An Algebraic Approach for the MIMO Control of Small Scale Helicopter

Transcription

1 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A A Algebrc Approch for the MIMO Cotrol of Smll Scle Helcopter A. Budyoo* d T. Sudyto *Ceter for Umed Sytem Stude Ittut Tekolog Bdug, Idoe e-ml: gu.budyoo@e.tb.c.d Aeroutc d Atroutc Deprtmet Ittut Tekolog Bdug, Idoe Abtrct The cotrol of mll-cle helcopter MIMO problem. To ue of clcl cotrol pproch to formlly olve MIMO problem, oe eed to come up wth multdmeol Root Locu dgrm to tue the cotrol prmeter. The problem wth the requred dmeo of the RL dgrm for MIMO deg h forced the deg procedure of clcl pproch to be coducted ccded mult-loop SISO ytem trtg from the ermot loop outwrd. To mplemet th cotrol pproch for helcopter, ptch d roll tttude cotrol ytem ofte ubordted to, repectvely, logtudl d lterl velocty cotrol ytem eted rchtecture. The requremet for th techque to work tht the er tttude cotrol loop mut hve hgher bdwdth th the outer velocty cotrol loop whch ot the ce for hgh performce m helcopter. To ddre the bove problem, lgebrc deg pproch propoed th work. The deged cotrol ug -CDM pproch demotrted for hoverg cotrol of mll-cle helcopter multeouly ubjected to plt prmeter ucertte d wd dturbce. Itroducto The cotrol for mll cle helcopter h bee deged ug vrou method. Durg the perod of 99, the clcl cotrol ytem uch gle-put-gle-output SISO proportol-dervtve (PD) feedbck cotrol ytem hve bee ued extevely. Ther cotroller prmeter were uully tued emprclly. Th trl-derror pproch to deg ''cceptble'' cotrol ytem however ot greeble wth complex mult-put multoutput MIMO ytem wth ophtcted performce crter. For more dvced multvrble cotroller ythe pproche, ccurte model of the dymc requred. To cotrol model helcopter complex MIMO ytem, pproch tht c ytheze cotrol lgorthm to mke the helcopter meet performce crter whle tfyg ome phycl cotrt requred. More recet developmet th re clude the ue of optml cotrol (Ler Qudrtc Regultor) mplemeted o mll erobtc helcopter deged t MIT []. Smlr pproch bed o -ythe h bee lo depedetly developed for rotor umed erl vehcle t UC Berkeley []. A dptve hgh-bdwdth helcopter cotroller lgorthm w ythezed t Georg Tech. [3]. To ddre MIMO problem, LQR d H re the mot populr cotrol deg procedure. However, thee method re ot up to expectto for prctcl pplcto eropce commuty, becue of the followg reo [];. Prmeter tug procedure re ot provded. Weght electo rule re ot etblhed 3. The cotroller order uecerly hgh. Robute gurteed oly for predefed oe 5. Some tme, trdtolly ccepted good cotroller re excluded 6. Exteo of g chedulg or cluo of proper turto of tte vrble dffcult 7. LQR d LQG deg ometme fl to produce robut cotroller for the plt wth flexblty Due to the bove lmtto, the clcl cotrol deg by expereced egeer tll commo the eropce commuty. However, the hertce of uch experece ofte dffcult, thu mprovemet of the method hghly dered. I prtculr, the drwbck of th pproch to be ued cotrol deg tool for mll cle helcopter c be elborted follow.. The cotrol of mll-cle helcopter MIMO problem. To ue of clcl cotrol pproch to formlly olve MIMO problem, oe eed to come up wth multdmeol Root Locu dgrm to tue the cotrol prmeter. Such dgrm however ot preetly vlble. The problem wth the requred dmeo of the RL dgrm for MIMO deg h forced the deg procedure of clcl pproch to be coducted ccded multloop SISO ytem trtg from the ermot loop ISBN ICIUS

2 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A outwrd. A how the deg exmple th work, th type of pproch uecerly cumberome 3. The ccded mult-loop SISO pproch h lmtto t mplemetto. To mplemet th cotrol pproch for helcopter, ptch d roll tttude cotrol ytem ofte ubordted to, repectvely, logtudl d lterl velocty cotrol ytem eted rchtecture. The requremet for th techque to work tht the er tttude cotrol loop mut hve hgher bdwdth th the outer velocty cotrol loop. For cl of hgh-performce helcopter, uch the X-Cell 6, or helcopter where th bdwdth eprto ot uffcet, multeou deg ecery[5].. The clcl cotrol pproch octed wth the ue of trfer fucto whch c become ccurte whe pole-zero ccellto occur due to ucotrollble d uobervble mode To ddre the bove problem, thrd pproch geerlly clled lgebrc deg pproch propoed th pper. Dymc of mll cle helcopter The dymc model of mll cle RUAV h bee elborted [6] for X-Cell 6 RC helcopter. The model developed ug frt prcple pproch. The mthemtcl model w developed ug bc helcopter theory ccoutg for prtculr chrctertc of the mture helcopter. Mot of the prmeter were meured drectly, everl were etmted ug collected dt from mple flght tet expermet volvg tep d mpule repoe vrou ctutor put. No forml ytem detfcto procedure re requred for the propoed model tructure.. Frt prcple pproch Legth Beyod the prevou work [], the clculto of tblty d cotrol dervtve to cotruct the ler model preeted detl. The lytcl model dervto llow the compreheve ly of reltve domce of vehcle tte d put vrble to force d momet compoet. Ad hece t fcltte the developmet of mmum complexty mll cle helcopter dymc model tht dffer from tht of t full-cle couterprt. I the preeted mplfed model, the ege drve-tr dymc d flow dymc re ot ecery to be tke to coderto. The ddtol rotor degree of freedom for cog d led-lg c be omtted for mll cle helcopter. It demotrted lytclly tht the dymc of mll cle helcopter domted by the trog momet produced by the hghly rgd rotor. The domt rotor force d momet lrgely overhdow the effect of complex tercto betwee the rotor wke d fuelge or tl. Th tedecy ubttlly reduce the eed for complcted model of ecod-degree effect typclly foud the lterture o full-cle helcopter... Ler model of mll cle helcopter The preeted pproch ot lmted to pecfc trm codto lke hover or forwrd flght d therefore c be ued to develop globl model of mll cle rotorcrft vehcle to the purpoe of prctcl cotrol deg. The developed model preeted the form of tte-pce wth te tte d four put. Subequetly, t w how by the Frobeu orm ly tht the couplg betwee logtudl d lterl drectol mll. Thu the cotrol deg ue the decoupled model logtudl d lterl drectol mode [6]. Log (rd) u (m/) w (m/) q (rd/) (rd) (rd) Tme Repoe (Logtudl) NoLer Ler t () Fgure Repoe compro betwee ler d oler model 3 Coeffcet Dgrm Method for Cotrol Deg I th tudy, ovel pproch poeered by Mbe [7] ug lgebrc repreetto ppled to polyoml loop the prmeter pce, propoed cotrol deg cddte. Wth th techque, uque cet dgrm ued the me to covey the ecery prctcl deg formto d the crter of good deg. The evetul pplcto of CDM th work the LQ deg frmework to be elborted the ext ecto. 3. Deg prcple Mthemtcl model. The mthemtcl model of the CDM deg decrbed geerl block dgrm how Fgure. I th fgure, r the referece put gl, u the cotrol gl, d the dturbce d the oe geerted by the meurg devce t the output; N() d D() re the umertor d deomtor polyoml of the plt trfer fucto, repectvely. A(), F() d B() re the polyoml octed wth the CDM cotroller whch re the deomtor polyoml mtrx of the cotroller, the referece d the feedbck umertor polyoml mtrx of the cotroller repectvely. For MIMO ce, the vrble d compoet re the form of vector d mtrce wth the pproprte dmeo. ISBN ICIUS

3 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A d Plt r + + u x y F() N() A() D() B() Cotroller + Fgure CDM block dgrm The plt equto gve by: y N() x N() y ( ud ) D () () x I y N () dja()[ F() r A() d B()] () z D() d (7) CDM cotroller deg. The deg prmeter CDM ' re the tblty dce, the tblty lmt dce * ' d the equvlet tme cott,. The tblty dex d the tblty lmt dex determe the ytem tblty d the tret behvor of the tme dom repoe. I ddto, they determe the robute of the ytem to prmeter vrto. The equvlet tme cott, whch cloely relted to the bdwdth, determe the rpdty of the tme repoe. Thoe prmeter re defed follow: whch fter ome lgebrc mpulto, c be completely wrtte : N()F() A()N() N()B() y= r d P() P() P() where P() the cloed-loop ytem polyoml mtrx expreed by: (),,,, ( ) *, (8) P () AD () () BN () () The chrctertc polyoml () det P() () gve by: (3) () where () re cet of the chrctertc polyoml. The equvlet tme cott of the -th order defed the me wy. (9) To wrte the put-output relto of the ytem, the expreo for the tte d the cotroller re eeded. The cotroller equto c be wrtte : Au () F() rb ()( y) (5) By ug the bove equto, the relto betwee c be wrtte : Where the tte equto c be obted by elmtg u d y from the cotroller d output equto follow: () Px () Fr () Ad () B () (6) Alo, by mple mpulto, c be wrtte : Combg the output, tte d cotroller equto, Eq.(), (5) d (6), the mtrx put-output equto c flly be expreed : ISBN ICIUS

4 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A, The chrctertc polyoml c the be expreed : () j j j The uffcet codto for tblty gve :. *.,,3,, Ad the uffcet codto for tblty :, for ome,, () () (3) () 3. Applcto to helcopter cotrol To cotrol model helcopter complex MIMO ytem, pproch tht c ytheze cotrol lgorthm to mke the helcopter meet performce crter whle tfyg ome phycl cotrt requred. Overll t lwy dered to hve cotroller tht c ccommodte the umodeled dymc or prmeter chge d perform well uch tuto. Coeffcet Dgrm Method (CDM) choe the cddte to ytheze uch cotroller due to t mplcty d coveece demotrtg tegrted performce meure cludg equvlet tme cott, tblty dce d robute. To ue CDM pproch, the dymc model hould be frt developed. I our ce, the dymc model of the mll cle helcopter h bee derved lytclly. Hover d Crue Cotrol Deg To demotrte the vblty of the lgebrc pproch, the hover tblzto d crue cotrol re tke ce tude. The well-kow hover cotrol problem repreet uque chllegg problem the rel world. My helcopter pplcto (both med d umed) requre the tble hoverg cpblty for dfferet mo: vdeo r urvellce, r photogrphy, preco trgetg etc. The prelmry tudy coducted by the uthor for R-5 Ymh helcopter hover cotrol w gve [8]. The cotrol durg crue lo mportt for dfferet type of mo d erve the b for utoomou cpblty uch wy-pot followg vgto d uto-plotg. A the bele cotrol deg, mult-loop SISO ytem bed o clcl pproch frt propoed. The CDM method the propoed mprovemet of uch pproch. For the ke of brevty, the peed cotrol tke exmple.. Clcl pproch to peed cotrol The clcl pproch to peed cotrol of mll cle helcopter the exteo of the SAS d Hold ytem ccded cotrol rchtecture. For the purpoe of llutrto, the reult preeted for the deg of forwrd peed u cotrol. Fgure 3 decrbe the root locu dgrm of peed cotrol to be ued for cotrol prmeter optmzto. The g elected for the deg ug the root locu dgrm ku = -.. The tme repoe dgrm for the peed cotrol ubject to tep gve Fgure. Imgry Ax Root Locu Fgure 3 Root locu peed cotrol ytem The tme repoe dgrm how tht the deg for the forwrd peed cotrol ug clcl pproch doe ot meet the expectto. The mult-loop ccded deg lo geerl cumberome d effectve to hdle the MIMO problem. Ampltude Step Repoe Tme (ec) Fgure Step repoe peed cotrol ytem ISBN ICIUS

5 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A. Coeffcet Dgrm Method for peed A deg exmple ug CDM, the followg ecto elborte the procedure for developg u cotrol d v cotrol. For deumertor d umertor polyoml the ptch cyclc chel for the logtudl vertcl model re clculted the followg: 5 3 ( ) ( ulo )( ) ( qlo )( ) ( lo)( ) ( wlo)( ) (5) 3 Coeff. chrctertc poly The correpodg cet dgrm for the bove umertor d deomtor polyoml re gve the followg fgure. The cotrol deg objectve th ce to chge the cet of polyoml order to tblze the ytem ug the pproprte feedbck. By obervto, t more effectve to ue lo to cheve the objectve ce col effectve oly whe the vertcl velocty feedbck ued. 6 Coeff. poly u- log Coeff. poly q- log Fgure 7 Coeffcet dgrm for deumertor polyoml -u cotrol 5 CDM Cotrol Sytem' Coeff. Chrctertc Poly Sytem' Chrctertc Trget Sytem' Chrctertc Coeff. poly thet- log Coeff. poly w- log Fgure 5 Coeffcet dgrm for umertor poly - lo Fgure 8 Coeffcet dgrm for deumertor polyoml -u cotrol Ug the CDM dgrm, t c be oberved tht we c chooe the PID cotroller uch tht: Coeff. poly u- col Coeff. poly q- col - Coeff. poly thet- col Coeff. poly w- 5 col 3 Fgure 6 Coeffcet dgrm for umertor polyoml - col k u [( k k ) u( k k ) k w] (6) lo r 3 The ew chrctertc polyoml P() the become: P () () ( k k)( u ) 3 lo ( k k )( ) k( w ) lo P() (7) lo Solvg the Dophte equto, ISBN ICIUS

6 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A -.8k3.65 5, -.8k - 8.9k9.k k, -8.9k 36.5k k 9.7k..53k 3,..6k 75.6k k36.5k k,.86k k -5.5k 75.6k 6.85k (8) we get the vlue for the g the followg: k k (9) k k k A mlr procedure mplemeted for the deg of de velocty cotrol. The reult of the deg demotrted the followg fgure, cludg the tet for the robute due to mpulve dturbce. The repoe of ut doublet the put d to mpule dturbce t t=35 re gve. The cotroller deg ug CDM ll how good dturbce rejecto wth zero tedy-tte error. The fgure lo how the robute of cotrol due to modelg ucertty. For the logtudl ce, the cotrol teted by llowg the tblty dervtve xu, x d m q to vry by 3%. u (m/) w (m/) t () x -3 oml -3% xu,x,mq +3% xu,x,mq t () Fgure 9 Forwrd peed cotrol deg wth ucertty xu, x d m q The uperorty of CDM deg over clcl pproch h bee demotrted through reult compro of peed cotrol deg. The reult further exteded for the deg of MIMO cotrol ytem. The followg ecto detl the mplemetto of CDM the optml cotrol frmework, ug the o-clled qured-coeffcet Dgrm Method (- CDM). The propoed techque exteo from prevou work by Mbe [7] where the mplemetto of -CDM gve for mple SIMO problem. 5 Squred-Coeffcet Dgrm Method (-CDM) for helcopter cotrol The motvto behd th pproch the lmtto of the extg LQR/LQG techque. The prmry cocer for LQR/LQG deg h bee the lytcl weght electo for uch techque. Numerou effort hve bee gve the lterture d the oly workble oluto to dte obted through terto. I ddto to th problem, LQR d LQG ometme fl to produce robut cotroller for the plt wth pole t the vcty of the mgry x. A typcl eropce vehcle hve th kd of behvor (.e. the phugod mode), the LQR or LQG hould be ued wth cuto. I wht follow, -CDM propoed cojucto wth LQ deg whch lytcl weght electo preeted. The robute of the cotrol wll be teted gt prmeter ucertte d dturbce. 5. LQR Frmework The dervto trted by troducg the tdrd LQR formulto. I LQR frmework, the plt expreed the tte pce expreo: x Ax Bu () where x vector of dmeo, d u put vector. LQR deg mde to mmze the performce dex J gve T T J x x u u dt Q R () where R potve defte, but Q ot ecerly g defte. The frt term repreet the regulto or trckg performce d the ecod term the mmzto of cotrol power. The cloed-loop pole of the ytem wth the feedbck cotrol re gve by the tble ege vlue of the Hmlto H, where o ege vlue le o the mgry x. - T A -R B H T -Q -A () Whe the chrctertc polyoml gve P the followg relto obted. (3) ISBN ICIUS

7 ICIUS 7 Oct -5, 7 Bl, Idoe P P det I H () Therefore, f P () deged by CDM uch doe the prevou ecto, the weght Q c be foud. O the cotrry, f Q pecfed d LQ deg mde, P () c be obted d t wll be eed term of CDM. 5. Squred polyoml -CDM For gve polyoml P (), P (-) P () polyoml - =, deoted PP (). PP () wll be clled the qured polyoml of P () herefter, d P () wll be clled the orgl polyoml of PP (). P P PP PP (5) The LQR deg for hover h the gol to mmze y devto from hover trm codto wth mmum cotrol effort. It formulted the followg cot fucto: where c p J [ quu qy y ] dt (9) qu ' d qy ' re clr cott d p the A () p. The weght of the trckg performce Q order of expreed Q dg([ qu... qu qu... qy qy ]) c p ICIUS7-A (3) If PP () h o potve rel root, there ext oe orgl polyoml P () whch tble. Th polyoml wll be clled the qure-root polyoml of PP (). Whe P () chrctertc polyoml, P () the qure-root polyoml of the qured polyoml PP () = P (-) P (), becue t tble. The cet of thee polyoml re elected follow. P, m, m, j PP q q q q q q q q m m j j (6) (7) I th wy the cet q of PP () expreed by the cet d the tblty dex of hgh order j, whch, tur, expreed by tblty dex. 5.3 Implemetto of -CDM for Hover Cotrol To mplemet -CDM for the hover cotrol the plt polyoml equto re rewrtte : c u uc A () y B () u p p (8) The weght for the cotrol R c the be determed by coderg the trde-off of trckg performce d cotrol mmzto. The reult of formulto for clcultg the weght mtrx Q follow PP( ) Q ( ) AA ( ) Q ( ) BB ( ) u p y p cp ( ) ( ) ( ) p p( ) p( ) p( ) mp p( ) p( ) p( ) c u( ) qu p y( ) qy PP P P q AA A A pq BB B B bq Q Q Here mp the order of the plt umertor polyoml B () p. I th pproch, f PP( ) obted the reult of Qu( ) d Qy( ) CDM deg, the weght polyoml c be obted. The clculted reult for cotrol deg for hoverg X- Cell 6 SE re preeted follow. The helcopter chrctertc polyoml for re expreed for the logtudl mode d lterl drectol mode follow: 5 3 LogVer ISBN ICIUS

8 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A 5 3 LtDr (3) The deged cloed-loop ytem chrctertc polyoml (CDM deg) re lo gve for both mode : PLogVer (3) P LtDr (33) I -CDM frmework, the expreo for the plt chrctertc qure polyoml re: 5 3 APLogVer (3) 5 3 APLtDr (35) Mewhle, we hve the expreo for the deged cloed-loop ytem chrctertc qure polyoml: K K LogVer - - LtDr -3 - Flly, the reult of the deg preeted the followg fgure: u (m/) w (m/) t () oml -3% xu,x,mq +3% xu,x,mq t () Fgure Step repoe ubjected to prmeter vrto u dturbce (m/) u (m/) -.. Sytem' Repoe Compro -Vrble Stte-Dturbce' Mgtude (Logtudl-Vertcl) Dturbce Ampltude = m/ Dturbce Ampltude = m/ Dturbce Ampltude = 5 m/ Dturbce Ampltude = m/ PPLogVer (36) PP (37) Pluggg the bove polyoml to Eq.(3), we c obt the weght of the CDM-LQ deg. The weght mtrce for the logtudl d lterl drectol mode re gve follow: Q Q 6 3 LtDr LogVer LtDr The correpodg CDM-LQ g mtrce re clculted the followg: w (m/) q (rd/) (rd) 5 x tme (ecod) Fgure Sytem repoe compro for dfferet mgtude of dturbce (logtudl mode) 6 Cocluo The pper preet the formulto of the cotrol for mllcle helcopter the lgebrc pproch frmework. I th pproch the chrctertc polyoml d the cotroller re deg multeouly wth due coderto to the performce pecfcto d cotrt mpoed to the cotroller. I CDM, the performce pecfcto rewrtte few prmeter (tblty dce ', the tblty lmt dce * ' d the equvlet tme cott, ). Thee deg prmeter determe the cet of the chrctertc polyoml whch re relted to the cotroller prmeter lgebrclly explct form. The ISBN ICIUS

9 ICIUS 7 Oct -5, 7 Bl, Idoe ICIUS7-A cotrol of mll cle helcopter re deged by CDM how to be robut gt model prmeter ucertte d exterl dturbce The elborto the pper clude the forml mplemetto of CDM for eropce MIMO problem by ug LQR frmework. I the propoed frmework, the coderto for tblty, robute d optmlty re ddreed multeouly for MIMO problem. Beyod the deg exmple vlble the lterture tht re lmted to SISO d SIMO pplcto, the work demotrte ucceful mplemetto of CDM-bed LQR techque wthout the eed of decompog MIMO problem to ere of SISO or SIMO problem. Flly the exteo of the ue of Squred Coeffcet Dgrm Method (-CDM) for MIMO problem preeted. To the bet of uthor kowledge, to dte the pplcto of -CDM lmted to mple SIMO problem. The work propoed the ue of -CDM cojucto wth LQ deg whch lytcl weght electo preeted. The deged cotrol ug -CDM pproch demotrted for hoverg cotrol of mll-cle helcopter multeouly ubjected to plt prmeter ucertte d wd dturbce. Referece Coeffcet Dgrm Method, Itertol Coferece o Emergg Sytem Techology, Seoul, Kore. [] Gvrlet, V. (3) : Autoomou Aerobtc Meuverg of Mture Helcopter, PhD the, Mchuett Ittute of Techology. [] Shm, D. () : Herrchcl Cotrol Sytem Sythe for Rotorcrft-Bed Umed Aerl Vehcle, PhD the, Uverty of Clfor, Berkeley. [3] Corb, J. E., Cle, A. J., Prd, J. V. R., Hur, J., d Km, N. () : Flght evluto of dptve hgh bdwdth cotrol method for umed helcopter, Proceedg of the AIAA Gudce, Nvgto d Cotrol, Amerc Ittute of Aeroutc d Atroutc. [] Mbe, S. () : Applcto of Coeffcet Dgrm Method to MIMO Deg Aeropce, 5th Trel World Cogre, IFAC, Brcelo, Sp. [5] Mettler, B., Tchler, M., d Kde, T. () : Sytem detfcto modelg of mll-cle umed rotorcrft for flght cotrol deg, Jourl of the Amerc Helcopter Socety, 7, [6] Budyoo, A, Sudyto, T d Lem, H. (7) Frt prcple pproch to modelg of mll cle helcopter, Itertol Coferece o Itellget Umed Sytem, ICIUS, Bl, Idoe [7] Mbe, S. (998) : Alytc weght electo for LQ deg, 8 th Workhop o Atrodymc d Flght Mechc, ISAS, July 3- [8] Budyoo, A. (5) : Obord Multvrble Cotroller Deg for Smll Scle Helcopter Ug ISBN ICIUS