CN105425206A - Steady least square positioning method in nonsynchronous wireless network - Google Patents

Abstract

The invention discloses a steady least square positioning method in the nonsynchronous wireless network. When measurement signals emitted by an unknown target source are spread to different sensors in the sensor network and then forwarded and returned to the unknown target source, measurement signal equivalent transmission distance measuring values based on round arrival time; according to the measurement signal equivalent transmission distance measuring value corresponding to each sensor, an equivalent distance measuring model corresponding to each sensor is obtained; the steady least square problem is established according to the re-described distance measuring models; optimization variables are introduced, the second-order cone relaxation technology is utilized, and the steady least square problem is relaxed into a second-order cone programming problem; and the second-order cone programming problem is solved in the interior point method to obtain the coordinate estimation value of the unknown target source. The method has the advantages that influence of clock drift and transfer time on the positioning precision can be effectively inhibited, the positioning precision is high, and the positioning stability is high.

Description

Robust least squares localization method in a kind of unsynchronized wireless networks

Technical field

The present invention relates to a kind of object localization method, especially relate to the robust least squares localization method in a kind of unsynchronized wireless networks.

Background technology

Target localization has indispensable effect in military field such as accurate military attack, modern society's mobile interchange epoch, along with the great development of the commercialization market application such as location Based service, also make efficient accurate target localization research obtain increasing concern; Meanwhile, Technology for Target Location all has broad application prospects in fields such as military surveillance, traffic monitoring, home automation, industrial or agricultural control, biologic medical, rescue and relief works, and therefore, goal in research localization method tool is of great significance.And as the good complement of the positioning systems such as GPS, the target localization in wireless network is a classical research topic.

In target localization, localization method based on the measured value of time of arrival account for greatly, but the prerequisite of this localization method realize target location is the whole fixer network of supposition Complete Synchronization in time, does not consider the asynchronous impact on locating effect of fixer network, and in fact, due to various factors such as hardware conditions, real network is normally asynchronous, in other words, impossible Complete Synchronization, therefore this localization method is difficult to be applied to real network.Other conventional Initial Transmission Time also requiring accurate known network based on localization method time of arrival existing, but the more difficult realization of this requirement cost is larger in other words.

For solving the technical matters existed based on the localization method of the measured value of time of arrival, the localization method in unsynchronized wireless networks arises at the historic moment.In localization method in unsynchronized wireless networks, because sensor clock exists clock jitter and clock drift, therefore when complete unknown clock jitter and clock drift, the problem of they and target location Combined estimator is difficult to solve.In order to overcome this problem, there has been proposed some schemes, that compares main flow has: one is the location based on time of arrival poor (TDOA); Another kind is the location based on round time of arrival (TW-TOA), and Fig. 1 gives typically based on the schematic diagram of the localizing environment of round time of arrival (TW-TOA).In the localization method based on difference time of arrival, clock jitter is removed, and final need be carried out Combined estimator to clock drift and target location, but it should be noted that this method still needs the sensor in fixer network synchronous in time; In the localization method based on round time of arrival, it does not need any inter-node synchronous in fixer network, but it is envisaged that the connecting time in measuring is on the impact of positioning result, the total feasible zone method proposed if any scholar and least square method, all first these two kinds of methods are estimated the connecting time, try to achieve target location estimated value eventually through to the Combined estimator of clock drift and target source position, but wherein can produce larger impact to the performance of location to the error of transfer time Estimate.

Summary of the invention

Technical matters to be solved by this invention is to provide the robust least squares localization method in a kind of unsynchronized wireless networks, and its location is based on round time of arrival, and positioning precision is high.

The present invention solves the problems of the technologies described above adopted technical scheme: the robust least squares localization method in a kind of unsynchronized wireless networks, is characterized in that comprising the following steps:

1. in unsynchronized wireless networks environment, a two-dimensional coordinate system or three-dimensional system of coordinate is set up as with reference to coordinate system, and there is the known sensor in a unknown object source and N number of position in hypothesis in unsynchronized wireless networks environment, and the coordinate of unknown object source in reference frame is x, the coordinate of N number of sensor in reference frame corresponds to s ₁, s ₂..., s _n, wherein, N>=n+1, n represent the dimension of reference frame, s ₁represent the coordinate of the 1st sensor in reference frame, s ₂represent the coordinate of the 2nd sensor in reference frame, s _nrepresent the coordinate of N number of sensor in reference frame;

2. in unsynchronized wireless networks environment, by unknown object source emission measurement signal, measuring-signal forwards through each sensor of propagation arrival again and turns back to unknown object source after transfer process, first determine that the measuring-signal launched in unknown object source is through propagating the mistiming of the time point arrived when each sensor forwards time point when turning back to unknown object source and unknown object source emission measurement signal again after transfer process, the measuring-signal launched of unknown object source is 2t through propagating the mistiming of time point when arrival i-th sensor forwards time point when turning back to unknown object source and unknown object source emission measurement signal again after transfer process _i, unit is second, then wherein, 1≤i≤N, w represents the clock drift in unknown object source, s _irepresent the coordinate of i-th sensor in reference frame, c is the light velocity, T _irepresent the connecting time needed for measuring-signal that i-th sensor transfer process unknown object source is launched, represent that the measuring-signal launched in unknown object source forwards through propagation arrival i-th sensor the variance turned back on the whole piece transmission path in unknown object source again and is after transfer process gaussian reflectivity mirrors, symbol " ‖ ‖ " is Euclid 2 norm, then calculate the measuring-signal launched in unknown object source to arrive each sensor after transfer process, forward the measuring-signal effective transmission distance measure based on round time of arrival when turning back to unknown object source again through propagating, the measuring-signal launched of unknown object source through propagate arrival i-th sensor again after transfer process the measuring-signal effective transmission distance measure based on round time of arrival forwarded when turning back to unknown object source be 2d _i, unit is rice, then $d_i=c\×t_i=c\×\left(w\×\left(\frac{\left|\right|x-s_i\left|\right|}{c}+\frac{T_i}{2}\right)+\frac{{\tilde{n}}_i}{2}\right)=w\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2},$ Wherein, $d_{T_i}=c\×T_i,$ represent T _ito d _iimpact, n _irepresent d _iin noise, n _igaussian distributed, and n _ivariance be ${\mathrm{\σ}}_i^2=c^2\×{\widetilde{\mathrm{\σ}}}_i^2.$

3. obtain the range observation model that each sensor is corresponding, for i-th sensor, the acquisition process of its corresponding range observation model is: make w=1+ δ, and require that δ satisfies condition | δ | and≤δ _max< < 1, and determine span be then combine with w=1+ δ, obtain $d_i=\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2},$ Combine again $d_i=\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2}$ With w=1+ δ, obtain then basis $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)=\mathrm{\&delta;}\left(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})}\right)$ With | δ |≤δ _max< < 1, obtains $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)=\mathrm{\&delta;}\left(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})}\right)\&ap;\mathrm{\&delta;}\left(d_i-\frac{n_i}{2}\right),$ Suppose again $\left|\frac{n_i}{2}\right|<<d_i,$ Then basis $\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})\&ap;\mathrm{\&delta;}(d_i-\frac{n_i}{2})$ With , obtain $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)\&ap;{\mathrm{\&delta;d}}_i;$ Combine afterwards $d_i=\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2}$ With $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)\&ap;{\mathrm{\&delta;d}}_i,$ Obtain $d_i\&ap;\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+{\mathrm{\&delta;d}}_i+\frac{n_i}{2},$ Right again $d_i\&ap;\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+{\mathrm{\&delta;d}}_i+\frac{n_i}{2}$ About equal sign both sides deduct intermediate value obtain $d_i-\frac{a_i+b_i}{4}\&ap;\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i+\frac{n_i}{2};$ Finally make ${\hat{d}}_i=d_i-\frac{a_i+b_i}{4},$ And make $e_i=\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i,$ Will $d_i-\frac{a_i+b_i}{4}\&ap;\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i+\frac{n_i}{2}$ Be reduced to and will as the range observation model that i-th sensor is corresponding; Wherein, δ represents the clock drift amount of unknown object source relative standard's clock, and symbol " || " is the symbol that takes absolute value, and symbol " < < " is much smaller than symbol, δ _maxrepresent the maximal value of the clock drift amount of unknown object source relative standard's clock, a _iand b _icorresponding expression the upper bound of value and lower bound, $\left|e_i\right|\&le;{\mathrm{\&rho;}}_i,{\mathrm{\&rho;}}_i=\frac{b_i-a_i}{4}+{\mathrm{\&delta;}}_{max}\&times;d_i;$

4. corresponding to each sensor range observation model is redescribed, for the range observation model that i-th sensor is corresponding to its detailed process redescribed be: will ${\hat{d}}_i\&ap;\left|\right|x-s_i\left|\right|+e_i+\frac{n_i}{2}$ Change into ${\hat{d}}_i-e_i\&ap;\left|\right|x-s_i\left|\right|+\frac{n_i}{2};$ Then right ${\hat{d}}_i-e_i\&ap;\left|\right|x-s_i\left|\right|+\frac{n_i}{2}$ About equal sign both sides to carry out square, and to suppose then omit n _iquadratic component obtain ${\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2\&ap;\left|\right|x-s_i|{|}^2+||x-s_i||\&times;n_i;$ Again will ${\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2\&ap;\left|\right|x-s_i|{|}^2+||x-s_i||\&times;n_i$ Change into: $n_i\&ap;\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|},$ Namely ${\hat{d}}_i\&ap;\left|\right|x-s_i\left|\right|+e_i+\frac{n_i}{2}$ Redescribe for $n_i\&ap;\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|};$

5. according to the range observation model after redescribing, set up a robust least squares problem, be described as: $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2};$ Then make $f\left(e_i\right)=\frac{\left|{\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ According to $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ With $f\left(e_i\right)=\frac{\left|{\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ Will $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Change into $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left(f\left(e_i\right)\right)}^2}{{\mathrm{\&sigma;}}_i^2};$ Basis again $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left(f\left(e_i\right)\right)}^2}{{\mathrm{\&sigma;}}_i^2}$ Robust least squares problem is described as: $\underset{x}{\min }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\&lsqb;{\max }_{e_i}f\left(e_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2};$ Wherein, $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Represent to get and make $\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ The minimum x of value, $\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Represent to get and make maximum { the e of value _i, { e _irefer to by e ₁, e ₂..., e _nthe set of composition, f (e _i) represent and get and make f (e _i) the maximum e of value _i;

6. f (e is determined _i) maximal value, if then f (e _i) maximal value be max (f (-ρ _i), f (ρ _i)); If then f (e _i) maximal value be then basis with f (e _i) maximal value, obtain diagram form on show, describe

For: $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f\left(-{\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right)\end{array};$ Wherein, symbol " || " is the symbol that takes absolute value, max () for getting max function, wherein $f(-{\mathrm{\&rho;}}_i)=\frac{\left|{(-{\mathrm{\&rho;}}_i)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ $f\left({\mathrm{\&rho;}}_i\right)=\frac{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ $f\left({\hat{d}}_i\right)=\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;{\hat{d}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|}=\left|\right|x-s_i\left|\right|,\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i$ Represent to get and make the minimum x of value, { η _i, η _ifor $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f\left(-{\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right)\end{array}$ I-th optimized variable of middle introducing, { η _ithe set of N number of optimized variable for introducing, " s.t. " expressions " condition that submits to is ";

7. combine $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f\left(-{\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right)\end{array}$ And $f\left(-{\mathrm{\&rho;}}_i\right)\frac{\left|{\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},f\left({\mathrm{\&rho;}}_i\right)=\frac{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|}$ With

$f\left({\hat{d}}_i\right)=\left|\right|x-s_i\left|\right|,$ Obtain $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\left|{\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|x-s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right).\end{array};$

8. exist $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\left|{\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|x-s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right).\end{array}$ Middle introducing optimized variable y, y=||x|| ², then utilize second order cone relaxing techniques by y=||x|| ²relax and be || x|| ²≤ y, obtains Second-order cone programming problem, is described as: $\begin{array}{cc}\underset{x,y,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \left|\right|\left[\begin{array}{c}2\left({\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|\left[\begin{array}{c}2\left({\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & y-2s_i^{T}x+\left|\right|s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right),\\ & \left|\right|\left[\begin{array}{c}2x\\ y-1\end{array}\right]\left|\right|\&le;y+1.\end{array},$ Wherein, represent to get and make the minimum x of value, y, { η _i, symbol " [] " is vector representation symbol, for s _itransposed vector;

9. interior point method technology pair is utilized $\begin{array}{cc}\underset{x,y,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \left|\right|\left[\begin{array}{c}2\left({\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|\left[\begin{array}{c}2\left({\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & y-2s_i^{T}x+\left|\right|s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right),\\ & \left|\right|\left[\begin{array}{c}2x\\ y-1\end{array}\right]\left|\right|\&le;y+1.\end{array}$ Solve, obtain x, y, { η _icorresponding estimated value, correspondence is designated as

Described step 3. in a _ivalue and b _ithe deterministic process of value be:

3. N number of sensor that in the unsynchronized wireless networks tested, location information is known-1, is supposed;

3.-2, by jth ' individual sensor to i-th ' individual sensor send measuring-signal, calculate jth ' the measuring-signal of individual sensor emission through propagating arrival i-th ' individual sensor forward after transfer process again turn back to jth ' individual sensor time time point and jth ' individual sensor initial transmissions measuring-signal time the mistiming 2t of time point _{j', i'}product with light velocity c, is designated as wherein, 1≤i'≤N, 1≤j'≤N, i' ≠ j';

-3 3. the positional information, according to N number of sensor, calculates the actual distance between jth ' individual sensor and i-th ' individual sensor, is designated as d _{j', i'}; Then calculate jth ' effective transmission distance in the measuring-signal of the individual sensor emission connecting time through propagating arrival i-th ' individual sensor after i-th ' individual sensor, be designated as d _{t, j', i'},

-4 3., from N number of sensor a selected sensor arbitrarily, suppose the sensor selected be i-th ' individual sensor, then make a _i=min (d _{t, j', i'}| 1≤j'≤N, j' ≠ i'), and make b _i=max (d _{t, j', i'}| 1≤j'≤N, j' ≠ i'), wherein, min () is for getting minimum value function, and max () is for getting max function.

Compared with prior art, the invention has the advantages that: compare existing total feasible zone method and least square method, the present invention utilizes Robust Least Squares Method clock drift and connecting time steadily and surely to be processed as an irrelevant variable, only go to estimate the coordinate figure of unknown object source in reference frame, thus can effectively suppress clock drift and connecting time on the impact of positioning precision, positioning precision is high; Meanwhile, utilize second order cone relaxing techniques the description of robust least squares problem to be relaxed as Second-order cone programming problem, can guarantee to obtain globally optimal solution like this and not by the impact of local convergence, thus have higher hi-Fix stability.

Accompanying drawing explanation

Fig. 1 is typically based on the schematic diagram of the localizing environment of round time of arrival (TW-TOA);

Fig. 2 is the overall procedure block diagram of the inventive method;

Fig. 3 be the inventive method and existing total feasible zone method and least square method in the measurements root-mean-square error with the change schematic diagram of noise size;

Fig. 4 be the inventive method and existing total feasible zone method and least square method in the measurements root-mean-square error with the change schematic diagram of sensor (anchor node) number.

Embodiment

Below in conjunction with accompanying drawing embodiment, the present invention is described in further detail.

Robust least squares localization method in a kind of unsynchronized wireless networks that the present invention proposes, as shown in Figure 2, it comprises the following steps its overall procedure block diagram:

1. in unsynchronized wireless networks environment, a two-dimensional coordinate system or three-dimensional system of coordinate is set up as with reference to coordinate system, and there is the known sensor in a unknown object source and N number of position in hypothesis in unsynchronized wireless networks environment, and the coordinate of unknown object source in reference frame is x, the coordinate of N number of sensor in reference frame corresponds to s ₁, s ₂..., s _n, wherein, N>=n+1, n represent the dimension of reference frame, n=2 or n=3, n=2 when namely reference coordinate is two-dimensional coordinate system, n=3, s when reference coordinate is three-dimensional system of coordinate ₁represent the coordinate of the 1st sensor in reference frame, s ₂represent the coordinate of the 2nd sensor in reference frame, s _nrepresent the coordinate of N number of sensor in reference frame.

2. in unsynchronized wireless networks environment, as shown in Figure 1, by unknown object source emission measurement signal, measuring-signal forwards through each sensor of propagation arrival again and turns back to unknown object source after transfer process, first determine that the measuring-signal launched in unknown object source is through propagating the mistiming of the time point arrived when each sensor forwards time point when being back to unknown object source and unknown object source emission measurement signal again after transfer process, the measuring-signal launched of unknown object source is 2t through propagating the mistiming of time point when arrival i-th sensor forwards time point when being back to unknown object source and unknown object source initial transmissions measuring-signal again after transfer process _i, unit is second, then wherein, 1≤i≤N, w represents the clock drift in unknown object source, unknown in the value of this w, s _irepresent the coordinate of i-th sensor in reference frame, c is the light velocity, T _irepresent the connecting time needed for measuring-signal that i-th sensor transfer process unknown object source is launched, at this T _ivalue unknown, represent that the measuring-signal launched in unknown object source forwards through propagation arrival i-th sensor the variance turned back on the whole piece transmission path in unknown object source again and is after transfer process gaussian noise, symbol " ‖ ‖ " is Euclid 2 norm, then calculate the measuring-signal launched in unknown object source to arrive each sensor after transfer process, forward the measuring-signal effective transmission distance measure based on round time of arrival when turning back to unknown object source again through propagating, the measuring-signal launched of unknown object source through propagate arrival i-th sensor again after transfer process the measuring-signal effective transmission distance measure based on round time of arrival forwarded when turning back to unknown object source be 2d _i, unit is rice, then $d_i=c\&times;t_i=c\&times;\left(w\&times;\left(\frac{\left|\right|x-s_i\left|\right|}{c}+\frac{T_i}{2}\right)+\frac{{\tilde{n}}_i}{2}\right)=w\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2},$ Wherein, represent T _ito d _iimpact, n _irepresent d _iin noise, n _igaussian distributed, and n _ivariance be (i.e. 2d _imiddle noise n _ipower),

3. obtain the range observation model that each sensor is corresponding, for i-th sensor, the acquisition process of its corresponding range observation model is: make w=1+ δ, and require that δ satisfies condition | δ | and≤δ _max< < 1, and determine span be then combine with w=1+ δ, obtain $d_i=\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2},$ Combine again $d_i=\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2}$ With w=1+ δ, obtain $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)=\mathrm{\&delta;}\left(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})}\right);$ Then basis $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)=\mathrm{\&delta;}\left(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})}\right)$ With | δ |≤δ _max< < 1, obtains $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)=\mathrm{\&delta;}\left(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})}\right)\&ap;\mathrm{\&delta;}\left(d_i-\frac{n_i}{2}\right),$ Suppose again then basis $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)\&ap;\mathrm{\&delta;}\left(d_i-\frac{n_i}{2}\right)$ With obtain $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)\&ap;{\mathrm{\&delta;d}}_i;$ Combine afterwards $d_i=\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+\frac{n_i}{2}$ With $\mathrm{\&delta;}\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)\&ap;{\mathrm{\&delta;d}}_i,$ Obtain $d_i\&ap;\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+{\mathrm{\&delta;d}}_i+\frac{n_i}{2},$ Right again $d_i\&ap;\left(\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}\right)+{\mathrm{\&delta;d}}_i+\frac{n_i}{2}$ About equal sign both sides deduct intermediate value obtain $d_i-\frac{a_i+b_i}{4}\&ap;\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i+\frac{n_i}{2};$ Finally make ${\hat{d}}_i=d_i-\frac{a_i+b_i}{4},$ And make $e_i=\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i,$ Will $d_i-\frac{a_i+b_i}{4}\&ap;\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i+\frac{n_i}{2}$ Be reduced to and will as the range observation model that i-th sensor is corresponding; Wherein, δ represents the clock drift amount of unknown object source relative standard's clock, and symbol " || " is the symbol that takes absolute value, and symbol " " is much smaller than symbol, δ _maxrepresent the maximal value of the clock drift amount of unknown object source relative standard's clock, at this δ _maxvalue known, a and b correspondence represent the upper bound of value and lower bound, a _ivalue and b _ivalue known, | e _i|≤ρ _i, ${\mathrm{\&rho;}}_i=\frac{b_i-a_i}{4}+{\mathrm{\&delta;}}_{max}\&times;d_i.$

In this particular embodiment, step 3. in a _ivalue and b _ithe deterministic process of value be:

-1 3., suppose N number of sensor that the unsynchronized wireless networks location information of testing is known, the positional information of sensor can be located by GPS and be obtained.

3.-2, by jth ' individual sensor to i-th ' individual sensor send measuring-signal, calculate jth ' the measuring-signal of individual sensor emission through propagating arrival i-th ' individual sensor forward after transfer process again turn back to jth ' individual sensor time time point and jth ' individual sensor initial transmissions measuring-signal time the mistiming 2t of time point _{j', i'}product with light velocity c, is designated as wherein, 1≤i'≤N, 1≤j'≤N, i' ≠ j'.

-3 3. the positional information, according to N number of sensor, calculates the actual distance between jth ' individual sensor and i-th ' individual sensor, is designated as d _{j', i'}; Then calculate jth ' effective transmission distance in the measuring-signal of the individual sensor emission connecting time through propagating arrival i-th ' individual sensor after i-th ' individual sensor, be designated as d _{t, j', i'},

-4 3., from N number of sensor a selected sensor arbitrarily, suppose the sensor selected be i-th ' individual sensor, then make a _i=min (d _{t, j', i'}| 1≤j'≤N, j' ≠ i'), and make b _i=max (d _{t, j', i'}| 1≤j'≤N, j' ≠ i'), wherein, min () is for getting minimum value function, and max () is for getting max function.

4. corresponding to each sensor range observation model is redescribed, for the range observation model that i-th sensor is corresponding to its detailed process redescribed be: will ${\hat{d}}_i\&ap;\left|\right|x-s_i\left|\right|+e_i+\frac{n_i}{2}$ Change into ${\hat{d}}_i-e_i\&ap;\left|\right|x-s_i\left|\right|+\frac{n_i}{2};$ Then right ${\hat{d}}_i-e_i\&ap;\left|\right|x-s_i\left|\right|+\frac{n_i}{2}$ About equal sign both sides to carry out square, and to suppose then n can be omitted _iquadratic component obtain ${\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2\&ap;\left|\right|x-s_i|{|}^2+||x-s_i||\&times;n_i;$ Again will ${\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2\&ap;\left|\right|x-s_i|{|}^2+||x-s_i||\&times;n_i$ Change into: $n_i\&ap;\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|},$ Namely ${\hat{d}}_i\&ap;\left|\right|x-s_i\left|\right|+e_i+\frac{n_i}{2}$ Redescribe for $n_i\&ap;\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|}.$

5. according to the range observation model after redescribing, set up a robust least squares problem, be described as: $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2};$ Then make $f\left(e_i\right)=\frac{\left|{\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ According to $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ With $f\left(e_i\right)=\frac{\left|{\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ Will $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Change into $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left(f\left(e_i\right)\right)}^2}{{\mathrm{\&sigma;}}_i^2};$ Basis again $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left(f\left(e_i\right)\right)}^2}{{\mathrm{\&sigma;}}_i^2}$ Robust least squares problem is described as: $\underset{x}{\min }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\&lsqb;{\max }_{e_i}f\left(e_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2};$ Wherein, $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Represent to get and make $\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ The minimum x of value, $\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Represent to get and make maximum { the e of value _i, { e _irefer to by e ₁, e ₂..., e _nthe set of composition, symbol " || " is the symbol that takes absolute value, f (e _i) represent and get and make f (e _i) the maximum e of value _i.

6. f (e is determined _i) maximal value, if then f (e _i) maximal value be max (f (-ρ _i), f (ρ _i)); If then f (e _i) maximal value be then basis with f (e _i) maximal value, obtain diagram form on show, be described as: $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f\left(-{\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right)\end{array};$ Wherein, symbol " || " is the symbol that takes absolute value, max () for getting max function, $f(-{\mathrm{\&rho;}}_i)=\frac{\left|{(-{\mathrm{\&rho;}}_i)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ $f\left({\mathrm{\&rho;}}_i\right)=\frac{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},$ $f\left({\hat{d}}_i\right)=\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;{\hat{d}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|}=\left|\right|x-s_i\left|\right|,\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i$ Represent to get and make the minimum x of value, { η _i, η _ifor $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f\left(-{\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right)\end{array}$ I-th optimized variable of middle introducing, { η _ithe set of N number of optimized variable for introducing, " s.t. " expressions " condition that submits to is ".

7. combine $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f\left(-{\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right)\end{array}$ And $f\left(-{\mathrm{\&rho;}}_i\right)=\frac{\left|{\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|},f\left({\mathrm{\&rho;}}_i\right)=\frac{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}{\left|\right|x-s_i\left|\right|}$ With $f\left({\hat{d}}_i\right)=\left|\right|x-s_i\left|\right|,$ Obtain $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\left|{\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|x-s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right).\end{array}.$

8. exist $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\left|{\left(-{\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;\left(-{\mathrm{\&rho;}}_i\right)+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \frac{{\left|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-\left|\right|x-s_i|{|}^2\right|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|x-s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N.\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right).\end{array}$ Middle introducing optimized variable y, y=||x|| ², then utilize second order cone relaxing techniques by y=||x|| ²relax and be || x|| ²≤ y, obtains Second-order cone programming problem, is described as: $\begin{array}{cc}\underset{x,y,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \left|\right|\left[\begin{array}{c}2\left({\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|\left[\begin{array}{c}2\left({\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & y-2s_i^{T}x+\left|\right|s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right),\\ & \left|\right|\left[\begin{array}{c}2x\\ y-1\end{array}\right]\left|\right|\&le;y+1.\end{array},$ Wherein, represent to get and make the minimum x of value, y, { η _i, symbol " [] " is vector representation symbol, for s _itransposed vector.

9. interior point method technology pair is utilized $\begin{array}{cc}\underset{x,y,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&eta;}}_i\\ s.t.& \left|\right|\left[\begin{array}{c}2\left({\left(-{\mathrm{\&rho;}}_i\right)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & \left|\right|\left[\begin{array}{c}2\left({\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-\left|\right|s_i|{|}^2\right)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]\left|\right|\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\\ & y-2s_i^{T}x+\left|\right|s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,\mathrm{...},N,\left(if\left|{\hat{d}}_i\right|\&le;{\mathrm{\&rho;}}_i\right),\\ & \left|\right|\left[\begin{array}{c}2x\\ y-1\end{array}\right]\left|\right|\&le;y+1.\end{array}$ Solve, obtain x, y, { η _icorresponding estimated value, correspondence is designated as

For verifying feasibility and the validity of the inventive method, l-G simulation test is carried out to the inventive method.

1) situation of change of performance with measurement noises size of the inventive method is tested.Suppose that use 8 sensors are measured, the method for measurement is: first set up a plane right-angle coordinate, and 8 sensors are respectively (40,40), (40 ,-40), (-40,40), (-40 ,-40), (40,0), (0,40), (40,0), (0 ,-40) place (unit: m), unknown object source is then randomly dispersed in (-40,40) × (-40,40) m ²coordinates regional in.In simulations, the clock drift w in unknown object source is then randomly dispersed in the scope of [0.99,1.01], is also the maximal value δ of the clock drift amount of unknown object source relative standard's clock _max=0.01, and then suppose that the scope of stochastic distribution is for [24,36] m, 1≤i≤8, in addition, suppose that the power of the noise in each self-corresponding measuring-signal effective transmission distance measure of all the sensors is identical, are wherein, σ refers to that the standard deviation of the noise of the horizontal ordinate representative in Fig. 3 is poor.

Fig. 3 give the inventive method and existing total feasible zone method and linear least square method in the measurements root-mean-square error with the change schematic diagram of noise size.As can be seen from Figure 3, in noise change procedure from small to large, the positioning performance of the inventive method is better than existing total feasible zone method (GTR) and linear least-squares (LLS) algorithm.Specifically, when the standard deviation of noise is 1m and 9m, root-mean-square error can reduce 0.5m and 2.1m.

2) situation of change that the positioning precision of testing the inventive method increases along with number of probes.Measure method be: in a plane right-angle coordinate hypothesis have 8 sensors and 8 sensors respectively in (40,40), (40 ,-40), (-40,40), (-40,-40), (40,0), (0,40), (40,0), (unit: m) of (0 ,-40) place, unknown object source is then randomly dispersed in (-40,40) × (-40,40) m ²coordinates regional in, get front 5 respectively and position test to front 8 sensors.In addition, suppose that the standard deviation of the noise in each self-corresponding measuring-signal effective transmission distance measure of all the sensors is identical, be σ ₁=σ ₂=...=σ ₈=4m.

Fig. 4 give the inventive method and existing total feasible zone method (GTR) and linear least square method (LLS) in the measurements root-mean-square error with the change schematic diagram of sensor (anchor node) number.In number of sensors by 5 to 8 change procedures increased gradually, the inventive method is better than existing total feasible zone method and linear least-squares algorithm in positioning precision.Specifically, when number of sensors is less than 8, the performance of total feasible zone method and linear least-squares is sharply deteriorated, so that cannot complete location, and the inventive method still can complete and locates more accurately.

As can be seen from upper simulation result, the inventive method has good performance.Compare with linear least square method with existing total feasible zone method, the inventive method effectively can reduce root-mean-square error, improve the precision of location, and the increase of noise power can't weaken the performance of location significantly, embodies the robustness of location; In addition, still relatively accurately can locate when sensor is fewer in a network, further illustrate feasibility and the validity of the inventive method.

Claims (2)

1. the robust least squares localization method in unsynchronized wireless networks, is characterized in that comprising the following steps:

1. in unsynchronized wireless networks environment, a two-dimensional coordinate system or three-dimensional system of coordinate is set up as with reference to coordinate system, and there is the known sensor in a unknown object source and N number of position in hypothesis in unsynchronized wireless networks environment, and the coordinate of unknown object source in reference frame is x, the coordinate of N number of sensor in reference frame corresponds to s ₁, s ₂..., s _n, wherein, N>=n+1, n represent the dimension of reference frame, s ₁represent the coordinate of the 1st sensor in reference frame, s ₂represent the coordinate of the 2nd sensor in reference frame, s _nrepresent the coordinate of N number of sensor in reference frame;

2. in unsynchronized wireless networks environment, by unknown object source emission measurement signal, measuring-signal forwards through each sensor of propagation arrival again and turns back to unknown object source after transfer process, first determine that the measuring-signal launched in unknown object source is through propagating the mistiming of the time point arrived when each sensor forwards time point when turning back to unknown object source and unknown object source emission measurement signal again after transfer process, the measuring-signal launched of unknown object source is 2t through propagating the mistiming of time point when arrival i-th sensor forwards time point when turning back to unknown object source and unknown object source emission measurement signal again after transfer process _i, unit is second, then wherein, 1≤i≤N, w represents the clock drift in unknown object source, s _irepresent the coordinate of i-th sensor in reference frame, c is the light velocity, T _irepresent the connecting time needed for measuring-signal that i-th sensor transfer process unknown object source is launched, represent that the measuring-signal launched in unknown object source forwards through propagation arrival i-th sensor the variance turned back on the whole piece transmission path in unknown object source again and is after transfer process gaussian reflectivity mirrors, symbol " || || " is Euclid 2 norm, then calculate the measuring-signal launched in unknown object source to arrive each sensor after transfer process, forward the measuring-signal effective transmission distance measure based on round time of arrival when turning back to unknown object source again through propagating, the measuring-signal launched of unknown object source through propagate arrival i-th sensor again after transfer process the measuring-signal effective transmission distance measure based on round time of arrival forwarded when turning back to unknown object source be 2d _i, unit is rice, then $d_i=c\&times;t_i=c\&times;(w\&times;(\frac{\left|\right|x-s_i\left|\right|}{c}+\frac{T_i}{2})+\frac{{\tilde{n}}_i}{2})=w\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\frac{n_i}{2},$ Wherein, $d_{T_i}=c\&times;T_i,$ represent T _ito d _iimpact, n _irepresent d _iin noise, n _igaussian distributed, and n _ivariance be ${\mathrm{\&sigma;}}_i^2=c^2\&times;{\widetilde{\mathrm{\&sigma;}}}_i^2.$

3. obtain the range observation model that each sensor is corresponding, for i-th sensor, the acquisition process of its corresponding range observation model is: make w=1+ δ, and require that δ satisfies condition | δ | and≤δ _max< < 1, and determine span be then combine $d_i=w\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\frac{n_i}{2}$ With w=1+ δ, obtain $d_i=\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\frac{n_i}{2},$ Combine again $d_i=\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\frac{n_i}{2}$ With w=1+ δ, obtain $\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})=\mathrm{\&delta;}(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})});$ Then basis $\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})=\mathrm{\&delta;}(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})})$ With | δ |≤δ _max< < 1, obtains $\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})=\mathrm{\&delta;}(\frac{d_i}{1+\mathrm{\&delta;}}-\frac{n_i}{2(1+\mathrm{\&delta;})})\&ap;\mathrm{\&delta;}(d_i-\frac{n_i}{2}),$ Suppose again then basis $\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})\&ap;\mathrm{\&delta;}(d_i-\frac{n_i}{2})$ With obtain $\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})\&ap;{\mathrm{\&delta;d}}_i;$ Combine afterwards $d_i=\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})+\frac{n_i}{2}$ With $\mathrm{\&delta;}\left(\right||x-s_i||+\frac{d_{T_i}}{2})\&ap;{\mathrm{\&delta;d}}_i,$ Obtain $d_i\&ap;\left(\right||x-s_i||+\frac{d_{T_i}}{2})+{\mathrm{\&delta;d}}_i+\frac{n_i}{2},$ Right again $d_i\&ap;\left(\right||x-s_i||+\frac{d_{T_i}}{2})+{\mathrm{\&delta;d}}_i+\frac{n_i}{2}$ About equal sign both sides deduct intermediate value obtain $d_i-\frac{a_i+b_i}{4}\&ap;\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i+\frac{n_i}{2};$ Finally make ${\hat{d}}_i=d_i-\frac{a_i+b_i}{4},$ And make $e_i=\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i,$ Will $d_i-\frac{a_i+b_i}{4}\&ap;\left|\right|x-s_i\left|\right|+\frac{d_{T_i}}{2}-\frac{a_i+b_i}{4}+{\mathrm{\&delta;d}}_i+\frac{n_i}{2}$ Be reduced to and will as the range observation model that i-th sensor is corresponding; Wherein, δ represents the clock drift amount of unknown object source relative standard's clock, and symbol " || " is the symbol that takes absolute value, and symbol " < < " is much smaller than symbol, δ _maxrepresent the maximal value of the clock drift amount of unknown object source relative standard's clock, a _iand b _icorresponding expression the upper bound of value and lower bound, $\left|e_i\right|\&le;{\mathrm{\&rho;}}_i,{\mathrm{\&rho;}}_i=\frac{b_i-a_i}{4}+{\mathrm{\&delta;}}_{max}\&times;d_i;$

4. corresponding to each sensor range observation model is redescribed, for the range observation model that i-th sensor is corresponding to its detailed process redescribed be: will ${\hat{d}}_i\&ap;\left|\right|x-s_i\left|\right|+e_i+\frac{n_i}{2}$ Change into ${\hat{d}}_i-e_i\&ap;\left|\right|x-s_i\left|\right|+\frac{n_i}{2};$ Then right ${\hat{d}}_i-e_i\&ap;\left|\right|x-s_i\left|\right|+\frac{n_i}{2}$ About equal sign both sides to carry out square, and to suppose then omit n _iquadratic component obtain ${\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2\&ap;\left|\right|x-s_i|{|}^2+||x-s_i||\&times;n_i;$ Again will ${\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2\&ap;\left|\right|x-s_i|{|}^2+||x-s_i||\&times;n_i$ Change into: $n_i\&ap;\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|},$ Namely ${\hat{d}}_i\&ap;\left|\right|x-s_i\left|\right|+e_i+\frac{n_i}{2}$ Redescribe for $n_i\&ap;\frac{{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left(e_i\right)}^2-\left|\right|x-s_i|{|}^2}{\left|\right|x-s_i\left|\right|};$

5. according to the range observation model after redescribing, set up a robust least squares problem, be described as: $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2};$ Then make $f\left(e_i\right)=\frac{|{\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2|}{\left|\right|x-s_i\left|\right|},$ According to $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|\left|\right)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ With $f\left(e_i\right)=\frac{|{\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2|}{\left|\right|x-s_i\left|\right|},$ Will $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Change into $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{\left(f\right(e_i\left)\right)}^2}{{\mathrm{\&sigma;}}_i^2};$ Basis again robust least squares problem is described as: wherein, $\underset{x}{\min }\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Represent to get and make $\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\frac{{({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ The minimum x of value, $\underset{\left\{e_i\right\}}{\max }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}\&ap;\frac{({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2)}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Represent to get and make $\underset{i=1}{\overset{N}{\&Sigma;}}\frac{{({\left(e_i\right)}^2-2{\hat{d}}_i\&times;e_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2)}^2}{{\mathrm{\&sigma;}}_i^2\&times;\left|\right|x-s_i|{|}^2}$ Maximum { the e of value _i, { e _irefer to by e ₁, e ₂..., e _nthe set of composition, represent to get and make f (e _i) the maximum e of value _i;

6. f (e is determined _i) maximal value, if then f (e _i) maximal value be max (f (-ρ _i), f (ρ _i)); If then f (e _i) maximal value be then basis with f (e _i) maximal value, obtain diagram form on show, be described as: $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f(-{\mathrm{\&rho;}}_i)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N.\left(if\right|{\hat{d}}_i|\&le;{\mathrm{\&rho;}}_i)\end{array};$ Wherein, symbol " | " is the symbol that takes absolute value, max () for getting max function, wherein $f(-{\mathrm{\&rho;}}_i)=\frac{|{(-{\mathrm{\&rho;}}_i)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2|}{\left|\right|x-s_i\left|\right|},$ $f\left({\mathrm{\&rho;}}_i\right)=\frac{|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2|}{\left|\right|x-s_i\left|\right|},$ $f\left({\hat{d}}_i\right)=\frac{|{\left({\hat{d}}_i\right)}^2-2{\hat{d}}_i\&times;{\hat{d}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2|}{\left|\right|x-s_i\left|\right|}=\left|\right|x-s_i\left|\right|,$ represent to get and make the minimum x of value, { η _i, η _ifor $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f(-{\mathrm{\&rho;}}_i)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N.\left(if\right|{\hat{d}}_i|\&le;{\mathrm{\&rho;}}_i)\end{array}$ I-th optimized variable of middle introducing, { η _ithe set of N number of optimized variable for introducing, " s.t. " expressions " condition that submits to is ";

7. combine $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{{\&lsqb;f(-{\mathrm{\&rho;}}_i)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{{\&lsqb;f\left({\mathrm{\&rho;}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{{\&lsqb;f\left({\hat{d}}_i\right)\&rsqb;}^2}{{\mathrm{\&sigma;}}_i^2}\&le;{\mathrm{\&eta;}}_i,i=1,...,N.\left(if\right|{\hat{d}}_i|\&le;{\mathrm{\&rho;}}_i)\end{array}$ And $f(-{\mathrm{\&rho;}}_i)=\frac{|{(-{\mathrm{\&rho;}}_i)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2|}{\left|\right|x-s_i\left|\right|},$ $f\left({\mathrm{\&rho;}}_i\right)=\frac{|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2|}{\left|\right|x-s_i\left|\right|}$ With $f\left({\hat{d}}_i\right)=\left|\right|x-s_i\left|\right|,$ Obtain $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{|{(-{\mathrm{\&rho;}}_i)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2{|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2{|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \left|\right|x-s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N.\left(if\right|{\hat{d}}_i|\&le;{\mathrm{\&rho;}}_i).\end{array};$

8. exist $\begin{array}{cc}\underset{x,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }& \underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i\\ s.t.& \frac{|{(-{\mathrm{\&rho;}}_i)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2{|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \frac{|{\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-||x-s_i|{|}^2{|}^2}{\left|\right|x-s_i|{|}^2}\&le;{\mathrm{\&sigma;}}_i^2\&le;{\mathrm{\&eta;}}_i,i=1,...,N,\\ & \left|\right|x-s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N.\left(if\right|{\hat{d}}_i|\&le;{\mathrm{\&rho;}}_i).\end{array};$ Middle introducing optimized variable y, y=||x|| ², then utilize second order cone relaxing techniques by y=||x|| ²relax and be || x|| ²≤ y, obtains Second-order cone programming problem, is described as: $\begin{array}{c}\underset{x,y,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i\\ s.t.||\left[\begin{array}{c}2({(-{\mathrm{\&rho;}}_i)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-|\left|s_i\right|{|}^2)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]||\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\\ ||\left[\begin{array}{c}2({\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-|\left|s_i\right|{|}^2)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]||\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\left(if\right|{\hat{d}}_i|\&le;{\mathrm{\&rho;}}_i),\\ ||\left[\begin{array}{c}2x\\ y-1\end{array}\right]||\&le;y+1.\end{array},$ Wherein, represent to get and make the minimum x of value, y, { η _i, symbol " [] " is vector representation symbol, for s _itransposed vector;

9. interior point method technology pair is utilized $\begin{array}{c}\underset{x,y,\left\{{\mathrm{\&eta;}}_i\right\}}{\min }\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{\mathrm{\&eta;}}_i\\ s.t.||\left[\begin{array}{c}2({(-{\mathrm{\&rho;}}_i)}^2+2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-|\left|s_i\right|{|}^2)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]||\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\\ ||\left[\begin{array}{c}2({\left({\mathrm{\&rho;}}_i\right)}^2-2{\hat{d}}_i\&times;{\mathrm{\&rho;}}_i+{\left({\hat{d}}_i\right)}^2-y+2s_i^{T}x-|\left|s_i\right|{|}^2)\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2-{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i\end{array}\right]||\&le;y-2s_i^{T}x+\left|\right|s_i|{|}^2+{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\\ y-2s_i^{T}x+\left|\right|s_i|{|}^2\&le;{\mathrm{\&sigma;}}_i^2\&times;{\mathrm{\&eta;}}_i,i=1,...,N,\left(if\right|{\hat{d}}_i|\&le;{\mathrm{\&rho;}}_i),\\ ||\left[\begin{array}{c}2x\\ y-1\end{array}\right]||\&le;y+1.\end{array}$ Solve, obtain x, y, { η _icorresponding estimated value, correspondence is designated as

2. the robust least squares localization method in a kind of unsynchronized wireless networks according to claim 1, is characterized in that a during described step 3. _ivalue and b _ithe deterministic process of value be:

3. N number of sensor that in the unsynchronized wireless networks tested, location information is known-1, is supposed;

3.-2, by jth ' individual sensor to i-th ' individual sensor send measuring-signal, calculate jth ' the measuring-signal of individual sensor emission through propagating arrival i-th ' individual sensor forward after transfer process again turn back to jth ' individual sensor time time point and jth ' individual sensor initial transmissions measuring-signal time the mistiming 2t of time point _{j', i'}product with light velocity c, is designated as wherein, 1≤i'≤N, 1≤j'≤N, i' ≠ j';

-3 3. the positional information, according to N number of sensor, calculates the actual distance between jth ' individual sensor and i-th ' individual sensor, is designated as d _{j', i'}; Then calculate jth ' effective transmission distance in the measuring-signal of the individual sensor emission connecting time through propagating arrival i-th ' individual sensor after i-th ' individual sensor, be designated as d _{t, j', i'},

-4 3., from N number of sensor a selected sensor arbitrarily, suppose the sensor selected be i-th ' individual sensor, then make a _i=min (d _{t, j', i'}| 1≤j'≤N, j' ≠ i'), and make b _i=max (d _{t, j', i'}| 1≤j'≤N, j' ≠ i'), wherein, min () is for getting minimum value function, and max () is for getting max function.