Zeros and Zero Dynamics for Linear, Time-delay System

The aim of this paper is to discuss a notion of Zero Module and Zero Dynamics for linear, time-delay systems. The Zero Module in the time-delay framework is defined using the correspondence between time-delay systems and system with coefficients in a ring, so to exploit algebraic and geometric methods. By combining the algebraic notion of Zero Module and the geometric structure of the lattice of invariant submodules of the state module, we point out a natural way to study the Zero Dynamics and its properties. In particular, stability of this latter is characterized using, in the ring framework, a formal notion of Hurwitz set. Application to the study of decoupling problems with stability, inversion problems and traking problems for timke-delay systems are illustrated.