From bccd6c4a3f4e1731c600cdff13983425d9f62099 Mon Sep 17 00:00:00 2001 From: Nolan Date: Mon, 20 Apr 2026 16:11:03 +0200 Subject: [PATCH] ajout figure et avance sujet --- BE_Beam.m | 61 +++++++++++++++++++++++++---- latex/Illustrations/Question12.pdf | Bin 0 -> 41465 bytes latex/Illustrations/Question16.pdf | Bin 0 -> 66325 bytes latex/Illustrations/Question19.pdf | Bin 0 -> 50715 bytes latex/contents.tex | 27 +++++++++++++ latex/main.pdf | Bin 300326 -> 471357 bytes 6 files changed, 81 insertions(+), 7 deletions(-) create mode 100644 latex/Illustrations/Question12.pdf create mode 100644 latex/Illustrations/Question16.pdf create mode 100644 latex/Illustrations/Question19.pdf diff --git a/BE_Beam.m b/BE_Beam.m index 38de810..623fd28 100644 --- a/BE_Beam.m +++ b/BE_Beam.m @@ -111,7 +111,7 @@ Question5 = figure ; plot(t, cmd, t, y); xlabel("Temps (s)"); ylabel("Amplitude"); -legend("u(t)", "y(t)"); +legend("$u(t)$", "$y(t)$", 'Interpreter', 'latex'); title("Simulation en boucle ouverte") ; if SAVE @@ -128,9 +128,9 @@ Cw = @(zeta) [zeta^2*(2*zeta^3-5*zeta^2+10)/20 ... Cw_L = Cw(L); %% -TransientTime = [] ; -for i = 0.01:0.01:10 - TransientTime(end+1) = stepinfo(feedback(sys, i)).TransientTime ; +TransientTime = zeros(size(0.01:0.01:10)) ; +for i = 1:size(TransientTime, 2) + TransientTime(i) = stepinfo(feedback(sys, i*0.01)).TransientTime ; end [minimum, min_index] = min(TransientTime) k = min_index * 0.01 @@ -186,11 +186,58 @@ H_lqr = inv(Cw_L*inv(-A+B*K)*B) ; sys_CL_lqr = ss(A-B*K, B*H_lqr, C, 0); [y, t, x] = lsim(sys_CL_lqr, cmd, t); -u = H_lqr*cmd'-k*y ; -figure ; +u = H_lqr*cmd'-(K*x')' ; +Question12 = figure ; plot(t, u, t, y, t, cmd, t, (Cw_L*x')'); xlabel("Temps (s)"); ylabel("Amplitude"); -legend("u(t)", "y(t)", "wc(t)", "w(t)"); +legend("$u(t)$", "$y(t)$", "$w_c(L,t)$", "$w(L,t)$", 'Interpreter', 'latex'); title("Simulation du retour d'états avec pré-gain") ; +if SAVE + exportgraphics(Question12, './latex/Illustrations/Question12.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]; + +q = -5 * double (t' >= 4); +sys_perturbations = ss(A-B*K, [B*H_lqr Bp], C, [0 0]); + +%%Question 16 +[y, t, x] = lsim(sys_perturbations, [cmd ; q], t'); +u = H_lqr*cmd'-(K*x')' ; +Question16 = figure ; +plot(t, u, t, y, t, cmd, t, (Cw_L*x')'); +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 et perturbations") ; + +if SAVE + exportgraphics(Question16, './latex/Illustrations/Question16.pdf') +end + +%% Question 17 +eig_lqr = eig(sys_CL_lqr); +eig_4 = [eig_lqr ; -2]; +A_aug = [A, zeros(8,1) ; -Cw_L, 0]; +B_aug = [B; 0]; +B_p_aug = [Bp; 0]; +B_ref = [zeros(8,1); 1]; + +%% +K_aug = place(A_aug, B_aug, eig_4) +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'); +u = -(K_aug*x')' ; +Question19 = figure ; +plot(t, u, t, y, t, cmd, t, ([Cw_L 0]*x')'); +xlabel("Temps (s)"); 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