T^P CHi KHOA Hpc & C6NG NGHe CAC TRU'ONG B^l HQC KV THU-f T 4 S6 89 - 2012
M p T PHirONG P H A P G I A I l i C H TI^M CAN Dfe PHAN XfCH ON OJNH BEN VCTNG CHO H ^ T H 6 N G T U Y ^ N XfNH C 6 T H 6 N G S 6 BAT flJNH
AN ASYMPTOTIC ANALYTICAL METHOD FOR CHECKING ROBUST STABILITY OF LINEAR SYSTEM WITH PARAMETERS UNCERTAINTY Phgm Vdn Minh, Hodng Duy Khang Nguyin Thi Thdng
Trudng Dgi hgc Cdng Nghiip Hd Ngi Trudng Dgi hgc Bdch khoa Hd Ngi Nhgn ngdy 30 thdng 11 ndm 2010, chdp nhgn ddng ngdy 8 thdng 5 ndm 2012
T 6 M TAT
Vi$c kiim tra 6n djnh cda mdt h$ thing du<?c md td bdi md hinh tuyin tinh cd thdng s6 bit djnh ddn t6i viic kiim tra sg-thda mdn cOa cdc bit ding thde cd chdv thdng s6 d^ng:g(q)^, VqeQ. Trong dd q=(qi, q2..., qif Id vdc ta thdng s6 bit djnh xuit hi$n trong md hinh. Q Id tip thdng s6 bit djnh.Hdm g(q) dlipc thdnh l$p nhC/ mdt tidu chuSn d^i s6 eg thi. Khd khdn chd yiu Id vi$c kiim tra s y thda mdn chdt cda bit ding th(rc g(q)^ vdi VqeQ.
Bdo cdo ndy trinh bdy mdt phuong phdp phdn tich dn djnh cua h$ tuyin tinh cd thdng s6 bit dinh nhd vi$c du^ diiu kiin 6n djnh vi mdt d^ng ridng, quan trgng Id hdm g(q) cd d^ng da thuc.
If /
g(g) ^ S ' ^ ' O ^ / ' - ' ^ ('^^- ^™"£' *^** ^(^^ ^ " ^ ' ^ ^ " ' ' '^P "^^ ' ' ^ " c f t " ^ " ^ o " * ^ ''o^c Hurwitz vd t$pQ cd dang sidu hdp Nhd' ddng phuong phdp phu tuyin tinh ta tim duoc mdt tri cue tiiu non tidm cdn cua ham g(q) trong tdp Q ta cd thi biit duoc st/thda mdn chdt cua (1)
ABSTRACT
Stability of a linear system with parameter uncedainty can be transformed as some Inequalities in the forrn:g{q)>0,\/q&Q (A), here q is parameter uncertainty vector q=(qi. qa.., q j ^ , Q is uncertainty set Function g(q) can be etabilised with the aid of some concretealgebric stability critenon.The mam diffculty for checking problem (A) is the checking of satisfy inequalty (A) with all value of qeQ.
This arliclepresentedoneanatytic method to check the robust stability of linear system with parameter uncertainty. Using the Routh or Hurwitz stability criteria, the stability of the system is led to the form (A) and g(q) is supposed in the polynomic form.g(q) = '^d,Y\9j^ ^ 0 (V. the Q set is a box. Using the linear overboud one can find anasymptoticunder minimal value of function g(q) in the Q set so that the strictly satisfactlonto the inequalty (1) can be checked.
1. Gidl THIEU CHUNG thieu thdng tin va thudng chua bidt trudc. Khi khong the mo ta dugc cau true ciia phan bat Su dung md hinh bat dinh va dieu khifin
ben vijng cho ta khac phuc phan nao hau qua do dinh AS, ta co mo hinh bat dinh khong co cau 1- u - " u- u •' - „ - u- u u'-, A- u triic. Mo hinh dang nay CO the md ta dudi lech ve mo hinh, vi vay mo hmh bat dinh . *• u u- t-X h - u- .• u r/;i roi dugc quan tam nhieu trong ly thuyet dieu khien cdng tinh, nhan tinh, hoac chia tinh [6],[9],[ 12].
hien nay. Mo hinh bat dinh cua mpt he thong Viec nghien cuu he thong co md hinh bat thuc CO the md ta mot each tong quat bang mot djnh khdng cau triic thudng duoc tidn hanh tap md hinh dang: trong khdng gian H^, liic dd can xac dinh duac
S=rS ASl fi) S'^' ^^^ ^^ '''^^" '''^^ ^^^ '^i"'^ AS(s). Trudng hgp sai lech AS cd cau triic ta cd the thong sd Trong do So la mot mo hinh chuan ^^^ ^5 j,!^ ijinh^ ta co mo hinh bdt djnh thong (nominal model) dugc xay dung tir nhung ^6. Mo hinh tuySn tinh vdi thong s6 bit djnh co thong tin xac djnh, sai lech AS gay ra bdi su th^ viSt dudi dang phuong trinh vi phan cho
P CHl KHOA HOC & C6NG NGHf CAC TRlfONG D^I HQC K^ THU-^T * S6 89-2011 biin trpng thdi hay dpng hdm truydn. Tir md
hinh ta xdc djnh dtrgrc da thi>c d^c trung cim hf:
Ms,q):=J^aXq)s';qeQ (3) Trong dd q=(qi Qi qj •• q i f 'A vfc to thdng sfi bit djnh, gid thi^t thay d6i d^c 19p ttvng tjip thdng s6 QeR^. Tftp thdng s6 Q thudng cd dpng da difn, d^ic bift Id dpng h^p (box, hypercube) thudng du?c quan tdm nhi^u:
Q = \q\g^^q,iqj^;j = \..L (4) T$p Q cd thi djnh nghTa tnng quat hmi, v(
dy dpng (2) trong [21 ]. Tuy theo dpng cQa ai(q) ta cd thi phdn lopi cAu true bit djnh ciia da thOrc A(s.q)nhusau:
Da thu-c khodng (Interval polynominal) cd cdc gid trj a, biin thifn d$c l$p trong khodng: a,^a,^a, (5) Da thuc bat djnh tuyen tinh (linear uncertainty structure polynominal): cd a, phy thuf e Ujyen tfnh vdo q,.
Da thue b4t djnh multilinear: cd a,(q) Id ham multilinear eua q (tue la ton tpi d d ^ g (6) vdi my=I). Thyc chat la dpng don gian phi tuyen cua q.
Da thuc bit djnh phi tuyen: a,(q) Id hdm phi tuyen eua q. Ddc bift trudng h^p a,(q) Id ham da thuc nhieu bien ciia q (polynomic parameter uncertainty) duo'c quan tdm nhieu:
"M)^Y.".Y\<iT
(6)trong do a,v la cdc hdng so.
Da thuc (3) dirge ggi Id dn djnh ben vihig (robust stability) khi nd dn djnh vdi VqeQ.
Lien quan den dn djnh ben vdng ta cd cdc bai loan sau:
BTI: Kiem tra on djnh ben vdng
BT2: Xde djnh dg dy trd dn djnh (stability robustness margins).
BTJ: Tdng hgp bp dicu khien vdi dn djnh ben vijng.
Bdi bdo ndy chi di e§p tdi bdi todn kiim tra in djnh bin vOng.
2. PHU'ONG P H A P NGHlfcN C<S\} 6 N D|NH BfeN VO*NC CUA H$ TUVfeN XfNH B A T D | N H T H d N G S 6
Di nghidn cihi 6n djnh hf lifn tyc tuyin tfnh cd thdng s6 bit djnh ta c6 thi dung phuong phdp dpi s6, phuang phdp tin s6 (hinh hgc) hay phuang phdp H„. Phuong phdp H„ khdng tifn dung cho cdch tiip c^n thdng so [9]. Nhiiu lieu chudn dn djnh dya vdo da thirc d^c trung (3) cua hf. Phuang phap nghifn cOru 6n djnh bin vflng ciia da thiire A(s, q) dpng (3) cd Ihi phdn thdnh cdc nhdm sau:
1. MOt s6 phuang phdp dira diiu ki£n cin vd dii di in djnh bin vQng cho da ihuc (3) ve dieu kifn dn djnh cua m$t s6 hihi hpn da thuc Ak(s), k= I ..M khdng chua thdng sd t^t djnh q, do dd ta cd thi di ddng kiem tra sy thoa mSn chft diiu kifn dn djnh cua (3).
Trudng hgp da thiic khoang (3), (3) tifu chuan Kharitanov |S] dSn tdi xil dn djnh ciia M=4 da thiire Ak(s) dugc I$p tir cdc gid trj bien a, ciJa (S). Theo hirdng ciia Kharitanov nhieu cdng trinh | l . 3. 7. 10, 13. ...j dua ra dieu kifn dn djnh bin vOng cho mgi sd trufmg hgp don gidn khde.
Trong 11 ],{131 gidi thif u diiu kifn cdn vd du de dn djnh ciia da tiiirc bit djnh tuyen tinh vdi tpp Q Id da difn cd C cpnh. Difu kifn cdn vd dii dc .\(s, q) dn djnh bin vihig Id tdt ca C da thiic cpnh phai dn djnh. Moi da thuc cpnh dugc xdy dyng theo mgt cpnh ciia da difn Q. Dd Id da Ihuc chi chira m^t thdng sd. Da thuc mgt thdng so dugc dua vf da thue khoang vd ta Ipi ed the diing lieu chuan Kharitanov.
Dc kiem tra dn djnh ben vdng cua A(s, q) vdi eau true dpng miltilinear. dpng da thirc polynomic, dpng phi tuyen ngudi ta cd the diing dieu kifn dpng dpi sd hay dieu kien tdn so (hinh hgc).
2. Phuang phdp tan so: dua ve viee vg hg ddc tinh tan trong mdt phdng phiic (value set) A(io), q) vdi VqeQ va V(0€(0, cc) theo nguyen tdc lopi trir diem zero (zero exclusion principle) [9, 14. 19] dua dieu kien dn dinh ben vung vi vife xac dinh:
T*P CHf KHOA HQC & C6NG NGHf: CAC TRllftNG D^l HQC KV THU^T * S6 89 - 2012 a) T6n tpi mOt q e Q di A(s, q) dn djnh.
b) OgA(ico, q) vdi VqeQ vdVwe(0,a)) Tdp A(j(a, q) rdt phiie tpp, Ipi khd ed thi vg d§c tinh tdn vdi VqeQ vd Vtoe(0, oo) nen ngudi ta thudng tim mgt tpp bao lay nd (over bound region). Nhd nhOng ky thu§t khdc nhau [14, 19, 21...].
Trudng hgp multilinear uncertainty vdi Q dang hdp nhd mapping theorem [15, 21] ta cd thS xay dyng m^t miin phu l6i (convex hull) dya vdo cdc dinh eiia Q, nhung vi nguyen tae van phdi x^t vdi Vcoe(0, OJ).
Thye te kiem tra 6n djnh theo phuang phdp t ^ sd ta ehi cd thi xdy dyng mdt tpp gid tri (Value set) vdi mOt s6 h(hi hpn diim q''€Q vd a>''e(0, x) vi v§y rdt ed thi bd sdt nhQng diem md d dd dieu kifn dn djnh khdng dugc thoa man (xem vi dy trong [25]). Nhu vdy theo each ndy se ed tdn tai la: khdng kiim tra dir^c su thoa mdn chat ciia dieu kien dn djnh dang tan sd.
3. Phuang phdp dai sd:
Dieu kifn dn dinh dugc dua ve eac rang budc ed the kiem tra duge bang cac phuang phap dai sd.
Vdi cau tnie a,(q) phi tuyen phucmg phap don gian nhat la dua ve da thiie khoang sau khi xac dinh dugc o, {q) = min o, {q), 0,(9)-maxa,(5) cac a,(q) bien thien trong khoang a, <a, <a, ta lai cd the diing tieu chuan Kharitanov. Tuy vay each lam nay chi diing duge khi dam bao tim dugc a^ la global minimum va a, la global maximum ciia a^q) tren tap Q. Hon niia each lam nay chi cho dieu kien dii ciia dn dinh va do do lam thu hep lap controller cua bai toan thiet ke bo dieu khien
Diing tieu chuan Routh hoac Hurwitz ta dugc diiu kien can va dii ciia dn dinh dang:
gj^ig) >0;\/q^Q;k^]..M;M^n + \ (7) Trudng hgp bit dinh da thirc (6) thi gk(q) cung dugc dua ve dang da thiic ciia q:
MOt s6 cdng trinh diing hdm Lyapunov [4],[8],[24] di dugc m^t diiu kifn dd dpng dpi so (7), (S) ciia 6n djnh vdi M tuj thu^c vdo dpng hdm Lyapunov ciia hf xft. Nhu v|y bdi todn kiim tra sy in djnh cua (3), (4) dSn din kiim tra tinh ducmg (positivity test) ciia (8), (4).
Khd khdn ciia nhifm vy ndy Id xft (8) cd thod man vdi VqeQ {thod mSn ch^i) hay khdng?.
De kiim tra diiu kifn ducmg ciia (8) Galoff. J;
Zettler [17, 18] dua g(q) vi dangti hgp convex ciia da thirc Bemtein (da thiie (6) trong [18]) vd riit ra mft dieu kifn dii eho positivity eiia (8).
Cdch Idm ndy thu h?p Idp A(s, q) thod mSn dieu kifn on djnh bin vQng.
Nhu v$y de kiim tra dn djnh ben viing ciia A(s, q) (3) bdng mgt phuang phdp tan sd hay mOt phuong phdp dpi sd ta diu phai xet vdi VqeQ. Trfin thyc ti neu chi x6t vdi mgt s6 hOn hpn diem q^'eQ vd (a''e(0,00) kit qua nhdn dugc se khdng thoa man chgt dieu kifn dn dinh.
Mdt hudng khac la diing qui hoach toan [22, 23] de nghien ciiu on dinh ben viing nhung mdi cho nhiing kit qua budc ddu.
Bdi bao nay gidi thieu mdt phuong phap tiem can de kiem tra sir thod mdn chat diiu kien can va dii ciia dn djnh dang dai sd (8), (4).
Diing true tiep tieu chuan Routh hoac Hurwitz ta di den dieu kien can va du cho dn djnh ben vung ciia A(s, q) (3) vdi (4), (6) dudi dang (n+!) bdt ddng thiic dang (8).
De kiem tra su thda man ciia cac bat dang thirc dang (8), (4) ta cd the diing phuang phap phii tuyen tinh da dugc trinh bay trong [11].
Dieu kien (8) cd the viet dudi dang:
M=min[g(q)]>0, VqeQ (9) Thay cho viec diing tri eye tieu toan the (global minimum M) ta cd the dung mot tri cue tieu non tiem can M^N [11].
-M (10) M„^<M, limM„^
S.(9) = Zg.IlC'
(8)N la sd khoang chia tren mdi canh ciia hop (4), neu MUN>0 thi chac chan bit dang thirc (8) dugc thoa man nghTa la dieu kien on dinh cua da thirc dac trung (3) dugc thda mdn chat.
Trudng hgp cac he so ciia A(s, q) la ham phi tuyen ciia cac tham so bat dinh q ta co the phan cac he sd ai(q) theo chuoi luy thira
TAP Cllf KMOA MQC & C 6 N G N G I I C C A C T R U ' O N G BAI HQC Kf THUAT • SO S9 - 2012 (Taylor-Maclourant) l§n c(ln m^t gid Irj chu4n
(nominal value), vl chuSi IQy t h ^ cQng thu^c d^ng (6) n£n ta l^i dfing duvc ptiirong phdp plii^
tuyen tlnli d a gidi tlii$u 6 tr£n.
3 . C A C V l D V
Muc ndy gi(Sri thi$u cdc vt du i ^ g dung mOt s6 phuong phdp da thdo lu|n 6 tr£n, d^ xdl 6n djnh b^n vQng cua m^t so d a thCrc chihi th6ng s6 b i t dinh.
yfdff I: X6t s u 6n dinh cOa da thi>c vdi cdc h$
s6 id hdm tuy^n tinh cua thdng s6 b4t djnh q.
A(s,i() = (?, - 2 ? , + 2 ) + (i), + l ) j + {2i;, -q,+4)s
= o . ( , ) + a,(»)J + " : ( 9 ) s ' (>'D1-1) e = { ? | - 0 , 5 S i 7 , s 2 ; - 0 , 3 5 ? , £ 0 , 3 } ( V D l - 2 )
VI cdc hdm a,(q) Id cdc hdm tuy^n tinh cua q ngn ta c6 t h i dua vk dang:
A(s, q) - ( 2 + s W ) + q i ( l +2s')+qj(-2+s-s') -*o(s)+qi*,(s)+q,(|.,(s) ( V D l - 3 ) Sii dung dieu ici^n on dinh cua da thuc cgnh ta phdi xet 2^.2/2=4 da thiic canh Ai, A,, Aj,A.:
4, . n M + « ; i ' , W * » , f t ( i ) - ( l . 5 * j - 3 ^ ) * , , ( - 2 t s - j ' ) 4,-«,WtW,M*9;«',(')-<2.6t0,7»*4,3«')-n,(lt2.') 4.=n(j)+9,P,U)+9,Xj)-(1.4 + '.3i*3.V) + 9,{l + 2i') ( V D l J l ) De xet on dinh cdn dua moi da thiic canh vk dang ( V D l - 5 ) , trong do cdc he so chi chiia mpt thong so, vi du xil Ai:
A, =l,5+(l+qj)s+(3Kii)s'
= aio+ai,(qi)s+ai,(qj)s' ( V D I - 5 ) T i i ( V D I - 2 ) t a c 6 :
a;„=ai„=\,5: 0 , 7 < u „ < l , 3 ; 2 . 7 < < i | , 5 3 , 3 ( V D I - 6 ) De xet 6n djnh ciia Ai ( V D I - 5 ) vol h? so bat dinh (VDI-6) dang "khodng" ta dung phuong phap Kharitanov, lap cac da thiic Kharitanov cho Ai:
( V D l - 7 )
B ^ e tiiu c h u i n Routh cd 4 d a thiic ( V D I - 7 ) deu 6n dinh, tuong ti/ ta cOng c6 cdc da thiic Ai, Aj, A,, A. trong ( V D l - 4 ) In djnh.
Vdy A(s, q) ciia ( V D l - l ) 6n djnh b i n vOng trong t « p O ( V D I - 2 ) ] .
y{ dl/ 2: Ddng phirong phdp phu tuyin tinh 6i xdt 6n dinh ciia h$ thdng dd cho dr vf dy I vdi cdc d a thirc ( V D l - 1 ) , ( V D l - 2 ) . Bi cdc thdng s6 c6 gid tri khdng dm ta th^c hi$n ph^p ddi biin s6 n h u sau:
( V D l - 8 )
a, a.
"r
a;
+ a + a + a + o
j + fl 5 + a s + a s + a s' s^
s^
s'
= 1.5 + 0.75
= 1.5 + ] . 3 J
= 1.5 + 0.7.v
= 1.5 + 1.35 + 3.35'
*2Js' + 2 . 7 r + 3 . 3 J '
q\ = 9 , + 0 , 5 - > 9 | =ci[ - 0 , 5
? , = ? , + 0 , 3 - * < j , = < , , - 0 , 3 Vdi ( V D I - 8 ) da thirc ( V D I - I ) trd thdnh:
i(s.,)=(2.i+,;-2,;)+(,,+o.7)5+(3.3t2,;-,;).' . o , ( ? ' | + o , ( ? ) « - f a , ( » ) j ' ( I D l - 9 ) Vdi khodng b i i n t h i ^ cua cdc thdng s i bat dinh mdi trong tdp Q Id;
Q' = y\0^q,i2,S: D S j ; £ 0 , 6 ) ( V D l - l 0 )
Bdi todn xet dn dinh cua ( V D I - t ) va ( V D l - 2 ) tuong duong vdi bdi todn xdt dn djnh cua ( V D l - 9 ) va (VDI-IO). Diing tieu chudn Routh d l cho da thirc ( V D I - 9 ) Id da thiic dn dinh ta phai cd:
S,(9) = <'.(«) = ( 2 . l + 9 ; - 2 9 , ) > 0 '
« ! ( 9 ' ) = a ; ( ? ) = (0,7 + ? ; ) > 0 S,(9) = o,(9) = (3,3 + 2?;-(;,)>0
Vq eQ ( r a i - l l ) DSt M i = m i n ( g i ( q ) ] , M!-min[g,(q')], M)=rnin[g,(q )j. d l h? d n dinh vdi Vq e Q cac bat ddng thiic sau phdi d u o c thoa mdn:
A/, = m i n [ g , ( , " ) ] > 0 : M , = m i n [ g , ( ? ' ) ] > D ; l M, =min[j,(?)]>0; VqeQ J
( V D l - 1 2 ) 6 day g i ( q ) , g . ( q ) , g3(q) la cac hdm tuyen tinh cua q nen khong gian Y (khdng gian phii trong [ 11 ] vd khdng gian Q triing nhau nen ta CO the de dang tim duoc cue tieu toan t h i cua ( V D l - 1 2 ) nhd phuong phap qui hoach tuyin tinh:
Ti^P CHl KHOA HQC & CdNG NGH$ CAC T R U 6 N G D ^ I HQC KV T H U ^ T * S6 89 - 2012 A/, = m i n ( 2 , I + ? ; - 2 ? ; ) = 2,1 + 1.0-2.0,6 = 0 , 9 > 0
M j = min(0,7 + q^) = 0,7 +1.0 = 0,7 > 0 M j = m i n ( 3 , 3 + 2 q ; - 9 ; ) = 2 . 0 - 1 . 0 , 6 + 3,3 = 2 . 7 > 0
VI cdc gid tri M | , Mj, M3 d^u duung nSn eac bdt ddng thiie ( V D l - l I) d^u ducmg do dd da thde A(s, q) eiia ( V D l - l ) 6n dinh b^n viing trong t§p thdng s6 Q (VDl-2). O ddy trudng hgp n=2, L=2 vdi da thiic bdt djnh tuyin tfnh p h u a n g phdp phu tuyin tinh cho k i t qud nhanh hon phucmg phdp da thiie c?nh. Tuy v^y trudng hgp n>3 thi diiu ki?n dn djnh theo Routh ho$e Hurwitz d i n tdi (8) n l u dimg phucmg phdp phii tuyin tinh cung duge nhung sg phde t^ip hon phuong phdp xir dung d a thire e ^ h .
Vl' df4 3: X&t 6n djnh eda da thure d (VD3-1) va (VD3-2) sau:
&(s,q) = {\0-3q,) + {5 + q,-2q,)s + [4 + g}-q,ql)s'
= a„{q) + a,{q)s+a,{q)s' (I'DS-I)
f U=q;=\<q,<3 = q;=0,;
[ \a,=q;=0<q,<2 = q;=/3, (VD3-2) Trong dd eac h e sd ak(q) cua da thire ( V D 3 - I ) , (VD3-2) khdng co dang tuyin tinh nen ta khdng the dp dung phuang phap d a thiic canh. Trong trudng hgp nay ta su dung phuang phap phii tuyin tinh, tir dieu kien can va dii theo tieu chuan Routh ta phai cd:
' & ( 9 ) = a<,(9) = ( " 0 - 3 ? . ) > 0 ( r a 3 - 3 ) g,(9) = o,(?) = (5 + g , + 2 9 , ) > 0 ( r a 3 - 4 ) .ft (?) = a, (?) = (4 + qf - 2q,ql) > 0 iVD3 - 5)
V ? e 0
Ddt M|-min[g_i(q)], M2=min[g2(q)], M3=min[g3(q)], d l he dn djnh vdi V q e Q cac bat d i n g thiic sau phai dugc thda man:
= mm[g,(<j)] = iiiina„(9) = m i n ( l 0 - 3 ^ , ) > 0 (roi-b) , - n i i n [ g , ( 9 ) ] = minfl,(9) = mm(5 + 9 , + 2 f l , ) > 0 (KD3-7) , = mm[«,(9)l = nima,(-7) = mm(4 + , 7 , ' - 2 9 , 7 j ) > 0 ( r a 3 - 8 )
Cac ham gi, gi la cac hdm tuyen tinh ciia q nen ta cd thi de dang tinh dugc cgc tri theo phuang phap qui hoach tuyen tinh, ket qua tinh d u a c nhu sau:
M , = 1 0 - 3 9 ; = I O - 3 . 3 = I > 0 ; M j = 5 + Iq; - 2 g ; = 5 +1.0 - 2.2 = 2 > 0
(VD3-9) Diiu ki^n (VD3-3). (VD3-4) dd duge thda mdn edn diiu ki§n (VD3-5) do cd d^ng da thiix: (8) khdng dp dvng dugc phuang phdp qui hogch tuyin tfnh nhung cd t h i dp dyng phuang phdp phu tuyin tinh [11], diiu ki^n (VD3-5) ed d?ng(S5):
. , ] / . . :
g,(9) = ''!{9) = Ss»n«7"'
= (4 + ,"-2,,,J) (^3-10)
Vdi:gja=4, g3i=l,g)!=-2,
V o - l , Y , - ( q i ) ' , Y , - q , ( q , ) ' (VD3-1I) Khi dd (VD3-10) dugc dua ve dang tuyen tinh ddi vdi Yi vd Y2 nhu sau:
ny)'f.g,.y,'g„y.+g,j. +g,M
C^{Y\Y,=Y,(q); qeQ] (W)3-12)
Vd Mj duoc xac dinh theo (VD3-13) De dat do chinh xac ta chia mdt canh ciia Q thanh N phdn bang nhau, khi dd cue tieu todn t h i chinh la cue tieu trong tdt cd cdc tap con cua Q " « :
M,ii„ = mm| ZSJ.*^ = 74? '^^"tllT"
(VD3-I3) M„ = min MZ' = min [ 4 + Y,""" - 2 n ' * " " ' ]
Chia mdi canh ciia Q lam N phdn deu nhau ta tinh duac M,u\, cu the la:
l ) X 6 t N - l , k , - l , k j = l :
I^IL'"LgLr"" (VD3-14)
I " , ' " " ' " - ( « " ) ' - I ' - L I T " " = ( » , " ) ' - i " - » yr"" 'iKiif-Hof =0. i ; - " " . « , ' ( « ; ) ' .3(2)'=12 Ta tinh dugc mot cue tieu non Msuitheo (VD3- I4)neu:g3, >0->?',*'*" =}•'-'*•*=;
TAP CHl KHOA Hpc i C6NG NCHf CAC TBITONG DAI HpC K* THUAT » S6 89-2012 g„ <0->r,*'" - >?*'"", ta tinh duoc ctrc tiiu + Tuong Hr ling vdi N-2, k,-l, kj-2, ta tinh non; M"«-M,i-4+l.1-2.12 -19 ''"d':: Wjjj^-ll
2)X«N-2.k|-l;2,k2-l;2liicndyl4pQdiiOc + Vdi N-2, k|-2,k,-l,ta tinh duoc: M',l,=2 chia ra thdnh 4 I4p con Q"", cu thi Id Q'\ Q",
0^', Q". Diing phirong phdp phii tuyin tInh + VdiN" 2, k|"2, k,-2,taduoc: d / 2 i = - l 6 [illxdtchotimgtftpcon: .^ j .- ,. > .-i . * .i.i IA
^ ' * "^ Ta xdc d)nh duoc eye lilu todn thi Id:
+ U'ng vdi N»2. ki"l, k:-l, trudc titn ta phdi ^^ ^^ ^^ ^ xdc djnh tdp con O" nhu sau: *',..- = min(W,.,,W,., ,*/,„,M„,)
„ „ , , =min(U-ll; 2;-l6) = - l 6
<>''=o,'=(z,+(*l-ll-'-^=l + (l-l)=—-=1
' 2 2 Do diiu ki4nM,.j>0 khdng thda mdn nAl
•••• „' „ . . l r i i . . n i l A - ° i 1 ^ , ' - ' 5 da thiic (VD3-I)vd diiu kiJn(VD3-2) khdng a, =,, = a , + [ ( * U I ) - l ] - ^ - H - l — . 2 ,haa mdn In djnh bin vlhlg.
i),"=<i;=a,+(»2-l)^l^^ = 0 + 0 ^ ^ = 0 Theo phuong phdp ndy thi Mjjg sj tiln 2 2 g^P (ti$m cdn) din M, nhung ludn thdanidn (/"•'=o'=a i l H 2 i l l l / ' ' ° ' Ol l^~° -1 M!.N<M,. Ta sS dimg lai d N dii Idn dl xdc
' ' ' ' ' 2 2 dinh xem G,(q)cd duong khdng.
Vdy lacd duoc tdp con: 4. KtT LU^N
p" ^Ll"'" ° ' ' ' °'^'i" ^2 = 9,"" = A" 1 • Bdo cdo ndy cho ta mOt cdch nhin tdng I ai'=(/!'= 1 SflJ'52-(/!"''=)3j'J quan vl cdc phirong phdp xet su dn djnh ben
vihig cua h$ tuyen tinh c6 thdng si bdt dinh.
Nhu d (VD3-11) „. ddt: Yo-I, Y|-(qi)', Bdo cdo cQng gidi thi^u mpt phuong phdp ti|m y, = q, (q,)' ta xdc dinh duoc: '^^ ^^ "B*"'*" 'V" *" '"!"'' "="" ''' ''""8 <^
tinh vdi cdc h^ sd cda da thuc ddc trung d^ng da y^""'=);•'".(,;•)'=i'=i. r;'"" = y;-"~{q!)' -z-=4 thiic ciia thdng sd bdt dinh nho dp dyng phuong r -"-r-=,•(,;)'=2(0)'=0 rr"' = r;-"g'M)'-2iU'-2 P'^^P P^^" ^"^^^ linh, nhd dd md kiim tra dugc
^'^'" ^' •' ' ''•^'"i sy thod mdn ch^t ttieu kifn cSn vd du cho s\/
Theo (VD3-I4) ta tinh dugc: 5„ jj^h.
f^ll =8v,+ S},h + gnYi = 4 +1 .y, - 2.^2
= 4 + 1 .>;'-"' - 2.rj*"" = 4 +1 - 2.2 = I
TAI LIfll THAM KHAO
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