Self-adaptive fractional extended Kalman filtering method with unknown multiplicative noise nonlinear system

Abstract

The invention discloses a self-adaptive fractional extension Kalman filtering method with an unknown multiplicative noise nonlinear system, and relates to the technical field of Kalman filtering. Establishing a dynamic model of a nonlinear time-varying system with unknown multiplicative noise to represent an original system; modeling unknown multiplicative noise, and constructing an augmented nonlinear Gaussian model to represent a new system by adopting a state augmentation method; performing fractional order division on the augmented nonlinear Gaussian model to obtain equivalent subsystems; calculating a state estimation value and a filtering error covariance matrix of each equivalent subsystem; fusing filtering results of the equivalent subsystems to obtain state estimation of a new system and a filtering error covariance matrix; and calculating the state estimation and filtering error covariance matrix of the original system. The defect that an existing filtering method cannot process unknown statistical information of multiplicative noise at the same time is overcome, and the robustness of system linearization errors is improved through fractional extended Kalman filtering.

Description

Technical Field

[0001] This invention relates to the field of Kalman filtering technology, specifically an adaptive fractional extended Kalman filtering method for nonlinear systems with unknown multiplicative noise. Background Technology

[0002] State estimation for nonlinear systems is a classic topic in the field of filtering. Because dynamic system filtering has broad application prospects in various fields such as navigation, environmental monitoring, and signal processing, and system nonlinearity is a common phenomenon in these fields, the filtering problem for nonlinear systems has received widespread attention.

[0003] For general nonlinear systems, linearizing the system equations is a common technique, but this process inevitably introduces linearization errors. To increase the robustness of filtering methods to linearization errors, fractional exponents can be introduced into the prior and likelihood functions to amplify the prediction error covariance and the measurement noise covariance, thereby increasing the overlap region between the prior and likelihood functions and improving the convergence conditions.

[0004] Furthermore, in signal processing and filtering theory, noise is typically modeled as additive noise, meaning the observed signal is a linear superposition of the desired signal and independent noise terms. However, this assumption does not hold true in many practical systems. In many physics, communication, biomedical, and remote sensing imaging systems, noise often acts on the signal multiplicatively, meaning the noise is correlated with the signal amplitude. This causes both the variance and mean of the signal to be affected simultaneously, thus violating the optimality assumption of linear filters. Therefore, for measurement outputs affected by multiplicative noise, employing effective processing methods is a crucial prerequisite for achieving good filtering performance.

[0005] However, traditional filtering schemes for nonlinear systems with multiplicative noise neglect the fact that the statistical characteristics of the multiplicative noise may be unknown or altered by environmental influences. Inaccurate prior information can lead to reduced filtering performance or even divergence. Therefore, adaptive filters are introduced to address this problem when the prior information on multiplicative noise is unknown or inaccurate.

[0006] Since existing filtering methods cannot effectively handle the filtering problem of nonlinear systems with unknown multiplicative noise, it is of practical significance to design an adaptive fractional extended Kalman filter method for nonlinear systems with unknown multiplicative noise. Summary of the Invention

[0007] To address the shortcomings of existing technologies, this invention provides an adaptive fractional extended Kalman filter method for nonlinear systems with unknown multiplicative noise. This method overcomes the limitation of existing filtering methods in handling unknown statistical information of multiplicative noise, and employs fractional extended Kalman filtering to improve robustness to system linearization errors.

[0008] To achieve the above objectives, the present invention adopts the following technical solution: an adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise, comprising the following steps:

[0009] Step 1: Establish a dynamic model representation of the original system for the nonlinear time-varying system with unknown multiplicative noise;

[0010] Step 2: Model the unknown multiplicative noise and use the state augmentation method to construct an augmented nonlinear Gaussian model to represent the new system;

[0011] Step 3: Apply fractional partitioning to the augmented nonlinear Gaussian model to obtain... An equivalent subsystem;

[0012] Step 4: Calculate the state estimate and filter error covariance matrix for each equivalent subsystem;

[0013] Step 5: Fuse the filtering results of the equivalent subsystems to obtain the state estimate and filtering error covariance matrix of the new system;

[0014] Step 6: Calculate the state estimate and filter error covariance matrix of the original system.

[0015] Furthermore, in step one, the state-space description of the dynamic model of the nonlinear time-varying system is as follows:

[0016] (1)

[0017] In the formula, and In the dynamic model respectively and The state variable at time t, It is a continuously differentiable nonlinear function. for The mean at time step is 0 and the covariance is The system has Gaussian white noise. for The measurement output at time, It is multiplicative noise. for The measurement matrix at time, for The mean at any given time is 0 and the variance is 0. Measurement of Gaussian white noise;

[0018] in,

[0019] (2)

[0020] In the formula, , For the first Multiplicative noise of each channel Indicates that the element The resulting diagonal matrix.

[0021] Furthermore, step two specifically includes:

[0022] The multiplicative noise is mapped as follows:

[0023] (3)

[0024] In the formula, for The image, Let the natural logarithm be denoted as . ,Right now ,but:

[0025] (4)

[0026] In the formula, Represented by natural constant Power functions with base 0. For about Vector functions;

[0027] Will The model is as follows:

[0028] (5)

[0029] In the formula, for Momentary multiplicative noise The image, This indicates that the mean is 0 and the covariance is 0. Gaussian white noise;

[0030] State and By applying augmentation, a new augmented system equation is constructed:

[0031] (6)

[0032] in,

[0033] (7)

[0034] (8)

[0035] In the formula, and They represent the augmented versions. and The state vector at time t, and The nonlinear function obtained through augmentation, To augment the process noise of the system, its covariance matrix is: .

[0036] Furthermore, step three specifically includes:

[0037] According to the system equations, we have:

[0038] (9)

[0039] In the formula, This represents the conditional probability density function. for The set of all measurement outputs, Describes a multivariate Gaussian distribution, where For random variables, The mean, It is the covariance matrix;

[0040] According to Bayes' theorem, we have:

[0041] (10)

[0042] In the formula, and Let f(x) be the likelihood function and the prior function of the augmented system equation (6), respectively. for exist The probability density function under the given conditions;

[0043] Applying fractional partitioning to the likelihood and prior functions, we obtain:

[0044] (11)

[0045] in,

[0046] (12)

[0047] (13)

[0048] In the formula, and These are the fractional exponents of the likelihood and the prior function, respectively;

[0049] and Consider the prior and likelihood functions of the following equivalent subsystems:

[0050] (14)

[0051] Thus obtained An equivalent subsystem.

[0052] Furthermore, step four specifically includes:

[0053] The filter designed using the extended Kalman filter method for the equivalent subsystem is represented as follows:

[0054] (15)

[0055] in,

[0056] (16)

[0057] In the formula, For equivalent subsystems exist One-step prediction of state at a given moment. and Equivalent subsystems exist and State estimation at time 10:00 For equivalent subsystems exist The error covariance matrix of the one-step prediction of the state at time step. For equivalent subsystems exist Error covariance matrix of state estimation at time step. For equivalent subsystems exist The filter gain matrix at time t. For equivalent subsystems exist One-step prediction output at any time. and Equivalent subsystems The Jacobian matrix of the corresponding function with respect to the variable.

[0058] Furthermore, step five specifically includes:

[0059] By fusing the filtering results of the equivalent subsystems, we obtain:

[0060] (17)

[0061] In the formula, To enhance the system in State estimation at time 10:00 for The time-time filtering error covariance matrix.

[0062] Furthermore, step six specifically includes:

[0063] remember:

[0064] (18)

[0065] The filtering result of the original system is:

[0066] (19)

[0067] In the formula, and The original system in State estimation and error covariance matrix at time 1. and For matrix The diagonal pieces, with orders respectively and .

[0068] Compared with existing technologies, the advantages of this invention are as follows: This invention addresses the state estimation problem of nonlinear systems with unknown multiplicative noise by modeling the unknown multiplicative noise and simultaneously estimating both the noise and the state augmentation. Furthermore, it introduces a fractional exponential partitioning of the system prior and likelihood function into the traditional extended Kalman filter method, making it more robust to linearization errors. Moreover, as a recursive method, it does not require multiple calls to previous data, or even the storage of complete data, saving space while being easy to solve and implement. Attached Figure Description

[0069] Figure 1 This is a flowchart of the method of the present invention;

[0070] Figure 2 This is a comparison chart of the actual values ​​of the target object position components in the embodiment and the estimated values ​​of the method of the present invention;

[0071] Figure 3 This is a comparison chart of the actual value of the target object's trajectory in the embodiment and the estimated value of the method of the present invention;

[0072] Figure 4 This is the root mean square error of the target object position component estimated by the method of the present invention in the embodiment. Detailed Implementation

[0073] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the invention, not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0074] An adaptive fractional extended Kalman filter method for nonlinear systems with unknown multiplicative noise, its process combined with Figure 1 As shown, it includes the following steps:

[0075] Step 1: Establish a dynamic model of the nonlinear time-varying system with unknown multiplicative noise;

[0076] The state space of a nonlinear system with unknown multiplicative noise is described as follows:

[0077] (1)

[0078] In the formula, and In the dynamic model respectively and The state variable at time t, For the real number field of the state of the dynamic model, Let be the dimension. It is a continuously differentiable nonlinear function. for The mean at time step is 0 and the covariance is The system has Gaussian white noise. for The measurement output at time, For the real number field output by the dynamic model, Let be the dimension. It is multiplicative noise. for The measurement matrix at time, for The mean at any given time is 0 and the variance is 0. Measurement of Gaussian white noise;

[0079] in,

[0080] (2)

[0081] In the formula, , express The The element is the nth element. Multiplicative noise of each channel, about and All are independent and follow a Gaussian distribution; the mean and variance of the multiplicative noise are unknown. Indicates that the element The diagonal matrix formed This indicates transpose.

[0082] Step 2: Model the unknown multiplicative noise by using the state augmentation method to construct an augmented nonlinear Gaussian model;

[0083] First, the multiplicative noise is mapped as follows:

[0084] (3)

[0085] In the formula, for The image, It represents the natural logarithm.

[0086] For ease of representation, let's remember... ,Right now ,but:

[0087] (4)

[0088] In the formula, Represented by natural constant Power functions with base 0. For about Vector functions.

[0089] Secondly, considering that the multiplicative noise changes slowly, The model is as follows:

[0090] (5)

[0091] In the formula, for Momentary multiplicative noise The image, This indicates that the mean is 0 and the covariance is 0. Gaussian white noise.

[0092] Then, regarding the state and By applying augmentation, a new augmented system equation is constructed:

[0093] (6)

[0094] in,

[0095] (7)

[0096] (8)

[0097] In the formula, and They represent the augmented versions. and The state vector at time t, and The nonlinear function obtained through augmentation, To amplify the process noise of the system, This represents an augmentation of the vector. Clearly, It is Gaussian white noise with zero mean, and its covariance matrix is... .

[0098] Step 3: Apply fractional partitioning to the new system to obtain... An equivalent subsystem;

[0099] Since the augmented system transforms into a classic nonlinear Gaussian system, according to the system equations, we have:

[0100] (9)

[0101] In the formula, This represents the conditional probability density function. for The set of all measurement outputs, Describes a multivariate Gaussian distribution, where For random variables, The mean, Let be the covariance matrix.

[0102] According to Bayes' theorem, we have:

[0103] (10)

[0104] In the formula, and Let f(x) be the likelihood function and the prior function of the augmented system equation (6), respectively. for exist The probability density function under the given conditions.

[0105] Applying fractional partitioning to the likelihood and prior functions, we obtain:

[0106] (11)

[0107] in,

[0108] (12)

[0109] (13)

[0110] In the formula, and Let be the fractional exponents of the likelihood and prior functions, respectively. The cumulative multiplication symbol is used. The sign is proportional. For summation operations.

[0111] So, and It can be regarded as the following equivalent subsystem Prior and likelihood functions:

[0112] (14)

[0113] Thus, we have obtained An equivalent subsystem.

[0114] Step 4: Calculate the state estimate and filter error covariance matrix for each equivalent subsystem. and ;

[0115] Based on the equivalent subsystem obtained in step three, the equivalent subsystem... The filter designed using the extended Kalman filter method is represented as follows:

[0116] (15)

[0117] in,

[0118] (16)

[0119] In the formula, For equivalent subsystems exist One-step prediction of state at a given moment. and Equivalent subsystems exist and State estimation at time 10:00 For equivalent subsystems exist The error covariance matrix of the one-step prediction of the state at time step. For equivalent subsystems exist Error covariance matrix of state estimation at time step. For equivalent subsystems exist The filter gain matrix at time t. For equivalent subsystems exist One-step prediction output at any time. and Equivalent subsystems The Jacobian matrix of the corresponding function with respect to the variable. For elements The reverse, To find the sign of the partial derivative, Indicates the function in The value at that location.

[0120] Step 5: Fuse the filtering results of the equivalent subsystems to obtain the state estimate and filtering error covariance matrix of the augmented system. and ;

[0121] right By fusing the filtering results of the equivalent subsystems, we obtain:

[0122] (17)

[0123] In the formula, To enhance the system in State estimation at time 10:00 for The time-time filtering error covariance matrix.

[0124] Step Six: Calculate the state estimate and filter error covariance matrix of the original system. and ;

[0125] The state estimate and error covariance matrix of the original system expressed by formula (1) are denoted as:

[0126] (18)

[0127] The filtering result of the original system is:

[0128] (19)

[0129] In the formula, and The original system represented by formula (1) is in State estimation and error covariance matrix at time 1. and For matrix The diagonal pieces, with orders respectively and .

[0130] Example

[0131] This embodiment takes a coordinated turning target tracking model with unknown multiplicative noise as an example, and uses the method of the present invention to estimate the trajectory of the target object. The system parameters of the selected coordinated turning target tracking model are set as follows:

[0132] The state of the target object, where and The x and y coordinates represent the position of the target object. and For the velocities in these two directions, For turning angle;

[0133] This is the state estimate of the filter;

[0134] Initial state of the target object and error covariance matrix Set to respectively and ;

[0135] Process noise Measurement noise The covariance matrices are as follows:

[0136]

[0137] nonlinear functions The settings are as follows:

[0138]

[0139] in for Moment State The fifth component, namely the turning angle, The sensor sampling interval is set here. The sensor's measurement matrix is Score Index and Set them to:

[0140]

[0141] The target location estimation results are shown in the attached figure:

[0142] Figure 2 and Figure 3 The true and estimated values ​​of the target object's position components and coordinates are given respectively. As can be seen from the figure, for a coordinated turning target tracking system with unknown multiplicative noise, the adaptive fractional extended Kalman filter of the present invention can effectively track the moving trajectory of the target object.

[0143] Figure 4 The root mean square error of the target object's position component is given, and experimental results verify the effectiveness of the proposed method.

[0144] In summary, the adaptive fractional extended Kalman filter method for nonlinear systems with unknown multiplicative noise proposed in this invention can effectively estimate the target state.

[0145] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered illustrative and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of the equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

[0146] Furthermore, it should be understood that although this specification describes embodiments, not every embodiment contains only one independent technical solution. This narrative style is merely for clarity. Those skilled in the art should consider the specification as a whole, and the technical solutions in each embodiment can also be appropriately combined to form other embodiments that can be understood by those skilled in the art.

Claims

1. An adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise, characterized in that: Includes the following steps: Step 1: Establish a dynamic model representation of the original system for the nonlinear time-varying system with unknown multiplicative noise; Step 2: Model the unknown multiplicative noise and use the state augmentation method to construct an augmented nonlinear Gaussian model to represent the new system; Step 3: Apply fractional partitioning to the augmented nonlinear Gaussian model to obtain... An equivalent subsystem; Step 4: Calculate the state estimate and filter error covariance matrix for each equivalent subsystem; Step 5: Fuse the filtering results of the equivalent subsystems to obtain the state estimate and filtering error covariance matrix of the new system; Step 6: Calculate the state estimate and filter error covariance matrix of the original system.

2. The adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise according to claim 1, characterized in that: In step one, the state space description of the dynamic model of the nonlinear time-varying system is as follows: （1） In the formula, and In the dynamic model respectively and The state variable at time t, It is a continuously differentiable nonlinear function. for The mean at time step is 0 and the covariance is The system has Gaussian white noise. for The measurement output at time, It is multiplicative noise. for The measurement matrix at time, for The mean at any given time is 0 and the variance is 0. Measurement of Gaussian white noise; in, （2） In the formula, , For the first Multiplicative noise of each channel Indicates that the element The resulting diagonal matrix.

3. The adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise according to claim 2, characterized in that: Step two specifically includes: The multiplicative noise is mapped as follows: （3） In the formula, for The image, Let the natural logarithm be denoted as . ,Right now ,but: （4） In the formula, Represented by natural constant Power functions with base 0. For about Vector functions; Will The model is as follows: （5） In the formula, for Momentary multiplicative noise The image, This indicates that the mean is 0 and the covariance is 0. Gaussian white noise; State and By applying augmentation, a new augmented system equation is constructed: （6） in, （7） （8） In the formula, and They represent the augmented versions. and The state vector at time t, and The nonlinear function obtained through augmentation, To augment the process noise of the system, its covariance matrix is: .

4. The adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise according to claim 3, characterized in that: Step three specifically includes: According to the system equations, we have: （9） In the formula, This represents the conditional probability density function. for The set of all measurement outputs, Describes a multivariate Gaussian distribution, where For random variables, The mean, It is the covariance matrix; According to Bayes' theorem, we have: （10） In the formula, and Let f(x) be the likelihood function and the prior function of the augmented system equation (6), respectively. for exist The probability density function under the given conditions; Applying fractional partitioning to the likelihood and prior functions, we obtain: （11） in, （12） （13） In the formula, and These are the fractional exponents of the likelihood and the prior function, respectively; and Consider the prior and likelihood functions of the following equivalent subsystems: （14） Thus obtained An equivalent subsystem.

5. The adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise according to claim 4, characterized in that: Step four specifically includes: The filter designed using the extended Kalman filter method for the equivalent subsystem is represented as follows: （15） in, （16） In the formula, For equivalent subsystems exist One-step prediction of state at a given moment. and Equivalent subsystems exist and State estimation at time 10:00 For equivalent subsystems exist The error covariance matrix of the one-step prediction of the state at time step. For equivalent subsystems exist Error covariance matrix of state estimation at time step. For equivalent subsystems exist The filter gain matrix at time t. For equivalent subsystems exist One-step prediction output at any time. and Equivalent subsystems The Jacobian matrix of the corresponding function with respect to the variable.

6. The adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise according to claim 5, characterized in that: Step five specifically includes: By fusing the filtering results of the equivalent subsystems, we obtain: （17） In the formula, To enhance the system in State estimation at time 10:00 for The time-time filtering error covariance matrix.

7. The adaptive fractional extended Kalman filter method for a nonlinear system with unknown multiplicative noise according to claim 6, characterized in that: Step six specifically includes: remember: （18） The filtering result of the original system is: （19） In the formula, and The original system in State estimation and error covariance matrix at time 1. and For matrix The diagonal pieces, with orders respectively and .

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