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Animation.m

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+function Animation(x,t)
+
+fs=20; lw=4; ms = 30;
+fst = 0.8*fs;
+x0screen=50;y0screen=200;WidthScreen=1300;HeightScreen=300;
+Blue    = [0 0.4470 0.7410];
+Orange  = [0.8500 0.3250 0.0980];
+Yellow  = [0.9290 0.6940 0.1250];
+Violet  = [0.4940 0.1840 0.5560];
+Green   = [0.4660 0.6740 0.1880];
+Cyan    = [0.3010 0.7450 0.9330];
+quality = '-r210';
+
+
+a = 0;
+b = 1;
+Cw1 = @(zeta) (zeta^2*(2*zeta^3 - 5*zeta^2 + 10))/20;
+Cw2 = @(zeta) -(zeta^4*(2*zeta - 5))/20;
+Cw3 = @(zeta) (zeta^3*(3*zeta^2 - 10*zeta + 10))/60;
+Cw4 = @(zeta) (zeta^4*(3*zeta - 5))/60;
+
+
+zeta_d = linspace(a,b);
+w = zeros(8,length(t));
+for i = 1:length(t)
+    for j = 1:length(zeta_d)
+        Cw = [Cw1(zeta_d(j)),Cw2(zeta_d(j)),Cw3(zeta_d(j)),Cw4(zeta_d(j)),0,0,0,0];
+        w(j,i) = Cw*x(:,i);
+    end
+end
+
+figure
+set(gcf,'units','points','position',[x0screen,y0screen,WidthScreen,HeightScreen])
+hold on
+% pu = plot(zeta_d,u(:,1),'LineWidth',lw);
+% pv = plot(zeta_d,v(:,1),'LineWidth',lw);
+pw = plot(zeta_d,w(:,1),'LineWidth',lw*4);
+legend({'$w(\zeta,t)$'},'Interpreter','latex','FontSize',fs,'Location','Northwest')
+xlabel({'space $\zeta$'},'Interpreter','latex','FontSize',fs)
+set(gca,'FontSize',fst)
+ylim([-0.7,0.7])
+xlim([a,b])
+% pause
+
+for i = 1:10:length(t)
+
+    % set(pu,'YData',u(:,i))
+    % set(pv,'YData',v(:,i))
+    set(pw,'YData',w(:,i))
+    pause(0.0001)
+end
+
+
+end

BE_Beam.m

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+%% [BE] Digital control of cantilever beam
+% Justin Bos
+% 
+% Wissal Guarni
+% 
+% Nolan Reynier Nomer
+% 
+% Aleksander Taban
+
+E = [13/35 9/70 11/210 -13/420 ;
+     9/70 13/35 13/420 -11/210 ;
+     11/210 13/420 1/105 -1/140 ;
+     -13/420 -11/210 -1/140 1/105] ;
+D = [-6/5 6/5 -1/10 -1/10 ;
+     6/5 -6/5 1/10 11/10 ;
+     -11/10 1/10 -2/15 1/30 ;
+     -1/10 1/10 1/30 -2/15] ;
+
+phi = @(zeta) [2*zeta^3-3*zeta^2+1 ; 
+            3*zeta^2-2*zeta^3 ; 
+            zeta^3-2*zeta^2+zeta ; 
+            zeta^3-zeta^2 ] ;
+L=1;
+%%
+A = [zeros(4), inv(E)*D ;
+    -inv(E)*(D'), zeros(4) ] ;
+phi_L = phi(L) ;
+B = [zeros(4,1) ; -inv(E)*phi_L] ;
+C = [zeros(1,4) , -phi_L'] ;
+%%
+Ts1 = 0.01 ;
+Ts2 = 0.02 ;
+Ts3 = 0.04 ;
+
+eig_cont = eig(A);
+sys = ss(A, B, C, 0) ;
+figure;
+subplot(1,3,1) ;
+plot(eig_cont, 'x');
+hold off;
+
+subplot(1,3,2);
+eig_tustin1 = eig(c2d(sys, Ts1, 'tustin')) ;
+eig_tustin2 = eig(c2d(sys, Ts2, 'tustin')) ;
+eig_tustin3 = eig(c2d(sys, Ts3, 'tustin')) ;
+plot(eig_tustin1,'x');
+hold on;
+plot(eig_tustin2,'x');
+plot(eig_tustin3,'x');
+hold off;
+
+subplot(1,3,3);
+eig_zoh1 = eig(c2d(sys, Ts1, 'zoh')) ;
+sys_zoh1 = c2d(sys, Ts1, 'zoh');
+eig_zoh2 = eig(c2d(sys, Ts2, 'zoh')) ;
+eig_zoh3 = eig(c2d(sys, Ts3, 'zoh')) ;
+plot(eig_zoh1,'x');
+hold on;
+pzmap(sys_zoh1);
+plot(eig_zoh3,'x');
+hold off;
+%% 
+% Conlsuion : On en conclut qqc.
+
+figure;
+t = 0:0.01:10;
+u = double (t >= 1) ;
+lsim(sys,u, t);
+title();
+legend();
+xlabel();
+ylabel();
+%% 
+% 
+
+Cw = @(zeta) [zeta^2*(2*zeta^3-5*zeta^2+10)/20 ...
+            -zeta^4*(2*zeta-5)/20 ...
+              zeta^3*(3*zeta^2-10*zeta+10)/60 ...
+              zeta^4*(3*zeta-5)/60 ...
+              0, 0, 0, 0];
+Cw_L = Cw(L);
+
+TransientTime = [] ;
+for i = 0.01:0.01:10
+    TransientTime(end+1) = stepinfo(feedback(sys, i)).TransientTime ;
+end
+[minimum, min_index] = min(TransientTime)
+k = min_index * 0.01;
+lsim(feedback(sys,k), u, t) ;
+H = -3
+%%
+sys_CL = feedback(sys,k);
+[y, tOut] = lsim(sys_CL, -3.*u, t);
+% plot(t, Cw(L)*y, t, u)
+eig_cont_CL = eig(sys_CL);
+figure;
+subplot(1,2,1) ;
+plot(eig_cont_CL, 'x');
+hold off;
+
+subplot(1,2,2);
+eig_zoh1 = eig(c2d(sys_CL, Ts1, 'zoh')) ;
+eig_zoh2 = eig(c2d(sys_CL, Ts2, 'zoh')) ;
+eig_zoh3 = eig(c2d(sys_CL, Ts3, 'zoh')) ;
+plot(eig_zoh1,'x');
+hold on;
+plot(eig_zoh3,'x');
+hold off;
+%%
+sys_CL_Ts = c2d(sys_CL, 10, 'zoh');
+figure
+step(sys_CL_Ts)
+%%
+K = lqr(sys, eye(size(A)), 1)
+sys_CL_lqr = ss(A-B*K, B*H, C, 0);
+lsim(sys_CL_lqr, u, t);