Data-Driven Distributed Spectrum Estimation for Linear Time-Invariant Systems

This paper tackles spectrum estimation of a linear time-invariant system by a multi-agent network using data. We consider a group of agents that communicate over a strongly connected, aperiodic graph and do not have any knowledge of the system dynamics. Each agent only measures some signals that are linear functions of the system states or inputs, and does not know the functional form of this dependence. The proposed distributed algorithm consists of two steps that rely on the collected data: (i) the identification of an unforced trajectory of the system and (ii) the estimation of the coefficients of the characteristic polynomial of the system matrix using this unforced trajectory. We show that each step can be formulated as a problem of finding a common solution to a set of linear algebraic equations which are amenable to distributed algorithmic solutions. We prove that, under mild assumptions on the collected data, when the initial condition of the system is random, the proposed distributed algorithm accurately estimates the spectrum with probability 1.