clear all
clc
close all

% For the process P=25/(s^2+15*s+50), design a digital controller by emulation 
% of a continuous design in order to satisfy the given requirments.

% plant to be controlled
s=tf('s');
P=25/(s^2+15*s+50);

%%%% REQUIREMENTS %%%%%%%%%%%%%%%%%%
% 1. error at steady state (e_inf) =0 wrt a step reference signal =>
%  include an integrator in the open loop function (a pole in the origin)

% 2. relative error at steady state (e_inf_r) <=0.05 wrt to disturbs. 
% in the range [0.01 0.1] rad/s => |F(jw)|>=1/0.05=20 in that range, i.e |F(jw)|>=26dB

% 3. no overshoot => zita_c>0.6 phase_margin>60 degrees 
% (first order approx. of the closed loop system)

% 4. ta5<=1 sec; ta5=3*tau, tau=1/wc <=> 3/wc<=1 <=> wc>=3
% Delta_ph: phase decrease due to discretization, i.e. wc*Ts/2,
% then with wc=3, Delta_ph is about 9 degrees
Ts=0.1;
wc=3;
Delta_ph=wc*Ts/2*180/pi;

% use an integral controller, increase the integral gain in order to satisfy
% the second requirement, i.e. |F(jw)|>=26dB for w in [0.01 0.1]

% integral gain ki
ki=6;
C1=ki/s; 
% open loop function with C1
F1=C1*P; 

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

% the cross. freq. of F1 is close to the desired wc,
% but it is needed to increase the phase margin

% introduce a zero in s=-3, then the tf of the controller is given by C2
C2=C1*(s/3+1);
% define the open loop function with C2
F2=C2*P;
% evaluate the cross. freq. and phase margin 
figure,
margin(F2)
grid on

% compare F1 and F2
figure
bode(F1,F2)
hold on
grid on
legend

% % evaluate the continuous closed loop system for the two cases (C1 and C2)
W1=feedback(F1,1);
W2=feedback(F2,1);
figure
hold on
grid on
step(W1,W2)
legend

% implement the digital controller
D1 = c2d(C2, Ts, 'tustin'); % Discretize using Tustin's method

% anti-aliasing filter - a pole at frequency wf such that wf<ws/2,
% check wf>>wc
ws=2*pi/Ts; % sampling freq.
wf=ws/2.1; % frequency of the pole of anti-aliasing filter
Fa=1/(s/wf+1);

scheme_exam_IA_25_Feb_2025