Distributed configuration of sensor networks for parameter estimation in spatio-temporal systems

The paper discusses an approach to configure a sensor network in a spatial domain that will be used to identify unknown parameters of a distributed system. Particularly, given a finite number of possible sites at which sensors may reside, we select gauged sites so as to maximize a criterion based on the Fisher information matrix associated with the estimated parameters. The technique is based on a relaxation of the orig- inal discrete optimization problem and amounts to operating on the density of sensors in lieu of individual sensor positions. Then, through an adaptation of some pairwise communication algorithms a numerical scheme is developed, which distributes the computational burden between the network nodes. As a result, a simple exchange algorithm is outlined to select the gauged sites in a decentralized fashion. The approach is verified on a numerical example involving sensor selection for a two- dimensional convective diffusion process. I. INTRODUCTION Measurement system design for spatio-temporal physical systems also called distributed-parameter systems (DPSs) is often related to an optimal choice of experimental conditions leading to the best information for estimating unknown parameters which can then be used in optimal control and forecasting. It has found special interest in many areas, e.g., air quality monitoring systems, groundwater-resources management, recovery of valuable minerals and hydrocar- bon, model calibration in meteorology and oceanography, chemical engineering, hazardous environments and smart materials, cf. (1)-(9). The impossibility to observe the system states over the entire spatial domain implies the question of where to locate discrete sensors so as to accurately estimate system parameters. The question of how to optimize sensor locations acquires especially vital importance in the context of recent advances in distributed sensor networks (10). information exchange. Simultaneously, numerous constraints inherent to network design and configuration have to be taken into account, e.g., limited computational power of nodes, energy consumption, time-varying network topology. The aim of the research reported here was to develop a practical approach to sensor selection which, while being independent of a particular model of the dynamic DPS in question, would be versatile enough to cope with practical monitoring networks consisting of many stationary sensors. Specifically, we consider N possible sites at which to locate a sensor, but limitations on the number of sensors at our disposal allow only n of them (typically, n is much smaller than N ) to be selected. A reason for not using all the available sensors could be the reduction of the total power consumption, system complexity and the cost of operation and maintenance (13). Consequently, the problem is to divide the N available sites between n gauged sites and the remain- ing N − n ungauged sites so as to maximize the determinant of the Fisher Information Matrix (FIM) associated with the parameters to be estimated. Selecting the best subset of sites to locate the sensors constitutes an inherently discrete large-scale resource allocation problem whose solution may be prohibitively time-consuming. As an approach which is suitable for situations when the numbers of candidate and gauged sites are rather large, a method whose idea is to operate on sensor densities per unit area instead of the individual sensor positions is presented. This convenient reformulation makes it possible to apply some powerful tools of convex analysis. Moreover, an extremely simple exchange algorithm is developed to determine best sensor configurations. Its idea is based on the class of so-called gossip algorithms in which each node communicates with no more than one neighbor at each time instant (17)-(19). The different nodes of the network independently calculate and store the desired quantities and their final estimates are obtained in a fully decentralized fashion. The advantages of this scheme of information exchange seem to be clear as no powerful information fusion center is necessary and it is robust with respect to individual sensor faults as the global estimate is recorded at all sensor nodes and easily can be recovered. Moreover, it may lead to simple implementations and efficient performance of the resulting procedures, cf. (18).