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- なお しどり
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1 25 22
2
3 1 [1] 3
4 3 [1] 3 [9] [13]
5
6 θ φ ε ε + ( ) θ + ( ) AC OA O O
7 6 ε ϕ CD EF BC CD C θ ε, θ, ϕ 3 0 M f, M b [kg] M c [kg] OA, AB, AC, CD L d, L c, L a, L e [m] DE DF L h [m] g [m/s 2 ] F f, F b [N] K m [N/V] V f, V b [V] F f = K m V f 0, F b = K m V b 0 ε, θ, ϕ η ε, η θ, η ϕ [kg m 2 /s] OA BC CD EF J ε ε = (M f + M b ) g L a cos (ε δ a ) + M c g cos (ε + δ c ) η ε ε cos δ a cos δ c +K m L a (V f + V b ) cos θ (2.1) J θ θ = Mf g L h cos (θ δ h ) + M b g cos (θ + δ h ) η θ θ cos δ h cos δ h +K m L h (V f V b ) (2.2) J ϕ ϕ = ηϕ ϕ Km L a (V f + V b ) sin θ (2.3) L h L c δ a = tan 1 {(L d + L e )/L a } δ c = tan 1 (L d /L c ) δ h = tan 1 (L e /L h ) J ε J θ J ϕ ε θ ϕ OA BC CD EF
8 7 2.2 u = [u P 1 u P 2 ] T =[V f + V b V f V b ] T x = [x P 1 x P 2 x P 3 x P 4 x P 5 x P 6 ] T =[ε ε θ θ ϕ ϕ] T ẋ = + x P 2 p 1 cos x P 1 + p 2 sin x P 1 + p 3 x P 2 x P 4 p 5 cos x P 3 + p 6 sin x P 3 + p 7 x P 4 x P 6 p 9 x P p 4 cos x P p p 10 sin x P 3 0 y = [x P 1 x P 3 ] T u (2.4) ε, θ, ϕ 2 3 ε θ p i (i = 1,..., 10) p 1 = [ (M f + M b )gl a + M c gl c ]/J ε p 2 = [ (M f + M b )gl a tan δ a M c gl c tan δ c ]/J ε p 3 = η ε /J ε p 4 = K m L a /J ε p 5 = ( M f + M b )gl h /J θ p 6 = ( M f M b )gl h tan δ h /J θ p 7 = η θ /J θ p 8 = K m L h /J θ p 9 = η ϕ /J ϕ p 10 = K m L a /J ϕ
9 8 (2.4) D = L g1 L ν 1 1 f h 1 (x) L gm L ν 1 1 f h 1 (x)..... L g1 L ν m 1 f h m (x) L gm L ν m 1 f h m (x) (2.5) [2] [4] ν 3 ε θ ν 1 = ν 2 = 2 D [ ] p4 cos x P 3 0 D = (2.6) 0 p 8 D = 0 ε
10 ε θ [ ] [ ] [ ] [ ] d ξ i ξ i = v P i (3.1) dt 1 ξ i 2 a i0 a i1 v P i y 1 = ε y 2 = θ ξ 1 1 ξ2 1 a 10 a 11 a 20 a 21 A i = [ 0 1 a i0 a i1 ] ξ i 2 i = 1 2 (3.2) u P = [ ] [ ] up 1 p4 cos x P 3 0 D = u P 2 0 p 8 f = [ ] [ f1 p1 cos x P 1 + p 2 sin x P 1 + p 3 x P 2 = f 2 p 5 cos x P 3 + p 6 sin x P 3 + p 7 x P 4 h = [ ] [ ] h1 a10 x P 1 a 11 x P 2 + v P 1 = a 20 x P 3 a 21 x P 4 + v P 2 h 2 ] (2.4) (3.1) f + Du P = h (3.3)
11 10 D D = 0 u P = D 1 ( f + h) (3.4) [e 1 e 2 ] = [x P 1 ξ 1 1 x P 3 ξ 2 1] { ẋp 2 ξ 1 2 = a 10 (x P 1 ξ 1 1) a 11 (x P 2 ξ 1 2) ẋ P 4 ξ 2 2 = a 20 (x P 3 ξ 2 1) a 21 (x P 4 ξ 2 2) { ë1 + a 11ė 1 + a 10e 1 = 0 ë 2 + a 21 ė 2 + a 20 e 2 = 0 e i (3.4) u E P,k = D 1 k ( f k + h k ) (3.5) 3.3 (3.5) x(k + 1) = x(k) + T ẋ(k) + T 2 ẍ(k) (3.6) 2 T w k u R P,k = u E P,k + T w k (3.7)
12 11 ε θ [ ] [ ] [ ] xp 1 (k + 1) xp 1 (k) xp 2 (k) = + T + T 2 [ ] s11,k x P 2 (k + 1) x P 2 (k) s 11,k 2 s 12,k T d T d 12 T d 11 + T 2 u P (k) (3.8) [ xp 3 (k + 1) x P 4 (k + 1) ] = 2 δ 11 T d 12 + T 2 2 δ 12 [ ] [ ] xp 3 (k) xp 4 (k) + T + T 2 x P 4 (k) s 21,k 2 T d T d 22 T d 21 + T 2 2 δ 21 T d 22 + T 2 2 δ 22 [ s21,k s 22,k ] u P (k) (3.9) u P,k (k) u P (k) = 0 (3.8) (3.9) s 1,k = s 2,k = D = = [ ] [ ] s11,k p1 cos x P 1 + p 2 sin x P 1 + p 3 x P 2 = s 21,k p 5 cos x P 3 + p 6 sin x P 3 + p 7 x P 4 [ ] [ s12,k p1 x P 2 sin x P 1 + p 2 x P 2 cos x P 1 + p 3 (p 1 cos x P 1 + p 2 sin x P 1 + p 3 x P 2 ) = s 22,k p 5 x P 4 sin x P 3 + p 6 x P 4 cos x P 3 + p 7 (p 5 cos x P 3 + p 6 sin x P 3 + p 7 x P 4 ) [ ] [ ] d11 d 12 p4 cos x P 3 0 = d 21 d 22 0 p 8 [ ] [ ] δ11 δ 12 p3 p 4 cos x P 3 p 4 x P 4 sin x P 3 0 = δ 21 δ 22 0 p 7 p 8 ] [ ] [ ] [ ξ i 1 (k + 1) ξ i = 1 (k) ξ i + T 2 (k) ξ2(k i + 1) ξ2(k) i ξ 2(k) i ] + T 2 2 [ ξi 2 (k) ξ i 2(k) ] (3.10) (3.10) ξ i 2(k), ξ i 2(k) = a i0 ξ i 1(k) a i1 ξ i 2(k) + v P i t ξ i 2(k) = a i0 ξ i 2(k) a i1 ξi 2 (k) + v P i (3.11) = a i0 ξ i 2(k) a i1 ( a i0 ξ i 1(k) a i1 ξ i 2(k) + v P i ) + v P i = (a 2 i1 a i0 )ξ i 2(k) + a i1 a i0 ξ i 1(k) a i1 v P i + v P i ξ i 2 g 2 (k) = [ g 1 2 (k) g 2 2(k) ] = [ ] (a 2 11 a 10 )x P 2 + a 11 a 10 x P 1 + a 11 v P 1 + v P 1 (a 2 21 a 20 )x P 4 + a 21 a 20 x P 3 + a 21 v P 2 + v P 2
13 12 w k s 2,k + u E P,k + 2Dw k = g 2 (k) (3.12) (3.8) (3.9) (3.7) (3.10) O(T 2 ) w k = 1 2 D 1 {g 2 (k) (s 2,k + u E P,k)} (3.13) 3.4 p 1 = p 2 = p 3 = p 4 = p 5 = 0.0 p 6 = p 7 = p 8 = p 9 = p 10 = a 11 = a 21 = 3 a 10 = a 20 = 2
14 13 T = 0.2 [s] ξ (0) = 0 [ ε ε θ θ ϕ ϕ ] = [ ] [ v1 v 2 ] = [ 0.3 sin(0.1t) 0.2 sin(0.2t) ]
15 14 (1) Reference Input v v Reference Input v v 2
16 15 4 Input u u Input u u 2
17 Output x ε;, 0.2 Output x θ;,
18 Error x ε ξ Error x θ ξ 2 1
19 18 (2) Emu Input u u Emu Input u u 2
20 19 Emu Output x ε;, 0.1 Emu Output x θ;,
21 20 0 Error x ε ξ Error x θ ξ 2 1
22 21 (3) Re Input u u Re Input u u 2
23 Re Output x ε;, 0.1 Re Output x θ;,
24 23 0 Error x ε ξ Error x θ ξ 2 1
25 24 (4) Input u u 1 ;, 0.2 Input u u 2 ;,
26 25 0 Error x ε ξ1 1 ;, Error x ε ξ1 1 ( );,
27 Error x θ ξ1 2 ;, Error x θ ξ1 2 ( );,
28 27 T = 0.1[s] T = 0.2[s] 0 Error x ε ξ1 1 ;, Error x ε ξ1 1 ( );,
29 Error x θ ξ1 2 ;, Error x θ ξ1 2 ( );,
30 29 (5) T = 0.2[s] T = 0.1[s] (T = 0.2[s]) ε ξ θ ξ (T = 0.1[s]) ε ξ θ ξ O(T 2 ) T = 0.2[s] 10 1 T = 0.1[s] 20 1 T = 0.1[s] T = 0.2[s] 0
31 4 3 3
32 2.
33 [1] D. S. Laila, D. Nešić and A. Astolfi: Sampled-data control of nonlinear systems; Advanced topics in control systems theory, Springer Verlag, pp , [2] H K Khalil Nonlinear Systems 3rd ed. Prentice Hall [3],,,,,,, 1993 [4],,,,,,, 1997 [5] [6] M. Ishitobi, M. Nishi and K. Nakasaki, Nonlinear adaptive model following control for a 3-DOF tandem-rotor model helicopter, Control Engineering Practice, Vol.18(8), pp , [7] K. Takamuku, M. Ishitobi, T. Kumada, M. Nishi and S. Kunimatsu, Redesign implementation of a nonlinear sampled-data controller for a 3-DOF model helicopter, Proc. of 2011 SICE Annual Conference, pp , [8] [9] [10] [11] 3, 2012.
34 33 [12] [13]
35 cont main.m 35 state equation cont.m 38 emu main.m 40 state equation emu.m 43 stable input.m 44 redes main.m 45 state equation redes.m 48 stable input2.m 49 simu.mdl Simulink Emu Controller 51 v emu 52 Re Controller 52 add input w 53 51
36 35 cont main.m clc clear all global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 a11 a10 a21 a20 dt v u x0 t; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mf=0.69; Mb=0.69; Mc=1.67; La=0.62; Lc=0.44; Ld=0.05; Le=0.02; Lh=0.177; Km=0.5; g=9.81; % etae=0.01; etap=0.01; etat=0.05; % Je=(Mf+Mb)*(La^2+Ld^2)+Mc*(Lc^2+Ld^2); Jp=(Mf+Mb)*(Lh^2+Le^2); Jt=(Mf+Mb)*La^2+Mc*Lc^2; % tan a, c, h tana = (Ld+Le)/La;tanc = Ld/Lc;tanh = Le/Lh; % p1=(-(mf+mb)*la+mc*lc)*g/je; p2=(-(mf+mb)*la*tana-mc*lc*tanc)*g/je; p3=-etae/je; p4=km*la/je; p5=(-mf+mb)*g*lh/jp; p6=(-mf-mb)*g*lh*tanh/jp; p7=-etap/jp; p8=km*lh/jp; p9=-etat/jt; p10=-km*la/jt; % a11=3; a10=2; a21=3; a20=2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37 36 % x0=[-0.1; 0; 0.1; 0; 0.2; 0]; min=0; max=90; begin=min; dt=0.2; xout=x0 ; tout=0; rout=[0 0]; uout=[0 0]; % for i=min:max/dt [t, x]=ode45( state_equation_cont, [begin,begin+dt], x0); x0=x(length(x),:); xout=[xout;x0]; Tend=t(length(t),:); begin=tend ; tout=[tout; begin]; vout=[vout; v ]; uout=[uout; u ]; end % eout1=vout(:,1)-xout(:,1); eout2=vout(:,2)-xout(:,3); % figure plot(tout,vout(:,1)); xlabel( time[s] );ylabel( Reference Input v1 );xlim([min,max]); figure plot(tout,vout(:,2)); xlabel( time[s] );ylabel( Reference Input v2 );xlim([min,max]);
38 37 figure plot(tout,uout(:,1)) xlabel( time[s] );ylabel( Input u1 );xlim([min,max]); figure plot(tout,uout(:,2)) xlabel( time[s] );ylabel( Input u2 );xlim([min,max]); figure plot(tout,vout(:,1), :,tout,xout(:,1)); xlabel( time[s] );ylabel( Output x1 );xlim([min,max]); figure plot(tout,vout(:,2), :,tout,xout(:,3)); xlabel( time[s] );ylabel( Output x3 );xlim([min,max]); figure plot(tout,eout1, r ); xlabel( time[s] );ylabel( Error x1 );xlim([min,max]); figure plot(tout,eout2, g ); xlabel( time[s] );ylabel( Error x3 );xlim([min,max]);
39 38 state equetion cont.m function[dx]=state_equation_cont(t,x) global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 a11 a10 a21 a20 v u ; % v1,v2 v1=0.3*sin(0.1*t); v1dot=0.3*0.1*cos(0.1*t); v1dot2=-0.2*0.1*0.1*sin(0.1*t); v2=0.2*sin(0.2*t); v2dot=0.2*0.2*cos(0.2*t); v2dot2=-0.2*0.2*0.2*sin(0.2*t); v=[v1; v2]; % u1,u2 u1=1/(p4*cos(x(3)))*(-p1*cos(x(1))-p2*sin(x(1)) -(p3+a11)*x(2)-a10*x(1)+a10*v1+a11*v1dot+v1dot2); u2=1/p8*(-p5*cos(x(3))-p6*sin(x(3)) -(p7+a21)*x(4)-a20*x(3)+a20*v2+a21*v2dot+v2dot2); u=[u1; u2]; % A=[x(2); p1*cos(x(1))+p2*sin(x(1))+p3*x(2); x(4); p5*cos(x(3))+p6*sin(x(3))+p7*x(4); x(6); p9*x(6)];
40 39 B=[0 0; p4*cos(x(3)) 0; 0 0; 0 p8; 0 0; p10*sin(x(3)) 0]; dx=a+b*u; end
41 40 emu main.m clc clear all global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 a11 a10 a21 a20 dt r u t dt v; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mf=0.69; Mb=0.69; Mc=1.67; La=0.62; Lc=0.44; Ld=0.05; Le=0.02; Lh=0.177; Km=0.5; g=9.81; % etae=0.01; etap=0.01; etat=0.05; % Je=(Mf+Mb)*(La^2+Ld^2)+Mc*(Lc^2+Ld^2); Jp=(Mf+Mb)*(Lh^2+Le^2); Jt=(Mf+Mb)*La^2+Mc*Lc^2; % tan a, c, h tana = (Ld+Le)/La;tanc = Ld/Lc;tanh = Le/Lh; % p1=(-(mf+mb)*la+mc*lc)*g/je; p2=(-(mf+mb)*la*tana-mc*lc*tanc)*g/je; p3=-etae/je; p4=km*la/je; p5=(-mf+mb)*g*lh/jp; p6=(-mf-mb)*g*lh*tanh/jp; p7=-etap/jp; p8=km*lh/jp; p9=-etat/jt; p10=-km*la/jt; % a11=3; a10=2; a21=3; a20=2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % x0=[-0.1; 0; 0.1; 0; 0.2; 0]; x=[-0.1; 0; 0.1; 0; 0.2; 0]; u=[0;0];
42 41 v=[0;0]; min=0; max=90; begin=min; dt=0.2; xout=x0 ; tout=0; rout=[0 0];uout=[0 0];vout=[0 0]; u=stable_input(begin,x0); % for i=min:max/dt; [t, x]=ode45( state_equation_emu, [begin,begin+dt], x0); x0=x(length(x),:); u=stable_input(begin,x0); xout=[xout;x0]; Tend=t(length(t),:); begin=tend ; tout=[tout; begin]; rout=[rout; r ]; uout=[uout; u ]; vout=[vout; v ]; end % eout1=rout(:,1)-xout(:,1); eout2=rout(:,2)-xout(:,3); % figure plot(tout,uout(:,1)) xlabel( time[s] );ylabel( Input u1 );xlim([min,max]);
43 42 figure plot(tout,uout(:,2)) xlabel( time[s] );ylabel( Input u2 );xlim([min,max]); figure plot(tout,rout(:,1), :,tout,xout(:,1)); xlabel( time[s] );ylabel( Output x1 );xlim([min,max]); figure plot(tout,rout(:,2), :,tout,xout(:,3)); xlabel( time[s] );ylabel( Output x3 );xlim([min,max]); figure plot(tout,eout1, r ); xlabel( time[s] );ylabel( Error x1 );xlim([min,max]); figure plot(tout,eout2, g ); xlabel( time[s] );ylabel( Error x3 );xlim([min,max]);
44 43 state equation emu.m function[dx]=state_equation_emu(t,x) global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 u; % A=[x(2); p1*cos(x(1))+p2*sin(x(1))+p3*x(2); x(4); p5*cos(x(3))+p6*sin(x(3))+p7*x(4); x(6); p9*x(6)]; B=[0 0; p4*cos(x(3)) 0; 0 0; 0 p8; 0 0; p10*sin(x(3)) 0]; dx=a+b*u; end
45 44 function[u]=stable_input(t,x) stable input.m global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 a11 a10 a21 a20 r v; % v1,v2 r1=0.3*sin(0.1*t); r1dot=0.3*0.1*cos(0.1*t); r1dot2=-0.3*0.1*0.1*sin(0.1*t); r2=0.2*sin(0.2*t); r2dot=0.2*0.2*cos(0.2*t); r2dot2=-0.2*0.2*0.2*sin(0.2*t); r=[r1; r2]; v=[a10*r1+a11*r1dot+r1dot2; a20*r2+a21*r2dot+r2dot2]; % u1,u2 u1=1/(p4*cos(x(3)))*(-p1*cos(x(1))-p2*sin(x(1))-(p3+a11)*x(2)-a10*x(1)+v(1)); u2=1/p8*(-p5*cos(x(3))-p6*sin(x(3))-(p7+a21)*x(4)-a20*x(3)+v(2)); u=[u1; u2]; end
46 45 redes main.m clc clear all global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 a11 a10 a21 a20 dt r u t v ru x0; %///////////////////////////////////////////////////////////////////// % Mf=0.69; Mb=0.69; Mc=1.67; La=0.62; Lc=0.44; Ld=0.05; Le=0.02; Lh=0.177; Km=0.5; g=9.81; % etae=0.01; etap=0.01; etat=0.05; % Je=(Mf+Mb)*(La^2+Ld^2)+Mc*(Lc^2+Ld^2); Jp=(Mf+Mb)*(Lh^2+Le^2); Jt=(Mf+Mb)*La^2+Mc*Lc^2; % tan a, c, h tana = (Ld+Le)/La;tanc = Ld/Lc;tanh = Le/Lh; % p1=(-(mf+mb)*la+mc*lc)*g/je; p2=(-(mf+mb)*la*tana-mc*lc*tanc)*g/je; p3=-etae/je; p4=km*la/je; p5=(-mf+mb)*g*lh/jp; p6=(-mf-mb)*g*lh*tanh/jp; p7=-etap/jp; p8=km*lh/jp; p9=-etat/jt; p10=-km*la/jt; % a11=3; a10=2; a21=3; a20=2; % x0=[-0.1; 0; 0.1; 0; 0.2; 0]; x=[0; 0; 0; 0; 0; 0]; v=[0;0]; ru=[0;0]; rv=[0;0];
47 46 min=0; max=90; begin=min; tout=0; vout=[0 0]; uout2=[0 0];vout2=[0 0]; %//////////////////////////////////////////////////////////////////// % dt=0.2; %for i=0 ru ru=stable_input2(begin,x0); vout=v ; xout2=x0 ; % for i=min:max/dt; %/////////////////////////////////////////////////////////// [t, x]=ode45( state_equation_redes, [begin,begin+dt], x0); x0=x(length(x),:); begin=t(length(t),:); xout2=[xout2; x0]; tout=[tout; begin]; %////////////////////////////////////////////////////////// ru=stable_input2(begin,x0); vout=[vout; v ]; uout2=[uout2; ru ]; end % ( ) eout3=vout(:,1)-xout2(:,1); eout4=vout(:,2)-xout2(:,3);
48 47 figure plot(tout,uout2(:,1)) xlabel( time[s] );ylabel( Input u1 );xlim([min,max]); figure plot(tout,uout2(:,2)) xlabel( time[s] );ylabel( Input u2 );xlim([min,max]); figure plot(tout,vout(:,1), :,tout,xout2(:,1)); xlabel( time[s] );ylabel( Output x1 );xlim([min,max]); figure plot(tout,vout(:,2), :,tout,xout2(:,3)); xlabel( time[s] );ylabel( Output x3 );xlim([min,max]); figure plot(tout,eout3, r ); xlabel( time[s] );ylabel( Error x1 );xlim([min,max]); figure plot(tout,eout4, g ); xlabel( time[s] );ylabel( Error x3 );xlim([min,max]);
49 48 state equation redes.m function[dx]=state_equation_redes(~,x) global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 ru; % A=[x(2); p1*cos(x(1))+p2*sin(x(1))+p3*x(2); x(4); p5*cos(x(3))+p6*sin(x(3))+p7*x(4); x(6); p9*x(6)]; B=[0 0; p4*cos(x(3)) 0; 0 0; 0 p8; 0 0; p10*sin(x(3)) 0]; dx=a+b*ru; end
50 49 function[ru]=stable_input2(t,x) stable input2.m global p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 a11 a10 a21 a20 r v dr dt; % a11=3; a10=2; a21=3; a20=2; % v1,v2 v1=0.3*sin(0.1*t); v1dot=0.3*0.1*cos(0.1*t); v1dot2=-0.3*0.1*0.1*sin(0.1*t); v1dot3=-0.3*0.1*0.1*0.1*cos(0.1*t); v2=0.2*sin(0.2*t); v2dot=0.2*0.2*cos(0.2*t); v2dot2=-0.2*0.2*0.2*sin(0.2*t); v2dot3=-0.2*0.2*0.2*0.2*cos(0.2*t); v=[v1; v2]; r=[a10*v1+a11*v1dot+v1dot2; a20*v2+a21*v2dot+v2dot2]; dr=[a10*v1dot+a11*v1dot2+v1dot3; a20*v2dot+a21*v2dot2+v2dot3]; % u1,u2 u1=(-p1*cos(x(1))-p2*sin(x(1))-(p3+a11)*x(2)-a10*x(1)+r(1))/(p4*cos(x(3))); u2=(-p5*cos(x(3))-p6*sin(x(3))-(p7+a21)*x(4)-a20*x(3)+r(2))/p8; u=[u1; u2]; % invd=[1/(p4*cos(x(3))) 0; 0 1/p8]; xi=[(a11^2-a10)*x(2)+a11*a10*x(1)-a11*r(1)+dr(1); (a21^2-a20)*x(4)+a21*a20*x(3)-a21*r(2)+dr(2)]; S=[-p1*x(2)*sin(x(1))+p2*x(2)*cos(x(1))+p3*(p1*cos(x(1))+p2*sin(x(1))+p3*x(2)); -p5*x(4)*sin(x(3))+p6*x(4)*cos(x(3))+p7*(p5*cos(x(3))+p6*sin(x(3))+p7*x(4))];
51 50 d=[p3*p4*cos(x(3))-p4*x(4)*sin(x(3)) 0; 0 p7*p8]; % w=invd/2*(xi-s-d*u); % ru=u+dt*w; end
52 51 C.1 simu.mdl C.2 Emu Controller
53 52 C.3 v emu C.4 Re Controller
54 53 C.5 add input w
all.dvi
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Gmech08.dvi
51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r
K E N Z OU
K E N Z OU 11 1 1 1.1..................................... 1.1.1............................ 1.1..................................................................................... 4 1.........................................
PSCHG000.PS
a b c a ac bc ab bc a b c a c a b bc a b c a ac bc ab bc a b c a ac bc ab bc a b c a ac bc ab bc de df d d d d df d d d d d d d a a b c a b b a b c a b c b a a a a b a b a
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9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
( ; ) C. H. Scholz, The Mechanics of Earthquakes and Faulting : - ( ) σ = σ t sin 2π(r a) λ dσ d(r a) =
1 9 8 1 1 1 ; 1 11 16 C. H. Scholz, The Mechanics of Earthquakes and Faulting 1. 1.1 1.1.1 : - σ = σ t sin πr a λ dσ dr a = E a = π λ σ πr a t cos λ 1 r a/λ 1 cos 1 E: σ t = Eλ πa a λ E/π γ : λ/ 3 γ =
5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1
4 1 1.1 ( ) 5 1.2, 2, d a V a = M (1.2.1), M, a,,,,, Ω, V a V, V a = V + Ω r. (1.2.2), r i 1, i 2, i 3, i 1, i 2, i 3, A 2, A = 3 A n i n = n=1 da = 3 = n=1 3 n=1 da n i n da n i n + 3 A ni n n=1 3 n=1
1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +
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通信容量制約を考慮したフィードバック制御 - 電子情報通信学会 情報理論研究会（IT） 若手研究者のための講演会
IT 1 2 1 2 27 11 24 15:20 16:05 ( ) 27 11 24 1 / 49 1 1940 Witsenhausen 2 3 ( ) 27 11 24 2 / 49 1940 2 gun director Warren Weaver, NDRC (National Defence Research Committee) Final report D-2 project #2,
(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (
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1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.
1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
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[1.1] r 1 =10e j(ωt+π/4), r 2 =5e j(ωt+π/3), r 3 =3e j(ωt+π/6) ~r = ~r 1 + ~r 2 + ~r 3 = re j(ωt+φ) =(10e π 4 j +5e π 3 j +3e π 6 j )e jωt
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3.4.7 [.] =e j(t+/4), =5e j(t+/3), 3 =3e j(t+/6) ~ = ~ + ~ + ~ 3 = e j(t+φ) =(e 4 j +5e 3 j +3e 6 j )e jt = e jφ e jt cos φ =cos 4 +5cos 3 +3cos 6 =.69 sin φ =sin 4 +5sin 3 +3sin 6 =.9 =.69 +.9 =7.74 [.]
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
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1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb
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r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)
A 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
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Gmech08.dvi
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1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π
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さくらの個別指導 ( さくら教育研究所 ) 1 φ = φ 1 : φ [ ] a [ ] 1 a : b a b b(a + b) b a 2 a 2 = b(a + b). b 2 ( a b ) 2 = a b a/b X 2 X 1 = 0 a/b > 0 2 a
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