Fundamentals of Control Theory : An Intuitive Approach from the Creator of Control System Lectures on YouTube

33 Pages·2015·8.565 MB·English

by  Brian Douglas

Fundamentals of Control Theory
An Intuitive Approach from the Creator of Control System Lectures on YouTube
Brian Douglas
Revision 1.0
Copyright©2016BrianDouglas
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Revision,1.0
PrintingDate,January23,2016
Contents
Preface i
I TheBigPicture 1
1 TheControlProblem 2
1.1 Whatisasystem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Thethreedifferentproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Thesystemidentificationproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Thesimulationproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Thecontrolproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Whydoweneedafeedbackcontrolsystem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Whatisacontrolsystem? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 TheFirstFeedbackControlSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 TryThis! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Appendices 24
A HowtoProvideFeedback 25
A.1 FillingouttheCreateissuescreen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Preface
WelcometotheFundamentalsofControlTheory! Thisbookisthedirectresultofmyonlinevideolectureson
controlsystemtheoryandtheoverwhelmingpositivefeedbackandencouragementI’vereceivedfrommyviewers
to write a book. I started my YouTube channel (https://youtube.com/ControlLectures) because I was
frustratedbythelackofstraightforwardandeasytounderstandvideosonthetopicandfeltthatwhatsomestudents
neededwasamorepracticalandintuitiveapproachtounderstandingthematerial. Thisbookisanextensionofthat
idea.
I’mreleasingthisbookonesectionatatime. SimilartohowIcreatenewvideosonamonthlybasisIwilladdnew
contenttothebookonamonthlyschedule. Thisallowsmetogettheearlychaptersouttherehelpingpeopleright
awaywhilebeingabletohaveyourreactionsandresponsesinfluencethelaterchapters.
So with that in mind, as I write this book there are four goals I hope to accomplish that I think will make it a
valuableresourceforanyaspiringcontrolsengineer.
1. Provide an intuitive understanding - Iwanttostartbysayingtherealreadyexistseveralfantastic
controlsystemtextbooks. Therefore,Idon’tthinkIwouldbeabletowriteausefulbookinthiscrowdedfield
by presenting the same information in the same formal format. So I’m not going to try to duplicate them,
insteadI’mcreatingabookthatisabitdifferent. Thelanguageisalittlelessformal-it’swrittenasthough
we’re having a conversation - and the mathematical proofs are a little more casual. However, I claim that
what you’ll learn from this book is just as useful as the existing textbooks because you’ll gain an overall
understandingoftheproblemandhowtoapproachit.
2. Update the book frequently - OneoftheluxuriesofmakinganeBookisthatIcandistributeupdates
quicklyandcheaply. Iamapproachingthisbookmorelikeadistributionofsoftware,wherebugfixesand
minorlayoutchangescanbeupdatedanddistributedasapointrelease(minorupdate)ratherthanamajor
new edition. With this model I can fix bugs and add content on a regular basis so that readers will always
havethemostuptodaterevision.
3. Allow the readers to participate in improving the book - How many times have you come
across an error in a textbook or a really confusing explanation and wished you had a way of providing
feedbackeasilytotheauthor? Iwanttohearthatfeedbackforthisbook! Itisyourfeedbackthatwilldrive
i
PREFACE ii
thequickpointreleasesandhelpmecreatethemostusefultextbookpossible. ThisiswhyIprovideasimple
ticketingsystemwhereyoucangivemefeedbackonerrorsincalculations,vagueorconfusingexplanations,
andmissingcontent. Iaskthatyoupleaseletmeknowanytimeyoucomeacrosssomethinginthebookso
thatIcanfixitanditwon’tconfusethenextroundofreaders. Fordetailsonhowtoprovidethatfeedback
seeAppendixA.
4. Make the book as inexpensive as possible - Lastly,collegetextbooksareexpensiveandifIwant
thisbooktoreallyhelpstudentsallovertheworldthenitneedstobeaffordable. Iunderstandthatstudents
don’thavemuchmoney1 andsohavingtobuyseveral$180bookseachsemesterisnothighonyourlistof
funactivities. ThatiswhyI’mreleasingthisbookundertheCreativeCommonsLicenseandgivingthebook
outforfreetoanyonewhosupportsmywork(whichincludesthevideosImake)throughkonoz.io. Youcan
gettomycreator’spageatkonozwiththislink.
https://konoz.io/briandouglas
For any amount of monthly support (even if it’s just $1 a month) you will have continuous access2 to the
bookandtoallfutureupdates. Ifyoudecideyounolongerwanttosupportmeyouwillstillgettokeepand
use the book you already have. So theoretically you could get the book for as little as $1. I think this is a
goodwayofallowingpeopletodecidehowmuchtheywanttosupportmewhilenotexcludingpeoplewho
reallywanttolearncontroltheorybutcan’taffordthebook.
Engineeringproblemsareinherentlymulti-disciplinaryandsoyouhaveyourchoiceoflearninganynumberof
specializedfieldsthatwillallowyoutocontributetoaproject. ButIthinkthebestreasontolearncontroltheoryis
thatitisthegluethatcombinesallotherengineeringfieldsandbyunderstandingthefundamentalsofcontroltheory
it opens the door for you to understand all of those other fields at a more basic level. It is actually a fascinating
subjectandthroughthisbookIhopetoinfectyouwiththesameenthusiasmforthesubjectthatIhave.
Chapter1 willdescribe thecontrol problem. This chapterwill setthestage forwhat we’re tryingto accomplish as
controlsystemengineersanddefinethetermsthatwe’llusethroughoutthisbook.
Oncewritten,therestofthebookwillcovertransferfunctions,howwerepresentsystemswithblockdiagrams,
andconceptslikesystemstability,time,frequency, discretedomains,andsystemidentification. We’llthencover
1WhenIwasincollegeIhadsolittlespendingmoneybytheendofthesemesterthatIwouldbuylargebagsofwhitericeandafewcondimentsand
theneatriceateverymeal;riceandhoney,riceandhotsauce,riceandmustard.
2AccessmeansyougettocopythePDFontoyourcomputer,putitonyoureReader,orprintitout,makecopiesofit,useitinyourpresentation,or
partofyourlecture,andevenshareitwithyourfriends!
PREFACE iii
howweusespecializedplottingtoolslikeRootLocus,Nyquistplots,andBodeplotstoanalyzeandunderstandour
system. Laterchapterswilldescribecompensationtechniqueslikeleadandlag,loopshaping,andPID.
By the end of this book I hope you realize that control system theory is so much more than just tuning a PID
controllerorgettinganinvertedpendulumtostandupright. It’sbuildingmodelsofyoursystemandsimulatingitto
makepredictions,it’sunderstandingthedynamicsandhowtheyinteractwiththerestofthesystem,it’sfiltering
outnoiseandrejectingoutsidedisturbances,it’sdesigningorselectingpropersensorsandactuators,andit’stesting
yoursystemtoensureit’llperformasexpectedinanunexpectedenvironment.
Now before you proceed any further I want to thank you for reading this book3 and for supporting me to keep
makingimprovementstothistext. Ihopeyougainabetterintuitionintocontroltheoryandultimatelyyoubecome
amorewell-roundedengineer.
Brian Douglas
3andthepreface!Whoreadstheprefaceanyway?

1 The Control Problem
In this chapter we’ll get an overview of the big picture problem that we’re trying to solve as control system
engineers. Thiswillgivecontexttoeverythingwe’llcoverinlaterchaptersandindoingsoIthinkwillhelpyou
understandwhyyouarelearningthetopicspresentedinthisbook.
1.1 What is a system?
To begin we describe exactly what a system is. The concept is really straight forward but since the term is so
generic we tend to apply the word to describe just about everything. This can get confusing to someone new to
the fieldwhen we refer tosomething called the controlsystem which isthen used to controlthe actual system and
whenputtogetherthetwopartsmakeyetanotherlargersystem. Assomeonelearningcontroltheorythequestion
becomeswhatsystemamIworkingon? Toanswerthislet’sstartwiththedefinitionandworkfromthere.
Asystemisacollectionofinterconnectedpartsthatformalargermorecomplexwhole.
Engineering projects aretypically complex. Dividing complex projects into smaller pieces, or systems, simplifies
the problem because it allows people to specialize in their functional area and not have to be a generalist in all
areas. Therefore,asaspecialistyoumightbeworkingonjustoneoftheinterconnectedpartsthatformtheentire
system. However,theremightbemanylayersofcomplexitysuchthatthesmallpartyouareworkingonisactually
acomplexsysteminitsownright!
Thesameistrueforspecialistsincontroltheory. Asacontrolengineeryourgoalistocreatesomethingthatmeets
the functional or performance requirements you set for the project. In general, we refer to the collection of the
interconnectedpartsthatarecreatedspecificallytomeettheserequirementsasthecontrolsystem. Foranyproject
otherthantheverysimplestones,however,thecontrolsystemagainmightbeacollectionofinterconnectedparts
thatrequirespecialistslikesensorexperts,actuatorsexperts,digitalsignalprocessingexperts,orstateestimation
experts.
2
Toillustratethislet’simaginethatyouhaveacceptedajobatanautomotivecompanyandyouwillbeworkingon
thebrakingsystem. Atfirstglanceyoumightsuspectthatyouwillbeinvolvedinallpartsrelatedtoslowingthe
vehicle. However,therearemanypartstothebrakingsystemonyourcarandittakesmanydifferentspecialiststo
designthecompleteproduct.
Themostobviouscomponentisthediscbrakeassemblyineachwheel. Thisisthepartthatisactuallyconverting
thecar’skineticenergyintoheatenergyandslowingdownthevehicle. Yetthediscbrakesaresmallsystemson
theirownbecausetheyaremadeupofrotors,calipers,brackets,shielding,fastenersandhoseswhichallowthe
discbrakestofunctioncorrectly.
Engagingthebrakesrequiresthebrakehydraulicsystemwhichisresponsiblefortransferringthepressureapplied
by your foot at the brake pedal through the power booster, dual master cylinder and the combination valve and
finallytothebrakecalipersateachofthefourwheels.
3
Thereis themechanicalparkingbrake systemthatbypassesthehydraulic systemwithasecondary cablepathto
the brakes and the brake light system that is responsible for lighting the tail lights and that annoying dashboard
lightthattellsyoutheparkingbrakeisengaged.
Finallythereareanynumberofelectronicbrakecontrolsystemsthatoverridethehumaninputtokeepthevehicle
fromskiddingonslicksurfacesoradistracteddriverfromcrashingintothecarinfrontofthem.
All of these smaller systems - the brakes, hydraulics, parking brake, lighting, and electronic controls - are the
interconnectedpartsthatformthelargerandcompletebrakingsystem. Furthermore,thebrakingsystemisjustone
ofmanyinterconnectedpartsthatcreatethecaritself.
Asacontrolspecialistinthebrakedepartmentyoumightberesponsibleforwritingandtestingthealgorithmfor
theelectronicbrakecontrolsystembuthaveverylittleimpacton,say,thecableroutingfortheparkingbrake.
4