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

Abstract

Two fundamental concepts of geometric control theory, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(A,B)</tex> -invariant and controllability subspaces, are discussed in terms of spaces spanned by closed-loop eigenvectors. Included is a characterization of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V^{\ast},R\Re^{\ast}</tex> , the supremal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(A,B)</tex> -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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V^{\ast},R\Re^{\ast}</tex> is not sufficient: certain subspaces of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V^{\ast},R\Re^{\ast}</tex> may be useless with respect to true design applications. Possible consequences of design based on these unreliable parts of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V^{\ast},R\Re^{\ast}</tex> are discussed. Finally, prototype algorithms for computing basis vectors for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V^{\ast},R\Re^{\ast}</tex> are given. Their strength is in the additional information which makes it possible to identify the reliable components of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V^{\ast},R\Re^{\ast}</tex> 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.