Nguyin Hmi Cong va Dtg T^p chi KHOA HOC & CONG NGHE 132(02): - 1 iTNG DVNG THUAT TOAN GIAM B^C CHO BAI TOAN DIEU KHIEN ROBOT HAI BANK Nguyin Hii-u Cong'", Vu Ngoc Ki^n^ D6 T r u n g H a i ^ Bui Manh Cu^ng^ Dai hgc Thdi Nguyen, Trudng Dgi hoc Kp thudl Cdng nghiep ~ DH Thai Nguyen TOM TAT Dilu khifin ckn bing robot hai b^nh hiSn dang dugc nhilu nha khoa hgc quan tam Mgt kho khan cho bai toan dilu khiln la d6i tugng dilu khiln la khau khong 6n djnh va luon bi nhilu tSc dOng D l giai quylt hki toin nay, c^c tac gia thudng sii dung thuat to^n dilu khiln bin vGng H„ Tuy nhien b6 dieu khiln ckn bang xe hai bknh theo thuat loan dilu khiln bin vung H„ thudng CO bic cao nen d^phiic t?.p Idn lap trinh cho bo dilu khiln va anh hudng tdi chit lugng qua trinh difiu khien thyrc Bai bdo dg dl xuit mgt thuat toan mdi vl giam bac mo hinh n6i chung vk ling dung cho viec gi^m bac hp dilu khiln dilu khiln can bing xe hai banh noi riSng ThuSt toan gi^m bSc mo hinh da d l xuit c6 thi ling dung cac ITnh vyc khdc nhu viln thfing, cong nghe thong tin va todn ling dung Til- khoa: Gidm bgc mo hinh, diiu khiin bin viing, xe hai bdnh DAT VAN DE Trong nhu-ng nam gin day, nghien ciiu vg robot hai banh tu cSn bang da dugc nhilu nha khoa hgc tren the gidi quan tam Trong do, mgt van dl kho khan la nghign ciiu dilu khiln can bang robot hai banh Viec dilu khign can bing cho robot hai banh co the dugc ling dung dg dieu khien cho robot di bang hai chan, nhu robot ASIMO vi nguygn tic dilu khign can bang la nhu Co nhieu nghien ciru vg dilu khiln c3n bang xe hai banh, vi du nhu robot Murata Boy dugc phat trien tai Nhat ban nam 2005 [7], Mgt so phuong phap dugc sii dung dl dieu khien can bang cho xe hai banh la: can bang nhd sii dyng mgt banh da, nhu cac nghign ciiu cua Beznos [1], Gallaspy nam 1999 [3], va Suprapto nam 2006 [9]; can bing bang each di chuyen tam trgng luc ciia Lee va Ham nam 2006 [4] va can bang nhd luc hudng tam cua Tanaka va Murakami nam 2004 [10] Trong so cac phuong phap do, can bang nhd su dyng banh da co uu diem la dap ling nhanh va co thi can bang ca xe khdng di chuyen Co nhigu thuat toan digu khien xe hai banh da dugc dl xuat nhu digu khien phi tuygn ciia Beznol nam 1998 [1], Lee va Ham nam 2002 [4], thilt ke bii bing each sur dung phuang phap tiep can quy dao g6c ciia Gallaspy nam 1999 [3] va dilu khien PD cua Surpato nam 2006 [9] Tuy nhien, nhu'ng thuat toan dilu khien khong bin viing, robot khdng thg mang tai vdi cac tai trgng bien doi, va khong the lam viec moi trudng c6 nhigu loan Vi vay cac thuat toan dieu khign bgn viing cho robot hai banh la rat can thigt Ly thuyet digu khiln bin viing H„ la mgt ly thuyet digu khiln hien dai cho viec thilt kl cac bg dieu khign t6i uu va ben vung cho cac doi tugng dieu khien co thdng so thay ddi hoac chju tac ddng ciia nhigu bgn ngoai Tuy nhign, phuong phap thilt ke H„ ma McFarlane va Glover lan dau tign dua vao nam 1992 [5] va kg ca cac nghien cuu sau vl ly thuyet dilu khiln H^ [8] bo dilu khiln thu duoc thudng co bac cao (bac cua bg dilu khiln dugc xac djnh la bSc cua da thiic mlu) Bac cua bd dieu khiln cao cd nhilu bit loi chiing ta dem thyc hien digu khiln trgn xe, vi ma chuang trinh phiic tap, thdi gian tinh toan lau ngn dap ling cua he thong se bi cham.Vi vay, viec giam bac bg dilu khien ma van dam bao chit lugng co mgt y nghTa thuc tiln Email conghn@tnu edu.v 95 Nguyin Huu Cong vd Dtg 132(02): 95-101 Tap chi KHOA HQC & CONG NGHg Trong bai bao nay, nhdm tac gia lua chgn phuang phap dilu khien can bing cho xe hai banh cd ling dung thuat toan giam bac mo hinh theo theo hai budc nhu sau: tien tdi khong Md ta chi tiet robot hai banh t\[ can bing thabien hinh nhu sau: Mo ta chi tigt cau tao xe hai banh cSn bangco hinh a, Thilt kl bg dilu khiln H„ dl dilu khiln can bing cho xe hai banh, bd dieu khien tim duoc ggi la bg dieu khien du bac r" b, DB xuit thuat toan giam bac bg dieu khien Ho du bac vl bg dilu khiln cd bac thap hon ma vin dam bao chit Iirgng Viec giam bac co y nghTa la giam thdi gian dap img ciia he Mo hinh dgng luc va mo hinh toan hgc robot hai banh tu can bang Mo hinh robot hai banh dugc xay dung dua trgn cd sd dinh luat bao toan dgng lugng co CO sd la: Ngu khong co mgt mo men xoan (mo men luc) ben ngoai nao tac dgng Ign mgt d6i tugng hay he thSng (ho^c t6ng mo men xoan - mo men lire tac dgng vao mgt doi tugng bing khong) thi ting momen dgng lugng ciia doi tugng se dugc bao toan Robot chuyen dgng bang banh, lech khoi vj tri can bang (tuang ling mgt goc nghieng theo phuang thang dung) thi trgng luc ciia robot tao mgt mdmen lam cho xe CO xu hudng xuong Dg tri d trang thai can bing chung toi dat tren robot mgt banh da boat ddng dua tren nguygn ly "con lac ngugc" Banh da se quay trdn xung quanh tryc (vdi gia t6c goc la a) va tao mgt mdmen de can bang vdi mdmen trgng luc ciia robot tao Dg dieu khign gia tdc cua banh da, chung tdi su dung mgt dgng co mdt chigu DC vdi dien ap dat len dgng co la U, ta dua bai toan dieu khien can bing xe vl bai toan dieu khien gdc nghigng (diu ra) bang each digu khien dien ap U (dau vao) d^t len ddng co DC Nhiem vu dat la phai thiet kl mdt bd dieu khien de giir cho robot can bang tiic la giir cho goc (dau ra) luon W,(5) _H{s) I Hinh I Md hinh chi tiit robot hai bdnh tu cdn ban M6 hinh hoa robot hai banh tu can bing voi cac th6ng s6 danh dinh, tac gia thu dugc mo hinh ham truyin danh dinh ciia he thSng can bing robot nhu sau: W(s) -0.223s _6{s) " U(s) s' + 4.7225' - 47.2s - 254 Bo dilu khien H„ du bac cua robot hai banh can bing Tir mo hinh ham truyin ciia he thong can bing robot cho ta thay d6i tuong dieu khiln la he thdng khong dn dinh, Ngoai ra, he thong can bang chiu nhilu tac dgng nhilu loan, Dong thdi tai trgng cua robot can bing ciing CO the thay doi nen dan tdi md hinh ciia h? thong can bang ciing thay doi Do vay thual toan dilu khiln bin viTng la toi uu nhat de dieu khiln he thong can bing robot Cau triic he thong dilu khiln nhu hinh 1: Controller Wo(s) Object •W(s) Q Hinh 2: Cdu trdc he thdng diiu khien cdn bdng robot hai bdnh Thiet kl bg dilu khiln he thdng can bing robot theo thuat toan dilu khiln bin viing H du bac theo phuong phap tham sd hda Youla va can bing md hinh theo tai lieu [8], bg diSu khien H„ dii bac dugc thilt kl nhu sau: (4) Nguyin H&u Cong vd Dtg T?p chi KHOA HQC & CONG NGHE 132(02): 95-101 i/(5) ^-2.23.10's'"-4.67.10^ j''-0.2665'"-22.96s"-IOO65''-2.853.10's"-5.837.10'5" -9.144.I0'5"-1.139.10*5"-1.158.10*5"-9.776.10'5'°-6.949.]0'"5"-4.199.10"5" -2.172.10"5" -9,663.10"5" -3.71.10"5'^ -1.231.lO'^s" -3.53.10"'s" -8.74.10"5" -1.862.10"5"-3.398.]0"5"'-5.276.10^5'-6.903.lO'^s"-7.511.I0"5'-6.676.10''5^ -4.721.10"5'-2.556.10"5''-9.953.10"5'-2.482.10"5'-2.977.10"5-0.00439 D(^) = 4.971.ia^"5"'+2,032.10^"'5"-I-2.663,10-'5"+1.221.10-^5''+9.72,10-'5''+0.39185"+10,145''' + I87.I5''+26125"+2.862.10'5"-1-2.523.10'5^"+1.82.10*5"+1.088.I0'5"+5.428.10's" + 2.273.10''5" +8.005.10*5" +2.372.10*5'* +5.9.10''5'^ +1.225.10'"5" +2.107.I0"'5" + 2.962.10'° 5'" +3.341.lO'^' +2.941.10'^' +1.931.10'° 5' +8.743,10's* +2.286,10's' + 1.519.10°s' -5.226.10'5' +3.6.10^s= +5.32.10-''5 Eg dieu khign du b|ic co bac 30 se din tdi nhilu bat loi chiing ta dem thyc hien dilu khign can bang xe vi ma chuong trinh phiic tap lam thdi gian xii ly se tang Ign, toe dap ling ciia he thong digu khign bi cham va khong dap irng t6t yeu cau ve thdi gian thuc cua bg digu khien va cd the lam he thdng can bang mat on djnh Chinh vi vay dg nang cao chit lugng bg dilu khiln cin phai thuc hien giam bac bd dilu khiln dl mk chuong trinh trd len don gian hon, giam thdi gian xd ly, tang tdc dap irng ma van thoa man dugc yeu ciu on dinh bgn viing cua he thong THUAT TOAN GIAM BAC MO HINH DUA THEO PHAN TICH SCHUR Bai toan giam bac mo hinh Cho mgt hi tuyen tinh, lien tuc, tham sd bat bign theo thdi gian, co nhigu dau vao, nhieu dau ra, md ta khong gian trang thai bdi he phuong trinh sau: x= Ax + Bu y = Cx (5) do, jceD", w e D ' ' , : v e O ^ ^ G a " ^ BeU""", C^W^'" Muc tigu ciia bai toan giam bac doi vdi md hinh mo ta bdi he phuang trinh da cho (5) la tim md hinh mo ta bdi he cac phuong trinh: x^ = A^x^ + B^u y^ = C,x, (6) do, x^eD', w^eD'',y, eD^^,eD™^ B^^U"", C,eD^-"'vdi rU n Sao cho md hinh md ta bdi phuong trinh (6) cd thg thay the md hinh md ta bdi phuang trinh (5) ling dyng phan ti'ch, thiet kl, dilu khiln he thdng Thuat toan giam bac mo hinh dua theo phan tich Schur Hiu bet cac thuat toan giam hkc mo hinh dugc cdng b6 trgn thi gidi deu chi ap dyng cho cac mo hinh tuyen tinh bac cao dn dinh (turc la cac nghiem ciia da thiic dac trimg luon CO phan thuc am), Tuy nhien thuc te, rat nhieu md hinh toan hgc bac cao la md hinh khdng dn djnh, nhu mo hinh bg digu khign bac cao muc ciia bai bao nay, vi the thuat toan giam bSc cSn giam bac dugc cho ca he tuygn tinh khong 6n dinh de co thg ap dung thuat toan giam bac cho mgi ddi tugng ciia bai toan giam bac (md hinh tuyen tinh on dinh hoac khong on dinh) De thuc hien giam bac cho he khong on djnh thi cd hai phuong phap co ban: Phuang phap giam bac gian tiep va phuang phap giam bac true tiep Trong ngi dung bai bao nay, tac gia de xuat mgt thuat toan giam bac mdi ap dung cho he khong on djnh theo phuong phap giam bac gian tiep dya trgn tai lieu [2], [6] Ndi dung cu thg ciia thuat toan nhu sau: Ddu vdo: He khdng dn dinh dugc mo ta (5) 97 Nguyin Hau Cong vd Dtg 132(02): 95- 10] Tap chi KHOA HQC & CONG NGH$ Bir&c 1: Phan tach he khong on dinh hai phan he la phan he In dinh va phan he khong In djnh theo cac budc nhu sau: Bie&c 1.1: Chuyin he thing vl dang tua tam giac ta thu dugc he th6ng c6 dang ^Q+Q^+(cuY {cu) = Q Bir&c 2.3: Phan tich Cholesky ciia Q = R^R, R la ma trSn tam giac trgn Bir&c 2.4: Tinh ma tr^n khdng suy hiln T = UR~^ Bir&c 2.5: Tinh(i,5,c)-(r-'^^„r,r-'5^,,c^,r) vdi A^^^ ^R"""" (vdi m la cac dilm cue In dinh), 4,2 e J?"'*"''"',^22 eT?'"-"'"'"""*, He iA,B,C\ giac tren cd ma tran A la ma tran tam Bie&c 3: Sip xgp cac diem cue ciia phan he khdng dn dinh theo cac budc nhu sau: Bir&c 1.2: Tinh Lyapunov sau tii phuang trinh 4nS-•5422+4,2=0 s S, = JtracQ[^f^ Bie&c 3.2: Chgn thudc tinh trgi Idn nhit 5, Btrdc 3.3: Sap xep lai diem eye X, (va li6n hgp ciia no X^ , neu can thigt) vj tri dau tign tren dudng cheo ciia ma tran A bang ma tran unitary (unitary matrix) Ui: / n-ri Vdi / „ va /„_„ tuong ung la ma tran dem vj Itichthuac mxm va ( n - m ) x ( n - m ) Bir&c 1.4: Tinh [A,B,C^ Bie&c 3.1: Vdi m6i dilm 2,, vdi i = \, n ta tinh toan-.thudc tinh troi tuong ung Bu&c 1.3: Xac dinh ma tran chuyen trang thai B&u vao: He Xi^ {AJ,BJ,CJ)^(W-'A,W,W'B„C,W) BarfciJ: Phan tach he (.4j,Sj,Cj,D) vSdang I /I,,: Q -[^,^1 Q2J * * * * \ * ** * ** * Bir&c 3,4: Tinh he thdng tuong duong mai [uiAV„U[B,CU,) Bit&c 3.5: Bd di hai hang va cot diu tieri ciia {U^AU,,U!B,CU,) Daura; He on djnh (-^rfUjS^pC^/,) ta thu dugc mgt h? thdng nho ( i , S , c ) vdi kich cd n - "" He khdng dn djnh {^j23,-5j2'Q2) 5wr?c Dua phan he 6n dinh {AJ^^,B^^,CJ^\ vl dang tam giac (ma tran A ciia he co dang tam giac) theo cac budc nhu sau: Bie&c 2.1: Tinh phan tich Schur ciia ma tran ^d\\ '• '^dw =U/!iU^, U la ma tran unitary va A la ma tran tam giac tren Bie&c 2.2: Tinh Gramian quan sat Q cua he tir phuong trinh Lyapunov 98 Buac 3.6: Lap lai qua trinh tren tir budc din cho he thong nhd {A,B,C\ va tilp tyc vong Igp cho din tit ca cac dilm cue dugc sip xgp lai theo Idn giam din cQa thudc tinh trgi Bau ra: Hg tuong duong (A,B,C) vdi cic cac diem cue dugc sip xgp lai theo Idn giam dan cua thudc tinh trgi Nguyin Huu Cong vd Dtg T?p chf KHOA HQC & CONG NGHE Diem mdi quan trgng nhit ciia bwdc la dua cong thiic xac dinh tinh trdi ciia dilm cue va kha nang sip xgp theo tinh chit trgi giam dan ciia cac digm cue tren dudng cheo chinh cua ma tran tam giac trgn A , dilu giiip cho thuat toan giam bac co kha nang bao luu cac digm cue trgi ciia md hinh gdc md hinh giam bac ddng thdi thu dugc sai sd giam b|ic nho BW&C 4: Giam bac phan he on dinh theo cac budc nhu sau: Dau vdo: He tuong duong (A,B,CJ [O A,,\' 132(02): 95-101 ;£= ' hc = [C, Q] [«,_ io All f, B"', B, s R"', Ci e B?" Bau ra: He iiit gon (.4,,,iJ],Q) Bau ra: He rut ggn ({A^, B„ C,) + {A,,„ B,„ C,;)) LTNG DUNG THUAT TOAN GIAM BAC M I CHO BAI TOAN DIEU KHIEN CAN B A N G R O B O T HAI BANH Ket qua giam bac bo dieu khien can bang robot hai banh Bg dilu khien H„ du bac dugc thilt kg nhu (4), dd la bd digu khign bac 30 Thuc hien giam bac bg digu khien H» dii bac theo thuat toan giam bac dugc de xuat myc II, ta dugc ket qua theo bang sau: M5 hinh ham truyen - Wcr(s) -4.485.10^5'-6.804,ia'5''-4.123.10^^-1.235.lQ'5^-1.816.10'5-1.09.10' 5^+20095'+1.833.l0'5'-19135'+2.165.10""5-2.804.10-'^ -4.485.10'5''-2.65.10'5^-1.141.10'5'-1.833.IO'5-1.176.10' 5' +2OOO5' -2O6.55' +2.369.10'"5-3.026.10'" Budc 4.1: Chgn s6 bac can nit ggn r cho r