Nguyen Dinh Hoa Tap chi KHOA HOC & CONG NGHE 122(08): 161 - 166
THIET KE BO DIEU KfflEN LQR DICH CHUYEN CO CHON LOC CAC DlfeM c u e
Nguyen Dinh Hba Dgi hgc Bdch khoa Ha Noi TOM TAT:
Bai bao nay irinh bay mot phuong phap de thiet ke bo dieu khien LQR dich chuyen co chon lpc cac diem cue cho cac h? luyen t(nh diing, De dich chuyen niol hoac mot so diem cue khong mong muon ciia he ho sang ben trai mat phang phiie trong khi cdc diem cue con lai khong bi thay doi, cac ma tran trgng so ciia phiem ham toi uu LQR ducrc chon dua Ir6n cac vector rieng ben trai tuong ling vm cac gia tn rieng khong mong muon Sau do, ehiing toi chi ra mien xac dmh eiia cdc diem cue mdi. Mot vf du minh hoa duoc giiji thieu de mmh chung eac ket qua ly ihuyei da de xuat.
Tir khoa: Phuang phap gan diem cue; Phuang phap LQR; Dich chuyen diem c 6 chon loc: He tuyen tinh dimg.
DAT VAN DE
Xet mot he tuyen tinh dimg co mo hinh trang
\-Bu I G W'.ue R'". (0,1) Hien nay, cac phitong phap pho bien de thiet ke bo dieu khien gan diem cue cho cac he tuyen tinh dimg (0.1) bao gom phuong phap tnrc tiep, phuong phSp Ackcrman, phuong phap modal [I] Moi phuoTig phiip lai CO uu hoac nhugc dicm rieng, chang han phuang ptiap tnic tiep thl rai thu cong va khfing tong quai, Phuong phap Ackcrman co cong thitc long qiiSl nhung chi ap dung dirge ciio cac he CO inol dau vao. Phitong phap modal co the ap dung cho h? CO nhicu dau vao nhirng can mgl gia thiet quan trong la ma tran he thong c6 the bien doi thanh dang dirong cheo (diagonal) hoac khoi duong cheo (block- diagonal). Ngoai ra phuong phap modal con en mol dac diem nira la nd chi c6 the dich chuyen du<?c mgt so luong cac diem cue khong vugt qua rank(Z?). nghTa la khong vugt qua m . Day co the coi la nhitgc diem ma cung co ihc coi la iru diem vi trong nhieu truong hgp ta khong can thiet dich chuyen hel lat ca cac diem cue ciia he het,
Ngoai cac phuong phap tren, ta con cd the thiet ke bo dieu khien gan diem cue dua ircn phirong phap LQR Dicu nay cd th5 thuc hien duoc bang each tntoc het thiet ke bg dieu khien LQR de dich chuyen e6 chon loc mol so diem cue nhu trong cac tai lieu 12-51, sau do dya vao cac ket qua ay de lim cac nia Iran trpng so sao cho cac dieni cue khong mong muon dugc dich chuyen chinh xae den cac vi ui bict intoc, Tuy nhien. cac phirong phap trong [2-5] Ion
tai mgt so nhugc diem nhit can mot so gia ihiel ve cac ma Iran trong so Irong phiem ham toi iru LQR, Hon nu-a, cac ket qua trong [2-5] moi chi xei den viec thiet ke bg dieu khien LQR de dich chuyen cd chon Igc cac diem cue khong mong muon va xac dmh viing ma chiing chuyen den, chu chua xet den viec tim cac ma tran trgng so de co the dich chuyen cac diem cue mot each cliinh xac den cae vj trf mong muon
Dc tien cho vice trinh bay, sau day chiing loi sc dmh nghia cu ihe hai bai loan khae nhau tuong iing vui hai budc a ticn dc thiet kc mgt bo dieu kJiien gan diem cue LQR.
Bai loan Ihir nhai: Thi6t ka bg dicu khicn LQR dc dich chuyen co chon Igc mgt so diem cue khong mong muon eCia he (0.1) sang ben trai mat phang phiic,
Bai loan thu hai: Cho trude mot so diem cue mong mu6n ben Crai mat phang phire. thiet kc bg dicu khiijn LQR de dich chuyen cd chon loc cac diem cue khimg mong muon eua he (0.1) toi cae vi tri hiet truije do.
Bai bdo ndy di xudt moi .so kei qua mdi Irong viec su dung phuang pbdp LQR di thiit ki ho diiu khien dich chuyen co chon lpc cdc diem cut elm hi (0.1). til do tao CO so d£ giai bai toan thu hai hay noi each khac la tao eo so cho buoc tiep iheo de giai bai loan thict H bo digu khien LQR gan diem cue Cu the ban. bai bdo chi ra cckh chon cdc ma irdn Uong
\o diu phiim ham idi uu LQR sao tho LUL diem tin khang mong muon cua hi (0 1) dugc dich chuyin cd clion hi .sang hen Irdi mdl phdng phut, mu kbang tdn din cdc gid ihiei nbu o trong 12-51 Tiep do.
mien hen irdi mdt phdng phuc md cdc diem cut ci
SDT :0949823777 Email hoa nguycndinh@husl-edu vn
Nguyen Dinh Hoa Tap chi KHOA HOC & CONG N G H E 122(08): 1 6 1 - 1 6 6
llie duac dich chuyen din duac chi ra mot cdch tudng minh. Do khuon kho eiia bai bao nen liri giai eho bai toan thir hai khong dugc trinh bay o day ma se dugc gioi thieu Irong cac bai bao khac.
Cac phan liep theo ciia bai bao dirge trinh bay nhu sau. Phan II gioi ihieu eac ket qua cho bai toan thuan voi ba muc con cho phan dich chuyen cd chgn igc mgt diem cue thuc, mgt cap diem cue phiie lien hgp va mgt cap diem cue thyc. Phan HI gidi thieu mot vf du minh hoa Phan IV la ket luan va cac huong mo rong ctia bai bao.
Trong bai bao co sii dung mgt so ky ty nhu sau.
<T(A\ bieu thl eho tap cac gia tri rieng ciia ma tran A. Ky hieu gach duoi, chang han x , la de ehi eae dai luong vector, Ky hieu gach tren, vi du v la de chi gia trj phiic lien hgp eiia v. Ngoai ra. Re [a) chi phan thuc eiia mgt so phiic a.
THIET KE BO BIEU KHIEN Xet phiem ham muc tieU'
J = l{x'Qx + a''Ru)dt,
trong do Qe R" la ma Iran doi xtmg, ban xac dinh duong, i?, G R'" ia ma tran doi xiing, xac dinh duong, Nhu da biet trong ly thuyet dieu khien toi uu [1], [6], bo dieu khien toi uu phan hoi trang ihai LQR eho he (0.1) dugc tfnh bang;
u = -Fx, (1.2) voi F - R~'B' P trong do P ia nghiem ciia
phuong trinh dai so Riccati:
PA + A'P ~ PDR^'B'T + Q^0. (13) Ngoai ra, de phuang crinh (1 3) cd nghiem duy nh5t ihi hai gia thiet sau phai dugc thoa man:
AI: {A.B'J la dieu khien dugc A2. '•: [Q''',A\ la quan sat duge.
Van de dai ra eho bai toan thuan la cim each ihiet ke bg dieu khien (1,2) sao eho chi mgt so cac diem cue khong mong muon duac dieh chuyfin sang ben trai mat phang phuc trong khi cac di6m cue khac duge giir nguyen, De don gian, ehiing toi chi trinh bay ke\ qua cho cac trucmg hop dieh chuyen mgt
dilm cite thirc va dich chuySn hai diem cyc phiic lien hgp hoac hai di^m cite thuc. Cac ket qua nay co dil dugc t6ng quat hoa cho truong hgp dich chuyen mgt so lugng bat ki cac diem cyc va se dugc gioi thieu trong cac bai bao sau Ngoai ra, ta cung cd the lap lai nhieu lan phuong phap dieti chuyen mgt hoac hai diem cyc nay de dich chuyen tat ca cac diem cue khong mong muon.
fl. Dich chuyen en chon lgc mgt diem arc ihuc Gia sCr A e M la diem cyc thuc khong mong muon Clia he (0 1). Ggi v'' f 0^ la vector rieng ben trai Clia A tuong iing voi A Ta chgn ma tran trgng so Q nhu sau:
Q = v!'%v.. % £ 0- (1.4)
Dinh ly sau chi ro bieu thiic ciia bg dieu khien LQR va tap cac diem cyc ciia he kin.
Dinh ly 2.1. Voi cac gia ihiet A1-A2 va ma tran Q chon 6(1,4), bg dicu khien LQR co dang:
trong do A -I- ^X^ + r^q^
tl5)
Dong thcfi, tap cac diem cue ciia he kin la:
<>(A- BF) = [-.j\' +r,,]u[a{A]\{>]]. (1.7) Chimg minh- Vtii ma Cran Q chgn 6 (1.4), de thay P - vp^v', p^ > 0 la nghiem ciia phuang trinh Riccati (1,3) u-ong do p^ tfnh theo (1.6) thu dugc bang each thay P vao (1.3) va giai phuong U-inh bac hai. Han nijra, vtii cac gia ihiet A1-A2, phuong Crinh (1 3) CO nghiem duy nhai, Dieu nay co nghia la gia tri Clia P tinh a tren chinh la nghiem duy nhat ay.
Tir do ta chu dugc bg dieu khien LQR nhu cr (1,5) Gia sii fv ? A la mot gia tri rieng bat ky ciia A va w la vector rieng ben phai tuong ling voi no Ta cd i;'^w = n . d o vay:
[A-BF]W^[A- P^BR-'B'VV') w.
NguySn Dinh Hda Tap chi KHOA HOC & CONG NGHE 122(08): 161 - 166 Dieu nay chiing to w ciing la vector tieng ben phai QI^U nay din din 7^, = 5^ D I tranh mom ra, ta ky ciia ma Iran he kin va tucmg iing la gia tri neng c
hieu 9 = 9,1 = 9j,,
Dinh ly sau chi rd bieu thiic cua bo dicu khien LQR va lap cac diem cite cua he kfn.
Dinh ly 2.2. Vdi cac gia thiel A1-A2 va ma tran Q chgn 6 (1.4), bg dieu khien LQR cd dang-
Irong do P^ la nghiem ctia phuong trinh Riceati P , r ^ + f [ P ^ - P ^ R . P , + ( ? , = 0 , (1 10) i c3ch khac{(T(>l) {A}) <ZU[A-BF]. Ngoai
11'" [A - BF) = v^ [A - p^BR-'B'^vv'').
= i/A- p^r^i/.
= {>'- P,r,) v^,
Do vay, vf cung la vector rieng ben tiai cua ma tran he kin va tuong iing 1^ gia tri rieng - JA^ + r^q^ . Kcl hgp ca hai ket luan tren ve cac gia tri rieng ciia he kin, ta Ihu dugc (1.7). ^ ™i
Tir Dinh ly 2.1 ta thay cac diem cue eiia he kfn p = gom D — 1 diem cue giong ven he ho va chi cd duy [
nhat mgl diem cue khong mong muon ciia he ho la Hon nua. cac dilm cue khong mong mu6n cua he ho bi may doi, Hon nua, gia tr, cua diem cue moi cho ^^^^ ^-^^^ ^j^ .^ , ^ ^.^ - ^^- ^^^^ ,i„^
thay no nam ben trai mat phang phuc, va cu the hon , - - ' - la ben trai Clia diem — A tren Inic thyc
,,,^ + ,L'^ = 2[Re(A^) + Re (7;,,,/,,^) + f/r], H Dich (.huyen cii i hon lot mol < ap diciii, la phih lien . ,j , ,2 , .
Gia sir ( A , A ) la mdl cap diem cyc phiic khong + 17- - k J ' ' - k , J • mong muon ciia he (0 I) va I c ' , ? ' j ia cap vector ,, . , . . , , - .
° - \- - I '^ Irong khi cac dicm euc khac dugc giu nguyen tai vi rieng ben Irai lien hgp cua A lucmg irng voi chiing. irf
, « , = - B « - ' B ' U
Ta chgn ma Iran trong SO Q nhu sau:
Irong do Q, la ma Iran Hermitian, ban : duong. Gia su
-fe d
<i,r'iu^'^Khi dd,
f? ~ 9^21;' -^ Qyi— -^ '^1:'^'^ + <l..^'l'' Do Q v& (IfjVJl'^ +9i,I^'^ d^u '3 cac ma tran thyc nen il^^i^vf + q.^Jhf cung phai la ma Iran Ihyc
Chimg minh- Vdi ma Iran Q chon 6 ( 1 8 ) , dc :i,8) thay P ^\v H l ' d ^ r la nghiem eiia (1,3) irong
,. . do P, la nghiem ciia (1.10). Do cd cac gia thiel Al- A2 de dam bao phuong trinh Riceaii (1 3) cd nghiem duy nhat nen gia in do cua P chi'nh lit nghiem duy nhat ay.
Tiep iheo, chimg mmh tuong tu nhu o Dmh ly I.
ta CO the chi ra rang chi co hai dicm cue A.A eua he ho la bl thay doi. eijn cac dicm cue khat thi khong bi anh huong.
Nguyen Dinh Hoa Tap chi KHOA HOC & CONG N G H E 122(08): 161 - 166
]l''BR~'B^v v_'^BR''i v^BR-'B^v 7/BR~'l
Ta thay cac phan t^ tren duong cheo chfnh cua R.^ la lien hgp voi nhau nhung do P , la ma tran Hermitian nen chiing phai la eac gia Cri thyc, dieu nay dan den cac phan tii tren duong cheo ciia P^ phai bang nhau.
•>0- De don gian, ta ky hieu Iai R^ =
sii diem cue khSng mong muon ciia he ho dugc dich chuyen tdi cac gia tri p.^, //^, Tir ly thuyet dieu khien toi uu [6], ta da biet ii^.ii^,-JJ.^,-Ji^ la cac gia tn neng eiia H Noi each khae,
det fs/ - Hj - is — i-tAis — p., j(s + JlAis + Ji,,j,
= •'•"' - ( / J l +i'^i)-'^^ -^ t'^il4 (1 12)
d e t ( . ; - f f ) d e t ° ^
= d e l ( s / - f f ) Ngoai ra,
det ' - , '
= det(Qj)det(R,-(»;-r,)a;'(s/ + r;)).
Do do, [a cd each khac de ti'nh d e t ( s / - / / ] nhu sail:
d e t ( . i / - f f )
= det [Q,) det [R, - (si - F , ) Q;' (,,; + r J ) )
Tiip theo, thay cac bi6u thiic ciia Q^.Rj vao (1.13), ta ti'nh dtroc
det(s7 - ff) = s' - 2 [ R e ( A ' ) + R e ( ; ; , ? „ ) + ?r]s"
+ |A|' +2gr|A|' +2Re(i;,i;,jA')
(I 14) So sanh (1.13) va (I 14) Ca thu dugc (1.11). ^
Vi (5„,P, la cac ma Iran ban xac djnh ducmg va xae dinh duong, nen q' > Q^o^y.-.^' S r^J\.^ Do vay,
,r>j,„][,;,j>Re(i;,5„),
5''H'-l'iih>lW'-'*''fe'i.*')
Vi the, lir ket qua ciia Dmh ly 2, ta thu duoc mien xac dmh ciia cac diem cue mdi /(.|, //,, nhu sau:
11^^ +P^- > 2 R e ( A ' ) , fi^pl > A .
(115)
L. Dith chuyen td thon Igt hm diem cue ihuc Gia SU (A|, A, j la hai diem cue thue khong mong muon cua he (0,1) va yv^ .'U^) la cac vector neng ben trai ciia A tuong ijng voi (A|,A,,j, Ta chon ma Iran trong so Q nhu sau:
%\Q. (1 16)
trong do Qj la ma Iran doi xiing, ban xac dinh duong. Gia sir
Dinh ly sau chi ro bieu thiic ciia bo dieu khien LQR va tap cac diem cue cua he kin,
Dinh ly 2.3. Voi cac gia thiet A1-A2 va ma tran Q chon cr(l.)6), bgdieu khien LQR co dang;
Nguyan Dinh Hoa Tap chf KHOA HOC & CONG NGHE 122(08): 161 - 166
trong do Pj la nghiem cua phucmg trinh Riccati
''X,+TlP,-P,R,P,+Q,=0. (1.18)
voi r - n
= 0 \
U a
Hon niia, cac diem cyc khong mong mu6n ciia he ho duoc djch ehuyen toi cac gia tn p.^,p,^ dugc tinh bing:
+ (li'"22-''il)(5n'?22-gL)' (i-19) Irong khi eac diem cyc khac duoc giii nguyen tai vi Iri.
Chimg minh. Phan chiing minh eiia Dinh !y nay hoan loan tuong tu nhu cua Dinh Iy 2,2, nen chiing
t6i khong trinh bay lai o day. ^ Tir linh x^c djnh ban duong va xac dmh duong
cua Oj,yi'j, ta cd ngay q^^q.^^ > q'^^^.r^j-^., > rjj. Do vay, theo dmh ly Cauchy-Schvartz, la co:
'•„';,, + r.^,G,, ^ ^ ^ ' n ^ n / ^ A . ^K'lu^
Vilhg, cir(l i9)taihudugc;
^1' + PI > X' +A^
(1.20) Hai bat dang ihiic Irong (1,20) cho ta mien xac dinh Clia cac diem cue moi.
Chu y rang cdc ket qud cua Binh ly 2.1, Dinh Iy 2.2 va Djnh ly 2.3 ciing nhu miin xdc dinh cua cdc gid trj rieng mdi Id gidng v&i cdc kit qud &14], 15j.
Tuy nhiin, chung thu dirac md khong cdn them bat cu- gid titiet ndo, trong khi cdc kit qud ff 14], [5]
can CO mpt so gid Ihiel khdc vi cdc ma Iran trgng
VTDU MINH HOA
Xet mgt he tuyen tfnh dimg mo la boi (0.1) voi:
0 1 3
. B = 0 0 1 (1.21)
Cac gia tri neng ciia ma tran A la -0.3283.1 6641 + L823I. Do vay, he la khong an djnh vi CO hai gia tri rieng nam ben phai mat phang phiic
Tiep theo, su dung phuong phap de xuat trong bai bao, chiing coi chiet ke bg dieu khien LQR d^
dieh ehuyen hai diem cue 1 6641 ±1.823i sang ben irai mat phang phiie trong khi diem cue con lai dugc giii nguyen. Cae ma tran trong so dugc chgn nhu sau. Q^ = /.fl = 10 Ket qua mo phong tren hinh H] cho chay diem cyc -0.3283 khdng bi thay doi boi bo dieu khien LQR trong khi hai diem cue khac da duoc dich chuyen sang ben irai mat phang phiic thanh hai diem cyc phiic lien hgp on dmh.
Cuoi cung, voi Irang thai dku ctia he la |-1;2,3|, hinh H2 bieu dien ket qua mo phdng thu duge va cho thay he km tro thanh on dinh
HI Su phSn bo cSc diem cyc ciia he ha (ky hieu bm hinh vuong mau do •) va cua he kin thu duoc bm bo dieu
khien LQR de xuat {kf hieu bm hinh iron mau xanh •)
Nguyen Dinh Hoa Tap chi KHOA HOC & CONG NGHE 122(08)- I6i - 166
H2 Dap ling eac trang thai ciia h? vai bg dieu khien LQR dieh ehuyen c6 chgn loc hai diem cue
KETLUAN
Bai bao nay da de xuat mgt phuong phap de thiet ke bo dieu khien LQR dieh ehuyen co chon !oc cac diem eye khong mong muon ciia cac he tuyen tinh dirng. Y tudng chu dao ct day la viec chgn cac ma tran trong so ciia phiem ham muc Cieu LQR dua Cren cac vector rieng ben trai ciia ma Cran A tuong iing
voi eac gja tri rieng khong mong muon. Day ia budc CO so d^ thi^t ki bg di^u khiin gan diem cue bang phuong phap LQR. Cac kit qua cho viec thiit ke bp diiu khiin LQR gan diim cyc se dugc gid'i Ihieu trong eae bai bao tiep theo.
TAI LIEU THAM KHAO
II] Nguyen Doin Phuoc. Ly ihuySt di6u khien luySn Itnh.
NXB Khoa hoc va Ky ihuat. Ha Noi. 2009.
(2| N Kawasaki, F Shimemura, -Determining quadratic weighting matrices lo locate poles in a specied region".
Auiomatica. vol, 19, pp 557-560, 1983 PI N Kawasaki. E. Shimemura. J,-W, Shin, "On [he quadralic
weighis of an LQ-problem shifting only the specified poles", Proceedings of the Society of Insirumeni and Conirol Engineers, vol 25(11). pp. I24S-I250, 1989 HI F Kraus, V Kucera. "Linear quadralic and pole placcmeni
Iterative design". Proc, of European Control Conference.
1999-
[5] J Cigler, V Kucera, "Pole-by-pole shifting via a linear- quadratic regulation". Proc of the 17ih Inlemational Conference on Process Control. 2009
|6| B, D- O. Anderson. J B, Moore. Oplimal Control: Linear Quadratic Methods, Englewood Cliffs, NJ: Prentice Hall, 1990
S U M M A R Y
LQR CONTROLLER DESIGN FOR SELECTIVE POLE SHIFT
This paper proposes a method for designing stale feedback controllers based on LQR approach for LTI systems. To selectively shift one or some undesirable poles of the open-loop system to the open left half complex plane while not affecting the other poles, che weighting matrices in the LQR performance index are chosen based on the left eigenvectors of the system matrix associated with che undesirable poles Then we pome ouC the region in -which the shifted poles must he in. A numerical example is prcsenced co demonstrate the theoretical results
Keywords: Pole placement, LQR method. Selective pole shift, LTI systems.
Ngdy nhdn bdi. 12/5/2014. Ngdy phdn biin 22/5/2014 Ngdy duyil ddng- 25/8/2014 Phdn biin klwa hoc: TS Nguyin Van Chi - Tru&ng Dai hgc Ky thuai Cong nghiip - DHTN