Accurate Solutions of Ill-Posed Problems in control theory

Computable, guaranteed error bounds are presented for controllable subspaces and uncontrollable modes, unobservable subspaces and unobservable modes, supremal $( A,C )$ invariant subspaces in ker D, supremal $( A,C)$ controllability subspaces in ker D, the uncontrollable modes within the supremal $( A,C )$ invariant subspace in ker D, and invariant zeros. In particular the bounds apply in the nongeneric case when the solutions are ill-posed. This is done by showing that all these features are eigenspaces and eigenvalues of certain singular matrix pencils, which means they may all be computed by a single algorithm to which a perturbation theory for general singular matrix pencils can be applied. Numerical examples are included.