Structural Stability of Polynomial Matrices Related to Linear Time-Invariant Singular Systems

We consider the set of quadruples of matrices defining singular linear time-invariant dynamical systems and show that there is a one-to-one correspondence between this set and a subset of the set of polynomial matrices of degree two. This correspondence preserves the equivalence relations introduced in both sets (feedback-similarity and strict equivalence): two quadruples of matrices are feedback-equivalent if, and only if, the polynomial matrices associated to them are also strictly equivalent. We characterize structurally stable polynomial matrices (stable elements under small perturbations) describing singular systems and derive a lower bound on the distance to the orbits of polynomial matrices with strictly lower dimension. Key-Words:singular systems, polynomial matrices, structural stability.