CN116047498A - Maneuvering target tracking method based on maximum correlation entropy extended Kalman filtering - Google Patents

Maneuvering target tracking method based on maximum correlation entropy extended Kalman filtering Download PDF

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CN116047498A
CN116047498A CN202310063017.0A CN202310063017A CN116047498A CN 116047498 A CN116047498 A CN 116047498A CN 202310063017 A CN202310063017 A CN 202310063017A CN 116047498 A CN116047498 A CN 116047498A
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model
target
state
probability
matrix
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李晓花
秦璐瑶
金海燕
蔡磊
杨秀红
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Xian University of Technology
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Xian University of Technology
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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
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/66—Radar-tracking systems; Analogous systems
    • G01S13/72—Radar-tracking systems; Analogous systems for two-dimensional tracking, e.g. combination of angle and range tracking, track-while-scan radar
    • G01S13/723—Radar-tracking systems; Analogous systems for two-dimensional tracking, e.g. combination of angle and range tracking, track-while-scan radar by using numerical data
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/16—Matrix or vector computation, e.g. matrix-matrix or matrix-vector multiplication, matrix factorization

Abstract

The invention provides a maneuvering target tracking method based on maximum correlation entropy extended Kalman filtering, which comprises the following steps: establishing a model set, a target state equation and a measurement equation of an interactive multi-model algorithm; initializing model probability, target state vector, covariance matrix and Markov state transition matrix; performing input interaction between models, and linearizing a nonlinear model by using an extended Kalman filter; carrying out one-step prediction on each model, and designing a cost function based on a maximum entropy criterion; obtaining posterior update values of the k+1 moment targets in the models through an iteration method; updating the model probability, correcting the transition probability, and finally fusing and outputting the tracking result. By the method, the nonlinear maneuvering target tracking problem under the non-Gaussian noise can be solved, and the target estimation accuracy can be improved.

Description

Maneuvering target tracking method based on maximum correlation entropy extended Kalman filtering
Technical Field
The invention belongs to the technical field of radar target tracking methods, and particularly relates to a maneuvering target tracking method based on maximum correlation entropy extended Kalman filtering.
Background
Maneuvering target tracking is a process of estimating the state of maneuvering objects based on sensor information, and the tracking technology of maneuvering targets has wide application in the fields of national defense science and technology and national economy. One challenging problem in the area of maneuver target tracking is that maneuver targets have uncertainty, often described by multiple models, as a single model is not sufficient to describe the target motion state, and interactive multimodal algorithms are one of the common multimodal estimation strategies. Under the framework of an interactive multi-model algorithm, when the target estimation state or the sensor measurement vector is nonlinear, a nonlinear filter such as extended Kalman can be used for tracking the maneuvering target. However, the conventional nonlinear filter is derived based on the minimum mean square error criterion and only contains the second-order information of the tracking error, so that a good tracking effect can be obtained under the condition of Gaussian noise, but the tracking effect is poor under the condition of non-Gaussian noise. In order to solve the problem, the invention combines an extended Kalman filtering algorithm based on the maximum correlation entropy criterion with an interactive multi-model algorithm, and can retain high-order moment information of errors because the maximum correlation entropy is adopted instead of the minimum mean square error criterion. Compared with the traditional nonlinear maneuvering target tracking algorithm, the algorithm can obtain better tracking effect under the condition of non-Gaussian noise.
Disclosure of Invention
The invention aims to provide a maneuvering target tracking method based on maximum correlation entropy expansion Kalman filtering, which solves the problem that the tracking effect of the traditional nonlinear filter is poor under the condition of non-Gaussian noise; the tracking effect is optimized, so that the effect is more accurate.
The technical scheme adopted by the invention is as follows: the maneuvering target tracking method based on maximum correlation entropy extended Kalman filtering comprises the following specific operation steps:
step 1, establishing a model set, a target state equation and a measurement equation of an interactive multi-model algorithm;
step 2, initializing parameters: initializing model probability, a target state vector, a covariance matrix and a Markov state transition matrix of each model at the moment k=1;
step 3, input interaction: calculating the mixing probability among the k moment models according to the model probability, the target state vector, the covariance matrix and the Markov state transition matrix of each model at the k moment to obtain a mixing state estimated value and a mixing covariance matrix of each model of the target at the k moment;
step 4, filter prior estimation: because the measurement equation is linear, the nonlinearity degree of the system is smaller, and therefore an extended Kalman filter with smaller calculation amount is used for linearizing nonlinear models in a model set, and one-step prediction is carried out on each model to obtain a priori predicted value of a k+1 moment target in each model;
step 5, updating a filter posterior: according to the target state equation obtained in the step 1, the measurement equation and the priori predicted value obtained in the step 4, designing a cost function based on the maximum entropy criterion, and obtaining posterior updated values of the k+1 moment targets in the models through an iteration method;
and 6, updating the model probability: calculating likelihood functions of the models, obtaining probability updating values of the models according to a Bayesian probability formula, and adopting the probability change rate of the models to construct a correction function to correct transition probabilities among the models;
step 7, fusing output results: fusing posterior update values and covariance of each model, and outputting state vector of target at k+1 time
Figure BDA0004061504940000021
And covariance matrix P k+1 ；
And 8, repeating the steps 3-7 until the target tracking process is finished.
The invention is also characterized in that:
the step 1 is specifically as follows:
step 1.1, the target state at time k is expressed as
Figure BDA0004061504940000022
The linear CV model of the uniform linear motion of the target is simulated, and a target state equation is constructed as follows:
x k+1 ＝F k x k +G k w k (1)
Figure BDA0004061504940000023
wherein the state vector x k Wherein x, y respectively represent the position of the target along the x direction and the position along the y direction;
Figure BDA0004061504940000036
is the speed of the target in the x-direction, the speed in the y-direction; t represents the sampling interval, w k Mean 0, covariance Q Gaussian white noise, F k Representing a state transition matrix, G k Representing a noise driving matrix;
step 1.2 to
Figure BDA0004061504940000031
The method comprises the steps of representing a target state at the moment k, simulating a nonlinear CT model of coordinated turning motion of a target with unknown angular velocity, and constructing a target state equation:
x k+1 ＝f(x k )+G k w k (3)
Figure BDA0004061504940000032
wherein the angular velocity omega k Not a constant;
step 1.3, taking the x-direction position and the y-direction position as observables, and establishing a measurement equation with non-Gaussian noise:
z k ＝H k x k +v k (5)
Figure BDA0004061504940000033
wherein ,zk Observation vector representing k time, measurement noise v k Is non-Gaussian noise obeying a mixed Gaussian distribution, and is specifically expressed as follows: v k ～(1-α)N(0,R 1k )+αN(0,R 2k ) Wherein alpha represents weight, the value range is 0-1, N (DEG,) represents normal distribution, R 1k To measure the noise covariance, R 2k To measure the covariance of noise when it is abnormally disturbed.
Step 2 is specifically implemented in the following manner:
initializing parameter values, including determining a probability that the target is in model i at time k=1
Figure BDA0004061504940000034
State estimation->
Figure BDA0004061504940000037
Covariance matrix->
Figure BDA0004061504940000035
And a Markov state transition matrix pi k ＝{π ij,k } M×M Wherein each element pi ij,k The transition probability of the system from the model i to the model j at the moment k is represented, and M represents the total number of the models.
The step 3 is specifically as follows:
step 3.1, according to the model probability
Figure BDA0004061504940000041
And a Markov state transition matrix pi k Calculating the mixing probability between the k moment models by the formulas (7), (8)>
Figure BDA0004061504940000042
Figure BDA0004061504940000043
Figure BDA0004061504940000044
wherein ,
Figure BDA0004061504940000045
representing the probability that the target is in the model j after the input interaction;
step 3.2, according to the target state at the time of k
Figure BDA0004061504940000046
Covariance matrix->
Figure BDA0004061504940000047
And the mixing probability between the models obtained in step 3.1 +.>
Figure BDA0004061504940000048
Obtaining the mixed state estimation +.A mixed state estimation of the k moment target in each model is obtained by the formulas (9) (10) respectively>
Figure BDA0004061504940000049
And hybrid covariance matrix->
Figure BDA00040615049400000410
Figure BDA00040615049400000411
Figure BDA00040615049400000412
wherein ,
Figure BDA00040615049400000413
is the hybrid state estimate for the object in each model at time k,>
Figure BDA00040615049400000414
is the time kThe target is at the hybrid covariance of each model.
Step 4 is specifically implemented in the following manner:
step 4.1, calculating a Jacobian matrix of a nonlinear system state equation according to formulas (11) and (12), and linearizing the nonlinear state equation (3):
Figure BDA00040615049400000415
wherein the partial derivatives of the state quantities over angular rate are:
Figure BDA0004061504940000051
wherein ,ωk Is angular velocity, T is sampling interval,
Figure BDA0004061504940000052
The speed in the x direction and the speed in the y direction are respectively;
step 4.2, estimating the mixed state of each model by using the k moment targets obtained in the step 3.2
Figure BDA0004061504940000053
And hybrid covariance matrix->
Figure BDA0004061504940000054
According to whether the state equation is linear, the prior prediction value is filtered by using the formula (13) or (14) respectively>
Figure BDA0004061504940000055
Solving for a priori covariance estimate +.>
Figure BDA0004061504940000056
Figure BDA0004061504940000057
Figure BDA0004061504940000058
Figure BDA0004061504940000059
The step 5 is specifically as follows:
step 5.1, filtering priori predicted values obtained in step 4.2 according to the target state equations (1) (3), the measurement equation (5)
Figure BDA00040615049400000510
and />
Figure BDA00040615049400000511
The system equation is extended as follows:
Figure BDA00040615049400000512
wherein ,
Figure BDA00040615049400000513
Figure BDA00040615049400000514
the covariance matrix of (2) is:
Figure BDA0004061504940000061
wherein ,
Figure BDA0004061504940000062
for a priori prediction covariance matrix->
Figure BDA0004061504940000063
Cholesky score was performedThe resulting lower triangular matrix is solved for,
Figure BDA0004061504940000064
for measuring covariance matrix R k+1 Performing Cholesky decomposition to obtain a lower triangular matrix; />
The two sides of the formula (16) are respectively multiplied by
Figure BDA0004061504940000065
The following expansion equation is obtained:
Figure BDA0004061504940000066
wherein
Figure BDA0004061504940000067
Wherein I represents an identity matrix;
step 5.2, designing a cost function based on maximum entropy:
Figure BDA0004061504940000068
wherein ,Gσ Representing a Gaussian kernel function, L is
Figure BDA0004061504940000069
And l=n+m, n representing the state dimension, m representing the measurement dimension, +.>
Figure BDA00040615049400000610
Representation->
Figure BDA00040615049400000611
The r element of (2)>
Figure BDA00040615049400000612
Representation->
Figure BDA00040615049400000613
R line of (2);
the optimal estimate of the target state is:
Figure BDA00040615049400000614
wherein ,
Figure BDA00040615049400000615
is->
Figure BDA00040615049400000616
R line of (2):
Figure BDA00040615049400000617
deriving the optimal target state for equation (21) is:
Figure BDA00040615049400000618
wherein ,
Figure BDA00040615049400000619
and has the following steps:
Figure BDA0004061504940000071
Figure BDA0004061504940000072
wherein ,
Figure BDA0004061504940000073
diagonal matrix representing kernel function structure related to target motion state prediction +.>
Figure BDA0004061504940000074
A diagonal matrix representing the construction of a kernel function related to target measurements,/->
Figure BDA0004061504940000075
Is->
Figure BDA0004061504940000076
Lines 1 to n, +.>
Figure BDA0004061504940000077
Is->
Figure BDA0004061504940000078
N+1th to n+mth rows;
step 5.3, let iteration number t=1, use the priori predicted value obtained in step 4.2 as the initial value of the iterative process:
Figure BDA0004061504940000079
and determining a kernel bandwidth value sigma and a threshold epsilon;
step 5.4, calculating the state estimation value obtained by the t-th iteration process by using the following formula
Figure BDA00040615049400000710
Figure BDA00040615049400000711
/>
wherein ,
Figure BDA00040615049400000712
Figure BDA00040615049400000713
Figure BDA00040615049400000714
step 5.5, comparing whether the estimated values of the t iteration and the t-1 iteration obtained in the step 5.4 meet the formula (31), and if not, repeating the step 5.4; if yes, ending the iterative process, and setting the iteration result of the t time
Figure BDA00040615049400000715
As a filtered posterior update value +.>
Figure BDA00040615049400000716
And updating covariance matrix +.>
Figure BDA00040615049400000717
Figure BDA00040615049400000718
Figure BDA00040615049400000719
Step 6 is specifically implemented in the following manner:
step 6.1, filtering a priori prediction value obtained according to step 4.2
Figure BDA00040615049400000720
And covariance matrix->
Figure BDA00040615049400000721
Calculating likelihood functions of the models:
Figure BDA0004061504940000081
step 6.2, updating the probability of each model according to a Bayesian probability formula by the following formula:
Figure BDA0004061504940000082
step 6.3, constructing a transition probability correction function of the model j:
Figure BDA0004061504940000083
Figure BDA0004061504940000084
wherein ,
Figure BDA0004061504940000085
representing the probability change rate of the model;
step 6.4, using the correction function of the model j to correct the transition probability of other models to the model:
Figure BDA0004061504940000086
normalizing the transition probability:
Figure BDA0004061504940000087
the step 7 is specifically as follows:
step 7.1, updating the value according to the filtering posterior obtained in step 5.5
Figure BDA0004061504940000088
And the updated probability of each model obtained in the step 6.2 is fused with the state estimation of each model:
Figure BDA0004061504940000089
step 7.2, updating the value of the covariance matrix obtained in step 5.5
Figure BDA00040615049400000810
Covariance of each model was fused:
Figure BDA00040615049400000811
the beneficial effects of the invention are as follows:
1. the invention provides a maneuvering target tracking algorithm based on maximum correlation entropy expansion Kalman filtering, which can solve the tracking problem of maneuvering targets under the nonlinear and non-Gaussian noise conditions.
2. Under the framework of the interactive multi-model algorithm, an extended Kalman filter based on the maximum entropy is adopted, and because the maximum correlation entropy is used instead of the minimum mean square error criterion, the high-order moment information of errors can be reserved, and under the non-Gaussian noise condition, the tracking result is superior to that of the traditional interactive multi-model algorithm based on the extended Kalman filter.
Drawings
FIG. 1 is a global flow chart of the maneuver target tracking method of the present invention based on maximum correlation entropy extended Kalman filtering;
FIG. 2 is a graph of a target tracking trajectory in embodiment 1 of the maneuver target tracking method of the present invention based on maximum correlation entropy extended Kalman filtering;
FIG. 3 (a) is a graph of position mean square error versus conventional method for example 1 of the maneuver target tracking method of the present invention based on maximum correlation entropy extended Kalman filtering;
fig. 3 (b) is a velocity mean square error comparison graph of the motorized target tracking method of the present invention based on the maximum correlation entropy-extended kalman filter with the conventional method.
Detailed Description
The present invention is described in detail below with reference to the drawings and specific embodiments so that the advantages and features of the present invention will be more readily understood by those skilled in the art.
Example 1
In order to verify the effectiveness of the invention, a maneuvering target is tracked, the radar sampling period is 0.1s, the Monte Carlo simulation times are 100 times, and the specific movement process of the target is as follows: starting from an origin point, performing uniform linear motion along the x direction at a speed of 5m/s, starting to perform left turning at 30 degrees after 3 seconds, stopping turning at the target after 5 seconds, continuing to perform uniform linear motion along the x direction at a speed of 5m/s, starting to perform right turning at 30 degrees after 4 seconds, stopping turning at the target after 6 seconds, continuing to perform uniform linear motion along the x direction at a speed of 5m/s, and stopping moving at the target after 2 seconds; the method is implemented according to the following steps:
and (3) executing the step (1) to establish a model set, a target state equation and a measurement equation of the interactive multi-model algorithm.
The method is implemented according to the following steps:
step 1.1, the target state at time k is expressed as
Figure BDA0004061504940000101
The linear CV model of the uniform linear motion of the target is simulated, and a target state equation is constructed as follows:
x k+1 ＝F k x k +G k w k (1)
Figure BDA0004061504940000102
wherein the state vector x k Wherein x, y respectively represent the position of the target along the x direction and the position along the y direction;
Figure BDA0004061504940000103
is the speed of the target in the x-direction, the speed in the y-direction; t represents the sampling interval, w k Mean 0, covariance Q Gaussian white noise, F k Representing a state transition matrix, G k Representing a noise driving matrix;
step 1.2 to
Figure BDA0004061504940000104
Representing the state of the target at the moment k, and coordinating unknown angular velocity by the simulation targetThe nonlinear CT model of turning motion is constructed as follows:
x k+1 ＝f(x k )+G k w k (3)
Figure BDA0004061504940000105
wherein the angular velocity omega k Not a constant;
step 1.3, taking the x-direction position and the y-direction position as observables, and establishing a measurement equation with non-Gaussian noise:
z k ＝H k x k +v k (5)
Figure BDA0004061504940000106
wherein ,zk Observation vector representing k time, measurement noise v k Is non-Gaussian noise obeying a mixed Gaussian distribution, and is specifically expressed as follows: v k ～(1-α)N(0,R 1k )+αN(0,R 2k ) Wherein alpha represents weight, the value range is 0-1, N (DEG,) represents normal distribution, R 1k To measure the noise covariance, R 2k To measure the covariance of noise when it is abnormally disturbed.
Step 2, initializing parameters: initializing model probability, a target state vector, a covariance matrix and a Markov state transition matrix of each model (the total number of the models is M) at the moment of k=1;
Figure BDA0004061504940000111
step 3, input interaction: and calculating the mixing probability among the k moment models according to the model probability, the target state vector, the covariance matrix and the Markov state transition matrix of each model of the k moment models, and obtaining the mixing state estimated value and the mixing covariance matrix of each model of the k moment target.
The method is implemented according to the following steps:
step 3.1, according to the model probability
Figure BDA0004061504940000112
And a Markov state transition matrix pi k Calculating the mixing probability between the k moment models by the formulas (7), (8)>
Figure BDA0004061504940000113
Figure BDA0004061504940000114
Figure BDA0004061504940000115
wherein ,
Figure BDA0004061504940000116
representing the probability that the target is in the model j after the input interaction;
step 3.2, according to the target state at the time of k
Figure BDA0004061504940000117
Covariance matrix->
Figure BDA0004061504940000118
And the mixing probability between the models obtained in step 3.1 +.>
Figure BDA0004061504940000119
Obtaining the mixed state estimation +.A mixed state estimation of the k moment target in each model is obtained by the formulas (9) (10) respectively>
Figure BDA00040615049400001110
And hybrid covariance matrix->
Figure BDA00040615049400001111
Figure BDA00040615049400001112
Figure BDA00040615049400001113
wherein ,
Figure BDA00040615049400001114
is the hybrid state estimate for the object in each model at time k,>
Figure BDA00040615049400001115
is the mixed covariance of the model with the target at time k.
Step 4, filter prior estimation is performed: because the measurement equation is linear, the system has smaller nonlinearity degree, and therefore, an extended Kalman filter with smaller calculation amount is used for linearizing nonlinear models in a model set, and one-step prediction is carried out on each model to obtain a priori predicted value of the target at the moment k+1 in each model.
The method is implemented according to the following steps:
step 4.1, calculating a Jacobian matrix of a nonlinear system state equation according to formulas (11) and (12), and linearizing the nonlinear state equation (3):
Figure BDA0004061504940000121
wherein the partial derivatives of the state quantities over angular rate are:
Figure BDA0004061504940000122
wherein ,ωk Is angular velocity, T is sampling interval,
Figure BDA0004061504940000123
Velocity sum in x-direction respectivelyVelocity in the y direction;
step 4.2, estimating the mixed state of each model by using the k moment targets obtained in the step 3.2
Figure BDA0004061504940000124
And hybrid covariance matrix->
Figure BDA0004061504940000125
According to whether the state equation is linear, the prior prediction value is filtered by using the formula (13) or (14) respectively>
Figure BDA0004061504940000126
Solving for a priori covariance estimate +.>
Figure BDA0004061504940000127
Figure BDA0004061504940000131
Figure BDA0004061504940000132
Figure BDA0004061504940000133
Step 5, updating a filter posterior: and (3) designing a cost function based on the maximum entropy criterion according to the target state equation obtained in the step (1), the measurement equation and the priori predicted value obtained in the step (4), and obtaining posterior updated values of the target in each model at the time k+1 by an iteration method.
The method is implemented according to the following steps:
step 5.1, filtering priori predicted values obtained in step 4.2 according to the target state equations (1) (3), the measurement equation (5)
Figure BDA0004061504940000134
and />
Figure BDA0004061504940000135
The system equation is extended as follows:
Figure BDA0004061504940000136
/>
wherein ,
Figure BDA0004061504940000137
Figure BDA0004061504940000138
the covariance matrix of (2) is:
Figure BDA0004061504940000139
wherein ,
Figure BDA00040615049400001310
for a priori prediction covariance matrix->
Figure BDA00040615049400001311
A lower triangular matrix obtained by Cholesky decomposition,
Figure BDA00040615049400001314
for measuring covariance matrix R k+1 Performing Cholesky decomposition to obtain a lower triangular matrix;
the two sides of the formula (16) are respectively multiplied by
Figure BDA00040615049400001312
The following expansion equation is obtained:
Figure BDA00040615049400001313
wherein
Figure BDA0004061504940000141
Wherein I represents an identity matrix;
step 5.2, designing a cost function based on maximum entropy:
Figure BDA0004061504940000142
wherein ,Gσ Representing a Gaussian kernel function, L is
Figure BDA0004061504940000143
And l=n+m, n representing the state dimension, m representing the measurement dimension, +.>
Figure BDA0004061504940000144
Representation->
Figure BDA0004061504940000145
The r element of (2)>
Figure BDA0004061504940000146
Representation->
Figure BDA0004061504940000147
R line of (2);
the optimal estimate of the target state is:
Figure BDA0004061504940000148
wherein ,
Figure BDA0004061504940000149
is->
Figure BDA00040615049400001410
R line of (2):
Figure BDA00040615049400001411
deriving the optimal target state for equation (21) is:
Figure BDA00040615049400001412
/>
wherein ,
Figure BDA00040615049400001413
and has the following steps:
Figure BDA00040615049400001414
Figure BDA00040615049400001415
wherein ,
Figure BDA00040615049400001416
diagonal matrix representing kernel function structure related to target motion state prediction +.>
Figure BDA00040615049400001417
A diagonal matrix representing the construction of a kernel function related to target measurements,/->
Figure BDA00040615049400001418
Is->
Figure BDA00040615049400001419
Lines 1 to n, +.>
Figure BDA00040615049400001420
Is->
Figure BDA00040615049400001421
N+1th to n+mth rows;
step 5.3, let iteration number t=1, use the priori predicted value obtained in step 4.2 as the initial value of the iterative process:
Figure BDA00040615049400001422
and determining a kernel bandwidth value sigma and a threshold epsilon;
step 5.4, calculating the state estimation value obtained by the t-th iteration process by using the following formula
Figure BDA00040615049400001423
Figure BDA00040615049400001424
wherein ,
Figure BDA0004061504940000151
Figure BDA0004061504940000152
Figure BDA0004061504940000153
step 5.5, comparing whether the estimated values of the t iteration and the t-1 iteration obtained in the step 5.4 meet the formula (31), and if not, repeating the step 5.4; if yes, ending the iterative process, and setting the iteration result of the t time
Figure BDA0004061504940000154
As a filtered posterior update value +.>
Figure BDA0004061504940000155
And updating covariance matrix +.>
Figure BDA0004061504940000156
Figure BDA0004061504940000157
Figure BDA0004061504940000158
And 6, updating the model probability: calculating likelihood functions of the models, obtaining probability updating values of the models according to a Bayesian probability formula, and adopting the probability change rate of the models to construct a correction function to correct transition probabilities among the models; the method is implemented according to the following steps:
step 6.1, filtering a priori prediction value obtained according to step 4.2
Figure BDA0004061504940000159
And covariance matrix->
Figure BDA00040615049400001510
Calculating likelihood functions of the models:
Figure BDA00040615049400001511
/>
step 6.2, updating the probability of each model according to a Bayesian probability formula by the following formula:
Figure BDA00040615049400001512
step 6.3, constructing a transition probability correction function of the model j:
Figure BDA00040615049400001513
Figure BDA00040615049400001514
wherein ,
Figure BDA0004061504940000161
representing the probability change rate of the model;
step 6.4, using the correction function of the model j to correct the transition probability of other models to the model:
Figure BDA0004061504940000162
normalizing the transition probability:
Figure BDA0004061504940000163
step 7, fusing output results: fusing posterior update values and covariance of each model, and outputting state vector of target at k+1 time
Figure BDA0004061504940000164
And covariance matrix P k+1 。
The method is implemented according to the following steps:
step 7.1, updating the value according to the filtering posterior obtained in step 5.5
Figure BDA0004061504940000165
And the updated probability of each model obtained in the step 6.2 is fused with the state estimation of each model:
Figure BDA0004061504940000166
step 7.2, updating the value of the covariance matrix obtained in step 5.5
Figure BDA0004061504940000167
Covariance of each model was fused:
Figure BDA0004061504940000168
and 8, repeating the steps 3-7 until the target tracking is finished, and obtaining a final target tracking effect as shown in fig. 2 and 3 (a) -3 (b).
As can be seen from fig. 2, under the non-gaussian noise condition, the maximum entropy extended kalman filtering algorithm based on the interactive multi-model is close to the real state of the target, which indicates that the algorithm can track the maneuvering target well, and as can be seen from fig. 3 (a) and fig. 3 (b), the position root mean square error and the speed root mean square error of the maximum entropy extended kalman filtering algorithm based on the interactive multi-model are smaller than the corresponding error value of the extended kalman filtering algorithm based on the interactive multi-model, which indicates that the accuracy of the maximum entropy extended kalman filtering algorithm based on the interactive multi-model is higher and the tracking effect is better.

Claims (8)

1. The maneuvering target tracking method based on maximum correlation entropy extended Kalman filtering is characterized by comprising the following specific steps of:
step 1, establishing a model set, a target state equation and a measurement equation of an interactive multi-model algorithm;
step 2, initializing parameters: initializing model probability, a target state vector, a covariance matrix and a Markov state transition matrix of each model at the moment k=1;
step 3, input interaction: calculating the mixing probability among the k moment models according to the model probability, the target state vector, the covariance matrix and the Markov state transition matrix of each model at the k moment to obtain a mixing state estimated value and a mixing covariance matrix of each model of the target at the k moment;
step 4, filter prior estimation: linearizing nonlinear models in the model set by using an extended Kalman filter, and predicting each model in one step to obtain a priori predicted value of each model of which the target is at the time of k+1;
step 5, updating a filter posterior: according to the target state equation obtained in the step 1, the measurement equation and the priori predicted value obtained in the step 4, designing a cost function based on the maximum entropy criterion, and obtaining posterior updated values of the k+1 moment targets in the models through an iteration method;
and 6, updating the model probability: calculating likelihood functions of the models, obtaining probability updating values of the models according to a Bayesian probability formula, and adopting the probability change rate of the models to construct a correction function to correct transition probabilities among the models;
step 7, fusing output results: fusing posterior update values and covariance of each model, and outputting state vector of target at k+1 time
Figure FDA0004061504930000011
And covariance matrix P k+1 ；
And 8, repeating the steps 3-7 until the target tracking process is finished.
2. The maneuvering target tracking method based on maximum correlation entropy expansion kalman filtering according to claim 1, wherein the step 1 is specifically as follows:
step 1.1, the target state at time k is expressed as
Figure FDA0004061504930000012
The linear CV model of the uniform linear motion of the target is simulated, and a target state equation is constructed as follows:
x k+1 ＝F k x k +G k w k (1)
Figure FDA0004061504930000021
wherein the state vector x k Wherein x, y respectively represent the position of the target along the x direction and the position along the y direction;
Figure FDA0004061504930000022
is the speed of the target in the x-direction, the speed in the y-direction; t represents the sampling interval, w k Mean 0, covariance Q Gaussian white noise, F k Representing a state transition matrix, G k Representing a noise driving matrix;
step 1.2 to
Figure FDA0004061504930000023
The method comprises the steps of representing a target state at the moment k, simulating a nonlinear CT model of coordinated turning motion of a target with unknown angular velocity, and constructing a target state equation:
x k+1 ＝f(x k )+G k w k (3)
Figure FDA0004061504930000024
wherein the angular velocity omega k Not a constant;
step 1.3, taking the x-direction position and the y-direction position as observables, and establishing a measurement equation with non-Gaussian noise:
z k ＝H k x k +v k (5)
Figure FDA0004061504930000025
wherein ,zk Observation vector representing k time, measurement noise v k Is non-Gaussian noise obeying a mixed Gaussian distribution, and is specifically expressed as follows: v k ～(1-α)N(0,R 1k )+αN(0,R 2k ) Wherein alpha represents weight, the value range is 0-1, N (DEG,) represents normal distribution, R 1k To measure the noise covariance, R 2k To measure the covariance of noise when it is abnormally disturbed.
3. The maneuvering target tracking method based on maximum correlation entropy expansion kalman filtering according to claim 1, wherein the step 2 is specifically implemented in the following manner:
initializing parameter values, including determining a probability that the target is in model i at time k=1
Figure FDA0004061504930000031
State estimation->
Figure FDA0004061504930000032
Covariance matrix->
Figure FDA0004061504930000033
And a Markov state transition matrix pi k ＝{π ij,k } M×M Wherein each element pi ij,k The transition probability of the system from the model i to the model j at the moment k is represented, and M represents the total number of the models.
4. A maneuvering target tracking method based on maximum correlation entropy expansion kalman filtering according to claim 3, wherein step 3 is specifically as follows:
step 3.1, according to the model probability
Figure FDA0004061504930000034
And a Markov state transition matrix pi k Calculating the mixing probability between the k moment models by the formulas (7), (8)>
Figure FDA0004061504930000035
Figure FDA0004061504930000036
Figure FDA0004061504930000037
wherein ,
Figure FDA0004061504930000038
representing the probability that the target is in the model j after the input interaction;
step 3.2, according to the target state at the time of k
Figure FDA0004061504930000039
Covariance matrix->
Figure FDA00040615049300000310
And the mixing probability between the models obtained in step 3.1 +.>
Figure FDA00040615049300000311
Obtaining the mixed state estimation +.A mixed state estimation of the k moment target in each model is obtained by the formulas (9) (10) respectively>
Figure FDA00040615049300000312
And hybrid covariance matrix->
Figure FDA00040615049300000313
Figure FDA00040615049300000314
/>
Figure FDA00040615049300000315
wherein ,
Figure FDA00040615049300000316
is the hybrid state estimate for the object in each model at time k,>
Figure FDA00040615049300000317
is the mixed covariance of the model with the target at time k.
5. The maneuvering target tracking method based on maximum correlation entropy expansion kalman filtering according to claim 1, wherein the step 4 is specifically implemented in the following manner:
step 4.1, calculating a Jacobian matrix of a nonlinear system state equation according to formulas (11) and (12), and linearizing the nonlinear state equation (3):
Figure FDA0004061504930000041
wherein the partial derivatives of the state quantities over angular rate are:
Figure FDA0004061504930000042
wherein ,ωk Is angular velocity, T is sampling interval,
Figure FDA0004061504930000043
The speed in the x direction and the speed in the y direction are respectively;
step 4.2, estimating the mixed state of each model by using the k moment targets obtained in the step 3.2
Figure FDA0004061504930000044
And hybrid covariance matrix->
Figure FDA0004061504930000045
According to whether the state equation is linear, the prior prediction value is filtered by using the formula (13) or (14) respectively>
Figure FDA0004061504930000046
Solving for a priori covariance estimate +.>
Figure FDA0004061504930000047
Figure FDA0004061504930000048
Figure FDA0004061504930000049
Figure FDA00040615049300000410
6. The maneuvering target tracking method based on maximum correlation entropy expansion kalman filtering according to claim 1, wherein the step 5 is specifically as follows:
step 5.1, filtering priori predicted values obtained in step 4.2 according to the target state equations (1) (3), the measurement equation (5)
Figure FDA0004061504930000051
and />
Figure FDA0004061504930000052
The system equation is extended as follows:
Figure FDA0004061504930000053
wherein ,
Figure FDA0004061504930000054
Figure FDA0004061504930000055
the covariance matrix of (2) is:
Figure FDA0004061504930000056
wherein ,
Figure FDA0004061504930000057
for a priori prediction covariance matrix->
Figure FDA0004061504930000058
Lower triangular matrix obtained by Cholesky decomposition>
Figure FDA0004061504930000059
For measuring covariance matrix R k+1 Performing Cholesky decomposition to obtain a lower triangular matrix;
the two sides of the formula (16) are respectively multiplied by
Figure FDA00040615049300000510
The following expansion equation is obtained:
Figure FDA00040615049300000511
wherein
Figure FDA00040615049300000512
Wherein I represents an identity matrix;
step 5.2, designing a cost function based on maximum entropy:
Figure FDA00040615049300000513
wherein ,Gσ Representing a Gaussian kernel function, L is
Figure FDA00040615049300000514
And l=n+m, n representing the state dimension, m representing the measurement dimension,
Figure FDA00040615049300000515
representation->
Figure FDA00040615049300000516
The r element of (2)>
Figure FDA00040615049300000517
Representation->
Figure FDA00040615049300000518
R line of (2);
the optimal estimate of the target state is:
Figure FDA0004061504930000061
wherein ,
Figure FDA0004061504930000062
is->
Figure FDA0004061504930000063
R line of (2); />
Figure FDA0004061504930000064
Deriving the optimal target state for equation (21) is:
Figure FDA0004061504930000065
wherein ,
Figure FDA0004061504930000066
and has the following steps:
Figure FDA0004061504930000067
Figure FDA0004061504930000068
wherein ,
Figure FDA0004061504930000069
diagonal matrix representing kernel function structure related to target motion state prediction +.>
Figure FDA00040615049300000610
A diagonal matrix representing the construction of a kernel function related to target measurements,/->
Figure FDA00040615049300000611
Is->
Figure FDA00040615049300000612
Lines 1 to n, +.>
Figure FDA00040615049300000613
Is->
Figure FDA00040615049300000614
N+1th to n+mth rows;
step 5.3, let iteration number t=1, use the priori predicted value obtained in step 4.2 as the initial value of the iterative process:
Figure FDA00040615049300000615
and determining a kernel bandwidth value sigma and a threshold epsilon;
step 5.4, calculating the state estimation value obtained by the t-th iteration process by using the following formula
Figure FDA00040615049300000616
Figure FDA00040615049300000617
wherein ,
Figure FDA00040615049300000618
Figure FDA00040615049300000619
Figure FDA00040615049300000620
step 5.5, comparing whether the estimated values of the t iteration and the t-1 iteration obtained in the step 5.4 meet the formula (31), and if not, repeating the step 5.4; if yes, ending the iterative process, and setting the iteration result of the t time
Figure FDA00040615049300000621
As a filtered posterior update value +.>
Figure FDA00040615049300000622
And updating covariance matrix +.>
Figure FDA0004061504930000071
Figure FDA0004061504930000072
Figure FDA0004061504930000073
7. The maneuvering target tracking method based on maximum correlation entropy expansion kalman filtering according to claim 1, wherein the step 6 is specifically implemented in the following manner:
step 6.1, filtering a priori prediction value obtained according to step 4.2
Figure FDA0004061504930000074
And covariance matrix->
Figure FDA0004061504930000075
Calculating likelihood functions of the models: />
Figure FDA0004061504930000076
Step 6.2, updating the probability of each model according to a Bayesian probability formula by the following formula:
Figure FDA0004061504930000077
step 6.3, constructing a transition probability correction function of the model j:
Figure FDA0004061504930000078
Figure FDA0004061504930000079
wherein ,
Figure FDA00040615049300000710
representing the probability change rate of the model;
step 6.4, using the correction function of the model j to correct the transition probability of other models to the model:
Figure FDA00040615049300000711
normalizing the transition probability:
Figure FDA00040615049300000712
8. the maneuvering target tracking method based on maximum correlation entropy expansion kalman filtering according to claim 1, wherein the step 7 is specifically as follows:
step 7.1, updating the value according to the filtering posterior obtained in step 5.5
Figure FDA0004061504930000081
And the updated probability of each model obtained in the step 6.2 is fused with the state estimation of each model:
Figure FDA0004061504930000082
step 7.2, updating the value of the covariance matrix obtained in step 5.5
Figure FDA0004061504930000083
Covariance of each model was fused:
Figure FDA0004061504930000084
/>
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CN117565877A (en) * 2024-01-19 2024-02-20 华东交通大学 Layered fusion estimation method for running state of distributed driving electric automobile

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CN117353705A (en) * 2023-10-08 2024-01-05 哈尔滨工业大学 One-step delay tracking filtering method and system with unknown delay probability
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