%% Robustness as the motivation for introduction of feedback

%% Uncertain system
% We define the two uncertain parameters, that is, the parameters whose
% values are only known to be in some given intervals.

ze = ureal('zeta',0.2,'Range',[0.1,0.3]);
wn = ureal('omega_n',1,'Range',[0.8,1.3]);

%%
% From now on we can work with these uncertain parameters as with any other
% Matlab parameters/variables. Therefore we can define an uncertain
% function in the following standard way

G = tf(wn^2*[1 2 1],[1 2*ze*wn wn^2]);

%% Design of feedforward controller
% Ideal feedforward compensator is always as close to the inverse of the
% system as possible. 

F0 = inv(G.NominalValue);

%% Design of a feedback controller for the nominal system
R0 = 2e9*tf([1 12],[1 0]);

%% Simulating the response of the uncertain system
subplot(3,1,1)
step(G,'c:',G.NominalValue,'r',20)
xlabel('t')
ylabel('h_a(t)')
title('Response of the nominal and uncertain models to a step input')

subplot(3,1,2)
step(G*F0,'c:',G.Nominal*F0,'r',20)
title('Response with a feedforward compensator')
xlabel('t')
ylabel('h_b(t)')

subplot(3,1,3)
step(feedback(G*R0,1),'c:',feedback(G.Nominal*R0,1),'r',20)
title('Response with a feedback compensator')
xlabel('t')
ylabel('y(t)')
%print -depsc '../obrazky/robustnost_jako_motivace_pro_zpetnou_vazbu.eps'