clear all
close all

% For the process P=3/(s^2+6*s+9), 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 => K(s)=k0/s
% - error at steady state (e_inf) <=0.1 wrt to a ramp signal r(t)=R0 t*1(t), R0=0.5
%  => e_inf=R0/F0<=0.1;  F0=k0*P0=k0/3 => k0>=15
% - s% < 30% (zita_c> 0.35, i.e. phase_margin > 35°)
% - ta5% <= 3 s (ta5%=3/zita_c/omega_c ==> omega_c >= 2.2 rad/s)
% - Ts = 0.05 s (delay at omega_c: in terms of phase, 
%  a decrase of phase margin (Delta_phi) of about omega_c*Ts/2, 
% for example if omega_c=5, Delta_phi=0.125 rad = 7.2 degrees)


% plant to be controlled
s=tf('s');
P=3/(s+3)^2;
k0=15;
C1=k0/s; % controller implementing an integral action with gain =15
% open loop function
F1=C1*P;

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

% introduce a zero in s=-3, then the tf of the controller is given by C2
C2=k0*(s/3+1)/s;
F2=C2*P;

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

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)
grid on
hold on
legend

% design the corresponding digital controller
Ts=0.05; % (omega_s=2*pi/Ts approx 125 rad/s, check if omega_s  is at least about 5-10 times omega_c

% delay at omega_c, in terms of phase, about  -omega_c*Ts/2=-3*0.05/2=0.07 rad 
% => decrase of phase margin of about 0.07 rad = 4-5 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_23_July_2024