Secure State Estimation for Multi-Agent Systems: On the Relationship between the Number of Agents and System Resilience

This paper addresses the problem of secure state estimation in multi-agent systems consisting of n agents. For the problem, the sparse observability index is a well-known measure of the system resilience. This index is defined as the largest integer delta for which the system observability is preserved even if any delta sensors are eliminated. The larger delta is, the more the system state can be reconstructed even if the system is subjected to more sensor attacks. In this paper, we analyze the relationship between the number of agents n and the sparse observability index delta especially when the network structure is path and cycle graphs. Intuitively, it is assumed that, as the number of agents n increases, the system resilience, i.e., d, increases accordingly. However, as this study provides, delta does not monotonically increase with an increase in n. In path graphs, delta depends on whether n + 1 is prime or composite, and the system resilience is enhanced when n + 1 is prime. In cycle graphs, delta obeys whether n is prime or composite, and the system is more resilient when n is prime.