classical feedback control with matlab® and simulink® 2nd ed - boris lurie, paul enright (crc,...

Electrical Engineering

K12752

Classical Feedback Control

Automation and Control Engineering Series

Boris J. Lurie and Paul J. Enright

With MATLAB and Simulink

SECOND EDIT ION

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SECOND EDIT ION

An ideal instructional tool, Classical Feedback Control: With MATLAB and Simulink, Second Edition lays out the techniques that the authors found most useful in their experience designing and analyzing control systems for industry, telecommunications, and space programs. Designed to prepare readers to conduct research in the area of high-performance nonlinear controllers, the book is logically organized in layers that build on each other, covering linear/nonlinear systems, feedback, modeling, and simulation.

New in the Second Edition: Integrates MATLAB throughout the text

Analyzes frequency response methods, including multi-loop extension of the Nyquist stability criteria Covers the Bode integral relations and their use in performance/stability trade-offs Describes effective methods for dealing with plant-flexible modes

This text addresses the inevitable nonlinearities involved in engineering control systems and describes both digital and continuous systems to clarify how they work in terms of theory and practical applications. Emphasizing the frequency-domain design and methods based on Bode integrals, loop shaping, and nonlinear dynamic compensation, the authors describe optimized techniques for the construction and implementation of high-performance feedback controllers in engineering systems.

They also discuss practical applications using illustrations and plots that employ MATLAB and Simulink software for simulation and design examples. The book contains a wealth of problems accompanied by actual solutions, and material is reinforced by case studies that deal with real-world situations.

K12752_Cover_mech.indd 1 8/26/11 4:04 PM



SECOND EDIT ION

AUTOMATION AND CONTROL ENGINEERINGA Series of Reference Books and Textbooks

Series Editors

Classical Feedback Control: With MATLAB and Simulink, Second Edition, Boris J. Lurie and Paul J. Enright

Synchronization and Control of Multiagent Systems, Dong Sun

Subspace Learning of Neural Networks, Jian Cheng Lv, Zhang Yi, and Jiliu Zhou

Reliable Control and Filtering of Linear Systems with Adaptive Mechanisms, Guang-Hong Yang and Dan Ye

Reinforcement Learning and Dynamic Programming Using Function Approximators, Lucian Busoniu, Robert Babuka, Bart De Schutter, and Damien Ernst

Modeling and Control of Vibration in Mechanical Systems, Chunling Du and Lihua Xie

Analysis and Synthesis of Fuzzy Control Systems: A Model-Based Approach, Gang Feng

Lyapunov-Based Control of Robotic Systems, Aman Behal, Warren Dixon, Darren M. Dawson, and Bin Xian

System Modeling and Control with Resource-Oriented Petri Nets, Naiqi Wu and MengChu Zhou

Sliding Mode Control in Electro-Mechanical Systems, Second Edition, Vadim Utkin, Jrgen Guldner, and Jingxin Shi

Optimal Control: Weakly Coupled Systems and Applications, Zoran Gajic, Myo-Taeg Lim, Dobrila Skataric, Wu-Chung Su, and Vojislav Kecman

Intelligent Systems: Modeling, Optimization, and Control, Yung C. Shin and Chengying Xu

Optimal and Robust Estimation: With an Introduction to Stochastic Control Theory, Second Edition, Frank L. Lewis, Lihua Xie, and Dan Popa

Feedback Control of Dynamic Bipedal Robot Locomotion, Eric R. Westervelt, Jessy W. Grizzle, Christine Chevallereau, Jun Ho Choi, and Benjamin Morris

Intelligent Freight Transportation, edited by Petros A. Ioannou

Modeling and Control of Complex Systems, edited by Petros A. Ioannou and Andreas Pitsillides

Wireless Ad Hoc and Sensor Networks: Protocols, Performance, and Control, Jagannathan Sarangapani

Stochastic Hybrid Systems, edited by Christos G. Cassandras and John Lygeros

Hard Disk Drive: Mechatronics and Control, Abdullah Al Mamun, Guo Xiao Guo, and Chao Bi

Autonomous Mobile Robots: Sensing, Control, Decision Making and Applications, edited by Shuzhi Sam Ge and Frank L. Lewis

FRANK L. LEWIS, Ph.D.,Fellow IEEE, Fellow IFAC

ProfessorAutomation and Robotics Research Institute

The University of Texas at Arlington

SHUZHI SAM GE, Ph.D.,Fellow IEEE

ProfessorInteractive Digital Media Institute

The National University of Singapore

CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York


Automation and Control Engineering Series


SECOND EDIT ION

Boris J. Lurie and Paul J. Enright

MATLAB is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This books use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB software.

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vContents

Preface............................................................................................................................................ xiiiTo.Instructors..................................................................................................................................xv

1. FeedbackandSensitivity.......................................................................................................11.1. Feedback.Control.System..............................................................................................11.2. Feedback:.Positive.and.Negative.................................................................................31.3. Large.Feedback...............................................................................................................41.4. Loop.Gain.and.Phase.Frequency.Responses.............................................................6

1.4.1. Gain.and.Phase.Responses..............................................................................61.4.2. Nyquist.Diagram..............................................................................................91.4.3. Nichols.Chart.................................................................................................. 10

1.5. Disturbance.Rejection................................................................................................. 121.6. Example.of.System.Analysis...................................................................................... 131.7. Effect.of.Feedback.on.the.Actuator.Nondynamic.Nonlinearity........................... 181.8. Sensitivity...................................................................................................................... 191.9. Effect.of.Finite.Plant.Parameter.Variations.............................................................. 211.10. Automatic.Signal.Level.Control................................................................................. 211.11. Lead.and.PID.Compensators.....................................................................................221.12. Conclusion.and.a.Look.Ahead...................................................................................23Problems...................................................................................................................................23Answers.to.Selected.Problems..............................................................................................33

2. Feedforward,Multiloop,andMIMOSystems................................................................352.1. Command.Feedforward..............................................................................................352.2. Prefilter.and.the.Feedback.Path.Equivalent.............................................................382.3. Error.Feedforward....................................................................................................... 392.4. Blacks.Feedforward.Method..................................................................................... 392.5. Multiloop.Feedback.Systems...................................................................................... 412.6. Local,.Common,.and.Nested.Loops..........................................................................422.7. Crossed.Loops.and.Main/Vernier.Loops.................................................................432.8. Manipulations.of.Block.Diagrams.and.Calculations.of...

Transfer.Functions.......................................................................................................452.9. MIMO.Feedback.Systems...........................................................................................48Problems................................................................................................................................... 51

3. FrequencyResponseMethods............................................................................................ 573.1. Conversion.of.Time.Domain.Requirements.to.Frequency.Domain..................... 57

3.1.1. Approximate.Relations.................................................................................. 573.1.2. Filters................................................................................................................ 61

3.2. Closed-Loop.Transient.Response..............................................................................643.3. Root.Locus.....................................................................................................................653.4. Nyquist.Stability.Criterion.......................................................................................... 673.5. Robustness.and.Stability.Margins............................................................................. 703.6. Nyquist.Criterion.for.Unstable.Plant........................................................................ 74

vi Contents

3.7. Successive.Loop.Closure.Stability.Criterion............................................................ 763.8. Nyquist.Diagrams.for.the.Loop.Transfer.Functions.with.Poles.at.

the.Origin...................................................................................................................... 783.9. Bode.Integrals...............................................................................................................80

3.9.1. Minimum.Phase.Functions...........................................................................803.9.2. Integral.of.Feedback....................................................................................... 823.9.3. Integral.of.Resistance.....................................................................................833.9.4. Integral.of.the.Imaginary.Part......................................................................853.9.5. Gain.Integral.over.Finite.Bandwidth...........................................................863.9.6. Phase-Gain.Relation.......................................................................................86

3.10. Phase.Calculations....................................................................................................... 893.11. From.the.Nyquist.Diagram.to.the.Bode.Diagram................................................... 913.12. Nonminimum.Phase.Lag............................................................................................ 933.13. Ladder.Networks.and.Parallel.Connections.of.m.p..Links.................................... 94Problems................................................................................................................................... 96Answers.to.Selected.Problems............................................................................................ 101

4. ShapingtheLoopFrequencyResponse.......................................................................... 1034.1. Optimality.of.the.Compensator.Design................................................................. 1034.2. Feedback.Maximization............................................................................................ 105

4.2.1. Structural.Design.......................................................................................... 1054.2.2. Bode.Step........................................................................................................ 1064.2.3. Example.of.a.System.Having.a.Loop.Response.with.a.......

Bode.Step........................................................................................................ 1094.2.4. Reshaping.the.Feedback.Response............................................................ 1184.2.5. Bode.Cutoff.................................................................................................... 1194.2.6. Band-Pass.Systems........................................................................................ 1204.2.7. Nyquist-Stable.Systems................................................................................ 121

4.3. Feedback.Bandwidth.Limitations............................................................................ 1234.3.1. Feedback.Bandwidth.................................................................................... 1234.3.2. Sensor.Noise.at.the.System.Output............................................................ 1234.3.3. Sensor.Noise.at.the.Actuator.Input............................................................ 1254.3.4. Nonminimum.Phase.Shift.......................................................................... 1264.3.5. Plant.Tolerances............................................................................................ 1274.3.6. Lightly.Damped.Flexible.Plants:.Collocated.and.

Noncollocated.Control................................................................................. 1284.3.7. Unstable.Plants.............................................................................................. 133

4.4. Coupling.in.MIMO.Systems..................................................................................... 1334.5. Shaping.Parallel.Channel.Responses...................................................................... 135Problems................................................................................................................................. 139Answers.to.Selected.Problems............................................................................................ 142

5. CompensatorDesign........................................................................................................... 1455.1. Accuracy.of.the.Loop.Shaping................................................................................. 1455.2. Asymptotic.Bode.Diagram....................................................................................... 1465.3. Approximation.of.Constant.Slope.Gain.Response............................................... 1495.4. Lead.and.Lag.Links................................................................................................... 1505.5. Complex.Poles............................................................................................................. 1525.6. Cascaded.Links.......................................................................................................... 154

Contents vii

5.7. Parallel.Connection.of.Links.................................................................................... 1575.8. Simulation.of.a.PID.Controller................................................................................. 1595.9. Analog.and.Digital.Controllers............................................................................... 1625.10. Digital.Compensator.Design.................................................................................... 163

5.10.1. Discrete.Trapezoidal.Integrator.................................................................. 1635.10.2. Laplace.and.Tustin.Transforms.................................................................. 1655.10.3. Design.Sequence........................................................................................... 1685.10.4. Block.Diagrams,.Equations,.and.Computer.Code................................... 1685.10.5. Compensator.Design.Example................................................................... 1705.10.6. Aliasing.and.Noise....................................................................................... 1745.10.7. Transfer.Function.for.the.Fundamental.................................................... 176

Problems................................................................................................................................. 177Answers.to.Selected.Problems............................................................................................ 182

6. AnalogControllerImplementation................................................................................. 1916.1. Active.RC.Circuits...................................................................................................... 191

6.1.1. Operational.Amplifier.................................................................................. 1916.1.2. Integrator.and.Differentiator....................................................................... 1936.1.3. Noninverting.Configuration....................................................................... 1946.1.4. Op-Amp.Dynamic.Range,.Noise,.and.Packaging................................... 1956.1.5. Transfer.Functions.with.Multiple.Poles.and.Zeros.................................. 1966.1.6. Active.RC.Filters............................................................................................ 1986.1.7. Nonlinear.Links............................................................................................200

6.2. Design.and.Iterations.in.the.Element.Value.Domain........................................... 2026.2.1. Cauer.and.Foster.RC.Two-Poles.................................................................. 2026.2.2. RC-Impedance.Chart.................................................................................... 205

6.3. Analog.Compensator,.Analog.or.Digitally.Controlled........................................ 2076.4. Switched-Capacitor.Filters........................................................................................ 208

6.4.1. Switched-Capacitor.Circuits........................................................................ 2086.4.2. Example.of.Compensator.Design............................................................... 208

6.5. Miscellaneous.Hardware.Issues.............................................................................. 2106.5.1. Ground........................................................................................................... 2106.5.2. Signal.Transmission..................................................................................... 2116.5.3. Stability.and.Testing.Issues......................................................................... 213

6.6. PID.Tunable.Controller............................................................................................. 2146.6.1. PID.Compensator.......................................................................................... 2146.6.2. TID.Compensator......................................................................................... 216

6.7. Tunable.Compensator.with.One.Variable.Parameter........................................... 2186.7.1. Bilinear.Transfer.Function........................................................................... 2186.7.2. Symmetrical.Regulator................................................................................ 2196.7.3. Hardware.Implementation.......................................................................... 220

6.8. Loop.Response.Measurements................................................................................ 221Problems.................................................................................................................................225Answers.to.Selected.Problems............................................................................................ 229

7. LinearLinksandSystemSimulation.............................................................................. 2317.1. Mathematical.Analogies........................................................................................... 231

7.1.1. Electromechanical.Analogies...................................................................... 231

viii Contents

7.1.2. Electrical.Analogy.to.Heat.Transfer...........................................................2347.1.3. Hydraulic.Systems........................................................................................235

7.2. Junctions.of.Unilateral.Links.................................................................................... 2377.2.1. Structural.Design.......................................................................................... 2377.2.2. Junction.Variables......................................................................................... 2377.2.3. Loading.Diagram..........................................................................................238

7.3. Effect.of.the.Plant.and.Actuator.Impedances.on.the.Plant.Transfer.Function.Uncertainty................................................................................................ 239

7.4. Effect.of.Feedback.on.the.Impedance.(Mobility)................................................... 2417.4.1. Large.Feedback.with.Velocity.and.Force.Sensors.................................... 2417.4.2. Blackmans.Formula..................................................................................... 2427.4.3. Parallel.Feedback.......................................................................................... 2437.4.4. Series.Feedback............................................................................................. 2437.4.5. Compound.Feedback................................................................................... 244

7.5. Effect.of.Load.Impedance.on.Feedback.................................................................. 2457.6. Flowchart.for.Chain.Connection.of.Bidirectional.Two-Ports.............................. 246

7.6.1. Chain.Connection.of.Two-Ports.................................................................. 2467.6.2. DC.Motors...................................................................................................... 2497.6.3. Motor.Output.Mobility................................................................................2507.6.4. Piezoelements................................................................................................ 2517.6.5. Drivers,.Transformers,.and.Gears.............................................................. 2517.6.6. Coulomb.Friction..........................................................................................253

7.7. Examples.of.System.Modeling.................................................................................2547.8. Flexible.Structures..................................................................................................... 257

7.8.1. Impedance.(Mobility).of.a.Lossless.System.............................................. 2577.8.2. Lossless.Distributed.Structures..................................................................2587.8.3. Collocated.Control........................................................................................ 2597.8.4. Noncollocated.Control................................................................................. 260

7.9. Sensor.Noise............................................................................................................... 2607.9.1. Motion.Sensors.............................................................................................. 260

7.9.1.1. Position.and.Angle.Sensors......................................................... 2607.9.1.2. Rate.Sensors................................................................................... 2617.9.1.3. Accelerometers.............................................................................. 2627.9.1.4. Noise.Responses............................................................................ 262

7.9.2. Effect.of.Feedback.on.the.Signal-to-Noise.Ratio...................................... 2637.10. Mathematical.Analogies.to.the.Feedback.System.................................................264

7.10.1. Feedback-to-Parallel-Channel.Analogy....................................................2647.10.2. Feedback-to-Two-Pole-Connection.Analogy............................................264

7.11. Linear.Time-Variable.Systems.................................................................................. 265Problems................................................................................................................................. 267Answers.to.Selected.Problems............................................................................................ 271

8. IntroductiontoAlternativeMethodsofControllerDesign....................................... 2738.1. QFT............................................................................................................................... 2738.2. Root.Locus.and.Pole.Placement.Methods.............................................................. 2758.3. State-Space.Methods.and.Full-State.Feedback......................................................277

8.3.1. Comments.on.Example.8.3..........................................................................2808.4. LQR.and.LQG............................................................................................................. 2828.5. H,.-Synthesis,.and.Linear.Matrix.Inequalities...................................................283

Contents ix

9. AdaptiveSystems................................................................................................................2859.1. Benefits.of.Adaptation.to.the.Plant.Parameter.Variations...................................2859.2. Static.and.Dynamic.Adaptation............................................................................... 2879.3. Plant.Transfer.Function.Identification....................................................................2889.4. Flexible.and.n.p..Plants..............................................................................................2889.5. Disturbance.and.Noise.Rejection............................................................................ 2899.6. Pilot.Signals.and.Dithering.Systems....................................................................... 2919.7. Adaptive.Filters.......................................................................................................... 292

10. ProvisionofGlobalStability............................................................................................ 29510.1. Nonlinearities.of.the.Actuator,.Feedback.Path,.and.Plant................................... 29510.2. Types.of.Self-Oscillation........................................................................................... 29710.3. Stability.Analysis.of.Nonlinear.Systems................................................................ 298

10.3.1. Local.Linearization....................................................................................... 29810.3.2. Global.Stability.............................................................................................. 299

10.4. Absolute.Stability.......................................................................................................30010.5. Popov.Criterion.......................................................................................................... 301

10.5.1. Analogy.to.Passive.Two-Poles.Connection.............................................. 30110.5.2. Different.Forms.of.the.Popov.Criterion.....................................................304

10.6. Applications.of.Popov.Criterion..............................................................................30510.6.1. Low-Pass.System.with.Maximum.Feedback............................................30510.6.2. Band-Pass.System.with.Maximum.Feedback...........................................306

10.7. Absolutely.Stable.Systems.with.Nonlinear.Dynamic.Compensation................30610.7.1. Nonlinear.Dynamic.Compensator.............................................................30610.7.2. Reduction.to.Equivalent.System................................................................. 30710.7.3. Design.Examples...........................................................................................309

Problems................................................................................................................................. 317Answers.to.Selected.Problems............................................................................................ 320

11. DescribingFunctions.......................................................................................................... 32311.1. Harmonic.Balance...................................................................................................... 323

11.1.1. Harmonic.Balance.Analysis........................................................................ 32311.1.2. Harmonic.Balance.Accuracy....................................................................... 324

11.2. Describing.Function.................................................................................................. 32511.3. Describing.Functions.for.Symmetrical.Piece-Linear.Characteristics................ 327

11.3.1. Exact.Expressions......................................................................................... 32711.3.2. Approximate.Formulas................................................................................330

11.4. Hysteresis.................................................................................................................... 33111.5. Nonlinear.Links.Yielding.Phase.Advance.for.Large-Amplitude........

Signals..........................................................................................................................33511.6. Two.Nonlinear.Links.in.the.Feedback.Loop..........................................................33611.7. NDC.with.a.Single.Nonlinear.Nondynamic.Link................................................ 33711.8. NDC.with.Parallel.Channels....................................................................................34011.9. NDC.Made.with.Local.Feedback.............................................................................34211.10. Negative.Hysteresis.and.Clegg.Integrator.............................................................34511.11. Nonlinear.Interaction.between.the.Local.and.the.Common.

Feedback.Loops..........................................................................................................34611.12. NDC.in.Multiloop.Systems......................................................................................348

x Contents

11.13. Harmonics.and.Intermodulation.............................................................................34911.13.1.Harmonics......................................................................................................34911.13.2. Intermodulation............................................................................................350

11.14 Verification.of.Global.Stability................................................................................. 351Problems.................................................................................................................................353Answers.to.Selected.Problems............................................................................................ 356

12. ProcessInstability............................................................................................................... 35912.1. Process.Instability...................................................................................................... 35912.2. Absolute.Stability.of.the.Output.Process................................................................36012.3. Jump.Resonance......................................................................................................... 36212.4. Subharmonics.............................................................................................................365

12.4.1. Odd.Subharmonics.......................................................................................36512.4.2. Second.Subharmonic.................................................................................... 367

12.5. Nonlinear.Dynamic.Compensation........................................................................368Problems.................................................................................................................................368

13. MultiwindowControllers.................................................................................................. 37113.1. Composite.Nonlinear.Controllers........................................................................... 37213.2. Multiwindow.Control................................................................................................ 37313.3. Switching.from.Hot.Controllers.to.a.Cold.Controller.......................................... 37513.4. Wind-Up.and.Anti-Wind-Up.Controllers.............................................................. 37713.5. Selection.Order...........................................................................................................38013.6. Acquisition.and.Tracking.......................................................................................... 38113.7. Time-Optimal.Control..............................................................................................38313.8. Examples.....................................................................................................................384Problems.................................................................................................................................388

Appendix1:FeedbackControl,ElementaryTreatment...................................................... 391

Appendix2:FrequencyResponses.........................................................................................405

Appendix3:CausalSystems,PassiveSystems,andPositiveRealFunctions................ 419

Appendix4:DerivationofBodeIntegrals............................................................................. 421

Appendix5:ProgramforPhaseCalculation.........................................................................429

Appendix6:GenericSingle-LoopFeedbackSystem..........................................................433

Appendix7:EffectofFeedbackonMobility........................................................................ 437

Appendix8:DependenceofaFunctiononaParameter..................................................... 439

Appendix9:BalancedBridgeFeedback................................................................................. 441

Appendix10:Phase-GainRelationforDescribingFunctions..........................................443

Appendix11:Discussions.........................................................................................................445

Appendix12:DesignSequence............................................................................................... 459

Appendix13:Examples............................................................................................................. 461

Contents xi

Appendix14:BodeStepToolbox.............................................................................................505

Bibliography................................................................................................................................. 529

Notation........................................................................................................................................533

xiii

Preface

Classical Feedback Control. presents. practical. single-input. single-output. and. multi-input.multi-output.analog.and.digital.feedback.systems..Most.examples.are.from.space.systems.developed.by.the.authors.at.the.Jet.Propulsion.Laboratory,.Pasadena,.California..The.word.classical.refers.to.the.systems.frequency.response.

The.system.analysis.must.be.simplified.before.the.system.is.synthesized..Here.the.Bode.integral.relations.play.a.key.role.for.formulating.the.optimal.system.parametersstabil-ity,.stability.margins,.nonlinear.stability,.and.process.stabilityand.designing.the. loop.compensators.

The. compensators. in. the. loop. responses. are. typically. linear.. The. improved. nonlinear compensators.are.mainly.included.for.adjusting.the.system.at.the.lowest.frequencies.and.for.using.multi-input.multi-output.systems.with.different.limit.frequencies.

The.whole.engineering.system.performance.would.be.optimal.if.each.controller.within.the.subsystems.were.optimal..Bode.integrals.allow.the.responses.to.be.optimized.in.the.final.design.stage,.thus.simplifying.the.whole.procedure.

MATLAB.with.Simulink.and.SPICE.are.used.for.the.frequency.responses..No.prelimi-nary.experience.with.this.software.is.presupposed.in.this.book.

The.first.six.chapters.support.linear.control.systems..Other.chapters.consider.nonlinear.control,.robustness,.global.stability,.and.complex.system.simulation.

It.was.the.authors.intention.to.make.Classical Feedback Control.not.only.a.textbook.but.also.a.reference.for.students.becoming.engineers,.enabling.them.to.design.high-perfor-mance. controllers. and. easing. the. transition. from. school. to. the. competitive. industrial.environment.

The.second.edition.introduces.various.updates,. in.particular.Simulink.calculations.of.the. stability. of. a. superheterodyne. optical. receiver,. discussions. of. a. systems. nonlinear.voltage-current.response,.a.systems.active.vibration.isolation,.design.of.a.longer.antenna.control,.attitude.regulation.errors,.and.a.multioutput.optical.receiver.

We.will.be.grateful.for.any.comments,.corrections,.and.criticism.our.readers.may.take.the.trouble.to.communicate.to.us.by.e-mail.([email protected]).or.addressed.to.B.J.Lurie,.2738.Orange.Avenue,.La.Crescenta,.CA.91214.

Acknowledgments..We.thank.AllaRoden.for.technical.editing.and.acknowledge.the.help.of.AsifAhmed..We.appreciate.previous.discussions.on.many.control.issues.with.Professor.Isaac. Horowitz,. and. collaboration,. comments,. and. advice. of. our. colleagues. at. the. Jet.Propulsion.Laboratory.

We.thank.Drs..AlexanderAbramovich,.John.OBrien,.DavidBayard.(who.helped.edit.the.chapter.on.adaptive.systems),.Dhemetrio.Boussalis,.DanielChang,.Gun-Shing.Chen.(who.contributed.Appendix.7),.AliGhavimi. (who.contributed. to.Section.A13.14.of.Appendix.13),.FredHadaegh.(who.coauthored.several.papers.in.Chapter13),. JohnHench,.Mehran.Mesbahi,.Jason.Modisette.(who.suggested.many.changes.and.corrections),.Gregory.Neat,.SamuelSirlin,.and.JohnSpanos..We.are.grateful. to.EdwardKopf,.who.told. the.authors.about.the. jump.resonance.in.the.attitude.control. loop.of.the.Mariner10.spacecraft,.and.to. George. Rappard,. who. helped. with. the. second. edition.. Suggestions. and. corrections.made.by.Profs..Randolph.Beard,.Arthur.Lanne,.RoySmith,.and.MichaelZak.allowed.us.

xiv Preface

to.improve.the.manuscript..Alan.Schier.contributed.the.example.of.a.mechanical.snake.control..To.all.of.them.we.extend.our.sincere.gratitude.

BorisJ.LuriePaulJ.Enright

xv

To Instructors

The.book.presents.the.design.techniques.that.the.authors.found.most.useful.in.designing.control.systems.for. industry,. telecommunications,.and.space.programs.. It.also.prepares.the.reader.for.research.in.the.area.of.high-performance.nonlinear.controllers.

Plant and compensator.. Classical. design. does. not. utilize. the. plants. internal. variables.for.compensation,.unlike.the.full-state.feedback.approach..The.stand-alone.compensator.achieves. the. feedback. loop.responses..These.are. the. reasons.why. this.book.starts.with.feedback,.disturbance.rejection,.loop.shaping,.and.compensator.design..The.plant.model-ing.is.used.for.the.system.without.the.feedback.

Book architecture.. The. material. contained. in. this. book. is. organized. as. a. sequence. of,.roughly.speaking,.four.design.layers..Each.layer.considers.linear.and.nonlinear.systems,.feedback,.modeling,.and.simulation,.including.the.following:

. 1..Control.system.analysis:.Elementary.linear.feedback.theory,.a.short.description.of.the.effects.of.nonlinearities,.and.elementary.simulation.methods.(Chapters.1.and.2).

. 2..Control.system.design:.Feedback.theory.and.design.of.linear.single-loop.systems.developed. in. depth. (Chapters 3. and. 4),. followed. by. implementation. methods.(Chapters5and.6).

This.completes.the.first.one-semester.course.in.control.

. 3.. Integration.of.linear.and.nonlinear.subsystem.models.into.the.system.model,.uti-lization. of. the. effects. of. feedback. on. impedances,. various. simulation. methods.(Chapter.7),. followed.by.a.brief.survey.of.alternative.controller.design.methods.and.of.adaptive.systems.(Chapters.8.and.9).

. 4..Nonlinear. systems. study. with. practical. design. methods. (Chapters 10. and. 11),.methods.of.elimination.or.reduction.of.process.instability.(Chapter12),.and.com-posite.nonlinear.controllers.(Chapter.13).

Each.consecutive. layer. is.based.on.the.preceding.layers..For.example,. introduction.of.absolute.stability.and.Nyquist.stable.systems.in.the.second.layer.is.preceded.by.a.primitive.treatment.of.saturation.effects.in.the.first.layer;.global.stability.and.absolute.stability.are.then.treated.more.precisely.in.the.fourth.layer..Treatment.of.the.effects.of.links.input.and.output.impedances.on.the.plant.uncertainty.in.the.third.layer.is.based.on.the.elementary.feedback.theory.of.the.first.layer.and.the.effects.of.plant.tolerances.on.the.available.feed-back.developed.in.the.second.layer.

This.architecture.reflects.the.theory.of.the.system.without.excessive.idealization.Design examples..The.examples.of.the.controllers.were.chosen.among.those.designed.by.

the.authors.for.the.various.space.robotic.missions.at.the.Jet.Propulsion.Laboratory:

. An.optical.receiver.for.multichannel.space.system.control.(Section.2.9.5).. A.prototype.for.the.controller.for.a.retroreflector.carriage.and.several.other.control-

lers.of.the.Chemistry.spacecraft.(Section4.2.3)..All.these.controllers.are.high-order.

xvi ToInstructors

and.nonlinear.control.plants.with.structural.modes,.and.include.a.high-order.lin-ear.part.with.a.Bode.step.

. A.nonlinear.digital.controller.for.the.Mars.Pathfinder.high-gain.antenna.pointing.to.Earth.(Section5.10.5).

. A.switched-capacitor.controller.for.the.STRVspacecraft.cryogenic.cooler.vibration.rejection.(Section6.4.2).

. Vibration.damping. in. the.model.of.a. space. stellar. interferometer. (Examples7.2.and.7.20).

. Mars. Global. Surveyor. attitude. control. (briefly. described. in. Example 3.3,.Section13.6).

. Microgravity. accelerometer. analog. feedback. loop. (described. in. detail. in.Example11.12).

. Cassini.Narrow.View.Camera.thermal.control.(described.in.detail.in.Section7.1.2.and.in.Example.3,.Section.13.6).

. More.design.examples.are.described.in.Appendix13.and.14.

Software..Most.design.examples.in.the.book.use.MATLAB.from.MathWorks,.Inc..Only.a.small.subset.of.MATLAB.commands.is.used,.listed.below.in.the.order.of.their.introduc-tion:.logspace,.bode,.conv,.tf2zp,.zp2tf,.step,.gtext,.title,.set,.grid,.holdon,.hold.off,.rlocus,.plot(x,y),.inv,.linspace,.lp2lp,.lp2bp,.format,.roots,.poly,.inv,.bilinear,.residue,.ezplot,.linmod,.laplace,.invlaplace,.impulse.

Additional.MATLAB.functions.written.by. the.authors.and.described. in.Appendix.14.may.complement.the.control.design.methods.

The.Simulink.problems.are.used.with.the.same.programs.as.the.MATLAB..Simulink.is.employed.for.repetition.and.adjustments.of.the.program..For.the.single.run.during.control.course.exercises.we.prefer.using.MATLAB.

Simulink.should.obtain.the.transient.responses.and.plot.Bode.diagrams.from.the.block.diagrams.(on.p..160).

Use.of.the.SPICE.is.taught.to.EE.majors.for.combining.conventional.design.and.control.system.simulation..Some.C.code.examples.are.demonstrated.in.Chapter5.

Frequency responses.The.design.methods.taught.in.this.book.are.based.on.frequency.responses..The.transforms.are.performed.numerically.with.MATLAB.commands.step.and.impulse.

EE.students.know.the.frequency.responses.from.the.signals.and.systems.course,.the.pre-requisite.to.the.control.course..Mechanical,.chemical,.and.aerospace.engineering.majors.know.frequency.responses.from.the.courses.on.dynamic.responses..If.needed,.frequency.responses.for.these.specialties.can.be.taught.using.Appendix.2,.either.before.or.in.parallel.with.Chapter.3..Appendix.2.contains.a.number.of.problems.on.the.Laplace.transform.and.frequency.responses.

Undergraduate course..The.first.sixchapters,.which.constitute.the.first.course.in.control,.also. include.some.material.better.suited. for.a.graduate.course..This.material.should.be.omitted.from.a.one-semester.course,.especially.when.the.course.is.taught.to.mechanical,.aerospace,.and.chemical.engineering.majors.when.extra.time.is.needed.to.teach.frequency.responses..The.sections.to.be.bypassed.are.listed.in.the.beginning.of.each.chapter.

Digital controllers..The.best.way.of.designing.a.good.discrete.controller.is.to.design.a.high-order.continuous.controller,.break.it.properly.into.several.links,.and.then.convert.each.link.to.a.digital.form;.thus,.two.small.tables.of.formulas.or.a.MATLAB.command.are.all.that.is.needed..The.accuracy.of.the.Tustin.transform.is.adequate.and.needs.no.prewarping.

ToInstructors xvii

The implementation of analog controller..The.analog.controllers.of.Chapter.6.are.commonly.easier.and.cheaper.to.design,.and.they.are.important.for.all.engineers..The.chapter.need.not.be.fully.covered,.but.can.be.used.for.self-study.or.as.a.reference.later,.when.the.need.for.practical.design.arises.

Second one-semester course..Chapters.713. are.used. for. the. control. course.and for. self-study.and.as.a.reference.for.engineers.who.took.only.the.first.course.

Chapter. 7. describes. structural. design. and. simulation. systems. with. drivers,. motors,.and.sensors..In.particular,.it.shows.that.tailoring.the.output.impedance.of.the.actuator.is.important.to.reduce.the.plant.tolerances.and.to.increase.the.feedback.in.the.outer.loop.

Chapter8.gives.a.short.introduction.to.quantitative.feedback.theory.and.H.control.and.the.time.domain.control.based.on.state.variables.

Adaptive. systems. are. described. briefly. since. the. control. systems. rarely. need. to. be.applied..But.the.need.for.adaptive.systems.does.exist..Therefore,.the.engineer.should.be.aware.of. the.major.concepts,.advantages,.and.limitations.of.adaptive.control..He.or.she.must.be.able.to.recognize.the.need.for.such.control.and,.at.the.same.time,.not.waste.time.trying.to.achieve.the.impossible..The.material.in.Chapter.9.will.enable.engineers.either.to.figure.out.how.to.design.an.adaptive.system.themselves,.or.to.understand.the.language.of.the.specialized.literature.

The. design. of. high-order. nonlinear. controllers. is. covered. in. Chapters. 1013.. These.design.methods.have.been.proven.very.effective.in.practice.but.are.far.from.being.final-ized..Further.research.needs.to.be.done.to.advance.these.methods.

Problems.. Design. problems. with. mechanical. plants. are. suitable. for. both. ME. and. EE.majors..Additional.problems.for.EE.majors.can.be.found.in.[9].

A.booklet.with.solutions.to.the.problems.is.available.for.instructors.from.the.publisher.

11FeedbackandSensitivity

Chapter.1.introduces.the.basics.of.feedback.control..The.purpose.of.feedback.is.to.make.the.output.insensitive.to.plant.parameter.variations.and.disturbances..Negative,.positive,.and.large.feedback.are.defined.and.discussed.along.with.sensitivity.and.disturbance.rejec-tion..The.notions.of.frequency.response,.the.Nyquist.diagram,.and.the.Nichols.chart.are.introduced..(The.Nyquist.stability.criterion.is.presented.in.Chapter.3.)

Feedback. control. and. block. diagram. algebra. are. explained. at. an. elementary. level. in.Appendix.1,.which.can.be.used.as.an.introduction.to.this.chapter..Laplace.transfer.func-tions.are.described.in.Appendix.2.

1.1 FeedbackControlSystem

It. is. best. to. begin. with. an. example. describing. the. explanation. of. feedback.. Figure 1.1a.depicts.a.servomechanism.regulating.the.elevation.of.an.antenna.

The.output.of.the.feedback.summer.is.called.the.error..The.error.amplified.by.the.com-pensator C.is.applied.to.the.actuator A,.in.this.case.a.motor.regulator.(driver).and.a.motor..The.motor.rotates.the.plant P,.the.antenna.itself,.which.is.the.object.of.the.control..The.com-pensator,.actuator,.and.plant.make.up.the.forward path.with.the.transfer.function.CAP.

Figure1.1b.shows.the.Simulink.block.diagram.made.of.cascaded.elements,. i.e.,. links..The.letters.here.stand.for.the.signals.Laplace.transforms.and.for.the.functions.of.the.linear.links..The.diagram.is.single-input single-output.(SISO).since.only.one.is.the.commanded.elevation.angle U1.and.one.is.the.actual.output.antenna.elevation.U2.

C

B

A PU1 U2

(a)

(b)

T

Compensator+

+

Feed back path,elevation angle sensor

Driver

Commandedelevation

ElevationMotor

BU2, or TE

Error

E

Measured elevation

Actuator

Plant

CE CAE

FIGURE 1.1Single-loop.feedback.system.

2 ClassicalFeedbackControl:WithMATLABandSimulink

It.is.referred.to.as.single loop..The.feedback path.contains.a.sensor.B.The.loop transfer function T.=.CAPB.is.also.called.the.return ratio of.BU2/E.The.return signal.that.goes.into.the.summer.from.the.feedback.path.is.TE.The.error.E =.U1..BU2.is.measured.at.the.output.of.the.summer..It.is.also.seen.as.the.signal

E.=.U1..ET (1.1)

The.error.is.therefore.expressed.as

. E UTUF= + =

1 1

1. (1.2)

where.F.=.T.+.1.is.the.return difference.The.magnitude.|F|.is.the.feedback..It.is.seen.that.when.the.feedback.is.large,.the.error.is.small.

If. the.feedback.path.were.not.present,. the.output.U2.would.simply.equal. the.product.CAPU1..The.system.would.be.referred.to.as.open loop.

Example1.1

A servomechanism for steering a toy car (using wires) is shown in Figure1.2. The command volt-age U1 is regulated by a joystick potentiometer. Another identical potentiometer (angle sensor) placed on the shaft of the motor produces voltage Uangle proportional to the shaft rotation angle. The feedback makes the error small, so that the sensor voltage approximates the input voltage, and therefore the motor shaft angle tracks the joystick-commanded angle.

The arrangement of a motor with an angle sensor is often called servomotor, or simply servo. Similar servos are used for animation purposes in movie production.

The system of regulating aircraft control surfaces using joysticks and servos was termed fly by wire when it was first introduced to replace bulky mechanical gears and cables. The required high reliability was achieved by using four independent parallel analog electrical circuits.

The telecommunication link between the control box and the servo can certainly also be wireless.

Example 1.2

A phase-locked loop (PLL) is drawn as Simulink in Figure1.3. The plant here is a voltage-con-trolled oscillator (VCO).

The VCO is an ac generator whose frequency is proportional to the voltage applied to its input. The phase detector combines the functions of phase sensors and input summer: its output is pro-portional to the phase difference between the input signal and the output of the VCO.

+

U1Joystick AnglesensorWires Uangle

Steeringmechanism

Toy carControl box

FIGURE 1.2Joystick.control.of.a.steering.mechanism.

FeedbackandSensitivity 3

Large feedback makes the phase difference (phase error) small. The VCO output therefore has a small phase difference compared with the input signal. In other words, the PLL synchronizes the VCO with the input periodic signal.

It is very important to also reduce the noise that appears at the VCO output.The PLLs are widely used in telecommunications (for tuning receivers and for recovering the

computer clock from a string of digital data), for synchronizing several motors angular positions and velocities, and for many other purposes.

1.2 Feedback:PositiveandNegative

The.output.signal.in.Figure1.1b.is.U2.=.ECAP,.and.from.Equation.1.2,.U1.=.EF..The.input-to-output.transfer.function.of.the.system.U2/U1,.commonly.referred.to.as.the.closed-loop transfer function,.is

.UU

ECAPEF

CAPF

2

1= = . (1.3)

or,.the.feedback.reduces.the.input-output.signal.transmission.by.the.factor.|F|.The.system.is.said.to.havenegative.feedback.when.|F|>1.(although.the.expression.|F|.

is.certainly.positive)..This.definition.was.developed.in.the.1920s.and.has.to.do.with.the.fact.that.negative.feedback reduces.the.error.|E|.and.the.output.|U2|,.i.e.,.produces.a.negative increment.in.the.output.level.when.the.level.is.expressed.in.logarithmic.values.(in.dB,.for.example).preferred.by.engineers.

The.feedback.is.said.to.be.positiveif.|F||U1|..Positive.feedback.increases.the.error.and.the.level.of.the.output.

We.will.adhere.to.these.definitions.of.negative.and.positive.feedback.since.very.impor-tant.theoretical.developments,.to.be.studied.in.Chapters.3.and.4,.are.based.on.them.

Whether.the.feedback.is.positive.or.negative.depends.on.the.amplitude.and.phase.of.the.return.ratio.(and.not.only.on.the.sign.at.the.feedback.summer,.as.is.stated.sometimes.in.elementary.treatments.of.feedback).

Lets.consider.several.numerical.examples.

Example 1.3

The forward path gain coefficient CAP is 100, and the feedback path coefficient B is 0.003. The return ratio T is 0.3. Hence, the return difference F is 0.7, the feedback is positive, and the closed-loop gain coefficient 100/0.7=143 is greater than the open-loop gain coefficient.

Compensator VCO

Phase of inputperiodic signal

Phaseerror Output periodic signalPhase

detector

Voltagecontrol

Phase ofthe signal

FIGURE 1.3Phase-locked.loop.


Example 1.4

The forward path gain coefficient is 100, and the feedback path coefficient is 0.003. The return ratio T is 0.3. Hence, the return difference F is 1.3, the feedback is negative, and the closed-loop gain coefficient 100/1.3=77 is less than the open-loop gain coefficient.

It is seen that when T is small, whether the feedback is positive or negative depends on the sign of the transfer function about the loop.

When |T|>2, then |T+1|>1 and the feedback is negative. That is, when |T| is large, the feed-back is always negative.

Example 1.5

The forward path gain coefficient is 1,000 and B = 0.1. The return ratio is therefore 100. The return difference is 101, the feedback is negative, and the closed-loop gain coefficient is 9.9.

Example 1.6

In the previous example, the forward path transfer function is changed to 1,000, and the return ratio becomes 100. The return difference is 99, the feedback is still negative, and the closed-loop gain coefficient is 10.1.

1.3 LargeFeedback

Multiplying.the.numerator.and.denominator.of.Equation.1.3.by.B.yields.another.meaning-ful.formula:

.UU B

TF B M

2

1

1 1= = . (1.4)

where

. M TFTT= = +1 . (1.5)

Equation.1.4.indicates.that.the.closed-loop.transfer.function.is.the.inverse.of.the.feed-back.path.transfer.function.multiplied.by.the.coefficient M..When.the.feedback.is.large,.i.e.,.when.|T|>>1,.the.return.difference.FT,.the.coefficient.M1,.and.the.output.becomes

. U BU2 11

. (1.6)

One. result. of. large. feedback. is. that. the. closed-loop. transfer. function.depends. nearly.exclusively. on. the. feedback. path,. which. can. usually. be. constructed. of. precise. compo-nents..This.feature.is.of.fundamental.importance.since.the.parameters.of.the.actuator.and.the.plant. in. the.forward.path.typically.have. large.uncertainties.. In.a.system.with. large.


feedback,.the.effect.of.these.uncertainties.on.the.closed-loop.characteristics.is.small..The.larger.the.feedback,.the.smaller.the.error.expressed.by.Equation.1.2.

Manufacturing.an.actuator.that.is.sufficiently.powerful.and.precise.to.handle.the.plant.without.feedback.can.be.prohibitively.expensive.or.impossible..An.imprecise.actuator.may.be.much.cheaper,.and.a.precise.sensor.may.also.be.relatively.inexpensive..Using.feedback,.the.cheaper.actuator.and. the. sensor. can.be. combined. to. form.a.powerful,.precise,. and.reasonably.inexpensive.system.

According.to.Equation.1.6,.the.antenna.elevation.angle.in.Figure1.1.equals.the.command.divided.by.B..If.the.elevation.angle.is.required.to.be.q,.then.the.command.should.be.Bq.

If.B.=.1,.as.shown.in.Figure1.4a,.then.the.closed-loop.transfer.function.is. just.M.and.U2U1;.i.e.,.the.output.U2.follows.(tracks).the.commanded.input.U1..Such.tracking sys-tems.are.widely.used..Examples.are.a. telescope. tracking.a.star.or.a.planet,.an.antenna.on.the.roof.of.a.vehicle.tracking.the.position.of.a.knob.rotated.by.the.operator.inside.the.vehicle,.and.a.cutting.tool.following.a.probe.on.a.model.to.be.copied.

Example 1.7

Figure1.4b shows an amplifier with unity feedback. The error voltage is the difference between the input and output voltages. If the amplifier gain coefficient is 104, the error voltage constitutes only 104 of the output voltage. Since the output voltage nearly equals the input voltage, this arrangement is commonly called a voltage follower.

Example 1.8

Suppose that T=100, so that M=T/(T+1)=0.9901. If P were to deviate from its nominal value by +10%, then T would become 110. This would make M = 0.991, an increase of 0.1%, which is reflected in the output signal. Without the feedback, the variation of the output signal would be 10%. Therefore, introduction of negative feedback in this case reduces the output signal variations 100 times. Introducing positive feedback would do just the oppositeit would increase the varia-tions in the closed-loop input-output transfer function.

Example 1.9

Consider the voltage regulator shown in Figure1.5a with its block diagram shown in Figure1.5b. Here, the differential amplifier with transimpedance (ratio of output current I to input voltage E) 10A/V and high input and output impedances plays the dual role of compensator and actuator. The power supply voltage is VCC. The plant is the load resistor RL. The potentiometer with the voltage division ratio B constitutes the feedback path.

CAP

+ U2 U2

U1 U1

Error

104

+

(a) (b)

FIGURE 1.4(a).Tracking.system..(b).Voltage.follower.


The amplifier input voltage is the error E=U1TE, and the return ratio is T=10BRL. Assume that the load resistor is 1 k and the potentiometer is set to B=0.5. Consequently, the return ratio is T = 5,000.

The command is the 5V input voltage (when the command is constant, as in this case, it is commonly called a reference, and the control system is called a regulator). Hence, the output voltage according to Equation 1.4 is 10 5,000/5,001= 9.998V. The VCC should be higher than this value; 12V to 30V would be appropriate.

When the load resistance is reduced by 10%, without the feedback the output volt-age will be 10% less. With the feedback, T decreases by 10% and the output voltage is 104,500/4,5019.99778, i.e., only 0.002% less. The feedback reduces the output voltage variations 10%/0.002% = 5,000 times.

This.example.also.illustrates.another.feature.of.feedback..Insensitivity.of.the.output.volt-age.to.the.loading.indicates.that.the.regulator.output.resistance.is.very.low..The.feedback.dramatically.alters.the.output.impedance.from.very.high.to.very.low..(The.same.is.true.for.the.follower.shown.in.Figure1.4b.).The.effects.of.feedback.on.impedance.will.be.studied.in.detail.in.Chapter.7.

1.4 LoopGainandPhaseFrequencyResponses

1.4.1 Gain and Phase Responses

The. sum. (or. the.difference).of. sinusoidal. signals.of. the. same. frequency. is. a. sinusoidal.signal.with.the.same.frequency..The.summation.is.simplified.when.the.sinusoidal.signals.are.represented.by.vectors.on.a.complex.plane..The.modulus.of.a.vector.equals.the.signal.amplitude.and.the.phase.of.the.vector.equals.the.phase.shift.of.the.signal.

Signal.u1=|U1|sin(t+1).is.represented.by.the.vector.U1=|U1|1,.i.e.,.by.the.complex.number U1=|U1|cos1+j|U1|sin1.

Signal.u2=|U2|sin(t+2).is.represented.by.the.vector.U2=|U2|2,.i.e.,.by.the.complex.number.U2.=.|U2|cos2+.j|U2|sin2.

The.sum.of.these.two.signals.is

u=.|U|sin(t.+.)=(|U1|cos1.+|U2|cos2)sint.+.(|U1|sin1.+.|U2|sin2)cost

U1

U2

VCC

10 A/V

B

RLI

TE

E

+

+

B

RLU1 U210

I

(a) (b)

FIGURE 1.5Voltage.regulator:.(a).schematic.diagram,.(b).block.diagram.


That.is,.ReU.=.ReU1.+.ReU2.and.ImU.=.ImU1.+.ImU2,.i.e.,.U =.U1.+.U2..Thus,.the.vector.for.the.sum.of.the.signals.equals.the.sum.of.the.vectors.for.the.signals.

Example 1.10

If u1=4sin(t+/6), it is represented by the vector 4/6, or 3.464+j2. If u2=6sin(t+/4), it is represented by the vector 6/4, or 4.243+j4.243. The sum of these two signals is represented by the vector (complex number) 7.707 + j6.243 = 9.920.681 rad or 9.9239.0.

Example 1.11

Figure1.6 shows four possible vector diagrams U1=E+TE of the signals at the feedback sum-mer at some frequency. In cases (a) and (b), the presence of feedback signal TE makes |E|>|U1|; therefore, |F|


The.frequency.responses.for.the.loop.gain in.dB,.20log|T(j)|,.and.the.loop.phase.shift.in.degrees,.(180/)argT(j),.can.be.plotted.over.the.0.1.to.100rad/s.range.with.the.software.package.MATLAB.from.Mathworks,.Inc..by.the.following.script:

w = logspace(1, 2);% log scale of angular% frequency wnum = 5000;den = [1 55 250 0];bode(num, den, w)

The.plots.are.shown.in.Figure1.7..The.loop.gain.rapidly.decreases.with.frequency,.and.the.slope.of.the.gain.response.gets.even.steeper.at.higher.frequencies;.this.is.typical.for.practical.control.systems.

The.loop.gain.is.0dB.at.9rad/s,.i.e.,.at.9/(2).1.4.Hz..The.phase.shift.gradually.changes.from.90.toward.270,.i.e.,.the.phase.lag.increases.from.90.to.270.

MATLAB.function.conv.can.be.used.to.multiply.the.polynomials.s, (s.+5),.and.(s+50).in.the.denominator:

a = [1 0]; b = [1 5]; c = [1 50];ab = conv(a,b); den = conv(ab,c)

More.information.about.the.MATLAB.functions.used.above.can.be.obtained.by.typing.help.bode,.help.logspace,.and.help.conv.in.the.MATLAB.working.window,.and.from.the.MATLAB.manual..Conversions.from.one.to.another.form.of.a.rational.function.can.be.also.done.using.MATLAB.functions.tf2zp.(transfer.function.to.zero-pole.form).and.zp2tf.(zero-pole.form.to.transfer.function).

101 100 101 10250

0

50

Frequency (rad/sec)

101 100 101 102Frequency (rad/sec)

Gain

dB

90

180

270

0

Phase d

eg

FIGURE 1.7MATLAB.plots.of.gain.and.phase.loop.responses.


Example1.13

The return ratio from Example1.12, explicitly expressed as a function of j, is

T jj j j

( ) ,( )( )

=

+ +5 0005 50

At frequency =2, T10110, and F9115o, so the feedback is negative. |T| reduces with frequency. At frequency = 9, T1160, and F 0.270, so the feedback is positive.

The.Bode.plot.of.|T|.for.a.typical.tracking.system.is.shown.in.Figure1.8.by.the.solid.line..The.loop.gain.decreases.with.increasing.frequency..The.diagram.crosses.the.0dB.line.at.the crossover frequency fb.where,.by.definition,.|T( fb)|=1.

The.frequency.response.of.the.feedback.|F|.is.shown.by.the.dashed.line..It.can.be.seen.that.the.feedback.is.negative.(i.e.,.20log|F| >.0).up.to.a.certain.frequency,.becomes.positive.in.the.neighborhood.of fb,.and.then.becomes.negligible.at.higher.frequencies,.where.F 1.and.20log|F|0.

The.input-output.closed-loop.system.response.is.shown.by.the.dotted.line..The.gain.is.0dB.(i.e.,.the.gain.coefficient.is.1).over.the.entire.bandwidth.of.large.feedback..The.hump.near. the.crossover. frequency. is.a.result.of. the.positive.feedback..This.hump,.as.will.be.demonstrated.in.Chapter3,.results.in.an.oscillatory.closed-loop.transient.response,.and.should.therefore.be.bounded.

In.general,.feedback.improves.the.tracking.systems.accuracy.for.commands.whose.dom-inant.Fourier.components.belong.to.the.area.of.negative.feedback,.but.degrades.the.sys-tems.accuracy.for.commands.whose.frequency.content.is.in.the.area.of.positive.feedback.

1.4.2 Nyquist Diagram

To.visualize.the.transition.from.negative.to.positive.feedback,.it.is.helpful.to.look.at.the.plot.of.T on.the.T-plane.as.the.frequency.varies.from.0.to...This.plot.is.referred.to.as.the.Nyquist diagram.and.is.shown.in.Figure1.9..Either.Cartesian.(ReT,.ImT).or.polar.coordi-nates.(|T|.and.arg.T).can.be.used.

Gain, dB

0f, log. scale

|T|

|F| |M|

Negative feedback,|F|>1

Positivefeedback,|F|>1fb

FIGURE 1.8Typical.frequency.responses.for.T,.F,.and.M.


The.Nyquist.diagram.is.a.major.tool.in.feedback.system.design.and.will.be.discussed.in.detail.in.Chapter.3..Here,.we.use.the.diagram.only.to.show.the.locations.of.the.frequency.bands.of.negative.and.positive.feedback.in.typical.control.systems.

At.each.frequency,.the.distance.to.the.diagram.from.the.origin.is.|T|,.and.the.distance.from.the.1.point.is.|F|..It.can.be.seen.that |F|.becomes.less.than.1.at.higher.frequencies,.which.means.the.feedback.is.positive.there.

The.Nyquist.diagram.should.not.pass.excessively.close.to.the critical point 1.or.else.the.closed-loop.gain.at.this.frequency.will.be.unacceptably.large.

In.practice,.Nyquist.diagrams.are.commonly.plotted.on.the.logarithmic.L-plane with.rectangular. coordinate. axes. for. the. phase. and. the. gain. of. T,. as. shown. in. Figure 1.10a..Notice.that.the.critical.point.1.of.the.T-plane.maps.to.point.(180,.0dB).of.the.L-plane..The.Nyquist.diagram.should.avoid.this.point.by.a.certain.margin.

Example1.14

Figure1.10b shows the L-plane Nyquist diagram for T=(20s+10)/(s4+10s3+20s2+s) charted with MATLAB script:

num = [20 10]; den = [1 10 20 1 0];[mag, phase] = bode(num, den);plot(phase, 20*log10(mag), r, 180, 0, wo)title(L-plane Nyquist diagram)set(gca,XTick,[270 240 210 180 150 120 90])grid

It is recommended for the reader to run this program for modified transfer functions and to observe the effects of the polynomial coefficient variations on the shape of the Nyquist diagram.

1.4.3 Nichols Chart

The. Nichols chart. is. an. L-plane. template. for. the. mapping. from. T. to. M,. according. to.Equation.1.5,.and.is.shown.in.Figure1.11..When.the.Nyquist.diagram.for.T.is.drawn.on.this.template,.the.curves.indicate.the.tracking.system.gain:.20log.|M|.

It.is.seen.that.the.closer.the.Nyquist.diagram.approaches.the.critical.point.(180,0dB),.the.larger.|M|.is,.and.therefore.the.higher.is.the.peak.of.the.closed-loop.frequency.response.in.Figure1.8..The.limiting.case.has.|M|.approaching.infinity,.indicating.that.the.system.

1

fbf

T-plane

0

f1

Im T

Re T

TF

T-plane

01801

(a) (b)

FIGURE 1.9Nyquist.diagram.with.(a).Cartesian.and.(b).polar.coordinates;.the.feedback.is.negative.at.frequencies.up.to.f1.


180 fb

fL-planeGain, dB

Phase, degr0 dB

(a) (b)270240210180150120 90

100

50

0

50

100L-plane Nyquist Diagram

FIGURE 1.10Nyquist. diagrams. on. the. L-plane,. (a). typical. for. a. well-designed. system. and. (b). MATLAB. generated. for.Example.1.14.

10

0 9030 60

10

dB

0

20

10

10

20

0

2020 4010 50 8070

15

5

15

5

15

5

15

5

0 B

10 dB

1 dB

2 dB

3 dB 2 dB 1 dB6 dB

8 dB10 dB

6 dB

15 dB

4 dB

4 dB

Deviation in Phase from 180

FIGURE 1.11Nichols.chart.


goes.unstable..Typically,.|M|.is.allowed.to.increase.not.more.than.two.times,.i.e.,.not.to.exceed.6dB.

Therefore,.the.Nyquist.diagram.should.not.penetrate.into.the.area.bounded.by.the.line.marked.6.dB.

Consider.several.examples.that.make.use.of.the.Nichols.chart.

Example 1.15

The loop gain is 15 dB, the loop phase shift is 150. From the Nichols chart, the closed-loop gain is 14dB. The feedback is 151.4=13.6dB. The feedback is negative.

Example 1.16

The loop gain is 1dB, the loop phase shift is 150. From the Nichols chart, the closed-loop gain is 6dB. The feedback is 16 = 5dB. The feedback is positive.

Example 1.17

The loop gain is 10dB, the loop phase shift is 170. From the Nichols chart, the closed-loop gain is 7dB. The feedback is 10(7)= 3dB. The feedback is positive.

1.5 DisturbanceRejection

Disturbances.are.signals.that.enter.the.feedback.system.at.the.input.or.output.of.the.plant.or.actuator,.as.shown.in.the.Simulink.diagram.in.Figure1.12,.and.cause.undesirable.signals.at.the.system.output..In.the.antenna.pointing.control.system,.disturbances.might.be.due.to.wind,.gravity,.temperature.changes,.and.imperfections.in.the.motor,.the.gearing,.and.the.driver..The.disturbances.can.be.characterized.either.by.their.time.history.or,.in.frequency.domain,.by.the.Fourier.transform.of.this.time.history,.which.gives.the.disturbance spectral density.

The.frequency.response.of.the.effect.of.a.disturbance.at.the.systems.output.can.be.calcu-lated.in.the.same.way.that.the.output.frequency.response.to.a.command.is.calculated:.it.is.the.open-loop.effect.(D1AP,.for.example).divided.by.the.return.difference.F.

In.Figure1.12,. three.disturbance.sources.are.shown..Since. in. linear.systems.the.com-bined.effect.at.the.output.of.several.different.input.signals.is.the.sum.of.the.effects.of.each.separate.signal,.the.disturbances.produce.the.output.effect

C

+ ++ +

B

A PCommand

D2Output

D1 D3

FIGURE 1.12Disturbance.sources.in.a.feedback.system.


.D AP D P D

F1 2 3+ +

The.effects.of.the.disturbances.on.the.output.are.reduced.when.the.feedback.is.negative.and.increased.when.the.feedback.is.positive..Disturbance.rejection.is.the.major.purpose.for.using.negative.feedback.in.most.control.systems.

There.exist.systems.where.there.is.no.command.at.all,.and.the.disturbance.rejection.is.the.only.purpose.of. introducing. feedback..Such.systems.are.called.homing systems..A.typical.example.is.a.homing.missile,.which.is.designed.to.follow.the.target..No.explicit.command.is.given.to.the.missile..Rather,.the.missile.receives.only.an.error.signal,.which.is.the.deviation.from.the.target..The.feedback.causes.the.vehicles.aerodynamic.surfaces.to.reduce.the.error..This.error.can.be.considered.a.disturbance,.and.large.feedback.reduces.the.error.effectively..Another.popular.type.of.a.system.without.an.explicit.command.is.the.active.suspension,.which.uses.motors.or.solenoids.to.attenuate.the.vibration.propagating.from.the.base.to.the.payload.

Example 1.18

The feedback in a temperature control loop of a chamber is 100. Without feedback, when the temperature outside the chamber changes, the temperature within the chamber changes by 6. With feedback, the temperature within the chamber changes by only 0.06.

Example 1.19

Gusty winds disturb the orientation of a radio telescope. The winds contain various frequency com-ponents, some slowly varying in time and others rapidly oscillating. The feedback in the antenna attitude control loop is 200 at very low frequencies, but drops with frequency (since motors cannot move the huge antenna rapidly), and at 0.1Hz the feedback is only 5. The disturbance components are attenuated by the feedback accordingly, 200 times for the effect of steady wind, and 5 times for the 0.1Hz gust components. Detailed calculations for a similar example will be given in the next section.

To further reduce the higher-frequency disturbances, an additional feedback loop might be introduced that will adjust the position not of the entire antenna dish, but of some smaller mirror in the optical path from the antenna dish to the receiver front end (or from the power amplifier of the transmitter).

1.6 ExampleofSystemAnalysis

We. proceed. now. with. the. analysis. of. the. simplified. antenna. elevation. control. system.shown.in.Figure1.1b..Assume.that.the.elevation.angle.sensor.function.is.1V/rad,.the.feed-back.path.coefficient.is.B =.1,.the.actuator.transfer.function.(the.ratio.of.the.output.torque.to.the.input.voltage).is.A.=.5,000/(s+10)Nm/V,.and.the.antenna.is.a.rigid.body.with.the.moment.of.inertia.J=5,000kgm2..The.plants.input.variable.is.torque,.and.the.output.vari-able. is. the.elevation.angle;. i.e.,. the.plant. is.a.double. integrator.with.gain.coefficient.1/J..Since.the.Laplace.transform.of.an.integrator.is.1/s,.the.transfer.function.of.the.plant.is


P(s).=.1/(Js2)

As.shown.in.the.Simulink.diagram.(Figure1.13),.the.torque.applied.to.the.antenna.is.the.sum.of.the.torque.produced.by.the.actuator.and.the.disturbance.wind.torque,.w.

It.is.known.that.for.large.antennas,.the.wind.torque.spectral.density.is.approximately.proportional.to

.1

0 1 2( . )( )s s+ +

The.spectral.density.of.the.disturbance.in.the.antenna.elevation.angle.is.therefore.pro-portional.to

.1

0 1 21 1

2 1 0 22 4 3 2( . )( ) . .s s s s s s+ + = + + . (1.7)

The.frequency.response.of.the.disturbance.can.be.plotted.using.MATLAB.with:

w = logspace(2,1);% frequency range% 0.01 to 10 rad/secnum = 1;den = [1 2.1 0.2 0 0];bode(num, den, w)

The.plot.is.shown.in.Figure1.14.but.normalized.to.100dB.at..=.0.01..The.spectral.density.is.larger.at.lower.frequencies..Then,.large.feedback.is.shown.approximately.by.a.dotted.line..It.shows.the.feedback.to.reject.the.disturbance.

The.compensator.transfer.function.(calculated.elsewhere).is

. C s s ss s( )( . )( . )

( )=+ +

+

50 0 05 0 55

(This. simple. compensator. makes. the. system. work. reasonably. well,. although. not. opti-mally.).The.loop.transfer.function.is

. T s CAP s ss s s( )( . )( . )

( ),

= =

+ +

+

+

50 0 05 0 55

5 00010

115 000 2, s

Compensator C+ +

1

ACommand Elevation angle

w 1Js2

FIGURE 1.13(a).Elevation.control.system.diagram..(b).Simulink.diagram.


i.e.,

. T s numdens ss s s( )( . )( . )( )( )= =+ +

+ +=

50 0 05 0 55 103

550 27 5 0 2515 50

2

5 4 3s ss s s

+ +

+ +

. .

The.return.difference.is

F(s).=.T(s).+.1.=.(num + den)/den (1.8)

The.closed-loop.transfer.function.M(s)=T/F=num/(num +.den).The.plots.of.the.gain.and.phase.for.the.loop.transfer.function T(j),.for.F(j),.and.for.

M(j).can.be.made.in.MATLAB.with:

w = logspace(1,1); % freq range 0.1 to 10rad/secden = [1 15 50 0 0 0];num = [0 0 0 50 27.5 0.25]; % equal length of the vectorsg = num + den; % makes the addition allowablebode(num, den, w) % for Thold onbode(g, den, w) % for Fbode(num, g, w) % for Mhold off

The.plots.are.shown.in.Figure1.15a..The.labels.are.placed.with.mouse.and.cursor,.one.at.a.time,.using.MATLAB.command.gtext(label)..The.feedback.is.large.at.low.frequencies.and.is.negative.up.0.8.rad/s.

The.closed-loop.gain.response.20log|M| is.nearly.flat.up.to.1.4rad/s,.i.e.,.up.to.0.2Hz..The.gain.is.peaking.at.0.8rad/s..The.hump.on.the.gain.response.does.not.exceed.6dB,.which.satisfies.the.design.rule.mentioned.in.Section.1.4.3..More.precise.design.methods.will.be.studied.in.the.following.chapters.

The.plot.for.disturbances.in.the.system.with.feedback,.the.dashed.line.in.Figure1.14,.is.obtained.by.subtracting.the.feedback.response.(in.dB).from.the.disturbance.spectral.den-sity.response,.or.directly.by.dividing.Equation.1.7.by.Equation.1.8..The.disturbances.are.greatly.reduced.by.the.feedback.

f, log. sc

dB

0

100

20

10.1

406080

20

Feedback in dB

40

FIGURE 1.14Spectral.density.of.the.elevation.angle.disturbance,.in.relative.units:.before.the.feedback.was.introduced,.solid.line;.with.the.feedback,.dashed.line.


The.mean.square.of.the.output.error.is.proportional.to.the.integral.of.the.squared.spec-tral.density.with.linear.scales.of.the.axes..The.plots.required.to.calculate.the.reduction.in.the.mean.square.error.can.be.generated.with.MATLAB,.and.the.areas.under.the.responses.found.graphically,.or.directly.calculated.using.MATLAB.functions.

The.L-plane.Nyquist.diagram.is.shown.in.Figure1.15b..The.diagram.avoids.the.critical.point.by.significant.margins:.by.20dB.from.below,.by.40dB.from.above,.and.by.42from.the.right.

This.system.has.a.better.feedback:.the.Nyquist.diagram.reaches.270.at.lower.frequen-cies..However,.the.system.is.not stable.over.the.full.range.of.input.signals.with.a.common.nonlinear. element.. The. nonlinear. element. reduces. the. plot. in. Figure 1.15b,. which. slips.down.and.encompasses.the.critical.point.

To.be.stable,.the.system.needs.a.dynamic.nonlinear.element,.including.also.a.dynamic.linear.element,.as.will.be.shown.in.Sections.11.711.9.

Using.Simulink.simplifies.the.addition.of.extra.linear.blocks.in.the.loop.(Figure1.15c)..The.step.response.is.obtained.by.writing.the.triangle.in.the.Simulink.

To.obtain.the.Bode.diagram,.we.linearize.the.system.by.replacing.the.nonlinear.matrices.by.linmod(model_name).and.write.it.in.MATLAB.

It.would.be,.however,.more.difficult.to.implement.in.Simulink.dynamic.nonlinear.loops..For.this.reason.we.prefer.using.MATLAB.

101 100 10150

0

50

Frequency (rad/sec)

101 100 101

Frequency (rad/sec)

Gain

dB

120

150

180

210

240

Phase d

eg

F

FT

M

M

T

T,F

(a) (b)

270 240 210 180 150 120100

50

0

50

100

150

200L-plane Nyquist Diagram

+ 5000s + 10

15000s2

Step 1Comp C A

(s + 0.5)(s + 0.05)s(s + 5)

Plant ScopeSaturation

5000s + 10

15000s2

50

Comp C A

(s + 0.5)(s + 0.05)s(s + 5)

PlantSaturation

(c)

(d)

[a b c d] = linmod(model_name);w = logspace(3,1);bode(a,b,c,d,1,w); grid on

Scope

1

1

In 1

Out 1

50

FIGURE 1.15(a).Loop.frequency.response.for.the.elevation.control.system:.at.lower.frequencies,.T.and.F.overlap;.at.higher.frequencies,.T.and.M.overlap..(b).L-plane.Nyquist.diagram..(c).Simulink.for.step.response..(d).Simulink.for.Bode.diagram.


Example 1.20

If the compensator gain coefficient in the system with the Nyquist diagram shown in Figure1.16a is increased five times, i.e., by 14dB, the Nyquist diagram shifts up by 14dB and the margin from below decreases from 20dB to 6dB.

If the loop gain is increased by (approximately) 20dB, the Nyquist diagram shifts up by 20dB, the return ratio becomes 1 at a certain frequency, and the closed-loop gain at this frequency therefore becomes infinite. As we already mentioned in Section 1.4.3, this is a condition for the system to become unstable. Similarly, if the loop gain is reduced by 40dB, the Nyquist diagram shifts down, at some frequency the return ratio becomes 1, and the system becomes unstable. Using the Nyquist diagram for stability analysis will be discussed in detail in Chapter 3.

The.transient.response.of.the.closed-loop.system.to.the.1radian.step command.(increase.instantly.the.elevation.angle.by.1radian).is.found.with

num = [0 0 0 50 27.5 0.25];den = [1 15 50 0 0 0];g = num + den; step(num, g)grid

The.step.response.is.shown.in.Figure1.16,.nominal.gain.case..The.output.doesnt.rise.instantly.by.1radian.as.would.be.ideal:.it.rises.by.1radian.in.less.than.2.s,.but.then.over-shoots.by.30%,.then.slightly.undershoots,.and.settles.to.1radian.with.reasonable.accuracy.in.about.10s..This.shape.of.the.closed-loop.transient.response.is.typical.and,.commonly,.acceptableif.the.nonlinear.element.is.dynamic.

Example 1.21

The effect of the Nyquist diagram passing closer to the critical point can be seen on the closed-loop transient response. The response generated with den=0.2*den is shown Figure1.16, curve loop gain increased. The response is faster, but it overshoots much more and is quite oscillatory.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (secs)

Amplitu

de

Nominal case

Loop gain increased

FIGURE 1.16Output.response.to.1.radian.step.command.


1.7 EffectofFeedbackontheActuatorNondynamicNonlinearity

Actuators.can.be.relatively.expensive,.bulky,.heavy,.and.power.hungry..Economy.requires.that.the.actuator.be.as.small.as.possible..However,.these.actuators.will.not.be.able.to.repro-duce. signals. of. relatively. large. amplitudes. without. distortion.. We. will. consider. these.issues.in.Chapter.3.and.later.in.Chapters.913.

Because.the.output.power.of.any.actuator.is.limited,.saturation.limits.the.amplitude.of.the.output.signal..The.hard.saturation.in.Figure1.17a.is.shown.by.the.dashed.line..In.many.actuators,.the.saturation.is.soft, as.shown.by.the.solid.curve.

Both.hard.and.soft. saturations.are.nondynamic,. i.e.,.not.depending.on. the. frequency.characteristic..Dynamic.saturation.is.exemplified.in.Sections.11.8.and.11.9.

When.the.amplitude.of.the.input.increases,.the.ratio.of.the.output.to.the.input.decreases..Loosely.speaking,.the.gain.coefficient.of.the.saturation.link.decreases..It.will.be.studied.in.detail.in.Section.11.1.

Large.feedback.about.the.actuator.changes.the.shape.of.the.input-output.characteristic..If.the.input.signal.level.is.such.that.the.slope.(differential.gain).of.the.saturation.curve.is.not.yet.too.flattened.out,.then.the.differential.feedback.may.remain.large.and.the.closed-loop.differential. transfer. function.can.be.quite. close. to.B 1..The.closed-loop.amplitude.characteristic.shown.by.the.dotted.line.is.therefore.a.segment.of.a.nearly.straight.line..(The.slope.of.the.line.is.shallow.since.the.feedback.reduces.the.input-output.differential.gain.).Therefore,.in.a.system.with.soft.saturation,.the.input-output.curve.appears.as.hard.satura-tion.when.the.feedback.is.large.

The.dead-zone.characteristic.is.shown.in.Figure1.17b..Large.feedback.reduces.the.dif-ferential.input-output.gain.coefficient.and.therefore.makes.the.input-output.characteristic.shallower,.as.shown.by.the.dotted.curve..Therefore,.for.any.large.amplitude.a of.the.output.signal,.increasing.the.feedback.will.cause.the.input.signal.amplitude.to.change.from.d.to.2d..The.value.d.that.causes.no.response.in.the.output.a.decreases.

In.other.words,.the.feedback.reduces.the.relative.width.of.the.dead.zone..This.feature.allows.the.achievement.of.high.resolution.and.linearity.in.control.systems.that.use.actua-tors. and. drivers. with. rather. large. dead. zones. (such. actuators. and. drivers. may. be. less.expensive.or.consume.less.power.from.the.power.supply.line,.like.push-pull.class.B.ampli-fiers.or.hydraulic.spool.valve.amplifiers,.briefly.described.in.Section.7.1.3,.or.the.motors.with.mechanical.gears).

Next,.consider.the.output.signal.distortions.caused.by.a.small.deviation.of.the.actuator.from.linearity..In.response.to.a.sinusoidal.input.with.frequency.f,.the.output.of.the.nonlin-ear.forward.path.consists.of.a.fundamental.component.with.amplitude.U2.and.additional.

Input

Output

Input

Output WithoutfeedbackWith

feedbackWithout feedback

With feedback,1/B

d/a d/aBDead zone

a

(a) (b)

FIGURE 1.17Input-output.characteristic.of.the.actuator.with.(a).soft.saturation.and.(b).dead.zone.


Fourier.components.called.nonlinear products..The.ratio.of.the.amplitude.of.a.nonlinear.product.to.the.amplitude.of.the.fundamental.is.the.nonlinear product coefficient.

Consider.one.of.these.products.having.frequency.nf.and.amplitude.U2(n)..If.the.forward.path.is.approximately.linear,.the.nonlinear.product.can.be.viewed.as.a.disturbance.source.added.to.the.output,.as.shown.in.Figure1.18.

Now,.compare.two.cases:.(1).the.system.without.feedback.and.(2).the.system.with.feed-back.and.with.the.input.signal.increased.so.that.the.output.signal.amplitude.U2. is.pre-served..In.the.second.case,. the.disturbance,. i.e.,. the.amplitude.of.the.nonlinear.p

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