Signed networks with Laplacian feedback: Observability and minimal sensor problem

Abstract

In many multi-agent networks that evolve according to specific dynamical rules, our direct access is often limited to only a small subset of agents, known as “sensors.” The remaining agents in the network, whose states are entirely inaccessible, are referred to as “actuators.” If it is possible to retrieve the complete states of the actuators from the knowledge of the sensor states, the network is said to be “sensor–actuator observable”; otherwise, it is deemed unobservable. This research explores the analysis of sensor–actuator observability property (i.e., observable or unobservable) in networks where agent communications encompass both cooperation and competition (i.e., signed networks). It is assumed that the network agents continuously update their states within R under the Laplacian protocol. We simplify the Kalman’s and the PBH rank criteria for evaluating sensor–actuator observability property into several verifiable algebraic tests, emphasizing the significance of the system matrices’ spectral properties. This property is also examined from the perspective of the network graph topology. Sensor–actuator observability property is significantly influenced by the nature of agent communications and often differs between signed and unsigned networks. However, we demonstrate that for a structurally balanced signed network, with specific set of sensors, its sensor–actuator observability property aligns with that of its unsigned variant. We present a formula utilizing Laplacian spectral information to determine the minimum number of sensors for an ensured sensor–actuator observable network. Applying the formula to path and cycle networks, it is found that paths are observable with one sensor, and cycles require two for observability.