CN112491393B - Linear filtering method based on unknown covariance matrix of observed noise - Google Patents
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Abstract
According to the linear filtering method based on unknown observation noise covariance matrix, when the observation noise covariance matrix cannot be obtained, a novel objective function is adopted aiming at the problem that the observation noise covariance matrix is unknown in a linear dynamic model, a linear filtering estimation method when the observation noise accords with a zero mean value condition but the covariance matrix 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 an unknown observed noise covariance matrix.
Background
The Kalman filter estimation method is a linear optimal filter in the sense of estimating minimum and maximum natural estimation of a bias covariance matrix. Compared with a wiener filter, the Kalman filter can process a one-dimensional stable random process and 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 a linear model obeys zero-mean and uncorrelated Gaussian distribution on the premise of conforming to Gaussian distribution, and covariance matrix of noise is accurately known. Such requirements are generally difficult to meet in practical applications due to factors such as operating environments, sensor accuracy variations, and the like. Therefore, when the linear model cannot meet the requirements of the kalman filtering method, how to obtain reliable estimation of the state to suppress the influence of the observed noise on the state becomes one of the research hotspots in the technical fields of linear filtering and observed noise error suppression.
The kalman filtering method requires that the covariance matrix of the observed noise be accurately known for error suppression of the observed noise. However, in applications such as radar tracking, the observed noise covariance matrix of the model cannot be accurately known due to the maneuvering of the carrier and the influence of the working environment, and the observed noise covariance matrix changes with time and position.
Aiming at the problems, the common practice is mainly divided into two types: the method is to obtain an observation noise covariance matrix estimated value based on observation data driving so as to meet the requirement of a Kalman filtering method on parameters. The method is effective when the observed noise covariance matrix is a constant matrix or a slow variable matrix. However, if the noise covariance matrix has a rapid 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 accuracy is reduced, and even the state estimation sequence diverges. In addition, another problem of this type of method is that the calculation process is generally complex, which is not beneficial to the real-time requirement of the filtering method, and another core disadvantage of this type of parameter time variation is that it is difficult to ensure 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 results in this field, no perfect theoretical method has been constructed for the state estimation and observed noise error suppression problems when the noise covariance matrix in the linear time-varying model is rapidly changing.
The other is to solve the problem that the observed noise covariance matrix cannot be obtained in advance, realize the transformation range of the assumed observed noise covariance matrix, and the filtering method designer sets the upper limit of the observed noise covariance matrix based on experience to replace the accurate observed noise covariance matrix in the Kalman filtering method. In theory, the upper limit technology of the observed noise covariance matrix amplifies the observed noise covariance matrix, reduces the reliability evaluation standard of observed information, and the obtained state estimation sequence can not completely meet the optimality standard of the Kalman filtering method. That is, strictly speaking, the kalman filtering method employing the error upper limit technique is no longer the optimal filtering method for state estimation and noise suppression. In practical application, we can also find that the state sequence accuracy obtained by the error upper limit technology has certain uncertainty, and the observed noise covariance matrix parameter has a certain degree of inaccuracy, so that the optimal recursion process of the Kalman filtering method is destroyed to a certain degree, and the problem that the accuracy of the estimated sequence is difficult to accurately evaluate by the filter parameter is solved.
In the fields of radar, sonar, wireless communication network control, navigation and the like, the problem of how to obtain an optimal state estimation sequence of a time-varying observation noise covariance matrix but an unknown linear time-varying model so as to reduce the influence of observation noise on the system longitude has become a problem which needs to be solved urgently in the technical fields of current linear filtering and noise error suppression, and even has become one of the technical problems which limit high-precision application.
Disclosure of Invention
In order to solve the problems, the invention provides a recursive linear filtering method, which aims to solve the problems of state estimation or noise error suppression when the observed noise covariance matrix is unknown and the identification is difficult to realize in a linear dynamic model, and provides a linear filtering method based on the unknown observed noise covariance matrix under the condition of a specific objective function for solving the engineering application problem.
The technical scheme adopted by the invention is as follows:
a linear filtering method based on observed noise covariance matrix unknowns, comprising:
step one: obtaining an initial estimation and an error covariance matrix corresponding to the initial estimation deviation;
step two: establishing a discrete time linear system dynamic model which is suitable for the method;
step three: performing one-step prediction recursive updating on the estimation and the error covariance matrix based on a system equation;
step four: based on the objective function requirement, performing error covariance matrix recursive updating of the post-experiment estimation;
step five: obtaining a gain matrix numerical solution of a filtering estimation method based on an error covariance matrix result of the post-experiment recursive estimation and a system observation equation;
step six: based on the one-step recursive result, the system observation vector and the filter gain matrix numerical solution, updating and solving the post-verification state vector estimation vector numerical solution.
In one embodiment, step one is:
the initial estimate vector and covariance matrix of the state based on the prior information are obtained as follows:
In one embodiment, step one is:
The estimated vector and covariance matrix of the initial state are obtained based on the observation sequence fitting as follows:
P0＝(HTH)-1
where 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:
Where xk is the system state vector at time k, xk-1 is the system state vector at time k-1, Φ k,k-1 is the state transition matrix, Γ k,k-1 is the input matrix of process noise, ω k-1 is the system process noise vector, hk is the observation matrix, vk is the system observation noise vector, zk is the system observation vector at time k.
In one embodiment, step three is:
Based on the initial state vector estimation value and the covariance matrix thereof, the next step of state vector is predicted and updated 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, Is one-step predictive vector,/>For the previous step of estimating the vector, P k,k-1 is the covariance matrix of the one-step prediction vector, and P k is the covariance matrix of the previous step of estimating the vector.
In one embodiment, the fourth step is:
according to the characteristics of the observed noise, the following objective function is selected:
According to the objective function requirement, the following post-test covariance matrix updating method is obtained:
Wherein, I n×n is an n-order identity matrix.
In one embodiment, 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, obtaining a filter gain matrix numerical solution meeting the requirement of minimizing the objective function is as follows:
Wherein P k is the covariance matrix numerical solution of the post-test estimate.
In one embodiment, the sixth step is:
based on the requirement of minimizing the objective function, correcting the one-step recursive result through a filter gain matrix numerical solution, thereby obtaining the following estimation after inspection:
Wherein, For one-step recursive predictive estimation, z k is the observation vector.
In one embodiment, the method further comprises:
step seven: checking whether the result obtained in the step six meets an expected value;
Step eight: the operation can be ended by checking that the obtained result accords with the expected value, and the result is transmitted to a radar or sonar or a wireless communication network control platform or a navigation processor input interface; or the obtained result does not accord with the expected value, repeating the operation from the steps until the obtained result accords with the expected value, and transmitting the result to a radar or sonar or a wireless communication network control platform or a navigation processor input interface.
According to another aspect of the invention, an application of a linear filtering method based on an unknown observed noise covariance matrix in the fields of radar, sonar, wireless communication network control and navigation is provided.
Compared with the prior art, when the observed noise covariance matrix cannot be obtained, the unknown problem of the observed noise covariance matrix exists for the linear dynamic model, and a novel objective function is adopted, wherein the requirement of the objective function on the observed noise covariance matrix is different from that of the Kalman filtering method, the known noise covariance matrix of the observed noise vector is not required, and the dependence of a filtering estimation algorithm on the covariance matrix of the observed noise vector is reduced;
From the implementation method, the filtering method provided by the invention adopts a numerical recurrence implementation form of a 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 the linear filtering method meeting the specific objective function under the unknown condition of the observed noise covariance matrix, reduces the requirements of the engineering application on the setting, the identification and the like of the system model related to the parameters, 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 trajectory, an observed trajectory, and an estimated trajectory of a non-cooperative target in accordance with one embodiment of the present invention;
FIG. 3 is a diagram of the target position error before and after filtering according to one embodiment of the present invention;
FIG. 4 is a 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 diagram of a filter horizontal velocity estimation error according to one embodiment of the present invention;
fig. 7 is a schematic diagram of a filter vertical velocity estimation error according to one 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 minimum objective function of the weighted square sum of the estimation deviation based on one-step recursive estimation of a system equation and the statistical distribution condition of an observation sequence. When the observed noise covariance matrix does not meet the known condition and cannot be identified offline or online through a general algorithm, we cannot use the objective function in the kalman filtering method to calculate the optimal filtering method. In order to complete the state vector estimation and the observation noise error suppression requirements, the filtering method provided by the invention constructs an objective function under the actual condition of the observation noise covariance matrix, then constructs a recursive loop implementation scheme similar to a Kalman filtering method under the minimum requirement of the objective function, and completes the design of the optimal linear filtering method under the minimum requirement of the new objective function.
The filtering method provided by the present invention will be described in detail with reference to the following examples and fig. 1 to 7.
Aiming at the problems of optimal state estimation and observation noise error suppression of unknown observation noise covariance matrix R k in the linear model, as shown in figure 1, the specific implementation steps of the filtering method provided by the invention are as follows:
step one, obtaining an initial estimation of a filtering method and an error covariance matrix corresponding to initial estimation deviation;
The filtering method provided by the invention can adopt two different methods to obtain initial estimation and covariance matrixes of deviation thereof; one approach is to obtain an initial estimate vector of states and its covariance matrix based on a priori information as follows:
another method is to obtain an estimated vector and a covariance matrix of an initial state based on observation sequence fitting for the case that the initial estimated vector and 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 meets the fully observable assumption condition, the coefficient matrix H is a non-singular matrix, and the estimated vector and covariance matrix of the initial state obtained by calculation based on the method only exist.
In the second step, in this embodiment, in order to facilitate description of the principle and implementation manner of the linear filtering method with unknown observed noise covariance matrix, we first provide a discrete time linear system dynamic model adopted by the method and a precondition assumption that model parameters need to be satisfied.
The discrete time linear model aimed at by the method is as follows:
Wherein x k is a system state vector at k moment, x k-1 is a system state vector at k-1 moment, phi k,k-1 is a state transition matrix, Γ k,k-1 is an input matrix of process noise, omega k-1 is a system process noise vector, H k is an observation matrix, v k is a system observation noise vector, and z k is a system observation vector at k moment;
Assuming that the state transition matrix Φ k,k-1, the process noise input matrix Γ k,k-1 and the observation matrix H k are precisely known, the process noise and the observation noise vector are uncorrelated, i.e
The process noise vector ω k-1 satisfies the zero-mean gaussian distribution and its covariance matrix is known, i.e
E(ωk-1)＝0
The observed noise vector v k satisfies a zero-mean Gaussian distribution, i.e
E(vk)＝0
The linear system model satisfies the fully observable condition, but the observed noise covariance matrix R k is unknown, which is generally assumed to be unable to be accurately identified and obtained by an offline or online method based on a finite time.
Step three, carrying out one-step prediction recursive updating on the estimation and the error covariance matrix based on a system equation;
Based on the initial state vector estimation value and the covariance matrix thereof, we can perform one-step prediction update on the next state vector according to the state equation in the system model and the Gaussian propagation principle of the random variable, and obtain the covariance matrix of the predicted vector as follows:
Wherein, Is one-step predictive vector,/>For the previous step of estimating the vector, P k,k-1 is the covariance matrix of the one-step prediction vector, and P k is the covariance matrix of the previous step of estimating the vector.
Step four, performing error covariance matrix recursive updating of estimation after the experiment based on the objective function requirement;
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 not applicable to the state estimation problem of the filter method design provided by the invention. Based on the characteristics of the observed noise, we selected the following objective function:
Compared with the objective function of the Kalman filtering method, the objective function does not use the observed noise covariance matrix as necessary information, but aims at minimizing the square sum of deviation corresponding to the estimated sequence, 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, I n×n is an n-order identity matrix. The covariance matrix updating method has the following advantages: 1) Covariance matrix information of the observed noise vector is no longer needed; 2) As the symmetry and the positive nature of the covariance matrix can be known, the post-test covariance matrix obtained by the updating method also has the symmetry and the positive nature.
Step five, obtaining a gain matrix numerical solution of a filtering estimation method based on an error covariance matrix result of post-test recursive estimation 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, a filter gain matrix numerical solution meeting the requirement of minimizing the objective function can be obtained as follows:
wherein P k is the covariance matrix numerical solution of the post-test estimate, which can be obtained from step three.
Step six, updating and solving the post-verification state vector estimation vector numerical solution based on the one-step recursive result, the system observation vector and the filtering gain matrix numerical solution
Based on the requirement of minimizing the objective function, the one-step recursive result can be corrected through a filter gain matrix numerical solution, so that the post-test estimation is obtained as follows:
Wherein, For one-step recursive predictive estimation, z k is the observation vector.
Step seven: checking whether the result obtained in the step six meets an expected value;
Step eight: the operation can be ended by checking that the obtained result accords with the expected value, and the result is transmitted to a radar or sonar or a wireless communication network control platform or a navigation processor input interface; or the obtained result does not accord with the expected value, repeating the operation from the steps until the obtained result accords with the expected value, and transmitting the result to a radar or sonar or a wireless communication network control platform or a navigation processor input interface.
The following will exemplify a radar scan target tracking problem to test the effective effect of the filtering method provided by the present invention in observed noise error suppression and state estimation applications.
Assuming that a non-cooperative target moves on a two-dimensional plane, the coordinate value of an initial position on the two-dimensional plane is (0 m, 0 m) (wherein m represents a distance unit m), the horizontal running speed of the target is 10m/s, the vertical direction speed is 20m/s, the radar scanning period T=1 second, the observed noise mean value of a radar positioning signal is 0m, and the standard deviation is 100m. The process noise is assumed to meet Gaussian distribution with zero mean value, the physical meaning is set as random motion of a non-cooperative target for avoiding radar tracking, and the smaller the covariance matrix is, the target motion track is approximately similar to uniform linear motion; otherwise, the target motion trail is curved motion. Assuming that the position information of the target including the observation noise can be obtained in real time through radar scanning, the noise component in the observation vector is restrained according to the linear dynamic model through a novel algorithm, and therefore the target position estimation with higher precision is obtained.
The system dynamic model used in the test application is as follows:
Wherein,
X k contains four states, horizontal position, horizontal velocity, vertical position and vertical velocity, respectively; omega k is the process noise vector; v k is the observation noise vector;
after the filtering and noise reduction by the new algorithm, the estimation and error results of the track are shown in figures 2-5,
From the filtering results, shown in schematic diagrams 2-5, we can find that: the novel filtering estimation algorithm realizes the suppression of detection noise in two directions of a horizontal position and a vertical position under the unknown condition of the observed noise covariance matrix, and improves the accuracy and reliability of a radar detection signal on a target detection result.
In addition, as can be seen from fig. 6 and 7, the filter effectively estimates the speed information of the non-cooperative target on the basis of obtaining the position estimation with higher precision, and obtains the target speed estimation result with certain precision and reliability.
From the implementation of the method, it can be seen that: the filtering method is simple in form, low in calculation complexity, easy to realize by a computer and beneficial to application and realization in engineering practice systems.
The technical features of the above-described embodiments may be arbitrarily combined, and all possible combinations of the technical features in the above-described embodiments are not described for brevity of description, however, as long as there is no contradiction between the combinations of the technical features, they should be considered as the scope of the description.
The foregoing examples illustrate only a few embodiments of the invention and are described in detail herein without thereby limiting the scope of the invention. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the invention, which are all within the scope of the invention. Accordingly, the scope of protection of the present invention is to be determined by the appended claims.
Claims (2)
1. A linear filtering method based on observed noise covariance matrix unknowns, comprising:
step one: obtaining an initial estimation and an error covariance matrix corresponding to the initial estimation deviation;
The estimated vector and covariance matrix of the initial state are obtained based on the 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;
step two: establishing a discrete time linear system dynamic model which is suitable for the method;
firstly, giving a discrete time linear system dynamic model adopted by the method and a precondition assumption that model parameters need to be met;
The discrete time linear model aimed at by the method is as follows:
Wherein x k is a system state vector at k moment, x k-1 is a system state vector at k-1 moment, phi k,k-1 is a state transition matrix, Γ k,k-1 is an input matrix of process noise, omega k-1 is a system process noise vector, H k is an observation matrix, v k is a system observation noise vector, and z k is a system observation vector at k moment;
Assuming that the state transition matrix Φ k,k-1, the process noise input matrix Γ k,k-1 and the observation matrix H k are precisely known, the process noise and the observation noise vector are uncorrelated, i.e
The process noise vector ω k-1 satisfies the zero-mean gaussian distribution and its covariance matrix is known, i.e
E(ωk-1)＝0
The observed noise vector v k satisfies a zero-mean Gaussian distribution, i.e
E(vk)＝0
Step three: and carrying out one-step prediction recursive updating on the estimation and the error covariance matrix based on a system equation: based on the initial state vector estimation value and the covariance matrix thereof, the next step of state vector is predicted and updated 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, Is one-step predictive vector,/>For the previous step of estimating the vector, P k,k-1 is the covariance matrix of the one-step prediction vector, and P k-1 is the covariance matrix of the previous step of estimating the vector;
Step four: based on objective function requirements, performing error covariance matrix recursive update of post-experiment estimation: according to the characteristics of the observed noise, the following objective function is selected:
According to the objective function requirement, the following post-test covariance matrix updating method is obtained:
wherein I n×n is an n-order identity matrix;
step five: obtaining a gain matrix numerical solution of a filtering estimation method based on an error covariance matrix result of the post-experiment recursive estimation and a system observation equation;
based on the requirement of the objective function, on the basis of the result of the step four and the observation equation, obtaining a filter gain matrix numerical solution meeting the requirement of minimizing the objective function is as follows:
wherein P k is the covariance matrix numerical solution of the post-test estimation;
step six: updating and solving a post-verification state vector estimation vector numerical solution based on the one-step recursive result, the system observation vector and the filtering gain matrix numerical solution;
based on the requirement of minimizing the objective function, correcting the one-step recursive result through a filter gain matrix numerical solution, thereby obtaining the following estimation after inspection:
Wherein, For one-step recursive predictive estimation, z k is the observation vector.
2. A linear filtering method based on observed noise covariance matrix unknowns according to claim 1, wherein: further comprises:
step seven: checking whether the result obtained in the step six meets an expected value;
step eight: the operation can be ended by checking in the seventh step, wherein the obtained result accords with the expected value; or the obtained result does not accord with the expected value, and then the operation is repeated from the step until the obtained result accords with the expected value.
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| CN104168005A (en) * | 2014-08-25 | 2014-11-26 | 北京理工大学 | Kalman filtering method for recursive estimation under condition that observation noise covariance matrix is unknown |
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