Positive Semi-definite Programming Localization in Wireless Sensor Networks

Transcription

1 Postve Sem-defnte Programmng Localzaton n Wreless Sensor etworks Shengdong Xe 1,, Jn Wang, Aqun Hu 1, Yunl Gu, Jang Xu, 1 School of Informaton Scence and Engneerng, Southeast Unversty, 10096, anjng Computer and Software Insttute, anjng Unversty of Informaton Scence and Technology, 10044, anjng Chna {xe-shd@163.com, wangjn@nust.edu.cn, aqhu@seu.edu.cn, guyunlly@hotmal.com, xj00055@sna.com.cn} Abstract. We propose an algorthm to locate an object wth unknown coordnates based on the postve sem-defnte programmng n the wreless sensor networks, assumng that the squared error of the measured dstance follows Gaussan dstrbuton. We frst obtan the estmator of the object locaton; then transform the non-convex problem to convex one by the postve sem-defnte relaxaton; and fnally we take the optmal soluton as the estmated locaton. Smulatons results show that our algorthm s superor to the R-LS algorthm. Keywords: Wreless Sensor etworks; Localzaton; Postve Sem-defnte Programmng; Convex Optmzaton 1 Introducton Wreless sensor network (WS) s manly used to montorng and trackng, and these two applcatons n most cases requre the locaton nformaton of the target node. And some route protocols and management mechansms desgned for the WS also need the nodes locatons. Exstng localzaton prncples can be roughly dvded nto the followng four categores: the frst s based on the receved sgnal strength (RSS) or energy [1]; the second s based on the sgnal tme of arrval (TOA) or tme dfference of arrval (TDOA) []; the thrd s based on the sgnal angle of arrval (AOA) [3]; and the fnal s based on the combnaton of above aspects [4]. However, the estmator of the object locaton s usually a non-lnear and nonconvex functon of the measured values of the dstances, and t s dffcult to drectly acheve ts global optmal soluton usng exstng optmzaton methods. To the best of our knowledge, the common concerned solutons could be dvded nto three categores: frst, to make the whole optmzaton problem nto a convexty by abandonng the non-lnear constrant parts [5]; second, to lnearze the optmzaton problems by plane ntersect, sphercal nterpolaton [6] or sphercal ntersecton [7]; and the thrd s based on the second order cone relaxaton [8] or the postve sem- 31

2 defnte cone relaxaton [9] to make the optmzaton problem convex. In above three methods, the thrd s commonly utlzed. In ths paper, we assume that the squared error of the measured dstance follows Gaussan dstrbuton and propose an algorthm to locate a sngle object based on the postve sem-defnte programmng. We compare our algorthm wth the R-LS [5] algorthm when the object s wthn and out of the convex hull composed of all the anchors, and the results show that our algorthm s superor to R-LS algorthm regardless of whether the object s located wthn the convex hull. Dstance Model We assume that n a d dmensons space, there are sensor nodes whch s called as anchors wth known coordnates denoted as column vectors s,, 1 s, and there s one object node wth unknown coordnates denoted as a column vector u. Then the measured dstance d between anchor and the object s expressed by [1]: d = u s + n, = 1,,. (1) where n s the measured nose whch follows Gaussan dstrbuton wth zero mean and varance δ. Based on the maxmum lkelhood prncple, the locaton estmator û of the object node s:. () = 1 uˆ = arg mn ( d u s ) u 3 Postve Sem-defnte Programmng Obvously, () s non-convex and we can hardly obtan ts global mnmum. So we need to change the expresson of () and take some relaxaton to make t convex. Let t = d u s, then () s equvalent to: 3

3 uˆ = arg mn ut, st.. t = d u s t = 1. (3) Contnue to let y t, then (3) could be rewrtten to: uˆ = arg mn st.. ut,, y = 1 y t = d u s y t. (4) T T ow, we defne a column vector u = [ u,1], whch contans d + 1 elements, and let U T = uu ; we also defne ( d 1) ( d 1) Id d s S =, 1,, T T s ss = reformulated as: + + matrx, and then the equaton constrant n (4) could be t = d trace( S U ). (5) Substtute (5) nto (4), we wll get: uˆ = arg mn st.. Ut,, y = 1 y t = d trace( SU ) y t T U = uu = 1,,. (6) 33

4 Obvously, U s a ( d + 1) ( d + 1) sem-defnte matrx. Usng semdefnte relaxaton,.e. substtutng U 0 for y t Y = t 1, (6) s changed to: U T = uu, and lettng uˆ = arg mn Y(1,1) Ut,, Y = 1 t = d trace( SU ) Y 0 U 0 st.. Y (, ) = 1 Ud ( + 1, d+ 1) = 1 t = Y(1, ) = 1,,. (7) Observng (7), we could easly fnd that t s a postve sem-defnte programmng problem, so t could be solved by SeDuM [1]. 4 Smulatons and Analyss We assume that n the D space, there are four anchors wth coordnates are (0, 0), (0, 10), (10, 10) and(10, 0)respectvely and one object node. In order to evaluate the performance of our algorthm, we compare t wth the R-LS algorthm [5]. Fg.1 shows the locaton error when the object node locatng at [5, 5] whch s wthn the convex hull comprsed by the four anchors and [15,15] whch s out of the convex hull, and the varance of the measure nose vares from 0.5m to 1.5m. From the fgure we could see that the performance of the two algorthms s better when object node s out of the convex hull than wthn the convex hull and our algorthm s superor to R-LS regardless of whether the object s located wthn the convex hull. 5 Conclusons In ths paper, we assume that the squared error of the measured dstance follows Gaussan dstrbuton and propose an algorthm to locate an object based on the postve sem-defnte programmng. We frst obtan the estmator of the object 34

5 locaton based on the maxmum lkelhood prncple; then consderng that the estmator s a non-convex functon wth respect to the measured dstances between the object and the anchors, we transform the non-convex optmzaton nto convex one by the postve sem-defnte relaxaton; and fnally we take the optmal soluton as the estmated value of the object locaton. We compare our algorthm wth the R-LS algorthm when the object s wthn and out of the convex hull, and the results show that our algorthm s superor to R-LS algorthm regardless of whether the object s located wthn the convex hull Locacton Error: m Our Algorthm [5,5] R-LS Algorthm [5,5] 0.03 Our Algorthm [15,15] R-LS Algorthm [15,15] Standard Varance of Measure ose: m Fg. 1. Locaton Estmaton Error under Dfferent Measure ose Acknowledgments Ths work s supported by the Chnese atonal Technology Support Program (o. 01BAH38B05), the ature Scence Foundaton of Jangsu Provnce (o. BK01461), the Jangsu Product-Study-Research Jont Creatve Foundaton (o. BY009149) and the anjng Unversty of Informaton Scence and Technology Foundaton (o. S ). Ths research was also supported by a grant (07- HUDP-A01) from Hgh-tech Urban Development Program funded by Mnstry of Land, Transport and Martme Affars of Korean government. References 1. Ouyang W.T, Wong A.K.S, Lea C.T. Receved Sgnal Strength Based Wreless Localzaton va Semdefnte Progarmmng: oncooperatve and Cooperatve Schemes. IEEE Transactons on Vehcular Technology, vol. 59, pp (010) 35

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