Group Observability of Two‐Time‐Scale Discrete‐Time Second‐Order Multi‐Agent Systems

ABSTRACT This paper investigates the group observability problem of discrete‐time second‐order multi‐agent systems with two‐time‐scale dynamics, which has not been systematically addressed in the existing literature. Conventional observability analysis often suffers from severe numerical ill‐conditioning in two‐time‐scale settings and neglects the inherent subgroup structure. To address these limitations, a unified analytical framework is developed by explicitly incorporating group‐level information. By exploiting singular perturbation theory, the original system is decomposed into fast‐ and slow‐time‐scale subsystems, leading to reduced‐order models suitable for group observability analysis. On this basis, tractable group observability criteria are established by integrating the Kalman rank condition, matrix‐theoretic techniques, and Wonham's geometric approach. The effect of the singular perturbation parameter on the numerical behavior of the proposed group observability analysis is also investigated, and numerical simulations demonstrate that the proposed framework effectively overcomes the limitations of conventional methods and provides reliable group observability assessment for two‐time‐scale multi‐agent systems.