A Guide to Feedback Theory

203 Pages·2021·1.855 MB·English

by  Joel L Dawson

AGuidetoFeedbackTheory
Do you need to understand feedback? Perhaps you’re a little rusty on theory basics?
Dig in to this self-contained guide for an accessible and concise explanation of the
fundamentals.
(cid:129) Distillstherelevantessenceoflinearsystemtheory,calculus,differentialequations,
linearalgebra,basicphysics,numericalmethods,andcomplexanalysisandlinks
thembacktoanexplanationoffeedbacktheory.
(cid:129) Providesatightsynthesisofanalyticalandconceptualunderstanding.
(cid:129) Maintainsafocusoncommonusecases.
Whether you are a struggling undergraduate, a doctoral student preparing for your
qualifyingexams,oranindustrypractitioner,thiseasy-to-understandbookinvitesyou
torelax,enjoythematerial,andfollowyourcuriosity.
joel l. dawson is an entrepreneur and former MIT professor. He is a 2009
recipient of the PECASE Award, the highest honor bestowed by the US government
on young scientists and engineers. His last start-up company, Eta Devices, Inc., was
aTechnologyPioneerofthe2015WorldEconomicForumandacquiredbyNokiain
2016.Hiscurrentstart-upcompanyisTalkingHeadsWireless,Inc.
“Feedbacktheoryisanintrinsicallymathematicaldisciplineinwhichonecanfeeleither
submergedbyformulaeordriventouseblindcomputersimulationsthathideinsight.
Dawson’s approach is to extract visceral meaning out of this tangle, arguing that a
deepunderstandingofdynamicstabilitycriteriacanfreethedesignerfrom“equational
overload” and lead to incisive selection of the right mathematical tool for the job at
hand.”
StephenD.Senturia,MassachusettsInstituteofTechnology
“Feedbackisperhapsthemostfoundationalconceptforelectronicsandcontrolsystems
ingeneral,butitisoftencoveredforspecificcircuitsfortheformer,andintermsofthe-
oreticalconceptsforthelatter.Thisbookprovidesuswithauniqueperspectiveonhow
feedbacktheoryingeneralrelatestopracticalsystemsandelectronicsapplications.”
LarryPileggi,CarnegieMellonUniversity
A Guide to Feedback Theory
Joel L. Dawson
TalkingHeadsWireless,Inc.
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Informationonthistitle:www.cambridge.org/9780521199216
DOI:10.1017/9781139018289
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Contents
Preface pagexi
Acknowledgments xiii
1 LinearSystems:WhatYouMissedtheFirstTime 1
1.1 DifferentialEquationsAreaNaturalWaytoExpressTime
Evolution 1
1.1.1 AFirst-OrderSystem 1
1.1.2 Higher-OrderSystems 8
1.1.3 For Those of You Bothered by the Numerical
FittinginSection1.1.1 12
1.2 ConvenientPropertiesofLinearDifferentialEquations 12
1.2.1 Superposition! 13
1.2.2 TheSpecialPlaceofExponentials 15
1.2.3 But...WhyComplexExponentials? 19
1.3 FrequencyDomainMethods:ABeautifulStrategy 22
1.3.1 FourierSeriesRepresentationofPeriodicSignals 23
1.3.2 TheFourierTransformandtheMeaningofIntegrals 26
1.3.3 TheStrategy 28
1.4 ImpulsesinLinear,Time-InvariantSystems 29
1.4.1 WhyImpulses? 29
1.4.2 TheFourierTransformandtheImpulseResponse 31
1.4.3 TheFourierTransformofDifferentialEquations 33
1.5 TheUnilateralLaplaceTransform 35
1.5.1 DynamicInterpretationofPoles 36
1.5.2 TheGeometricViewofPolesandZeros 40
1.5.3 InitialandFinalValueTheorems 44
1.5.4 InvertingtheLaplaceTransform 46
1.6 ConvolutionandtheSpecialPlaceofExponentials 47
vii
viii Contents
1.7 Discrete-TimeFormalism:SameIdeas,DifferentNotation 47
1.7.1 DifferenceEquationsAreaReallyNatural
ExpressionofTimeEvolution 47
1.7.2 TheFourierTransforminDiscreteTime 49
1.7.3 The Z-Transform, the Impulse Response, and
ConvolutioninDiscreteTime 50
1.8 ChapterSummary 50
2 TheBasicsofFeedback 52
2.1 FillingaGlasswithWater 52
2.2 Open-versusClosed-LoopControlinBlockDiagrams 55
2.3 AnatomyofaFeedbackLoop 58
2.3.1 BlockDiagrams 58
2.3.2 SensorsandActuators 61
2.3.3 LoopTransmission,NegativeFeedback,and
StableEquilibria 66
2.3.4 Black’sFormula 69
2.4 DelayComplicatesEverything 70
2.4.1 PhaseResponseasaFrequency-DependentDelay 71
2.4.2 TheFundamentalOscillationCondition 74
2.4.3 PolesintheRight-HalfPlaneAreBad 75
2.5 RootLocusTechniques 77
2.5.1 TheProblemWe’reTryingtoSolve 79
2.5.2 TheAmazingThingsYouCanDowithTwo
SimpleConditions 82
2.5.3 RootLocusasaDesignTool 89
2.5.4 RootLocusinDiscreteTime 99
2.5.5 AUsefulLimitofDT 102
2.6 CommonControlStrategies 105
2.6.1 GainReduction 105
2.6.2 DominantPolesandIntegrators 107
2.6.3 LagandLeadCompensators 108
2.6.4 PIDControl 111
2.7 AnswerstoSampleProblems 113
3 TheNyquistStabilityCriterion 116
3.1 AnAuthoritativeTestofStability 116
3.1.1 TrueDelayandRootLocus 117
3.2 ANoteonConformalMapping 117
3.3 Cauchy’sPrincipleoftheArgument 119
Contents ix
3.4 AndNow... theNyquistStabilityCriterion 122
3.5 BodePlotsHelpwithNyquist 126
3.6 NyquistPlotExamples 131
3.7 PhaseMargin:WhyYouNeverReallyLearnedNyquist 137
3.7.1 TheStabilityMarginConcept 138
3.7.2 PhaseMarginDefinition 139
3.7.3 PhaseMargin,Overshoot,Ringing,and
MagnitudePeaking 145
3.8 NyquistandBodeTechniquesforDTSystems 146
4 SomeCommonLooseEnds 147
4.1 “ButinControlTheory,TheyUseLotsofLinearAlgebra...” 147
4.2 TheProblemof“SinusoidsRunningAroundLoops” 150
4.3 Discrete-TimeControlofContinuous-TimeSystems 157
4.3.1 DTProcessingofCTSignals 158
4.3.2 Don’tKidAround:JustOversample 163
4.3.3 Relationshipbetweenzands inMixed-SignalControl 165
4.3.4 DTCompensatorsforCTSystems 169
4.3.5 TheOtherUsefulExtreme:SlowSampling 169
4.3.6 ANoteontheBiastowardCTMethods 169
4.3.7 Sometimes,Real-TimeComputerControlIsHopeless 170
5 FeedbackintheRealWorld 172
5.1 FindingLoopTransmissions 172
5.1.1 IstheSignRight?AUsefulCheck 174
5.2 ACommonApplication:HowlingSpeakersandMicrophones 175
6 ConclusionandFurtherReading 181
Index 183