The pencil (sE-A) and controllability-observability for generalized linear systems: A geometric approach

In this paper we adopt a geometric approach to study the pencil (sE-A). The relationship between a certain subspace and a polynomial basis for ker (sE-A) is established. An alternative characterization for the finite-zero structure of a singular pencil is presented. Necessary and sufficient conditions for the columns or the rows of the singular pencil (sE-A) to be linearly independent over the ring of the polynomials are also given. The main geometric properties of a regular pencil are presented, including the identification of the subspace in which the impulsive response of the autonomous generalized linear system Ex = Ax takes place. The generalized linear system Ex = Ax + Bu; y = Cx is also considered: necessary and sufficient conditions for the infinite-zeros of the regular pencil (sE-A) to be controllable and observable are shown.