%%% Initialisation
clc;
clear;
close all;

% Poles et zéros de X(z)
%%%%%% X(z)=(z^2 - z)/(z^3 - z^2 +2z +1)
num = [1 -1 0];
den = [1 -1 2 1];
z=roots(num);
p=roots(den);
zplane(z,p);
disp('zeros=');disp(z);
disp('pôles=');disp(p);

%%
% Poles et zéros de X(z)
%%%%%% Y(z)/z = (z^2 + z)/(z^3 - 2z^2 + (3/2)z - 1/2)

% B(z)      R(1)         R(2)               R(n)
% ----- = --------- + --------- + ... + ---------- + K(z)
% A(z)     z - P(1)     z - P(2)          z - P(n)

B = [1 1 0];
A = [1 -2 3/2 -1/2];
[R,P,K] = residuez(B,A)

% 
%% Tustin and BOZ

%Transformée inverse
%H(z)= (z-0.5)/(z^2+z+0.3); T=0.1s

Hd = tf([1 -0.5], [1 1 0.3],0.1);
Hc1 = d2c(Hd, 'zoh')
Hc2 = d2c(Hd, 'tustin')
% 
%% exemple du cours slide 47
% 
%H(s)= (s+1)/(s^2 + 4s + 5); 
Hc = tf([1 1], [1 4 5]);
% periode d'echan.
Ts=0.1;
Hd1 = c2d(Hc, Ts, 'zoh');
Hd2 = c2d(Hc, Ts, 'tustin');
%Hd1.Variable = 'z^-1'

step(Hc)
hold on
step(Hd1)
hold on
step(Hd2)
legend('$Hc$','$Hd1$','$Hd2$','Interpreter','latex')