Asymptotic mean ergodicity of average consensus estimators

Dynamic average consensus estimators suitable for the decentralized computation of global averages of constant or slowly-varying local inputs include the proportional (P) and proportional-integral (PI) estimators. We analyze the convergence properties of these estimators when run on i.i.d. random graphs which are connected and balanced on average, but need not be connected or balanced at each time step. The statistics of the steady-state process are found using the Kronecker product covariance and an ergodic theorem is used to determine whether the steady-state process is mean ergodic. We show that for constant inputs the P estimator is asymptotically mean ergodic only for systems with non-zero forgetting factor which do not have zero steady-state error on average. The PI estimator has both the asymptotic mean ergodicity property and zero steady-state error in expectation for constant inputs independent of initial conditions, proving that the time-averaged output of each agent robustly converges to the correct average.