Mathematical Simulation on Self-tuning Fuzzy Controller for Small Cylindrical Object Navigating near Free-surface

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1 Availabl onlin at Procdia Enginring () 9 96 SREE Confrnc on Enginring Modlling and Simulation (CEMS ) Mathmatical Simulation on Slf-tuning Fuzzy Controllr for Small Cylindrical Objct Navigating nar Fr-surfac Shao Zhiyu a Fang Dongyang b Fng Shunshan c a* a,c Stat Ky Laboratory of Explosion Scinc and Tchnology, Bijing Institut of Tchnology, Bijing 8, China b School of Mchatronical Enginring, Bijing Institut of Tchnology, Bijing 8, China Abstract Basd on th adaptiv fuzzy control thory, a slf-rgulating fuzzy controllr is dsignd for a small cylindrical objct navigating nar fr-surfac to minimiz wav disturbanc and to kp th objct mov in th dsird dpth. First, th nar fr-surfac dynamics modl of th cylindrical objct is prsntd with th first and scond ordr wav forcs. Scond, math simulation is introducd to tstify th validity of th control mthod. Th rsults obtaind by a sris of mathmatical simulations dmonstrat that th slf-tuning fuzzy dpth controllr prforms wll in stabilizing th nar-surfac navigating objct and in kping it in th xpctd dpth during th fr-surfac wav disturbanc. Publishd by Elsvir Ltd. Opn accss undr CC BY-NC-ND licns. Slction and/or pr-rviw undr rsponsibility of Socity for Rsourcs, Environmnt and Enginring fuzzy control; fr-surfac; wav forc; mathmatical simulation. Introduction A kind of small cylindrical objct navigating nar fr-surfac is ndd in som civil or martial uss. On of th primary challngs confrontd during th dsign of its control systm is to ovrcom th big wav disturbancs suffrd by th nar-surfac navigating objct. Th wav forcs and momnts can induc oscillatory motion of th objct, which might caus th motion of th objct to b unstabl. * Corrsponding author. Tl.: ; fax: addrss: shaozhiyu@bit.du.cn Publishd by Elsvir Ltd. Opn accss undr CC BY-NC-ND licns. doi:.6/j.prong..5.3

2 9 Shao Zhiyu t al. / Procdia Enginring () 9 96 Consquntly, it is ncssary to find an fficint dpth control mthod to rduc th wav disturbanc and to kp th objct navigat in th dsird dpth. PID controllrs ar most widly usd in industrial control systms. But paramtrs in a classic PID controllr ar fixd, which maks it infficint in conditions of xtrnal disturbancs such as wavs or currnts. Fuzzy control has bn succssfully usd in th control of undrwatr vhicl which oprats in strict and tough conditions. Study in articl [] shows that fuzzy control mthod can b usd to compnsat paramtrical uncrtaintis of th mathmatic modl of som kind of undrwatr vhicl and can prvnt viscous nvironmntal influncs. In articl [] a nw slf-rgulating fuzzy control rul is applid to dsign a trajctory control systm for an undrwatr vhicl with high nonlinarity. Simulation rsults show that th trajctory control systm has nic dynamic proprtis and strong adaptability to changs in initial conditions and systm paramtrs. According to articl [3], fuzzy motion controllr can b usd to nhanc th stability and prcision of th undrwatr vhicl. In this papr, a nw control mthod using slf-tuning fuzzy control rul is prsntd to dsign an fficint and robust dpth controllr for a kind of small cylindrical nar-surfac navigating objct, and th validity of th slf-tuning fuzzy control mthod is vrifid by th mathmatical simulation rsults.. Mathmatical Modl of th Cylindrical Nar-surfac Navigating Objct Two coordinat rfrnc frams ar usd in dfining th motions of th 6-DOF cylindrical narsurfac navigating objct: th body-fixd rfrnc fram X B Y B Z B and th arth-fixd (inrtial) rfrnc fram X E Y E Z E. Th origin of th body-fixd fram is at th cntr of th buoyancy of th objct and th origin of th arth-fixd fram is at th cntr of buoyancy of th objct at tim t=. For fram X B Y B Z B X B axis points toward th had of th objct, Y B axis points toward th starboard sid, and th Z B axis points downward in accordanc with th right-hand rul. For fram X E Y E Z E, X E and Y E axs ar dirctd toward th north and th ast dirction rspctivly, whil Z E axis points downward in accordanc with th right-hand rul. Whn dsigning th dpth control systm, what w mainly concrn is th vrtical motion of th cylindrical objct that movs clos to th fr-surfac. Th mathmatical modl of th vrtical motion of th undrwatr objct is dscribd through Nwton laws of linar and angular momntum. 3. Wav spctrum and forc Th wav spctrum which is normally usd in th study of th influnc on nar-surfac undrwatr objct is th standard Pirson-Moskowitz spctrum, which is writtn as 5 4 S( ) A xp( B ) () With A=8. -3 g -, B=3./Hs. S( ) dnots th wav spctrum dnsity, is th wav frquncy, Hs is th significant wav hight. Th spctrum should b modifid whn th watr is flowing. Th wav frquncy in quation () should b rplacd by th ncountr frquncy, U cos( ) () g Whr U is th vlocity of th undrwatr objct, g=9.8m/s, and is th wav incidnc angl in radians. Th wav incidnc angl is th angl btwn th movmnt dirction of th wav and th vlocity of th moving objct. 8 mans that th movmnt dirction of th wav is coincidnt with

3 Shao Zhiyu t al. / Procdia Enginring () th vlocity of th moving objct, which is thortically th most difficult navigation condition for th navigating objct to kp its dpths. Th controllr will b dsignd undr this condition. According to articl [4], th wav ffcts on a submrgd objct can b split mathmatically into first and scond ordr forc. Th componnt of th first ordr forc is oscillatd in high frquncy and its amplitud is proportional to th wav amplitud. Th amplitud of th scond ordr forc is proportional to th squar of th wav amplitud. Th scond ordr forc is minor compard to th first ordr forc, and it has th ffct to drag th undrwatr objct onto th surfac. Whn th undrwatr objct approachs th watr surfac ths sction forcs incras xponntially and produc a dstabilizing ffct on th alrady stabl systm. As givn in articl [5], th instantanous vrtical forcs of th wavs acting on th undrwatr objct Z wav can b xprssd as Zwav Z() t Z() t ( ) ( sin (. cos ) ( sin )) N Z t Cz U F i wit i 4 N N 3sin i j i j 4 sin z i i j g Z ( t) ( F F sin w tsin w t C (.4Ucos )) (3) Whr Z () t and Z () t ar th first and scond ordr wav forcs rspctivly in th vrtical plan, C Z, C Z ar hydrodynamic cofficints rspctivly. Similarly, th instantanous pitch momnt disturbanc M wav can b xprssd as Mwav () t M() t M() t N M( t) CMLV cos( ) (.Ucos( ) F i cos( wit)) i M CMLZ() t (4) Whr M (t) and M (t) ar th first and scond ordr wav pith momnt disturbancs rspctivly. L is th ovrall lngth of th undrwatr objct. C M, C M ar hydrodynamic cofficints. For this dsign, w hav C Z =.58, C Z =., C M =.85, C M =.5. is th pitch angl, is th watr dnsity, dnots th volum of th objct. F i is givn by F a h t g (5) i ii xp( i ( ) ) / Whr ai ( S( i) ). i can b obtaind by quation (3), in which substitut for i and substitut for i, whr i i, i=,, 3,, N. is th width of th computing bands into which th frquncy rang of intrst of th spctrum S( ) is dividd to calculat th wav forc and momnt. 4. Fuzzy Dpth Controllr Dsign Th fuzzy dpth control systm of th cylindrical objct navigating nar fr-surfac is shown in figur. Th pitch angl is slctd as th innr loop fdback signal to nhanc th dynamic proprtis of th control systm.

4 94 Shao Zhiyu t al. / Procdia Enginring () 9 96 H d Amplifir Elvato H H Fuzzy Dpth Controllr u r Wav Disturbanc Z wav M wav Udrwatr Objct H Fig.. block diagram of fuzzy dpth control systm of th undrwatr objct Th fuzzy dpth controllr prsntd in fig. is dsignd using a nw control mthod applying slftuning fuzzy control ruls, which could hlp to ovrcom dfcts such as stady-rror and fluttr phnomnon. Th structur of th fuzzy controllr is shown in fig.. H du dt K c K c E EC Rul Bas U K u Ki Fuzzy Dpth Controllr u Fig.. structur of th fuzzy dpth controllr with slf-tuning fuzzy control ruls Whr E K, EC =K c c, U E( ) EC. K, K c ar scaling gains and K u, K i ar proportion and intgration cofficints. In this dsign xampl, th dpth rror signal and th chang in rror signal c hav th sam mmbrship function for linguistic input variabls shown in fig. 3(a). is th on-lin sarching adjusting factor which srvs to chang th fuzzy ruls and thrfor to chang th output of th fuzzy dpth controllr according to th two input variabls and c. Th mmbrship function of is shown in fig. 3(b) Dgr Mmbrship Dgr Mmbrship Dpth rror or Chang-in-rror c Fig. 3. (a) mmbrship function of dpth rror (or chang-in-rror c); (b) mmbrship function for output

5 Shao Zhiyu t al. / Procdia Enginring () Th rational for th choics for th rul bass is, for xampl, whn thr is a big positiv chang-inrror, thn th output of th controllr will b wightd havir on c than (a small ) and dcrass th rror rapidly. As th chang-in-rror dcrass, th output of th controllr will b wightd havir on than c (a big ) to avoid ovrshoot. Th othr part of tabl can b xplaind similarly. 5. Simulation Rsults Som mathmatical simulations hav bn mad using Matlab/Simulink to prov th validity of th dsignd fuzzy dpth controllr. Th simulations ar basd on th modl of th small cylindrical narsurfac navigating objct. Th simulation tim is sconds, with a simulation stp of. scond. Bfor simulation, an off-lin optimization for K, K c, K u and K i has bn mad according to intgral-oftim-multiplid absolut-rror (ITAE) critrion as dscribd in articl [6]. Th dpth curvs of th nar-surfac objct which is undr fuzzy dpth control and navigating at diffrnt sa stat (H S =) and 4 (H S =) with 8 incidnc angl ar shown in fig. 4. Th simulation rsults of PID dpth controllr ar also givn in fig. 4 for comparison..3. Fuzzy Dpth Control PID Dpth Control.3. Fuzzy Dpth Control PID Dpth Control.. Dpth rror [ m ] -. Dpth rror [ m ] Wav Hight = m Wav Hight = m Tim [ sc ] Tim [ sc ] Fig. 4. curvs of dpth rror of th nar-surfac objct undr diffrnt sa stats From th rsults of th simulation, w can obsrv that th dsignd fuzzy dpth controllr can succssfully kp th nar-surfac objct navigat at th dpth of 3m undr th two diffrnt sa stat lvls with maximum stady stat rror of lss than.5m. Th fuzzy dpth controllr shows a strong ability to rjct th wav disturbanc and can satisfy th nds for an fficint and robust dpth control for th cylindrical nar-surfac navigating objct, whil th prformanc of th PID dpth controllr is not as good as th fuzzy on in prvnting th wav disturbanc. Th maximum dpth rror of th narsurfac navigating objct with PID dpth controllr is about.3m at th sa stat 4, which dos not mt th prformanc rquirmnts. 6. Conclusions In this papr a nw control mthod applying slf-tuning fuzzy control ruls is prsntd in th dpth control of th cylindrical nar-surfac navigating objct, which hlps to liminat th stady-rror and th fluttr phnomnon xistd in PID controllr and convntional fuzzy controllr. Th rsults of th

6 96 Shao Zhiyu t al. / Procdia Enginring () 9 96 mathmatical simulation show that th slf-tuning fuzzy dpth control mthod can b succssfully applid to rduc th wav disturbanc and to maintain th cylindrical nar-surfac objct navigat at th xpctd dpth. Rfrncs [] M. Santhakumar, T. Asokan. A slf-tuning proportional-intgral-drivativ controllr for an autonomous undrwatr vhicl, bas on Taguchi mthod. Journal of Computr Scinc,, 6(8): [] Duan Fuhai. Dsign of a fuzzy control systm for nonlinar vhicl using on-lin slf-rgulating fuzzy control ruls. Acta Aramntrii, 999, ():-4 [3] LI Y. A fuzzy motion control of AUV basd on apry intllignc. Chins Control and Dcision Confrnc. Guilin: CCDC, 9: 36-3 [4] Lucia Morira. H and H dsigns for diving and cours control of an autonomous undrwatr vhicl in prsnc of wavs. IEEE Journal of Ocanic Enginring, 8, 33(): [5] Grrit. M. H submarin dpth and pitch control. Procdings of th 3nd confrnc on dcision and control. Txas: 993: 5-56 [6] Bing Chn. Adaptiv fuzzy output tracking control of MIMO nonlinar uncrtain systms. IEEE Transactions on fuzzy systms, 7, 5(): 87-3