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Decentralized Computation and Communication in Stabilization of Distributed Control Systems

We provide a necessary and sufficient condition for the existence of decentralized controllers for stability of linear systems under the decentralized information structure under any class of control policies, where the control policies can be non-linear and time-varying. The approach adopted is information theoretic, yet constructive. Combining this finding with a result by Khargonekar and Özgüler, we show that a linear decentralized system is stabilizable if and only if it is stabilizable by linear time-varying periodic control laws. Building on our result, we revisit the problem of information minimization leading to stabilizability, when the eigenvalues can also be repeated and the topology of the decentralized system can consist of external sensors in addition to controllers. We show that when the eigenvalues are repeated, cut-set type lower bounds for a given mode might not be tight; whereas when the eigenvalues are distinct the rate analysis is significantly simpler. As such, we present a unified view on a number of results that have appeared in the literature.