CN112491393A - Linear filtering method based on observation noise covariance matrix unknown - Google Patents
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Abstract
According to the linear filtering method based on the observation noise covariance matrix unknown, when the observation noise covariance matrix can not be obtained, aiming at the problem that the observation noise covariance matrix is unknown in a linear dynamic model, a novel objective function is adopted, a linear filtering estimation method for observing noise which meets the zero mean condition but is unknown is obtained under the requirement of the objective function, and the filtering effect is verified through radar target tracking application.
Description
Technical Field
The invention belongs to the technical field of linear filtering and noise error suppression, and particularly relates to a linear filtering method based on observation of unknown noise covariance matrix.
Background
The Kalman filtering estimation method is a linear optimal filter in the sense of estimating the minimum and maximum natural estimation of a deviation covariance matrix. Compared with a wiener filter, the Kalman filter can process a one-dimensional stable random process and still has an effect on a high-dimensional non-stable random process. Therefore, the Kalman filter is widely applied to the fields of radar, sonar, communication, control, navigation and the like.
The Kalman filtering method is a special linear optimal filter, and requires that the process and observation noise in a linear model obey zero mean and non-relevant Gaussian distribution on the premise that the linear model accords with the Gaussian distribution, and the covariance matrix of the noise is accurately known. Due to the influence of factors such as working environment, sensor precision change and the like, the requirements are generally difficult to meet in the practical application process. Therefore, when the linear model cannot meet the requirement of the kalman filtering method, how to obtain reliable estimation of the state to suppress the influence of the observation noise on the state becomes one of the research hotspots in the technical field of linear filtering and observation noise error suppression.
The kalman filtering method requires that the covariance matrix of the observation noise is accurately known for the observation noise error suppression. However, in applications such as radar tracking, due to the influence of the maneuvering and working environment of the carrier, the observation noise covariance matrix of the model cannot be accurately known, and the problem that the observation noise covariance matrix changes along with time and position exists.
For the problems, the general methods are mainly divided into two categories: one is to obtain an observation noise covariance matrix estimation value based on observation data drive so as to meet the requirement of a Kalman filtering method on parameters. The method is effective for observing that the noise covariance matrix is a constant matrix or a slow-varying moment matrix. However, if the noise covariance matrix has a fast time-varying problem, the noise covariance matrix estimation is difficult to converge, so that the dynamic process does not conform to the model assumption, the state estimation precision is reduced, and even the state estimation sequence diverges. In addition, another problem of this kind of method is that its computation process is complex in general, which is not favorable for the real-time requirement of the filtering method, and another core disadvantage of this kind of parameter time variation is that it is difficult to guarantee the convergence and reliability of the unknown parameter estimation sequence and the state estimation sequence. In recent years, although researchers and engineering designers have achieved some success in this field, a perfect theoretical approach to the problem of state estimation and observation noise error suppression when the noise covariance matrix in a linear time-varying model changes rapidly has not been established.
And the other type aims at the problem that the observation noise covariance matrix can not be obtained in advance, the transformation range of the assumed observation noise covariance matrix is realized, and a filter method designer sets the upper limit of the observation noise covariance matrix based on experience to replace the accurate observation noise covariance matrix in the Kalman filter method. Theoretically, the observation noise covariance matrix upper limit technology artificially amplifies an observation noise covariance matrix, the reliability evaluation standard of observation information is reduced, and the obtained state estimation sequence does not completely meet the optimality standard of the Kalman filtering method any more. That is, strictly speaking, the kalman filtering method using the error upper limit technique is no longer the optimal filtering method for state estimation and noise suppression. In practical application, the problem that the precision of a state sequence obtained by an error upper limit technology is difficult to evaluate accurately through filter parameters due to the fact that the observation noise covariance matrix parameters are inaccurate to a certain degree is solved.
In the fields of radar, sonar, wireless communication network control, navigation and the like, how to obtain an optimal state estimation sequence with a time-varying observation noise covariance matrix but an unknown linear time-varying model to reduce the influence of observation noise on system longitude has become a problem which needs to be solved urgently in the technical field of current linear filtering and noise error suppression, and even becomes one of the technical problems of limiting high-precision application.
Disclosure of Invention
In order to solve the problems, the invention provides a recursive linear filtering method, aims to solve the problems of state estimation or noise error suppression when an observation noise covariance matrix is unknown and identification is difficult to realize in a linear dynamic model, and provides a linear filtering method based on the observation noise covariance matrix unknown under a specific objective function condition for solving the engineering application problems.
The technical scheme adopted by the invention is as follows:
a linear filtering method based on observation noise covariance matrix unknown comprises the following steps:
the method comprises the following steps: obtaining an error covariance matrix corresponding to the initial estimation and the initial estimation deviation;
step two: establishing a discrete time linear system dynamic model adopted by the method;
step three: performing one-step prediction recursive updating on the estimation and the error covariance matrix thereof based on a system equation;
step four: based on the objective function requirement, carrying out recursive updating of the error covariance matrix estimated after the experiment;
step five: obtaining a gain matrix numerical solution of a filtering estimation method based on an error covariance matrix result of recursive estimation after an experiment and a system observation equation;
step six: and updating and solving the post-test state vector estimation vector numerical solution based on the one-step recursion result, the system observation vector and the filtering gain matrix numerical solution.
In one embodiment, the first step is:
the initial estimation vector of the state and its covariance matrix based on the prior information are obtained as follows:
in one embodiment, the first step is:
the estimation vector and covariance matrix for the initial state are obtained based on observation sequence fitting as follows:
P0＝(HTH)-1
wherein n is the dimension of the initial state vector, and the coefficient matrix H is a non-singular matrix.
In one embodiment, the second step is:
modeling a system:
wherein xk is the system state vector at time k, xk-1 is the system state vector at time k-1, phik,k-1Being a state transition matrix, Γk,k-1Input matrix, ω, being process noisek-1The system process noise vector is shown, Hk is an observation matrix, vk is a system observation noise vector, and zk is a k-time system observation vector.
In one embodiment, the third step is:
based on the initial state vector estimated value and the covariance matrix thereof, according to a state equation in a system model and a Gaussian propagation principle of a random variable, carrying out one-step prediction updating on the next state vector, and obtaining the covariance matrix of the prediction vector, wherein the method comprises the following steps:
wherein,
in order to predict the vector in one step,
estimating the vector, P, for the previous stepk,k-1Covariance matrix, P, for one-step prediction vectorkThe covariance matrix of the vectors is estimated for the previous step.
In one embodiment, the fourth step is:
according to the characteristics of the observed noise, the following objective function is selected:
according to the requirement of the objective function, the following updating method of the covariance matrix after the test is obtained:
wherein, In×nIs an n-order identity matrix.
In one embodiment, step five is:
based on the requirement of the objective function, on the basis of the result of the step four and the observation equation, the numerical solution of the filter gain matrix meeting the requirement of the minimized objective function is obtained as follows:
wherein, PkIs the covariance matrix numerical solution of the post-test estimate.
In one embodiment, step six is:
based on the requirement of minimizing the objective function, the one-step recursion result is corrected through the numerical solution of the filter gain matrix, so that the posterior estimation is as follows:
wherein,
predicting the estimation result for one step recursion, zkIs an observation vector.
In one embodiment, the method further comprises the following steps:
step seven: checking whether the result obtained in the step six meets an expected value;
step eight: the operation can be finished when the result obtained by the check in the seventh step meets the expected value, and the result is transmitted to a radar or sonar or wireless communication network control platform or a processor input interface of navigation; or, repeating the operation from the steps until the obtained result accords with the expected value, and transmitting the result to a processor input interface of a radar or sonar or a wireless communication network control platform or navigation.
According to another aspect of the invention, an application of a linear filtering method based on observation noise covariance matrix unknown in the fields of radar, sonar, wireless communication network control and navigation is provided.
Compared with the prior art, the observation noise covariance matrix unknown-based linear filtering method provided by the invention has the advantages that when the observation noise covariance matrix cannot be obtained, aiming at the problem that the observation noise covariance matrix is unknown in a linear dynamic model, a novel objective function is adopted, the requirement of the objective function on the observation noise covariance matrix is different from that of a Kalman filtering method, the noise covariance matrix of an observation noise vector is not required to be known, and the covariance matrix dependence of a filtering estimation algorithm on the observation noise vector is reduced;
from the realization method, the filtering method provided by the invention continues to use the numerical recursion realization form of the Kalman filtering method, is a time domain filter, and is easy to realize by a computer. In addition, the filtering method provided by the invention realizes a linear filtering method which meets a specific objective function under the condition that an observation noise covariance matrix is unknown, reduces the requirements of setting, identifying and the like of a system model related to the parameter in engineering application, has higher universality, is easy to popularize and apply, and has good engineering application value.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a diagram of a real, observed, and estimated trajectory of a non-cooperative target in accordance with one embodiment of the present invention;
FIG. 3 is a schematic diagram of target position errors before and after filtering according to one embodiment of the present invention;
FIG. 4 is a schematic diagram of horizontal position error before and after filtering according to one embodiment of the present invention;
FIG. 5 is a schematic diagram of vertical position error before and after filtering according to one embodiment of the present invention;
FIG. 6 is a schematic diagram of filter horizontal velocity estimation error according to one embodiment of the present invention;
FIG. 7 is a diagram illustrating an exemplary filter vertical velocity estimation error according to an embodiment of the present invention.
Detailed Description
The basic principle of the filtering method provided by the invention is as follows:
the basic principle of the Kalman filtering method is an observation noise error suppression method for solving the optimal estimation vector of the current state under the condition of the minimum objective function of the weighted square sum of estimation deviation based on one-step recursion estimation of a system equation and the statistical distribution condition of an observation sequence. When the observation noise covariance matrix does not meet the known conditions and the offline or online identification cannot be performed through a general algorithm, the optimal filtering method cannot be obtained by using the target function in the Kalman filtering method. In order to complete the requirements of state vector estimation and observation noise error suppression, the filtering method provided by the invention constructs an objective function under the actual condition of an observation noise covariance matrix, then constructs a recursion cycle implementation scheme similar to a Kalman filtering method under the minimum requirement of the objective function, and completes the design of an optimal linear filtering method under the minimum requirement of a new objective function.
The filtering method provided by the present invention is described in detail below with reference to the embodiments and shown in fig. 1 to 7.
The existence of an observation noise covariance matrix R for the linear modelkFor the unknown problem of optimal state estimation and observation noise error suppression, as shown in fig. 1, the filtering method provided by the present invention has the following steps:
step one, obtaining an initial estimation of the filtering method provided by the invention and an error covariance matrix corresponding to the initial estimation deviation;
the filtering method provided by the invention can adopt two different methods to obtain the covariance matrix of the initial estimation and the deviation thereof; one method is to obtain the initial estimation vector of the state based on the prior information and its covariance matrix as follows:
another method is to obtain an initial estimation vector and a covariance matrix of an initial state based on observation sequence fitting for a situation where the initial estimation vector and the covariance matrix of the state cannot be obtained through prior information, and the implementation method is as follows:
P0＝(HTH)-1
where n is the dimension of the initial state vector. Since the linear system dynamic model satisfies the fully observable assumption condition, the coefficient matrix H is a nonsingular matrix, and the estimated vector and the covariance matrix of the initial state calculated and obtained based on the above method exist uniquely.
In the second step, in this embodiment, in order to facilitate description of the principle and implementation of the linear filtering method for observing unknown noise covariance matrix, we first provide a dynamic model of a discrete-time linear system used in the method and a premise assumption that model parameters need to be satisfied.
The discrete time linear model aimed at by the method is as follows:
wherein xkIs the system state vector at time k, xk-1Is the system state vector at time k-1, phik,k-1Being a state transition matrix, Γk,k-1Input matrix, ω, being process noisek-1Is a systematic process noise vector, HkTo observe the matrix, vkObserve the noise vector, z, for the systemkSystem observation vectors at the k moment;
assuming a state transition matrix phik,k-1Process noise input matrix gammak,k-1And an observation matrix HkPrecisely known, the process noise and the observed noise vector are uncorrelated, i.e.
Process noise vector ωk-1Satisfy a zero mean Gaussian distribution and whose covariance matrix is known, i.e.
E(ωk-1)＝0
Observing a noise vector vkSatisfy a zero mean Gaussian distribution, i.e.
E(vk)＝0
The linear system model satisfies the complete observable condition, but observes the noise covariance matrix RkUnknown, it is generally assumed that it cannot be accurately identified and acquired by either off-line or on-line methods based on a limited time.
Thirdly, performing one-step prediction recursive updating on the estimation and error covariance matrix based on a system equation;
based on the initial state vector estimation value and the covariance matrix thereof, the next state vector can be predicted and updated in one step according to the state equation in the system model and the Gaussian propagation principle of the random variable, and the covariance matrix of the predicted vector is obtained as follows:
wherein,
in order to predict the vector in one step,
estimating the vector, P, for the previous stepk,k-1Covariance matrix, P, for one-step prediction vectorkThe covariance matrix of the vectors is estimated for the previous step.
Step four, based on the requirement of the objective function, carrying out recursive updating of the estimated error covariance matrix after the experiment;
the objective function of the kalman filtering method is as follows:
due to the lack of covariance matrix information of the observed noise vector, the objective function is no longer applicable to the state estimation problem designed by the filtering method provided by the invention. According to the characteristics of observed noise, the following objective functions are selected:
compared with an objective function of a Kalman filtering method, the objective function does not use an observation noise covariance matrix as necessary information, but the objective function is to minimize the deviation square sum corresponding to the estimation sequence as a target, and has certain practicability and reliability.
According to the new objective function requirement, the following post-test covariance matrix updating method can be obtained:
wherein, In×nIs an n-order identity matrix. The covariance matrix updating method has the following advantages: 1) covariance matrix information of the noise vector does not need to be observed any more; 2) as the symmetry and the positive nature of the covariance matrix are known, the post-test covariance matrix obtained by the updating method also has the symmetry and the positive nature.
Fifthly, obtaining a gain matrix numerical solution of the filtering estimation method based on an error covariance matrix result of the recursive estimation after the check and a system observation equation;
based on the requirement of the objective function, on the basis of the result of the step three and the observation equation, the numerical solution of the filter gain matrix satisfying the requirement of the minimized objective function can be obtained as follows:
wherein, PkThe covariance matrix numerical solution for the post-test estimation can be obtained from step three.
And sixthly, updating and solving the numerical solution of the post-test state vector estimation vector based on the one-step recursion result, the system observation vector and the numerical solution of the filter gain matrix
Based on the requirement of minimizing the objective function, we can correct the one-step recursion result by filtering gain matrix numerical solution, so as to obtain the posterior estimation as follows:
wherein,
predicting the estimation result for one step recursion, zkIs an observation vector.
Step seven: checking whether the result obtained in the step six meets an expected value;
step eight: the operation can be finished when the result obtained by the check in the seventh step meets the expected value, and the result is transmitted to a radar or sonar or wireless communication network control platform or a processor input interface of navigation; or, repeating the operation from the steps until the obtained result accords with the expected value, and transmitting the result to a processor input interface of a radar or sonar or a wireless communication network control platform or navigation.
In the following, we take the target tracking problem of radar scanning as an example to test the effective effect of the filtering method provided by the present invention in observation noise error suppression and state estimation applications.
Assuming that a non-cooperative target moves on a two-dimensional plane, the coordinate value of the initial position on the two-dimensional plane is (0m, 0m) (where m represents a distance unit m), the horizontal moving speed of the target is 10m/s, the vertical direction speed is 20m/s, the radar scanning period T is 1 second, the mean value of the observation noise of the radar positioning signal is 0m, and the standard deviation is 100 m. Assuming that the process noise meets Gaussian distribution with a mean value of zero, setting the physical meaning as random motion of a non-cooperative target for avoiding radar tracking, wherein the smaller the covariance matrix is, the motion trajectory of the target is approximately close to uniform linear motion; otherwise, the target motion track is a curvilinear motion. Supposing that the position information of the target including observation noise can be obtained in real time through radar scanning, noise components in observation vectors are restrained through a novel algorithm according to a linear dynamic model, and the estimation of the target position with higher precision is obtained.
The system dynamic model adopted in the test application is as follows:
wherein,
xkthe method comprises four states, namely a horizontal position, a horizontal speed, a vertical position and a vertical speed; omegakIs a process noise vector; v. ofkTo observe the noise vector;
after filtering and noise reduction by the new algorithm, the estimation and error results of the trajectory are shown in figures 2-5,
from the filtering results shown in fig. 2-5, we can easily find that: the novel filtering estimation algorithm realizes the suppression of detection noise in the horizontal position and the vertical position under the unknown condition of the observation noise covariance matrix, and improves the precision and the reliability of radar detection signals to target detection results.
In addition, as can be seen from fig. 6 and 7, the filter realizes effective estimation of the speed information of the non-cooperative target on the basis of obtaining a position estimation with higher precision, and obtains a target speed estimation result with certain precision and reliability.
From the implementation of the method it can be seen that: the filtering method has simple form and low computational complexity, is easy to realize by a computer and is beneficial to application and realization in an engineering practice system.
The technical features of the embodiments described above may be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the embodiments described above are not described, but should be considered as being within the scope of the present specification as long as there is no contradiction between the combinations of the technical features.
The above-mentioned embodiments only express several embodiments of the present invention, and the description thereof is more specific and detailed, but not construed as limiting the scope of the present invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the inventive concept, which falls within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.
Claims (10)
1. A linear filtering method based on observation noise covariance matrix unknown is characterized by comprising the following steps:
the method comprises the following steps: obtaining an error covariance matrix corresponding to the initial estimation and the initial estimation deviation;
step two: establishing a discrete time linear system dynamic model adopted by the method;
step three: performing one-step prediction recursive updating on the estimation and the error covariance matrix thereof based on a system equation;
step four: based on the objective function requirement, carrying out recursive updating of the error covariance matrix estimated after the experiment;
step five: obtaining a gain matrix numerical solution of a filtering estimation method based on an error covariance matrix result of recursive estimation after an experiment and a system observation equation;
step six: and updating and solving the post-test state vector estimation vector numerical solution based on the one-step recursion result, the system observation vector and the filtering gain matrix numerical solution.
2. The linear filtering method based on the observation noise covariance matrix unknown as claimed in claim 1, wherein the first step is:
the initial estimation vector of the state and its covariance matrix based on the prior information are obtained as follows:
3. the linear filtering method based on the observation noise covariance matrix unknown as claimed in claim 1, wherein the first step is:
the estimation vector and covariance matrix for the initial state are obtained based on observation sequence fitting as follows:
P0＝(HTH)-1
wherein n is the dimension of the initial state vector, and the coefficient matrix H is a non-singular matrix.
4. The linear filtering method based on the observation noise covariance matrix unknown as claimed in claim 2 or 3, wherein the second step is:
modeling a system:
wherein xk is the system state vector at the moment k, xk-1 is the system state vector at the moment k-1, phik,k-1Being a state transition matrix, Γk,k-1Input matrix, ω, being process noisek-1The system process noise vector is shown, Hk is an observation matrix, vk is a system observation noise vector, and zk is a k-time system observation vector.
5. The linear filtering method based on the observation noise covariance matrix unknown as claimed in claim 4, wherein the third step is:
based on the initial state vector estimated value and the covariance matrix thereof, according to a state equation in a system model and a Gaussian propagation principle of a random variable, carrying out one-step prediction updating on the next state vector, and obtaining the covariance matrix of the prediction vector, wherein the method comprises the following steps:
wherein,
in order to predict the vector in one step,
estimating the vector, P, for the previous stepk,k-1Covariance matrix, P, for one-step prediction vectorkThe covariance matrix of the vectors is estimated for the previous step.
6. The linear filtering method based on the observation noise covariance matrix unknown as claimed in claim 5, wherein the fourth step is:
according to the characteristics of the observed noise, the following objective function is selected:
according to the requirement of the objective function, the following updating method of the covariance matrix after the test is obtained:
wherein, In×nIs an n-order identity matrix.
7. The linear filtering method based on the observation noise covariance matrix unknown as claimed in claim 6, wherein the fifth step is:
based on the requirement of the objective function, on the basis of the result of the step four and the observation equation, the numerical solution of the filter gain matrix meeting the requirement of the minimized objective function is obtained as follows:
wherein, PkIs the covariance matrix numerical solution of the post-test estimate.
8. The linear filtering method based on the observation noise covariance matrix unknown as claimed in claim 7, wherein the sixth step is:
based on the requirement of minimizing the objective function, the one-step recursion result is corrected through the numerical solution of the filter gain matrix, so that the posterior estimation is as follows:
wherein,
predicting the estimation result for one step recursion, zkIs an observation vector.
9. The method of claim 8, wherein the method comprises a linear filtering method based on an observation noise covariance matrix unknown, the method comprising: further comprising:
step seven: checking whether the result obtained in the step six meets an expected value;
step eight: the operation can be finished if the result obtained by the check in the step seven meets the expected value; or, repeating the operation from the step to the step until the obtained result is in accordance with the expected value.
10. The application of the linear filtering method based on the observation of unknown noise covariance matrix according to claim 9 in the fields of radar, sonar, wireless communication network control and navigation.
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