Counterexample to "RGA=I implies that the plant is one-way coupled"

The counterexample given in Johnson and Shapiro (1986) is
	 g = [1 1 0 0; 0 1 1 1; 1 1 1 0; 0 0 1 1] 	
This plant has RGA=I but it cannot be reordered to be one-way coupled.

It is possible to change some of the entries and still have RGA=I.
For example, the matrix 
	 g1 = [1 1 0 0; 0 fa 1 1; 1 1 fb 0; 0 0 1 1] 	
also has RGA=I for any nonzero value of fa and fb.

I will now give a simple physical example of a process which has a transfer
matrix g1: Consider a mixing process where six streams are mixed together:
m1, m2, m3, m4, ma and mb. 
	- m1 and m2 are mixed to form a stream with flowrate y1
        - m3 and m4 are mixed to form a stream with flowrate y4
	- y4 is mixed with stream ma to form a stream with flowrate y2
	- y1 is mixed with stream mb to form a stream with flowrate y3
	- ma=fa*m2 (ratio controller installed and part of process)
	- mb=fb*m3 (ratio controller installed and part of process)
In other words we have
	m1 + m2 = y1
	fa*m2 + m3 + m4 = y2
	m1 + m2 + fb*m3 = y3
	m3 + m4 = y4

With inputs u=[m1 m2 m3 m4] and outputs y=[y1 y2 y3 y4] the gain matrix
for this process is EXACTLY as given by g1.

This process has RGA=I, but it is not one-way coupled. 
Comment: It would have been nice if it were one-way coupled because, 
then the stability of the individual loops would always guarantee 
the stability of the overall system, but unfortunately, this is not how it is.... :-( 

This process does not have any dynamics, but we could, for example,
add dynamics to all the inputs (m1, m2, m3 m4) and to the outputs (e.g.
measurement delays) without changing the RGA.

To show that we may get stability problems because G is not one-way decoupled 
we may consider the eigenvalues of the "relative interaction matrix" 
E=(G-Gdiag) Gdiag^-1.
Eigenvalues of E larger than 1imply that we may get instability of the 
overall system with stable individual loops.

Let use consider a plant corresponding to fa=fb=0.1:
g2 =
     1     1     0     0
     0    0.1   1     1
     1     1    0.1   0
     0     0     1     1
g2diag=diag(diag(g2));
e2 = (g2-g2diag)*inv(g2diag) =
     0    10     0     0
     0     0    10     1
     1    10     0     0
     0     0    10     0
eig(e2) =
   10.9161
   -8.8730
   -1.1270
   -0.9161

Two of the eigenvalues are much larger than 1, which indicates
that we can have the case where stable individual diagonal loops 
gives an unstable overall system. (indeed, designing nominally
identical loops L=l/s, gives overall instability for fa,fb less than
about 0.1).