Parametric Iteration Method for Solving Linear Optimal Control Problems

Transcription

1 Applied Mahemaics,, 3, hp://dx.doi.org/.436/am Published Olie Sepember (hp:// Parameric Ieraio Mehod for Solvig Liear Opimal Corol Problems Abdolsaeed Alavi, Aghileh Heidari Deparme of Mahemaics, Payame Noor Uiversiy, ehra, Ira Deparme of Mahemaics, Payame Noor Uiversiy, Mashhad, Ira Received Jue 3, ; revised Augus 7, ; acceped Augus 4, ABSRAC his aricle preses he Parameric Ieraio Mehod (PIM) for fidig opimal corol ad is correspodig rajecory of liear sysems. Wihou ay discreizaio or rasformaio, PIM provides a sequece of fucios which coverges o he exac soluio of problem. Our emphasis will be o a auxiliary parameer which direcly affecs o he rae of covergece. Compariso of PIM ad he Variaioal Ieraio Mehod (VIM) is give o show he preferece of PIM over VIM. Numerical resuls are give for several es examples o demosrae he applicabiliy ad efficiecy of he mehod. Keywords: Parameric Ieraio Mehod; Opimal Corol Problem; Poryagi s Maximum Priciple; He s Variaioal Ieraio Mehod. Iroducio Cosider liear sysem described by Ax Bu,, x x. m where, u are he sae ad corol vecor, m respecively. A, B are cosa marix ad x is he iiial sae. he Opimal Corol Problem (OCP) is o fid a corol law u which miimizes he quadraic cos fucioal () f J x f Sf x Qu Rud. () where SQ, are symmeric posiive semi-defiie marices ad R mm is symmeric posiive defiie marix. I geeral he problem ca be rasformed o he Riccai differeial equaio [], alhough solvig he Riccai equaio arised from OCP is o very simple. Aoher proposal for direcly solvig he OCP is discreizig he origial problem ad solvig i umerically. Herei, he specral collocaio mehods differ from oher compuaioal mehods i heir special discreizaio a carefully seleced odes for example, he so-called Legedre- Gauss-Lobao odes. he he differeial equaios of he OCP are approximaed by algebraic equaios []. Alhough hese mehods are flexible ad for programmig wih compuer are compaible, bu hey have heir weakesses for isace hey reac quie sesiively o he selecio of ime-sep size [3]. Accordig o he classic opimal corol heory, as poied ou i [4], by usig Poryagi s maximum priciple, we ca obai he followig wo-poi Boudary Value (PBV) problem ABR B, x (3) Q A, f Sf. ad he opimal corol law for OCP ca be wrie as u R B where is kow as he cosae variable. Aalyic soluios ca rarely be foud for such PBV problem ad auhors ofe solve i approximaely for example Yousefi, Dehgha ad aari [5] applied He s Variaioal Ieraio Mehod (VIM) o fid he opimal soluios. I his paper, we are goig o solve (3) by use of he Parameric Ieraio Mehod (PIM) wih emphasis o preferece of PIM over VIM.. Parameric Ieraio Mehod PIM is a approximaio mehod for solvig liear ad oliear problems ad a begiig i was proposed for solvig oliear fracioal differeial equaios [6], by modifyig He s variaioal ieraio mehod [7]. he idea of PIM is very simple ad sraighforward. Cosider he followig differeial equaio: Au. (4) Copyrigh SciRes.

2 6 A. ALAVI, A. HEIDARI where A is a oliear operaor, deoes he ime, ad u is a ukow variable. o explai he basic idea of PIM, we firs cosider Equaio (4) as below: Lu Nu g,. where L deoes a liear differeial operaor wih respec o u, N is a oliear operaor wih respec o u ad g is he source erm. We he cosruc a family of ieraive formulas as: hh Au L u u (5) (6) where h ad H deoe he so-called auxiliary pa rameer ad auxiliary fucio respecively. Now by use of L which is a weighed iegral operaor, we have: u u L hh Au Accordigly, he successive approximaios will be readily obaied by choosig he zeroh compoe u saisfyig he geeral propery. u,. u u, I order o solvig he OCP described by () ad (), he PIM cosrucs he followig sequeces o direcly ap- Oe logical guess for u ca be sablished by solvig is correspodig liear homogeeous equaio L u. Aoher choice is u u accordig o he iiial codiio. Oherwise i ca be freely chose wih possible ukow cosas. Noe ha choosig u ca affec o he form of he soluios. he auxiliary parameer h is a acceleraig facor which ca be ideified opimally by he echique proposed i his paper. We show ha a suiable value of h, direcly improves he rae of covergece. he auxiliary fucio H prepare us o have various basis fucios o chage he soluio erms o a desired form. Relaio (6) shows ha he sequece cosruced by PIM is depede o h ad H, ad his direcly ables us o ideify ad corol he domai ad rae of covergece ad his is he mai preferece of PIM over VIM. I should be emphasized ha hough we have he grea freedom o choose he liear operaor L, he auxiliary parameer h, he auxiliary fucio H, ad he iiial approximaio u, which is fudameal o he validiy ad flexibiliy of PIM, we ca also assume ha all of hem are properly chose so ha soluio of (6) exiss, as will be show i his paper laer. Fially, he exac soluio may be obaied by usig 3. Soluio of Opimal Corol Problem via PIM (7) u lim u. (8) proximae he soluios of he PBV problem (3), x x hhsx () s Ax sbr B s ds (9) h H s s Qx s A ()d s s () Sarig wih x x ad as iiial approximaios, x ad calculae from above ieraio formulas. Covergece of hese sequeces o he opimal soluio of he problems ( ) ad () is guaraeed by he followig heorem. A similar heorem for oliear chaoic Geesio sysem ca be foud i [8]. Covergece heorem: if sequece (9) cosruced by PIM coverges o, he is he opimal rajecory of sysem (), ad if is he limi of (), he he opimal corol fu cio u is u R B Proof: Aalyically, as meioed i [4,5], by havig he aswers of he sysem (3), i.e. ad, we ca esablish he opimal corol law u R B of OCP () - () ad i s correspo dig opimal rajecory x. Hece if we show ha he limis of he ieraio formulas (9) ad () are he aswers of he sysem (3), he he proof is complee. o his ed, suppose ha X lim x, lim. (). Also cosider ha ad be uiformly coverge. his hypohesis is i order o guaraee covergece of sequece of derivaives o derivaive of he limi i.e. X lim, lim. () Now lim x x hh slim sax sbr B s ds lim hh slim sqx s A s ds ad sice h, H, we have: lim BR B Ax Qx A lim Copyrigh SciRes.

3 A. ALAVI, A. HEIDARI 6 Now by subsiuig () ad () we have: Also X (3), because: X AX BR B QX A ad saisfy i codiios of sysem x x X lim f f S f lim. his shows X ad are he a swers of sysem (3), ad his complees he proof. Remark. Uforuaely he secod codiio of sysem (3) i.e. f Sf, is o a iiial codiio, so he iiial approximaio for ieraio formula () is o available. o overcome his difficuly we use a echique likes shooig mehod, such ha firs we le s where s is a cosa ad calculae usig (), ex we apply he codiio f Sf ad solve his equaio due o s as a ukow o fid ou s. Fially we reur o ieraio formula () wih s as a iiial approximaio. Remark. Fidig a opimal h: h is a parameer i his mehod which has effec o he rae of covergece. If h his mehod is coicidig o He s variaioal ieraio mehod. Bu we show by several examples ha a suiable value of h, direcly improves he rae of covergece. A opimal value of he covergece acceleraig parameer h ca be deermied by he residual error f Resh LX h ; N X h ; g d. (3) Oe ca easily miimize (3) by imposig he re-. dres h quireme dh 4. Illusraive Examples I his secio, we solve several examples by he PIM o show he efficiecy ad usefuless of he mehod idicaig o he ifluece of parameer h o decreasig he ieraios ad icreasig he covergece rae ad accuracy of approximaios. Wheever he form of approximaios has o imporace, we ake H. As poied ou i secio 3, we solve OCPs by solvig he correspodig PBV problems (3). Example. Cosider he followig opimal corol sysem [4]: u,, mi J x u d. he PIM cosrucs he followig sequeces o approximae he soluios: x x h x () s x s s ds h s x s s ds he exac soluios are: x u e e e e 3 3 ( ) e ( ) e e 3 3 e Figure, shows he approximae resuls obaied from he above ieraio formulas for =. As show i fig- ure whe h approximaios are o so good. o improve he accuracy we have o icrease ieraios, whereas by chagig he auxiliary parameer we ca accelerae he covergece ad esablish good esimaios by lower ieraio s. his shows he flexibiliy ad excellece of he PIM. Figure is plo of he error for various ieraios. I is clear ha accuracy of PIM is higher ha VIM. Example. Cosider he followig sysem: mi J x d, x u so. : x u, x.9. Accordig o [4,5], u k x. I Figure 3, he approximae value for k ad is exac value are ploed for h a d opimal value h.83. he exac value of k is k 5cosh 5 sih 5 5 cosh 5 3sih 5 Example 3. Cos ider a secod-order sysem as follows: π mi J x x u d, 4 so. : u, x, x. Accordig o Equaios (9) ad (), he ieraio for- mulas are: x xh x s xs s ds, Copyrigh SciRes.

4 6 A. ALAVI, A. HEIDARI Figure. Plo of exac ad approximaio soluios. Figure. Plo of errors, lef: VIM, righ: PIM. Figure 3. Exac ad approximae soluio ad affecio of h. Copyrigh SciRes.

5 A. ALAVI, A. HEIDARI 63 Figure 4. Plo of firs coordiae for various h. Figure 5. Plo of secod coordiae for various h. h s xs sds 4 he exac soluios are: e π π π π x e e cos 3e si π π π e e e cos e 3e si e π π π e e e cose si x e e cos e si π π π π e π π π e 4 cose e si π π u e e e 4 cos si Figures 4 ad 5 show he exac ad approximae soluios. his problem was solved by VIM i [5] ad heir preseed soluios are oly i a small regio [.4,.7]. 5. Coclusio here are various mehods for solvig liear OCPs, bu i pracice, he preferred mehod is ha which be execuable by compuers ad he PIM is oe of hem, because, moreover i s simple srucure, i has a acceleraor parameer h which direcly icreases he covergece rae Copyrigh SciRes.

6 64 A. ALAVI, A. HEIDARI ad decreases he umber of ieraios ad his abiliy will be ieresig for usig i he sofwars. Oe idea o esimae opimal h meioed i he paper. I geeral fidig opimal auxiliary parameer h ad auxiliary fucio H, are ope problems. his easy o use mehod ca be used for oliear sysems oo. REFERENCES [] L. Nogramazidis ad A. Ferrae, O he Soluio of he Riccai Differeial Equaio Arisig from he LQ Opimal Corol Problem, Sysems & Corol Leers, Vol. 59, No.,, pp. 4-. doi:.6/j.syscole.9..6 [] P. Williams, A Gauss-Lobao Quadraure Mehod for Solvig Opimal Corol Problems, ANZI Joural, Vol. 47, 6, pp. C-C5. [3] M. Yamagui ad S. Ushiki, Chaos i Numerical Aaly- sis of Ordiary Differeial Equaios, Physica D: Noliear Pheomea, Vol. 3, No. 3, 98, pp doi:.6/67-789(8)944- [4] C. K. Chui ad G. Che, Liear Sysems ad Opimal Corol, Spriger-Verlag, Berli, Heidelberg, 989. doi:.7/ [5] S. A. Yousefi, M. Dehgha ad A. Lofi, Fidig he Opimal Corol of Liear Sysems via He s Variaioal Ieraio Mehod, Ieraioal Joural of Compuer ad Mahemaics, 9. [6] A. Ghorbai, oward a New Aalyical Mehod for Solvig Noliear Fracioal Differeial Equaios, Compuer Mehods i Applied Mechaics ad Egieerig, Vol. 97, No. 49-5, 8, pp doi:.6/j.cma [7] J. H. He, Variaioal Ieraio Mehod A Kid of Noliear Aalyical echique: Some Examples, Iera- he Noliear ioal Joural of No-Liear Mechaics, Vol. 34, 999, pp [8] A. Ghorbai ad J. S. Nadjafi, A Piecewise-Specral Parameric Ieraio Mehod for Solvig Chaoic Geesio Sysem, Mahemaical ad Compuer Modelig, Vol. 54, No. -,, pp doi:.6/j.mcm...44 Copyrigh SciRes.