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PHUONG PHAP TONG HOP LUAT DIEU KHIEN CAN TOI UtJ THEO TAC DONG NHANH CHO CAC HE D O N G HOC NHIEU DAU VAO Phan Nguyin Hdi, Hgc vien ky thugt qudn sir Tdm tdt: Bdi bdo gidi thieu vai trd ciia dieu khien can tdi uu theo tdc dgng nhanh vd phucmg phdp de tdng hgp ludt dieu khien tdi uu theo tdc ddng nhanh cho cdc ddi tugnig nhiiu ddu vdo Abstract: This article introduces the role of quasi-time-optimal control and one method to synthesize quasi-time-optimal control law for multi-input dynamic systems I MOf DAU Bai toan tdng hgp ludt dieu khien tdi uu theo tdc dgng nhanh Id mdt nhirng bai todn cdp thilt nhdt ciia ly thuylt dieu khien ty dgng, nhien viec gidi bdi todn ndy gap nhieu vdn de Thiir nhdt, Idi gidi gidi tfch chi cd the tim dugc mgt sd ft tmdng hgp vd chi doi vdi nhirng ddi tugng dieu khien don gidn Thir hai, theo nguyen ly eye dai Pontryagin nhiing Idi giai tim dugc la nhirng luat dieu khien gidn doan vdi hdm ddu sign[) cau tnic, nhirng ludt rdt khd dem dp dung thuc te Nhung sai sd ludn tdn tai cua thiet bj ky thudt cdng vdi sy bien thien dot ngdt ciia hdm sign[j co the Idm cho ludt dieu khien thyc te cd khdc biet Idn so vdi ly thuyet vd he qud Id tfn hieu dieu khien khdng cd tfnh tiem can, he thdng khdng dap img dugc myc dfch de Ndi mdt cdch khdc, nhirng ludt dieu khien tdi uu theo tdc ddng nhanh ly thuyet tren thyc te trd khdng tdi uu Vi nhihig vdn de tren, de gidi bdi todn tdng hgp dieu kien tdi uu theo tdc ddng nhanh, cdc phuang phap xdp xi gan dung thudng dugc sii dyng, va ludt dieu khien tdng hgp dugc dp dyng cdc phuang phap dugc ggi Id ludt dieu khien can toi uu theo tdc dgng nhanh (quasi-time-optimal) (QTO) Nhirng ludt dieu khien can tdi uu theo tdc ddng nhanh dem lai kit qud khdc biet khdng nhieu so vdi ket qua ciia dieu khien tdi uu ly thuyet, nhung he thdng dilu khiln vdi nhirng luat cd nhieu uu diem nhu tfnh on djnh ti?m can vd sy ben vung Trong thdi gian tir 1999-2007, dudi sy hudng ddn cua ^iao su R.A Neydorf ddai hoc ky thudt isdng Ddng Lien bang Nga, viec tdng hgp dilu khien QTO da thu dugc nhilu kit qud qu&n trgn^ nhu da xdy dyng nen khdi niem dilu khiln QTO vdi tham sd dac trung cho mire can tdi uu [3], tire mire dg gdn vdi toi uu theo tdc dgng nhanh ly thuylt, xdy dpg dugc cdc phuang phdp ddi chilu de quy, phuong phdp vi phdi (diffeomorphism) [4] dk ting hgp dieu khien QTO cho cdc ddi tugng dilu khien mgt dau vao bdc bdt ky vdi md hinh toan hgc cd dang Jordan Bai bao la sy phdt triln tilp tyc ly thuylt dilu khiln QTO, xay dyng phuang phap tdng hgp dieu khien QTO cho cdc ddi tugng dilu khiln nhilu ddu vdo Phuang phdp dya tren sy phan tdch he phuang trinh vi phdn md hinh todn hgc ciia ddi tugng nhilu ddu vao cac he cd dang Jordan vd su dyng ket qud cua viec dp dyng phuang phdp ddi chilu de quy hay phuang phdp vi phdi vdo cdc he de tdng hgp ludt dilu khiln IT- DAT V A N D E 702 Cho he nhieu ddu vao dieu khien dugc bdc n voi m tfn hieu dieu khien (m < n) ^=f{x^,x,, ,x,,,u^,u,, ,u,^,),i = l,n, H) d day cdc hdm /, khd vi theo tdt cd cac biln cua minh vd — ^ = 0, V/ = 1, n, \/j = 1, m, m>i dii-j Cdn tim vecta dilu khiln u = (», ( A ) , « , (A), ,«„, (A-)),A = (A,,A,,K AJ, cd nhiem vy dua he (1) tir trang thai ban ddu (.if, A°,K AJ) ndo dd vl gdc he tga dg sau mdt thdi gian r^,„^, gdn vdi thdi gian nhd nhdt ly thuylt vdi r^,„^, cd thi thay ddi dugc bdng cdch thay doi cdc tham sl ndo dd, dieu kien phdi bdo ddm han chl dii vdi dao hdm ciia mgt sl biln trang thai 4N4->//e(l.2 n) (2) in NHlTNG TIEN Bt VE BltV KHifeN CAN T I UtJ THEO T A C DONG NHANH Ly thuylt dilu khiln can tdi uu theo tdc dgng nhanh dugc bdt dau tu viSc xem xet viec tdng hgp ludt dilu khien cdn toi uu cho ddi tugng bdc don gidn nhdt JS^U, , (3) vdi myc tieu dua A tir trang thdi ban ddu x^ bdt ky vl vd phdi bdO ddm dilu kien |4^^ MQ • \ ^ 0 • Ddi vdi (3), ludt dieu khien tdi uu theo tdc dgng nhanh ly thuyet cd dang u"^ =-u,sign{x) (4) Ddi vdi ham sign[), bai bdo [3], gido su R.A Neydorf da xdp xi nd bang hdm khd vi • = sign[x) voi f > 0, vd he qud la ludt dieu khien QTO cho (3) cd dang ui^op< = _ „ ^ (5) °V7+^^ Ap dung ludt (5), ddi tugng (3) trd thdnh ^-"0-7=^ (6) ^Jx- + £•' Ddi tugng (6) ben viing tiem can vdi ham Lyapunov x', ban nua ludt di8u khien (5) la hdm khd vi va cd tfnh tiem can tai dilm myc tieu Tham sd e dac trung cho miic gdn ciia (6) so vdi chuyin ddng tdi uu theo tdc ddng nhanh ciia (3), e cang nhd thi (6) cdng gdn vdi tdi uu ly thuylt Nlu ky hieu T"""" (e) thdi gian dilu khiln (3) vdi ludt (5), T""' - thdi gian dilu khiln (3) bdng ludt (4), thi T"'""' {e) khdng chdm hon nhilu so vdi T°" va \iinr"'""'ie) = T">" (7) e-*0 Ngoai ham - j = dk xdp xi vdi hdm sign (A) cdn cd the su dyng cdc ham th /x' + e' arctang - \£ , Cac hdm cd khd nang xdp xi sign{x) dugc xem xet chi tilt [1] Doi vdi ddi tugng dilu khiln mdt ddu vao bac cao ban I, chiing ta chi xem xet cdc ddi tugng vdi md hinh toan hgc cd dang Jordan [2] 4=/(^P-^2.Kx,.„) , / = l , n - l ; = /«h'^2.K.v„) + g(A,,A2, ,A„)«, (8) 703 ddaycacham /" kha vi theo cac biln v,,.v,,K ,v,, , cdn ^l V ;^0,V/ Dl tono hop luat diC^u khiln QTO cho (8), giao su R.A Neydorf va tien sy Tr§n Nouyen Nagc da xay dung nen phuang phap dii chilu dc quy va phuang phap vi phoi j4], hai phirang phlp bao dam cho dao ham ciia bi(in trang thai dau tien he dugc tong hep ndm mot gidi han yeu cau dat |,iSf| < 4,™N ] Phuong phan doi chieu dc ciuy Phirang phap dii chilu dc quy dugc xay dung tren gia dinh rdng ludt dieu khien QTO " r r ( ' ' -^ = {x^,x,,K ,v„_,f cho he bac (/i-l) sau da dugc xac djnh €i= /;.(A,,A„K A,„) / = l,/J-2; ^ , = «„_„ (^^ dd ;i - ham vl phdi ddu tien, cac gid trj ban ddu (.t° v_?,K xl) va dilu kien | j ^ < ^ ciia he triing vdi cac ylu to tuong img ciia he (9) Theo cdch xay dung, h? (9) se la mgt he mau dgng Id mgt diem ma (8) se phdi di qua dl tiln vl goc tga dg vdi thdi gian can eye tilu va ludt dilu khien u (8) se dugc ting hgp nhu Id tac dgng dua he (8) tiln din he (9) cung theo mdt tiln trinh can tdi uu theo tac ddng nhanh Vdi each tilp can ndy ludt dilu khiln QTO cho (8) se tim theo cdng thiic dequy (/ -JTl-lf/.^M^^ u'"""(x,e) = 1=1 'a/.-,^ V -I •g -\ ^x„ J f.-8-\ m dday h(fi,e) la mgt ham tron vd /z(//,f)-//, ' ^ i n - , = ^„(3'«.^J % y ^ O , / l , ( , £ , ) = Theo each xdy dyng, bieu thirc cua cdc bien cua he (11) tim dugc theo cdng thiic de quy sau >'i=^i ; Xx = - ' ' / ( > ' , ' ^ , ) i = ln-l, vd tir ciia phuang trinh cudi 38^ = h„{y„,£„) se nhdn dugc ludt dieu khien QTO cho (8) theo cdng thiic f u= 704 fay, M ^Xj ^xj •8~' -fn-8 -1 m IV PHUONG PHAP TONG HOP DIEU KHIEN CAN TOI UU THEO TAC DQNG NHANH CHO CAc HE DONG HOC NHIEU DAU VAO _^ Tren ca so nguyen ly phan tach cac bdi todn ting hgp dilu khiln cho cdc he da chieu va nhirng kit qua da dat dugc baj todn ting hgp dilu khiln QTO cho cdc he mgt dau vdo vdi md hinh todn hgc dang Jordan, bdf todn ting hgp dilu khiln QTO cho he nhilu , ddu vdo (1) se dugc gidi theo phuang phdp sau: nghien ciru khd ndng phan tdch he (1) cdc he cd dang Jordan; trirdng hgp phdn tdch thdnh cdng, lira- chgn mgt phuang dn phdn tdch va nghien ciru khd nang ting hgp ludt dilu khiln QTO cho timg he bdng phuang phdp ddi chilu de quy hodc phuong phdp vi phdi; tren co sd kit gud tdng hgp ludt dieu khiln QTO nhan dyge^cho tirng he rut vecto dilu khiln QTO cho he ban ddu (1) dudi dang gidi tfch rd rdng,^ Van de nghien ciru khd ndng phan tdch he (1) cdc he cd dang Jordan.da dugc gidi quyet d [5,6], cd nghia Id d budc cd thi gid djnh he (1) phdn tdch dugc m he cd dang Jordan sau 4.i=/y.i( xj., ); dfj., dfj_i n, ^ *^^y ^—'~ '^ ^ ' ;N 9^ 0,y = 1,»I , / = 1,^^ - , ^k J =n; biln trang thdi cua he ndy "Xjj^i aUj J^^ khdng la bien trang thdi ciia he khdc Ap dung phuang phdp ddi chieu de quy hodc phuang phdp vi phdi vdo he (13) de ting hgp ludt dilu khiln QTO, kit qua nhan dugc se cd dang Hj (J:,M^ , ) = Ldm mdt cdch tuang ty ddi ydi cdc he khdc, ket qud nhdn dugc Id he m phuang trinh cd m biln la m tin hieu dieu khien cua he (1) H^{x,u^, ) = 0; Hj{x,Uj, ) = 0; (14) H„{x,u,„, ) = Nghiem cua he phuang trinh (14) chi'nh Id ludt dieu khien QTO cho he (1) Neu he (1) phdn tdch dugc cdc he cd dang Jordan toan eye thi (14) Id he phuang trinh tiiyen tfnh vd cd the gidi bdng phuang phdp Cramer de nhdn dugc vecto ludt dieu khien QTO cho he (1) NIU he (1) chi cd khd ndng phdn tdch dugc cdc he cd dang Jordan eye bg thi he (14) se cd dao hdm cua mdt sd tfn hieu dieu khien Trong trudng hgp ndy khdng cd phuang phdp chung dk gidi, nhien ddi vdi mdt sd trudng hgp rieng van cd khd nang gidi dugc bdng thudt todn gidi timg phdn sau: Lya chgn trono he (14) he phuang trinh gdm cdc phuang trinh khdn^ chua d^o ham ciia cac tfn hieu dieu khiln Gidi he phuang trinh ndy, tim dugc mdt sd ludt dieu khiln QTO Thay cac ludt nhdn dugc vdo he (14), riit ggn lai he (14) thdnh he phuong trinh cua cac dieu khien chua tim dugc vd quay lai budc Phuang phdp tdng hgp dieu khiln QTO ndy cd the sur dyng cho rdt nhilu he thdng ky thudt nhilu ddu vdo, sd dd cd thi kl din he dilu chinh dn djnh chuyen ddng gdc cua ve 705 tinh, he dieu khien hieu dien the va cdng suat co hgc ciia may phdt tua bin, he dieu khien dc CO dien mot chieu [5\ Phuong phap cung da dugc hien thyc tren may tfnh, phan mem nhan dugc giiip cho viec long hgp, md phdng, tfnh loan tham sd ludt dieu khien QTO trd nen rat de dang Xet vf du dgng co dien mgt chieu vdi phan kfch thfch dgc lap [5]: dco^C,„0(l,)l, M^ dl, ^C0(f)co U, I,R, dl, _ U, I.R, L, dt L, • + dt J J dt L, L, L (15) a day co - v^n toe goc cua rotor dgng co; /, - ddng dien ciia phdn irng; / - ddng dien kfch thfch; I^yLj ~ ^9 ^M" ^^^ '^'^^ P'^^" '^"S ^^ P'^^" ^^^^ thich; /?,,/?, - cdc dien trd; U,,U2 dilu khiln va ( / J = 1.129-1.129^-'*"- Cdn tdng hgp dilu khiln QTO, cd nhiem vudua van tdc gdc co va ddng dien kfch thfch / , tir gid tri ban ddu ndo ve dfch i^aij,l2j) Phuang phap da xdy dung giup f/, ,f/-, tim dugc dudi dang gidi tfch, cdng thiic ciia chiing khdng the hien d ddy rdt ddi Hinh thi hien qud trinh qud ciia OJ vd /, vdi (y(0) = / , ( ) = , / , ( ) = ; ft;, = /,,, = ; C,„ = M^ = J = C^ = R, = R,=l , I^= L, =1 Cdc qua trinh hodn todn khdng cd tfnh dao dgng y 1 1• I r >M 1—r-t—i r i -/-i 1 ^ 1 1 1 1 > 1 1 1 1 1 1 1 — 4—!——-1 —^_4 ! 4—J II 11 II -.,.J (• Ui iti 1 1 int 1 1 1 1 KH TH -la 1 1 sn 1 1 1 1 ^m »• mi II 11 II II 1 I 1 I 1 1 ! I 1 1 :iii i Al II II 1 1 1 1 II II 11 i 1 1 1 ! 1 II •• Itt III IS U< la 1 IE M !• Hinh I Qud dg cua (15) V K E T LUAN Bdi bao trinh bdy mgt phuang phdp tdng hgp dilu khiln can tdi uu theo tdc ddng nhanh, phuong phdp ndy cd the dp dyng cho mgt Idp rdt rgng cdc he dgng hgc nhilu dau vdo, dem lai Igi fch thiet thyc cho viec thilt ke cdc he thdng ky thudt 70d TAI LIEU THAM K H A O BojiKOB P.B KBa3HonTHMH3amifl GbicTpoxtei^ciBHa aciiMnxoTHHecKH ycxoHHiiebix CHcxcM ynpaBJieHHH / P.B BOJIKOB // /^HCC i
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