Necessary and sufficient conditions for balancing unstable systems

Necessary and sufficient conditions are given for the existence of balancing transformations for minimal state-space realizations
(A, B, C)
where
A
may be unstable. These conditions are expressed in terms of the real diagonalizability of the product of the reachability Gramian and the observability Gramian. For symmetric realizations these conditions can be reformulated in terms of the real diagonalizability of the cross Gramian, and we show that minimal symmetric systems can be internally balanced if the associated Hankel matrix is positive semidefinite. Examples are given of minimal systems, including symmetric systems, which cannot be balanced.