NGHIEN CQU KHOA HOC
Md hlnh hoa va dieu khien tdi iTu tUthe
1^'ve tinh nhd dung ba thanh tiT life
Modeling and Attitude Control of Small Satellite Using Three Magnetic Torqrods
Pham ThiroTig Cat, Pham Minh Tuan, Nguyen Van Tinh
Vien Cong Nghe Thong Tin, Vien Khoa Hpc va Cong Nghe Viet Nam Email: ptcat&ioit.ac.vn
Ngutri phan bien: TS. Biii Trong Tuyen - Vien Cong nghe Vii Tru
Abstract:
The report presents ttie modeling methodology and proposed methods of optimal attitude control for small satellite in low orbit with preset stability degree. The problems of modeling attitude kinematic and dynamic systems of the satellite using 3 magnetic torqrods, magnetic field sensors, space noise torques, building control algoritlims for the attitude control of satellite with preset stability degree mentioned in the report.
The simulation results on MatLab are analyzed and 3D animation of satellite's attitude motions are given.
Tom tat:
Bao cao trinh bay phuong phap mo hinh hoa va de xuat phuong phap dieu khien toi mi tu the ve tinh nho tren quy dao thap vdi do du trii on dinh cho tnroc. Cac van de ve xay dimg mo hinh he dong hoc, dong lire hoc tu the ve tinh dung 3 thanh tu luc, mo hinh cac cam bien tir truang, mo hinh mo men nhieu moi trudng vii tru, xay dung thuat toan dieu khien tu the ve tinh vdi do du tru on dinh cho trudc duoc de cap trong bao cao.
Cac ket qua mo phong tren Matlab dupe phan tich va trinh bay vdi ehuong trinh mo ta chuyen dong ciia tu' the ve tinh trong khong gian 3D.
1. Mffdau
Dac diem chinh cua ve tinh nho la khoi luong cua no <500kg va bay d quy dao thSp tu 600km dSn 800 km xung quanh trai dat. Cac ve tinh nho thudng mang theo camera, cac thiet bi do ludng va truyen thong khac de phuc vu cho cac ung dung giam sat moi truang phong chdng thien tai, cuu ho cuu nan va thong tin ve tinh. Nghien cim, thiet kg va chS tao ve tinh nho bay
tren quy dao thap dang la van de thdi su d nudc ta hien nay.
Qua trinh hoat dong, ve tinh thudng phai thay doi tu the cua minh theo lenh nhan dupe tu tram mat dat hoac phai tu on djnh tu the khi bi cac md men nhieu vQ tru ngau nhien tac dong lam lech tu the hoat dong can thiet. Trong cac trudng hpp nay ve tinh can tinh dugc cac sai lech tu the, xac dinh dupe cac mo men dieu khien can thiet va tu dieu chinh lai tu the ciia minh qua viec dieu khien cac mo men cua co cau chap hanh.
Van de dieu khien tu the ve tinh la van de cot loi
trong suot thdi gian hoat dong cua ve tinh tren quy dao
bao dam cho ve tinh thuc hien duoc cac nhiem vu ciia
minh. De c6 the dieu khien duoc ve tinh ta can xay
dung dugc mo hinh cua cac dau do, co cau chap hanh,
he dong hoc va dong luc hgc cua ve tinh va tim phuang
phap dieu khien phu hgp. Co nhieu phuang phap dieu
khien tu the ve tinh truyen thong da dugc cong bo va
xuat ban trong thdi gian qua [3], [4], [5]. Tuy nhien do
ve tinh la mot doi tugng phi tuyen, c6 nhieu tham so
bat dinh va nhieu ngau nhien tac dong nen cac nghien
CUU ve cac phuang phap dieu khien hien dai van la chii
de dugc nhieu tac gia quan tam nghien cuu hien nay
[6], [7], [8], [9], [10]. Da s6 cac nghien ciiu hien nay
tap trung vao xay dung phuang phap dieu khien tu the
ve tinh bao dam he thdng dn dinh con cac chi tieu chat
lugng it dugc de cap den nhu qua trinh qua do, thdi
gian on dinh va dat cac chi tieu chat lugng tdi uu. Van
dS di6u khiSn tdi uu dugc ling dung nhieu d dieu khien
qua trinh cong nghe nhung trai lai d ITnh vuc dieu
khign tu the ve tinh con ft dugc cong bd. Ta c6 th6
didm qua mot s6 kSt qua ve ITnh vuc dieu khiSn tdi uu
ve tinh trong thdi gian qua. [12] Cong bd phuong phap
dilu khiin tu thk ve tinh toi uu da mue tieu tdi thiSu
hda thdi gian qua do va toi thieu hoa binh phuang sai
lech cac goc [13] Gidi thieu phuong phap dieu khien
phi tuyen toi uu tu thi ve tinh tdi thieu hda mo men trong khi bao dam toe do hdi tu tdi da. Tinh on dinh eua he thdng dugc chiing minh bang phuong phap dn dinh Lyapunov. [14] D6 xuat mot thuat toan dieu khien tdi uu thdi gian dn dinh tu thi ve tinh dung cac thanh tir luc.
Bao cao nay de xuat phuang phap dieu khien tdi uu tu the ve tinh nho tren quy dao thSp vdi do du trii dn dinh cho trudc. Cac van de ve xay dung md hinh he ddng hgc, ddng luc hgc tu the ve tinh dung 3 thanh tu luc, md hinh cac cam bien tu trudng, md hinh md men nhieu mdi trudng vu tru, xay dung thuat toan dieu khien tu the ve tinh tdi thieu hda chuan toan phuang vdi do du trQ dn dinh cho trudc dugc de cap trong bao cao. Cae ket qua md phdng kiem chiing dugc thuc hien tren Matlab Simulinh chay tren PC.
2. Xay dung mo hinh dieu khien tu* the ve tinh vol 3 thanh tir lure
Trong bao cao nay ta coi ve tinh la mot khdi cung bay quanh quy dao hinh trdn xung quanh trai dat. Tat nhien cac ve tinh thuc ra la mot he phuc tap cd nhieu phan he chuyen ddng cue bg va khdng hoan toan la mot khdi Cling. Vi du cac tam pin mat trdi la cac tam deo cd do dao ddng nhat dinh, nhien lieu cho cac dng phut la cac chat long tieu hao w . . . Tuy nhien, viec gia thiet ve tinh la mot vat ran la budc dau tien cho viec phan tich va xac dinh cac dac tinh chuyen ddng co ban ciia ve tinh. De khao sat chuyen ddng cua ve tinh, thdng thudng ngudi ta su dung hai he true toa do quy chieu.
Mot la he toa do quan tinh va hai he tga do gan tren ve tinh (vat ran). He tga do quan tinh la he tga do dung yen khdng chuyen ddng trong dd cac dinh luat chuyen ddng cua Newton dugc dinh nghia. Ngugc lai, he tga do gan cd dinh tren ve tinh va chuyen ddng dudi sy tac ddng ciia ngoai luc va md men len ve tinh. Ta chgn he tga do quan tinh la he ECI {EartJi Centered Intertial Frame) ^^ cd gdc tga do dugc dat d tam trai dat, vdi cac hudng x', y', z' trong do z' trung vdi true quay cua trai dat. He tga do ve tinh 3 {body frame) cd gdc tga do nam d tam khdi ve tinh va cac true x', y ' , z thudng dugc chgn tning vdi cac true chinh cua md men quan tinh cua ve tinh. He 3^ dugc gin cd dinh tren ve tinh.
Vdi viec chgn nhu tren, ta de dang md ta cac phuang trinh chuyen ddng ddng cua ve tinh. Phep quay xung quanh cac true x ' , y ' , z ' d u g c ggi la cac phep quay nghieng (j) (roll), chue ngoc 6 (pitch), va lai \ff (yaw).
Mac du hai he true tga dp quan tinh 3 va ve tinh 3^ ciing du de md ta chuyen dong cua ve tinh, nhung no khdng phan anh rd dugc thuc te la trong qua trinh bay quanh quy dao, mot sd md men tac ddng len ve tinh phu thudc vao tu the ve tinh. Tren quy dao ve tinh
bay quanh trai dat, hudng ciia gia tdc trgng trudng ludn chi tdi tam cua trai dat va tac ddng cua luc trpng trudng len ve tinh se bi thay ddi phu thudc vao tu the ciia ye tinh nhu the nao so vdi trudng trpng luc cua trai dat.
Do do ta dinh nghia them mpt he true tpa dp quy chieu thii 3 la he 3 chay tren qiiy dao ve tinh cd gdc tning vdi gdc tpa dp ciia he tga do ve tinh 3 , cd true z tning vdi hudng xuyen tam trai dat (nadir), true x°
tning vdi hudng tdc do cua ve tinh va y°la hudng vudng gdc vdi mat phang quy dao. He tga do 3^
thudng duge gpi la he tpa dp quy dao ve tinh (orbit frame) hay he tga dp LVLH (local vertical - local horizontal). Do he tpa dp 3 ludn cd true z° tning vdi hudng cua vec to gia tdc trgng trudng, nen neu biet tu the ciia ve tinh so vdi he tga do 3 thi ciing cd the tinh dugc tu the ve tinh so vdi he tga do quan tinh 3 . Can luu y la he true tga do 3^ la he tga do quay theo quy dao ve tinh nen khdng phai la he tga do quan tinh.
He ddng luc ciia ve tinh khdi ran dugc dan dat tii dinh luat Euler cd dang [3]:
I to-I-CO I (0 (1) trong do T la tdng md men tac ddng len ve tinh tinh trong 3 ; l'' =< f,f,I^ > Ikma. tran dudng cheo md men quan tinh cua ve tinh, <a la toe dp quay cua ve tinh trong he quan tinh 3 dugc md ta trong 3 . Ddi vdi ve tinh cd co cau chap hanh la 3 thanh tu luc thi tdng md men tae ddng len ve tinh T bao gdm md men dieu khien eiia 3 thanh tir lue T^ dugc dat tren 3 true cua khung tpa dp 3^ va tdng eac md men nhieu Xj tac dpng len ve tinh. Nhu vay he dpng luc cua ve tinh vdi thanh tu luc dugc md ta nhu sau ;
If) • K -w- D b ^ - t V
CO = - c o I CO + T^ + T_, ( 2 )
Ve tinh nhd hoat ddng trong mdi trudng vii tru bao bpc xung quanh trai dat nen chiu anh hudng cua tu trudng trai dat, gia toe trpng trudng, mat dp khi quyen, biic xa mat trdi va nhieu anh hiing cua eae hien tugng vii tru khac. Cac anh hudng nay tao nen cac md men nhiiu tac ddng len he dpng lue eua ve tinh lam sai lech dp dn dinh cua tu the ve tinh tren quy dao chuyen ddng. Md men nhieu trgng trudng T^ xuat hien do su thay ddi cua gia tdc trgng trudng trong khdng gian. Do luc trgng trudng thay ddi ty le nghieh vdi binh phuong ciia khoang each tinh tu tam trai dat, nen phan nao gin trai trai dat hon se bi hut vdi luc Idn hon. Xet v l miic do anh hudng thi md men nhilu trgng trudng cd gia tri gap nhieu lan cac md men nhieu khac nen ta cd thi x4p
xi tdng md men nhieu tuong duong vdi md men nhieu trgng trudng tinh dugc nhu sau:
: 3co"c'I c. (3)
trong do Oijj la tdc dp quay cua quy dao ve tinh va Cj la vec to cot thii 3 cua ma tran quay R,^^ tii khung 3 sang 3 . Phep quay tu 3 sang 3^ duge md ta theo trinh tu quay mpt gde (j) (Roll) xung quanh true z ° , rdi quay tiep mpt gdc 6 (pitch) xung quanh true y° (mdi) va cudi cung la quay mpt gdc \)/ (yaw) quanh true x°
(mdi). Ta ed ma tran quay theo cac gdc roll, pitch, yaw nhu sau:
R . . = R , ( ^ ) R , ( ^ ) R W
m
Can luu y la ma tran quay R^^ cua phep quay roll, pitch,
K
yaw cd diem suy bien d gdc 6 = ± —. Dieu nay se dan 2
den cae diem suy bien trong md hinh toan hpc ciia he dpng hpc ve tinh. Cae diem suy bien nay ed the tranh duge khi ta han che gdc quay 9. Tuy nhien, ve tinh cd the quay d tat ca eac hudng nen phuang phap md ta tu the theo 3 gde roll, pitch, yaw RPY khdng phai la phuong phap ly tudng. Mae dii vay eac gde quay nay van duge su dung do kha nang bieu dien hinh tugng ciia vat quay trong khdng gian hinh hpc de dang. Do phep quay khdng the hoan vi nen vdi thii tu quay cac gdc Euler khac nhau ta cd cac ma tran quay khac nhau.
Do cac chudi quay theo cac gdc RPY khdng phii hgp cho md ta he phuang trinh dong lire cua ve tinh nen ngudi ta tiep tuc tim cac dang md ta tu the khac. Gdc Euler (t> va true quay Euler a la mot dang dugc phat trien tren co sd dinh ly Euler md ta phep quay tong quat cua mot vat ran xung quanh mot diem co dinh co the mo ta bdng phep quay mot goc (P xung quanh mot true CO dinh co vec ta dan vi Id a . Vec to don vi a cd 3 thanh phan a = [a,, a,, a^ ] va ||a|| = 1. Nhu vay thay vi phai quay mdt chudi eae phep quay, ta chi can quay mot lan xung quanh mot true cd dinh. Ta cd the bieu dien ma tran quay R^. tu he tga do 3 sang 3 xung quanh true a mpt gdc 0. Ngoai eac phep bieu dien tu the ve tinh qua ma tran quay R hay gde va true quay Euler (<I>, a ) ngudi ta edn su dung phep md ta tu the quaternion. Quatemion la mpt phuong phap md ta tu the thong dung trong ky thuat ve tinh do nd khdng bi suy bien va ed the md ta gdc quay Idn. Quatemion gdm 4 thanh phan bao gdm mpt vec to 3x1 v v a mpt bien vd hudng q4 nhu sau:
q = [ v ' ? , J ; y = hq^qj (5)
Ta CO the bieu dien q dudi dang gdc va true Euler ( 0 , a ) nhu sau:
q = a. sin— i = 1,2,3 2
9, =cos — 2
(6)
Nhu vay quatemion bilu diln tu thi/phep quay tir he tpa dp nay sang he tpa do khac. Vdi 4 tham sd, quaternion khdng cd eae dilm suy biln, tuy nhien no ludn can thoa man rang budc don vi sau:
q q = ?, + ?2 = 1 (7)
Mdi quan he giiia tdc do quay eua ve tinh vdi tu the ve tinh dugc md ta qua he phuang trinh ddng hpc tu the ve tinh nhu sau:
1 q = - 2
T
- V
1
trong do
fi(co): -co
T
- c o CO
0)
0
n ( c o ) q (8)
(9) Thay the (3) vao (2) ta dugc md hinh ddng luc ciia ve tinh vdi 3 thanh tir lue nhu sau
CO = -co 1 CO + T -I- 3a) c,I c. (10) Phuang trinh (10) ed y nghia vat ly la md men T^ eua thanh tir luc tae dpng len ve tinh phai tang toe dugc md men quan tinh I ca , thang dugc md men ly tam/mo men coriolis -co'l'co va thang duge md men nhieu gia tdc trong trudng tae ddng len ve tinh. Md men dieu khien T^ ciia cac thanh tir luc dugc tinh theo cdng thiie:
ra x B (11)
trong dd B la vec to tir trudng ciia trai dat do dugc;
m la md men tir trudng cua thanh tir luc tinh trong 3 ^ . Do md men dieu khien la tich cd hudng cua md men tir trudng m ' vdi vee to tir trudng B ' nen no ludn cd chieu vudng gdc vdi tir trudng trai dat. Thdng thudng 3 thanh tir luc dugc dat song song vdi 3 true tpa do eua he 3 ^ . Liie nay ta cd vec to m ' dudi dang:
m
iN A i N A
y y ) '
iN A
(12)
Trong dd ij la ddng dien chay trong cudn day thanh tit luc i, Ni la sd vdng day cua thanh tir luc i; Ai la dien tich trung binh cua vdng day ciia thanh tir lue /. Nliu vay dl tao md men dieu khien eiia thanh tir luc ta ckn
cap ddng dien i^, i^^, i^, cho cudn day eiia cac thanh tir lire. Vdi cac thanh phSn cua tir trudng B' =[B', B' , B' ] ' md men dilu khiln eua 3 thanh tir luc T^ cdn ed the tinh theo cong thirc:
trong do
B_m^ -B m Bm_ - Bm^
B'm -B m
(13)
De tinh dugc md men dieu khien T^ ta can xac dinh dugc tir trudng trai dat B ' . Ta cd the dung cam biln tir trudng (magnetometer) de do dugc B ' . Ngoai ra de xac dinh duge he ddng luc tu the ve tinh ta can biet tdc do quay cua ve tinh ft). Hien nay cac ve tinh thudng sir dung eon quay (gyroscope) hoac khdi quan tinh IMU (Initial Measurement Unit) de do tdc do quay eua ve tinh. Dang tiec la cac phep do nay thudng bi can nhieu va phai dimg eae phuong phap Ipc ciing hoac eac phuang phap udc lugng trang thai phi tuyen nhu Ige Kalman md rdng, bg Ige mode trugt .. .de ed gia tri do chinh xac hon. Cac van de ve xac dinh tu the ve tinh tir cac tin hieu do can nhieu khdng nam trong khudn khd bao eao nay.
3. Dieu khien toi uu tu the ve tinh vol 3 thanh tu \\xc
3.1 Phircmg trinh trang thai he digu Ichiln tir the ve tinh vdi 3 thanh tir lire
Tren co sd tuyen tinh hoa he ddng luc ve tinh (10) xung quanh tu the ve tinh q = [0,0,0, l]^ dugc dan djt trong [1] taed :
ci={\-k)coq,-Akciyq+ (B'm -B'm]
1 ~ ~3^;^„'?2 + (•^°'"- ~^°'^,) 11
q, =-{\-k )co q^-k co^q-i- (B'm -B°m)
If ^ ' ' ' ''
(14) trong do
K = ^-^>K =^^K =~^. Dinh
nghia vec to trang thai '^ = {q^,q,,q,,q,,q^,q^'\ va vec to dilu khiln u = [/w^, w_, m_ J, ta nhan duge he phuong trinh trang thai tuyen tinh sau
x = A x ( t ) + B ( t ) u ( t ) (15)
0 -Ak^
0 0 0 0 - ( 1
1 0 0 0 0 -k^)Q)„
0 0 0 -3A:^,(u;
0 0
0 0 1 0 0 0
0 0 0 0 0 -kxf^l
0 ( 1 - ^ , 0 0 1 0
m
B(r) = 0 0 1 ,
B 2/, '
0
— B, 1 ,
— B'' 2/_ '
0 0 0 1 ,
B
B' 2/_ '
0 1 ,
B 2/,, '
0 0
2/ 2 /
(17)
Ta thay he (15) la he tuyen tinh nhung cd ma tran dau vao B(?)thay ddi do ve tinh quay quanh quy dao ed gia trj tir trudug do dugc thay ddi tuy theo vi tri cua ve tinh. Tir trudng nay thay ddi cd tinh tuan hoan khi ve tinh quay het mpt vdng quanh trai dat.
3.2 Thuat dieu Idiien tu- the ve tinh toi ini voi do du- trir on djnh A, cho tru-ffc
Bai toan dieu khien bam tdi uu toan phuang LQR (Linear Quadratic Regulator) la tim thuat dieu khien u (t) tdi uu sao cho trang thai x(t) eua he ddng luc ve tinh (15) bam theo trang thai mong mudn x^ (t) va tdi thieu hoa ham muc tieu sau :
m i n J ( u ) = - j [ x ' ^ Q i i + u^Pu]c?f (18)
trong do X = x_^ (t) - \(t) la vee to sai leeh tu thi ta muon triet tieu, Q, P la eac ma tran trgng sd, trong dd
Q la ma tran xac dinh khdng am, P la ma tran xac dinh duong. De cd the dieu khien dugc, he (15) phai cd tinh dieu khien duge hoan toan. Viec chgn cac ma tran Q, P anh hudng den chat lugng va dp dn dinh eua he kin. Khdng phai vdi mpi Q, P deu bao dam he kin dn dinh. De he kin dn dinh ta can chpn ma tran
Q = FF'^ vdi (A,F) la mpt cap quan sat dugc hoan toan [2]. Van de ta quan tam d day khdng chi bao dam he tdi uu ludn dn dinh, ma phai bao dam he dn dinh vdi miic du trii dn dinh X cho trudc. Dieu nay ed nghia la ta phai tim dugc tin hieu dieu khien tdi uu u'(/) sao cho he kin cd tat ca diem cue nam d ben trai mat phang phiie va each true ao tdi thieu mdt khoang each
X. Ta tim ldi giai eho bai toan tdi uu toan phuong
trong hai trudng hgp on dinh ve tinh xung quanh trang thai can b5ng x^(t)=0 va hdm sdt trang thai mong mudn Xj (t) = constant T^ 0 vdi dp du trii dn dinh X cho trude .
a) Bdi todn on dinh tw tfie toi uu x^ (t) = 0
Vdi Xj (0 = 0 ta cd x = x^ (0 - x(t) = -x(t). Luc nay ldi giai ciia bai toan dn dinh tdi uu toan phuang (18) cho ta tin hieu dieu khien phan hdi trang thai vdi tham sd thay doi theo thdi gian nhu sau [2] :
u (t) =-P-'B'(t)R(t)x(t)
(19) trong do R(t) la nghiem cua phuong trinh ma tran Riecati vi phan:
R(r) = R(0A + A''R(0 - R(0B(/)P-'B'' (t)K(t) + Q (20) Ldi giai tdi uu (19) cd dp phiic tap tinh toan cao do phai giai phuong trinh ma tran Riecati vi phan (20) d mdi vi tri cua ve tinh va phai udc lugng duge toan bg trang thai x(t). De khac phue khdi lugng tinh toan Idn ta tiep tuc trien khai mot sd gian ude. Do tir trudng trai dat ed tinh tuan hoan theo quy dao ve tinh, nen ta cd the sir dung gia tri trung binh cua vec to B' va liic nay phuang trinh trang thai cua he ddng lue ve tinh (15) trd thanh he hang
x = Ax(t) + Bu(t) * (21) Vdi ham mue tieu
min7(u) = -J[x''Qx-i-u''Pu](i^ (22) ta cd ldi giai tdi uu ma tran phan hdi trang thai hang:
u*(0 = - P ' B ' R X ( 0 (23)
trong dd R la nghiem ddi ximg, xae dinh duong eiia phuang trinh ma tran Riecati dai sd:
A'R + RA - RBP
B ' R+
Q= 0 m
Vdi viee chpn ma tran Q = FF'^ trong do cap (A,F) la mpt cap ma tran quan sat dugc hoan toan thi ta ludn cd cd ldi giai tdi uu (24) bao dam he dieu khien dn dinh tiem can lim x(t) = 0 .
De ed duge ldi giai tdi uu bao dam he ed do du trii dn dinh ta ap dung cac bude duge md ta trong [2] tren eo sd eac gia tri A, B, 1 cho trudc nhu sau:
• Chgn Q = FF'^ trong dd cap (A,F) la mdt cap ma tran quan sat dugc hoan toan.
• Chgn P„ = P^ > 0 la ma tran ddi xirng, xae dinh duong.
• Tim nghiem R ddi xiing xac dinh duong cua phuang trinh Riecati dai so siia ddi
(A'^ -^• /II)R + R(A + Al) - RBP;'B^R + Q„ = 0 (25)
• Tinh ma tran phan hdi tdi uu va tin hieu dieu khien tdi uu
K = P
B ' Ru (t) = -Kx(0
(26)
b) Bdi todn dieu kliien bdm trang tiiai toi mi:\j (t) = constant T^ 0 ;
Ldi giai eua bai toan dieu khien he ddng luc ve tinh (21) bam trang thai tdi uu x^(t} = constant ?i 0 theo ham muc tieu dang toan phuong (22) cho ta tin hieu dieu tdi uu dang dieu khien phan hdi trang thai vdi tham sd hang nhu sau [11]:
u'(0 = P"'B'Rx(0-K,x,
(27)trong do R la nghiem xac dinh duong cua phuang trinh ma tran Riecati dai sd
A ' R + RA-^RBP B R + Q = 0 (28) Va Kj la nghiem cua phuang trinh ma tran
[ A - BK^ ] x_, = 0 . Thuat dieu khien (27) se toi thieu hda ham mue tieu (22) va ed trang thai dimg bam theo gia tri trang thai mong mudn lim x(t) = X^.
t->co
De cd dugc ldi giai tdi uu bao dam he cd do du trii dn dinh X eho trudc, tuong tu nhu bai toan dn dinh tu the ve tinh ta trien khai eae budc tinh toan nhu sau:
• Chgn Q = FF"^ trong do cap (A, F) la mot cap ma tran quan sat dugc hoan toan.
• Chgn P_ = Pj > 0 la ma tran ddi xung, xac dinh duong.
• Tim nghiem R ddi xiing xac dinh duong cua phuong trinh Riecati dai sd
(A'^ + /II)R + R( A + II) - RBP;'B'^R + Q„ = 0 (29)
• Tinh ma tran K^ thoa man phuong trinh
[ A - B K J X „ = 0 (30)
• Tinh ma tran phan hii tdi uu
K = P"'B'R (31)
• Tinh tin hieu dieu khien
u'(0 = Kx(0-K,x„ (32)
4. Mo phong he dieu khiln t6i mi LQR tir the ve tinh vol 3 thanh tir lire
Md hinh he dilu khien tdi uu LQR tu thi ve tinh dimg 3 thanh tir lire duge md ta trong Hinh 1. Dau vao la tu the mong mudn cua ve tinh dugc cho dudi dang cac goc roll, pitch, yaw mong mudn, va tdc do quay mong mudn.
0 0 0 ,427.10"'°
0 1,057.10"'
0 -1,861.10"''
0 0 0 3,64.10"'°
0 5,283.10"
0 - 2 , 4 8 1 . 1 0
0 0
Md hiuli OM tiicii iihicu HI mdi Inrimg Nhieu ni<> men Nhieu inc
tryng u m m s men khac
M6 hinh d^nc h^c \'a d(mg lire ciia v^ linh voi 3
thanh \it l\ri:
Nhieu
1
1X ejm hitn huiVnt: lu
Iruanp
" T "
Nhitfu
Hinh 1: So dd he thdng dieu khien tdi uu tu the ve tinh dung 3 thanh tir lire.
Thuat dieu khien duge phat trien tren he ddng luc cua ve tinh dung quatemion do vay ta can chuyen ddi tin hieu Roll-Pitch-Yaw mong mudn sang dang quatemion.
Tren ea sd Igc cac tin hieu cua thiet bi do tir trudng, cam bien hudng tir trudng, va cam bien tdc dp quay gyro ma ta xac dinh dugc sai leeh ve tu the va toe dp.
Dua vao cac sai leeh nay may tinh tren ve tinh se tinh ra tin hieu dieu khien tdi uu de cap cho cac thanh tir luc tao ra md men dieu khien can thiet de quay ve tinh theo hudng sao eho tu the ve tinh tiep can den tu the mong mudn tdi uu nhat. Phuong phap dieu khien tdi uu toan phuang LQR vdi dp du trii dn dinh eho trudc tren co sd md hinh tuyen tinh hoa he dpng luc tu the ve tinh duge sir dung. Phan xae dinh tu the ve tinh su dung phuang phap quan sat theo bd Ipe Kalman md rpng nhung do khudn khd bai bao cd han se duge trinh bay ehi tiet trong mdt bao cao khac.
Ket qua mo pliong
Ve tinh md phdng la ve tinh nhd bay quanh trai d4t d do eao 600 km tuong duong vdi toe do quay C0o = - 0,0011 rad/s.
Ma tran md men quan tinh ciia ve tinh cd cac gia tri 200 0 0
kgrn^ • Cae tham sd ciia he 0 140 0
0 0 200
phuang trinh trang thai ve tinh dugc tinh ra nhu sau 0
1,21.10"' 0 0 0 0
1 0 0 0 0 8,25.10""
0 0 0 2,42.10"'*
0 0
0 0 1 0 0 0
0 0 0 0 0 -3,025.10"'
0 8,25.10
0 0 1 0
a) Mo phong bdi todn on dinh toi uu tu the ve tinh
Cae gia tri ihx vao Y_, = [ 0 , 0 , 0 ] ^ co,, = [ 0 , 0 , 0 ] \ X = [o 0 0 0 0 o ] ' . Cae gia trj ban dau x(0) = [o,i 0 0,1 0 0,3 o ] . Chgn eac ma tran P, Q cho phuong trinh Ricatti P = 10''l,^,, Q = 10^1^^^
trong dd 1 la ma tran dan vj. Giai phuang trinh Riecati
ta duge R va K
2,856.10"
1,629.10'- -4,591,10"
-2,699,10'"
-1,335.10'°
-6,202,10"
1,629,10'- 1,701,10"
2,021,10'"
1,92,10''' 8,377.10'- 4,62,10'-
-4,591,10"
2,021,10'°
7,871,10' 5,509.10'=
4,171,10'°
3,334,10'°
-2,699,10'"
1,92,10'-' 5,509,10'=
6,97,10"
1,782,10'"
1,049.10'"
-1,335,10'°
8,377,10'=
4,171,10'°
1,785,10'"
7,173,10'=
3,991.10'=
-6,202,10' 4,62,10'=
3,334,10'°
1,051,10'"
3,99,10'=
4,023,10'=
Kit qua mo phong khi dieu khien vdi do du tru on dinh X = \(Hinh2):
Ta thay cac sai leeh x(l),x(3) va x(5) khdng ed dp qua dieu chinh, sai leeh tTnh nam trong pham vi 2.10"', thdi gian qua dp la 300s.
-0.01 -0.02 -0.03 -0.04 -0.05 -0,06 -0,07
0,06|
0,05 0,04 0,03
—data1
•— data2 - data3
200 400 600 800 1000
a)Do thi cac sai leeh x(l),x(3) va x(5)
^ ^ • \
• 0 20 40 60 80 100 120 140 160 180 200
b)Dd thi cac sai leeh x(2), x(4) va x(6) Hinh 2 : Cac sai lech ve tu the va tdc dp khi dn dinh
tu the ve tinh vdi X = \
Bien thien ve tdc do quay cua ve tinh va tin hieu dilu khien tdi uu dugc md ta trong Hinh 3.
a) Do thi cac thanh phan dieu khien u
-0,02 -0,04 J),06 .0,08 -0,1
— O M E G A .
„-, OMEGA.
— O M E G A ,
b) Do thi cua w
Hinh 3: Bien thien ve tin hieu dieu khien toi uu va toe dp quay cua ve tinh khi /I = 1
So sanh vai tnrdng hgp dieu khien khong co dp du trii on dinh A = 0 thi cac sai lech x(I), x(3) va x(5) c6 dp qua dieu chinh la 0,16, sai lech tinh nam trong pham vi 4.10"^ , thai gian qua do la S.lO^s (Hinh 4)
0 ons
-0.1 0,15 -0,^
1 /
— error X - --errorX -
—error X
-
2 4 6 8 1
a) Do thi cac sai lech x(l), x(3) va x(5)
^xioj_
-error X ., error X.
4 - error X.,
b) Do thi cac sai lech x(2), x(4) va x(6) Hinh 4 : Cac sai lech ve tu the va toe do khi on dinh tu
the ve tinh vdi A = 0.
b) Dieu khien bdm LQR vai do du trii on dinh alio tnr&c
Gia trj tu the mong muon J 7Z 7t -K _36 18 12.
CO, = [O 0 O] . Tir day ta tinh dugc gia tri trang thai x_ =[-0,1337 0 0,08066 0 0,05445 o]'. Gia tri ban diu x(0) = [ 0 0 0 0 0 0]'''. Chpn cac ma tran P, Q cho phuang trinh Riecati P = lO'^lj^j,
Q = 10"*lj^j, trong do 1 la ma tran dem vi, Giai phuang trinh Riecati ta dugc ma tran R va K_, :
2 856,1*"
1,629,16=
-4,591,10"
-2,69910'"
-1,33510"
-6,20210"
' 2,261 656,5 0,2211
1,629,1(4^
1,701.10"
2,021,10'°
1,9210'"
8,377,10'=
4,6110'=
-7880 -4,591,10"
2,021,l(y°
7,871,10' 5,509,10'=
4,171,10""
3,334lrf°
-2,699,10'"
1,9210'"
5,509,10'=
6,97, iri' 1,78210'"
1,04910'"
-96,28 0 -2,188,10'' -2,796,10" 0
- 7 7 0 , 9 -984,8 0 -1,335,10'°
8,37710'=
4,17110'°
1,785,10'"
7,173.10'=
3,99110'=
2,89 8,024 0,2827
-6,20210' 4,6210'=
333410'°
1,051,10'"
3,99,10'=
4,02310'=
1541 4,476,10^
150,8
Ket qua mo phong khi dieu khien bam vai dp du tru on dinh /I = 1 dugc mo ta trong Hinh 5, Ta thay cac sai lech x(l), x(3) va x(5) khong co dp qua dieu chinh.
sai lech tinh nam trong pham vi 2,10"' gian qua dp la 200s.
, 3.10-\thm
""'^0 Toob 2000 3000 4000 5000 6000
a) Do thi cac sai lech x(l), x(3) va x(5)
0,15
—ex.
0 20 40 60 80 100
b) D6 thi cac sai lech x(2), x(4) va x(6)
Hinh 5 : Cac sai lech ve tu the va t6c dp khi 6n dinh tu thS ve tinh vdi A = \
Cac mo phong cho ta thiy phuang phap dilu khiln ve tinh vdi du tru 6n dinh cho trudc bao dam cac sai lech tu thi va tic do bi triet tieu vai sai so tinh rit nho va thdi gian xac lap nhanh, co the kiem soat dugc.
5. Ket luan
Bai bao giai thieu phuong phap xay dung mo hinh he thong dieu khien tu the ve tinh nho bay tren quy dao thap tir cac phuang phap mo ta tu the ve tinh, he dong hpc, cam bien, ca cau chap hanh, he dpng luc tu the ve tinh tai viec xay dung bai toan dieu khien toi uu toan phuong. Bao cao de xuat cac thuat toan dieu khiln tii uu tu the ve tinh voi dp du trii on dinh cho trudc cho ca hai trudng hgp on dinh tu the vi tinh toi uu va diku khien bam toi mi. Cac mo phong tren Matlab Simulink vai ve tinh nho bay tren quy dao 600 km quanh trai dit cho cao ket qua on dinh, dam bao dp chinh xac cao va phii hgp vai cac ket qua mong dgi cua ly thuylt. So vai cac phuang phap dieu khiln toi uu phi tuyin, phuang phap de xuat de dang thuc hien tren ve tinh do cac ma tran phan hoi Uang thai toi uu co the tinh trudc va nap vao bp nha cua may tinh tren khoang. Nhugc diem cua phuang phap de xuat la sir dung mo hinh dpng luc cua ve tinh da dugc tuyen tinh hoa nen phuong phap de xuat chi CO the cho ket qua tin cay trong vimg thay doi tu the nho xung quanh diem can bang.
Tai lieu tham khao
[1] Pham Thugng Cat, Pham Minh Tuin: "5ao cao nghien citu mot so van di xac dinh va diiu khien tu the vi tinh nho trin quy dao thap". Bao cao ket qua nghien cuu ve dieu khien tu the ve tinh, Phong cong nghe Tu dpng hoa, Vien Cong nghe thong tin 10/2010.
[2] Somlo Janos, Pham Thuong Cat: "Computer Aided Design of Linear and Nonlinear Control System". Academy Publishing House, Budapest Hungary 1983 (in Hugarian).
[3] Wiley J. Larson and James R. Wertz: ''Space Mission Analysis and Design." Kluwer Academic Publisher 2005.
[4] Scott A. Kowalchuck, 'Lnvestigation of nonlinear control strategies using GPS Simulator and Spacecraft attitute control simulator". Ph.D Thesis Virginia Polytechnic Institute and State University, 7/2007.
[5] Vladimir A. Chobotov,: " Spacecraft Attitude Dynamics and ControV . Krieger Publishing Company, Malabar Florida, 1991.
[6] Morten Pedersen Topland and Jan Tommy Gravdahl, "Nonlinear attitude control of the micro-satellite ESEO" , Proceedmgs of the 55th
Intemational Astronautical Congress 2004 - Vancouver, Canada , pp, l-ll.
[7] Pooya Sekhavat, Andrevv' Fleming and I, Michael Ross: "Time-Optimal Nonlinear Feedback Control for the NPSATl Spacecraft'\ Proceedings of the 2005 lEEE/ASME Intemational
Conference on Advanced Intelligent
Mechatronics Monterey, California, USA, 24-28 July, 2005, pp. 843-850.
[8] Dongeun Seo and Maruthi R. Akella: "High- Performance Spacecraft Adaptive Attitude- Tracking Control Through Attracting-Manifold Design", JOURNAL OF GUIDANCE,
CONTROL, AND DYNAMICS Vol. 31, No. 4, July-August 2008, pp. 884- 891.
[9] Aaron Dando: "Spacecraft Attitude Maneuvers using Composite Adaptive Control with Invariant Sliding Manifold", Proceedings of Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R.
China, December 16-18, 2009, pp. 4535- 4540.
[10] All Heydari, Seid H. Pourtakdoust: "Closed Loop Near Time Optimal Magnetic Attitude Control
Using Dynamic Weighted Neural NetworK\
Proceedings of 16th Mediterranean Conference on Control and A
[11] Ashish Tewari: "Modern Control Design with Matlab and Simulink', John Wiley & Son, Ltd , West Sussex England 2002.
[12] Zhang Fan, Shang Hua, Mu Chundi, Lu Yuchang:
"An Optimal Attitude Control of Small Satellite with Momentum Wheel and Magnetic Torqrods", Proceedings of the Q World Congress on
• Intelligent Control and Automation June 1&14, 2002, Shanghai, P.R.China, pp. 1395-1398.
[13] Nadjim M. HORRI, Philip L. Palmer, Mark R.
Robert: "Optimal Satellite Attitude Control: a Geometric Approach", Proceedings of the 2009 IEEE aerospace conference. Big Sky, Montana, 7- 14 march 2009, pp,l-ll,
[14] Pooya Sekhavat, Andrew Fleming and I. Michael Ross: "Time-Optimal Nonlinear Feedback Conh-olfor the NPSATl Spacecraft", Proceedings of the 2005 lEEE/ASME
Intemational Conference on Advanced Intelligent Mechatronics, Monterey, Califomia, USA, 24-28 July, 2005, pp,843-850.
