Phân tích tổng quát những dấu hiệu của dạng Jordan ở hệ động học đa chiều

PHAN TICH TONG QUAT NHUNG DAU HIEU CUA DANG
JORDAN Of HE DONG HOC DA CHIEU
Phan Nguyin Hdi
Hgc vien ky thudt qudn sir
Tdm tdt:
Di xudt phuong phdp gidi tich hiiu qua cho viec nghien ciru nhirng hi ddng hgc da chiiu
vdi muc dich tim kiim nhirng hi con cd dgng Jordan. Xdy dung md hinh todn hgc vd thugt
todn ddnh gid su tdn tgi nhung ddu hi?u cua dgng Jordan a hi da chiiu, tren ca sd dd cd thi
thuc hiin dugrc viec phdn tdch chung thdnh cdc hi con cd dgng Jordan.
Abstract:
An efficient analytical method of studying multidimensional systems with a purpose of extracting controllable sub-systems of the Jordan forms is presented. A mathematical model and
algorithms for estimation of the presence of "Jordan" signs in controllable multidimensional
systems on the basis of which a decomposition of these multidimensional systems into Jordan
sub-systems lias been built.
L MOf DAU
Trong ly thuyet dieu khien, dan^ Jordan (Jordan form) Id mgt dang cdu tnic cua cdc he
ddng hgc mdt ddu vdo [1,2], dang cdu tnic nay cd vai trd rat Idn trong ITnh vyc tdng hgp ludt
dieu khien cho cdc he thdng ky thudt. Nhirng he cd dang ndy Id nhiing he dieu khien dugc, va
ludt dilu khiln cho chiing ludn de dang tdng hgp dugc [1,5,6]. Dang Jordan, vl mat md hinh
todn hgc, bilu diln mdt ldp vd ban cdc he dgng hgc, mat khdc trong thyc te ciing cd rdt nhieu
he thdng ky thudt cd dang ndy. Tir nhung vai trd to Idn tren, viec nghien ciiu tim hieu sdu
them nhdm muc dfch phdt triln, tdng qudt hda ly thuylt vl dang Jordan cd y nghia rdt quan
ygng. Mdt hudng phdt triln cd thi thdy n^ay dugc Id vdn dl xay dyng ly thuylt ve dang Jordan trong tmdng hgp he ddng hgc cd nhieu ddu vdo. Mdt each tiep can de phyc vy cho viec
gidi quyet vdn dk ndy Id xdy dyng khdi niem vl he con dang Jordan, cdch lira chgn ra chiing tir
he nhilu ddu vdo va nghien ciiru nhung khd nang phan tdch mdt he nhieu ddu vdo ra nhiing he
con cd dang Jordan.
II. DAT VAN DE.
Cho he da chilu bdc n vdi m ddu vdo [mj^= f{x^,x^_,...,x„,Ui,u,,...,u„,)
d day x^,x,,...,x„

Id cdc biln trang thdi, u^,u,,...,u^

V/ = i ^ , j = \^-^f{0)

= 0,

d-f{o)./du]=0,

, / = l,/i.

(1)

Id cdc tfn hieu dilu khien,

cdc ham / , ( o ) khd vi theo cdc biln

cua minh.
Cdn tim kilm khd ndng bilu diln he (I) dudi dang tdng hda cua cdc he con cd dang Jordan cua he mdt ddu vdo [1,2], cd nghia Id cd dang sau.
4 =./; (•^l;'^2v'K X(,„j.) , /• = lUj - 1 ;
4jj=fnMj^''2j^^
(2)

Xnj)+gMj^X2j,...,X„j)-Uj(x);

_
j = lm.

697

d day n^ la bac ciia he con, .\ .^ ,.v,^ , . . . , A ; ^ ,U^ dugc ggi la cac biln trang thai va tfn hieu
dieu khiln ciia nd. ctic biln trang thai vii tfn hieu dieu khien khac ciia he ban dau (I) dirge coi
la cdc tac dgng ngoai, V/ < n^ -^ df^ (o)/3.Vj^^|j^ ^ 0
NIU he (1) cd mdt dau vao (m = [) va dugc bieu dien bdi he phuong trinh vi phdn co
dang Jordan, thi mo hinh loan hgc ciia nd se cd dang
.i^=./;(.v,,.v,,K.v,,) , / = l,/;-l ; j8f, = f,{x,,x„K
x„)-i-g{x,,x,,...,x„)-u{x),
(3)
d day V/ < n ^ df (o)/3.v^^., 9^ 0 va v, dugc ggi Id bien trang thdi dau tien.
Trong bai toan ndy, mo hinh toan hgc neu ra ben tren cho he con dang Jordan (2) bieu
dien mgt Idp he con rat rgng va md hinh nay chua phdn anh dugc vai trd vd phan tfch dugc sir
dnh hudng cua cac bien trang thai khac, cung nhu ciia cdc dieu khien khac, vi vay de giai bai
toan tmdc het phdi chfnh xdc hda khai niem ciia he con cd dang Jordan cua he (1).
HI. KHAI NIEM HE CON DANG JORDAN
Trong [7] tac gia da xac djnh vd chirng minh dugc ddu hieu hay tieu chudn cd dang Jordan ciia he dgng hgc mgt ddu vdo, ddu hieu ndy nhu sau.
Gid sir co he bac n vdi mgt tfn hieu dieu khien u(^x) dugc cho dudi dang he phuang
trinh vi phan cua cac bien trang thai theo mgt thir tu tiiy y
j8f.= Fi(x,u),i = l,n ; x = {x^,x,,...,x„)
(4)
He (4) cd dang Jordan [1,2] vdi biln trang thdi ddu tien x^ khi vd chi khi dilu kien sau
duac thda man:
ir' g arg

^cTx^'
—
^0,
au V ^'" J

, / = 0, 1, ... ;

d day ic'' Id dai hdm bdc / ciia u , arg

(5)

U-^x, ^

in-

Id ky hieu tap cdc bien ciia

dt n-\ ;
dao ham cdc bdc ciia ;c^ dugc tfnh theo cdng thirc de quy

dx,

dx,

dt = Ff.\x,u) ; — - =

^

a

>

—

con

ti' ' , 1 = 2 , ...

j^jI'^Q,.,

trong dd i2,_i la tap cdc biln cua

dt n-\

d

X

(Q

OU

=aro

dt'-'
dt'-^
Tren co sd tieu chudn (5) khdi niem cua cdc he con dang Jordan dugc djnh nghTa nhu sau.
Dinh nghia 1. He con dang Jordan bdc n^ vai diku khiln Uj cua he (I) cd dang
•*^.l ~ //;.! [XpA^ X^,,

K j;

•*?.: ~ . V : ( - ^ ; . i ' -^'p.;. x^j,

X^ff.n^ ~ fl>.n^\Xp.\,

698

Kj;

Jf,,.2. K ^/i.«,,

(6)

"y>

Kj


0 day x^^,Xp._,...,x^^^^ 6

(A-,,A-,,...,J:„)

dugc coi la cdc bien tr^ng thai ciia he con nay.

Trong he con (6) cd the co cac bien trang thai ciing nhu dieu khien khac ciia he (1), nhung
nhiing dai lugng nay dugc coi la cdc tham so tac dgng tir ben ngodi.
Djnh nghTa 2. He con (6) dugc ggi Id cd dang Jordan cue bg (Local Jordan Forni- LJ)
'

^

r

vdi bien trang thdi ddu tien A^ ,, neu nhu
f /»,.-!

f ,n

V.I , / = 0,1,K , / i - 2 ;
uj'^ 6 arg
^ at
J

d"^>P.I

du.

dt"

^0.

(7)

Dinh nshJa 3. He con (6) dugc ggi Id cd dang Jordan toan cue (Global Jordan FormGJ) vdi bien trang thdi ddu tien x^,, neu nhu
r.7"
d"''xp.i
Vr = 1,;77,—> u/'' g arg
^ 0 . (8)
= 0, i = 0 , l , K , n - 2 ;
9M,
, dt""''
dt"-"
0

%

0

Cdc he con cd dang U vd GJ cd vai trd quan trgng trong viec dp dyng hieu qud cdc
phuang phdp tdng hgp luat dieu khien Uj cho hf con (6), ddc biet Id GJ, tren co sd dd cd the
tim dugc ludt dieu khien cho he (1) [7]. Vi vay cdn nghien ciiru cdc khd ndng va xdy dyng
thudt todn phan tdch he da chieu (I) ra cdc he con cd dang Jordan todn eye
IV. THUAT T O A N NGHlfeN ClTU KHA NANG PHAN TACH CAC HE DA CHIEU
RA CAC HE CON C 6 DANG JORDAN TOAN CyC®
Thuat todn nghien cihi khd ndng phdn tdch nd ra cdc he con cd dang Jordan todn eye dugc
xay dyng tren y tudng vet todn bd cdc phuang dn phdn tdch. Khi phdn tdch, ngoai viec phdi
cd cdu tnic cua dang Jordan todn cue, cdc he con cdn phdi tudn theo cdc dieu kien sau.
Dieu kien 1. Hgp cua cdc he con phdi triing vdi he ban ddu (I).
Dieu kien 2. Cac biln trang thai cua cdc he con phdi khdng dugc triing nhau, nghTa la
\/jk, fl = 1, m ; ik = I, Uj^; U = 1, n^., -^ jk ^ fl ; Xj^,^ ^ x^.,,,,
d day jk, fl Id cac sd thii ty, cdn n^^, n^, tuang iing la cdc bdc cua cdc he con.
Theo tieu chudn (8) vd tren co sd cdc dilu kien 1 vd 2, thudt todn vet tdt cd cdc phuang an
phan tdch gdm cac budc chii ySu sau.
Budc 1. Doi vdi timg dilu khiln Uj lap tap cdc he con cd dang Jordan todn eye cd dilu
khiln u. theo tieu chudn (8). Kit qud cua budc nay Id viec nhan dugc m tap hgp (tuong iing
vdi sd dilu khien cua he (1)).
Budc 2. Tir cdc tap nhdn dugc lap tdt cd cac td hgp m h? con (lap tap tfch Bk cdc ciia cdc
tdp), mdi td hgp mang y nghla Id mdt phuang dn phdn tdch he (1) cd thi. Tir tap cdc td hgp
nhaii dugc loai bd cdc phdn tir khdng thda man cac dilu kien I vd 2. NIU nhu tap con cdn lai
rong, thi he (I) khdng phan tach dugc ra cdc he con cd dang Jordan todn eye, ngugc lai tap
con nay chird tdt cd cdc phuang an plian tdch he (1) ra cdc he con cd dang Jordan toan eye.

'6 dSy gia sir rSng, tfn hieu dilu khiln cua he con thii / la Uj, j = l,m

699

Til' y tudng ciia thuat toan nay, thudt toan nghien ciru kha ndng phan tach he nhieu dau
vao ra cac he con co dang Jordan cue bo dirge xay dung mdt each tuong ty, chi bdng each
thay tieu chudn (8) cua thual loan tren bdi lieu chuan (7).
V HIEN THUC H 6 A THUAT T O A N PHAN T A C H T R ^ N M A Y TINH
TrSn CO sd ly thuylt da xay dung, tac gid da lap chuang trinh mdy tfnh trg giiip cho viec
nghien ciru phan tach cac he da chieu ra cac he con cd dang Jordan (cd LJ vd GJ). Hon thi
nira, chuang trinh con giiip cho viec ting hgp luat dieu khien can tdi uu theo tdc ddng nhanh
tir kit qua phan tach va mo phdng luat dieu khien dd.
Vl du minh hoa. Cho he bac 5 vdi 2 ddu vao.
.iSf = -A, - .v, ; A = -'V, + A, - A, + x^ ; 4 = "i ; 4 = -^3 + ^3 •' 4 = ": •
(9)
Vdi sy trg giup ciia chuang trinh, he (9) cd cdc phuang dn phdn tdch sau.
Bang 1
Phuang an 1
He con dang GJ vdi dieu khien M.
He con dang GJ vdi dieu khien u
.1^ = - A , - A,

•

J8^

[ 4 = ^3+^5 '

;

= -A, + A, - A, +

AJ

1-4 = "2

;

A = "i
Phuong dn 2
He con dang GJ vdi dieu khien u^
He con dang GJ vdi dieu khien M.

jSf=-x,-x,

fjSp = A3+A5 ;

14 = ".

•

;

J ^ = - A , + A-, -

AJ

+ Xj ;

Theo [1,3], ddi vdi he dgng hgc cd dang Jordan vdi mdt ddu vdo cd thi de dang tdng hgp
lu?it dieu khien va theo phuang phdp ddi chieu de quy trong [4-6] cd thi tdng hgp dugc ludt
dieu khien can tdi uu theo tdc ddng nhanh. Ddi vdi he con bdt ky trong bdng tren cung cd thi
dp dyng y tudng ciia phuang phdp nay dk tdng hgp ludt dilu khiln can tdi uu theo tdc dgng
nhanh, tir do nhdn dugc ludt dilu khiln can tdi uu tijeo tdc ddng nhanh cho he (9). Ta cd thi

sii dung phuong dn phdn tdch 2 dl tdng hgp ludt dilu khiln cho u^ vd u, (cdc bilu thiirc khdng
dugc thi hien d ddy do rdt ddi). Tren hinh 1 thi hien kit qud md phdng luat dilu khiln can tii
uu theo tdc ddng nhanh dd tdng hgp dugc vdi gid trj ban dau ciia cdc biln trang thai:
A, (0) = 1, X, (0) = 0.5 - AJ (0) = A, (0) = AJ (0) = 0 trong dilu kien ban chl cua cdc tfn hieu
dieu khien I M,|< 1, «J<1,
Cdc he con cd dang Jordan (7),(8) cung vdi thudt todn phan tach cdc he da chilu ra chiing
CO the sir dyng trong viec tdng hgp ludt dilu khiln can tdi uu theo tdc ddng nhanh trong cac
h$ thdng ky thudt cd nhilu ddu vdo, trong sd dd cd thi kl din he dilu cliinh dn djnh chuyin
dgng^^gdc cua ve tinh, he dilu khiln hieu dien thi vd cdng sudt ca hgc cua mdy phdt tua bin,
he dieu khiln dgng co dien mgt chilu [7],... Phuang phdp tdng hgp ludt dilu khien can tdi uu
theo tdc ddng nhanh cho he nhilu ddu vdo dya tren y tudng phan tdch he dd ra cdc he con
dang Jordan dugc mieu td chi tilt trong [7].

700


1.000

0,500

-

0,000

-0.500

-1.000
0,00

0,S0

1.20

1,80

2,40

3.00

3,60

4,20

4,80

5,40

6,00

Hinh 1. Qua trinh qua d$ cua hf (9)
VI. KET LUAN
Trong bdi bdo de xuat mgt each tiep can mdi cho viec phdn tfch dang Jordan cua cac he
ddng hgc vd cua cac he con ciia cdc he da chieu trong ITnh vyc dilu khiln ty ddng. Cdch tilp
can ndy ludn chi ra dugc khd ndng phdn tdch he dilu khien da chilu ra cdc he con cddang
Jordan thuan Igi cho viec tdng hgp ludt dieu khien, neu nhu he da chilu cd khd nang dd.

TAI LIEU THAM KHAO
1. FaitoyK A.P. Hoaaa jmpaBJiaeMaH 4>opMa HejiHHeHHwx ypaBHeuHH AHnaMUHecKHX
odtcKTOB / A.P TamyK IIC6. xp. MHK. MMTTI8. T.2. Kaaani,. -2005, c. 88-90.
2. FaiiayK A.P. HpHBefleHHe ypaBHenHii odieKTOB xpexbero nopsAKa K ynpaanaeMOH

4)opMe "Kopnima I A.P Fan^tyK // C6. xp. MHK. MMTT19. T.2. Bopoue^c , -2006, c. 115118.
3. FaitoyK A.P. PemeuHBie H HepemeuHbie MaxeMaxHHecKHe npodjieM&i coBpeMeHHOH
xeopHH ynpaBJieuHa// CHCXCMHWH auajiHS, ynpaBJieuHe H odpadoxKa uHtJiopMai^HH: cdopuHK
HayHHBix cxaxeii/ Oo^t o6m. pe;i. npoiji. P.A. He^jxop^a - PocxoB-na-^OHy: .UTTY.
Taraupor: TTH KXDV. 2007.- C. 18-27.
4. HefiwpHa ocHOBCflHHaMHHecKOHanopoKCHMauHn/ P.A. HeH;top(|), H.H. Han // C6. xpyflOB
MeacAynap. nayn. KOH(J). "MaxeMaxHHecKHe MCXOAW B xexHHKC H xexHOJioraflx MMTT-19".
BopoHe)K, -2006, T. 2, - C.l lO-113.
5. Hah H.H. KoMnosHimoHHwii cHHxe3 KaasHonxHMajibHbix no dbicxpoACHCXBHio CHCXCM
ynpaBJieuHa BbicoKoro nopH^tKa /H.H. Hau, P.A. HeHflop(^ // BecxHHK AoH.roc.xexH.yn-xa.2007.^24.
6. Hau H.H. CHHxe3 saKOHOB KBasHonxHMaJibHoro no dbicxpo^eHcxBHio ynpaBJieuHH
o6beKxaMH BbicoKoro nopa^Ka/ H.H. Han// Jl^vicc. ua COHCK. yn. exen. KaHfl. xexH. nayK.
05.13.01, PocxoB-Ha-;5oHy.- 2007 r.
7. Oau H.X. Mexoflbi pacnosHaaaHMH ynpaBJiacMbix )Kop;taHOBbix ^opu AHHaMHHecKHX
CHcxcM H Hx ;^eKOMno3HitHH Ha jKopaauoBbi noflCHCxeMbi B saaanax cuuxesa
KBasHonxHMajibHbix no dbicxpoflCHCXBHio saKOHOB ynpaBJicHHa/ H.X. Oan// ^HHCC. ua COHCK.
yn. exen. KaH^. xexH. nayK. 05.13.01, PocxoB-na-^^oHy.- 2008 r.

701