On Boundedness of Error Covariances for Kalman Consensus Filtering Problems

In this paper, the uniform bounds of error covariances for several types of Kalman consensus filters (KCFs) are investigated for a class of linear time-varying systems over sensor networks with given topologies. Rather than the traditional detectability assumption, a new concept called collectively uniform detectability (CUD) is proposed to address the detectability issues over sensor networks with relaxed restrictions. By using matrix inequality analysis techniques, the conditions for the newly proposed CUD concept are established, and then, the explicit expressions of the uniform upper/lower bounds are derived for error covariances of several commonly used KCF algorithms. Consequently, a comparison is conducted between the obtained bounds so as to reveal their relationships. Finally, a numerical example is provided to calculate and further compare the bounds of interest in order to demonstrate the practical usefulness of the developed theory.