Evaluating Robustness of Consensus Algorithms Under Measurement Error over Digraphs

Consensus algorithms constitute a powerful tool for computing average values or coordinating agents in many distributed applications. Unfortunately, the same property that allows this computation (i.e., the nontrivial nullspace of the state matrix) leads to unbounded state variance in the presence of measurement errors. In this work, we explore the trade-off between relative and absolute communication (feedback) in the presence of measurement errors. We evaluate the robustness of first and second order integrator systems under a parameterized family of controllers (homotopy), that continuously trade between relative and absolute feedback interconnections, in terms of the H
₂
norm of an appropriately defined input-output system. Our approach extends the previous H
₂
norm based analysis to systems with directed feedback interconnections whose underlying weighted graph Laplacians are diagonalizable. Our results indicate that any level of absolute communication is sufficient to achieve a finite H
₂
norm, but purely relative feedback can only achieve finite norms when the measurement error is not exciting the subspace associated with the consensus state. Numerical examples demonstrate that smoothly reducing the proportion of absolute feedback in double integrator systems smoothly decreases the system performance (increases the H
₂
norm) and that this performance degradation is more rapid in systems with relative feedback in only the first state (position).