Tgp chi Khoa bgc Tnrang Dgi ligc Cdn Tba Pb^n A Kboa hgc Tif nbien. Cong nghi vd Moi tnrdng 29 (2013): 8-14
Tap chl Khoa hpc Tru'dng Dai hpc Can Thd website: sj.ctu.edu.vn
DIEU KHIEN CHONG LAC H ? CAN CAU CONTAINER CO BU MA SAT Ngo Quang Hieu'
' Khoa Cong nghi, Truang Dai hoc Cdn Tlio Thong tin chung:
Ngdy nhgn-23/05/2013 Ngdy chdp nhdn: 24/12/2013 Title:
Anti-Sway control of container cranes in the presence of friction Td khda:
Dieu khiin chdng ldc, udc lugng ma sdt, phuangphdp binh phugng cue liiu hdi qui Keywords:
Anti-sway control, Friction estimalion. Recursive least squares method
ABSTRACT
In this paper, an anti-sway control scheme for a container crane that is used for vessel-to-truck and truck-to-vessel loading and unloading oj containers at a container terminal is investigated. The control objectives are to move a container to a desired position and to suppress its transverse vibration produced by trolley motion in the presence of the friction force and the control input saturation In this study, friction coefficients are estimated using the recursive least squares (RLS) method.
and the friction force is incorporated into a nonlinear control law. The designed control input for control of the trolley motion is modified to satisfy the saturation condition of the actuator (electrical motor). The nonlinear control law guarantees the asymptotical stability of the closed- loop system. Simulation and experimental results verify the efficiency gj the proposed algorithm
T 6 M TAT
Trong bdi bdo ndy, mgt phugng thitc dieu khiin chong ldc cho cdn cdu container dang dugc sir dung di van chuyin container tgi cdc cdng container dugc phdt trien. Muc tiiu diiu khiin Id di chuyen container den vi tri mong mudn vd ngdn chgn viec dao ddng ngang cita container trong qud trinh di chuyen vdi su tdn tgi ciia luc ma sdt. Trong nghien ciru ndy, hi sd ma sdt duac udc linh bdng phuong phdp binh phuang cyc tieu hdi qui (RLS), vd lire ma sdt dugc tich hgp trong lugt diiu khiin phi tuyin.
Lugt diiu khien phi tuyin ddm bdo su dn dinh liem can cita hi diiu kliiin vong kin. Kit qud md phdng vd thuc nghiem xdc nhgn tin hiiu qud cita thudt todn de xudt.
1 GIOI THIEU
Cin ciu dugc sir dyng rpng rii trong cfing nghiep de van chuyin hdng hod tir noi nay den noi kfaic trong nhieu dia diem kfaac nfaau: ben cang.
kho hang, nhi raiy, cdng trudng, co sd hat nfain va nhung ngudi khac. Khi chuyin dpng cua gian xe day la nguyen nhdn gay ra vi|c lie lu ciia tii frpng khi vgn chuyen. ddc bill li khi ting tdc dau hinh frinh va khi giira tdc cufii hdnh trinh, thi vile logi bfi chuyin ddng lac nham ldm tdng thdi gian hogt dfing hi|u qud ciia can cdu Id mfit van dl quan frpng trong viec sir dyng can cau. Do do. nhilu nhi nghien ciiu lim vile frong linh vyc dilu khiln can
ciu da lufin ludn nham myc tiiu tri|t tiiu dao ddng lie ciia tai.
Nhu da nlu, chuyin ddng lie phit sinh khi tang tfic ddu hinh frinh va gidm toe cudi hinh trinh ciia xe day. Chinh dieu nay lam giim hieu qua hogt ddng cua can cau trong cdc nginh cfing nghiip giao thong van tdi va xdy dyng, ki ca trudng hgp can cau tuang ddi don gidn dugc sii dyng. Do do, cic nha nghien ciiu frong dilu khiln cdn cdu da tip trung vdo cdc giai phip dilu khien fri|t lilu dao dpng mpt cdch nhanh chdng, lien tyc vd hieu qui.
Cac phuang phdp dilu khien can cau di dugc phit trien bao gfira hieu chinh ngo vao (Blackburn et al.,
Tap chi Khoa hgc Tnrcmg Dgi hoc Can Tho Phdn.4: Khoa boc Tunhien. Congn^ vd Moi tnr&ng: 29 (2013) S-14 2010: Hong, Huh. & Hong, 2003; Hong, Park. &
Lee. 2000; Huey, Sorensen, & Singhose, 2008;
Singhose, Perter, Kenison, & Krrikku, 2000;
Sorensen, & Singhose, 2008, Sorensen, Singhose,
& Dickerson, 2007; Sung & Singhose. 2009), dieu khiln tfii uu (AI-Gami, Moustafa, & Javeed Nizami, 1995; Terashima, Shen, & Yano, 2007), dilu khiin tuyin tinh/phi tuyen (Kim, Hong. &
Suk, 2004; Klosinski, 2005; Messineo, Celani, &
Egeland, 2008; Park, Cfawa. & Hong. 2007:
Sawodny, Ascheraann, & Lahres, 2002), diiu khiln bien lir quan diem xem he thdng cin cdu la he van chuyin lien tuc (d'Andrea-Novel & Coron, 2000; Kim & Hong, 2004: Ngo, Hong, & Jung, 2009), dieu khiin frirgt (Almutairi & Zribi, 2009;
Bartolini, Pisano & Usai, 2002), diiu khiin md (Benhidjeb & Gissinger. 1995; Chang & Chiang, 2008; Liu, Yi, Zhao, & Wang, 2005) va diiu khiin thich nghi (Cheng & Chen, 1996; Hua & Shine, 2007; Messineo & Serrani, 2009; Mizuraoto et al., 2007).
Trong nghien ciiu nay, mdt thudt loan dilu khiln dya frln mfi hinh phi tuyen tinh cua cdn cdu container da dugc thilt kl. Trong do, luc ma sat dugc linh din vi xac djnh dya trin md hinh lyc raa sat Coulomb (Olsson et al., 1998; Lu et al, 2009).
Lyc ma sat xudt hien frong he thdng lam tang sai sfi xac lap trong bdi loan dilu khiln vi fri. Neu xet den viec dieu khien chdng ldc thi chat lugng dilu khiln nhu vay Id khdng the chip nhgn. Vi viy, ma sat phai dugc udc tinh, va sau dd la vile bii ma sat phii dugc ap dyng dl loai bd tdc ddng ciia nd. Dl udc lugng lyc raa sdt, rapt mfi hinh todn hpc ciia no dugc sir dyng, vd cac he sd ma sit ciia cic rafi hinh dugc uoc tinh bang cdch sii dyng phuang phdp binh phuang cyc tieu hdi qui (RLS).
Bai bdo dugc frinh biy nhu sau: he phuang frinh toan hpc mfi ta d^ng lyc cua mdt cin ciu container dugc thiit lap vi cic mfi hinh toin hpc de udc tinfa ma sat dugc cdng bfi trong phdn 2: frong phdn 3. mfit luat dilu khiln phi tuyln dugc de xuat;
phdn 4, kit qui md phdng va thyc nghiim cho cd hai phuang phap dilu khiln myln tinh vd phi tuyln dugc thao ludn: cufii cimg, kit ludn dupe nit ra frong phin 5.
2 MO TA H$ CAN CAU CONTAINER 2.1 Dpng l i ^ fapc bf can ciu
Xem xet cdn cau container dugc minh hpa frong Hinh 1. Container dugc giir bdi ngdm kep container (spreader) vi ci hai dupe lien kit vdi xe day (xe goong) bing rapt day cd chieu dai /. Khdi lugng ciia xe day va tdi frpng tuong iing Id m, va
mp. Mpl lyc dieu kfaien _^ dugc tic ddng vdo xe di\
ddng thdi xe day bj anh hufing boi rapt lyc raa sat fr. Vile di chuyin ciia xe day den vi fri mong mudn se gay ra dao ddng lie lu ciia container. Trong tfayc ti, mdt cin cau container su dung bdn sgi diy thimg dk lien kit ngam kep vdi gidn xe diy. Tuy nhiin, frong nghien ciiu niy, de dan gidn, can ciu container dugc md hinh hoi Ii mdt sgi day Hen kit giiia tdi \ a xe diy. Gii thilt ring chuyen ddng ciia xe day va chuyen dfing lie ciia tai cimg nira frong mdt mat phdng, cd nghia Id, mat phang XY (xem Hinh 1). Ddt x li vj fri xe ddy frln tryc X, 9 la gdc lie vi g Ii gia tfic frpng trudng.
Xem cac chuyin dpng ciia he thfing frong mpt mat phdng hai chiiu khfing ma sit, dfing ndng T va thi ning U cua loan bfi he tfafing dugc xdc dinfa bdi:
1 .1 1
— n-" ' 2
— I . 1 2
W \-2 1 P 1 + m /xsin 0 + mplx6cos9,
U =-m pgl cos 0. (2) T = -m,x- + -m Jicos 9 - W sin df
2 ' 2 ^^ / - m p ( x ^ + / s i n ^ + / ^ c o s ^ f
-{m,+mp)x^ +-mpP-\--mp!^0^
Dgt ^ = (.11:,^) li toa dp suy rfing tuang ung vdi cac lyc tac dfing frong log dp suy rfing, / = ( / j , O), va sir dyng phuang frinh Lagrange
d_(dT_]d7^^8U_^ •
= / , / = !, 2. (3) thi phuong frinh chuyen ddng dugc xic djnh nhu
Hinh 1: Mfi hinh toan hoc hi can cau container
Tap chi Khoa hgc Tnr&ng Dai hgc Cdn Tha
fx =("'; + nip}x-^mpi sm6-i-mpWcos6 + 2mpi9 cos e - mpl9^ sin 9, 0 = Mplxcos 9 + mpl^9
+ 2m pli9-\-mpgl sin 9.
Vdi viec tfin tai cua lyc ma sdt, f, thi phuong frinh (4) dugc viet lai:
f^-fr= (m, + mp)c+mpl sin 9
Ph&n A: Khga bgc Tu nhien. Cong nghi va Mdi tnr&ng: 29 (2013): 8-14
(4)
(5)
+ mpl9 cos 9 + 2m pW cos 9 -mJ9^sin9.
(6)
Trong phan tiep theo, rapt md hinh todn hpc ciia luc raa sat cimg vdi pfauong phap udc lugng he sd raa sdt se dugc trinh bay.
2.2 Udc lirgng ma sat
Ma sit ludn tfin tai frong cdc he ca khi va la nguyin nfaan ciia nfaiiu faien tugng nhu chuyin dpng theo tre (fracking lags), sai Ilch tinfa (steady- state) hoic la chuyin ddng nhdy (stick-slip raotion). Rit nhilu mfi hinh ma sat da dugc nghiin ciru, frong dfi, mfi fainh frong Hinh 2 dugc gioi thilu bfii Olsson et al (1998) dugc su dyng phd biln. Ap dyng md hinh niy, raa sal giura xe diy vi duong ray frln can cau container dugc cho nhu sau:
6 diy./, Ia lyc ma sat nghi tai vi tri diing yIn, ^va Cv la he sd ma sat Coulomb va h | sfi nhdt, va g- dugc dinh nghia la chilu ciia van tfic, g = sign(x).
Lyc ma sdt tinh tac ddng nfau la phin lyc kfai lyc pfaat dpng kfadng ldn faon lyc ma sat Coulomb.
Vi vay, luc ma sat ttnh cd thi Id bit ky gid tri nao frong khoing f vi ^" frong mien vgn tfic kfafing (zero-velocity). Dl triet tieu lyc ma sat tinh, lyc phit ddng cin phai dio efaiiu sau khi qua khdi vimg van Idc khfing. dilu nay se tao ra chuyin dfing quay vdng theo chu ky (limit cycles). Dl khac phyc hiln tugng ndy, lyc ma sdt ttnh dugc xip xi nhu la mpt ham lien tuc frong raiin van tdc khfing. nhuHinh 2(b), Khi do, md hinh ma sdt mdi dugc dl xuit trong dilu kfaien cin ciu li
+ ff (7)
/. =
(,-,(£±i)(r-:i)
(i-.(^](r-c,:4
l ^ " ! ^ "
(8) a day dv la giai han tren ciia vung v|n t6c khong va rj la he so duac dinh nghTa nhu sau:
| i | < A ,
|ij>£/v. (9)
Trong vile udc lugng ma sit Coulomb va ma sat nfaot. xe ddy dupe thilt kl dl di chuyin ra ngoai viing van tdc khdng. Vi vgy, lyc ma sal tinh dugc bd qua frong md hinfa udc lugng. Bin cgnh do, dip irng ciia ddng co tfai nhanfa han ddp iing cua xe diy. Cho nin, ddng co dieu khien chuyen dpng ciia xe day dugc md hinh hoa nhu la md hinh tuyen tinh vdi hang sd chua bilt, km-
L=K"'
(10)
d diy. u Ii ngo vdo dilu kfaien dpng ca.
Friction force (^
Velocity X
(a) Friction force (fr)
Velocity X
(b) Hinfa 2: Mfi fainfa ma sat
Tgp chi Kboa hgc Tru&ng Dgi hgc Can Tha Phuong trinh chuyin ddng eiia dpng co dugc viet lai frong trudng hgp tdn tai lyc ma sit Id:
i=h{0,0)+gx{e'if,-f^ (11) s = h{<>,eVgi(0lf,-fr\ (12)
Phdn .4. Khoa hgc Tu nhien. Cong ngh? vd Moi irir&ng' 29 (2013): 8-14
i[e,e)
sM
gi{e)
m, + m sin 9 - m„l sin 9 + rrtpW^ sin 0
m, + nip sin 9
-[nip + m,)gsm9 + mpi s i n ^ c o s ^ m,l + mpism~ 0 mpl9~ sin ^ c o s ^ 210 mfl + mpi s'm 0 I
1 m, + mp sin 0
c o s ^ mJ + mpl sin 0
Thay phuong frinh (8) vi (10) vdo (11) vd bd qua ma sdt tinfa dugc kit qui sau:
x-hi{0.
[r-c;i
(13)dddy
4)=M')-/('-iW'-i), pW=(/-^((y(r-i)H-i)/i,
A. la he sd suy nhd (forgetting factor), vi / li ma frin dan vi. Dl iing dung phuong phip RLS, fr'n hieu tham chilu dugc cfapn dl thoa man dilu kiln kich tfaich liln tyc (persistent excitation - PE). (5 da\', gia tdc vi van tfic ciia frolley dugc gii dinh Id do ludng hay udc lugng dugc lir thfing sfi vi fri do dugc. Vl vdy, hdm sfi ^(0 trong (14) dugc xdc dinh. Cdc thi nghiim da dugc thyc failn de ddnh gid phuang phap ndy va cac kit qui dugc trinh bdy frong phin 4.
3 DIEU KHIEN PHI TUYfiN
Trong tfayc te, lufin tdn tai mfit gid fri eiia ngo vdo dilu khiln de he phuang trinh (II) vd (12) thoa mdn dilu kiln:
•x=h{9,9)-^g,{0){f,-f,)
= x^-k,i{x-Xj)-k,2(x-xA tI6) 0 = h2{9,9)+g2{9lf,-fr)
= -2kg9-k^9, f^'^) d ddy, Xd la vj fri mong mufin ciia xe day. Khi dilu
kiln (16) va (17) cimg thda man thi he tfafing Id fin djnh tiem can vi hfii ty vl frgng thai mong mudn, q^ ={Xj,0). Dl thilt kl ngo vdo dilu khiln, phuong frinh (16) vd (17) duge vilt lai dudi dang ma fran sau:
h,(l
hA0,9
Phuong trinh (13) Id tuyln tinh theo cac thdng sfi chua bilt 0 = [A„ ^^ ^~ c~ c~] nen phuong phdp binh phuang cyc tieu hdi qui dugc sir dung ^e:
udc lupng cdc thfing sd frln. Cac ham sd dupe djnh nghia nhu sau:
hi))=[x-h,(e.eygAo)'
/(/)=[£/ -04f+l} -0.4-1) -(i.ig+\)x 0.is-\)x], a, = - vd vi vay.
A(r) = / ( / ) 0 . (14) + - p Su dung phuong phap uac luong RLS (Astrom
& Wittenmark. 1994), vec ta thong so chua biet duac xac dinh la:
0 ( , ) = 0 ( ' - ! ) + ' ' • ( ' > ( ' ) . (15)
,S2(e).
\{f.-fr)-
(18)6 day, V, = x ^ - i , , ( i - i ^ ) - * , j ( x - i ^ ) va VT = -2kgO -kgO. Vi vay, ng6 vao dieu khien Uc duac thiet ke la:
gAt
-hAe.e]
g}(e)+8l(0) gKeygKe) + fr
(19)
Hang so dpng ca va cac he so raa sat da uoc luang trong phan 2 duoc dua vao trong giai thuat dieu khien.
Tgp chi Khoa hge Tnrang Dgi hoc Can Tha Phan.4- Khoahgc Tunhien. Cong ngh? va Mdi iru&ng. 29(2013). 8-14 4 KET QUA MO PHONG VA T H U C
NGHIDM
4.1 Udc lugng ma sat
Mfit can cdu hai chieu diing trong pfaong thi nghiim dugc sii dung di thyc hien cdc thi nghiim. Dl thfia man dilu kiln PE [30], tin hieu tham chilu dugc chpn nhu sau: Xd = 0.5sin(n//2) + 0.25sin(ji//3) + 0.25sin(Tuf/6). IChfii lugng ciia xe ddy vd tdi dugc cho nhu sau:
m,=I.67kg, vd mp=0.73kg. Khi ngo vdo dieu kfailn tdc dfing vdo xe day, xe day se di chuyen theo ci hai chieu de ham sd frong (13) dugc xac dinh vdi mpi thdng sfi. Thdng sd ude lugng dugc xic dinfa theo (15) vd dugc trinh bay trong Hinh 3. Tdt cd cdc thfing sfi diu hpi ty vl gii tri thyc ciia he thdng khi vgn tfic thay dfii bao gfim chieu chuyen dfing am va duong. Tuy nhien, be mat tilp xiic giira dudng ray va xe diy khdng dugc nhan mfit each ly tuong nen vdn cdn tfin tai sy sai lech nhd. Trong thyc tl, gii frj sai Ilch nay la khfing ding kl va chdp nh|n dugc dfii vdi mfi hinh thyc nghiim. Vi vgy, gia trj trung binh (bao gfim ci sai l|ch) ciia he sd ma sat dugc diing cho myc dich dilu khiln.
PosiliTC direction [ c^ ) NcgatiTC direclion ( c| )
4.2 Gidi thuit diiu kfaien phi tuyen Vdi cac hi sd ma sat dugc uo^ lugng mpt each chinh xac thi lyc ma sdt hoan todn dugc tinh toan theo phuong trinh (8). Mdt khi lyc ma sit va hdng so dfing CO km dugc xac dinh Ihi luat dilu khiin (18) dugc ip dyng dl dilu khiln cin cdu. Vile ip dung ludt dieu kfailn pfai tuyen cd bii ma sit mang Iai kit qud rat khd quan. Xe day dat dugc vj tri mong mudn vdi dao ddng rdt nfad cua tai, theo Hinh 4. Tuy nhiin. gdc ldc ddng ca vdn cdn dao ddng vdi bien dp nhd ngay kfai xe ddy dtmg lai a vj tri mong mudn. Dao dpng nay la do bfii xe day di vao viing van tdc kfadng vd tai do lyc ma sit thay dfii chilu dot ngpt. Tuy nfailii, dao dpng nay la khfing dang kl nlu so vdl gdc lac cua tai khi kfaong cd dieu khiln.
5 KET LUAN
Lyc raa sal lufin tfin tgi frong cic he thfing can ciu va cac he sfi ma sat thudng li khfing xac dinh, dilu nay gdy kho khan frong vile thill kl h$ thfing dieu khiin chinh xac vi fri ciia xe diy va chfing ik cho tii. Tuy nfailn, bdi bdo gioi thilu mfit phuong phap udc lugng cac he sd ma sdt sir dyng phuong phdp binh phuang cue tieu hdi qui, dfing tfadi thiet ke luat dieu khien phi tuyen cd bu ma sat de dieu khien xe diy din vi fri mong mu6n vdi gdc lie dugc friet tllu d cufii hdnh trinh. Kit qua thyc nghiem da cho thay tinh hieu qud ciing nhu kha nang fin dinh ciia ludt dieu khiln phi luyln cfi bii ma sat.
/
3 4 6 g 10 1! 14 16 IE 2 0 Time [sec]
(a) Vi tri xe diy
(c)
Hinh 3: Thong so dugc uwc lugng
(b) Goc ldc ciia tii
Hinfa 4: Dap ung cua hf tbdng kfai dieu khiln
Tap chi Khoa hgc Tnr&ng Dgi hoc Cdn Tho PhdnA: Khoa bgc Tunhien. Congnghe vd .\l6i tnr&ng. 29 (2013). 8-14 L0I CAM TA
Tac gjd chdn tfadnfa cdm cm Trudng Dgi hgc Cdn Tho da cip kinh phi de thyc hiln nghiin ciru ndy. Ngfaien ciru dugc thyc hiln dudi sy tai trg tir De tai nghiin ciiu cap Truong ndm 2013 (Ma so di tdi: T20I3-06).
TAI LifU THAM KHAO
1. Al-Gami, A. Z.. Moustafa. K. A. F.. &
Javeed Nizami. S. S. A. K. (1995). Optimal control of o\'ehead cranes. Control Engineering Practice. 3(9). 1277-1284.
2. Afanulairi, N. B. & Zribi. M. (2009).
Slidmg mode confrol ofa three- dimensional overhead crane. Joumal of Vibration and Control, 15(11), 1679-1730.
3. Asttdm, K. J., & Wittenmark. B. (1994).
Adaptive control, Addison-Wesley, 2"^ ed.
4. Bartolini, G., Pisano, A., & Usai, E.
(2002). Second-order sliding-mode confrol of container cranes. .4uiomatica, 38(10), 1783-1790.
5. Benhidjeb, A., & Gissinger. G. L. (1995).
Fuzzy control of an overhead crane performance comparison with classical control. Control Engineering Practice.
3(12), I687-I696.
6. Blackburn, D., LawTence, J., Danielson, J.. Singhose. W.. Kamoi, T.. & Taura, A.
(2010). Radial-motion assisted command shapers for nonlinear lower crane rotational slewing. ContiolEngineering Practice. In press.
7. Chang, C. Y., & Chiang, K. H. (2008).
Fuzzy projection confrol \z\\ and its . application to the overhead crane.
Mechatronics, 18(10). 607-615.
8. Cfaeng. C. C . & Chen, C. Y. (1996).
Confroller design for an overhead crane system with uncerlainl;'. Control Engineering Practice, 4(5). 645-653.
9. d'Andrea-NoveL B.. & J. M. Coron.
(2000). Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach. Automatica, 36(4). 587-593.
10. Hong. K. T.. Huh. C. D..&Hong. K.-S.
(2003). Command shaping confrol for limiting tfae transient swa\ angle of crane sN'stems. Internal ionalJoumal of Control.
Automation, and Systems. 1(1), 43-53.
11. Hong, K.-S.. Park, B. J..& Lee. M. H.
(2000). Two-stage confrol for container cranes. JSME IntemationalJoumal, Series C, 43(2), 273-282.
12. Hua, Y. J., & Shine, Y. K. (2007).
Adaptive coupling control for overhead crane systems. Mechatronics, 17(2-3), 143-152.
13. Huey, J. R., Sorensen, K. L., & Singhose, W. (2008). Useful applications of closed- loop signal shaping confrollers. Control Engineering Practice, 16 (7), 836-846.
14. Kim. C. S., & Hong, K.-S. (2009).
Boundary confrol of container cranes from perspective of controlling an axially moving string system.Intemational Journal of Control, Automation, and Systems. 7(3). 437-445.
15. Kim, Y. S., Hong, K.-S.,& Sul, S. K.
(2004). Anti-sway control of container cranes: inclinometer, observer, and state feedback. IntemationalJoumal of Control, Automation, and Systems, 2(4), 435-449.
16. Klosinski, J. (2005). Swing-free stop confrol of the slewing motion ofa mobile crane. Control Engineering Practice, 13, 451-460.
17. Liu, D., Yi, J., Zhao, D., & Wang, W.
(2005). Adaptive sliding mode fuzzy confrol for a two-dimensional overhead crane. Mechatronics, 15(5). 505-522.
18. Lu. L.. Yao. B.. Wang, Q., & Chen, Z.
(2009). Adaptive robust confrol of linear motors with dynamic friction compensation using modified LuGre model. Automatica. 45(12). 2890-2896.
19. Messineo, S., Celani, P., & Egeland, O.
(2008). Crane feedback confrol in offshore moonpool operations. Control Engineering Practice, 16(3). 356-364.
20. Messineo, S., & Serrani, A. (2009).
Offshore crane confrol based on adaptive external models. Automatica, 45(11), 2546-2556.
21. Mizumoto, 1.. Chen, T.. Ohdaira, S- Kumon. M.. & Iwai, Z. (2007). Adaptive output feedback confrol of general MIMO systems using muliirate sampling and its application to a cart-crane system.
.Automatica. 43(12). 2077-2085.
Tap cbi Khoa bgc Tnt&ng Dgi boc Cdn Tho Phdn A Khoa hgc Tif nhien. Cong nghi vd Mdi tnr&ng: 29 (2013): 8-N 22. Ngo, Q. H., Hong, K.-S., & Jung. I. H.
(2009). Adaptive confrol of an axially moving system. Joumal of Mechanical Science and Technology, 23(11), 3071-3078.
23. Olsson. H., Asfrdm. K. J., Canudas de Wit, C, GafVert, M.. & Lischinsky, P.
(1998). Friction models and friction compensation. European Journal of Control, 4(3), 176-195.
24. Park, H., Cfawa, D., & Hong, K.-S.
(2007). A feedback linearization confrol of container cranes: varying rope length.
International Joumal ofConti-ol, Automation, and Systems. 5(4), 379-387.
25. Sawodny, O., Aschemann. H., & Lafares, S. (2002). An automated gantry crane as a large workspace robot. Control Engineering Practice, 10(12), 1323-1338.
26. Singhose, W., Perter, L.. Kenison, M., &
Krrikku, E. (2000). Effects of hoisting on the input shaping confrol of gantry cranes.
Control Engineering Practice, 8( 10), 1159-1165.
27. Sorensen, K. L. & Singhose, W. (2008).
Command-induced vibration analysis using input shaping principles.
Automatica, 44(9), 2392-2397.
28. Sorensen, K.L., Singhose, W., &
Dickerson, S. (2007). A confroller enabling precise positioning and sway reduction in bridge and gantty cranes. Control Engineering Practice, 15(7), 825-837.
29. Sung, Y. G., & Singhose, W. (2009).
Limited-state commands for systems with two flexible modes. Mechatronics, 19(5), 780-787.
30. Terashima. K., Shen, Y.. & Yano. K.
(2007). Modeling and optimal confrol ofa rotary crane using the straight fransfer fransformation method. Control Engineering Practice, 15(9), 1179-1192.
