686 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 26, NO. 4, MAY 2008 Zero-Error Target Tracking with Limited Communication Hua Li, Patricia R. Barbosa, Student Member, IEEE, Edwin K. P. Chong, Fellow, IEEE, Jan Hannig, and Sanjeev R. Kulkarni, Fellow, IEEE Abstract—We study the problem of target tracking in a sensor network environment. In particular, we consider a target that moves according to a Markov chain, and a tracker that queries sets of sensors to obtain tracking information. We are interested in finding the minimum number of queries per time step such that a target is trackable under three different requirements. First we investigate the case where the tracker is required to know the exact location of the target at each time step. We then relax this requirement and explore the case where the tracker may lose track of the target at a given time step, but it is able to “catch-up” at a later time, regaining up-to-date information about the target’s track. Finally, we consider the case where tracking information is only known after a delay of d time steps. We provide necessary and sufficient conditions on the number of queries per time step to track in the above three cases. These conditions are stated in terms of the entropy rate of the target’s Markov chain. Index Terms—Target tracking, Markov chain, adaptive sens- ing, entropy rate, Huffman coding, causal source coding. I. I NTRODUCTION W E EXPLORE the similarities between the target track- ing problem in a sensor network setting and R´ enyi- Ulam games [1], [2], [3], of which the game of “twenty questions” can be considered a subclass. Our goal is to find theoretical bounds on the number of queries per time step a tracker is required to ask a set of sensors to track a target. To the best of our knowledge, this work is the first to investigate the interplay between these two problems, providing the notable simplicity of our tracking scheme. Perhaps the initial driving force in sensor networks research [4], the target tracking problem is revisited in this work. While much of recent research in sensor networks explores networking issues like energy consumption optimization [5], time synchronization [6], [7], sensor localization [8], [9], and routing [10], [11], additional communications problems such as data compression and message complexity have become increasingly important as the number of small and inexpensive networked sensing devices continues to grow. Moreover, the Manuscript received May 15, 2007; revised November 11, 2007. The material in this paper was presented in part at the 44th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Illinois, September 27–29, 2006. H. Li, P. R. Barbosa, and E. K. P. Chong are with the Department of Elec- trical and Computer Engineering, Colorado State University, Ft Collins, CO 80523 USA (e-mail: {hua.li, patricia.barbosa, edwin.chong}@colostate.edu). J. Hannig is with the Department of Statistics, Colorado State University, Ft Collins, CO 80523 USA (e-mail: jan.hannig@colostate.edu). S. R. Kulkarni is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: kulkarni@princeton.edu). Digital Object Identifier 10.1109/JSAC.2008.080510. reliability and the capacity of the channel available for com- munication with the tracker lead to restrictions on the amount of data sent over such networks. As a consequence, the tracker needs to make judicious decisions when selecting sensors to send data, so that communication with the sensor network is kept to a minimum. Related problems include those in the area of control under communication constraints. In particular, Tatikonda and Mitter [12], [13], [14] examined a control problem with a noiseless digital communication channel connecting a sensor to a controller, and provided upper and lower bounds on the channel rate required to achieve different control objectives, namely, asymptotic observability and asymptotic stabilizability [15]. Sahai and Mitter [16], [17], [18] investigated the problem of tracking and controlling unstable processes over noisy channels and demonstrated that Shannon’s classical notion of capacity was insufficient to characterize noisy channels for this purpose. Furthermore, they identified a novel characterizing quantity called anytime capacity and showed that it is both necessary and sufficient for channels to have a certain amount of anytime capacity such that unstable processes can be tracked and stabilized. We propose a sensor model in which sensors are capable of sending only one-bit messages to a tracker. These messages are used to gather tracking information about a moving target. In the literature, one-bit-message sensor networks are called binary sensor networks and have been previously considered for target tracking [19], [20], [21]. In [22], Evans et al. ana- lyzed the problem of optimal sensor selection; their approach is to formulate the problem as a partially observed stochastic control problem, where sensors are not constrained to one-bit messages, and the tracker also controls the channel data rate so that mean squared errors are bounded. The remainder of this paper is organized as follows. Sec- tion II formalizes the tracking problem under three different definitions. In Section III, the main theorems are stated; they are later proved in Section IV. Finally, Section V concludes this work. II. PROBLEM FORMULATION Consider a target moving around in an area. Suppose that the area is partitioned into a number of non-overlapping regions, referred to as locations. The target motion can be described by a directed graph G =(X ,E), where the set of nodes X , with finite cardinality, represents target locations, and the set of edges E describes each neighborhood, that is, possible target motion. If there exists an edge (i, j ) ∈ E, 0733-8716/08/$25.00 c  2008 IEEE