MODELING AND STABILITY ANALYSIS OF POWER ELECTRONICS BASED SYSTEMS. A Thesis Submitted to the Faculty of Purdue University by Steven F.

Transcription

1 MODELING AND STABILITY ANALYSIS OF POWER ELETRONIS BASED SYSTEMS A Thss Submttd to th Faculty of Purdu Unvrsty by Stvn F. Glovr In Partal Fulfllmnt of th Rqurmnts for th Dgr of Doctor of Phlosophy May 3

2 - - To hryl Jula and Elzabth

3 - - AKNOWLEDGEMENTS I would lk to prss my apprcaton to my commtt mmbrs for thr gudanc and ncouragmnt. In partcular a spcal thanks to my major advsors Dr. Scott Sudhoff and Dr. Stanslaw Żak for thr advc rgor ncouragmnt and patnc and Ph.D. commtt mmbrs Dr. Olg Wasynczuk and Dr. Martn J. orlss for thr nsght and wsdom. I am dply ndbtd to Dr. Sudhoff for hs gudanc and support profssonally as wll as acadmcally. Dr. Żak provdd trmndous nsght and many opportunts to larn from hs prnc. Th opportunts to work wth and larn from Dr. Wasynzcuk hav gratly shapd my profssonal vws and provdd many prncs that wll b ongong assts n my profssonal carr. Th profssonalsm and knowldg of nonlnar systms provdd by Dr. orlss through hs nonlnar systms class add many aspcts of th progrsson through ths rsarch. Many popl hav hlpd to mak ths rsarch possbl. Dr. Paul Kraus affordd many opportunts profssonally and acadmcally through convrsatons and th opportunty to work for P..K.A. Bran Kuhn provdd prts n systm bhavor programmng and computr systms. Hs nsghtfulnss and humor hav bn contnuously apprcatd. Bnjamn Loop wth hs mathmatcal sklls and nsght was a plasur to work wth n th study of polytopc systms. Donysos Alprants offrd ncouragmnt wsdom and humor whnvr ndd. Ths and othr collagus srvd as a walth of ncouragmnt and das hav my ndurng grattud. Th opportunty to prform ths rsarch s gratly apprcatd. It was mad possbl through grants from th offc of Naval Rsarch ( Polytopc Modl Basd Stablty Analyss and Gntc Dsgn of Elctrc Warshp Powr Systms grant N Kathrn Drw program managr) and Naval Sa Systms ommand ( Naval ombat Survvablty grant N4--NR-647 Dana Dlsl program managr).

4 - v - TABLE OF ONTENTS Pag LIST OF TABLES... v LIST OF FIGURES... v ABSTRAT... INTRODUTION.... Mthods of Stablty Analyss for PEB Systms.... Summary...8 EXAMPLE SYSTEMS.... Nonlnar Powr Elctroncs Basd Systm..... Sourc componnt modl..... Load componnt modl Sth-Ordr NLAM appromaton...7. Systm....3 Systm....4 Systm Systm Systm Systm LOAL MODELING Taylor Srs Appromaton Tra-Żak Appromaton Altrnatv Appromatons Local modl wth prscrbd valu of th drvatv Kronckr product form Dagonal form...4

5 - v - Pag 3.4 onstructng Local Modls wth Pr-assgnd Equlbrum Pars Gnralzd Tra-Żak transformaton Matr nvrson basd transformaton as Study : Systm Lnar Wth Rspct to Input as Study : Systm Nonlnar Wth Rspct to Input as Study 3: GTZ Transformaton to Forc Prscrbd Equlbrum POLYTOPI MODELING Polytopc Modl Structur as study : Systm lnar n trms of th nputs as study : omparson btwn th numbr of local modls as study 3: Systm nonlnar n trms of th nputs Equlbrum Stat lassfcaton Polytopc modls wth non-concdnt qulbrum stats Polytopc modls wth concdnt qulbrum stats as study 4: Smulaton of a concdnt polytopc modl DEENTRALIZED POLYTOPI MODELING Intrconncton wth a Sngl Local Modl Matchd Local Modls Spcal cas: On nonlnar systm componnt as Study : Matchd Local Modls onstant omponnt Intrconnctons as Study: onstant omponnt Intrconnctons Gnral as as Study: Gnral as... 6 STABILITY ANALYSIS OF POLYTOPI MODELS oncdnt Polytopc Modl Global asymptotc stablty Asymptotc stablty Dcntralzd oncdnt Polytopc Modls...5

6 - v - Pag 7 STABILITY ANALYSIS OF TRUTH MODELS Lyapunov Functon anddat Sarchng for th Rgon of Asymptotc Stablty by Optmzaton Rctangular Rgon of Intal ondton Dtrmnng Intal Input ondtons as Study: S Ordr Systm ONLUSIONS AND REOMMENDATIONS onclusons Rcommndatons...6 LIST OF REFERENES...65 APPENDIX: HYPERBOXES AND HYPERELLIPSOIDS...69 A. Hyprbos Boundd by Hyprllpsods...7 A.. Vrt on th mnor as of th hyprllpsod and facs paralll wth th ~ -coordnat as...7 A.. Mamum volum...73 A..3 Facs paralll wth th ~ -coordnat as and mamum volum...74 A. Hyprbos Boundng Hyprllpsods...74 A.. Mnmum volum...75 A.. Mnmum volum and facs paralll to th coordnat as...76 A.3 Hyprllpsods Boundd by Hyprbos...77 VITA...79

7 - v - LIST OF TABLES Tabl Pag. Sourc componnt paramtr valus.... Sourc componnt control paramtrs Load componnt paramtr valus Sth-ordr systm componnt valus....5 Systm paramtr valus....6 Systm currnt lmts....7 Systm 5 paramtrs valus Statstcal rsults from th grd sarch analyss...56

8 - v - LIST OF FIGURES Fgur Pag. onstant powr load.... Systm dagram...5. Subsystm usd for analyss.... Sourc componnt schmatc....3 ontrol fltrs Ovr currnt protcton Slw rat lmtr contour Sourc componnt proportonal ntgral fdback Sourc componnt drvatv fdback Sourc componnt phas angl convrson Sourc componnt nonlnar avrag valu modl...6. Load componnt schmatc...7. Load componnt nonlnar avrag valu modl...7. Svnth-ordr systm schmatc Equvalnt capactor frquncy rspons comparson Sth-ordr systm schmatc....5 Systm crcut dagram....6 Systm 3 crcut dagram Systm 5 crcut dagram Systm 6 crcut dagram Local and truth modl comparson u Local and truth modl comparson u Local and truth modl trajctors corrspondng to 4 / 3 and u...5 π

9 - - Fgur Pag 3.4 Local and truth modl comparson u Local and truth modl comparson u Local and truth modl trajctors corrspondng to 4 / 3 and u π 3.7 Plot of th stat drvatv vrsus stat for th truth and Taylor srs basd local modls Plot of th stat drvatv vrsus stat for th shftd truth and Taylor srs basd local modls Plot of th stat drvatv vrsus stat for th truth and transformd local modls Plot of th stat drvatv vrsus stat for th truth and gnralzd Tra- Żak basd local modls Wghtng functon usd by Takag and Sugno Wghtng functon usd by ao Rs and Fng Local modl wghtng functons for th Taylor srs and th Tra-Żak basd polytopc modls Smulaton comparson for th truth and polytopc modls Smulaton rror comparson for th polytopc modls Taylor srs basd modl trajctory plot wth th local modl qulbrum ponts Tra-Żak basd polytopc modl trajctory plot wth th local modl qulbrum ponts Gnralzd Tra-Żak basd polytopc modl trajctory plot wth th local modl qulbrum ponts Smulaton comparson for th truth and polytopc modls Smulaton rror comparson for th polytopc modls Smulaton comparson for th truth and polytopc modls Smulaton rror comparson for th polytopc modls Taylor srs basd systm trajctory plot wth th local modl qulbrum ponts Gnralzd Tra-Żak basd polytopc modl trajctory plot wth th local modl qulbrum ponts Smulaton comparson for th truth and polytopc modls Smulaton rror comparson for th polytopc modls...8

10 - - Fgur Pag 5. omponnt systm ntrconnctons Systm currnt trajctors Systm currnt trajctors Intal rgon Γ surroundng Epanson of th rgon Γ Rgon Γ dfnng th α st of local modls Lvl st contours of V plottd on top of V & Lvl st contours of V plottd on top of V & Lvl st contours of V 69 plottd on top of V & Lvl st contours of V T plottd on top of V & Lvl st contours of V GTZ plottd on top of V & GTZ Lvl st contours of V MI plottd on top of V & Lvl st contours of V Tα plottd on top of V & Tα Lvl st contours of V GTZα plottd on top of V & GTZα Lvl st contours of V MIα plottd on top of V & MIα Boundary lmts for th -th dmnson Appromaton of th rgon Γ along th -th dmnson Unformly spacd ponts n Γ along th -th dmnson Th α st of local modls along th -th dmnson Lvl st contours of V plottd on top of V & Boundars to th valu of V for th stmatd RAS Rgon of asymptotc stablty for systm Rctangular rgon of ntal condton Input spac projcton Input spac projcton for systm Input stp tst for systm Input spac projcton for systm T MI

11 - - Fgur Pag 7. Systm 4 stat trajctors Systm 4 stat trajctors Valu of th Lyapunov functon valuatd along th stat trajctory Prcntag of smulatons rachng qulbrum Systm 3 sourc and load componnts Sourc output mpdanc Load nput admttanc Systm mmttanc basd stablty analyss...59 A. Ellps n th ~ -coordnats...7 A. Ellps n th y -coordnats...7 A.3 Bo wth th vrt on th mnor as...7 A.4 Bo wth mamum volum and boundd by th llps...73 A.5 Bo orthogonal to th coordnat as wth th largst volum...74 A.6 Bo wth th mnmum volum n th y -coordnats...75 A.7 Bo wth th mnmum volum n th ~ -coordnats...76 A.8 Bo wth facs paralll to th ~ coordnat as and havng mnmum volum...77

12 - - ABSTRAT Glovr Stvn F. Ph.D. Purdu Unvrsty May 3. Modlng and Stablty Analyss of Powr Elctroncs Basd Systms. Major Profssors: Scott D. Sudhoff and Stanslaw H. Zak. Systms consstng of powr lctroncs basd convrtrs for powr dstrbuton hav bcom ncrasngly promnnt n th mltary as wll as n ndustry. Stablty analyss of all but th smplst of ths systms s oftn basd on modl lnarzaton (ntrprtd thr n stat-spac or frquncy doman) and tm doman smulaton. Th nformaton obtand wth ths tools s lmtd to small-sgnal bhavor or to spcfc trajctors. Th followng rsarch tnds prsnt thors and analyss tools nablng larg-sgnal stablty analyss of powr-lctroncs-basd systms. As an altrnatv to lnar systm analyss nonlnar analyss may b conductd usng Lyapunov tchnqus. Th dffculty n such an approach s n calculatng a sutabl Lyapunov functon canddat. Th structur of polytopc modls whch ar basd on conv combnatons of lnar or affn local modls provds a framwork that can b usd to stablsh a Lyapunov functon canddat. ondtons for th polytopc modl to b globally unformly asymptotcally stabl ar dntfd. Thn th Lyapunov functon canddat constructd usng polytopc modls s mployd to stmat rgons of attracton of th truth modl. Ths mthods ar llustratd usng scond- and sth-ordr nonlnar systm ampls. Fnally th tnson of tchnqus proposd hrn to dcntralzd analyss s addrssd. onvrson of a dcntralzd polytopc modl to a mor tractabl form s prsntd allowng tchnqus dvlopd for th basc polytopc modl structur to b utlzd on ntrconnctd polytopc modls.

13 - -. INTRODUTION Powr-lctroncs-basd (PEB) systms ar promnnt n both mltary and ndustral applcatons. Dvcs such as IGBT s (Insulatd Gat Bpolar Transstors) BJT s (Bpolar Juncton Transstors) and thyrstors hav contnud to mprov n swtchng spds ffcncy and strss factors thrby ncrasng th numbr of fasbl applcatons. Eampls of powr lctroncs applcatons nclud motor drvs lghtng and powr suppls. Ths thss concrns th applcatons of powr lctroncs n submarns shps lctrc and hybrd lctrc vhcls and arplans whrn powr lctroncs ar usd for powr dstrbuton and propulson. As a rsult of hgh swtchng spds and hgh-spd procssors for control mplmntaton powr convrtrs can b mad to hbt narly dal rgulatory capablty. Howvr powr convrtrs dsgnd wth such hgh bandwdth rgulaton bhav as constant-powr loads whn vwd from thr nput trmnals. Thus ths convrtrs hbt a ngatv mpdanc charactrstc whch can b dtrmntal from a systm prspctv [SUD98A] [MID76]. To undrstand how ths arss consdr th constant powr load n Fgur.. Th currnt nto a constant powr load s dfnd as whr th commandd powr s dnotd as P ( P v) v cp P and th voltag across th constant powr load s dnotd by v. Takng th partal drvatv of th currnt cp ( P v) (.) wth rspct to th voltag and nvrtng th nput mpdanc to th constant powr load s dtrmnd to b

14 - - cp v ( P v) P P v P v v (.) whch s ngatv as long as th commandd powr P s postv. Hnc th constant powr load looks lk a ngatv rsstanc to th systm clarly a dstablzng ffct. Fgur.. onstant powr load. Du to th constant powr loads n ths systms stablty analyss s crtcal and oftn prformd usng tm-doman smulaton or frquncy-doman mmttanc analyss. Ths mthods provd nsght nto th bhavor of th systm but do not provd rgorous proof of th sz or shap of a rgon of stablty. As an altrnatv th drct mthod of Lyapunov charactrzs th rgon of stablty thrby allowng stmaton of th sz and shap of th rgon. Howvr th complty of ths systms oftn prohbts th analytcal dtrmnaton of Lyapunov functon canddats. Ths thss advancs th ara of stablty analyss of PEB systms by takng advantag of th drct mthod of Lyapunov and utlzs polytopc modlng as a mans of dtrmnng Lyapunov functon canddats.. Mthods of Stablty Analyss for PEB Systms In ths scton stablty analyss of systms havng th form & F( u) s rvwd. For ths systms th stat tm drvatv of th stat and th nput ar dnotd by & and u rspctvly. Hrn symbols rprsntng vctors or matrcs ar placd n bold typ. An opratng pont of ths systm s consdrd stabl f t satsfs th followng dfnton ([ZAK3] [ANT97] [KHA96] and [BRO9]).

15 - 3 - Dfnton.: onsdr th systm ~ & F( ~ u~ ) whr th stat ~ and nput u ~ ar obtand from th coordnat transformatons qulbrum stat corrspondng to th constant nput ~ s: ~ and u u u ~ usng th u. Th qulbrum pont. Stabl f for ε > thr sts a δ ( ε t o ) > such that f ( t ) < δ thn ~ ( t ) < ε for all t t. ~. Asymptotcally stabl (AS) f t s stabl and thr sts η such that ~ ( t ) as t for all ( t ) < η. ~ 3. Globally asymptotcally stabl (GAS) f t s asymptotcally stabl for any choc of η. For lnar tm-nvarant systm modls stablty s radly dtrmnd by chckng th gnvalus of th modl. Howvr for nonlnar systms stablty analyss s consdrably mor nvolvd. Prhaps th smplst and oftn th frst tst of nonlnar systm bhavor s tm doman smulaton. An ntal condton s chosn for th modl and thn th smulaton s cutd n ordr to dtrmn th stat trajctors as tm volvs. In trms of stablty th only concluson that can b mad from smulaton s that th truth modl valuatd along a partcular trajctory s thr wll bhavd or not. That s f th modl s wll bhavd whn startng from a partcular ntal condton thr s not a guarant that th sam would b tru f a dffrnt ntal condton whr chosn. To vrfy stablty of an qulbrum stat Lyapunov suggstd th us of lnarzaton ([BRO9] [KHA96] and [ŻAK3]). Thorm. [KHA96]: (Indrct mthod of Lyapunov) Lt b an qulbrum pont for th nonlnar systm & F() whr n F : D R s contnuously F dffrntabl and D s a nghborhood of th orgn. Lt A ( ). Thn

16 th orgn s asymptotcally stabl f R λ < for all gnvalus of A. th orgn s unstabl f R λ > for on or mor gnvalus of A. If an qulbrum stat s dtrmnd to b AS by Thorm. thn thr sts a rgon of asymptotc stablty (RAS) n th nghborhood of th qulbrum. Howvr an ndcaton to th sz and shap of th RAS s not provdd. Analyss basd on Thorm. s a common mans of dtrmnng stablty of a partcular qulbrum stat for ampl th radr s rfrrd to [HAD9] and [AS93]. In som systms act lnarzaton va control allows for stablty to b dtrmnd n a global sns usng Thorm. [LIE93]. Loop gan analyss has bn mployd as wll n analyzng powr convrtrs and th rsultng gnvalus usd for stablty analyss [KI98]. Othr tchnqus ar also avalabl for chckng th stablty of partcular qulbrum ponts but ar basd on Nyqust thory [DOR9]. In PEB systms ths analyss s mplmntd n trms of mmttancs. To furthr undrstand how mmttancs may b usd to analyz th stablty of a PEB systm consdr th block dagram dpctd n Fgur. consstng of two componnts rprsntd by Thvnn quvalnts. Th small-sgnal output mpdanc of th sourc and th nput mpdanc of th load ar dfnd as and Z Z s L v s v rspctvly. Ths mpdancs ar ratonal functons of th form and L N Z s D s s N Z L L. D L (.3) (.4) (.5) (.6)

17 - 5 - Fgur.. Systm dagram. To nvstgat th stablty of th systm n Fgur. th quaton for th bus voltag v may b wrttn as ZL Z v v s s + vl. Z + Z Z + Z s L Substtutng quatons (.5) and (.6) nto (.7) th quaton for th bus voltag may b rformulatd as whch s rprsntd as N LDsvs + N sd v v N D + N D v L s s s L L L L N LDsvs + N sdlv N LDs ( + ZsYL ) whr Y L s th load admttanc dfnd as / Z L. From quaton (.9) t can b sn that f th sourc output mpdanc transfr functon not dos not hav any pols n th closd rght-half sd of th compl plan and load nput mpdanc transfr functon dos not hav any zros n th closd rght-half sd of th compl plan thn th stablty of th systm may b dtrmnd by th locaton of th zros n th trm + Z s Y L. By th Nyqust thory th transfr functon of th bus voltag wll not hav any pols n th closd rght hand sd of th compl plan.. th systm s stabl f th product of th sourc mpdanc and th load admttanc Z syl valuatd along th Nyqust contour dos not ncrcl -. Mddlbrook was on of th frst to addrss stablty analyss of a PEB systm from an mmttanc prspctv rsultng n th Mddlbrook crtra [MID76]. Mor rcntly mpdanc analyss has pandd nto th gnralzd mmttanc analyss L (.7) (.8) (.9)

18 - 6 - ([SUD] [SUD98B] and [SUD]) and volvd nto a mthod for analyzng th stablty of larg-scal systms. Through th us of gnralzd mmttancs sts of qulbrum stats may b analyzd togthr. Howvr th rsults of ths analyss ar an ndcaton of th stablty of ach qulbrum pont n th st. Ths dos not charactrz th rgon of stablty about ach qulbrum pont. To stmat th rgon of stablty Lyapunov s drct mthod s avalabl ([BRO9] [KHA96] or [ŻAK3]) and statd n th followng thorm whr th notaton D {} dnots th rgon D wth th orgn rmovd. Thorm. [KHA96]: (Drct mthod of Lyapunov) Lt b an qulbrum pont for & F() and n D R b a doman contanng. Lt V : D R b a contnuously dffrntabl functon such that Thn s stabl. Morovr f thn s asymptotcally stabl. V ( ) and V ( ) > n D {} V & ( ) n D V & ( ) < n D {} As can b sn ths thorm stats condtons for an qulbrum stat to b stabl and n dong so dtrmns charactrstcs dfnng a rgon of stablty surroundng th qulbrum. Takng advantag of ths charactrzaton allows for th dtrmnaton of an stmat of th rgon of stablty. Th us of th drct mthod of Lyapunov n PEB systms dd not bgn untl th lat 97 s wth th work of Edwards [EDW78]. Ths work along wth th mor rcnt fforts of [HE9] [KAW95] [BER] and [BER] all focus on th stablty analyss of ndvdual swtchng rgulators drvng analytcal forms of Lyapunov functon canddats. Th dffculty wth ths approach s that for hgh-ordr systms th analytcal dtrmnaton of a Lyapunov functon canddat s mpractcal.

19 - 7 - To crcumvnt th dffcults assocatd wth dtrmnng a Lyapunov functon canddat a numrcal approach s avalabl basd on lnar matr nqualts (LMI) [BOY94]. To dmonstrat how th LMI s obtand consdr th unforcd systm dfnd by & A and th quadratc Lyapunov functon canddat dfnd as V ( ) P. Th tm drvatv of V () valuatd ovr th trajctors of th systm (.) s thn V & ( ) [ A P + PA]. (.) If th matr P P > can b found satsfyng th LMI A P + PA < thn V & () s ngatv dfnt and th systm dfnd by (.) s globally unformly asymptotcally stabl accordng to Thorm.. Algorthms to solv th LMI (.3) ar radly avalabl for ampl n MATLAB [MAT]. For nonlnar systms an approach to dtrmnng ths LMI s s through th us of th Takag-Sugno (TS) modl [TAK85]. On form of ths modl s r ( θ) A & w whr r s th numbr of local modls θ rprsnts a stat nput output or paramtr and th wghtng functon ( θ) w s constrand to (.) (.) (.3) (.4) and ( θ) w r w ( ) θ. (.5) (.6) Th appal of ths partcular form ls n th fact that th TS modl has bn provn to unvrsally appromat smooth nonlnar systms to th frst drvatv to any dsrd accuracy [WAN] and that such a systm can radly b tstd for GAS by sarchng for a matr P P > that satsfs

20 - 8 - A P + PA < for K r whch was provn for dscrt systms by Tanaka and Sugno n 99 [TAN9]. Although quaton (.4) ntally appars qut gnral and th advantags of ths form from a stablty analyss prspctv s clar t should b notd that lnarzng a nonlnar modl to obtan a local modl oftn rsults n an affn rathr than lnar modl n th orgnal coordnats. Ths rsults n th TS modl of th form whr r w φ ( θ)[ A + B u ] & + φ s a constant affn trm assocatd wth th -th local modl stat quaton. A modl of th form gvn by quaton (.8) s rfrrd to hrn as a polytopc modl.. Summary (.7) (.8) Th abov mthods of analyss hav all bn appld to th study of PEB systm stablty. Ths thss tnds th capablts of ths mthods by provdng a numrcal approach to dtrmnng Lyapunov functon canddats. Ths thss bgns by frst ntroducng a st of ampl systms whch ar basd upon th nonlnar avrag valu modl of a powr supply and thr phas nvrtr. Som of th ampl systms dvat from th drvd modl to ad n th ntroducton of varous concpts. Local modlng tchnqus ar nt ntroducd startng wth Taylor srs appromaton. Othr mthods of constructng local modls ar st forth along wth mthods of forcng partcular charactrstcs. Th local modls ar nt usd to construct polytopc modls. Th polytopc modls ar dmonstratd to b nonlnar and capabl of appromatng nonlnarts nhrnt to som truth modls. haractrstcs of polytopc modls whch ad n stablty analyss ar nvstgatd as wll. Th tnson of polytopc modls to a dcntralzd modlng procss s addrssd and a tchnqu for smplfyng a dcntralzd polytopc modl to th standard form s prsntd.

21 - 9 - Stablty analyss of polytopc modls s thn consdrd wth condtons for dtrmnng global asymptotc stablty dntfd. Through th us of polytopc modls a procdur for skng Lyapunov functon canddats s dntfd. Th Lyapunov functon canddats ar thn mployd to analyz stablty of systm truth modls. A mthod for stmatng a rgon of stablty s ntroducd and a fnal cas study on a sth-ordr systm s prsntd.

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23 - -. EXAMPLE SYSTEMS Th ampl systms usd n ths thss ar rprsntatv of componnts from th Naval ombat Survvablty (NS) tstbds [PEK] locatd at Purdu Unvrsty and th Unvrsty of Mssour Rolla. In ordr to gan nsght nto th tchnqus and analyss wthn ths thss a subsystm was cratd comprsng of componnts from th NS systms and appromatd by nonlnar avrag valu modls (NLAM) [GLO97].. Nonlnar Powr Elctroncs Basd Systm Th rducd systm consstng of a thr phas sourc D powr supply (PS) shp srvc nvrtr modul (SSIM) and a load bank (LB) s shown n Fgur.. For mor nformaton on ths componnts s [PEK]. In ths systm th thr phas sourc s assumd to b dal and s ncludd as part of th sourc componnt modld n Scton... In addton th shp srvc nvrtr modul (SSIM) and th load bank (LB) wll b combnd nto a sngl load componnt as dscussd n Scton... Fgur.. Subsystm usd for analyss... Sourc componnt modl A dagram of th systm sourc s dpctd n Fgur. [SUD] whrn t s shown that th sourc componnt conssts of an dal thr phas sourc a stp-down transformr a controlld rctfr and an output fltr. Th paramtr valus for ths componnt ar lstd n Tabl..

24 - - Fgur.. Sourc componnt schmatc. Tabl.. Sourc componnt paramtr valus. paramtr valu dscrpton V A 56 V Ln to ln supply voltag (rms) ω p6 rad/sc Supply frquncy N sp.79 Transformr turns rato scondary ovr th prmary (wy-quvalnt) L M 3. H Transformr magntzng nductanc rfrrd to th prmary. R lk. Ω Transformr total lakag rsstanc rfrrd to th scondary. L lk 66 µh Transformr total lakag nductanc rfrrd to th scondary. V D.98 V Dod voltag drop R D. mω Dod rsstanc R Ldc.8 Ω D lnk nductor rsstanc. L dc mh D lnk nductanc. R dc.4 Ω D lnk capactor ESR at 36 Hz. dc 46 µf D lnk capactanc at 36 Hz.

25 - 3 - Th control of th powr supply conssts of th followng parts. Th frst part of th control dpctd n Fgur.3 fltrs th masurd nductor currnt L and masurd output voltag v rsultng n th fltrd quantts dnotd as î L and ˆv. Fgur.3. ontrol fltrs. Th nt part of th control llustratd n Fgur.4 provds ovr currnt protcton. If th masurd output currnt cds th thrshold valu of th currnt lmtr I thr thn th commandd output voltag V s modulatd by a rducton factor. If th output currnt cds th mamum short crcut currnt I sc thn th modulaton factor gos to zro. To prvnt cssvly rapd changs n th modulatd commandd voltag t s slw rat lmtd. Th slw rat lmtr (SRL) s a frst-ordr low pass fltr wth tm constant τ srl and a boundd drvatv. Th drvatv s boundd usng th functon whr pv mn and v& tan π pv mn f v pv π mn f ( v& ) v& ma tan π pv f v > pv π ma pv ma ar th lowr and uppr bounds on th rat of chang to In Fgur.5 f ( v& ) s plottd vrsus v&. Not that th contour s scald to MV/s n th lowr lft quadrant and to kv/s n th uppr rght quadrant. (.) v.

26 - 4 - Fgur.4. Ovr currnt protcton. Fgur.5. Slw rat lmtr contour. Fgur.6 llustrats th proportonal-ntgral voltag fdback and proportonal currnt fdback. Th ntgrator s mplmntd wth an ant wndup fatur basd on th frng angl α of th thyrstors. If α s at th mamum and th nput to th ntgrator s postv or f α s at th mnmum and th nput to th ntgrator s ngatv thn th ntgrator nput s st to zro. As can b sn ths part of th control also ncluds a proportonal currnt fdback trm. Drvatv currnt fdback s prformd basd on th dagram gvn n Fgur.7. Ths stratgy can b shown to produc a sgnal appromatly proportonal to th drvatv of th output currnt. Th rsults from ths subcontrols ar combnd as llustratd n Fgur.8 to dtrmn th thyrstor frng angls. Th sourc componnt control paramtrs ar lstd n Tabl..

27 - 5 - Fgur.6. Sourc componnt proportonal ntgral fdback. Fgur.7. Sourc componnt drvatv fdback. Fgur.8. Sourc componnt phas angl convrson. Tabl.. Sourc componnt control paramtrs. τ v.65-3 s pv -.6 V/s mn R ff 56 mω τ L.65-3 s τ srl.-4 s L ff.3-3 H V 5. V K v. V/V α ma. I sc 55. A τ v. s α mn -.9 I thr 35. A K p. V/A V R 597. V pv 5. V/s K ma L.5 H

28 - 6 - Th sourc componnt s nt rducd to th nonlnar avrag valu modl (NLAM) quvalnt shown n Fgur.9 usng mthods st forth n [KRA]. For ths modl t s assumd that th transformr magntzng nductanc s larg nough that t can b nglctd. Th paramtrs n th NLAM modl ar dfnd as Vs ( α ) Vs cos( α) V D and Th constant V s s dfnd by 3L lk ϖ R RLdc + RD + Rlk + π L L dc + L lk. (.) (.3) (.4) 3 V A N sp Vs π and th frng angl α s obtand from th control dscrbd abov. (.5) Fgur.9. Sourc componnt nonlnar avrag valu modl... Load componnt modl Th load componnt s dpctd n mor dtal n Fgur. whrn th SSIM conssts of a D lnk capactor wth an quvalnt srs rsstanc thr phas nvrtr L fltr and th load bank. Hrn t s assumd that th SSIM control s dsgnd to tghtly rgulat th fltrd output and that th nvrtr bhavs lk a constant powr load from th D lnk sd. Vrfcaton of ths assumpton can b found n [SUD]. In addton to provdng a hghly rgulatd output th nvrtr control montors th D bus voltag and turns th nvrtr off f th bus voltag rachs valus outsd th rang of Vmn v Vma. (.6)

29 - 7 - Basd upon ths obsrvatons th systm load componnt s appromatd by th NLAM dpctd n Fgur. and havng th componnt paramtrs lstd n Tabl.3. In Fgur. th constant powr load currnt s dfnd as whr P V v V cp ( u) mn ma v othrws P s commandd powr. Tradtonal control structur would allow for th load componnt to turn back on whn th voltag v rturns to an approprat valu howvr for th purpos of dmonstraton th systm hrn s rqurd to mantan th D lnk voltag btwn th two lmts. (.7) Fgur.. Load componnt schmatc. Fgur.. Load componnt nonlnar avrag valu modl...3 Sth-Ordr NLAM Appromaton Whn th sourc and load componnts dpctd n Fgur.9 and Fgur. ar combnd a svnth-ordr modl shown n Fgur. s obtand (four stats ar n th sourc control). Ths svnth-ordr modl can b rducd to a sth-ordr modl by

30 - 8 - combnng th capactors dc and n and rsstors R dc and R n to form an quvalnt capactanc and rsstanc R. Tabl.3. Load componnt paramtr valus. paramtr valu dscrpton V mn 45 V Mnmum D lnk voltag for opraton. V ma 55 V Mamum D lnk voltag for opraton. R n 7 mω D lnk capactor srs rsstanc. n 59 µf D lnk capactanc. Fgur.. Svnth-ordr systm schmatc. In ordr to rduc th systm ordr th mpdanc of th parallld R componnts s prssd by s Rdc Z R s Rndcn + s( Rdcdc + Rnn ) ( R + R ) + s( + ) dc n At low frquncs mpdanc may b appromatd as Z R Lf s dc n ( + ) dc n dc n +. ndcatng that th capactors domnat th mpdanc. At hgh frquncs th scondordr trms domnat th mpdanc charactrstcs allowng Z R to b appromatd as Z R Hf Rdc Rn Rdc + Rn (.8) (.9) (.)

31 - 9 - ndcatng that th quvalnt srs rsstancs domnat th hgh frquncy rspons. Rqurng th mpdanc of th rducd R crcut to b dntcal to th paralll R crcuts at both low and hgh frquncs th mpdanc may b appromatd as Z R R + (.) s whr Rdc Rn R (.) Rdc + Rn and dc + n. (.3) Fgur.3 llustrats th mpdanc of th parallld R crcuts dfnd n (.8) along wth th mpdanc of th quvalnt R crcut dfnd n (.). Th rsultng sthordr appromaton of th subsystm n Fgur. s shown n Fgur.4 and has th componnt paramtr valus lstd n Tabl.4. As can b sn th frquncy rspons of th rducd systm s an cllnt match to th full systm. Fgur.3. Equvalnt capactor frquncy rspons comparson. Basd upon th sth-ordr modl of th systm dpctd n Fgur.4 th systms usd by th ampls throughout ths thss ar dscrbd blow. Som of ths

32 - - systms dvat from th abov modl by changs n paramtrs whl othrs dvat by th addton or rmoval of componnts. Fgur.4. Sth-ordr systm schmatc.. Systm Tabl.4. Sth-ordr systm componnt valus. paramtr R L R valu 53 mω.3 mη 83 mω.5 mf Ths scond-ordr nonlnar systm dpctd n Fgur.5 conssts of an dal sourc connctd to a constant powr load through an RL fltr. Th paramtr valus for ths systm ar lstd n Tabl.5. Th truth modl usd for ths systm s and whr Vs L R v & F( u) L L cp( u) L y h( u) cp( u) (.4) (.5)

33 - - P cp ( u). v Th nputs and stats for ths systm ar chosn as u [ V s P ] (.6) (.7) and [ ] L v. (.8) Tabl.5. Systm paramtr valus. Paramtr Valu L mh R.3 Ω 4 mf Fgur.5. Systm crcut dagram..3 Systm Systm s dntcal to systm cpt that th constant-powr load s currnt lmtd as dfnd by whr th functon bound( ) s dfnd as P cp ( u) bound cp mn cpma v a c a bound( a b c) b c b and a < b c othrws (.9) (.)

34 - - and th currnt lmts ar gvn n Tabl.6. Ths lmt on th nput currnt to th constant powr load s not typcal but t dos ntroduc addtonal nonlnarts to th systm for llustraton purposs n th chaptr on polytopc modls. Tabl.6. Systm currnt lmts. Paramtr Systm cpmn A cpma A.4 Systm 3 Systm 3 dpctd n Fgur.6 ncluds quvalnt srs rsstanc for th capactor as wll as a dod for prvntng ngatv currnt flow nto th D sourc. Ths systm s scond-ordr allowng vsualzaton and mprovd undrstandng of many of th stablty concpts ntroducd. If th nductor currnt L s nonzro or L and Vs LR v > thn th truth modl stat quaton s wrttn as othrws t s wrttn as Vs LR v & F( u) L L cp( u) & F( u) cp ( u ) whr th constant-powr load currnt s dfnd as P cp ( u). v Th nputs stats and outputs for ths systm ar chosn as [ L v ] (.) (.) (.3) (.4) u [ V s P ] (.5) and

35 - 3 - cp ( u) y h( u). v Th paramtrs for ths systm ar dfnd n Tabl.4. (.6) Fgur.6. Systm 3 crcut dagram. Physcally ths systm rprsnts th full systm lss th control..5 Systm 4 Systm 4 s dfnd as th sth-ordr systm dpctd n Fgur.4 and havng th paramtrs lstd n Tabl. and Tabl.4. Th opratonal lmt of th constant powr load s dfnd by nqualty (.6) wth th lmts Tabl.3. Th nput for ths systm s u [ V s P ] whr V s s th sourc voltag dfnd by quaton (.5) and powr. Th stats and output for ths systm ar chosn to b and [ L ˆ L ˆ v v v vrror ] y cp ( u). V mn and V ma dfnd n (.7) P s th commandd (.8) (.9).6 Systm 5 Systm 5 dpctd n Fgur.7 rprsnts a two componnt systm. Th truth modl for ths systm s

36 - 4 - wth nputs & Vs LR v L L L v LR v L L cp( u) y [ L v ] u [ V s P ] (.3) (.3) (.3) stats and a constant powr load currnt dfnd as [ v v ] L L P cp ( u). v Th paramtr valus for ths systm ar lstd n Tabl.7 (.33) (.34) Fgur.7. Systm 5 crcut dagram..7 Systm 6 Ths systm s dntcal to systm 5 wth th cpton of th dal dod addd to componnt. Thrfor th nputs stats and outputs for ths systm ar th sam as

37 - 5 - dfnd for systm 5. Addtonally th truth modl gvn for systm 5 s vald for ths systm whn L > or L and V s > v othrws th truth modl s wrttn as L & v LR v. L ( ) L cp u (.35) Tabl.7. Systm 5 paramtr valus. Paramtr R L R L Systm.56 Ω.3mΗ.5mF.56 Ω 3.µΗ.5mF Fgur.8. Systm 6 crcut dagram. Ths complts th ntroducton of th sampl systms usd throughout th thss. Addtonal dscrpton of ths systms s only gvn as ncssary n th followng

38 - 6 - chaptrs. Othrws th radr wll b rfrrd back to ths chaptr. Now that th nonlnar ampl systms hav bn ntroducd th followng chaptr dscusss tchnqus for constructng local modls of ths systms.

39 LOAL MODELING A broad class of nonlnar systms can b modld usng th quatons & F( u) y h( u) (3.) (3.) whr n R s th stat vctor m u R s th nput vctor and p y R s th output vctor. Th abov modl s rfrrd to as th truth modl of th nonlnar systm. In gnral ffctv stablty analyss of modls of th typ (.-.) can b arduous f not mpossbl. Thrfor rsarchrs oftn utlz tools ffctv for appromatng th systm dynamcal bhavor. On tool frquntly usd to appromat th bhavor of nonlnar systms s lnar appromaton. That s a nonlnar systm s lnarzd about a partcular modlng pont ( u ) whr and u ar a partcular stat and nput combnaton. Lnarzaton of a nonlnar systm at an qulbrum par lads to a lnarzd modl of th systm havng th form n th coordnat systm dfnd by ~ & A ~ + Bu~ ~ y ~ + Du~ ~ and u u u ~ whr ) s an ( u qulbrum par. Hrn th qulbrum par s assumd to b a pont. For A systms ths s not always th cas. Howvr undr som condtons usng a qd-transformaton [KRA] ths can stll b obtand. In th orgnal coordnats th rsultng modl s n gnral affn and dfnd as & A + Bu + φ (3.3) (3.4) (3.5) y + Du + φ y (3.6) whr φ and φ y ar constant affn trms. Ths modls appromat th truth modl

40 - 8 - for valus of stat and nput that ar clos to th qulbrum pont ) thrfor ( u ladng to th frquntly usd trm local modl. How clos th stat and nput nd to b to th modlng pont dpnds on th nonlnarts n th truth modl and th mthod of prformng th lnarzaton. In ths chaptr thr mthods of formng local modls from th truth modl ar prsntd. On of ths mthods allows chocs n th constrants placd upon th local modl whch allows smplfcatons to th stablty analyss of som systms. On charactrstc of partcular bnft s obtand by assgnmnt of th qulbrum stats. Thrfor two procdurs for forcng stng local modls to hav spcfc qulbrum stats ar also prsntd. Fnally to llustrat th dffrncs btwn th typs of local modls thr cas studs ar dscrbd. 3. Taylor Srs Appromaton On of th most popular mans of formng th local modl of a nonlnar systm s th us of Taylor srs appromaton (s for ampl [BRO9] or [BER84]). Taylor srs panson of th rght-hand sd of quatons (3.) and (3.) about ) ylds and rspctvly whr F( u) F( ) F( ) ~ F( ) ~ u + D u + Du u u + u u u u h( u) h( ) h( ) ~ h( ) ~ u + D u + Du u u + u u u u ( u HOT (3.7) HOT (3.8) D F( u) s th Jacoban matr of F( u) takn wth rspct u u to at th modlng pont ) HOT dnots hghr ordr trms ~ s dfnd by ( u ~ (3.9) and u ~ s dfnd by u~ u u. (3.) Lt th suprscrpt dnot th transpos oprator. Thn th Jacoban matr D F( u) may b wrttn as u u

41 - 9 - F ( u) u u D F( u) M u u F ( ) n u u u (3.) whr F ( u) s th gradnt of th th componnt of F( u) wth rspct to u u at th modlng pont ). Not that th gradnt s a column vctor. Th matrcs D F( u) u u u ( u D h( u) u u and D h( u) ar dfnd n a u u u smlar fashon. Taylor srs appromaton assums that th rght-hand sd of th truth modl quatons s contnuously dffrntabl. In ordr to valuat systms wth dscontnuts th drvatv at th dscontnuty s dfnd to b th avrag of th drvatvs takn on ach sd of th dscontnuty. Th Taylor srs panson of (3.7) and (3.8) rsult n a modl that s lnar n trms of ~ and u ~ f th modlng pont ( u ) s an qulbrum par and th hghr ordr trms ar nglctd. Th lnar modl may b wrttn n stat spac form as ~ & A ( u ) ~ B ( u ) u~ T + (3.) T or mor smply as ~ y ( u ) ~ D ( u ) u~ T + T ~ & A ~ B u~ T + T ~ y ~ D u~ T + T whr ) s droppd for brvty and th subscrpt T dnots th Taylor srs ( u appromaton. Th matrcs n (3.4) and (3.5) ar dfnd as A D F( u) T u u (3.3) (3.4) (3.5) (3.6) and BT Du F( u) u u D h( u) T u u (3.7) (3.8)

42 - 3 - D D h( u) T u. u u In gnral th abov modl s affn n trms of and u unlss th modlng pont at whch th local modl was constructd s an qulbrum pont of th truth modl and locatd at th orgn or crtan qualts ar satsfd. Rformulatng th modl n trms of and u rsults n & AT + BT u + φt y T + DT u + φyt (3.9) (3.) (3.) whr th affn trms ar dtrmnd to b and φ T φ yt F( u A B u ) h( u ) T D T u. T T (3.) (3.3) A modl dfnd by (3.) and (3.) s lnar n trms of and u f (3.) and (3.3) valuat to zro. Othrws f th Taylor srs basd local modl s affn thn othr tools ar avalabl for constructng local modls that ar lnar n trms of and u. 3. Tra-Żak Appromaton Th Tra-Żak appromaton [TEI99] may b usd to construct a lnar modl for nonlnar systms havng truth modls of th form & F( u) f( ) + G( ) u (3.4) y h( u). (3.5) Not that ths class of systms s lnar n trms of th nput u. Th rsultng local modl has th form & ATZ + BTZ u (3.6) y TZ + DTZ u + φytz (3.7) whr th subscrpt TZ dnots th Tra-Żak appromaton. Two constrants ar placd upon ths local modl. Thy ar:. Th rght-hand sd of th local modl must qual th rght-hand sd of th

43 - 3 - truth modl at th modlng pont for any u that s A + B TZ u f( ) + G( ) u TZ. Th local modl must appromat th truth modl n th vcnty of th opratng stat that s (3.8) A TZ + B TZ u f( ) + G( ) u. (3.9) Th frst constrant s satsfd f and Ths mmdatly dtrmns th matr A f( ) TZ B u G( ) u TZ. B TZ as B G( ) TZ whch s qual to th B matr of th Taylor srs appromaton n (3.7). Ths lavs th scond constrant to b satsfd by whch s quvalntly wrttn as ATZ f() a f ( ) for K n (3.34) whr a s th th row of A TZ and f ( ) s th th componnt of f( ). Epandng f ( ) about th stat and nglctng all trms that ar scond ordr and hghr rsults n To dtrmn th rows of a ( )( ) f ( ). f + A TZ th functon E f ( ) a. s mnmzd subjct to th constrant a ( ) s [TEI99] rsultng n f (3.3) (3.3) (3.3) (3.33) (3.35) (3.36)

44 - 3 - a f ( ) f ( + ) f ( ) (3.37) for. Ths s quvalntly rprsntd n matr-vctor format as f( ) D f( ) A f( ) + TZ D. (3.38) Th constructon of th output quaton (3.7) for ths modl s undfnd by Tra-Żak. Hrn t s dfnd usng th Taylor srs appromaton thrby rsultng n TZ h( u) u u D (3.39) D TZ u h( u) u u D (3.4) and φ ytz h( u ) TZ D TZ u. (3.4) If th systm nputs ar assumd constant thn th truth modl taks th form & F( u + u ) f( ) G( ) (3.4) y h( u) (3.43) whch can b appromatd by whr & A TZ y TZ + φytz (3.44) (3.45) F( u ) D F( u) u u A F( ) + TZ D u u u (3.46) TZ s dfnd n (3.39) and φ yt h( u ) TZ. (3.47) Th Tra-Żak basd appromaton llustrats that mor than on local modl

45 may b constructd from th truth modl at a partcular modlng pont. In fact dpndng upon th constrants placd upon a local modl durng constructon an nfnt numbr of local modls may b possbl. 3.3 Altrnatv Appromatons Th prvous two mthods constructd local modls from th truth modl. Ths suggsts th followng quston: How many local modls ar actually avalabl to dscrb th bhavor of a nonlnar systm at a partcular modlng pont? To answr ths quston t s shown n ths scton that fndng a local modl possssng crtan proprts can b rducd to solvng a systm of quatons. Thrfor f th systm of quatons s undrdtrmnd an nfnt numbr of local modls ar avalabl. In th followng drvatons t s assumd that a local modl n th form of (3.5) of th nonlnar systm dfnd by (3.) s dsrd and that th stat drvatv s known for q modlng ponts ) and ( u. Addtonally th numbr of nputs and stats ar n u n rspctvly. Equaton (3.6) s not drvd but t can b dfnd usng smlar tchnqus Local modl wth prscrbd valu of th drvatv Ths drvaton s partcularly smpl and compact n form. It s bgun by frst takng th transpos (3.5) to obtan Rarrangng (3.48) th quaton & A + u B + φ. A & [ u ] B φ (3.48) (3.49) s obtand whch dfns th matrcs A B and vctor φ n trms of th nput stat and drvatv. hoosng q modlng ponts for th local modl to satsfy a systm of quatons of th form V ZM s constructd whr th matrcs V Z and M ar dfnd by (3.5)

46 and & & V M & q u u Z M M q u q A M B. φ To solv quaton (3.5) for th matr M th ffctv numbr of quatons and unknowns must frst b dtrmnd. Th numbr of quatons s qual to th numbr of lmnts n V that s qn qn M (3.5) (3.5) (3.53) n and th numbr of unknowns s qual to th numbr of lmnts n M that s n n n ) n unk ( + u +. If unk nq n th matr Z wll b squar. If t s also full rank thn th soluton for M s M Z V. Howvr f n < thn Z s rctangular and th systm of quatons s unk nq ovrdtrmnd. A soluton for M sts f V Rang( Z) and can b dtrmnd as L M Z V whr L ( Z Z) Z Z (3.54) s th lft nvrs of Z. Th lft nvrs If unk nq L Z sts f and only f Z has mamum rank. n > thn Z s agan rctangular. Howvr th systm of quatons s now undrdtrmnd nfntly many solutons st and th mnmum lngth last squars soluton can b found as R M Z V whr Z R ( Z ) Z Z (3.55) s th rght nvrs of Z. Th rght nvrs R Z sts f and only f Z has mamum

47 rank. Othr solutons can b found by addng any matr from th nullspac of Z to M. Th nullspac of Z dnotd as N (Z) s th subspac contanng all matrcs M satsfyng th homognous quaton Th dmnson of th N (Z) s qual to ZM. dm( N ( Z)) nunk n qn. (3.56) (3.57) Modfcatons to (3.5) can b mad n ordr to forc partcular charactrstcs n th local modl. For ampl to forc a partcular valu for th affn trm th matr quaton can b constructd as & φ & φ M M & q φ q u u A. M B u q (3.58) If th affn trms ar chosn to b zro vctors thn lnar rathr than affn local modls ar obtand. Eampl 3.: In ths ampl an affn local modl satsfyng th followng constrants s constructd: Th tm drvatv of th stat must b & [ ] at th pont [3.3 99] u [6 ] and th tm drvatv of th stat must b & [ ] at th pont [ 4] u [6 ]. Th matrcs V and Z ar dfnd by & V & and u Z. (3.6) u Th numbr of quatons s n qn qn 4 and th numbr of unknowns s (3.59) n unk ( n + n + ) n. Addtonally Z has a rank of vrfyng that th rows ar u

48 lnarly ndpndnt. Thrfor th systm s undrdtrmnd and an nfnt numbr of solutons st. Usng th rght nvrs a soluton s dtrmnd to b M (3.6) Th matrcs A B and and φ can now b dtrmn from M as A B φ. (3.64).543 It s asly vrfd that th obtand local modl satsfs th dsrd charactrstcs. In partcular and as dsrd. A A 3.3. Kronckr product form + Bu + φ + Bu + φ Ths drvaton bgns smlarly to th drvaton n th prvous scton. Takng th transpos of (3.5) and thn rarrangng th quaton A & [ u ] B φ (3.6) (3.63) (3.65) (3.66) (3.67)

49 s agan obtand. Equaton (3.67) s thn rwrttn usng th Kronckr product kron () and th stackng oprator (:) to obtan ( M) () M A & [ kron ( I ] n ) kron( Inu u ) In B φ whr th Kronckr product and th stackng oprator ar dfnd as and X() Y X() Y kron( X Y) M X( m) Y X(:) X() Y X() Y M X( m) Y X() X() M X( m) X() X() M. X X X X ( m) M ( n) ( n) M ( m n) In quatons (3.69) and (3.7) th matr X has dmnson L L M L X( n) Y X( n) Y M X( m n) Y m n and th notaton X ( m n) s usd to dntfy th lmnts of th matr X n ths cas t s th lmnt occupyng th m-th row and th n-th column. (3.68) (3.69) (3.7) Agan choosng q modlng ponts for th local modl to satsfy a systm of quatons of th form V ZM s constructd whr th matrcs V Z and M ar dfnd by (3.7)

50 q V & M & & I u I I I u I I I u I I Z ) kron( ) kron( ) kron( ) kron( ) kron( ) kron( q q M M M and ( ) () φ B A M : :. Th structur of quatons (3.7) (3.73) and (3.74) maks th numbr of quatons and th numbr of unknowns much asr to vsualz snc V and M ar now both column vctors. Th numbr of quatons s qual to th numbr of lmnts n V that s qn qn n and th numbr of unknowns s qual to th numbr of lmnts n M that s u unk n n n n ) ( + +. Th soluton to quaton (3.7) s found smlarly as was dscussd n Scton Modfcatons can b mad n ordr to forc partcular charactrstcs n th local modl. For ampl to forc a partcular valu for th affn trm th matr quaton can b constructd as (:) (:) ) kron( ) kron( ) kron( ) kron( ) kron( ) kron( B A u I I u I I u I I φ φ φ q q q M M & M & &. In addton t s smpl to plac qualty constrants upon ndvdual lmnts or groups of lmnts n A B and φ. For ampl to st th frst lmnt of A to - th matr quaton (3.7) (3.73) (3.74) (3.75)

51 & kron( I & kron( I M M & q kron( I q ) ) ) [ L ] kron( I u ) kron( I u ) M kron( I u q ) I I A (:) M B (:) I φ (3.76) can b solvd. Eampl 3.: In ths ampl an affn local modl satsfyng th followng constrants s found. Th tm drvatv of th stat must b & [ ] at th pont [3.3 99] u [6 ] and th tm drvatv of th stat must b & [ ] at th pont [ 4] u [6 ]. Th matrcs V and Z ar thn dfnd by & V (3.77) & and kron( I ) Z kron( I ) kron( I u ) kron( I u ) I I. Th numbr of quatons s n qn qn 4 and th numbr of unknowns s n ( n + n + ) n. Addtonally Z has a rank of 4 vrfyng that th rows of unk u (3.78) Z ar lnarly ndpndnt. Thrfor th systm s undrdtrmnd and an nfnt numbr of solutons st. Usng th rght nvrs on soluton s dtrmnd to b

52 M Th matrcs A B and φ ar now dtrmnd from M as A B and.45 φ ;.543 whch ar th sam as n Eampl 3.. (3.79) (3.8) (3.8) (3.8) Dagonal form Forcng th matr A to b dagonal can rsult n on of th smplst forms of local modls. Howvr ths sgnfcantly rducs th numbr of unknowns whch n turn lmts th numbr of constrants that th local modl s capabl of satsfyng. Ths may sgnfcantly affct how wll th local modl appromats th truth modl. To forc th matr A to b dagonal frst quaton (3.5) s agan transposd and rarrangd to obtan A & [ u ] B. (3.83) φ Nt sttng all of th off dagonal trms of A to zro and rarrangng quaton (3.83) s

53 - 4 - placd nto th form A() A() M & [ dag( ) kron( I u ) I] A( n n ) B (:) φ whr th Kronckr product and stackng oprator ar dfnd n Scton 3.3. and th dagonal oprator s dfnd as for a vctor of lngth m. () dag( ) () O ( m) (3.84) (3.85) Agan choosng q modlng ponts for th local modl to satsfy a systm of quatons of th form V ZM s constructd whr th matrcs V Z and M ar dfnd by and dag( ) dag( Z ) M dag( q ) & & V M & q kron( I u ) kron( I u ) M kron( I u q ) I I M I (3.86) (3.87) (3.88)

54 - 4 - φ B A A A M (:) ) ( () () n n M. Th numbr of quatons quals th numbr of lmnts n V that s q n n qn. Howvr th numbr of unknowns th lmnts n M s now rducd to ( ) u unk n n n +. Th radr s rfrrd to Scton 3.3. for dscusson on how to solv quaton (3.86). Modfcatons can b mad n ordr to forc partcular charactrstcs n th local modl. For ampl to forc a partcular valu for th affn trm th matr quaton can b constructd as (:) ) ( () () ) kron( ) dag( ) kron( ) dag( ) kron( ) dag( B A A A u I u I u I φ φ φ q q q n n M M M & M & &. It s smpl to plac qualty constrants upon ndvdual lmnts or groups of lmnts n A B and φ. For ampl to st th trac of A to - th matr quaton [ ] φ B A A A I u I I u I I u I (:) ) ( () () ) kron( ) dag( ) kron( ) dag( ) kron( ) dag( q q q n n M L M M M & M & & s formd. For systms that hav th sam numbr of nputs as stats t may also b possbl to forc both A and B to b dagonal by solvng th quaton (3.89) (3.9) (3.9)

55 & dag( ) & dag( ) M M & q dag( q ) dag( u) dag( u ) M dag( uq ) A() A() M I A( n I n ) B(). M B() I M B( n n ) φ (3.9) Howvr to prvnt th systm of quatons from bng ovrdtrmnd a mamum of thr ponts n th stat spac can b usd n formng th st of quatons. Eampl 3.3: In ths ampl an affn and dagonal local modl satsfyng th followng constrants s constructd. Th tm drvatv of th stat must b & [ ] at th pont [3.3 99] u [6 ] and th tm drvatv of th stat must b & [ ] at th pont [ 4] u [6 ]. Th matrcs V and Z ar thn dfnd by & V (3.93) & and dag( ) dag( u) I Z. (3.94) dag( ) dag( u ) I Th numbr of quatons ar n qn qn 4 and th numbr of unknowns ar n n + n 6. Addtonally Z has a rank of 4 vrfyng that th rows ar unk u lnarly ndpndnt. Thrfor th systm s undrdtrmnd and an nfnt numbr of solutons st. Usng th rght nvrs on soluton s dtrmnd to b

56 M Th matrcs A B and φ ar now constructd from M as A (3.96) B (3.97) and. φ. (3.98).443 It s asy to vrfy that ndd th abov local modl satsfs th dsgn spcfcatons and as dsrd. A A + Bu + φ + Bu + φ (3.95) (3.99) (3.) Thus far t has bn shown that local modls may b constructd satsfyng a lmtd numbr of constrants ncludng th dsrd qulbrum stat. As a mattr of computatonal convnnc t should b notd that many smulaton packags such as ASL [ADV99] hav th provson to numrcally dtrmn Taylor srs basd local modls. Howvr ths modls do not ncssarly possss th dsrd qulbrum pont. In ordr to tak advantag of ths numrcal analyss transformaton of th local modls s ncssary n ordr to obtan local modls possssng th dsrd qulbrum. Ths transformaton s consdrd n th nt scton.

57 onstructng Local Modls wth Pr-assgnd Equlbrum Pars Whn constructng a local modl usng thr Taylor or Tra-Żak basd appromatons th locaton of th local modl qulbrum pont s not consdrd. Howvr as can b sn n th followng chaptrs t s convnnt f th local modl has a partcular qulbrum par ). Thrfor two mthods of transformaton ar ( u ntroducd ach satsfyng th followng constrants:. Th rght-hand sd of th nw local modl must qual th rght-hand sd of th orgnal local modl whn valuatd at th modlng pont ). That s and A + B u + φ A + B u + φ + Du + φy + Du + φy ( u whr th subscrpt dsgnats th orgnal local modl and subscrpt dsgnats th nw local modl.. Th rght-hand sd of th local modl must qual zro whn valuatd at th dsrd qulbrum pont ). That s ( u + A + Bu φ. (3.) (3.) (3.3) 3.4. Gnralzd Tra-Żak transformaton Ths transformaton though not drvd by optmzaton s nsprd by th Tra-Żak appromaton prvously dscussd. Th gnralzd Tra-Żak mthod modfs stng local modls of th form dfnd by (3.5) and (3.6) to gnrat nw local modls wth prscrbd qulbrums. Dpndng upon th choc of th qulbrum th rsultng local modl dfnd by and & A + B u + φ GTZ GTZ GTZ y GTZ + DGTZ u + φygtz (3.4) (3.5) may b lnar or affn. Th subscrpt GTZ dnots th gnralzd Tra-Żak

58 appromaton. To partally satsfy th frst constrant th matrcs D GTZ and vctor ygtz φ ar dfnd as B GTZ GTZ and B GTZ B GTZ D GTZ D φ ygtz φ y (3.6) (3.7) (3.8) (3.9) automatcally forcng th satsfacton of quaton (3.) and lavng th matr and vctor φ GTZ to b drvd. To bgn ths drvaton t s rqurd that f (3.4) s shftd by ~ A GTZ (3.) and u~ u u (3.) thn th rsultng quaton s lnar thrfor ~ & A ~ B u~ GTZ + GTZ. Ths rqurmnt forcs th satsfacton of constrant (3.3). To forc satsfacton of th constrant (3.) quaton (3.3) s frst shftd usng quatons (3.) and (3.) to obtan ~ & A ~ + Bu~ + φ~ (3.3) whr th affn trm φ ~ s dfnd as φ ~ φ + A + Bu. To satsfy constrant (3.) th rght-hand sd of quatons (3.) and (3.3) must qual whn valuatd at th modlng pont rsultng n A ~ B u~ A ~ Bu~ φ~ GTZ + GTZ + + (3.5) whr and ~ (3.) (3.4) (3.6)

59 u ~ u. Usng th dfnton of B GTZ n (3.6) quaton (3.5) may b smplfd to A ~ A ~ φ~ GTZ + and rwrttn as Th matr A GTZ s thn dfnd by u ~ ~ ~ φ A ~ GTZ A +. ~ ~ A φ~ ~ GTZ A +. ~ ~ Notc that f ~ that s th modlng pont s qual to th dsrd qulbrum pont thn A GTZ s chosn as A GTZ A whch rsults n th two local modls bng dntcal. If th nput s constant thn an unforcd stat quaton of th form ~ & A ~ GTZ (3.) may b obtand by sttng To dntfy th affn trm coordnats rsultng n A [ Bu~ + φ~ ] ~ GTZ A +. ~ ~ φ GTZ th quaton (3.) s shftd back to th orgnal φ GTZ A Bu. GTZ To vrfy that th local modl dfnd by (3.6) (3.3) and (3.4) has th prscrbd qulbrum par and that th rght-hand sd of (3.4) valuatd at th modlng pont ) quals th rght-hand sd of (3.5) valuatd at th modlng ( u pont ) th followng drvaton s provdd. ( u Frst to show that th rght-hand sd of (3.4) s zro at th dsrd qulbrum (3.7) (3.8) (3.9) (3.) (3.) (3.3) (3.4)

60 par ) substtut ) nto (3.4) to obtan ( u ( u & A + Bu + φ. GTZ GTZ (3.5) Nt substtutng (3.4) nto (3.5) th quaton & AGTZ + Bu AGTZ Bu (3.6) s obtand. It s radly apparnt that th rght-hand sd of (3.6) valuats to zro for th qulbrum par ). ( u To prov that th rght-hand sd (3.4) s qual to th rght-hand sd of (3.5) for th opratng par ) substtut th opratng par ) nto (3.4) to obtan ( u ( u & +. AGTZ + Bu φgtz (3.7) Thn substtutng (3.4) nto (3.7) th quaton & AGTZ + Bu A GTZ Bu (3.8) s obtan whch s smplfd to & A ~ + B u ~ GTZ (3.9) usng (3.6) and (3.7). Substtutng (3.) nto (3.9) ylds & A ~ Bu~ φ~ + + (3.3) whch pands to & Bu + φ + A + Bu A A + Bu (3.3) usng (3.4) (3.6) and (3.7). Smplfcaton of (3.3) rsults n & A + Bu + φ (3.3) whch s th ntal local modl valuatd at th modlng pont ). ( u It s also possbl to forc a dsrd qulbrum par upon an stng local modl wthout havng to prform coordnat transformatons. Ths s consdrd n th nt scton Matr nvrson basd transformaton In th prvous scton a coordnat translaton and a nonlnar transformaton wr usd to obtan local modls havng partcular qulbrum stats. As an altrnatv

61 mthod for prscrbng qulbrums th tchnqus dscrbd n Scton 3.3 may b usd to construct th local modls as long as both th dsrd qulbrum par and th modlng pont ar usd as constrants. If a local modl has alrady bn found usng for ampl Taylor srs appromaton portons of that local modl may b rtand n th fnal modl. Prhaps t would b dsrabl to only chang th matr A to satsfy th two constrants smlar to th procdur dfnd n Scton Thn for an ntal local modl of th form gvn by quaton (3.5) a nw modl of th form & A MI + B MI u + φmi y MI + D MI u + φymi s constructd whr th subscrpt MI rfrs to th matr nvrson A MI A + A (3.35) B MI B (3.36) and th matrcs φ MI MI D MI φ ymi ar dfnd smlar to B MI. Th matr A s found by solvng quaton (3.5) whr A u B φ V and Z M A. Th nullspac of quaton (3.5) has th dmnson dm( N ( Z)) n n n thrfor th soluton M s not unqu f th systm s of thrd ordr or hghr. If th systm s of scond ordr thn th soluton s unqu. If th systm s of frst ordr thn mor than th coffcnts of A must b modfd bcaus (3.37) bcoms ovrdtrmnd wth nsuffcnt rank. In som stuatons t may b mor prudnt to modfy B or φ nstad of A. If th systm s sngl nput thn a soluton rqurs th modfcaton of B and dfnng (3.33) (3.34) (3.37) (3.38) (3.39) φ by

62 - 5 - A V (3.4) & A u Z (3.4) u and B M. (3.4) φ Howvr f th systm has two nputs or mor thn th modfcaton of B s suffcnt to mt th two constrants. In ths cas A φ V (3.43) & A φ and u Z u M B (3.45) can b solvd for a nw matr B. Th followng cas studs contan ampls usng th varous local modl constructon tchnqus dscussd abov. 3.5 as Study : Systm Lnar Wth Rspct to Input In ordr to vsualz th dffrncs btwn th local modl appromatons for systms lnar n nput consdr th frst-ordr systm gvn by & F( u) sn( ) + u whch fts th form dfnd by quaton (3.4). Th Taylor appromaton rsults n an affn modl dfnd by [ cos( ) + u ] + u + [ sn( ) + cos( ) u ] & o Th Tra-Żak appromaton ylds th lnar modl. (3.44) (3.46) (3.47) sn( ) & + u (3.48)

63 - 5 - and th gnralzd Tra-Żak appromaton rsults n whn th dsrd qulbrum par of ( ) sn( ) & + u (3.49) u s chosn. Not that th Tra- Żak modl s qual to th gnralzd Tra-Żak modl for ths ampl. For th local modl obtand usng matr nvrson th stat quaton form drvd n Scton 3.3. s usd. Th constrants chosn ar for th rght-hand sd of th rsultng local modl to qual th rght-hand sd of th truth modl at th modlng pont and for th orgn to b th qulbrum pont for th local modl. Ths constrants dfn th matrcs and whch ar usd to solv V F ( u ) u Z u M Z V. Th soluton to (3.5) rsults n th local modl sn( u ) + &. R Th rght-hand sds of quatons (3.46) through (3.49) and quaton (3.53) ar plottd n Fgur 3. and Fgur 3. for th nomnal modlng pont of 4 / 3 π and nputs of u and u. 5 rspctvly. In Fgur 3. and Fgur 3. all appromatons ar qual at th modlng pont about whch th appromaton was appld and all of th local modls ar obsrvd to b unstabl vn though th truth modl has at last th orgn as a stabl qulbrum stat n both cass. To furthr dmonstrat th dffrncs n th bhavor of th truth and corrspondng local modls smulatons ar prformd for th nput u and th ntal condton 4 / 3. Notc n Fgur 3.3 that th trajctors of th local modls π (3.5) (3.5) (3.5) (3.53)

64 - 5 - dvrg wth tm whl th trajctory of th truth modl approachs an qulbrum stat. Fgur 3.. Local and truth modl comparson u. Fgur 3.. Local and truth modl comparson u. 5. Fgur 3.3. Local and truth modl trajctors corrspondng to 4 / 3 and u. π

65 as Study : Systm Nonlnar Wth Rspct to Input In ordr to vsualz th dffrncs btwn th thr local modl appromatons for systms nonlnar wth rspct to nputs consdr th frst-ordr systm gvn by 3 & sn( ) + u. Th Taylor srs appromaton rsults n an affn modl dfnd by 3 & [ cos( ) + u ] + 3 u u + [ sn( ) + cos( ) 4 u Th Tra-Żak appromaton mthod rqurs a fd nput snc th systm dos not ft nto th approprat class of systms whn varabl nputs ar allowd. Th rsultng local modl s 3 ]. (3.54) (3.55) sn( + u & ). Gnralzd Tra-Żak appromaton rsults n th local modl 3 (3.56) sn( & ) u whn th dsrd qulbrum par of ( u ) u u s chosn. For th local modl obtand usng th altrnatv mthods th form drvd n Scton 3.3. s usd. Th constrants chosn ar for th rght-hand sd of th rsultng local modl to qual th rght-hand sd of th truth modl at th modlng pont and for th orgn to b th qulbrum pont for th local modl. Ths constrants dfn th matrcs V F ( u (3.58) ) and u Z (3.59) u whch ar usd to solv R M Z V. Th soluton to (3.6) rsults n th local modl (3.57) (3.6)

66 sn( u ) + &. Th rght-hand sds of quatons (3.54) through (3.55) and quaton (3.6) ar plottd n Fgur 3.4 and Fgur 3.5 for th nomnal modlng pont of 4 / 3 and nputs of u and u. 4 3 (3.6) π rspctvly. In Fgur 3.4 and Fgur 3.5 all appromatons ar qual at th modlng pont about whch th lnarzaton was appld and all of th local modls cpt for th GTZ basd local modl n Fgur 3.5 ar obsrvd to b unstabl vn though th truth modl has at last th orgn as a stabl qulbrum stat n both cass. Fgur 3.4. Local and truth modl comparson u. Fgur 3.5. Local and truth modl comparson u. 4.

67 To furthr dmonstrat th dffrncs n th bhavor btwn th truth and corrspondng local modls smulatons ar prformd for th nput u. 4 and th ntal condton 4 / 3. Notc n Fgur 3.6 that th trajctors of all th modls π dvrg wth tm cpt for th GTZ basd local modl. Fgur 3.6. Local and truth modl trajctors corrspondng to 4 / 3 and u. 4. π 3.7 as Study 3: GTZ Transformaton to Forc Prscrbd Equlbrum Th gnralzd Tra-Żak transformaton can b usd to forc all of th local modls to hav concdnt qulbrum pars ( u ). That s all of th local modls shar th sam qulbrum stat for a partcular nput. As an ampl consdr th frst ordr systm n as Study dscrbd by 3 & sn( ) + u F( u). (3.6) Th Taylor srs appromaton for ths systm s gvn by whr and & A + B u + φ T T T 3 ) u A T cos( + B T 3 u (3.63) (3.64) (3.65)

68 φ T ) + cos( ) 4 3 u sn(. A constant nput s chosn to b u u. and th local modls ar constructd at th modlng ponts Th rght-hand sds of th truth and local modls (3.66) vrsus stat ar plottd n Fgur 3.7. Notc that fv of th local modls hav postv slops ndcatng thr nstablty. Fgur 3.7. Plot of th stat drvatv vrsus stat for th truth and Taylor srs basd local modls. In Fgur 3.7 th qulbrum stats of th local modls ar dntfd as valus of at whch th lns ntrsct wth &. Th qulbrum pont of th truth modl around whch th local modls ar constructd s Th coordnat transformaton of th modls s prformd usng th chang of coordnats dfnd by ~ (3.67) and u ~ u u (3.68) rsultng n a truth modl of th form and local modls dfnd by whr ~ & sn( ~ + ) + ( ~ + ) ( u ~ + u ~ & A ~ T + BT u~ + φ~ T ) 3 (3.69) (3.7)

69 φ ~ A + B u + φ. T T Fgur 3.8 contans plots of th shftd truth and local modl rght-hand sds vrsus stat. Th qulbrum pont of ntrst s now locatd at th orgn. T T (3.7) Fgur 3.8. Plot of th stat drvatv vrsus stat for th shftd truth and Taylor srs basd local modls. Nt transformng th local modls usng (3.) rsults n local modls of th form ~ & A ~ GTZ + BT u~. (3.7) Plottng th rght-hand sd of (3.7) n Fgur 3.9 along wth th rght-hand sd of (3.69) t s shown that all of th local modls ar now lnar and shar th orgn as an qulbrum stat. Fgur 3.9. Plot of th stat drvatv vrsus stat for th truth and transformd local modls.

70 If th lnar local modls ar now shftd back to th orgnal coordnats usng th coordnat transformatons and ~ + u u ~ + u (3.73) (3.74) thn th fnal local modls n th orgnal coordnats hav th form of & A + B u + φ GTZ T (3.75) whr φ A B u. GTZ T (3.76) Th fnal local modls agan bcom affn n th orgnal coordnats unlss (3.76) happns to quat to zro howvr thy hav th concdnt qulbrum par of ( ) (6.644.) as llustratd n Fgur 3.. u Th local modlng tchnqus drvd hrn rsult n local modls that ar not suffcntly accurat for analyzng a nonlnar systm usng a sngl modl. Howvr whn ths local modls ar combnd togthr to form a polytopc modl nonlnarts of th systm can b capturd. Ths wll b dmonstratd n th nt chaptr. Fgur 3.. Plot of th stat drvatv vrsus stat for th truth and gnralzd Tra-Żak basd local modls.

71 POLYTOPI MODELING In th prvous chaptr mthods for constructng local modls wr dscussd. In ths chaptr a mthod of combnng th local modls nto a polytopc modl that rprsnts th nonlnar systm ovr a wd rang of opratng condtons s st forth. It s notd that th truth modl alrady has ths proprty and s gnrally mor computatonally ffcnt to smulat than th polytopc modl. Howvr th polytopc modl dfnd hrn has a structur partcularly amnabl to sarchng for Lyapunov functon canddats. Ths chaptr dfns th form of th polytopc modl usd as a bass for stablty analyss of th truth modl n latr chaptrs. In addton som polytopc modl charactrstcs ar dscussd and ampls provdd. 4. Polytopc Modl Structur Hrn polytopc modls ar constructd from r local modls whr th -th local modl s dfnd as & A + B u + φ y + Du + φy. Th polytopc modl s a conv combnaton of th local modls combnd togthr by wghtng functons ( θ) w n a form basd on th TS modl [TAK85] and dfnd n haptr as r & w ( θ) A + Bu + φ whr r [ ] [ + D u + φ ] y w ( θ) y (4.) (4.) (4.3) (4.4)

72 - 6 - w ( θ) (4.5) r w ( θ ) and θ may b a functon of u or systm paramtr. Many dffrnt forms of wghtng functons ar possbl. Th choc of wghtng functons can affct not only th complty of th modl but also th locaton of th qulbrum ponts as llustratd by Eampl 4.. An ampl of a wghtng functon ( ) µ n on varabl s st forth n [TAK85] and has th structur: ( ) µ for ( ) µ for µ ( ) for or. Ths functon s llustratd n Fgur 4. for modlng pont. 5. Anothr possbl wghtng whch has bn proposd n [AO96A] and t has th form m( b) µ ( ) + m( + b) + and s dpctd n Fgur 4. for. 5 m 6 and b. 5. (4.6) (4.7) (4.8) (4.9) (4.) Fgur 4.. Wghtng functon usd by Takag and Sugno.

73 - 6 - Fgur 4.. Wghtng functon usd by ao Rs and Fng. Eampl 4.: To llustrat how wghtng functons can affct th qulbrum pont consdr th followng polytopc modl whr and [ A + φ ] + ( [ A φ ] & µ ( ) µ ) + A 4 φ 5 9 A 3 (4.) (4.) (4.3) (4.4) 9 φ. (4.5) 5 Thr ar at last thr possbl qulbrum stats for ths systm all of whch may or may not st dpndng upon how th wghtng functons ar dfnd. Possbl qulbrum stats ar

74 - 6 - and 4.44 f µ ( ) and µ ( ) f µ ( ) and µ ( ) f µ ( 3). 5 and µ ( 3). 5. (4.8).857 Not that s an qulbrum stat of th frst local modl and t s not an qulbrum stat of th scond local modl whl s an qulbrum stat of th scond local modl and t s not an qulbrum stat of th frst local modl and 3 s nthr an qulbrum stat of th frst nor th scond local modl. (4.6) (4.7) hoosng whch typ of wghtng functon to us for a partcular systm s stll unclar and a potntal ara of furthr rsarch. Hrn wghtng functons of th form gvn n (4.) ar chosn. To us ths wghtng functon n hghr dmnsons ondmnsonal wghtng functons ar multpld togthr to obtan µ K + ( θ θ ) + m m( θ ( ) ( ) θθb m θ + θ b K θ+ b) m( θ + ) + θ b whr θ may b a stat nput or paramtr. To nsur that condton (4.6) s satsfd th wghtng functon for th -th local modl s calculatd as w ( θ) r µ j ( θ) µ j ( θ) Th dcson of how many dmnsons th wghtng functon should vary s systm dpndnt and chosn hrn basd on th stats nputs or paramtrs for whch th systm modl vars nonlnarly.. (4.9) (4.)

75 In ordr to obtan furthr nsght nto how polytopc modls ar constructd and to confrm that polytopc modls ar abl to ffctvly modl nonlnar charactrstcs th followng thr cas studs ar provdd. 4.. as study : Systm lnar n trms of th nputs As an ampl of polytopc modlng of systms lnar n trms of th nputs consdr systm dscrbd n haptr. Polytopc modls usng Taylor srs Tra-Żak and gnralzd Tra-Żak basd local modls ar constructd and compard for ths systm. Usng th Taylor srs appromaton drvd about th modlng pont [ ] L v [ ] P V s u th local modl can b prssd as + + v L v L L R cp cp u & + + cp cp v v u y whr cp v P. Th Tra-Żak basd local modls ar drvd by frst obsrvng that th systm truth modl can b rwrttn as + + ) G( ) f( P V v L L v R s L L u & ) ( ) h( u u y cp L. Ths systm fts nto th format sutabl for th Tra-Żak basd local modl appromaton whch rsults n local modls of th form (4.) (4.) (4.3) (4.4) (4.5)

76 u + v L L L R & + + cp cp v v u y. Th last local modl to b drvd s basd on th gnralzd Tra-Żak appromaton whch ylds u ) ( ) ( L cp L L cp v L v v v P v L L R & + + cp cp v v u y whn th dsrd qulbrum par s chosn as ( ) u. Th stat s th qulbrum stat for th -th local modl corrspondng to th nput u. Th wghtng functons usd to combn th Taylor srs basd local modls as wll as th Tra-Żak basd local modls togthr ar dfnd n (4.) whr ) ( ) ( ) ( ) ( ) ( p p p p v v v v b P P m b P P m b v v m b v v m P v µ and th modlng pont of th -th local modl s dfnd by v and P. Th othr constant trms n (4.3) ar dfnd as 3 v m v 3 m P P (4.6) (4.7) (4.8) (4.9) (4.3) (4.3) (4.3)

77 v bv P b P whr v s th dffrnc n voltag btwn th modlng pont valus and P s dfnd smlarly for th modlng pont powr. Evry possbl combnaton of v [ 35V] and P [ 5kW kw] 5V ach dvdd nto 4 qually spacd ponts was usd to gnrat th local modls. Th wghtng functons for all of th local modls usd n th Taylor srs and th Tra-Żak appromaton ar llustratd n Fgur 4.3. Th wghtng functons usd for th gnralzd Tra-Żak mthod ar chosn to vary n thr dmnsons and dfnd n (4.) whr m ( ) v ( ) + v v bv µ v P L mv ( ) + v v + bv mp ( P P bp ) ( ) + m + L L L b L. ( ) ( + ) mp P P + b m p L L b + + L L Evry possbl combnaton of v [ 35V] P [ 5kW kw] [ A] 5V and L A ach dvdd nto 4 qually spacd ponts was usd to gnrat th local modls. Th othr constants usd n (4.35) ar 3 m L and L b L whr L s th chang n nductor currnt modlng pont valu. L (4.33) (4.34) (4.35) (4.36) (4.37)

78 Fgur 4.3. Local modl wghtng functons for th Taylor srs and th Tra-Żak basd polytopc modls. Fgur 4.4 dpcts smulaton rsults for th thr polytopc modls and th truth modl. Varabls dpctd nclud th nductor currnt L capactor voltag v and th constant powr load currnt cp. In ths study th sourc voltag V s s stppd from 5V to 3V at.5 sconds and th commandd powr P s stppd from 5 kw to kw at.5 sconds. Th rsults from th four studs ar so clos that thy cannot b dstngushd n ths fgur. Th modl rrors ar plottd n Fgur 4.5. It s ntrstng to not that th stady stat rror for th Tra-Żak and gnralzd Tra-Żak basd modls s largr than th Taylor srs basd modl for all varabls cpt th constant powr load currnt n th stady stat. Fgur 4.6 through Fgur 4.8 dpct th trajctors for all thr polytopc modls plottd on top of th local modl qulbrum ponts. Obsrv that th polytopc modl trajctory may convrg to an qulbrum stat that s not ncssarly an qulbrum stat of th local modls. Ths ssu s furthr dscussd n th chaptr on stablty.

79 Fgur 4.4. Smulaton comparson for th truth and polytopc modls. Fgur 4.5. Modl rror comparson for th polytopc modls. Fgur 4.6. Taylor srs basd modl trajctory plot wth th local modl qulbrum ponts.

80 Fgur 4.7. Tra-Żak basd polytopc modl trajctory plot wth th local modl qulbrum ponts. Fgur 4.8. Gnralzd Tra-Żak basd polytopc modl trajctory plot wth th local modl qulbrum ponts. 4.. as study : omparson btwn th numbr of local modls As on mght pct th numbr of local modls usd to construct th polytopc modl can affct how wll th polytopc modl rprsnts th truth modl. To llustrat that ths s n fact dos occur cas study s rpatd hr wth vrythng th sam cpt that th numbr of modlng ponts s doubls along ach dmnson. In ths study vry possbl combnaton of v [ 35V] and P [ 5kW kw] 5V ach dvdd nto ght qually spacd ponts was usd to gnrat th local modls for th Taylor srs and th Tra-Żak basd modls. For th gnralzd Tra-Żak basd

81 modl vry possbl combnaton of v [ 35V] P [ 5kW kw] [ A] 5V and L A ach dvdd nto 8 qually spacd ponts was usd to gnrat th local modls. Fgur 4.9 dpcts smulaton rsults for th thr polytopc modls and th truth modl. Varabls dpctd nclud th nductor currnt L capactor voltag v and th constant powr load currnt cp. In ths study th sourc voltag V s s stppd from 5V to 3V at.5 sconds and th commandd powr P s stppd from 5 kw to kw at.5 sconds. Th rsults from th four studs ar so clos that thy cannot b dstngushd n ths fgur. Th modl rrors ar plottd n Fgur 4.. omparng Fgur 4. to Fgur 4.5 t s radly apparnt that th ncras n th numbr of local modls n a fd rgon of th stat-spac rducs th rror btwn th polytopc modls and th truth modl as study 3: Systm nonlnar n trms of th nputs For an ampl of polytopc modlng of systms nonlnar n trms of th nputs consdr systm. Polytopc modls usng Taylor srs and gnralzd Tra-Żak basd local modls ar constructd and compard for ths systm. Fgur 4.9. Smulaton comparson for th truth and polytopc modls.

82 - 7 - Fgur 4.. Smulaton rror comparson for th polytopc modls. Th Taylor srs appromaton s drvd about th modlng pont [ L v ] u [ ] V s P. As a rsult of currnt lmtng th prssons for th paramtrs of th local modls hav two forms. If th systm s n th currnt lmtng mod thn th local bhavor can b modld by R L L + & + L u cp (4.38) and y + u + (4.39) cp whr cp cp mn or cp cp ma. If th systm s not n currnt lmt thn th local bhavor can b appromatd by (4.) and (4.) n cas study. For th cas whn th rght-hand sd of th truth modl s non-dffrntabl at a modlng pont.. btwn currnt lmtng and not currnt lmtng th avrag of (4.38) and (4.) as wll as (4.39) and (4.) ar takn by summng th rght-hand sds of th quatons and dvdng by two. Ths rsults n th local modl

83 v L v L L R cp cp u & and + + cp cp v v u y whr mn cp cp or ma cp cp. Bcaus ths systm s nonlnar n nput du to currnt lmtng t dos not ft nto th class of systms sutabl for th Tra-Żak basd appromaton. Thrfor th gnralzd Tra-Żak basd local modls ar consdrd. If th systm s currnt lmtng thn th gnralzd Tra-Żak appromaton mthod producs a modl of th form u ) ( ) ( L v v v L L R L cp L L cp & + + cp u y whr mn cp cp or ma cp cp. If th systm s not n currnt lmt thn th local bhavor can b appromatd by (4.8) and (4.9) as n cas study. For th cas whn th rght-hand sd of th truth modl s non-dffrntabl at a modlng pont.. btwn currnt lmtng and not currnt lmtng th avrag of (4.4) and (4.8) as wll as (4.43) and (4.9) ar takn by summng th rght-hand sds of th quatons and dvdng by two. Ths rsults n th local modl u ) ( ) ( L cp L L cp v L v v v P v L L R & and (4.4) (4.4) (4.4) (4.43) (4.44)

84 - 7 - whr cp cp mn or cp cp ma. y cp + + u. v cp v (4.45) Th wghtng functons usd to combn th Taylor srs basd local modls togthr to form a polytopc modl wr chosn to b of th form gvn n (4.3). Evry possbl combnaton of v [ 4V] and [ 5kW kw] 5V P dvdd nto ght qually spacd ponts was usd to gnrat th local modls. Th wghtng functons usd for th gnralzd Tra-Żak mthod ar chosn to vary n thr dmnsons and gvn by quaton (4.35) whr vry combnaton of v [ 4V] P [ 5kW kw] and [ A] 5V L A ach dvdd nto ght qually spacd ponts was usd to gnrat th local modls. Fgur 3. dpcts th smulatd rspons of th systm prdctd by th two polytopc modls as wll as th truth modl. Varabls dpctd nclud th nductor currnt L capactor voltag v and th constant powr load currnt cp. In ths study th sourc voltag V s s stppd from 5V to 3V at.5 sconds and th commandd powr P s stppd from 5 kw to kw at.5 sconds. As can b sn th rsults of th thr smulatons cannot b radly dstngushd. Th modl rrors ar plottd n Fgur 4.. Fgur 4.3 and Fgur 4.4 dpct th trajctors for both polytopc modls plottd on top of th local modl qulbrum ponts.

85 Fgur 4.. Smulaton comparson for th truth and polytopc modls. Fgur 4.. Smulaton rror comparson for th polytopc modls. Fgur 4.3. Taylor srs basd systm trajctory plot wth th local modl qulbrum ponts.

86 Fgur 4.4. Gnralzd Tra-Żak basd polytopc modl trajctory plot wth th local modl qulbrum ponts. 4. Equlbrum Stat lassfcaton In Fgur 4.6 through Fgur 4.8 as wll as n Fgur 4.3 and Fgur 4.4 t can b sn that th qulbrum stat for th polytopc modl dos not ncssarly concd wth th qulbrum stats of th local modls. Ths gvs rs to two classs of polytopc modls whch ar dscussd n th followng subsctons. 4.. Polytopc modls wth non-concdnt qulbrum stats A par ( u ) dos not hav to b an qulbrum par of ach local modl to b an qulbrum par of th olytopc modl as obsrvd n Eampl 4. and cas studs and 3. That s can b satsfd vn f r w [ A + Bu + φ ] ( θ ) A + Bu + φ for som. (4.46) (4.47) A polytopc modl that satsfs (4.46) and (4.47) s rfrrd to as a non-concdnt polytopc modl. Ths charactrstc s furthr dmonstratd n th followng ampl.

87 Eampl 4.: Ths ampl llustrats th stuaton whr qulbrum pars of th polytopc modl ar not ncssarly qulbrum pars of th local modls. Suppos that r A A b b φ -φ and w ( u) w( u) such that w ( u ). (4.49) Thn th stat pard wth any u wll b an qulbrum par for th polytopc modl whl not ncssarly an qulbrum par for local modls. (4.48) In gnral unlss spcfc stps ar takn a polytopc modl wll hbt non concdnt qulbrum pars. Howvr wth propr constructon of th local modls as dscussd n haptr 3 t s possbl for a polytopc modl to hav concdnt qulbrum pars. 4.. Polytopc modls wth concdnt qulbrum stats Bfor furthr nvstgatng th qulbrum stats of a polytopc modl t s approprat to nvstgat th qulbrum stats of a sngl local modl. Th followng rsult [STR88] wll b ncssary for th drvatons to follow. Lmma 4.: A ncssary and suffcnt condton for a systm of quatons hav a soluton s that s b rang(a). rank( A) rank[ AMb] A b to (4.5) Th condtons that charactrz th qulbrum pars ( u ) of a local modl of th form & A + Bu + φ (4.5)

88 can now b analyzd. Th st of all qulbrum pars for ths systm ar obtand by solvng th algbrac quaton A + Bu + φ. Equaton (4.5) can b quvalntly rprsntd as [ AM B] φ u. (4.53) By Lmma a soluton sts f and only f (ff) rank [ AM B] rank [ AMBMφ]. An qulbrum stat corrspondng to a spcfc constant nput u u s dfnd by A Bu φ (4.54) whch sts by Lmma ff rank A rank [ AM Bu φ]. To tnd ths rsults to a conv combnaton of local modls consdr th polytopc modl dfnd n (4.3) to (4.6) whch has an qulbrum stat that satsfs r w ( θ ) [ A + Bu + φ ]. (4.55) If th qulbrum pars for all of th local modls ar concdnt that s A B u + φ + (4.5) (4.56) for all K r thn th polytopc modl s rfrrd to as a concdnt polytopc modl. Notc that th satsfacton of (4.56) automatcally forcs satsfacton of (4.55) as s dmonstratd n Eampl 4.3 blow. In gnral (4.56) wll not hold for a polytopc modl constructd from Taylor srs basd local modls. Howvr usng th tchnqus dscussd n haptr 3 ths condton can b nforcd. To llustrat concdnt qulbrums consdr th followng ampl. Eampl 4.3: onsdr a polytopc modl of th form gvn by (4.55) whr r and th local modls ar gvn by 3 A 6 (4.57)

89 B 4 9 φ 3 8 A B and 49 φ. Both modls hav an qulbrum pont at [ ] f u. That s th par ( ) u satsfs (4.56) for. To gnralz th abov lnar systm analyss to a concdnt polytopc modl consstng of r local modls suppos that th par ( ) u s an qulbrum par of ach ndvdual local modl that s ths par satsfs th quatons r r r φ u B A φ u B A M. Equatons (4.63) can b rprsntd n th matr-vctor format as r r r φ φ u B A B A M M M whr by Lmma 4. a soluton sts ff r r r r r φ B A φ B A B A B A rank rank M M M M M as llustratd by th followng ampl. (4.58) (4.59) (4.6) (4.6) (4.6) (4.63) (4.64) (4.65)

90 Eampl 4.4: Utlzng th local modls from Eampl 4.3 t s vrfd that 3 rank B A B A and 3 rank φ B A φ B A thrfor a soluton sts to th systm of quatons. Obsrv that f all th local modls ar lnar (that s r... φ ) thn at last on soluton sts whch s th par u. To dtrmn th stnc of an qulbrum stat corrspondng to a spcfc constant u th followng st of r quatons can b solvd r r r φ u B A φ u B A M whch by Lmma 4. has a soluton ff r r r r φ u B A φ u B A A A rank rank M M M as llustratd by th followng ampl. Eampl 4.5: Utlzng th local modls from Eampl 4.3 t s vrfd that rank A A and rank φ u B A φ u B A (4.66) (4.67) (4.68) (4.69) (4.7) (4.7)

91 for u. Thrfor th qulbrum stat for th systm of quatons sts and s [ ]. In chaptr 3 mthods for constructng local modls that forc concdnt qulbrum pars wr prsntd. Th followng ampl compars concdnt and nonconcdnt polytopc modls as study 4: Smulaton of a concdnt polytopc modl In th prvous cas studs th qulbrum pars for th local modls wr non concdnt. In ths ampl a non-concdnt polytopc modl s constructd for systm 3 usng Taylor srs appromaton. Usng th local mods n th non-concdnt polytopc modl two concdnt polytopc modls ar constructd. On usng th gnralzd Tra-Żak transformaton and on usng th matr nvrson basd transformaton. Taylor srs basd local modls for ths systm ar drvd at all combnatons of th modlng ponts gvn by L [A 5A] dvdd nto 5 qually spacd ponts and v [3V 7V] dvdd nto 4 qually spacd ponts. Th nputs for ths u 5V kw. modl ar assumd to b constant and qual to [ ] Usng th GTZ transformaton tchnqu dscrbd n scton 3.4. and th matr nvrson tchnqu dfnd by quatons (3.33) through (3.39) two nw sts of local modls ar constructd all of whch hav qulbrums concdnt wth th truth modl.4395a V qulbrum [ ]. Ths two nw sts of local modls ar usd to construct two concdnt polytopc modls. Th wghtng functon usd to combn th local modls s th sam for all thr polytopc modls and s dfnd by quaton (4.) whr

92 - 8 - whr m v ( ) ( ) + + v v bv m L L b L L µ ( v ) L ( + ) ( L L + ) m v v b m b + + v v L L v and dfn th modlng pont of th -th local modl. Th othr L constant trms n (4.3) ar dfnd as 3 mv v 3 m L L v bv L b L (4.76) whr v s th dffrnc n voltag btwn th modlng pont nomnal valus and L s dfnd smlarly for th modlng pont nductor currnt. Th truth modl and th thr polytopc modls wr gvn an ntal condton of ( 35A v 575V ) and smulatd. Rsults from th smulaton ar plottd n L (4.7) (4.73) (4.74) (4.75) Fgur 4.5 and th rror btwn th truth modl and th polytopc modls ar plottd n Fgur 4.6. Th rrors n ach of th polytopc modls ar appromatly qual as can b obsrvd by th ndstngushabl tracs n Fgur 4.6 thrby llustratng th accuracy of th polytopc modls obtand usng local modls constructd by th gnralzd Tra-Żak and th matr nvrson basd tchnqus. In th ampls provdd n ths chaptr t can b sn that polytopc modls ar nonlnar and can captur th nonlnar charactrstcs of systm truth modls t s notd that th truth modl s gnrally mor computatonally ffcnt to smulat than th polytopc modl. Thrfor th motvaton for constructng polytopc modls s to us thm as a mans to sarch for Lyapunov functon canddats as llustratd n latr chaptrs. Bfor ths howvr th polytopc modlng of ntrconnctd systms s addrssd n th followng chaptr.

93 - 8 - Fgur 4.5. Smulaton comparson for th truth and polytopc modls. Fgur 4.6. Smulaton rror comparson for th polytopc modls.

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95 DEENTRALIZED POLYTOPI MODELING For smpl systms t s convnnt to formulat th systm modl as a whol. Howvr for compl systms comprsd of many componnts t s oftn asr to construct and valdat modls on a componnt rathr than a systm bass. In addton many systms nclud componnts dsgnd and constructd by dffrnt groups. Oftn th dsgns ar consdrd proprtary or th systm componnt modls ar constructd wth dffrnt smulaton languags. For ths rasons ths chaptr consdrs ntrconnctd polytopc modls. Hsao and Hwang [HSI] may b th frst to addrss ntrconnctd polytopc modls spcfcally focusng on TS modls constructd from unforcd autonomous local modls. Ths chaptr pands th class of systms applcabl to ntrconnctd analyss by provdng condtons that rsult n managabl ntrconnctd systm modls basd upon th mor gnral polytopc modl. Suppos that th systm undr consdraton conssts of q componnts whos truth modls hav th form & F ( a ) for k K q k k k k b h ( a ) for k K q k k k whr th subscrpt k dsgnats th k-th componnt. Addtonally k a k and k b k ar th stat nput and output of th k-th componnt rspctvly. Th componnt nput a k and output (5.) (5.) b k may b composd of a subst of th systm lvl and componnt lvl nputs and outputs rspctvly. Aggrgatng th nputs and outputs of th componnt modls togthr th nput and output vctors a a M a q (5.3)

96 and b b M (5.4) b q ar constructd allowng for th lnar ntrconncton schm found n [DE8] [BLO83] and partally utlzd n [HSI] to b usd. Ths structur s dfnd by a b L (5.5) y u whr L s th constant ntrconncton matr dfnd by L L L. (5.6) L L To ntroduc th ntrmdat modls and th procss of ntrconnctng componnt modls a systm comprsd of componnts rprsntd by a sngl local modl s frst consdrd. Thn two cass usng polytopc modls ar prsntd ach allowng smplfcatons to th procss of constructng th systm polytopc modl. Thn last a tchnqu for convrtng an ntrconnctd polytopc modl to th form dfnd n haptr 4 s prsntd. 5. Intrconncton wth a Sngl Local Modl In ths scton an ntrconnctd systm modl s formd assumng that ach componnt s rprsntd by a sngl local modl. Ths allows th ntroducton of th noton of ntrmdat modls as wll as a rlatvly smpl form of th ntrconnctd systm modl. Frst ach systm componnt s appromatd by a local modl to obtan th k-th componnt modl dfnd as & A + B a + φ k k k k k k k k k k k b + D a + φ. Th compost componnt () modl s thn constructd by aggrgatng th q componnt modls nto a block dagonal form. Ths procss ylds bk (5.7) (5.8)

97 cc cc cc φ a B A + + & cc cc cc b φ a D b + + whr th subscrpt cc dnots a modl matr. Th modl matrcs and vctors ar dfnd as q cc A A A O q cc B B B O q cc φ φ φ M q cc O q cc D D D O and q cc b b b φ φ φ M. Th stat for th modl s dfnd as q M and th nputs a and outputs b ar gvn by (5.3) and (5.4). (5.9) (5.) (5.) (5.) (5.3) (5.4) (5.5) (5.6) (5.7)

98 Th compost systm (S) modl s obtand by ntrconnctng th modl as llustratd n Fgur 5.. To dtrmn th stat quaton th nput to th modl s dtrmnd frst by substtutng quaton (5.) nto (5.5) and smplfyng to obtan [ I L D ] [ L + L u L φ ] a cc cc + bcc. Equaton (5.8) s thn substtutd nto (5.9) rsultng n th stat quaton whr matrcs and & Acs + Bcsu + φcs A cs B cs and vctor φ cs ar dfnd as cc[ I LDcc] L cc [ I L D ] Acs Acc + B cs B cs Bcc cc L cc [ I L Dcc] Lφbcc φcc φ B +. (5.8) (5.9) (5.) (5.) (5.) Fgur 5.. omponnt systm ntrconnctons. Th S modl output quaton can b found n a smlar fashon by substtutng (5.) nto (5.5) and solvng for y to obtan [ + D a + φ ] L u y L cc bcc + cc. Substtutng (5.8) nto (5.3) rsults n th output quaton y cs + Dcsu + φycs (5.3) (5.4) whr th matrcs cs D cs and vctor φ ycs ar dfnd as cs L[ cc + Dcc[ I LDcc ] Lcc ] (5.5)

99 and cs L Dcc[ I LDcc ] L L D + φ ycs L[ Dcc[ I LDcc ] Lφbcc + φbcc ]. (5.6) (5.7) As can b sn th procss for ntrconnctng componnt modls nto a S modl s rlatvly straghtforward whn componnt modls consst of a sngl modl. In th followng sctons ths ntrconncton stratgy s tndd to polytopc modls. 5. Matchd Local Modls In ths scton a spcal cas of polytopc modls s consdrd whch rsults n a partcularly smpl systm modl. Th polytopc modls satsfy th condtons:. All of th systm componnts must hav th sam numbr of local modls that s r r L rq r whr r k s th numbr of local modls for th k-th systm componnt. (5.8). Th -th local modl of all q systm componnts uss th sam wghtng functon thrfor satsfyng th condton w ( θ) w ( θ) L wq ( θq) w( θ) (5.9) whr w k ( θ k ) s th wghtng functon for th -th local modl of th k-th systm componnt. Th varabl θ k may b a stat nput output or paramtr such that θ L θq θ. Undr ths condtons th -th local modl from all of th systm componnts can b ntrconnctd frst to form th -th local S modl usng th procdur st forth n Scton 5.. Thn th local S modls may b combnd to form th S polytopc modl.

100 Spcal cas: On nonlnar systm componnt Ths class of systms conssts of q componnts that can b appromatd wth affn modls (5.7) and (5.8) and on nonlnar componnt appromatd by th polytopc modl & q r w q ( θ) [ A + B a + φ ] q q q [ + D a + φ ] r b q wq ( θ) q q bq Formng th -th local modl for ths class of systms rsults n th quatons & A + B a + φ cc cc cc (5.3) (5.3) (5.3) b + D a + φ cc cc bcc (5.33) whr th matr A cc and th vctor φ cc ar dfnd as and Th rmanng matrcs A cc A φ B cc cc cc O A φ M φq φq q. A q D cc and vctor φ bcc ar dfnd smlarly. Th constructon of th local S modl s thn dntcal to th constructon prsntd n Scton 5. rsultng n & Acs + Bcs a + φcs (5.34) (5.35) (5.36) y cs + Dcs a + φycs (5.37) whr th matrcs A cs cs B cs D cs and vctors φ cs and φ ycs ar dfnd by

101 and A cs Acc + Bcc [ I LDcc ] Lcc Bcs Bcc [ I LDcc ] L φ cs Bcc [ I LDcc ] Lφbcc + φcc cs [ cc cc cc cc L + D [ I L D ] L ] D cs LDcc [ I LDcc ] L + L φ ycs L[ Dcc [ I LDcc ] Lφbcc + φbcc ].. (5.38) (5.39) (5.4) (5.4) (5.4) (5.43) Th S modl s compltd by combnng th local S modls usng wghtng functons to obtan r & w( θ) [ Acs + Bcs a + φcs ] (5.44) [ + D a + φ ] r y w( θ) cs cs ycs. Th formaton of a systm polytopc modl for ths class of systms s llustratd n th followng cas study. 5.3 as Study : Matchd Local Modls onsdr systm 5 dscrbd n haptr. Th nputs outputs and stats ar dfnd as and u [ P ] V s [ L v ] y [ v v ] L L (5.45) (5.46) (5.47) (5.48) rspctvly. Ths systm conssts of two componnts. omponnt on s rprsntd by th truth modl

102 - 9 - whr th nputs and stats ar dfnd as Vs LR v L & L L [ L v ] b [ V s ] [ L v ] a L. Placng quatons (5.49) and (5.5) n stat-spac form rsults n a lnar componnt modl wrttn as & A + Ba + φ b + D a + φ b whr matrcs A B D and vctors φ and φ b ar dfnd as (5.49) (5.5) (5.5) (5.5) (5.53) (5.54) and R L L A L B D [ ] φ [ ] φ b. omponnt s rprsntd by th truth modl (5.55) (5.56) (5.57) (5.58) (5.59) (5.6)

103 - 9 - whr Th nputs and stats ar & v LR v L cp( u L ) L b v P cp( u). v (5.6) (5.6) (5.63) v a (5.64) P and L. (5.65) v Usng Taylor srs appromaton th -th local modl s constructd from quatons (5.6) and (5.6) to obtan & A + B a + φ (5.66) b + D a + φ b (5.67) whr th matrcs A B D and vctors φ and φ b ar dfnd as R L L A P v L B v (5.68) (5.69)

104 - 9 - and φ P v D b [ ] φ. To form th -th local modl th modls dfnd by quatons (5.53)-(5.54) and (5.66)-(5.67) ar combnd rsultng n and & Acc + Bcc a + φcc b cc + Dcc a + φbcc. (5.7) (5.7) (5.7) (5.73) (5.74) (5.75) whr th matrcs as and A cc cc B cc D cc and vctors φ cc and φ bcc ar dfnd A A cc A B B cc B φ φ cc φ cc D D cc D (5.76) (5.77) (5.78) (5.79) (5.8)

105 φb φ bcc. (5.8) φb Th nt stp s to dtrmn th nputs of th -th local modl n trms of th systm stats and nputs. onstructng quaton (5.5) basd on (5.46) (5.47) (5.5) (5.5) (5.6) and (5.64) rsults n th ntrconncton matr L dfnd by L (5.8) L (5.83) L (5.84) and L. (5.85) Substtutng quaton (5.75) nto quaton (5.5) and solvng for a rsults n whch smplfs to bcaus a L + L D a + L u + L φ cc cc bcc a Lcc + Lu + Lφbcc D cc s a zro matr. Th modl nput quaton (5.87) s thn substtutd nto quaton (5.74) to dtrmn th local S stat quaton & A + B a + φ. cs cs cs Th local S output quaton s dtrmnd smlarly by substtutng (5.87) and (5.75) nto (5.5) to dtrmn y cs + Dcs a + φycs. (5.86) (5.87) (5.88) (5.89)

106 Th matrcs b A cs cs B cs D cs and vctors φ cs and φ bcs ar dtrmnd to R L L A cs R L L L P v (5.9) and L B cs v φ cs P v cs D cs (5.9) (5.9) (5.93) (5.94) φ ycs. (5.95) Th local systm modl dfnd by (5.9)-(5.95) s th sam local systm modl that would hav bn obtand f th polytopc modlng constructon tchnqus prsntd n haptr 4 had bn usd.

107 ombnng th local S modls usng wghtng functons th polytopc S modl s dtrmnd to b and r [ A + B a + φ ] & w ( θ) cs cs cs [ + D a + φ ] r y w( θ) cs cs ycs. (5.96) (5.97) 5.4 onstant omponnt Intrconnctons Ths class of systms allows mor flblty n slctng modlng ponts at th pns of furthr rstrctons on th componnt modl ntrconnctons and a mor compl form for th S modl wghtng functon. Th constrants for ths class of systms ar:. All of th systm componnts must hav th sam numbr of local modls r r L rq r whr r k s th numbr of local modls for th k-th systm componnt.. Th ntrconnctons of th systm componnts satsfy th followng qualts L cc L L cc q constant (5.98) (5.99) L Dcc L LD cc q constant (5.) and r L w w q ( θ ) φb φb M L M constant. q φbq φbq ( θ ) Th wghtng functon for th -th local modl of th k-th systm componnt w k ( θ k ) s a functon of th varabl θ k whch may b a stat nput output or paramtr assocatd wth th k-th systm componnt. (5.)

108 Usng polytopc modlng tchnqus ach systm componnt s appromatd by r local modls. Th k-th componnt modl s dfnd by & k r w k ( θk ) [ A + B a + φ ] k k k k k r b k wk ( θk )[ k k + Dk ak + φbk ] whr th subscrpt ndcats th local modl numbr. Th wghtng functons ar rstrctd to satsfyng and w ( ) r k θ k wk ( θ k ) whr th vctor θ k may b a functon of k a k b k or of componnt modl paramtr. Th componnt modls ar now aggrgatd nto a block dagonal form to construct th modl dfnd by & r W [ A + B a + φ ] ( θ K θq) cc cc cc r b Wb ( θ K θq) [ cc + Dcc a + φbcc ]. (5.) (5.3) (5.4) (5.5) (5.6) (5.7) Th rqurmnt that all of th componnts hav r local modls allows th wghtng functons to b formulatd nto wghtng functon matrcs dfnd as W ( θ K θ q I ) w ( θ ) O I q w q ( θ q ) (5.8) and

109 whr and W b ( θ I b w ( θ) K θq) O Iqb wq ( θq) r W ( θ K θq ) I n n (5.9) (5.) r W b ( θ K θq ) I n b n. b (5.) Th dntty matr I k has dmnson n k by n k and th matr I k b has dmnson n kb by n kb whr n k and n kb ar th numbr of stats and outputs n th k-th componnt modl rspctvly. Lkws n and n b ar th numbr of stats and outputs for th modl rspctvly. Th modl matr ar dfnd as A cc and vctor φ cc and A Acc O A n (5.) Th matrcs B cc cc cc φ φcc M. φn D and vctor φ bcc ar dfnd smlarly. Intrconncton of th modl n quatons (5.6) and (5.7) rsults n th S modl. To dtrmn th coffcnts for th S modl stat quaton th modl nput a s frst dtrmnd. Substtutng (5.7) nto (5.5) rsults n th quaton r [ + D a + φ ] L u a L Wb ( θ K θq) cc cc bcc + whch may b rwrttn as (5.3) (5.4)

110 a r L L L D W cc cc b bcaus th block dagonal structur of W W b b ( θ K θ ( θ K θ ( θ K θ q ) φ q q bcc cc and D cc ) + ) a + + L u allow thm to commut wth th wghtng functon matr. Bcaus th vctors and a ar ndpndnt of and bcaus of condtons (5.99) through (5.) quaton (5.5) may b smplfd to a L + LDcc a + Lφbcc + Lu cc (5.5) (5.6) whr can b any valu. Rarrangng (5.6) and solvng for a rsults n [ I L D ] [ L + L u L φ ] a cc cc + bcc. (5.7) It s assumd that [ I LD cc ] s nvrtbl and hnc a s wll dfnd. Th S stat modl s now dtrmnd by nsrtng (5.7) nto (5.6) rsultng n r & W ( θ K θq) [ Acs + Bcs u + φcs ] (5.8) whr th matrcs A cs B cs and vctor φ cs may b prssd as [ I LDcc ] L cc Acs Acc + Bcc [ I LDcc ] Bcs Bcc L (5.9) (5.) [ I LDcc ] Lφbcc φ cc φ cs Bcc + Smlarly th S output quaton s dtrmnd by substtutng (5.7) nto (5.7) and thn th rsult nto (5.5) obtanng. (5.) [ D [ I LD ] L ] cc + cc cc cc + r y Wy ( θ θ ) [ D [ I LD ] L ] u K q cc cc + + Lu D [ I LD ] Lφb + φ cc cc cc bcc whch may b wrttn as (5.)

111 [ cc + Dcc [ I LDcc ] Lcc ] + r + ( y W ) + y ( θ K θq ) [ Dcc [ I LDcc ] L L LL L] u D + cc [ I LDcc ] Lφbcc φbcc provdd that L L s nvrtbl. Equaton (5.3) may b wrttn mor compactly as whr W y ( θ K θ ) s dfnd by q [ + D u + φ ] r y Wy ( θ K θq) cs cs ycs (5.3) (5.4) Wy ( θ K θq) LWb ( θ K θq) (5.5) and th matrcs cs cs D and φ ycs ar dfnd as and [ I LDcc ] L cc cs cc + Dcc [ I LDcc ] L + L ( LL ) Dcs Dcc L [ I LDcc ] Lφbcc φ cc φ D + ycs cc b. (5.6) (5.7) (5.8) Th followng ampl llustrats th us of th abov formulaton. 5.5 as Study: onstant omponnt Intrconnctons To llustrat th constant componnt ntrconncton class of systms consdr systm 6 dscrbd n haptr whch conssts of a currnt lmtd voltag sourc connctd to a constant powr load. Th nputs outputs and stats of th systm ar u [ V s whr V s s th commandd nput voltag and V P y cp ( u) [ v v ] L L ] P s th commandd nput powr. omponnt s nonlnar and has two mods of opraton. If L > or ( L and ( ) s v ) thn th truth modl can b prssd as (5.9) (5.3) (5.3)

112 - - othrws th truth modl s ( a ) F Vs LR v L L L G ( a ) L. Th nputs stats and outputs for ths componnt ar dfnd as and a [ V s L] [ L v ] b v rspctvly. Th polytopc modl for componnt s wrttn as & r w ( θ) [ A + B a + φ ] r b w ( θ ) [ + D a + φb ] (5.38) whr th wghtng functons ar rstrctd accordng to (5.4) and (5.5). Th matrcs A B φ D and φ b ar dfnd dpndng upon th mod of opraton. Whn th opratng condton of th componnt s such that F ) n ( a (5.49) charactrzs th componnt bhavor th matrcs A B and φ for th local modl stat quaton ar dfnd as: R L L A (5.3) (5.33) (5.34) (5.35) (5.36) (5.37) (5.39)

113 - - and L B [ ] φ. (5.4) (5.4) Othrws whn th opratng condton of th componnt s such that G ) n ( a (5.33) charactrzs th componnt bhavor th matrcs A B and φ for th local modl stat quaton ar dfnd as: and A [ ] B [ ] φ. Th componnt modl output quaton matrcs ar dfnd th sam for both mods of opraton as and to b w whr ( [ ] [ ] D b φ. Th wghtng functon usd to combn th componnt local modls s chosn L v ) and L + + m ( ) L L b L L m L ( + ) L L b L + + m ( v ) v b v v mv ( v + ) v bv v dfn th modlng pont of th -th local modl. Evry possbl combnaton of th nductor currnt L (5.4) (5.43) (5.44) (5.45) (5.46) (5.47) (5.48) dvdd nto qually spacd ponts ovr th

114 - - and th capactor voltag v dvdd nto qually spacd rang L [ A 4A] ponts ovr th rang [ 55V] v 4V s usd to gnrat th local modls. Th othr constant trms n (5.48) ar dfnd as 3 m L (5.49) and L 3 mv v b L L (5.5) (5.5) v b v (5.5) whr L s th dffrnc n currnt btwn th modlng pont valus and v s dfnd smlarly for th modlng pont voltag. Th truth modl for componnt has th form v LR v L & L cp( u) b [ L cp ( u)] (5.53) (5.54) whr th constant powr load currnt nputs and stats ar dfnd as and rspctvly. Th rsultng matrcs P cp( u) v v P a [ ] [ L v ] A B φ D and φ b ar: (5.55) (5.56) (5.57)

115 - 3 - v P L L R A v L B v P φ v P v D and v P b φ. Th wghtng functon usd to combn th componnt local modls s chosn to b ) ( ) ( ) ( ) ( ) ( w p p p p v v v v b P P m b P P m b v v m b v v m P v whr v and P dfn th modlng pont of th -th local modl. Evry possbl combnaton of th capactor voltag dvdd nto 4 qually spacd ponts ovr th rang [ ] 55V 4V v and th commandd powr dvdd nto 5 qually spacd ponts (5.58) (5.59) (5.6) (5.6) (5.6) (5.63) (5.64)

116 - 4 - ovr th rang [ 95W 5W] P s usd to gnrat th local modls. Th othr constant trms n (5.64) ar dfnd as 3 mv v and 3 m P P v b v P b P whr v s th dffrnc n voltag btwn th modlng pont valus and (5.65) (5.66) (5.67) (5.68) P s dfnd smlarly for th modlng pont powr Aggrgatng th componnt modls th modl s constructd as & r W [ A + B a + φ ] ( θ θ) cc cc cc r Wb ( θ θ) cc cc bcc whr th wghtng functon matrcs ar dfnd as and [ + D a + φ ] b w ( θ) W ( ) θ θ w ( θ) w ( θ) w ( θ) (5.69) (5.7) (5.7) Th matr w ( θ) W b ( θ θ) w ( θ). w ( θ) A cc and vctor φ cc ar dfnd as (5.7)

117 - 5 - and Th matrcs A A cc A φ φ cc. φ B cc cc D cc and vctor φ bcc ar dfnd smlarly. Aftr constructng th modl th nt stp s to dtrmn th nputs to th modl n trms of th systm stats and nputs. onstructng quaton (5.5) basd on quatons (5.9) (5.3) (5.34) (5.36) (5.54) and (5.56) rsults n th ntrconncton matr L dfnd by and L L [ ] L [ ] L. (5.73) (5.74) (5.75) (5.76) (5.77) (5.78) Th nput to th modl s dtrmnd from quaton (5.7) f condton s satsfd. To vrfy that condton s satsfd for ths systm th quatons (5.99) (5.) and (5.) ar vrfd blow: L cc P v (5.79)

118 - 6 - D L cc v φ L b b v P φ. Intrconnctng th modl rsults n th S modl [ ] + + r cs cs cs ) ( φ u B A θ θ W & [ ] + + r cs cs cs ) ( y y φ u D θ θ W y whr th matrcs cs A cs B cs φ cs cs D and ycs φ ar dtrmnd to b th sam as was found n cas study whn th currnt L s nonzro or t s drvatv s postv. Othrws th matrcs ar dtrmnd to b cs v P L L R L A cs v B (5.8) (5.8) (5.8) (5.83) (5.84) (5.85)

119 - 7 - and φ cs P v cs P v D cs v (5.86) (5.87) (5.88) φ ycs. P v Fnally th S output quaton wghtng functon matr s dtrmnd to b W y θ ( θ θ ) [ w ( )]. (5.89) (5.9) To valdat ths modl aganst th truth modl a tm doman smulaton s prformd. Th nputs and th ntal condtons ar chosn as Vs 5V (5.9) P kw and

120 - 8 - v v L L 5A 5V A 53V Fgur 5. contans plots of th currnt trajctors for th truth and polytopc modls whch ar shown to match vry wll. Not that th rsultng S modl s constructd of local modls. If ths systm and bn constructd usng th tchnqus dscussd n haptr 4 th sam modlng ponts would hav rsultd n 4 local modls.. (5.9) 5.6 Gnral as Fgur 5.. Systm currnt trajctors. In gnral f th condtons st forth n Sctons 5. and 5.4 ar not mt and th S modl s constructd usng th approachs usd n thos sctons th rsultng S modl wll b an ntractabl quaton of nstd summatons. Howvr ths may b avodd by propr rformulaton of th modl as s dmonstratd blow. Usng polytopc modlng tchnqus ach systm componnt s appromatd by r k local modls. Th k-th componnt modl s dfnd by r & k k wk ( θ k k ) k [ A + B a + φ ] k k k k k k k k (5.93)

121 - 9 - [ ] + + k k k k k k r k k k k k k k k ) ( w b φ a D θ b whr th nd k ndcats th local modl numbr. Th wghtng functons ar rstrctd to satsfyng ) ( w k k k θ and k k k r k k ) ( w θ whr th vctor k θ may b a functon of k k a k b or of componnt modl paramtr. Th componnt modls ar aggrgatd to construct a modl of th form [ ] [ ] q q q q q r q q q q q q q q r ) ( w ) ( w φ a B A θ φ a B A θ M & [ ] [ ] q q q q q q r q q q q q q q r ) ( w ) ( w b b φ a D θ φ a D θ b M. Equatons (5.97) and (5.98) may b rwrttn as [ ] [ ] q q q q q q q q q q q r q q r ) ( w ) ( w φ a B A φ a B A θ θ M L & (5.94) (5.95) (5.96) (5.97) (5.98) (5.99)

122 r r q b w ( θ) L w q q q - - ( θq ) [ + D a + φ ] [ + D a + φ ] bcaus th k-th rows ar ndpndnt of all nds cpt for k and th constrants gvn n quatons (5.95) and (5.96). Th wghtng functons may b groupd for smlar rasons thrby smplfyng th modl to whr r r q & L w K b q q ( θ) q q q M q [ A + B a + φ ] q q b bq [ A + B a + φ ] q q q M q q q q [ + D a + φ ] r r q b L w K ( θ) q M q [ + + ] q q q Dq a q q φbq q w K ( θ) w ( θ) Lw q ( θ q q q ) q q (5.) (5.) (5.) (5.3) and θ θ K θ q. (5.4) Groupng lk matrcs th modl s placd n th form [ Acc K + Bcc K a φcc K ] r r q & L w K ( θ) + q q q q q [ + D a φ ] r r q b L w K ( θ) q cc K q cc K + q bcc K q q (5.5) (5.6) whch s smlar n form to th modl gvn by quatons (5.3) and (5.33) wth th cptons that th wghtng functons ar ncludd hr and th modl matrcs ar dpndnt upon q nds. Th stat nputs a and outputs b ar gvn by (5.7) (5.3) and (5.4). Th modl matrcs and vctors ar dfnd as

123 - - and A B D cc cc A K q O A q q B K q O B q q φ cc cc cc φ K q M φ q q K q O q q D K q O D q q (5.7) (5.8) (5.9) (5.) (5.) φ bcc φ b K q M. φ bq q Th nstd summatons ar nt lmnatd by convrtng th multpl nds to a sngl nd rsultng n a modl of th form r [ A + B a φ ] & w ( θ) + cc cc cc r b w ( θ) cc cc + bcc whr r r r L [ + D a φ ] r q (5.) (5.3) (5.4) (5.5)

124 - - and th nd convrson s + ( ) r r3 L rq + ( ) r3 r4 Lrq + L + ( q ) rq q. (5.6) Th nd convrson may b wrttn n th oppost drcton as K q q floor + r r3 r L q ( ) r r3 Lrq floor r3 r4 Lrq K + (5.7) ( ) r r3 Lrq L ( q ) rq rq floor rq ( ) r r3 Lrq L ( q ) rq + whr th functon floor () rounds down to th narst ntgr. Th modl gvn by quatons (5.3) and (5.4) satsfs th condtons for th cas of matchd local modls. Thrfor smlar to th cas for th matchd local modls th -th local modl may b ntrconnctd to form th -th local S modl. Th r local S modls ar thn combnd to form th S polytopc modl of th form r & w ( θ) [ A cs + Bcs u + φcs ] (5.8) r y w ( θ) [ cs + Dcs u + φbcs ] (5.9) whr th S modl matrcs ar dfnd by quatons (5.38) through (5.43). Ths s llustratd furthr n th followng cas study. 5.7 as Study: Gnral as Agan consdr systm 6 usd n Scton 5.5. Howvr ths tm th modlng ponts for componnt ar chosn to b vry possbl combnaton of th nductor currnt dvdd nto 4 qually spacd ponts ovr th rang [ 4A] voltag L L A and th capactor v dvdd nto 4 qually spacd ponts ovr th rang [ 55V] v 4V.

125 - 3 - For componnt th modlng ponts ar chosn to b vry possbl combnaton of th capactor voltag dvdd nto 4 qually spacd ponts ovr th rang [ 55V] v 4V and th commandd powr dvdd nto 5 qually spacd ponts ovr th rang [ 95W 5W] P. Th S modl s put nto th form gvn by (5.8) and (5.9). Th total numbr of local S modls s r 3 and th nd convrson s floor + ( ). (5.) To valdat ths modl aganst th truth modl a tm doman smulaton s prformd usng th two modls. Th nputs and th ntal condtons ar chosn as th valus gvn n (5.9) and (5.9). Fgur 5. contans plots of th currnt trajctors for both modls whch ar shown to match vry wll. Fgur 5.3. Systm currnt trajctors. Th goal of ths chaptr s to ntroduc dcntralzd polytopc modlng. Th rsultng polytopc modls ar dmonstratd to accuratly rprsnt th truth modls from whch thy wr drvd howvr t s mphaszd that th prmary focus for polytopc modls n ths thss s not as a smulaton tool but as a tool for dtrmnng Lyapunov functon canddats. Th followng chaptr addrsss ths ssu by amnng stablty of polytopc modls.

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127 STABILITY ANALYSIS OF POLYTOPI MODELS Ths chaptr addrsss th stablty analyss of concdnt polytopc modls. For a brf rvw of stablty dfntons th radr s rfrrd to haptr. 6. oncdnt Polytopc Modl For th spcal class of concdnt polytopc modls t may b possbl to dtrmn a rgon of asymptotc stablty basd on analyss of th local modls. Bfor amnng th stablty of concdnt polytopc modls t s hlpful to amn th smlarts btwn concdnt polytopc modls and lnar modls of th form & A + Bu. (6.) Thr ar two charactrstcs of th lnar systm that ar of partcular ntrst hr. Th frst s global asymptotc stablty (GAS) of a partcular qulbrum stat. That s f an qulbrum stat of th lnar modl s asymptotcally stabl thn t s GAS. Th scond proprty of ntrst s th ablty to ascrtan th stablty of all qulbrum pars basd on th stablty of th qulbrum par ( u ). Ths mpls that f th qulbrum stat corrspondng to u s stabl thn so s any othr qulbrum stat corrspondng to any constant u such that A Bu. + (6.) To justfy th abov statmnt assum that th matr A s asymptotcally stabl and lt ~ and u ~ u u. Substtuton of ths dfntons nto (6.) ylds ~& A ~ + Bu~ + A + Bu (6.3) whch may usng (6.) b rducd to ~ & A ~ + Bu~. (6.4)

128 - 6 - In th nw coordnats th qulbrum stat ~ corrspondng to u ~ s asymptotcally stabl by assumpton. Howvr th qulbrum stat ~ corrsponds to th qulbrum stat. In othr words corrspondng to u s asymptotcally stabl f and only f corrspondng to som u s asymptotcally stabl. Ths proprty dos not hold n gnral for polytopc systms whch ar nonlnar n natur and thrfor may hav multpl qulbrum stats corrspondng to a gvn constant nput as has bn shown n haptr 4. Howvr th followng scton dntfs condtons for whch ths condtons hold for concdnt polytopc modls 6.. Global asymptotc stablty Undr som condtons t s possbl to dtrmn GAS of th qulbrum stat. To dtrmn ths condtons suppos that thr sts a matr P such that whch satsfs th lnar matr qualts for som r P P > A P + PA M A P + PA r Q Q Q Q > K r. onsdr an qulbrum par ) of th polytopc modl gvn by quaton (4.3) ( u and coordnat translatons of th form r ~ u~ u u whch rsult n ~ + (6.5) (6.6) (6.7) (6.8) and u u ~ +. u (6.9) Substtutng (6.8) and (6.9) nto (4.3) and rarrangng ylds

129 - 7 - whr ~ & + r r w ( ~ + w ( ~ + u + u u + u [ A ~ + B u~ ] ) )[ A + B u + ] A B u + φ for all + bcaus all local modls hav concdnt qulbrum pars. Hnc th polytopc modl n th nw coordnats taks th form r ~ & w( ~ + u~ + u ~ ) [ A + B ~ u ]. φ Not that th abov modl has an qulbrum par ( ~ u~ ) and obsrv that and u ~ u u Now to nvstgat th stablty proprts of th qulbrum stat ~ consdr ~. th followng Lyapunov functon canddat V ~ P ~ and valuat t s tm drvatv V & on th trajctors of (6.) (6.) (6.) (6.3) (6.4) (6.5) ~ r & w ( ~ + u~ + u ) A ~. (6.6) Substtuton of (6.6) nto (6.5) ylds r V & w ( ~ u~ u ) ~ [ A P PA ] ~ (6.7) From (6.6) coupld wth th constrant that th wghtng functons ar nonngatv for all valus of thr argumnts and that th sum of th wghtng functons valuats to on w hav V & <. Thus th followng proposton has bn provn.

130 - 8 - Proposton 6.: If thr sts a common P P > such that A P + < for... r PA thn any qulbrum stat of th polytopc modl obtand by solvng s GAS n th sns of Lyapunov. A B u + φ for... r + (6.8) (6.9) Rmark 6.: Proposton 6. parallls th smlar rsult for lnar tm nvarant systms of th form & A + Bu + φ. Th abov proposton can b rstatd n an quvalnt fashon as follows. If an qulbrum stat of th polytopc modl obtand by solvng A B u + φ for... r s not globally asymptotcally stabl n th sns of + Lyapunov thn thr s no common P P > such that A P + < for... r. PA Eampl 6.: onsdr systm 3 whr local modls ar constructd frst usng Taylor srs appromaton and thn transformd usng th GTZ transformaton and th truth modl qulbrum [.4395A V] corrspondng to u [ 5V kw]. Th local modls ar constructd at all combnatons of th modlng ponts dfnd by L [.833A 5.A] dvdd nto 4 qually spacd ponts and v [38.53V 7.V] dvdd nto 3 qually spacd ponts. Usng th MATLAB LMI toolbo [MAT] a common matr P (6.) s found satsfyng all 768 of th LMI s gvn by quaton (6.8). Thrfor th polytopc modl constructd from ths local modls s GAS. 6.. Asymptotc stablty If at last on local modl has a pol n th closd rght hand sd of th compl plan thn a common P satsfyng (6.8) cannot b found. In fact th asymptotc stablty of th local modls s only a ncssary condton for th stnc of a common

131 - 9 - P. In othr words asymptotc stablty of th local modl A s s only ncssary and not suffcnt for th qulbrum stat at th orgn of th polytopc modl to b asymptotcally stabl for any choc of w as th followng ampl shows. ' Eampl 6.: onsdr th polytopc modl and lt and A & w A 4 A. (6.3) 4 Both Aand A ar asymptotcally stabl. Suppos that w w. 5 thn th polytopc modl taks th form & w A (6.4) rsultng n a lnar systm whos gnvalus ar locatd at and.566 whch rndrs th systm unstabl. Obsrv that th matrcs A and A do not hav a common P that solvs (6.8). Ths follows from th thorm that can b found for ampl n [ŻAK3] on pag 47. (6.) (6.) Whn a common P dos not st satsfyng all r local modls t may stll b possbl to dtrmn a rgon of asymptotc stablty (RAS) for th polytopc modl. Bfor contnung to dfn th RAS th followng dfntons ar st forth. Dfnton 6.: Th α st of local modls s dfnd as th st of all local modls n a polytopc modl havng modlng ponts contand by th closd rgon n Γ R whr:

132 - -. Th qulbrum stat s contand by Γ that s Γ.. All r α local modls constructd at modlng ponts nsd Γ satsfy whr rα r. A P + < for... rα and P P > PA (6.5) Dfnton 6.: Th β st of local modls s dfnd as th st of all local modls that ar not n th α st of local modls. Idntfcaton of a rgon Γ s not straghtforward. Algorthm 6. gvs a mthod for dtrmnng Γ but t s rqurd that th modlng ponts for th polytopc modl form a unformly spacd grd as dpctd n Fgur 6.. Algorthm 6.: Appromatng th rgon Γ and dtrmnng an α st of local modls.. Form a unformly spacd grd of modlng ponts throughout a rgon contanng.. Fnd th modlng pont that mnmzs qulbrum stat of th polytopc modl. whr s th 3. Dfn th smallst rgon Γ surroundng th modlng pont that has cornrs dfnd by modlng ponts as llustratd n Fgur 6.. Th rgon Γ s n th shap of a bo. In hghr dmnsons Γ bcoms a hyprbo wth n facs n n R. For a dfnton of a hyprbo s Appnd A. Th rgon Γ s closd wth facs paralll to th grd lns of th modlng pont msh. 4. Sarch for a common matr P satsfyng (6.5) for all local modls that hav modlng ponts n Γ. If a common P cannot b found th local modls hav bn dvlopd too far from th qulbrum and must b r-drvd n closr promty. Othrws go to stp 5.

133 Pck a fac of Γ and mov t on grd spacng away from thrby ncrasng th sz of Γ. 6. Sarch for a common matr P satsfyng (6.5) for all local modls that hav modlng ponts n Γ. 7. If a matr P s not found mov th fac of Γ back on grd spacng towards. 8. Pck a nw fac of Γ and go to stp 5 rpttvly pandng Γ as dpctd n Fgur 6. untl th largst Γ s dtrmnd. Th st of local modls havng modlng ponts n Γ dfn th α st of local modls. Whl Algorthm 6. dos not ncssarly mamz th numbr of local modls n th α st t dos provd a straghtforward approach for dtrmnng an α st. For furthr llustraton of ths tchnqu s Eampl 6.3. Fgur 6.. Intal rgon Γ surroundng. Eampl 6.3: onsdr systm 3 whr local modls ar constructd frst usng Taylor srs appromaton and thn transformd usng th GTZ transformaton and th truth modl qulbrum [.4395A V] corrspondng to u [ 5V kw]. Local modls ar constructd at all combnatons of th modlng

134 - - ponts dfnd by L [A 5.A] dvdd nto 5 qually spacd ponts and v [3.V 7.V] dvdd nto 4 qually spacd ponts. Fgur 6.. Epanson of th rgon Γ. In Fgur 6.3 th locaton of th modlng ponts for all of th local modls ar plottd along wth th qulbrum stat markd by th crcl. Modlng ponts corrspondng to local modls wth an gnvalu n th closd rght half sd of th compl plan ar markd by an and all othr modlng ponts ar markd by a dot. Th rgon Γ dtrmnd usng Algorthm 6. s markd by th shadd rgon and th common matr P found for th α st of local modls s P. (6.6) Although th α st dtrmnd by Algorthm 6. s not optmal n th sns that a largr st of local modls could probably b found f th local modls wr addd and subtractd from th st on a modl by modl bass th shap of th rgon Γ dtrmnd n Algorthm 6. has attractv faturs n dtrmnng a rgon of stablty. Ths s mad apparnt n th followng drvatons but bfor plorng ths da furthr som dfntons must b st forth.

135 - 3 - Fgur 6.3. Rgon Γ dfnng th α st of local modls. Dfnton 6.3: Th support of a functon s th rgon of th stat-spac n whch th functon s nonzro that s Supp f { R : f ( ) } n. Dfnton 6.4: Th rgon of nflunc wghtng functon for that local modl. R OI for a sngl local modl s th support of th Dfnton 6.5: Th rgon of nflunc support of all wghtng functons of th local modls n th st. R OI for a st of local modls s th unon of th Dfnton 6.6: Th krnl of a functon s th rgon of th stat-spac n whch th functon s zro that s kr f { R : f ( ) } n. Dfnton 6.7: Th rgon of non-nflunc R ONI for a sngl local modl s th krnl of th wghtng functon for that local modl. Ths s quvalntly wrttn as R R R. ONI n OI

136 - 4 - Dfnton 6.8: Th rgon of non-nflunc R ONI for a st of local modls s th ntrscton of th krnls of all wghtng functons of th local modls n th st. Ths s quvalntly wrttn as R R R. ONI n OI Now consdr a polytopc modl and a Lyapunov functon canddat V dfnd by (6.5). Th tm drvatv of V along th trajctors of th systm s V & ~ P~ & whch may b rwrttn substtutng (4.3) nto (6.5) to obtan and rarrangd as r V& ~ P w ( ~ + u~ + u ) + V& ~ P + ~ P rα r w ( ~ + w j j rα + ( ~ + u~ + u u~ + u [ A ~ + B u φ ] [ A ~ + B u + φ ] ) )[ A ~ + B u + φ ] whr th α and β sts of local modls hav bn groupd nto sparat summatons. By dfnton th wghtng functons for all of th local modls n th β st valuat to j j j (6.7) (6.8) zro n th rgon R ONI for th β st whch s dnotd as R ONI. Lkws by β dfnton th wghtng functon of at last on local modl n th α st s nonzro vrywhr n th rgon R OI for th α st whch s dnotd as R OI. Thrfor V & α s ngatv dfnt vrywhr n th rgon Ψ ROI R α ONI. Th largst lvl β st of th Lyapunov functon canddat V that fts n th rgon Ψ thrfor bounds a rgon of asymptotc stablty. Thus th largst lvl st of th Lyapunov functon canddat (6.5) that fts n th rgon dfnd by Ψ ROI R α ONI β (6.9) s a RAS for th polytopc modl.

137 Dcntralzd oncdnt Polytopc Modls Stablty analyss of ntrconnctd polytopc systms s th sam as th analyss for a sngl polytopc modl f th compost systm modl was gnratd usng matchd local modls or placd n th gnral form dfnd n Scton 5.6. Th frst part of ths chaptr addrsss ths cas. Thrfor stablty analyss of polytopc modls wth constant componnt ntrconnctons (I) and qual numbrs of local modls ar addrssd n ths scton. Stablty analyss basd on th I basd polytopc modl allows for fwr lnar matr nqualts (LMI s) to b solvd but at th pns of ncrasng th consrvatsm of th rsults. Ths drvaton has th followng rqurmnts.. Th qulbrum stat s locatd at th orgn that s ~ whr ~.. Th nput u s constant and u u thrby u ~ u u. 4. Th compost systm modl s a concdnt polytopc modl. 5. Th wghtng functon matr and th compost componnt modl matr must commut that s A. Ths condton s cc W W A cc automatcally satsfd du to th dfnd structur of th matrcs. 6. A common matr P P > must b found that also satsfs P W W P. 7. Th ntrconncton matr must satsfy A L constant. con A con q Th compost systm modl may thn b wrttn as whr A cs s dfnd as ~ r ~ & W ( θ + θ ~ ) Acs A +. cs Acc Acon (6.3) (6.3) Th compost componnt modl matr A cc s dfnd n quaton (5.) and th ntrconncton matr A con s dfnd as

138 - 6 - [ I LDcc ] L cc Acon Bcc from quaton (5.9). hoosng th Lyapunov functon canddat to b V ~ P ~ th tm drvatv of V valuatd ovr th trajctors of th modl may b wrttn as r V & ~ ~ ~ [( A cs W ( θ + θ ) P + PW ( θ + θ ) Acs )] ~. Substtutng quaton (6.3) nto quaton (6.34) and rarrangng V & may b wrttn as V& Snc th matrcs r r ~ ~ ~ ~ [ A W ( θ + θ ) P + PW ( θ + θ ) A ] cc ~ ~ [ A W ( θ + θ ) P + PW ( θ + θ ) A ] ~. con A cc and functon V & may b smplfd to V& r r ~ W ~ cc con ~ + W commut as wll as th matrcs P and W th ~ ( θ + θ ) [ A P + PA ] ~ + ~ ~ [ A PW ( θ + θ ) + W ( θ + θ ) PA ] ~. con cc Furthr smplfcaton s possbl snc th product cc con PA con s a constant matr and (6.3) (6.33) (6.34) (6.35) (6.36) r ~ W ( θ + θ ) I rsultng n r ~ V & ~ W ( θ + θ ) cc cc con + con whch s qual to V& ( A P + PA ) ~ + ~ ( A P PA ) ~ r [ ~ ~ ~ W ( θ + θ )( A cc P + PAcc ) ~ + ~ W ( θ + θ )( A conp + PAcon ) ~ ]. (6.37) (6.38) (6.39)

139 - 7 - Groupng trms V & can b prssd as r ~ V & ~ W ( θ θ )[( A P PA ) ( A P PA )] ~ + cc + cc + con + con (6.4) whch can b rwrttn as r ~ V & ~ W ( θ θ )[ A P PA ] ~ + cs + cs. (6.4) Now to amn quaton (6.4) t s quadratc n trms of ~ and W thrfor th > stat ~ and th matr W cannot chang th sgn of V &. Ths lavs th trm [ A cs P + PAcs ] to dtrmn th sgn of V &. If [ A cs P + PAcs ] s ngatv dfnt for all thn V & s ngatv dfnt. Ths provs th followng proposton. Proposton 6.: Th polytopc modl (6.3) s globally asymptotcally stabl f th matr A con s ndpndnt of that s Acon L A con q constant and a matr P P > can b found such that P W W P and [ P + PA ] cs cs < A for all. (6.4) (6.43) Sarchng for othr mthods of valuatng systm stablty va Lyapunov tchnqus s an ongong study. Obtanng smpl rsults that ar not ovrly consrvatv can prov to b qut dffcult. In th followng haptr stablty analyss of truth modls s addrssd.

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141 STABILITY ANALYSIS OF TRUTH MODELS haptr 6 prsnts condtons undr whch polytopc modls ar asymptotcally stabl. Howvr th ultmat goal of stablty analyss s to dtrmn a rgon of stablty for th systm bng modld. Analyss of th truth modl should thrfor provd a bttr stmat of th rgon of systm stablty. Analytcal stablty analyss of a nonlnar truth modl s oftn mpractcal largly bcaus of th dffculty n gnratng an approprat Lyapunov functon. Hrn t s proposd that th Lyapunov functon canddat obtand durng th analyss of a polytopc modl b usd n conjuncton wth th truth modl. A mthod of mprovng th slcton of modlng ponts for th polytopc modl s ntroducd as wll as a mthod for stmatng rgons of asymptotc stablty (RAS). Fnally a cas study of a sth-ordr systm s prformd. 7. Lyapunov Functon anddat To facltat th sarch for a Lyapunov functon th quadratc form V ~ P ~ (7.) usd for th polytopc modls n haptr 6 s chosn. Ths facltats a numrcal approach to sarchng for Lyapunov functon canddats by frst constructng th polytopc modl and thn usng Algorthm 6. to dtrmn th rgon Γ and an α st of local modls. Th common P satsfyng th lnar matr nqualts (LMI s) dfnd by A P + PA < for K r α (7.) for all r α local modls n th α st s thn usd to construct th Lyapunov functon canddat for th truth modl. Howvr an ndcaton of how th modlng ponts usd by th polytopc modl should b slctd was not mntond. Locaton of th modlng

142 - 3 - ponts drctly affct th rsultng matr P. Ths n turn can affct th rgon of th stat-spac n whch V & s ngatv dfnt as llustratd n th followng ampl. Eampl 7.: To llustrat th snstvty of P and hnc V & wth rspct to th locaton of th modlng ponts and th rgon of th stat spac spannd by th modlng ponts consdr systm 3. Thr dffrnt chocs of local modl modlng ponts ar usd to dtrmn a Lyapunov functon canddat and th rgons n whch V & s postv smdfnt ar dntfd. In all thr cass Taylor srs appromaton s usd to construct th local modls and MATLAB s LMI toolbo [MAT] s usd to sarch for a common P satsfyng (7.). Th frst matr P s dtrmnd basd on a sngl local.4395a V corrspondng to modl drvd at th qulbrum stat [ ] [ 5V ] u kw and rsultng n P. (7.3) Nt local modls ar constructd at all combnatons of th modlng ponts dfnd by L [.573A 6.68 A] dvdd nto qually spacd ponts and by v [ V V] dvdd nto 6 qually spacd ponts rsultng n local modls. Th common P found s P. (7.4).56.3 Th last st of local modls for th systm ar constructd at all combnatons of th modlng ponts dfnd by L [.3934A 6.68 A] dvdd nto 3 qually spacd ponts and by v [43.43V V] dvdd nto 3 qually spacd ponts rsultng n 69 local modls. Th common P found s P 69. (7.5) Usng th matrcs P P and P 69 th Lyapunov functons V ~ P ~ V P ~ and V P ~ ar constructd. Ths Lyapunov functons and ~ ~ 69 69

143 - 3 - thr tm drvatvs ar valuatd ovr a rgon of th stat-spac and th rsults plottd n Fgur 7. through Fgur 7.3. In ach fgur dark gray shadng ndcats rgons whr V & th llptcal contours mark lvl sts of V th crcl marks th qulbrum stat of th truth modl and th dots mark th modlng ponts of th local modls. In ths ampl th largr th rgon of stat spac that th local modls span th furthr away th rgon satsfyng V & s from th qulbrum Fgur 7.. Lvl st contours of V plottd on top of V &. Fgur 7.. Lvl st contours of V plottd on top of V &.

144 - 3 - Fgur 7.3. Lvl st contours of V 69 plottd on top of V &. 69 From Eampl 7. t s clar at last for ths ampl that th largr th rgon of stat-spac spannd by Γ th furthr th rgon for whch V & s from th qulbrum stat. Th typ of local modl usd to construct th polytopc modl can also affct th rsultng matr P. Ths ffct s llustratd n th followng ampls. Eampl 7.: To llustrat th snstvty of P wth rspct to th local modl typ consdr systm 3. In ths ampl thr sts of local modls ar constructd. Each st s cratd usng a dffrnt constructon tchnqu but th sam modlng ponts. A common matr P satsfyng th LMI s n (7.) s thn dtrmnd for ach st of local modls usng MATLAB s LMI toolbo [MAT]. Lyapunov functon canddats ar thn constructd th rsultng P matrcs and valuatd ovr a rgon of th stat-spac for comparson. For ach st of local modls th truth modl nput s st to u [ 5V kw] and th modlng ponts dfnd by all combnatons of L [.3934A 6.68A] dvdd nto 3 qually spacd ponts and v [ V V] dvdd nto qually spacd ponts ar chosn. Th frst st of local modls s constructd from th truth modl usng Taylor srs basd appromaton rsultng n

145 P T. (7.6) Th scond st s constructd usng th GTZ transformaton wth th qulbrum stat chosn as th truth modl qulbrum [.4395A V] corrspondng to u [ 5V kw] thrby forcng a concdnt polytopc modl. Th common P found s P GTZ. (7.7).58.9 Th last st of local modls s constructd from th Taylor srs basd local modls usng th matr nvrson tchnqu dfnd by quatons (3.33) through (3.39) wth th qulbrum stat chosn as th truth modl qulbrum [.4395A V] corrspondng to u [ 5V kw] concdnt polytopc modl. Th common P found for ths st s Usng th matrcs P T ~ GTZ thrby forcng a P MI (7.8) and P MI th Lyapunov functons V T ~ P ~ P GTZ V P ~ and V P ~ ar constructd. Ths Lyapunov functons and GTZ MI ~ MI thr tm drvatvs ar valuatd ovr a rgon of th stat-spac and th rsults plottd n Fgur 7.4 through Fgur 7.6. In ach fgur dark gray shadng ndcats rgons whr V & th llptcal contours mark lvl sts of V th crcl marks th qulbrum stat of th truth modl and th dots mark th modlng ponts of th local modls. Ths ampl ndcats that th matr nvrson basd local modls rsult n th smallst rgon for whch V &. Howvr ths rsult was obtand usng th sam modlng ponts for all thr sts of local modls. If th modlng ponts ar allowd to b dffrnt for ach st of local modls thn th stuaton changs as llustratd n th nt ampl.

146 Fgur 7.4. Lvl st contours of V T plottd on top of V &. T Fgur 7.5. Lvl st contours of V GTZ plottd on top of V & GTZ. Fgur 7.6. Lvl st contours of V MI plottd on top of V &. MI

147 Eampl 7.3: To furthr llustrat th snstvty of P wth rspct to th local modl typ agan consdr systm 3. In ths ampl thr sts of local modls ar constructd. Each st s cratd usng a dffrnt constructon tchnqu but th sam modlng ponts. Thn for ach st of local modls an α st s dtrmnd usng Algorthm 6.. Lyapunov functon canddats ar thn constructd from th P matrcs of th α sts and valuatd ovr a rgon of th stat-spac for comparson. Th prmary dffrnc btwn ths ampl and Eampl 7. s th dffrnc n th modlng ponts and th numbr of local modls usd to formulat th LMI s n th sarch for th matr P. For ach st of local modls th truth modl nput s st to u [ 5V kw] and th modlng ponts dfnd by all combnatons of L [.573A 6.68A] dvdd nto qually spacd ponts and v [ V V] dvdd nto 3 qually spacd ponts ar chosn. Th frst st of local modls s constructd from th truth modl usng Taylor srs basd appromaton. Th matr P found satsfyng LMI (7.) for th α st of local modls s P Tα. (7.9) Th scond st of local modls s constructd from th Taylor srs basd local modls usng th GTZ transformaton wth th qulbrum stat chosn as th truth modl qulbrum [.4395A V] corrspondng to [ 5V kw] u thrby forcng a concdnt polytopc modl. Th matr P found satsfyng th LMI s n (7.) for th α st of local modls s P GTZα. (7.) Th last st of local modls s constructd from th Taylor srs basd local modls usng th matr nvrson tchnqu dfnd by quatons (3.33) through (3.39) wth th qulbrum stat chosn as th truth modl qulbrum [.4395A V] corrspondng to u [ 5V kw] thrby forcng a

148 concdnt polytopc modl. Th matr P found satsfyng th LMI s n (7.) for th α st of local modls s P MIα. (7.) Usng th matrcs P Tα P GTZα and P MIα th truth modl th Lyapunov functons V ~ P ~ Tα Tα V ~ P ~ GTZα GTZα and V ~ P ~ MIα MIα ar constructd. Ths Lyapunov functons and thr tm drvatvs ar valuatd ovr a rgon of th stat-spac and th rsults plottd n Fgur 7.7 through Fgur 7.9. In ach fgur dark gray shadng ndcats rgons whr V & th llptcal contours mark lvl sts of V th crcl marks th qulbrum stat of th truth modl and th dots mark th modlng ponts of th α st of local modls. Ths ampl ndcats that th matr nvrson basd local modls actually rsult n a largr rgon for whch V &. Ths may b assocatd wth th mthod of constructng th local modls. In addton th Taylor srs basd local modls and th GTZ basd local modls rsult n comparabl rgons for whch V & howvr fwr Taylor srs basd local modls ar ncssary to obtan th smlar rsults. Ths can b bnfcal from a computatonal pont of vw snc fwr local modls rsults n fwr LMI s to b solvd. Fgur 7.7. Lvl st contours of V Tα plottd on top of V & Tα.

149 Fgur 7.8. Lvl st contours of V GTZα plottd on top of V & GTZα. Fgur 7.9. Lvl st contours of V MIα plottd on top of V & MIα. As dmonstratd by th prvous ampls th choc of modlng ponts can sgnfcantly affct th stmat of th rgon of stablty by drctly affctng th matr P that s found. Thrfor th choc of ths st s qut mportant. In haptr 6 Algorthm 6. dtald a mthod to dtrmn an α st of local modls from a prdtrmnd st of local modls. Howvr thr s n gnral no way to justfy th choc of th prdtrmnd st of local modls. In ordr to crcumvnt ths dffculty th followng algorthm s provdd. Ths algorthm assums that th nput to th truth modl s constant and that th local modl paramtrs vary only wth stat.

150 Algorthm 7.: Algorthm for dtrmnng a rgon Γ dfnng an α st of local modls. Ths algorthm mploys a bscton sarch mthod to dtrmnng th boundars of th rgon Γ. Snc Γ s n dmnsonal smultanous sarchs for th uppr and lowr bounds of Γ ar prformd n ach dmnson rsultng n n smultanous sarchs bng prformd. Th ntalzaton for th sarch s prformd by stps -5 and thn th sarch s cutd n stp 6.. Shft th truth modl so that th qulbrum stat s ~ usng th translaton ~ and u u u ~.. Dfn th largst rgon of th stat-spac that may b boundd by Γ. Ths rgon s appromatd by a hyprbo dfnd as ~ n Γ { ~ ~ ~ ma R mn ma} whr ~ ~ mn < ma. (7.) Ths appromaton may b basd upon practcal bounds of th truth modl and opratonal lmts of th systm. Th fnal rgon Γ wll b contand by ths ntal appromaton. 3. Intalz lmts that confn th boundars of Γ. Along ach dmnson of Γ ma st four lmts dfnd as l ll - stat lowr bound lowr lmt l lu - stat lowr bound uppr lmt l ul - stat uppr bound lowr lmt l uu - stat uppr bound uppr lmt. and llustratd n Fgur 7.. Th ntal valus of ths lmts ar st to: ll ~ mn l from stp l th qulbrum valu of stat lu ~ l th qulbrum valu of stat ul ~ uu ~ ma l from stp.

151 Fgur 7.. Boundary lmts for th -th dmnson. 5. St th convrgnc tolranc ε for ach stat whr ε and l ll l lu l l ε must both b satsfd along th -th dmnson of Γ. u l uu 6. A bscton mthod s now mployd to fnd a rgon Γ. Th stps of th bscton mthod ar as follows: A. Appromat th rgon Γ by a hyprbo of th form whr and ~ n Γ R : ~ ~ ~ } { mn ma l + l ~ ll mn lu (7.3) (7.4) ~ l ul + l ma uu. (7.5) Ths s llustratd n Fgur 7. along on dmnson of th stat-spac. Fgur 7.. Appromaton of th rgon Γ along th -th dmnson. B. Dtrmn a grd of unformly spacd modlng ponts n Γ. Ponts along th -th dmnson ar dpctd n Fgur 7..

152 - 4 - Fgur 7.. Unformly spacd ponts n Γ along th -th dmnson.. onstruct a local modl at ach modlng pont. D. Us Algorthm 6. to fnd an α st of local modls. E If th α st contans all of th local modls that s constrants r α r and th l ll l lu ε (7.6) and l ul l uu ε (7.7) ar satsfd for all thn a rgon Γ has bn dtrmnd go to stp 7. Othrws go to stp F. F. Along ach dmnson of th α st fnd th mnmum and mamum valu ~ α mn ~ α of th modlng ponts dnotd by and ma rspctvly as dpctd n Fgur 7.3. Fgur 7.3. Th α st of local modls along th -th dmnson. G. Adjust th lmts for th lowr boundars. For th -th dmnson prform th followng stps. a. If l ll l lu ε and mn mn ~ > ~ α th algorthm has potntally stppd out of th soluton rgon (possbly du to th chang n lmts

153 - 4 - along anothr dmnson). Thus Hrn lu mn l ll and l lu must b modfd. ~ l.5 ~ and l ll mn ar arbtrarly chosn to pand th sarch rgon. b. If l ll l lu > ε and mn mn c. If l ll l lu > ε and mn mn ~ > ~ α thn st l ~ ll mn. ~ ~ α thn st l ~ l u mn. H. Adjust th lmts for th uppr boundars. For th -th dmnson prform th followng stps. a. If l ul l uu ε and ~ ~ α ma < ma th algorthm has potntally stppd out of th soluton rgon (possbly du to th chang n lmts along anothr dmnson). Thus Hrn ul ma l u l and l uu must b modfd. ~ l.5 ~ and l ma ar arbrarly chosn to uu pand th sarch rgon. b. If l ul l uu > ε and ~ ~ α ma < ma thn st l ~ uu ma. c. If l ul l uu > ε and ~ ~ α ma ma thn st l ~ u l ma. I. Itrat th sarch agan. Go to stp A. 7. A rgon Γ has bn dntfd. Us P found for th α st of local modls to construct th Lyapunov functon canddat V. Th abov procdur dtrmns an α st of local modls from whch th common matr P may b usd to construct a Lyapunov functon canddat. To furthr llustrat consdr th followng ampl. Eampl 7.4: onsdr systm 3. Algorthm 7. s ntalzd by choosng th rgon Γ ma to b Γma n A L 3A R : (7.8) V v 9V

154 - 4 - ach stat to b dvdd nto 3 qually spacd ponts and th convrgnc tolranc to b ε. for all stats. Th local modls wr constructd usng Taylor srs appromaton and th algorthm tratd untl th fnal rgon Γ s found to b n 4.66A L A Γ R : V v V Th matr P obtand s (7.9) P. (7.) To tst f th matr P s snstv to th numbr of modlng ponts ach dmnson of Γ s dvdd nto qually spacd ponts and local modls constructd at all combnatons of th modlng ponts. Th matr P obtand from ths st of local modls was found to b qual to th matr P n (7.) to th dsplayd prcson. Usng th truth modl th Lyapunov functon V ~ P ~ and t s tm drvatv s valuatd ovr a rgon of th stat-spac and th rsults plottd n Fgur 7.4. Th dark gray shadng ndcats rgons whr V & th llptcal contours mark lvl sts of V th crcl marks th qulbrum stat of th truth modl and th dots mark th modlng ponts of th local modls. Fgur 7.4. Lvl st contours of V plottd on top of V &.

155 Sarchng for th Rgon of Asymptotc Stablty by Optmzaton Evn for scond-ordr systms stmatng th RAS can b rlatvly nvolvd. For hghr ordr systms ths task bcoms vn mor dffcult. Thrfor ths scton provds a mans to stmat th RAS usng constrant optmzaton mthods. onsdr th Lyapunov functon V for th systm ~ & F( ~ ) vald for th qulbrum stat ~ whr ~. Thn t s notd that. Th mnmum valu of V valuatd along th constrant V & n th rgon n { ~ } R dntfs th valu of V dnotd by V for th largst lvl st nclosd by th rgon n whch V &. Ths lvl st dfns an stmatd RAS.. Th mamum valu of V & valuatd along th contour V V s zro. A procdur usd to sarch for th contour dscrbd n th followng algorthm. V V basd upon and abov s Algorthm 7.: Sarch for an stmat of th RAS Ths algorthm uss a combnaton of bscton sarch mthods and constrand optmzaton to sarch for an stmatd RAS. Th ntalzaton for th sarch s prformd by stps -5 and thn th sarch s cutd n stp 6.. Shft th truth modl so that th qulbrum stat s ~ usng th translaton ~ and u u u ~.. Dfn th largst rgon of th stat-spac that may contan an stmatd RAS. Appromat ths rgon by a hyprbo dfnd as ~ n HB { R ~ ~ ~ mn ma} whr ~ ~ mn < ma. (7.)

156 Th dfnton of ths rgon may b basd upon practcal bounds of th truth modl or opratonal lmts of th systm. Th fnal stmaton of th RAS wll b contand by HB. 3. Dtrmn th largst lvl st of V contand by HB and dtrmn th valu of V whn valuatd on ths lvl st. S Appnd A for a mthod for fndng th lvl st. 4. St th lmts for th sarch by bscton. Th lowr lmt V l s ntalzd to. Th uppr lmt V u s st to th rsult from stp St th convrgnc tolranc ε for th sarch by bscton. Th nqualty V u V < ε must b satsfd for th fnal soluton. l 6. A bscton mthod s now mployd to fnd an stmat of th RAS. Th stps of th bscton mthod ar as follows: A. Appromat th RAS by th lvl st of V dfnd by V + Vu V l as dpctd n Fgur 7.5. V V whr (7.) Fgur 7.5. Boundars to th valu of V for th stmatd RAS. B. Sarch for th mamum valu of V & along th contour V V. Snc V & may hav local mamums ths optmzaton should b prformd a numbr of tms wth th sarch ntalzd dffrntly ach tm. Ths rsults n th st of mamums [ K V& ] V& ma ma n whr th -th mamum n (7.3) dnotd by optmzaton problm V& ma (7.3) s a soluton of th

157 V & ma subjct to V V. (7.4) ~ Ths optmzaton problm s ntalzd to a stat of th systm on th contour V V.. Adjust th lmts of th sarch by bscton by prformng th followng tsts. a. hck to s of a contour dfnd by V & has bn found that s f V & ma for any thn prform ths stp. Othrws go to stp b. Sarch along th contour V & to fnd a nw valu of V u. Ths s accomplshd by solvng th optmzaton problm mnv subjct to V & V u ~ (7.5) whr ~ s ntalzd to th valu of ~ for whch V & ma. If Vl V u thn st Vl. 5V u to contnu th sarch by bscton. Go to stp A. b. Dtrmn th mamum of th rsults from stp B whch s dfnd as [ V& K V ] & ma ma &. V ma ma n c. If V & ma < and V u V ε thn an stmatd RAS has bn dtrmnd go to stp 6. l d. If V & ma < and V > ε V u thn st l V l V.. If V & ma and V u V l thn th rgon that th sarch convrgd to s not a RAS. Ths may occur du to local mnmzrs or mamzrs of V &. Thrfor th sarch s contnud by pandng th sarch rgon. Hrn ths s accomplshd by sttng V. 5V f. If V & ma and V u V l ε thn st V u Vl. g. If V & ma and V > ε h. Go to stp B. V u thn st l V u V. l u. (7.6)

158 An stmat of th RAS s th rgon boundd by th lvl st of V dfnd by V V. Snc th systms consdrd hrn ar nonlnar t s possbl for V & to hav local mnmzrs and mamzrs. As a rsult car must b takn n slctng th ntal condtons for th optmzatons. To llustrat ths algorthm th followng ampl s provdd. Eampl 7.5: For systm 3 th abov tchnqu s mplmntd n an attmpt to stmat th RAS. Th ntal hyprbo s chosn to b whr ~ n A 3A HB R : ~ + V 9V ~ and [ ] [ 5V ].4395A V corrspondng to u kw and th convrgnc tolranc s st to ε.. Th rsultng rgon of attracton s boundd by th lvl st of V havng th valu V Ths lvl st s plottd n Fgur 7.6. Th shadd rgon ndcats whr V &. (7.7) (7.8) Fgur 7.6. Rgon of asymptotc stablty for systm 3.

159 If an stmatd RAS s dtrmnd usng Algorthm 7. t n gnral has th shap of a hyprllpsod. Occasonally t s usful to bound th stats of th systm by a hyprbo dscussd n th nt scton. 7.3 Rctangular Rgon of Intal ondton Th stmatd RAS found usng a Lyapunov functon of th form V ~ P ~ wll n gnral b llptcal n shap. Howvr n som cass t may b convnnt to dntfy a rgon of th stat spac n whch th m stats hav bounds ndpndnt of ach othr of th form ~ ~ Xmn ~ Xma M. (7.9) ~ ~ ~ X X n mn n n ma Ths typ of bound on th stats s a hyprbo n th stat spac surroundng th qulbrum havng facs paralll wth th coordnat as and boundd by th lvl st. Lmts of ths form would allow a smpl st of constrants to b dntfd for whch th systm s guarantd to b wthn th RAS. Th radr s rfrrd to Appnd A for a dscusson on dtrmng hyprbos boundd by hyprllpsods. Eampl 7.6: onsdr systm 3 and th stmatd RAS dtrmnd n Eampl 7.5. Usng th procdur dfnd n Scton A..3 th bo of mamum volum nscrbd nsd th lvl st s dfnd by 7.384A L A. (7.3) V v 53.8V Fgur 7.7 contans a plot of th bo and th lvl st. As can b sn f th stat of th systm s wthn th hyprbo thn t s also wthn th RAS. 7.4 Dtrmnng Intal Input ondtons Th tchnqus dscussd abov ar nvolvd wth dntfyng rgons of stablty n trms of th stats of th systm. Howvr for many systms spcally powr lctroncs basd systms th nputs vary as wll. Ths lads to th followng quston: From whch stabl stady stat nput condtons can th systm b stppd wth a

160 guarant that th stats asymptotcally approach th nw qulbrum stat? Rqurng th systm to b n qulbrum at th ntal nput condton allows th us of th followng algorthm. Fgur 7.7. Rctangular rgon of ntal condton. Algorthm 7.3: Dtrmnaton of ntal nput condtons. Dtrmn th valu of V dntfyng th lvl st boundng a RAS dnotd by V. S Fgur Pck a rgon of th nput spac surroundng th qulbrum u and dtrmn th qulbrum stats assocatd wth ach nput u of ntrst. 3. Solv th Lyapunov functon V V ( ) at ach qulbrum stat. 4. Th systm may b stppd from all nput condtons for whch th corrspondng qulbrum stat rsults n V V. For ampl n Fgur 7.8 all nputs btwn u and u hav qulbrum stats for whch V V. Thrfor th truth modl nput may b stppd from u u u to u and th stat of th modl wll asymptotcally approach.

161 Howvr t should b notd that th rvrs stp u to u cannot b pctd to b stabl wthout prformng th abov analyss basd on a RAS dntfd for ( u ). Ths procdur s addtonally attractv for systms wth a low ordr nput-spac. Fgur 7.8. Input spac projcton. Eampl 7.7: Ths analyss s prformd usng th matr P obtand n Eampl 7.4 for systm 3. Th rgon of th nput-spac surroundng th nput [ ] u 5V kw chosn for valuaton s dpctd n Fgur 7.9. Th rgon of th nput spac n whch V ( ) V whr V s th valu of V along th lvl st found n Eampl 7.5 s not shadd. Thrfor any nput n th non-shadd rgon of Fgur 7.9 f th systm s n qulbrum may b stppd to u. From Fgur 7.9 an nput u [ 5V kw] s chosn and t s qulbrum stat s dtrmnd to b [ A 5V]. Intalzng th truth modl to and u th truth modl s smulatd. At t th nput s u stppd to [ ] 5V kw. In Fgur 7. th rsultng stat trajctory s plottd on top of th lvl st dtrmnd n Eampl 7.5. It can b sn that th stat of th systm asymptotcally approachs.

162 - 5 - Ths concluds th dscrpton of th tchnqus usd hrn for analyzng th nonlnar truth modl. Th followng cas study utlzs ths tchnqus on a hghr ordr modl. Fgur 7.9. Input spac projcton for systm 3. Fgur 7.. Input stp tst for systm as Study: S-Ordr Systm Th ampls abov basd on th scond-ordr systm ar usful for ntroducng and vsualzng th concpts. Howvr ths cas study s st forth to dmonstrat th abov tchnqus on a mor practcal systm. onsdr systm 4 dscrbd n haptr.

163 - 5 - Th truth modl for ths systm s constructd n ASL [ADV99] allowng automatd local modl gnraton. Th ndrct mthod of Lyapunov s usd frst to chck th stablty of th qulbrum par ) whr th qulbrum stat s ( u corrspondng to th nput [ A A 5V 5V 5V 9.9 ] V [ V ] u kw. (7.3) Usng functons bult nto ASL th Taylor srs appromaton to th truth modl s constructd at ) and th gnvalus of th rsultng Jacoban matr ar dtrmnd to b ( u - - g( A ) All s gnvalus ar locatd n th opn lft half sd of th compl plan thrfor by th ndrct mthod of Lyapunov ths s an asymptotcally stabl qulbrum stat. Hnc a RAS sts about. Th nt stp n analyzng ths systm s to fnd a quadratc Lyapunov functon canddat of th form gvn by (7.). Algorthm 7. s mployd to dtrmn an α st of local modls and thrby a matr P to construct V. Th rgon Γ ma s ntally dfnd as th hyprbo (7.3) (7.33) Γ ma A L 55A A ˆ L 55A ~ ˆ n R ~ 45V v 55V : V v 7V 44V v 56V -39V vrror 697V (7.34)

164 - 5 - whr th bounds ar chosn basd on pctd practcal lmtatons of th systm and lmtatons of th control. Thr modlng ponts ar chosn along ach dmnson Taylor srs basd local modls ar constructd at ach modlng pont and th convrgnc tolranc s st to ε. for all. Th rgon found from th sarch s Γ ~ R and th rsultng matr P s n : ~.4A L.4A ˆ L 45.5V vˆ V v 48.57V v V v 54.96A 34.39A 56.65V 5.9V V 44.9V rror P To sarch for a RAS th Lyapunov functon canddat s constructd by substtutng P from (7.36) nto quaton (7.) and thn cutng Algorthm 7.. Th convrgnc tolranc s chosn to b ε. and th ntal hyprbo s chosn to b Γ ma A L 55A A ˆ L 55A ~ n R ~ 45V vˆ 55V :. 4V v 7V 44V v 56V -39V vrror 697V Th valu of V along th lvl st found cutng Algorthm 7. s Basd upon V V th rctangular rgon of ntal condton s dtrmnd to b (7.35) (7.36) (7.37) (7.38)

165 A L.68A 8.74A ˆ L.6A 498.4V vˆ 5.6V V v 5.7V 497.4V v 5.76V 5.46V vrror.58v An uppr bound to th dvaton of th stats n th RAS can b mad usng th hyprbo of mnmum volum wth facs paralll to th coordnat as and contanng th RAS. Gvn that th stat of th systm s n th RAS thn t s boundd by.56a.74a 46.3V V 458.8V -7.73V v L 9.44A ˆ L 9.6A vˆ V. v 54.6V v 54.7V rror 55.78V It should b notd that th rgon dfnd n (7.4) dos not dfn th RAS nstad (7.4) bounds th largst curson gvn that th ntal condton satsfs (7.38). Th last ssu dscussd n ths chaptr s a mthod to dtrmn th largst stp chang from a prvous stady stat qulbrum pont. To dtrmn ths rgon for systm 4 th qulbrum stats for a st of nputs ar frst dtrmnd. Thn th Lypaunov functon constructd usng th matr P gvn n (7.36) s valuatd for ach qulbrum stat. Th rsultng valus of V ar plottd n Fgur 7. vrsus th nputs of th systm. Th unshadd rgon marks th st of nputs for whch th corrspondng qulbrum stats rsultd n [ 594.5V ] V V. As an ampl from Fgur 7. th nput u.83kw s chosn as th nput condton to stp from. Th qulbrum stat corrspondng to ths nput s [ 3.66A 3.66A 5V 5V 5V ] 34.76V. Fgur 7. and Fgur 7.3 llustrat th trajctors for th systm whn th nput s stppd to u [ 5V kw] (7.39) (7.4)

166 at t. Fgur 7.4 contans a plot of th valu of th Lyapunov functon valuatd along th stat trajctory. Th valu of V s markd by th ln. Fgur 7.. Input spac projcton for systm 4. Fgur 7.. Systm 4 stat trajctors. Ths concluds th us of tchnqus prsntd n ths chaptr for analyzng ths systm. Th nt two mthods of analyss ar prsntd as a mans of vrfyng th RAS boundd by th lvl st for whch V V. In ordr to vrfy ths analyss th systm bhavor s studd nt usng a grd sarch approach basd on tm doman smulaton. Ths analyss s strctly tm doman basd and prformd by ntalzng th systm stat and smulatng untl thr a voltag lmt s rachd or all of th stats ar wthn on prcnt of thr qulbrum valus.