%% [BE] Digital control of cantilever beam

% Justin Bos
% Wissal Guarni
% Nolan Reynier Nomer
% Aleksander Taban
clear;
clc;
close all;

set(0, 'DefaultLineLineWidth', 1)

%% Question 3

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 ; % Longueur de la poutre
phi_L = phi(L) ;

A = [zeros(4), inv(E)*D ;
    -inv(E)*(D'), zeros(4) ] ;
B = [zeros(4,1) ; -inv(E)*phi_L] ;
C = [zeros(1,4) , -phi_L'] ;

A, B, C

%% Question 4
Ts1 = 0.01 ;
Ts2 = 0.02 ;
Ts3 = 0.04 ;

sys = ss(A, B, C, 0) ;

eig_cont = eig(sys);
eig_tustin1 = eig(c2d(sys, Ts1, 'tustin')) ;
eig_tustin2 = eig(c2d(sys, Ts2, 'tustin')) ;
eig_tustin3 = eig(c2d(sys, Ts3, 'tustin')) ;
eig_zoh1 = eig(c2d(sys, Ts1, 'zoh')) ;
eig_zoh2 = eig(c2d(sys, Ts2, 'zoh')) ;
eig_zoh3 = eig(c2d(sys, Ts3, 'zoh')) ;

figure;
subplot(1,3,1) ;
plot(eig_cont, 'x');
grid on;
xline(0, 'r--');
xlabel('Axe Réel');
ylabel('Axe Imaginaire');
title('Pôles du système');

subplot(1,3,2);
hold on;
plot(eig_tustin1,'x', 'LineWidth',2);
plot(eig_tustin2,'x', 'LineWidth',2);
plot(eig_tustin3,'x', 'LineWidth',2);

theta = linspace(0, 2*pi, 100);
plot(cos(theta), sin(theta), 'k--', 'HandleVisibility','off');
xlim([-1.1, 1.1]);
ylim([-1.1, 1.1]);
grid on;
title('Pôles dans le plan en Z (Discret)');
xlabel('Réel');
ylabel('Imaginaire');
hold off;

subplot(1,3,3);
hold on;
plot(eig_zoh1,'x', 'LineWidth',1);
plot(eig_zoh2,'x', 'LineWidth',1);
plot(eig_zoh3,'x', 'LineWidth',1);

legend("Eigen Values Ts1", "Eigen Values Ts2", "Eigen Values Ts3");
theta = linspace(0, 2*pi, 100);
plot(cos(theta), sin(theta), 'k--', 'HandleVisibility','off');
xlim([-1.1, 1.1]);
ylim([-1.1, 1.1]);
grid on;
title('Valeurs propres avec un bloqueur d''ordre 0 (ZOH)');
xlabel('Réel');
ylabel('Imaginaire');
hold off;

% Conlsuion : On en conclut qqc.


%% Question 5

t = 0:0.01:10;
cmd = double (t >= 1) ;
[y, t] = lsim(sys,cmd,t) ;

figure ;
plot(t, cmd, t, y);
xlabel("Temps (s)");
ylabel("Amplitude");
legend("u(t)", "y(t)");
title("Simulation en boucle ouverte") ;

%% Question 7

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 

%% Question 8 

H = inv(Cw_L*inv(-A+B*C*k)*B)
[y, t, x] = lsim(feedback(sys,k), H*cmd, t) ;
u = H*cmd'-k*y ;
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)");
title("Simulation du retour de sortie avec pré-gain") ;

%% Question 9

sys_CL = feedback(sys,k);
[y, tOut] = lsim(sys_CL, -3.*cmd, 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)

%% Question 11

K = lqr(sys, eye(size(A)), 1);
H_lqr = inv(Cw_L*inv(-A+B*K)*B) ;

%% Question 12

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 ;
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)");
title("Simulation du retour d'états avec pré-gain") ;