On Decentralized Stabilization and Communication Complexity for Decentralized Computation

In this paper, a communication complexity view on the problem of decentralized controllability is presented, which uses ideas from information theory and distributed computation. To this end, we present a new approach to revisit a result on the characterization of the necessary and sufficient conditions for the existence of decentralized controllers for stability of linear systems under any class of admissible decentralized control policies. It is observed that one could either adopt a decentralized communication point of view to identify controllers to control certain modes, or perform stabilization decentrally by a decentralized computation of control actions. One message is that, decentralized computation is in general more efficient than the designation of a particular controller acting on a mode when the eigenvalues are repeated. When the eigenvalues are not repeated, however, designation of a particular controller is optimal. Furthermore, we show that when the eigenvalues are repeated, cut-set type lower bounds for a given mode might not be tight.