Giảm bậc mô hình dựa trên phân tích Schur cho bài toán điều khiển robot hai bánh tự cân bằng

^ - - ^ GIAM BAC MO fflNH DlTA TREN PHAN TIC H SCHU R

'• ^f e ^" ^^^ ^^^ TOAN DIEU KHIEN ROBOT HAI BANH Ty" CAN BANG

^ JT-" ' MODEL ORDER REDUCTION BASED ON SCHUR ANALYSIS

FOR TWO-WHEELED BALANCED MOBILE ROBOT CONTROL

Bill Trung Thanh', VQ Ngoc Kien-, Nguyin Hiru Cong^

/ Trudng Dgi hpc Supham Ky thudt Hung Yen

2 Truong Dai hoc Ky thugt Cdng nghiep Thdi Nguyen

3 Dai hpc Thdi Nguyen

Tom tit:

Trong nhOng nam gdn ddy dd eo nhieu nghiin cuu trong linh vuc gidm bde md hinh. Dd co nhieu

thugt todn gidm bgc dugc giai thieu trong do 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 trong do cdc diim cue duac sdp xip gidm ddn vi tinh

quan trong 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 ra tinh hiiu qud cita thugt todn.

Tir kh6a: gidm bdc md hinh, phdn tich Schur, Robot hai bdnh tu edn bdng.

1. 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 ra 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 trong he giam bac hay khong.

Tuy nhien trong 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

trong qua trinh giam bac, dong thai img dung thuat

toan nay vao giam bac bg dieu khien ciia bai toan

dieu khien can bang robot di dgng hai banh.

2. 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 trong khong gian trang thai bdi he phuang

trinh sau:

x-A x + Bu (I)

y-C x

trong 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 trong (I) la

tim mo hinh mo ta bdi he cac phuang trinh.

Xr — ArXr + BrU ,-,

yr - ax,

Khoa hoc & Cong nghe - So 02/2014

trong 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

trong (1) ling dung trong 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 do y tudng ehinh eiia no cd the

phan chia lam 2 budc: budc 1 chuyen doi he thong

goc bac cao ve he thong tuang duong bang mpt

chuyen doi khdng suy bien trong khdng gian trang

thai, bude 2 xoa mpt so hang va mpt sd cot cua h$

thong tucmg duang de tao ra 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

nay 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 trong

he thong giam bac. Tuy nhien, eac diim cue quan

trgng la bat bien trong he thdng thuc do vay no cin

dugc bao toan trong qua trinh giam bac.

Tu nhihig nhugc diem do tac gia Minh H.B

[6] ciing nhdm nghien cuu da dua ra y tudng chinh

eua thuat toan la chuyen ddi ma tran A ciia he thdng