Some comments on the paper by Alstad et al. (JPC, 2009)

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A. A trivial derivation of the explicit expression for the nullspace method:

1. Measurement relationship (eq.3)
   y = Gtildey [u d ]'
Inverting this:
   [u d]' = Gtildey-1 y
2. Condtion for minimum then gives (eq.2)
   [Juu Jud]  [u d]' = [Juu Jud] Gtildey-1 y =0
This gives eq.8:
   H = [Juu Jud] Gtildey-1

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B.
When we use inputs as part of the measurements, 
     c = H y, where y = [x u] with noise ny = [nx  nu],
then the noise nu is mathematically equivalent to an input disturbance du.
  utrue = u + nu
or
  utrue = u + du
where u is the input computed by the controller

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% C
% TOY EXAMPLE FROM ALSTAD et al. (JPC, 2009)
% J = (u-d)^2
% y1=0.1*(u-d)
% y2=20*u
% y3=10*u-5d
% y4=u
nu=1
nd=1

Gy = [0.1 20 10 1]'
Gyd= [-0.1 0 -5 0]'
Juu=2, Jud=-2
F=[0 20 5 1]'
Wd=0*eye(1)
Wn=eye(4)
Y=[F*Wd Wn]

% Single measurements
H1 = [1 0 0 0]; H2 = [0 1 0 0]; H3=[0 0  1 0]; H4=[0 0 0 1];
%M = sqrtm{Ju}*inv(H*Gy)*H*Y
H=H1; M = sqrtm(Juu)*inv(H*Gy)*H*Y; L1 = 0.5*svd(M)^2 %=100 (WORST)
H=H2; M = sqrtm(Juu)*inv(H*Gy)*H*Y; L2 = 0.5*svd(M)^2 %=1.0025
H=H3; M = sqrtm(Juu)*inv(H*Gy)*H*Y; L3 = 0.5*svd(M)^2 %=0.26 (BEST)
H=H4; M = sqrtm(Juu)*inv(H*Gy)*H*Y; L4 = 0.5*svd(M)^2 %=2

% Nullspace combination of y2 and y3
ny=2;[u,s,v]=svd([20 5]'), Hnull=u(:,[nd+1:ny])' % = [-0.2425  0.9701]
H=[0 Hnull 0]; M = sqrtm(Juu)*inv(H*Gy)*H*Y; Lnull = 0.5*svd(M)^2 %=0.0425
% Nullspace, alternative explicit expression (here using all measurements):
Ha = [Juu Jud]*pinv([Gy Gyd]); Ha = Ha/norm(Ha)  % = [0.0206 -0.2419 0.9700 -0.0121]
H=Ha; M = sqrtm(Juu)*inv(H*Gy)*H*Y; Lnull2 = 0.5*svd(M)^2 %=0.0425


% Exact local method (loss method)
He = (inv(Y*Y')*Gy)'; He = He/norm(He)  % [0.0206  -0.2317   0.9725  -0.0116]
H=He; M = sqrtm(Juu)*inv(H*Gy)*H*Y; Le = 0.5*svd(M)^2 %=0.0405