Zero dynamics normal form and disturbance decoupling of commensurate and distributed time-delay systems

The zero dynamics of linear systems with commensurate delays is studied. The zero dynamics is meant to be the dynamical equation that remains after compelling, by the input of the system, the system output to be identically equal to zero. Conditions are found out allowing to transform the system into a normal form that reveals its zero dynamics. By a straightforward use of the obtained decomposition, the disturbance decoupling problem is solved readily. An explanation is given on how to extend the obtained results to systems with distributed delays. Two examples of physical systems illustrate the effectiveness of the presented study.