Controllability and Condition Number

Consider following System (A, B) given as follows:

A =
-205.5237 198.3209 7.2028 0 -0.3256 0 0
198.3209 -205.5237 0 7.2028 0 0 0
0.0646 0 -0.0646 0 0 0 0
0 0.0646 0 -0.0646 0 0 0
0 0 0 0 0 0 0
463.7826 0 0 0 0 -43.4783 0
0 463.7826 0 0 0 0 -43.4783

B =

0
0
0
0
0.5600
0
0


Lets check the controllability of this system:

>> rank(ctrb(A,B)) = 4

You get a different result when you apply PBH (Popov-Belevitch-Hautus) test. In this test we check for the rank of [\lambda*I-A, B] matrix for each eigenvalue lambda. Following code (written by Gopal) performs the PBH test


% %%Controllabilaty test
Eigen_A=eig(A);
Data_Con = zeros(7,2);
for i=1:length(Eigen_A)
S=Eigen_A(i)*eye(length(A) );
D=S-A;
Control_A=[D B];%Formation of controllability matrix.
Rank_Con = rank(Control_A);
Data_Con(i,:)= [Eigen_A(i), Rank_Con];
end
Data_Con
%%%%%%%%%%%%%%%%%%
This gives

Data_Con =

-43.4783 6.0000
-43.4783 6.0000
-403.8457 7.0000
-7.2674 7.0000
0.0000 7.0000
-0.0634 7.0000
0 7.0000

where first column gives the eigen values and the second gives the corresponding rank.

As far as I know the controllability information obtained in either way should be equivalent. That means both methods should give same information about the controllability.

On little analysis I found that the matrix A is highly ill-conditioned with a condition number

>> cond(A) = 4.3373e+20

Could this be the reason for this anomoly. In other words, you can not rely on the controllability tests as it is ill-conditioned. Gopal tells me that this system pertains to a real system (related to a nuclear reactor) . In that case, how do you deal with such a system. Is there any process by which we can make it well-conditioned so that one can carry out computer simulations with good accuracy.