Rotational Speed Control of Multirotor UAV's Propulsion Unit based on Fractional-order PI Controller

Transcription

1 Roonl Speed Conol of Muloo UAV's opulson Un bsed on Fconl-ode Conolle Wocech Genck, Tl Sdll, Josłw Goślńsk, o ozesk,, João. Coelho, Sš Sldć oznn Unvesy of Technology, oowo 3 See, oznn, olnd, Fculy of Eleccl Engneeng, nsue of Conol nd nfomon Engneeng Fculy of Compung, Ch of Conol nd Sysems Engneeng, Dvson of Sgnl ocessng nd Eleconc Sysems nsuo olécnco de Bgnç, Cmpus de Sn Apolón, Apdo 34, Bgnç, ougl Fculy of Engneeng, Unvesy of Rek, Vukovsk 58, 5 Rek, Co E-ml: wocech.genck@pu.poznn.pl conolle ype. should be noed h hs s he fses conol loop n he checue of muloo UAVs, whch leds o he hghes me equemens nd suons o be me. Absc n hs ppe he synhess of oonl speed closed-loop conol sysem bsed on fconl-ode popoonl-negl (FO) conolle s pesened. n pcul, s poposed he use of he SCoMR-FO pocedue s he conolle unng mehod fo n unmnned el vehcle s populson un. n hs fmewok, boh he Heme-Behle nd onygn heoems e used o pedefne sbly egon fo he conolle. Sevel smulons wee conduced n ode o y o nswe he quesons s he FO conolle good enough o be n lenve o moe complex FOD conolles? n wh ccumsnces cn be dvngeous ove he ubquous D? How obus hs fconl-ode conolle s egdng he pmec unceny of consdeed populson un model?. eywods fconl-ode conolle; speed conolle; FO conolle; lne BDC moo model; populson un; moo-oo model; Heme-Behle nd onygn heoems. NTRODUCTON Thee e wo mn obecves when desgnng conolle sysem: o ensue he conol sysem sbly nd subsequenly o povde he hghes possble conol pecson n he pesence of dsubnces nd uncenes. Those subecs hve been ddessed by mny eseches wokng on unmnned el vehcles (UAVs) especlly whn sevel conexs such s he conol lgohms [],[9], opmzon of he UAV s consucon [3] o ensung ful-olen conol [2]. Mny of hose soluons e nsped on des used fo fxed-wng UAVs [7],[27]. n he ppe [4], he uho poposed conolle n he λdμ (FOD Fconl-Ode D) sucue fo he poblem descbed n Secons 2-3. The mn dffculy (bu especvely, s s shown, lso he possbly) n he poposed soluon s he conolle unng wh fve sepe degees of feedom (conolle ses). Accodng o he defned cos funcon by usng pcle swm opmzon lgohm (see [3]), s possble o ensue he ckng of efeence sgnl (oonl speed of moo-oo un model used n UAVs) n moe effcen wy hn fo he D nd CDM (Coeffcen Dgm Mehod) conolles. Resuls e pesened n []. Beng n mnd he cscde conol sucue used n UAVs, s mpossble o hnk bou msson plnnng [8] o flghs uonomy [6],[2] whou ensung ppope soluons n he lowe lyes of conol checue (poson nd oenon conol). A hs boom level one cn fnd he sysem known s Eleconc Speed Conol (ESC). Ths module s esponsble o dve he DC moo v ppopely fs chnges of ulse Wdh Modulon (WM) duy cycles, ledng o he geneed hus nd oque. Theefoe, s mpon o use ppope moo-oo uns [28], wh he pope szes [5], pope ngemens [2] nd ppope /7/$3. 27 EEE ROBEM STATEMENT The oonl speed conol s deeply nvesged poblem, bu s sll n open e egdng he pplcon of moden echnques bsed on conol heoy. Fo UAVs hee e myd of dsnc conol sysems ppoches, some moe complex nd ohes smple bu wh hghe effcency. n he leue bgge enon s gven o hs second ype due o sevel esons such s s poenl pplcbly n he lowes lye of oonl speed conol nd smplcy egdng s mhemcl fomulon. Regdng hs concep nd he cuenly used soluons, he mn de oses: why no o exend he hgh flexbly nd smplcy of opoonlnegl-devve (D) conol o fconl-ode s counep o mpove he conol quly by usng s ledng dvnge. Nmely he possbly of moe ccue shpe fng of closed loop sysem s fequency chcescs, no esced only o 2log(gn). Ths cn be ned by he noducon of fconl-ode n he noon of nego nd dffeeno polynomls ps of conolle. Thus, one obns wo ddonl desgn pmees (odes of he nego nd dffeeno: λ nd μ). The choce depends on he shpe of me nd fequency chcescs of he closed-loop conol sysem [3]. 993

2 The wo mn poblems obseved n he pevous soluons wee he need o une he fve conolle pmees nd he lck of cle poposls fo mehods of he sbly ssunce, nlyss nd ssessmen n he pocedue fo he conol sysem synhess wh he poposed conolle ype. n ode o mee hese poblems, n he pesen wok, smple conolle sucue s used: λ (o lenvely nmed FO), o vefy f by usng s dvnges, s possble o popose n effcen conolle, wh bee pefomnce hn D nd FOD. The second ssue s o nswe o he queson: how obus s hs conolle on pmec unceny? Such compve nlyss nd esech e poposed fo he fs me nd e he novely nd conbuon of hs ppe. The ppe s ognzed s follows n Secon, SCoMR- FO pocedue s defned nd explned. A synhec nfomon bou populson un model poposed fo ess s povded n Secon V. n Secons V nd V, heoy bou fconl-ode conolles nd Heme-Behle nd onygn Theoems, s pesened. Secon V descbes FO conolle synhess nd he sysem sbly nlyss. s wo secons povde nfomon bou conduced ess nd concluson.. SCOMR-FO ROCEDURE The mos genel fom of he pocedue/lgohm fo modelng moo-oo uns nd fo he closed-loop sysem synhess, hee nmed SCoMR-FO (Modelng nd Roonl Speed Conol of Moo-Roo Un) wh he FO ype conolle, ssumes he fom:. Recod he possble me chcescs of moo-oo un fo vous ypes of efeence/npu sgnls. 2. By usng pmec esmon mehods (.e. üpfmülle, Sec o gphcl mehod) defne/clcule he pmees vlues fo he pedefned pln model sucue (.e. nsfe funcon sucue) nd pefom he model vldon on ohe ecoded d ses fom es bed. 3. Fo he conolle n he FO sucue descbed by he nsfe funcon, use he Heme-Behle nd onygn heoems o fnd he conolle gns of, nd degee of nego p s polynoml λ, fo whch he closed-loop conol sysem s sble. 4. Fo he desgned sech spce (, nd λ) use he chosen lgohm o fnd he opmum vlue of he cos funcon (fo exmple: mnmum of he negl of Absolue Eo AE o mnmum of he negl of Squed Eo SE). 5. Genee me couses fo some of bes esuls (hose wh smll vlues of cos funcon). 6. Selec he soluon (, nd λ se vlue) fo whch he me couses mee he expecons (fo exmple: one h povdes smll/no oveshoo, sho lg/selng me, smll/no conol eo, ec.). NOTE: Becuse he conolle se sech s bsed on negl quly ndcos nd hey do no ke decly no ccoun he shpe of he geneed me couses of closed-loop sysem (only descbe he numecl devon beween he oupu sgnl nd he efeence one), hee s song need fo use he seps 5 nd 6 of he SCoMR-FO lgohm. n he followng secons, moe dels on ech lgohm sep, e pesened. V. MODE OF ROUSON UNT n ode o fulfl he needs of oonl speed conol n el ESC sysem, specl es bed ws conduced he nsue of Conol nd nfomon Engneeng of he oznn Unvesy of Technology. Bsed on he ecoded me couses (elon beween he moo ngul speed nd he ppled volge []), one cn ssume he exsence of moo-oo un mhemcl model h descbes well enough he devce dynmcs. Regdng pevous sudes nd expeences, numbe of dffeened models wee poposed, subsequenly condonng he conol lw nd he conolle ype, whch cn be used o buld he conol sysem. Exmples cn be found n sevel woks. n [] n ppoxmon of moo-oo un dynmcs by lne second-ode nel model ws poposed. Auhos of efeences [4-6] desgned fs-ode ne model wh pue me dely, whch cn be ppoxmed by he second-ode nel model. n [24] he second-ode nel pln model wh me dely ws developed. n [2] fuzzy model of coxl populson un (fs-ode nel elemen wh he vyng me consn) ws defned. ws decded o ecod chcescs of he AX 284/2 GOD NE BDC moo fom T-Model Moos compny wh hee-blded popelle GWS-HD95x3-SW 9x5. Moe dels bou he used es bed, dedced DYNO Temnl sofwe nd cquson woks, e povded n []. Also, n [3], compehensve physcl chcesc nd usfcon fo he choce of h pcul moo-oo un, e povded. Fom he ess pefomed o obn he nsfe funcon of he moo-oo model (elons beween ngul speed Ω nd ppled volge V), one decded o use fs-ode nel model wh nspo dely : G () s () s () s Ω b s = = e, () V s + nd wo ppoxmon ppoches wh he use of: - DENT lby of MATAB sofwe, - d-hoc gphcl mehod (üpfmülle) nd by selecon of he model pmees. Fom vldon expemens, bee cuve fng nd, s esul, pmees of moo-oo un model s nsfe funcon, povded he second mehod. V. FO CONTROER THEORETCA BACGROUND Theoecl bscs of fconl clculus e pesened n [] nd [9]. Exmples of fconl-ode conol pplcons n elon o poson/oenon conol of muloo UAVs cn be found n [26] nd moe exmples of such conolles synhess n he wok []. n he cles [22] nd [23], nfomon bou me nd fequency domn nlyss e, especvely, povded. 994

3 Accodng o [] fconl clculus s genelzon of dffeenon nd negon o non-nege ode fundmenl (connuous nego-dffeenl) opeo D defned s: D = d R() d R() ( dτ ) R( ) > =, < whee nd e opeon lms nd s he opeon ode (usully s el vlue). Fom he Remnn-ouvlle defnon of he fconl dffeo-negl: f ( τ ) ( τ ) n D () = ( n ) + Γ (2) d d τ (3) n α d fo n < < n whee Γ () s he Eule s Gmm funcon, n nlogy o nege-ode sysems, fo non-nege ode sysems, one my use he plce nsfom of equon (3), whch s defned s: s n k k () d = s F () s s D f () e D f (4) k = = fo n < < n, whee s denoes he plce nsfom vble nd s=. Fom he equon (4), s hence possble o se he sucue of FO conolle ype o he fom of nsfe funcon: C λ () s = + s, (5) whee s he popoonl gn, negon gn nd λ s posve el numbe. The qus-polynoml (s), whch descbes he closed-loop chcesc equon of he sysem fom Fg. s gven by: λ s () s = ( b + b s ) e + s. (6) + n ode o fnd he conolle pmees ses, Heme-Behle nd onygn heoems e used. V. HERMTE-BEHER AND ONTRYAGN THEOREM Theoem. Heme-Behle Theoem ([8],[7]). e be complex funcon of descbed by equon: ( ) ( ) + ( ), = (7) whee ( ) nd ( ) ps of ( ). The ( ) ) ( ) nd ( ) e nelced; epesen he el nd mgny s sble f: only hve smple el oos nd hese ' 2) ( ) ( ) ( ) ( ) > ' n (,+ ),, fo some = ' Whee ' ( ) nd ( ) e he devves of ( ) nd ( ) wh espec o. An mpon sep s o ensue h ( ) nd ( ) only hve el oos. Ths cn be cheved by pplyng he onygn Theoem. Theoem 2. onygn Theoem e () s be descbed by he equon (7) ssumng s=. To ssue h ( ) = nd ( ) = only hve el oos, mus be ssued h n nevls 2 lπ + η 2lπ + η l =,2,3,..., (8) ( ) nd ( ) hve excly 4lN+M oos. Fo suons whee chcesc equon s of fconl ode, mus hve 4l([N]+)+[M]+ oos, he ( ) nd ( ) whee [ ] epesens he nege p, nd N nd M e ken s degee of he numeo nd denomno polynomls of he nege p. oofs cn be found n [8]. As s shown n [24], s necessy o ewe he (s) qus-polynoml s: λ s () s b s + b + ( s ) e s. = (9) λ + Assumng h g=s nd λ=/b: g / ( g ) = b ( g / ) + b + ( g / + ) e ( g / ). () b Fg.. Block dgm of closed-loop conol sysem wh he FO conolle ype nd moo-oo model Thus, fo g=, ( ) becomes: + ( ) = b ( / ) + b ( / + ) e ( / ). () Replcng he e wh cos()+sn(), he el ( ) nd mgny ( ), ps of ( ) cn be descbed by:, 995

4 Re m / ( ) = b + cos ( ) ( ) ( ) + b cos ( ) + sn ( ) b m sn ( ) sgn ( ), ( ) = b + cos ( ) Re( ). sn ( ) ( ) sgn ( ) cos ( ) + sn ( ) (2) (3) Accodng o he onygn Theoem ( ) nd ( ) =. The pmee cn be descbed by: = V. CONTROER SYNTHESS AND SYSTEM STABTY ANAYSS n ode o deemne he pemssble spce of conolle ses (, nd λ) fo whch he closed-loop conol sysem wh FO conolle s sble, s poposed o use he negl quly ndces (AE nd SE) nd, especvely, equon (4) nd nequly (8) o deemne he nge of nd pmees. The sysem s sble n ccodnce wh Bounded npu Bounded Oupu (BBO) ceon, when he condons e me:. The duon of smulon mus be he sme s he desed smulon me se he pocess begnnng. 2. Only successfully conduced smulons e consdeed: one needs o vefy f he consecuve peks of he oupu sgnl y() e decesng by mesung he dffeence beween hem. n ffmve cse, he sysem s ssumed o be sble. cos b = b b cos ( ) + sn ( ) b ( ) + sn ( ) Re m ( ) ( ) + sgn ( ). (4) One cn ewe he ( ) s: ( ) b m( ) b( ), = (5) m( ) = Re ( ), (6) b( ) = cos ( ) + sn ( ) Re( ) / b (7) sn ( ) + cos ( ) m( ) sgn ( ). Fg. 2. Sbly egons of pmees, fo chnge of λ vlues fom. o fo closed-loop conol sysem wh pln model (): vlue fom -2 o 2 (sep: 4 d/s). The sech of conolle ses fom Fg. 2 n he desgned spce (fo whch he closed-loop conol sysem s sble), o fnd he bes ckng quly (, nd λ ses whch ensue he possbly fs ckng nd sgnl wh mnml o no oveshoo), my be bsed on ecoded vlues of quly ndcos (n hs ppe he AE), o on nohe cos funcon, n ccodnce wh seleced opmzon mehods. n ccodnce wh [4], he nge of pmees h ssues closed-loop sbly mus me he condons: mx < mn Snce ( ) ( m ( ) b ( )) m ( ) b ( ) ( ) <,. =,,2,... (8) s n odd funcon, hs he oo =. Thus, fo = =: ( ) = b. (9) + To ensue he nelce popey beween ( ) nd ( ) one mus mpose: Fg. 3. Refeence sgnl (SET) ckng n dsubed (DST) sysems (wh CDM, D, FOD nd FO conolles) conol sgnl consns (u mx=±6). ( ) > => b + > => > / b. (2) 996

5 Fg. 4. AE vlues s funcon of λ fo FO conolle ype. Fg. 8. AE nd SE vlues due o he chnge of nd populson un model pmees. Fg. 5. vlues s funcon of λ fo FO conolle ype. Fg. 6. Sbly egons of pmees, fo chnge of λ vlues fom. o fo closed-loop conol sysem wh nomnl pln model (): vlue fom -π o π (sep: 2π/). Fg. 7. Compson of ckng effecveness fo wo FO conolles: FO ( =32.27, =.9, λ=.49, AE=.3) nd FO2 ( =2.86, =4.4, λ=., AE=.3). V. COMARATVE TESTS RESUTS n hs ppe, esuls e pesened fo he model () fom [], n whch b =, =, =.4, =.35. Tes Bsed on eve sech n sbly egon fom Fg. 2, ecoded ckng me couses nd AE vlues fo he poblem llused n Fg. 3, FO conolle ses wee seleced ( =32.27, =.9, λ=.49), fulfllng he ssumpons fom he pevous secon. The FO conolle effecveness ws comped wh ohes fom [] nd [4] s depced n Fg. 3. The mn m ws cheved: bee ckng quly ws obned (n he pesence of sep ype dsubnces nd conol sgnl suon) fo he bes unes of D (dels n [4]) nd ype conolles. As dsdvnge one my egd ppeng oscllons. f he conolle ses sech eled solely on nlyss of AE vlues, one my conclude h smple conolle ype s enough fo ckng conol puposes (83 fom bes AE vlues wee obned fo = nd λ fom. o ). Howeve, me couses show h oveshoo s no decesed o zeo. n he emnng 7 vns, lso he use of he nego does no gunee he elmnon of sedy-se eo ( should be emembeed h one mus del hee wh fconl-ode fo nego p of conolle). Bsed on Fg. 4 nd Fg. 5, fo sml nge of vlues, n lowe nge of λ, s possble o enfoce he mpovemen (decese) of he AE (by he ncese of ). n pcce, n he conolle ses sech, useful nge of vlues ncluded: fom 24 o 33, fom o 4 nd λ fom.49 o.79. Tes Relve o Tes, nge of vlues ws chnged fom π o π (sep: 2π/) [25]. ws evlued whehe he decese of vlue (nd heeby spce of ses fo whch closed-loop sysem s sble), wll cuse dcl deeoon n he efeence sgnl ckng quly. The bes ckng pefomnce nd lowes vlue of AE, wee obned fo ses: =2.86, =4.4 nd λ=.. Despe he educon on he sech spce sze fo FO conolle ses (see Fg. 6), hee ws no loss of ckng quly (see Fg. 7). Tes The conolle obned n Tes ws vlded n conex of obusness on pmec unceny (see Fg. 8). Model pmees nd wee chnged up o 25% 997

6 of nomnl vlues. The obusness s smlle hn fo nomnl model, only fo smulneously bgge vlues of nd, h s, AE nd SE vlues gow. X. CONCUSON n hs ppe, he SCoMR-FO pocedue fo conol sysem synhess wh he FO conolle nd model of UAV s populson un s poposed, by usng he Heme-Behle nd onygn heoems. Ths mehod povdes he bee ckng quly hn clsscl D nd CDM conolles. Howeve, s pefomnce s less effcen hn he fconl ode D conol. The educon of he conolle pmees numbe fom fve n FOD ype conolle (,, D, λ, μ) o hee n FO, hs pos nd cons. One my use Heme- Behle nd onygn heoems o pedefne sbly egon fo conolle ses sech. The possbly o use wo moe ses (degees of feedom: D, μ) on FOD conolle genees ncesed compuonl complexy. n hs feld, FO conolle my be n lenve. n fc, povdes hgh obusness nd my be used o conol populson un model, whch pmees vy fom el pln pmees. Fuhe esech wll nclude dscee veson of he poposed FO conolle by Ousloup ppoxmon nd opmzon echnques o shoen he unng me. REFERENCES [] V.S. Akknpll, G.. Flcon, nd F. 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