We describe an SDP relaxation based method for the position estimation problem in wireless sensor networks. The optimization problem is set up so as to minimize the error in sensor positions to fit distance measures. Observable gauges are developed to check the quality of the point estimation of sensors or to detect erroneous sensors. The performance of this technique is highly satisfactory compared to other techniques. Very few anchor nodes are required to accurately estimate the position of all the unknown nodes in a network. Also the estimation errors are minimal even when the anchor nodes are not suitably placed within the network or the distance measurements are noisy.

An SDP relaxation based method is developed to solve the localization problem in sensor networks using incomplete and inaccurate distance information. The problem is set up to find a set of sensor positions such that given distance constraints are satisfied. The nonconvex constraints in the formulation are then relaxed in order to yield a semidefinite program which can be solved efficiently. The basic model is extended in order to account for noisy distance information. In particular, a maximum likelihood based formulation and an interval based formulation are discussed. The SDP solution can then also be used as a starting point for steepest descent based local optimization techniques that can further refine the SDP solution. We also describe the extension of the basic method to develop an iterative distributed SDP method for solving very large scale semidefinite programs that arise out of localization problems for large dense networks and are intractable using centralized methods. The performance evaluation of the technique with regard to estimation accuracy and computation time is also presented by the means of extensive simulations. Our SDP scheme also seems to be applicable to solving other Euclidean geometry problems where points are locally connected.

Abstract-A sensor network localization problem is to determine the positions of the sensor nodes in a network given incomplete and inaccurate pairwise distance measurements. Such distance data may be acquired by a sensor node by communicating with its neighbors. We describe a general semidefinite programming (SDP) based approach for solving the graph realization problem, of which the sensor network localization problems is a special case. We investigate the performance of this method on problems with noisy distance data. Error bounds are derived from the SDP formulation. The sources of estimation error in the SDP formulation are identified.The SDP solution usually has a rank higher than the underlying physical space, which when projected onto the lower dimensional space generally results in high estimation error. We describe two improvements to ameliorate such a difficulty. First, we propose a regularization term in the objective function that can help to reduce the rank of the SDP solution. Second, we use the points estimated from the SDP solution as the initial iterate for a gradient-descent method to further refine the estimated points. A lower bound obtained from the optimal SDP objective value can be used to check the solution quality. Experimental results are presented to validate our methods and show that they outperform existing SDP methods.

We propose a distributed algorithm for solving Euclidean metric realization problems arising from large 3D graphs, using only noisy distance information, and without any prior knowledge of the positions of any of the vertices. In our distributed algorithm, the graph is first subdivided into smaller subgraphs using intelligent clustering methods. Then a semidefinite programming relaxation and gradient search method is used to localize each subgraph. Finally, a stitching algorithm is used to find affine maps between adjacent clusters and the positions of all points in a global coordinate system are then derived. In particular, we apply our method to the problem of finding the 3D molecular configurations of proteins based on a limited number of given pairwise distances between atoms. The protein molecules, all with known molecular configurations, are taken from the Protein Data Bank. Our algorithm is able to reconstruct reliably and efficiently the configurations of large protein molecules from a limited number of pairwise distances corrupted by noise, without incorporating domain knowledge such as the minimum separation distance constraints derived from van der Waals interactions.

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Abstract-Network coverage of wireless sensor network (WSN) means how well an area of interest is being monitored by the deployed network. It depends mainly on sensing model of nodes. In this paper, we present three types of sensing models viz. Boolean sensing model, shadow-fading sensing model and Elfes sensing model. We investigate the impact of sensing models on network coverage. We also investigate network coverage based on Poisson node distribution. A comparative study between regular and random node placement has also been presented in this paper. This study will be useful for coverage analysis of WSN.

Piecewise affine (PWA) systems are useful models for describing nonlinear and hybrid systems. They also result from LTI systems subject to constrained optimal control. Recently, stability analysis of PWA systems has received increased interest since it can help to obtain explicit feedback control laws of low complexity (Grieder et al., 2003b;Grieder and Morari, 2003;Grieder et al., 2004). A wide range of methods with varying degrees of conservativeness are available, for analyzing stability of PWA systems. This survey introduces the most promising existing methods and adapts them to discrete-time PWA systems. Specifically, we investigate the computation of common quadratic, common quartic, piecewise affine, piecewise quadratic and piecewise quartic Lyapunov functions using linear programming, semi-definite programming and sum-of-squares techniques. Subsequently, the different methods are compared regarding the likelihood of finding a certificate of stability and computation run time. The objective is to provide the reader with a practical 'recipe' for analyzing PWA systems. To this end, an extensive case study is performed.

The problem of position estimation in sensor networks using a combination of distance and angle information as well as pure angle information is discussed. For this purpose, a semidefinite programming relaxation based method that has been demonstrated on pure distance information is extended to solve the problem. Practical considerations such as the effect of noise and computational effort are also addressed. In particular, a random constraint selection method to minimize the number of constraints in the problem formulation is described. The performance evaluation of the technique with regard to estimation accuracy and computation time is also presented by the means of extensive simulations.