^- -^ GIAM BAC MO fflNH DlTA TREN PHAN T I C H S C H U R '• ^ f e ^" ^^^ ^JT-"' MODEL ORDER REDUCTION BASED ON SCHUR ANALYSIS FOR TWO-WHEELED BALANCED MOBILE ROBOT CONTROL ^^^ TOAN DIEU KHIEN ROBOT HAI BANH Ty" CAN BANG Bill Trung Thanh', VQ Ngoc Kien-, Nguyin Hiru Cong^ / Trudng Dgi hpc Supham Ky thudt Hung Yen Truong Dai hoc Ky thugt Cdng nghiep Thdi Nguyen Dai hpc Thdi Nguyen Tom tit: Trong nhOng nam gdn ddy dd eo nhieu nghiin cuu linh vuc gidm bde md hinh Dd co nhieu thugt todn gidm bgc dugc giai thieu gia Igt cdc diem cue quan trpng cua hi ban ddu Bdi bdo ndy trinh bdy mot thudt todn gidm bae md hmh m&i dua tren phdn tich Schur Thugt todn chuyen ma trdn A cita he bgc cao ban ddu vi dang ma trgn tam gidc trin cdc diim cue duac sdp xip gidm ddn vi tinh quan tren dudng cheo ehinh euar ma trgn Ddng thdi nhom nghien ciru cUng edi tien thudt todn de chuyen ede thdng so eiia hi dirge gidm bdc tie dgng so phue ve dgng sd thuc Tinh hiiu qud cua thudt todn gidm bgc md hinh mai dugc minh hga qua bdi todn gidm bdc bg dieu khien cua he Robot hai bdnh tu cdn bdng Ket qud md phdng chi tinh hiiu qud cita thugt todn Tir kh6a: gidm bdc md hinh, phdn tich Schur, Robot hai bdnh tu edn bdng Gidi thieu chung Cl bai bao trudc [8] nhom tac gia da gidi thieu thuat toan giam bac can bang cua Moore vdi y tudng chinh ciia thuat toan la dua vao cac trang thai ddng gop nhieu hay it vao dap ung qua dp dau eua he goc (cac gia tri Hankel suy bien) de thuc hien giam bae ma khong quan tam den cae diem cue cua he goc CO dugc bao luu he giam bac hay khong Tuy nhien thue te thi cae diem cue quan trpng ctia he goc la bat bien Do do, ngi dung bai bao tiep theo nay, chiing toi gidi thieu mgt thuat toan mdi, thuat toan giam bac dua theo phan tich Schur, dua tren y tudng giir cac diem cue quan trpng ciia he goc qua trinh giam bac, dong thai img dung thuat toan vao giam bac bg dieu khien ciia bai toan dieu khien can bang robot di dgng hai banh ThuSt toan giam bac md hinh dira theo phan tich schur 2.1 Bai toan giam bSc md hinh Cho mgt he tuyen tinh, lien tuc, tham so bat bien theo thdi gian, co nhieu dau vao, nhilu diu ra, mo ta khong gian trang thai bdi he phuang trinh sau: x - A x + Bu (I) y-Cx do, X e R", u e RP, y e Ri, A e R""", B e R"~p, C G Ri™ Muc tieu eua bai toan giam bac doi vdi md hinh md ta bdi he phucmg trinh da cho (I) la tim mo hinh mo ta bdi he cac phuang trinh Xr — ArXr + BrU ,-, yr - ax, do, x_ G R', u G RP, y^ G K\ \ G R-^^ B^ G R'^p, C^ G Ri"; vdi r < n; ' Sao cho mo hinh md ta bdi phuang trinh (2) CO the thay the mo hinh mo ta bdi phuang trinh (1) ling dung phan tich, thiet ke, dieu khien he thong 2.2 Thuat to^n giam b^c md hinh dua theo phSn tich Schur Thuat toan giam bac md hinh dua theo phan tich Schtu dugc phat trien bdi nhom nghien eiiu dua tren ea sd ky thuat eat ngan va phan tieh Schur Ky thuat cat ngan la mpt phuang phap giam bac don gian Trong y tudng ehinh eiia no cd the phan chia lam budc: budc chuyen doi he thong goc bac cao ve he thong tuang duong bang mpt chuyen doi khdng suy bien khdng gian trang thai, bude xoa mpt so hang va mpt sd cot cua h$ thong tucmg duang de tao he thong giam bac Hai thuat toan tieu bieu nhat cho ky thuat eat ngan la eat ngan can bang [8] va cat ngin mo hinh nhieu loan [12] Tuy nhien nhugc diem cua ca hai phuong phap la viee su dung phan tich gia tn suy biln (SVD) can ed rat nhieu dieu kien rang bugc, mat khac viec cat ngan he thong la dua vao cae gia tri Hankel suy bien (trang thai tuang iing vdi gia tri Hankel suy bien nho thi loai bd) nen dan tdi cac diem cue quan trpng eua he thdng gdc khdng dugc bao toan he thong giam bac Tuy nhien, eac diim cue quan trgng la bat bien he thdng thuc vay no cin dugc bao toan qua trinh giam bac Tu nhihig nhugc diem tac gia Minh H.B [6] ciing nhdm nghien cuu da dua y tudng chinh eua thuat toan la chuyen ddi ma tran A ciia he thdng Khoa hoc & Cong nghe - So 02/2014 (I) vl dang ma tran tam giac tren dua theo phan tich Schur (tranh phan tich SVD), tren ca sd cae gia tri diem cue dugc sip xip theo tinh chit quan trgng giam din tren dudng cheo chinh cua ma tran tam giac tren A Sau thuc hien bude eua ky thu^t cat ngan, nhu the cac diim cue quan trgng dugc bao toan he th6ng giam bac Dilm mdi quan trgng nhat eua thuat toan la kha nang sip xip theo tinh chat quan trgng giam din cua cae dilm cue tren dudng cheo chinh cua ma tran tam giac tren A va kha nang bao luu eac diem cue quan trgng ciia mo hinh goc mo hinh giam bac Trinh tu thuat todn nhu sau: Dau vao: He thong goc mo ta khong gian trang thai (A,B,C,D) Bu&c 1: Tinh toan Gramian quan sat cua he thong gdc tir phuang trinh Lyapunov A*Q -i- QA +C*C = Bu&c 2: Tinh phan tich Cholesky ciia Q = R*R Budc 3: Tinh phan tich Schur ciia RAR': RAR"'=U A U*, U la ma tran true giao va A la ma tran tam giae tren Chgn U dugc thuc hien theo eae bude sau: Budc 3.1: Tinh toan Gramian dieu khien dupc cua he thdng goc tir phuang trinh A*P+ PA+ BB* = Btrde 3.2: Phan tich gia tri suy bien eua RPR" I = V* L^ V Dat v \k cot thii i ciia ma tran V* Budc 3.3 Tinh n vec ta gia tri rieng x,, x^, ,x^ciia RAR' Budc 3.4: Cho i chay tir den n chpn x^ cho I vjxi I dat cue dai Dat u, = x^ la egt dau tien ctia ma tran U Buac 3.5: Cau tnic ma tran U, vdi U|^la edt dSu tien Khi ma tran U|(RAR-')Uj co the dupc phan U^* U,*(RAR-')Uj U = A, * M, Bude 3.9- Tinh n-i-Hl vecta gia tri rieng z,, Zj, , z^^^^ c i i a M Buac 3.10: Choi chay tir denn-i+1 ehgnz cho )v'Ui U,col(0 z,}| d?t euc dai Dat u =^ U, U col(0 z) la cot thli i eiia ma tran U Ket qua cudi ciing ta duoc ma tran U = U| Bu&c 4: Tinh ma tran khong suy bien T = R'U Bu&c 5: Tinh toan (A^, B^, C^, D^) = (T'AT, T'B,CT,D) Bu&c 6: cit ngin (A^, B^, C^, D^) vl dang (A^, B^ C^, D^) dya theo thuat toan can bing cua Moore Diu ra: Mpt he giam bac (A^, B^ C_, D_) Ndi dung chi tiet cua thuat toan dupc trinh bay [6] Tuy nhien thuat toan giam bae tren vin eon mgt nhugc diem la thuat toan dupc thuc hien tren trudng so phuc, dan tdi cac ket qua dau la he giam bac (A^ B_, C_, D^) cd thong so phue Diiu gay khd khan cho viec chuyen doi he thong tir mo hinh trang thai sang dang mo hinh ham truyen de thuc hien mo phdng he thong d dang ham truyen, ciing nhu ve cac dap iing bude nhay cua he thong Matlab va Simulink De hoan thien thuat toan, nhom tac gia da thuc hien chuyen doi ket qua giam r-i, bae tir trudng sd phuc ve trudng so thuc Ngi dung cu the thuc hien theo thuat toan sau' u,.(RAR-OU, = [„ J Cac budc tu I den giong nhu thuat toan Budc 3.6: Tmh n-\ vec ta gia tn rieng y,, y^, ban dau , y^, ciia Mj Bir&c 4m: Tinh ma tran khong suy bien T Biedc 3.7: Cho i chay tir I den n-l chpn y^ - R 'U^ cho I V2 Ui col (0 y,) | dat cue dai D^t u, = U, Chgn U dugc thuc hien nhu sau: col(0 y ) la cot thii ciia ma tran U Bi«7c 4.7; Phan tich Schur cua RAR': RAR'=U_ At U^*, dd U^ la ma tran true giao va At Tir day ta se tim duge cae cot tiep theo eua la ma tran lam giae tren theo thuat toan Schur thong U theo each sau: thudng (lenh schur phan mem Matlab va simBudc 3.8: Cau triic ma tran V vdi u la cot ulik) Budc 4.2- So sanh vi tri diem cue quan trgng ri,dau tiSn Dat U •trgn dudng cheo ciia U vdi vi tri cac diem cue tren At Ket qua so sanh tra mgt ma tran cot K, vdi Budc 3.6: Tinh n-l vecta gia tri rieng y,,yj, cae gia tn ciia cot la vi tri cac diem cue quan trpng duae sap xep tren A, ,, y J cua M^ Khoa hgc & Cong nghe - So 02/2014 17 dilu khiin tli uu va bin viing cho cac doi tugng diiu khiin cd thong s6 thay dfli hoSc ehju tac ddng ciia nhieu ben ngoai Tuy nhien, phuang ph^p thilt kl H^ ma McFarlane va Glover lin dau tien dua vao nam 1992 [5] va ke ca cac nghien ciiu sau vl ly thuyit dilu khiin H^ [2] bg dilu khiin thu duge thudng cd bae cao (b|e cua bg dieu khien dugc xac dinh la bac cua da thurc mau) Bac cua b6 dieu khien cao cd nhieu bat lgi ehiing ta dem thuc hien dilu khien tren robot, vi ma chuang trinh phde tap, thdi gian tinh toan lau nen dap iing ciia he thing s£ bi cham.Vi v3y, viec giam bac b6 dilu khiin ma vin dam bao chat lugng ed mgt y nghia thye tien Trong bai bao nay, nhdm tac gia lua chgn phuang phap dieu khien can bang cho robot di dpng hai banh eo dng dung thuat toan giam bSc mo hinh theo theo hai budc nhu sau: a Thill kl bp dilu khiin H^ d^k dieu khien iTng dung thuat toan giam bac cho bai toan cSn bang cho robot di ddng hai banh, bg dieu khien dieu khiSn cSn bang robot hai banh Trong nhung nam gan day, nghien cim ve ro- tim dugc goi la bd dieu khien dii bac b, Be xuit thuat toan giam bac bg dieu khien bot di dgng (mobile Robot) da dugc nhieu nha khoa hgc tren the gidi quan tam Trong dd, mgt van de H dii bae ve bp dieu khien co bac thap han ma van kho khan la nghien ciru dieu khien can bang robot dam bao chat lugng Viec giam ble co y nghia hai banh, Viee dieu khien can bang eho robot hai la giam thdi gian dap img cua he banh co the dupe img dung de dieu khien cho robot di bing hai chan, nhu robot ASIMO vi nguyen tac 3.1 Mo hinh d^ng lyx! va mo hinh toan hoc robot diiu khien can bang la nhu di d$ng hai banh Co nhieu nghien cuu ve dieu khien can bang Mpt mo hinh robot di dpng hai banh can cho robot di dgng hai banh, vi du nhu robot Murata bang dugc trien khai tai Phdng thi nghiem Ca - dien tu Vien Cong nghe Chau A (AIT), Thai Lan nhu la Boy duoc phat trien tai Nhat ban nam 2005 [7] Mpt so phucmg phap dugc sit dung de dieu ca sd de kiem tra hieu nang ciia thuat toan dieu khkhi^n c3n bang cho robot hai banh la: can bang nhd ien dugc phat trien bdi Thanh B.T [11] Co the mo su dung mgt banh da, nhu cae nghidn eihi ciia ta ngan ggn rao hinh ddng lyc ciia robot di ddng hai Beznos [I]- Gallaspy nam 1999 [3], va Suprapto banh can bang nhu sau: robot chuyen d5ng bang nam 2006 [9]; can bang bang each di ehuyen tam banh, lech khdi vi tri can bang (tuang iing mgt trpng lye ciia Lee va Ham nam 2006 [4] va can bang goc nghieng y theo phuang thang diing) thi trgng nhd lue hudng tam ciia Tanaka va Murakami nam lyc ciia robot tao mdt momen lam cho robot e6 xu hudng xudng De tri d trang thai can bing 2004 [10] Trong sd cac phuang phap dd, can bang nhd ngudi ta dat tren robot mgt banh da boat ddng dya tren nguyen ly "con quay hoi chuyen" Banh da sir dung banh da cd uu diem la dap iing nhanh va CO the can bang ca robot khong di chuyen se quay tron xung quanh true (vdi gia tdc goc la a Co nhieu thuat toan dieu khien da dugc de ) va tao mgt mdmen de can bang vdi momen xuat nhu dieu khien phi tuyen ciia Beznol nam 1998 trpng lyc cua robot tao D I dilu khiln gia tic [1], Lee va Ham nam 2002 [4], thill kl bu bing eiia banh da, ta sir dung mdt dpng co mgt chilu DC each sir dung phuang phap tiep can quJ dao gdc vdi dien ap dat len dgng ca la U, ta dua eiia Gallaspy nam 1999 [3] va dieu khien PD cua bai toan dieu khien can bang robot ve bai toan dieu Surpato nam 2006 [9] Tuy nhien, nhihig thuat toan khien goc nghieng y (dau ra) bing each dilu khiln dieu khien dd khong ben vung, robot khong the dien ap U (dau vao) dat len ddng co DC Nhiem vu mang tai vdi cac tai trgng bien ddi, va khdng the dat la phai thiet ke mgt bg dieu khien dl giii cho lam viec moi trudng co nhieu loan Vi vay cac robot can bing tuc la giu cho gdc y (diu ra) ludn thuat loan dieu khien ben vung cho robot di ddng tien tdi khong Mo ta chi tiet cau tao robot hai banh can bang co [11] Md hinh ham truyin cua hai banh la rat can thiet hg thing can bing robot dugc mo t^ [11] nhu Ly thuyet dieu khien ben vimg H^ la mgt ly sau: thuyet dieu khien bien dai cho viec thiet ke cac bg Buac 4.3: Chgn bac r ctia he thong giam bac Budc 4.4- Tinh ma tran cot K_ bing each cat di n-r hang ciia ma tran K Birdc 4.5: Tinh ma tran cot E(l,n), dd cac hang co gia tri bang la gia tri ciia K^, cac hang lai cogiatriO Bu&c 4.6 Phan tieh Schur ciia RAR"'theo ma tran E' RAR"'=U^ A \}*, dd U^ la ma tran true giao va A la ma tran tam giac tren theo thuat toan Schur ed sip xip (lenh ordschur phan mem Matlab va simulik) Budc 4.7 Tinh ma tran khong suy bien T = R-'U^ Bude va hoan toan giong thuat toan ban dau Dau ra: Mdt he giam bae (A^ B_, C^, D^) co thong sd thuc 18 Khoa hoc & Cong ngh? - So 02/2014 W(s) y(s) U(s) dieu khien he thdng can bang robot Cau true he thdng dieu khien nhu hmh 4887 " s"" + 683 3s' + 1208s^ + 109700s - (3) 3.2 B6 dilu khien H^ du bic cua robot di ddng hai banh c3n bing Tir md hinh ham truyen eua he thong can bing robot eho ta thay ddi tugng dilu khiln la he thong khdng on dinh Ngoai ra, he thong can bang chiu nhieu tac dpng nhilu loan Ddng thdi tai trpng eua robot can bang eiing co the thay dSi nen dan tdi mo hmh ciia he thdng can bing cung thay ddi Do vliy thuat toan dieu khien bin viing la tli uu nhit dl Wc (s) = Bg dieu khien ^ He Robot W(s) Hinh Cdu true he ihdng dieu khien cdn bdng robot di ddng hai bdnh Thiet ke bg dieu khien he thing can bang robot theo thuat toan dieu khien bin virng H^ du bac duge chi chi tiet [10], bp dilu khiln H^ du bae dugc thiet ke nhu sau: 1275s' + • 695 • lO^s' + 5.151 lO's' + 1.359 lO's' + 2.435 lO^s + 1.091 • lO' * + 715.7s' + 2.355.10S' + 2.789.10's^ + 3.802.10's' + 6.519.10's + 2.872.10' Bp dieu khien du bac cd bac se dan tdi nhieu bat lgi chiing ta dem thyc hien dieu khien can bang robot vi ma ehuong trinh phiic tap lam thdi gian xu ly se tang len, toe dap iing ciia he thong dieu khien bi eham va khdng dap iing tdt yeu eau ve thdi gian thyc ciia bg dieu khien va co the lam he thong can bing mat on dinh Chinh vi v|y de nang cao ehat lugng bg dieu khien can phai thyc hien giam bac bg dieu khien de raa chuang trinh trd len don gian ban, giam thdi gian xii ly Bac Wc(s) tang toe dp dap iing ma van thoa man dugc yeu cau on dinh ben vimg cua he thing 3.3 Ket qua giam bac bd dilu khien cho robot di dong Bp dieu khien H^ dii bac dugc thiet ke nhu (4), la bg dieu khien bac Thue hien giam bac bg dieu khien H^ du bac theo thuat toan giam bac da neu tren, ta dugc ket qua theo bang sau: Mo hinh ham tniyfin - W (s) Sai so |W.(S)-W„{S)||H 1275s' + e s ' + e s ' + 1.359e8s + 1.209e7 s ' + s ' + e s ' + 2.768e5s' + 777e6s + 3.183e5 5.1995e-005 1275s' + I s ' + 1.993e5s + 1.773e4 ,' + 33 « , ' + 3q7 q = ' + 5540^ + ISK 4.3560e-004 1275s'+ s + 9 e s ' + 33 7Ss' + 395s + 55nfi 1.7910 1130s + 247.6 s" + 30.25s + 94.43 37.2364 I 1006 s + 26.71 38.1419 * De danh gia mo hinh giam bac, nhdm nghien ciru da mo phdng dap iing qtia cua bg dieu khien dii b$c va eac bd dieu khien da giam bae Ket qua mo phong tren Matlab - Simulink nhu hinh (trang ben) Tir kit qua md phdng ta thay: So vdi dap ung h(t) cua bd dilu khien du bae thi dap ung h(t) cua bg dilu khiln giam bac 5, triing khdp hoan toan; dap ung h(t) ciia bg dieu khien giam bac co sai khac rit nhd; dap iing h(t) eua bg dilu khien giam bac 2, bac sai khac rat nhieu Do ta co the dung bp dieu khien giam bac: 5,4, thay the bg dieu khien du bac Tat nhien, d day ta ehgn bp dieu khien bac thay the eho bp dieu khien gde bac 3.4 Sudung bo dieu khien giam bac dieu khiln Robot di ddng hai banh Su dung bd dieu khien giam bac d bang I de dieu khien he thdng can bang cho robot di dgng hai banh co mo hinh ddi tugng dieu khien nhu (3) Khoa hpc & Cfing nghe - So 02/2014 19 D I thay ro chat lugng, ta so sanh vdi bo dieu khiln du bae (bac 6) Viec md phong nhd Matlab/ Simulink, ket qua mo phong nhu hinh 3, 3.5 Nhan xet ket qua - Coi bp dilu khiln (4) la mgt md hinh toan hpc tuyin tmh bae 6, ta hoan toan co the giam ve md hinh bac 5, 4, ma dap iing diu gin nhu khdng thay dli ta tae dgng dau vao la ham l(t) nhu hinh Dilu khdng nhumg co y nghTa ky thuat ma cdn co y nghTa ITnh vyc toan iing dung va viee giam bac phuang trinh vi phan tuyen tinh - Kit qua mo phdng hinh cho thay: sir dung bg dilu khiln giam bac dieu khien robot di dgng hai banh can bing cho ket qua dap iing h(t) tuang duang nhu bp dilu khien dii bac Nhu vay ta CO the thay the bp dieu khien du bac bang bg dilu khiln giam bac ma chit lugng bp dieu khien van dugc dam bao Hinh Kit qud md phdng bo diiu khien du bdc vd Ket luln cdc bd diiu khiin gidm bdc Bai bao da de xuat va hieu chinh thuat loan giam bae mo hinh dya theo phan tich Schur va ap dung cong bai loan dieu khien can bang robot di dpng hai banh: ehuyfin bp dieu khien dii bae theo H (bac 6) ve bg dieu khien giam bac Ilet 1UI12 ilieii hliieii ciiii bails lobot t)i dons vdi chat lugng he thong dieu khien van dugc dam hniili ihma bo (tiai kliiai 20c bac bao Sit dung bp dieu khien giam bac se lam ma chuang trinh dan gian han, tang toe dp tinh toan, thdi gian xu ly nhanh han va dam bao tinh thdi gian thue ciia he thdng dieu khien robot can bang Cac ket qua md phdng the hien tinh dung dan cua thuat toan da de xuat Mgt sd van de ma nhom nghien ciiu se cong bo them bai bao tiep theo, la: Vdi mgt mo hinh cho trudc, chiing minh bang ly thuyet toan hpc de khang dinh viec giam mo hinh den dau ma van dam bao mgt sai so eho trude va ddng thdi so sanh hieu qua eua phuong phap giam bac eSn bing va phuong phap giam bac dya theo phan tich Schur viec giam bac doi tugng dieu khien, iing dung Hinh Ket qud md phdng he thong dieu khien cdn bdng robot di ddng hai bdnh sir dung bd diiu khienthuat loan giam bac md hinh cho cac ITnh vuc khae nhu vien thdng va edng nghe thong tin dii bae vd bd diiu khiin gidm bde \ , : T^i Min (ham khao [1] Beznos AV, Formalsky AM, Gurfinkel EV,Jieharev DN, Lensky AV, Savitsky K V, et al (1998) "Control of autonomous motion of two-wheel bicycle with gyroscopic stabilization, " In: Procee ings of the IEEE intemational conference on robotics and automation 1998, p 2670-5 [2] Chu YC, Glover K, Dowling AP (2003) "Control of combustion oscillations via H loop shaping, fi-analysis and integral quadratic constraints, " Automatica 2003; 39(2): 219-31 [3] Gallaspy JM (1999) "Gyroscopic stabilization of an unmanned bicycle " M.S Thesis, Auburn University [4] Lee S, Ham W (2002) "Self-stabilizing strategy in tracking control of unmanned electric bicycle with mass balance," IEEE intemational conference on intelligent robots and systems 2002, p 2200-5 [5] McFarlane D, Glover K (1992) "A loop shaping design procedure using H^ synthesis, " IEEE Trans Automat Contr 1992; 37(6): 759-69 20 Khoa hoc & Cong nghe - S6 02/2014 [6] Mmh H.B and Kiyolsuga Takaba (2011) "Model reduction in Schur basic with pole retention and H^-norm error bound, "In: Proceedingsof international workshop on Modeling, Systems, and Conrol2011 [7] Murata Boy Robot (2005) (www.murataboy.com) [8] Nguyen Hiru Cong, Vii Ngoc Kien, Dao Huy Du, Nghien cuu thudt todn gidm bde md hinh theo phuong phdp cdn bdng Tap chi Khoa hgc va Cdng nghe eac trudng Dai hgc Ky thuat, so 80, trang 34-39,nam2011 [9] Suprapto S (2006) "Development of a gyroscopic unmanned bicycle, " M.Eng Thesis, Asian Institute of Technology, Thailand [10] Tanaka Y, Miuakami T (2004) "Self sustaining bicycle robot with steering controller, " In: Proceedings of intemational workshop on advanced motion control 2004, p 193-7 [ I I ] Thanh B.T, and Manukid Pamichkun (2008) "Balancing control of Bicyrobo by particle swarm optimization - based strueture-specified mixed H2/ H^ control " Intemational Journal of Advanced Robotic Systems 2008; 5(4): 395- 402 [12] Y Liu, and B D, O.Anderson (1989), Singular Perturbation Approximation of Balanced Systems, InKmaXionalSoumal of ContTol^Vol 50, pp 1379-1404, 1989 Abstract: In recent years, many researches have been done in the area of the model order reduction There have been many order reduction algorithms introduced in which retaining the important poles of the original system This paper presents a new model order reduction algorithm, which is based on Sehur analysis It is based on the idea of keeping the important poles of the original system m the order reduction process This algorithm transforms matrix A of the higher-order original system to upper - triangle matrix on which the poles are arranged in descending important properties on the main diagonal of the upper - triangle matrix At the same time, the authors also have improved the algorithm in order to transform the parameters of the reduced system fi-om the complex numbers to the real number The effectiveness of the new model order reduction algorithm is illustrated by reducing order of higher-order controller of the robot balanced system The simulation results show the correctness of the proposed algorithm Keywords: model reduction, Schur analysis, two-wheeled balanced robot Ngirdi phan bien: GS.TSKH Banh Tiin Long Khoa hoc & C6ng nghe - So 02/2014 21 ... hinh toan hoc robot diiu khien can bang la nhu di d$ng hai banh Co nhieu nghien cuu ve dieu khien can bang Mpt mo hinh robot di dpng hai banh can cho robot di dgng hai banh, vi du nhu robot Murata... giam ble co y nghia hai banh, Viee dieu khien can bang eho robot hai la giam thdi gian dap img cua he banh co the dupe img dung de dieu khien cho robot di bing hai chan, nhu robot ASIMO vi nguyen... cho robot di dpng hai banh eo dng dung thuat toan giam bSc mo hinh theo theo hai budc nhu sau: a Thill kl bp dilu khiin H^ d^k dieu khien iTng dung thuat toan giam bac cho bai toan cSn bang cho