CS:4980Founda/onsofEmbeddedSystems

Copyright 20014-16, Rajeev Alur and Cesare Tinelli. Created by Cesare Tinelli at the University of Iowa from notes originally developed by Rajeev Alur at the University of Pennsylvania. These notes are copyrighted materials and may not be used in other course settings outside of the University of Iowa in their current form or modified form without the express written permission of one of the copyright holders. During this course, students are prohibited from selling notes to or being paid for taking notes by any person or commercial firm without the express written permission of one of the copyright holders.

DynamicalSystemsPartIII

ControlDesignProblem

PlantmodelasCon/nuous-/meComponentH

UncontrolledInputs ObservableOutputs

q WewanttodesignacontrollerCsothatC||Hisstableq  Isthereamathema/calwaytocheckwhenasystemisstable?

q  IsthereinfactawaytodesignCsothatC||Hisstable?q  Yes,iftheplantmodelislinear

ControllerC

ControllableInputs

LinearComponent

Alinearexpressionovervariablesx1,x2,…xnisoftheforma1x1+a2x2+…+anxn

wherea1,a2,…arera/onalconstants

Acon/nuous-/mecomponentHwithstatevariablesS,inputvariablesI,andoutputvariablesOislinearif

§  foreverystatevariablex,thedynamicsisgivenbydx=fx(S,I),wherefxisalinearexpression

§  foreveryoutputvariabley,yisdefinedbyy=hy(S,I),wherehyisalinearexpression

Examples

§  linear:heaYlow,car,helicopter§  nonlinear:pendulum

Con/nuous-/meComponentCar2

v

F

dx=vdv=(F–kv–mgsin(θ))/m

realxL<=x<=xUvL<=v<=vU

q  Right-handsideofdvequa/onnotlinear

q  Easyfix:replacedisturbanceθbyanothervariablesθ=sinθ

real[–π/6,π/6]θ

Con/nuous-/meComponentCar2

v

F

dx=vdv=(F–kv–mgsθ)/m

realxL<=x<=xUvL<=v<=vU

real[sin(–π/6),sin(π/6)]sθ

Rewri/ngtonormalform:

dx/dt=0x+1v+0F+0sθdv/dt=0x+(–k/m)v+(1/m)F+(–g)sθ

v=0x+1v+0F+0sθ

Matrix-basedrepresenta/on:

S=(xv)TI=(Fsθ)TO=(v)

dS/dt=AS+BI

O=CS+DI

A=01B=000-k/m1/m-g

C=(01)D=(00)

(A,B,C,D)Representa/onofLinearComponents

Supposealinearcon/nuous-/mecomponenthas§  nstatevariablesS={x1,x2,…xn}§  minputvariablesI={u1,u2,…um}§  koutputvariablesO={y1,y2,…yk}

Thenthedynamicsisgivenby

dS/dt=AS+BIandO=CS+DIwhere

Aisann×nmatrix Cisak×nmatrix

Bisann×mmatrix Disak×mmatrix

Therateofchangeofi-thstatevariableandthevalueofj-thoutputare

dxi/dt=Ai,1x1+Ai,2x2+…+Ai,nxn+Bi,1u1+Bi,2u2+…+Bi,mum

yj=Cj,1x1+Cj,2x2+…+Cj,nxn+Dj,1u1+Dj,2u2+…+Dj,mum

Input-OutputLinearity

Con/nuous-/meComponentH

InputsI OutputsO

Withafixedini/alstate,acon/nuous-/mecomponentHmapsinputsignalsI(t)tooutputsignalsO(t)

Theorem:IfHislinear,thenbothofthefollowinghold.§  Scaling:IftheoutputresponseofHtotheinputsignalI(t)is

O(t),thenforeveryconstantα,theoutputresponseofHtotheinputsignalαI(t)isαO(t)

§  Addi,vity:IftheoutputresponsesofHtotheinputsignalsI1(t)andI2(t)areO1(t)andO2(t),thentheoutputresponseofHtotheinputsignal(I1+I2)(t)is(O1+O2)(t)

ResponseofLinearSystems

Consideraone-dimensionallinearsystemwithnoinputs:dx/dt=ax;ini/alstatex0

Itsexecu/onisgivenbythesignalx(t)=x0eat

§  Recallthatea=1+∑n>0an/n!

§  Verifythatsolu/onx(t)sa/sfiesthedifferen/alequa/on§  Seetextbookonhowsolu/onisfound

ResponseofLinearSystems

GeneralCase(noinputs)

q  StatesetS

q  Dynamicsisgivenby dS/dt=AS

ini/alstates0

q  ForeachinputsignalI(t),execu/onisgivenbythesignal S(t)=eAts0

§  At=scalarproductofAandt§  eM=I+∑n>0Mn/n!§  I=iden/tymatrix(Ii,j=if(i=j)then1else0)

GeneralCase(withinputs)

q  StatesetS,inputsetI

q  Dynamicsisgivenby dS/dt=AS+BI

ini/alstates0

q  ForeachinputsignalI(t),execu/onisgivenbythesignal S(t)=eAts0+∫0t(eA(t–τ)BI(τ)dτ)

ResponseofLinearSystems

MatrixExponen/al

q Matrixexponen/aleA=I+A+A2/2+A3/3!+A4/4!+…

q  Eachterminthesumisann×nmatrix

q  HowdowecomputeeA?

q  IfAk=0forsomek,thesumisfiniteandcanbecomputeddirectly

q  IfAisadiagonalmatrixD(a1,a2,…,an)(Aij=if(i=j)thenaielse0),theneA=D(ea1,ea2,…,ean)

q  Ingeneral,thesumofthefirstktermswillgiveanapproxima/on(whosequalityispropor/onaltok)

q Otherwise,wecanuseanaly/calmethodsbasedoneigenvaluesandsimilaritytransforma/ons

EigenvaluesandEigenvectors

q  LetAbeann×nmatrix,λ ascalarvalueandxann-dimensionalnon-zerovector.Iftheequa/onAx=λxholds,thenxisaneigenvectorofA,andλisthecorrespondingeigenvalue

q  Howtocomputeeigenvaluesandeigenvectors?

q Wesolvethecharacteris,cequa,onofA:

det(A–λI )=0

q  Recall:thedeterminantdet(M)ofa2×2matrixMisM1,1M2,2–M1,2M2,1

Theeigenvaluesofann×nmatrixAaretherootsofthecharacteris/cpolynomialp=det(A–λI)

Note:

q  Themul/plicityofaneigenvalue(asarootofp)canbe>1

q  Aneigenvaluecanbeacomplexnumberq  IfAisadiagonalmatrixthenthediagonalentriesarethe

eigenvaluesq  Foragiveneigenvalueλ,wecancomputethecorresponding

eigenvector(s)bysolvingthelinearsystemAx=λ x,withunknownvectorx

q  IfalleigenvaluesofareAdis/nct,thenthesetofcorrespondingeigenvectorsislinearlyindependent

EigenvaluesandEigenvectors

SimilarityTransforma/on

q  ConsiderdynamicalsystemHwithdynamicsgivenby:

dS/dt=AS;ini/alstates0q  LetPbeaninver/blen×nmatrix

q  ConsidersystemH’withstatevectorS’=P-1Sanddynamics

d/dtS’=d/dt(P-1S)=P-1dS/dt=P-1AS=P-1APS’=JS’

q  ThematrixJ=P-1APissaidtobesimilartoA

q  Theini/alstateoftransformedsystemH’isS’(0)=P-1s0

q  TheresponseofthetransformedsystemH’isgivenby

S’(t)=eJtP-1s0

q  TheresponseoftheoriginalsystemisS(t)=PeJtP-1s0

q Whenisallthisuseful?

q WhenwecanchoosePsothatJisdiagonal!

SimilarityTransforma/onusingEigenvectors

q  ConsidersystemHwithdynamicsgivenby: dS/dt=AS;ini/alstates0

q  Calculateeigenvaluesλ1,…,λnandsupposetheyarealldis/nct

q  Calculatecorrespondingeigenvectorsx1,…,xn(whichmustbelinearlyindependent)

q  Considerthen×nmatrixP=(x1x2…xn)

q  FinditsinverseP-1(whichmustexistinthiscase)

q  Claim:ThematrixJ=P-1APisthediagonalmatrixD(λ1,…,λn)

q  Execu/onofthesystemisgivenby

S(t)=PD(eλ1t,…,eλnt)P-1s0

Example:ResponseofLinearSystemsConsider2-dimensionalsystemwithdynamicsgivenby

ds1=4s1+6s2 ini/alstate(s1,s2)=(1,1)T

ds2=s1+3s2

1.  Computeeigenvaluesλ1andλ2ofA=((41)T(63)T)§  λ1=6andλ2=1

2.  Computeeigenvectorsx1andx2§  x1=(31)Tandx2=(2-1)T

3.  Choosethesimilaritytransforma/onmatrixP=(x1x2)=((31)T(2-1)T)

4.  ComputetheinverseP-1ofP§  P-1=((-1-1)T(-23)T)/(-3-2)=((1/51/5)T(2/5-3/5)T)

5.  VerifythatJ=P-1APisdiagonalmatrixD(λ1,λ2)=((60)T(01)T)§  J=P-1AP=((6/51/5)T(12/5-3/5)T)((31)T(2-1)T)=((60)T(01)T)

6.  Desiredsolu/onisS(t)=PD(eλ1t,eλ2t)P-1(1,1)T§  S(t)=((31)T(2-1)T)((e6t0)T(0et)T)((1/51/5)T(2/5-3/5)T)(1,1)T=…

BacktoEquilibriaandStability

q  ConsideraclosedlinearsystemHwithdynamicsgivenby: dS/dt=AS

q  Recall:astatesisanequilibriumstateofHifAs=0

q  Howtocomputeequilibria:solvesystemoflinearequa/ons

q  Claim1:State0isanequilibrium

q  Claim2:IfAisinver/ble,then0isthesoleequilibrium

q  Ifstatesisanon-zeroequilibriumofH,considerthetransformedsystemH’withstateS’=S–s§  Theequilibria0ofH’andsofHhavethesameproper/es

BacktoEquilibriaandStability

Henceforth,wewillfocusonclosedlinearsystemsHandtheirequilibriumstate0

Defini;on:

1.  Hisstableifstate0isstable2.  Hisasympto,callystableifstate0isasympto/callystable

Stability:One-DimensionalSystem

q  Consideraone-dimensionallinearsystemHwithdynamicsgivenby:dx/dt=axwithsomeini/alstates0

q  Recall:Hisasympto/callystableiff1.  (Stable)Foreveryε >0,thereisaδ >0suchthatforallini/alstates

swith||s||<δandforall/mest,||eats||<ε

2.  (Asympto/cally)Thereisaδ >0suchthatforallini/alstatesswith||s||<δ,||eats||goesto0astgoesto∞

q  Casea<0:eatsconvergesexponen/allyto0astgoesto∞,regardlessofs.Asympto/callystable

q  Casea=0:dynamicsisdx/dt=0.Thestatestaysequaltotheini/alstates.Stablebutnotasympto/callystable

q  Casea>0:eatsgrowsexponen/allyastincreases,andthus,statedivergesawayfrom0.Unstable!

Stability:DiagonalStateDynamics

q  Considern-dimensionallinearsystemHwithdynamicsgivenbydS/dt=AS,whereAisthediagonalmatrixD(a1,…,an)

q  Eachdimensionevolvesindependently:thei-thcomponentofS(t)iseaits0iwheres0istheini/alstatevector

q  Allcoefficientsai<0:Stateconvergestotheequilibrium0nomasertheini/alstate.Asympto/callystable

q  Allcoefficientsai<=0:Stablebutnotasympto/callystableifsomecoefficientaj=0(j-thstatecomponentstaysunchanged)

q  Somecoefficientai>0:Somestatecomponentgrowsunboundedlyawayfromequilibrium0.Unstable!

SimilarityTransforma/onsandStability

q  ConsidersystemHwithdynamicsgivenby:dS/dt=ASq  LetPbeaninver/blen×nmatrix,andconsiderJ=P-1AP

q  ConsidersystemH’withstateS’=P-1S(andnotethatS=PS’)

q  ResponsesignaloftransformedsystemH’: S’(t)=eJtP-1s0q  ResponsesignaloforiginalsystemH: S(t)=PeJtP-1s0

q  Note:responseofH’isalineartransforma/onofresponseofH

q Ifasignalisbounded,soisitslineartransforma/on

q Ifasignalconvergesto0,sodoesitslineartransforma/on

q  Claim:HisstableiffH’isstable

q  Claim:Hisasympto/callystableiffH’isasympto/callystable

EigenvaluesandStability

q  ConsiderHwithdynamicsisgivenby:dS/dt=ASq  Supposealleigenvaluesλ1,…,λnarerealanddis/nctq  Thenthesetofeigenvectors,x1,…,xnisguaranteedtobe

linearlyindependent

q  Choosen×nmatrixP=(x1x2…xn)forsimilaritytransforma/on

q  ThematrixJ=P-1APisthediagonalmatrixD(λ1,…,λn)

q  Ifalleigenvaluesarenega/ve,thenthetransformedsystemH’isasympto/callystable,andsoisH

q  Ifalleigenvaluesarenon-posi/ve,thenH’isstable,andsoisH

Theorem:AlinearsystemHwithdynamicsdS/dt=ASisasymptot-icallystableiffeacheigenvalueofAhasanega/verealpart

Con/nuous-/meComponentCar

vF dx=v

dv=(F–kv)/m

q  LetS=(xv)Tq  ThematrixAis((00)T(1-k/m)T)

q  Eigenvalues:0and-k/mq  Stablebutnotasympto/callystable

q  Ifweconsideronlythedimensionv,thenasympto/callystable

Exercise:SetF(t)=0forallt,andanalyzewhathappensifweperturbthesystemfromtheequilibrium(00)T

LyapunovStabilityvsBIBOStability

q  ConsiderlinearcomponentHwithdynamicsgivenby dS/dt=AS+BIO=CS+DI

q  BIBOstability:Star/ngfromini/alstate0,iftheinputisaboundedsignal,outputmustbeaboundedsignal

q  Theorem:Forlinearcomponents,asympto/cstabilityimpliesBIBOstability

q Note:Asympto/cstabilitydependsonlyontheproper/esofmatrixA

q  Proofofthetheoremreliesofanalysisofdynamicalsystemsusingtransferfunc/ons

Credits

NotesbasedonChapter6ofPrinciplesofCyber-PhysicalSystemsbyRajeevAlurMITPress,2015