clear all
close all

% For the process P=1/(s^2+4*s+4), design a digital controller by emulation 
% of a continuous design in order to satisfy the following requirments.

%%%% REQUIREMENTS %%%%%%%%%%%%%%%%%%
% - error at steady state =0 wrt to a step disturbance d
% - s% < 30% (zita_c> 0.35, i.e. phase_margin > 35°)
% - ta5% <= 3 s (ta5%=3/zita_c/omega_c ==> omega_c >= 2.86 rad/s)
% - Ts = 80 ms (delay at omega_c: in terms of phase, 
%  a decrase of phase margin of about omega_c*Ts/2)

% plant to be controlled
s=tf('s');
P=1/(s^2+4*s+4);

C1=1/s; % controller implementing an integral  action with gain =1
% open loop function
F1=C1*P;

% evaluate the cross. freq and phase margin
figure,
margin(F1)

% increase the integral gain by 10 and introduce a zero in s=-0.4, then the tf of the controller is given by C2
C2=10*(s/0.4+1)/s;
F2=C2*P;

% evaluate the cross. freq and phase margin
figure,
margin(F2)

figure
bode(F1,F2)
hold on
grid on
legend

% evaluate the closed loop system for the two different cases (C1 and C2)
W1=feedback(F1,1);
W2=feedback(F2,1);

figure
step(W1,W2)
legend

% design the corresponding digital controller
Ts=0.08; % (omega_s=2*pi/Ts approx 80 rad/s, check if omega_s  is about 5-10 times omega_c

% delay at omega_c (about 4 rad/s), in terms of phase, about  -omega_c*Ts/2=-4*0.04=-0.16 rad 
% => decrase of phase margin of about 0.16 rad = 9.2 degrees

% discret equivalent tf by using Tustin
D1=c2d(C2,Ts,'tustin');

% anti-aliasing filter - a pole at frequency omega_f such that omega_f<omega_s/2 and
% omega_f>>omega_c
omega_s=2*pi/Ts; % sampling freq.
omega_f=omega_s/2.1; % frequency for the pole of anti-aliasing filter
Fa=1/(s/omega_f+1);

scheme_exam_IA_13_June_2024