EE5/6102: Multivariable Control Systems - NUS UAVuav.ece.nus.edu.sg/~bmchen/courses/EE5102.pdf · EE5102/6102 PART2~ PAGE1 BENM. CHEN, NUS ECE EE5/6102: Multivariable Control Systems

EE5102/6102PART 2 ~PAGE 1EE5102/6102PART 2 ~PAGE 1 BEN M.CHEN,NUSECEBEN M.CHEN,NUSECE

EE5/6102:MultivariableControl SystemsPart2:TimeDomainApproaches

EE5/6102:MultivariableControl SystemsPart2:TimeDomainApproaches

BenM.Chen

Professor&Provost’sChairDepartmentofElectrical&ComputerEngineering

Office:E4‐06‐08,Phone:6516‐2289Email:[email protected]://www.bmchen.net

BenM.Chen

Professor&Provost’sChairDepartmentofElectrical&ComputerEngineering

Office:E4‐06‐08,Phone:6516‐2289Email:[email protected]://www.bmchen.net


1.IntroductiontoPart21.IntroductiontoPart2


1.1 CourseOutline

• Revision: Generalintroductiontocontrolsystems;demonstrationof

actualcontrolexample;reviewofbasiclinearsystemtheory.

• Propertiesoflinearquadraticregulation(LQR)control;returned

differences;guaranteedgainandphasemargins;Kalmanfilter;

linearquadraticGaussian(LQG)designtechnique.

• Introductiontomoderncontrolsystemdesign;H2 andHoptimal

control;solutionstoregularandsingularH2 andHoptimalcontrol

problems;solutionstosomerobustcontrolproblems.

• Robust&perfecttracking(RPT)controltechnique.

• Looptransferrecovery(LTR)designtechnique.


1.2ReferenceTextbooks

• B.M.Chen,Z.Lin,Y.Shamash,LinearSystemsTheory:AStructuralDecompositionApproach,

Birkhauser,Boston,2004.

• F.L.Lewis,L.XieandD.Popa,OptimalandRobustEstimation,CRCPress,2007.

• A.Saberi,P.Sannuti,B.M.Chen,H2OptimalControl, PrenticeHall,London,1995.

• B.M.Chen,Robustand HControl, Springer,NewYork,2000.

• A.Saberi,B.M.Chen,P.Sannuti,LoopTransferRecovery:AnalysisandDesign, Springer,

London,1993.

• J.M.Maciejowski,MultivariableFeedbackDesign,AddisonWesley,NewYork,1989.

• G.Cai,B.M.Chen,T.H.Lee,UnmannedRotorcraftSystems,Springer,NewYork,2011.

• B.M.Chen,T.H.Lee,K.Peng,V.Venkataramanan,HardDiskDriveServoSystems,2ndEdn.,

Springer,NewYork,2006.


1.3HomeworkAssignments

Therewillbethree(3)homeworkassignments(forEE5102,itistodesign

controlsystemsforanHDDservosystem,whereasforEE6102,itisto

designaflightcontrolsystemforanunmannedhelicopter),whichrequire

computersimulations.Allstudentsareexpectedtohaveknowledgein

MATLABTM (ControlToolboxandRobustControlToolbox)andSIMULINKTM

aftercompletingtheseassignments.Homeworkassignmentsaretobe

markedandcountedasacertainpercentageinyourfinalgrade.

EE5102studentscanchoosetodotheEE6102assignmentsinstead!

Youarewelcometodobothassignmentsifyoulike


1.4FinalGradesforPart2

FinalGrade=70% FinalexammarksforPart2(max=50)+

30% Homeworkassignmentsmarks(max=50)


2.Revision:BasicControlConcepts2.Revision:BasicControlConcepts


Controller

2.1Whatisacontrolsystem?

Systemtobecontrolled

DesiredPerformanceREFERENCE

INPUTtothesystem

Informationaboutthesystem:

OUTPUT

+–

DifferenceERROR

Objective: TomakethesystemOUTPUT andthedesiredREFERENCEasclose

aspossible,i.e.,tomaketheERROR assmallaspossible.

KeyIssues: (1)Howtodescribethesystemtobecontrolled?(Modeling)

(2)Howtodesignthecontroller?(Control)

aircraft,missiles,economicsystems,

cars,etc


2.2SomeControlSystemsExamples

System to be controlledController+

–

OUTPUTINPUTREFERENCE

EconomicSystemDesired

PerformanceGovernmentPolicies


2.3Uncertainties,NonlinearitiesandDisturbances

Therearemanyotherfactorsoflifehavetobecarefullyconsideredwhen

dealingwithreal‐lifeproblems.Thesefactorsinclude:

R (s)

+ U(s))(sG)(sK

Y(s)–

E (s)

disturbances noisesuncertainties

nonlinearities

?


2.4ControlTechniques– ABriefViewThefollowingismypersonalviewontheclarificationofcontroltechniques…

ClassicalcontrolPIDcontrol,developedin1940sandusedheavilyforinindustrialprocesses.

OptimalcontrolLinearquadraticregulatorcontrol,Kalmanfilter,H2control,developedin1960stoachievecertainoptimalperformance.

RobustcontrolH control,developedin1980s&90stohandlesystemswithuncertaintiesanddisturbancesandwithhighperformances.

NonlinearcontrolDevelopedtohandlenonlinearsystemswithhighperformances.

Multi‐agentsystems&cooperativecontrol

Itisahottopicatmoment.

Intelligentcontrol(withalinktodeeplearning…)Knowledge‐basedcontrol,adaptivecontrol,neuralandfuzzycontrol,etc.,developedtohandlesystemswithunknownmodels.


2.5AnAccidentduetoControlFailure

GunterStein


2.6AnActualControlSystemDesignExample

Command

RCJoystick

AvionicSystem

GroundStation

BareHelicopter

Measurement

ControlSignal

Real‐timeData

ManualOperation

– HeLion

ThefirstfullyautonomousunmannedhelicopterconstructedatNUS


Modeling– DataCollection

DataCollectionProcedure:

1).Chirp‐likesignalissued

insinglechannel;

2).Chirp‐likesignalissued

inmulti‐channels;

3).Step‐likeandrandom

signalsissuedforvalidation.

Exampleofchirp‐likesignal

Chirp‐likesignalandcorrespondingresponses


Modeling– TestFlights

Flighttestingformodelingpurpose


HoverModelofHeLion

Modeling– ModelStructure


Physicalmeaningsoftheplantparameters


1).Angularratedynamics; 2).Horizontalvelocitydynamics;

3).Yawdynamics; 4).Heavedynamics.

Modeling– ParameterIdentification


FlightControlSystemStructure

Inner‐loopControl

Outer‐loopControl

MissionPlanning

Inner‐loopcontrolistostabilizetheoverallaircraftandtoproperlycontrolitsattitude.

Theouter‐loopcontrolistocontrolthepositionoftheaircraftandatthesametimetogeneratenecessarycommandsfortheinner‐loopcontrolsystem…


20142014 20162016

2012201220102010

NUSresearchteam&unmannedsystemsplatforms…NUSresearchteam&unmannedsystemsplatforms…


VideodemoofafullyautomaticUAVflightcontrolsystems

NUSNUS

UAVUAV

hit&run

ALLALL ONEONEinin

IndoorNavigation&ControlIndoorNavigation&Control InsideForestNavigationInsideForestNavigationFirefightingFirefighting

CargoTransportationCargoTransportationHybridUAVsHybridUAVs UAVCalligraphyUAVCalligraphy

FlightFormationFlightFormation

MicroUAVMicroUAV

Hit‐and‐RunUAVsHit‐and‐RunUAVs


3.BriefReviewofBasicLinearSystems

Theory

3.BriefReviewofBasicLinearSystems

Theory

…Chapter3…


Givenalineartime‐invariantsystem

3.2DynamicalResponses


MarginallyStable

Unstable

Stable

3.3SystemStability

Alineartime‐invariantsystemissaidtobeasymptoticallystableifallits

closed‐looppolesarelocatedontheleft‐halfcomplexplane(LHP),unstable

ifatleastofitspolesareontheright‐halfplane(RHP)…


LyapunovStability

Considerageneraldynamicsystem,withf(0)=0.

Ifthereexistsaso‐calledLyapunovfunctionV(x),which

satisfiesthefollowingconditions:

1.V(x)iscontinuousinx andV(0)=0;

2.V(x)>0(positivedefinite);

3.(negativedefinite),

thenwecansaythatthesystemisasymptoticallystableatx=0.Ifinaddition,

thenwecansaythatthesystemisgloballyasymptoticallystableatx=0.Inthis

case,thestabilityisindependentoftheinitialconditionx(0).

)(xfx

0)()( xfxVxV

( ) , as V x x

Aleksandr Lyapunov1857–1918


LyapunovStabilityforLinearSystems

Consideralinearsystem,.Thesystemisasymptoticallystable(i.e.,the

eigenvaluesofmatrixA areallintheopenLHP)ifforanygivenappropriate

dimensionalrealpositivedefinitematrixQ =QT >0,thereexistsarealpositive

definitesolutionP =PT >0 forthefollowingLyapunovequation:

Proof. DefineaLyapunovfunction.Obviously,thefirstandsecond

conditionsonthepreviouspagearesatisfied.Nowconsider

Hence,thethirdconditionisalsosatisfied.Theresultfollows.

Notethat thecondition,Q =QT >0,canbereplacedbyQ =QT 0and

beingdetectable.

xAx

QPAPA T

xPxxV T)(

( ) ( ) 0T T T T T T TV x x P x x P x A x P x x P Ax x A P PA x x Qx

2

1, QA


3.4ControllabilityandObservability



Recallthegivensystem(3.1.1),whichhasatransferfunction

3.5SystemInvertibility


Here

3.6NormalRankandInvariantZeros

which is known as the so-called Rosenbrock system matrix.

HowardH.Rosenbrock1920–2010


+r

)(sG)(sKy

–

e

3.7FrequencyResponses

Considerthefollowingfeedbackcontrolsystem,

BodePlots arethemagnitudeandphaseresponsesoftheopen‐looptransfer

function,i.e.,K(s)G(s), withs beingreplacedbyj.Forexample,fortheballand

beamsystemwithaPDcontroller,whichhasanopen‐looptransferfunction

222

3.27.33.27.31023.037.0)()(

js

ss

ssGsKjsjs

js

1807.3

3.2tan)()(,)3.2(7.3)()( 12

22

jGjKjGjK

Hendrik WadeBode1905–1982


gaincrossoverfrequency

phasecrossoverfrequency

gainmargin

phasemargin

Gainandphasemargins


Nyquist Plot

InsteadofseparatingintomagnitudeandphasediagramsasinBodeplots,Nyquist

plotmapstheopen‐looptransferfunctionK(s)G(s) directlyontoacomplexplane,

e.g.,

HarryNyquist1889–1976


–1

PM

1GM

Gainandphasemargins

ThegainmarginandphasemargincanalsobefoundfromtheNyquist plotbyzooming

intheregionintheneighbourhoodoftheorigin.

1 , ( ) ( ) 180( ) ( ) p p p

p pK j G j

K j G j

GM where issuchthat

Mathematically,

( ) ( ) 180 , ( ) ( ) 1PM where issuchthatg g g g gK j G j K j G j

Remark: Gainmarginisthemaximum

additionalgainyoucanapplytotheclosed‐

loopsystemsuchthatitwillstillremain

stable.Similarly,phasemarginisthe

maximumphaseyoucantoleratetothe

closed‐loopsystemsuchthatitwillstill

remainstable.


Sensitivityfunctions

Considerthetypicalfeedbackcontrolscheme

Thesensitivityfunctionisdefinedastheclosed‐looptransferfunctionfromthereferencesignal,r,tothetrackingerror,e,andgivenby

Thecomplimentarysensitivityfunctionisdefinedastheclosed‐looptransferfunctionbetweenthereference,r, andthesystemoutput,y,andisgivenas

Clearly,


Agoodcontrolsystemdesignshouldhaveasensitivityfunctionthatissmallatlowfrequenciesforgoodtrackingperformanceanddisturbancerejectionandisequaltounityathighfrequencies.Ontheotherhand,thecomplementarysensitivityfunctionshouldbemadeunityatlowfrequencies.Itmustrolloffathighfrequenciestopossessgoodattenuationofhigh‐frequencynoise.

GunterStein


4.PropertiesofLQRControl4.PropertiesofLQRControl


LinearQuadraticRegulator(LQR)

Consideralinearsystemcharacterizedby

where(A,B)isstabilizable.Wedefineacostindex

andisdetectable.Thelinearquadraticregulationproblemistofind

acontrollawu =– Fx suchthat(A – BF)isstableandJ isminimized.The

solutionisgivenby,withP beingapositivesemi‐definitesolution

ofthefollowingRiccatiequation:

x A x B u

0,0,)(),,,(0

RQdtRuuQxxRQuxJ TT

1/ 2( , )A Q

1F R B P T

01 QPBPBRPAPA TT

JacopoFrancescoRiccati1676–1754

(SeethereferencebySaberietal,1995,forthemethodsonhowtosolveRiccatiequations)


IfwearrangetheLQRcontrolinthefollowingblockdiagram,

wecanfinditsgainmarginandphasemarginaswehavedoneinclassical

control.Itisclearthattheopen‐looptransferfunction,

Theblockdiagramcanbere‐drawnasfollows,

Fx Ax Bu –

1 1 1( ) ( )TOpenlooptransferfunction F sI A B R B P sI A B

– BAsIPBR 11 )( T


ReturnDifferenceEqualityandInequality

ConsidertheLQRcontrollaw.Thefollowingso‐calledreturndifferenceequalityholds:

Thefollowingiscalledthereturndifferenceinequality:

Proof.Recallthat

Thenwehave

])([])([)()( 1T1TT11TT BAIjFIRFAIjBIBAIjQAIjBR

RBAIjFIRFAIjBI ])([])([ 11 TTT

P BRF T1 01 QPBPBRPAPA TT

0)()( 11 QPBRRPBRPAIPjPAIPj TT

QRFFPAIjAIjP TT )()(


BAIjQAIjBBAIjRFFAIjBBAIjPAIjAIjBBAIjAIjPAIjB1111

1111

)()()()()()()())(()(

TTTTT

TTTTT

Multiplyingitontheleftbyandontherightby,

BAIjQAIjBBAIjRFFAIjBBAIjPBPBAIjB

11

1111

)()()()()()(

TT

TTTTTT

Notingthefactthat

wehave

RFPBRFPBP BRF TTT &1

BAIjQAIjBBAIjRFFAIjBBAIjRFRFAIjB

11

1111

)()()()()()(

TT

TTTTTT

BAIjQAIjBRBAIjFIRFAIjBI 11TT1T1TT )()(])([])([

R

R

1)( TT AIjB BAIj 1)(


SingleInputCase

Inthesingleinputcase,thetransferfunction

isascalarfunction.Then,thereturndifferenceequationisreducedto

1( )Openlooptransferfunction f sI A b

1 1 1 1( ) ( ) [1 ( ) ][1 ( ) ]T T T T Tr b j I A Q j I A b r b j I A f f j I A b

211 ( ) 0wherer r f j I A b

rbAIjfr 21)(1

211 ( ) 1 ReturnDifferenceInequality...f j I A b


Graphically, implies

–160

PM 60

1)01()(1)(1 121 jbAIjfbAIjf

Clearly,thephasemarginresultingfromtheLQRdesignisatleast60deg.

– 2

Thegainmarginisfrom[0.5, ).

* R.E.Kalman,Whenisalinearcontrolsystemoptimal?JournalofBasicEngineering,TransactionsoftheASME,SeriesD,Vol.86,No.1,pp.51–60,1964.

–1


Example: Consideragivenplantcharacterizedby

SolvingtheLQRproblemwhichminimizesthefollowingcostfunction

weobtain

whichresultstheclosed‐loopeigenvalues at.Clearly,the

closed‐loopsystemisasymptoticallystable.

uxx

10

1110

1.0,0001

,)(),,,(0

RQdtRuuQxxRQuxJ withTT

1.37342.3166and0.13730.23170.23170.6872

FP

3814118671 . j .


Frequency (rad/sec)

Pha

se (

deg)

; M

agni

tude

(dB

)

Bode Diagrams

-20

-10

0

10From: U(1)

100 101-100

-80

-60

-40

-20

0

To: Y

(1)

PM=84 GM=


5.KalmanFilter5.KalmanFilter

RudolfE.Kalman,1930–2016


Kalman‐Bucy Filter?Kalman‐Bucy Filter?

Kalman andBucy (1977)


Review:RandomProcess

Arandomvariable X isamappingbetweenthesamplespaceandtherealnumbers.

Arandomprocess (a.k.a stochasticprocess)isamappingfromthesamplespace

intoanensembleoftimefunctions(knownassamplefunctions).Toeverymemberin

thesamplespace,therecorrespondsafunctionoftime(asamplefunction)X(t).

X(t)

time


Mean,Moment,Variance,CovarianceofRandomProcess

Letf(x,t)betheprobabilitydensityfunction (p.d.f.)associatedwitharandomprocess

X(t).Ifthep.d.f.isindependentoftimet,i.e.,f(x,t)=f(x),thenthecorrespondingrandom

processissaidtobestationary.Wewillfocusourattentiononlyonthisclassofrandom

processesinthiscourse.Forthistypeofrandomprocesses(RP),wedefine:

1)mean (orexpectation): 2)moment(j‐th ordermoment)

3)variance 4)covarianceoftworandomprocesses

TwoRPsv andw aresaidtobeindependent iftheirjointp.d.f.

( )m E X x f x dx

( )j jE X x f x dx

2 2 2( ) ( ) ( )E x m x m f x dx

( , ) ( [ ])( [ ])v w E v E v w E w con

( , ) ( ) ( ) [ ] [ ]E vw vwf v w dvdw vf v dv wf w dw E v E w

( , ) ( ) ( )f v w f v f w


AutocorrelationFunctionandPowerSpectrum

Autocorrelationfunctionisusedtodescribethetimedomainpropertyofarandom

process.Givenarandomprocessv,itsautocorrelationfunction isdefinedas

follows:

Ifv isawidesensestationary(WSS)process,

NotethatRx(0)isthetimeaverageofthepowerorenergyoftherandomprocess.

Powerspectrum ofarandomprocessistheFouriertransformofitsautocorrelation

function.Itisafrequencydomainpropertyoftherandomprocess.Tobemore

specific,itisdefinedas

1 2 1 2( , ) ( ) ( )xR t t E v t v t

1 2 2 1( , ) ( ) ( ) ( , ) ( ) ( )x x x xR t t R t t R R t t E v t v t

deRS jxx

)()(

EE5102/6102PART 2 ~PAGE 51EE5102/6102PART 2 ~PAGE 51 BEN M.CHEN,NUSECEBEN M.CHEN,NUSECE51

WhiteNoise,ColoredNoiseandGaussianRandomProcess

WhiteNoiseisarandomprocesswithaconstantpowerspectrum,andanautocorrelation

function

whichimpliesthatawhitenoisehasaninfinitepowerandthusitisnonexistentinreal

life.However,manynoises(ortheso‐calledcolorednoises,ornoiseswithfiniteenergy

andfinitefrequencycomponents)canbemodeledastheoutputsoflinearsystemswithan

injectionofwhitenoiseintotheirinputs,i.e.,acolorednoisecanbegeneratedbyawhite

noise

GaussianProcessv isalsoknownasnormalprocesshasap.d.f.

( ) ( )xR q

whitenoise colorednoiseLinearSystem

variancemean,

22)(

,2

1)( 2

2

v

evf


KalmanFilterforaLinearTimeInvariant(LTI)System

ConsideranLTIsystemcharacterizedby

Assume:(1)(A,C)isobservable

(2)v(t)andw(t)areindependentwhitenoiseswiththefollowingproperties

(3)isstabilizable(toguaranteeclosed‐loopstability).

( ) istheinputnoise( ) isthemeasurementno e

is

x Ax Bu v t vy Cx w t w

T T[ ( )] 0, [ ( ) ( )] ( ), 0,E v t E v t v Q t Q Q

T T[ ( )] 0, [ ( ) ( )] ( ), 0E w t E w t w R t R R

2

1, QA

TheproblemofKalman Filter istodesignastateestimatortoestimatethestatex(t)

bysuchthattheestimationerrorcovarianceisminimized,i.e.,thefollowing

indexisminimized:

)(ˆ tx

ˆ ˆ[{ ( ) ( )} { ( ) ( )}]eJ E x t x t x t x t T


ConstructionofSteadyStateKalman Filter

Kalman filterisastateobserverwithaspeciallyselectedobservergain(orKalman

filtergain).Ithasthedynamicequation:

withtheKalman filtergainKe beinggivenas

wherePe isthepositivedefinite solutionofthefollowingRiccati equation,

Let.Wecanshow(seenext)thatsuchaKalman filterhasthefollowing

properties:

ˆ ˆ ˆ ˆ( ), (0)ˆ ˆ

ex Ax Bu K y y xy Cx

isgiven

1e eK PC R T

1 0e e e eP A AP PC R CP Q T T

lim lim [ ( ) ( )]T tracee et tJ E e t e t P

xxe ˆ

ˆlim ( ) lim ( ) ( ) 0,t t

E e t E x t x t


KalmanFilterandLQR– TheyAreDual

Recalltheoptimalregulatorproblem,

TheLQRproblemistofindastatefeedbacklawu =– Fx suchthatJ isminimized.Itwas

shownthatthesolutiontotheaboveproblemisgivenby

andtheoptimalvalueofJ isgivenby.Notethatx0 isarbitrary.Letusconsider

aspecialcasewhenx0 isarandomvectorwith.

Then,wehave

0

0

(0)

, 0 0

x Ax Bu x x

J x Qx u Ru dt Q Q R R

T T T T

given

and

1F R B P T 0,0 TT1T PPQPBPBRPAPA

P xxJ 0T0

IxxExE ][,0][ T000

n

iii

n

i

n

jjiij

n

i

n

jjiij PpxxEpxxpEPxxEJE

11 100

1 1000

T0 trace][][][


TheDuality

LinearQuadraticRegulator Kalman Filter

P BRF T1 1T RCPK ee

0T1T QPBPBRPAPA 01TT QCPRCPAPAP eeee

ePJ traceoptimal PJ traceoptimal

Thesetwoproblemsareequivalent(ordual)ifwelet

PFBA

T

T

T

e

e

PKCA


ProofofPropertiesofKalmanFilter

RecallthatthedynamicsofthegivenplantandKalman filter,i.e.,

( )( )

x Ax Bu v ty Cx w t

ˆ ˆ ˆ( )ˆ ˆ

ex Ax Bu K y yy Cx

&

Wehave

with

Next,itisreasonabletoassumethatinitialerrore(0)andd(t)areindependent,i.e.,

ˆ ˆ ˆ( ) [ ( ) ]ˆ( )( ) ( ) ( )

( )

e

e

e

e

e

e x x Ax Bu v t Ax Bu K Cx w t CxA K C x x v

A K

t Kvw t

e eC AI K d tw

( [ ( )] 0[

)( )

(] 0

[ ( )] 0) e ee

E v tE E I K I K

E w tv t

d t I Kw t

[ (0) ( )] [ (0)] [ ( )] 0T TE e d t E e E d t


Furthermore,

where.

T TT

TT

T

T

T

[ ( ) ( )] [ ( ) ( )][ ( ) ( )]

[ ( ) ( )] [ ( ) ( )]

( ) 00 ( )

( )

( )

e e

ee

ee

IE v t v E v t wE d t d I K

KE w t v E w t w

IQ tI K

KR t

tK

t

Q RK

A

lim [ ( ) ( ) ]Tet

E e t e t P

Wewillnextshowisasymptoticallystableand

T 0 e eQ K RK


Recallthatand

Wehave

SinceQ =QT 0 andisassumedtobestabilizable,itfollowsfromLyapunov

stabilitytheorythatmatrixisasymptoticallystable.

01TT QCPRCPAPAP eeee

T T 1T 1 T 1 0e e ee e ee eP A AP PCPC R CP PCR CP R CP Q

T T 0 e e e eP P K RK QA A

2

1, QA

eA A K C

1T T 1T 1T 0 e e e ee eR CP PC RP A C A C P PC R CP Q

T 1e eK PC R

( )te Ae d

t

tAtA ddeeete0

)( )()0()(

Recallalsothesolutionto,i.e.,

T T T 1T 1 0 e ee e e eKP A C A C P PC R CPR QK R


Notingthatisdeterministic,wehave

T T

T T

T

( ) ( )

0 0

T ( ) T

0

T ( ) ( ) T (

0

T (0) ( ) (0) ( )

[ (0) (0)] [ ( ) (0)]

[ (

( ) [

0) ( )] [

( ) ( )]

( ) ( )]

t tAt A t At A t

tAt A t A t A t

tAt A t A t A t

E e e e d d e e e d d

e E e e e e E d e e d

e E e d e d e d E d

P t E e t e t

d e

T T

T T T T

)

0 0

T ( ) ( )

0 0

T ( ) ( ) T

0 0

[ (0) (0)] ( )

[ (0) (0)] [ (0) (0)]

t t

t tAt A t A t A t

t tAt A t A t A t At A t A A

d

e E e e e e d e d

e E e e e e e d e E e e e e e d

tAe

Sinceisstable,wehave.Thus,A 0, as Ate t

0

T

)( deeP AA

0

0


WenextshowthatP()=Pe ,i.e.,thesolutiontotheKalman filterARE. Let

Inviewof,wehave

Next,wehave

Thus,wehaveforeverygivenz(0),

TT , (0) given ( ) (0), ( ) 0A tz A z z z t e z z

ee PAAP T

zzzPzdtdzzzPzzPz

zzzPAzzAPzzzzPAAPz

eee

eeee

TTTTT

TTTTTTT

)(

][

T TT T T T

0 0 0

(0) (0) (0) (0) (0) ( ) (0)At A t At A tz zdt z e e z dt z e e dt z z P z

)0()0(0)0()0()()()()()( TTT

0

T

0

T zPzzPzzPztzPtzdtzPzdtd

eeeee

TT T

0

(0) (0) (0) ( ) (0) ( ) A Ae ez P z z P z P P e e d


Itisnowsimpletoseethat

Finally,wehave

lim [ ( ) ( )] ( ) lim [ ( ) ( )]T T tracee et tE e t e t P P E e t e t P

( )

0

lim [ ( )] lim [ (0)] [ ( )] 0

t

At A t

t tE e t e E e e E d d

Example: Consideragivenplantcharacterizedbythefollowingstatespacemodel,

SolvingtheKalman filterARE,weobtain

)(0001.0

)()]()([),(10

1110 T

ttQvtvEtvuxx

)(2.0)()]()([),(01 T ttRwtwEtwxy

,0.03140.03430.03430.0792

eP

0.1715 0.3962

eK

xCyyyKBuxAx e

ˆˆ)ˆ(ˆ


RelatedTopics:ExtendedKalmanFilter(EKF):Inestimationtheory,theEKFisthenonlinear

versionoftheKalmanfilter,whichlinearizes aboutthecurrentmeanandcovariance.

TheEKFhasbeenconsideredthedefacto standardinthetheoryofnonlinearstate

estimation,navigationsystemsandGPS.

UnscentedKalmanfilter(UKF):Whenthestatetransitionandobservationmodels,

thepredictandupdatefunctionsarehighlynonlinear,theextendedKalmanfiltercan

giveparticularlypoorperformance.Thisisbecausethecovarianceispropagated

throughlinearizationoftheunderlyingnon‐linearmodel.UKFusesadeterministic

samplingtechniqueknownastheunscentedtransformtopickaminimalsetof

samplepoints(calledsigmapoints)aroundthemean.Thesesigmapointsarethen

propagatedthroughthenon‐linearfunctions,fromwhichthemeanandcovarianceof

theestimatearethenrecovered.Theresultisafilterwhichmoreaccuratelycaptures

thetruemeanandcovariance.


6.LinearQuadraticGaussian(LQG)6.LinearQuadraticGaussian(LQG)


ProblemStatement

Itisveryoftenincontrolsystemdesignforareallifeproblemthatonecannotmeasure

allthestatevariablesofthegivenplant.Thus,thelinearquadraticregulator,althoughit

hasveryimpressivegainandphasemargins(GM= andPM>60degrees),is

impracticalasitutilizesallstatevariablesinthefeedback,i.e.,u=– Fx.Inmostof

practicalsituations,onlypartialinformationofthestateofthegivenplantisaccessible

orcanbemeasuredforfeedback.Thenaturalquestionsonewouldask:

• Canwerecoverorestimatethestatevariablesoftheplantthroughthepartially

measurableinformation? Theanswerisyes.ThesolutionisKalmanfilter.

• Canwereplacex thecontrollawinLQR,i.e.,u=– Fx, bytheestimatedstateto

carryoutameaningfulcontrolsystemdesign?Theanswerisyes.Thesolution

iscalledLQG.

• DowestillhaveimpressivepropertiesassociatedwithLQG?Theanswerisno.

Anysolution?Yes.Itiscalledalooptransferrecovery (LTR)technique.


LinearQuadraticGaussianDesign

Consideragivenplantcharacterizedby

wherev(t)andw(t)arewhitewithzeromeans.v(t),w(t)andx(0)areindependent,and

Theperformanceindexhastobemodifiedasfollows:

TheLinearQuadraticGaussian (LQG)controlistodesignacontrollawthatonly

requiresthemeasurableinformationsuchthatwhenitisappliedtothegivenplant,the

overallsystemisstableandtheperformanceindexisminimized.

( )( )

x Ax Bu v t vy Cx w t w

istheinputnoiseisthemeasurementnoise

0[ ( ) ( )] ( ), 0, [ ( ) ( )] ( ), 0, [ (0)]e e e eE v t v Q t Q E w t w R t R E x x T T

0

1lim ( ) , 0, 0T TT

TJ E x Qx u Ru dt Q R

T

(A,B)stabilizable

(A,C)detectable


SolutiontoLQGProblem– SeparationPrinciple

Step1. DesignanLQRcontrollawu=– Fxwhichsolvesthefollowingproblem,

i.e.,compute

Step2. DesignaKalman filterforthegivenplant,i.e.,

where

Step3. TheLQGcontrollawisgivenby,i.e.,

uBxAx 0,0,)(),,,(0

RQdtRuuQxxRQuxJ TT

PBRF .T1,0,0T1T PQPBPBRPAPA

xCyyyKBuxAx e ˆˆ),ˆ(ˆˆ

.0,01TT eeeeeee PQCPRCPAPAP,1T RCPK ee

xFu

ˆ

)ˆ(ˆˆ

xFuxCyKuBxAx e

xFuyKxCKBFAx ee

ˆˆ)(


BlockDiagramImplementationofLQGControlLaw

PLANT

KALMAN FILTERLQRCONTROL

Reference

x

u yr

G

1 12[ ( ) ]G C A BF B

MatrixC2 isrelatedtooutputvariablesofinterest,say

z =C2 x

where z istotrackthereferencer.

LQGControlLaw


Closed‐LoopDynamicsofGivenPlanttogetherwithLQGController

Recalltheplant:andcontroller

Wedefineanewvariableandthus

and

)(

)(

twCxytvBuAxx

GrxFuyKBGrxCKBFAx ee

ˆˆ)(

ˆe x x

)()()()()()ˆ()ˆ()(ˆˆˆ)(ˆˆ

twKtveCKAtwKtvxxCKxxAtwKCxKBGrxCKxBFxAtvBGrxBFAxxxe

eeee

eee

)()()()()(ˆ)(

tvBGrBFexBFAtvBGrexBFAxtvBGrxBFAxtvBuAxx

Clearly,theclosed‐loopsystemischaracterizedbythefollowingstatespaceequation,

wKv

vvvr

BGex

CKABFBFA

ex

ee

~,~00

0 wex

Cy

Theclosed‐looppolesaregivenby,whicharestable.)()( CKABFA e


HomeworkAssignment1:

UsingtheLQRdesign,theKalmanfilterdesignandtheircombination,i.e.,theLQGcontrolmethodtodesignanappropriatemeasurementfeedbackcontrollawthatmeetsallthedesignspecificationspecifiedinthe(HDDorhelicopter)problem.

ShowallthedetailedcalculationandsimulateyourdesignusingMATLAB andSimulink.Giveallthenecessaryplotsthatshowtheevidenceofyourdesign.


7.IntroductiontoRobustControl7.IntroductiontoRobustControl

GeorgeZames1934–1997


ARealControlProblem

responsedisturbances

sensor noise

Controllercommands

measurementscontrolinput

ControllerObjective: Toprovidedesiredresponsesinfaceof

• Uncertainplantdynamics+Externalinputs

Plant

disturbancessensornoisecontrolinput


RepresentationofUncertainPlantDynamics

Perturbation

NominalPlantresponse

measurements

disturbance

sensornoisecontrolinputs

• NominalPlantisanFDLTISystem

• PerturbationisMemberofSetofPossiblePerturbations


AnalysisObjectives

• NominalPerformanceQuestion (H2 OptimalControl):

Areclosedloopresponsesacceptablefordisturbances?sensornoise?

• RobustStabilityQuestion (H OptimalControl):

Isclosed loopsystemstablefornominalplant?forallpossibleperturbations?

• RobustPerformanceQuestion (MixedH2/H OptimalControl):

Areclosedloopresponsesacceptableforallpossibleperturbationsandall

externalinputs?Simultaneously?


CompletePictureofRobustControlProblem

K

P


StandardFeedbackLoopsintermsofGeneralInterconnectionStructure

K Gr u

d

+ –

e

K

Ge

? ?+r

u


8. H2 OptimalControlandH Control8. H2 OptimalControlandH Control


IntroductiontotheProblems

Considerastabilizableanddetectablelineartime‐invariantsystemwithaproper

controllerc

c

u

w z

y

where

www

DE

uuu

D

B

xxx

CCA

zyx

00: 1

22

1

statevariable controlinput

measurement & disturbance

controlledoutput controllerstate

n m

p l

q k

x R u R

y R w R

z R v R

yy

DB

vv

CA

uv

c

c

c

cc

:


TheproblemsofH2 andH optimalcontrolaretodesignapropercontrollawc such

thatwhenitisappliedtothegivenplantwithdisturbance,i.e.,,wehave

• Theresultingclosedloopsystemisinternallystable(thisisnecessaryforany

controlsystemdesign).

• Theresultingclosed‐looptransferfunctionfromthedisturbancew tothe

controlledoutputz, say,,isassmallaspossible,i.e.,theeffectofthe

disturbanceonthecontrolledoutputisminimized.

• H2 optimalcontrol:theH2‐normofisminimized.

• H optimalcontrol:theH‐normofisminimized.

)(sTzw

)(sTzw

)(sTzw

Note: Atransferfunctionisafunctionoffrequenciesrangingfrom0to.Itishardto

tellifitislargeorsmall.Thecommonpracticeistomeasureitsnormsinstead.H2‐

normandH‐normaretwocommonlyusednormsinmeasuringthesizesofatransfer

function.


ClosedLoopTransferFunctionfromDisturbancetoControlledOutput

Recallthat

www

DE

uuu

D

B

xxx

CCA

zyx

00: 1

22

1

and y

yDB

vv

CA

uv

c

c

c

cc

:

ww

DE

yDvC

yDvC

D

B

xxx

CCA

zyx

cc

cc

1

22

1

)(

)(

11222

11

wDxCDDvCDxCzEwwDxCBDvBCAxx

cc

cc

wDDDvCDxCDDCz

wDBDEvBCxCBDAx

ccc

ccc

122122

11

)()()(

yDDvCDxCzEwyBDvBCAxx

cc

cc

222

wDBxCBvAwDxCBvAv

ccc

cc

11

11 )(

wDxCwDDDvx

CDCDDCz

wBxAwDB

DBDEvx

ACBBCCBDA

vx

cc

c

c

c

c

c

cc

clcl2

clcl

~][

~

12122

1

1

1

1


Thus,theclosed‐looptransferfunctionfromw toz isgivenby

clclclcl DBAsICsTzw 1)(

Remark: Forthestatefeedbackcase,C1 =I andD1 =0,i.e.,allthestatesofthegiven

systemcanbemeasured,c canthenbereducedtou =Fx andthecorresponding

closed‐looptransferfunctionisreducedto

Theresultingclosed‐loopsystemisinternallystableifandonlyiftheeigenvaluesof

cc

cc

ACBBCCBDA

A1

1

cl

areallinopenlefthalfcomplexplane.

EBFAsIFDCsTzw1

22)(

Theclosed‐loopstabilityimpliesandisimpliedthatA+BFhasstableeigenvalues.


H2‐normandH‐normofaTransferFunction

Definition: (H2‐norm) GivenastableandpropertransferfunctionTzw(s),itsH2‐normis

definedas

21

2)()(

21

djTjTT zwzwzwHtrace

Graphically,

Note: TheH2‐norm isthetotalenergycorrespondingtotheimpulseresponseof

Tzw(s).Thus,minimizationoftheH2‐norm ofTzw(s)isequivalenttotheminimization

ofthetotalenergy fromthedisturbancew tothecontrolledoutputz.

|Tzw(j)|H2‐norm


Definition: (H‐norm) GivenastableandpropertransferfunctionTzw(s),itsH‐normis

definedas )(sup max

0

jTT zwzw

wheremax[Tzw(j)] denotesthemaximumsingularvalueofTzw(j).Forasingle‐input‐

single‐outputtransferfunctionTzw(s),itisequivalenttothemagnitudeofTzw(j).

Graphically,

Note: TheH‐norm istheworstcasegaininTzw(s).Thus,minimizationoftheH‐norm

ofTzw(s)isequivalenttotheminimizationoftheworstcase(gain)situationonthe

effect fromthedisturbancew tothecontrolledoutputz.

H‐norm

|Tzw(j)|


Infima andOptimalControllers

Definition: (Theinfimum ofH2 optimization)Theinfimum oftheH2 normofthe

closed‐looptransfermatrixTzw(s)overallstabilizingpropercontrollersisdenoted

by2*,thatis *

2 2: inf internally stabilizes .zw cT

Definition: (Theinfimum ofH optimization)Theinfimum oftheH‐normofthe

closed‐looptransfermatrixTzw(s)overallstabilizingpropercontrollersisdenotedby

*,thatis * : inf internally stabilizes .zw cT

Definition: (TheH2 optimalcontroller)Apropercontrollerc issaidtobeanH2optimalcontrollerifitinternallystabilizes and.*

22zwT

Definition: (TheH ‐suboptimalcontroller)Apropercontrollerc issaidtobeanH

‐ suboptimalcontrollerifitinternallystabilizes and. *

zwT


CriticalAssumptions:RegularCasevs SingularCase

MostresultsinH2 andH optimalcontroldealwithaso‐calledaregularproblemor

regularcasebecauseitissimple.AnH2 orH controlproblemissaidtoberegular if

thefollowingconditionsaresatisfied,

1.D2 isofmaximalcolumnrank,i.e.,D2 isatallandfullrankmatrix

2.Thesubsystem(A,B,C2,D2)hasnoinvariantzerosontheimaginaryaxis;

3.D1 isofmaximalrowrank,i.e.,D1 isafatandfullrankmatrix

4.Thesubsystem(A,E,C1,D1)hasnoinvariantzerosontheimaginaryaxis;

AnH2 orH controlproblemissaidtobesingular ifitisnotregular,i.e.,atleastone

oftheabove4conditionsisnotsatisfied.

Note:Forstatefeedbackcontrol,Conditions1and2aresufficientfortheregularcase.


SolutionstotheStateFeedbackProblems:theRegularCase

ThestatefeedbackH2 andH controlproblemsarereferredtotheproblemsinwhich

allthestatesofthegivenplant areavailableforfeedback.Thatisthegivensystemis

wE

u

u

D

B

xxx

C

A

zyx

22

:

where(A,B)isstabilizable,D2 isofmaximalcolumnrankand(A,B,C2,D2)hasno

invariantzerosontheimaginaryaxis.

Inthestatefeedbackcase,wearelookingforastaticcontrollaw

xFu


SolutiontotheRegularH2 StateFeedbackProblem

SolvethefollowingalgebraicRiccatiequation(H2‐ARE)

0221

222222 PBCDDDDCPBCCPAPA TTTTTT

forauniquepositivesemi‐definitestabilizingsolutionP 0.TheH2 optimalstate

feedbacklawisthengivenby

xPBCDDDxFu TTT 22

122 )(

Itcanbeshowedthattheresultingclosed‐loopsystemTzw(s)hasthefollowing

property:

Itcanalsobeshowedthat.Notethatthetrace ofamatrixis

definedasthesumofallitsdiagonalelements.

.*22 zwT

1* T 22 trace( )E PE


Example: Considerasystemcharacterizedby

uxz

xy

wuxx

111

21

10

4325

:

A B E

C2 D2

SolvingthefollowingH2‐AREusingMATLAB,

weobtainapositivedefinitesolution

and

164040144

P

1741 F1833.19*

2

Theclosed‐loopmagnituderesponse

fromthedisturbancetothecontrolled

output:

Theoptimalperformanceorinfimum

isgivenby


ClassicalLQRProblemisaSpecialCaseofH2 Control

Itcanbeshownthatthewell‐knownLQRproblemcanbere‐formulatedasanH2optimalcontrolproblem.Consideralinearsystem,

TheLQRproblemistofindacontrollawu=Fx suchthatthefollowingindexis

minimized:

whereQ 0isapositivesemi‐definitematrixandR>0isapositivedefinitematrix.

TheproblemisequivalenttofindingastaticstatefeedbackH2 optimalcontrollaw

u=Fx for

0)0(, XxuBxAx

T T

0J x Q x u Ru dt

uRxQ

z

xywXuBxAx

0

0 21

21

0


SolutiontotheRegularH StateFeedbackProblem

Given >*,solvethefollowingalgebraicRiccatiequation(H‐ARE)

forauniquepositivesemi‐definitestabilizingsolutionP 0.TheH ‐suboptimalstate


xPBCDDDxFu TTT 22

122 )(

Theresultingclosed‐loopsystemTzw(s)hasthefollowingproperty:

Remark: ThecomputationofthebestachievableH attenuationlevel,*,isingeneral

quitecomplicated.Forcertaincases,* canbecomputedexactly.Therearecasesin

which* canonlybeobtainedusingsomeiterativealgorithms.Onemethodistokeep

solvingtheH‐AREfordifferentvaluesof untilithits* forwhichandany <*,the

H‐AREdoesnothaveasolution.PleaseseethereferencebyChen(2000)fordetails.

.zwT

0)()(/ 221

22222

22 PBCDDDDCPBPPEECCPAPA TTTTTTT


Example: Again,considerthefollowingsystem

uxz

xy

wuxx

111

21

10

4325

:

A B E

C2 D2

ItcanbeshowedthatthebestachievableH

performanceforthissystemis.

SolvingthefollowingH‐AREusingMATLAB

with =5.001,weobtainapositivedefinite

solution

1.266798.1100288.1100285.330111

P

1.366808.110029 F

5*



output:

Clearly,theworsecasegain,occurred

atthelowfrequencyisroughlyequal

to5 (actuallybetween5 and 5.001)


Whyarezerosmoreinteresting?…

Considerthesystem

withbeingsquareandfullrank,i.e.,itisnonsingular.Wecanthenapplyapre‐

feedbacktothegivensystem,whichyields

andtheRosenbrock systemmatrixofthesubsystemfromv toz isgivenby

2 2

:x A x B u E wy xz C x D u

2D1 1

2 2 2u D C x D v

1 12 2 2( )

0

x A BD C x BD v E wy xz x I v

( )0

sIP s

IA B

Alltheeigenvalues of aretheinvariantzerosofthesystem!

0

x x v E wy xz x

B

I

A

v

A


If(A,B,C2,D2 )isofminimumphase,i.e.,allitsinvariantzerosarestable,or

equivalently isastablematrix,theitscorrespondingH2‐ARE,i.e.,

canbesimplifiedas

Then,itcanbeseenthatP =0istherequiredsolution!Theoptimalsolutionis

givenby

andthesolutionintermsoftheoriginalcontrolinputisgivenby

0221

222222 PBCDDDDCPBCCPAPA TTTTTT

T T 0A P PA PB B P

T 1 T T( ) 0 0 0v F x I I I B x

1 1 12 2 2 2 2u D C x D v D C x

A


Similarly,thecorrespondingH∞‐ARE,i.e.,

canbesimplifiedas

Again,P =0istherequiredsolution.Theoptimalsolution(forthisspecialsituation,

theH∞ controlhasanoptimalsolution)isgivenby

andthesolutionintermsoftheoriginalcontrolinputisgivenby

Inbothcases,theclosed‐looptransferfunctionmatrixfromw toz is

0)()(/ 221

22222

22 PBCDDDDCPBPPEECCPAPA TTTTTTT

T T 2 T/ 0A P PA PEE P PBB P

T 1 T T( ) 0 0 0 v F x I I I B x

1 1 12 2 2 2 2u D C x D v D C x

1( ) 0 ( ) 0!zwT s sI A E

andallthestableinvariantzeros of(A,B,C2,D2)aretheclosed‐loopsystempoles!


If(A,B,C2,D2 )hasallitsinvariantzerostobeunstable,orequivalently isananti‐

stablematrix,theitscorrespondingH2‐ARE,i.e.,

hasasolutionP =0too.But,itdoesnotgiveastabilizingcontrollaw(why?).However,

itcanbeconvertedintoaLyapunovequation

FromtheLyapunov stabilitytheorem,ithasauniquepositivedefinitesolution.The

optimalsolutionisgivenby

andtheresultingclosed‐loopsystemmatrix

T T 0A P PA PB B P

T 1 T T T( ) 0 v F x I I I B P x B P x 11 T2 2 2 2

u D C D D BP x

A

T1 1 TP A A P B B T1 T P A P A BB P

Themirrorimagesoftheunstableinvariantzeros of(A,B,C2,D2 ),i.e.,are

theclosed‐loopsystempoles!

T TT 1 1 A BF A BB P A P A P A P A P

A


Similarly,thecorrespondingH∞‐ARE,i.e.,

canbere‐writtenas

whichcanbesolvedbysolvingtwoLyapunov equations:

Itcanbeshowedthat

The‐suboptimalsolutionisgivenas

MoregeneralresultsforthesingularcasecanbefoundinChenetal(1992).

T T 2 T/ 0A P PA PEE P PB B P

1 T 1 T T 2/P A AP B B EE

T T T TandS A A S B B T A AT EE

1* 1 *

max 2

1a d( ) 0,nTS P S T

11 T T2 2 2 2

u D C D D B P x

IanPetersen


Q:Whathappensifthegivensystemhasbothstableandunstableinvariantzeros?

A: Forthecasewhenhasbothstableandunstableeigenvalues,thereexistsa

similaritytransformationT suchthat

Onecanthendealwitheachpartseparately.ThesolutiontotheAREcorrespondingto

thestablepartis0andthesolutiontotheAREcorrespondingtotheunstablepartcan

becalculatedbysolvingLyapunov equationsasonthepreviouspage.

Q:Itcanbeseenthatwhenthegivensystemisofnonminimum phase,theoverall

performanceoftheclosed‐loopsystemislimited.Canwerelocatethezerosastheway

thatwehavechangedthelocationsofpoles?

A: Yes.Itinvolvesrelocationsofthesensorsand/oractuators

ofthegivensystem.Itiscalledsensor/actuatorplacement.

A

stable.-anti stable, ,0

01

AA

AA

TAT


SolutionstotheStateFeedbackProblems– theSingularCase

Considerthefollowingsystemagain,

wE

u

u

D

B

xxx

C

A

zyx

22

:

where(A,B)isstabilizable,D2 isnotnecessarilyofmaximalrankand

(A,B,C2,D2)mighthaveinvariantzerosontheimaginaryaxis.

Solutiontothiskindofproblemscanbedoneusingthefollowingtrick

(orso‐calledaperturbationapproach):Defineanewcontrolled

output

uI

DxI

C

ux

zz

0

0

~22

Clearly,if =0.zz ~

small

perturbations

PKhargonekar

KeminZhou


Nowletusconsidertheperturbedsystem

uDxCzxy

wEuBxAx

22

~~~:~

where

I

DDI

CC

0:~

0:~

2

2

2

2 and

Obviously,isofmaximalcolumnrankandisfreeofinvariantzerosfor

any >0.Thus,satisfiestheconditionsoftheregularstatefeedbackcase,andhence

wecanapplytheproceduresforregularcasestotheperturbedsystemtofindtheH2andH controllaws.

Example:

2

~D )~,~,,( 22 DCBA

~

uxz

xy

wuxx

011

21

10

4325

: :~uxz

xy

wuxx

000

0

01

00

1

21

10

4325


SolutiontotheGeneralH2 StateFeedbackProblem

Givenasmall >0,SolvethefollowingalgebraicRiccati equation(H2‐ARE)

1T T T T T T

2 2 2 2 2 2 2 2 0A P PA C C PB C D D D D C B P

forauniquepositivedefinitesolution >0.Obviously,isafunctionof . TheH2suboptimalstatefeedbacklawisthengivenby

T 1 T T2 2 2 2( )u F x D D D C B P x

Itcanbeshowedthattheresultingclosed‐loopsystemTzw(s)has

Itcanalsobeshowedthat

*22 as 0zwT

1T *2

2trace( ) as 0.E PE

P~ P~



uxz

xy

wuxx

011

21

10

4325

:

SolvingthefollowingH2‐AREusingMATLAB

with =1,weobtain

186.1968 46.2778

,46.2778 18.2517

P 46.2778 18.2517F

225.1*2



output:


isgivenby

21.2472 4.9311

,4.9311 1.8975

P 49.3111 18.9748F

• =0.1

• =0.0001

1.6701 0.0424

,0.0424 0.0112

P 423.742 112.222F

= 1

= 0.1

= 0.0001


SolutiontoGeneralH StateFeedbackProblem

Step1: Givena >*,choose =1.

Step2: Definethecorresponding

Step3: SolvethefollowingalgebraicRiccatiequation(H‐ARE)for

Step4: If,gotoStep5.Otherwise,reducethevalueof andgotoStep2.

Step5: Computetherequiredstatefeedbackcontrollaw

Itcanbeshowedthattheresultingclosed‐loopsystemTzw(s)has:

MoregeneralresultsforthesingularcasecanbefoundinChen(2000).

2 2

2 2: and : 00

C DC I D

I22

~~ DC and

1T T T 2 T T T T2 2 2 2 2 2 2 2/ ( ) ( ) 0A P PA C C PEE P PB C D D D D C B P

:P

0~ P

T 1 T T2 2 2 2( )u F x D D D C B P x

.zwT


Example: Again,considerthefollowingsystem

uxz

xy

wuxx

011

21

10

4325

:



SolvingthefollowingH‐AREusingMATLAB

with =0.6and =0.001, weobtaina

positivedefinitesolution

and

15.1677 0.98740.9874 0.0981

P

987.363 98.1161F

5.0*



output:


atthelowfrequencyisslightlyless

than0.6.Thedesignspecificationis

achieved.


SolutionstoOutputFeedbackProblems– RegularCase

Recallthesystemwithmeasurementfeedback,i.e.,

where(A,B)isstabilizableand(A,C1)isdetectable.Also,itsatisfiesthefollowing

regularityassumptions:


2.Thesubsystem(A,B,C2,D2)hasnoinvariantzerosontheimaginaryaxis


4.Thesubsystem(A,E,C1,D1)hasnoinvariantzerosontheimaginaryaxis

1 1

2 2

:x A x B u E wy C x D wz C x D u


SolutiontoRegularH2 OutputFeedbackProblem

SolvethefollowingalgebraicRiccati equation(H2‐ARE)

1T T T T T T2 2 2 2 2 2 2 2 0A P PA C C PB C D D D D C B P

forauniquepositivesemi‐definitestabilizingsolutionP 0,andthefollowingARE

T 1 T T2 2 2 2( )F D D D C B P

1T T T T T T1 1 1 1 1 1 0QA AQ EE QC ED D D D E C Q

forauniquepositivesemi‐definitestabilizingsolutionQ 0.TheH2 optimaloutput


1:c

v A BF KC v K yu F v

1T T T

1 1 1 1K QC ED D Dandwhere

Furthermore,

1

2* T T T2 2 2trace( ) traceE PE A P PA C C Q



uxz

wxy

wuxx

111

110

21

10

4325

:

SolvingthefollowingH2‐AREsusingMATLAB,

weobtain

andanoutputfeedbackcontrollaw,

144 4040 16

P 41 17F

*2 347.3



output:


isgivenby

49.7778 23.333323.3333 14.0000

Q

24.333316.0000

K

5 22.3333 24.333338 29 16:

41 17c

v v y

u v


SolutiontoRegularH OutputFeedbackProblem

Givena >*,solvethefollowingalgebraicRiccatiequation(H‐ARE)

forapositivesemi‐definitestabilizingsolutionP 0,andthefollowingARE

T 1 T T2 2 2 2( ) ,F D D D C B P

1T T T 2 T T T T

2 2 1 1 1 1 1 1/ 0QA AQ EE QC C Q QC ED D D D E C Q

forapositivesemi‐definitestabilizingsolutionQ 0.Infact,these

P andQ satisfytheso‐calledcouplingcondition:.The

1T T T

1 1 1 1 .K QC ED D D

where

1T T T 2 T T T T

2 2 2 2 2 2 2 2/ ( ) ( ) 0A P PA C C PEE P PB C D D D D C B P

12 2 2cmp 1 1

T TA A EE P BF I QP K C D E P

12cmp cmp,B I QP K C F

andwhere

cmp cmpcmp

cmp

:v A v B yu C v

2)( PQ

JohnDoyle

KeithGlover

PKhargonekar

BruceFrancis

GileadTadmor

H ‐suboptimaloutputfeedbacklawisthengivenby[DGKF](1989)



uxz

wxy

wuxx

111

110

21

10

4325

:



SolvingthefollowingH‐AREsusingMATLAB

with =97,weobtain

144.353 40.116840.1168 16.0392

P



output:

49.8205 23.355623.3556 14.0118

Q

cmp

38.814 1848.66 1836.5859.414 914.112 894.227:41.116 17.039

v v y

u v

32864.96*


atthelowfrequencyisslightlyless

than97.Thedesignspecificationis

achieved.


SolutionstoOutputFeedbackProblems– SingularCase

Forgeneralsystemsforwhichtheregularityconditionsarenotsatisfied,itcanbe

solvedagainusingtheperturbationapproach.Wedefineanewcontrolledoutput:

andnewmatricesassociatedwiththedisturbanceinputs:

TheH2 andH controlproblemsforsingularoutputfeedbackcasecanbeobtained

bysolvingthefollowingperturbedregular systemwithsufficientlysmall :

2 2

00

C Dzz x I x u

u I

1 1[ 0 ] and [ 0 ].E E I D D I

1 1

2 2

:x A x B u E wy C x D wz C x D u

Remark:Perturbationapproachmight

haveseriousnumericalproblems!


SideNotesonH SingularCase


2.(A,B,C2,D2)hasnoinvariantzerosontheimaginaryaxis


4.(A,E,C1,D1)hasnoinvariantzerosontheimaginaryaxis

AntonStoorvogel &coworkers CarstenScherer BugsBunny&coworkers

Constructionofclosed‐formsolutionsandcomputationof*etc…


SideNoteson(Almost)DisturbanceDecoupling

1. If 2*=0,thenthecorrespondingH2 optimalcontrolproblemisalsocalledan

H2 (almost)disturbancedecouplingproblem.ItcanbeshowedthattheH2almostdisturbancedecouplingproblemissolvableifthefollowing

conditionsaresatisfied:

• (A,B )isstabilizableand(A,C1 )isdetectable

• (A,B,C2,D2 )isrightinvertibleandhasnoinvariantzerosonopenRHP

• (A,E,C1,D1 )isleftinvertibleandhasno invariantzerosonopenRHP

Necessaryandsufficientconditionsforthesolvabilityofthealmost

disturbancedecouplingproblemisavailableintheliterature.However,they

canonlybeexpressedintermsofcertaingeometricsubspacesonthegiven

system…


2. If ∞*=0,thenthecorrespondingH∞ optimalcontrolproblemisalsocalledanH∞almostdisturbancedecouplingproblem.ItcanbeshowedthattheH∞ almost

disturbancedecouplingproblemissolvableifthefollowingconditionsaresatisfied:

• (A,B )isstabilizableand(A,C1 )isdetectable

• (A,B,C2,D2 )isrightinvertibleandof minimumphase

• (A,E,C1,D1 )isleftinvertibleandofminimumphase

Studiesondisturbancedecouplingproblemsledtothedevelopmentofthegeometric

theoryinlinearsystems…

CarstenSchererJanC.Willems1939–2013

W.M.Wonham BugsBunny

……


Applications:SomeRobustControlProblemsApplications:SomeRobustControlProblems


RobustStabilizationofSystemswithUnstructuredUncertainties

Consideranuncertainplantwithanunstructuredperturbation,

c

u

w z

y

w z

zwT

SmallGainTheory(!)

Ifisstableand,thentheinterconnectedsystemisstable.

1

M

M

Assume.Thenthesystemwith

unstructureduncertaintyif

zwT

11

zwT


RobustStabilizationwithAdditivePerturbation

Consideranuncertainplantwithadditiveperturbations,

m

e

u y

m hasatransferfunction mmmmm DBAsICsG 1)()(

e isanunknownperturbation.

m and em havesamenumberofunstablepoles.

Givenaa >0,theproblemofrobuststabilizationforplantswithadditiveperturbations

istofindapropercontrollersuchthatwhenitisappliedtotheuncertainplant,the

resultingclosed‐loopsystemisstableforallpossibleperturbationswiththeirL‐norm

a.(ThedefinitionofL‐normisthesameasthatofH‐normexceptforL‐norm,the

systemneednotbestable.) SuchaproblemisequivalenttofindanH ‐suboptimal

controllaw(with =1/a )for

m m

add m m

0:

0

x A x B u wy C x D u I wz x I u


RobustStabilizationwithMultiplicativePerturbation

Consideranuncertainplantwithmultiplicativeperturbations,

m

e

u y

m hasatransferfunction mmmmm DBAsICsG 1)()(

e isanunknownperturbation.

m andem havesamenumberofunstablepoles.

Givenam >0,theproblemofrobuststabilizationforplantswithmultiplicative

perturbationsistofindapropercontrollersuchthatwhenitisappliedtotheuncertain

plant,theresultingclosed‐loopsystemisstableforallpossibleperturbationswiththeir

L‐norm m.Again, suchaproblemisequivalenttofindanH ‐suboptimalcontrollaw

(with =1/m )forthefollowingsystem,

m m m

multi m m m:0

x A x B u B wy C x D u D wz x I u



UsingbothH2 andH controltechniquestodesignappropriatemeasurementfeedbackcontrollawsthatmeetallthedesignspecificationspecifiedinthe(HDDorhelicopter)problem.



9.Robust&PerfectTracking9.Robust&PerfectTracking


Therobustandperfecttracking(RPT)controltechniquedevelopedbyChenandhisco‐workersistodesignacontrollersuchthattheresultingclosed‐loopsystemisstableandthecontrolledoutputalmostperfectlytracksagivenreferencesignalinthepresenceofanyinitialconditionsandexternaldisturbances.

OneofthemostinterestingfeaturesintheRPTcontrolmethodisitscapabilityofutilizingallpossibleinformationavailableinitscontrollerstructure.Suchafeatureishighlydesirableforflightmissionsinvolvingcomplicatedmaneuvers,inwhichnotonlythepositionreferenceisuseful,butalsoitsvelocityandevenaccelerationinformationareimportantorevennecessarytobeusedinordertoachieveagoodoverallperformance.

TheRPTcontrolrendersflightformationofmultipleUAVsatrivialtask.

2000

RobustandPerfectTrackingControl

BugsBunny


Problemformulation

Robusttodisturbance&initialcondition

Robusttodisturbance&initialcondition

PerfectinTrackingPerfectinTracking

RPTCONTROLRPTCONTROL


Solvabilityconditions:

Thesolvabilityconditionforthegeneralmeasurementfeedbackcaseisrathercomplicated.Pleaserefertothereferencetextfordetails(Theorem9.2.1).


Solution:


SpecialCase…

Forthespecialcasewhenthegivenplantisofadoubleintegrator,i.e.,

wherep isthepositionandv istheacceleration,assumingthereferenceposition(pr),velocity(vr)andacceleration(ar)areallavailable,itcanbeshownthattheRPTcontrollawcanbecalculatedinthefollowingclosed‐form

where isthedampingratioandn isthenaturalfrequencyoftheclosed‐loopsystem,and isthetuningparameter.

Wenotesuchaplantisverycommoninrealapplicationsincludingtheouterloopflightcontrolsystems.Infact,theRPTcontrol isveryeffectiveinimprovingflightperformanceforUAVs.

vp

zvp

ywEuvp

vp

01,,10

0010

rrn

r2

2nn

2

2n 22 avp

vp

u


CaseStudy…UnmannedHelicopterFlightControlSystems

Measured Signal

Inner Loop Control

Inner Loop Control

Outer Loop Control

Outer Loop Control

Trajectory Generation

Path Planning

Mission Management

AnunmannedsystemAnunmannedsystem

InnerLooptostabilizeUAVattitude

o PIDControl(commonlyused)o OptimalControlo RobustControlo NonlinearControlo ……

InnerLooptostabilizeUAVattitude

o PIDControl(commonlyused)o OptimalControlo RobustControlo NonlinearControlo ……

OuterLooptocontrolposition/velocity

o PIDControl(commonlyused)o Poleplacemento RPTControlo RobustControlo ……

OuterLooptocontrolposition/velocity

o PIDControl(commonlyused)o Poleplacemento RPTControlo RobustControlo ……


H controlH control

WinddisturbanceWinddisturbance

Detailedcontrolstructure


Inner‐loopcontrolsystemdesignsetup


Inner‐looplinearizedmodelathover

pqr

y

out

h

Onecanusethetechniquescoveredearlier,i.e.,H2 control,H∞ control,orLQGtodesignanappropriateinner‐loopcontrollerfortheabovesystem.


Inner‐loopcommandgenerator

Theinner‐loopcommandgeneratorisgivenas

r

r b,r

r

0 0.0019 0.04770 0.1022 0.0037

0.1022 0 0.0001

a


Outer‐loopcontrolsystemdesignsetup

VIRTUAL

ACTUATOR

VIRTUAL

ACTUATOR


Propertiesofthevirtualactuator

XChannel

YChannel

ZChannel

UnstableZeros!

Frompracticalpointofview,itissafe

toignorethemsolongastheouter‐

loopbandwidthiswithin1rad/sec…

Frequencyresponseofthevirtualactuator…


Propertiesoftheouter‐loopdynamics

Itcanalsobeverifiedthatcouplingamongeachchanneloftheouterloopdynamicsisveryweakandthuscanbeignored.Asaresult,allthex,yandzchannelsoftherotorcraftdynamicscanbetreatedasdecoupledandeachchannelcanbecharacterizedby

wherep* istheposition,v* isthevelocityanda* istheacceleration,whichistreatedacontrolinputinourformulation.

Forsuchasimplesystem,itcanbecontrolledbyalmostallthecontroltechniquesavailableintheliterature,whichincludethemostpopularandthesimplestonesuchasPIDcontrol…

* **

* *

0 1 00 0 1

p pa

v v


Outer‐loopRPTcontrollaw


SimulationofRPTcontrolwith =0.7&n=1…SimulationofRPTcontrolwith =0.7&n=1…

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

Time (sec)

Posi

tion

red: actual response; blue: reference

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

Time (sec)

Velo

city


0 5 10 15 20 25 30-1

-0.5

0

0.5

1

Time (sec)

Acce

lera

tion


0 5 10 15 20 25 30-0.6

-0.4

-0.2

0

0.2

Time (sec)

Trac

king

err

or

Position

0 5 10 15 20 25 30-1

-0.5

0

0.5

Time (sec)

Trac

king

err

or

Velocity

0 5 10 15 20 25 30-0.5

0

0.5

1

1.5

Time (sec)

Trac

king

err

or

Acceleration


SimulationofRPTcontrolwith =0.7&n=1(cont.)SimulationofRPTcontrolwith =0.7&n=1(cont.)

0 5 10 15 20 25 300

5

10

15

Time (sec)

red: actual response; blue: referenceP

ositi

on

0 5 10 15 20 25 30-0.5

0

0.5

1

1.5

2

Time (sec)

Vel

ocity

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

Time (sec)

Acc

eler

atio

n

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

Trac

king

err

or

Position

0 5 10 15 20 25 30-2

-1.5

-1

-0.5

0

0.5

Trac

king

err

or

Velocity

0 5 10 15 20 25 30-0.5

0

0.5

1

1.5

2

Time (sec)

Trac

king

err

or

Acceleration


10.LoopTransferRecovery10.LoopTransferRecovery


IsLQGControllerRobust?

Itisnowwell‐knownthatthelinearquadraticregulator(LQR)hasveryimpressive

robustnessproperties,includingguaranteedinfinitegainmarginsandatleast60⁰phase

marginsinallchannels.Theresultisonlyvalid,however,forthefullstatefeedbackcase.

IfobserversorKalmanfilters(i.e.,LQGregulators)areusedinimplementation,no

guaranteedrobustnesspropertieshold.Stillworse,theclosed‐loopsystemmaybecome

unstableifyoudonotdesigntheobserverofKalmanfilterproperly.Thefollowing

examplegiveninDoyle(1978)showstheunrobustness oftheLQGregulators.

Example: Considerthefollowingsystemcharacterizedby

1 1 0 1,

0 1 1 1x x u v

[1 0]y x w

wherex,u andy denotetheusualstates,controlinput&measured

output,andw andv arewhitenoiseswithintensities1& >0,

respectively.

JohnDoyle1955–


TheLQGcontrollerconsistsofanLQRcontrollaw+aKalmanfilter.

LQRDesign:Supposewewishtominimizetheperformanceindex

Itisknownthatthestatefeedbacklawu=– Fxwhichminimizethe

performanceindexJ isgivenby

Forthisparticularexample,wecanobtainaclosed‐formsolution,

ItcanbeverifiedthattheopenloopofLQregulatorwithanyq >0hasan

infinitegainmarginandaphasemarginover105degrees.Thus,itisvery

robust.

T T

0

1( ) , 1, 1 1 , 01

J x Qx u Ru dt R Q q q

1 T ,F R B P T 1 T 0, 0.PA A P PBR B P Q P

[1 1]4 1 .2 [ 1]F q f


ItcanalsobeshownthattheKalmanfiltergainforthisproblemcanbeexpressedas

whichtogetherwiththeLQRlawresultanLQGcontroller,

Supposethattheresultingclosed‐loopcontrollerhasascalargain1+ (nominallyunity)

associatedwiththeinputmatrix,i.e.,

Tediousmanipulationsshowthatthecharacteristicfunctionoftheclosed‐loopsystem

comprisingthegivensystemantheLQGcontrollerisgivenby

11

21

41

kK

ˆ( )ˆ

x A BF KC x K yu F x

or

0(1 )

1B

theactualinput matrix

4 3 2 2( ) 4 1K s s s s sk f k f k f

1( )u F sI A BF KC Ky


Anecessaryconditionforstabilityisthat

Itiseasytoseethatforsufficientlargeq and,theclosed‐loopcouldbeunstablefora

smallperturbationinB ineitherdirection.Forinstance,letuschooseq= =60.Thenit

issimpletoverifytheclosed‐loopsystemremainsstableonlywhen– 0.08< <0.01.

TheaboveexampleshowsthattheLQGcontrollerisnotrobustatall!

Whatiswrong?

Theansweristhattheopen‐looptransferfunctionoftheLQRdesignandtheopen‐loop

transferfunctionoftheLQGdesignaretotallydifferentandthus,alltheniceproperties

associatedwiththeLQRdesignvanishintheLQGcontroller.Itcanbeseenmoreclearly

fromtheprecisemathematicalexpressionsofthesetwoopen‐looptransferfunctions,

andthisleadstothebirthoftheso‐calledLoopTransferRecoverytechnique.

2 4 0 1 0k f k f k f and


Open‐LoopTransferFunctionofLQR

Open‐looptransferfunction:Whentheloopisbrokenattheinputpointofthe

plant,i.e.,thepointmarked×,wehave

Thus,thelooptransfermatrixfromu toisgivenby

WehavelearntfromourpreviouslecturesthattheopenlooptransferLt(s)have

veryimpressivepropertiesifthegainmatrixF comesfromLQRdesign.

BuAxx

– F

xr = 0 u u

1ˆ ( )u F sI A Bu

u

1( ) ( )tL s F sI A B


Open‐LoopTransferFunctionofLQG

Open‐looptransferfunction:Whentheloopisbrokenattheinputpointofthe

plant,i.e.,thepointmarked×,wehave

Thus,thelooptransfermatrixfromu toisgivenby

Clearly,Lt(s)andLo(s)areverydifferentandthatiswhyLQGingeneraldoesnot

havenicepropertiesasLQRdoes.

u

BuAxx

– F ( sI – A + B F + K C )–1 K

xr = 0 u Cyu

1 1ˆ ( ) ( )u F sI A BF KC KC sI A Bu

1 1( ) ( ) ( )oL s F sI A BF KC KC sI A B


Doyle‐SteinConditions: ItcanbeshownthatLo(s)andLt(s)are

identicaliftheobservergainK satisfies

whichisequivalenttoB=0(proveit!).Thus,itisimpractical.

LoopTransferRecovery

TheaboveproblemcanbefixedbychoosinganappropriateKalmanfiltergainmatrix

KsuchthatLt(s)andLo(s)areexactlyidenticaloralmostmatchedoveracertainrange

offrequencies.SuchatechniqueiscalledLoopTransferRecovery.

TheideawasfirstpointedoutbyDoyleandSteinin1979.Theyhadgivenasufficient

conditionunderwhichLo(s)=Lt(s).Theyhadalsodevelopedaproceduretodesignthe

KalmanfiltergainmatrixK intermsofatuningparameterq suchthattheresulting

Lo(s) Lt(s)asq,forinvertibleandminimumphasesystems.

1 1 1( ) ( ) , ( )K I C K B C B sI A

GunterStein


ClassicalLTRDesign

ThefollowingprocedurewasproposedbyDoyleandSteinin1979forleftinvertibleand

minimumphasesystems:Define

whereQ0 andR0 arenoiseintensitiesappropriateforthenominalplant(infact,Q0 canbe

chosenasazeromatrixandR0= I ),andV isanypositivedefinitesymmetricmatrix(V

canbechosenasanidentitymatrix).Thentheobserver(orKalmanfilter)gainisgivenby

where Pisthepositivedefinitesolutionof

Itcanbeshownthattheresultingopen‐looptransferfunctionLo(s)fromtheabove

observerorKalmanfilterhas

20 0,qQ Q q BVB R R T

1K PC R T

1 0qAP PA Q PC R CP T T

( ) ( ), .o tL s L s as q



withand

Thissystemisofminimumphasewithoneinvariantzeroats =– 2.TheLQRcontrol

lawisgivenby

Theresultingopen‐looptransferfunctionLt(s)hasaninfinitygainmarginandaphase

marginover85°.WeapplytheDoyle‐SteinLTRproceduretodesignanobserverbased

controller,i.e.,

whereK iscomputedasonthepreviouspagewith

0 1 0 35,

3 4 1 61x x u v

[2 1]y x w

[ ( ) ( )] [ ( ) ( )] ( ).E v t v E w t w t [ ( )] [ ( )] 0E v t E w t

[50 10]u F x x

1 1[ ]u F BF KC K y

22

35 0 1225 2135[35 61] [0 1] .

61 1 2135 3721qQ qq


targetLo(s)with

q2=500

Lo(s)with

q2=10000

Lo(s)with

q2=100000


NewFormulationforLoopTransferRecovery

Considerageneralstabilizableanddetectableplant,

ThetransferfunctionisgivenbyAlso,letF beastate

feedbackgainmatrixsuchthatunderthestatefeedbackcontrollawu= – Fx hasthe

followingproperties:

• theresultingclosed‐loopsystemisasymptoticallystable;and

• theresultingtargetloopmeetsdesignspecifications(GM,PM).

SuchastatefeedbackcanbeobtainedusingLQRdesignoranyotherdesignmethodsso

longasitmeetsyourdesignspecifications.Usually,adesiredtargetloopwouldhavethe

shapeasgiveninthefollowingfigure.

x A x B uy C x D u

1( ) , ( ) .P s C B D sI A

( )tL s F B


Typicaldesiredopen‐loopcharacteristics…


Theproblemoflooptransferrecovery(LTR)istofindastabilizingcontroller

suchthattheresultingopen‐looptransferfunctionfromuto,i.e.,

iseitherexactlyorapproximatelyequaltothetargetloopLt(s).Letusdefinetherecovery

errorasthedifferencebetweenthetargetloopandtheachievedloop,i.e.,

Then,wesayexactLTRisachievableifE(s)canbemadeidenticallyzero,oralmostLTRis

achievableifE(s)canbemadearbitrarilysmall.

P(s)

C(s)

r u y

u

( )u C s y

u

( ) ( ) ( )oL s C s P s

( ) ( ) ( ) ( ) ( )t oE s L s L s F B C s P s


ObserverBasedStructureforC(s)

DynamicequationsofC(s):

Transferfunctionof

Achievedopen‐loop:

P(s)

D

K

C

r = 0

F

B– +

+–++

u y–

FullOrderObserverBasedControlleru

x

ˆ ˆ ˆ ˆ ˆ( ),x A x B u K y Cx D u u u F x

1 1( ) ( ) ( )oC s C s F BF KC KDF K

1 1( ) ( ) ( ) ( ) ( )o oL s C s P s F BF KC KDF K C B D


Notethatwehaveused . Thus,

Lemma: Recoveryerror,Eo(s),i.e.,themismatchbetweenthetargetloopandthe

resultingopen‐loopoftheobserverbasedcontrollerisgivenby

Proof.

)()()(),()()()( 111 KDBKCFsMBFIsMIsMsEo

)]([)]([)]()([)]([

])()([)]([})(])([{)]([

])()([)]([)()()]()([)()(])()([

)()()()()(

1

111

11111

111111

11111

11111

11111

11

sMBFsMIKDBKCFBFsMI

KDKCFBKCFBFsMIKDKCFBKCIFsMI

KDKCFBKCKCFsMIDBCKKCFKDBKCFIDBCKKCFKDBKCIF

DBCKKDFKCBFFsPsCsL oo

1 1 1 1 1( ) ( )KC KC I KC

1 1[ ] [ ] [ ] ( ).o t oE L L F B I M F B M I M M I F B

CredittoGCGoodmaninamasterthesis

conductedatMITin1984

CredittoGCGoodmaninamasterthesis

conductedatMITin1984

MichaelAthans1937–


LoopTransferRecoveryDesign

Itissimpletoobservefromtheabovelemmathatthelooptransferrecoveryis

achievableifandonlyifwecandesignagainmatrixKsuchthatM(s)canbemade

eitheridenticallyzeroorarbitrarilysmall,where

Letusdefineanauxiliarysystem

1 1( ) ( ) ( ).M s F KC B KD

uDxBzxy

wFuCxAx

TT

TTT

aux :

+ xKu T

Closed‐looptransferfunctionfromw toz is 1( )( ) ( ).B D K sI A C K F M s T T T T T T T T

Thus,LTRdesignisequivalenttodesignastatefeedbacklawfortheaboveauxiliarysystemsuchthatcertainnormoftheresultingclosed‐looptransferfunctionismadeeitheridenticallyzeroorarbitrarilysmall.Assuch,theH2 andH optimizationtechniquescanbeusedtosolvetheLTRproblem.Thereisnoneedtorepeatalloveragainoncethisisformulated.


AstorybehindanewcontrollerstructureforLTR…

AstorybehindanewcontrollerstructureforLTR…

W

BernardFriedland


DynamicequationsofC(s):

Transferfunctionof

Achievedopen‐loop:

ˆ( ) ,v A KC v Ky u u F v

KKCFsCsC c11 )()()(

1 1( ) ( ) ( ) ( ) ( )c cL s C s P s F KC K C B D

P(s)r = 0 u y

u+

–

F K–

C

v

CSSBasedController

ProposedbyChen,Saberi andSannuti in1991,theCSSbasedcontroller

hasthefollowingcharacteristics:

AliSaberi1949–

PeddaSannuti1941–

C

LTRDesignviaCSSArchitectureBasedController


Lemma: Recoveryerror,Ec(s),i.e.,themismatchbetweenthetargetloopandtheresultingopen‐loopoftheCSSarchitecturebasedcontrollerisgivenby

Proof.

)()()()( 11 KDBKCFsMsEc

)()()(

])[()()()(

)()()(

11

111

11

sMKDBKCBKCBKCF

KDBKCBKCKCFDBCKKCFBF

sLsLsE ctc

ItisclearthatLTRviatheCSSarchitecturebasedcontrollerisachievableifandonlyifonecandesignagainmatrixKsuchthattheresultingM(s)canbemadeeitheridenticallyzeroorarbitrarilysmall.ThisisidenticaltotheLTRdesignviatheobserverbasedcontroller.Thus,onecanagainusingtheH2andH techniquestocarryoutthedesignofsuchagainmatrix.

• B.M.Chen,A.Saberi andP.Sannuti,Anewstablecompensatordesignforexactandapproximatelooptransferrecovery,Automatica,Vol.27,No.2,pp.257–280,March1991.CollectedinBibliographyonRobustControlbyP.Dorato,R.Tempo,G.Muscato inAutomatica,Vol.29,No.1,January1993.


WhatistheAdvantageofCSSStructure?

Answer:Theorem. Considerastabilizableanddetectablesystem characterizedby

(A,B,C,D)andtargetlooptransferfunctionLt(s)=FB.Assumethat isleft

invertibleandofminimumphase,whichimpliesthatthetargetloopLt(s)is

recoverablebybothobserverbasedandCSSarchitecturebasedcontrollers.Also,

assumethatthesamegainK isusedforbothobserverbasedcontrollerandCSS

architecturebasedcontrollerandissuchthatforall ,where issomefrequency

regionofinterest,

Then,forall ,

Proof. SeeChen,SaberiandSannuti,Automatica,vol.27,1991.

max0 [ ( )] 1,M j 1min min[ ( )] [ ( ) ] 1tL j F j I A B

max max[ ( )] [ ( )].c oE j E j


Remark: Inordertohavegoodcommandfollowinganddesireddisturbancerejection

properties,thetargetlooptransferfunctionLt(j)hastobelargeandconsequently,

theminimumsingularvalueshouldberelativelylargeintheappropriate

frequencyregion.Thus,theassumptionintheabovetheoremisverypractical.


LetthetargetloopLt(s)=FB becharacterizedbyastatefeedbackgain

UsingMATLAB,weknowthattheabovesystemhasaninvariantzeroats =– 2.Hence

itisofminimumphase.Also,itisinvertible.Thus,thetargetloopLt(s)isrecoverable

byboththeobserverbasedandCSSarchitecturebasedcontrollers.

UsingtheH2 optimizationmethod,weobtainmatrix

0 1 0, 2 1 0

3 4 1x x u y x u

50 10 .F

6.9.

84.6K

)]([min jLt


CSS

Target

Observer


CSS

Observer



ThetargetloopLt(s)=FB ischaracterizedbyanLQRstatefeedbackgainwithQ =I7and0.001 I3.TheobserverandCSScontrollergainmatricesarechosenthruthe

classicalLTRdesigntechniquewithtuningparametersummarizedinthetablebelow.

, ,




Usingthelooptransferrecoverycontroltechniquetodesignappropriatemeasurementfeedbackcontrollawsthatmeetallthedesignspecificationspecifiedinthe(HDDorhelicopter)problem.



11.ConcludingRemarksSomePersonalViewpointsonControlSystemsDesign

11.ConcludingRemarksSomePersonalViewpointsonControlSystemsDesign


在上一期“有问必答”栏目里，我们刊登了华东交通大学杨辉教授团队提出

的一个常常问起的控制问题：如今各类型的先进控制算法这么多，为什么实际应

用最常见的还是PID控制？

答（B.M.C.）: 在实际应用当中，因为PID控制简单易调，自然就成为工程设计人员

的首选控制器。如果被控系统无法被PID所控，设计人员一般首先考虑是重新设计

被控的系统（如重新设计机械架构或重新调配驱动器和传感器），而不是直接采

用先进的控制方法，如此这般，PID就成了实际应用中常见的控制器。不过PID也

非万能，由于PID控制器结构上的缺陷，在PID控制的系统当中，我们一般无法优

化系统的整体性能；在诸多多变量系统中，PID往往也是束手无策的。在设计多变

量控制系统时，同样因为简易，LQR则是最常见的控制方法。顺便提一句，在实

际应用中，许多所谓的先进控制方法其实都是大同小异的（此处可以有石头、鸡

蛋飞来）。

中国〖TCCT通讯〗2014年第4期


Question:WhyisPIDcontrolstilldominatinginpracticalapplicationseventhoughthere

aresomanyadvancedcontroltechniquesoutthere?

Answer(B.M.C.): PIDcontrolisalwaysthefirstchoiceofpracticingengineersbecauseit

isstructurallysimpleanditiseasytotune.IfasystemcannotbecontrolledbyaPID

controller,thefirstthingthatengineerswoulddoistoredesignthesystem(suchasto

restructurethesystemmechanicalpartortoreselectand/orreplacethesystemsensors

andactuators), insteadoftryinganadvancedcontroltechnique.Assuch,PIDis

dominatinginpracticalapplications.However,thisdoesnotmeanthatPIDissuperior.

Becauseofitsstructurallimitation,itisgenerallydifficulttopushforanoptimal

performanceofthePIDcontrolledsystem.Furthermore,manymultivariablesystems

cannotevenbestabilizedbyPIDcontrollaws.ForMIMOsystems,LQRcontrolonthe

otherhandisthemostpopularchoiceamongallthecontroltechniques.

Bytheway,manyadvancedcontrolmethodsdonotmakemuchdifferenceincontrolling

practicalsystems.

TCCTNewsletter,April2014


Ipersonallybelievethatagoodcontrol

systemdesignshouldnotstartfrom

differentialequationsbutshouldbedowntoearthandstartfromthehardwarelevel,

includingtheselectionand

placementofsensorsandactuators.BMC

Ipersonallybelievethatagoodcontrol

systemdesignshouldnotstartfrom

differentialequationsbutshouldbedowntoearthandstartfromthehardwarelevel,

includingtheselectionand

placementofsensorsandactuators.BMC

03.201303.2013 09.201309.2013 03.201403.2014

2017201720162016


Amoreadvancedcourseinlinearsystemsandcontrol…Amoreadvancedcourseinlinearsystemsandcontrol…

Coursematerialisavailableonlineatwww.bmchen.netCoursematerialisavailableonlineatwww.bmchen.net


Thecourseisaimedtoanswerthefollowingquestions:

WhyisthecommonlyusedPIDabadcontroller?

Whatcontrolperformancecanoneexpectfromagivensystem?

Why aresystemnonminimumphasezerosbadforcontrol?

Whatelsearebadtobecontrolled?

Whenanairplanepassesthroughturbulences,whycanitmaintainitspositionwhileitsbodyisshakingbadly?

Whenandhowcandisturbances,uncertaintiesandnonlinearitiesbeattenuatedthroughpropercontrolsystemdesign?

Whatisthebestwaytodesignacontrolsystem?o todesignagoodcontrollaw?oro todesignagoodsystem?

Howtodesignagoodsystemthroughsensorandactuatorselection?

WhyisPIDnotbadatallafterall?

Howtoimprovecontrolperformance?


That’sall,folks!

ThankYou!

That’sall,folks!

ThankYou! BugsBunny1940–

Documents

EE5/6102: Multivariable Control Systems - NUS UAVuav.ece.nus.edu.sg/~bmchen/courses/EE5102.pdf · EE5102/6102 PART2~ PAGE1 BENM. CHEN, NUS ECE EE5/6102: Multivariable Control Systems