Computation of supremal (A,B)-invariant and controllability subspaces

Two fundamental concepts of geometric control theory,
(A,B)
-invariant and controllability subspaces, are discussed in terms of spaces spanned by closed-loop eigenvectors. Included is a characterization of
V^{\ast},R\Re^{\ast}
, the supremal
(A,B)
-invariant and controllability subspaces contained in the kernel of some map. Applying ideas found in numerical analysis literature, it is shown that, for design purposes, knowledge of
V^{\ast},R\Re^{\ast}
is not sufficient: certain subspaces of
V^{\ast},R\Re^{\ast}
may be useless with respect to true design applications. Possible consequences of design based on these unreliable parts of
V^{\ast},R\Re^{\ast}
are discussed. Finally, prototype algorithms for computing basis vectors for
V^{\ast},R\Re^{\ast}
are given. Their strength is in the additional information which makes it possible to identify the reliable components of
V^{\ast},R\Re^{\ast}
Numerical stability and efficiency are "built in" to the algorithms through the use of routines which have been implemented, tested thoroughly, and recommended by recognized experts in numerical analysis.