ajout figure et avance sujet

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 = [] ;
+TransientTime = zeros(size(0.01:0.01:10)) ;
-for i = 0.01:0.01:10
+for i = 1:size(TransientTime, 2)
-    TransientTime(end+1) = stepinfo(feedback(sys, i)).TransientTime ;
+    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 ;
+u = H_lqr*cmd'-(K*x')' ;
-figure ;
+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)");
+ylabel("Amplitude");
+legend("$u(t)$", "$y(t)$", "$w_c(L,t)$", "$w(L,t)$", 'Interpreter', 'latex');
+title("Simulation du retour d'états avec action intégrale et perturbations") ;
+if SAVE
+    exportgraphics(Question19, './latex/Illustrations/Question19.pdf')
+end

latex/contents.tex

@@ -204,6 +204,30 @@ Pour :
 	\end{aligned}\right. \Rightarrow d\acute{e}cla \ A, \ B, \ C \ en \ MATLAB
 \end{equation*}
 
+\begin{equation*}
+	A = \begin{pmatrix}
+		0 & 0 & 0 & 0 & 114 & 6 & 12 & -2 \\
+		0 & 0 & 0 & 0 & -54 & 6 & -2 & 4 \\
+		0 & 0 & 0 & 0 & -1188 & -12 & -114 & 6 \\
+		0 & 0 & 0 & 0 & -828 & -12 & -54 & 0 \\
+		6 & 54 & 4 & -2 & 0 & 0 & 0 & 0 \\
+		-6 & -114 & -2 & 12 & 0 & 0 & 0 & 0 \\
+		-12 & -828 & -6 & 54 & 0 & 0 & 0 & 0 \\
+		-12 & -1188 & -6 & 114 & 0 & 0 & 0 & 0 \\
+	\end{pmatrix}, \  B = \begin{pmatrix}
+		0 \\
+		0 \\
+		0 \\
+		0 \\
+		4 \\
+		-16 \\
+		-60 \\
+		-120 \\
+	\end{pmatrix}, \ C = \begin{pmatrix}
+		0 & 0 & 0 & 0 & 0 & -1 & 0 & 0
+	\end{pmatrix}
+\end{equation*}
+
 
 \subsection{Exercice 4}
 \begin{equation*}
@@ -299,6 +323,7 @@ $C_1$ et $C_2$ ?????????????????
 
 
 \subsection{Exercice 12}
+\includegraphics[width=\textwidth]{Illustrations/Question12}\begin{align*}
 
 
 \subsection{Exercice 13}
@@ -314,6 +339,7 @@ $C_1$ et $C_2$ ?????????????????
 
 
 \subsection{Exercice 16}
+\includegraphics[width=\textwidth]{Illustrations/Question16}\begin{align*}
 
 
 \subsection{Exercice 17}
@@ -323,6 +349,7 @@ $C_1$ et $C_2$ ?????????????????
 
 
 \subsection{Exercice 19}
+\includegraphics[width=\textwidth]{Illustrations/Question19}\begin{align*}
 
 
 \subsection{Exercice 20}