The Analysis of Feedback Systems

198 Pages·1971·7.934 MB·English

by  Jan C. Willems

TheA nalysoifFs e edbaScyks tems
TheA nalysoifFs e edbaScyks tenls
Jan CW.i llems
ResearMcohn ograpNho .6 2
THE M.LT.P RESS
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ISBN0 262230416 (hardcover)
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Contents
Preface xi
Chapte1:rI ntroduction 1
1.1O.r ientation 1
1.2M.a thematiPcraell iminaries 2
1.3T:r ansfoTrhme ory 6
Chapt2er:N onlineOpaerr ators 10
2.1I.n troduction 10
2.2O.p eratoGresn:e ralities 11
2.3E.x tendSepda ces 12
2.4C.a usaOlp erators 14
2.5A.l gebraGse:n eralities 16
2.6O.p eratAolrg ebras 19
2.7C.o ntractiCoonnsi,Oc p erators, POopseirtaitvoer s 24
2.8C.o nditifoonrIs n vertiboifNJ ointlyi neOapre rators 28
2.9C.o nditifoonrNs o ninvertiobfiN lointlyi neOapre rators 40
Chapt3er:P ositive Operators 45
3.1.I ntroduction 45
3.2P.o sitiCvoen volutiOopne ratoarnsd PositiMveem oryless
Operators 49
3.3M.e moryleTsism e-InvarNioannlti neOapre rators 51
3.4P.e riodGiaci n 53
3.5P.o sitiOvpee ratowrist hM onotonoer O dd-MonoonteN on-
linearities 57
vii
viii CONTENTS
3.6F.r equency-PRoewleart ifoonrNs o nlineRaers istors 66
3.7F.a ctorizaotfOi poenr ators 72
3.8P.o sitOipveer atoornsE xtendSepda ces 81
Chapt4er:F eedbaScyks tems 86
4.1I.n troduction 86
4.2M.a thematiFcraalm ework 87
4.3W.e ll-PosedonfFe esesd baScyks tems 93
4.4S.t abilaintdIy n stability 101
4.5C.o ntinuaintdyD iscontinuity 110
4.6C.o ncludRienmga rks 114
Chapt5er:T heN yquiCsrti terainodtn h eC ircle Criteri1o2n2
5.1M.a thematiDceaslc riptoifto hne F eedbaScyks tem 122
5.2W.e ll-Posedness 124
'
5.3S.tabilaintdIy n stabiilnti hteyT ime-InvarCiaasnet 125
5.4S.t abilaintdIy n stabiilnti hteyT ime-Varying Case 130
Chapt6e:rS tabiClriittye Orbitaa inUesdi nMgu ltipliers 136
6.1I.n troduction 136
6.2T.r ansformatoifto hneFs e edbacLko op 138
6.3A. StabilCirtiyt erifoonrL ineaFre edbacSky stemwsi tha
PeriodGiaci ni nt heF eedbaLcok op 140
6.4A. StabilCirtiyt erfioorFn e edbaScyks tewmist ah M onotonoer
anO dd-MonotoNnoen lineairnit thyeF eedbaLcok op 155
Chapt7e:rL inearizaantdiS otna bility 161
7.1L.i nearization 161
7.2L.i nearizaSttiaobni,l aintdyC ,o ntinuity 166
7.3T.h eD escribFiunngc titohne,T otaGla inL inearizaatnidto hne,
IncremenGtaailnL inearization 169
7.4A.v eragiTnhge ory 173
7.5C.o unterexamtpoAl iezse rmaCno'nsj ecture 177
Index 185
Foreword
Thiiss t hes ixty-sevcoolnudmi en t heM .I.TR.e searMcohn ograph
Seripeusb lisbhyet dh eM .LT.P ressT.h eo bjectoiftv hei sse riiests o
contribtuott heep rofessilointaelr aat nuurmeb eorf s ignificpainetc es
ofr esearlcahr,g ienrs copteh anj ournaarlt icbluetsn ormayl lless
ambititohuasfin n ishbeodo ksW.e belietvheas tu csht udideess erav e
widre circulatthiaoncn a nb e accomplisbhyei dn formcahla nnels,
andw e hopet hatth ifso rmo fp ublicawtiilomlna ket hemr eadily
accessitbol ree searocrhg anizatliiobnrsa,r ainesd, i ndependent
workers.
HowarWd .J ohnson
Preface
Theraer tew ow aysd,i stiinnpc rti nciopfml aet,h ematidceaslclryi bing
physiscyaslt eTmhse.fi rsotn ei sc alltehde" input-oudtepsuctr iption"
sinciter elaetxetse rvnaarli abTlheesm .a thematmiocdaeltl h euns ually
taketsh ef ormof a ni ntegerqaula ti(otnhG er eenf'usn ctaipopnr oach)
orm oreg eneraolfla yn,o perateoqru atieoxnp resstihnerg e latshiiopn
betwetehne i nput(st hvea riabwlheisc cha nb em anipulaatneddt )h e
outpu(ttsh vea riabolfei sn terest-tytphierc eaaldliyn ogfas seto f
sensorSsu)c.ha ni nput-oudtepsuctr ipctainou ns ualbleyo btained
froms omer epresenteaxtpievriem eTnhtisas.p proarcehq uimriensi mal
knowledogfte h ep hysilcaawlgs o verntihnegs ysteamn do ft hei nter­
connectwiiotnhsti hne " blacbko x."
Fort he" interdneaslc riptoifao pnh"y siscyaslt eomn,t hec ontrary,
oneu setsh espeh ysilcaawlas n di nterconneacstt ihoebn ass iosft he
mathematimcoadle lT.h isg eneratlalkye tsh ef ormo fa n ordinary
differenetqiuaalt ioorna partidailff erenetqiuaalt iIonnt .h ep rocess
leaditnogt himso deoln ew orkwsi tah seotf i ntermedviaartiea bles,
relatteotd h ec onceopfts tatTeh.e raer teh utsw op arttsom athemtiacal
modelosf i nternadlelsyc ribseysdt emas :d ynamicpaalr t-which
descritbheees v olutoifot nh es tatuen dre thei nfluenocfet hei nputs,
anda memorylepsasr t-whircehl attehseo utputto t hes tat(ea nd
sometimteots h ei nstantanveaoluuoseft hei npuatsw ell).
Whicho ft het wod escriptiismo onrsec onvenideenpte ndosn t he
applicaatnidto hnep urpoosfet hae nalyosrit sh sey nthesiosn eth haast
inm indM.o dersny stetmh eorreyl iheesa violnyt hes taftoer mulation
xi
xii PREFACE
fors ynthetseicsh niqausee sx emplibfiyet dh eh ighligohfmt osd ern
contrtohle orPyo:n tryagmianx'ism upmr incitphlreee ,g ulaptroorb lem
forl inesayrs temasn,dt heK alman-Buficlyt ertihnego ry.
Int hea nalyosfic so ntrsoyls teomnse u sualilnyv estiqguaetsetsi ons
relatteosd t abilciotnyt,i nuaintyds, e nsitoifva i ctlyo sed-slyosotpe m.
Thesqeu esticoannsb et reatferdo mb otha ni nput-ouotrpa u stt ate­
spacpeo inotfv iewb,u ti ti so nly vreercye ntthlayst u ccesrsefsuull ts
havbee eonb tainaenddt haat s ufficiengteInyerfarla mewohraksb een
developteotd r eatth emi na ni nput-ousteptutti Tnhge.p ioneering
worki nt hidse velopmheansbt e epne rformbey1d .W .S andbeartBg e ll
LaboratoarnidGe .sZ amesa tN ASA-ERTeh.e saeu thofrosr mulated
thes tabiqluietsyt i(otnhm eo sti mportdaensti cgonn strianif nete dback
contrionla )n i nput-ousteptutti Tnhge.i deoaf i nput-ousttpaubti lity
findsi tsr ootisn t hec oncepotf bounded-inbpouutn,d ed-output
stabialnidit ynt hweo rko fN yquisNty.q uitsatk etsh fien itien tegrability
oft hei mpulrsees ponasste h eb asirce quiremfeonrst t abilwihteyr,e as
thec onceopftb ounded-inbpouutn,d ed-ousttpaubti rleiqtuyi trheast
boundeidn puptrso duce bouonudtepdu tTsh.ei deoaf N yquigsitv es
ane xcellteynptoe f s tabibluittu yn fortunaaptpelliyoe nsl tyo a very
restriccltaesdos f s ystemtsh,el inetairm e-invariantT hsey stems.
conceopftb ounded-inbpouutn,d ed-ousttpaubti lnietvyeh ra sh ad
muchs ucceasns,dv erfye ws pecirfiecs ulhtasv eb eenb aseodn i t.
Moreoveirth, a sb eenno s impmlaet tteora nalyfzeee dbascyks teimns
thicso nteixntw ,h icthh eayr ed escribbyei dm pliecqiuta tioTnhsek. e y
int hgee neralizoaftt hieosmnee thodtsof eedbascyks tehmassb eetnh e
introducotfie oxnt endsepda ceTsh.i sw ilble emphasiziendt he
subsequcehnatp teorfts h imso nograph.
Thimso nograipsah na ttemtpotd evelfoupr thaenrdr efinmee thods
baseodn input-oudtepsuctr iptfiooran nsa lyzfienegd bascyks tems.
Contratropy r eviwooursk i nt hiasr eat,ht er eatmheenatv ielmyp hasizes
ande xplotihtesc ausaloiftt yh eo perationrvso lvTehdi.sb rintghse
worki ntcol osceorn tawcitt thh et heoroyfd ynamicsayls teamnsd
automat(aI.fn a cittc, a nb ea rguetdh atth ev erdye finitioofsn tsa bility
ande xtendsepda caersei ll-concuenilveetsdhs eo perationrvso lvaerde
explicaistsluym etdob ec ausal.)
Them onograpihs b uilatr ounCdh apte4r, w heret her elevant
concepotfsw eII-posedsnteasbsi,lc iotnyt,i nuaintdys ,e nsitiavriet y
introduTcheedm .a thematfiocuanld atfioortn hsi sst udwyi lble f ound
inC hapte2r asn d3 .I nC hapt2e nro nlinoepaerr ataorreis n troduced
andg enercaoln ditifoonrts h ei nvertiobrit lhietn yo ninvertiobfi lity
nonlinoepaerr ataorreds e rivTehde.s ceo nditiroenlhsye aviolnyt he
PREFACE xiii
theoorfyB anacahl gebarnadse xplociatu salciotnys ideraitngi roenast
detaCihla.p t3ei rs f ort hem ostp ardte votteodt hee stablisohfma e nt
serioefsi nequaliatniu ensp,l eastaansttk h amta thematicuisaunasl ly
leavteo a pplimeadt hematicainadne sn gineeIrnse.q ualiatriete hse
workhorosfae nsa lysist-hanids monoigsrn aoep xhc epttioto hnre u le,
sincteh esien equalairteei sesse nitnigarle difeonrtt hses pecisfitca bility
andi nstabiclriittyed reisac riibneC dh apte5r as nd6 .T heu seo f
lineariztaetcihonni qiunse tsa bitlhietoyir sty h edni scusisnCe hda pt7e.r
Them onograipsih n tendperdi marfiolrrye searcihnse yrsst tehme ory.
Thea uthohro petsh aittw ilall sboe e njoybeydc ontreonlg inewehros
aree agetrofi nda u nifiemdo dertnr eatmoefnt th aen alyosfif se edback
systemasn,d b ym athematicwihaon asp precitahteae p plicaotfi on
relatiavdevlayn cmeadt hemattieccahln iqtuoee sn gineeqruiensgt ions.
Partosft himso nograapphp earientd h ea uthodro'cst ordailss erta­
tioenn tit"lNeodn linHeaarrm oniAcn aylsisa"n ds ubmittienJd u ne
196t8o t heD epartmeonfEt l ectrEincgailn eeroiftn hge M assachusetts
InstitoufTt eec hnoloCgaym,b ridgMea,s sachusetts.
Novemb1e9r6 9 JanW illems