Optimal Infinite Horizon Decentralized Networked Controllers With Unreliable Communication

We consider a decentralized networked control system (DNCS) consisting of a remote controller and a collection of linear plants, each associated with a local controller. Each local controller directly observes the state of its colocated plant and can inform the remote controller of the plant's state through an unreliable uplink channel. The downlink channels from the remote controller to local controllers were assumed to be perfect. The objective of the local controllers and the remote controller is to cooperatively minimize the infinite horizon time average of expected quadratic cost. The finite horizon version of this problem was solved in our prior work ("Optimal local and remote controllers with unreliable uplink channels," IEEE, May 2019). The optimal strategies in the finite horizon case were shown to be characterized by coupled Riccati recursions. In this article, we show that if the link failure probabilities are below certain critical thresholds, then the coupled Riccati recursions of the finite horizon solution reach a steady state and the corresponding decentralized strategies are optimal. Above these thresholds, we show that no strategy can achieve finite cost. We exploit a connection between our DNCS Riccati recursions and the coupled Riccati recursions of an auxiliary Markov jump linear system to obtain our results. Our main result in Theorem 1 explicitly identifies the critical thresholds for the link failure probabilities and the optimal decentralized control strategies when all link failure probabilities are below their thresholds.