Beyond the Classical Performance Limitations Controlling Uncertain MIMO Systems: UAV Applications

Transcription

1 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio Mario Garcia-Saz, Ir Euioa, Jor Elo Automatic Cotrol ad Computr Scic partmt Public Uivrity of avarra. Campu Arroadia 36 Pamploa SPAI ABSTRACT Thi papr ummariz a w mthodoloy to di qutial o-diaoal QFT cotrollr for multiiput-multi-output MIMO ytm with ucrtaity, which i a ctral iu i UAV cotrol ytm. It alo dmotrat th faibility of that mthodoloy to cotrol th poitio ad attitud of a 6x6 MIMO paccraft with lar flxibl appda. Th lat part of th papr itroduc a w practical mthodoloy to di robut cotrollr that work udr a witchi mchaim, oi byod th claical liar limitatio ad ivi a olutio for th wll-kow robut-prformac trad-off.. ITROUCTIO Cotrol of multivariabl ytm multipl-iput-multipl-output, MIMO with modl ucrtaity i till o of th hardt problm that cotrol ir hav to fac i Umad Air Vhicl UAV ralworld applicatio. Iput-output dirctioality, coupli amo cotrol loop, tramiio zro, pairi, tc. ar om of th mai complxiti that dfi a MIMO ytm. Morovr, modl ucrtaiti ubtatially icra uch difficulti, maki mor rtrictiv th ihrt prformac limitatio of th cotrol ytm. I th lat fw dcad a vry iificat amout of work i liar MIMO ytm ha b do. Th firt tchiqu that mad a quatitativ ythi of MIMO ytm, taki ito accout quatitativ boud o th plat ucrtaity ad quatitativ tolrac o th accptabl clod-loop ytm rpo, wa th Quatitativ Fdback Thory QFT []. I th lat fw yar om w mthod for o-diaoal full matrix multivariabl QFT robut cotrol ytm di hav b itroducd. Th firt part of th papr itroduc a w mthodoloy [-6] that improv th currt odiaoal MIMO QFT cotrol tchiqu. Th cod part validat th w tchiqu by applyi thm to cotrol th poitio ad attitud of a 6x6 paccraft with lar flimy appda [7]. Combii robut di ad tabl witchi, th cotrol traty could optimiz th tim rpo of th ytm by fat adaptatio of th cotrollr paramtr duri th trait rpo accordi to crtai rul bad o th amplitud of th rror. Th lat part of th papr itroduc a mthodoloy to di a family of robut cotrollr abl to o byod th claical liar prformac limitatio. Th mthodoloy i bad o both a w raphical tability critrio for witchi liar ytm ad th robut quatitativ fdback thory QFT [8].. O-IAGOAL MIMO QFT COTROL ESIG METHOOLOGY [-7] Cotrol of multivariabl ytm multipl-iput-multipl-output, MIMO with modl ucrtaity ar till o of th hardt problm that th cotrol ir ha to fac i ral-world applicatio. Thr of th RTO-E-SCI-95 -

2 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio mai charactritic that dfi a MIMO ytm ar th iput ad output dirctioality -diffrt vctor to actuat U ad to maur Y-; th coupli amo cotrol loop -om output y i ca b iflucd by vral iput u i, ad om iput u i ca ifluc vral output y i ; ad th tramiio zro of th plat matrix. I th lat fw dcad a vry iificat amout of work i MIMO ytm, too umrou to lit hr, ha b do. Ui MIMO QFT, Horowitz propod to tralat th oriial x MIMO problm ito parat quatitativ multipl-iput-il-output MISO problm, ach with plat ucrtaity, xtral diturbac ad clod-loop tolrac drivd from th oriial problm []. Two diffrt approach, th o-calld qutial ad o-qutial mthod, coidr i ucciv itrativ tp a quivalt plat that ithr tak alo ito accout th cotrollr did i th prviou tp, or oly dal with th plat rpctivly. Howvr, althouh uch oriial MIMO QFT mthod tak th coupli amo loop ito accout, thy oly propo th u of a diaoal cotrollr G to ovr th MIMO plat. Thi tructur ca b improvd ui o-diaoal cotrollr. I fact, a fully populatd matrix cotrollr allow th dir much mor di flxibility to cotrol MIMO plat tha th claical diaoal cotrollr tructur. Th u of th o-diaoal compot ca alo a th diaoal cotrollr di problm. I th lat fw yar om w mthod for o-diaoal multivariabl QFT robut cotrol ytm di hav b itroducd. For th ak of clarity, thi ctio ummariz a prviou work [-7] that xtd th claical QFT diaoal cotrollr di for MIMO plat with ucrtaity to th fully populatd matrix cotrollr di. Th work tudi thr ca: th rfrc tracki, th xtral diturbac rjctio at plat iput ad th xtral diturbac rjctio at plat output. It prt th dfiitio of thr pcific coupli matric c, c, c 3, o for ach ca, ad itroduc a qutial di mthodoloy for o-diaoal QFT cotrollr.. Th Coupli Matrix Th objctiv of thi ctio i to dfi a maurmt idx th coupli matrix that allow o to quatify th loop itractio i MIMO cotrol ytm. Coidr a x liar multivariabl ytm - Fi. -, compod of a plat P, a fully populatd matrix cotrollr G, a pr-filtr F, a plat iput diturbac trafr fuctio P di, ad a plat output diturbac trafr fuctio P do, whr P IP, IP i th t of poibl plat du to ucrtaity, ad, P p p p p p p p p p ; G ; F f f f f f f f f f Th rfrc vctor r ad th xtral diturbac vctor at plat iput d i ad plat output d o ar th iput of th ytm. Th output vctor y i th variabl to b cotrolld. It i dotd P a th plat ivr o that, p p - [ ] P P p Λ B p p RTO-E-SCI-95 -

3 Byod th Claical Prformac Limitatio G G d Gb 3 whr Λ i th diaoal part ad B i th balac of P ; ad G d i th diaoal part ad G b i th balac of G. Th xt pararaph itroduc a maurmt idx to quatify th loop itractio i th thr claical ca: rfrc tracki, xtral diturbac at plat iput, ad xtral diturbac at plat output. That idx i calld th coupli matrix ad, dpdi o th ca, how thr diffrt xprio: C, C, C 3 rpctivly. d i d o P di P do r r u F G P - d i d o y T Y/R Fi. Structur of a r of Frdom MIMO Sytm.. Tracki Th trafr fuctio matrix of th cotrolld ytm for th rfrc tracki problm, without ay xtral diturbac, ca b writt a how i Eq. 4, y I P G P G r T r T F r' y / r y / r 4 Ui Eq. ad 3, Eq. 4 ca b rwritt a, T y/r r I Λ- G Λ- G r I Λ- G Λ- G r B G T r d d d b b y/r 5 I th xprio of th clod-loop trafr fuctio matrix of Eq. 5, it i poibl to fid two diffrt trm: i. A diaoal trm T y/r_d, I Λ G - d Λ Gd T - y/r_d 6 that prt a diaoal tructur. ot that it do ot dpd o th o-diaoal part of th plat ivr B, or o th o-diaoal part of th cotrollr G b. It i quivalt to rfrc tracki SISO ytm formd by plat qual to th lmt of Λ - wh th corrpodi part of a diaoal G d cotrol thm, a how i Fi. a. RTO-E-SCI-95-3

4 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio ii. A o-diaoal trm T y/r_b, T y/r_b I Λ - G - [ ] - - d Λ Gb B Gb Ty/r I Λ G d Λ C 7 that prt a o-diaoal tructur. It i quivalt to th am prviou ytm with itral diturbac c r at plat iput Fi. b. j I Eq. 7, th matrix C i th oly part that dpd o th o-diaoal part of both th plat ivr B ad th cotrollr G b. Hc, it compri th coupli, ad from ow o C will b th coupli matrix of th quivalt ytm for rfrc tracki problm, C Gb B Gb Ty/r 8 Each lmt c of thi matrix oby, c δ p t δ 9 k ik ik kj ik whr δ ki i th dlta of Krockr that i dfid a, δki k i δ ki δ k i ki r i - ii u i p ii y i a c r j j - u i ii p ii y i b Fi. i-th quivalt SISO ad MISO ytm.. iturbac rjctio at plat iput Th trafr matrix from th xtral diturbac at plat iput i Eq., ' d i to th output y ca b writt a how RTO-E-SCI-95-4

5 Byod th Claical Prformac Limitatio y I P G P d T d T P d ' i y / di i y / di di i ad th, T y/di d i I Λ Gd Λ di I Λ Gd Λ B Gb Ty /di di I that xprio -Eq. - it i poibl to fid two diffrt trm: i. A diaoal trm T y/di_d, y/di_d - - I Λ Λ T G 3 d Aai, Eq. 3 i quivalt to rulator MISO ytm, a how i Fi. 3a. ii. o diaoal trm T y/di_b T y/di_b I Λ Gd Λ B Gb Ty/di I Λ Gd Λ C 4 that prt a o-diaoal tructur which i quivalt to th am prviou ytm with xtral diturbac c di at plat iput, a how i Fi. 3b. j I Eq. 4, th matrix C compri th coupli, ad from ow o C will b th coupli matrix of th quivalt ytm for xtral diturbac rjctio at plat iput problm, C B G b Ty/di 5 di i - ii u i p ii y i a c di j j - b u i y i ii p ii Fi. 3 i-th quivalt MISO ytm RTO-E-SCI-95-5

6 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio Each lmt c of thi matrix oby, c pik ik tkj δ ik 6 k whr i th dlta of Krockr dfid i Equatio. δ ki..3 iturbac rjctio at plat output Th trafr matrix from th xtral diturbac at plat output how i Eq. 7, y I P G d T d T P d' o y / do o y / do do o ' d o to th output y ca b writt a 7 ad th, T y/do d o I Λ- G d - o I Λ G Λ- B B G Ty/do do d d b 8 I that xprio -Eq. 8- it i poibl to fid two diffrt trm: i. A diaoal trm T y/do_d, y/do_d I Λ T - G 9 d Oc mor, Eq. 9 i quivalt to th rulator MISO ytm howd i Fi. 4a, ii. o diaoal trm T y/do_b T y/do_b I Λ Gd Λ [ B B Gb Ty/do ] I Λ Gd Λ C3 that prt a o-diaoal tructur. It i quivalt to th am prviou ytm with xtral diturbac c 3 do j at plat iput, a how Fi. 4b. I Eq., th matrix C 3 compri th coupli, ad from ow o it will b th coupli matrix of th quivalt ytm for xtral diturbac rjctio at plat output problm, B Gb Ty/do C3 B Each lmt of th coupli matrix, c 3 oby, c 3 p δ pik ik tkj δ ik k whr i th dlta of Krockr a dfid i Equatio. δ ki RTO-E-SCI-95-6

7 Byod th Claical Prformac Limitatio do i - ii u i p ii y i a c 3 do j j - b u i y i ii p ii Fi. 4 i-th quivalt MISO ytm. Th Coupli Elmt I ordr to di a MIMO cotrollr with a low coupli lvl, it i cary to tudy th ifluc of vry o-diaoal lmt o th coupli lmt c, c ad c 3, dfid i Eq. 9, 6 ad. Th lmt ca b implifid to quatify th coupli ffct. Th it will b poibl to aalyz th loop dcoupli ad to tat om coditio ad limitatio ui fully populatd matrix cotrollr. To aalyz th coupli lmt, o Hypothi i tatd. Hypothi H: uppo that i Eq. 9, 6 ad, t jj >> pik ik tkj, for k j, ad i th badwidth of t jj p 3 ot that th abov xprio i cal ivariat ad i typically fulfilld oc th MIMO ytm ha b ordrd accordi to appropriat mthod lik th Rlativ Gai Aalyi, tc. Th th diaoal lmt t jj will b much larr that th o-diaoal o t kj, t jj t kj, for k j, ad i th badwidth of t jj >> 4 ow, two implificatio ar applid to facilitat th quatificatio of th coupli ffct c, c, c 3. Simplificatio S: Ui th Hypothi H, Eq. 9, 6 ad, which dcrib th coupli lmt i th tracki problm, diturbac rjctio at plat iput ad diturbac rjctio at plat output rpctivly, ar rwritt a how Tabl I. Simplificatio S: Th lmt t jj ar computd for ach ca from th quivalt ytm drivd from Eq. 6, 3 ad 9. Th rult ar how i Tabl I. RTO-E-SCI-95-7

8 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio Tabl I. Simplificatio to quatify th coupli ffct Rfrc tracki Extral diturbac at plat iput Extral diturbac at plat output Simplificatio S c t jj p ; i j 5 c jj p ; i j t 6 c 3 p t jj p ; i j 7 Simplificatio jj pjj t jj 8 S p jj jj jj jj pjj p t jj 9 t jj 3 jj p jj u to Simplificatio S ad S, th coupli ffct c, c, c 3 ca b computd a, Tracki jj p p jj c ; i j 3 jj iturbac rjctio at plat iput p p jj c ; i j 3 iturbac rjctio at plat output jj jj p p jj p c p ; i j 33 3 jj.3 Th Optimum o-diaoal Cotrollr Coidr o-diaoal cotrollr to rduc th coupli ffct ad diaoal cotrollr that hlp to achiv th loop prformac pcificatio. Th optimum o-diaoal cotrollr for th thr ca tracki ad diturbac rjctio at plat iput ad output ca b obtaid maki th loop itractio of Eq. 3, 3 ad 33 qual to zro. p ot that both lmt, ad, of th quatio ar ucrtai lmt of P. Evry ucrtai plat p p jj ca b ay plat rprtd by th family, { p } p, p, for i, j,, whr p i th omial plat, ad p th maximum of th o-paramtric ucrtaity radii. 34 p p jj Th omial plat ad that will b cho for th optimum o-diaoal cotrollr will RTO-E-SCI-95-8

9 Byod th Claical Prformac Limitatio follow th xt rul: a If th ucrtai paramtr of th plat how a uiform Probability itributio Fi. 5a which i typical i th QFT mthodoloy-, th th lmt p ad p jj for th optimum o-diaoal p p jj cotrollr will b th omial plat ad, which miimi th maximum of th o- paramtric ucrtaity radii ad p that compri th plat tmplat Fi. 5b. p jj b If th ucrtai paramtr of th plat how a o-uiform Probability itributio Fi. 5c, th th lmt p ad p jj for th optimum o-diaoal cotrollr will b th omial plat p ad p jj, who t of paramtr maximiz th ara of th Probability itributio i th rio [ a ε, a ε ] ad [ a ε, a ε ] paramtr a, b,, a jj, b jj rpctivly. jj jj ow, maki Eq. 3, 3 ad 33 qual to zro ad ui Eq. 34, th optimum o-diaoal cotrollr for ach ca i obtaid. Uiform P Paramtr a a a _mi a _max Ima p p p p b R o-uiform P Paramtr a A max c a -ε a a Fi. 5 Probability itributio of th paramtr a, ad o-paramtric ucrtaity radii th plat tmplat p that compri RTO-E-SCI-95-9

10 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio.3. Tracki opt p Fpd jj, for i j 35 pjj.3. iturbac rjctio at plat iput opt F pd p, for i j iturbac rjctio at plat output opt p Fpd jj for i j, 37 jj p whr th fuctio F pd A ma i vry ca a caual ad tabl propr fuctio mad from th domiat pol ad zro of th xprio A..4 Th Coupli Effct Th miimum achivabl coupli ffct -Eq. 38, 4, 4- ca b computd ubtituti th optimum cotrollr of Eq. 35, 36 ad 37 i th coupli xprio of Eq. 3, 3 ad 33 rpctivly, ad taki ito accout th ucrtaity radii of Eq. 34. Aaloouly, th maximum coupli ffct without ay o-diaoal cotrollr -pur diaoal cotrollr ca- ca b computd ubtituti i th Eq. 3, 3 ad 33 rpctivly -Eq. 39, 4, 43-. That i to ay,.4. Tracki c opt ψ c ψ jj jj jj iturbac rjctio at plat iput c ψ opt c ψ iturbac rjctio at plat output c 3 opt 3 ψ c ψ jj jj jj 4 43 RTO-E-SCI-95 -

11 Byod th Claical Prformac Limitatio whr, ψ p 44 jj pjj jj ad th ucrtaity i: jj p, jj p, for i, j,, jj Th coupli ffct, calculatd i th pur diaoal cotrollr ca, rult i thr xprio 39, 4 ad 43 that till prt a o-zro valu wh th omial-actual plat mimatchi du to th ucrtaity diappar: ad. Howvr, th coupli ffct obtaid with th optimum odiaoal cotrollr -Eq. 38, 4 ad 4- td to zro wh that mimatchi diappar..5 i Mthodoloy Th propod cotrollr di mthodoloy i a qutial procdur cloi loop with four tp [-7]: Stp A: Cotrollr tructur, iput-output pairi ad loop ordri. Firt, th mthodoloy idtifi th cotrollr tructur miimum rquird lmt of th cotrollr matrix ad th iput-output pairi by ui th frqucy-dpdt Rlativ Gai Array RGA- [-]. Th, th matrix P i roraizd o that [p ] ha th mallt pha mari frqucy, [p ] th xt mallt pha mari frqucy, ad o o to uarat th xitc of a olutio []. Aftr that, th qutial di tchiqu compod of ta, a may a loop, prform th followi two tp B ad C for vry colum of th matrix compator G from k to Fi. 6. G k k k k k k k kk k k Stp B ad C a to C Stp B ad C a to C Fi. 6 Stp for cotrollr di Stp B ad C a to C Stp B: i of th diaoal compator kk. Th diaoal lmt kk i calculatd throuh tadard QFT loop-hapi [] for th ivr of th quivalt plat [p kk ] k i ordr to achiv robut tability ad robut prformac pcificatio [3-4]. Th quivalt plat atifi th rcuriv rlatiohip 45 [3], which i a xtio for th o-diaoal ca of th rcuriv xprio propod by Horowitz [] a th Improvd di tchiqu, alo calld Scod mthod by Houpi t al. []. [ p ] [ ] [ ] [ ] [ ] [ ] p ik- i k- k- j kj p p k k k k ; k k [ p k-k- ] [ ] k k- k- k 45 i, j k; [ P ] P k RTO-E-SCI-95 -

12 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio y/r ii ii ii If th cotrol ytm rquir tracki pcificatio a a ω t jω b ω th, bcau y/r t t t -Eq.5-, th tracki boud bii ad a ii will hav to b corrctd with th coupli ii rii cii pcificatio τ cii, o that: c c b ii b ii - τ cii, a ii a ii τ cii 46 t w c τ 47 cii ii ii cii c c ii rii ii a ω t jω b ω 48 Th ar th am corrctio propod oriially by Horowitz alo []. Howvr, with th propod o-diaoal mthod th corrctio will b l dmadi. Th coupli xprio t cii w ii c ii i ow mior tha i th prviou diaoal mthod compar Eq. 38 ad 39-. Th off-diaoal lmt i j of th matrix cotrollr will attuat or cacl that cro coupli. Th th diaoal lmt kk of th o-diaoal mthod will d l badwidth tha th diaoal lmt of th prviou diaoal mthod. Stp C: i of th - o-diaoal lmt ik i k, i,,. Th ik i k lmt of th k-th compator colum ar did to miimiz th o-diaoal lmt of th cro-coupli matric accordi to diffrt purpo: rfrc tracki 3, 35; diturbac rjctio at plat iput 3, 36; ad diturbac rjctio at plat output 33, 37. Th rulti compator ik hav to b caual ad tabl, ad iclud th domiat dyamic. Th off-diaoal cotrollr lmt ca b allocatd ot oly to rduc th coupli ffct of th MIMO ytm, but alo to rach complmtary objctiv, uch a to rmov RHP riht-half pla tramiio zro itroducd duri th cotrollr di [5], improv ytm itrity [3] ad tability mari, rduc cotrollr ffort, tc. Stp : i of th prfiltr. Th di of th prfiltr F do ot prt ay additioal difficulty bcau th fial trafr fuctio that rlat R to Y how l loop itractio thak to th fully populatd compator di. Thrfor, th prfiltr F ca rally b a diaoal matrix..6 Stability Coditio Clod-loop tability of a MIMO ytm with a o-diaoal cotrollr did by ui a qutial procdur i uaratd by th followi ufficit coditio [4]: c. c. c.3 c.4 ach L i ii [p ii ] i, i,,, atifi th yquit circlmt coditio, o RHP pol-zro cacllatio occur btw ii ad [p ii ] i, i,,, o Smith-McMilla pol-zro cacllatio occur btw P ad G, ad o Smith-McMilla pol-zro cacllatio occur i P G..7 Rmark It i importat to ot that th calculatio of th quivalt plat [p kk ] k, 45, uually itroduc om xact pol-zro cacllatio. That opratio could b prcily prformd by ui ymbolic mathmatical tool []. Howvr, fictitiou pol ad zro may b itroducd wh ui umrical calculu du to th typical roudi rror of th computr. Additioally, it i dd to dtrmi th RTO-E-SCI-95 -

13 Byod th Claical Prformac Limitatio ivr of th plat matrix, which ca alo b umrically o-rliabl. I thi papr, th problm ar ovrcom throuh a w frqucy rpo computatio mthod. That i, for ach frqucy of itrt ω ad for vry t of paramtr withi th rio of ucrtaity, ach lmt p jω of th plat trafr fuctio matrix i tralatd ito a complx matrix P frq_ that rprt th frqucy rpo of vry plat lmt withi th ucrtaity. Thu, thi complx matrix ha a may row a diffrt ca ratd du to th ucrtaity ad a may colum a frquci 49. All th abovmtiod calculatio ar th prformd o th bai of thi t of complx matric by ui lmt-by-lmt matrix opratio. A a rult, pottial impdimt rlatd to practical computatio ar avoidd. P frquci ω frq_ Ca a a Ca a Ca k Ca m am ω ω r ω a am Plat k 49 a kr R a kr j Ima a kr Tmplat at ω r At th am tim, arbitrarily picki th wro ordr of th loop to b did ca rult i th oxitc of a olutio. Thi may occur if th olutio proc i bad o atifyi a uppr limit of th pha mari frqucy ω φ, for ach loop. Hc, Loop i havi th mallt pha mari frqucy will hav to b cho a th firt loop to b did. Th loop that ha th xt mallt pha mari frqucy will b xt, ad o o []. Althouh vry rmot, thortically thr xit th poibility of itroduci RHP tramiio zro du to th compator di. Thi udirabl ituatio ca ot b dtctd util th multivariabl ytm di i compltd. To avoid it th propod mthodoloy Stp A, B ad C i irtd i a procdur itroducd by Garcia-Saz ad Euioa [5]. Oc th matrix compator G i did, th tramiio zro of P G ar dtrmid ui th Smith-McMilla form ad ovr th t of poibl plat IP du to ucrtaity. If thr xit w RHP tramiio zro apart from tho iitially prt i P, thy ca b rmovd by ui th o-diaoal lmt placd i th lat colum of th matrix G. 3. MIMO QFT COTROL FOR A SPACECRAFT WITH LARGE FLEXIBLE APPEAGES [7] Thi ctio ummariz th di of a robut o-diaoal MIMO QFT cotrollr to ovr th poitio ad attitud of a arwi-typ paccraft with lar flxibl appda. Th atllit i o of th flyr of a multipl paccraft cotllatio for a futur ESA miio. It prt a 6x6 hih ordr MIMO modl with lar ucrtaity ad loop itractio itroducd by th flxibl mod of th low-tiff appda. Th citific objctiv of th atllit rquir vry dmadi cotrol pcificatio for poitio ad attitud accuracy, hih diturbac rjctio, loop-coupli attuatio ad low ordr cotrollr. Thi ctio dmotrat th faibility of qutial o-diaoal MIMO QFT trati cotrolli th arwi paccraft ad compar th rult with a prviou H-ifiity di. RTO-E-SCI-95-3

14 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio 3. criptio Th arwi miio coit of thr to ix tlcop arrad i a ymmtric cofiuratio flyi i formatio aroud a matr atllit or ctral hub Fi. 7. arwi will mploy ulli itrfromtry to dtct ad aalyz throuh appropriat pctrocopy tchiqu th atmophr of rmot plat clo to a briht tar. Th ifrard liht collctd by th fr flyi tlcop will b rcombid iid th hubatllit i uch a way that th liht from th ctral tar uffr dtructiv itrfrc ad i caclld out, allowi thi way th much faitr plat air to tad out. Th itrfromtry rquir vry accurat ad tabl poitioi of th paccraft i th cotllatio, which put hih dmad o th attitud ad poitio cotrol ytm. arwi will b placd furthr away, at a ditac of.5 millio kilomtr from Earth, i th oppoit dirctio from th Su Earth-Su Laraia Poit L Fi.8. Fi. 7 arwi paccraft Artit' viw. ESA courty. θ L 4 L 3 Su L L Earth ARWI Lara poit L 5 Su-Earth itac AU 5,, km Fi. 8 Earth-Su Laraia Poit ad arwi paccraft locatio Each tlcop flyr i cylidrically hapd m diamtr, m hiht ad wih 5 k. I ordr to protct th itrumt from th uliht, it i quippd with a uhild modld with 6 lar flxibl bam 4 m lo ad 7 k attachd to th riid tructur Fi. 9; bam d-poit coordiat i brackt. RTO-E-SCI-95-4

15 Byod th Claical Prformac Limitatio Th mai mchaical charactritic of th arwi-typ Flyr ar ummarizd i Tabl II. For vry bam Fi. 9, two diffrt frquci for th firt mod alo Y ad Z bam ax ar coidrd. Thir frqucy ca vary from.5 Hz to.5 Hz, with a omial valu of. Hz, ad thir dampi ca vary from.% to %, with a omial valu of.5%. A rard paccraft ma ad irtia, th corrpodi ucrtaity aroud thir omial valu i of 5%. Bad o th prviou dcriptio ad ui a mchaical modli formulatio for multipl flxibl appda of a riid body paccraft, th op-loop trafr fuctio matrix rprtatio of th arwi-typ Flyr i iv i 5 ad Fi. : p x p y p 3 z P U p 4 ϕ p 5 θ ψ p 6 p p3 p4 p5 p p 3 p 4 p 3 p 33 p 43 p 4 p 34 p 44 p 5 p 35 p 45 p 5 p 6 p 53 p 63 p 54 p 64 p 55 p 65 p6 p 6 p 36 p 46 p 56 p 66 Fx F y Fz Tϕ Tθ Tψ 5 whr x, y, z ar th poitio coordiat; ϕ, θ, ψ ar th corrpodi attitud al; Fx, Fy, Fz ar th forc iput; Tϕ, Tθ, Tψ ar th torqu iput; ad whr vry p, i, j,,6, i a 5th ordr Laplac trafr fuctio with ucrtaity. Fi. 9 arwi typ 6 OF atllit modl Th Bod diaram of th plat Fi. how th dyamic of th 36 matrix lmt. Each of thm ad th MIMO ytm matrix ar miimum pha. Th flxibl mod itroducd by th appda cod-ordr dipol affct all th lmt aroud th frquci ω [.9, ] rad/c. Th diaoal lmt pii, i,,6, ad th lmt p5, p5, p4 ad p4 ar maily doubl itrator plu th flxibl mod. Tabl II. Mchaical charactritic of th arwi-typ Flyr modl RTO-E-SCI-95-5

16 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio Paramtr Satllit body ma Cylidr dimio Irtia tor of atllit i cotrol fram at atllit Ctr of Ma without rflctor Irtia tor of atllit i cotrol fram at atllit Ctr of Ma with rflctor Poitio of Ctr of Ma i cotrol fram at atllit Ctr of Ma Suhild ma Bam lth Valu 5 k m diamtr, m hiht 5 Cot 5 k m 5 [ I ] arwi_body CoM [ I ] 59 k m arwi_body CoM Cot 59 [,, ] m 7 k 6 bam 4 k 4 m 684 To: Out From: I From: I Frqucy B od-plat From: I3 From: I4 From: I5 From: I6 To: Out Maitud db To: Out4 To: Out To: Out To: Out Frqucy rad/c Fi. arwi-typ flyr dyamic Th block diaram of th cotrol ytm i how i Fi.. Th or modul rprt both th OP Optical Pathlth iffrc Fri Trackr or ad th FPM Fi Poiti Mtroloy or, which maur th atllit poitio ad attitud, rpctivly. Th actuator, FEEP Fild Emiio Elctric Propulio thrutr, ar a typ of lctrotatic propulio that provid vry mall ad prci actuatio Tabl III. RTO-E-SCI-95-6

17 Byod th Claical Prformac Limitatio Fi. Gral 6x6 atllit cotrol loop Th xtral diturbac acti o th atllit ravity radit ad olar prur, althouh vry mall, ar alo modld a forc ad torqu alo th 3 ax. Th ravity radit i modld a a cotat bia ad th olar prur i rprtd a a whit oi prturbatio Tabl III. Tabl III. Charactritic of or, actuator ad xtral diturbac am Charactritic Valu W FPM Attitud whit oi. For vry prci rlativ attitud Fi Poiti maurmt PS of.66 ma/ Hz alo th 3 ax Mtroloy FPM θ ma θ tru W FPM Attitud ra:[-4; 4] arcc Sampli frqucy: Hz Optical Pathlth iffrc OP Fri Trackr FEEPS actuator Solar Prur Gravity Gradit For prci 3-axi maur of poitio X ma X tru W OP For vry mall ad prci actuatio F FEEP F commadd W FEEP T FEEP T commadd W T_FEEP F Su B F_Su W F_Su T Su B T_Su W T_Su F Grav T Grav θ attitud. X poitio. F Forc. T Torqu W OP Poitio whit oi. PS of m/ Hz alo th 3 ax Attitud ra:[-; ] µm Sampli frqucy: Hz Forc modl: W FEEP Forc whit oi. PS of. µ / Hz alo th 3 ax, which ca vtually vary up to.5 µ / Hz. Forc ra:[-5; 5] µ Torqu modl: W T_FEEP Torqu whit oi. PS of. µm / Hz alo th 3 ax, which ca vtually vary up to.5 µm / Hz. Torqu ra:[-5; 5] µm Sampli frqucy: Hz Forc modl: W F_Su Forc whit oi. PS of.5 µ / Hz alo th 3 ax B F_Su µ Torqu modl: W T_Su Torqu whit oi. PS of. µm / Hz alo th 3 ax B T_Su µm Forc modl: F Grav.3 µ alo th 3 ax Torqu modl: T Grav.3 µm alo th 3 ax Th oriial dyamic bchmark imulator, providd by ESA ad implmtd udr Matlab/Simulik, itrat all tho lmt cotituti th whol atllit cotrol ytm: or, actuator, dyamic, diturbac, tc. Fi.. For ach prformac valuatio campai, 3 radom dyamic withi th ucrtaity Mot-Carlo aalyi ar ratd to valuat th prformac of th cotrollr. RTO-E-SCI-95-7

18 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio Tabl IV. arwi-typ Flyr rquirmt Atroomical Rquirmt Eiri Rquirmt Cotrol Rquirmt Objctiv Poitio accuracy Poiti accuracy Badwidth Saturatio limit Rjctio of hih frqucy oi from maurmt ad actuatio Stability mari Loop itractio Rjctio of flxibl mod Cotrollr complxity ad ordr umrical Rquirmt Maximum abolut valu: µm for all ax Stadard dviatio:.33 µm for all ax Maximum abolut valu: 5.5 ma for all ax 3 σ Stadard dviatio: 8.5 ma for all ax σ. Hz for all ax Maximum forc: 5 µ Maximum torqu: 5 µm Hih roll-off aftr th badwidth maxt ω max S jω < jω < ω Miimum Maximum Miimum 3. Cotrol objctiv Th mai objctiv of th paccraft i to fulfill om atroomical rquirmt that dmad to kp th flyi tlcop poiti at both th obrvd pac tart ad th ctral hub-atllit. Thi t of pcificatio lad to om additioal iri rquirmt badwidth, aturatio limit, oi rjctio, tc. ad alo d om complmtary cotrol rquirmt tability, low loop itractio, low cotrollr complxity ad ordr, tc. Tabl IV o-diaoal MIMO QFT Cotrollr i Th qutial o-diaoal MIMO QFT mthodoloy prviouly dcribd i Sctio [-7] i applid hr to cotrol th poitio ad attitud of th arwi-typ Flyr Rlativ Gai Array Itractio Aalyi Stp A- Th Rlativ Gai Array RGA of a o-iular quar matrix P i a cal-ivariat maur of itractio. It oriial dfiitio itroducd by Britol [] wa oly propod for tady tat ω rad/c. Howvr, th RGA ca alo b computd frqucy-by-frqucy 5 ad ud to a th itractio at frquci othr tha zro []. I mot ca, th valu of RGA at frquci clo to croovr i th mot importat o. RGA [ ] jω jω T jω λ jω P P 5 whr dot lmt-by-lmt multiplicatio Schur product. Furthr iformatio o how to itrprt th RGA rult ad lct pairi ca b foud at [, ]. Th 6x6 poitio ad attitud dyamic modl of th arwi-typ paccraft with lar flimy appda ha b aalyzd by ui th RGA mthod a a fuctio of frqucy ad for th whol t of paramtr ucrtaity. Althouh th matrix obtaid by ma of 5 i a complx o, oly th RTO-E-SCI-95-8

19 Byod th Claical Prformac Limitatio abolut valu ar prtd. By xamii th corrpodi matric at ach frqucy, it i obrvd that th tady tat rult xtd throuh low frqucy up to.9 rad/c. A a rprtativ xampl withi thi ra, 5 how th rult for th mot coupld plat withi th ucrtaity at ω rad/c. Accordi to it, th pairi hould b do throuh th mai diaoal of th matrix, which cotai poitiv RGA lmt, ad th lmt 5, 4, 4, 5 hould alo b coidrd rlvat RGA 4 5 ω 6.8 rad/c If th aalyi i prformd at hih frqucy, it produc th am cocludi rult i th tir pctrum tarti at 3.5 rad/c. So far, th rtaid compator lmt would b tho of th RGA matrix markd i bold i 5. vrthl, a aformtiod, th RGA lmt icra ad mor itractio ar foudd i th itrval of frquci whr th flxibl mod of th atllit motly affct i.. ω [.9-3.5] rad/c, a ca b i 53 ad 54 for th mot coupld plat at ω.8 rad/c ad rad/c, rpctivly RGA ω.8 rad/c RGA ω rad/c Cotrollr Structur I accordac with th abov RGA rult ad taki ito accout th rquirmt of miimum cotrollr complxity ad ordr Tabl IV, a firt compator tructur coiti of ix diaoal lmt ad four off-diaoal lmt i cho a th mot uitabl o G RTO-E-SCI-95-9

20 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio From thi, four idpdt compator di problm hav b adoptd, two SISO ad two x MIMO problm: [ 33 ] ad [ 66 ]; [ 5 ; 5 55 ] ad [ 4 ; 4 44 ], rpctivly. Th SISO problm will b coidrd from th claical SISO QFT poit of viw, whil th two x MIMO ubytm will b tudid throuh th o-diaoal MIMO QFT mthodoloy. Th coupli dtctd i th ra of frquci of th flxibl mod will b coidrd i th cour of th di procdur throuh mor dmadi pcificatio with rpct to diturbac rjctio. Providd th prformac rult wr ot atifactory, th th compator tructur hould b filld up with additioal off-diaoal compator coitt with 53 ad 54: 34, 35, 43, 45, 53 ad 54 lmt Robut Clod-Loop Spcificatio Th applid o-diaoal ad diaoal MIMO QFT tchiqu di ach loop idividually, icludi th multivariabl charactritic by ma of thir corrpodi quivalt plat. So, th robut clodloop pcificatio ar dfid i trm of SISO xprio for both SISO ad MIMO ubytm. Sic th mthodoloi ar frqucy domai tchiqu, thr i obviouly a d for tralati tim domai rquirmt Tabl IV ito th frqucy domai. Th oriial pcificatio i Tabl IV impo maximum ad tadard dviatio valu o th poitio ad attitud tim rror ial, a wll a actuator forc ad torqu. Thir tralatio ito th frqucy domai i bad o cotrol-ratio modl [9], ad tak ito accout th xpctd xtral diturbac o th arwi-typ flyr, th paccraft flxibl mod ad th coupli amo loop. A a rult, four Typ of pcificatio ar dfid to calculat th QFT boud: Typ : Robut tability; Typ : Robut itivity; Typ 3: Robut diturbac rjctio at plat iput; ad Typ 4: Robut cotrol ffort attuatio. Th otatio ud for th ial i th followi dcriptio of pcificatio rfr to th chm of th ric MIMO ubytm prtd i Fi.. Th compator hav b did withi th t of frquci of itrt ω [6.8-4, 6.8] rad/c. Typ : Robut Stability pcificatio Fi. Structur of a r of Frdom MIMO Sytm Thi pcificatio, how i 56, i tatd to uarat a robut tabl cotrol. All th rquird valu, diplayd loop by loop i 57 ad 58, imply at lat db ai mari ad at lat 49.5º pha mari. Th pcificatio corrpod ot oly to th clod-loop trafr fuctio y i /r i, but alo to trafr fuctio y i / i ad u i /v i. Hc thi coditio additioally impo th rquirmt o or oi attuatio, diturbac rjctio at plat iput ad flxibl mod. [ pii ] i ii p [ ii ] ii i δ ω 56 whr [p ii ] i - i th ivr of th quivalt plat 45, which corrpod to p ii i SISO di. RTO-E-SCI-95 -

21 Byod th Claical Prformac Limitatio Loop, ad 3: δ ω.85 ; ω 57 Loop 4, 5 ad 6:.687 δ ω ; ω Typ : Sitivity rductio Th mai objctiv of thi pcificatio, 59 ad 6, i or oi attuatio ad rductio of th ffct of th paramtr ucrtaity o th clod-loop trafr fuctio. It corrpod to i / i ad [dt ii /t ii ] / [dp ii /p ii ] trafr fuctio. [ pii ] ii i δ ω 59 Typ 3: Rjctio of diturbac at plat iput All loop: δ ω ; ω 6 Solar prur prturbatio ad ravity radit ar coidrd to affct at plat iput i th form of both forc ad torqu. Th purpo of thi pcificatio 6, which corrpod to i /v i ad y i /v i trafr fuctio, i to attuat th ffct of plat iput diturbac o th cotrol rror ad th output ial. Thu, a hih ai i rquird i th low frqucy bad, 6 to 64. Bid, ic v i alo rprt th flxibl mod, pcial atttio i paid to thir frqucy ra maily to accomplih th attitud rquirmt. Loop 4, 5 ad 6: Loop ad : ω Loop 3: ω 3 ω Typ 4: Cotrol ial [ pii ] p [ ii ] ii i i δ ω δ ; ω δ ; ω δ ; ω 64 Bcau of aturatio limit, cotrol ial movmt hould b kpt raoably mall dpit diturbac comi from actuator ad or. Thi pcificatio, 65, corrpod to u i / i trafr fuctio ad i dpictd i ii [ pii ] ii i δ ω 4 65 RTO-E-SCI-95 -

22 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio Loop 3: Loop 4, 5 ad 6: Loop ad : Rduci coupli ffct a much a poibl δ ω ; ω δ ω ; ω δ ω ; ω 68 4 Th coupli ffct from othr ax ca b coidrd a part of th diturbac acti at th iput of th quivalt SISO plat. Th way of dii th o-diaoal lmt of th matrix compator dal with th aim of miimizi th off-diaoal lmt of th coupli matrix SISO i Problm: 33, 66 Compator 33 ad 66 ar idpdtly did by ui claical SISO QFT [] to atify th robut tability ad robut prformac pcificatio tatd i Sctio for vry plat withi th t of ucrtai plat. Th corrpodi QFT boud ad th omial ca of th did op-loop trafr fuctio L ii p ii ii, i 3, 6, ar plottd o th ichol Chart for om of th mot rlvat frquci i Fi. 3a ad 3b for loop 3 ad 6 rpctivly. Both di atify ot oly thir rpctiv boud but alo th yquit circlmt coditio, ad o RHP pol-zro cacllatio occur btw 33 ad p 33, or btw 66 ad p 66. Th Bod plot of ach compator ca b foud i Sctio 3.6, whr 33 [Fi. 8a] ad 66 [Fi. 8b] ar rprtd i olid li i compario with th H-ifiity di dahd li itroducd i Sctio 3.5. Th QFT compator xprio ar prtd i Sctio 3.5. a b Fi. 3 Loop-hapi a L 33 p b L 66 p RTO-E-SCI-95 -

23 Byod th Claical Prformac Limitatio Firt MIMO Problm:, 5, 55, 5 i Th compator for thi x MIMO problm ha b did by applyi th o-diaoal MIMO QFT mthodoloy dvlopd by Garcia-Saz t al. [-7] ad outlid i Sctio. I thi particular ca, th plat to b cotrolld i compod of th followi lmt comi from th oriial 6x6 arwityp paccraft modl P 5 [p p 5 ; p 5 p 55 ], who ivr matrix i P 5 [P 5 ] - [p p 5 ; p 5 p 55 ]. Stp A: Arramt of th ytm Firt, th mthodoloy adopt th tructur ad th pairi of iput ad output iv by th RGA tchiqu i 55 ad arra th plat ivr matrix P 5 o that th ivr of th firt diaoal lmt i thi matrix ha th mallt pha mari frqucy []. I om ca, arbitrarily picki th wro ordr of th loop could lad to th o-xitc of a olutio. I th prt problm, th badwidth of th loop i quit imilar. Th, ay ordr ca b lctd to di th o-diaoal MIMO QFT compator. Stp B: i of th diaoal compator Th diaoal compator i did throuh tadard QFT loop-hapi [] for th ivr of th quivalt plat [p ] p to fulfill th robut tability ad robut prformac pcificatio dtrmid i Sctio for vry plat withi th t of ucrtai plat. Fi. 4a how th omial ca of th did op-loop trafr fuctio L [p ] i bold olid li, which atifi th QFT boud, alo rprtd i th fiur. Additioally, th di fulfil th firt two ufficit tability coditio c. ad c. Sctio.6. That i, L [p ] atifi th yquit circlmt coditio ad o RHP pol-zro cacllatio occur btw ad [p ]. Th Bod plot for th obtaid compator i prtd i Fi. 9a olid li tothr with th di of th H-ifiity approach. a b Fi. 4 Loop-hapi a L [p ]. b L 55 [p 55 ] 55 Stp C : i of th o-diaoal compator 5 Th o-diaoal compator 5 i did to miimiz th 5, lmt of th coupli matrix i th ca of diturbac rjctio at plat iput 3, which iv th followi xprio: RTO-E-SCI-95-3

24 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio p opt 5 69 whr dot th plat that miimiz th maximum of th o-paramtric ucrtaity radii comprii th plat tmplat o th ichol Chart. u to th ucrtaity, th xprio [-p 5 ] dtrmi a rio i th maitud ad pha plot, whr th compator 5 i hapd followi th ma valu at vry frqucy ω [,.] rad/c [ Fi. 5 with 5 itrpolati th plot]. Th compator Bod plot i compard i Fi. 9c with that of th 5, lmt of th H-ifiity compator itroducd i Sctio Fi. 5 Maitud plot of [-p 5 ] with ucrtaity ad 5 bold olid li- Stp B : i of th diaoal compator 55 A i tp B, th diaoal compator 55 i did to cotrol th ivr of th quivalt plat, [p 55 ], which tak th compator prviouly did ito accout 45. [ p ] p 55 [ ] [ p ] [ ] 5 p 5 5 [ p ] [ ] [ ] 55 7 O th bai of th robut pcificatio dfid i Sctio for [p 55 ], ad alo taki ito accout th plat ucrtaity, th QFT boud ar computd. Th, th compator i did by claical loop-hapi o th ichol Chart, a i how i Fi. 4b. ot oly do th di fulfil th boud but alo th firt two tability coditio of c. ad c. from Sctio.6. I othr word, L 55 [p 55 ] 55 atifi th yquit circlmt coditio ad o RHP pol-zro cacllatio occur btw 55 ad [p 55 ]. Th Bod plot of 55 i prtd i Fi. 9d. Stp C : i of th o-diaoal compator 5 u to th rquirmt of miimum cotrollr complxity ad ordr Tabl IV, th o-diaoal compator 5 ha b t to zro. Ayway, th quivalt xprio to th o ud i 69, 5 opt -p 5, could b applid to cacl itractio i both dirctio i th MIMO ubytm. At thi poit, oc th whol cotrollr of th MIMO ubytm ha b dtrmid, th lat two tability coditio mtiod i Sctio.6, c.3 ad c.4, ar chckd. Th ytm i tabl. Fially, RTO-E-SCI-95-4

25 Byod th Claical Prformac Limitatio th o-xitc of RHP tramiio zro of P G i chckd by ui th Smith-McMilla form ovr th t of poibl plat IP du to ucrtaity [5]. Th o-diaoal MIMO QFT compator xprio ar prtd i Sctio Scod MIMO Problm:, 4, 44, 4 i Th cod MIMO problm coit of th followi lmt: P 4 [p p 4 ; p 4 p 44 ]. From th x plat ivr matrix P 4 [P 4 ] - [p p 4 ; p 4 p 44 ] ad taki ito accout th robut tability ad robut prformac pcificatio Sctio 3.3.3, th o-diaoal MIMO QFT mthodoloy i quivaltly prformd by followi th tp dtaild i Sctio. Th loop-hapi for th diaoal compator lmt ad 44 ar how i Fi. 6a ad 6b, rpctivly. Th Bod plot for th four compator ar how i Fi. a, b, c ad d for, 4, 4, 44, rpctivly. Th x MIMO ubytm i foud to b tabl accordi to th ufficit tability coditio Sctio.6. Fially, it i alo chckd that o additioal RHP zro hav b itroducd by th compator [5]. Th o-diaoal MIMO QFT compator xprio ar prtd i Sctio 3.6. a b Fi. 6 Loop-hapi a L [p ]. b L 44 [p 44 ] iaoal MIMO QFT Cotrollr i For th ak of compario, th qutial diaoal MIMO QFT mthodoloy dvlopd by Horowitz [] i alo applid to cotrol th poitio ad attitud of th arwi-typ Flyr. Bad o th am robut clod-loop pcificatio dfid i Sctio 3.3.3, thi tchiqu u a qutial procdur imilar to th o dtaild i Sctio.5 Stp B, whr th rcuriv xprio of th quivalt plat i a implifid ca of 45, with i j. For th arwi-typ Flyr, th loop-hapi tp of th diaoal mthod rquir th am diaoal compator ii a th o-diaoal o. Thi happ bcau, i thi ca, i th middl ad hih frqucy ra th off-diaoal lmt i j of th o-diaoal cotrollr hav l rlativ ifluc tha th corrpodi p lmt i th quivalt plat 45. iffrc btw both MIMO QFT cotrollr ari i th low frqucy ra, a ca b obrvd i Fi. 7. Th C 5, lmt of th coupli matrix for diturbac at plat iput 3 i plottd for a rprtativ plat ca RTO-E-SCI-95-5

26 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio ad for th thr cotrollr: o-diaoal ad diaoal MIMO QFT ad H-ifiity di. For th frqucy ra ω [,.] rad/c it i how that C 5, o-dia QFT < C 5, dia QFT < C 5, H-ifiity, which xplai why th o-diaoal MIMO QFT improv th diaoal MIMO QFT ad th H-ifiity cotrollr rult udr low frqucy xtral diturbac Sctio Fi. 7 Elmt 5, of th coupli matrix C : o-diaoal MIMO QFT i olid li, diaoal MIMO QFT i dottd li, H-ifiity i dahd li 3.5 Cotrollr Th otatio adoptd for trafr fuctio xprio dot th tady tat ai a a cotat without parthi; impl pol ad zro a ω, which corrpod to /ω domiator ad umrator, rpctivly; pol ad zro at th orii a ; cojuat pol ad zro a [ξ ; ω ], with /ω ξ/ω domiator ad umrator, ach; -multiplicity of pol ad zro a a xpot. Th o-diaoal MIMO QFT compator coit of th followi iht lmt: { } / { [.676;.479]} ; 5-4 {4.4 } / {.3 3 }; 5 4 ; 33 { [.79;.3565] [;.]} / { [.8;.34]}; 44 {.4.34 [.8644;.74] [.3;.344] [.45;.64] [.879;.593] [.7;.5] [;.7]} / { [.45;.78] [.97;.673] [-.5;.54] [.4;.5] [.7;.45]}; 55 { [.79;.799] [.769;.34] [.3;.593] [.7;.5] [;.7]} / { [.45;.78] [-.8;.54] [.3;.5] [.7;.45]}; 66 { [.8644;.74] [.3;.344] [.45;.64] [.879;.593] [.7;.5] [;.7]} / { [.45;.78] [.97;.673] [-.7;.54] [.687;.4] [.7;.45]}. Th diaoal MIMO QFT compator coit of th am diaoal lmt ii a th o-diaoal compator abovmtiod, ad, i j. Th mai lmt of th -OF H-ifiity compator ar how i Fi. 8-. Thir dc ai tay withi th ra [-5 db, 6 db]. Th rmaii 6 lmt prt a vry low ai, oi from -6 db to -33 db. RTO-E-SCI-95-6

27 Byod th Claical Prformac Limitatio a Fi. 8 Bod iaram Compator: o-diaoal ad diaoal MIMO QFT i olid li, H-ifiity i dahd li. a 33, b 66 b a b c Fi. 9 Bod iaram Compator: o-diaoal MIMO QFT i olid li [alo diaoal MIMO QFT for ad 55 ], H-ifiity i dahd li. a, b 5, c 5, d 55 d RTO-E-SCI-95-7

28 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio a b c d Fi. Bod iaram Compator: o-diaoal MIMO QFT i olid li [alo diaoal MIMO QFT for ad 44 ], H-ifiity i dahd li. a, b 4, c 4, d Comparativ valuatio Thi ctio how a comparativ aalyi of th qutial o-diaoal MIMO QFT cotrollr, did abov for th 6x6 arwi-typ Flyr, with both qutial diaoal MIMO QFT ad H-ifiity cotrollr. Firt, comparativ Bod plot of th compator ar how. Th, tim prformac rult ar prtd atroomical rquirmt, followd by op-loop badwidth, ad forc ad torqu compario iri rquirmt. Fially, th tability objctiv ad th ordr of ach compator ar aalyzd cotrol rquirmt Compator Bod Plot Th Bod plot ar prtd for th compator of th o-diaoal MIMO QFT olid li i compario with tho of th H-ifiity dahd li. ot that, i thi ca, th diaoal MIMO QFT mthod yild th am diaoal compator a th o-diaoal MIMO QFT tchiqu. Fi. 8 prt th rult for th two SISO ubytm 33 ad 66, a ad b rpctivly. Fi. 9 plot RTO-E-SCI-95-8

29 Byod th Claical Prformac Limitatio th compator of th x MIMO ubytm compod of, 5, 5 ad 55 lmt. Th, 4, 4 ad 44 compator lmt that coform th othr x MIMO ubytm ar how i Fi.. ot that 5 ad 4 hav b t to zro i th o-diaoal MIMO QFT di. Additioally, accordi to th RGA rult i 55 th rmaii lmt of th cotrollr matrix G did with thi tchiqu qual zro. By cotrat, tho 6 lmt prt a o-zro, althouh vry mall, maitud rpo wh thy ar did with th H-ifiity tchiqu. Fially, th off-diaoal lmt of th diaoal MIMO QFT ar obviouly zro. Tabl V. Tim imulatio prformac with th thr cotrollr Spcificatio Rquirmt Bchmark Maximum Poitio Error X µm Maximum Poitio Error Y µm Maximum Poitio Error Z µm Maximum Attitud Error X ma Maximum Attitud Error Y ma Maximum Attitud Error Z ma Std. viatio of Poitio Error X µm Std. viatio of Poitio Error Y µm Std. viatio of Poitio Error Z µm Std. viatio of Attitud Error X ma Std. viatio of Attitud Error Y ma Std. viatio of Attitud Error Z ma Maximum Actuator Forc Commad X Maximum Actuator Forc Commad Y Maximum Actuator Forc Commad Z Maximum Actuator Torqu Commad X m Maximum Actuator Torqu Commad Y m Maximum Actuator Torqu Commad Z m < µm < µm < µm < 5.5 ma < 5.5 ma < 5.5 ma <.33 µm <.33 µm <.33 µm < 8.5 ma < 8.5 ma < 8.5 ma <.5-4 <.5-4 <.5-4 <.5-4 m <.5-4 m <.5-4 m B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B o-diaoal MIMO QFT Cotrollr iaoal MIMO QFT Cotrollr H-ifiity Cotrollr RTO-E-SCI-95-9

30 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio 3.6. Atroomical Rquirmt Tim imulatio ar prformd for 3 radom dyamic withi th ucrtaity ra MotCarlo aalyi i th oriial ESA bchmark imulator B dcribd i Sctio 3. ad i a complmtary bchmark B, both dvlopd udr Matlab/Simulik. Th lattr jut add to B a low frqucy diturbac forc at plat iput alo th X-axi maitud: µ, frqucy: ω.5 rad/c i ordr to coidr diturbac rjctio ad coupli attuatio at low frquci. I ach imulatio, th critria appari i Tabl IV ar computd ovr th tir imulatio tim i.. 5 c. I ordr to charactriz th miimum prformac obtaid, th wort rult rachd by vry cotrollr ar prtd i Tabl V. I othr word, for ach cotrollr, th ratt valu ovr th 3 ucrtai ca i how for th maximum ad th tadard dviatio of poitio ad attitud rror, a wll a for maximum actuator commad, i all ax. Th, it i poibl to vrify whthr th wort prformac till rpct th rquirmt. Th bold umbr i vry row of Tabl V rprt th bt rult bt cotrol traty for vry particular pcificatio. Poitio rror,,3,7,8,9 Tabl V By ipcti Tabl V, it i foud that th prformac obtaid i tim imulatio i vry ood cocri poitio accuracy, ic th rquirmt ar aily fulfilld a improvmt of two ordr of maitud with rpct to th pcificatio i achivd i bchmark B, ad at lat o ordr of maitud i bchmark B for maximum ad tadard dviatio valu. Th o-diaoal MIMO QFT di ithr qual or lihtly improv th diaoal MIMO QFT. Both QFT cotrollr improv th H- ifiity rult for th two bchmark. Attitud rror 4,5,6,,, Tabl V Th pcificatio for th hiht attitud rror i hardr to mt maily bcau of th ffct of th flxibl mod. Som of th maximum attitud valu of th H-ifiity v xcd th 5.5 ma rquird: bchmark B 4 Tabl V ad bchmark B 4,5 Tabl V. Aai, th MIMO QFT mthodoloi improv th rult of th H-ifiity cotrollr i th ix attitud rror ca 4,5,6,,, Tabl V. Oc mor, th o-diaoal MIMO QFT ithr qual or improv th diaoal QFT cotrollr rult. Th ratt diffrc btw both cotrollr ca b obrvd at th Attitud Error alo th Y-axi 5, Tabl V, pcially for bchmark B. Thr, th o-diaoal di dcra th tadard dviatio attitud rror by.85 % Tabl V ad th maximum attitud rror by 5.5 % 5 Tabl V with rpct to th valu rachd by th diaoal compator. Th improvmt could tur out to b rlvat to th atroomical miio. Thir achivmt i du to th fact that th off-diaoal compator hav b did to miimiz th coupli at low frquci, which ar pricipally trd i th cod bchmark Eiri Rquirmt Saturatio limit. Actuator commad 3,4,5,6,7,8 Tabl V A ca b i Tabl V, actuatio i vry mall ad far blow th aturatio limit. Th rult for th thr cotrollr rmai at imilar valu 3,4,5,6,7,8 Tabl V. Op-loop Badwidth Compario Th op-loop cro-ovr frqucy rult of th ix SISO loop ubytm ar how i Tabl VI. Th maur corrpod to th mallt frquci i Hz whr th trafr fuctio of th op-loop of ach SISO ubytm p ii ii [without th coupli lmt p, i j] ar qual to db. Th RTO-E-SCI-95-3

31 Byod th Claical Prformac Limitatio miimum prformac for ach of th thr di ha b tablihd a th miimum badwidth valu ovr th 3 radom atllit dyamic. Obviouly, th badwidth rult for th two MIMO QFT di coicid ic thir diaoal compator ar th am. A valu of. Hz i coidrd a ood compromi choic for badwidth. Sic th frquci of th firt flxibl mod ar withi th ra [.5,.5] Hz, th op-loop cro-ovr frquci for attitud ad poitio ar tud to b a hih a poibl whil imultaouly prvti th flxibl mod from diturbi th ytm output prformac. Tabl VI. Frqucy prformac with th thr cotrollr Rquirmt o-diaoal MIMO QFT Cotrollr iaoal MIMO QFT Cotrollr H-ifiity Cotrollr Poitio Badwidth X Hz. Hz Poitio Badwidth Y Hz. Hz Poitio Badwidth Z Hz. Hz...5 Attitud Badwidth X Hz. Hz Attitud Badwidth Y Hz. Hz... Attitud Badwidth Z Hz. Hz max T jω db.73 Max.5 Max 4.7 Max <6 db ω 5.64 Ma 5.55 Ma 4.69 Ma max S jω db 3. Max.5 Max 6.4 Max <6 db ω 6.69 Ma 6.6 Ma 5.48 Ma For th poitio trafr fuctio, th thr cotrollr xcd th. Hz rcommdatio two ad v four tim dpdi o th cotrollr ad th axi. Howvr, th flxibl mod motly affct th attitud trafr fuctio ad do ot impo uch tro cotrait o th poitio trafr fuctio. Coqutly, it i poibl to o ovr. Hz for th poitio loop, a i provd by th atifyi tim domai rult i Tabl V. For th attitud trafr fuctio, th op-loop cro-ovr frquci ar aroud. Hz for th H- ifiity ad for both th o-diaoal ad diaoal MIMO QFT di Cotrol Rquirmt Stability Objctiv Stability ad prformac pcificatio ar tially dcribd a mathmatical xprio rady to b ud duri th di proc of th cotrollr. Th xprio uually diffr from o cotrol mthodoloy to aothr providd thy ar bad o ditict approach, which i th ca of H-ifiity ad QFT-bad mthodoloi. I thi papr th tability pcificatio hav b dfid i two diffrt way: a Stability coditio of Sctio.6 for MIMO QFT yquit critrio for qutial mthod. jω max S jω for H-ifiity claical critrio for MIMO ytm. b Mari o maxt ad ω ω Th o-diaoal ad th diaoal MIMO QFT cotrollr fulfill th tability coditio for qutial procdur dfid i Sctio.6. Th H-ifiity compator fulfill th mari of T ad S dfid i Tabl VI. With rpct to thi claical itrprtatio of robut tability, th QFT approach rpct thm i mot of th ca ma, but ot i vral ca max. Thi i du to th fact that th itrprtatio of th tability mari which ar idd a mari of a mari ar ot itratd a a di pcificatio i th cor of QFT tchiqu. RTO-E-SCI-95-3

32 Byod th Claical Prformac Limitatio Cotrolli Ucrtai MIMO Sytm: UAV Applicatio Sic tability ad prformac pcificatio ar oly itrprtatio of th fuctioal rquirmt atroomical ad iri rquirmt Tabl IV-, th dir hould b awar of which tradoff d to b mad. Etially, th itrprtatio of rality i trm of a particular thory ca vr rplac th ral world itlf. I th abc of th ral ytm implmtatio or a uitabl prototyp to b ud itad, th dir mut maa tim domai imulatio i ordr to vrify th cotrol ytm bhavior []. That itrprtatio wa do ad uccfully validatd i th prviou ctio of th papr. Additioally, th claical mari o maxt jω ad jω ω max S ar tability MIMO mari, but thy do ot iclud pha iformatio. Thi fact mak thm ufficit, but ot cary coditio ad could yild vry corvativ cotrollr i om ituatio. Althouh th thr mthod ar robut tabl accordi to thir ow rquirmt ad to tim domai imulatio, futur rarch work to r-itrprt both typ of robut tability coditio ad mari cotitut o of th xt rarch objctiv. Cotrollr Complxity ad Ordr Th umbr of opratio that hav to b prformd pr ampli priod may plac rtrictio o th compator di. Th implmtatio of a cotrollr bad o th tat pac rprtatio diffr from that bad o trafr fuctio. Th formr appar to hav a commo domiator for vry lmt of th compator wh it i traformd ito trafr fuctio dcriptio ad th lattr do ot actually d it. Idd, th xprio of ach cotrol ial i a trafr fuctio matrix dpd o it corrpodi row of th compator matrix. But v thr, commo domiator ar ot dd. Th cotrol ial u i i computd a th um of ial ratd by vry compator i th i-th row. I ordr to mak a ralitic compario of th computatioal cot of th diffrt cotrollr o-diaoal MIMO QFT, H-ifiity ad diaoal MIMO QFT, th umbr of um ad multiplicatio computd i ach ampl at th fial implmtatio ar aalyzd. Followi th am dicrtizatio proc, th valu i Tabl VII ar obtaid. Th compator matrix of th H-ifiity di xprd i trafr fuctio dcriptio prt 36 lmt havi 4 d ordr. Th diaoal MIMO QFT di coit of ix diaoal compator oi from 5 th to 4 th ordr. Th o-diaoal MIMO QFT di coit of iht compator oi from 3 rd to 4 th ordr. Tabl VII. Computatioal cot pr ampli for th thr cotrollr Cotrollr umbr of Multiplicatio umbr of Sum o-diaoal MIMO QFT 3 4 H-ifiity iaoal MIMO QFT 6 ω 4. COMBIIG SWITCHIG & ROBUST QFT COTROL STRATEGIES TO IMPROVE CLASSICAL COTROL [8] Thi ctio itroduc a mthodoloy to di a family of robut cotrollr abl to o byod th claical liar limitatio. Combii robut di ad tabl witchi, th w cotrollr optimiz th tim rpo of th ytm by fat adaptatio of th cotrollr paramtr duri th trait rpo accordi to crtai rul bad o th amplitud of th rror. Th mthodoloy i bad o both a w raphical tability critrio for witchi liar ytm ad th robut quatitativ fdback thory QFT. RTO-E-SCI-95-3