Nguyn Phc Hng-20701015-VP2007CT BO CO TIU LUN MN: IU KHIN CC C CU THA HNH THNG MINH ------------------------o0o------------------------ CH : IU KHIN H THNG BALL BEAM DNG LQR V LQG GVHD: TS. Trn Vit Hng SVTH: Nguyn Phc Hng ----------------------------------------------------------------------------------------------------------- I.GII THIU CHUNG: H qu bng v thanh cn c gi l h thng cn bng qu bng trn thanh Ball and Beam. H thng ny c s dng nh mt bi th nghim hu ht cc trng i hc k thut trn th gii v n kh gn gi vi cc h thng iu khin thc nh vic nh h thng cn bng my bay theo phng ngang khi h cnh di tc ng hn lon ca cc dng kh. Mc ch ca h thng l iu khin v tr ca bng trn thanh sao cho n c cn bng trn thanh di cc nh hng t nhiu bn ngoi c th c thc hin bng vic y nh.Mt iu tt nhin l h thng ca chng ta l h h, phi tuyn v khng n nh. gii quyt bi ton khng n nh, ta cn s dng phng php gn ng tuyn tnh ho h. v vy gc alpha ca thanh hp vi phng ngang phi tng i nh ( 30 30o oo s s ). Nu gc alpha nm ngoi min trn th kt qu thu c t phng php xp x tuyn tnh s khng cn chnh xc. Trong trng hp bt buc ta phi dng n cc phng php iu khin phi tuyn. bi bo co ny, chng ta s ln lt i n cc phn sau: -Xy dng phng trnh ng lc hc v phng trnh trng thi cho h thng trn. -Kho st p ng h thng vng h bng cc phng php trn phn mm Matlab. -ng dng cc lnh trn phn mm Matlab thit k b iu khin LQR, LQR c b quan st v LQG cho h thng. -Nhn xt cc kt qu p ng. -Gii thiu v giao din GUI ca h thng trn. Nguyn Phc Hng-20701015-VP2007CT -M hnh ton ca h thng trn nh sau: -Cc thng s ca h thng: STTTham sK hiun vGi tr 1Khi lng bngmKg0.11 2Bn knh bngRm0.005 3Mmen qu tnh bngJKg.m.m 510 4Chiu di thanhLm1 5Khong cchdm0.15 6Gc quay ca thanh o rad 7Gc quay ca ng cu rad 8Li ca bngrm -Mc tiu iu khin ca h thng trn: STTTham s cn iu khinGi tr 1Thi gian qu 3( ) s s2 qu iu chnh5(%) s Nguyn Phc Hng-20701015-VP2007CT II.XY DNG PHNG TRNH NG LC HC V PHNG TRNH TRNG THI: 1.Phng trnh ng lc hc theo nh lut II Lagrange: Xt h dao ng qu bng v thanh l mt h kn trong h quy chiu ton cc: ng nng ca h: ( ) ( ) ( ) ( )22 2 2 221 1 1 1 12 2 2 2 2ball thanh thanhr JT I m r I m r IR Ro o-- - - -| || |= + + = + + ||\ .\ . Th nng ca h (Chn gc th nng ti O): sin( )* V mg x o = Lc th trong trng hp ny bng 0: 0 Q = Phng trnh nh lut II Lagrange: ( )2(1) sin 02cos * 0i i iithanhT T V dQdt r r rJm r mgRI mg xoo o-----c c c | | + = |c c c\ . | |+ = | \ . ` ` ) = ) Vi: sind dL Lo o u o u- -= = Ta suy ra: 2J dm r mgR Lu--| |+ = |\ . Nguyn Phc Hng-20701015-VP2007CT 2.Phng trnh trng thi ca h thng: n y, ta c th xy dng phng trnh trng thi theo 2 cch ln lt ph thuc vo s quy nh ca tn hiu u vo: -Theo theta: 200 10 0r rmgdr J rL MRu-- -- ( ( ((( ( = + (((| |( + | (\ . -Theo alpha: 20 1 0 000 0 0000 0 0 110 0 0 0r rmgr J rmuRo oo o-- ---- -- ( ( ((( ( (((| |( (((+= + | ( ((( \ . ( ((( ( ( | | 1 0 0 0rryoo-- ( ( (= ( ( Nguyn Phc Hng-20701015-VP2007CT III.P NG VNG H CA H THNG TRN: 1.Dng hm truyn lin tc trong Matlab: 2.Dng phng trnh trng thi trong Matlab: Nguyn Phc Hng-20701015-VP2007CT 3.Dng m phng Simulink trong Matlab: a.Xy dng m hnh cho h thng Ball and Beam: b.S khi chnh ca h thng: c.p ng vng h ca h: Nguyn Phc Hng-20701015-VP2007CT IV.THIT K B IU KHIN LQR CHO H THNG TRN: 1.iu khin LQR s: a.M hnh: Cho h thng : Ng ra y l ng ra hi tip v o c. Ng ra z l iu khin c. Tn hiu nhiu w l nhiu h thng v v l nhiu o. Tn hiu v v w l nhng tn hiu nhiu trng. Trng thi ban u ca x(0) c gi s l mt vecto ngu nhin. Nhiu s gi s khc nhau nh ngha trng thi x(t) v ng ra iu khin c z(t) l nhng tn hiu ngu nhin. Biu thc sai s ton phng: l mt qu trnh ngu nhin. Vn ca iu khin h thng l gi tr mong i ca tch phn: L nh. y l vn iu khin tuyn tnh nhiu lon. Khong thi gian [0 T] l xc nh nhng tht s chng ta xem xt trng hp . Ti bt k thi gian t ton b tn hiu o c qu kh y(s) ( s> %Thiet lap phuong trinh trang thai roi rac voi nhung thong so sau: m = 0.111; R = 0.015; g = 9.8; J = 10^-5; %Cac ma tran cua phuong trinh trang thai A=[0 1 0 0;0 0 m*g/(J/R^2+m) 0;0 0 0 1;0 0 0 0]; B=[0 0 0 1]'; C=[1 0 0 0]; D=[0]; Ts=1/100; sys=ss(A,B,C,D) sysd=c2d(sys,Ts,'zoh') %Xet tinh quan sat duoc va tinh dieu khien duoc co=ctrb(A,B); ob=obsv(A,C); Controllability=rank(co) Observability=rank(ob) %Dieu khien LQR %Chon cac thong so sau %R=1,N=0,Q=[1 0 0 0;0 0 0 0;0 0 1 0;0 0 0 0] [F,G,H,I]=ssdata(sysd); T=0:0.01:10; U=0.2*ones(size(T)); Q=[5000 0 0 0;0 0 0 0;0 0 100 0;0 0 0 0]; R=1; [K,S,e]=dlqr(F,G,Q,R) syscl=ss(F-G*K,G,H,I,0.01) [Y,T,X]=lsim(syscl,U,T); plot(T,Y) a =x1 x2 x3 x4 x10100 x2006.9980 x30001 x40000

b = u1 Nguyn Phc Hng-20701015-VP2007CT x1 0 x2 0 x3 0 x4 1

c = x1x2x3x4 y1 1 0 0 0

d = u1 y1 0

Continuous-time model.

a = x1x2x3x4 x1 10.01 0.00034991.166e-006 x2 0 1 0.06998 0.0003499 x3 0 0 10.01 x4 0 0 0 1

b = u1 x12.916e-009 x21.166e-006 x35e-005 x40.01

c = x1x2x3x4 y1 1 0 0 0

d = u1 y1 0

Sampling time: 0.01 Nguyn Phc Hng-20701015-VP2007CT Discrete-time model. Controllability = 4 Observability = 4 K = 66.4432 37.5248 74.1533 12.0723 S = 1.0e+005 * 2.82380.78330.88070.0708 0.78330.31260.42220.0396 0.88070.42220.68880.0775 0.07080.03960.07750.0125 e = 0.9801 + 0.0423i 0.9801 - 0.0423i 0.9576 + 0.0183i 0.9576 - 0.0183i a =x1 x2 x3 x4 x11 0.010.0003497 1.131e-006 x2-7.749e-00510.069890.0003358 x3-0.003322-0.001876 0.9963 0.009396 x4-0.6644-0.3752-0.7415 0.8793 b = u1 x12.916e-009 x21.166e-006 x35e-005 x40.01 c = x1x2x3x4 y1 1 0 0 0 d =Nguyn Phc Hng-20701015-VP2007CT u1 y1 0

Sampling time: 0.01 Discrete-time model. c.p ng: Nguyn Phc Hng-20701015-VP2007CT 2.Thit k b quan st cho h thng: a.M hnh: Xem xt h thng quan st : H thng c xt l h thng khng c nhiu h thng w v nhiu o v. Trng thi x ca h thng khng th s dng trc tip nh trn bi v ch c ng ra y l o c. Xy dng li trng thi vi s chnh xc tu bi vic kt ni b quan st: Tn hiu ^x l mt c lng ca trng thi x(t). N tho mn phng trnh vi phn trng thi ca h thng vi thnh phn thm vo. L l ma trn li quan st cn c la chn ph hp. Sai s quan st l s khc nhau gia ng ra c thc t y(t) v ng ra . Thnh phn thm vocung cp mt s iu chnh ch ng ngay khi sai s ca s quan st l khc 0. b.Cc lnh trong Matlab: >> %Tinh cac cuc cua he thong poles=eig(syscl) poles = 0.9801 + 0.0423i 0.9801 - 0.0423i 0.9576 + 0.0183i 0.9576 - 0.0183i Nguyn Phc Hng-20701015-VP2007CT >> %Cac cua cua bo quan sat duoc dat xa ben trai de dat dap ung nhanh hon %voi P=[-0.3 -0.31 -0.32 -0.33] P=[-0.3 -0.31 -0.32 -0.33]; >> L=place(A',C',P)' L = 1.2600 0.5951 0.0178 0.0014 >> %Cac lenh ve dap ung: Nbar=20; >> Fce=[F-G*K G*K;zeros(size(F)) F-L*H]; >> Gce=[G*Nbar;zeros(size(G))]; >> Hce=[H zeros(size(H))]; >> Jce=[0]; >> [Y,X]=dlsim(Fce,Gce,Hce,Jce,U); >> stairs(T,Y) >> legend('Ball(y)') title('Dap ung cua he thong') c.p ng: Nguyn Phc Hng-20701015-VP2007CT V.THIT K B IU KHIN LQG (LINEAR QUADRATICGAUSSIAN): 1.M hnh: 2.Cc lnh trong Matlab: %Thiet lap phuong trinh trang thai roi rac voi nhung thong so sau: m = 0.111; R = 0.015; g = 9.8; J = 10^-5; %Cac ma tran cua phuong trinh trang thai A=[0 1 0 0;0 0 m*g/(J/R^2+m) 0;0 0 0 1;0 0 0 0]; B=[0 0 0 1]'; C=[1 0 0 0]; D=[0]; sys=ss(A,B,C,D)% Phuong trinh trang thai

a =x1 x2 x3 x4 x10100 x2006.9980 x30001 x40000

b = u1 x1 0 x2 0 x3 0 x4 1

c = x1x2x3x4 y1 1 0 0 0

Nguyn Phc Hng-20701015-VP2007CT d = u1 y1 0

Continuous-time model. % Tinh K K=lqry(sys,10,1) %u=-Kx min J(u)

K =

3.16233.8099 16.06125.6677

P=sys(:,[1,1]); % Buoc tren tach ngo vao dieu khien u va d %Tinh mo hinh Kalman Kest Kest=kalman(P,1,0.01)

a = x1_ex2_ex3_ex4_e x1_e-7.558 1 0 0 x2_e-28.56 0 6.998 0 x3_e-9.035 0 0 1 x4_e -10 0 0 0

b =u1 y1 x1_e07.558 x2_e028.56 x3_e09.035 x4_e1 10

c = x1_ex2_ex3_ex4_e y1_e 1 0 0 0 x1_e 1 0 0 0 x2_e 0 1 0 0 x3_e 0 0 1 0 x4_e 0 0 0 1

d = u1y1 y1_e 0 0 x1_e 0 0 x2_e 0 0 x3_e 0 0 x4_e 0 0 Nguyn Phc Hng-20701015-VP2007CT

Input groups: NameChannelsKnownInput1Measurement 2 Output groups: Name Channels OutputEstimate 1 StateEstimate 2,3,4,5 Continuous-time model. %Tao bo dieu khien LQG=LQ gain+Kalman filter F=lqgreg(Kest,K)

a = x1_ex2_ex3_ex4_e x1_e-7.558 1 0 0 x2_e-28.56 0 6.998 0 x3_e-9.035 0 0 1 x4_e-13.16 -3.81-16.06-5.668

b =y1 x1_e7.558 x2_e28.56 x3_e9.035 x4_e 10

c = x1_ex2_ex3_ex4_e u1-3.162 -3.81-16.06-5.668

d = y1 u1 0

Input groups: NameChannels Measurement 1

Output groups:NameChannelsControls 1

Continuous-time model. Nguyn Phc Hng-20701015-VP2007CT %Tao vong hoi tiep duong cho he thong clsys=feedback(sys,F,+1)

a =? ? ? ?x1_ex2_ex3_e ?0 1 0 0 0 0 0 ?0 0 6.998 0 0 0 0 ?0 0 0 1 0 0 0 ?0 0 0 0-3.162 -3.81-16.06 x1_e 7.558 0 0 0-7.558 1 0 x2_e 28.56 0 0 0-28.56 0 6.998 x3_e 9.035 0 0 0-9.035 0 0 x4_e10 0 0 0-13.16 -3.81-16.06

x4_e ?0 ?0 ?0 ? -5.668 x1_e 0 x2_e 0 x3_e 1 x4_e-5.668

b = u1 ?0 ?0 ?0 ?1 x1_e 0 x2_e 0 x3_e 0 x4_e 0

c =? ? ? ?x1_ex2_ex3_ex4_e y1 1 0 0 0 0 0 0 0

d = u1 y1 0

Continuous-time model. T=0:0.01:10; U=0.2*ones(size(T)); Nguyn Phc Hng-20701015-VP2007CT [Y,T,X]=lsim(syscl,U,T); plot(T,Y) 3.p ng: VI.NHN XT: -p ng cho h thng trn vi cc phng php iu khin iu tho mn vi cc yu cu t ra u bi v thi gian p ng v vt l. -Vi phng php iu khin LQR, ta c p ng ca h thng tt v thi gian p ng v sai s v tr khi chn cc thng s ca ma trn Q ph hp v ln. Vi bi ny, ta chn ma trn Q c cc thng s nh sau: 5000 0 0 00 0 0 00 0 100 00 0 0 0Q ( ( (= ( ( -Nu nh chn ma trn Q vi cc thng s nh hn ta s c thi gian p ng chm hn mc d sai s v v tr t c l tt. -Vi phng php iu khin LQR c thm b quan st, ta thy p ng u ra c s sai khc v v tr xc lp so vi phng php trn. V mt iu c bit l sai s v v tr tng ln trong kh thi gian p ng dng nh khng i. S c bit ny c l l do cch chn h s Nbar cho b iu khin. y, ta chnNbar = -61.55. -Vi phng php iu khin LQG, ta thy p ng rt tt v cng tng t nh p ng ca h thng nu s dng b iu khin LQR. Nguyn Phc Hng-20701015-VP2007CT VII.THIT K GIAO DIN GUIDE CHO H THNG: VIII.TI LIU THAM KHO: 1.http://www.engin.umich.edu/class/ctms/examples/ball/bbss.htm 2.http://www.mathworks.com/ 3.Matlab v ng dng trong iu khin - Nguyn c Thnh. 4.L thuyt iu khin hin i - Nguyn Th Phng H. 5.http://www.mathworks.com/help/toolbox/control/ref/lqg.html