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