In higher dimensional systems, it is possible to pair on negative RGA values and still have a stable system, something of an impossibility in a $22$ system. However, for such systems, it has been shown $($see also Gagnepain and Seborg $)$ that if the loop which is paired on the negative RGA element is opened, the lower dimensional subsystem will be unstable.

$K=G(0)=\left[\begin{array}{ccc}1& 1& -0\mathrm{.}1\\ 0\mathrm{.}1& 2& -1\\ -2& -3& 1\end{array}\right]$

with a determinant, $|K|=0\mathrm{.}53.$ The RGA for this system is:

$=\left[\begin{array}{ccc}-1\mathrm{.}89& 3\mathrm{.}59& -0\mathrm{.}7\\ -0\mathrm{.}13& 3\mathrm{.}02& -1\mathrm{.}89\\ 3\mathrm{.}02& -5\mathrm{.}61& 3\mathrm{.}59\end{array}\right]$

from which one finds that the only feasible pairing has to involve a negative RGA element for the first loop because the other pairing configurations are disqualified by Rule #$2.$

The suggested $1-1/2-2/3-3$ pairing gives rise to a configuration for which the system, according to Niederlinski's theorem, is not structurally unstable despite pairing on the negative $(1,1)$ RGA element.

However, if the first loop is opened $($i$.$e$.,$ the $y_1-m_1$ variables are dropped from the process model$)$ the resulting subsystem will have a steady$-$state gain matrix relating the remaining two output variables $y_2$ and $y_3,$ to the remaining two input variables, $m_2$ and $m_3,$ given by:

tilde$(K)=\left[\begin{array}{cc}2& -1\\ -3& 1\end{array}\right]$

and it is easy to verify that the $y_2-m_2,y_3-m_3$ pairing will violate the Niederlinski theorem and thus give rise to an unstable system.

Such a system that is stable when all loops are closed, but that goes unstable should one of them become open, is said to have a low degree of integrity:

"Integrity: A system has integrity if$,$ after one or more loops are placed in manual, the remaining closed$-$loop system can be stable without changing the signs of any feedback controller gains remaining in automatic. $($we would like multiloop system to have integrity!$)$