Computational Aids in Control Systems Using MATLAB - Saadat.us

200 Pages·2000·1.68 MB·English

by  Dr. Hadi Saadat

COMPUTATIONAL AIDS IN CONTROL
SYSTEMS USING MATLAB
Hadi Saadat
ProfessorofElectricalEngineering
MilwaukeeSchoolofEngineering
Milwaukee, Wisconsin
Computational Aids inControl Systems UsingMATLAB
Copyright c
(cid:13)
CONTENTS
PREFACE vi
1 INTRODUCTION TOMATLAB 1
1.1 INSTALLINGTHE TEXTTOOLBOX . . . . . . . . . . . . . . . . 2
1.2 RUNNINGMATLAB. . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 OUTPUTFORMAT . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 CHARACTERSTRING . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 VECTOROPERATIONS . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 ELEMENTARYMATRIXOPERATIONS . . . . . . . . . . . . . . . 10
1.7.1 UTILITYMATRICES . . . . . . . . . . . . . . . . . . . . . 12
1.7.2 EIGENVALUES . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 COMPLEXNUMBERS . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 POLYNOMIALROOTS
ANDCHARACTERISTICPOLYNOMIAL . . . . . . . . . . . . . . 15
1.9.1 PRODUCTANDDIVISIONOFPOLYNOMIALS . . . . . . 16
1.9.2 POLYNOMIALCURVEFITTING . . . . . . . . . . . . . . 17
1.9.3 POLYNOMIALEVALUATION . . . . . . . . . . . . . . . . 18
1.9.4 PARTIAL-FRACTIONEXPANSION . . . . . . . . . . . . . 18
1.10 GRAPHICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.11 GRAPHICSHARDCOPY . . . . . . . . . . . . . . . . . . . . . . . 21
1.12 THREE-DIMENSIONALPLOTS . . . . . . . . . . . . . . . . . . . 27
1.13 HANDLEGRAPHICS . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.14 LOOPSANDLOGICALSTATEMENTS . . . . . . . . . . . . . . . 30
1.15 SIMULATIONDIAGRAM . . . . . . . . . . . . . . . . . . . . . . . 36
1.16 INTRODUCTIONTOSIMULINK . . . . . . . . . . . . . . . . . . 38
1.16.1 SIMULATIONPARAMETERSANDSOLVER . . . . . . . 39
1.16.2 THESIMULATIONPARAMETERSDIALOGBOX. . . . . 40
1.16.3 BLOCKDIAGRAMCONSTRUCTION . . . . . . . . . . . 41
1.16.4 USINGTHETOWORKSPACEBLOCK . . . . . . . . . . . 47
iii
iv CONTENTS
1.16.5 LINEARSTATE-SPACE
MODELFROMSIMULINKDIAGRAM . . . . . . . . . . . 47
1.16.6 SUBSYSTEMSANDMASKING . . . . . . . . . . . . . . . 49
2 Mathematical Modelsof Systems 51
2.1 Differential EquationsofPhysicalSystems. . . . . . . . . . . . . . . 51
2.2 NumericalSolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5 TransferFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5.1 PolynomialRootsandCharacteristic Polynomial . . . . . . . 59
2.5.2 PolesandZerosofaTransferFunction . . . . . . . . . . . . 60
2.5.3 Partial-Fraction Expansion . . . . . . . . . . . . . . . . . . . 61
3 State-Space Representation 63
3.1 State-Variable Modeling . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Equations ofElectricalNetworks . . . . . . . . . . . . . . . . . . . . 64
3.3 TransferFunctiontoState-SpaceConversion . . . . . . . . . . . . . 66
3.4 State-SpacetoTransferFunctionConversion . . . . . . . . . . . . . 66
3.5 SimilarityTransformation . . . . . . . . . . . . . . . . . . . . . . . . 67
3.5.1 Diagonalization oftheAMatrix . . . . . . . . . . . . . . . . 67
3.5.2 Transformation toPhase-variable . . . . . . . . . . . . . . . 68
3.6 SolutionoftheStateEquation . . . . . . . . . . . . . . . . . . . . . 70
3.7 LaplaceTransformofStateTransitionMatrix, . . . . . . . . . . 70
3.8 Evaluation of fromtheCharacteristic Valu(cid:8)es(so)fA . . . . . . . . 71
3.8.1 Cayley(cid:30)-(Ht)amilton Method . . . . . . . . . . . . . . . . . . . 71
3.9 NumericalSolutionoftheStateEquation . . . . . . . . . . . . . . . 73
3.10 BlockDiagramReduction . . . . . . . . . . . . . . . . . . . . . . . 74
4 SystemResponses 77
4.1 TheResponseofSecond-Order Systems . . . . . . . . . . . . . . . . 77
4.2 Time-DomainPerformanceSpecifications . . . . . . . . . . . . . . . 78
4.3 EffectsofAdditionalPolesandZeros . . . . . . . . . . . . . . . . . 79
4.3.1 AdditionofaZero . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.2 AdditionofaPole . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Frequency ResponseofSystems . . . . . . . . . . . . . . . . . . . . 83
4.5 ControlSystemToolbox
LTIModelsandLTIViewer . . . . . . . . . . . . . . . . . . . . . . 91
4.5.1 LTIModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5.2 TheLTIViewer . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 ControlSystemCharacteristics 94
5.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 TheRouth-HurwitzStabilityCriterion . . . . . . . . . . . . . . . . . 95
5.2.1 SpecialCases . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
CONTENTS v
5.4 Steady-State ErrorandSystemType . . . . . . . . . . . . . . . . . . 100
6 Root-Locus Analysis andDesign 104
6.1 Root-LocusMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 SummaryofGeneralRulesforConstructing Root-Loci . . . . . . . . 106
6.3 Root-LocusDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 GainFactorCompensation orPController . . . . . . . . . . . . . . . 118
6.5 Phase-LeadDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.6 Phase-LagDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.7 PIDDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.7.1 PDController . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.7.2 PIController . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7.3 PIDController . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.8 Minor-Loop FeedbackControl . . . . . . . . . . . . . . . . . . . . . 132
6.9 RateFeedbackorTachometer-Feedback Control . . . . . . . . . . . . 132
6.10 Feedback Compensation usingPassiveElements. . . . . . . . . . . . 135
6.11 GUIProgramforRoot-locus Design . . . . . . . . . . . . . . . . . . 137
7 FrequencyResponse AnalysisandDesign 142
7.1 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.1.1 BodePlot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.1.2 PolarPlot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.1.3 Log-magnitude versusPhasePlot . . . . . . . . . . . . . . . 145
7.2 RelativeStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.1 GainandPhaseMargins . . . . . . . . . . . . . . . . . . . . 146
7.2.2 NyquistStabilityCriterion . . . . . . . . . . . . . . . . . . . 148
7.2.3 SimplifiedNyquistCriterion . . . . . . . . . . . . . . . . . . 148
7.3 Closed-Loop FrequencyResponse . . . . . . . . . . . . . . . . . . . 150
7.3.1 NicholsChart . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4 Frequency-Response Design . . . . . . . . . . . . . . . . . . . . . . 152
7.5 GainFactorCompensation or Controller . . . . . . . . . . . . . . 154
7.6 Phase-LeadDesign . . . . . P. . . . . . . . . . . . . . . . . . . . . . 157
7.7 Phase-LagDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.8 PIDDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.8.1 PDController . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.8.2 PIController . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.8.3 PIDController . . . . . . . . . . . . . . . . . . . . . . . . . 167
8 Modern Control Design 170
8.1 Pole-Placement Design . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.3 ObserverDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.4 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.5 CombinedController-Observer Design . . . . . . . . . . . . . . . . . 178
8.6 OptimalRegulatorDesign . . . . . . . . . . . . . . . . . . . . . . . 180
PREFACE
Thistextisintended toprovide assistance insolving computational problems associ-
ated with the study and application of linear control systems. It is written expressly
to support the use of MATLABas a part of an introductory course in automatic con-
trol systems. MATLAB, developed by Math Works, Inc., is an interactive system for
scientific and engineering computation and must be purchased separately from this
book.
TheobjectiveistointroducetheusertosomeofthecapabilitiesofMATLAB,and
theassociated ControlSystem Toolbox, sothatitcanbeused toaidinthedesign and
analysis ofcontrol systems. Thetextcontains sufficient detail andover75examples.
These examples are designed with two objectives in mind. First, to teach the student
howtowriteMATLABprograms, andsecond, toallow theuser toadvance rapidly in
solving more advanced control system problems. The organization of this text is as
follows:
Chapter 1 is an introduction to MATLABand its capabilities. It describes sev-
eral examples dealing with matrix manipulations, complex algebra, and the graphics
featuresofMATLAB.
Chapter2showshowtoobtainthenumericalsolutionofdifferentialequations,
polynomialrootsandcharacteristicpolynomials,polesandzerosoftransferfunctions,
andpartialfractionexpansion.
Chapter 3 explains how to perform state-space transformations such as state-
space to transfer function, state-space to zero-pole, transfer function to state-space,
conversion ofblockdiagramstostate-space modelsandreduced transferfunctions.
Chapter4dealswithtimeresponse ofsecond-order systems, timedomainper-
formance specifications, and the effects of adding poles and zeros to the closed-loop
transfer function. Anexampleofmodelreduction isalsopresented. Examplesoffre-
quencyresponse, including somefrequency domainspecifications, aregiven.
Chapter5covers someoftheessential characteristics offeedback control sys-
temssuchasstability, sensitivity, andsteady-state errors.
Chapter 6 includes Routh-Hurwitz Stability Criterion, root-locus analysis, de-
signofphase-lead, phase-lag, andPIDcontrollers byroot-locus techniques.
vi
PREFACE vii
Chapter7dealswithgainandphasemargin,Bode,Nyquist,Nicholsplots,and
designofcontrolsystemsinfrequency domain.
Chapter 8 deals with the design of control systems based on modern control
theory. Pole-placement, stateestimatorandoptimalregulatordesigns arepresented.
I wish to express my sincere appreciation to my colleague Professor George
Smith,ofMilwaukee SchoolofEngineering forhiscareful andmeticulous reviewof
the manuscript and his numerous constructive suggestions, which improved its con-
tent.Iamespecially gratefultoProfessorRayPalmer,chairmanofthedepartment of
ElectricalEngineeringandComputerScience,MilwaukeeSchoolofEngineering,for
his aid and encouragement to begin the project in the first place. I express my pro-
found gratitude toMs.LynnKallaswhospent manyhours editing andcorrecting the
manuscript. I am appreciative of the assistance given by the reviewers of this book:
Professor Ashok Ambardar, Michigan Technological University; Professor Thomas
J. Csermely, Syracuse University; Professor Shankar P.Bhattacharyya, Texas A&M;
Professor Lee Keel, Tennessee State University; Professor Surjit S. Mahil, Purdue
UniversityCalumet.Theconstructive commentsandencouragement provided bymy
editor Anne Brown of McGraw-Hill are greatly appreciated. Finally, I express my
gratitudeandlovetomyfamily,withoutwhomthisundertaking wouldnothavebeen
possible.
HadiSaadat
CHAPTER
1
INTRODUCTION
TO MATLAB
MATLAB,developedbyMathWorksInc.,isasoftwarepackageforhighperformance
numerical computation and visualization. The combination of analysis capabilities,
flexibility, reliability, and powerful graphics makes MATLAB the premier software
package forelectrical engineers.
MATLABprovides aninteractive environment withhundreds ofreliable andac-
curate built-in mathematical functions. These functions provide solutions to a broad
rangeofmathematicalproblemsincluding matrixalgebra, complexarithmetic, linear
systems, differential equations, signal processing, optimization, nonlinear systems,
andmanyothertypesofscientificcomputations. ThemostimportantfeatureofMAT-
LABisitsprogramming capability, whichisveryeasytolearnandtouse, andwhich
allows user-developed functions. It also allows access to Fortran algorithms and C
codesbymeansofexternalinterfaces.Thereareseveraloptionaltoolboxeswrittenfor
special applications suchassignalprocessing, control systemsdesign, system identi-
fication, statistics, neural networks, fuzzy logic, symbolic computations, and others.
MATLABhasbeenenhancedbytheverypowerfulSIMULINKprogram.SIMULINKis
agraphicalmouse-drivenprogramforthesimulationofdynamicsystems.SIMULINK
enablesstudentstosimulatelinear,aswellasnonlinear,systemseasilyandefficiently.
The following section describes the use of MATLAB and is designed to give
a quick familiarization with some of the commands and capabilities of MATLAB.
For a description of all other commands, MATLABfunctions, and many other useful
features, thereaderisreferred totheMATLABUser’sGuide.
1
2 1. INTRODUCTION TOMATLAB
1.1 INSTALLING THE TEXT TOOLBOX
The software diskette included with the book contains all the developed functions
andchapter examples. Thefilenamesforchapter examples beginwiththeletters ch.
For example, the M-file for Example 2.4 is ch2ex04. Create a subdirectory, such as
HS, where the MATLABR11toolbox resides. Copy all the files on the diskette to the
subdirectory MATLABR11 TOOLBOX HS.
IntheMATLAB5.3CommandWindowopenthePathBrowserbyselectingSet
n n
Path from the File menu. Press to open the AddtoPath window. Open the toolbox
folder and double-click on the HS folder. Choose Add to Back option and click on
SaveSettingstosavethenewpathpermanently.
1.2 RUNNINGMATLAB
MATLABsupports almost every computational platform. MATLABfor WINDOWSis
started by clicking on the MATLAB icon. The Command window is launched, and
aftersomemessagessuchasintro,demo,helphelp,info,andothers, theprompt“
”isdisplayed.Theprogramisinaninteractivecommandmode.Typingwhoorwhos
(cid:29)
displays a list of variable names currently in memory. Also, the dir command lists
allthe filesonthedefault directory. MATLABhasanon-line help facility, and itsuse
ishighlyrecommended. Thecommandhelpprovides alistoffiles,built-in functions
andoperators forwhichon-linehelpisavailable. Thecommand
function name
help
willgiveinformationonthespecifiedfunctionastoitspurposeanduse.Thecommand
help help
willgiveinformation astohowtousetheon-linehelp.
MATLAB has a demonstration program that shows many of its features. The
command demobrings up amenuoftheavailable demonstrations. Thiswillprovide
a presentation of the most important MATLAB facilities. Follow the instructions on
thescreen–itisworthtrying.
MATLAB5.3 includes a Help Desk facility that provides access to on line help
topics,documentation,gettingstartedwithMATLAB,onlinereferencematerials,MAT-
LABfunctions,real-timeWorkshop,andseveraltoolboxes.Theonlinedocumentation
is available in HTML, via either Netscape Navigator or Microsoft Internet Explorer.
The command helpdesk launches the Help Desk, or you can use the Help menu to
bringuptheHelpDesk.
If an expression with correct syntax is entered at the prompt in the Command
window, it is processed immediately and the result is displayed on the screen. If an
expression requires more than one line, the last character of the previous line must
1.2.RUNNINGMATLAB 3
contain three dots “...”. Characters following the percent sign are ignored. The ( )
may be used anywhere in a program to add clarifying comments. This is especia%lly
helpfulwhencreatingaprogram.ThecommandclearerasesallvariablesintheCom-
mandwindow.
MATLABisalsocapableofexecutingsequencesofcommandsthatarestoredin
files, known as script files or M-files. Clicking on File, Open M-file, opens the Edit
window.AprogramcanbewrittenandsavedinASCIIformatwithafilenamehaving
extension .m in the directory where MATLABruns. To run the program, click on the
Command window and type the filename without the .m extension at the MATLAB
command “ ”. You can view the text Edit window simultaneously with the Com-
mand window. That is, you can use the two windows to edit and debug a script file
(cid:29)
repeatedly andrunitintheCommandwindowwithouteverquittingMATLAB.
InadditiontotheCommandwindowandEditwindowaretheGraphicwindows
orFigurewindowswithgrey(default) background. Theplotscreated bythegraphic
commandsappearinthesewindows.
Another type of M-file is a function file. A function provides a convenient way
to encapsulate some computation, which can then be used without worrying about
its implementation. In contrast to the script file, a function file has a name following
the word “function” at the beginning of the file. The filename must be the same as
the “function” name. The first line of a function file must begin with the function
statementhaving thefollowingsyntax
function [output arguments]=function name(inputarguments)
Theoutputargument(s)arevariablesreturned.Afunctionneednotreturnavalue.The
input arguments are variables passed tothe function. Variables generated in function
filesarelocaltothefunction.Theuseofglobalvariablesmakedefinedvariablescom-
monandaccessiblebetweenthemainscriptfileandotherfunctionfiles.Forexample,
the statement global R S T declares the variables , , and to be global without
the need for passing the variables through the inpuRt liSst. ThisTstatement goes before
anyexecutablestatementinthescriptandfunctionfilesthatneedtoaccessthevalues
oftheglobalvariables.
Normally, while an M-file is executing, the commands of the file are not dis-
played on the screen. The command echo allows M-files to be viewed as they exe-
cute. echo off turns off the echoing of all script files. Typing what lists M-files and
Mat-filesinthedefault directory.
MATLABfollowsconventional Windowsprocedure. Information fromthecom-
mand screen can be printed byhighlighting the desired text withthe mouse andthen
choosing the print Selected ... from the File menu. If no text is highlighted the en-
tireCommand window isprinted. Similarly, selecting printfrom the Figurewindow
sendstheselectedgraphtotheprinter.Foracompletelistandhelpongeneralpurpose
commands,typehelpgeneral.