Realization of Acausal Weighting Patterns with Boundary - Value Descriptor Systems *

This paper examines the realization of acausal weighting patterns with two-point boundary-value descriptor systems (TPBVDSs). Attention is restricted to the subclass of TPBVDSs that are stationary, so that their input-output weighting pattern is shift-invariant, and extendible, i.e., their weighting pattern can be extended outwards indefinitely. Then, given an infinite acausal shift-invariant weighting pattern, the realization problem consists of constructing a minimal TPBVDS over a fixed interval, whose extended weighting pattern matches the given pattern. The realization method that is proposed relies on a new transform, the (s, t)-transform, which is better adapted to the analysis of descriptor dynamics than the standard z-transform, since it handles zero and infinite frequencies in a symmetric way. This new transform is used to determine the dimension of a minimal realization, and then to construct a minimal realization by obtaining state-space representations for two homogeneous rational matrices in s and obtained from the causal and anticausal components of the weighting pattern.