CN109991570B - Moving target direct positioning method based on quasi-Newton method - Google Patents

Moving target direct positioning method based on quasi-Newton method Download PDF

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CN109991570B
CN109991570B CN201910273392.1A CN201910273392A CN109991570B CN 109991570 B CN109991570 B CN 109991570B CN 201910273392 A CN201910273392 A CN 201910273392A CN 109991570 B CN109991570 B CN 109991570B
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薛文丽
张花国
范曦丹
高岚
廖红舒
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University of Electronic Science and Technology of China
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S5/00—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations
    • G01S5/02—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using radio waves
    • G01S5/04—Position of source determined by a plurality of spaced direction-finders
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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Abstract

The invention belongs to the technical field of direct positioning in passive positioning, and relates to a moving target direct positioning method based on a quasi-Newton method. The method mainly uses a quasi-Newton method to simplify the grid points of the possible positions of the search target; by solving the gradient of the objective function, the unknown quantity is iterated by using a quasi-Newton method, so that the search process is simplified, and the quick positioning is realized.

Description

Moving target direct positioning method based on quasi-Newton method
Technical Field
The invention belongs to the technical field of direct positioning in a passive positioning technology, and relates to a moving target direct positioning method based on a quasi-Newton method.
Background
Passive localization is a way to track and locate objects using existing radio frequency signals. Passive positioning is generally divided into passive positioning based on self radiation and passive positioning based on an external radiation source, wherein the passive positioning is performed by receiving a signal radiated by a target, and the passive positioning is performed by irradiating the target through a non-cooperative external radiation source, receiving an echo signal of the target and positioning the target. The direct positioning technology in passive positioning does not need intermediate parameter estimation, directly utilizes signal data domain information to position a target, can effectively utilize the correlation among observation stations, improves the positioning accuracy, and is one of the hot spots of domestic and foreign research at present. The existing direct positioning technology mostly adopts a method based on grid search, and the calculation amount is huge. Therefore, finding a direct positioning method for rapidly solving the target position becomes an important development direction of the direct positioning technology.
Disclosure of Invention
The invention aims to solve the problem of huge calculation amount caused by estimating the initial position and the speed of a target by grid search, researches a moving target positioning technology based on a quasi-Newton method, and reduces the calculation amount.
For ease of understanding, the related art adopted by the present invention will be described first:
the basic idea of the quasi-Newton method is to use a Hesse matrix on the basis of the basic Newton method
Figure BDA0002019136650000011
Of a certain approximation matrix B k Substituted G k . In general, B k The following three characteristics should be provided:
(one) has B in a certain sense k ≈G k The direction generated by the corresponding algorithm is made to approximate to a Newton direction, so as to ensure that the algorithm has a faster convergence speed.
(II) for all k, B k Is symmetrically positive such that the direction generated by the algorithm is the function f at x k In the descending direction.
(III) matrix B k The update rule is relatively simple, i.e. usually corrected using a rank 1 or rank 2 matrix.
Common quasi-Newton algorithms include a BFGS algorithm, a DFP algorithm, a Broyden algorithm and the like, and the quasi-Newton algorithm is mainly based on the BFGS.
The correction formula of the BFGS algorithm is as follows:
Figure BDA0002019136650000021
if the exact search or Wolfe search criteria are adopted in the BFGS algorithm, then
Figure BDA0002019136650000022
Can ensure B k And the symmetry is positive.
The Armijo search criteria are generally not guaranteed
Figure BDA0002019136650000023
To guarantee the matrix sequence B when the Armijo criterion is adopted k The symmetry positive characterization can adopt the following correction mode:
Figure BDA0002019136650000024
wherein: s is k ＝x k+1 -x k ，y k ＝g k+1 -g k ，
Figure BDA0002019136650000025
The technical scheme of the invention is as follows:
the positioning scene of the invention is that the existing uniform motion target to be positioned has the speed v and the initial position p 0 . The radio signals radiated by the target can be intercepted by the L observation stations at the same time. Assuming that each observation station samples a target radiation signal in K time slot sections, and the time interval between two adjacent time slot sections is T d And the target position of the first slot segment is set as its initial position. To achieve direct localization of moving objects, the following assumptions are made: within each sample slot segment (usually short in time), the instantaneous position and velocity of the target remain unchanged, so the position vector of the target in the kth slot segment is: p is a radical of formula k ＝p 0 +v*(k-1)*T d . And the location vector for the ith observation station is noted as: ql.
A moving object direct positioning technology based on quasi-Newton comprises the following steps:
s1, given parameter delta epsilon (0,1), sigma epsilon (0,0.5) and initial point x 0 ＝(p',v')∈R n (p ', v' is any initial value on the real number set, R n A set of real numbers), the number of iterations M iter Initial symmetric positive definite matrix B 0 (usually taken as a unit matrix), let k =0.
Wherein f (p) 0 And v) is a cost function,
Figure BDA0002019136650000026
Figure BDA0002019136650000027
Figure BDA0002019136650000028
in the formula, r l,k ＝[r l,k (t 1 ),…,r l,k (t N )] T For the received signal of each observation station, r l,k ＝b l,k A l,k F l,k s k +n l,k ，l＝1,2,…,L。
Figure BDA0002019136650000029
f l,k Represents the frequency of the down-converted signal intercepted by the ith observation station during the kth observation time slot, which can be modeled as: f. of l,k ＝f c μ l,k Wherein f is c Is the carrier frequency and is,
Figure BDA0002019136650000031
s k (t) represents the complex envelope of the signal in the k-th slot segment, sampled as s k ＝[s k (t 1 ),…,s k (t N )] T (N is the number of samples of the signal in each time slot). F l,k Is a time shift matrix, i.e.: f l,k s k Denotes a general formula s k Displacement of
Figure BDA0002019136650000032
Figure BDA0002019136650000033
The time delay of the target radiation signal at the k time slot relative to the l observation station is obtained.
S2, calculating
Figure BDA0002019136650000034
(
Figure BDA0002019136650000035
To solve the gradient for the objective function).
Figure BDA0002019136650000036
(x, y are horizontal and vertical coordinate components of the target initial position, v x ，v y Respectively, horizontal and vertical coordinate components of ocular velocity)
Each of which is:
Figure BDA0002019136650000037
Figure BDA0002019136650000038
Figure BDA0002019136650000039
Figure BDA00020191366500000310
in the formula (I), the compound is shown in the specification,
Figure BDA00020191366500000311
Figure BDA00020191366500000312
Figure BDA00020191366500000313
Figure BDA00020191366500000314
Figure BDA00020191366500000315
Figure BDA0002019136650000041
Figure BDA0002019136650000042
Figure BDA0002019136650000043
wherein s is k ＝[s k (t 1 ),…,s k (t N )] T ，
Figure BDA0002019136650000044
N =1,2, … N is s (t) n -τ l,k ) Of the gradient of (c).
Figure BDA0002019136650000045
Figure BDA0002019136650000046
d l,k ＝||p k -ql | (| |. | | is taken as a two-norm)
S3, solving a linear equation set B k d＝-g k To obtain solution d k
S4, setting m k Is the smallest non-negative integer m that satisfies the following inequality:
Figure BDA0002019136650000047
wherein delta E (0,1), sigma E (0,0.5), let
Figure BDA0002019136650000048
x k+1 ＝x k +α k d k
S5, determining B by a correction formula k+1
S6, let k = k +1, if k>M iter Then x is output k As approximate extreme points, i.e. positioning results; otherwise, go to S2.
The method of the invention was verified and compared as follows:
and calculating a Cramer-Rao bound of the target direct localization variance, wherein the Cramer-Rao bound is the inverse of Fisher information quantity.
The cramer-Rao Bound-CRB of the parameter estimation variance gives a lower Bound for any unbiased estimation variance, and a closed form expression for the cramer Bound is derived below. The Cramer-Rao kingdom is the inverse of the Fisher information content. For complex gaussian data, the unknown parameter is in its mean, not the variance. The amount of Fisher information is given by:
Figure BDA0002019136650000051
Figure BDA0002019136650000052
Figure BDA0002019136650000053
Figure BDA0002019136650000054
[J] 2,1 ＝[J] 1,2
Figure BDA0002019136650000055
Figure BDA0002019136650000056
Figure BDA0002019136650000057
[J] 3,1 ＝[J] 1,3
[J] 3,2 ＝[J] 2,3
Figure BDA0002019136650000058
Figure BDA0002019136650000059
[J] 4,1 ＝[J] 1,4
[J] 4,2 ＝[J] 2,4
[J] 4,3 ＝[J] 3,4
Figure BDA00020191366500000510
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00020191366500000511
respectively representing the derivatives of the Doppler frequency shift and the time shift to the target position, and the rest of the same principles;
Figure BDA00020191366500000512
Figure BDA00020191366500000513
Figure BDA00020191366500000514
comparison of algorithm complexity
Here the computational complexity of the quasi-newton based direct positioning method and the grid search based direct positioning method are compared (under the same simulation platform). To give the computational complexity of both methods, some of the following parameters need to be defined first:
(a) The number of iterations of the quasi-Newton iteration is recorded as M iter ；
(b) In the grid search, the search interval of the target position and speed in each dimension is respectively recorded as
Figure BDA0002019136650000061
And
Figure BDA0002019136650000062
(c) In the grid search, the search step length of the target position and the speed in each dimension is respectively recorded as
Figure BDA0002019136650000063
And
Figure BDA0002019136650000064
based on the above symbolic definitions, table 1 and table 2 give the computational complexity (both in complex multiplication) of the two positioning methods, respectively.
TABLE 1 computational complexity of grid search based direct positioning method
Figure BDA0002019136650000065
TABLE 2 computational complexity of direct positioning method based on quasi-Newton method
Figure BDA0002019136650000066
In the present invention, moving object localization in a two-dimensional plane is considered. In the grid search, four-dimensional search is required, and taking the example that only 10 grids are searched for in each dimension, calculation needs to be performed in 10000 grids. The direct positioning method based on quasi-Newton only needs to iterate for 60 times at most, search is carried out from three different initial points, the target position finally obtained by comparing the three initial points is optimal, only 180 grid points need to be calculated in total, and the calculation amount is far smaller than that of the direct positioning method based on grid search.
Drawings
FIG. 1 is a geometric position distribution diagram of an observation station and a target;
FIG. 2 is a plot of root mean square error of initial position estimate as a function of signal-to-noise ratio;
figure 3 is a plot of the root mean square error of the velocity estimate against the signal-to-noise ratio.
Detailed Description
The invention is described in detail below with reference to the figures and examples
Examples
Since the whole moving track of the target can be directly determined by the initial position and speed of the target, the estimation accuracy of the initial position and speed of the target determines the estimation accuracy of the moving track of the target. Here, the estimation performance of the unknown quantity is compared, and a grid search-based direct positioning method and a quasi-newton-based direct positioning method are compared.
Assume an initial position vector of the target radiation source is p 0 ＝[4000,4000] T Velocity vector v = [200,20]The signal radiated by the target is a pulse train signal, the carrier frequency of the pulse train signal is 1GHz, the type of the pulse train signal can be intercepted by three existing stationary observation stations, and the signal is sampled every second in a time slot section (total 10 time slots), and the position of the observation station and the motion track of the target are shown in fig. 1.
FIGS. 2 and 3 show the simulation results of the present example, where the number of sample points in each time slot segment is fixed to 512, the number of quasi-Newton iterations is 50, and the number of initial points is 3. FIG. 2 shows the variation curve of the root mean square error of the target initial position estimation with the SNR; figure 3 shows the root mean square error of the target velocity estimate as a function of the signal-to-noise ratio.
As can be seen from fig. 2 and fig. 3, the positioning performance of the quasi-newton based direct positioning method provided by the present invention is equivalent to that of the grid search based direct positioning method, and the performance curves of the quasi-newton based direct positioning method can gradually approximate to the corresponding cralmelo curve, but the quasi-newton based direct positioning method can estimate the initial position and speed of the target by only calculating 150 grid points, whereas the grid search based direct positioning method needs to perform four-dimensional search, and 10000 grid points are also calculated if each dimension only searches 10 grid points. However, under the condition of insufficient prior knowledge, the search amount of 10 grids in each dimension is far from the requirement, and the searched position point is far from the speed and the real position and the speed of the target.

Claims (1)

1. A method for directly positioning a moving target based on quasi-Newton sets the real speed of a uniform-speed moving target to be positioned as v and the initial real position as p 0 (ii) a Radio signals radiated by the target can be intercepted and captured by L observation stations at the same time; assuming that each observation station samples a target radiation signal in K time slot segments, and the time interval between two adjacent time slot segments is T d Setting the target position of the first time slot segment as the initial position; simultaneously defining: in each sampling time slot segment, the instantaneous position and speed of the target are kept unchanged, and then the position vector of the target in the kth time slot segment is: p is a radical of k ＝p 0 +v*(k-1)*T d And the location vector for the ith observation station is noted as: q. q.s l (ii) a The positioning method is characterized by comprising the following steps:
s1, given parameter delta epsilon (0,1), sigma epsilon (0,0.5), and joint variable x of initial point position and speed 0 ＝(p',v')∈R n P ', v' is a real number set R n At any initial value of (1), number of iterations M iter Initial symmetric positive definite matrix B k Let k =0, then B 0 Is a unit array; definition f (p) 0 V) is the cost function:
Figure FDA0003962060320000011
Figure FDA0003962060320000012
Figure FDA0003962060320000013
in the formula, r l,k ＝[r l,k (t 1 ),…,r l,k (t N )] T For the received signal of each observation station, r l,k ＝b l,k A l,k F l,k s k +n l,k ，l＝1,2,…,L；
Figure FDA0003962060320000014
f l,k Represents the frequency of the down-converted signal intercepted by the l observation station during the k observation time slot, which is modeled as: f. of l,k ＝f c μ l,k Wherein f is c Is the frequency of the carrier wave and,
Figure FDA0003962060320000015
c is the speed of light; s k (t) represents the complex envelope of the signal in the k-th slot segment, sampled as s k ＝[s k (t 1 ),…,s k (t N )] T N is the number of samples of the signal in each time slot, F l,k Is a time shift matrix, i.e.: f l,k s k Denotes a general formula s k Displacement of
Figure FDA0003962060320000016
Figure FDA0003962060320000017
Time delay of a target radiation signal relative to the ith observation station at the kth time slot;
s2, calculating
Figure FDA0003962060320000018
Figure FDA0003962060320000019
x and y are respectively the horizontal and vertical coordinate components of the initial position of the target, v x ，v y Respectively are the horizontal and vertical coordinate components of the target speed; each of which is:
Figure FDA0003962060320000021
Figure FDA0003962060320000022
Figure FDA0003962060320000023
Figure FDA0003962060320000024
in the formula (I), the compound is shown in the specification,
Figure FDA0003962060320000025
Figure FDA0003962060320000026
Figure FDA0003962060320000027
Figure FDA0003962060320000028
Figure FDA0003962060320000029
Figure FDA00039620603200000210
Figure FDA00039620603200000211
Figure FDA00039620603200000212
wherein s is k ＝[s k (t 1 ),…,s k (t N )] T ，
Figure FDA00039620603200000213
Is s is k (t n -τ l,k ) A gradient of (a);
Figure FDA00039620603200000214
Figure FDA00039620603200000215
d l,k ＝||p k -q l ||
s3, solving a linear equation set B k d k ＝-g k To obtain solution d k ；
S4, setting m k Is the smallest non-negative integer m that satisfies the following inequality:
Figure FDA0003962060320000031
wherein
Figure FDA0003962060320000032
x k+1 ＝x k +α k d k ，f(x k ) Is a cost function f (p) 0 V) simplified expression of;
s5, determining B by a correction formula k+1 The correction formula is:
Figure FDA0003962060320000033
wherein: s k ＝x k+1 -x k ，y k ＝g k+1 -g k ，
Figure FDA0003962060320000034
S6, let k = k +1, if k > M iter Then x is output k As a result of the positioning; otherwise, go back to step S2.
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