Consider the transfer function
G(s)=(s-3)/(s+5)(s-3)
it is noted that there are pole and zero that can be cancel.
How to find a UNCONTROLLABLE realization of the system????

If you cancel the (s-3) term, then G(s) is in its lowest term (the num
and the den are co-primes polynomials), and hence it has only a
minimal realization, hence by definition the realization is both
controllable and observable. Hence there is no uncontrollable
realization.
If you do not do any cancellation, then you can use the
controllability structure theory to convert this into a controllable
and uncontrollable parts. Use similarity transformation
Abar=T*A*inv(T). Need to find T, which is obtained from the
controllability matrix.
The new A matrix will be in the form [A11 A12;0 A22]. and the B matrix
will be [B1;0]. In this (A11,B1) is the controllable system, and A22
is the uncontrollable part.
If you have matlab, do help on the function ctrbf, this will do all
the work for you.
Nasser

Polytechforum.com is a website by engineers for engineers. It is not affiliated with any of manufacturers or vendors discussed here.
All logos and trade names are the property of their respective owners.