%% Analysis of robust stability for a distillation column in DV configuration
% We will show the conservativeness of the test of robust stability based
% on singular values, in contrast with the more tailored test based on
% structured singular values.

%% Linearized model of a distillation column in DV configuration 
% We take it from Skogestad's book, where it appears throughout the text.
% For example, page 314 at the bottom (in the second edition of the book).

s = tf('s');
T = 75;                                       % time constant
G0 = [-87.8, 1.4; -108.2, -1.4]/(T*s+1);      % nominal model

%% Inverse-based controller
% The dynamics of the controller are given by the inverse of the model. 
% An integrator is appended to make it proper. Scaling for each pair of inputs 
% and outputs is added too. The resulting controller is actually a PI controller.

K = tf([T 1],[1 0]) * [-0.0015, 0; 0, -0.075];                        

%% Modeling the uncertainty

tau = 1.0;
r0 = 0.2;
rinf = 2.0;

wi = (tau*s+r0)/(tau/rinf*s+1) 
wi = ss(wi);  

%wi = makeweight(r0,1/tau,rinf);

%% Closed-loop transfer functions
% We can find all closed-loop transfer function using a single command in
% Matlab

C0 = loopsens(G0,K);                            

%% Computing the structured singular values of complementary sensitivity function
omega = logspace(-3,2,100);                     % frequency
BlkStruct = ones(2,2);                          % diagonal uncertainty structure

muT = mussv(frd(C0.Ti,omega),BlkStruct);        % computing SSV of T

%% Plotting the singular and structured singular values of T
% The robust stability test now consists in comparing the plots of singular
% values and structured singular values of T with the plot of the (inverse
% of) the uncertainty weighting filter W.

sigma(C0.Ti, muT(1), inv(wi), omega),        
legend('\sigma(Ti)', '\mu(Ti)','1/Wi'),

title('');
ylabel('Magnitude')
xlabel('Frequency')
grid on

%%
% The obvious conclusion of such comparision is that if the ordinary
% singular values are used in the test, the robust stability of the
% closed-loop system cannot be confirmed since the plot of $\sigma(T)$ 
% is not below the plot of $1/|W|$ for all frequencies. If, on the other hand, the more
% tailored structured singular values are looked at, closed-loop robust
% stability can be confirmed.