Analytical Solution to Optimal Control by Orthogonal Polynomial Expansion

Transcription

1 Proceedngs o he World Congress on Engneerng and Compuer cence WCEC, Ocober -,, an Francsco, UA Analycal oluon o Opmal Conrol by Orhogonal Polynomal Expanson B. ous,. A. avallae,. K. Yadavar Nravesh Absrac In hs paper he use o orhogonal polynomals or obanng an analycal approxmae soluon o opmal conrol problems wh a weghed quadrac cos uncon, s proposed. he mehod consss o usng he Orhogonal Polynomals or he expanson o he sae varables and he conrol sgnal. hs expanson resuls n a se o lnear equaons, rom whch he soluon s obaned. A numercal example s provded o demonsrae he applcably and eecveness o he proposed mehod.. Index erms Opmal conrol, Orhogonal Polynomals, pecral ehod, egendre Polynomals, Rcca ehod. I. INRODUCION he goal o an opmal conroller s he deermnaon o he conrol sgnal such ha a speced perormance creron s opmzed, whle a he same me specc physcal consrans are sased. any deren mehods have been nroduced o solve such a problem or a sysem wh gven sae equaons. he mos popular s he Rcca mehod or quadrac cos uncons however hs mehod resuls n a se o usually complcaed derenal equaons whch mus be solved recursvely [. In he las ew decades orhogonal uncons have been exensvely used n obanng an approxmae soluon o problems descrbed by derenal equaons [-. he approach, also nown as he specral mehod [, s based on converng he derenal equaons no an negral equaon hrough negraon. he sae and/or conrol nvolved n he equaon are approxmaed by ne erms o orhogonal seres and usng an operaonal marx o negraon o elmnae he negral operaons. he orm o he operaonal marx o negraon depends on he parcular choce o he orhogonal uncons le Walsh uncons [, bloc-pulse uncons [, aguerre seres [, Jacob seres [-, Fourer seres [, Bessel seres [, aylor seres [, shed egendre [, Chebyshev polynomals [, and Herme B. ous s wh he Deparmen o Elecrcal and Compuer Engneerng, abrz Unversy, P.O. Box abrz, Iran (e-mal: bous@abrzu.ac.r ).. A. avallae( correspondng auhor ) s wh he Deparmen o Elecrcal and Compuer Engneerng, abrz Unversy, P.O. Box abrz, Iran (Phone: +; ax: +; e-mal: a.avallae@homal.com).. K. Yadavar Nravesh s wh he Deparmen o Elecrcal and Compuer Engneerng, AmrKabr Unversy o echnology, P.O. Box - ehran,iran (e-mal: nravesh@au.ac.r). polynomals [ and Wavele uncons [. In hs paper apar rom he shed egendre Polynomals, new se o Orhogonal Polynomals are consdered based on he requremen o he problem. hs mehod proves o be arly precse rom smulaon resuls and may be expanded o a vas range o cos uncons. nce only lnear sysems are consdered n hs paper, he sae space equaons and he cos uncon are consdered n he ollowng ormas: X ( ) AX ( ) + Bu( ) In whch A and B are consan marces. X () vecor and u() s he conrol sgnal. And he cos uncon: J [ X ( ) QX ( ) + ru ( ) d () s he sae In whch Q s a posve dene marx, and r and are consan values. s he nal me and speced. As can be seen n equaon () he ne horzon cos uncon consss o he weghng uncon or he sae vecor. In he presened mehod here, X () and u () are expanded based on orhogonal polynomals.he man reason or he use o such an expanson s ha resuls n he smplcaon o he cos uncon J, hs s due o he ac ha he negral o he mulplcaon o non-dencal orhogonal erms s zero. One reasonable approach s expanson based on shed egendre polynomals, however because o he presence o he erm n J new weghed orhogonal polynomals mus be obaned n order o be useul or solvng such a problem. In hs paper rs some useul properes o orhogonal polynomals and means o obanng hem based on hese properes are presened n secon. In secon, a mehod o obanng an analycal approxmae answer o he opmal conrol problem dened by equaons and s presened based on he specral mehod. he resuls o he presened mehod are provded or a numercal example and compared wh hose o he classcal Rcca mehod n secon and nally a concluson o he overall wor s gven n secon. () IBN:---- WCEC

2 Proceedngs o he World Congress on Engneerng and Compuer cence WCEC, Ocober -,, an Francsco, UA II. OVING OPIA CONRO PROBE BY HE UE OF ORHOGONA POYNOIA In he presened mehod, u() and X () mus be expanded based on orhogonal polynomals n order o solve he opmal conrol problem. However due o he presence o he erm n J, X () may no be presened by he shed egendre Polynomals. In order o solve hs problem, weghed orhogonal polynomals wh he weghng uncon, are dened whch wll be represened wh (). hen X () s expanded based on hese new orhogonal polynomals { ( )}. ome o he characerscs o orhogonal uncons are recapulaed nex. A. Orhogonal polynomals he denon o orhogonal polynomals ψ n () and some o her eaures are presened below: b χ W ( ) ψ ( ) ψ ( ) d () a In whch W () s he wegh uncon. he expanson o an arbrary uncon () on he regon s as ollows: [, N ( ) C ψ ( ) () In whch: C χ ( ) ψ ( ) d One propery o orhogonal polynomals s [ : ψ ( ) κψ ( ) () Now or a wegh uncon W ( ), we have he shed egendre polynomals: { P ( )} { ψ ( )} () And: PP d γ () In whch: P ) γ () ( ) [ P ( ), P ( ),..., P n ( () dag[ γ, γ,..., γ n () γ P ( ) d () Now or a wegh uncon W ( ), we dene he polynomals { ( )} shown below: { ( )} { ψ ( )} d α () In whch: [ ( ), ( ),..., n( ) dag α, α,..., α () α [ n () α ( ) d () he negral expanson o he weghed orhogonal polynomals () based on he se { ( )} s as ollows: d D () he mehod or obanng D s explaned n he appendx. I s worh nong ha when dealng wh uncons and her dervaves, he propery menoned n equaon () s o maor mporance. B. Obanng orhogonal polynomals Deren mehods may be used o oban orhogonal polynomals, namely, mos commonly, he Graham-chmd mehod [.However hs mehod s compuaonally cumbersome or large ses and may produce naccurae resuls. Here, anoher mehod s nroduced whch s based on he properes o orhogonal polynomals, or W ( ). he presened mehod s compuaonally eecve and precse compared o he Graham chmd mehod due o he ac ha approxmaons n numercal negraon needed or he Graham chmd mehod are no requred or he presened mehod. I s assumed ha: () Hence: d d ( In whch: d) Y () n [,,..., IBN:---- WCEC

3 Proceedngs o he World Congress on Engneerng and Compuer cence WCEC, Ocober -,, an Francsco, UA and K + K + Y K + n+ K + n + K + K + K + n+ K + n + K + n+ K + n + K + n+ K + n + ) ) () Now because Y s real symmercal and posve dene can be ransormed o he orm below by he Cholesy mehod: Y () Now based on he denon o orhogonal polynomals, we can assume ha : Y I () In whch I s an ( n + ) ( n + ) deny marx. hs would mean ha: () o we can now oban, and nally. III. FORUAION OF AN OPIA CONRO PROBE UING WEIGHED ORHOGONA POYNOIA Now he opmal conrol problem descrbed n equaons () and () are ormulzed by he use o he orhogonal polynomals descrbed n secon. Frs x() wll be expanded based on he se { ( )} up o degree n : x e X E In whch: e e... e n em em emn m ) x( )... ( ) x m () E, X () m Where m s he order o he sysem. I mus be noed ha E s unnown and wll be obaned laer on. And u () he conrol sgnal wll be expanded based on he se { P ( )} up o degree n : u β P () In whch: β β, β,..., β [ n ) Noe ha each o he uncons P () may be expanded based on he se { ( )} or vce versa n equaon (). hereore we have: P c () In whch c s a ( n + ) ( n + ) square marx. ubsung no equaon () we have: u β c () β s also unnown and wll be obaned laer on. Now by he use o equaons (), () and () we can wre: X ED + VP X ( ED + Vc ) () In whch: x V x m... () m ) Where x n equaon () s he nal condon or he sae varable x (). By replacng equaons (), () and () n o he sae equaon () we have: E A( ED + Vc ) + Bβ c () hereore he ollowng equaon holds or all values o : E A( ED + Vc ) Bβ c () And he marx I may be dened as: I Δ Vec[( E A( ED + Vc ) Bβc ) () o I. By replacng he expansons or u and X as ormulaed n equaons () and () respecvely, n he expresson or J n equaon () we have: J ( ED + Vc ) Q( ED + Vc )d + r β γβd () By he use o he properes o orhogonal uncons P () and (), he unconal J aes he smpler orm o: J { race[ α ( ED + Vc Q( ED + Vc ) + rβ γβ} () For mnmzng J wh he resrcon n equaon () or () we can use he agrange coecens and mnmze he ollowng expresson nsead: η( E, β, λ) J ( E, β ) + λ. I( E, β ) () In whch λ s he agrange coecen and: λ [ λ, λ,..., λm n () Now η( E, β, λ) mus be mnmzed, whch s done by solvng he ollowng equaons: IBN:---- WCEC

4 Proceedngs o he World Congress on Engneerng and Compuer cence WCEC, Ocober -,, an Francsco, UA η η η,, e β λ () Aer perormng he above derenaons and smplcaon [, he ollowng mporan equaons are obaned: race( αd E QED) Q ( DαD ) vec( () λ. vec( D E A ) ( A D) λ vec( () race( αd E QVc ) vec( DαcV Q) vec( () λ. vec( D E A ) vec( D E A ) λ () vec( D E A ) ( A D ). vec( () And he se o lnear equaons are nally obaned: Q I ( Dα D ) I A D Vec( A D B c β rγ B c λ Vec[ DαcV Q Vec[ cv A () In whch s he symbol o Kronecer mulplcaon and by Vec (marx) we mean placng he columns o he marx n consecuve order n one vecor. he unnown varables λ, β and e n () are o rs order and hence are easly obanable rom solvng he lnear se o equaons n (). By olvng he se o lnear equaons n () he coecens e and β whch were used n he expansons represened n equaons () and () are obaned. hereore we have now obaned he ollowng soluon o our orgnal problem: u β P, X ED + VP () IV. NUERICA EXAPE In hs secon he resul ormulaed n equaon () s appled and compared wh he resul o he classcal Rcca mehod or a specc sysem wh he ollowng speccaons: A, B, Q (), r. () And we wsh o mnmze he cos uncon: J [ X ( ) Q( ) X ( ) + ru ( ) d () he problem s solved or and, wh n.he resuls are shown n Fgures and respecvely. For, and n he ollowng soluons are obaned: X (. ) + (. (. ) + (. ). +. (. ). +.. X (. ) (. ) + (. ) (. ) u (. ) (. ) + (. ) For, and n he ollowng soluons are obaned: X (. ) (. ) + (. ) X (. ) + (. ) (. ) u (. ) + (. ) (. ) As can be seen rom he smulaon o he resuls presened mehod and he classcal Rcca mehod, he resuls o boh have allen upon one anoher when ploed as shown n Fgure and Fgure. ) IBN:---- WCEC

5 Proceedngs o he World Congress on Engneerng and Compuer cence WCEC, Ocober -,, an Francsco, UA V. CONCUION In hs paper we have presened an alernave mehod or obanng an analycal approxmae soluon o opmal conrol problems wh me varan wegh n he cos uncon. he presened mehod maes use o he properes o orhogonal polynomals and ransorms he problem no a lnear se o equaons. he resuls o he presened mehod proved o be accurae by comparson wh ha o he classcal Rcca mehod. u X d D A- In whch:, n [,,..., A- Where A- N We now ha: d d λ... X Fg.. Depced numercal resuls or,n, o boh he presened mehod and ha o he Rcca mehod. N N, N N - - u X X Fg.. Depced numercal resuls or,n, o boh he presened mehod and ha o he Rcca mehod. APPENDIX In hs Appendx a numercally ecen mehod s nroduced or obanng he D marx requred n equaon. b a α W ( ) ( ) ( ) d n n n ) + n A- ( a a a A- λ A- N And we now ha: A- hereore: d λ D λ A- REFERENCE [ D.Kr, Opmal Conrol heory: An Inroducon, Prence-Hall Englewood Cls: NJ, [ I. ade,. Bohar, Opmal conrol o a parabolc dsrbued parameer sysem va orhogonal polynomals, Opmal Conrol Appl. ehods () - [ R. Chang,. Yang, oluons o wo-pon boundary-value problems by generalzed orhogonal polynomals and applcaons o conrol o lumped and dsrbued parameer sysems, Inerna. J. Conrol () -. [. Razzagh, A. Arabshah, Opmal conrol o lnear dsrbued parameer sysems va polynomal seres, Inerna. J. ysems c. () - [ A. Alpanah,. Razzagh,. Dehgan, Nonclasscal pseudospecral mehod or he soluon o brachsochrone problem, Chaos, oluons and Fracals () - [ C.F. Chen, C. H. Hsaso, A sae-space approach o Walsh seres soluons o lnear sysems, Inerna. J. ysems c. ()() - IBN:---- WCEC

6 Proceedngs o he World Congress on Engneerng and Compuer cence WCEC, Ocober -,, an Francsco, UA [ N.. Hsu, B. Cheng, Analyss and opmal conrol o me varyng lnear sysems vaa bloc pulse uncons, Inerna. J. Conrol () - [ D. hh, F. Kung, C. Chao, aguerre seres approach o he analyss o a lnear conrol sysem ncorporang observers, Inerna. J. Conrol () - [. Razzagh, hed Jacob seres drec mehod or varaonal problems, Inerna. J. ysem c. () -. [ C.u, Y. hn, ysem analyss, parameer esmaon and opma regular desgn o lnear sysems va Jacob seres, Inerna. J. Conrol () - [ C. Yang, C Chen, Analyss and opmal conrol o me-varyng lnear sysems va Fourer seres, Inerna J. ysems c. () -. [ P.N. Parashevopoulos, P. avounos, G.Gh. Georgou, he operaon marx o negraon or Bessel uncons, J Franln Ins. () - [.Razzagh, aylor seres drec mehod or varaonal problems, J. Franln Ins. () -. [. Razzagh, G. Elnager, A egendre echnque or solvng me-varyng lnear quadrac opmal conrol problems, J. Franln Ins. () -. [.. Nagura,. Wang, A Chebyshev-based sae represenaon or lnear quadrac opmal conrol. rans. AE Dynam. ys. eas. Conrol () - [ K.K. hyu, Analyss o dynamc sysems usng Herme polynomals, aser hess, Deparmen o Elecrcal Engneerng Naonal Cheng Kung Unversy, nan, awan, Republc o Chna,. [ I. ade,. Abualrub,. Abuhaled, A compuaonal mehod or solvng opmal conrol o a sysem o parallel beams usng egendre waveles, ahemacal and Compuer odelng () - [ E. Kreyszg, Advanced Engneerng ahemacs, John Wley & ons Inc,. [ Neva P. Orhogonal polynomals: heory and pracce. Norwell, A: Kluwer Academc/Cambrdge Unversy press;. IBN:---- WCEC