A linear matrix inequality approach for designing a family of observers suitable for systems with variable communication topologies is presented. In particular, the observer is composed of blocks associated to the status of the communication links, providing increasing performance as more links are enabled. The error boundaries for topology switchings are analyzed both for the presented observer and for a specific Kalman filter for each topology. Finally, a simulation example is used to illustrate the feasibility of the proposed method. I. INTRODUCTIONIn the last decades, distributed schemes have gained relevance in the field of control and estimation due to their wellknown advantages such as scalability, modularity, ease of implementation, and robustness, which make them appropriate for large-scale networked systems [1], and other complex approaches [2]. Distributed architectures are characterized for being composed of a set of subsystems, governed by agents that have access to local information. Consequently, communication is needed to gather information from the rest of the system to carry out estimation and control tasks in the most efficient way [3]. Reshaping the topology of the communication network according to the necessities of the system provides additional flexibility to optimize performance. In this way, only communication channels that really contribute to the improvement of the task are enabled (see, e.g., [4], [5] for cooperative approaches).Through the observers, variables such as the system state, disturbances, noises, etc., can be estimated in the cases in which they cannot be measured directly [6]. Some classical estimators, e.g., the Luenberger filter [7], settled the basis for most observers designed nowadays. Also, the Kalman filter, which introduced white noise in the filtering formulation to offer optimal solutions [8], [9], has been widely studied and extended over the years [10]. Recently, both approaches have been adapted to noncentralized systems requirements. Some distributed Luenberger approaches for linear systems are explored in: [11], with a given fixed communication architecture between subsystems; [12], where time-varying communication topologies are considered; and [13], where consensus between subsystems is used to estimate the nonlocally detectable portion of the state. More details about