MODEL REFERENCE ROBUST CONTROL FOR MARINE PROPULSION SYSTEMS WITH MODEL UNCERTAINTY CAUSED BY HULL DEFORMATION

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1 4 Journal of Marin Scinc and Tchnology, Vol., No. 4, (3) DOI:.69/JMST MODEL REFERENCE ROBUST CONTROL FOR MARINE PROPULSION SYSTEMS WITH MODEL UNCERTAINTY CAUSED BY HULL DEFORMATION Zhixiong Li, Xining Yan, Li Qin, Chngqing Yuan, and Zhongxiao Png 3 Ky words: marin roulsion systm, modl uncrtainty, modl rfrnc adativ control, shi sd rgulation, hull dformation. ABSTRACT A marin roulsion systm can b strongly could with th hull dformation in a larg-scal shi. Th intraction of this couling dynamics incrass th nonlinarity and comlxity of th control modl. Traditional control mthod basd on th mirical PID (roortion-intgration-diffrntiation) aramtrs has difficulty in daling with th modl uncrtainty causd by th hull dflction. Howvr, u to dat, limitd work has bn don to addrss th modl uncrtainty roblm of th marin roulsion systm. It is thrfor imrativ to dvlo fasibl and ffctiv control systms that tak th modl uncrtainty into account. In this study, a nw mthod basd on th modl rfrnc adativ systm (MRAS) has bn roosd for th activ and accurat control of th marin roulsion systm couling with th hull dformation. A finit lmnt modl of th shi shaft lin was stablishd to invstigat th uncrtainty boundaris of th roulsion systm. Morovr, th sa trial was carrid out on th hydraulic drdg namd Changjing to masur th main ngin owr loss undr hull dformation. Th finit lmnt analysis (FEA) and shiboard masurmnt rsults showd a fiducial intrval of [.% %] for th modl uncrtainty of th marin roulsion systm. Ths uncrtainty boundaris wr addd into th shi sd control systm modl, and th MRAS controllr was dsignd basd on th Lyaunov thory Par submittd 6/8/; rvisd 3/9/; acctd 5/8/. Author for corrsondnc: Xining Yan (-mail: lzx 5@63.com). School of Mchatronic Enginring, China Univrsity of Mining and Tchnology, Xuzhou, China. Rliability Enginring Institut, School of Enrgy and Powr Enginring, Wuhan Univrsity of Tchnology, Wuhan, China. 3 School of Mchanical and Manufacturing Enginring, Th Univrsity of Nw South Wals, Sydny, NSW, Australia. to mitigat th advrs ffcts of modl uncrtainty. Th stability of th MRAS has bn rovn by th Lyaunov stability critrion. Numrical valuations using MATLAB / Simulink softwar for th Changjing shi ngin aramtrs hav showd high ffctivnss of th MRAS structur for sd tracking undr th hull dflction couling. This study has dmonstratd that th nwly roosd control systm can work stably with various shi orating conditions, and its rformanc is surior to th traditional PID controllr. I. INTRODUCTION Th govrnors of marin roulsion systms ar usually dsignd using mirical rocss modls without considring th influnc of hull dformation [8]. Ths govrnors tyically nglct th modl uncrtainty but work fin in ractic for th sd control of small-scal shis. Howvr, with th raid dvlomnt in th shiing industry, larg vssls such as mammoth tankrs/monstr tankrs, mining and storag shis for marin rsourcs, aircraft carrirs and dradnaughts, tc., bcom biggr and biggr in siz, thrby lading to an vidntial incras of th intraction btwn th roulsion systm and hull. Th rformanc of traditional controllrs in this situation is not as good as dsird [3]. Nonthlss, litratur rviw shows that littl rsarch has considrd th ffcts of uncrtain aramtrs on th modling of th marin roulsion systms. It is thrfor critical to conduct furthr rsarch so that th modl uncrtainty causd by hull dformation is considrd in th control of th marin roulsion systms of larg-scal shis. As wll known, th shi shaft lin is on of th most imortant comonnts to transfr th nrgy from th shi ngin to thrustrs, and to rciv th axial thrust of th rollr to driv th shi. It is th indisnsabl art for th normal running of th marin roulsion systm that onc abnormalitis occur, th systm roulsiv fficincy may dcras gratly. This is bcaus whn a hull dformation occurs, th dislacmnt of th suort barings of th shaft can b significant,

2 Z.-X. Li t al.: Modl Rfrnc Robust Control for Marin Proulsion Systms 4 lading to a misalignmnt of th shaft lin. Th torsional and transvrs vibration of th shaft lin causd by th misalignmnt may incras th modl uncrtainty of th marin roulsion systm, which cannot b ignord in th systm control. Convntional PID controllrs hav bn rovn to b ffctiv for th shi sd control in ratd conditions and hav bn widly usd bcaus of thir rliability, simlicity and univrsality [3]. Howvr, if th oration conditions dviat th systm ratd conditions or th systm ncountrs disturbancs, thir rformanc may b unsatisfactory and vn out of function bcaus th PID aramtrs ar fixd in advanc. To solv ths roblms, advancd controllrs hav bn introducd to control th marin roulsion systm adativly. Brzózka [3] rsntd a nw combination of th modl following control (MFC) and intrnal modl control (IMC) to control a shi ngin sd. Th numrical simulation showd th high ffctivnss of MFC/IMC structur for sdking and sd-changing in th shi rgulation. Xiros [9] roosd an imrovd PID tuning mthod basd on snsitivity H -norm scification for marin disl ngin govrnors to srv against svr rollr load fluctuation. In ordr to imrov th stability of shi sd and main ngin (ME) load, Zhang and Rn [-7] dsignd a shi sd rgulating simulation systm, which consistd of a comutr, PLC, frquncy control motor, srvo motor and cntrifugal um. Th adativ nural ntworks wr usd as th rror modls to comnsat th systm modl uncrtaintis and nhanc th shi sd control vrsus th PID controllr. Th simulation rsults showd that th intllignt control rocss of shi sd was smooth and stabl whn th systm modl had som uncrtaintis. Although ths rcntly dvlod mthods can ovrcom th disadvantags of traditional PID controllrs, thy also suffr som drawbacks including comlx and tim-consuming []. Furthrmor, th uncrtaintis and rturbations introducd into th systms wr slctd randomly, and th uncrtainty caus and boundaris wr not xlaind and discussd in thir work. Du to th lack of th insight into th imact of th aramtr uncrtainty on th systm dsign,,th influnc of th modl uncrtainty of th marin roulsion systm causd by hull dflction has not bn invstigatd yt. This ar rsnts a nw systm to control th marin roulsion systm taking th hull dformation into account. In th rvious work [] th adativ controllrs wr invstigatd in th shi sd rgulation with modl uncrtaintis and hav bn rovn fficint for daling with th systm disturbancs. Basd on th abov dvlomnt, a modl rfrnc adativ robust control schm is roosd for th marin roulsion systm with modl uncrtainty causd by hull dformation in this ar. Th boundaris of th systm dynamical uncrtainty about th intraction of roulsion systm and hull dflction is dtrmind using both finit lmnt analysis (FEA) simulation and sa trial masurmnts, and th quantification of th roulsion systm modl uncrtainty is hnc obtaind. To accommodat Dry Dock In Srvic-Watrborn Fig.. Th shaft lin misalignmnt causd by hull dformation [4]. th modl uncrtainty, th modl rfrnc adativ systm (MRAS) is rsntd to track th roulsion systm sd in th closd control loo. By doing so, th disturbancs can b comrssd. Numrical studis ar carrid out in this ar to valuat th fficincy of th roosd control systm. This ar is organizd as follows. In Sction, th man valu modl of th marin roulsion systm with modl uncrtainty causd by hull dflction is stablishd. Th invstigation on th modl uncrtainty is conductd by FEA simulation and shiboard masurmnt in Sction 3. Th uncrtainty boundaris ar stimatd according to th simulation and xrimnt rsults. In Sction 4 th robust controllr dsign is rsntd basd on th MRAS. Th stability of th control systm is analyzd via th Lyaunov stability critrion. Th ffctivnss of th roosd mthod is assssd by numrical studis in Sction 5. Finally, conclusions ar drawn in Sction 6. II. MEAN VALUE MODEL OF MARINE PROPULSION SYSTEM Th dynamics charactristics of th marin roulsion systm rsnt strong nonlinarity and tim varianc, scially for th shi disl ngin. Thus it is vry difficult to modl th systm rcisly [, ]. Man valu modl is on of th imortant and fasibl mans to dsigning and simulating th marin roulsion systm [8]. Th xisting mthods for modl dsign and simulation of th marin roulsion systm usually considr th imact factors of th wavs and loads on th modl rformanc [4, 5], but sldom tak into account th couling ffct of th hull dformation on th roulsion systm. Howvr, th accuracy of shaft lin alignmnt may b influncd significantly by th hull dflction. Chris [4] rortd in his rsarch that th shaft lin alignmnt dviats from th rfrnc lin in srvic. Fig. shows th comarison of shaft lin alignmnt in dry dock and srvic [4]. It is noticabl that th shaft lin dflcts svrly with th hull dflction. Dong t al. [5] statd that th maximum offst of th shaft lin could b u to 8 mm, and th minimum offst of th strn barings was. mm for a larg

3 4 Journal of Marin Scinc and Tchnology, Vol., No. 4 (3) Rfrnc sd Fdback sd Controllr Load indicator notch valu Intraction Disl ngin Intraction Prollr tional inrtia of th shaft lin. Th additional torqu is causd by th hull dflction modl and is calculatd as Ms = kt M (4) Hull dformation couling modul Suort bas of th roulsion systm Intraction Hull dflction Fig.. Shi sd control structur with considring hull dformation. containr shi. Furthrmor, Zhou [8] ointd out that a.8 mm lift of th intrmdiat baring of th larg scal shis may lad to th baring void, which may incras th shaft vibration and nrgy loss, affct th systm rformanc and vn caus a systm brakdown. Hnc, it is imrativ to invstigat th intraction of hull and shaft lin in th modling of th roulsion systm for larg scal shis. In this Sction, th fusion of hull dformation into th modling of th marin roulsion systm is discussd. Th shi sd control structur is shown in Fig.. Th systm consists of th controllr modul, disl ngin modul, rollr modul and hull dformation couling modul, tc. Th man valu modl of th marin roulsion systm is brifly dscribd as follows.. Disl Engin Modul Normally, th outut torqu of th shi disl ngin can b aroximatly rgardd as a function involvd with th oil injction Q and ngin sd ω, i.. M = f(q, ω). Th diffrntial quation is givn by [3] M Q M ω M = Q+ whr dnots th incrmnts of th variabls. Th oil injction Q is drivd by artially diffrntiating th ngin sd ω and th strok of th ful um rack L: Q Q Q= ω + L ω L. Shaft Lin Modul According to Nwton s scond law for rotational motion, th dynamics quation of th shaft lin modul can b xrssd as ω π J M Ms M () () d ω = (3) dt M = ω + ω ζ M M ζ whr k t is th ngin outut torqu loss indx bcaus of th couling ffct of th hull dflction, and ζ dnots th rollr load fluctuation. 3. Marin Proulsion Systm Modul Substituting Eqs. (), (), (4), and (5) for M, M s and M in Eq. (3), th marin roulsion systm modl is drivd as: d ω M M πj + ω ( kt ) ω dt ω ω M M = ( kt ) L ζ L ζ whn th disl ngin working in th stabl oration rocss, Eq. (6) can b rwrittn as M R dωr ( kt) LR M M πj + ωr = Lr ζ ωr K dt ωr K L ωr K ζ (7) whr ω R, M R and L R dnot th ratd ngin rotational sd, ratd ngin torqu and maximum rack strok, rsctivly. ω r and L r dnot th rlativ variations of ngin sd and M M rack strok, rsctivly. K quals ( kt ) ω ω. 4. Shi Sd Rgulator Modl M In th stabl oration rocss, th trms of K, L and M in Eq. (7) can b aroximatd according to th ngin ζ oration charactristic curvs [3]. Howvr, th torqu loss indx k s is unknown. It is only fasibl to dtrmin its ur and lowr boundaris via obsrving th contribution of th shaft lin vibration to th ngin torqu loss. Thus, th mathmatical modl of th transfr function of th marin roulsion systm in th stabl oration rocss can b xrssd as []: (5) (6) whr M s and M dnot th additional ngin torqu loss and rollr load torqu, rsctivly. J dnots th rota- G () s = f( kt ) K f( k ) Ts+ t (8)

4 Z.-X. Li t al.: Modl Rfrnc Robust Control for Marin Proulsion Systms 43 M 3 M s + M Baring suort Baring suort Baring suort Fig. 3. Shi roulsion shaft lin modl consisting of 3 shafts labld as, and 3. whr f(k t ) dnots th modl uncrtainty of th torqu loss indx causd by hull dformation, K dnots th magnification cofficint of th rgulator channl, and T dnots th ngin tim constant. Th transfr function of th actuator for th shi ngin control is givn by [] Fig. 4. Th FEA analysis rsult of th shi shaft lin. K G () s = Ts+ (9) whr K dnots th magnification cofficint, and T dnots th actuator tim constant. Considring th hystrsis roblm of th disl ngin sd signal in th transfrring rocss, a hystrtic link τ s should b addd into th shi sd control systm. In addition, a sd dtcting link G 3 (s) = is also ndd. Hnc, th mathmatical modl of th shi sd control systm is () () () () f( k ) K K τs t τs Gs = G sg sg3 s = ( f( kt ) Ts + )( Ts+ ) () It can b noticd from Eq. () that th modl uncrtainty trm f(k t ) causd by th hull dflction incrass th systm comlxity. For a robust dsign of th control systm, it is thrfor ssntial to invstigat th dgr of th modl uncrtainty and dtrmin its boundaris. III. INVESTIGATION ON THE MODEL UNCERTAINTY Th modling uncrtainty may influnc th accuracy of th ossibl rsonss of th systm. To wll comnsat th systm rturbations, it nds to dtrmin th robability distributions of th uncrtainty in advanc. Hnc, rior to th dsign of robust control for th marin roulsion systm with a considration of th hull dformation couling ffct, th finit lmnt analysis (FEA) and th shiboard masurmnt hav bn imlmntd to assss th uncrtainty dgr of th modl.. Finit Elmnt Analysis (FEA) In this study, w hav simulatd th dformation of th shaft bas which is causd by th hull dformation to invstigat its influnc on th ngin owr transfr. A shi shaft lin transfrring th thrust forc to th thrust baring (s Fig. 3) is stablishd using FEA softwar. Fig. 5. Th Changjing shi alid to th sa trial. In Fig. 3, th shaft lin modl consists of th first shaft, intrmdiat shaft, strn shaft 3 and suort barings. Th lngth of th first shaft is 45 mm with a diamtr of mm. Th lngth of th intrmdiat shaft is 6 mm with a diamtr of 8 mm, and strn shaft is 5 mm in lngth with and 8 mm in diamtr. Th ntir shaft is hollow with an intrnal diamtr of 5 mm. In th analysis, th ngin torqu of 6 N m is alid to th shaft lft nd and th load torqu of 6 N m is alid to th othr nd. A distributd load of 45 N/m acting as th shaft lin gravity is distributd along th shaft. In addition, a comarison has bn don to invstigat th hull dflction ffct on th ngin torqu loss via two diffrnt simulation stats. On is th dry dock stat whr thr concntratd loads with th sam valu of 5 kn ar addd on th suort barings of shaft, and 3, rsctivly. Th othr on is to simulat th svr hull dflction situation whr thr concntratd loads of 35 kn, 55 kn and 75 kn ar alid to th suort barings of shaft, and 3, rsctivly. Th FEA analysis rsult of stat is shown in Fig. 4. It can b sn from Fig. 4 that th shaft lin is dflctd obviously whn introducing th hull couling ffct on it, and th ngin owr loss could rach to 4.5% of th inut owr according to th calculation rsult using th stat as th baslin. If doubl th thr concntratd loads in stat, th ngin owr loss could rach to 9.8%.. Shiboard Masurmnt Th shiboard masurmnt is carrid out in th shi of Changjing (s Fig. 5), which is an 8 m 3 hydraulic drdg. Th normal ratd owr of th main ngin is 63 kw, and th ratd rotation sd is 6 rm. Th tmratur

5 44 Journal of Marin Scinc and Tchnology, Vol., No. 4 (3) Tabl. Shaft owr tst rsults for Changjing. Load Shaft Sd Shaft Powr Shaft Powr Rlativ (rm) A (kw) B (kw) Error (%) 5% % % % % of sa watr is C in th sa trial and th wind sd is 4~7 MPH. Th torqu mtr is installd on th shaft to masur th shaft owr. In th actual masurmnt rocss, th shaft lin may suffr from th bnding momnt and axial forc tc., which aris mainly from th hull dflction. Tabl shows th sa trial rsults of th shaft owr. In Tabl, th shaft owr A dnots that th masurmnt has nglctd th hull dflction couling ffct, whil th shaft owr B has considrd this ffct. It can b sn that an uncrtainty ranging from.3% to 3.87% is rsntd for shaft owr B with rsct to th valu of shaft owr A undr th diffrnt load conditions. Thus, th finit lmnt analysis (FEA) and th shiboard masurmnt rsults show that thr is a fiducial intrval of [.% %] for th modl uncrtainty of th marin roulsion systm. This uncrtainty rgion should b incororatd in th controllr dsign of th shi sd rgulator to nsur th ffctivnss of th control systm. IV. ROBUST DESIGN AND TEST FOR THE MARINE PROPULSION SYSTEM As mntiond in Sction 3, th control dsign of th shi sd systm ncountrs mismatch btwn th actual lant dynamics and th dynamics of th modl du to hull dformation. Fortunatly, th closd loo control tchniqu can nsur th robustnss of th control systm within crtain bounds of th modl uncrtaintis. Th modl rfrnc adativ systm (MRAS) is on of th most romising closd loo aroachs [6, 7]. Th tyical aralll schm of th MRAS is givn in Fig. 6, which consists of th rfrnc modl, th adjustabl modl and th adativ mchanism. Th gnral ida bhind th MRAS is to dvlo a closd loo controllr with aramtrs that can b udatd to incororat th modl uncrtainty of th systm. Th control aramtrs ar modifid basd on an rror vctor that is drivd from th diffrnc btwn th oututs of th two dynamic modls. Th MRAS aims to liminat th rror and to match th rsons of th rfrnc modl.. Modl Rfrnc Adativ Control of Shi Sd Th nonlinar modl of marin roulsion systm in Eq. () can b rroducd in th diffrntial form: y+ f ( k ) y + f ( k ) y = u f( y, u) () t t Control inut u Rfrnc modl Adativ mchanism Adjustabl modl Dsird outut ŷ + Error - Actual outut y Fig. 6. Th tyical aralll schm of th MRAS. whr symbol y dnots th ngin sd and symbol u is th control inut of th rack strok. f (k t ), f (k t ) and f( y, u) dnot th modl uncrtaintis. f (k t ) and f (k t ) ar unknown aramtrs involvd with k t but satisfis a f( kt ) b c f( kt ) d () whr a, b, c and d ar known constants. And f( y, u) is th nonlinar function involvd with th ngin torqu loss. f( y, u) = κα ( )sgn( y ) + κβ ( )x( γ y )sgn( y ) (3) whr κ(α) α and κ(β) β; α, β and γ (- < γ < ) nd to b scifid. So Eq. () rsnts th actual marin roulsion modl with th hull dflction couling ffct. Th rfrnc modl is dfind as: y+ y + qy = r (4) whr ( > ) and q (q > ) ar th dsird constants to guarant th tracking of dsird rack strok r. Subtract Eq. () from Eq. (4) to yild th rror + + q= r u+ f( y, u) + ( f( kt) ) y + ( f( kt) q) y (5) whr = y y. Rwrit Eq. (5) as = = q + r u + f( y, u) + ( f( kt) y ) + ( f( kt) qy ) (6) T Dfin th stat variabl vctor x = [ ]. Eq. (6) can b xrssd as

6 Z.-X. Li t al.: Modl Rfrnc Robust Control for Marin Proulsion Systms 45 x Ax = + ( u r ) (7) whr = f( y, u) + ( f( kt) ) y + ( f( kt) q) y, and A =. q Sinc A is a stabl matrix, according to Lyaunov stability critrion, thr xists th ositiv dfinit matrix P for any ositiv dfinit matrix Q to satisfy th Lyaunov quation: T AP+ PA= Q (8) whr B= k y+ k y + k. 3 Choos Lyaunov candidat function Dfin C = B T Vtx (, ) = xpx (3), thn x = Ax+ C T T Vtx (, ) = xpx + xpx, which yilds Dfin th auxiliary signal as = [ ] Px= [ ] Thus th control law is givn by 3 3 (9) u = r+ k y+ k y + k () Whr th gain functions ar dfind as k = { k [ sgn( y )] + [ + g sgn( y )]} k = { kd [ sgn( y )] + gd [ + sgnsgn( y )]} () k3 = α sgn( ) + β sgn( ) whr T T = ( Ax + C) Px + x P( Ax + C) T T T T = x ( A P+ PA) x+ ( C Px+ x PC) (4) C Px= [ B] 3 4 T = ( B ) T x PC = [ ] 3 B (5) whr k, k d, g and g d ar th gain constants. Hnc, th control law givn by Eq. () uss th gain functions to comnsat th modl uncrtaintis, i.. uss th itms k y + kẏ+ k 3 to stimat th modl uncrtainty itm in Eq. (7). By doing so, th rror can b liminatd ffctivly. Sinc th controllr is drivd via th Lyaunov stability critrion, onc A is stabl and th ositiv dfinit matrixs P and Q ar obtaind, th control systm is global asymtotically stabl. Th roof is givn blow.. Th Stability Proof of th Controllr Thorm. As to th uncrtain nonlinar systm () and th rfrnc systm (4), if th gains of th modl rfrnc adativ robust control law u () and () ar givn by k = c q, g = d q, k d = a and g d = b, thn for any uncrtain f( y, u) and uncrtain aramtrs f (k t ) and f (k t ), th rror can b mad to lim( y y) =. t Proof of Thorm. Substitut Eq. () for u in Eq. (7) to yild th closd control loo: x = Ax+ B () [ ] B = = ( B ) Substitut (8), (5) and (6) in (3) to obtain λ Vt x xqx B xt B (6) T m (, ) = + ( ) () + ( ) (7) whr λ m dnots th minimum ignvalu of Q. As to th itm of ( B ), unfold it as whr, ( B ) = [ f( yu, ) + ( f( k) y ) t + ( f ( k ) q) y k y k y k ] t 3 = [ f( y, u) k ] + [( f ( k ) ) k ] y + [( f ( k ) q) k ] y 3 t t (8)

7 46 Journal of Marin Scinc and Tchnology, Vol., No. 4 (3) [ f( y, u) k ] = 3 ( κα ( ) + κβ ( )x( γ y )) ( α+ β) y > > ( κα ( ) + κβ ( )x( γ y )) + ( α + β) y < ( α + β) ( κ( α) + κ( β)x( γ y )) y > < ( α + β) + ( κ( α) + κ( β)x( γ y )) y > f( kt) gd = f( kt) b y > [ f( kt ) k] = f( kt) kd = f( kt) a y < f( kt) q g = f( kt) d y > [ f( kt ) q k] = f( kt) q k = f( kt) c y< So [ f( ẏ, u) k 3 ], [(/f(k t ) ) k ] y and [(/f(k t ) q) k ] y. Hnc, ( B ), and (7) satisfis λ V t x x t x m (, ) () (9) Eq. (9) indicats that for any givn f( ẏ, u), f (k t ) and f (k t ), th control systm is asymtotically stabl. Hnc, this comlts th roof. V. NUMERICAL EXPERIMENTS AND RESULTS Th roulsion systm of Changjing shi is takn as th control objctiv in this work. Th DAIHATSU DKM- 36F havy slow disl ngin is slctd in th numrical xrimnts. Th ngin tim constant T is 5., th magnification cofficint K is 46.8, th hystrtic tim τ is.3, th magnification cofficint K and tim constant T of th actuator ar.5 and.3, rsctivly. Thus, th roulsion modl is 49.4 f( kt ) Gs () = ( f( kt)5.s+ )(.3s+ ) t.3s (3) According to th uncrtainty invstigation in Sction 3, th variation intrval of f(k t ) is btwn.9 and.999. Hnc, in th diffrntial form (), w can dduc that a = 3.55 f( kt ) b= c =.64 f( kt ) d =.7 (3) Choos κ(α) =.5, κ(β) =.4 and γ =.. Th rfrnc modl is st as: Sd tracking (rad/s).5.5 Zoomd ara in Fig. 8 Rfrnc modl outut Th roosd controllr Traditional PID controllr Fig. 7. Th ngin sd tracking rformanc of th two mthods. Sd tracking (rad/s) Rfrnc modl outut Th roosd controllr Traditional PID controllr Fig. 8. Th zoomd ictur of th markd ara in Fig. 7. y y +.64y = 3.5r (3) whr = 3.56 and q =.64. Thus A = St th ositiv dfinit matrix b Q =. Thn us th Lyaunov quation to calculat matrix P, and P = Two kinds of tim-varying rfrnc of th rack strok r ar usd to valuat th robustnss and sd tracking ability of th roosd control systm in th numrical tsts. On is st rfrnc ordr, and th othr is sinusoidal rfrnc ordr.. Cas Study In this cas study, w choos r = sgn(sin(.t)) to simulat th ahad and astrn orations. Th roosd MRAS mthod is comard with th traditional PID controllr in th xrimnts. Th comarison of th ngin sd tracking ability of th two controllrs is shown in Fig. 7. Fig. 8 is th local zoomd ara of Fig. 7. Th fficincy of th sd tracking of th roosd controllr is valuatd by th tracking rror in Fig. 9, and Fig. givs th controllr outut.. Cas Study In this cas study, w choos r = sin(.t) to simulat a

8 Z.-X. Li t al.: Modl Rfrnc Robust Control for Marin Proulsion Systms 47 Sd tracking rror (rad/s) Fig. 9. Th ngin sd tracking rror of th MRAS. Sd tracking (rad/s) Plant outut Rfrnc modl outut Fig.. Th zoomd ictur of th markd ara in Fig.. Th control inut u (mm) Fig.. Th control inut u of th MRAS. Sd tracking rror (rad/s) Fig. 3. Th ngin sd tracking rror of th MRAS. Sd tracking (rad/s) 5 4 Plant outut 3 Rfrnc modl outut Zoomd ara in Fig Fig.. Th ngin sd tracking rformanc of th MRAS. Th control inut u (mm) Fig. 4. Th control inut u of th MRAS. svr nonlinar disturbanc in th marin nvironmnt. Th ngin sd tracking ability of th roosd controllr is shown in Figs. -. Th fficincy of th sd tracking of th roosd controllr is valuatd by th tracking rror in Fig. 3 whil Fig. 4 givs th controllr outut. In ordr to highlight th ffctivnss of th roosd controllr, w hav incrasd th modl uncrtainty u to a = 3.5, b= 4.47 c =.4, d =. (33) Th rformancs of th MRAS in this condition ar shown in Figs Th outut data has dmonstratd that th controllr works wll vn with a high lvl uncrtainty. 3. Discussion Th numrical xrimnt rsults of th two cas studis dmonstrat that th roosd controllr works stably and can track th givn rfrnc ordr rcisly with modl uncrtainty causd by th hull dformation. W can s from Figs. 7-8 that th MRAS outrforms th convntional PID controllr with rsct to th sd tracking ability. Th PID controllr dviats from th rfrnc sd sriously, and its tracking accuracy is low whn ncountring th situations with hull dformation. Th man squar rror is.5 rad/s in cas study using th PID controllr. It can infr that th PID controllr cannot adat th modl uncrtainty du to that th PID aramtrs hav bn rdtrmind and cannot b changd in th control rocss.

9 48 Journal of Marin Scinc and Tchnology, Vol., No. 4 (3) Sd tracking (rad/s) Zoomd ara in Fig. 6 Rfrnc modl outut Plant outut Tim(s) Fig. 5. Th ngin sd tracking rformanc of th MRAS. Sd tracking rror (rad/s) Fig. 7. Th ngin sd tracking rror of th MRAS. Sd tracking (rad/s) Rfrnc modl outut Plant outut Fig. 6. Th zoomd ictur of th markd ara in Fig. 5. Th control inut u (mm) Fig. 8. Th control inut u of th MRAS. Howvr, taking th advantag of slf-adatation, th MRAS can follow th dsird sd outut accuratly undr varying rfrnc ordrs, and its tracking rror is much smallr than that of th convntional on. In cas study, th tim-varying rfrnc ordr incrass th control comlxity. Howvr, Figs. -3 show that th roosd MRAS is suitabl and stabl to control th shi sd. Morovr, in th nhancd xrimnt, although th modl uncrtainty has incrasd significantly, th control systm can still work wll, and th sd tracking rror is acctabl. In cas, a comarison to th PID controllr is not carrid out. This is bcaus th traditional PID cannot follow th tim-varying rfrnc ordr, and th control systm is radily to divrg. Hnc, th robustnss of th roosd controllr has bn vrifid by th numrical xrimnts, which shows th ffctivnss of th rsntd control systm. VI. CONCLUSIONS Th larg-scal shis nhanc th fficincy of th watrborn transort, which bnfits th national and intrnational social conomy significantly. Howvr, th raid incras of th shis in siz brings th couling roblm of th roulsion systm and th hull. Svr hull dformation may intnsify th shaft lin vibration, cost xtra ngin owr, and vn damag th roulsion systm. As a rsult, th modl uncr- tainty of th roulsion systm which oftn incrass with th sizs of th shis, has bcom an urgnt issu. Traditional PID controllr limits itslf on th mirical aramtr sttings, and cannot achiv high control rformanc whn daling with th marin roulsion systm with th modl uncrtainty. Basd on th marin roulsion nonlinar systm modl, a novl control structur via modl rfrnc adativ robust control algorithm is rsntd in this ar. Th modl uncrtainty invstigation is imlmntd through both finit lmnt analysis and sa trial to rovid insight into th intraction of th roulsion systm and th hull dformation. Th advantags of this invstigation ar to dtrmin rliabl uncrtainty boundaris for th roulsion systm. Thus, th Lyaunov aroach can b usd to obtain asymtotically stabl MRAS. Finally, th marin roulsion control systm basd on th roosd MRAS is tstd and valuatd using th Changjing shi. Two cas studis show th ffctivnss and th salint rformanc of th roosd control aroach on th systm uncrtainty. Th roosd MRAS can orat robustly on th dsign objctiv, and th comarison btwn th roosd mthod and th traditional PID dmonstrats that th nwly dvlod control systm outrforms th traditional on. In ordr to validat th roosd control modl clos to a ractical nvironmnt, th influnc of th shi motion will b takn into account in our futur work. This is bcaus in a ral sa nvironmnt, th wav loads may caus not only th hull

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