Realization of autoregressive equations in pencil and descriptor form

A linear system described by autoregressive equations with a given input/output structure cannot be transformed to standard state-space form if the implied input/output relation is nonproper. Instead, a realization in descriptor form must be used. In this paper, it is shown how to obtain minimal descriptor realizations from autoregressive equations without separating finite and infinite frequencies, and without going through a reduction process. External equivalence is used, so that even situations in which there is no transfer matrix can be considered. The approach is based on the so-called pencil representation of linear systems, and it is shown that there is a natural realization of autoregressive equations in pencil form. In this way, the link between the realization theories of Willems and Fuhrmann can also be clarified.