On the Zero-freeness of Tall Multirate Linear Systems

TL;DR

This paper investigates the zeros of tall multirate linear systems, showing that generically they lack finite nonzero zeros but may have zeros at zero or infinity depending on system parameters.

Contribution

It provides a comprehensive analysis of the zero structure of tall multirate systems and clarifies conditions leading to zeros at special locations.

Findings

Tall blocked systems generally have no finite nonzero zeros.

Zeros at the origin or infinity depend on blocking delay and system dimensions.

Zeros are characterized for generic parameter matrices.

Abstract

In this paper, tall discrete-time linear systems with multirate outputs are studied. In particular, we focus on their zeros. In systems and control literature zeros of multirate systems are defined as those of their corresponding time-invariant blocked systems. Hence, the zeros of tall blocked systems resulting from blocking of linear systems with multirate outputs are mainly explored in this work. We specifically investigate zeros of tall blocked systems formed by blocking tall multirate linear systems with generic parameter matrices. It is demonstrated that tall blocked systems generically have no finite nonzero zeros; however, they may have zeros at the origin or at infinity depending on the choice of blocking delay and the input, state and output dimensions.