Poles and Zeros of Quaternionic Systems: Coincidence Without Cancellation

A scalar quaternionic transfer function can have coincident pole and transmission-zero classes while remaining coprime and minimal. We classify a first-order family on every nonreal similarity sphere and derive its minimal realization and complex and real Smith–McMillan structures. At each complex representative, pole and zero directions coexist; one quaternionic state corresponds to four real states. A higher-order family permits independent pole and zero multiplicities, and a strictly proper second-order example shows that feedthrough is unnecessary. We prove an exact two-block cascade criterion: reversing the order can change the minimal degree. Three fixed blocks give identity transfer in one order and minimal quaternionic degree three in reverse. A matrix-fraction extension connects these results with classical multivariable systems. For stable members, closed-form Gramians and the exact frequency response quantify near cancellation and the error of static replacement. A four-channel real implementation distinguishes modal blocking from the response from rest. Author preprint of a manuscript submitted to the 26th International Conference on Process Control (Process Control 2027), Štrbské Pleso, Slovakia. Submitted manuscript; not the version of record. The manuscript content is unchanged; only a preprint notice has been added in the upper page margins. This work was co-funded by the European Union under the project ROBOPROX (reg. no. CZ.02.01.01/00/22_008/0004590).