Predictive Controller for Pitch Controller Missile. Faculty of Electrical Engineering, Khorasan University, Mashhad, Iran

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1 Prdictiv Controllr for Pitch Controllr issil Amir Torabi (Corrsponding author), Sobhan Salhi, Ali Karsaz, Ebrahim Tarsayi. Faculty of Elctrical Enginring, Khorasan Univrsity, ashhad, Iran. Faculty of Elctrical Enginring, Islamic Azad Univrsity, ashhad, Iran. Faculty of Elctrical Enginring, Khorasan Univrsity, ashhad, Iran. Faculty of Elctrical Enginring, Khorasan Univrsity, ashhad, Iran Abstract: This papr xhibits a comparativ assssmnt basd on tim rspons spcification prformanc btwn fuzzy and odl prdictiv control (PC) for a pitch control systm of an aircraft systm. Th dynamic modling of pitch control systm is considrd on th dsign an autopilot that controls th pitch angl. It starts with a drivation of suitabl mathmatical modl to dscrib th dynamics of an aircraft. For gt clos to actual conditions. Th whit nois disturbanc applid to th systm. Th prformancs of pitch control systms ar invstigatd and analyzd basd on common critria of stp s rspons in ordr to idntify which control stratgy dlivrs bttr prformanc with rspct to th dsird pitch angl..th dsign of PC gav rspons lss quality than that was givn from Fuzzy controllr but accptabl rsponss. Finally, It is found from simulation, prdictiv controllr proposd givs th bst prformanc compard to fuzzy controllr. [Amir Torabi, Sobhan Salhi, Ali Karsaz, Ebrahim Tarsayi. Prdictiv Controllr for Pitch Controllr issil. J Am Sci ;9(7):7-5]. (ISSN: 55-).. Kywords: controllr, Fuzzy, odl prdictiv, pitch controllr. Introduction Today s aircraft dsigns rly havily on automatic control systm to monitor and control many of aircraft s subsystm. Th dvlopmnt of automatic control systm has playd an important rol in th growth of civil and military aviation. odrn aircraft includ a varity of automatic control systm that aids th flight crw in navigation, flight managmnt and augmnting th stability charactristic of th airplan. To rduc th complxity of analysis, th aircraft is usually assumd as a rigid body and aircraft s motion consist of a small dviation from it is quilibrium flight condition [].Th pitch of aircraft is control by lvator which usually situatd at th rar of th airplan running paralll to th wing that houss th ailrons. Pitch control is a longitudinal problm, and this work Givs on dsign an autopilot that controls th pitch of an aircraft. Autopilot is a pilot rlif mchanism that assists in maintaining an attitud, hading, altitud or flying to navigation or landing rfrncs []. Th combination of nonlinar dynamics, modling uncrtaintis and paramtr variation in charactrizing an aircraft and its oprating nvironmnt ar th on major problm of flight control systm. This work is attmptd to Survy th control stratgis rquird to addrss th complx longitudinal dynamic charactristics of such aircraft. any th rsarch works has bn don in [], [5], [6], [7] and [8], to control th pitch or longitudinal dynamic of an aircraft for th purpos of flight stability. This rsarch is still rmains an opn issu in th prsnt and futur fforts [9]. In this papr usd prdictiv controllr to improv prformanc systm.in th past papr, Disturbanc ffct havn t apply to systm but in this papr it applid. Th simulation rsults shown that th dynamic charactristics of control systms can b improvd by this mthod. Problm Statmnt Control of dynamic systms with prsnt day sophistication and complxitis has oftn bn an important rsarch ara du to th difficultis in modling, nonlinaritis, and uncrtaintis, particularly whn thr is a constant chang in systm dynamics. It is also known that th rspons of a dynamic nonlinar plant cannot b trackd into a dsird pattrn with a linar controllr. Thus, a changing dynamic controllr is important to control such a plant. [] Pitch is dfind as a rotation around th latral or transvrs axis, which is paralll to th wings, and is masurd as th angl btwn th dirction of spd in a vrtical plan and th horizontal lin. Changs of pitch ar causd by th dflction of th lvator, which riss or lowrs th nos and tail of th aircraft. Whn th lvator is raisd (dfind as ngativ valu), th forc of th airflow will push th tail down. Hnc, th nos of th aircraft will ris and th altitud of th aircraft will incras. On of th targts of a pitch control systm is to control or hlp a pilot to control an 7

2 aircraft to kp th pitch attitud constant, that is, mak th aircraft rturn to dsird attitud in a rasonabl lngth of tim aftr a disturbanc of th pitch angl, or mak th pitch follow a givn command as quickly as possibl []. odling of a Pitch Control This sction provids a brif dscription on th modling of pitch control longitudinal quation of aircraft, as a basis of a simulation nvironmnt for dvlopmnt and prformanc valuation of th proposd controllr tchniqus. Th systm of longitudinal dynamics is considrd in this invstigation and drivd in th transfr function and stats pac forms. Th pitch control systm considrd in this work is shown in Figur whr b,y b and Z b rprsnt th arodynamics forc componnts., Ф and δ rprsnt th orintation of aircraft (pitch angl) in th arth-axis systm and lvator dflction angl. Th quations govrning th motion of an aircraft ar a vry complicatd st of six nonlinar coupld diffrntial quations. Although, undr crtain assumptions, thy can b dcoupld and linarizd into longitudinal and latral quations. Aircraft pitch is govrnd by th longitudinal dynamics. In this xampl w will dsign an autopilot that controls th pitch of an aircraft. Th basic coordinat axs and forcs acting on an aircraft ar shown in th figur givn blow. Figur.Dscription of pitch control systm. Figur shows th forcs, momnts and vlocity componnts in th body fixd coordinat of aircraft systm. Th arodynamics momnt componnts forroll, pitch and yaw axis ar rprsnt as L, and N. Th trm p, q, r rprsnt th angular rats about roll, pitch and yaw axis whil trm u, v, w rprsnt th vlocity componnts of roll, pitch and yaw axis. α and β ar rprsnts as th angl of attack and sidslip. A fw assumption nd to b considrd bfor continuing with th modling procss. First, th aircraft is stady stat cruis at constant altitud and vlocity, thus th thrust and drag ar cancl out and th lift and wight balanc out ach othr. Scond, th chang in pitch angl dos not chang th spd of an aircraft undr any circumstanc. Figur : Dfinition of forc, momnts and vlocity in body fixd coordinat. Rfrring to th Figur and Figur, th following dynamic quations includ forc andmomnt quations ar dtrmind as shown in quation (), () and (). Rfrring to th Figur and Figur, th following dynamic quations includ forc and momnt quations ar dtrmind. Th longitudinal stability drivativs paramtr usd ar dnotd in Tabl []. Tabl.Longitudinal Drivativ Stability Paramtrs Componnts Dynamics Prssur and Dimnsional Drivativ Q = 6.8lb/ft,QS= 677lb, QS c = 8596ft.lb, ( c / u ) =.6s Rolling vlocitis Yawing vlocitis Angl of attack Pitching rat Elvator dflction Z-Forc, (F - ) Pitching omnt, (FT - ) Pitching omnt, (FT - ).5 Z u W W.6 u Z Z Z W W. 55. u W W Z.5 a a Z a 8

3 () () () It is rquird to compltly solvd th aircraft problm with considring th following assumption: () rolling rat () yawing rat,q=,()pitching rat, r=, () Pitch Angl, (5) roll Angl, and (6) Yaw Angl, Equation (), () and () should b linarizd using small disturbanc thory. Th quations ar rplacd by a variabl or rfrnc valu plus a prturbation or disturbanc, as shown blow. (9) () Thrfor th transfr function of th pitch control systm is obtaind in () and () rspctivly. () Transfr function To find th transfr function of th abov systm, w nd to tak th Laplac transform of th abov modling quations. Rcall that whn finding a transfr function, zro initial conditions should b assumd. Th Laplac transform of th abov quations ar shown blow. [] p=p + q=q + r=r + = + = +Y Z=Z + For convninc, th rfrnc flight condition is assumd to b symmtric and th propulsiv forcs ar assumd to rmain constant. This implis that, =. Aftr linarization th (), (5) and (6) ar obtaind. () (5) (6) By manipulating th (), (5), (6) and substituting th paramtrs valus of th longitudinal stability drivativs, th following transfr function for th chang in th pitch chang in th pitch rat to th chang in lvator dflction angl is shown as (7) obtaind. (7) Th transfr function of th chang in pitch angl to th chang in lvator angl can b obtaind from th chang in pitch rats to th chang in lvator angl in th following way. (8) Ths valus ar takn from th data from on of Boing's commrcial aircraft. Th Dsign of Fuzzy Controllr Fuzzy control is basd on th artificial xprinc. Thrfor, for thos control problms which can t b rsolvd by traditional mthods can oftn b rsolvd by th fuzzy control tchnology. By th fuzzy control tchnology, it dos not know th mathmatical modl of th plant and asy to control uncrtain systms or nonlinar control systms and can rstrain th strong disturbanc. Input chaptr Fuzzy controll r Snsor Figur. Th basic structur of fuzzy control systm [6] Th only diffrnc is to control th dvic by fuzzy controllr to achiv th dsird prformanc. Fuzzy slf-tuning PID controllr is a convntional PID rgulator basd on th fuzzy st thory, undr th absolut control rror and dviation chang and th absolut valu of th rat, on-lin automatically adjusting th proportional cofficint KP, intgral cofficint of KI and diffrntial factor KD of th fuzzy controllr. Fuzzy controllr is a Output chaptr 9

4 nonlinar control dvic, using fuzzy rasoning algorithm. Th sampl data of th controlld procss ar takn as th clar amount of input to th controllr, and thn aftr input quantization factor calculation, ar transfrrd into fuzzy valus, so thy can b usd for fuzzy rasoning by fuzzy languag and ruls. To th othr part of procss, th rasoning rsults ar firstly transfrrd into clar valus by antifuzzy infrnc and thus driv th control output with quantifid factor calculation usd as th control valu for th controlld procss. Basd on th ATLAB fuzzy logic toolbox, th abov control algorithm can b asily implmntd [7]. Dsign of nominal fuzzy controllr In ordr to dsign th PID paramtrs basd-on fuzzy controllr, at first th simplst structur of two-input singl output nominal fuzzy controllr is givn. At any givn tim instanc n with a sampling tim Ts, th two input variabls of fuzzy controllr, rror stat variabl and rror chang ar dfind as (n) =y(n)-r(n) () (n)=(n)-(n-). () And its output variabl u(n) is th control signal of procss. Without loss th gnrality, th systm is assumd to hav r inputs dnotd by th r- dimnsional vctor U ( KT ) u ( KT )... u ( ) T r KT and s outputs dnotd by th s-dimnsional vctor y( KT ) y ( KT ) y ( KT ) T. ost oftn th inputs s to th fuzzy controllr ar gnratd by som function of th plant output y(kt)and rfrnc input yr(kt). Th inputs to th fuzzy controllr ar th rror ( KT ) ( KT ) ( ) T s KT and changs in rrorc ( KT ) C ( KT ) C ( KT ) T dfind as s () c (kt) = (5) Whr dnots th dsird plant output is sampl priod. For gratr flxibility in fuzzy controllr implmntation, th univrss of discours for ach plant input ar normalizd to th intrval by mans of constant scaling factors. Th gains g, g c and g u wr mployd to normaliz th univrs of discours for th rror (kt) and changs in rror c(kt), and controllroutput u(kt) rspctivly. With th plant input is gnratd from IF- THEN control ruls of th form If is and is thn is Whr and c dnot th linguistic variabls associatd with controllr inputs and c rspctivly.u dnots th linguistic variabl associatd with th controllr output u, E and C j dnot th linguistic valus rspctivly and j i U dnots th consqunt linguistic valu[ i.ths ar 9 ruls that hav bn utilizd as a closd-loop componnt in dsigning th FLC for maintaining pitch angl of aircraft systm as dfind in Tabl [8]. Tabl. Fuzzy control ruls / NL N NS ZR PS P PL PL NS ZR PS P PL PL PL PS N NS ZR PS P PL PL ZR NL N NS ZR PS P PL NS NL NL N NS ZR PS P N NL NL NL N NS ZR PS NL NL NL NL NL N NS ZR Implmntation and rsults In this sction, th proposd of control schms ar implmntd and th corrsponding rsults ar. Th mmbrship functions for rror and control surfac of fuzzy ar shown as fig.7 and 8 rspctivly. Dgr of mmbrship NL N P NS ZR PS PL Figur. mbrship function of input and Δ. 5

5 Y(t)= Figur 5. Control surfac of fuzzy logic controllr proposd. - Gnralizd Prdictiv Control Thory for Nonlinar Systms Th gnralizd prdictiv control approach givs an analytic solution for tracking problms of multivariabl nonlinar systms in trms of a gnralizd prdictiv control prformanc indx. A novl guidanc law is dvlopd in th following sction by mploying this algorithm. Th gnralizd prdictiv control givs th approximation of th tracking rror in th rcding horizon by its Taylor-sris xpansion to any spcifid ordr. A closd-form optimal prdictiv controllr is obtaind by minimizing a quadratic prformanc indx with intgral action. On-lin optimization is not rquird and stability of th closd-loop systm is guarantd. For mor dtail, th radr is rfrrd to Rfs. [,]. Considr th nonlinar systm (6) Whr, and ar th stat, control and output vctors, rspctivly. It is assumd that ach of th systm output has th sam wll-dfind rlativ dgr, th output and th rfrnc trajctory ar sufficintly many tims continuously diffrntiabl with rspct to t and th control ordr is chos to b r. Th futur output is approximatly prdictd by its Taylor-sris xpansion up to ordr, givn by (7) (8) (9) + () Whr, rprsnt th Li drivativ with rspct to f and g, is nonlinar in both u(t),...,u i+ (t) and x(t) for i=,...,r. In th moving tim fram, th rfrnc trajctory is also approximatd by th Taylor xpansion of up to th ordr, givn by () = () Th rcding-horizon prformanc indx with built-in intgral action is givn by () Whr T is th prdictiv priod. Th actual control input u(t) givn by th initial valu of th optimal control input,, which minimizs th prformanc indx by stting () Whr,is dscribd as is givn by (5) K is th first rows of matrix which ar th submatrics of, givn by 5

6 (6) As w can s, th optimal prdictiv control law (6) is a nonlinar tim invariant stat fdback law. Th control gain K is constant, which only dpnds on th prdictiv tim T, th control ordr r, and th rlativ dgr.5.5 prdictiv Rfrnc input Input amplitud prdictiv rfrnc input Fuzzy Rfrnc input Tim missil trajctory missl trajctory prdictiv missil trajctory Tim 5 - prdictiv missl trajctory Tim Tim(s) Figur 6: Comparing btwn prdictiv controllr with rfrnc input, two missil trajctory and fuzzy controllr Simulation Th proposd control schms hav bn implmntd within simulation nvironmnt in atlab and Simulink. In th prvious sction, th controllrs wr introducd which usd to control th procsss. As xpctd, controllr fuzzy in compar with othr controllrs dspit th svr disturbanc on pitch systm (causd by svr storms, rainy wathr, (.اtc had th dsirabl stp rspons. Svr disturbancs (mans high amplitud disturbanc ) to proof th robustnss of th fuzzy controllr has bn applying on pitch systm. Prformanc of th control schms has bn valuatd in trm of tim domain spcification. Conclusion A nw control approach to pitch-rat command tracking off lightr aircraft has bn proposd in this papr. odling is don on an aircraft pitch control and prdictiv controllr is proposd succssfully. Th proposd control schms hav bn implmntd within simulation nvironmnt in atlab and Simulink. Prformanc of th control schms has bn valuatd in trm of tim domain spcification. Th rsults obtaind, dmonstrat that th ffct of th disturbancs in th systm can succssfully b handld by prdictiv controllr. PC controllr with constraints will b dvlopd and abl to compnsat for constraints that rprsnt physical limits of actuators in pitch angl. Th dsign of PC gav rspons lss quality than that was givn from Fuzzy controllr but accptabl rsponss. 5

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