INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volue 0, 06 Design nd erifiction of roust controller for te twin rotor MIMO syste Petr Clup, Jku Noák, Jn Přikryl Astrct Te pper dels wit control design nd erifiction of roust controller of te Twin Rotor MIMO Syste rel-tie lortory plnt y Feedck Ltd. Te plnt pysicl ppernce s well s its eior reseles elicopter. It consists of two propellers indiidully controlled y externl controller. An initil nonliner teticl odel is deried using first principles odeling nd furter used for siultion erifiction of te designed controllers. Besides, seerl liner lck ox odels re identified y pplying rious input courses to te plnt. Resulting set of odels is used for roust control design. Te designed roust controllers wit proising eior in siultions re erified y rel-tie control of te lortory odel. Keywords First principle odelling, Rel-tie control, Roust control I. INTRODUCTION OST of current control lgorits re sed on odel Mof controlled plnt []. It is oious tt soe infortion out controlled plnt is required to llow for design of controller wit stisfctory perfornce. A plnt odel cn e lso used to inestigte properties nd eior of te odeled plnt witout risk of dge or iolting tecnologicl constrints of te rel plnt. Te re two sic pproces of otining plnt odel: te lck ox pproc nd te first principles odeling (teticl-pysicl nlysis of te plnt). Te lck ox pproc to te odeling [], [3] is sed on nlysis of input nd output signls of te plnt. Te in dntge of te lck ox pproc consists in te possiility of usge te se identifiction lgorit for wide set of rious controlled plnts [4], [5]. Contrry, te first principle odeling proides generl odels lid for wole rnge of plnt inputs nd sttes. A odel is creted y nlyzing te Tis work ws supported y te Ministry of Eduction, Yout nd Sports of te Czec Repulic witin te Ntionl Sustinility Progre project No. LO303 (MSMT-7778/04). Tis work ws supported y te Europen Regionl Deelopent Fund under te project CEBIA-Tec No. CZ..05/..00/03.0089. Tis support is gretly cknowledged. Petr Clup is wit te Tos Bt Uniersity in Zlin, Fculty of Applied Infortics, Regionl Reserc Centre CEBIA-Tec, n. T. G. Msryk 5555, 760 0 Zlin, Czec Repulic (pone: +40 57603 536, e-il: clup@fi.ut.cz). Jku Noák is wit te Tos Bt Uniersity in Zlin, Fculty of Applied Infortics, Regionl Reserc Centre CEBIA-Tec (eil: jnok@fi.ut.cz). Jn Přikryl studied ster degree t te Tos Bt Uniersity in Zlin, Fculty of Applied Infortics. odeled plnt nd coining pysicl lws [6]. But, tere re usully ny unknown constnts nd reltions wen perforing nlysis of plnt. Terefore first principle odels re suitle for siple controlled plnts wit sll nuer of preters. First principle odeling cn e lso used for otining sic infortion out controlled plnt (rge of gin, rnk of suitle sple tie, etc.). Soe siplifictions ust e used to otin resonle results in ore coplicted cses. Tese siplifictions ust relte wit te purpose of te odel. Te first principle odeling is lso referred to s wite ox odeling. Te pper coines ot etods. Bsic reltions etween plnt inputs nd outputs re deried using first principles. Te otined odel is furter iproed on te sis of esureents. Tis pproc is known s grey ox odeling [7]. Te gol of te work ws to otin teticl odel of te Twin Rotor MIMO Syste [8], design te odel in MATLAB-Siulink enironent nd use tis odel for design of dptie controller. Te Twin Rotor MIMO Syste ws deeloped y Feedck Instruents Ltd. nd seres s rel-tie odel of nonliner ultidiensionl syste. A odel, wic represents te plnt well, cn considerly reduce testing tie of different control pproces. Ten only proising control strtegies re pplied to te rel plnt nd erified. Te pper is focused on roust control design [9]. Roust control pproc llows otining stilizing controller not only for noinl controlled syste ut lso for wider set of controlled systes. Te set cn e defined in rious wys. Tis pper dels wit pretric uncertinty, nely interl uncertinty [0]. Te pper is orgnized s follows. Section presents te odelled syste Twin Rotor MIMO Syste. A derition of te nonliner odel nd its ipleenttion in te MATLAB / Siulink enironent crried out in Section 3. Roust control design is presented in Section 4. As prt of roust design, liner odels of te two single input single output decoupled susystes re identified nd presented lso in Section 4. Te erifiction of te designed roust controller using ot nonliner Siulink odel nd te rel-tie lortory plnt is descried in Section 5. II. TWIN ROTOR MIMO SYSTEM A potogrp of te Twin Rotor MIMO Syste is presented in Fig.. Te syste is used to deonstrte te ISSN: 998-4464 7
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volue 0, 06 principles of non-liner MIMO syste, wit significnt cross-coupling. Its eior reseles elicopter ut contrry to ost flying elicopters te ngle of ttck of te rotors is fixed nd te erodynic forces re controlled y rying te speeds of te otors. Significnt cross-coupling is osered etween te ctions of te rotors, wit ec rotor influencing ot ngle positions []. Tere re two propellers drien y DC-otors t ot ends of e, wic is pioting on its se. Te joint llows te e to rotte in suc wy tt its ends oe on spericl surfces. Tere is counter-weigt fixed to te e nd it deterines stle equiliriu position. Te controls of te syste re te otors supply oltges. Te esured signls re position of te e in te spce, i.e. two position ngles []. til otor counter- weigt z x in otor Fig.. Scetic front iew of te Twin Rotor MIMO Syste wit grittion forces. Fig.. Twin Rotor MIMO Syste. III. MODEL OF THE TWIN ROTOR MIMO SYSTEM A nonliner odel of te plnt is deried in tis section. Te odel is sed on first principle odeling [], [3], nd [4]. More detils concerning odeling of te Twin Rotor MIMO Syste cn e found in [5]. Tere re two outputs of te plnt: position ngle in te erticl plne eletion (i.e. ngle wit respect to orizontl xis) nd position in te orizontl plne ziut (i.e. ngle wit respect to erticl xis). First erticl plne will e considered, nd ten te orizontl ngle will e focused on. A scetic front iew of te free e nd connected prts of te Twin Rotor MIMO Syste is depicted in Fig.. Te grittion forces tking effect re presented s well. Constnt g represents grittionl ccelertion, preters l t, l nd l c stnd for te lengt of te til prt of te e, te lengt of te in prt of te e nd te lengt of te counter-weigt e respectiely. Te ss of te til otor wit til rotor, te ss of te til sield nd te ss of te til prt of te e re represented y tr, ts, nd t respectiely. Constnts nd c represent te ss of te counter-weigt e nd te ss of te counter-weigt respectiely. Te ss of te in otor wit til rotor, te ss of te in sield nd te ss of te in prt of te e re represented y r, s, nd respectiely. Finlly stnds for te pitc ngle of te e eletion A. Initil odel Te derition of oents in te erticl plne is sed on Newton s second lw of otion: d M J () dt were M is su of coponents of oent of forces nd J is su of oents of inerti reltie to orizontl xis of indiidul prts of te plnt: M J Mi () J (3) i Te oents present in te orizontl plne cn e deried in te siilr wy s te oents in te orizontl plne. It is possile to derie stte equtions of te wole syste: ds dt ( ) cos sin + ( ω ) g A B C lf J Ω ( A+ B+ C) sin +Ωk J (4) ISSN: 998-4464 7
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volue 0, 06 ( ) ds lf t ωt cos Ω k (5) dt J Jtrωt Jrω sin Ω S + ; Ω S + (6) J J were A, B, nd C re constnts deried fro te pysicl preters of te plnt; S nd S re te ngulr oentu in erticl plne for te e nd te ngulr oentu in orizontl plne for te e respectiely. Te oent of inerti in DC-otor til propeller susyste nd te oent of inerti in DC-otor in propeller susyste re represented y J tr nd J r respectiely. Tese equtions descriee dependencee of output ngles (eletion nd ziut ) on rottions of te in nd te til otors ω nd ωt respectiely. Te otors re controlled y control oltge ccording to te following cointions of liner dynics nd sttic non-linerity: du ( u dt Tr ω P u ( ) du ( u dt Ttr ω P u ( ) t + u + u ), ), were T r nd T tr re te tie constnt of te in otor propeller syste nd te tie constnt of te til otor propeller syste. Functions P () nd P () descrie te sttic nonlinerity of te in otor propeller syste nd te sttic nonlinerity of te til otor propeller syste. Inputs u nd u represent te control oltge of te i otor nd te control oltge of te til otor respectiely. B. Ennced odel Docuenttion [] proides preters nd reltions of te Twin Rotor MIMO Syste wic were presented in preious susection. Howeer rel-tie experients sowed tt tese preters nd equtions sould e refined or reised. Tis susection is focused on odifiction of te initil odel nd preters gien in [] in order to otin etter correspondence of te teticl odel nd rel tie syste. ) Refineent of te diensions Te diensions of te odeled Twin Rotor MIMO Syste were esured to refine constnts gien in docuenttion []. Especilly te lengt of te counter-weigt e is different fro te lue gien in docuenttion. ) Nonliner sttic functions Te ode contins seerl nonliner functions e.g. equtions (4), (5), (7) nd (8). Tese functions e to e deterined to design te finl odel. A penoenologicl pproc ws used for teir identifiction. A polynoil (7) (8) pproxition ws used witout deep study of te pysicl fundentls of te reltion: 6 5 4 P ( u ) 90.99u + 599.73u 6.6u (9) 3 38.64u + 63.45u + 83.4u F P F ( ) 3.87 5 9 4 ω 0 ω 4.096 0 ω + 6 3 +.385 0 ω +.34 0 ω + 0..799ω 5 4 94.696u 5 ( u ) 00u 3 483.5u + 6.7 u + 3796.83 u 5 9 4 0 ωt 9.8440 ωt + 7 3 4 +.785 0 ωt +.7300 ωt + 0.79ω t ( ω ) 9.496 t 3) Cross coupling trnsfers Te cross coupling cn e osered in te Twin Rotor MIMO Syste. Te rottion of te til otor sligtly ffects eletion ngle wile in otor strongly ffects not only eletion ut lso ziut. Te influencee of til otor to eletion ws odeled s liner function of til rotor rottions. Te dependence of ziut on rottions of te in otor is ore coplicted to odel. An exponentil function of te M oent ws used to cope wit tis prole. A Siulink scee of tis reltion is presented in Fig. 3. Fig. 3. Model of influence of in otor to te ziut. 4) Clewys A clewy etween te fixed se of te Twin Rotor MIMO Syste nd its e plys significnt role especilly in cse of low rottion speed of te til otor. Due to te cle wy te syste does not ee s n integrtiee ut proportionl eior cn e osered. Te effect of te clewy is odelled s nonliner functionn of te ziut. IV. DESIGN OF ROBUST CONTROLLER A design of roust controllers for te Twin Rotor MIMO Syste is sed on pretric uncertinty pproc. Te pretric uncertinty cn e used for ot continuous-tiotin n interl uncertinty of te liner odel of te syste. For te control purposes, te Twin Rotor MIMO Syste ws considered s nd d discrete systes [6]. Te controlled plnt ws nlyzed to twoo independent systes control oltge of te in rotor (input) eletion (output) control oltge of te til rotor (input) ziut (output) Te cross-couplings re considered s disturnces in tis cse. Vrious step cnges of te control input were pplied to te plnt, tie responses of plnt were esured nd (0) () () ISSN: 998-4464 73
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volue 0, 06 corresponding liner odels were identified. A. Identifiction of in rotor eletion susyste Step cnges of te control oltge of in rotor usully led n oscilltory response of te eletion output. Tis syste ws odeled y 3 rd order liner syste: ox in te 3D spce. Contrry to experients not ll trnsfer functions fored fro te preters inside tis ox re stle. Te sitution is presented in Fig. 5. G 0 ( s) 3 3 + + + s s s (3) Identifiction ws perfored in lest squres sense using MATLAB function finserc. Results re surized in Tle I. TABLE I. INTERVALS OF THE MAIN ROTOR ELEVATION MODEL Preter Minil lue Mxil lue 0 60 57 3 0.3.30 0..35 0.8.74 Coon nottion for writing trnsfer functions of systes wit interl uncertinties uses squre rckets: G ( s) [ 60;57] [ ] 3 [ ] [ ] (4) 0.3;.30 s + 0.;.35 s + 0.8;.74 s+ Model eior especilly its stility is defined y position of trnsfer function poles i.e. roots of te denointor. Te interl odel ws otined fro 9 liner odels wic were identified fro step responses. Poles of tese 9 odels re presented in Fig. 4. Te poles re rked y sterisk nd poles of ec odel re connected y line. Fig. 5. Stility of interl odel of te in rotor eletion susyste Four crosscuts of te ox re presented. Te crosscuts re prllel to x plne. Te upper nd te lower crosscut correspond to te xil nd te inil lue of te 3 preter respectiely. Te oter two crosscuts correspond to one tird nd two tirds etween te iniu nd xiu of 3. Te red res correspond to unstle systes wile green res correspond to stle systes. Te gent circles correspond to odels identified fro step responses. B. Identifiction of til rotor ziut susyste Te til rotor ziut susyste ws identified in te siilr wy s te in rotor eletion susyste. Te in difference consists in fct tt nd order odels were used, ecuse teir ccurcy ws good enoug. G 0 ( s) + ts+ s (5) Te step responses were te plnt reced ckstop were oitted fro te identifiction. Te positions of poles re presented in Fig. 6. As ll te poles presented in Fig. 6 re in te left lf-plne, ll te odels re stle nd ost of te re oscilltory. Liits of te odel preters re surized in Tle II. TABLE II. INTERVALS OF THE TAIL ROTOR AZIMUTH MODEL Fig. 4. Poles of te in rotor eletion odels It cn e seen tt ll odels re stle (rel prts re sller tn zero). Moreoer ll odels re oscilltory ecuse ec odel s pir of coplex conjugted poles. Interls of denointor of te interl odel (4) for Preter Minil lue Mxil lue 0 75 76.63 6.04.49 4.55 Resulting interl odel cn e written in te following for: ISSN: 998-4464 74
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volue 0, 06 G ( s) [ 75; 76] [ ] s [ ].63; 6.04 +.49; 4.55 s + (6) w Interls of te denointor define rectngle in x plne. As te denointor is nd order degree polynoil nd ot, re lwys positie roots of denointor e lwys negtie rel prt nd terefore ll te odels re stle. Soe of te odels re oscilltory wile te oters re periodic. Te sitution is presented in Fig. 7. Te gent circles correspond to odels identified fro step responses. controller - + u y Fig. 8. DOF controller Control signl is clculted ccording to te following eqution: R( s) Q( s) ( ) ( ) ( ) ( ) ( ) G ( s) W ( s) G ( s) Y( s) U( s) W s Y s K s P s P s R Q (7) Fig. 6. Poles of te til rotor ziut odels Fig. 7. Beior of interl odel of te til rotor ziut susyste C. Design of roust DOF controller Seerl controller types were tested nd results of te DOF (Two Degree Of Freedo) controllers re presented in tis pper. Te scee of te control loop wit DOF controller is presented in Fig 8. were U(s), W(s), nd Y(s) re Lplce trnsfors of u(t), w(t), nd y(t) signls respectiely. Pole-plceent etod ws used to clculte controllers polynoils R(s), P(s), Q(s) nd K(s). Detil cn e found for exple in []. Pole plceent etod is sed on fixing poles of closed loop to te desired positions. For roust control of syste wit interl uncertinties te controller is required to gurntee stility of closed loop for ny cointion of preters fro te gien interls. Unfortuntely coefficients of te crcteristic polynoil of te closed loop re not independent. Terefore Kritono polynoils cnnot e used for stility testing [7]. Soling te roust closed loop stility is teoreticlly coplicted tsk. Hence siplified pproc ws used:. Controller polynoils were clculted wit respect to gien position of poles nd noinl syste.. Coefficient lues eqully spred troug n interl were generted for ec uncertinty interl. 3. Stility of te closed loop ws tested for ec cointion of coefficient lues. Coefficients of te noinl syste were defined s idpoints of te uncertinty interls. Fie coefficient lues were generted for ec interl. It leds to 5 4 65 cointions (systes) for te in rotor eletion susyste nd 5 3 5 cointions (systes) for te til rotor ziut susyste. Pole positions were defined in te siplest possile wy. One ultiple rel pole ws used. Hence te crcteristic polynoil of te closed loop s te following for: D( s) ( s+ ) n (8) Wole tsk of coputing te roust controller cn e seen s n optiiztion prole wit te gol of stilizing ll te ISSN: 998-4464 75
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volue 0, 06 generted systes nd te pole position s tuning preter. Tis pproc does not gurntee stility of ll possile coefficient cointions ut its results in rel-tie enironent were good no unstle eior of te closed loop ws osered. V. REAL-TIME EXPERIMENTS Tis section presents seerl rel-tie experients fro uge set of experients perfored. A spling period of T 0 0.0s ws used for sensors nd ctutors in ll experients. Eletion control nd ziut control ws perfored siultneously to erify effect of cross-couplings. Control design consists in funding optil position of te closed loop poles i.e. find optil coefficient presented in eqution (8). Te gol of te optiiztion procedure ws to stilize ll te 65 systes for te in otor eletion susyste nd ll 5 systes for te til otor ziut susyste s entioned in te preious section. Optil lue of pole position for eletion control ws.739 (i.e. ultiple pole in position -.739) nd optil lue for ziut control ws 0.6 (i.e. ultiple pole in position -0.6). Courses of eletion control re presented in Fig. 9. Control of ot rel-tie plnt nd nonliner odel deried in Section 3 re presented. Fig. 0. Aziut control.739, 0.6 It cn e osered, tt cross-coupling plys iportnt role in te ziut control. Pole position defined y preter ws cnged to otin sooter course of control signl for te in otor. A lue of.7 ws used nd te control courses of te eletion control re presented in Fig.. Fig. 9. Eletion control.739, 0.6 It cn e seen tt ery good reference trcking ws reced ut course of control signl ws ery oscilltory in rel-tie conditions contrry to siultion. Courses of ziut control re presented in Fig. 0. Control of ot rel-tie plnt nd nonliner odel deried in Section 3 re presented. Fig.. Eletion control.7, 0.6 Reference trcking ws sligtly worse copring to Fig. 9 ut course of control signl ws uc sooter. Courses of ziut control of tis experient re presented in Fig.. ISSN: 998-4464 76
INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volue 0, 06 A siple roust control tecnique ws presented nd successfully erified in rel-tie conditions. Furter iproeent cn e cquired y ipleenting etter pole plceent. Usge of seerl different poles would led to een etter control courses tn te courses presented in te pper. Furter work will e focused on otining een etter perfornce of roust control nd susequent coprison of results of roust control nd n dptie control. Moreoer, te presented odel will e used to design nd erify odel sed predictie control of te Twin Rotor MIMO Syste. Fig.. Aziut control.7, 0.6 A stisfctory control eior wit good reference trcking ws osered. Te gretest difference etween nonliner odel nd rel-tie plnt ws otined for control signl of te til rotor were non-odeled eior ws osered. Control courses were copred not only y isul coprison of figures ut lso y qudrtic criteri. Te results re presented in Tle III. TABLE III. COMPARISON OF CONTROL COURSES Controller Eletion Aziut e Δu e Δu.739 Siultion.470e+7 0.0707 4.3537e+8 0.093 (Fig. 9, Fig. 0).7 Siultion.9766e+7 0.006 4.559e+8 0.0 (Fig., ).739 Rel plnt.346e+7 3.407e+3 4.5479e+8 0.35 (Fig. 9, Fig. 0).7 Rel plnt (Fig., ).9948e+7 4.463 4.554e+8 0.557 REFERENCES [] V. Boál, J. Bö, J. Fessl nd J. Mcáček. Digitl Self-tuning Controllers: Algorits, Ipleenttion nd Applictions. London: Springer-Verlg, 005. [] G. P. Liu, Nonliner identifiction nd control A neurl network Approc. London: Springer-Verlg, 00. [3] L. Ljung, Syste identifiction: teory for te user. Upper Sddle Rier, NJ: Prentice Hll PTR, 999. [4] J. Mikleš nd M. Fikr. Process odelling, identifiction, nd control. Berlin: Springer-Verlg, 007. [5] B. Codrons, Process odelling for control: unified frework using stndrd lck-ox tecniques. London: Springer-Verlg, 005. [6] D. M. Hiellu nd J. B. Riggs, Bsic principles nd clcultions in ceicl engineering. Upper Sddle Rier, N.J.: Prentice Hll, 004. [7] K.C. Tn nd Y. Li, Grey-ox odel identifiction i eolutionry coputing, Control Engineering Prctice, ol. 0, pp. 673 684, 00. [8] Feedck Instruents, Twin Rotor MIMO Syste 33-007-PCI, Croworoug, 03. [9] M. Morri nd E. Zfiriou. Roust process control. Englewood Cliffs, N.J.: Prentice Hll, 989. [0] R. S. Sáncez-Peñ nd M. Sznier. Roust systes teory nd pplictions. New York: Wiley, 998. [] Twin Rotor MIMO. Feedck [online]. Aille: ttp://www.feedckinstruents.co/products/eduction/control_instruenttion/ twin_rotor_io. [] Feedck, Twin Rotor Mio Syste Adnced Tecing Mnul 33-007-4M5. [3] D. Rotondo, F. Nejjri nd V. Puig, Qusi-LPV odeling, identifiction nd control of twin rotor MIMO syste, Control Engineering Prctice, ol., pp. 89 846, 03. [4] A. Ride, M.H. Seed nd H. J.C. Huijerts, Dynic odelling of TRMS using nlyticl nd epiricl pproces, Control Engineering Prctice, ol. 6, pp. 4 59, 008. [5] P. Clup, J. Přikryl nd J. Noák Modelling of Twin Rotor MIMO Syste, Procedi Engineering, ol. 00, pp. 49 58, 05. [6] R. Mtušů Roust Stility Anlysis of Discrete-Tie Systes wit Pretric Uncertinty: A Grpicl Approc, Interntionl Journl of Mteticl Models nd Metods In Applied Sciences, ol. 8, pp. 95-0, 04. [7] R. Mtušů nd R. Prokop. Control of Interl Systes Using DOF Configurtion, Procedi Engineering, ol. 00, pp. 340 344, 05. In rel-tie experients, cnging lue of te preter led to lower lues of criterion wic sus squres of control signl differences wile reference trcking criterion reined lost uncnged. VI. CONCLUSION A odel of nonliner rel tie syste Twin Rotor MIMO Syste ws deried using first principle odeling nd ten ennced to correspond etter wit te rel plnt. ISSN: 998-4464 77