Assessment of He's Homotopy Perturbation. Method for Optimal Control of. Linear Time-Delay Systems

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1 Alie Mahemaical Scieces Vol. 7 3 o Assessme o He's Homooy erurbaio Meho or Oimal Corol o iear ime-delay Sysems Arma Sargolzaei Kag K. Ye Shiri Noei 3 a Hamireza Ramezaour 4 Dearme o Elecrical Egieerig Floria Ieraioal Uiversiy Miami Floria USA a.sargolzaei@gmail.com 3 abriz Uiversiy abriz Ira 4 übige Uiversiy übige Germay Absrac his aer reers a oimal regulaor or a sysem wih sae ime-elay a quaraic erormace ie. Firs by meas o Maimum ricile a ecessary oimal coiios a coiues-ime wo-oi bouary-value roblem BV icluig boh ime-elay a ime-avace is erive. he usig Homooy erurbaio Meho HM he obaie BV is rasorme io a sequece o liear ime-ivaria BV wihou ay ime-elay or imeavace. Solvig he resule liear BV sequece i a recursive maer coclues ha he oimal corol law i he orm o rai coverge series. A he e a ieraive algorihm wih low comleiy i comuaio a as covergece rae is roose o achieve a accurae eough suboimal corol law. I's worh oig ha simulaio base o alicaio o a harmoic oscillaor is couce o emosrae eiciecy o his meho. Keywors: Coiues-ime Oimal Corol roblem OC ime-delay wo-oi Bouary Value roblem Homooy erurbaio Meho Iroucio he goal o oimal corol heory is o eermie corol laws or a give yamic rocess i a sysem subec o oimizig oe or more reeie

2 35 Arma Sargolzaei e al crieria. ime elay i biology chemisry elecroics a mechaics omais [ 3 4 5] as oe level o iiculy i he esigig hase o he corol sysem or he rocess. A valuable amou o researches i he corol heory have bee se o oimal corol heory o ime-elay sysems. Diere aroaches have bee roose each wih is ow ros a cos rom he accuracy a comuaioal comleiy oi o view i heory a racice a also he ye o alicaio omai i which he meho is imlemee [7 8 9 ]. he coveioal echiques or solvig oimal corol roblems ca be caegorize io wo broa classes: irec mehos where he roblem is beig raserre io a oliear rogrammig sace by aramerizaio a iscreizaio oeraors [5 6] a iirec mehos which i he soluio by alyig a yamic rogrammig o Hamilo-Jacobi-Bellma HJB equaio [7] or alyig oryagi's maimum ricile i a wo-oi bouary-value roblem BV [8]. Sice solvig a oliear HJB arial iereial equaio is comle i mos cases mos researchers ry o avoi usig i. A ecelle lieraure review o HJB is rovie i [9]. O he oher ha oe o he mos accurae mehos or solvig oimal corol roblems is o se a series o oimal irs-orer ecessary coiios base o he oryagi's Maimum ricile. his meho yiels more accurae soluios a rovies more coiece wih he resoses obaie ha he ohers. However he oryagi s coiios oly chage o suicie uer seciic circumsaces o rouce a sysem o oliear bouary-value iereial equaios ha are iicul o be solve. Obaiig a aalyical aswer also a iicul ask; hereore iig suboimal resose or ime-elay oimal corol roblems is he sae o he ar aroach. As meioe above BV resule by oryagi s maimum oes o rovie a aalyical soluio so researchers are ryig o i a aroimae soluio or hese roblems a irouce Sesiiviy Aroimaio Aroach[] a Successive Aroimaio Aroach SAA[]. Here we roose o aly Homooy erurbaio Meho HM o solve he wo-oi bouary-value roblem BV [8]. he iea is o irouce a Homooy arameer which akes is values rom rage [ ]. Whe he homooy arameer value is he se o equaios are reuce o a simle orm a as he arameer value icreases he sysem goes hrough a sequece o eormaios. he soluio o each sage o eormaio is close o he oe o he revious sage. I brie he sysem has he origial orm o equaio a a rogressively by ecreasig he value o a he ial sage o eormaios he we have he avorable soluio. HM gives us a accurae soluio i a ew umbers o erurbaios a hereore see u he covergece. Our roose aroach cosiers quaraic cos ucio as he crierio a rasorms he ime-elay oimal corol roblem io a ime elay a ime avace i he shae o a wooi bouary-value roblem BV a ially covers i o a se o liear ime-ivaria BV by alyig Homooy erurbaio Meho. Ulike raiioal aroaches [7 8 9 ] he resee echique oly requires solvig a sequece o liear BV's which eecively reuces he require

3 Assessme o He's homooy erurbaio meho 35 amou o comuaios a makes i more racical comarig wih aroimaio aroaches alie i coveioal laorms. Secio irouces he olieariy a ime elay i oimal corol heory a Secio iscusses he iea o Homooy erurbaio Meho HM. roose algorihm o i a suboimal soluio base o HM is resee i Secio 3. Resuls are escribe i Secio 5 a we coclue i Secio 6.. roblem Saeme Cosier a liear sysem wih ime-elay i sae is escribe below: & A A Bu φ X wih is quaraic erormace ie J Q Q u Ru m where R a u R are he sae a corol ucio resecively. Marices A B a A are cosa wih aroriae imesios. Also le φ be a give iiial vecor ucio X be a iiial value a be a ime-elay a osiive m m ieger. Mari Q R is osiive semi-eiie a R R is osiive eiie. he oimal corol roblem is o i he oimal corol u which miimizes he erormace ie a subec o sysem yamic escribe i. I accorace o he oryagi s maimum ricile he oimal coiio is aaie as he ollowig oliear BV coaiig ime-elay a imeavace erms: & A A BR B & Q A χ A φ < Q 3 where χ 4 So he oimal corol law ca be sae as: u R B 5 Sice i is very iicul o solve such roblems aroimaig soluios o he wo-oi bouary-value roblems BV i 3 or soluio which will

4 35 Arma Sargolzaei e al rouce suboimal corol will be sough. Hece HM will be cosiere as he meho o solve he BV escribe i 3 a eails are ollowe.. roose Meho he moel i 3 has bee boh ime-elay a ime-avace erms. A ew meho base o HM irouce here will geerae a liear sequece o ihomogeeous BV wih o elay a avace i he sae a co-sae variables. o show he basic iea o his meho le us eie a oeraor F i 6a or a i 6b or : Δ & A A BR B F & Q A A 6a Δ & A A BR B F & Q A 6b Now le us searae he oeraor F io wo ars: F N 7 Here a N rerese he liear a oliear erms resecively. Base o 7 Homooy echique resee i [6] ca be wrie as he ollowig: H Q F 8 which is equivale o: H N 9 q where Q q q a [ ] is a embeig arameer which is ame Homooy arameer. hus i a way ha agrees wih Homooy equaios i 9 he erurbe vecors o a become as ollows: :[ ] [] R :[ ] [] R Obviously rom 9 we have: H ˆ ˆ H I N F a b

5 Assessme o He's homooy erurbaio meho 353 While varies i he sa o [ ] he soluio o liear ars o ˆ a ˆ as i a o he eac aswers o a as b. I oology we call i eormaio. A eaile illusraio o choosig liear oeraors is give i []. I regar o he roblem wo oeraors are selece as ollows: i liear oeraor: A Q B BR A ii oliear oeraor: A A N χ 3 Firsly embee arameer is use as a small arameer a is assume ha he soluios o 6a a 6b ca be eresse as ower series i ˆ ˆ 4 Seig i he above series yiels ˆ ˆ 5 where ˆ! a ˆ!. Subsiuig 4 io 6a a 6b we have ˆ ˆ ˆ ˆ ˆ ˆ g g g F 6 From 6 a rearragig base o orer o a equaig all erms o wih same orer we i:

6 354 Arma Sargolzaei e al < ˆ ˆ : ˆ ˆ : : g g Q φ M 7 where!! ˆ ˆ ˆ ˆ N N g N N g 8 Solvig all above liear ime-ivaria roblems i a recursive maer leas o a or all ; a makig use o 5 hels us ear ˆ a ˆ which are he aroimaio o he eac aswer. hereore he oimal corol law ca be eresse i he orm below: B R u 9 heorem.. Suicie coiio o covergece Suose ha X a Y are wo Baach saces a le s eie a coracive oliear:

7 Assessme o He's homooy erurbaio meho 355 Δ N [ ˆ ˆ ] N N [ ˆ ˆ ] where N is maig X o Y ha u u~ ; N u N u ~ γ u u~ where u a u ~ are members o X a γ is bewee a. Base o Baach s ie oi heorem he coracive oliear has a uique ie oi U which N U U he ollowig sequece ca be geerae base o he HM ˆ U N U i i where Δ U. Suose ha U u B u where B u { u X u u < [ ]} he we have U B u 4 a lim U u 5 roo. e i 3 rom we have U u N U N u γ u u 6 For assume ha u γ u as a iucio hyohesis he U u N U N u γ U u γ u u. 7 Usig 7 we have U u γ u u γ 8 Equaio 8 coclues ha U B u. he equaio 4 has bee rove. From 7 we ou U u γ u u a also we kow ha lim γ so lim u which is U have bee rove. lim u a he roo is comlee. he equaio 5 U

8 356 Arma Sargolzaei e al 3. Suboimal Corol Sice i is imossible o i he corol law rom a iiie series as escribe i 9 we have o search or iie oes. o kee he irs k erms i h 9 we have he k -orer suboimal corol law as escribe by k u R B. 9 k Here he value o k ca be ecie base o he recisio require. Wih he suboimal corol law escribe i 9 he quaraic erormace ie ca be calculae by usig J k Q Q uk Ruk 3 where is he corresoig sae raecory obaie rom alyig u k o he origial oliear sysem i wih φ we le: J k J k < ε. 3 h I he error bou ε > is chose small eough he he k -orer suboimal corol law i 9 will be very close o he oimal corol law u * he value o erormace ie J k will be very close o is oimal value J * a he bouary coiio will also be saisie ighly. 4. Algorihm o iig suboimal corol law Se : Cosruc Homooy as escribe i 8 9 a a esimae he iiial aroimaios. Se : Subsiu 4 a 5 io 9 a a arrage hem wih resec o he orer o. Se 3: e. Se 4: Make he coeicies o zero a solve he resule liear BV. h Se 5: e k a calculae he k orer suboimal corol law u k accorig o 4. he we aly his corol law o he oliear sysem as escribe by o corresoig sae raecory i a he cos ucio J k usig o 5. Se 6: I he imroveme saisies J k J k < ε which ε is he selece hreshol a suiciely small osiive arameer he go o Se 8. Se 7: Icreme by he go o Se 5.

9 Assessme o He's homooy erurbaio meho 357 Se 8: So 5. Numerical Eamle Noliear oscillaors usually escribe by a se o oliear oriary iereial equaios a some iiial coiios lay a imora role i high echologies such as biology mechaics oics a aricularly i elecroic circuis. o show high accuracy a eiciecy o he roose meho we aly he eveloe meho o a harmoic oscillaor wih reare amig. [7] Cosier he roblem: & 3 & u wih he give iiial coiios 33 he quaraic cos ucio o be miimize is escribe by: J 5 u 34 o show he valiaio o he roose meho we irs le he elay be zero i.e.. he resuls rom he roosal meho a hose rom he collocaio meho are show i Figures -3. A he value o he cos ucio o his harmoic oscillaor esig wih he collocaio meho is J.637 a he value J is.63 aer 5 ieraios wih he roose meho. o urher ivesigae he eiciecy o his meho we icrease he amou o elay i he sae a simulae he erormace o he sysem. he calculae corol iu u o he sysem wih. 3 is give i Figures 4. he values o he cos ucio or hree iere values o elay are give i able. 6. Coclusio o oimize liear ime-elay corol sysems his aer oers a eicie ieraive meho base o he HM o eermie he oimal corol law i he orm o iiie series wih easy comuable erms. Desie o he oher oular mehos such as Successive Aroimaio Aroach a Sesiiviy Aroach his meho avois he iiculy o solvig a sequece o liear ime-varyig BVs or o solvig irecly he oliear BV or he HJB equaio. Isea i oly ees solvig a sequece o liear ime-ivaria BVs which rovies

10 358 Arma Sargolzaei e al a more racical way ha he above-meioe aroimae mehos i asec o comuaioal comleiy. 4 3 Oe ieraio wo ieraios Five ieraios Collocaio meho U Figure Suboimal Corol aw. 8 Oe ieraio wo ieraios Five ieraios Collocaio meho 6 X Figure Sae raecory.

11 Assessme o He's homooy erurbaio meho Oe ieraio wo ieraios Five ieraios Collocaio meho - -3 X Figure 3 Sae raecory Oe ieraio wo ieraios Five ieraios.5.5 U Figure 4 Suboimal Corol aw.3

12 36 Arma Sargolzaei e al able Cos Fucio Values or Diere Delays o Diere Ieraios Ieraio Case Delay Reereces [] M. Jamshii a C.M. Wag A comuaioal algorihm or large-scale oliear ime-elay sysems IEEE ras sysems Ma Cybere SMC [] Herma A Oimal corol o he amosheric reery o a sace shule by a homooy meho. Oim. Corol Al. Meh. 3: [3] uus R.; Zhag X.; Harig F.; Keil F. J. Use o iecewise iear Coiuous Oimal Corol or ime-delay Sysems. I. Eg. Chem. Res [4] YOUNG J. N. Alicaio o Oimal Corol heory o Civil Egieerig Srucures Joural o he Egieerig Mechaics Divisio o he ASCE [5] O. Sryk a R. Bulirsch Direc a iirec mehos or raecory oimizaio Aals o Oeraios Research vol. 37 o [6] C. J. Goh a K.. eo Corol arameerizaio: a uiie aroach o oimal corol roblem wih geeral cosrais Auomaica [7] R. Bellma O he heory o yamic rogrammig roceeigs o he Naioal Acaemy o Scieces USA vol. 38 o [8]. S. oryagi Oimal corol rocesses Usekhi Maemaicheskikh Nauk vol [9] ar. W. Bear G. N. Sariis a J.. We Galerki aroimaios o he geeralize Hamilo-Jacobi-Bellma equaio Auomaica vol. 33 o [] ag G.-Y. Su H.-Y. a iu Y.-M. OIMA RACKING CONRO FOR DISCREE IME-DEAY SYSEMS WIH ERSISEN DISURBANCES. Asia Joural o Corol 8: 35 4.

13 Assessme o He's homooy erurbaio meho 36 [] ag G Zhao Y. Oimal corol o oliear ime-elay sysems wih ersise isurbaces. Joural o Oimizaio heory a Alicaios 7; 3:37 3. [] Jaarmi A Ramezaour H Sargolzaei A Shaaei Oimal corol o oliear sysems usig he homooy erurbaio meho: Iiie horizo case Ieraioal Joural o Digial Coe echology a is AlicaiosJDCA vol.4 o.9.4- [3] J.H. He Homooy erurbaio echique Comuaioal Mehos i Alie Mechaics a Egieerig [4] J.H. He A coulig meho o a homooy echique a a erurbaio echique or o-liear roblems Ieraioal Joural o No-iear Mechaics [5] S. Abbasbay Alicaio o He s homooy erurbaio meho o ucioal iegral equaios Chaos Solios Fracals 3 7 [6] Shakeri F a Dehgha M 8 Soluio o elay iereial equaios via a homooy erurbaio meho Mah. Comu. Moellig [7] F. Khella Oimal corol o liear ime-elaye sysems by liear egere muliwaveles J. Oim. heory Al [8] Jaarmi A Ramezaour H Sargolzaei A Shaaei Oimal corol o Noliear Sysems usig Homooy erurbaio Meho roceeigs o Ieraioal Colloquium o Comuig Commuicaio Corol a Maageme CCCM Volume 4. Receive: Seember