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

Abstract

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 D_em ^e[z, R _em ^e(z)] and D_em ^f[zb centred at an AZ and ADZ respectively are defined. It is shown that D_em ^e[R _em ^e(z)] contains at least one zero of all systems derived from G(s) under squaring down and D_em ^f [z, R _em ^f (z)] contains at least one pole of all systems derived from G(s) under constant output feedback. A criterion for determining upper bounds for R _em ^e(z) and R _em ^f(z)is given. These results show that AZs act as ‘ nearly fixed ’ zeros under squaring down, and that ADZs act as ‘ nearly fixed ’ closed-loop poles under constant output feedback. Systems which under constant post and feedback compensation have their zeros and poles, respectively, ‘ trapped ’ in the right-half complex plane are examined and a criterion for testing such properties is given. This work reveals the AZs and ADZs as ‘ strong ’ invariants characterizing families of systems derived from G(s) and not just the particular minimal system defined by G(s).