilled the followingprerequisites: introductory courses on differential equations, Laplace transforms, vectormatrix analysis, circuit analysis, mechanics, and introductory thermodynamics.The main revisions made in this edition are as follows: The use of MATLAB for obtaining responses of control systems to various inputshas been increased. The usefulness of the computational optimization approach with MATLAB has beendemonstrated. New example problems have been added throughout the book. Materials in the previous edition that are of secondary importance have been deletedin order to provide space for more important subjects. Signal flow graphs weredropped from the book. A chapter on Laplace transform was deleted. Instead,Laplace transform tables, and partial-fraction expansion with MATLAB are presented in Appendix A and Appendix B, respectively. A short summary of vector-matrix analysis is presented in Appendix C; this will helpthe reader to find the inverses of n x n matrices that may be involved in the analysis and design of control systems.This edition of Modern Control Engineering is organized into ten chapters.The outline ofthis book is as follows: Chapter 1 presents an introduction to control systems. Chapter 2ix
deals with mathematical modeling of control systems. A linearization technique for nonlinear mathematical models is presented in this chapter. Chapter 3 derives mathematicalmodels of mechanical systems and electrical systems. Chapter 4 discusses mathematicalmodeling of fluid systems (such as liquid-level systems, pneumatic systems, and hydraulicsystems) and thermal systems.Chapter 5 treats transient response and steady-state analyses of control systems.MATLAB is used extensively for obtaining transient response curves. Routh’s stabilitycriterion is presented for stability analysis of control systems. Hurwitz stability criterionis also presented.Chapter 6 discusses the root-locus analysis and design of control systems, includingpositive feedback systems and conditionally stable systems Plotting root loci with MATLAB is discussed in detail. Design of lead, lag, and lag-lead compensators with the rootlocus method is included.Chapter 7 treats the frequency-response analysis and design of control systems. TheNyquist stability criterion is presented in an easily understandable manner.The Bode diagram approach to the design of lead, lag, and lag-lead compensators is discussed.Chapter 8 deals with basic and modified PID controllers. Computational approachesfor obtaining optimal parameter values for PID controllers are discussed in detail, particularly with respect to satisfying requirements for step-response characteristics.Chapter 9 treats basic analyses of control systems in state space. Concepts of controllability and observability are discussed in detail.Chapter 10 deals with control systems design in state space. The discussions includepole placement, state observers, and quadratic optimal control. An introductory discussion of robust control systems is presented at the end of Chapter 10.The book has been arranged toward facilitating the student’s gradual understandingof control theory. Highly mathematical arguments are carefully avoided in the presentation of the materials. Statement proofs are provided whenever they contribute to theunderstanding of the subject matter presented.Special effort has been made to provide example problems at strategic points so thatthe reader will have a clear understanding of the subject matter discussed. In addition,a number of solved problems (A-problems) are provided at the end of each chapter,except Chapter 1. The reader is encouraged to study all such solved problems carefully;this will allow the reader to obtain a deeper understanding of the topics discussed. Inaddition, many problems (without solutions) are provided at the end of each chapter,except Chapter 1. The unsolved problems (B-problems) may be used as homework orquiz problems.If this book is used as a text for a semester course (with 56 or so lecture hours), a goodportion of the material may be covered by skipping certain subjects. Because of theabundance of example problems and solved problems (A-problems) that might answermany possible questions that the reader might have, this book can also serve as a selfstudy book for practicing engineers who wish to study basic control theories.I would like to thank the following reviewers for this edition of the book: Mark Campbell, Cornell University; Henry Sodano, Arizona State University; and Atul G. Kelkar,Iowa State University. Finally, I wish to offer my deep appreciation to Ms.Alice Dworkin,Associate Editor, Mr. Scott Disanno, Senior Managing Editor, and all the people involved in this publishing project, for the speedy yet superb production of this book.Katsuhiko OgataxPreface
1Introductionto Control Systems1–1 INTRODUCTIONControl theories commonly used today are classical control theory (also called conventional control theory), modern control theory, and robust control theory. This bookpresents comprehensive treatments of the analysis and design of control systems basedon the classical control theory and modern control theory.A brief introduction of robustcontrol theory is included in Chapter 10.Automatic control is essential in any field of engineering and science. Automaticcontrol is an important and integral part of space-vehicle systems, robotic systems, modern manufacturing systems, and any industrial operations involving control of temperature, pressure, humidity, flow, etc. It is desirable that most engineers and scientists arefamiliar with theory and practice of automatic control.This book is intended to be a text book on control systems at the senior level at a college or university. All necessary background materials are included in the book. Mathematical background materials related to Laplace transforms and vector-matrix analysisare presented separately in appendixes.Brief Review of Historical Developments of Control Theories and Practices.The first significant work in automatic control was James Watt’s centrifugal governor for the speed control of a steam engine in the eighteenth century. Othersignificant works in the early stages of development of control theory were due to1
Minorsky, Hazen, and Nyquist, among many others. In 1922, Minorsky worked onautomatic controllers for steering ships and showed how stability could be determined from the differential equations describing the system. In 1932, Nyquistdeveloped a relatively simple procedure for determining the stability of closed-loopsystems on the basis of open-loop response to steady-state sinusoidal inputs. In 1934,Hazen, who introduced the term servomechanisms for position control systems,discussed the design of relay servomechanisms capable of closely following a changing input.During the decade of the 1940s, frequency-response methods (especially the Bodediagram methods due to Bode) made it possible for engineers to design linear closedloop control systems that satisfied performance requirements. Many industrial controlsystems in 1940s and 1950s used PID controllers to control pressure, temperature, etc.In the early 1940s Ziegler and Nichols suggested rules for tuning PID controllers, calledZiegler–Nichols tuning rules. From the end of the 1940s to the 1950s, the root-locusmethod due to Evans was fully developed.The frequency-response and root-locus methods, which are the core of classical control theory, lead to systems that are stable and satisfy a set of more or less arbitrary performance requirements. Such systems are, in general, acceptable but not optimal in anymeaningful sense. Since the late 1950s, the emphasis in control design problems has beenshifted from the design of one of many systems that work to the design of one optimalsystem in some meaningful sense.As modern plants with many inputs and outputs become more and more complex,the description of a modern control system requires a large number of equations. Classical control theory, which deals only with single-input, single-output systems, becomespowerless for multiple-input, multiple-output systems. Since about 1960, because theavailability of digital computers made possible time-domain analysis of complex systems, modern control theory, based on time-domain analysis and synthesis using statevariables, has been developed to cope with the increased complexity of modern plantsand the stringent requirements on accuracy, weight, and cost in military, space, and industrial applications.During the years from 1960 to 1980, optimal control of both deterministic and stochastic systems, as well as adaptive and learning control of complex systems, were fullyinvestigated. From 1980s to 1990s, developments in modern control theory were centered around robust control and associated topics.Modern control theory is based on time-domain analysis of differential equationsystems. Modern control theory made the design of control systems simpler becausethe theory is based on a model of an actual control system. However, the system’sstability is sensitive to the error between the actual system and its model. Thismeans that when the designed controller based on a model is applied to the actualsystem, the system may not be stable. To avoid this situation, we design the controlsystem by first setting up the range of possible errors and then designing the controller in such a way that, if the error of the system stays within the assumedrange, the designed control system will stay stable. The design method based on thisprinciple is called robust control theory. This theory incorporates both the frequencyresponse approach and the time-domain approach. The theory is mathematically verycomplex.2Chapter 1 / Introduction to Control Systems
Because this theory requires mathematical background at the graduate level, inclusion of robust control theory in this book is limited to introductory aspects only. Thereader interested in details of robust control theory should take a graduate-level controlcourse at an established college or university.Definitions. Before we can discuss control systems, some basic terminologies mustbe defined.Controlled Variable and Control Signal or Manipulated Variable. The controlledvariable is the quantity or condition that is measured and controlled. The control signalor manipulated variable is the quantity or condition that is varied by the controller soas to affect the value of the controlled variable. Normally, the controlled variable is theoutput of the system. Control means measuring the value of the controlled variable ofthe system and applying the control signal to the system to correct or limit deviation ofthe measured value from a desired value.In studying control engineering, we need to define additional terms that are necessary to describe control systems.Plants. A plant may be a piece of equipment, perhaps just a set of machine partsfunctioning together, the purpose of which is to perform a particular operation. In thisbook, we shall call any physical object to be controlled (such as a mechanical device, aheating furnace, a chemical reactor, or a spacecraft) a plant.Processes. The Merriam–Webster Dictionary defines a process to be a natural, progressively continuing operation or development marked by a series of gradual changesthat succeed one another in a relatively fixed way and lead toward a particular result orend; or an artificial or voluntary, progressively continuing operation that consists of a series of controlled actions or movements systematically directed toward a particular result or end. In this book we shall call any operation to be controlled a process. Examplesare chemical, economic, and biological processes.Systems. A system is a combination of components that act together and performa certain objective. A system need not be physical. The concept of the system can beapplied to abstract, dynamic phenomena such as those encountered in economics. Theword system should, therefore, be interpreted to imply physical, biological, economic, andthe like, systems.Disturbances. A disturbance is a signal that tends to adversely affect the valueof the output of a system. If a disturbance is generated within the system, it is calledinternal, while an external disturbance is generated outside the system and isan input.Feedback Control. Feedback control refers to an operation that, in the presenceof disturbances, tends to reduce the difference between the output of a system and somereference input and does so on the basis of this difference. Here only unpredictable disturbances are so specified, since predictable or known disturbances can always be compensated for within the system.Section 1–1/Introduction3
1–2 EXAMPLES OF CONTROL SYSTEMSIn this section we shall present a few examples of control systems.Speed Control System. The basic principle of a Watt’s speed governor for an engine is illustrated in the schematic diagram of Figure 1–1. The amount of fuel admittedto the engine is adjusted according to the difference between the desired and the actualengine speeds.The sequence of actions may be stated as follows: The speed governor is adjusted such that, at the desired speed, no pressured oil will flow into either side ofthe power cylinder. If the actual speed drops below the desired value due todisturbance, then the decrease in the centrifugal force of the speed governor causesthe control valve to move downward, supplying more fuel, and the speed of theengine increases until the desired value is reached. On the other hand, if the speedof the engine increases above the desired value, then the increase in the centrifugal force of the governor causes the control valve to move upward. This decreasesthe supply of fuel, and the speed of the engine decreases until the desired value isreached.In this speed control system, the plant (controlled system) is the engine and thecontrolled variable is the speed of the engine. The difference between the desiredspeed and the actual speed is the error signal. The control signal (the amount of fuel)to be applied to the plant (engine) is the actuating signal. The external input to disturb the controlled variable is the disturbance. An unexpected change in the load isa disturbance.Temperature Control System. Figure 1–2 shows a schematic diagram of temperature control of an electric furnace. The temperature in the electric furnace is measured by a thermometer, which is an analog device. The analog temperature is convertedPowercylinderOil underpressurePilotvalveFigure 1–1Speed controlsystem.4CloseOpenFuelControlvalveChapter 1 / Introduction to Control SystemsEngineLoad
furnaceProgrammedinputFigure 1–2Temperature controlsystem.RelayAmplifierInterfaceHeaterto a digital temperature by an A/D converter. The digital temperature is fed to a controller through an interface. This digital temperature is compared with the programmedinput temperature, and if there is any discrepancy (error), the controller sends out a signal to the heater, through an interface, amplifier, and relay, to bring the furnace temperature to a desired value.Business Systems. A business system may consist of many groups. Each taskassigned to a group will represent a dynamic element of the system. Feedback methodsof reporting the accomplishments of each group must be established in such a system forproper operation. The cross-coupling between functional groups must be made a minimum in order to reduce undesirable delay times in the system. The smaller this crosscoupling, the smoother the flow of work signals and materials will be.A business system is a closed-loop system. A good design will reduce the managerial control required. Note that disturbances in this system are the lack of personnel or materials, interruption of communication, human errors, and the like.The establishment of a well-founded estimating system based on statistics is mandatory to proper management. It is a well-known fact that the performance of such a systemcan be improved by the use of lead time, or anticipation.To apply control theory to improve the performance of such a system, we must represent the dynamic characteristic of the component groups of the system by a relatively simple set of equations.Although it is certainly a difficult problem to derive mathematical representationsof the component groups, the application of optimization techniques to business systems significantly improves the performance of the business system.Consider, as an example, an engineering organizational system that is composed ofmajor groups such as management, research and development, preliminary design, experiments, product design and drafting, fabrication and assembling, and tesing. Thesegroups are interconnected to make up the whole operation.Such a system may be analyzed by reducing it to the most elementary set of components necessary that can provide the analytical detail required and by representing thedynamic characteristics of each component by a set of simple equations. (The dynamicperformance of such a system may be determined from the relation between progressive accomplishment and time.)Section 1–2/Examples of Control Systems5
liminarydesignExperimentsProductdesign igure 1–3Block diagram of an engineering organizational system.A functional block diagram may be drawn by using blocks to represent the functional activities and interconnecting signal lines to represent the information orproduct output of the system operation. Figure 1–3 is a possible block diagram forthis system.Robust Control System. The first step in the design of a control system is toobtain a mathematical model of the plant or control object. In reality, any model of aplant we want to control will include an error in the modeling process. That is, the actualplant differs from the model to be used in the design of the control system.To ensure the controller designed based on a model will work satisfactorily whenthis controller is used with the actual plant, one reasonable approach is to assumefrom the start that there is an uncertainty or error between the actual plant and itsmathematical model and include such uncertainty or error in the design process of thecontrol system. The control system designed based on this approach is called a robustcontrol system.苲Suppose that the actual plant we want to control is G(s) and the mathematical modelof the actual plant is G(s), that is,苲G(s) actual plant model that has uncertainty (s)G(s) nominal plant model to be used for designing the control system苲G(s) and G(s) may be related by a multiplicative factor such as苲G(s) G(s)[1 (s)]or an additive factor苲G(s) G(s) (s)or in other forms.Since the exact description of the uncertainty or error (s) is unknown, we use anestimate of (s) and use this estimate, W(s), in the design of the controller. W(s) is ascalar transfer function such that冟冟 (s)冟冟q 6 冟冟W(s)冟冟q max 冟W(jv)冟0 v qwhere 冟冟W(s)冟冟q is the maximum value of 冟W(jv)冟 for 0 v q and is called the Hinfinity norm of W(s).6Chapter 1 / Introduction to Control Systems
Using the small gain theorem, the design procedure here boils down to the determination of the controller K(s) such that the inequalityßW(s)ß1 K(s)G(s)6 1qis satisfied, where G(s) is the transfer function of the model used in the design process,K(s) is the transfer function of the controller, and W(s) is the chosen transfer functionto approximate (s). In most practical cases, we must satisfy more than one suchinequality that involves G(s), K(s), and W(s)’s. For example, to guarantee robust stability and robust performance we may require two inequalities, such asßWm(s)K(s)G(s)ß1 K(s)G(s)6 1for robust stabilityqßWs(s)ß1 K(s)G(s)6 1for robust performanceqbe satisfied. (These inequalities are derived in Section 10–9.) There are many differentsuch inequalities that need to be satisfied in many different robust control systems.(Robust stability means that the controller K(s) guarantees internal stability of allsystems that belong to a group of systems that include the system with the actual plant.Robust performance means the specified performance is satisfied in all systems that belong to the group.) In this book all the plants of control systems we discuss are assumedto be known precisely, except the plants we discuss in Section 10–9 where an introductory aspect of robust control theory is presented.1–3 CLOSED-LOOP CONTROL VERSUS OPEN-LOOP CONTROLFeedback Control Systems. A system that maintains a prescribed relationshipbetween the output and the reference input by comparing them and using the differenceas a means of control is called a feedback control system. An example would be a roomtemperature control system. By measuring the actual room temperature and comparingit with the reference temperature (desired temperature), the thermostat turns the heating or cooling equipment on or off in such a way as to ensure that the room temperature remains at a comfortable level regardless of outside conditions.Feedback control systems are not limited to engineering but can be found in variousnonengineering fields as well. The human body, for instance, is a highly advanced feedback control system. Both body temperature and blood pressure are kept constant bymeans of physiological feedback. In fact, feedback performs a vital function: It makesthe human body relatively insensitive to external disturbances, thus enabling it to function properly in a changing environment.Section 1–3/Closed-Loop Control versus Open-Loop Control7
Closed-Loop Control Systems. Feedback control systems are often referred toas closed-loop control systems. In practice, the terms feedback control and closed-loopcontrol are used interchangeably. In a closed-loop control system the actuating errorsignal, which is the difference between the input signal and the feedback signal (whichmay be the output signal itself or a function of the output signal and its derivativesand/or integrals), is fed to the controller so as to reduce the error and bring the outputof the system to a desired value. The term closed-loop control always implies the use offeedback control action in order to reduce system error.Open-Loop Control Systems. Those systems in which the output has no effecton the control action are called open-loop control systems. In other words, in an openloop control system the output is neither measured nor fed back for comparison with theinput. One practical example is a washing machine. Soaking, washing, and rinsing in thewasher operate on a time basis. The machine does not measure the output signal, thatis, the cleanliness of the clothes.In any open-loop control system the output is not compared with the reference input.Thus, to each reference input there corresponds a fixed operating condition; as a result,the accuracy of the system depends on calibration. In the presence of disturbances, anopen-loop control system will not perform the desired task. Open-loop control can beused, in practice, only if the relationship between the input and output is known and ifthere are neither internal nor external disturbances. Clearly, such systems are not feedback control systems. Note that any control system that operates on a time basis is openloop. For instance, traffic control by means of signals operated on a time basis is anotherexample of open-loop control.Closed-Loop versus O
ventional control theory), modern control theory, and robust control theory.This book presents comprehensive treatments of the analysis and design of control systems based on the classical control theory and modern control theory.A brief introduction of robust control theory is included in Chapter 10.
Materials Science and Engineering, Mechanical Engineering, Production Engineering, Chemical Engineering, Textile Engineering, Nuclear Engineering, Electrical Engineering, Civil Engineering, other related Engineering discipline Energy Resources Engineering (ERE) The students’ academic background should be: Mechanical Power Engineering, Energy .
Control theories commonly used today are classical control theory (also called con-ventional control theory), modern control theory, and robust control theory.This book presents comprehensive treatments of the analysis and design of control systems based on the classical control theory and modern control theory.A brief introduction of robust
Careers in Engineering Guide the brighter choice. Contents ABOUT LSBU 4–5 BUILDING SERVICES ENGINEERING 6–7 CHEMICAL AND PETROLEUM ENGINEERING 8–9 CIVIL ENGINEERING 10–11 ELECTRICAL AND ELECTRONIC ENGINEERING 12–13 MECHANICAL ENGINEERING 14–15 MECHATRONICS ENGINEERING 16–17 PRODUCT DESIGN ENGINEERING 18–19 An engineering degree is a big challenge to take on. There is no denying .
OLE MISS ENGINEERING RECOMMENDED COURSE SCHEDULES Biomedical engineering Chemical engineering Civil engineering Computer engineering Computer science Electrical engineering General engineering Geological engineering Geology Mechanical engineering Visit engineering.olemiss.edu/advising for full course information.
on the classical control theory and modern control theory.A brief introduction of robust control theory is included in Chapter 10. Automatic control is essential in any field of engineering and science. Automatic control is an important and integral part of space-vehicle systems,robotic systems,mod-
on the classical control theory and modern control theory.A brief introduction of robust control theory is included in Chapter 10. Automatic control is essential in any field of engineering and science. Automatic control is an important and integral part of space-vehicle systems,robotic systems,mod-
Civil Engineering 30 Computer Systems Engineering 32 Engineering Science 34 Electrical and Electronic Engineering 36 Mechanical Engineering 38 Mechatronics Engineering 40 Software Engineering 42 Structural Engineering 44 Course descriptions 46 APPENDIX 84 Find out more 88. 2 Dates to remember 06 Jan Summer School begins 12 Jan Last day to add, change or delete Summer School Courses 01 Feb .
ACCOUNTING 0452/22 Paper 2 October/November 2018 1 hour 45 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction .
