Study On Ill-posed Problem In Wireless Sensor Networks Nodes Localization

Wireless sensor networks(WSN) has a wide range of applications in military and national defense, environmental monitoring, intelligent transportation and medicine. Localization information is an important indicator and one of the underlying technologies of WSN applications. There are currently in-depth studies and many results about the 2-dimensional(2-D) nodes localization. With the development of WSN,3-dimensional(3-D) nodes localization is needed more and more, but new situations and problems come out in 3-D nodes localization. This paper mainly researches the WSN 3-D nodes localization in detail.Proposing the method of calibrating the ranging based on received singal strength indicator(RSSI). The external environment impacts on the signal loss and makes the precision of ranging based on RSSI is coarse. Ranging is one of the prerequisites of localization. In order to improve the accuracy of ranging, this paper proposes the method of calibrating the ranging by combining both path loss model and empirical data. The experiment results show that determining the ranging model through curve fitting can improve the ranging accuracy to some extent.This paper proposes diagnosing ill-posed problem in WSN 3-D nodes localization by condition number. When using traditional least square(LS) estimation based on distance information to solve the results of 3-D localization, there is unstable phenomenon some time, and it is an ill-posed problem. LS estimation fluctuates substantially when the ill-posed problem existed. The analysis shows that the cause of 3-D nodes localization ill-posed problem is the unreasonable choice of reference nodes, which leads to multicollinear relationship of the design matrix column vectors, so analyzing multicollinearity and diagnosing ill-posed is necessary before 3-D nodes localization. Through researching the typical methods of multicollinear analysis found that the condition number method is suitable for diagnosing ill-posed problem in WSN 3-D nodes localization.This paper researches the methods of increasing the amount of information, ridge estimation, truncated singular value decomposition(TSVD) and modified singular value decomposition(MSVD) to overcome the ill-posed. The results show that increasing the amount of information can improve the design matrix structure, so that the LS estimate is more stable. Increasing the amount of information locates ultimately through the LS estimate, requires moderate operations and is suitable for higher real-time requirements. The choice of ridge parameter is the key point of ridge estimate, and using the L-curve and generalized cross-validation to determine the ridge parameters increases the ridge estimate stability greatly. Compared with the LS estimate, localization accuracy is significantly improved, but the computation will also increases accordingly, so it is suitable for less demand of real-time off-line localization. The TSVD and MSVD reduce the impact on the localization result by truncating or modifing the smaller singular values, so the localization accuracy improves to some extent, but require more matrix operations, need large amount of calculation.Proposing solution of 3-D localization based on the above study. First, diagnosing the ill-posed in 3-D localization, then taking different localization methods according to the two kind of diagnosis results, so improving the credibility and accuracy of localization. In the first case, if the ill-posed is weak, localizating the nodes by LS estimate. In the second case, if the ill-posed is serious, taking measures to overcome the problem:(1)If the high real-time is required, increasing the amount of information and then localizing through the LS; (2)If the high precision is needed, using the methods of ridge estimation, TSVD or MSVD to overcome the ill-posed.