CN115221768A - Improved unscented particle filter direct tracking method based on multi-nested array - Google Patents
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Abstract
The invention discloses an improved unscented particle filter direct tracking method based on a multi-nested array, which comprises the steps of establishing a multi-nested array model, an observation equation and a state equation; initializing; calculating a filtering estimated value and a variance; sampling importance, sampling from a suggested distribution function, and updating the sampling particles; calculating importance weight and carrying out normalization processing; judging whether resampling is carried out or not, calculating the effective particle capacity, and carrying out resampling when the effective particle capacity is not larger than a threshold value; when the effective particle capacity is larger than the threshold value, performing the next iteration; resampling by adopting a system resampling algorithm; outputting a state estimate; an MH-based MCMC step; and performing the next iteration and returning importance samples. The improved unscented particle filter direct tracking method based on the multi-nested array has higher tracking precision and improves the number of received signals.
Description
Technical Field
The present invention relates to the field of direct tracking algorithms, and more particularly, to an improved unscented particle filter direct tracking method based on a multi-nested array.
Background
The target tracking is more and more widely applied in civil and military fields. Common tracking methods comprise Extended Kalman Filtering (EKF), unscented Particle Filtering (UPF) and the like, the EKF solves the problem that standard Kalman filtering cannot be applied to a nonlinear system, but the EKF has the defects that the nonlinear system cannot solve Jacobian matrix, taylor series linearization only has first-order precision, noise obeys Gaussian distribution and the like; the filtering precision of the UPF is higher under the nonlinear and non-Gaussian conditions, but the requirements of high-precision and high-dynamic target tracking cannot be met. Meanwhile, particle filter tracking has the defects of particle degradation, particle depletion, low filter precision and the like. The commonly used observation array is a uniform array, the number of observation source is limited, and the positioning precision and the degree of freedom are not high.
Disclosure of Invention
To solve the above problems, the present invention provides an improved tracking method (I-UPF) based on multi-nested arrays, which uses multi-nested arrays to receive signal sources to improve the degree of freedom and combines system scale symmetric unscented particle filtering (SPSUPF) and markov chain monte carlo algorithm (MCMC).
In order to achieve the purpose, the technical scheme of the invention is as follows:
an improved unscented particle filter direct tracking method based on a multi-nested array comprises,
s1, establishing a multi-nested array model, an observation equation and a state equation;
step S2, initialization is carried out, and prior probability distribution p (d) is subjected to 0 ) Sampling to generate N obeys p (d) 0 ) Distributed sampling particle set
And weighting all the sampling particles
Set to 1/N, K =1, 2.., K being the total number of iterations, i =1, 2.., N being the total number of sampled particles,
for the motion state of the ith sampled particle of the kth iteration,
the weight of the ith sample particle of the kth iteration;
step S3, calculating a filtering estimated value
Sum variance
Step S4, importance sampling, from the suggested distribution function
Middle sampling, updating the sampling particles, z k A signal received for a kth iterative observation station;
s5, calculating an importance weight and carrying out normalization processing;
s6, judging whether resampling is carried out or not, calculating the effective particle capacity, and carrying out resampling in the step S7 when the effective particle capacity is not larger than a threshold value; when the effective particle capacity is larger than the threshold value, performing step S10;
s7, resampling by adopting a system resampling algorithm;
s8, outputting state estimation;
step S9, MH-based MCMC step;
step S10 returns to step S4 with k = k + 1.
The multi-nested array in the step S1 includes an array element number N 1 The first subarray and the second subarray with the array element number of N2, the array element distance of the first subarray is r 0 The array element spacing of the second subarray is (N) 1 +1)r 0 The distance between the array elements r 0 λ/2, λ being the operating wave wavelength.
The observation equation in step S1 above is:
l =1, 2.. Wherein L, L is the total number of observation stations, K =1, 2.. K, K is the total number of observation times, i.e. the total number of iterations, z l，k In the form of a matrix of the signals received by the l observation station at the k observation instant,
is a Doppler shift matrix, G is a discrete DFT matrix, G H Is the conjugate transpose of the discrete DFT matrix G,
is a time delay matrix, s k In discrete form of the signal, n l，k Complex white Gaussian noise with zero mean value and generalized stationarity l And (p) is a direction matrix.
The discrete DFT matrix G and the Doppler shift matrix
The delay matrix
Respectively as follows:
number v of sampling points k ＝[1，2，...V k ] T ，V k Is the total number of sample points, j represents the imaginary part, T s Is a sampling period, f l，q Is Doppler frequency shift, tau, caused by the relative displacement of the ith observation station and the qth uniform linear motion target l，q And Q =1, 2.. For the time delay of the qth signal source to the ith observation station, Q is the total number of signal sources.
The above-mentioned direction matrix A l (p) is:
A l (p)＝[a l (p 1 )，...，a l (p Q )]，
q =1, 2.. Q, Q is the total number of signal sources, p q Is the coordinates of the qth signal source.
The state equation in the step S1 is:
d k+1 ＝Fd k +σ k ，
d k f is a state transition matrix, sigma, of the motion state of the uniform linear motion target at the kth observation moment k Is zero mean white gaussian noise.
In the above step S3, the set of sampling particles is calculated by UKF algorithm
Said filtered estimate of
And the variance
Sigma points were chosen using the SPSUPF algorithm.
In the above step S4, the proposed distribution function is generated using the SPSUPF algorithm.
In the above step S5, the importance weight is calculated according to a formula
And (6) carrying out normalization processing.
In the above step S6, the effective particle volume N eff The expression of (a) is:
the above threshold value N th Is N th ＝N/3。
In the step S8, the outputting the state estimation includes outputting the filter estimation value
And the variance
In step S9, the likelihood function in the MCMC step is:
the improved unscented particle filter direct tracking method based on the multi-nested array has higher tracking accuracy, and a fixed observation station is used for capturing signal waveforms to track the mobile transmitter without intermediate parameter estimation; the signal source is received by using the multi-nested array, the degree of freedom is expanded, and the number and the resolution of the received signals are improved; the SPSUPF algorithm and the MCMC algorithm are combined, so that the positioning accuracy is improved.
In order to make the aforementioned and other features and advantages of the invention more comprehensible, embodiments accompanied with figures are described in detail below.
Drawings
FIG. 1 is a flow chart of an improved unscented particle filter direct tracking method based on a multi-nested array according to the present invention.
FIG. 2 is a schematic diagram of one embodiment of a multiple nested array.
FIG. 3 is a schematic diagram of one embodiment of an observation station and a signal source.
FIG. 4 is a schematic diagram of tracking a track by the I-UPF method and the PF-TDOA method according to the present invention.
FIG. 5 is a diagram showing the signal-to-noise ratio of the I-UPF method, UPF-MCMC, EPF-MCMC, PF-TDOA method of the present invention.
Fig. 6 is a graphical illustration of the population of the I-UPF method and the supf-MCMC method for homogeneous arrays of the present invention.
In the drawings, like reference numerals refer to the same drawing elements.
Detailed Description
In order to make the purpose and technical solution of the embodiments of the present invention clearer, the technical solution of the embodiments of the present invention will be clearly and completely described below with reference to the drawings of the embodiments of the present invention. It should be apparent that the described embodiments are only some of the embodiments of the present invention, and not all of them. All other embodiments, which can be derived by a person skilled in the art from the described embodiments of the invention without any inventive step, are within the scope of protection of the invention.
As shown in fig. 1, the invention relates to an improved unscented particle filter direct tracking method based on a multi-nested array, which comprises,
s1, establishing a multi-nested array model, an observation equation and a state equation.
Step S2, initialization is carried out, and prior probability distribution p (d) is subjected to 0 ) Sampling to generate N obeys p (d) 0 ) Distributed sampling particle set
And weighting all the sampled particles
Is set to 1/N.
Step S3, calculating a filtering estimated value
Sum variance
Step S4, importance sampling, from the suggested distribution function
And (5) middle sampling, and updating sampling particles.
And S5, calculating the importance weight and carrying out normalization processing.
S6, judging whether resampling is carried out or not, calculating the effective particle capacity, and carrying out resampling in the step S7 when the effective particle capacity is not larger than a threshold value; when the effective particle capacity is larger than the threshold value, step S10 is performed.
And S7, resampling by adopting a system resampling algorithm.
And step S8, outputting the state estimation.
Step S9, MCMC step based on MH;
step S10 returns to step S4 with k = k + 1.
More specifically, the method of establishing the nested array model, the observation equation and the state equation in step S1 is as follows.
In a two-dimensional space, a uniform-speed linear motion target to be tracked exists, and the uniform-speed linear motion target sends out 2 uniform-speed linear motion targets with the bandwidth of W and the carrier frequency of f c Of the signal of (1). The motion state of the uniform linear motion target at the k-th observation moment is expressed as
Wherein,
representing the position component of the uniform linear motion object at the k-th observation moment,
and representing the speed component of the uniform linear motion target at the k-th observation moment.
In a two-dimensional space, L observation stations are further arranged, the observation stations are fixed and are synchronous in time and frequency, a multi-nested array is arranged on each observation station, the multi-nested array is a two-stage nested array, the multi-nested array comprises M array elements, and each array element is provided with a sensor and used for receiving signals sent out by the uniform linear motion target. The multi-nested array is formed by respectively setting two array elements as N 1 And N 2 Wherein M = N 1 +N 2 . The dense uniform linear subarray has an array element number of N 1 Distance r between array elements 1 ＝r 0 (ii) a The array element number of the sparse subarray is N 2 Distance r between array elements 2 ＝(N 1 +1)r 0 Wherein r is 0 = λ/2, λ being the wavelength of the operating wave. The position set L of the array elements is as follows:
L＝{n 1 r 0 |0≤n 1 ≤N 1 -1}∪{(n 2 (N 1 +1)-1)r 0 |1≤n 2 ≤N 2 where A is an integer set.
FIG. 2 is a schematic diagram of one embodiment of a multiple nested array. As shown in FIG. 2, N 1 ＝3，N 2 ＝3，M＝N 1 +N 2 =6, the number of array elements of the dense uniform linear sub-array 21 is 3, and the array element spacing is r 0 The positions of 3 array elements of the dense uniform linear subarray 21 are 0 0 ，2r 0 (ii) a The sparse subarray 22 has an array element number of 3 and an array element spacing of 4r 0 The positions of 3 array elements of the sparse subarray 22 are 3r in sequence 0 ，7r 0 ，11r 0 。
The uniform linear motion target sends Q signal sources outwards, and the position coordinate of each signal source is p q ＝[x q ，y q ] T Wherein Q =1, 2. The observation station intercepts the signal of the signal source within K observation moments u l，k ＝[u l，x ，u l，y ] T The position coordinates of the L-th observation station at the k-th observation time are indicated, wherein L =1, 2. Fig. 3 is a schematic diagram of an embodiment of the observation station and the signal source. As shown in FIG. 3, in the xy coordinate system, there are two signal sources, and the position coordinates of the signal sources are p respectively 1 、p 2 The number of observation stations is 4, and the position coordinates of the observation stations are u 1 、u 2 、u 3 、u 4 And the observation station receives the signal sent by the signal source.
The signals of the signal sources are all far-field narrow-band signals, and the signal of the qth signal source is s l，q (t) the number of sampling points is v k The signal z received by the ith observation station at the kth observation time l，k Is composed of
A l (p)＝[a l (p 1 )，...，a l (p Q )]， (2)
Wherein j represents the imaginary part, T s For a sampling period, a l (p q ) Is a direction vector, τ l，q For the time delay of the q signal source to the l observation station, n l (v k ) The complex Gaussian white noise is generalized and stable with zero mean value, and the covariance matrix is gamma l，k ＝σ n 2，A l (p) is a direction matrix, f l，q The Doppler frequency shift caused by the relative displacement of the first observation station and the uniform linear motion target is obtained.
More specifically, the time delay tau of the q signal source to the l observation station l，q Is composed of
Doppler frequency shift f caused by relative displacement of the ith observation station and the qth uniform linear motion target l，q Is composed of
Where c is the propagation velocity of the signal, f c For the carrier frequency, | | · | |, represents the euclidean norm.
The covariance matrix of the signal received at the l-th observation station is
n l (t) is a time domain representation of zero mean, generalized stationary complex white Gaussian noise.
Carrying out vectorization operation on the covariance matrix to obtain
Wherein, delta l A power vector formed by the power of the signals received by the l-th observation station, I M Is an M × M dimensional identity matrix, σ 2 Is the variance, σ q The qth source power.
For y l Sorting to remove redundancy to obtain y l1 ＝A l1 δ l +σ 2 I′ M Wherein A is l1 Is A l (p) * ⊙A l (p) deleting duplicate rows and sequencing 2 A + 2M-2)/2 xK dimensional directional matrix with the array element position distribution of (-M) 2 /4-M/2+1)r 1 To (M) 2 /4+M/2+1)r 1 (ii) a The total array element number of the virtual array is N = (M) 2 + 2M-2)/2, wherein the virtual array means that after vectorization, holes, namely spaces, of the sparse sub-array are filled up and changed into a longer uniform linear array; i' M Represents removing M 2 The/4 + M/2 element is 1, and the other elements are N × 1 dimensional column vectors of 0.
In the invention, the time delay information of the signal
Using Discrete Fourier Transform (DFT) and inverse transform (IDFT), without ignoring
Thereby avoiding the introduction of quantization errors. Number v of said sampling points k ＝[1，2，...V k ] T Time delay information for signals
Obtaining a signal z received by the l observation station at the k observation time after applying discrete Fourier transform and inverse transform l，k In the form of a matrix
Wherein s is k In discrete form of signal source signal, G is discrete DFT matrix, G H Is the conjugate transpose of the discrete DFT matrix G,
is a matrix of doppler frequency shifts and is,
is a delay matrix. More specifically, the discrete DFT matrix G and the Doppler shift matrix
The time delay matrix
Are respectively as
The signal z received by all observers at the k-th observation instant k Is as follows
z k ＝A l1 (p)H k s k +n k ， (14)
Wherein,
the state equation of the uniform linear motion target is expressed as follows:
d k+1 ＝Fd k +σ k ， (16)
where F is the state transition matrix, σ k Is zero-mean white gaussian noise.
A group of uniform linear motion targets to be tracked is given, the total number of particles is P, the total iteration number is K, and the kth iteration, namely the motion state of the ith sampling particle at the kth observation moment is
The weight of the ith sample particle of the kth iteration is
In step S2, k =0 is initialized, and the prior probability distribution p (d) is calculated 0 ) An initial sampling is performed, N particles are extracted from the known initial distribution, and N obedients p (d) are generated 0 ) Distributed sampling particle set
Where k =0. At the same time, the weights of all the sampled particles are weighted
Is set to 1/N.
In the step S3, the sampling particle set is calculated by using a UKF algorithm
Filtered estimate of
Sum variance
And using SPSUPF algorithm to select Sigma pointsThe precision of UT conversion is improved.
In the step S4, importance sampling is performed, and the specific method includes:
at the kth iteration, the posterior probability density function is
According to the theorem of large numbers, the posterior probability density function p (d) is given when the total number of particles V is sufficiently large k |z k ) Will converge to the true posterior probability. Weight of the sampling particle
The calculation formula of (a) is as follows:
wherein,
is a suggested distribution function. The SPSUPF algorithm is used to generate the suggested distribution function, real-time observation data can be merged into the system, the latest observation data is not lost, and the expression of the suggested distribution function is as follows
Sampling from the proposed distribution function, and updating the particles.
In the SPSUPF algorithm, the process of generating the proposed distribution function adopts a proportional symmetric sampling method, and the weight coefficient selection in the UT conversion is changed, and the first-order and second-order weight coefficients corresponding to the UT conversion in the invention are
Wherein, gamma = beta 2 (n + k) -n, k being the scaling parameter and n being the system state dimension. In general, semi-positivity of the posterior covariance should be ensured, and for the case of gaussian distribution, when the state variable is univariate, κ ≠ 0; when the state variable is multidimensional, κ =0 is typically selected. Beta is a scaling factor, and the distance from a Sigma point to a central point can be adjusted by adjusting the value of beta. Therefore, the proportional symmetric sampling method can effectively solve the problem that the distance from the Sigma point to the central point is farther and farther along with the increase of the dimension of the system state vector to generate the non-local effect of sampling. Epsilon represents prior distribution information of the state vector, and the approximation precision of the covariance is improved by combining the dynamic differences of high-order terms in the covariance.
In said step S5, an importance weight calculation is performed, more specifically, an importance weight of the sample particle is calculated according to formula (18), and said importance weight is calculated according to formula (21)
Normalization processing is carried out to obtain a new importance weight
In step S6, it is determined whether resampling is performed, for which a threshold N needs to be set th When the effective particle capacity is not greater than the threshold value N th When the importance value of the particle is higher than the importance value of the particle, the importance value of the particle is higher than the importance value of the particle. The SPSUPF algorithm of the invention adopts a system resampling algorithm to resample, thereby improving the filtering precision and inhibiting the degradation phenomenon.
More specifically, the effective particle capacity N is set eff The expression of (a) is as follows:
and, the threshold value N th Is N th And (4) = N/3. If the effective particle capacity is less than or equal to the threshold value, N eff ≤N th Then the sample particle is seriously degraded, and the sample particle set needs to be processed
Performing resampling in the step S7 to obtain a new sampling particle set; if the effective particle capacity is greater than the threshold value, N eff ＞N th If it is determined that the resampling is not necessary, step S10 is performed, and k = k +1 is set, and the process returns to step S4.
In said step S8, the state estimate is output, i.e. the filter estimate is output
And variance
The specific expressions are respectively as follows:
in order to avoid the problem of particle depletion, in the step S9, an MCMC algorithm based on MH is used to perform importance sampling to obtain a group of predicted particles at the k-th observation time
In step S10, let k = k +1, the process returns to step S4.
The I-UPF method and the PF-TDOA method provided by the invention are respectively used for tracking 2 irrelevant far-field narrow-band signals in a two-dimensional space. As shown in fig. 4, in a two-dimensional space with a coordinate system xy, an asterisk indicates a position of an observation station, the number of the observation stations is 4, and a trajectory of a real path is a solid line. As can be seen from FIG. 4, the I-UPF method of the present invention has higher tracking accuracy than the PF-TDOA method. FIG. 5 is a diagram showing the signal-to-noise ratio of the I-UPF method and UPF-MCMC, EPF-MCMC, PF-TDOA methods of the present invention, where the horizontal axis is the signal-to-noise ratio and the vertical axis is the root of mean square error. As shown in FIG. 5, the signal-to-noise ratio of the I-UPF method of the present invention is lower compared to UPF-MCMC, EPF-MCMC, PF-TDOA methods. FIG. 6 is a graphical representation of the particle number for the I-UPF method of the present invention and for a uniform array, with the horizontal axis being the particle number and the vertical axis being the root mean square error. As shown in FIG. 6, the I-UPF method for nested arrays of the present invention is more accurate than the I-UPF method for uniform arrays.
Although the present invention has been described with reference to particular embodiments, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (12)
1. An improved unscented particle filter direct tracking method based on a multi-nested array is characterized by comprising the following steps,
s1, establishing a multi-nested array model, an observation equation and a state equation;
step S2, initialization is carried out, and prior probability distribution p (d) is subjected to 0 ) Sampling to generate N obeys p (d) 0 ) Distributed set of sampling particles
And weighting all the sampling particles
Set to 1/N, K =1, 2.., K being the total number of iterations, i =1, 2.., N being the sampling particleThe total number of the first and second batteries,
for the motion state of the ith sampled particle of the kth iteration,
the weight of the ith sample particle for the kth iteration;
step S3, calculating a filtering estimated value
Sum variance
Step S4, importance sampling, from the suggested distribution function
Middle sampling, updating the sampling particles, z k A signal received for a kth iterative observation station;
s5, calculating an importance weight and carrying out normalization processing;
s6, judging whether resampling is carried out or not, calculating the effective particle capacity, and carrying out resampling in the step S7 when the effective particle capacity is not larger than a threshold value; when the effective particle capacity is larger than the threshold value, performing step S10;
s7, resampling by adopting a system resampling algorithm;
s8, outputting state estimation;
step S9, MH-based MCMC step;
step S10 returns to step S4 with k = k + 1.
2. The method of claim 1 wherein the method further comprises using a multi-nested array to improve unscented particle filteringThe wave direct tracking method is characterized in that the multi-nested array in the step S1 comprises an array element number N 1 The number of the first sub-array and the array element is N 2 The array element spacing of the first subarray is r 0 The array element spacing of the second sub-array is (N) 1 +1)r 0 The array element spacing r 0 λ/2, λ being the operating wave wavelength.
3. The method for improved direct tracking of unscented particle filter based on multi-nested array as claimed in claim 2, wherein the observation equation in step S1 is:
l =1, 2.. Wherein L, L is the total number of observation stations, K =1, 2.. K, K is the total number of observation times, i.e. the total number of iterations, z l，k In the form of a matrix of the signals received by the l observation station at the k observation instant,
is a Doppler shift matrix, G is a discrete DFT matrix, G H Is the conjugate transpose of the discrete DFT matrix G,
is a time delay matrix, s k In discrete form of the signal, n l，k Complex white Gaussian noise with zero mean, generalized stationarity, A l And (p) is a direction matrix.
4. The method for improving the direct tracking of the unscented particle filter based on the multi-nested array as claimed in claim 3, characterized in that the discrete DFT matrix G, the Doppler shift matrix G
The time delay matrix
Respectively as follows:
number v of sampling points k ＝[1，2，...V k ] T ，V k Is the total number of sample points, j represents the imaginary part, T s Is a sampling period, f l，q Is Doppler frequency shift, tau, caused by the relative displacement of the ith observation station and the qth uniform linear motion target l，q For the delay of the qth signal source to the l observatory, Q =1, 2.
5. The method for improving the direct tracking of the unscented particle filter based on the multi-nested array of claim 4, wherein the direction matrix A l (p) is:
A l (p)＝[a l (p 1 )，...，a l (p Q )]，
q =1, 2.. Q, Q is the total number of signal sources, p q The coordinates of the qth signal source.
6. The method for improved direct tracking of unscented particle filter based on multi-nested array as claimed in claim 5, wherein the state equation in step S1 is:
d k+1 ＝Fd k +σ k ，
d k f is a state transition matrix, sigma, of the motion state of the uniform linear motion target at the kth observation moment k Is zero mean white gaussian noise.
7. The improved unscented particle filter direct tracking method based on the multi-nested array as claimed in claim 1, characterized in that in step S3, the sampled particle set is calculated by the UKF algorithm
Said filtered estimate of
And the variance
Sigma points were chosen using the SPSUPF algorithm.
8. The method for improved direct tracking of unscented particle filters based on multi-nested arrays according to claim 1, characterized in that in step S4 the proposed distribution function is generated with SPSUPF algorithm.
9. The method for improved direct tracking of unscented particle filter based on multi-nested array as claimed in claim 1, wherein in step S5, the importance weight is calculated according to formula
And carrying out normalization processing.
10. The method for improved direct tracking of unscented particle filters based on multiple nested arrays of claim 1, whereinIn the step S6, the effective particle volume N eff The expression of (c) is:
the threshold value N th Is N th ＝N/3。
11. The method as claimed in claim 1, wherein in step S8, said outputting the state estimate comprises outputting the filter estimate
And the variance
12. The method for improved direct tracking based on multi-nested particle filter as claimed in claim 1, wherein in the step S9, the likelihood function in the MCMC step is:
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| CN115932907B (en) * | 2022-12-16 | 2024-02-13 | 南京航空航天大学 | Single-target direct tracking method based on Kalman filter and particle swarm optimization |
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