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Scheduling for Distributed Sensor Networks

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Submitted Workshop on Information Processing in Sensor Networks IPSN 04 http www cds caltech edu murray papers 2003x gchm04 ipsn html Scheduling for Distributed Sensor Networks Vijay Gupta Timothy Chung Babak Hassibi and Richard M Murray Division of Engineering and Applied Science Caltech Pasadena CA 91125 USA gupta timothyc hassibi murray caltech edu Abstract We examine the problem of distributed estimation when only one sensor can take a measurement per time step The measurements are then exchanged among the sensors The problem is motivated by the use of sonar range finders used by the vehicles on the Caltech Multi Vehicle Wireless Testbed We solve for the optimal recursive estimation algorithm when the sensor switching schedule is given Then we investigate several approaches for determining an optimal sensor switching strategy We see that this problem involves searching a tree in general and propose and analyze two strategies for pruning the tree to keep the computation limited The first is a sliding window strategy motivated by the Viterbi algorithm and the second one uses thresholding We also study a technique that employs choosing the sensors randomly from a probability distribution which can then be optimized The performance of the algorithms are illustrated with the help of numerical examples 1 Introduction and Motivation Recently there has been a lot of interest in networks of sensing agents which act cooperatively to obtain the best estimate possible e g 1 and the references therein While such a scheme admittedly has higher complexity than the strategy of treating each sensor independently the increased accuracy often makes it worthwhile If all the sensors exchange their measurements the resulting estimate can be better even than the sensor with the least measurement noise were no information exchange happening The advantages of forming sensor networks are even greater if the sensors are heterogenous The increased complexity arises from the communication infrastructure atop every sensor and the algorithmic changes needed for fusing the measurements from other sensors to obtain a better estimate Because of the above mentioned advantages there has been a lot of attention on data fusion of heterogeneous sensor measurements as in 2 Works such as the EYES project 3 WINS 4 and Smart Dust 5 are examples of systems implementing such networks The assumption usually made in the analysis of such systems is that all the sensors take measurements at the same time Thus the main issue is multi sensor data fusion One example of many sensor fusion algorithms can be found in 6 The sensor management issues if present at all are in the context of energy efficiency 7 8 imperfect localization of sensor platforms 9 optimal coverage of a given region 10 9 and efficient networking and communication protocols 11 2 However in some applications the use of one sensor places restrictions on the use of other sensors This situation exists whenever simultaneous use of sensors causes interference in measurements We face this situation in our own work related to the Caltech Multi Vehicle Wireless Testbed MVWT 12 When the individual vehicles are using sonar range finding devices only one sensor can be active at any time In such a case apart from the issue of optimal multisensor data fusion there is the additional issue of optimally scheduling the sensor measurements so as to minimize the state estimate error covariance In this paper we study this problem of coming up with the optimal sensor schedule when only one sensor is allowed to take the measurement at every time step While optimization of sensor schedules have been examined using optimal or stochastic control theory techniques as in 13 14 solutions to Ricatti differential equations and even information theoretic methods as in 15 we pursue two simpler methods sliding window and thresholding for determining an optimal sensing schedule These methods trade computation memory requirements for sub optimality however they seem to work well on the simulation examples In addition we also study a method that involves simply choosing the sensors randomly according to some probability distribution The probability distribution can then be optimally chosen so as to minimize the expected error covariance The paper is organized as follows In the next section we set up the problem and solve for the optimal data fusion algorithm for a given sensor schedule We briefly consider the degradation in the performance when this scheme is used for the case when communication noise is present Then we consider the question of choosing the optimal sensor schedule which is the focus of the paper We present some methods that obtain sub optimal sensor schedules but have the advantage of being much simpler to use We demonstrate these algorithms with the help of examples and end with conclusions and scope for future work 2 Modeling and Problem Formulation Consider the system evolving as follows x k 1 Ax k Bw k 1 x k Rn is the process state at time step k and w k is the process noise The process noise is assumed white Gaussian and zero mean with covariance matrix Q The process state is being observed by N sensors with the measurement equation for the i th sensor being yi k Ci x k vi k 2 where yi k Rs is the measurement The measurement noises vi k s for the sensors are assumed independent of each other and of the process noise Further the noise vi k is assumed to be white Gaussian and zero mean with covariance matrix Ri It is assumed that only one sensor can be used at any time However unless stated otherwise we assume that the measurements are communicated 3 to all the sensors in an error free manner The estimate of i th sensor given the measurements till time steps k 1 is denoted by x i k k 1 or in short as x i k We first pose the question Assuming the sensor switching sequence to be given what is the optimal filtering for the i th node It is fairly obvious that the innovation for the i th node at time step k is given by ei k yj k Cj x i k k 1 3 where we have assumed that the j th sensor takes the measurement at time step k Then following the standard derivation see e g 16 we obtain the recursive optimal filtering equation as x i k 1 k Ax i k k 1 Kki ei k where Kki APi k k 1 CjT Re 1 j k Rei k Cj Pi k k 1 CjT Rj and Pi k k 1 s evolve as Pi k 1 k A Kki Cj Pi k k 1 A Kki Cj T BQB T Kki Rj Kki T 4 Assuming the initial state x 0 has mean zero and covariance 0 the initial covariance


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