The University of Sheffield

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1 The Uversy of Sheffeld Deprme of uom Corol d Sysems Egeerg Sldg ode Corol of Noler Sysems Houm Dlll.S. Corol Sysems ugus 7 Supervsor: Professor Sephe P. Bks dssero submed prl fulflme of he requremes for he degree of ser of See Corol Sysems

2 BSTRCT Sldg mode orol s re of resg eres orol egeerg. Ths mehod s proved o be robus gs dsurbes d dsrepes bewee he physl pl d s mheml model. However, hs mly bee ppled o ler sysems d s pplo o oler sysems s bsed o ulsg ler sldg surfes. Ths dssero s oered wh sldg mode orol desg usg oler sldg surfes. Our objeve s o pply hs mehod o oler sysems of y dmeso by replg he orgl sysem wh sequee of ler me vryg sysems. I f, sbly lyss of sldg surfes s mjor pr of he desg. Therefore, severl sbly lyss mehods hve bee vesged d ew suffe odo for sbly of ler sysems s lso rodued bsed o he dey of Euld orm of ses veor. Ths mehod offers sgf dvges omprso wh esg sbly odos. Cosequely, we hve desged oler sldg mode orols for usble oler sysems of rbrry dmeso. I ddo, fuure dreo s suggesed for furher reserh o ob he oler sldg surfes hrough opmso heory. I

3 EXECUTIVE SURY INTRODUCTION Sldg mode orol mehods hve bee used wdely se hey provde robusess gs prmeer vros d dsurbes. Sldg mode orol s kow o be prulr ype of vrble sruure orol. s he me suggess, vrble sruure orol heory s bsed o hgg sruure of he sysem uder sudy by mes of feedbk lw. IS ND OBJECTIVES Sldg mode orol desg hs bee mosly pprohed by ulsg ler sldg surfes. s rve robus pproh s pplo o oler sysems s sgf. Therefore, he m of hs proje s o desg oler sldg mode orol for usble oler sysems of rbrry dmeso d bsed o reely rodued ero ehque. CHIEVEENTS I hs dssero, heory of sldg mode orol for ler me vr, ler me vryg d oler sysems s preseed. Sbly Issue of ler me vryg sysems hs bee vesged. To heve sbly ew suffe odo s rodued bsed o he dey of Euld orm of ses veor s lerve for pole ssgme. oler sldg mode orol s desged d jusfed. I prulr, he oler sldg mode orol s ppled o hree well kow oler sysems. Nmely, V der Pol s osllor s sysem wh qudr oler erm, Duffg equo wh ub oler erm d Lorez ror wh hree dmesos d wo qudr oler erms. CONCLUSIONS ND RECOENDTIONS Noler sldg mode orol mehod preseed hs dssero, hs he poel of beg ppled o lrge lss of usble oler sysems subje o uery d bouded dsurbes. I ddo, fuure dreo s suggesed o ob opml oler sldg surfes bsed o he desg rer. II

4 CKNOWLEDGEENTS Ths dssero s deded o my fmly, for her uflg love d suppor hroughou my lfe. Espelly my moher for beg pllr of sregh h empowers me o lve o my poel. I wsh o epress my sere grude o my supervsor Professor Sephe P. Bks for hs os suppor, pee d eourgeme. Hs sghful heorel d ehl dve ws essel o he ompleo of hs proje. I m greful for he kowledge he shred wh me d he vluble lessos he ugh me for my fuure dem lfe. III

5 Tble of Coes Iroduo o Sldg ode Corol.... Theory d pplo.... Robusess of sldg mode orol Cherg Elmo ddol sldg mode orol desgs pprohes Robus Corol desg bsed o Lypuov mehod Fuzzy- Sldg mode orol... Sldg ode Corol of Ler Tme Ivr Sysems.... Iroduo..... Se-Spe represeo of Dyml sysems..... Phse ple lyss Numerl soluo of dfferel equos.... Sldg ode Corol wh slr pus.... Sldg ode Corol wh Veor pus Sbly lyss of me vr sldg surfes Pole Pleme Lypuov pproh ew suffe odo for sbly of ler sysems....5 Emple of desgg sldg mode orol for seod order sysem... 6 Sldg ode Corol of Ler Tme Vryg Sysems Iroduo Sldg ode Corol of LTV sysems wh slr pu Sldg ode Corol of LTV sysems wh veor pu Sbly lyss of me vryg sldg surfes Pole Pleme Lypuov lyss of Tme vryg sysems ew suffe odo for sbly of ler sysems IV

6 .5 Emples of desgg sldg mode orol for LTV sysem Pole pleme emple Emple of pplyg he ew suffe odo Noler Sldg ode Corol Iroduo Iero ehque for oler dyml sysems Sldg mode orol of V der Pol Osllor Sldg mode orol of Duffg Dfferel Equo Sldg mode orol of Lorez ror Coluso d Reommedos Coluso Fuure dreos Referees V

7 Ls of Fgures Chper : Fgure -:... Fgure -:...4 Fgure -:...6 Fgure -4:...8 Fgure -5:...8 Fgure -6:...9 Chper : Fgure -:...5 Fgure -:...8 Fgure -:...9 Fgure -4:...9 Chper : Fgure -:...46 Fgure -:...5 Fgure -:...54 Fgure -4:...56 Fgure -5:...56 Fgure -6:...58 Fgure -7:...59 Fgure -8:...6 Fgure -9:...6 Fgure -:...6 Fgure -:...6 Chper 4: Fgure 4-:...65 Fgure 4-:...66 VI

8 Fgure 4-:...66 Fgure 4-4:...67 Fgure 4-5:...7 Fgure 4-6:...7 Fgure 4-7:...7 Fgure 4-8:...7 Fgure 4-9:...7 Fgure 4-:...74 Fgure 4-:...75 Fgure 4-:...75 Fgure 4-:...76 Fgure 4-4:...77 Fgure 4-5:...79 Fgure 4-6:...8 Fgure 4-7:...8 Fgure 4-8:...8 Fgure 4-9:...8 Fgure 4-:...8 VII

9 Chper Iroduo o SC Iroduo o Sldg ode Corol. Theory d pplo Robus orol desg des orge from he f h oroller desg d mplemeo do o shre equl rumses. Th s oroller desg s bsed o mheml represeo of he pl where s, s ppled o he ul physl pl. Therefore, s rul o oe he dsrepes bewee he pl d he mheml model used he desg. smh bewee he model d he ul pl sems from delzo proess of modellg, model reduo, umodelled dyms, prmeer vros d prmeer ueres. Thus robus orol desg mples h good performe should be gureed whe he oroller s mplemeed presee of modellg errors. Oe of he robus orol desg mehodologes s Sldg ode Corol. Sldg modes our sysems desrbed by ordry dfferel equo wh dsouous rgh hd sde. Rgh hd sde of he dfferel equo s dsouous se he forg fuo hs elemes pog oppose dreos eghbourhood of surfe lled sldg surfe. The forg pu s used o brg he sysem ses o he sldg surfe d preserve he sysem moo o he surfe. Sldg mode orol s spel ype of vrble sruure orol VSC. s he me suggess VSC Sysems re bsed o hgg sruure of sysem by mes of deso rule or swhg fuo order o heve desred performe. VSC sysems be regrded s ombo of severl fed sruure subsysems eh wh spef propery h whe ulsed ombo, d oe eh s me, resul ew behvour whh o of he fed sruure subsysems possess. I Sldg ode Corol SC he sme prple holds. The sub-sruures re desged o mke sldg surfe rve o he sysem ses d o drve hem from y l odo owrd he sldg surfe d osr hem he eghbourhood

10 Chper Iroduo o SC of he sldg mfold oe hey h. I should be poed ou h he sldg surfes re desged o be sble so h he ere sysem wll beome globlly sympo sble. Ths heory ws orged former Sove Uo bk le 95 s d s sll rve re of reserh orol egeerg. To llusre hs de emple osder he followg double egror pl doped from Edwrds Spurgeo 998, p.: y u. Cosder pplyg he slr feedbk lw: u ky. Noe h k s posve slr. To llusre he lyss phse ple le us mulply boh sdes of. by y d subsue. o.. y y kyy. kg egrl of. resuls : y ky.4 u k y u k y y y b b y y b b Fgure -: Phse porr equo.4 geered by wo ds feedbk lws.

11 Chper Iroduo o SC I fgure. wo fed sruures of feedbk sysems re llusred where < k < < k bu oe of hem re sympolly sble. Neverheless by rodug he feedbk lw equo.5 we ob sympolly sble sysem from wo osllory feedbk sysems fgure.. k y f yy < u k y oherwse.5 Equo.5 s obed by osderg he followg Lypuov fuo: V y y kg he dervve of V d subsug from. d. we ge: V y y y y y y y ky yy k V < herefore, k y u k y f f k < f yy < k > f yy > yy < where k < yy > where k > To lyse he resulg rjeory of VSCS we hve dvded he phse ple o rego d b fgure.. Whe ses re rego, he orol lw employed where s rego b he orol lw wll ge loser o he org fe me. u u k y s k y s ulsed. The resulg moo

12 Chper Iroduo o SC Sprl rjeory of he ses for l vlue of [; ] d k.5, k s llusred fgure.. Fgure -: Phse porr of double egror sysem orolled by VSC lw. I ddo Uk 978, p.7 suggess ombg sddle po d sprl phse porr s wo fed sruures o yeld vrble sruure sysem wh sldg moo. Desg of sldg mode orol volves wo sges. Sge oe volves desg of sble sldg surfe h ssfes he desg rer d sge wo volves obg he orol pu deso rule h wll mke he ses rve o he sldg surfe. There re wo dvges ssoed wh he sldg moo: Sldg surfe equos hve redued dmeso. Depedg o wheher he orol pu s slr or veor he redued dmeso sysem wll be or m where m s he sze of he veor pu. 4

13 Chper Iroduo o SC b Sldg moo s robus d sesve o prmeer vros mpl vrous pplos. Ths s deped he e seo by emple. I s mpor o oe h he rel physl proess mgh ehb some udesred hrerss suh s hyseress, dely or some hgh frequey dyms h s o modelled our desg. Esee of suh o-deles he sysem wll led o hgh frequey moo kow s herg. Ths pheomeo s kow s he moo of ses osllg he eghbourhood of sldg mfold. If fe swhg were possble we would hve del sldg moo rjeory whh s desrbed by redued order sysem. However due o rel world lmos suh s hyseress or dely uors or elero drves, herg beomes mpor ssue Sldg ode Corol d should be ddressed. Ths udesred moo uses wer d dmge o he uors proess uder orol. 5

14 Chper Iroduo o SC. Robusess of sldg mode orol Edwrds d Spurgeo 998, p.8 llusre he ssue of robusess by omprg he double egror pl d pedulum. The double egror s ssumed o be ler ppromo of he pedulum. Illy we wll esblsh he pedulum relos: θ legh l Fgure - : Shem of pedulum The vrble θ s he gulr dspleme d θ s he gulr veloy. I modellg he pedulum we ssume h here s o fro d lso he mss of he rod s eglgble omprg o he mss. hus he dfferel equo goverg he moo of hs sysem s: l θ sθ u g l.6 Normlsg he equo.6 we ob: y s y u.7 The double egror of he equo. be ssumed o be he pedulum levg ou he susodl erm. By pplo of sldg mode orol s show h whe he sldg moo s rehed boh sysems behve he sme wy d her rjeores re obed from he sldg mode equo.8. y my.8 6

15 Chper Iroduo o SC Therefore he susodl erm be ssumed o be eerl dsurbe o he double egror pl whh s olly suppressed by desg of sldg mode orol. s resul, Sldg mode orol s osdered o be oe of he robus orol desg mehodologes.. Cherg Elmo s meoed he prevous seo beuse of he rel world o-deles he rjeory of ses ppers o her sed of sldg o he predefed surfe. If we hk herg hrough urs ou h hs pheomeo s mly used by prs dyms or swhg me delys. Youg e l 999, p. llusre hs ssue by smple rely orol sysem emple. The ru of hs emple s kg he sesor dyms o ou whh hs se s of seod order. Ths udesred dyms s referred o s prs dyms. The mos wdely eped soluo o elme herg s he so lled Boudry Lyer Corol. Ide of hs mehodology s bsed o repleme of he dsouous orol pu wh peewse ler or smooh ppromo of swhg fuo eghbourhood of he sldg surfe. Followg suro fuo be used sed of sg fuo: s s s Φ Φ f f f s Φ s < Φ s > Φ Grphl represeo of suro fuo s llusred fgure.4 d.5. 7

16 Chper Iroduo o SC Fgure -4: use of suro fuo boudry lyer orol Fgure -5: orol pu sgl boudry lyer orol However, Youg e l 999, p. rgues h he mehod s o effeve f prs elemes es he orol loop. 8

17 Chper Iroduo o SC Youg e l 999, p.4 rodue Observer-bsed Sldg ode Corol s oher lerve o boudry lyer orol mehod. I hs pproh sympo observer s ulsed o osru hgh frequey by pss loop o vod he hgh frequey oe of he swhg eleme reh he pl. I oher words herg s osred d lolzed sde hgh frequey loop whh by psses he pl. I hs mehod he m hllege s o desg sympo observer wh he observo error overgg o zero sympolly. The mos rgug spe of hs desg s o overome he problem of prs dyms. Blok dgrm of hs pproh s llusred fgure.6. Fgure -6: Blok dgrm of observer-bsed Sldg ode Corol. Refer o Youg 999, p.4 I fgure.6 he hgh frequey bypss loop s show. I osrs he hgh frequey swhg from rehg he pl. I s mpor o oe h o be he effes of ukow bu bouded dsurbe, he observer g h mus be resed. However resg observer g ffes he ble bewee observer g h d he feedbk g g, herefore by resg h, g mus lso be ued he feedbk loop o re he ble. deled se spe llusro of Uk Observer d Wlo-Zk observer re preseed by Edwrds d Spurgeo 998, p.7. 9

18 Chper Iroduo o SC I ddo, Youg e l 999, p.5 rell dsurbe rejeo desg pproh whh s proposed bsed o he de of observer bsed sldg mode orol. I hs mehodology sldg mode dsurbe esmor s used o esme he ukow dsurbes d prmeer ueres. If he dsurbe s eued eough here wll be o eed o desg epl sldg mode orol. However he sldg mode orol desg s mpl he sldg mode dsurbe esmor.

19 Chper Iroduo o SC.4 ddol sldg mode orol desgs pprohes.4. Robus Corol desg bsed o Lypuov mehod ordg o Youg e l 999, p.8, Lypuov bsed pproh hs bee rodued 97 s h dels wh oler pls wh uer dym models. s he me suggess, ulses Lypuov fuo of he oml pl. lhough, sldg mode does o eplly pper he desg proedure, sll he resulg losed loop sysem ludes sldg moo s he sysem rjeory se spe rehes he equlbrum po..4. Fuzzy- Sldg mode orol Fuzzy sldg mode oroller s ew mehod of pprohg sldg mode orol desg. Wg 997, pp rgues h sldg mode orol d fuzzy sysems opere smlrly. Ths seme s proved for SISO oler uer sysems of y dmeso. Dervos preseed re bsed o forg he rkg error o zero d hee, he resuls be used o rk desred referee rjeory. However, sldg mode, he orol pu s of dsouous ure h mkes dfful o be used fuzzy sysems. I ddo, wll use herg he sysem oupu. Ths problem s ddressed by use of ombo of boudry lyer orol suro fuo d fuzzy log orol. The ru of hs de s h eher pror kowledge of he sysem model or pror kowledge of fuzzy membershp fuos s requred. Hwg d L 99 suggesed fuzzy bsed sldg mode orol of oler sysems h provdes sympo sbly d ddresses he herg problem fed sldg mode orol. Ipus of he fuzzy oroller re he sldg surfe σ d s dervve σ whh re fuzzfed before beg ppled o fuzzy oroller d he oroller oupu s fuzzy vlue d mus be defuzzfed before beg ppled o he sysem. oreover, herg s ddressed by desgg he membershp fuo o

20 Chper Iroduo o SC djus he orol pu lose o he sldg surfe. The sudy s oluded by emple of orollg vered pedulum. Hvg rodued he bs oeps of sldg mode orol he ls wo mehods ddress he problem of Noler Sldg ode Corol dffere wys. We wll pproh hs problem by lly sudyg he ler me vryg sysems sldg mode heory d he geerlsg hs de o oler sysem by ulsg reely rodued ero ehque by Toms-Rodrguez d Bks, pp

21 Chper Sldg ode Corol of LTI Sysems Sldg ode Corol of Ler Tme Ivr Sysems. Iroduo ler sysem s defed s oe h ssfes prple of superposo. The prple of superposo ses h he respose obed by pplo of wo ds pu fuos s he sum of he dvdul resposes. Thus, for he ler sysem, he respose o severl pus be luled by reg oe pu me d ddg he resuls. Ths prple llows oe o develop omple soluos o he ler dfferel equo from dvdul smple soluos. I hs hper we sr wh rellg some defos se spe d sbly heory, he seo d we rodue he sldg mode orol heory for ler me vr sysems wh slr d veor pus d seo 4 we vesge vrous sbly mehods o be ulsed desgg sble sldg surfes d flly he heory s supplemeed wh emples of ler me vr sysems... Se-Spe represeo of Dyml sysems I hs dssero ll sysems re preseed se spe d ll he dervos wll be obed by se spe lyss. ordg o Bks 986, p. some mpor oeps of se spe lyss re relled hs seo. dym ler sysem s of he form y Bu C Du. T m Where, K, R s he se veor, u u,, u m R pu veor. P T sysem he sysem respose s gve by vro of oss formul. T K s he orol or y, K, y p R s he oupu. For sgle pu-sgle oupu ler

22 Chper Sldg ode Corol of LTI Sysems ep ep[ ] b u τ τ dτ. However, geerl soluo of he sysem ses o be obed eplly wh he bove formul. I oher words, f he sysem s Ler wh me vryg prmeer or Noler we should use umerl mehods suh s Euler or Ruge-Ku mehods o ob he soluo. Euler mehod s effeve ool our lyss wll be dsussed more del seo... I s eessry o rell he followg defos h re used he e seos. For furher dels refer o Bks 986, p.4. Two mpor oeps ler sysems re hose of orollbly d Observbly. I he former se, we sy h sysem y Bu C Is Corollble f, gve R me suh h f s he soluo of he.9 srg.9,, here ess peewse ouous orol, he u d,.e. we drve y l po o y oher po. Smlrly,.9 s observble f me be foud suh h gve he oupu y d he pu u over he ervl [, ] l se be deermed. If we form he mres, R R o B C T B T C T B L C B T T L T T C The.9 s orollble f d oly f rk R 4

23 Chper Sldg ode Corol of LTI Sysems d.9 s observble f d oly f rk R o I ddo s sbly lyss for oler sysems s mpor pr of hs dssero followg defos of sbly re represeed hs seo. For furher dels refer o Bks 986, p.75. The oo of eress R be epressed erms of he blls B ε defed by { R } B ε : ε. Hvg rodued hs oep, ow we rodue prese defo of sbly. Sbly: The equlbrum se e of he sysem f, here ess δ > whh depeds o ε d suh h, B, s sble f for y ε > Bδ e ; ε e for ll.4 If δ s depede of, he sbly s sd o be uform. See fgure.: e B ε B δ e e Fgure -: Illusrg sbly 5

24 Chper Sldg ode Corol of LTI Sysems Bouded soluo: he soluos of f, > depedg o δ d suh h re bouded f for yδ >, here ess, B, B δ e ; e for ll.5 If s depede of, he soluos re uformly bouded. sympolly sbly: The se e s sympolly sble f s sble d for gve µ >, here ess δ > depede of µ d me T T µ, δ, suh h, B, Bδ e ; µ e for ll T.6 I oher words, gve y µ, o mer how smll, we fd suffely lrge T so h he soluo s I he bll B µ e of rbrry smll rdus µ. If he umber δ he bove defo be ke rbrrly lrge he e s sd o be sympolly sble he lrge. Epoel sbly: sysem s lled epoelly sble f here ess L, λ > suh h λ φ, Le for ll.7 where φ, des he rso mr of sysem. Lypuov fuo: fuo properes V : R R ssoed wh he sysem f, whh ssfes he V, > f d V, for ll V, b V V f, < 6

25 Chper Sldg ode Corol of LTI Sysems s lled Lypuov fuo. Lypuov s m sbly heorem: Cosder he sysem defed by he dfferel equo f,.8 d ssume h lol soluos es for ll l odos,. Suppose h he org s soled equlbrum po, so h f, for ll, d ssume h here ess fuo V : R R R suh h V s dffereble d ssfes V, s posve defe,.e. V, d here ess α C suh h V, > α > for ll d ll, b V V, V f < s egve defe so h here ess C d ll, here ess C d α s s s β suh h, < V β for ll γ suh h V, γ < The he org s uformly sympolly sble he lrge... Phse ple lyss Ths s useful ool lyss of wo-dmesol sysem se spe. phse porr s grphl represeo of seod-order sysem, where oe s he phse ple s poso d he oher s veloy. ordg o Bks 986, p.6 followg dervos re represeed. To rodue he phse ple osder he seod-order sysem. f,. 7

26 Chper Sldg ode Corol of LTI Sysems d defe where we my hk of s poso d s veloy. v he Provdg we kow he soluo of. wh l odos, we plo grphs of poso d veloy gs me. I geerl, however he soluos of. wll o be kow d so he rsformo below s rodued d he he equvle sysem. s obed. f,. The mp,, f., Is lled veor feld d ssgs o eh po he veor, f, R., soluo of. hrough gve po, s jus fuo, d we my plo suh soluo, spe s urve prmeerzed by. The wo dmesol ple, ossg of hese prmeerzed soluos s lled he phse ple. The fudmel propery of he phse ple s h f we plo he veor feld. he ple by drwg he veor, f, ge o he soluo rjeory d bse,. Phse ple lyss hs severl dvges. Frsly, vsulses he behvour of seod order sysem h helps wh beer udersdg of s properes. Seodly, be ppled o hghly oler sysems d does o hold y ssumpo ulsg 8

27 Chper Sldg ode Corol of LTI Sysems hs mehod. Flly d mos mporly, my prl sysems re ppromed by seod order dyml sysem whh mkes hs mehod ovee wy of orol sysems lyss. frequely eouered oep phse ple lyss s he so lled sgulr po. sgulr po s smply equlbrum po whh rs sysem s ses d s obed by: f, f, X. f, f, I lyzg orol sysems vrous ypes of odes re rodued. If he sysem hs wo rel egve egevlues sble ode wll be formed. If hs wo rel posve egevlues usble ode s formed. If oe of he egevlues s rel posve d he oher s rel egve sddle po s formed. d flly f he egevlues hve mgry pr depedg o wheher he rel pr s egve, posve or zero sble fous, usble fous or eer po s formed respevely. Furhermore, Phse Ple Trjeores erseo s o llowed d he followg dsusso holds: Suppose wo oupled dfferel equos, d / d f, y dy / d g, y.4 The sedy se soluo ssfy f, y, g, y dy dy / d d d / d g, y f, y dy d g, y f,, y y y 9

28 Chper Sldg ode Corol of LTI Sysems Theorem: rjeores ross oe oher oly sedy-se pos. dy Proof: phse spe, eep for he se where fg sedy-se pos, slope s d uquely spefed. If rjeores ross,, here would be more h oe slope y h po. Therefore, o rjeory rossg s llowed eep sedy-se pos... Numerl soluo of dfferel equos ly soluo of dfferel equos s o possble geerl. Therefore, we rell umerl mehod used hs proje. ssume h dfferel equo s represeed by: d d f,,, [, ].5 f The Tylor seres epso of.5 bou rbrry po me s: d d d L.6 d! d! d subsug.5.6: d f, L.7!! d f, f, ppromg f we hoose he frs wo erms of.7 : f,.8 I umerl smulo of equo.8, we ssume he sep sze o be h. Therefore: h h f, h.9

29 Chper Sldg ode Corol of LTI Sysems Equo.9 s lled Euler s mehod whh s wdely used solvg dfferel equos umerlly. Bslly, o ob soluo of he sysem me, over he me ervl of, ], provde we kow he l odo of he dfferel [ f equo, we e he soluo by subsug he l odo o.9 d solvg for he e sep ul we reh he fl po he ervl.

30 Chper Sldg ode Corol of LTI Sysems. Sldg ode Corol wh slr pus Cosder he followg dmesol ler me vr sysem wh slr pu. u L O L L. Le us represe equo. s: u η η η. Whe orol s slr: η. The sldg equo s defed by:... σ. Solvg he equo. for we ge:....4 Subsug.4 d. o he equo. we ge:

31 Chper Sldg ode Corol of LTI Sysems [ ] L [ ] L L O L L L O L L L O L L Hee we ob he followg redued order sysem: L O L L d bref form: ~.5 Equo.5 s mjor sep desg of sldg surfes. Sbly of hs redued order sysem s of prme mpore. pplyg vrous sbly ess o sysem.5 leds o eessry d suffe odo o sldg surfe oeffes h sblse he oml usble sysem.

32 Chper Sldg ode Corol of LTI Sysems 4 The sbly ssue of sysem.5 wll be ddressed he seo four of hs hper. I f, pplo of hree sbly lyss mehods sldg mode orol s demosred seo.4. ssumg h we hve obed he sbly odo we proeed o rodue he sldg mode orol h mus be ppled o he oml sysem. Rell he sldg equo.:... σ To ob suble pu ssume h sysem ses re o o he sldg surfe sgm. Therefore sgm s posve or egve. To brg he ses o hs surfe where sgm s zero d he moo s sympolly sble, dervve of sgm s se equl o mus sg of sgm.... σ σ sg Subsug from. for we ob: σ σ sg u Solvg for orol pu we ge: sg u... σ.6 Subsug he orol pu o he oml usble sysem resuls sble oler sysem o smule. sg L L σ.7

33 Chper Sldg ode Corol of LTI Sysems 5. Sldg ode Corol wh Veor pus Cosder he followg dmesol ler me vr sysem wh veor pu. u I η η.8 herefore, m m m where η η.9 I se of veor pus he sldg surfes wll be defed s followg: Σ m m m m m m m m m m m m L L L σ σ σ Or more omp form he sldg surfe s defed s: Σ η C.

34 Chper Sldg ode Corol of LTI Sysems 6 Solvg he equo. for η we ge: m m m m m m m m m L O L L I ose form η C So he dfferee here wh slr se s h ow we hve mr of oeffes defg our sldg surfes. To ob he redued order sysem, osder he followg dervo. η C C ~ The redued order sysem s of dmeso m d he sbly lyss should be ppled o hs sysem o ob he odos for sble sldg surfe. To ob he orol pu, osder he proedure below: Σ Σ sg C η where, Σ m sg sg sg sg σ σ σ Subsug from se spe equos o he bove equos we ge:

35 Chper Sldg ode Corol of LTI Sysems C η η U sg Σ Rerrgg he bove equo we ob he veor of pus: U Σ C η η sg. Flly he Sldg ode Corolled sysem equos re s followg: η η sg Σ C η. Noe h he oml sysem hs ler me vr equos bu he sldg mode orol sysem s oler..4 Sbly lyss of me vr sldg surfes I hs seo hree mehods of lysg sbly wll be vesged. Nmely, Pole pleme, Lypuov pproh d ew suffe odo for sbly of ler sysems s rodued..4. Pole Pleme I hs seo desg mehod lled pole pleme wll be used obg sble sldg surfes. Ths s oe of he desg mehods for ler sysems whh s bsed o plg losed loop poles desred loos. I s ssumed h he oml usble sysem ssfes he Corollbly d Observbly properes. Le us ssume h he desred losed loop poles re o be s µ, µ, K s m where m s dmeso of he redued order sysem. By hoosg ppropre g mr for sldg surfe, s 7

36 Chper Sldg ode Corol of LTI Sysems possble o fore he redued order sysem whh represes he sldg surfes o hve losed loop poles desred loos. I hs seo pplo of pole pleme o seod redued order sysem s represeed d he dsusso s oluded by emple seo.5. Cosder he followg redued order sysem: ~. ssume h he desred poles re loed λ,λ. Ths wll led o he followg hrers equo. where λ p d q λ. λ ~ de λi λ λ λ λ λ λ pλ q subsug ~ from. o he hrers equo, we ge: λ de λ pλ q λ Rerrgg he equos bove d pplyg Crmer s rule resuls : 8

37 Chper Sldg ode Corol of LTI Sysems p q λ λ λ λ p q pplyg Crmer s rule for obg he ukow sldg surfe oeffes we ge: p q.4 q p.5 emple s gve seo 5 of hs hper o llusre use of pole pleme for desgg sldg mode orol. However s lmos re lso llusred whh moves he ew suffe odo for sbly proposed seo.4.. Toms- Rodrguez d Bks 6, pp hve show he use of pole pleme mehod for oler sysems wh use of ero ehque. 9

38 Chper Sldg ode Corol of LTI Sysems.4. Lypuov pproh s sed he roduo f we fd Lypuov fuo he he sysem s sble. I hs seo we rell mehod of geerg Lypuov fuo. Le us osder he followg ler me vr sysem:. I Lypuov pproh we defe he followg fuo: V P The bove fuo s qudr form d we would lke o hve he dervve o be egve defe. V P P P P P P Gve y posve defe symmer mr Q, The equlbrum se of hs sysem s sympolly sble f d oly f here ess posve defe symmer mr P suh h P P Q We solve for P eplly he bove equo o ob he Lypuov fuo V. Noe h P s posve defe symmer mr so o lule P for dmesol mr ~ we should ob ukows. To demosre use of Lypuov mehod obg sldg modes emple s provded. Suppose we would lke o desg sldg mode orol for he followg hree dmesol ler me vr sysem b u Where, b d re oss d s dffere from he C used sldg surfes.

39 Chper Sldg ode Corol of LTI Sysems The sldg surfe s defed s:... σ Where s he oml sysem dmeso. Here he dmeso s so he sldg surfe s: σ Redued order sysem s: C ~ ~ pplyg he Lypuov pproh: p p p p p p p p p p p p p p p p p p p p Q P P Hee P s posve defe symmer p p p p p p p p p p p p p p p Therefore,

40 Chper Sldg ode Corol of LTI Sysems P So he problem of obg he odo o sldg oeffes redues o pplyg odo of posve defeess o mr P. Defo of posve egve defe mr s gve s: If j s mr he defe k j k k j The s posve defe f d oly f de > k k The s egve defe f d oly f de > k k k Cosderg he odo of P beg posve defe we ob he followg odo o sldg surfe oeffes: > > de de y oeffe ssfyg odos oe d wo obed he Lypuov pproh wll resul sble sldg moo. However, he desger should hoose he sldg oeffes h orrespod o he desg rer. Tme vryg verso of Lypuov mehod wll be osdered he hper hree.

41 Chper Sldg ode Corol of LTI Sysems.4. ew suffe odo for sbly of ler sysems Pole pleme d Lypuov pproh re boh very useful pprohes ler me vr sysem bu se of ler me vryg sysem we o pply hem wh ou pug odos o he sysem beg bouded d slowly vryg. Thus, ew suffe odo s rodued hs seo o ob sbly odo o he sldg surfes. I he -dmesol Cse he followg equos hold:... : : : : : : The sldg equo s... σ [ ] K K K K K O K K d ose form:

42 Chper Sldg ode Corol of LTI Sysems ~ ~ ~ ~ ~ ~ K K O K ~ ~ ~ So f we fore dervve of he Euld orm of se veor of he redued order sysem o be egve defe we hve obed he sbly for hs sysem. d d T ~ [... ] ~ ~ : ~ ~ ~ : ~ ~ ~ ~ : : fer mulplo we ob: ~ ~ ~ ~... ~ ~... ~.6 ~ ~ ~ ~ s meoed before, he de here s o fore dervve of he Euld orm of he ses veor equo.6 o be egve defe. So f he dgol erms he mr ~ re egve defe d he sum of symmer off-dgol erms s zero he hve ssfed he odo of os dey of he sysem ses owrd he equlbrum po. pplyg hs de resuls obg he followg wo se of odo for sbly of -dmesol sysem: 4

43 Chper Sldg ode Corol of LTI Sysems Codo Se I: ~ <,,...,.7 Codo Se II: ~ j ~ j j j j j,...,, j,...,, j.8 Codo se I wll resul equos. equly d odo se II wll resul I should be sressed h o derve he bove odo o ssumpo s mde bou he ler sysem prmeers. 5

44 Chper Sldg ode Corol of LTI Sysems.5 Emple of desgg sldg mode orol for seod order sysem For he purpose of llusro osder he followg seod order sysem: b u Where, b re oss. Le The sldg le be defed s: We w he sysem o be globlly sympo sble, herefore he ses mus reh he sldg le fe me d oe hey h he sldg le, sy o. Oe he ses h he sldg surfe hey move owrd he org. Therefore, he frs ple he sldg le s defed d subsequely he sbly odo for he redued order sysem s obed. Subsug from he se spe equos o he sldg le equo we ob he equo below whh s used o ob he sbly odo: Ths equo wll be sble for ll vlues of f >. Oe we re ssured of sbly o he sldg surfe he e sep s o desg he orol pu o fore he ses o h he sldg surfe. The followg proedure s followed o ob he suble orol pu. Le σ 6

45 Chper Sldg ode Corol of LTI Sysems To ob suble pu we osdered he f h f sysem ses re o o he sldg le he sgm s posve or egve. To brg he ses o hs le where sgm s zero, dervve of sgm s se o mus sg of sgm. dσ.e. sgσ or σ sg σ d Ths s equvle o: σ σ < Subsug from he se spe equos dervve of sldg surfe obed: σ. b u sg σ Therefore u σ b sg So f we subsue u o he oml sysem we ge he followg oler sysem: sg σ I geerl hs be wre s: X f, sg σ To solve he bove oler dfferel equo bove umerl mehod suh s Euler or Ruge-Ku should be ppled. 7

46 Chper Sldg ode Corol of LTI Sysems Smulg he sysem bove wh hoe of we ob he followg phse porr. Noe h oler dfferel equo bove s solved wh Euler umerl mehod. Fgure -: Phse porr of sldg mode orolled equo wh I he fgure - herg pheomeo h ours sldg mode s deped. I resuls from fs swhg behvour of he ses o sldg surfe. 8

47 Chper Sldg ode Corol of LTI Sysems Fgure -: Cherg pheomeo for rjeory srg from l odo of [; ] Ths udesred moo of ses he eghbourhood of sldg surfe be del by rodug suro fuo or vrous oher mehods. If we osder he orol o urs ou h he hgh frequey orol o oly kes ple whe rjeory hs he sldg surfe. Fgure -4: Sldg mode orol pu. 9

48 Chper Sldg ode Corol of LTV Sysems 4 Sldg ode Corol of Ler Tme Vryg Sysems. Iroduo I Ths Chper we wll be oered wh he ssue of pplyg sldg mode orol o ler me vryg sysems whh forms he foudo for delg wh oler sysems. s mjor pr of desgg sldg surfes, sbly of me vryg sysems wll be vesged. Sbly of me vryg sysems s muh more dfful h he me vr oes d s of rul mpore my res of orol sysems heory ludg dpve sysems d lyss of Noler Corol sysems. me-vryg orol sysem by defo s sysem for whh oe or more of he prmeers of he sysem my vry s fuo of me. I hs hper we sr wh eedg sldg mode orol heory o ommode ler me vryg sysems wh slr d veor pus. I seo.4 we vesge sbly of ler me vryg sysems d o olude wo emples re gve o demosre he heory.. Sldg ode Corol of LTV sysems wh slr pu Cosder he sysem below h s represeed se spe form: Bu X X where oe or more elemes of mr my vry wh me. u K O K K. Equo. be epressed s:

49 Chper Sldg ode Corol of LTV Sysems η u η herefore η. Se orol s slr: η. The sldg equo s:... σ.4 Here he sldg surfe vres wh me. Solvg he equo.4 for we ge:....5 Subsug.5 d. o he equo. we ge: [ L ] Hvg md h ll he eleme he mres below re me vryg we om he me rgume for smply. 4

50 Chper Sldg ode Corol of LTV Sysems 4 [ ] L O L L L O L L L L O L L Hee we ob he followg redued order sysem: L O L L From hs po we should be oered wh sbly of he followg sysem: ~.6 The ssue of sbly of me vryg sysems wll be ddressed he seo.4 bsed o Pole pleme, Tme vryg verso of Lypuov d he suffe odo for sbly of ler sysems. Wh ssumpo of hvg he sldg surfes desged, we shll proeed o ob sldg mode orol pu. Rell he sldg equo.4,... σ Noe h sldg surfe oeffes re ow fuos of me d we eed o osder hs dffereo: σ σ sg Subsug from se spe equo. for we ob:

51 Chper Sldg ode Corol of LTV Sysems 4... σ σ sg u Solvg for orol pu we ge:... sg u σ.7 Subsug he orol pu o he oml usble sysem resuls sble oler sysem. sg L L σ.8 Se of Equos.8 re he sldg mode orolled sysem h s used he smulos. Ths heory wll be suppored by severl emples provded seo 5 of hs hper.

52 Chper Sldg ode Corol of LTV Sysems 44. Sldg ode Corol of LTV sysems wh veor pu ssume h he oml usble sysem s epressed s: u K O K K.9 I omp form: u I η η The redued order sysem s: η. Dmeso of he veor pus s m where s he dmeso of oml sysem d m s dmeso of. Rellg he sldg surfe defo from equo.: Σ η C However for me vryg sysems he mr C os me vryg elemes whh should be osdered he dervos. Dffereg he sldg surfes we ge: Σ Σ sg C C η Subsug from se spe equos o he bove equos we ge: Σ sg U C C η η Rerrgg he bove equo we ob he veor of pus:

53 Chper Sldg ode Corol of LTV Sysems U Σ C η C η sg. Flly he Sldg ode Corolled sysem s gve by.. η η sg Σ C η C. Ths se of equos smules usble LTV sysem whh s orolled d sblsed by sldg mode orol..4 Sbly lyss of me vryg sldg surfes Sbly lyss for ler me-vryg sysems s of resg eres orol heory. Ths ssue rses severl pplos suh s dpve orollers, me vryg properes of ommuo hels d sudy of oler sysems bsed o ppromo wh sequee of ler me vryg oes whh s he subje of eres hs dssero. So fr, mos mpor resuls obed hs feld s by osderg slowly vryg sysem s oed by Desoer 969,pp. 78- or by pug bouds o he me vryg pr or re of hge of he dgolsg mr s s sed he e seo. I hs hper we wll lso vesge pplo of pole-pleme, Tme vryg Lypuov pproh d he ew mehod rodued he seod hper..4. Pole Pleme The de of ssgg desred poles o he redued order sysem s sed hper wo d hs seo we ry o eed he de o ommode me vryg ler sysems. I hs seo we osder sbly for he followg redued order me vryg sysem. ~ Pole ssgme volves solvg he followg equo, 45

54 Chper Sldg ode Corol of LTV Sysems 46 d d d I λ λ λ λ λ λ λ K ~ de. where d d d λ λ λ K, re he desred egevlues. I s eessry o emphsse h pole ssgme s ppled o he redued order sysem o ob sble sldg surfe. I LTI sysems hvg lef hlf ple poles gurees sbly bu hs s o he se LTV sysems d hs s show by he followg emple. Cosder he followg wo dmesol sysem: e.4 Obg he egevlues we ge: de λ λ λ λ λ λ λ e I Ths sysem hs lef hlf ple egevlues bu smulg he sysem, we ge he followg swer whh shows he sysem s usble. Fgure -: Usble soluo of me vryg sysem.4 wh egve egevlues.

55 Chper Sldg ode Corol of LTV Sysems I be show h LTV sysems f he egeveors vry slowly we pply pole pleme o sblse he sysem. Cosder he me vryg sysem gve below: X P Y X P Λ P X Y PX PX Y PP Y PP Y Λ Y PP Y Y ordg o bove dervos f hge P s o muh he he dgol pr wll be dom d h s wh s me by slowly vryg sysems. f he sysems we smule ssfy he odo below, hey be sblsed by pole pleme. P P << Λ.5 Whe he dgol pr s dom he bove mr problem wll redue o severl slr me vryg equos whh ssfy he followg oluso. Suppose he followg slr me vryg sysem. Sbly s heved f: e s. ds s. ds Wh he bove dervos me vryg slr equo wll be sble. Noe h sldg mode orol whe he redued order sysem s o fuo of me we del wh he sbly se smlr o LTI sysems. Emples of pplyg hs mehod re gve he seo.5. 47

56 Chper Sldg ode Corol of LTV Sysems.4. Lypuov lyss of Tme vryg sysems s show he prevous seo hvg ll he egevlues of he sysem lef hlf ple does o guree sbly for me vryg sysems. Sloe d L 99, p.4 suggess he followg de for sudyg sbly of ler me vryg sysems se spe. ordg o hs pproh ler me vryg sysem s sympolly sble f T egevlues of he symmer mr sys srly he lef hlf ple,.e. T λ >,,, λ λ.6 Ths s proved by usg Lypuov fuo V T V T T T T T λ λv whh shows h he dervve of Lypuov fuo s egve defe. Suh resul be eremely helpful pole pleme desg of sldg surfes. O he oher hd we show h he oveol mehod of fdg epl Lypuov fuo for me vr sysems does o hold he me vryg se. ssume he followg LTV sysem: ~ Defe he Lypuov fuo s: T V P Tkg derve d seg o be egve defe: V T T P P P T ~ T T T ~ P P P Rerrgg he equo: V T ~ T ~ P P P ~ T ~ P P P I.7 48

57 Chper Sldg ode Corol of LTV Sysems 49 Noe h P s obed from elemes of ~, d se elemes of ~ re fuos of me, P wll lso deped o me. Obg epl soluo for dfferel equo.7 wh ukow me vryg prmeers s eremely ompled d eourges us o vesge umerl mehods for s soluo..4. ew suffe odo for sbly of ler sysems Ths s he mos effeve mehod vesged hs proje for sbly lyss of me vryg ler sysems. Le us brefly rell he resuls we obed hper wo. I he -dmesol Cse he followg equos hold:... : : : : : : The sldg equo s... σ he resulg redued order sysem ose form s: ~ ~ ~ ~ ~ ~ ~ ~ ~ K O K K So ~ For smply we om he me rgume he followg represeos. ~ ~ ~... ~ ~ ~ ~ ~... ~ ~ ~ < < d d

58 Chper Sldg ode Corol of LTV Sysems Codo Se I: ~ <,,...,.8 Codo Se II: ~ ~ j j j j j j Codo se I wll resul equos.,...,, j,...,,.9 equly d odo se II wll resul Noe h hs ewly rodued suffe odos does o hold y ssumpo o me vryg prmeers, hus y me vryg ler sysem ssfyg he bove se of odos s sblzble. Ierpreo of he bove mehod gudes us hoosg he bes sldg oeffes by orporg he odo se I. Oe should oe h f he mkes hs eleme o derese fser, fser respose wll resul. Ths s show he emple seo of hs hper. j 5

59 Chper Sldg ode Corol of LTV Sysems.5 Emples of desgg sldg mode orol for LTV sysem Now, le us pply pole pleme d he ew mehod o he sysem gve below o ompre he effeveess of hese mehods..5. Pole pleme emple Now osder he followg hree dmesol me vryg sysem: s ω s ω os ω u. For smply le us osderω, s s os The sldg equo s defed s: σ ssume h he desred egevlues re: λ, λ Subsug elemes of he formule.4 d.5 we ge: q p s s s s s s s fer smplfo he frs oeffe s: 5

60 Chper Sldg ode Corol of LTV Sysems s s d 6 Cos s s s d for he seod oeffe we ge: q p s s s s s s s s s s d os s Noe h wh he bove sldg surfe obed he redued order sysem s: s. s 5

61 Chper Sldg ode Corol of LTV Sysems 5 Before obg he sldg mode orol pu le us smule he sldg surfe obed. See fgure.. Fgure -: smulo of sldg surfe. obed by plg he sysems poles - d -. I shows h lhough he sldg surfe equo ssfes he desred egevlues bu he me vryg pr s dom d produes he spkes show he fgure. Now le us ob he sldg mode equos o smule he sysem: os s s σ σ σ σ sg u sg Therefore, os s s sg u σ So he sysem o be smuled s gve by: s s sg s s σ.

62 Chper Sldg ode Corol of LTV Sysems The resul of smulo s gve fgure. Fgure -: llusro of he sldg mode orolled sysem. whh s o sblsed. Resul of smulo shows h he me vryg pr s dom hs se d so we o orol hs sysem wh pole pleme. I ddo s show h f we rese he mgude of os prmeers relve o he me vryg prmeers equo., he we ob suessful sldg mode orol for he ew sysem. s ω s ω os ω u. desred hrers equo s: λ p λ q λ λ Subsug elemes of he formule.4 d.5 we ge: p q 54

63 Chper Sldg ode Corol of LTV Sysems 7 s s s s s s s fer smplfo he frs oeffe s: d 6 Cos s s s s s d he seod oeffe s: q p 7 s s s s s 7 s s s s 7 d 7 os s s The redued order sysem s: s.4 s Smulg hs sysem o vesge s sbly we ob fgure.4. 55

64 Chper Sldg ode Corol of LTV Sysems Fgure -4: smulo of sldg surfe.4 obed wh pole pleme for sysem.. The resul of smulg he sldg mode orolled verso of sysem. s deped fgure.5. Fgure -5: deps he sble sldg mode orolled of he meded me vryg sysem. Ths shows he sysem s sble d he os pr s dom o he me vryg pr. However, here ess some sedy se error oe of he sysem ses. 56

65 Chper Sldg ode Corol of LTV Sysems.5. Emple of pplyg he ew suffe odo The Sysem s defed by he followg equos: s ω s ω os ω u wh ω d s s os α ~ [ ] Therefore: ~ s s s ordg o odo se II: d s s os s s s d bsed o odo se I: < s s < 57

66 Chper Sldg ode Corol of LTV Sysems so we hoose: 4 s.5 d os s 4 The redued order sysem obed wh he seleed vlues for sldg surfe oeffes s: s s.6 s 4 s Smulo of sldg surfe obed wh hs mehod s: Fgure -6: smulo of redued order sysem.6 obed by he ew mehod. 58

67 Chper Sldg ode Corol of LTV Sysems I ddo o rese he speed of overgee we hoose he seod sldg surfe oeffe s:.7 4 s Equo.7 resuls beer sldg moo. Fgure.7 s obed for l odo of [;-] Fgure -7: smulo of he seod sldg surfe.7 obed by he ew mehod. Smulo of hs sysem for he frs sldg surfe s gve fgure.8. 59

68 Chper Sldg ode Corol of LTV Sysems Fgure -8: sldg mode orol of he orgl me vryg sysem usg.5. Smulo resuls, by ulsg he seod sldg surfe s llusred fgure.8. he dfferee speed of respose s obvous d suggess he lose reloshp bewee hoe of sldg surfe oeffes d he hrerss of sysem respose. Fgure -9: Sldg mode orol of he orgl me vryg sysem usg.7. 6

69 Chper Sldg ode Corol of LTV Sysems 6 Impore of hoe of he sldg surfe oeffes s demosred hs emple. Thus o ob he performe rer requred from he desger he should follow he gudele provded he heory. Ths resul jusfes he effeveess of hs mehod omprg o pole pleme d provdes us wh good lerve obg sble sldg surfes he lss of sysems h re o slowly vryg d does o ssfy he pole pleme mehod rer. I would lke o po ou h hs mehod desger hs more freedom hoosg he desred sldg surfes s we ob he odos form of equles so depedg o he pplo oe hoose he bes sldg moo mog my sble sldg surfes. I ddo, emple of 4 h order sysem usg he ew mehod s provded hs seo. For he purpose of llusro osder he followg 4 h order me vryg sysems: 4 4 os os s s e e ω ω ω ω So obg ~ we wre: _ os s s ~ e ω ω ω Fgure. represes resul of smulo for l odo of [;;]:

70 Chper Sldg ode Corol of LTV Sysems Fgure -: smulo of he redued order sysem represeg he sldg moo. Ths shows h he sldg surfe desged s sble. Resul of smulo of he 4 h order sldg mode orolled sysem for he l odo of [ ;;; ] s deped fgure.. Fgure -: sldg mode orol of he fourh order sysem. These resuls show h he usble 4 h order me vryg sysem s sblsed. 6

71 Chper 4 Noler Sldg ode Corol 4 Noler Sldg ode Corol 4. Iroduo Noler orol heory s re of resg eres orol sysems egeerg. os ools ulsed ler sysems heory o be used for orol of globl behvour of oler sysems. I hs hper we wll ulse reely developed heory for orol of oler sysems. Ths heory s rodued by Toms-Rodrguez d Bks, pp. 89- d s bsed o replg he orgl oler sysem wh seres of ler me vryg sysems whh fer few eros wll eveully overge o he orgl oler sysem. Oe of he mos rgug spes of hs mehod s h ler orol heory be ulsed o del wh y omple oler sysem h ssfes mld Lpshz odo. I hs hper frs we rodue he ero ehque seo oe d he we demosre s use orol of hree well-kow oler sysems, V der Pol Osllor, Duffg equo d Lorez ror seos hree, four d fve respevely. 4. Iero ehque for oler dyml sysems I hs seo we rodue reely developed ehque for lyss of oler dyml sysems whh he oler equos s repled by sequee of ler me-vryg equos. If mld Lpshz odo s ssfed, hs soluo overges he spe of ouous fuos o he soluo of orgl oler sysem. Lol d globl overgee of hs mehod s proved by Toms-Rodrguez d Bks, pp Cosder he oler sysem of he form: f, R 6

72 Chper 4 Noler Sldg ode Corol The oly odo o s o hold he Lpshz odo whh s mld odo d my physl sysems ssfy. Lpshz odo s epressed s: y α y where α s posve slr. Therefore he bove sysem s ppromed by he followg sequee of LTV sysems: [ ], [ ], For he soluos of hs sequee of LTV equos overge o he soluo of he oler sysem preseed bove. I s mpor o oe h re of overgee of eros depeds o hoe of l odo. hs sge hvg suded he sldg mode orol ppled o ler me vryg sysems prevous hper d rodug hs mehod we pply sldg mode orol o oler sysems. 64

73 Chper 4 Noler Sldg ode Corol 4. Sldg mode orol of V der Pol Osllor I hs seo we beg wh rodug V der Pol osllor equos d s phse porr. Subsequely, we proeed o ulse ero ehque o solve V der Pol Osllor equo d ompre wh he orgl oler soluo. Ths s followed by, pplyg he sldg mode orol represeg he resuls. V der Pol s equo s he form of µ where µ s os prmeer d we wll hoose o be oe our dervos. V der Pol s Osllor se spe form s: u 4. Phse porr of V der Pol s Osllor for vrous l odos s gve below: Fgure 4-: V der Pol s Osllor phse porr for µ. pplyg ero ehque o equo 4. we ob he followg resul. 65

74 Chper 4 Noler Sldg ode Corol 66 [] [] [ ] [] [] [] [] [ ] [] [] [],, u u Resul of smulo ompres phse porr of orgl oler equo wh he ero ehque for 4 d eros s deped fgure 4. d 4. respevely. Fgure 4-: solvg V der Pol s equo wh ero ehque for l odo of [; ]. Fgure 4-: V der Pol s equo wh ero for l odo of [; ].

75 Chper 4 Noler Sldg ode Corol 67 I fgure 4-, he sold urve represes oler sysem d he dshed urve s he h ero of he sysem for l odo of [; ]. If we drw mmum orm of error eh ero bewee he orgl oler sysem d he ppromo, he we would ob he followg fgure whh mples overgee of hs pproh. Ths resul s llusred fgure 4.. Fgure 4-4: mum orm of error versus eh ero. Resuls obed mply h he ered soluo ppromes he oler sysem very losely. pplyg he sldg mode orol o V der Pol osllor: [] [] [ ] [] [] [] [] []. sldg equo u σ d he slr redued order sysem s: [] [ ] [] []

76 Chper 4 Noler Sldg ode Corol [] [] [] [] [ ] [] [] [] [ ] For sbly of he redued order model ordg o odo se I: [ ] [] k k [] [ ] k > k [] [ ] k k < [] [ ] k k [ ] k I should be oed h, se he smulo s beg performed by dgl ompuer so we should oe h he lulos re dsree sequee of umbers o ouous fuos of me. I hs seo he frs phse of he desg s performed d sble sldg surfe s obed. Subsequely, we eed o ob he sldg orol pu phse wo of he desg. Corol pu: σ σ [] [] [] [] [ ] sg σ [] [] [] [] [] k k k k k sg σ d flly he sldg mode orol s: [ ] [] [ ] [ k k k ] k [ ] k u sg σ 4. Sysem o be smuled s: 68

77 Chper 4 Noler Sldg ode Corol [] [] [] [ ] [] [] [] [] [ sg k k k ] σ [] [] 4. Frs osder he resul llusred fgure 4.5 obed wh wo eros me dom. I be observed h here s eglgble devo he ppromed swer ompred o he oler sysem s soluo. I fgure 4.6 s four eros re used, he ppromed swer s more prese d ompleely resembles he oler sysem s soluo. The oler soluo boh ses s obed by pplyg he sldg surfe obed he ls ero o he orgl oler sysem. Furhermore o llusre globl sympo sbly of hs mehod, phse porr of hs sysem s represeed for wo vrous sldg surfes. Illy we hve hose [] [ ] d he smulo resul s gve fgure 4.7. I he seod se, he sldg surfe ppled s hged o [] [ ] d he orrespodg resul s deped fgure 4.8. The dfferee bewee hese wo sldg surfes be observed lerly. 69

78 Chper 4 Noler Sldg ode Corol Fgure 4-5: Sldg mode orolled V der Pol Osllor wh l odo of [; ]. Fgure 4-6: Sldg mode orolled V der Pol Osllor wh l odo of [; ]. 7

79 Chper 4 Noler Sldg ode Corol Fgure 4-7: phse porr of he oler S. orolled sysem wh s sldg surfe. Fgure 4-8: phse porr of he oler S.. orolled sysem wh d sldg surfe. Wh hoe of he ler C fgure 4.8 smooher phse porr s obed ompred o fgure 4.7. Therefore, depedee of he sysems oupu respose o hoe of sldg surfe s lerly observed hs se. 7

80 Chper 4 Noler Sldg ode Corol Sldg mode orol of Duffg Dfferel Equo Duffg Dfferel Equo s oe of he well kow oler sysems pble of geerg hos. Duffg equo epresses mss, sprg, dmper sysem d s used sudy of mehl vbros. Duffg equo: u y d d m l y K d dy /... y Le u d d m l K d d /... Desrpo of he sysem bove se spe s s followg: [] [] [ ] [ ] [] u m l m K m. / / / 4.4 [] [] [] [ ] [ ] [ ] C C.. σ Obg he odo for sbly: [] [] [ ] [ ] [] [] [] [] [] [] [] [] [] [ ] [] [] [ ] [] [] [] [ ] [] [] [] [ ]. / /. / / /.. / /. /. / /. /... / / / m l m K C C m l m K m C m l m K m m l m K m C C u m l m K m σ

81 Chper 4 Noler Sldg ode Corol Therefore he Redued order sysem s defed s followg: [] K / m l / m [] [ ] / m. hs sge we ob he pu o he sysem o smule he SC Duffg Equos: σ C. [] [] [] [] [. C ] sg σ Subsug from se spe equos: u [] [] [ ] [ ] [ k sg σ C k. k C k ] k C [] k C [] k C [] k Therefore: [] K / m l / m. [] [] [ ] / m [] [] [] [] [] sg σ C k. k C k 4.5 The phse porr of he orgl oler sysem s preseed fgure 4.8. Fgure 4-9: Phse porr of Duffg dfferel equo. 7

82 Chper 4 Noler Sldg ode Corol Ulsg ero ehque he followg resul mples he overgee of he soluo o he orgl oler sysem. I fgure 4. he oler sysem soluo s esmed by s, egh d welve eros h re dsgushed by dshed pk le, doed blue le d dsh-doed red le respevely. ordg o fgure 4. ero welve lmos resembles he oler sysem s soluo. Fgure 4-: Covergee of eros o he orgl oler soluo. Le us pply he sldg mode orol for wo l odos of [; ] d [-; -]. Subsequely we pply he resulg sldg surfe o he oler sysem. Ths resul s deped fgure 4-. Noler sldg surfes re obed by hree, s d welve eros h re dsgushed wh gree, pk d red olours respevely. Sldg surfe of he welfh ero s ppled o he orgl oler sysem whh s lmos fed o he ppromed sldg mode orol obed wh welve eros. I fgures 4. d 4. he swer obed by hree eros shows he mmum devo from 74

83 Chper 4 Noler Sldg ode Corol he orgl oler soluo. I ddo he sh ero demosres he red of overgee o he orgl oler sysem s soluo. Fgure 4-: Comprso of S.. orolled oler sysem wh sequee of LTV sysems. Fgure 4-: Comprso of S. orolled oler sysem wh sequee of LTV sysems. 75

84 Chper 4 Noler Sldg ode Corol To llusre globl sbly of he sldg mode orolled Duffg dfferel equo phse porr s preseed fgure 4.. Ths phse porr s obed by usg ero ehque for vrous l odos. Fgure 4-: Phse porr of sldg mode orolled Duffg dfferel equos. I s mpor o oe h phse porr of fgure 4- llusres he esee of severl sldg rjeores ledg o org. Ths s resul of hvg he sldg surfes desged bsed o l odo d soluo of he prevous ero, whle usg he ero ehque. 76

85 Chper 4 Noler Sldg ode Corol Sldg mode orol of Lorez ror Oe of he mos well-kwo oler dyml sysems pble of geerg hos s Lorez ror. Ths emple s osdered s emple for hrd order oler sysem. Equo of hs dyml sysem geerl form s:,,, < > > b XY bz Z r XZ Y rx Y Y X X σ σ σ 4.6 The sble vr se of Lorez ror s gve by he equo: u 5 By defo vr se s se for whh eh soluo srg he se sys h se for ll mes.lorez ror se spe form s: u To beome more fmlr wh hs sysem he phse porr of he sysem s: Fgure 4-4: Phse porr of Lorez ror for l odo of [; ; ].

86 Chper 4 Noler Sldg ode Corol To desg he sldg mode orol for he sysem we follow he seps below. Sldg surfe s defed s: σ For sbly we mus form he redued order sysem: 5 5 ~ 5 Codo Se I: ~ ~ 5 5 / Codo Se II: ~ ~ < < < < Frs of ll we smule he orgl Lorez equos usg he ero ehque d he ompre o he orgl oler soluo. To use he ero ehque he sysem s preseed he followg form: 78

87 Chper 4 Noler Sldg ode Corol 79 [] [] [] [ ] [] [] [] [] [] [ ] [ ] [] [] [] The followg ppromo of he oler sysem s obed fer eros d s losely rg he orgl oler mfold. Fgure 4-5: Phse porr obed by eros ompred wh oler sysem for [;;]. Error eh ero wll derese so h eveully fer few eros he soluo overges o he orgl oler sysem.

88 Chper 4 Noler Sldg ode Corol Fgure 4-6: mum error versus umber of ero. I be see from he resuls bove h ero ehque s overgg o he orgl oler sysem soluo fer few umber of eros. Seodly, we smule he redued order sysem o hek he sbly odo of he sldg surfe. The redued order sysem fer subsug he hose sldg surfe oeffes s: 4.9 Fgure 4.5 obed for dffere l odos llusres sbly of sldg mode orol surfe. 8

89 Chper 4 Noler Sldg ode Corol 8 Fgure 4-7: Phse porr of he redued order sysem 4.9 hs sge we derve he sldg mode orol pu d he sysems equos o smule he resulg orolled sysem. The sldg surfe for hs hree dmesol oler dyml sysem s defed s: σ kg he dervve we ge: σ σ sg Subsug from se spe equos we ge: 5 σ σ sg u Therefore, he orol pu s obed s: 5 sg u σ 4.

90 Chper 4 Noler Sldg ode Corol Subsug he sldg mode orol pu o he sysem we ge: 5 sg σ 5 4. Smulo resul of hs sysem frs, seod d hrd ero s llusred fgures 4.6, 4.7 d 4.8. Fgure 4-8: Frs ero of Lorez ror sldg mode orol smulo for [; ; ]. Fgure 4-9: Seod ero of Lorez ror sldg mode orol smulo for [; ; ]. 8

91 Chper 4 Noler Sldg ode Corol Fgure 4-: Fourh ero of Lorez ror sldg mode orol smulo for [; ; ]. We see h fer four eros he swer s overgg o he oler sysem d he ses re drve o zero. s meoed erler hoe of sldg surfe oeffes s mpor hevg he performe rer requred from he orolled sysem. 8

92 Chper 5 Coluso d Reommedos 5 Coluso d Reommedos 5. Coluso I hs dssero, we hve osdered pplo of sldg mode orol heory ler me vr, ler me vryg d oler sysems of rbrry dmeso whou holdg y pror ssumpo bou he ype or lss of he oler sysem. ordgly, sldg mode orol of ler me vryg sysems d he omplos ssoed wh hem s llusred hper hree. I f, our eres sldg mode orol of ler me vryg sysems sems from he reely rodued pproh lled erve ehque by Toms-Rodrguez d Bks, pp Ths pproh provdes he foudo for sudyg of oler sysems whle ulsg ler orol heory mehodologes. I vew of hs, s he orgl sysem s repled by sequee of ler me vryg sysems, sbly sudy of me vryg sysems beomes rul. Erler reserh hs re employs pole pleme ehque whh ssumes h he sysem o be orolled possesses slowly vryg prmeers. I oher words, hs mehod oly be ppled o lmed lss of ler me vryg sysems. However, we hve rodued ew suffe odo s lerve for pole pleme o ob sbly o lrger lss of ler sysems. I should be emphszed h hs ew odo does o mpose y lmos o he pe of sysem s prmeer vro. For he purpose of llusro severl emples re gve o orol fs prmeer vryg sysem, where pole pleme o be used bu he ew suffe odo provdes very ssfory resul. Cosequely, hvg preseed he bkgroud kowledge uderlyg ero ehque hper hree, we hve ppled he sldg mode orol o hree well kow oler sysems, mely V der Pol s Osllor, Duffg dfferel equo d Lorez ror. Se he pproh used orol of oler sysems s provdg us wh ssfory resuls, fd wde rge of pplos orol egeerg where employg robus orol ehques s essel. 84