Matrix pencil characterization of almost ( A, Z) -invariant subspaces : A classification of geometric concepts

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Abstract
The equivalence between the algebraic, matrix pencil, characterization of the sub-spaces of the ‘ extended ’ geometric theory and their dynamic characterization 13 established. As a result, a complete classification of Almost ( A, B) -invariant, ( A, B) -invariant, Almost controllability and controllability subspaces is derived in terms of matrix pencil invariants. The frequency propagation aspects of infinite spectrum ( A, B)-invariant subspaces are investigated and it is shown that they are limits of closed-loop eigenspaces with arbitrarily largo eigenvalues. Finally, the importance of the infinite frequency subspaces in the study of the asymptotic behaviour of the closed-loop eigenspaces and eigenvalues under scalar gain output feedback is discussed.