# Distributed Cooperative Online Estimation With Random Observation Matrices, Communication Graphs and Time-Delays

We analyze convergence of distributed cooperative online estimation algorithms by a network of multiple nodes via information exchanging in an uncertain environment. Each node has a linear observation of an unknown parameter with randomly time-varying observation matrices. The underlying communication network is modeled by a sequence of random digraphs and is subjected to nonuniform random time-varying delays in channels. Each node runs an online estimation algorithm consisting of a consensus term taking a weighted sum of its own estimate and delayed estimates of neighbors, and an innovation term processing its own new measurement at each time step. By stochastic time-varying system, martingale convergence theories and the binomial expansion of random matrix products, we transform the convergence analysis of the algorithm into that of the mathematical expectation of random matrix products. Firstly, for the delay-free case, we show that the algorithm gains can be designed properly such that all nodes' estimates converge to the real parameter in mean square and almost surely if the observation matrices and communication graphs satisfy the stochastic spatial-temporal persistence of excitation condition. Especially, this condition holds for Markovian switching communication graphs and observation matrices, if the stationary graph is balanced with a spanning tree and the measurement model is spatially-temporally jointly observable. Secondly, for the case with time-delays, we introduce delay matrices to model the random time-varying communication delays between nodes, and propose a mean square convergence condition, which quantitatively shows the intensity of spatial-temporal persistence of excitation to overcome time-delays.

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