181 lines
3.8 KiB
Matlab
181 lines
3.8 KiB
Matlab
%% [BE] Digital control of cantilever beam
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% Justin Bos
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% Wissal Guarni
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% Nolan Reynier Nomer
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% Aleksander Taban
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clear;
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clc;
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close all;
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set(0, 'DefaultLineLineWidth', 1)
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%% Question 3
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E = [13/35 9/70 11/210 -13/420 ;
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9/70 13/35 13/420 -11/210 ;
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11/210 13/420 1/105 -1/140 ;
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-13/420 -11/210 -1/140 1/105] ;
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D = [-6/5 6/5 -1/10 -1/10 ;
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6/5 -6/5 1/10 11/10 ;
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-11/10 1/10 -2/15 1/30 ;
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-1/10 1/10 1/30 -2/15] ;
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phi = @(zeta) [2*zeta^3-3*zeta^2+1 ;
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3*zeta^2-2*zeta^3 ;
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zeta^3-2*zeta^2+zeta ;
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zeta^3-zeta^2 ] ;
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L = 1 ; % Longueur de la poutre
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phi_L = phi(L) ;
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A = [zeros(4), inv(E)*D ;
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-inv(E)*(D'), zeros(4) ] ;
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B = [zeros(4,1) ; -inv(E)*phi_L] ;
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C = [zeros(1,4) , -phi_L'] ;
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A, B, C
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%% Question 4
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Ts1 = 0.01 ;
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Ts2 = 0.02 ;
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Ts3 = 0.04 ;
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sys = ss(A, B, C, 0) ;
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eig_cont = eig(sys);
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eig_tustin1 = eig(c2d(sys, Ts1, 'tustin')) ;
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eig_tustin2 = eig(c2d(sys, Ts2, 'tustin')) ;
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eig_tustin3 = eig(c2d(sys, Ts3, 'tustin')) ;
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eig_zoh1 = eig(c2d(sys, Ts1, 'zoh')) ;
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eig_zoh2 = eig(c2d(sys, Ts2, 'zoh')) ;
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eig_zoh3 = eig(c2d(sys, Ts3, 'zoh')) ;
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figure;
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subplot(1,3,1) ;
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plot(eig_cont, 'x');
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grid on;
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xline(0, 'r--');
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xlabel('Axe Réel');
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ylabel('Axe Imaginaire');
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title('Pôles du système');
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subplot(1,3,2);
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hold on;
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plot(eig_tustin1,'x', 'LineWidth',2);
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plot(eig_tustin2,'x', 'LineWidth',2);
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plot(eig_tustin3,'x', 'LineWidth',2);
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theta = linspace(0, 2*pi, 100);
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plot(cos(theta), sin(theta), 'k--', 'HandleVisibility','off');
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xlim([-1.1, 1.1]);
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ylim([-1.1, 1.1]);
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grid on;
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title('Pôles dans le plan en Z (Discret)');
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xlabel('Réel');
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ylabel('Imaginaire');
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hold off;
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subplot(1,3,3);
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hold on;
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plot(eig_zoh1,'x', 'LineWidth',1);
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plot(eig_zoh2,'x', 'LineWidth',1);
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plot(eig_zoh3,'x', 'LineWidth',1);
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legend("Eigen Values Ts1", "Eigen Values Ts2", "Eigen Values Ts3");
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theta = linspace(0, 2*pi, 100);
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plot(cos(theta), sin(theta), 'k--', 'HandleVisibility','off');
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xlim([-1.1, 1.1]);
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ylim([-1.1, 1.1]);
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grid on;
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title('Valeurs propres avec un bloqueur d''ordre 0 (ZOH)');
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xlabel('Réel');
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ylabel('Imaginaire');
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hold off;
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% Conlsuion : On en conclut qqc.
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%% Question 5
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t = 0:0.01:10;
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cmd = double (t >= 1) ;
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[y, t] = lsim(sys,cmd,t) ;
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figure ;
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plot(t, cmd, t, y);
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xlabel("Temps (s)");
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ylabel("Amplitude");
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legend("u(t)", "y(t)");
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title("Simulation en boucle ouverte") ;
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%% Question 7
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Cw = @(zeta) [zeta^2*(2*zeta^3-5*zeta^2+10)/20 ...
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-zeta^4*(2*zeta-5)/20 ...
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zeta^3*(3*zeta^2-10*zeta+10)/60 ...
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zeta^4*(3*zeta-5)/60 ...
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0, 0, 0, 0];
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Cw_L = Cw(L);
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TransientTime = [] ;
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for i = 0.01:0.01:10
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TransientTime(end+1) = stepinfo(feedback(sys, i)).TransientTime ;
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end
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[minimum, min_index] = min(TransientTime)
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k = min_index * 0.01
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%% Question 8
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H = inv(Cw_L*inv(-A+B*C*k)*B)
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[y, t, x] = lsim(feedback(sys,k), H*cmd, t) ;
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u = H*cmd'-k*y ;
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figure ;
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plot(t, u, t, y, t, cmd, t, (Cw_L*x')');
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xlabel("Temps (s)");
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ylabel("Amplitude");
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legend("u(t)", "y(t)", "wc(t)", "w(t)");
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title("Simulation du retour de sortie avec pré-gain") ;
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%% Question 9
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sys_CL = feedback(sys,k);
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[y, tOut] = lsim(sys_CL, -3.*cmd, t);
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% plot(t, Cw(L)*y, t, u)
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eig_cont_CL = eig(sys_CL);
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figure;
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subplot(1,2,1) ;
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plot(eig_cont_CL, 'x');
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hold off;
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subplot(1,2,2);
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eig_zoh1 = eig(c2d(sys_CL, Ts1, 'zoh')) ;
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eig_zoh2 = eig(c2d(sys_CL, Ts2, 'zoh')) ;
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eig_zoh3 = eig(c2d(sys_CL, Ts3, 'zoh')) ;
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plot(eig_zoh1,'x');
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hold on;
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plot(eig_zoh3,'x');
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hold off;
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%%
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sys_CL_Ts = c2d(sys_CL, 10, 'zoh');
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figure
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step(sys_CL_Ts)
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%% Question 11
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K = lqr(sys, eye(size(A)), 1);
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H_lqr = inv(Cw_L*inv(-A+B*K)*B) ;
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%% Question 12
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sys_CL_lqr = ss(A-B*K, B*H_lqr, C, 0);
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[y, t, x] = lsim(sys_CL_lqr, cmd, t);
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u = H_lqr*cmd'-k*y ;
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figure ;
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plot(t, u, t, y, t, cmd, t, (Cw_L*x')');
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xlabel("Temps (s)");
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ylabel("Amplitude");
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legend("u(t)", "y(t)", "wc(t)", "w(t)");
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title("Simulation du retour d'états avec pré-gain") ;
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