Optimal Local and Remote Controllers With Unreliable Uplink Channels

Abstract

We consider a networked control system 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 co-located plant and can inform the remote controller of the plant's state through an unreliable uplink channel. We assume that the downlink channels from the remote controller to local controllers are perfect. The objective of the local controllers and the remote controller is to cooperatively minimize a quadratic performance cost. We provide a dynamic program for this decentralized control problem using the common information approach. Although our problem is not a partially nested problem, we obtain explicit optimal strategies for all controllers. In the optimal strategies, all controllers compute common estimates of the states of the plants based on the common information obtained from the communication network. The remote controller's action is linear in the common state estimates, and the action of each local controller is linear in both the actual state of its co-located plant and the common state estimates.We illustrate our results with numerical experiments using randomly generated models. Contributions of the PaperThe main contributions of the paper are as follows.1) We investigate a decentralized stochastic control problem in which local controllers send their information to a remote controller over unreliable links. To the best of our knowledge, this is the first paper that solves an optimal decentralized control problem with unreliable communication between controllers (in contrast to problems in networked control systems and remote estimation problems where the unreliable communication is between sensors/encoders and controller or between controllers and actuators).2) The information structure of our problem is not partially nested, hence we cannot a priori restrict to linear strategies for optimal control. We use ideas from the common information approach of [43] to compute optimal controllers. Since the state and action spaces of our problem are Euclidean spaces, the results and arguments of [43] for finite spaces cannot be directly applied. We provide a complete set of results to adapt the common information approach to our linear-quadratic setting with non-partially nested information structure.Our rigorous proofs carefully handle the issues of measurability constraints, the existence of well-defined value functions and infinite dimensional strategy spaces.3) We show that the optimal control strategies of this problem admit simple structures-the optimal remote control is linear in the common estimates of system states and each optimal local control is linear in both the common estimates of system states and the perfectly observed local state. The main strengths of our result are that (i) it provides a simple strategy that is proven to be optimal: not only is the strategy in Theorem 3 linear, it uses estimates that can be easily updated; (ii) it provides a tractable way of computing the gain ma...

Optimal Local and Remote Controllers With Unreliable Uplink Channels: An Elementary Proof

“…In this section, we restate the main results of [1] but we present them in a slightly different manner.…”

“…We follow the standard notation here. • In [1], it was assumed that the performance of a strategy profile is…”

“…In order to present the main result of [1], we recursively define matrices {P t } T t=1 as follows: P T = Q T and for t ∈ {T − 1, . .…”

“…It will be clear from the proof that the steps extend to general N . For the most part, we broadly follow the notation and terminology of [1], but we occasionally deviate from it to be consistent with the standard notation used in linear systems.…”

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Recently, a model of a decentralized control system with local and remote controllers connected over unreliable channels was presented in [1]. The model has a non-classical information structure that is not partially nested.

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