%% Structured vs. unstructured uncertainty 
% Example 8.5 from Skogestad. We consider just a constant (over frequency)
% system with two inputs and two outputs modelled by a matrix

M = [2 2; -1 -1]

%%
% SVD decomposition of the matrix gives not only the highest singular value
% (directly displaying the Hinf norm of the modelled system) but also the
% corresponding singular vectors.

[U,S,V] = svd(M)

V1 = V(:,1)
U1 = U(:,1)

sigma1 = S(1,1)

1/sigma1

%% Unstructured uncertainty (Delta matrix full)
% The largest singular value and the corresponding singular vectors give
% the smallest destabilizing uncertainty assuming arbitrary structure

Delta = 1/sigma1*V1*U1'
det(eye(2,2)-M*Delta)

hinfM = sigma1

%% Structured uncertainty (Delta matrix diagonal)
% If the structure of the uncertainty block is fixed to a diagonal one, 
% the smallest destabilizing uncertainty is a bit larger. Here we did not 
% compute the actual value but just copied from the referenced text and
% demonstrate that indeed the stability is lost.

Delta = 1/3*diag([1,-1])   
det(eye(2,2)-M*Delta)

muM = 3

1/muM