MOT SO PHUOTVG PHAP THIET KE BO DIEU KHIEN BEN VlTNG CHO MOT LOP DOI TUQWG TUYEN TINH
'C6 THONG SO BAT DINH
Phgm Vdn Minh, Hodng Duy Khang '"^, Nguyin The Thdng ^*^
(a) Khoa Diin-Dgi hgc Cong nghiep Hd ngi (b) Dgi hgc Bdch khoa Hd ngi.
Tdm tdt:
Bdo cdo ndy dica ra thdo lugn mgt so phuang phdp xit su on dinh vd ddnh ^id chdt lugng dicac dp dung cho bdi todn thiet ke bg dieu khien ben vieng doi vai mgt lap doi tirgnig cd mo hinh bdt dinh thdng so. Bang viec sic dung thich hgp cdc tiiu chudn ddnh gid chdt lugmg ta co the dica bdi todn xdc dinh thong so bg dieu khien ve bdi todn toi icu dgng qui hogch phi tuyen hay qui hogch niea vo hgn. Nghiem ciia bdi todn toi mi dugtc xdc dinh bai phuang pMp ddm bdo SVC thoa mdn chat doi vai cdc rdng biigc.
Abstract:
This paper to discuss about some methods to guarantee for stability and performance.
They were applied to robust controller design problems for linear class plants with parameter uncertainty. By using suitable for stability criteries and performance, we can be translate them into the nonlinear programming problem or semi-infinite optimization. The solution of optimal problem were difined by strickly satisfy for constrains.
L GI61 THIEU CHUNG
Vifc xay dung bd dilu khiln cho mdt ddi tugng thudng dua tren md hinh, giira md hinh v^
ddi tugng that cd su sai Ifch (nguyen nhan do dimg phuang phdp gan diing, do thieu thdng tin khi nhan dang...). De khdc phuc phan ndo hau qud do cd sai Ifch md hinh ta cd the xay dpg bd dieu khien ben viing dua tren md hinh bat dinh.
Md hinh bit djnh cd the md td bdng mgt tap md hinh cd cau tnic ho?ic khdn^ cau true, mo hinh khdng cau tnic thudng dugc ddnh gid qua gidi ban chdn, cdn md hinh c6 cau true thudng dugc thdng sd hda vd dugc ddnh gid qua t^p bat djnh cua thdng sd.
Thiet ke phai dam bdo cdc chi tieu chat lugng mong mudn cho mgi md hinh trong tap mo hinh khdo sdt.On djnh Id yeu cau co bdn cua mgt hf dilu khiln, ngodi ra cdc chi tieu chat lugng khdc ciing dugc x6t den nhu: Su bdm dau vdo tdt (good set point tracking), giam thieu su tdc ddng ciia nhieu (noise attenuation), cdc chi sd vl dO du trii dn djnh, ve s\r tdt cua qua trinh qud do, ve do qud dieu chinh...phdi du tot hay phai ndm trong mgt gidi ban cho phep.
Hifn nay, chua cd mgt phuang phdp thilt kl bg dilu khiln dk cd the dam bao dugc tit cacdc chi tieu chat lugng mong mudn ke tren.
Bdo cdo ndy gidi thifu mgt so phuang phdp vd ddng thdi de nghj mgt phuomg phdp giai tfch de thilt kl bg dilu khien d\ra tren md hinh tuyin tfnh cd thdng sd bit djnh.
Xet mdt hf thing lien tuc dang SISO vdi md hinh tuyin tfnh cd thdng sd bit dinh co the md td bdng md hinh tdi gian hinh 1.
X ) N e C(s) P(s, q)
^
Hinhl
Trong dd: y Id tfn hifu ra, r la tfn hifu ddt, d Id nhieu, C(s) Id bg dilu khiln. P(s, q) Id ham truyin cua dii tugng cd thdng sl bat djnh q cd dang:
P{s.q) = N,{s.q) DAs.q)
t.b^q)s'
y-o
i;«.('7)-^'
/•o
(1)
Vdi q=[qi, q2 qtf Id v€c ta thdng sd bit djnh qeQ, tap Q thudng cd dang sieu hgp
(hypercube, box): Q = jq| ai<qi<|3i}, l=l..L (2) C(s) Id ham truyin ciia bd dilu khiln, thudng cd ciu true dugc chgn trudc cd dang (bac ng,
nic dugc chgn trudc):
C(s) = NAs,q) D^s.q)
- ' • "
J'9
(3)
P la cdc hdm truyen hgp thiic (proper). Cdc Ddt x=[Ci, dj] Id vec to an sd can xdc djnh de hf
• » r y
thdng kfn thda man chat lugng mon^ mudn vdi mgi qeQ (2). Dieu ndy gay khd khdn Idn cho bdi todn phan tfch ciing nhu thiet ke hf dieu khien ben vun^. Su dn djnh Id dilu kien cin dk mgt hf thdng tu ddng lam vifc, vi vay mgi phuang phdp thiet kl bd dilu khiln bin virng phai ke den su dn djnh, ngodi ra cd mdt sd phuang phdp cdn ke them dugc mdt sd chi tieu chit lugng khdc.
Cd the thiet ke bd dieu khien C(s, x) nhd phuang phdp HL, tuy vay theo phuang phdp Hx tat cd anh hudng cua do bat djnh trong md hinh dugc the hifn qua hdm chdn K(s) dugc xdc djnh trudc tir khi nhan dang ddi tugng. 6 md hinh bat djnh thdng so, tap bat djnh thdng sd Q thudng dugc xdc djnh thdng qua bdn chat vat ly hodc yeu cau cdng nghf cua ddi tugng. Tii tap Q xdc dinh hdm chdn K(s) Id mdt vifc khd khdn nen vifc dimg phuang phdp H^ Id khdng hgply[3,8i.
O dang tan sd ve nguyen tdc cd the dan ra tieu chuan dn djnh ben vung, dn djnh trong mien D. Tieu chuan chit lugng bin vung nhu: Bdm ddu vdo, gidm tdc ddng cua nhieu [6, 7]...
Nhiing yeu cau tren din tdi vifc xet su dn djnh cua hg da thiirc chiia thdng sd, vf du, xet da thiic eg dang sau:
\|/s(ja), q, ri, ai)=rie'"'Nwi(J9)DcGto) Dp(jco, q)+ Dwi(jto)<|)(jfo. q) (4) De dam bdo do bam diu vdo thi (4) phdi dn djnh vdi:
VqeQ, V(ri, tti) thda man: 0<ri<l, 0<ai<27C (5) Trong dd (j)(j(0, q) Id da thiic ddc trung cua hf thdng kin dang:
(jiCjo), q)= Dp(jto, q)DcOa), x)+ Np(ja), q)NcGo), x) (6) Vd hdm trgng lugng:
y,^^!L^^ (7)
D.MO))
La hdm chdn tren cua hdm do nhay S(j(o) dugc chgn trudc theo do bdm cho trudc [8]. Vifc xet dn djnh cua (4) rit khd khan chi dugc giai quyet trong trudng hgp hf thdng cd ai(q), bj(q) la hdm affine ciia q [8].
Co thi lap iticu chuiin chit lugng clang dai sd, dilu kifn dn djnh bin virng dugc thdnh lap dua tren da thirc dac tnmg. Co Ihl vilt da thirc dac trung (6) dudi dang:
(|)(s, q, x)=Yo(q, x)+yi(q, x)s+...+ y„(q, x)s" ^ (8) Co thi diing each xet mgt sl huu han dilm q"", h=I..N ta cd mgt hg gdm N md hinh vdco
thi thilt kl bcrdilu khiln C(s, x) lam md hinh nay dn djnh bdng cac phuang phap vf du nhu Konigorski, phuong phap Modal ciia Roppenecker [5]. Tuy vay each Idm nay khdng dam bao chat che dilu kifn on djnh khi Q Id mft tap lien tyc. , , .
Trudng hgp cac he sl yh(q, x)=q''" theo tieu chuin Kharitanov [4] dieu kifn dn djnh bin virng din tdi dn dinh cua 4 da thirc khdng chira thdng sd bat djnh dang:
F(s,x)=fo(x)+f,(x)s+...+fn(x)s" (9) Tmdng hgp cac he sd Yh(c^, x) la hdm affine ciia q dilu kifn dn djnh theo cdc djnh ly ciia da
thirc canh [1] cung dan din on djnh cua L''=2*'-*" da thirc khdng chiira thdng sd bat djnh dang (9). ' , ^ ^ ,
Bdng mdt tieu chuin dai sl vf dy ti6u chuin Routh, moi da thirc khdng chira thdng so F(s, x) ta cd dugc rdng bugc doi vdi bien x de tir dd xdc djnh x.
Ngodi ra dilu kifn dn djnh bin virng dudi dang dai so, rit khd lap cdc dieu kifn cho cac yeu ciu chit lugng khac nhu dg bam diu vdo, giam thilu tac dgng ciia nhieu, dg qud dilu chinh... , , , . , . .
Dudi day gidi thifu mgt phuang phdp thilt kl bg dilu khien bin vihng theo sa dd cau tnic d dang tdi gian nhu tren hinh 1, ddi tugng cd hdm truyin dat P(s, q) (1) thdng sd bit djnh q (2) vd bd dilu khiln C(s) (3) vdi ciu triic Id bac nc, mc chgn trudc. Xac djnh cdc hf sd Cj, dj (i=0..nc, j=0..mc) theo cdc yeu cau chat lugng sau:
(1-1) Hfdn djnh bin virng vdi VqeQ.
(1-2) Qud trinh qud do tdt vdi hf sl tdt a: Q<a~ <a<a* (10) Vdi a' vd a^ da cho.
m,
(1-3) Bdm tifm can vdi tfn hifu vdo r(t) dang chudi luy thira: r{t) = ^r.t' (11)
1=0
(1-4) Tdi uu hda theo nghTa cue tieu mgt ham mgc tieu gan vdi ddi tugng cy the vi du qua trinh qud do tdt nhanh nhat, do ben viing vdi ddi thdng sd rgng nhat...
Bdi todn thilt ke bd dieu khien dugc dua ve bdi todn tdi uu dang qui hoach toan hgc (Dang qui hoach phj tuyen hay qui hoach nua vd ban). Trudng hgp cac hf sd a,, bj ciia ddi tugng (1) Id hdm affine ciia q ta cd the dua bdi todn xdc djnh bg dilu khien C(s) vl dang bdi todn qui hoach phi tuyen dang:
min/(.x)
A€C (A)
G = {;:|^(x)>0}J
Trudng hgp Np(s, q), Dp(s, q) phy thugc phiic tap vdo q trong tap Q (2) ta tim cdch dua vl bdi todn qui hoach nira vd han (semi infinite programming) d?ing B.
min/(jc)
.l€C
G = ^ W > 0 (1) (B)
V9e(2(jf) (2)
Trong dd x Id vec to an cin tim gdm cdc hf sd Ci, dj trong bd dilu khiln C cung cd thi gIm cdc do do chit lugng hodc cdc thdng so gia cin dimg khi tfnh todn. Vec ta ham g(x, q) dugc xay dung tir dilu kifn dn djnh cung nhu cdc yeu cau chit lugng (1-1) din (1-4). Dudi day muc 11 trinh bay cdch din ra bdi toan (B) vd thao luan each giai bdi todn (B). Muc III se giai thifu mdt vf dy img dyng.
IL LAP BAI TOAN A
Trong cdng trinh [10] da gidi thieu phuang phap cho mgt tiudng hgp don gian, d dd ddi tugng cd cac hf sd aj, bj la cdc tham so bit djnh va chi ke din cdc yeu ciu chit lugng (1-1) va (1-4) cdn yeu ciu chit lugng (1-3) chi xet vdi r(t)=const, vd bg dilu khiln d dang PID.
Bdo cdo nay xet cho tnrdng hgp tdng qudt ban: Hf sd ai, bj Id hdm ciia thdng sd bit djnh q, ldp tfn hifu vao r(t) rgng ban, kl din hf sd tdt ciia qud trinh qud dg, bd dilu khiln C cd dang tong qudt ban.
1. Rang budc ve su on dinh
Phuang trinh ddc tnmg cua hf hinh 1 c6 dang (8)
•(|)(s, q, x)= Dp(s, q)Dc(s, x)+ Np(s, q)Nc(s, x ) = Yo(q, x)+Yi(q, x)s+ ...+ Y„(q, x)s" (12) De ke dugc hf sd tdt a ta dimg tieu chuin dn djnh hdm mii ddn tdi vifc \6t su dn djnh ciia da thiic Da:
»
0a(^,q.x) = (p{s,q,x)\^^^_^ =J^0,(^q,x,a)s'' (13)
« ^ « mr ' ">
Dung tieu chuan Routh hodc Hurwitz dieu kifn dn djnh cua Da dugc dan tdi cdc bat ddng thiic dang (1), (2) cua (B). X6t trudng hgp rieng quan trgng ciia dii tugng (1), d dd ai, bj Id hdm dang chudi luy thira cua q, vf du nhu vilt cho a;:
a,ix,a) = f^H„,ix,a)Y[q";'^^ (14)
/i=0 j=\
mm f \ mm f W
De thay rang 9k(q, x, a) cung cd dang chudi liiy thiia ciia q giong nhu d (14), vi du viet cho ham gi(q, x, a) d rang budc (1) cua (B):
g,(x,a,q) = f H„(x,a,q)X\ ( / J " ' > 0
V ^ E Q
Trudng hgp neu do ban chat vat ly hoac cdng nghf khi nhan dang ddi tugng cdc hf sd ai, bj la hdm phiic tap cua q, ta cd the khai trien cdc hf sd ndy ra thdnh chuoi Taylor-Maclaurent tai lan can gid trj chuan q° khi nhan dang.
2. Yeu cau chSt lu-gng (1-2): da dugc ti'nh din khi diing tieu chuan dn djnh hdm mii cho da thirc (13), (2), (10) din tdi rdng budc dang (15).
3. Chat lugng bam d4u vao (1-3)
Anh Laplace cua tfn hifu vdo dang (11):
«W = i.['-(')]=i;r,-^ (16)
1=0 P
Sai Ifch e dang todn tir Laplace dugc xdc djnh nhu sau:
1 DAs)D(s,q) E(s) = -J—Ris) = ' ' R(s) = W„R(s)
^ 1 + CP D,D^+N^N^
Dilu kifn bam diu vdo tifm can din tdi:
e(oo) = \ime(t) = lim sEis) = \ims-—-—R(s)
^ ,-*- s-tO s-*o i + CP
DJs)DJs,q).;^ 1
= lim.W./?(.) = l i m . / ) ^ ^ ^ ' ; ^ : ^ S : , - ^ = 0 Tdch Dcvd Dp thdnh:
(17)
.V-.0 - .,->o D,.D^+N,N^U ' s'
mc2 inpi
D,.=s""'Y.d,s\ D^=s"'>"Y,d.s^ (18)
(=0 ;=0
s""'ya:s'.s"""ya:S^
im^ ' i—l ' nil- I
e{o°) = lim —— —— Zj'i ~T D,.D^+N,.N,. fo P D I cd E(°=)=0 vdi VqeQ ta phai cd md+mpi > mr+1, do dd:
mci>mr-mpi+l (19) Cdc sd mu mci, mc2, mpi, mp2, m,. khdng am. Vf dy vdi r ( t ) = r o ^ m r = 0 ^ (mp,=0, nicFl),
(mpi=l, mci=0). Nhd vay dk dam bao dg bdm tifm can dau vdo ta phdi chgn cau true cua bo dilu khiln theo (18), (19).
4. H a m muc tieu , , , , Hdm myc tieu I(x) dugc thdnh lap tuy theo bdi todn cy the, vf dy ta cd the coi hf sd tat a la mft thdnh phin cua v6c ta in x vd ham muc tieu cd thi chgn: I(x)=-a-»min (20)
Luc dd ta cd qud trinh qud df tdt nhanh nhit, cung cd thi liy cdc can tti, pj trong (2) phu
thugc vdo in x: a i ( x ) < qi < Pi(x) (21) Vd cd thi chgn can, vf dy Pi(x) Idm hdm myc tieu
I(x) =-|3i(x)-^ min ^ (22) Liic dd tdi uu dugc bilu theo nghia khoang bit djnh cua thdng sd qi dugc rgng nhit.
Trong cdc cdng trinh [9, 10, 11] cdc vin dk v l gidi bdi todn qui hoach nira vd han dang (B) da dugc d l cap tdi vd khdng trinh bdy l^i d day.
IIL v i DV
Hf dilu khiln hinh 1, cd ddi tugng P:
P(, .)=^iii:ll = _ ^ _ = ij , (VDI)
Di,(s,q) aQ + a^s a^ + (6 + q^ + q{q,)s
Vd bd dilu khien C chgn d dang:
Cis) = ^^^^^ = ^2 , (VD2) Deis) d^+d^s + djS^
Cac thdng sd bat djnh Q:
a,=l<q,<5 = fi,
Q = \i (VD3)
or, = 2 < 9j < 10 = /?,J
Xdc djnh thdng sd ciia bg dilu khien: Co, do, d|, d2 de hf bdm tiem c | n vdi tin hifu vao r(t)=const dn djnh vdi hf sd tdt a (a^ao>0) Idn nhit (tiro Id qud trinh qud dg tdt nhanh nhSt vdi VqeQ.
Tir dieu kifn bdm tifm can vdi tfn hifu vdo r(t)=const ta suy ra bg dieu khien phai co dang:
Cis)= . / \ . (VD4)
Phuang tnnh ddc trung cua hf khi dd Id:
(l)(s, q)=DcDp+ NcNp=s(di-Ki2s)(ao+a|S)+cobo=d2aiS^+(diai+d2ao)s^+diaos+cobo Ddt xi=di, X2=d2, X3=co, X4=a, khi d6: <|)'(s, q, a)=(|)(s, q)|s=s-oFf3S^+f2S^+fiS+fo Vdi:
fi =^2«i =-^2(6 + «72 -^ qUi) = ^^i-^ Ri^i-^ ^^Ri^i
fl = «o^2 + (-^1 - 2^:4-^2 ) ( 6 + ^2 + ^iQi)
/, = -2A-^ jr,ao + (Ixlx^ - 2X4^, )(6 + (73 + 9,^92)
/o = -x^x^aQ + xlx^ag + Xj^, + {-x^x^ + xlx2)(6 + ^^ + qfq^) Theo tieu chuan Routh, hf dn djnh hdm mu vdi hf sd tdt a dan tdi:
8} (-v,^) = f\ = 6x, + q,x, + qfq.x. > 0
gAx,q) = f^_ =aoA% +(.v, -2.v,.v.)(6 + (/3 +r/ffy,)>0
A./"i - / 3 . / 0 = -^^'2^1 ('V, fy) > 0 - » A-3 > 0
Sl ('^''9) = - 6 ( - a^XiX, + a(,x,xl -dx.xl + 6A-,A;)-6A-,^, + \aQX^x^ + a^x^x; Jcy, + ArWi^y.
+ (-Xjxl +x^xl)q; + 2(-A-,A-J -h A-,A4)<ry,-<ry; -{•(-aQX^x^ +«O-^2^4'AI'I'<?2 + ^yqUi + {-x-,xl -I- A, .\l )q*ql > 0
Cac ham gi, g2, g3 dkw cd dang chudi liiy thira (15), vf du vilt lai gs dang (15):
2 2
<?3 ('^'. ^) = E y'i' n *??' = ^-^2 + ^'2^2 + ^2'7l''?2
Vdi: Y30=6x2, m3oi=0, m302=0; Y3I=X2, m3ii=0, m3i2=0; Y32=X2, m32i=2, m322=l Vifc xdc djnh thdng sd x dan den bdi todn nira vd h^n sau:
minI(x)=min(-X4)
gi(x,q)-8>0.i=1..3,Vq€Q g4=X4-6>0.
g5=X2-6>0.
X4-ao-feO,
0 day 0 Id mdt sd duang dii nhd.
Bang phuang phdp phii tuyen tfnh bdi todn niia vd han tren dugc dua ve bdi todn qui hoach phi tuyin. Trong qua trinh tim nghifm, tai mgt diem x nghien ciiu ta cd the tim dugc mgt trj cue tieu non MiuN cua gi(x, q) vd do do ta biet dugc gi(x, q)>0 cd dugc thda man hay khdng.
IV. KET LUAN
Bdo cdo da cho mdt nhan djnh chung ddi vdi mdt sd phuang phdp thiet ke bd dieu khien bin viing cho hf thdng cd md hinh tuyin tfnh vdi tham sd bat djnh. Bdo cdo ciing de xuat mdt phuang phdp thilt k l bg dilu khiln bin viing bdng cdch chuyen ve bdi todn qui hoach niia vd han nhd nhiing phuang phdp ddnh gid chat lugng quen thudc. Phuang phdp dugc de xuat cho nghifm thda man chat dieu kifn dn djnh ben virng.
TAI LIEU THAM KHAO
[I] A.C. Bariett; C.V Hollot: L. Huang: "Root location of an entire polytope of polynomial: it suffice to check the edges", Mathematics of control, signal and systems Vol 1. 1988, pp.
61-17.
[2] Bhattacharyya; H. Chapellat and Kul: "Robust control: The parametric approach", Printice Hall 1995.
[3] Grimbele M. J: "Robust Industrial control-optimal design approach for polynomial systems", Printice Hall Intemational (UK) limited, 1994.
[4] V. L. Kharitonov: "Asymptotic stability of an equilibrium position of a family of systems of differential equation", Differentialnye Uravniya Vol.11, pp.2086-2088,1978.
[5] Nguyin Doan Phudc, Phan Xudn Minh: "Dieu khiln tdi uu bin virng", NXB KHKT, 1999
[6] A. Ranzer: "Stability conditions of polytope of polinomials", IEEE trans. Aut.control Vol AC-37 Noi, pp.79-89, 1992.
[7] Comptes rendus de lAcademie bulgare des sciences. Tome 62, No 8, 2009
[8] Theodore E. Djaferis: "Robust control design: A polynomial approach", Kluwer academic Publishers, 1995.
[9] Nguyin Thi Thing, Trin Vdn Tuin, Pham Van Minh: "Mgt phuomg phdp phii tuyin tfnh cho mgt Idp bdi todn vdi rdng bugc cd chii:a thdng sd", VICA5, pp.367-373, 2002.
[10] Nguyin Thi Thing, Trin Vdn Tuin, Ph^m Vdn Minh: 'Thiet kl bg dieu khiln bin virng cho mgt Idp hf tuyin tfnh cd thdng sd bit djnh", VICA6, pp. 153-157, 2005.
[II] Nguyen 'The Thdng, Le Vdn Bdng: "Algorithm for parameter optimization problems of nonlinear discrete systems", Proceeding of the fifth intemational conference on analysis and optimization of the systems, Versailes (France) Dec.l4-17.Lecture note in control and information sciences. 44, pp.869-884, 1982.
[12] Zapiriou E. Morari M: "Robust process control". Prentice Hall, 1989.
