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

Two fundamental concepts of geometric control theory, -invariant and controllability subspaces, are discussed in terms of spaces spanned by closed-loop eigenvectors. Included is a characterization of , the supremal -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 is not sufficient: certain subspaces of may be useless with respect to true design applications. Possible consequences of design based on these unreliable parts of are discussed. Finally, prototype algorithms for computing basis vectors for are given. Their strength is in the additional information which makes it possible to identify the reliable components of 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.