On Grassmann Invariants and Almost Zeros of Linear Systems

Abstract The canonical-IR [s]- Grassmann representative, g(X), of a rational vector space is defined as a new complete invariant of X. For the rational vector spaces X c , X b associated with a linear system, g(X c ), g(X b ) are used to define the new concepts of Almost Zeros (AZ) and Almost Decoupling Zeros (ADZ). The AZs and ADZs are natural extensions of the concepts of multivariable zeros and de coupling zeros respectively, and they are defi ned as local minima of a norm function defined on g(X c ), g(X b ) correspondingly. The dis tribution in the complex plane, the invariance properties and the significance for control problems of AZs and ADZs is discussed. The families of Strongly Zero Nonass ignable (SZNA) and Strongly Pole Nonassignable (SPNA) systems are defined. It is shown, that for SPNA (SZNA) systems the ADZs (AZs) define the centres of disks “trapping” closed-loop poles (zeros) of systems under any constant output feedback (squaring down). Criteria for determining upper bounds for the radii of the “trappining” disks are given. These results show that the AZs act as “nearly” fixed zeros under constant squaring down, and that the ADZs act as “nearly” fixed poles under constant output feedback.