Optimal Communication Logics in Networked Control Systems Yonggang Xu Jo˜ ao P. Hespanha Dept. of Electrical and Computer Eng., Univ. of California, Santa Barbara, CA 93106 Abstract— This paper addresses the control of spatially dis- tributed processes over a network that imposes bandwidth con- straints and communication delays. Optimal communication policies are derived for an estimator-based Networked Control System architecture to reduce the communication load. These policies arise as solutions of an average cost optimization problem, which is solved using dynamic programming. The optimal policies are shown to be deterministic. I. I NTRODUCTION In recent years Networked Control Systems (NCS) emerged as a playground for classic information theory and control theory, in which feedback is realized by information flow over data networks. Information theory provides upper bounds on the amount of information a channel can trans- mit. However, the optimal coding schemes may introduce long delays, which are not acceptable for feedback control. To provide adequate control performance, in general there is a minimum requirement of information exchange among the elements of a control system [6, 7, 9]. For example, Nair and Evans [7] investigated the exponential stabilizability of LTI plants with limited feedback data rates and gave the minimum stabilizing bit-rate, which is a logarithm function of the plant poles. In a previous paper [10], we considered an NCS struc- ture for spatially distributed processes in continuous-time. In this control architecture, originally proposed by Yook et al. [11], local controllers interchange information from time to time. To reduce communication requirements, each local controller estimates the remote processes’ states and coordinates its effort with others based on these estimates. Meanwhile, each controller broadcasts its local process’ state to the others according to a protocol specified by a communication logic. We proposed communication logics based on doubly stochastic Poisson processes (DSPP), in which the decision to broadcast data is triggered by increments of a DSPP with intensity (understood as a Poisson rate) related to the “necessity” to communicate. By choosing different DSPP intensity functions, we obtained different trade-offs between packet exchange rate and control performance. The threshold rule used by Yook et al. [11] can be viewed as a limiting case of this type of communication logics. The objective of this paper is to find the optimal com- munication logic. We restrict our attention to the discrete- time domain to avoid some of the technicalities that arise This research was supported by the National Science Foundation under the grants: CCR-0311084 and ECS-0093762. with continuous-time jump diffusion processes. We consider stochastic communication logics for which at each time instant, a probability u ∈ [0, 1] is assigned to whether or not data should be sent. These logics also include deterministic rules as a special case for which u ∈{0, 1}. Since any data network induces delays, we assume that it takes τ time steps for data from one local controller to reach another. The optimal communication problem is posed as a long term average cost (AC) minimization, and solved using dynamic programming (DP). The optimization index includes both communication and control (or estimation) performance penalties. Specifically, the per-stage cost of the AC optimization problem includes a quadratic term on the estimation error and a linear term on the communication cost. The optimal policy turns out to be deterministic, which is consistent with the observation in [10] that threshold rules appeared to be the most efficient ones. The average cost criterion is mathematically more dif- ficult to analyze than the discounted cost one, especially when the state and/or action spaces are Borel and the per- stage cost function is unbounded [1], which occurs in our case. We take advantage of some recurrent properties of the process and pose it as an essentially bounded cost per-stage problem, which makes results from [2, 4, 5] applicable. In Section II, the control-communication architecture is briefly described with further details in [10]. Optimal communication logic are derived in Section III. Section IV contains conclusions and directions for future work. II. NCS ARCHITECTURE In our setting, the dynamics of a spatially distributed pro- cesses are completely decoupled but the control objective is not. We view each local process with the associated local controller as a node. The overall control system consists of a certain number of nodes connected via a data network. Fig. 1 depicts the internal structure of the ith node. Each node consists of a local process,a local controller,a bank of local estimators that predicts both the local and remote process states, and a communication logic that schedules when to transmit data to the other nodes. The estimators run open-loop most of the time but are reset to the “correct” state when its value is received through the network. It is the responsibility of each node to broadcast to the network the state of its local process. The communication logic determines when this should occur. For simplicity of notation, we only consider two nodes but the results can be extended to any number of nodes.