Top chi Khoa hoc 2012 21a 30-36 ^nO^e Bai hoc Cin Tha
BIEU KHIEN TRirgrr DlfA TREN HAM TRlTOT KIEU PID
Nguyen Hoang Dung
ABSTRACT
Chattering phenomenon around sliding surface and sliding control law makes some drawbacks for control problems while using sliding mode controller. In practice, some actuators are unable to satisfy from the phenomenon. Therefore the paper presents a method using sliding mode controller with type PID (Proportional integral derivative) based sliding suiface for controlling nonlinear system. The solution is proposed for reducing chattering phenomenon surrounding sliding surface and sliding control law.
The algorithm is applied to control one degree of freedom robot manipulator. Simulation results are implemented basing on Simulink software of MATLAB as following: the response of one degree of freedom manipulator tracking desired signal with overshoot 0.02%, settling time 3 Is. steady-state error being not worth considering, the chattering phenomenon round sliding surface and sliding control law being completely eliminated.
Keywords: Sliding mode controller, PID, robot manipidator Title: PID type sliding surface based sliding mode controller
T O M TAT
Hi$n tugng dao dgng quanh mgt trugt cSng nhu trong luat dieu khien truat dd gay nhiSu kho khan cho bdi todn dieu khien khi su dung bg dieu khien ti'uat. Trong thuc ti, hi$n tuang nay co the lam cho cdc thiet bj chap hdnh khong the dap txng dugc. Do do, bdi bao trinh bay mgt giai phdp su dung bg dieu khien trugt de dieu khien doi tugng phi tuyen vdi ham tru0 dugc thiet ke dua tren PID. Gidi phdp tren dugc di nghi nhdm giam thiiu hien tugng dao dgng quanh mdt trugt vd dao d$ng trong lugt diiu khien. Gidi thugt nay dugc dp dung de dieu khien doi tuang h? tay mdy mot bdc tu do Kit qud mo phong dua tren phdn mim Simulink ciia MATLAB cho thdy: Ddp ung ciia he tay mdy bam theo tin hi$u mong mudn vdi dg vgt 16 0.02%. thdi gian xdc Igp 3.1s, sai s6 xdc Igp khong ddng kS, logi bo dugc hifn tugng dao dgng quanh mgt trucrt vd trong ludt diiu khiin tru0.
Tir khda: Bp diiu khien trufrt, PID, hi tay mdy
I G l 6 l T H l £ U
iTu diem n6i_ bac ciia bo khiln trucft la tinh 6n dinh va bin vGng ngay ca khi he thong CO nhilu hoac khi thong so cua ddi tugng thay d6i theo thoi gian. Tuy nhien, neu bien dO cua lu^t dilu khiln thay ddi qua lan co thi lam cho hS thSng dao dgng (chattering) va khong on dinh. Dl k h k phuc hi?n tugng tren, hk\ bao d l nghi sii dyn^ bo dieu khien trugt vcri ham tnrgt co dang phuang trinh cua cua bO dieu khien PID. Va ham trugt nay dugc ggi la ham trugt kilu PID. Giai thuat nay loai bo dugc hien tugng dao dpng khi bian do cua lu$t dilu khiln truot tang. Va gik thu^t dugc ap dung de dieu khiln doi tugng phi tuyln-he tay may" mot bac tu do.
Doi VOI hien nrgng dao dgng khi su dyng bg dilu khiln trugt, nhilu nha khoa hgc da nghien cuu va dua ra nhieu giai phap kh5c phuc khac nhau. Li Jian-jun (2010) de nghi su dung logic mo nhu mgt ngo ra lien tuc d6i vai bg dieu khiln trugt va ' Khoa Cong Ngh^. Tnrang Dai hgc Cln Tho
Tgp chi Khoa hgc 2012:21a 30-36 Truong Dgi hgc Cdn Tha
dimg mang noron de thay the ham sign trong bo dieu khien nay. Voi giai thuat de nghi, Li Jian-jun da minh chung dugc viec loai bo dugc hien tugng dao dgng quanh m | t trugt. Zhang Yuzeng (2010) sii dung vung gioi han cho qua trinh chuyen doi cua mat trugt. Phuang phap nay da giam thieu dugc dao dgng quanh mat trucrt. Tuy nhien viing gioi ban nay k h o i ^ phai la duy nhat cho tat ca cac doi tugng. Ben canh do ZhixiongHou (2003) da uoc Itigng cac thanh phan khong xac d|nh va bien d6 ciia luat dieu khien trugt dua tren he mof. Giai thuat nay khong yeu cau sii duiig vimg gioi han cho cac thanh phin bat dinh. Va giai thuat da chung minh tinh on dinh dua tren Lyapunov. Han t h i nvta, Yangmin Li (2010) thilt lap m | t trugt dua tren lu$t dieu khiln PID. Vai phuang phap nay luat dilu khiln trugt kieu PID da dieu khiln t6t doi tugng la tay may Piezo-Driven va loai bo dugc hi$n tugng dao d6ng khong mong muon. Tity nhien, dg vgt 16 va thoi gian xac l|p ciia dap ung Ion.
Dya tren kit qua nghien cuu ciia Yangmin Li (2010), bai bao d l nghj thilt k l bg dieu khien trugt dua tren ham tnrgt ki6u PID. Trong do cac tham s6 ciia ham trugt dugc dieu chinh dya tren phuong phap thu sai. De minh chung kha nang lam giam hi$n tugn^ dao dOng tren mat trugt, giai thuat nay dugc ap dung de dieu khien he tay may mgt beic tu do.
2 M O H I N H TOAN H E TAY MAY M O T BAC TV" DO
He tay may mgt b§c ty do dugc mo ta nhu Hinh 1. He tay may nay co th6 quay quanh mgt true nhoi vao moment u(t) tac dung len true. Vi tri cua canh tay e(t) dugc xac djnh la goc hgp bofi true thing dung va phuong cua canh tay. u(t) la tin hi$u ngo vao va e(t) la tin hiSu ng5 ra. Gia sur a thoi diem ban diu he tay may chua gap v | t n^ng m. Sau khoang thoi gian t nao do, he tay may sS gap vat nang m. Dua tren viec phan tich phuong trinh Eluer-Larange, phuong trinh dgng hgc cua h6 tay may mgt bac t u do dugc thiet lap nhu sau:
{j + mr)9(t)+B0it) + {ml-i-Ml^)gsm(0it)) = u(t) (1)
trong do J = Ml; la moment quan tinh ciia canh tay, cac thong so con l^i dugc m6 ta chi tiet trong bang 1.
Dat jc, = e(t) va x. = 0(t)
Phuong trinh (1) co thi bilu dien dudi dang phuong trinh trang thai nhu sau:
x,it) = x^it)
uit)-Bx.il)~{ml + MlJgsm{x,(t)) (2) J + m
Vangoray(t)=e(t)=x,(t).
r=^'>= J.ml^
Tgp chi Khoa hoc 2012:210 30-36 Tnimig Dgi hoc On Tha
Hinh 1: Mo hinh dong hgc h | tay may iii9t b a c t y d o Bang 1: Cac thong so dgDg hgc cua b | tay may mdt b | c tir d o
Ky hieu Y nghia G i a t r j Bern vi M
m 1 1, B S
Khoi liiong cua canh tay Khoi lugng vat n$ng Chieu dai canh tay
Khoang each tir trpng tam canh tay den true quay He so ma sat nhdt
Gia toe trpng truong
1 0.1 0.4 0.15 0.2 9.81
Kg Kg m m Kg.m=/s m/s^
3 THIET KE BO DIEU K H i t N TRlTOT BANG CACH C H O N M A T TRLTglT KlfiU PID
3.1 Thiet ke b$ dieu khien trirgt kinh di^n
Do phuong trinh (2) la he b^c 2 nen ham trugt dugc chpn nhu sau:
j(,) = e(()+je(/) (3) trong do e(t)=y(t)-r(t) (4) vdi r(t) la tin hieu mong mu6n, y(t) la u'n hieu ngo ra, X la hSng so duong chon
truoe.
Khi ham mrgrt s(t)=0, nghi$m ciia phucmg trinh (3) co dang e(r) = expf - - 1 . Do do kW r -> CO, e(0 -» 0. Voi X dupe xem la thai hSng ciia ham truot. X cang nho, he thong cang chgm tien vl mat trupt va ngupc lai.
Thay (4) vao (3), ham mipt dupe vilt lai:
J(() = x,CO-r(0+i(j:,(0-KO) (5) Theo ly thuylt 6n dinh Lyapunov, chpn mpt ham xac dinh duong
T u ( 6 ) s u y r a K = s{t).s(t)
D i ^(,) xac dinh am, chpn i{0 =-t.,/g„(.(0). Thay m-k.sisni.m vao Hm cho ham K(,) = -sM..ign(m, vcri k la hSng sa duong chon trudc
NhSn xet: neu sCt)>0 thi ^(,)<0. Nlu s(t)<0 thi K(0<0. N l u s(t)=0 thi (>(r)=0 TO P W^Tl' T^-''°,,K-9'' ^ * ™ « ' ' ^ " * • * * e o tieu chuto Lyapunov.
Tu (2). (5) luat dieu khien trupt u(t) dupe thilt k l nhu sau
Tgp chi Khoa hgc 2012:21a 30-36 Truong Dai hgc Cdn Tha
« ( 0 = BJC^ (J) + {ml + M , )g sin(xj (/)) +
+(j+me]^^,t)+J{m'^^i.t))+k.sign{sit)•)\ ^'^^
Ket qua mo phor^ cho thiy, neu su dung bg dieu khiln trugt kinh dien thi dap ung cua h8 tay may bam_ theo tin hieu mong mu6n voi do vgt 16 la 3.5% va thai gian xac lap la 5.8s, sai s6 xac lap khoi^ dang k l (xem Hinh 2).
3.2 Thiet ke bo dieu khi^n trirot dira tren ham tmr^t kieu PH)
Luat dieu khien trugt dugc thilt k l dya tren ham trugt. Ham trugt dugc xay dung dua tren bac ciia mo hinh ddi tugng. Doi voi doi tugng he tay may mgt bac t\r do, ham tnrgt dugc chgn nhu phuang trinh (3). Tuy nhien de loai bo hien tugng dao dgng quanh m&t trugt khi bien do cua luat dieu khien trugt thay doi loti, ham tnigt CO the dugc chgn nhu sau (Yangmin Li, 2010):
•^(0 = m + ^e{t) + /L Je(r)t/(r) (8) trong do e(t) la sai so giua dap ling ngo ra va tin hieu mong muon dugc thiet lap
nhu (4). X-i va Xj la hai hing so duong. Hai gia tri nay dugc chgn sao cho phuong trinh d|ic tinh s' ^A^s + X^^Q phai Hurwitz. Trong do s la biln phirc va nghiem ciia phuong trinh s^ -\- A^s-¥ ^=0 phai nam a nua bo trai ciia mat phang phuc.
Phuang trinh (8) co thi dugc viet lai nhu sau:
s{t) = y{t) - m + ^ iyil) - r(t)) + X,\iy(T) - riT))d(T) (9) Theo ly thuylt on dinh Lyapunov, chgn mgt ham xac dinh duong
nt) = ^s^-(t) (10) Phuang trinh (10) CO thi dugc viet lai K = s(t).s(t) ( U )
De V{t) xac dinh am, chgn sit) =-ksign^sit)). Thay s(t) ^-k.sign(s(t)) vao F(/)se cho ham: y(t) = ~sk.sign(s(t)), vai k la hing so duong chgn truoc. Do do luat dilu khiln trugt dira tren ham trugt kieu PID dugc thiet ke nhu sau:
u(t) = B.x^ (f) + (ml + Ml, )g sin(Xi if)) +
+ ( j + mP \rit) + X, (r{t) - x^ (/)) + XMt) - x, (/)) + k.sign{s(t))\ ^^^^
So sanh giiia luat dilu khiln trugt a bieu thuc (7) va luat dieu khien trugt dua tren ham trugt kieu PID (12) cho thay: lu|.t dilu khien trugt dua tren ham tnrgt kilu PID dugc them vao mgt thanh phan la A.2e(t).
Ket qua mo phong khi su diing b6 dieu khien trugt voi ham trugt kilu PID cho dap umg cua he tay may bam theo tin hieu mon§ mu6n vai dfi vgt 16 la 0.02%, thai gian xac lap la 3.1s. sai s6 xac lap khong dang ke (xem Hinh 2).
4 KET QUA M 6 PHONG VA THAO LUAN
Ba bg dieu khien khac nhau gom bg dieu khien trugt, bg dieu khien trugt vai ham trugt kieu PID va bg dieu khien PID ciing luc ap dung de dilu khien mgt d6i tugng duy nhat la h? tay may mgt bac tu do. Ket qua mo phong tren Simulink ciia
Tgp chi Khoa hgc 20122 la 30-36 Tru&ng Dgi hgc Cdn Tha
MATLAB cho thiy neu su dung bg dilu khien trugt kinh dien thi thai gian xac lap (5.8s) va do vgt 16 Ion nhit (3.5%) trong s6 dap ling cua ba bg dilu khiln trSn. Ben canh d6 luat dilu khiln va mat trugt diu co hi?n tugng feo dgng. Mat frugt dao dgng voi bien do 0.14 (xem Hinh 3) va lugt dilu khien trugt dao dgng voi bien dp 1.28 (xem Hinh 2). Nlu luat dilu khiln dao dgng nhu thi nay se lam anh huong den thai gian dap ling ciia d6i tugng dugc dieu khien (vi du nhu dgng c a mgt chilu chang hgn). Trong khi do bo dilu khien trugt vai ham trugt kieu PID cho kit qua t6t hon voi dp vol l6 va thai gian xac lap be nhit. Dp vgt 16 la 0.02% va thai gian xac lap la 3.1s (xem Hinh 2). Dac biet mat trugt va luat dieu khiln khong co hien tugng dao dgng voi bien dg lan (xem Hinh 2 va Hinh 4). Ket qua mo phong thgc te cho thiy nlu su dung bg dieu khien tnrgt vai ham trugt kieu PID thi mat trugt it c6 hien tugng dao dgng hon so vcri bg dieu khien trugt kinh dien (xem Hinh 3).
Ben cEmh hai bg dilu khien noi tren, bg dieu khien PID ciing dugc dieu khien ding thai voi cung mgt d6i tugng la he tay may mgt bac tu do voi cac hang s6 ciia b6 dieu khien nay dirge chgn theo phuong phap thu sai. Ket qua m6 phong cung chi ra rang dap ung co dg vpt 16 tuong doi thap 1.8% (xem Hinh 2) va thai gian xac lap la 4.9s. Tat ca cac tieu chuin chat lugng ciia cac bg dilu khiln dugc trinh bay trong bang 2.
Response tracking desired signal
O w s h o o l of SMC 3 499%
Oierehoc* o t P I D 1 8 0 1 % Settling lir Settling l>r
s of S M C 5,
! of PID. 4 9:
S M C - • — - RID
P l D - b a a e d ^ M C
i I Oiershoot of PID-SMC 0 022557% Selllino t i m e of PID-ShflC. 3.l£
i U - , ' » - _
\U
0^ . — • 1 1 I
Time [sec]
Control signal
u-SMC
him t r „ ^ "'*" ™'"-SMC: .In hif„ ail„ khllo , r „ ^ „.pm: tu. hlSu dilu khilu Pli), u- riu-SMC: lui hi«u dieu khien tnrM vfti ham tru^t kiju PID )
T<^ chi Khoa hgc 2012:21a 30-36 Truong Dgi hoc Cdn Tho
-10
1
1
w
'
SMC PfD-SMC
10 15 20 25 30 35
Time [seel
ITinb 3 : M | t triFot cua bo d i l u khiln truwt (SMC) vd m^t trvQt ciia b$ dieu khiln t n r g t dira tren h i m t r u o t kilu PID (PID-SMC) Bang 2: Chfit hrqra% ciia c i c by d i l u khiln (SMC: Bo d i l u khiln truot, PID: B$ dieu khiln
PID, P I D - S M C : BO d i l u khiln t r u v t vAI ham t r u o t kieu PID)
L o g i d i l u k h i l n P g v g t \i ( % ) T h M g i a n xdc l a p (s) T h M g i a n t a n g (s) S M C
P I D P I D - S M C
3.5 1.8 0.02
5.8 4.9 3.1
1 2.7 1.7
5 KfiTLUAN
Bai bao ap dimg luat dilu khiln trugt vai ham trugt dugc thiet ke dua tren PID da loai bo dugc hi$n tugng dao dgng quaiih mat trugt va lu|t dilu khiln trugt. Giai thu^t nay dugc ap dyng de dieu kliien h6 tay may mgt bac tu do. Kit qua mo phong tren MATLAB da minh chiing dugc rSng: vai giai thuat nay, dap ung he tay may bam theo tin hieu mong mu6n voi dg vgt 16 khong dang k l 0.02%, sai so xac lap bang kh6ng va thai ^ian xac lap la 3.1s. Neu he tay may dugc dilu khien bang bg dieu khien trugt truyen thong tW ket qua m6 phong cho thay do vgt !6 tang len 3.5%, lu|it dieu khien co Men tugng dao dgng vai bien dg bdng 1.28 va m^t trugt dao dgng voi bien d6 0.14.
D I ket qua dieu khien toi uu ban niia khi sii dung bg dieu khien trugt vai ham trugt kilu PID. bai bao dugc d l nghi nhu sau: cac tham s6 cua ham trugt va bien dg ciia luat dilu khien trugt dugc uoc lugng true tuyIn dua tren mang noron (Ming-guang et al, 2005), (Hui Peng el al, 2003), (Huang Yijun et al, 2010) ho&c
Tgp chi Khoa hgc 2012:21a 30-36 Tru&ngDgi hgc CdnTha
logic ma. Voi each lam nay se lam cho bg dilu khiln se thich nghi hon voi mgi d6i tugng cung nhu mgi loai nhilu can thiep vao he th6ng.
TAI LIEU THAM K H A O
Ming-guang, Zhang Xing-gui, Wang Man-qiang,Liu, 2005. Adaptive PID Control Based on RBF Neural Network Identification Proceedings of the 17th IEEE International Conference on Tools with Artificial Intelligence (ICTAFOS), pp. 681-683 ZhixiongHou, QuntaiShen, HeqingLi, 2003. Nonlinear System Identification Based on
ANFIS. IEEE International Conference Neural Networks & Signal Processing Nanjing, China, December 14-17, pp. 510-512
Hui Peng, Tohru Ozaki, Valerie Haggan-Ozaki and Yukihiro Toyoda, 2003. A Parameter Optimaization Method for Radial basis Function Type Models. IEEE Transactions On Neural Network, Vol.14, No.2, pp. 432-438
Huang Yijun, Niu Wu, 2010. Application ofRBF Network in System Identification for Flight Control Systems. IEEE, pp. 67-69
Li Jian-jun, 2010. Application of self tuning PID controller based on RBF network. IEEE, pp.
544-546
Zhang Yuzeng, Song Jianxin, Song Shuhan, Yan Mingyin, 2010. Adaptive PID Speed Controller Based on RBF for Permanent Magnet Synchronous Motor System. IEEE pp 425-428
Yangmin Li, 2010. Adaptive Sliding Mode Control With Perturbation Estimation and PID liding Surface for Motion Tracking of a Piezo-Driven Micromanipulator. IEEE Transactions On Control Systems Technology, VOL. 18, NO. 4, pp.798-810.
