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*, R*, 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*, R* is not sufficient: certain subspaces of V*, R* may be useless with respect to true design applications. Possible consequences of design based on these unreliable parts of V*, R* are discussed. Finally, prototype algorithms for computing basis vectors for V*, R* are given. Their strength is in the additional information which makes it possible to identify the reliable components of V*, R*. 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.