VQ Thj Thuy Nga vd Dtg Tap chi KHOA HQC & CONG NGHE 135(05): 219-224 DIEU K H I E N ON DINH T H O I GIAN HUtJ HAN (FTS) TREN NEN T O I UtJ T A C DONG NHANH VQ Thj Thiiy Nga*, Nguyen Dodn Phir6c Dai hpc Bach khoa Hd Ndi TOM T A T Bai bao gidi thieu mgt phuang phdp dieu khiln 6n dinh he phi tuyln vdi khodng thdi gian In djnh hiru hgn (FTS) Phuang phap dieu khiln cua bai bdo dugc xay dung dya tren nin nguydn Iy cyc dai de dieu khiln toi uu tdc dgng nhanh Nhd ta cd thi dilu khiln dugc he tir miln trang thdi bj chdn cho trudc v6 d6n g6c tpa sau khodng thdi gian bi chan tren bdi mgt gid trj hOu han tiiy y cho trudc cung nhu sd ldn chuyin ddi gid trj tin hifu dilu khiln tli da chi mgt ldn Tir khda: Toi uu tdc dong nhanh; Dieu khiin tuyen tinh hoa chinh xdc, Dieu khiin FTS DAT VAN DE Trong nhirng nam gdn ddy, d ITnh vuc dieu khiln phi tuyln, xudt hifn mdt sd cdng trinh nghien cuu ve dieu khien dn dinh tifm can toan cue vdi khoang thdi gian hiru hgn, duoc ggi tdt la bdi toan dilu khien FTS (finite time stabilization) cho he phi tuyen affine bac hai mgt ddu vao a, tuc la he cd hai bien trang thai x = {xi, 3:2)^, mota bdi: dx = f(x) + h(x)uvai /(0) = (44) dt vdi la ky hieu goc tpa khdng gian trang thdi R^ Mpt sd cdng trinh tieu bieu ve vdn 6h dilu khien FTS cd the ke den Id 0,0,0 Y nghTa ung dung ciia bdi toan dieu khiln FTS khdng chi dan thuan dirng lgi d vifc dilu khien he ve tdi gdc tpa sau khoang thdi gian hiru han, md xa hon Id bdi toan trung gian lam cau ndi cho viec xdy dung bg dieu khiln trugt bac cao vdi nhiem vu dieu khien quy dgo trgng thai cua he ve den mat trugt sau khoang thdi gian hiiu hgn Cdng cu nin tang ciia nhirng nghien cdu ndy van la ly thuyet Lyapunov 0, tuc la van di theo hudng tim mpt ham xac djnh duong V"(i), tron, dan difu tang theo |x|, cho vdi no ludn tdn tai it nhat mdt quan he u(x), dugc hiiu la md hinh ciia bd dieu khiln phdn h6i trang thdi, de dgo ham theo thdi gian ciia nd: 'Email nga.vuthilhuy@husledu.vr dV _dV dx_dV (/(x)+/i(i)u(i)) dt dx dt dx thda man: dV J r (45) (46) T Id mdt gia trj hiru han Gid trj T cung chinh Id thdi gian dn djnh hiJu hgn cua he (44) bd dieu khien phan hdi trang thai u(x) mang lgi, vi theo djnh ly LaSalle, dd cung phdi cd x(t) = vdi t>T Mac dil vdy, xu hudng giai quyet bdi todn nhu tren Igi gap phai van de mudn thiia ciia ly thuyet Lyapunov la di tim ham V(x) thich hgp Dd la hgn che chfnh ciia cdc phuang phap da cd Bai bao se gidi thieu mdt xu hudng khde de giai quyet bai toan dieu khien FTS ma khdng can su dung dfn ly thuyet Lyapunov Phuang phap giai quylt ciia bdi bao se dya tren nen ly thuyet dieu khiln tdi uu tac dpng nhanh nguyen ly cue dgi ciia Pontryagin 0, kit hgp vdi dieu khien tuyen tinh hda chinh xac nhd cdng cu hinh hpc vi phdn cua Sussmann va Isidori ciing cac cdng sy 0,0 Tdt nhien ta cdn cd thf thdy dugc rang bg dieu khien tdi uu tdc ddng nhanh ciia bai bdo se khdng nhirng lam hf dn djnh sau khoang thdi gian hiru hgn, ma cdn la vdi thdi gian huu han T nhd nhdt tdt cd cac bd dieu khien FTS cd the cd ctia hf Hon the nii'a ta cdn cd the thiet ke bd dieu khien FTS vdi khoang thdi gian 6n djnh T bi chdn tren bdi mpt gia trj hifu han T^^^ cho trudc 219 Vu Thi Thiiy Nga vd Dig Tap chi KHOA HOC & CONG NGHE Toan bg cac budc thiet ke bd dieu khien FTS se dugc trinh bay d chuang II Chuang la mpt vi du minh hga cho phuang phap ciia bai bao THIET KE B O DIEU KHIEN FTS TREN NEN DIEU KHIEN TOI UD T A C D O N G NHANH Phuong phap thiet ke bp dieu khien FTS trinh bay sau day chi gidi hgn cho hf phi tuyen affine bgc hai (44) Mac dii vgy no cung hoan todn md rgng dugc mdt each tuong ty cho ca nhirng he phi tuyen affine bac cao, dieu ma cac c6ng trinh cdng bd trudc ddy dua tren nen ly thuyet Lyapunov cua Bhaty, Bemsteinz 0, hay ciia Hong hodc cua Moulay, Perruquetti chua lam dugc Tuyen tinh hda chinh xdc he phi tuyin affine bdc hai Gia thiet he (44) Id dieu khien dugc Khi dd, theo 0, nd ludn dieu khien tuyen ti'nh hda chinh xdc dugc nhd phep ddi bien vi phoi ^ = •m(x) vd m^t bd dieu khien tTnh, phan hdi trang thai u(x,v) -(::)• 135(05): 219-224 (48) m(x) = '[L,J.lx)j la nghjeh dao dugc, hay z^ = m(x) la phep doi bien vi phdi Sd dung phep ddi biln vi phdi (48) ta co: dzi dA(x)r.^ X / ^ = LrA(x) H- Lf^A(x)u = LfX(x) = zj dz2 dLfA(x) = ~J^—-[f(x) dt + h(x)u] = LjA(x) + L}^LjX(^u Do dd, su dyng bp dieu khien: V = LfA(x) + Lf^LfA(x)u V~L]A(X) o u= = (49) Mgt each cu the thi he (44) la bgc hai va hf (44) cho ban dau se trd hf tuyen tinh toan bp khdng gian trang thai dudi dgng khau tich phan bac hai: dieu khien dugc, nen cd: Rankfft(x), adjh(x)\ = 2, Vx Ky hifu ham md rdng: A(x) = span(/t(3;)) ta thay: dh(x),, , dh(x)^, , „ -^^=^h(x)—=^^/i(i) = OeA(i) ox dx nen A(x) la ham md rgng xodn Dilu khdng djnh rdng phdi tdn tai ham vd hudng A(x) de cd 0: Bd dt6u khlAn (49) LhM¥.) = f> va Lf^LfMx):^0, Vx dA{x) ^ -A(x) = dx Tu ^.(x) ta djnh nghta vector ham: m(x): ( Hx) [L,_A(x) se thay 0: ax Hinh bieu dien cau tnic hf thdng dieu khien tuyln tinh hda chinh xac cho he phi tuyen affine bac hai ban ddu (44) he tuyen tinh (50) nhd bp dilu khiln tTnh phan hoi trang thai (49) va phep d6i biln vi phdi (48) (47) Hiphi tuy6ii(44) Sdi bien T ^48)_ Hinh Dieu khien tuyen tinh hoa chinh xdc Bieu khien toi uu tdc dgng nhanh he tuyen tinh bac hai Xet hf tuyln tinh dgng khau tich phan bac hai (50) Gia sir rang hf cd tin hieu vao bj chan |u|^ thi 2i(i) phai tang theo t dt (k) dt va ngugc Igi 22 < thi 2] (f) phai giam, , -I (5 Iglai thi quy dao trang thai tdi im di qua goc la: => Zl = — Z + C h o a c 2| = — ^ T + C T 2fc 2k 2] = — ^ ^^ ^2^^ va vdi cj, C2 la cac hdng sd dugc xac dinh tu 2fc trgng thai dau ZQ ciia he jk p2>0 Thdi diem ddi dang quy dao trang thai tdi uu md ta bdi phuang trinh (51) ciing la thdi diem ma tin hieu dieu khien tdi uu u ddi dau va la thdi diem ma tai dd cd P2it) = Nhung vi bien ddng trang thai p(t) cdn la nghiem cua phuong trinh vi phan Euler-Lagrange: Z2 22 > 2fc se dugc viet chung lai thdnh: i:)r p2(t) = at+b dt~ (dz) dd a, b la hai hang sd, nen p2(t) = chi cd the cd nhilu nhdt mgt nghifm i| > Do dd qu5' dao trgng thai tdi uu'ciing chi cd thi ddi tu dgng dgng khde (51) nhilu nhat la mpt ikn Hinh Dgng quy dao trang thdi toi uu Idc dpng nhanh Vu Thj Thiiy Nga vd Dtg Tgp chi KHOA HOC & CONG NGHE Vi tin hieu dieu khien tdi uu v chi chuyen ddi gid tri nhieu nhat la mpt lan, nen d6 thj (z)^0 ^ (^) = - -fcsgn(2|) ^ ( ) - , 2|7t0 135(05): 219-224 \-ks^g^(x)Uhi v{m(x))=\-ks^{A(x)) p/(x)*0 khig/(x) = 0, MM)^0 x = ip'(x) ^ ^(m(x)) = A(x) +—LjMx)^LfM^)\ Ta thay bp dieu khien FTS cdn cd chua tham sd fc > tiiy chpn va dd ta cd the chpn fc de thdi gian dn dinh T khdng Idn ban mgt gia trj cho trudc, dilu ma cac phuang phap dilu khiln FTS tren nin Lyapunov rat khd thyc hien dugc De xac djnh dugc thai gian on dinh T, ta gia sd he cd trgng thai 6ku ung vdi: (53) XQ=m (2Q)vaa*0 z - dd ham ^(z) dugc dinh nghTa theo Thiet ke bp dieu khiin FTS cho hiphi tuyen bac hai dieu khien duac Bg dieu khien FTS cho he nguyen ban gdc ban ddu (44), xay dung tren nen t6i uu tac dgng nhanh, se dugc xay dyng theo nguyen ly cascade gdm hai bp dieu khien tuyen tinh hda chinh xac (48), (49) va bd dieu khien tdi uu tdc dpng nhanh (53) cho he tuyen tinh Hinh bieu dien nguyen ly '*ieu khien cascade ^ B9 dilu u khi^n (49^ Bp di^u khifin (53) Hephl uySn(44 222 sf ddi dau (yk 0 thicfi hpp de he 6n djnh vdi khoang thdi gian T theo (54) thda man T < T^^ MO PHONG KIEM CHU"NG Df minh hga phuang phap, ta se xay dung bd dieu khien FTS cho hf phi tuyen bac hai: i4i;A(x) = ( x ? , l ) [ ° ] = I Do dd bd dieu khifn FTS ciia he se Id: "= , ", = " - i , (I, +X2)-i,i2 (56) Lf,LjA.(x) do: -A:sgn^'(3:) Ithi (f>'{x)*fi -^sgnX] Ichi ^{x) = 0.x^^O ^J^'^Ovoix^hl li' [x^xl+u) (55) Ichi = yxij So voi mo hinh (44) thi he d§ cho co: (D'(I) = X I + — ( x f + X j j p + i j j va ad.ft(x) = I2x,i2; Rank! ft(x), orff^(x) = Rank ^1 2X1X2; Hinh Quy dgo trgng thdi mat phdng pha Vgy nd la dieu khien dugc De tim phep doi bien vi phoi - m ( x ) , ta can phai xac djnh ham A(x) tii (47): o4^M,^,(,,= I, 9x| 0X2 J "I Sxi ' Si2 JuJ Hinh Bieu diin quy dao trgng thai theo Ih&i gian 223 Vu Thj Thuy Nga vd Dig Tgp chi KHOA HOC & CONG NUHE Hinh la ket qua md phdng, bieu dien thj quy dao trgng thai ciia he kin gdm ddi tugng phi tuyen bac hai (55) da cho va bd dieu khien FTS (56) thu dugc theo phuang phap de xuat ciia bdi bao De so sanh va kiem chdng chat lupng ma bp difu khien FTS mang lai, hinh cdn bieu dien ca ouy dao trang thai ciia he kin su- dyng bd dieu khien dn dinh hda: U = - X | - ( X j -1-X2) duac thiet ke td ham dieu khien Lyapunov (CLF): V(x) = 2x1 + i^\ -^ 2:2 )^ + (X| - Xj^ - X2 f Hinh la thj quy dao trang thai ciia ca hai bd dieu khien FTS thiet ke theo tdi uu tac dgng nhanh va bg dieu khien dn djnh tifm can thiet ke nhd Lyapunov Cd hai dd thj deu dugc bieu dien theo thai gian Mdt lan niia d day ta thay ro dugc kha nang lam he dn djnh sau khoang thdi gian hdu hgn ciia bg dieu khien FTS vdi khoang thdi gian dn dinh nhd hon 0.85 KET LUAN Bai bao da dua mpt phuong phdp dieu khien dn djnh he phi tuyen vdi khoang thdi gian dn djnh hiiu han dya tren nen nguyen ly cyc dgi de dieu khien tdi uu tac dpng nhanh, Ket qua md phong da chi rdng, so vdi phuang phap thiet ke dn djnh dua theo ham Lyapunov, phuong phap de xudt bai bao dua cac biln trang thai tu dilm ban dSu vl diem can bdng nhanh han hdn 135(05): 219-224 Phuong phap ciia bai bao ciing hoan toan md rpng dugc cho nhirng he phi tuyfn bgc Idn han 2, ma ta da xac djnh dugc bd dilu khien tdi uu tac dgng nhanh phan hdi trgng thai cho he tuyen tinh dang tich phan bgc cao TAI LIEU THAM KHAO Bhaty, S.P and Bemsteinz, D.S (2000); FiniteTime Stabilization of continuous autonomous Systems SIAM J of Control Optimi Society for Industrial and Applied Mathematics, Vol 38, No 3, pp.751-766 Brockett, R.W; Millmann, R.S and Sussmann, H.J (1983); Differential Geometric Control Theory Verlag Birhduse Basel Bostron Chalet, B and Levine, J (1989): On dynamic feedback linearization Systems & Control Letters, vol.13, pp.143-151 Hong, V (2002): Finite-Time Stabilization and stabilizability of a class of controllable Systems Systems and Controll Letters, Vol.46, No.4, 2002, pp.231-236 Isidori, A, (1999): Nonlinear Control Systems // Springer Verlag Levanl,A (2003): High-order sliding mode, differentiation and output-feedback control Int Journal of Control, Voi.76, No.9, pp, 924-941, Moulay, E and Perruquetti, W, (2005): Lyapunov-based approach for finite time stabilit and stabilization Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, December 12-15, 2005, pp 4742-4747 Pontryagin, L.S.; Boltjanskij, V.G,; Gamkrelidze, R.V und Miscenko, E.P (1964): Mathematische Theorie optimaler Prozesse VE Verlag Technik Berlin SUMMARY FINITE TIME STABILIZATION BASED ON TIME OPTIMAL CONTROL Vu Thi Thuy Nga , Nguyen Doan Phuoc Hanoi Umversity ofScience and Technolog This paper proposes afinitetime stabilization control method for nonlinear system based on time optimal control principle By using this method, the trajectoiy ofthe system will converge to zero with an upper bounded finite time Moreover, the conirol signal changes its value maximum only one time Keywords: lime optimal controlfinitetime stabilization Ngdy nhdn bdi:27/4/20I5; Ngdy phdn biin: 20/5/2015; Ngdy duy4i ddng: 31/5/2015 Phdn biin khoa hoc: PGS TS Lgi Khde Ldi - Dgi hpc Thdi Nguyen Email: nga.vulhtthuy@husl.edu.v 224 ... dgng nhanh va bg dieu khien dn djnh tifm can thiet ke nhd Lyapunov Cd hai dd thj deu dugc bieu dien theo thai gian Mdt lan niia d day ta thay ro dugc kha nang lam he dn djnh sau khoang thdi gian. .. the chpn fc de thdi gian dn dinh T khdng Idn ban mgt gia trj cho trudc, dilu ma cac phuang phap dilu khiln FTS tren nin Lyapunov rat khd thyc hien dugc De xac djnh dugc thai gian on dinh T, ta... dung tren nen t6i uu tac dgng nhanh, se dugc xay dyng theo nguyen ly cascade gdm hai bp dieu khien tuyen tinh hda chinh xac (48), (49) va bd dieu khien tdi uu tdc dpng nhanh (53) cho he tuyen tinh