Controllability is handled in textbooks in a rather academic way: > yes or no.
Well, apart from these yer/no tests you can use controllability gramian based tests that also give some information on relative controllability of each mode. Actually some model order reduction techniquies are based on these tools (balanced truncation).
[snip] I would transform the system into Jordan-block form and see whether
> each block has an input of reasonable size. If the input acts with
> factor 0.00001 for the respective block, then the system is theoreti-
> cally controllable but practically not.
> This is a modal transformation and each mode should be controllable.
Transformation into Jordan form is very (!) tricky process from numerical point of view. It serves as a theoretical and educational tool but is not suited for real use. Think a bit about implications of the following Matlab code snippet:
> A = [1 0;0 1]
A =
1 0 0 1
> epsilon = 1e-12; A = A + [0 epsilon;0 0]
A =
1.0000 0.0000 0 1.0000
> jordan(A)
ans =
1 1 0 1Best regards, Zdenek Hurak