Grassmann invariants, almost zeros and the determinantal zero, pole assignment problems of linear multivariable systems

The strucxpects of the rationxr spaces x c, x fwhere x c is the columxof the transfer function matrix G(s) and x f, is the space associated with right matrix fracxcriptions of G(s), are investigated. For x f gcanonixqb;s] Grassmgesentax g(x c) and g(x f) arxd, and are shown xmplete basis free invariants for x c and x f respectively. The almost zeros (AZ) and almost decoupling zeros (ADZ) of G(s)gdefinex local minima of a normgn defix g(x c) and g(x f) respectively. The computation, and certain aspects of the distribution in the complex plane of AZs and ADZs are examined. The role of AZs and ADZs in the determinantal zero and pole assignment problems respectively is examined next. Two important families of systems are defined : the strongly zero non-assignable (SZNA) and the strongly pole non-assignable (SPNA) systems. For SZNA and SPNA systems minimal radius discs Dem e[z, R em e(z)] and Dem f[zb centred at an AZ and ADZ respectively are defined. It is shown that Dem e[R em e(z)] contains at least one zer...