From 7da7e9becdebbff501a1fe96d90e2b5c517df185 Mon Sep 17 00:00:00 2001 From: Nolan Date: Wed, 6 May 2026 20:11:23 +0200 Subject: [PATCH] Avancement avant reu --- BE_Beam.m | 133 +++++++++++++++++++++++------ latex/Illustrations/Question10.pdf | Bin 44986 -> 47107 bytes latex/Illustrations/Question12.pdf | Bin 41465 -> 40815 bytes latex/Illustrations/Question13.pdf | Bin 0 -> 45115 bytes latex/Illustrations/Question14.pdf | Bin 0 -> 41296 bytes latex/Illustrations/Question16.pdf | Bin 66325 -> 46101 bytes latex/Illustrations/Question19.pdf | Bin 50715 -> 50895 bytes 7 files changed, 106 insertions(+), 27 deletions(-) create mode 100644 latex/Illustrations/Question13.pdf create mode 100644 latex/Illustrations/Question14.pdf diff --git a/BE_Beam.m b/BE_Beam.m index 7c1f7fd..38ef7ba 100644 --- a/BE_Beam.m +++ b/BE_Beam.m @@ -55,6 +55,7 @@ eig_zoh2 = eig(c2d(sys, Ts2, 'zoh')) ; eig_zoh3 = eig(c2d(sys, Ts3, 'zoh')) ; %% Question4 = figure ; + subplot(1,3,1) ; plot(eig_cont, 'x'); axis square; @@ -75,10 +76,10 @@ plot(cos(theta), sin(theta), 'k--', 'HandleVisibility','off','LineWidth',0.25); xlim([-1.1, 1.1]); ylim([-1.1, 1.1]); grid on; -title({'Valeurs propres en temps discret',' avec la méthode de Tustin'}); legend("Ts1", "Ts2", "Ts3"); xlabel('Réel'); ylabel('Imaginaire'); +title({'Valeurs propres en temps discret',' avec la méthode de Tustin'}); hold off; subplot(1,3,3); @@ -88,7 +89,6 @@ plot(eig_zoh2,'x', 'LineWidth',1); plot(eig_zoh3,'x', 'LineWidth',1); axis square; legend("Ts1", "Ts2", "Ts3"); -% theta = linspace(0, 2*pi, 100); plot(cos(theta), sin(theta), 'k--', 'HandleVisibility','off','LineWidth',0.25); xlim([-1.1, 1.1]); ylim([-1.1, 1.1]); @@ -135,12 +135,12 @@ end [minimum, min_index] = min(TransientTime) k = min_index * 0.01 -%% Question 8 - H = inv(Cw_L*inv(-A+B*C*k)*B) + +%% Question8 [y, t, x] = lsim(feedback(sys,k), H*cmd, t) ; u = H*cmd'-k*y ; -%% + Question8 = figure ; plot(t, u, t, y, t, cmd, t, (Cw_L*x')'); xlabel("Temps (s)"); @@ -154,35 +154,40 @@ end %% Question 9 -sys_CL = feedback(sys,k); -% [y, tOut] = lsim(sys_CL, H.*cmd, t); -% plot(t, Cw(L)*y, t, u) -eig_cont_CL = eig(sys_CL); + +eig_cont_CL = eig(feedback(sys,k)); +eig_zoh1_CL = eig(feedback(c2d(sys, Ts1, 'zoh'), k)) ; +eig_zoh2_CL = eig(feedback(c2d(sys, Ts2, 'zoh'), k)) ; +eig_zoh3_CL = eig(feedback(c2d(sys, Ts3, 'zoh'), k)) ; + +%% + Question9 = figure; subplot(1,2,1) ; + plot(eig_cont_CL, 'x'); +hold on; axis square; grid on; xline(0, 'r--'); title({'Valeurs propres en temps continu'}); xlabel('Réel'); ylabel('Imaginaire'); +hold off; subplot(1,2,2); -eig_zoh1 = eig(feedback(c2d(sys, Ts1, 'zoh'), k)) ; -eig_zoh2 = eig(feedback(c2d(sys, Ts2, 'zoh'), k)) ; -eig_zoh3 = eig(feedback(c2d(sys, Ts3, 'zoh'), k)) ; -plot(eig_zoh1,'x', 'LineWidth',1); + +plot(eig_zoh1_CL,'x'); hold on; -plot(eig_zoh2,'x','LineWidth',1); -plot(eig_zoh3,'x', 'LineWidth',1); +plot(eig_zoh2_CL,'x'); +plot(eig_zoh3_CL,'x'); axis square; plot(cos(theta), sin(theta), 'k--', 'HandleVisibility','off','LineWidth',0.25); xlim([-1.5, 1.5]); ylim([-1.5, 1.5]); + grid on; legend("Ts1", "Ts2", "Ts3"); - title({'Valeurs propres en temps discret',' avec un bloqueur d''ordre 0'}); xlabel('Réel'); ylabel('Imaginaire'); @@ -193,18 +198,31 @@ if SAVE end %% -sys_CL_Ts = feedback(c2d(sys, 0.037, 'zoh'),k); -t_d = 0:0.037:10 ; -cmd = double (t_d >= 1) ; -[y, t_d, x] = lsim(sys_CL_Ts, H*cmd, t_d) ; -u_d = H*cmd'-k*y ; + +for Ts = Ts2:0.001:Ts3 + if all(abs(eig(feedback(c2d(sys,Ts,'zoh'),k))) < 1) + ; + else + Ts; + break; + end + +end +% Maximum Sampling period +Ts = Ts-0.001 +%% +sys_CL_Ts = feedback(c2d(sys, Ts, 'zoh'),k); +t_d = 0:Ts:10 ; +cmd_d = double (t_d >= 1) ; +[y, t_d, x_d] = lsim(sys_CL_Ts, H*cmd_d, t_d) ; +u_d = H*cmd_d'-k*y ; Question10 = figure ; -plot(t_d, u_d, t_d, y, t_d, cmd, t_d, (Cw_L*x')'); +plot(t_d, u_d, t_d, y, t_d, cmd_d, t_d, (Cw_L*x_d')'); xlabel("Temps (s)"); ylabel("Amplitude"); legend("$u(t)$", "$y(t)$", "$w_c(L,t)$", "$w(L,t)$", 'Interpreter', 'latex'); -title({'Simulation du retour de sortie avec pré-gain'},{'avec un bloqueur d''ordre 0'}) ; +title({'Simulation du retour de sortie avec pré-gain','avec un bloqueur d''ordre 0'}) ; if SAVE exportgraphics(Question10, './latex/Illustrations/Question10.pdf') @@ -213,6 +231,7 @@ end %% Question 11 K_lqr = lqr(sys, eye(size(A)), 1); +K_lqr H_lqr = inv(Cw_L*inv(-A+B*K_lqr)*B) ; %% Question 12 @@ -231,13 +250,71 @@ if SAVE exportgraphics(Question12, './latex/Illustrations/Question12.pdf') end +%% Question 13 + +for Ts = 0.01:0.0001:0.02 + if all(abs(eig(c2d(sys,Ts,'zoh').A-c2d(sys,Ts,'zoh').B*K_lqr)) < 1) + ; + else + break; + end +end +Ts = Ts-0.0001 +%% +sys_d = c2d(sys, Ts, 'zoh'); +Ad = sys_d.A; +Bd = sys_d.B; +Cd = sys_d.C; +sys_d_CL = ss(Ad-Bd*K_lqr, Bd*H_lqr, Cd, 0, Ts); + +t_d = 0:Ts:10 ; +cmd_d = double(t_d > 1); +[y, t_d, x_d] = lsim(sys_d_CL, cmd_d, t_d) ; +u_d = H*cmd_d'-k*y ; + +Question13 = figure ; +plot(t_d, u_d, t_d, y, t_d, cmd_d, t_d, (Cw_L*x_d')'); +xlabel("Temps (s)"); +ylabel("Amplitude"); +legend("$u(t)$", "$y(t)$", "$w_c(L,t)$", "$w(L,t)$", 'Interpreter', 'latex'); +title({'Simulation du retour d''états avec pré-gain','avec un bloqueur d''ordre 0'}) ; + +if SAVE + exportgraphics(Question13, './latex/Illustrations/Question13.pdf') +end + +%% Question 14 + +sys_d = c2d(sys, 0.01, 'zoh'); +Ad = sys_d.A; +Bd = sys_d.B; +Cd = sys_d.C; +sys_d_CL = ss(Ad-Bd*K_lqr, Bd*H_lqr, Cd, 0, 0.01); + +t_d = 0:0.01:10 ; +cmd_d = double(t_d > 1); +[y, t_d, x_d] = lsim(sys_d_CL, cmd_d, t_d) ; +u_d = H*cmd_d'-k*y ; + +Question14 = figure ; +plot(t_d, u_d, t_d, y, t_d, cmd_d, t_d, (Cw_L*x_d')'); +xlabel("Temps (s)"); +ylabel("Amplitude"); +legend("$u(t)$", "$y(t)$", "$w_c(L,t)$", "$w(L,t)$", 'Interpreter', 'latex'); +title({'Simulation du retour d''états avec pré-gain','avec un bloqueur d''ordre 0'}) ; + +if SAVE + exportgraphics(Question14, './latex/Illustrations/Question14.pdf') +end + + %% Question 15 Bp = [0; 0; 0; 0; 1/2*L^4-L^3+L; L^3-1/2*L^4; 1/4*L^4-2/3*L^3+1/2*L^2; 1/4*L^4-1/3*L^3]; q0 = -5 * double (t' >= 4); sys_perturbations = ss(A-B*K_lqr, [B*H_lqr Bp], C, [0 0]); -%%Question 16 +%% Question 16 [y, t, x] = lsim(sys_perturbations, [cmd ; q0], t'); u = H_lqr*cmd'-(K_lqr*x')' ; Question16 = figure ; @@ -259,11 +336,13 @@ B_aug = [B; 0]; B_p_aug = [Bp; 0]; B_ref = [zeros(8,1); 1]; -%% +%% Question 18 K_aug = place(A_aug, B_aug, eig_4) + +%% Question 19 sys_integral = ss(A_aug - B_aug*K_aug, [B_ref B_p_aug], [C, 0], [0 0]); -[y, t, x] = lsim(sys_integral, [cmd ; q], t'); +[y, t, x] = lsim(sys_integral, [cmd ; q0], t'); u = -(K_aug*x')' ; Question19 = figure ; plot(t, u, t, y, t, cmd, t, ([Cw_L 0]*x')'); @@ -277,7 +356,7 @@ end %% sys_integral_d = c2d(sys_integral, 0.5, 'zoh'); -t_d = 0:0.5:10; % Define the time vector for discrete simulation +t_d = 0:0.5:10; cmd_d = double (t_d >= 1) ; q0_d = -5 * double(t_d >= 4); [y_d, t_d, x_d] = lsim(sys_integral_d, [cmd_d ; q0_d], t_d); diff --git 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