Unknown noise distribution time-varying time-lag nonlinear system filtering algorithm
By fusing artificial immunoparticle filtering algorithm and adaptive variable window wide kernel density estimation algorithm, complex network filtering problems with time-varying delay, nonlinear, non-Gaussian noise or unknown noise distribution are solved, and efficient filtering effect is achieved.
Patent Information
- Application Number
- CN202411703069.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-26
- Publication Date
- 2025-03-18
AI Technical Summary
The prior art is difficult to effectively deal with complex network filtering problems with time-varying delay, nonlinearity, non-Gaussian noise or unknown noise distribution.
A nonlinear system filtering algorithm for time-varying time-defying unknown noise distribution is proposed. By fusing artificial immunoparticle filtering algorithm with adaptive variable window wide kernel density estimation algorithm, the problems of time-varying time-varying, nonlinear systems, non-Gaussian noise and noise distribution are solved.
Effective filtering of complex networks with time-varying delay, nonlinearity, non-Gaussian noise or unknown noise distribution is realized, and the adaptability and accuracy of the filtering algorithm are improved.
Smart Images
Figure CN119652287A_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of digital signal processing, and in particular relates to a filtering algorithm for a time-varying and time-delay nonlinear system with unknown noise distribution. Background Art
[0002] Nonlinear systems are common systems in nature. For nonlinear system filtering problems, commonly used methods include: nonlinear minimum mean square error filtering, Kalman filtering, wavelet transform, adaptive filtering, etc. Most of the above methods use linearization methods to transform nonlinear systems into linear systems, and then filter according to the transformed linear systems. When the system structure does not meet the linearization requirements, the filtering algorithm is no longer applicable. Among them, some filtering algorithms require that the distribution of noise must be Gaussian noise. If the noise is non-Gaussian or the noise distribution is unknown, the filtering method is no longer applicable.
[0003] In order to solve the problem of non-Gaussian noise, the particle filter algorithm is often used. By using a set of weighted random samples, the state of the system under the posterior probability density distribution is approximated. At the same time, since the Monte Carlo simulation method is based on the law of large numbers, the particle filter algorithm can be easily extended to high-dimensional situations. This method has good adaptability and scalability in solving practical filtering problems. However, for nonlinear system filtering with time-varying and time-delay characteristics, the system changes from a first-order Markov process to a non-first-order Markov process, and the particle filter algorithm is generally targeted at first-order Markov processes. Therefore, for nonlinear systems with time-varying and time-delay characteristics and non-Gaussian noise, the particle filter algorithm will no longer be applicable.
[0004] Therefore, if the system has the characteristics of nonlinearity, time-varying delay, non-Gaussian noise or unknown noise distribution, the above methods are no longer applicable, and there is currently no applicable method.
[0005] In view of the shortcomings of the above-mentioned various filtering algorithms, the present invention proposes for the first time a filtering algorithm for time-varying and time-delay nonlinear systems with unknown noise distribution. It is based on the fusion of an artificial immune particle filtering algorithm and an adaptive variable window width kernel density estimation algorithm to solve the filtering problem of complex networks with time-varying and time-delay, nonlinear systems, non-Gaussian noise, and unknown noise distribution. Summary of the invention
[0006] The present invention proposes a filtering algorithm for a nonlinear system with unknown noise distribution and time-varying and time-delay, so as to solve the problems that nonlinear minimum mean square error filtering, Kalman filtering, wavelet transform, adaptive filtering, particle filtering algorithm and other methods are not applicable to non-Gaussian noise, cannot solve time-varying and time-delay, and cannot handle complex network filtering technology with unknown noise distribution.
[0007] In order to achieve the above object, the present invention solves the above technical problems through the following technical solutions:
[0008] A filtering algorithm for a time-varying, time-delayed nonlinear system with unknown noise distribution, including: theoretical proof of the usability of particle filtering, an artificial immune strategy optimization filtering algorithm, a convergence description of the particle filtering algorithm with artificial immune strategy, and an adaptive window width kernel density estimation algorithm.
[0009] The time-varying and time-delay nonlinear system is reconstructed to obtain a Bayesian network model, and a graph theory analysis method is used to obtain a recursive formula for a particle filter algorithm under the time-varying and time-delay nonlinear system.
[0010] The weight recursion is:
[0011] Time-varying and time-delay nonlinear system process equation:
[0012] Measurement equation: y k =h(x k )+v k
[0013] Among them, g(·) is the state sequence function, X 0:k is the state {x0,…,x k}, is the sample drawn from the posterior probability for the i-th time, is the normalized weight. k is the system status, y k is the system output, n k is the system process noise, v k is the system measurement noise, f(·) is the propagation function, and h(·) is the measurement function.
[0014] The artificial immune strategy optimizes the filtering algorithm:
[0015] S1: Generate particles that obey probability distribution by using Metropolis-Hastings algorithm
[0016] S2: Calculation of repulsive forces between antibodies
[0017] S3: Calculation of antibody affinity
[0018] S4: Calculate the number of clones knum k (i) = round(N·cos(n / 2·fit k (i)))
[0019] S5: Antibody Variation
[0020] S6: Indicator calculation
[0021] S7: Adaptive formula
[0022] in, is the predicted particle, To predict particle weights, is the mutated particle, is the particle weight after mutation, A a is a fixed weight.
[0023] The convergence of the particle filter algorithm of the artificial immune strategy is shown in Figure 1. For a complex network with time-varying delays, if:
[0024] The particle filter algorithm based on the artificial immune strategy estimates the system state to satisfy:
[0025]
[0026] That is, the estimated state of N particles converges to the state Xs as the number of particles N increases.
[0027] Among them, X s is the real state, y s is the real output, Estimated status.
[0028] The adaptive window width kernel density estimation algorithm, the kernel density estimation function is:
[0029]
[0030] The empirical distribution function is:
[0031]
[0032] Newton interpolation:
[0033]
[0034] Resample the empirical distribution function after Newton interpolation and perform Taylor expansion, then the window width can be corrected as follows:
[0035]
[0036] in, is the kernel density estimation function, is the window width, is the kernel function, is the noise set, G u (·) is the empirical distribution function.
[0037] Based on the implementation of the above technical solution, the present invention can achieve the following technical effects:
[0038] 1. The present invention provides a filtering algorithm for a time-varying, time-delay nonlinear system with unknown noise distribution. Aiming at the time-varying, time-delay system, the present invention adopts a graph theory analysis particle filtering algorithm to solve the theoretical basis problem of a time-varying, time-delay nonlinear complex network.
[0039] 2. The present invention provides a filtering algorithm for a time-varying, time-delay nonlinear system with unknown noise distribution. Aiming at the problem of particle depletion in the particle filtering algorithm, the present invention adopts an artificial immune strategy to optimize the filtering algorithm, enrich the particle diversity, and study the convergence of the algorithm.
[0040] 3. The present invention provides a filtering algorithm for a time-varying and time-delayed nonlinear system with unknown noise distribution. In scenarios with unknown noise distribution or non-Gaussian noise, an adaptive window width kernel density estimation algorithm is used to solve the noise distribution estimation problem.
[0041] 4. The present invention provides a filtering algorithm for a time-varying, time-delayed nonlinear system with an unknown noise distribution. Aiming at the problem that the fixed window width in kernel density estimation cannot always fit the data, an adaptive window width change formula is designed. At sample points where the density function is steep, the window bandwidth is small and local is preferred. At sampling points where the density function is relatively flat, the window bandwidth is large and global is preferred, thereby improving the estimation accuracy. BRIEF DESCRIPTION OF THE DRAWINGS
[0042] In order to more clearly illustrate the specific implementation methods of the present invention or the technical solutions in the prior art, the drawings required for use in the specific implementation methods or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some implementation methods of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.
[0043] Figure 1 The Bayesian network model of the time-varying and time-delay nonlinear system provided by the present invention;
[0044] Figure 2 is a distribution diagram of process noise and measurement noise provided by the present invention;
[0045] Figure 3 are the true and estimated probability density distribution functions of process noise and measurement noise provided by the present invention;
[0046] Figure 4 are the actual state and estimated state curves provided by the present invention;
[0047] Figure 5 It is the error curve of the estimated state provided by the present invention. DETAILED DESCRIPTION
[0048] The following embodiments of the present invention are described in further detail in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.
[0049] In the description of the embodiments of the present invention, it should be noted that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate positions or positional relationships based on the positions or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the embodiments of the present invention and simplifying the description, and do not indicate or imply that the devices or elements referred to must have a specific orientation, be constructed and operate in a specific orientation, and therefore cannot be understood as limitations on the embodiments of the present invention. In addition, the terms "first", "second", and "third" are used for descriptive purposes only and cannot be understood as indicating or implying relative importance.
[0050] In the description of the embodiments of the present invention, it should be noted that, unless otherwise clearly specified and limited, the terms "connected" and "connection" should be understood in a broad sense, for example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be a direct connection or an indirect connection through an intermediate medium. For ordinary technicians in this field, the specific meanings of the above terms in the embodiments of the present invention can be understood according to specific circumstances.
[0051] In the embodiments of the present invention, unless otherwise clearly specified and limited, a first feature being "above" or "below" a second feature may mean that the first and second features are in direct contact, or the first and second features are in indirect contact through an intermediate medium. Moreover, a first feature being "above", "above" or "above" a second feature may mean that the first feature is directly above or obliquely above the second feature, or simply means that the first feature is higher in level than the second feature. A first feature being "below", "below" or "below" a second feature may mean that the first feature is directly below or obliquely below the second feature, or simply means that the first feature is lower in level than the second feature.
[0052] In the description of this specification, the description with reference to the terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the embodiment of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art may combine and combine the different embodiments or examples described in this specification and the features of the different embodiments or examples, without contradiction.
[0053] Combine the following Figure 1-Figure 5 The present invention provides a filtering algorithm for a time-varying, time-delayed nonlinear system with unknown noise distribution:
[0054] Combination Figure 1 As shown, in an embodiment of the present invention, a filtering algorithm for a time-varying and time-delay nonlinear system with unknown noise distribution includes: theoretical proof of the usability of particle filtering, an artificial immune strategy optimization filtering algorithm, a convergence description of the particle filtering algorithm of the artificial immune strategy, and an adaptive window width kernel density estimation algorithm.
[0055] Furthermore, for nonlinear systems with time-varying delays, non-Gaussian noise, and unknown noise distribution, we conduct an usability analysis of the particle filter algorithm to solve the theoretical basis for the use of the particle filter algorithm in nonlinear systems with time-varying delays and unknown non-Gaussian noise. For particle concentration and binding problems, we develop artificial immune algorithm fusion technology to solve the particle depletion problem in the particle filter algorithm and enrich particle diversity. For unknown non-Gaussian characteristics, we conduct research on adaptive variable window width kernel density estimation algorithms to solve the kernel density estimation problem of unknown noise and improve the estimation accuracy through adaptive parameter methods.
[0056] Combination Figure 1 As shown, the time-varying and time-delay nonlinear system is reconstructed to obtain a Bayesian network model, and a graph theory analysis method is used to obtain a recursive formula for a particle filter algorithm under a time-varying and time-delay nonlinear system.
[0057]
[0058] The weight recursion is:
[0059] Time-varying and time-delay nonlinear system process equation:
[0060] Measurement equation: y k =h(x k )+v k
[0061] Among them, g(·) is the state sequence function, X 0:k is the state {x0,…,x k}, is the sample drawn from the posterior probability for the i-th time, is the normalized weight. k is the system status, y k is the system output, n k is the system process noise, v kis the system measurement noise, f(·) is the propagation function, and h(·) is the measurement function.
[0062] Furthermore, the artificial immune strategy optimizes the filtering algorithm:
[0063] S1: Generate particles that obey probability distribution by using the Metropolis-Hastings algorithm.
[0064] S2: Calculation of repulsive forces between antibodies
[0065] S3: Calculation of antibody affinity
[0066] S4: Calculate the number of clones knum k (i) = round(N·cos(π / 2·fit k (i)))
[0067] S5: Antibody Variation
[0068] S6: Indicator calculation
[0069] S7: Adaptive formula
[0070] in, is the predicted particle, To predict particle weights, is the mutated particle, is the particle weight after mutation, A a is a fixed weight.
[0071] Further, the specific algorithm is:
[0072] Algorithm 1 Particle filter algorithm based on artificial immune strategy
[0073] Furthermore, the convergence of the particle filter algorithm of the artificial immune strategy shows that for a time-varying and time-delay complex network, if:
[0074] The particle filter algorithm based on the artificial immune strategy estimates the system state to satisfy:
[0075]
[0076] That is, the estimated state of N particles converges to the state Xs as the number of particles N increases.
[0077] Among them, x s is the real state, y sis the real output, Estimated status.
[0078] Furthermore, the adaptive window width kernel density estimation algorithm, the kernel density estimation function is:
[0079]
[0080] The empirical distribution function is:
[0081]
[0082] Newton interpolation:
[0083]
[0084] Resample the empirical distribution function after Newton interpolation and perform Taylor expansion, then the window width can be corrected as follows:
[0085] in, is the kernel density estimation function, is the window width, is the kernel function, is the noise set, G u (·) is the empirical distribution function.
[0086] The present invention proposes a nonlinear complex network filtering problem with time-varying lag, non-Gaussian noise or unknown noise distribution, and the specific method is as follows:
[0087] Aiming at the time-varying and time-delay system, the theoretical basis of particle filter algorithm applicable to time-varying and time-delay nonlinear system is given by using graph theory.
[0088] Aiming at the problem of particle depletion in particle filtering, an artificial immune strategy is used to optimize the filtering algorithm, enrich the diversity of particles, and study the convergence of the algorithm.
[0089] In order to solve the problem of using particle filter algorithm in scenarios with unknown noise distribution or non-Gaussian noise, kernel density estimation method is used to estimate the noise distribution.
[0090] To solve the problem that the fixed window width in kernel density estimation cannot always fit the data, an adaptive window width change formula is designed. At the sampling points with steep density functions, the window bandwidth is small and local is preferred. At the sampling points with flat density functions, the window bandwidth is large and global is preferred.
[0091] The invention provides a filtering algorithm for a time-varying and time-delay nonlinear system with unknown noise distribution, which effectively solves the filtering problem of a time-varying and time-delay nonlinear system with non-Gaussian noise or unknown noise distribution for the first time.
[0092] In order to verify the feasibility of the present invention, simulation examples are used to verify the effectiveness of the optimized particle filter algorithm based on artificial immune strategy and the effectiveness of the kernel density estimation method with adaptive variable window.
[0093] Combination Figure 2-Figure 5 As shown, Figure 2 The sample distribution diagram of process noise and measurement noise is shown, as well as the curve of corresponding noise changing with time. Figure 3 The kernel density estimation diagram of process noise and measurement noise is shown. Figure 2 and Figure 3 , indicating that the kernel density estimation is a true and good estimation of the noise distribution. It can be seen that the estimated probability density function is very close to the probability density function of the real noise, and can well reflect the change process of the probability density function of the real noise, verifying the effectiveness of the kernel density estimation algorithm with variable window width. Figure 4 A graph showing the true state and the estimated state, from Figure 4 (a) It can be seen that the state value estimated by the improved particle filtering scheme is a good approximation of the true state value, which is better than the measured state curve obtained by the traditional method. Figure 4 (b) The estimated state distribution is within a certain range of the true state, which verifies the effectiveness of the algorithm from another perspective. Figure 5 (a) shows the standard error of the estimated state, Figure 5 (b) shows the root mean square error. This state estimation error caused by kernel density estimation is also small within the allowable range, indicating that the true state is a good estimate.
[0094] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. A filtering algorithm for time-varying and time-delay nonlinear systems with unknown noise distribution, characterized in that: include: Theoretical proof of the feasibility of particle filtering, optimization of filtering algorithm with artificial immune strategy, convergence of particle filtering algorithm with artificial immune strategy and adaptive window width kernel density estimation algorithm.
2. According to claim 1, the filtering algorithm for unknown noise distribution time-varying and time-delay nonlinear system is characterized by: The time-varying and time-delay nonlinear system is reconstructed to obtain a Bayesian network model, and a graph theory analysis method is used to obtain a recursive formula for a particle filter algorithm under the time-varying and time-delay nonlinear system. The weight recursion is: Time-varying and time-delay nonlinear system process equation: Measurement equation: y k =h(x k )+v k Among them, g(·) is the state sequence function, X 0:k is the state {x0,…,x k }, is the sample drawn from the posterior probability for the i-th time, is the normalized weight. k is the system status, y k is the system output, n k is the system process noise, v k is the system measurement noise, f(·) is the propagation function, and h(·) is the measurement function.
3. According to claim 2, the filtering algorithm for unknown noise distribution time-varying and time-delay nonlinear system is characterized by: The artificial immune strategy optimizes the filtering algorithm: S1: Generate particles that obey probability distribution by using Metropolis-Hastings algorithm S2: Calculation of repulsive forces between antibodies S3: Calculation of antibody affinity S4: Calculate the number of clones knum k (i) = round(N·cos(π / 2·fit k (i))) S5: Antibody Variation S6: Indicator calculation S7: Adaptive formula in, is the predicted particle, To predict particle weights, is the mutated particle, is the particle weight after mutation, A a is a fixed weight.
4. According to claim 3, the filtering algorithm for unknown noise distribution time-varying and time-delay nonlinear system is characterized in that: The convergence of the particle filter algorithm of the artificial immune strategy is shown in Figure 1. For a complex network with time-varying delays, if: The particle filter algorithm based on the artificial immune strategy estimates the system state to satisfy: That is, the estimated state of N particles converges to the state Xs as the number of particles N increases. Among them, x s is the real state, y s is the real output, Estimated status.
5. According to claim 4, the filtering algorithm for unknown noise distribution time-varying and time-delay nonlinear system is characterized in that: The adaptive window width kernel density estimation algorithm, the kernel density estimation function is: The empirical distribution function is: Newton interpolation: Resample the empirical distribution function after Newton interpolation and perform Taylor expansion, then the window width can be corrected as follows: in, is the kernel density estimation function, is the window width, is the kernel function, is the noise set, G u (·) is the empirical distribution function.