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...

Abstract-We consider a decentralized optimal control problem for a linear plant controlled by two controllers, a local controller and a remote controller. The local controller directly observes the state of the plant and can inform the remote controller of the plant state through a packet-drop channel. We assume that the remote controller is able to send acknowledgments to the local controller to signal the successful receipt of transmitted packets. The objective of the two controllers 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 LQG problem, we obtain explicit optimal strategies for the two controllers. In the optimal strategies, both controllers compute a common estimate of the plant state based on the common information. The remote controller's action is linear in the common estimated state, and the local controller's action is linear in both the actual state and the common estimated state.

No abstract

We introduce the epistemic neural network (ENN) as an interface for uncertainty modeling in deep learning. All existing approaches to uncertainty modeling can be expressed as ENNs, and any ENN can be identified with a Bayesian neural network. However, this new perspective provides several promising directions for future research. Where prior work has developed probabilistic inference tools for neural networks; we ask instead, 'which neural networks are suitable as tools for probabilistic inference?'. We propose a clear and simple metric for progress in ENNs: the KL-divergence with respect to a target distribution. We develop a computational testbed based on inference in a neural network Gaussian process and release our code as a benchmark at https://github.com/deepmind/enn. We evaluate several canonical approaches to uncertainty modeling in deep learning, and find they vary greatly in their performance. We provide insight to the sensitivity of these results and show that our metric is highly correlated with performance in sequential decision problems. Finally, we provide indications that new ENN architectures can improve performance in both the statistical quality and computational cost.1 Epistemic uncertainty relates to knowledge (ancient Greek episteme↔knowledge), as opposed to aleatoric uncertainty relating to chance (Latin alea↔dice) (Kendall and Gal, 2017).Preprint. Under review.

We consider a decentralized system with multiple controllers and define substitutability of one controller by another in open-loop strategies. We explore the implications of this property on the optimization of closed-loop strategies. In particular, we focus on the decentralized LQG problem with substitutable actions. Even though the problem we formulate does not belong to the known classes of "simpler" decentralized problems such as partially nested or quadratically invariant problems, our results show that, under the substitutability assumption, linear strategies are optimal and we provide a complete state space characterization of optimal strategies. We also identify a family of information structures that all give the same optimal cost as the centralized information structure under the substitutability assumption. Our results suggest that open-loop substitutability can work as a counterpart of the information structure requirements that enable simplification of decentralized control problems.