# uncontrollable system?

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????

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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