CN117110984A - TOAs/FOAs closed type co-location method under time-frequency synchronization error condition of sensor at receiving and transmitting ends - Google Patents
TOAs/FOAs closed type co-location method under time-frequency synchronization error condition of sensor at receiving and transmitting ends Download PDFInfo
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Abstract
The invention discloses a TOAs/FOAs closed type cooperative positioning method under the condition of time-frequency synchronization errors of sensors at two ends of receiving and transmitting, and provides a TOAs/FOAs cooperative positioning mode for multiple radiation sources aiming at the positioning scene of the time-frequency synchronization errors of the sensors at two ends of receiving and transmitting. The method has higher calculation efficiency, can effectively inhibit the influence of time-frequency synchronization errors of the sensors at the two ends of the receiving and transmitting, and can obtain higher cooperative gain.
Description
Technical Field
The invention relates to a TOAs/FOAs co-positioning method for multiple radiation sources, in particular to a closed positioning method under the condition that time-frequency synchronization errors exist in sensors at two ends of receiving and transmitting, which has higher calculation efficiency, can effectively inhibit the influence of the time-frequency synchronization errors of the sensors at the two ends of receiving and transmitting, and can obtain higher co-gain by carrying out co-positioning on the multiple radiation sources.
Background
As is well known, radiation source positioning technology has been widely used in many industrial and information technology fields such as wireless communication, object monitoring, aerospace, security management, etc., and plays an increasingly critical role therein. The radiation source positioning technology generally extracts information layer parameters from wireless signals, and then uses the parameters to calculate the position/speed parameters of the radiation source. Dividing the radiation source positioning technology into two major categories according to the number of sensors, namely a single-sensor positioning technology and a multi-sensor positioning technology, wherein the latter can obtain more observables generally, which is beneficial to improving the positioning accuracy, and mainly researching a multi-sensor-based radiation source positioning method.
In the radiation source positioning system, the Time of Arrival (TOA-Time of Arrival) is a relatively common positioning observed quantity, and under the condition of known signal propagation speed, the TOA observed quantity can be converted into the distance observed quantity between the radiation source and the sensor, and higher positioning precision can be obtained. For a moving radiation source or a motion sensor, the observed quantity of the arrival frequency (FOA-Frequency of Arrival) and the observed quantity of the TOA can be combined and positioned, and the observed quantity of the FOA can be converted into the observed quantity of the change rate of the distance between the radiation source and the sensor. The combined TOA/FOA is used for positioning, so that not only can the speed parameter of the radiation source be estimated, but also the estimation accuracy of the position parameter of the radiation source can be improved. TOA/FOA parameters can be obtained by a maximum likelihood estimation method [ Wang, wu Siliang, tian Jing ], clamerlo world [ J ]. Electronic journal, 2011,39 (12): 2761-2767 ] ], and the TOA/FOA combined positioning method based on multiple motion sensors is studied mainly aiming at a motion radiation source.
In the TOA/FOA positioning system, besides TOA/FOA observation errors and sensor position/speed priori observation errors, time-frequency synchronization errors between the sensors at the two ends of the receiving and transmitting are also important factors influencing positioning accuracy. On the other hand, in an actual positioning scenario, there may be multiple radiation sources in the monitored area, where the multiple radiation sources should be treated as a whole for co-positioning to obtain a co-gain. In fact, when there is a model error (e.g. sensor position/velocity a priori observation error, etc.), even if there is co-locating multiple uncorrelated radiation sources, the positioning accuracy of each radiation source can be significantly improved, because the model error can cause a statistical correlation between observables corresponding to different radiation sources, which can provide a theoretical basis for obtaining a cooperative gain. The patent mainly researches a multi-radiation source co-location method under the condition that time-frequency synchronization errors exist in the sensors at the receiving and transmitting ends.
The existing TOA/FOA positioning method is mostly realized by parameter searching or iteration, wherein the parameter searching or iteration is carried out on [ Liu R, wang Y L, yin J X, wang D, wu Y.passive source localization using importance sampling based on TOA and FOA measurements [ J ]. Frontiers of Information Technology & Electronic Engineering,2017,18 (8): 1167-1179 ] [ Jia C G, yin J X, yang Z Y, zhang L.position and velocity estimation using TOA and FOA based on Lagrange programming neural network [ A ]. Proceedings of the 3rd International Conference on Data Mining,Communications and Information Technology[C ]. Beijing, china: IOP Publishing, may 2019:012664 ] ], which generally has higher computational complexity and can also have problems of local convergence, iterative divergence and the like.
Disclosure of Invention
Aiming at the positioning scene and the practical problem, the invention discloses a TOAs/FOAs closed type cooperative positioning method under the condition of time-frequency synchronization error of a sensor at the receiving and transmitting ends. The new method not only can effectively inhibit the influence of time-frequency synchronization errors between the sensors at the receiving and transmitting ends, but also can obtain more remarkable cooperative gain.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
step 1: TOAs observables obtained by M motion sensors for N motion radiation sourcesConversion to distance observance +.>FOAs observance amount->Conversion to observed distance change rate
Step 2: under the condition of time-frequency synchronization error, aiming at N motion radiation sources in sequence, obtaining a time domain pseudo-linear observation equation and a frequency domain pseudo-linear observation equation, and utilizing the position priori observables of M motion sensorsAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time domain pseudo-linear observation matrix>And frequency domain pseudo-linear observation matrix>Time domain pseudo-linear observation vectorAnd frequency domain pseudo-linear observation vector->
Step 3: under the condition of time-frequency synchronization error, constructing a time-frequency domain pseudo-linear observation matrix for N motion radiation sources in sequenceAnd time-frequency domain pseudo-linear observation vector->
Step 4: under the condition of time-frequency synchronization error, the time-frequency domain pseudo-linear observation matrix is utilized for N motion radiation sources in turnAnd time-frequency domain pseudo-linear observation vector->Obtaining asymptotic unbiased estimation value of the dimension expansion parameter
Step 5: under the condition of time-frequency synchronization error, the position priori observed quantity of M motion sensors is utilized for N motion radiation sources in sequenceAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a distance/distance change rate observation error disturbance matrix>And a disturbance matrix of a priori observed errors of the position/speed of the motion sensor
Step 6: under the condition of time-frequency synchronization error, aiming at N motion radiation sources, the method is respectively based on a time-frequency domain pseudo-linear observation matrixAnd time-frequency domain pseudo-linear observation vector->Constructing a time-frequency domain pseudo-linear observation matrix facing multi-radiation source co-location>And time-frequency domain pseudo-linear observation vector oriented to multi-radiation source co-location>
Step 7: time-frequency synchronization error existence barUnder the part, aiming at N moving radiation sources, the error disturbance matrix is observed based on the distance/distance change rate respectivelyAnd a disturbance matrix of a priori observed errors of the position/speed of the motion sensorConstructing a distance/distance change rate observation error disturbance matrix facing multi-radiation source co-location>And a motion sensor position/velocity a priori observed error disturbance matrix oriented to multi-radiation source co-location>And further calculate a weighting matrix
Step 8: based on the condition of time-frequency synchronization errorAnd->Calculating a weighted least squares estimate of the spread parameters of the multiple radiation sources>And its mean square error matrix>
Step 9: under the condition of time-frequency synchronization error, N motion radiation sources are sequentially utilizedObtaining weighted least squares of dimension-expanding parametersEstimate->And based on->Constructing an estimation error equation constraint equation observation matrix>And estimation error equation constraint equation observation vector +.>Thereby obtaining a weighted least squares estimation error { Δθ } n,WLS } 1≤n≤N Equation constraint equations;
step 10: in the presence of time-frequency synchronization error, for N moving radiation sources, respectively utilizingAndconstructing an estimated error equation constraint equation observation matrix facing multi-radiation source co-location>And an estimated error equation constraint equation observation vector for multi-radiation source co-localization>
Step 11: under the existence of time-frequency synchronization error, utilizing And->Constructing an optimization model containing estimation error equation constraint, and obtaining a closed solution of the estimation error by a Lagrange multiplier method>
Step 12: under the existence of time-frequency synchronization error, utilizingAnd->Calculating final estimated value of multi-radiation source dimension expansion parameter +.>And thus obtaining final estimates of the N moving radiation source positions +.>Final estimate of the velocity of N moving radiation sources +.>Final estimate of time domain synchronization error parameter +.>And final estimate of frequency domain synchronization error parameter +.>
Further, in the step 1, firstly, the M (1.ltoreq.m.ltoreq.m) th motion sensor is used for obtaining TOA observance quantity of the N (1.ltoreq.n) th motion radiation sourceConversion to distance observance +.>As shown below
Wherein c represents a signal propagation speed;
then, the M (M is less than or equal to 1) th motion sensor is aimed at the FOA observed quantity obtained by the N (N is less than or equal to 1) th motion radiation sourceConversion to a distance rate of change observation +.>As shown below
F in 0 Representing the signal carrier frequency.
Further, in the step 2, under the condition that the time-frequency synchronization error exists, a time domain pseudo-linear observation equation and a frequency domain pseudo-linear observation equation are obtained, and for the nth (1.ltoreq.n) moving radiation source, the position priori observed quantity of the M moving sensors is utilized firstAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time domain pseudo-linear observation matrix>And frequency domain pseudo-linear observation matrix>Respectively as follows
Then using the position priori observables of M motion sensorsAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time-domain pseudo-linear observation vector>And frequency domain pseudo-linear observation vector->Respectively as follows
Further, in the step 3, in the presence of a time-frequency synchronization error, for the nth (1N) moving radiation source, a time-frequency domain pseudo-linear observation matrix is first constructedAs shown below
Then constructing a time-frequency domain pseudo-linear observation vectorAs shown below
Further, in the step 4, in the presence of a time-frequency synchronization error, for the nth (1N) moving radiation source, a time-frequency domain pseudo-linear observation matrix is utilizedAnd time-frequency domain pseudo-linear observation vector->Obtaining the asymptotically unbiased estimated value of the dimension expansion parameter +.>As shown below
Further, in the step 5, in the presence of a time-frequency synchronization error, for the nth (1N) moving radiation source, distance observables of M motion sensors are first utilizedAnd distance rate of change observance +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a distance/distance change rate observation error disturbance matrix>As shown below
In the middle of
Wherein the method comprises the steps ofRepresenting the distance observation vector of the nth (1.ltoreq.n) moving radiation source;an observation vector of the distance change rate of the nth (1.ltoreq.n) moving radiation source is represented; />Representing an M x M order identity matrix I M M (1.ltoreq.m.ltoreq.m) th column vector in (1); o (O) m×n Representing an m×n order all-zero matrix;
then using the position priori observables of M motion sensorsAnd velocity a priori observables +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a disturbance matrix of position/speed priori observation errors of a motion sensorAs shown below
In the middle of
0 in 3 Representing 3 x 1 order all zero column vectors.
Further, in the step 6, in the presence of a time-frequency synchronization error, for the N moving radiation sources, a time-frequency domain pseudo-linear observation matrix for multi-radiation source co-positioning is first constructedAs shown below
Then constructing a time-frequency domain pseudo-linear observation vector oriented to multi-radiation source co-locationAs shown below
Further, in the step 7, in the presence of time-frequency synchronization errors, for N moving radiation sources, a distance/distance change rate observation error disturbance matrix facing the co-location of multiple radiation sources is first constructedAs shown below
Then constructing a motion sensor position/speed priori observation error disturbance matrix facing multi-radiation source co-locationAs shown below
Finally, calculating a weighting matrixAs shown below
In E (v) Representing a motion sensor position/speed prior observation error covariance matrix;representing a multi-source distance/distance rate of change observed error covariance matrix, which can be written as
Wherein the method comprises the steps ofRepresenting an observation error covariance matrix of the distance/distance change rate of the nth (N is more than or equal to 1 and less than or equal to N) moving radiation source.
Further, in the step 8, a weighted least square estimation value of the spread parameters of the multiple radiation sources is calculated in the presence of the time-frequency synchronization errorAs shown below
Then calculate the estimated valueMean square error matrix>As shown below
Further, in the step 9, a weighted least square estimated value of the dimension expansion parameter is obtained for the nth (1. Ltoreq.n) motion radiation source in the presence of the time-frequency synchronization errorAs shown below
Then utilizeConstructing an estimation error equation constraint equation observation matrix>As shown below
In the middle ofRepresentation vector->Column vectors composed of the 1 st to 3rd elements in (a); />Representation vector->Column vectors composed of the 4 th to 6 th elements in (a); />Representation vector->The 7 th element of (b);representation vector->The 8 th element of (b);
then use is made ofConstructing an estimation error equation constraint equation observation vector +.>As shown below
Finally obtainEstimate error of +.>Equation constraint equation->Wherein θ is n Representing the true value.
Further, in the step 10, in the presence of time-frequency synchronization errors, for N moving radiation sources, an estimation error equation constraint equation observation matrix for multi-radiation source co-localization is first constructedAs shown below
Then constructing an estimated error equation constraint equation observation vector for multi-radiation source co-locationAs shown below
Further, in the step 11, in the presence of the time-frequency synchronization error, an optimization model containing the constraint of the estimation error equation is first constructed, as shown in the following formula
Then solving the optimization model by Lagrangian multiplier method and obtaining a closed-form solution of the estimation errorAs shown below
Further, in step 12, in the presence of a time-frequency synchronization error, a final estimated value of the spread-spectrum parameters of the multiple radiation sources is calculatedAs shown below
Then, aiming at the nth (N is more than or equal to 1 and less than or equal to N) moving radiation source, the final estimated value of the position of the nth moving radiation source is obtainedAnd final estimate of speed +.>Respectively as follows
In the middle ofRepresentation vector->Column vectors composed of 10 (n-1) +1 to 10 (n-1) +3 elements; />Representation vector->Column vectors composed of 10 (n-1) +4 to 10 (n-1) +6 elements;
finally, aiming at the nth (N is more than or equal to 1 and less than or equal to N) moving radiation source, the final estimated value of the time domain synchronization error parameter is obtainedAnd final estimate of frequency domain synchronization error parameter +.>Respectively as follows
In the middle ofRepresentation vector->10 (n-1) +7 elements; />Representation vector->The 10 th (n-1) +8 th element in (a).
Compared with the prior art, the invention has the beneficial effects that:
aiming at the positioning scene that the time-frequency synchronization errors exist in the sensors at the two ends of the transceiver, the invention provides the TOAs/FOAs co-positioning method for the multiple radiation sources, which has higher calculation efficiency, can effectively inhibit the influence of the time-frequency synchronization errors of the sensors at the two ends of the transceiver, and can obtain higher co-gain.
Drawings
FIG. 1 is a schematic block diagram of a TOAs/FOAs closed co-location method with time-frequency synchronization errors of the sensors at the transceiver ends;
FIG. 2 is a plot of root mean square error of multiple source position estimates as a function of parameter r;
FIG. 3 is a plot of root mean square error of multiple source velocity estimates as a function of parameter r;
FIG. 4 is a graph of root mean square error versus standard deviation sigma for source 1 position estimation 2 Is a change curve of (2);
fig. 5 is a graph of root mean square error of the radiation source 1 velocity estimate with standard deviation sigma 2 Is a change curve of (2);
FIG. 6 is a graph of root mean square error of radiation source 2 position estimate with standard deviation sigma 2 Is a change curve of (2);
FIG. 7 is a graph of root mean square error of radiation source 2 velocity estimation with standard deviation sigma 2 Is a change curve of (2);
FIG. 8 is a graph of root mean square error versus standard deviation sigma for a radiation source 3 position estimate 2 Is a change curve of (2);
fig. 9 is a graph of root mean square error of radiation source 3 velocity estimation with standard deviation sigma 2 Is a change curve of (a).
Detailed Description
The invention is further illustrated by the following description of specific embodiments in conjunction with the accompanying drawings:
the invention discloses a TOAs/FOAs closed type co-location method under the condition of time-frequency synchronization errors of a transceiver sensor. Firstly, converting TOAs/FOAs observables obtained by a sensor into distance/distance change rate observables; then, under the condition that time-frequency synchronization errors exist, a time domain pseudo-linear observation equation and a frequency domain pseudo-linear observation equation are sequentially obtained for each motion radiation source, a time-frequency domain pseudo-linear observation matrix and a time-frequency domain pseudo-linear observation vector are obtained based on the time domain pseudo-linear observation equation and the frequency domain pseudo-linear observation equation, and then the time-frequency domain pseudo-linear observation matrix facing the multi-radiation source co-location and the time-frequency domain pseudo-linear observation vector facing the multi-radiation source co-location are obtained; then, a first-order error analysis method is utilized to sequentially obtain a distance/distance change rate observation error disturbance matrix and a motion sensor position/speed priori observation error disturbance matrix for each motion radiation source, and further obtain a multi-radiation source co-location-oriented distance/distance change rate observation error disturbance matrix and a multi-radiation source co-location-oriented motion sensor position/speed priori observation error disturbance matrix; then, calculating a weighting matrix oriented to multi-radiation source co-location, and obtaining a weighted least square estimated value of multi-radiation source dimension expansion parameters and a mean square error matrix thereof based on the weighting matrix; then, under the condition that time-frequency synchronous errors exist, a weighted least square estimation error equation constraint equation is sequentially obtained for each motion radiation source, an estimation error equation constraint equation observation matrix and an estimation error equation constraint equation observation vector are obtained based on the weighted least square estimation error equation constraint equation, and then an estimation error equation constraint equation observation matrix facing multi-radiation source co-location and an estimation error equation constraint equation observation vector facing multi-radiation source co-location are obtained; then, under the condition that the time-frequency synchronization error exists, constructing an optimization model containing the constraint of an estimation error equation, and obtaining a closed solution of the estimation error through a Lagrangian multiplier method; and finally, obtaining final estimated values of the multi-radiation source position, the multi-radiation source speed, the time domain synchronous error parameter and the frequency domain synchronous error parameter by using a weighted least square estimated value and a closed solution of the estimated error.
As shown in fig. 1, the method specifically includes:
step 1: TOAs observables obtained by M motion sensors for N motion radiation sourcesConversion to distance observance +.>FOAs observance amount->Conversion to observed distance change rate
Step 2: under the condition of time-frequency synchronization error, aiming at N motion radiation sources in sequence, obtaining a time domain pseudo-linear observation equation and a frequency domain pseudo-linear observation equation, and utilizing the position priori observables of M motion sensorsAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time domain pseudo-linear observation matrix>And frequency domain pseudo-linear observation matrix>Pseudo-linear observation vector in time>And frequency domain pseudo-linear observation vector->
Step 3: under the condition of time-frequency synchronization error, constructing a time-frequency domain pseudo-linear observation matrix for N motion radiation sources in sequenceAnd time-frequency domain pseudo-linear observation vector->
Step 4: under the condition of time-frequency synchronization error, the time-frequency domain pseudo-linear observation matrix is utilized for N motion radiation sources in turnAnd time-frequency domain pseudo-linear observation vector->Obtaining asymptotic unbiased estimation value of the dimension expansion parameter
Step 5: under the condition of time-frequency synchronization error, the position priori observed quantity of M motion sensors is utilized for N motion radiation sources in sequenceAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a distance/distance change rate observation error disturbance matrix>And a disturbance matrix of a priori observed errors of the position/speed of the motion sensor
Step 6: in the presence of time-frequency synchronization error, for N moving radiation sources, respectively utilizingAndconstructing a time-frequency domain pseudo-linear observation matrix facing multi-radiation source co-location>And time-frequency domain pseudo-linear observation vector oriented to multi-radiation source co-location>
Step 7: in the presence of time-frequency synchronization errors, the method is respectively based on N motion radiation sourcesAndconstructing a distance/distance change rate observation error disturbance matrix facing multi-radiation source co-location>And a motion sensor position/velocity a priori observed error disturbance matrix oriented to multi-radiation source co-location>And further calculate a weighting matrix
Step 8: based on the condition of time-frequency synchronization errorAnd->Calculating a weighted least squares estimate of the spread parameters of the multiple radiation sources>And its mean square error matrix>
Step 9: under the condition of time-frequency synchronization error, N motion radiation sources are sequentially utilizedObtaining a weighted least squares estimate of the dimension-expanding parameter>And based on->Constructing an estimation error equation constraint equation observation matrix>And estimation error equation constraint equation observation vector +.>Thereby obtaining a weighted least squares estimation error { Δθ } n,WLS } 1≤n≤N Equations constraint equations.
Step 10: in the presence of time-frequency synchronization error, for N moving radiation sources, respectively utilizingAndconstructing an estimated error equation constraint equation observation matrix facing multi-radiation source co-location>And an estimated error equation constraint equation observation vector for multi-radiation source co-localization>
Step 11: under the existence of time-frequency synchronization error, utilizing And->Constructing an optimization model containing estimation error equation constraint, and obtaining a closed solution of the estimation error by a Lagrange multiplier method>
Step 12: under the existence of time-frequency synchronization error, utilizingAnd->Calculating final estimated value of multi-radiation source dimension expansion parameter +.>And thus obtaining final estimates of the N moving radiation source positions +.>Final estimate of the velocity of N moving radiation sources +.>Time domain synchronization error parametersFinal estimate +.>And final estimate of frequency domain synchronization error parameter +.>
Further, in the step 1, firstly, the M (1.ltoreq.m.ltoreq.m) th motion sensor is used for obtaining TOA observance quantity of the N (1.ltoreq.n) th motion radiation sourceConversion to distance observance +.>As shown below
Where c represents the signal propagation velocity. Then, the M (M is less than or equal to 1) th motion sensor is aimed at the FOA observed quantity obtained by the N (N is less than or equal to 1) th motion radiation sourceConversion to a distance rate of change observation +.>Shown by the following formula->
F in 0 Representing the signal carrier frequency.
Further, in the step 2, under the condition that the time-frequency synchronization error exists, a time domain pseudo-linear observation equation and a frequency domain pseudo-linear observation equation are obtained, and for the nth (1.ltoreq.n) moving radiation source, the position priori observed quantity of the M moving sensors is utilized firstAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time domain pseudo-linear observation matrix>And frequency domain pseudo-linear observation matrix>Respectively as follows
Then using the position priori observables of M motion sensorsAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time-domain pseudo-linear observation vector>Sum frequencyDomain pseudo-linear observation vector->Respectively as follows
Further, in the step 3, in the presence of a time-frequency synchronization error, for the nth (1N) moving radiation source, a time-frequency domain pseudo-linear observation matrix is first constructedAs shown below
Then constructing a time-frequency domain pseudo-linear observation vectorAs shown below
Further, in the step 4, in the presence of a time-frequency synchronization error, for the nth (1N) moving radiation source, a time-frequency domain pseudo-linear observation matrix is utilizedAnd time-frequency domain pseudo-linear observation vector->Obtaining the asymptotically unbiased estimated value of the dimension expansion parameter +.>As shown below
Further, in the step 5, in the presence of a time-frequency synchronization error, for the nth (1N) moving radiation source, distance observables of M motion sensors are first utilizedAnd distance rate of change observance +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a distance/distance change rate observation error disturbance matrix>As shown below
In the middle of
Wherein the method comprises the steps ofRepresenting the distance observation vector of the nth (1.ltoreq.n) moving radiation source;representing the nth (1. Ltoreq.n) moving radiation sourceA distance change rate observation vector; />Representing an M x M order identity matrix I M M (1.ltoreq.m.ltoreq.m) th column vector in (1); o (O) m×n Representing an mxn order all zero matrix.
Then using the position priori observables of M motion sensorsAnd velocity a priori observables +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a disturbance matrix of position/speed priori observation errors of a motion sensorAs shown below
In the middle of
0 in 3 Representing 3 x 1 order all zero column vectors.
Further, in the step 6, there is a time-frequency synchronization errorUnder the condition, aiming at N moving radiation sources, firstly constructing a time-frequency domain pseudo-linear observation matrix oriented to multi-radiation source co-positioningAs shown below
Then constructing a time-frequency domain pseudo-linear observation vector oriented to multi-radiation source co-locationShown by the following formula->
Further, in the step 7, in the presence of time-frequency synchronization errors, for N moving radiation sources, a distance/distance change rate observation error disturbance matrix facing the co-location of multiple radiation sources is first constructedAs shown below
Then constructing a motion sensor position/speed priori observation error disturbance matrix facing multi-radiation source co-locationAs shown below
Finally, calculating a weighting matrixAs shown below
In E (v) Representing a motion sensor position/speed prior observation error covariance matrix;representing a multi-source distance/distance rate of change observed error covariance matrix, which can be written as
Wherein the method comprises the steps ofRepresenting an observation error covariance matrix of the distance/distance change rate of the nth (N is more than or equal to 1 and less than or equal to N) moving radiation source.
Further, in the step 8, a weighted least square estimation value of the spread parameters of the multiple radiation sources is calculated in the presence of the time-frequency synchronization errorAs shown below
Then calculate the estimated valueMean square error matrix>As shown below
Further, in the step 9, a weighted least square estimated value of the dimension expansion parameter is obtained for the nth (1. Ltoreq.n) motion radiation source in the presence of the time-frequency synchronization errorShown by the following formula->
Then utilizeConstructing an estimation error equation constraint equation observation matrix>As shown below
In the middle ofRepresentation vector->Column vectors composed of the 1 st to 3rd elements in (a); />Representation vector->Column vectors composed of the 4 th to 6 th elements in (a); />Representation vector->The 7 th element of (b);representation vector->The 8 th element of (b).
Then use is made ofConstructing an estimation error equation constraint equation observation vector +.>As shown below
Finally obtainEstimate error of +.>Equation constraint equation->Wherein θ is n Representing the true value.
Further, in the step 10, in the presence of time-frequency synchronization errors, for N moving radiation sources, an estimation error equation constraint equation observation matrix for multi-radiation source co-localization is first constructedAs shown below
Then constructing an estimated error equation constraint equation observation vector for multi-radiation source co-locationAs shown below
Further, in the step 11, in the presence of the time-frequency synchronization error, an optimization model containing the constraint of the estimation error equation is first constructed, as shown in the following formula
Then solving the optimization model by Lagrangian multiplier method and obtaining a closed-form solution of the estimation errorAs shown below
/>
Further, in step 12, in the presence of a time-frequency synchronization error, a final estimated value of the spread-spectrum parameters of the multiple radiation sources is calculatedAs shown below
Then, aiming at the nth (N is more than or equal to 1 and less than or equal to N) moving radiation source, the final estimated value of the position of the nth moving radiation source is obtainedAnd final estimate of speed +.>Respectively as follows
In the middle ofRepresentation vector->Column vectors composed of 10 (n-1) +1 to 10 (n-1) +3 elements; />Representation vector->Column vectors composed of 10 (n-1) +4 to 10 (n-1) +6 elements.
Finally, aiming at the nth (N is more than or equal to 1 and less than or equal to N) moving radiation source, the final estimated value of the time domain synchronization error parameter is obtainedAnd final estimate of frequency domain synchronization error parameter +.>Respectively as follows
In the middle ofRepresentation ofVector->10 (n-1) +7 elements; />Representation vector->The 10 th (n-1) +8 th element in (a).
To verify the effect of the present invention, the following specific examples are performed:
the TOAs/FOAs information (namely distance/distance change rate information) is obtained by 11 motion sensors to position a plurality of uncorrelated motion radiation sources, the position/speed values of the sensors in a 3-dimensional space are shown in a table 1, and the N (1 is less than or equal to N) th motion radiation source distance/distance change rate observation error covariance matrix isIs a motion sensor position/velocity a priori observed error covariance matrix +.>Wherein sigma is 1 Sum sigma 2 All are corresponding standard deviations. There are 3 uncorrelated moving radiation sources to be positioned, where the position vector and the velocity vector of the moving radiation source 1 are u respectively 1 ＝[-170 80 140] T (m) and->The position vector and the velocity vector of the moving radiation source 2 are u respectively 2 ＝[-150 -130 -80] T (m) and->The position vector and the velocity vector of the moving radiation source 3 are u respectively 3 ＝[100 -40 -120] T (m) andsetting all time domain synchronization error parameters of 3 motion radiation sources as beta 1 ＝β 2 ＝β 3 Frequency domain synchronization error parameters are set to +.0.4r (m)>Wherein the parameter r varies from 0 to 20, the standard deviation is fixed to σ 1 =2.5 (m) and σ 2 =0.5 (m). Fig. 2 and 3 show the variation of the root mean square error of the multi-source position and multi-source velocity estimates with the parameter r, respectively.
Table 1 sensor position coordinates and velocity values (units: m and m/s) in 3-dimensional space
| Sensor serial number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| X-axis coordinates | 400 | -600 | 300 | 600 | 600 | -400 | -600 | -400 | 900 | -700 | 800 |
| Y-axis coordinates | 500 | 400 | -400 | 300 | -400 | 500 | -500 | -300 | 600 | -900 | -600 |
| Z-axis coordinates | 600 | 300 | 600 | -400 | -500 | -300 | 400 | -500 | 800 | 800 | 900 |
| Speed of X-axis | 6 | -11 | 12 | 8 | 12 | -11 | -9 | -8 | -8 | -10 | 8 |
| Y-axis speed | 8 | 8 | -8 | 8 | -12 | 12 | -8 | -9 | 9 | -7 | 6 |
| Z-axis velocity | 11 | 7 | 9 | -12 | -8 | -8 | 9 | -8 | -12 | 12 | -10 |
As can be seen from fig. 2 and 3: (1) The estimation accuracy of the positioning method without considering the time-frequency synchronization error gradually increases along with the increase of the parameter r; (2) The estimation accuracy of the positioning method disclosed by the patent can be always approximate to the corresponding CRB and cannot be changed along with the change of the parameter r, so that the positioning method disclosed by the patent is verified to have asymptotic statistical optimality; (3) When the parameter r is small, the magnitude of the time-frequency synchronization error is also small, the time-frequency synchronization error does not have a substantial influence on the positioning method, and the estimation accuracy of the positioning method without considering the time-frequency synchronization error is slightly better than that of the positioning method disclosed by the patent, because the positioning method disclosed by the patent considers the time-frequency synchronization error parameter as an unknown quantity, the degree of freedom of the whole observation system is reduced compared with the time-frequency synchronization error; (4) When the parameter r exceeds a certain threshold, the advantages of the positioning method disclosed by the patent are gradually revealed, and the performance difference of the parameter r and the positioning method is gradually increased along with the increase of the parameter r, so that the influence of time-frequency synchronization errors can be effectively restrained by the positioning method disclosed by the patent.
The position vectors of the 3 moving radiation sources are set as follows:
u 1 ＝1000[cos(α)cos(γ)sin(α)cos(γ)sin(γ)] T (m)，
u 2 ＝2000[cos(α)cos(γ)sin(α)cos(γ)sin(γ)] T (m)，
u 3 ＝3000[cos(α)cos(γ)sin(α)cos(γ)sin(γ)] T (m)；
Wherein the parameters alpha=110° and gamma=40°, the velocity vectors of the 3 moving radiation sources are the same as above, and the time domain synchronization error parameters of the 3 moving radiation sources are respectively set as beta 1 ＝220(m)、β 2 ＝150(m) Beta 3 = -180 (m), frequency domain synchronization error parameters of 3 moving radiation sources are respectively set as And +.>Standard deviation sigma 1 Fixed as sigma 1 =0.5 (m), standard deviation σ 2 Ranging from 0.2 to 4. Figures 4 and 5 show the root mean square error of the position of the source 1 and the velocity estimate of the source 1, respectively, with the standard deviation sigma 2 Is a change curve of (2); figures 6 and 7 show the position of the source 2 and the estimated root mean square error of the source 2 velocity with standard deviation sigma, respectively 2 Is a change curve of (2); figures 8 and 9 show the position of the source 3 and the estimated root mean square error of the source 3 velocity with standard deviation sigma, respectively 2 Is a change curve of (a).
As can be seen from fig. 4 to 9: (1) Compared with a non-cooperative positioning method, the positioning method disclosed by the patent can obtain obvious cooperative gain, and the standard deviation sigma is followed 2 The gain of the obtained synergy is larger and larger; (2) The estimation accuracy of the positioning method disclosed by the patent can be always close to the corresponding CRB, so that the asymptotic statistical optimality of the positioning method is verified again.
The foregoing is merely illustrative of the preferred embodiments of this invention, and it will be appreciated by those skilled in the art that changes and modifications may be made without departing from the principles of this invention, and it is intended to cover such modifications and changes as fall within the true scope of the invention.
Claims (10)
1. The TOAs/FOAs closed type cooperative positioning method under the condition that the sensors at the receiving and transmitting ends have time-frequency synchronization errors is characterized by comprising the following steps:
step 1: multiple time-of-arrival observations obtained for M motion sensors for N motion radiation sourcesConversion to distance observance +.>Multiple arrival frequency observables +.>Conversion to a distance rate of change observation +.>
Step 2: under the condition of time-frequency synchronization error, aiming at N motion radiation sources in sequence, obtaining a time domain pseudo-linear observation equation and a frequency domain pseudo-linear observation equation, and utilizing the position priori observables of M motion sensorsAnd velocity a priori observables +.>And +.>And->Constructing a time domain pseudo-linear observation matrix>And frequency domain pseudo-linear observation matrix>Pseudo-linear observation vector in time>And frequency domain pseudo-linear observation vector
Step 3: under the condition of time-frequency synchronization error, N motion radiation sources are respectively utilized in sequenceAndand->Constructing a time-frequency domain pseudo-linear observation matrix>And time-frequency domain pseudo-linear observation vector->
Step 4: under the condition of time-frequency synchronization error, N motion radiation sources are sequentially utilizedAnd->Obtaining the asymptotically unbiased estimated value of the dimension expansion parameter +.>
Step 5: under the condition of time-frequency synchronization error, N motion radiation sources are sequentially utilizedAndand->And +.>Constructing a distance/distance change rate observation error disturbance matrix>And a disturbance matrix of a priori observed errors of the position/speed of the motion sensor +.>
Step 6: in the presence of time-frequency synchronization error, for N moving radiation sources, respectively utilizingAnd->Constructing a time-frequency domain pseudo-linear observation matrix facing multi-radiation source co-location>And time-frequency domain pseudo-linear observation vector oriented to multi-radiation source co-location>
Step 7: in the presence of time-frequency synchronization errors, the method is respectively based on N motion radiation sourcesAndconstructing a distance/distance change rate observation error disturbance matrix facing multi-radiation source co-location>And a motion sensor position/velocity a priori observed error disturbance matrix oriented to multi-radiation source co-location>And then calculate the weighting matrix
Step 8: based on the condition of time-frequency synchronization errorAnd->Calculating a weighted least squares estimate of the spread parameters of the multiple radiation sources>And its mean square error matrix>
Step 9: under the condition of time-frequency synchronization error, N motion radiation sources are sequentially utilizedObtaining a weighted least squares estimate of the dimension-expanding parameter>And based on->Constructing an estimation error equation constraint equation observation matrix>And estimation error equation constraint equation observation vector +.>Thereby obtaining a weighted least squares estimation error { Δθ } n,WLS } 1≤n≤N Equation constraint equations;
step 10: in the presence of time-frequency synchronization error, for N moving radiation sources, respectively utilizingAndconstructing an estimated error equation constraint equation observation matrix facing multi-radiation source co-location>And an estimated error equation constraint equation observation vector for multi-radiation source co-localization>
Step 11: under the existence of time-frequency synchronization error, utilizingAnd->Constructing an optimization model containing estimation error equation constraint, and obtaining a closed solution of the estimation error by a Lagrange multiplier method>
Step 12: under the existence of time-frequency synchronization error, utilizingAnd->Calculating final estimated value of multi-radiation source dimension expansion parameter +.>And thus obtaining final estimates of the N moving radiation source positions +.>Final estimate of the velocity of N moving radiation sources +.>Final estimate of time domain synchronization error parameter +.>And final estimate of frequency domain synchronization error parameter +.>
2. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization errors of two sensors according to claim 1, wherein in said step 1, the arrival time observed quantity obtained by the mth motion sensor for the nth motion radiation source is first obtainedConversion to distance observance +.>As shown below
Wherein c represents a signal propagation speed;
then the m-th motion sensor obtains the observed quantity of the arrival frequency for the n-th motion radiation sourceConversion to a distance rate of change observation +.>As shown below
F in 0 Representing the signal carrier frequency.
3. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization errors of two sensors according to claim 2, wherein in said step 2, under the presence of time-frequency synchronization errors, a time-domain pseudo-linear observation equation and a frequency-domain pseudo-linear observation equation are obtained, and for an nth motion radiation source, first, the position priori observables of M motion sensors are utilizedAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time domain pseudo-linear observation matrix>And frequency domain pseudo-linear observation matrix>Respectively as follows +.>
Then using the position priori observables of M motion sensorsAnd velocity a priori observables +.>Distance observance +.>And distance rate of change observance +.>Constructing a time-domain pseudo-linear observation vector>And frequency domain pseudo-linear observation vector->Respectively as follows
4. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization error of a sensor at both ends of a receiver and transmitter according to claim 1, wherein in said step 3, in the presence of time-frequency synchronization error, for an nth moving radiation source, a time-frequency domain pseudo-linear observation matrix is first constructedAs shown below
Then constructing a time-frequency domain pseudo-linear observation vectorAs shown below
5. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization error of a sensor at both ends of a receiver/transmitter according to claim 1, wherein in said step 4, in the presence of time-frequency synchronization error, for an nth moving radiation source, a time-frequency domain pseudo-linear observation matrix is utilizedAnd time-frequency domain pseudo-linear observation vector->Obtaining the asymptotically unbiased estimated value of the dimension expansion parameter +.>As shown below
6. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization errors of two sensors according to claim 1, wherein in step 5, in the presence of time-frequency synchronization errors, for the nth motion radiation source, distance observables of M motion sensors are first utilizedAnd distance rate of change observance +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a distance/distance change rate observation error disturbance matrix>As shown below
In the middle of
Wherein the method comprises the steps ofA distance observation vector representing an nth moving radiation source;an observation vector representing a range rate of the nth motion radiation source; />Representing an M x M order identity matrix I M An mth column vector of (a); o (O) m×n Representing an m×n order all-zero matrix;
then using the position priori observables of M motion sensorsAnd velocity a priori observables +.>And an asymptotically unbiased estimate of the dimension-expanding parameter +.>Constructing a disturbance matrix of a priori observing error of the position/speed of the motion sensor +.>As shown below
In the middle of
0 in 3 Representing 3 x 1 order all zero column vectors.
7. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization error of a sensor at both ends of a transceiver according to claim 1, wherein in step 6, for N moving radiation sources, a time-frequency domain pseudo-linear observation matrix for co-location of multiple radiation sources is constructed firstAs shown below
Wherein O is 2M×10 Representing a 2M x 10 order all zero matrix;
then constructing a time-frequency domain pseudo-linear observation vector oriented to multi-radiation source co-locationAs shown below
8. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization errors of a transceiver sensor according to claim 1, wherein in said step 7, N movements are performed in the presence of time-frequency synchronization errorsThe radiation source firstly constructs a distance/distance change rate observation error disturbance matrix facing to the multi-radiation source co-locationAs shown below
Wherein O is 2M×2M Representing a 2M x 2M order all-zero matrix;
then constructing a motion sensor position/speed priori observation error disturbance matrix facing multi-radiation source co-locationAs shown below
Finally, calculating a weighting matrixAs shown below
In E (v) Representing a motion sensor position/speed prior observation error covariance matrix;representing a multi-source distance/distance rate of change observed error covariance matrix, which can be written as
Wherein the method comprises the steps ofRepresenting an nth moving source distance/distance rate of change observed error covariance matrix.
9. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization error of a sensor at both ends of a receiver/transmitter according to claim 1, wherein in said step 8, a weighted least squares estimation of spread parameters of multiple radiation sources is calculated in the presence of time-frequency synchronization errorAs shown below
Then calculate the estimated valueMean square error matrix>As shown below
10. The method for closed co-location of TOAs/FOAs in the presence of time-frequency synchronization error of a sensor at both ends of a receiver/transmitter according to claim 1, wherein in said step 9, in the presence of time-frequency synchronization error, a weighted least square estimation of a dimension-expanding parameter is obtained for an nth moving radiation sourceAs shown below
Wherein the method comprises the steps ofRepresenting an N-th order identity matrix I N N-th column vector of (a), I 10 Representing a 10×10 order identity matrix;
then utilizeConstructing an estimation error equation constraint equation observation matrix>As shown below
In the middle ofRepresentation vector->Column vectors composed of the 1 st to 3rd elements in (a); />Representation vector->Column vectors composed of the 4 th to 6 th elements in (a); />Representation vector->The 7 th element of (b);representation vector->The 8 th element of (b);
then use is made ofConstructing an estimation error equation constraint equation observation vector +.>As shown below
Finally obtainEstimate error of +.>Equation constraint equation->Wherein θ is n Representing the true value;
preferably, in the step 10, in the presence of time-frequency synchronization errors, for N moving radiation sources, an estimation error equation constraint equation observation matrix for multi-radiation source co-location is first constructedAs shown below
Then constructing an estimated error equation constraint equation observation vector for multi-radiation source co-locationAs shown below
Preferably, in the step 11, in the presence of the time-frequency synchronization error, an optimization model containing the constraint of the estimation error equation is first constructed, as shown in the following formula
Then solving the optimization model by Lagrangian multiplier method and obtaining a closed-form solution of the estimation errorAs shown below
Preferably, in step 12, in the presence of a time-frequency synchronization error, a final estimated value of the multi-radiation-source dimension-expanding parameter is calculated firstAs shown below
Then for the nth moving radiation source, a final estimated value of the position of the nth moving radiation source is obtainedAnd final estimate of speed +.>Respectively as follows
In the middle ofRepresentation vector->Column vectors composed of 10 (n-1) +1 to 10 (n-1) +3 elements; />Representation vector->Column vectors composed of 10 (n-1) +4 to 10 (n-1) +6 elements;
finally, aiming at the nth motion radiation source, obtaining the final estimated value of the time domain synchronization error parameterAnd final estimate of frequency domain synchronization error parameter +.>Respectively as follows
In the middle ofRepresentation vector->10 (n-1) +7 elements; />Representation vector->The 10 th (n-1) +8 th element in (a).
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